studies in physical and theoretical chemistry 83
STRUCTURE, FLUCTUATION, AND RELAXATION IN SOLUTIONS
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studies in physical and theoretical chemistry 83
STRUCTURE, FLUCTUATION, AND RELAXATION IN SOLUTIONS Proceedings of t h e Y a m a d a C o n f e r e n c e X X X X l l , held at N a g o y a , J a p a n , 11 - 15 D e c e m b e r 1994
Edited by HIROYASU NOMURA
and F U M I O K A W A I Z U M I
Department of Chemical Engineering, School of Engineering, Nagoya University, Nagoya 464-01, Japan and JACK YARWOOD
Materials Research Institute, Sheffield Hal/am University, Pond Street, Sheffield $1 1WB, United Kingdom Reprinted from the Journal of Molocular Liquids, Volumes 65 and 66
A m s t e r d a m - L a u s a n n e - N e w York - O x f o r d
ELSEVIER - S h a n n o n - T o k y o 1995
ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands
ISBN 0-444-82384-0 9Elsevier Science Publicatiosl and Yamada Science Foundation, 1995. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without the prior written permission from the publisher. Published by Elsevier Science B.V., Sara Burgerhartstraat 25, P.O. Box 211, 1000 AE Amsterdam, The Netherlands Special regulations for readers in the U.S.A.- This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the copyright owners, Elsevier Science B.V. and Yamada Science Fondation, unless otherwise specified. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper. Printed in The Netherlands
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YAMADA CONFERENCE XXXXII ON STRUCTURE, FLUCTUATION, AND RELAXATION IN SOLUTIONS, 11-15 December 1994
vii PREFACE
The 4 2 n d Yamada Conference on "Structure, Fluctuation, and Relaxation in Solution" was held at Nagoya Trade and I n d u s t r y Center (Fukiage Hall), Nagoya, Japan from December 11 to December 15, 1994. The meeting was attended by more than 110 scientists from nine countries.
During last four years(1990-1993) the special research project "Molecular Approaches to Non-equilibrium Process in Solutions" was carried out in Japan u n d e r the financial support of the Ministry of Education of Science and Culture. After the successful ending of this research project we have recognized that it is now our d u t y to tackle the problems emerged form these resent results. These are summarized as follows: (1) solvation dynamics, (2) Relaxation, fluctuation, and reaction dynamics, and (3) dynamic structure and reaction mechanisms in solutions. These topics are the main concern of this conference.
In planning this conference with m y colleagues of the organizing committee, we have hoped that this conference be an opportunity to find out a break-through in theories and experimental techniques of the solution chemistry for the coming century. On this intention the organizing committee has arranged the p r o g r a m to be composed of as wide fields as possible. In addition, the general discussion session was set after several presentations of the related subjects.
viii
In this publication the order of the papers follows the p r o g r a m of the Conference.
This is because we would like the readers of this
Proceedings to understand our above-mentioned intention to open this conference at this time as well as to feel the a t m o s p h e r e of the Conference.
This Conference could only be possible with the support of Yamada Science Foundation in financial and moral aspects. His support has led us to the success of this international meeting.
All
the participants will never forget the generosity of Yamada Science Foundation.
Finally, I would like to express m y sincere thanks to all invited speakers who gave excellent lectures, those who presented v e r y interesting posters, chairpersons and discussion leaders who conducted the general discussion section in marvelous fashion, and those who contributed to the vigorous discussion in the Conference. I must add also my deep appreciation to the cooperation of all the participants not only in the running of the Conference itself b u t also in submitting and revising their manuscript for this Proceedings in due time as requested by the organizing committee in spite of their very heavy schedule.
The Organizing Committee of the 42nd Yamada Conference Syflz6 SEKl(Advisory), Hitoshi OHTAKI(Advisory) Hiroyasu NOMURA (Chairman), Fumio KAWAIZUMI (Secretary) Shinobu KODA, Fumio HIRATA, Masaharu NAKAHARA, Tadashi OKADA Kiyoshi SAWADA, Toshio YAMAGUCHI LIST OF PARTICIPANTS Barnes, A. C. Barthel, Josef Buchner, Michael Burger, K~lm~in Button, Joanne Chong, Song-Ho Dore, John Elich, Karsten Espenson, James H. Fujimura, Yuichi Fujita, Yoshitaka Fushiki, Makoto Harada, Makoto Haymet, A.D.J. Heinzinger, Karl Hiram, Fumio Hiram, Yoshinori Hirota, Noboru Hiwatari, Yasuaki Ibuki, Kazuyasu Iida, Masayasu Ikawa, Shunichi Ikeuchi, Haruko Irisa, Masayuki Itoh, Sumiko Iwadate, Yasuhiko Iwata, Koichi Kai, Yasunobu Kameda, Yasuo Kaneko, Yutaka
University of Bristol University of Regensburg University of Bielefeld A.Jozsef University Kyoto University Kyoto University University of Kent at Canterbury Osaka University Iowa State University Tohoku University National Research Laboratory of Metrology National Institute of Materials and Chemical Research Osaka University University of Sydney Max-Planck Institute for Chemistry Kyoto University Osaka University Kyoto University Kanazawa University Doshisya University Nara Women's University Hokkaido University Sophia University ERATO JRDC Cavendish Laboratory Chiba University Kanagawa Academy of Science and Technology Osaka University Yamagata University Kyoto University
Kanekubo, Mitsuhiro Kaneto, Hirotaka KatS, Minoru Kato, Satoko Kawaizumi, Fumio Kimum, Yoshifumi Kinoshita, Shuichi Kinugawa, Kenichi Kitazawa, Minoru Koda, Shinobu Komatsuzaki, Tamiki Ma, Jangseok Maeda, Hiroshi Maroncelli, Mark Masuda, Akiko Masuda, Yuichi Mataga, Noboru Matsumoto, Mitsuru Matsuoka, Tatsuro Matsuura, Hiroatsu Miyasaka, Hiroshi Morita, Akio Munakata, Toyonori Murakami, Masataka Muranaka, Tadasi Nagaoka, Masataka Nakaham, Masaru Nakai, Takehiro Nakanishi, Koichiro Nakashima, Satoru Nomura, Hiroyasu Ohtaki, Hitoshi Okada, Isao Okada, Tadashi Okazaki, Susumu Okuyama, Naoto Olsson, Lars-Fride Ozutsumi, Kazuhiko
Sophia University Kyoto Institute of Technology Ritsumeikan University Seibo Jogakuin Jr. College Nagoya University Kyoto University Osaka University University of Osaka Prefecture Osaka University Nagoya University Institute for Fundamental Chemistry University of South Carolina Kyushu University University of Pennsylvania Tokyo Institute of Technology Ochanomizu Women's University Insitute for Laser Technology Nagoya University Nagoya University Hiroshima University Kyoto Institute of Technology University of Tokyo Kyoto University Natinal Institute for Physiological Science Aichi Institute of Technology Institute for Fundamental Chemistry Kyoto University Ritsumeikan University Kyoto University Osaka University Nagoya University Ritsumeikan University Tokyo Insitute of Technology Osaka University Tokyo Insitute of Technology Dicel Chemicals Gambro AB University of Tsukuba
Radnai, Tamas Raineri, Fernando O. Rick, Steven Rossky, Peter J. Sfmcez, Francisco Sasaki, Yoichi Sato, Gen Sawada, Kiyoshi Seki, Kazuhiko Seki, Syflz6 Sex), Yoshiteru Shushin, Anatoli I. Suga, Hiroshi Sumi, Hitoshi Suzuki, Satoshi Swaddle, Thomas W. Tabata, Masaaki Takagi, Hideo Takasu, Masako Tanaka, Hideki Tanaka, Motoharu Tanida, Hajime Taniguchi, Yoshihiro Tarumi, Toshiyasu Terazima, Masahide
Hungarian Academy of Science State University of New York Columbia University University of Texas at Austin Seville University Hokkaido University Sophia University Niigata University National Institute of Materials and Chemical Research The Japan Academy Natinal Institute for Physiological Scicnce Kyoto University Kinki University University of Tsukuba University of Tokyo University of Calgary Saga University Nagoya University Kanazawa University Kyoto University Nagoya University Osaka University Ritsumeikan University Nagoya University
Tominaga, Keisuke Tominaga, Toshihiro Tominaga, Yasunori Tran-Cong, Qui Troe, Jurgen Tsukahara, Keiichi Verrall, Ronaid E.
Kyoto University Institute for Molecular Science Okayama University of Science Ochanomizu Women's University Kyoto Institute of Technology University of Gt)ttingen Nara Women's University University of Saskatoon
Watanabe, Iwao
Osaka University
Weing~rtner, Hermann Yamabe, Tokio Yamada, Shinkichi Yamaguchi, Toshio Yamatera, Hideo
University of Karlsruhe Kyoto University Shizuoka University Fukuoka University Daido Institute of Technology
xii Y ano, Y os hi ko
Tokyo National College of Technology
Yasuoka, Kenji
Nagoya University
Yokoyama, Haruhiko Yoshida, Naoto
Yokohama City University Institute for Fundamental Chemistry
Yoshiham, Keitaro
Institute for Molecular Science
Yoshimori, Akim
Nagoya University
xiii Welcome Address December 14,1994
Good Evening, Ladies and Gentlemen!
It is my great honor and pleasure to extend the most cordial welcome address to all the participants of this meeting, the 42nd Yamada Conference on "Structure, Fluctuation and Relaxation in Solutions". In particular my special gratitude must go to those who traveled long distances to here from overseas; Australia, Canada, Germany, Hungary, Russia, Spain, Sweden, United Kingdom and United States of America. On behalf of Yamada Science Foundation we are very happy and privileged to sponsor the present Conference.
Yamada Science Foundation was established in
1977 as a result of the donation of the private holding of the late Kiro Yamada who was the former President of Rhoto Pharmaceutical Company in Osaka. Mr. Yamada recognized that creative basic research in Natural Science is indispensable for the welfare and prosperity of mankind.
Hence, the Foundation has been supporting
innovative research project in physics, chemistry and biology, assisting the international exchange of scientists and also the international conference called Yamada Conference with full financial support, two times every year recently. We are convinced that all the past conferences have been rewarding, stimulating and successful. Unfortunately I could not give a welcome address at the opening session, the day before yesterday, clue to inevitable circumstances.
This evening, however, I am
very happy to be here with all the participants. Incidentally, I had been an experimental physical chemist in the field of structural chemical thermodynamics and working, for a long time on phase transitions in the solid by main use of calorimetry. Although I am not the specialist in the field of Solution Chemistry discussed in the present conference, I am very much convinced that the selected subjects of "Structure, Fluctuation and Relaxation
xiv
in Solutions" are one of the most fundamental, fascinating and frontier topics for the essential progress of liquid and solution science which is associated with physics, chemistry and also biology interdisciprinarily. This is just in accordance with the original aim of Yamada Conference. I hear from Professor Ohtaki and Professor Nomura that the Conference is going in quite an active and marvelous fashion up to now.
I do hope such a
wonderful situation will continue and that aH the participants will enjoy a friendly and free contact to cherish mutual friendship with each other and have successful final results for the conference. Last but not least, I would like to express my thanks to the member of the organizing committee, particularly to Professor Nomura, who have made excellent arrangements for the success of the Conference. Thank you for your attention.
~
~
Syuzo Seki Director General of Yamada Science Foundation
xv
YAMADA S C I E N C E F O U N D A T I O N AND T H E S C O P E OF Y A M A D A C O N F E R E N C E Yamada Science Foundation was established in February 1977 in Osaka through the generosity of Mr. Kiro Yamada. Mr. Yamada was President of Rohto Pharmaceutical Company, Limited, a well-known manufacturer of medicines in Japan. He recognized that creative, unconstrained, basic research is indispensable for the future welfare and spenty of mankind and he has been deeply concerned with its promotion. erefore, funds for this Foundation were donated from his private holdings.
p~
The principal activity of the Yamada Science Foundation is to offer financial assistance to creative research in the basic natural sciences, particularly in !nterdisciplinary domains that brid e established fields. Projects which promote... international cooperation are also ~avored. By assisting the exchange of vmmng scientists and encouraging international meetings, this Foundation intends to greatly further the progress of science in the global environment. In this context, Yamada Science Foundation sponsors international Yamada Conferences once or twice a year in Japan. Subjects to be selected by the Foundation should be most timely and stimulating. These conferences are expected to be of the highest international standard so as to significantly foster advances in their respective fields.
xvi Executive M e m b e r s of F o u n d a t i o n
Officers: Board of Directors
Leo ESAKi Kenichi FUKUI Osamu HAYAISHI Junjiro KANAMORI Kiyoshi SF-GAMt
Standing Director
SyOz6 SEKI
Director General
metuo SHIBA
Tomoji SUZUKI Ikuo TAKEUCHi
Yasusada YAMADA Auditors Seturo EBASH!
Hisaya NARA
Advisors:
Nobur6 KAMIYA Takeo NAGAMIYA Shuntaro OGAWA
xvii
Contents
Special Issue: Structure, Fluctuation, and Relaxation in Solutions m Proceedings of the Yamada Conference XXXXII, Nagoya, Japan, 11-15 December 1994 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Organizing Committee and List of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Welcome Address . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Yamada Science Foundation and the Scope of Yamada Conference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Executive Members of Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi Papers not included in this special issue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi Perspective of solution chemistry of Japan. The historical background and recent trends H. Ohtaki (Shiga, Japan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Molecular theory of solvation processes in dipolar and non-dipolar solvents H.L. Friedman, F.O. Raineri, B.-C. Perng (Stony Brook, NY, USA) and M.D. Newton (Upton, NY, USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 An interaction-site representation of the dynamic structure factor of liquid and the solvation dynamics F. Hirata, T. Munakata (Kyoto, Japan), F. Raineri and H.L. Friedman (Stony Brook, NY, USA) . 15 The interplay of dielectric and mechanical relaxation in solvation dynamics B.J. Schwartz and P.J. Rossky (Austin, TX, USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Fluctuating charge force fields for aqueous solutions S.W. Rick, S.J. Stuart, J.S. Bader and B.J. Berne (New York, NY, USA) . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Solvation dynamics of dye molecules in polar solvents studied by time resolved hole burning spectroscopy K. Nishiyama, Y. Asano, N. Hashimoto and T. Okada (Osaka, Japan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Ultrafast measurements of the dynamics of soivation in polar and non-dipolar solvents J. Gardecki, M.L. Horng, A. Papazyan and M. Maroncelli (University Park, PA, USA) ............ 49 Solvent and nuclear dynamic effects on intermolecular electron transfer K. Yoshihara, Y. Nagasawa, A. Yartsev, A.E. Johnson and K. Tominaga (Okazaki, Japan) ..... 59 Theory on rates of solution reactions influenced by slow fluctuations in viscous solvents, and its experimental confirmation H. Sumi (lbaraki, Japan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Dynamics of Brownian motion under a potential and theory on the rate of chemical reactions A. Morita (Tokyo, Japan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Structural characteristics of hydrogen-bonded networks in water and ice systems J.C. Dore and D.M. Blakey (Canterbury, UK) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Structure and dynamics of supercooled and glassy aqueous ionic solutions T. Yamaguchi, M. Yamagami, H. Wakita (Fukuoka, Japan) and A.K. Soper (Didcot, UK) ........ 91 The structure and dynamics of aqueous solutions containing complex molecules A.C. Barnes, G.W. Neilson and J.E. Enderby (Bristol, UK) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Liquid-liquid phase separations and critical behavior of electrolyte solutions driven by long-range and short-range interactions H. Weing&rtner (Karlsruhe, Germany) and W. Schr6er (Bremen, Germany) . . . . . . . . . . . . . . . . . . . . . 107 Slow relaxation in pure and binary frozen-in systems H. Suga (Higashi-Osaka, Japan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
... XVlII
Slow dynamics in supercooled fluids Y. Hiwatari, J. Matsui (Kanazawa, Japan), T. Muranaka (Toyota, Japan) and T. Odagaki (Fukuoka, Japan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Density functional theory of solution dynamics T. Munakata (Kyoto, Japan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dissociation and solvation in water and aqueous solutions A.D.J. Haymet (Sydney, N.S.W., Australia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of solvent, temperature, and pressure on hydrogen bonding and reorientation of water molecules M. Nakahara and C. Wakai (Kyoto, Japan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Collective vibrational dynamics of simple liquids M. Buchner and T. DorfmQIler (Bielefeld, Germany) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanisms of dephasing in femtosecond time-resolved coherent Raman scattering from molecules in liquids Y. Fujimura (Sendal, Japan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ion solvation and ion association studied by infrared and microwave methods J. Barthel (Regensburg, Germany) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamical aspects of water by low-frequency Raman scattering Y. Tominaga, Y. Wang, A. Fujiwara (Tokyo, Japan) and K. Mizoguchi (Osaka, Japan) .......... Ultrasonic relaxation study of fast exchange processes in mixed micelle systems of alcoholdecyltrimethylammonium bromide-water R.E. Verrall (Saskatoon, Sask., Canada), D.J. Jobe (Pinawa, Man., Canada) and E. Aicart (Madrid, Spain) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Organometallic free radicals in solution J.H. Espenson (Ames, IA, USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The structure of metal complexes of small models of glycoproteins in aqueous solution K. Burger, L. Nagy and B. Gyuresik (Szeged, Hungary) ............................................. Kinetic evidence for short-lived intermediates in metalloporphyrin formation M. Tabata (Saga, Japan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure and dynamics of solvated ions: new tendencies of research T. Radnai (Pusztaszeriut, Hungary) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The role of solvent in inorganic reaction mechanisms, as elucidated by high pressure studies T.W. Swaddle (Calgary, Aita., Canada) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EXAFS of bromide ion in solvents and at air/solution interface I. Watanabe (Osaka, Japan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Redox properties of oxo-bridged di- and tri-nuclear complexes of heavy transition elements: stability of mixed-valence state Y. Sasaki (Sapporo, Japan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Study of the reduction of Co(NH3)4(pzCO2)2§ by Fe(CN)6'- in binary aqueous mixtures: an interpretation of solvent effects based on spectroscopic data P. Perez-Tejeda (Sevilla, Spain), J. Benko (Bratislava, Slovak Republic), M.L. Moya and F. Sanchez (Seviila, Spain) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ion pair of tetraalkylammonium picrates K. Sawada and F. Chigira (Niigata, Japan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ion-pairing effects on the SgCoelectric field gradients in the NMR relaxations of tripositive cobalt(Ill) complex ions M. lida, T. Nakamori, Y. Mizuno (Nara, Japan) and Y. Masuda (Tokyo, Japan) ................... Concentration dependence of S"Co relaxation rates of tris(acetylacetonatao)-cobalt(ill)in some organic solvents M. Kanakubo, H. Ikeuchi and G.P. Sat6 (Tokyo, Japan) ............................................. Mechanistic study of oxidation reactions of hydroquinone, catechol, and L-ascorbic acid by dicyanobis(1,10-phenanthroline) iron(Ill) in dimethyl sulfoxide H.D. Takagi, N. Kagayama, M. Matsumoto, T. Tarumi and S. Funahashi (Nagoya, Japan) ......
123 131 139 149 157 169 177 187
195 205 213 221 229 237 245 253
261 265 269 273 277
xix Molecular orientation in fluids near solid-liquid interface as studied by the density functional method Y. Teramoto and K. Nakanishi (Kyoto, Japan) ........................................................ The stability and dynamics of clathrate hydrates H. Tanaka (Kyoto, Japan) ............................................................................... Deviation of the TST rate constant through the nonlinear couplings M. Nagaoka (Kyoto, Japan), N. Yoshida (Hyogo, Japan) and T. Yamabe (Kyoto, Japan) ........ Spatial correlations in reaction-diffusion systems in nonequilibrium conditions K. Seki (Ibaraki, Japan) and K. Kitahara (Tokyo, Japan) ............................................. Nonlinear effects of solvation dynamics A. Yoshimori (Nagoya, Japan) .......................................................................... Solvation dynamics studied by ultrafast transient hole burning J. Ma, D.V. Bout and M. Berg (Columbia, SC, USA) ................................................. Hydrogen-bonded structure in aqueous sulfuric acid solutions Y. Kameda, K. Hosoya, S. Sakamoto, H. Suzuki, T. Usuki and O. Uemura (Yamagata, Japan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Raman spectroscopic study of rotational and vibrational relaxation of CF3H in the supercritical site S. Okazaki, N. Terauchi and I. Okada (Yokohama, Japan) .......................................... Anomalous conformational behavior of short poly(oxyethylene) chains in water: an FT-IR spectroscopic study H. Matsuura and T. Sagawa (Higashi-Hiroshima, Japan) ........................................... Time-resolved resonance Raman study of the recombination dynamics photodissociated carbon monoxide to sperm whale myoglobin and its mutants S. Nakashima (Toyonaka, Japan), T. Kitagawa (Okazaki, Japan) and J.S. Olson (Houston, TX, USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solvent viscosity dependence of bimolecular reaction rate constant of the excited 9cyanoanthracene quenched by 1,3-cyctohexadiene M.Kitazawa, T. Yabe, Y. Hirata and T. Okada (Osaka, Japan) ...................................... Extremely slow reorientation dynamics of molecular tracers in glassy polymers Q. Tran-Cong, H. Kanato and S. Chikaki (Kyoto, Japan) ............................................ Dynamics near a liquid surface: mechanisms of evaporation and condensation K. Yasuoka, M. Matsumoto (Nagoya, Japan) and Y. Kataoka (Koganei, Japan) .................. The effect of ionic atmosphere on the tracer diffusion of micelles T. Tominaga and M. Nishinaka (Okayama, Japan) ................................................... Effect of addition of surfactant on the micelle-monomer exchange process T. Matsuoka, T. Shibata, S. Koda and H. Nomura (Nagoya, Japan) ................................ Dissipative structure in aqueous polymer solutions H. Maeda, S. Sasaki, T. Mashita, K. Nakamura, H. Ojima, T. Gotoh, T. Fukuda and M. Yamanaka (Fukuoka, Japan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-linear effects in the free energy.profile of electron transfer reaction: an integral equation approach S.-H. Chong and F. Hirata (Kyoto, Japan) ............................................................. Induction time of precipitation of calcium carbonate L.-F. Olsson (Lund, Sweden) ........................................................................... Photophysical properties of viologen-linked N-alkylporphyrins and their zinc(ll) complexes in solutions K. Tsukahara, N. Sawai, S. Hamada, T. Nakazawa (Nara, Japan) and R. Nakagaki (Ishikawa, Japan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conductometric study of hydrophobic properties of tris(1,10-phenanthroline) and tris(2,2'bipyridine complexes H. Yokoyama, K. Shinozaki, S. Hattori, F. Miyazaki and M. Goto (Yokohama, Japan) ........... Structure of the monohalogenocadmium(ll)complexes in N,N-dimethylformamide K. Ozutsumi (Tsukuba, Japan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
281 285 289 293 297 301 305 309 313
317 321 325 329 333 337 341 345 349
353 357 361
xx Effect of pressure on the viscosity of aqueous cesium chloride solution at 25~ T. Nakai, S. Sawamura and Y. Taniguchi (Shiga, Japan) ............................................ Melting behaviour in hexagonal CeCI3 and monoclinic ErCI3 crystals Y. Iwadate, N. Okako, Y. Koyama, H. Kubo and K. Fukushima (Chiba, Japan) ................... Polarization effect on the static and dynamic properties of molten lithium iodide S itoh (Cambridge, UK) and P.A. Madden (Oxford, UK) ............................................ Evsluation of thermodynamic quantities of fluid mixtures by the PY equation F. Kawaizumi, Y. Amakasu, M. Ohba and H. Nomura (Nagoya, Japan) ........................... $olvation free energy of protein reproduced by the combination of the extended scaled particle theory and the Poisson-Boltzmann equation M. Irisa (Osaka, Japan), T. Takahashi (Tsukuba, Japan), F. Hirata (Kyoto, Japan) and T. Yanagida (Osaka, Japan) ............................................................................... Transport properties of a spherical molecule in solvent with position-dependent viscosity K. Ibuki and M. Ueno (Kyoto, Japan) ................................................................... Liquid dynamics studied by higher order non-linear optical spectroscopy K. Tominaga, G.P. Keogh and K. Yoshihara (Okazaki, Japan) ..................................... Temperature dependence of singlet excitation energy migration in liquid benzene as revealed by picosecond laser photolysis H. Miyasaka (Kyoto, Japan) and N. Mataga (Osaka, Japan) ........................................ Solvent effect on ultra-fast electron transfer in mixed-valence biferrocene monocation A. Masuda, Y. Masuda and Y. Fukuda (Tokyo, Japan) .............................................. Translational diffusion of transient radicals studied by the transient grating method M. Terazima, K. Okamoto and N. Hirota (Kyoto, Japan) ............................................. Intra and intermolecular energy transfer in thermal dissociation in the gas and liquid N20, T. Kato (Kyoto, Japan), S. Hayashi (Kyoto-fu, Japan) and K. Machida (Kyoto, Japan) ........... Photoelectron emission spectroscopy applied to bromide ion in solvents H. Tanida and I. Watanabe (Osaka, Japan) ........................................................... Direct comparison between femtosecond optical Kerr effect and high-resolution light scattering measurements Y. Kai, S. Kinoshita (Osaka, Japan), M. Yamaguchi and T. Yagi (Sapporo, Japan) .............. Vibrational cooling process in solution probed by picosecond time-resolved Raman spectroscopy. Analysis of the cooling kinetics Ko Iwata and H. Hamaguchi (Kawasaki, Japan) ...................................................... Rotational relaxation of rod like molecules: diphenylacetylene in various solvents Y. Hirata, Y. Kanemoto, T. Okada (Toyonaka, Japan) and T. Nomoto (Mie, Japan) .............. Ground state recovery of phenol blue in fluids near the gas-liquid critical density Y. Kimura and N. Hirota (Kyoto, Japan) ............................................................... Dynamics of two-component liquids near the glass transition Y. Kaneko (Kyoto, Japan) and J. Bosse (Berlin, Germany) .......................................... Relaxation characteristics of intracellular (Na § as measured by double quantum filtered (DQf) NMR Y. Seo and M. Murakami (Okazaki, Japan) ...........................................................
365 369 373 377
381 385 389 393 397 401 405 409 413 417 421 425 429 433
Author Index ..................................................................................................
437
Subject Index .................................................................................................
441
Following papers were presented in the Conference. However, these hhhpapers are not included in this special issue.
0-9
Collisional Energy Transfer of Vibrationally Highly Excited Molecules from Gaseous to Liquid Environments J.Troe (Universit~it GiSttingen, Germany)
A-13 High-Pressure FIR and Molecular Dynamics Study of Collision-induced Absorption of Non-dipolar Liquids Y.Fujita, and S.Ikawa (Hokkaido University) B-9
Calculation of Solvation Effect on the Ionization Potential Using Continuum Dielectric Model M.Harada, and K.Sakai, and l.Watanabe (Osaka University)
B-23 Solvent Effects on the Molecular Conformational Equilibrium of Chloroacetone M.Kato, Y.Nanba, and Y.Taniguchi (Ritsumeikan University)
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journal of
MOLECULAR
LIQUIDS ELSEVIER
Journal of Molccular Liquids, 65/66 ~1995) I-6
Perspective of Solution Chemistry of Japan. - The Historical B a c k g r o u n d and Recent Trends Hitoshi Ohtaki Department of Chemistry, Faculty of Science and Engineering, Ritsumeikan University, 1916 Noji-cho, Kusatsu, Shiga 525, Japan
Abstract The development of solution chemistry in Japan is reviewed in connection with the history of solution chemistry in Europe and in the United States. Studies on solution chemistry from molecular aspects, as well as those by means of thermodynamics and statistical thermodynamics, axe discussed from experimental and theoretical sides. Recent trends in solution chemistry, especially those in Japan, are overviewed from static and dynamic points of view. 1. Introduction Solution chemistry in Japan has a relatively long history from the beginning of the 20th century. After a rapid development of physical chemistry, inorganic chemistry and organic chemistry around the turn of the century to the 20th, the term "solution chemistry" is occasionally used among physical and coordination chemists who are interested in properties of electrolyte and non-electrolyte solutions in the 1950's. In fact, most of works by van't Hoff, Kohlrausch, Arrhenius and Ostwald, should be included in solution chemistry in the present sense. Classical analytical chemistry and electrochemistry represented by the works of Noyes, Treadwell, Heyrovsky and Kolthoff may also be considered as applied fields of solution chemistry. Marine chemistry and geochemistry are acceptable as fields closely related to solution chemistry. Even environmental chemistry in recent years should be considered to be an interdisciplinary area of solution chemistry and environmental science.
2. Streams of Solution Chemistry Solution chemistry has, from the origin of the traditional studies of modern chemistry, two main streams: one is concerned with physicochemical studies of solutions, and the other includes studies on chemical reactions in solution. The former has been investigated mainly by physical chemists, and the latter is closely related to coordination chemistry. New techniques such as laser Raman spectrophotometry, NMR spectroscopy and X-ray and neutron diffraction methods, as well as EXAFS and XANES spectroscopy, provide us tools to observe solution phenomena from the microscopic point of view on the bases of structural chemistry and reaction dynamics. Thus, structural and dynamic studies of solutions have been developed as new streams of solution chemistry. 3. Studies of Solution Chemistry in Europe and in the United States One might say that electrochemical studies achieved by Michael Faraday around the 1830's is the starting point of modern solution chemistry. If we focus our attention on basic solution chemistry, the work by Arrhenius (1886) 1 can be said as the most remarkable landmark of solution 0167-7322/95/$09.50 ,~ 91995 Elsevier Science B.V. All rights reserved. SSDI 0167-7322 (95) 00838-!
chemistry. Studies carried out by van't Hoff, Ostwald and Kohlrausch at almost the same period as Arrhenius' work, established an equation, "physical chemistry = thermodynamics = solution chemistry", in the early 1900's. This period may be classified to be the first golden period of solution chemistry, although the word "solution chemistry" has not yet been created as a scientific terminology. The second golden time of solution chemistry started by the work of Debye and Hiickel in 19232, which was based on statistical thermodynamics, one step further advanced from thermodynamics. This work was immediately followed by N. Bjerrum 3, whose idea of ion-pair formation was induced by the Debye-Hiickel theory. In these treatments they accepted the existence of ions and molecules with finite sizes as solutes, but solvent was regarded as a homogeneous continuum, and no molecular aspect was introduced in their concept. In this respect the second period can be said as the dawn of the molecular solution chemistry developed in the late 20th century. In this section, we have to mention the work by Bernal and Fowler (1933) 4, which will be referred to again later. In 1960's the molecular aspect was well introduced to solvents, as well as to solutes, and solution chemistry entered the third period. This period was sometimes called "Renaissance of Solution Chemistry". The structure of water and other liquids was discussed in various ways, and such theoretical considerations were highly supported by the development of high-speed computers. Computer simulation studies appeared as a new way of studying structure and dynamics of solutions. Besides the physicochemical studies on solution chemistry, investigations for chemical reactions in solution were extensively developed in the latter half of the 20th century. Studies on chemical reactions in solution were, in principle, closely related to Werner's synthetic works of coordination compounds, and thus, these studies were carried out mostly by coordination chemists rather than physical chemists. Accumulation of knowledge on chemical reactions through studies on synthetic and decomposition reactions led to construction of more reasonable scheme for chemical reactions in solution. Spectrophotometric and optical rotatory dispersive methods, which had been employed in the structural investigations for inert complexes in solution, became important techniques in studies of solution chemistry including reactions of labile complexes. An epoch-making event in solution chemistry happened in 1957. Frank and Wen5 from USA and Samoilov 6 from Russia presented their papers concerning ionic hydration in the journal, "Discussion of Faraday Society". A new concept on the structure and dynamics of ionic hydration was established in this year, although their models were rather primitive according to the present understanding. Nevertheless, their models are still well accepted by many solution chemists as simple but reasonable ones. Interestingly the two important papers on ionic hydration were reported in an English journal by an American and a Russian chemists in the period of the cold war. In the scientific community we have neither border nor war. Studies on ionic hydration and complex formation reactions in solution in the 1960's quite actively proceeded in Europe, especially in northern Europe. J. Bjerrum 7, a son of N. Bjerrum in Denmark, established a method for determination of stepwise formation constants of complexes with simple monodentate ligands in solution by the spectrophotometric method, which was soon modified to the method by potentiometry. Sillen in Sweden extended the method to polynuclear complex formation reactions and Schwartzenbach in Switzerland, who is a pioneer of chelate chemistry, applied this method to multidentate ligand complexes Studies on reactions in solution expanded from equilibrium to non-equilibrium state, and the most remarkable study done in this period was that by Eigen 8 in Germany. Thus, the northern Europe has a tradition for electrolyte solution chemistry since Arrhenius. In USA at the same time, Taube9 introduced a new concept of inner- and outer-sphere reaction mechanisms. Solution chemistry in USA yielded many interesting works,especially in thermodynamic and
statistical thermodynamic studies. The names of Lewis the the very early 1900's, and then, Pitzer, Guggenheim, Hildebrand, and Onsager (although he was originally Norwegian) easily came up to our mind. Coordination solution chemistry was also developed in USA since the 1930's under the leadership by J. Bailer, Jr. In a period when all scientists in Europe and Asia had a hard time during and after the War II, US chemists established the Renaissance of Coordination Chemistry. In the field of solution chemistry a number of names can be picked up from the literature in US in this period,e.g., Basolo, Calvin, and Taube. Statistical thermodynamics was thoroughly investigated in this period and the results were conveyed to the late 20th century. Landmark works by, e.g., Lennard-Jones-Devonshire, Eyring, Yvon-Born-Green, and Kirkwood were connected to the creation of the Ornstein-Zernike, Hypernetted-chain, and Percus-Yevick equations, and new theories such as Scaled Particle and Perturbation Theories were presented. In these theories particles are usually noncharged and longrange ion-ion and dipole-dipole electrostatic interactions are cut off by the introduction of a shielding function in order to eliminate complicated long-range electrostatic interactions. These theories depicted a system in terms of particles of the solute and solvent, and thus, the picture is essentially different from that of the Debye-H iJckel theory. These statistical mechanical theories established in the middle 1900's are especially powerful for systems constructed by non-charged particles with a spherical shape. The development of high-speed computers helps the utilities of the theories by solving equations by means of numerical calculations, which were not soluble until the middle 1900's in mathematical ways. Development of high-speed computers provided a new field of solution chemistry. Molecular mechanics, Monte-Carlo, inverse Monte Carlo, molecular dynamics simulations are those typical approaches in this area. These methods have never been thought in the past until a numerous iterative calculations can be done by computers. Imagine that a radial distribution function of a system containing non-charged spherical particles was simulated by black and white agar-agar balls in a glass vessel and by taking a number of photographs to calculate baU-ball distances by Hildebrand. Since computer simulations reproduce not only experimental results but also reveal unknown facts not accessible by any kind of presently available experimental techniques This is why studies by means of Monte Carlo and molecular dynamics simulations became so fashionable especially for young scientists. The multibody and polarization problems have not been well introduced in these calculations yet, and thus, the simulations are still far from ideal for the elucidation of solution phenomena. These difficulties should be overcome by the effort of physicochemical solution chemists in the 21 st century.
4. Studies of Solution Chemistry in Japan How did solution chemistry in Japan develop? It is said that the Japanese scientist who was first interested in solution chemistry in the present sense was Joji Sakurai (1858-1936). It seemed quite natural that Sakurai, a leading scientist in Japan, was attracted to this field, because he was a physical chemist and the equation "physical chemistry = thermodynamics = solution chemistry" holds rather strictly in this period due to the works by Arrhenius, Ostwald, van't Hoff, and others in Europe. Furthermore, Kikunae Ikeda (1864-1935), who discovered sodium glutamate ("Aji-nomoto") and then changed his interest from physical chemistry to organic chemistry, Yukichi Ohsaka (1867-1950), both of them studied physical chemistry in the laboratory of Ostwald in Germany, and Masao Katayama (1877-1961) significantly contributed to the development of solution chemistry of Japan from the stand-point of thermodynamics and thermochemistry. Although Sakurai's and Ikeda's interests were directed later to fields other than solution chemistry, solution chemistry was one of the most fashionable chemistry and an active branch of
physical chemistry in Japan in the early 20th century, and solution chemistry has been an important subject of research in some imperial universities in the 1930's. Thus, studies on solution chemistry were started rather early in Japan only a little later than Europe. However, originality of chemical researches in Japan was not highly evaluated, because they were mostly studies imitated solution chemistry in Europe. Accompanying the development of chemistry in other fields, researches unique to Japanese solution chemistry were grown. Investigations on properties of solutions by thermodynamics and statistical thermodynamics, chemical equilibria and reaction kinetics became to be the main fields of physical and inorganic solution chemistry in Japan. Even after a break of scientific investigations due to the War II we still had difficulties to communicate with scientists outside Japan. Not only solution chemists but also scientists in other fields in Japan had to tread a thorny path for several years in the 1940's-1950's. Experimental works which do not need expensive equipments could compete with works in the world rather early compared with other fields of solution chemistry. Complex formation studies by the pH-metry using glass electrodes, studies on weak ion-ion interactions by the conductometry, investigations of properties and structures of solutions by the UV-visible and IR spectrophotometry, and oxidation-reduction studies on ionic species by using polarography are typical examples in solution chemistry studies in Japan in the early stage after the second world war. Polarography created by Heyrovsky in 1929 was introduced to Japan by Masuzo Shikata, a coworker of Heyrovsky, and this technique quickly attracted interest of analytical chemists and electrochemists in Japan. N. Tanaka and his coworker R. Tamamushi made a remarkable achievement in the field of oxidation-reduction reactions of metal complexes in solution by polarography. Studies on ion-pair formation initiated by N. Bjerrum were extended theoretically by Onsager and Fuoss, and the theory of ion-pair formation almost completed in the 1960's. Since they were in USA, the studies on ion-pair formation were more actively carried out in USA than other parts of the world. Since the study does not need expensive tools, the work could be done in Japan by using the Onsager-Fuoss theory. Spectrophotometric studies in Japan has a long tradition since Yuji Shibata, a coworker of A. Werner, came back from Switzerland in 1913. The spectroscopic method for the determination of the composition of complexes formed in solution, which is now referred to as Job's method, was really reported by Shibata much earlier than Job. Studies on the spectrophotometric series (1938-1939) by Tuchida is one of the most important achievement in spectroscopic studies made in Japan in this period. Complex formation has also been studied in Japan from view-points of chemical equilibria, reaction kinetics and structures of complexes in solution. Works influenced by J. Bjerrum, Sillen and Schwarzenbach and others in North Europe formed an important stream in equilibrium studies in solution. Reaction kinetics were investigated on the basis of the absolute reaction rate developed by Eyring and fast reactions were analyzed by the method established by Eigen. Many young Japanese scientists went to Europe, USA and Canada to accept new ideas and to learn new methods of investigation there. Structural chemistry of solutions started in Japan rather late compared with other studies of solutions, although crystallographic investigations were very highly developed and actively investigated in Japan. This fact may be due to lack of the concept of structure of liquids in most Japanese physical chemists. The concept of "structure" was soundly applied to solids and molecules in the gas phase, but not to liquids and solutions. X-Ray diffraction studies on liquids and amorphous substances were already examined in 1916 by Debye and S herrer 10, immediately after the first work of Debye for the X-ray diffraction. Studies on liquid metals and molten salts by using X-
ray and neutron diffraction methods were examined by some workers in Japan, but this technique was quite unfamiliar to most Japanese and it had applied neither to water and aqueous solutions nor nonaqueous solutions until Ohtaki applied the solution X-ray diffraction method to cadmium iodide and copper perchlorate solutions in 197411. Combination of a diffraction experiment with molecular dynamics simulation was fin'st made in Japan in 198212, and this combination is now well accepted as one of the most general ways of studies on solution structures. Bioinorganic chemistry was extended in Japan by amalgamating solution chemistry, coordination chemistry and biochemistry. From such a historical background, biochemistry in Japan has still a strong interrelationship with solution chemistry. Everybody recognizes a high activity of bioinorganic chemistry in Japan. Organic chemists are interesting in reactions in solutions, but most of them pay much more attention to the play of solutes than the role of solvents in reactions. The society of solution chemistry in Japan is mostly supported by physical chemists, inorganic chemists and analytical chemists. In 1978 the National Symposium on Solution Chemistry in Japan has started and the meeting has been organized every year at various places in Japan since then. With the background of the activity of Japanese solution chemists, the Association of Japanese Solution Chemists has been established in 1983. The activity of the association increased year by year, and the number of participants of the national meeting on solution chemistry steadily increased We had about 250 participants at the 17th symposium in Okayama held in November, 1994. In this decade we had two big three-year projects concerning solution chemistry under the sponsorship of the Ministry of Education, Science and Culture of Japan. One is the project entitled "Microscopic and Dynamic Aspects of Solute-Solvent Interactions", headed by H. Yamatera and H. Ohtaki was the secretary, which run from 1985 to 1987 fiscal years. The other is "Molecular Approaches to Nonequilibrium Processes in Solutions", which was performed from April, 1990 to March, 1993. The project leader and the subleader were H. Ohtaki and H. Nomura, respectively. Japanese solution chemists have been highly encouraged by these projects. The results of the former project has been published from Elsevier Science Publishers 13. 5. Molecular Aspects and the Perspective of Solution Chemistry As mentioned above, solution chemistry was born at the end of the 19th century and developed on the basis of thermodynamics and statistical thermodynamics. Main fields supporting the solution chemistry were physical chemistry and coordination chemistry, which also absorbed some other parts of chemistry in order to establish an interdisciplinary field of chemistry in 19501960. Even in this period, however, solvent, e.g, water, was recognized as a continuum in most studies except for some works on ionic hydration, and thus, molecular picture of water was not clearly recognized. In modern solution chemistry, ion-solvent and ion-ion interactions should be depicted more clearly at the molecular level. The first molecular picture of water was drawn by Bernal and Fowler in 19334. Their work was based on X-ray diffraction data of water reported by Meyer 14, Steward 15, and Amidi 16. Therefore, we can say that the molecular aspect of water and solution was already established in 1930's, but proposition of the concept of component molecules by Bernal and Fowler is no doubt the most important historical event in structural solution chemistry. Statistical thermodynamics is also changing in recent years. Newly developed RISM (Reference Interactions Site Model) theory has no restriction on the shape of solute species, in contrast to old theories in which spherical species are usually assumed. Ab initio calculations are being combined with molecular dynamic simulations. This combination becomes possible because of the improvement of high-speed computers. The polarization effect and multibody problem will be
adequately introduced in MD calculations in the not too distant future. Dynamic properties of solvent and solute molecules in the condensed phase will be better represented by theories and experiments in the nano second or even shorter time scale in the near future. Structures of short-lived reaction intermediates and even activated complexes will be directly determined by experiments, although indirect observations are sometimes possible even now. Solution chemistry should be further developed by concurrent studies of thermodynamics-statistical thermodynamics, solution reaction studies, structural studies on liquids and solutions, and studies on solution dynamics. At the beginning of the 21st century we shall enjoy fruits grown in the field of solution chemistry irrigated by these four disciplines. References 1 S. Arrhenius, Z. phys. Chem., 1~ 631 (1887). 2 P. Debye and E. HiJckel, Phys. Z., 24, 185 (1923). 3 N. Bjerrum, Kungl. Danske Videnskab. Selskab. Math.-fys., Medd., 7_, 1 (1926). 4 J.D. Bernal and R. H. Fowler, J. Chem. Phys., 1_, 515 (1933). 5 H.S. Frank and W.-Y. Wen, Disc. Faraday Soc., 24, 133 (1957). 6 0 . Ya. Samoilov, Disc. Faraday Soc., 24, 141 (1957). 7 J. Bjerrum, Kungl. Danske Videnskab. Selskab. Math.-fys., Medd., 11, 1 (1931). 8 M. Eigen and K. Tamm, Z. Electrochem., 66, 107 (1962) and others. 9 H. Taube and H. Meyers, J. Am. Chem. Soc., 76, 2103 (1955). 10 P. Debye and P. Sherrer, Nachr. Ges. Wiss. G~ttingen, 1, (1916). 11 a) H. Ohtaki, M. Maeda and S. Ito, Bull. Chem. Soc. Jpn., 47, 2217 (1974); b) H. Ohtaki and M. Maeda, Bull. Chem. Soc. Jpn., 47, 2197 (1974). 12 I. Okada, Y. Kitsuno, H.-G. Lee, and H. Ohtaki, "Application of Molecular Dynamics Simulation to the Structural Analysis of X-Ray Diffraction Data on a Concentrated LiCI Aqueous Solution" in "Ions and Molecules in Solution" eds. N. Tanaka, H. Ohtaki, and R. Tamamushi, Elsevier, Amsterdam, p. 81 (1983). 13 H. Ohtaki and H. Yamatera, eds., "Structure and Dynamics of Solutions" Elsevier, (1992). 14 A. W. Meyer, Ann. Phys., 5, 701 (1930). 15 G. W. Steward, Phys. Rev., 35, 1462 (1930); 37, 9 (1931). 16 E. Amidi, Phys. Z., 32, 914 (1931).
journal of MOLECULAR
LIQUIDS Journ',dof Molecularliquids. 65/66(1995)7-14
ELSEVIER
MOLECULAR THEORY OF SOLVATION PROCESSES IN D I P O L A R A N D N O N - D I P O L A R S O L V E N T S H a r o l d L. F r i e d m a n , F e r n a n d o O. Raineri, B a w - C h i n g P e r n g
Department of Chemistry, State University of New York at Stony Brook, Stony Brook, New York 11794-3400, USA and M a r s h a l l D. N e w t o n
Chemistry Department, Brookhaven National Laboratory Upton, New York 11973, USA Abstract
We describe a molecular theory of solvation dynamics which provides a detailed description of the mechanism of solvent response following a sudden change in the charge distribution of a solute molecule. The theory is based on a new renormalized linear response development which incorporates nonlinear aspects of equilibrium solvation. The solute-solvent coupling is formulated in terms of intermolecular interactions, without recourse to macroscopic cavity concepts. Both the solute and solvent are represented by interaction site models, while the dynamical aspects are treated with our previously developed Reference Memory Function Approximation. The results for solvation dynamics in water and acetonitrile compare very favourably with MD simulation results for the same model systems. The same renormalized dynamical theory of solvation provides a very useful framework for estimating free energy profiles and associated parameters (average vertical energy gap, reorganization energy, change in solvation free energy) for electron transfer reactions. The new results demonstrate that a reasonably detailed molecular theory that deals with models incorporating the charge distribution of the molecules explains the solvation properties relevant to the theory of electron transfer reactions in both dipolar and non-dipolar (polar) solvents. I. I n t r o d u c t i o n In view of the great importance of chemical reactions in solution, it is not surprising that basic aspects (structure, energetics, and dynamics) of elementary solvation processes continue to motivate both experimental and theoretical investigations. Thus, there is growing interest in the dynamical participation of the solvent in the events following a sudden redistribution of the charges of a solute molecule. 1 These phenomena control photoionization in both pure liquids and solutions, the solvation of electrons in polar liquids, the time-dependent fluorescence Stokes shift, and the contribution of the solvent polarization fluctuations to the rates of electron transfer in oxidation-reduction reactions in solution. We describe a classical statistical mechanical theory of solvation dynamics, formulated for general molecular interaction site models (ISM) of the solute and solvent species. 2 Ba0167-7322/95/509.50
9 1995 Elsevier Science B.V. All rights reserved.
SSD!0167-7322(95)00839-X
sed on a surrogate Hamiltonian model, the theory gives an approximate nonequilibrium distribution function that describes the state of the solvent as it adapts to a change in the distribution of charges on the solute molecule. The surrogate Hamiltonian is expressed in terms of renormalized solute-solvent interactions, a feature that leads to a simple and natural linear response description of the solvent dynamics in the vicinity of the solute. In addition to the measurable solvation time correlation function (tcf), we can also calculate observables needed to elucidate the detailed mechanism of solvation response, such as the evolution of the solvent polarization charge density around the solute. The dynamical theory also provides a framework for the study of the diabatic free energy profiles as functions of the reaction coordinate required in the theory of nonadiabatic electron transfer reactions. We illustrate this new applicatiort by calculating the free energy profiles in solvents coveting a wide range of polarity. II. Basic H a m i l t o n i a n M o d e l a n d S u r r o g a t e D e s c r i p t i o n We consider a genetic donor-acceptor complex solute at infinite dilution in a polyatomic solvent. Both the solute and solvent molecules are represented by rigid and nonpolarizable ISM models. In the ISM models the potential energy of interaction between two molecules is a sum of pairwise-additive site-site terms, including Coulombic interactions between partial charges located at the molecular sites. Throughout the paper the subscript A refers to interaction sites of the solute, while the subscript aj refers to interaction site j of solvent molecule a. The solute has two electronic states, the "precursor" (P) and "successor" (S) states, with different charge distributions characterized as sets of partial charges {QP} and {Qs} at the interaction sites. We regard the solute as the source of an "external" field which is fixed in space, and we concentrate on the solvent dynamics driven by this field. When the solute is in state D the Hamiltonian 7/D of this system (solute plus solvent) is 7"lD = E D + H D = E D + Hw + ~tD , D = P,S , (1) where E D is the energy of the solute electronic state D in vacuum and H D is the Hamiltonian of the solvent in the field of the fixed solute. The second equality expresses H D in terms of Hw, the Hamiltonian of the pure solvent, and the solute-solvent potential energy of interaction ~tD = ~_,,~ ~ a j u~(rA,aj), where the (solute site .\)-(solvent site 3) interaction potential
~ JD( ~ , o : )
= ~ : ( ~9 , ~ : )
~_ QD q1 r~,aj
,
O = P, S
(2)
comprises both a short range (non-Coulombic) part u*~j(r), and a long range (Coulombic) part. The dependence of the latter on the solute electronic state is explicit in Eq. (2). Furthermore, rA,a3 is the distance between solute site A and site j of solvent molecule a, while qj is the partial charge at an interaction site of type j in the solvent molecules. We now introduce a dynamical variable that is central in this work: the solvent
contribution to the vertical energy gap l~ -= H S -
HP = ~S _ ~P __ ~
AQ~ r~,aJqJ ,
~ aj
(3)
where AQ), - Q s _ QP. Thus, for a given configuration of the solvent around the solute. the energy difference between the S and P states of the system is 7-/S - ~ P = A E + ~, where A E is the energy difference between the two electronic states in vacuum. Unfortunately, theories starting from the basic (or factual 2) Hamiltonian, Eq. (1), unavoidably lead (without further approximation) to expressions for the dynamic solvation properties that require the knowledge of two-time many-point correlation functions. 2 To avoid this difficulty we turn to an alternative description of the system, in terms of the surrogate (subscript X') Hamiltonians 2 7"lD - E D + Y D = E D + Y w + ~D
,
D=P,S.
(4)
The difference between ~/D and 7"~D is that the former is defined in terms of renormalized solute-solvent interactions ~D, which need to be specified. While there is no unique criterion for choosing ~D, the selection must lead to an accurate theory of solvation dynamics without invoking two-time many-point correlation functions. We have found that this goal can be achieved with a new theory for the nonequilibrium distribution function in which the renormalized solute-solvent interactions enter linearly. 2 In this theory ~ and ~S are chosen such that the renormalized linear response theory accurately describes the essential solute-solvent static correlations that rule the equilibrium solvation both at t = 0 (when solvent is in equilibrium with the initial charge distribution of the solute) and at t = oc (whcn the solvent has reached equilibrium with the new solute charge distribution). 2 In this work we discuss a Renormalized Dielectric Theory (RDT), that is obtained with the choice 2 ~D = ~ / d3r pA,~(r)~D,A(r) , (5) A where ~A,~(r) is the microscopic solvent polarization charge density at position r relative to site A of the solute. For the expression for 15A,~,(r) the reader is referred to our previous reports. 2 D In Eq. (5) ~2:,A(r) represents the renormalized electrostatic potential at point r relative to the solute site A. It is defined in terms of the equilibrium RISM solute-solvent D ) when the solute is in the electronic state D site-site direct correlation functions cAj(r (RISM refers to the Reference Interaction Site Method of Chandler and Andersen 3) D D ~E,~(k) = ~E,~(k) = -kBT ~ 9j(k)c~(k) , J
(6)
where c~j(k) and ~D,x(k) are, respectively, the spatial Fourier transforms of c ~ ( r ) and D ~Z,~(r). Furthermore, kB is the Boltzmann constant and T is the temperature. The
charge number factors Oj(k) only depend on the average structure of the homogeneous solvent. 2 Notice that in Eqs. (5) and (6) the solute-solvent coupling is formulated in terms of intermolecular interactions, without recourse to macroscopic cavity concepts. III. S o l v a t i o n D y n a m i c s For the discussion of solvation dynamics we identify the electronic states P and S with, respectively, the ground state and an excited electronic state of the solute. In the solvation dynamics experiment the solute undergoes a P ---, S Franck-Condon transition produced by photoexcitation. We assume that the solute in the S state returns to the P state by fluorescence on a time scale that is comparable with the structural relaxation rate of the solvent around the solute. At any instant the fluorescence frequency v(t) is given by the relation by(t) = AE + ((~:; t)), where ((~:; t)) is the nonequilibrium average (under the surrogat~ estimate of the solvent nonequilibrium distribution function) of the surrogate solvent contribution to the vertical energy gap ~ = ~ - ~'P [compare with Eq. (3)]. The solvation time correlation function, which monitors the solvation response to the sudden (at t = 0) P - . S transition, is given by
~,(t) - ~,(oo) z(t)
=
~,(o) - ~,(oo) =
((~; t/) - ((~; oo)) ((t; 0)) - ((t; ~ ) )
(7)
The RDT estimate of the solvation tcf is found to be z~(t)
=
f
a:3k c~,(k,t)BE(k) .t a3k S,,(k) 8z(k) '
(S)
C~,(k, t) is the time correlation function of the solvent polarization charge density, 2 S~,(k) "- C~,(k, t= 0), and where we define
where
Bz(k)
= ~
,.,~,~,(k)a~:,~(k) a,~z,~,(k).
(9)
Here w;~;v(k) is the Fourier transform of the solute intramolecular correlation function, 3 S P and AqoE,,~(k) = qaE,,~(k ) - qav,,~(k) According to Eq. (7), it is the dielectric dynamics of the homogeneous solvent, as expressed in Cu(k, t), that is the source of the time dependence of the estimate Z~(t) of the solvation tcf. In the RDT approximation the effect of the solute-solvent interactions is carried by the static coupling function BE(k). This "factorization" (to a function of the homogeneous solvent dynamics times a function of the static solute-solvent structure) is a characteristic feature of the RDT theory. The renormalized character of the coupling function allows us to bypass the two-time many-point correlation functions that would necessarily appear in a dynamical theory that explicitly addressed the inhomogeneity of the solvent in the neighborhood of the solute particle. As an application, in Figure 1 we compare the prediction Z~(t) of the surrogate theory and the result of nonequilibrium MD simulation. The solute is a spherical particle with Lennard-Jones (L J) parameters E/kB =2668 K, a -6.975/It, which undergoes ionization from QP = 0 to Q s = 1 e (where e is the proton charge). The simulation results are
taken from the nonequilibrium Molecular Dynamics (NEMD) study by Maroncelli and Fleming. 4 The dielectric tcf C ~ ( k , t ) was calculated with the Reference Memory Function Approximation. 2,s The required solute-solute and solute-solvent site-site correlation functions were calculated under the RISM-HNC approximation. 2 As the figure shows, Z ~ ( t ) agrees very well with the nonequilibrium MD response. z h ( t ) displays (i) a very fast initial decay (~ 60% of the total decay within 20-30 fs), (ii) a strong oscillatory feature due to the collective librational motion (optical mode 5) of water, and (iii) a final slower relaxation of diffusive character. 1.0
,
,
1
, W A T E R
Z (f)
- - - NEMD
o
RDT
o
s~ 0
I
!
o
l
"
tO s)
,1
~
t'
"
0.5
Figure 1" Solvation tcf of an ion in water. See the text for details. The RDT theory of solvation dynamics is capable of describing the solvation tcf as well as the evolution of various ancillary observables. 2 IV. Free E n e r g y Profiles The diabatic free energy profile for the electronic state D is defined by the equation -fiFO(r1) -- In
1/ / 9
dr2
dFe-
6(71 - U)
}
,
D = P,S ,
(10)
where "HD is the basic Hamiltonian introduced in Eq. (1). f2 represents the set of Euler angles that specify the solute orientation, fl = ( k B T ) -1 , and 9 is a parameter (with dimensions of energy) independent of the electronic state D. 6 The Dirac delta function in Eq. (10) selects those configurations for which the solvent contribution to the vertical energy gap/d has the numerical value rl (the reaction coordinate). We apply the RDT theory to calculate an estimate FD(rl) of the free energy profile. The central quantity in this analysis is a generalized characteristic function g D ( z ) = ((exp(iz~))) D of complex argument z. Here ((--"/)D indicates an average over the surrogate solvent equilibrium distribution function [defined in terms of the surrogate Hamiltonian
7"/D, cf. Eq. (4)] when the solute is in state D. Both FD(r]) and the surrogate estimate A A ~ - A~ - A P of the equilibrium Helmholtz free energy difference can be expressed in terms of gD(z). Evaluating gP(z) and gS(z) with the RDT theory leads to parabolic precursor and successor free energy profiles
F~:(V) = --U ~E ( ~ - v~ )2 FS(r/) = A F ~ + T~E (77 - r / ~ )2 ,
(11~) (llb)
where A F E = A E + AAE. These equations correspond to a particular choice for the dimensional parameter 0. Both profiles have the same curvature ~E which, in view of the effective linearity of the theory, is related as ~ z = (2AE) -1 to the surrogate estimate A,F of the solvent contribution to the reorganization energy. The theory gives detailed expressions for AA,F and AE in terms of solute-solvent and solvent-solvent sitesite correlation functions that can be calculated under the RISM-HNC approximation. 2 To illustrate this theory we calculate the diabatic free energy profiles FP(rl) and F~(r/) of a complex donor-acceptor solute in two solvents of different polarity. We also examine the dependence on the charge distribution of the S state. The aniline-like solute molecule comprises seven interaction sites, which are labeled as in the scheme below.
All seven sites have the same LJ parameters: a =3.75/~ and e/kB =55.36K, while all of the bond lengths are 1.4/~,. The interaction site 1 and the 6-membered ring are, respectively, the donor and acceptor groups. In the P state the partial charge at the donor site is QP = - 0 . 2 e , while the acceptor charge is locMized at the interaction site 2, Q2P = 0.2e; the charges at the remaining acceptor sites are zero. We consider two electronic transitions that differ in the succesor states, S~ and SH, reached. The two transitions are generated by transfer of charge from the donor to the acceptor sites. The transferred charge is distributed among the acceptor sites according to the unoccupied molecular orbitals e2ul (state S~) and e2u2 (state Su) of benzene. Thus in S~ the donor charge is Q] = 0.5 e, while the charges at the acceptor sites are Q~ = Q~ = 0, and Q~ = Q~I = Q~ = Q~r = - Q ~ / 4 . In state S,, Q~ = 0.5 e, while the acceptor charges are Q~m = Q~ = -V]~/3, and Q"=3 (~1i = Q~ = Qp = -Q~'/12. In Figure 2 we show the P and S free energy profiles for the transitions P ~ S~ and P ~ S~x in acetonitrile. Figure 3 shows the corresponding results in benzene. The ISM models of Edwards et al.7 and Jorgensen and Severance 8 were used to represent the
solvents. The necessary corrections for electronic polarizability of the solvent are also taken into account. 6 Also we have taken A E - 0, so that the vacuum contribution still needs to be added to the profiles in the figures. 10
l
'
i
,, \
\
1'//
_
FEP
-2 -4
m
t
-6 -30
-20
~
-10
i
J
0
10
r/
29
Figure 2: Free energy profiles [Eqs. (11)] in acetonitrile as functions of the reaction coordinate. Solid lines: P --. S~ transition; dashed lines: P ---. S~ transition. FD(r/) and 7/ are in units of kcal mo1-1 . Both figures illustrate the difference between the free energy profiles of the P - . S~ and P ~ Sx~ transitions, signaling the dependence of AA L, and of the reorganization energy ~ v (different curvature) on the P and S charge distributions of the solute molecule. This sensitivity of ,kZ' with the solute charge distribution is only rarely invoked in the analysis of experimental results. Another point of interest is the location of the minimum of the F~(r/) profiles in the two solvents, connected with the different solvatochromic shift of the P --, S transitions in acetonitrile and benzene, s Also of special relevance is the finite curvature of the free energy profiles in benzene (Figure 3), which indicates, in line with experiment, that the solvent reorganization energy is sizeable in this solvent. This is in marked contrast with the conclusion from the naive application of the conventional dielectric cavity theory (based on local electrostatics), that predicts negligible solvent reorganization energy in benzene. We conclude noting that by representing the solute and solvent with ISM models, the RDT theory and its extensions s appear capable of describing the equilibrium and nonequilibrium solvation effects in charge transfer reactions in solvents covering a wide range of polarity, including non-dipolar ("multipolar") solvents comprising molecules without permanent dipole moment. A more complete report on the solvent effects on the energetics of non-adiabatic electron transfer reactions will be presented elsewhere. 6
14
-~ \
\
_l,
l
I
,, ', i
~l
\l
f
\, V,I
/
:
I:
:: !
FS V~,
i i
............... \ - " , .................... " W .
-1 -2
""-',.2---_=/ -8
J
-6
I
1
i
1
-4
-2
0
2 77 4
Figure 3: Same as Figure 2, but in benzene. Acknowledgment. F. O. R. gratefully acknowledges a fellowship from the Consejo Nacional de InvestigacionesCientfficasy Tdcnicas (CONICET) de la Repdblica Argentina. This work was made possible by the support provided by the National Science Foundation of the U.S. under Grant No. CHE-9321963 (Stony Brook) and by the Division of Chemical Sciences, Officeof Basic Energy Sciences of the U.S. Department of Energy under contract DE-ACO2-76CH00016 (Brookhaven Natl. Lab.) References
1. a) B. Bagchi, Ann. Rev. Phys. Chem. 40, 115 (1989); b) P. F. Barbara and W. Jarzeba, Adv. Photochem. 15, 1 (1990); c) M. Maroncelli, J. Mol. Liquids, 57, 1 (1993); d) S. Roy and B. Bagchi, J. Chem. Phys. 101, 4150 (1994); e) N. Nandi, S. Roy, and B. Bagchi, J. Chem. Phys. 102, 1390 (1995). 2. a) F. O. Rained, H. Resat, B.-C. Perng, F. Hirata, and H. L. Friedman, J. Chem. Phys. 100, 1477 (1994); b) F. O. Raineri, B.-C. Perng, and H. L. Friedman, Chem. Phys. 183, 187 (1994); c) H. L. Friedman, B.-C. Perng, H. Resat, and F. O. Ra~neri, J. Phys.: Condens. Matter 6, A131 (1994); d) H. Resat, F. O. Rained, B.-C. Perng and H. L. Friedman, in Hydrogen-Bond Networks, edited by M.-C. Bellissent-Funeland J. C. Dore; NATO-ARW series, (Kluwer, Dordretch, 1994); e) H. L. Friedman, F. O. Raineri, F. Hirata, and B.-C. Perng, J. Statistical Phys. 78, 239 (1995). 3. D. Chandler and H. C. Andersen, J. Chem. Phys. 57, 1930 (1972). 4. M. Maroncelli and G. R. Fleming, J. Chem. Phys. 89, 5044 (1988). 5. H. Resat, F. O. Ra.~neri, and H. L. Friedman, J. Chem. Phys. 98, 7277 (1993). 6. B.-C. Perng, M. D. Newton, F. O. Raineri, and H. L. Friedman, to be submitted. 7. D. M. F. Edwards, P. A. Madden, and I. R. McDonald, Mol. Phys. 51, 1151 (1984). 8. W. L. Jorgensen and D. L. Severance, J. Am. Chem. Soc. 112, 4768 (1990).
journal of
MOLECULAR
LIQUIDS ELSEVIER
Journal of MolecularLiquids,65166(1995) 15-22
An Interaction-Site Representation of the Dynamic Structure Factor of Liquid and the Solvation Dynamics Fumio Hiratal), T. Munakata2), F. Raineri3), and H.L. Friedman 3) 1) Detmrtment of Chemistry, Faculty of Science, Kyoto University, Kyoto 606, Japan 2) Department of Applied Mathematics and Physics, Faculty of Engineering, Kyoto University, Kyoto 606, Japan 3) Department of Chemistry, State University of New York at Stony Brook, Stony Brook, New York 11794-3400 Abstract A recent development in the theory for the dynamics structure factor of molecular liquids, which employs the interaction-site model, is outlined. The theory is applied for a description of the solvation dynamics associated with a p h o ~ x c i t a t i o n of a molecule in polar liquid. Preliminary results of the solvation time correlation functions for an atomic molecule in a variety of solvents are presented.
L Introduction The space-time density fluctuations in polar liquid and their coupling with a variety of chemical processes have attracted much attention in recent years. The photo-induced electronic excitation of a solute and the thermal electron transfer reaction are among most well studied examples in such chemical processes. 1 In this brief note, we report our recent effort to build theoretical tools for investigating such phenomena based on the integral equation method in the statistical mechanics of liquids.2-5 For a microscopic description of solvent dynamics, it is necessary to specify a model of solvent molecules. The traditional model for such description is the electric multipoles located at center of a hard-sphere. Such model leads to a rot-translational description of a rigid body typically represented by a Debye-Fick type diffusion equation. Another representation for a solvent molecule is the interaction site model (ISM) which leads to a correlated atomic diffusion model. In that model, the solvent dynamics is described as a correlated translational diffusion of each atom: correlated in terms of not only intermolecular force but also intra-molecular constraints. We have been developing an interaction-site representation for the solvent dynamics based on the generalized Langevin equation (GLE) 6 combined with the RISM equation which describes the static structure of molecular liquids in terms of the positional correlation between a pair of interaction sites. 7,s The GLE theory provides a systematic way of projecting degrees of freedom which are less interesting onto few dynamic variables of interest. It is therefore essential for the theory to choose proper dynamic variables on which other degrees of freedom are projected. Two different choices for the collective variables have been examined: the polarization charge density and the site-number density. The p o l ~ o n charge density representation naUuaHy leads to a dielectric description of the liquids, while the site-number density representation gives a description for the atomic diffusion. It is worthwhile to persue the site-number density representation since it has prospect for developing the desoiption of the solvent dynamics from more fundamental level including translational motion. The relation between the two representations has been clarified in the earlier work. 5 Here, we just concentrate on the sitenumber density formalism. The paper is organized as follows. In the following section, the site-site Smoluchowsky -Vlasov equation for the solvent dynamics is briefly discussed. In section HI, a preliminary results for the solvation dynamics of an ion in a variety of solvent is presented. II. Dynamics of Solvent - Site-Site Smoluchowsky-Vlasov (SSSV) Equation .3 We chose the local density (~SPcx)and the momentum density (Aa) of atoms as the dynamic variables: 0167-7322/95/$09.50 ~" 1995 ElsevierScience B.V. All rights reserved. SSD! 0167-7322(95) 00840-3
Aa(r,t)= ~p~(r-
r~)
(2)
where a and M specify atoms and molecules, respectively. Then the standard treatment by GLE leads to a continuity equation and an equation of motion for atoms: ~p(r,t) = -m _tV -A(r,t) (3)
OA(r,t) Ot
Ot
t
kBTVOf O(rr
o
- r ) * A(r)dr + m-~F(t)
(4)
where ~(r,r') is the inverse density correlation matrix defined by ~(r,r')=~Sp(r)~p(r')> -1, K(t) and F(0 are the friction kernel and the random force, respectively. In the derivation, the equal time correlations of momentum between intra-molecular atom pairs are neglected. The inverse density correlation matrix can be calculated in terms of the RISM theory of liquid, r [~0(k)! -l -c(k) (5) where t0(k) and c(k) are the Fourier transforms of the intra-molecular and direct correlation matrices, respectively. Neglecting the inertia term and solving Eqs. (3) and (4) for ~0, we get a diffusion equation in the Markov limit, which we call 'site-site Smoluchowsky-Vlasov equation,' since it is a Smoluchowsky-type diffusion equation for atoms (sites) with a Vlasovtype mean field term expressed by the RISM direct correlation matrix.
c~p(kd) Ot
_k2/](k) O(i~k)~p(k ,t),
(6)
where D(k) is the diffusion constant matrix. The Smoluchowsky-Vlasov type equation for molecular liquids can be derived9 also by combining the dynamic density fimctional theory developed for multicomponent simple liquids 10 with the flee-energy density-functional for molecular liquids developed by Chandler and coworkers, l I The derivation begins with the two hydrodynmnic equations similar to Eqs. (3) and (4) but with a genemliz~ force field - p(r,t) VSF/~p(r,t) in place of the first term in Eq. (4), where FIp(r)] is a density functional of an atom (or a site) and ~/~P signifies the functional derivative with respect to the density field. Up to the first order in the Taylor expansion with respect to the density field, the free energy functional for the ISM liquid can be written as, F[O]- Z f d r p ~ ( r ) { l n o ~ ( r ) - l } __1 Zfdrfdr,~o~(r)C~u~,.(~r_rl)~Ovu(e)
2 a,v where CaMvM'(Ir-fl) is the direct correlation function defined by -I
I
Cojlvu.(r)-c~.(r)+?~uuO~ {/-to(r)- }
(7) (8)
It can be easily shown that the functional derivative 8F/Spa leads to the first term in Eq. (4). Dynamic Structure F a c t o r T h e (intermediate) dynamic structure factor (DSF) is defined by F(k,0= <Sp(-k,0)Sp(k,0>/N, which describes the time evolution of the spacial correlation of an atom (self) and a pair of atoms (distinct). Multiplying F-~I.(6) by ~p(-k,0) and taking ensemble average, we have the diffusion equation for F(k,t), -
0F(k,t) = _ k 2 D(k) pdp(k)F(k,t) (9) (gt Taking the Fourier-Laplace transform of Eq. (9), the dynamic structure factor can be obtained
as, S(k,s) =[st+ kZD(k){co(k) -l - oc(k) } ~ ~F(k,t =0)
(10)
where I denotes the n-by-n unit matrix. Eq. (1 0) can be inversely transformed to give, F(k,t) ffi L(k,t)F(k, tffiO) where L(k,0 is defined by, 1
"
L( k,t) ffi - ~ t _ ~ .
+ A(k) l-i e*'ds
(11) (12)
and A(k) is defined as, A(k)--- k 2D(k)[to(k) -' - pc(k)]. The integral in Eq. (12) can be performed to lead for the (a,l~)-element of the matrix L, ~j (-1)p+a A ~ (sj)e*/ L~ = .
H,i.j)(s~ = s , )
(13) '
where {si} are the eigen values of the matrix sI+A, and A(xl3denotes the minor of the (a,l~)element of the determinant IIsI+AII.It is worthwhile to note that Fcttl(k,t) is a superposition of a number of exponentials with different time constants. In the following section, we will observe some interesting consequence of the multiexponential behavior of solvent dynamics, which is manifested in the solvation dynamics of ions. 1 0"9l~
0.8 ~ 0.7 ~i\ 0.6i ~ o.4i
Co)
o.~ ~
",,.',, <'-,,
0.5 1~ ~
....... ~
..................................................................
(c) 0.5 1 ~
"
~
......... ~ ' ~ " : ......... ; ......... : " ' " - = .........
~s
Fig. 2. 7-(0 for Cl ~:)r process. Solvents:H20 (~); MeOH (~); MeCl (...); MeCN(-.-)
(d) 0..5 ~ ~ . . . . .
..........
|
o
0,
. . . . . . .
~
~
-21 -4!
(e) -8! ~c.w ...................... ,. . . . . . . . . .
~"~= -10: -12 -14
00 0.5
1 1.5 2 2.5......5" 3:5 - 7{ 4.-5" " t
Fig. 1 Intermediate Structure Factor for the hydrogen atompair of water. Circles, MD simulation; solid lines, SSSV. (a) k=0.3374, (b) k=0.7545, (c) k=1.0670, (d) k=1.3913, (e) k=1.8171, (f) k=2.33860.
-16 -18 -2C
f0 t's 20 2'5 3'0 3'5 4'0 4's t/ps
Fig. 3. lnZ(t) for CI ~>CI- process. Solvents:H20 (--); MeOH (~); MeCI (...); MeCN(-.-)
As an example of the solvent dynamics calculated from SSSV, F(k,t) for the SPC model of water is shown in Fig. l, compared with the simulation results. Only those for the H-H pair are shown for the different values of k. The simulation results were obtained from the 200 ps trajectory for a system of 216 water molecules by directly taking the average, 12
where the subscripts M,M' label molecules in the system. Due to the finite box size used in the simulation, the least value of k is 0.3374 A -l. The diffusion constant from the MD simulation, D1~3.9"10 -5 cm2/sec, is used as an input parameter for D(k)(=6apDr) in Eq.(9), which the single particle Ix~slational motion of water. III. Soivation Dynamics -site n u m b e r density representation- 5 Here, we give a brief review of a method presented in the earlier paper to describe the solvation dynamics associated with the photo excitation of a molecule in polar liquids, which can be p r o l ~ by the time resolved Stokes shifts. 5 The quantity which relates the dynamics theory with the experiment is the solvation time correlation function (STCF) defined by,
~(t)- ~(oo) z(t) = r~(0)- ~(oo)
(~5)
where ~(t)=~(t)-<4E> is the solvation energy change at time t after the photo-excitation. By virtue of the fluctuation dissipation theorem, Eq. (15) can be written as,
z(t)- c l t ) , g~(t~(0))
(16)
c(0) g~(0)~(0))'
where the canonical ensemble average <---> is taken over the equilibrium solvent configurations. The solvation energy fluctuation ~(t) can be represented in terms of the solvent site-density fluctuation 5pT,j(k,t) as follows, 1
dk
with ~4~,j(k)=d~7,j(k)-4~,j(k), where ~, and j Slx~fy solute and solvent sites, respectively, and 4~u),j(k) and 4~7,j(k) denote a solute-solvent coupling in equilibrium before and after the photo excitation.
r
(21 ,I,n,f~ ~.(~,jfk.t,~.r(W))S4~,(-k)A,~.r(-g,.
(18)
The final equation for Z(t) can be obtained from F_qs. (16)-(18) after average over the solute orientation in <--->,
f dkk2~ JJFj:(k't)BjI(k ) Z(t) - f dld~jjSjj (k)Bjf(k) '
(19)
nj:(k) = ~ ~o~.(k)~j(k)~.j.(k)
(20)
where Fjj(k,t) is the dynamic structure factor of the homogeneous solvent, and Bjj(k) and Sjj,(k) are defined by following equations,
Sjz(k). Fg(k,t = 0) (21) The Sjj(k) can be calculated from the RISM equation and Fjj(k,t) from the SSSV equations descnl~ed in the previous section. The solute-solvent coupling ~D?,j(k) (D: p or s) which was left unspecified in Eq. (17) can be identified as the solute-solvent direct correlation functions from the requirement such that
the solvent density response to the perturbation at t--0 and t=oo should satisfy the corresponding RISM equation in the equilibrium, namely, t~(k) = -kT%(k) . (22) In another word, the solute-solvent coupling so defined satisfies proper boundary conditions at the initial and final equilibrium states. -solvation dynamics of an ionIn the previous paper, 5 we have given just a formalism of STCF for the site number density representation, which has been outlined above. In what follows, a very preliminary numerical results of STC~ for a CI~>CI- process is presented based on the theories described in the previous sections. The calculation has been carried out for a variety of polar liquid as solvent including methyl chloride (MeCI), acetonitdle 0VIeCN), methanol 0VleOI-I) and water. Methyl chloride and acetonitrile represent a class of simple aprotic dipolar liquids while water does those liquids which feature the extensive hydrogen-bond network. The alcohol shows characteristics in between those two classes of fiquids. The caladafion is performed at the room temperature (298 K) for all solvents except MeCl. For MeCl, its liquid temperature (249 K: 1 arm) is chosen. The Edward-McDonald (EM), SPC and TIPS models are chosen for MeCN, water and MeOH, respectively, 13-15 while the parmneters determined by Jorgensen et. al. is employed for MeCI. 16 Those models use the same functional form for the intermolecular sitesite interaction, namely
U~(r) = Z[4~
-
+ q"qa r
(23)
with the standard combining rules, where a and [~ denote sites on either solute or solvent molecules and e, a, q have usual meanings. We use a single parameter, a (translational) diffusion constant DT of an atom, in order to characteriz~ the solvent dynamics in the long time limit. We assign the same value, 3.9* l0 -5 cm2/sec, 3 to D r for all solvents examined. This is partly because the diffusion constants of those solvent are not largely different from each other and also because our main focus in the present study is on the effect of liquid structure on the solvation dynamics. The solvation time correlation functions defined by Eq. (15) and its logarithm are plotted in Figs. 2 and 3, respectively. As seen in the figures, all solvents examined show a multi-exponential decay. Methyl chloride and acetoniU'ile as well as water show esseatially a biexponential decay while methanol clearly exhibits a tri-exlxmential decay. The time constants of the decay can be obtained from the slope of straight lines which gives a best fit with the linear portion of In Z(t) curves. Those are listed in Table 1. Since we have employed the same value for the translational diffusion constant for solvent atoms, the large difference in the behavior of STCF among the solvents should have come thoroughly from the discrepancy in the solvent structure. Especially, the difference between the simple polar liquids and the hydrogen bonded liquids is striking. The results seem qualitatively in accord with the simulation result obtained by Maroncelli et. al., 17,1s in which acetonitrile shows a much slower decay compared to water in the solvation time correlation function. In what follows, we give brief discussions of the results for each solvent, although they are still very preliminary in quantitative sense. Simple Polar Liquid: Methyl Chloride and AcetonitrUe Acetonitrile shows a hiexponential decay. The first decay which is characterized with the time constant 0.15 ps can be attributed to the rot-translational relaxation of solvent molecules in the first solvation shelL The second decay with the time constant 34.6 ps corresponds to the structural reorganization beyond the first cxx~dination shell, which brings the solvation structure to the new equilibrium. There has been a simulation study carried out for acetonitrile by Maroncelli. The result exhibits a decay with a dual character:, a rapid initial decay with oscillations, which is characterized by ~ 1 ps decay time. The first quick component has been assigned by the author to the inertial decay which involves mostly the solvent molecules in the first coordination shell around the solute molecule. Although the present theory does not reproduce the Gaussian character of the
20 initial decay as well as the oscillation, it gives the similar time constant for the relaxation. This is because the initial part of the relaxation is dominated by the intermolecular interactions between solute and solvent as well as those among solvent molecules, which are taken into account in the present theory through the effective local field represented in terms of the direct correlation functions. The theory does not reproduce the Gaussian character and the oscillation, because the SSSV equation neglects the inertial term as well as the memory in the friction kernel. As to the second component of the decay, it is important to point out that Maroncelli's results show different behavior for the second part between the charging process ( S 0 - > S +) and the opposite process (S + >SO).While the second component begins after about 1/4 of the energy is relaxed in the S~ + process, it begins after almost 100 percent of the energy is relaxed in the S + E > S 0 process. Our current theory does not distinguish the two process due to the symmetric ~ of Eq. (19). Also noteworthy in the simulation result is that the nonequilibrium results in short time resemble the linear response results averaged over the equilibrium ensemble at the initial state, while those in later time come close to the linear response results at the final state. Our result shows closer resemblance to the linear response results sampled from the equilibrium ensemble prepared with SOthe neutral solute, which turns out to be similar to the non-equilibrium result for the S+E>SO process in overall behavior. This can be understood by considering the fact that our treatment uses the pure solvent dynamics at equilibrium although the solute pemtrbation is taken into aca~unt at the initial and the final states. The equilibrium ensemble prepared at the S O state has less perturbation to the solvent, and thereby it is closer to our theoretical model. Although the time constant for the second component of the S+-->S 0 process is not explicitly given in the literature, it would be much longer than - 1 ps, and would be closer to our result. Methyl chloride shows a similar hiexponential decay with acetonitrile with time constants, 0.21 and 82 ps. Methyl chloride shows the slower initial decay than acetonitrile despite of its less bulky strucUtre. This may be because the temperature of MeCI is significantly lower. Nonetheless, the general feature of the relaxation behavior for both solutes is almost identical in harmony with the simulation re$ults. 18 Water STCF for water features a bi-exponential behavior with very fast initial decay characteriz~ by the time constant 0.01 ps and a slow decay of time constant 36 ps. The fast decay can be assigned to the rotational relaxation of water molecules in the first coordination shell from their initial orientation to the final state in which the hydrogen atom points toward the chloride ion. The following slow decay can be attributed to the reorganization of water molecules in the second and further shells in order to adapt the new situation. A similar behavior in the initial decay has been found in the simulation study by Maroncelli and Flemingl7 except for the pronounced oscillation observed in the simulation. The oscillation is caused by a combination of the two factors, the inertial motion and restoring force due to the hydrogen-bonded structure. The theory does not reproduce the oscillation because it does not include the inertia term although it does take account for the hydrogen-bonded structure. 3 2.5
/
S 1.5
5 4
31 0..5
I
2
//'i
21 11
r
~
~
Fig.4(a) Solvent-sitedistributionaround CIwithout charge.CI-O (~); CI-Me (...);CI-H (--)
o.L
1
3
4
~
6
r
Fig.4(b) Solvent-site distribution around CI- ion. CI-O (--); CI-Me(...); CI-H (--)
Of more interest is the much faster decay compared to acetonitrile. A possible explanation for the phenomena is that water molecules are spherical in shape and thereby the friction due to collision is very small for the rotational motion. Recently, Nakahara and coworkers have measured the rotational diffusion rate of a single water molecule in a variety of solvent based on NMR relaxation time and have observed no-correlation between the rate and the viscosity of solvent. 19 Instead they have found a quite good correlation between the rate a n d the interaction between the solute (water) and solven~ The observation implies that as long as the rotational motion of water molecules is concerned friction due to the collision is less important. Methanol STCF for methanol is characterizedby a tri-exponential decay. Although fu~her study including the molecular dynamics simulation is required in order to clarify the detailed mechanism of each decay, we give here a speculative interpretation regarding a possible mechanism which gives rise to the tri-exponential decay. In Fig. 4, the radial distributions of atoms around chloride ion in equilibrium before and after the abrupt production of the charge on the solute are shown. Before the change, the solute is primarily coordinated either by the methyl group or by the oxygen site. After the change, the solute is predominantly coordinated by the hydrogen site. The first decay c h a r a c ~ by the time constant 0.03 ps is attributed to the initial rotation which brings the methyl group in the first coordination shell, which has a positive partial charge, into the intermediate position to form a weak Coulombic bond with C1-. This tm3cess will be quick because the solute was already well coordinated by the methyl group before the change. The solvent molecules in the first coordination shell further rotate to brmg the hydrogen atom into the final equilibrium position in which the hydrogen atom and the ion are 'bonded' by the strong electrostatic interaction. This process gives the second decay with the time constant 1.69 ps in the lnZ(t) plot- After the solvent molecules in the first coordination shell settled down in the final orientation, the molecules in the second and timber shells must be reorganized cooperatively to adapt themselves in the new equilibrium state. The prtx3ess will be very slow since it involves the relaxation of many molecules. The third decay characterized by the time constant 61.2 ps is assigned to that reorganization process. If the mechanism stated above is correct, the behavior should be different when the solute is a cation instead of an anion, because in that case, the primarily coordination site of the solvent after the abrupt change in charge state of the solute, is the oxygen atom. There is no such intermediate coordination structure as in the case of CI-. The behavior of lnZ(t) for Na + in fact shows essentially biexponential decay as shown in Fig.5. The different behavior in lnZ(t) between the CI-->CI- and Nam>Na + suggests that the multi-exponential behavior is caused not only by the solvent dynamics itself (Fjj'(k,t)) but also by the solute-solvent coupling represented by Bjj'(k) in Eq. (20). O
o%'I 0.15 T1
MeCI
MeCN MeOH Water
.
0.03 0.01
T2
T3
.0
34.6 1.69 36.3
.
.
.
,-6 ""
61.2
.
2
Table 1. Relaxation Times.
~= -10 - -12 ! -14~
.............
-16 -18
-2
~
lb 1'5 2'o 23 3'o 3'5 ~ 4'5 t/ps Fig. 5 in Z(t) for Na -->Na + and Clm>Clprocesses in MeOH; Cl, (m); Na, (--). Concluding R e m a r k s We have presented a method of describing the dynamics of liquids and the solvation dynamics based on the interaction-site model. The method has exhibited its ability to account for the diversity in the relaxation behavior of solvent, such as the multi-exvonential decay.
22 observed in the polar and hydrogen-bonded solvents. An important implication of the results is that the detailed microscopic description of solvent structure and the solute-solvent coupling is essential for studying the solvation dynamics. The results presented here are still preliminary in quantitative sense. The problem which can be easily recognized is the much faster initial decay in every case than what is observed in the experiments (or simulations). Two remedies for the problem are under investigation: (1) replacing the translational diffusion constant by more elaborated model for the single particle dynamics which explicitly includes the rigid body rotation, and (2) inclusion of the nonlinearity in the density field in the SSSV equation.
Acknowledgment Authors ate gateful to the refen~es for invaluable comments on the manuscript.
References 1. Dynamics and Mechanics of Photoinduced Electron Transfer and Related Phenomena, Proceedings of Yamada Conference XXIX, N. Mataga, eds., Elsevier Science Publication and Yamada Science Foundation, 1992. 2. F. O. Raineri, Y. Zhou, H. L. Friedman, Chem. Phys., 152, 201 (1991). 3. F. tFuma, J. Chem. Phys., 96, 4619 (1992). 4. F. O. Raineri, H. Resat, B-C Perng, F. Hirata and H. L. Friedman, J. Chem. Phys. 100. 1477-1491(1994). 5. H. L. Friedman, F. O. Raineri, F. Hirata and B-C Pemg, J. Star. Mech., 78, No 1/2, 239 (1995). 6. H. Mori, Prog. Theor. Phys., 33, 423 (1965). 7. D. Chandler and H. C. Andersen, J. Chem. Phys., 57, 1930 (1972). 8. F. Hirata and P. J. Rossky, Chem. Phys. Leg., 83, 329 (1981); F. Hirata, P. J. Rossky, and B. M. Pettitt, J. Chem. Phys., 78, 4133 (1983). 9. F. Hiram and T. Munakata, in preparation. 10. T. Munakata, J. Phys. Soc. Jptt, 58, 2434 (1989). 11. D. Chandler, J.D. McCoy and SJ. Singer, J. Chem. Phys., 85, 5971(1986); 85, 5977(1986). 12. S. Okazaki, Y. Miyamoto, and I. Okada, Phys. Rev., 1145, 2055 (1992). 13. B. Bigot, B. J. Costa-Cabral and J. L. Riv'~l, J. Chem. Phys., 83,3083 (1985). 14. D. M. Edwards, P. A. Madden, and I. R. McDonald, Mol. Phys., 1141 (1984). 15. H. J. C. Berendsen, J.P.M. Postma, W.F. van Gunsteren, and J. Hermans, in Intermolecular Forces, ed. B. Pullman (Reidel, Dordrecht, 1981). 16. W. L. Jorgensen, J. Am. Chem. Soc., 103, 335 (1981); J. Chandrasekhar, D. C. Sellmeyer, W. L. Jorgensen, J. Am. Chem. Soc., 106, 903 (1984). 17. M. MaronceUi and G. R. Fleming, J. Chem. Phys., 89, 5044 (1988). 18. M. Maroncelli, J. Chem. Phys., 94, 2084 (1991). 19. M. Nakahara and C. Wakai, Chem. Left., 809 (1992); J. Chem. Phys., 97,4413 (1992).
journal of
~IOLECULAR
LIQUIDS Journal of Molecular Liquids, 65/66 (1995) 23-30
ELSEVIER
The Interplay of Dielectric and Mechanical Relaxation in Soivation Dynamics Benjamin J. Schwartz and Peter J. Rossky Department of Chemistry and Biochemistry University of Texas at Austin Austin, TX 78712-1167 USA Abstract Quantum molecular dynamics simulations are used to explore the possible coupling between dielectric solvation, the response of the solvent to a change in charge distribution of the solute, and mechanical solvation, the response of the solvent to a change in solute size or shape. The hydrated electron is chosen as a solvation probe, due to its large increase in spatial extent upon photoexcitation and significant contraction in size upon non-adiabatic relaxation. The strong displacement of translational solvent modes upon excitation hampers the effectiveness of individual solvent molecule rotations in providing relaxation, decreasing the relative amplitude of the inertial response. Following non-adiabatic relaxation, solvent molecules can freely translate and reorient, leading to rapid, effective initial solvation. These results suggest that in many situations where solutes undergo changes in both charge distribution and size, solvation can become rate-limited by the relatively slow viscoelastic solvent response.
I. Introduction Whenever the electronic states of a solute are coupled strongly to the surrounding environment, the dynamics of the solvent can play a critical role in determining the fate of condensed phase chemical species. Indeed, solvent fluctuations define the reaction coordinate for electron transfer and many other types of chemical reactions, l This has prompted an explosion of recent interest in solvation dynamics: the study of the relaxation of the solvent following a sudden perturbation due to a change in the solute. 2 At the heart of the issue are the specific solvent motions that lead to relaxation. When the solvent rearranges to accommodate the change in the solute, are the motions of individual solvent molecules important or should the response be viewed as inherently collective? In polar solutions, are reorientational or translational motions of the solvent molecules more effective in lowering the energy of the perturbed solute? Are there different types of solvent responses to changes in solute charge distribution versus changes in solute size and shape? A great deal of progress has been made recently in answering these questions for dielectric relaxation, that is, the response of the solvent accompanying a change in charge distribution of the solute. 2 Theory predicts that the earliest time motions of the solvent relaxing th e perturbed solute are inertial in character. 3 This inertial relaxation has now been observed in many simulations and by experiment, which are approaching generally good agreement. 4 Simulations have established that this early time relaxation can be ascribed to individual molecular behavior, 5 and theoretical developments have linked these motions to an instantaneous normal mode description of the solvent. 6 Less well examined, however, are the molecular details for the solvent mechanical response, that is, 0167-7322/95/$09.50 91995 Elsevier Science B.V. All rights reserved. SSDi 0167-7322 (95) 00841-1
24 the relaxation of the solvent accompanying a change in solute size and/or shape. Transient hole-burning experiments using non-polar solutes have established that the mechanical relaxation of the solvent behaves in a manner qualitatively similar to the dielectric response. 7 Molecular dynamics simulations changing the dispersion interaction between the solute and the solvent also show similar behavior. 8 Since real chemical solutes undergo changes in both size and shape and charge distribution upon photoexcitation or chemical relaxation, it is imperative to study the potential interplay between these mechanical and dielectric solvent responses. In this paper, we present the results of quantum molecular dynamics simulations aimed at a preliminary exploration of the coupling between mechanical and dielectric solvation dynamics. We have chosen the hydrated electron as our solute probe, since the hydrated electron is known to undergo large changes in both size and shape and charge distribution upon photoexcitation, and since experimental results are available for direct comparison. We find that translational motions of the solvent are of key importance in accommodating the change in size of the solute, and that relaxation by solvent rotational motions may in fact be significantly altered by coupling to the mechanically-induced solvent translations. II. Methodology The non-adiabatic quantum simulation procedures 9 we employ have been well described previously in the literature, 10 so we describe them only briefly here. The model system consists of 200 classical SPC flexible water molecules, II and one quantum mechanical electron interacting with the water molecules via a pseudopotential. 12 The equations of motion were integrated using the Verlet algorithm with a 1 fs time step in the microcanonical ensemble, and the adiabatic eigenstates at each time step were calculated with an iterative and block Lanczos scheme. 9 Periodic boundary conditions were employed using a cubic simulation box of side 18.17/~ (water density 0.997 g/ml). Twenty configurations from a 35 ps ground state adiabatic trajectory, chosen to be on resonance with the laser bandwidth corresponding to the experiments, were selected as the starting points for non-adiabatic excited state trajectories, lo A corresponding set of trajectories was run in D20, with a model identical in all respects to the work described previously except that the mass of the H atom was changed from 1 to 2 amu, and preliminary results of the behavior in D20 are included here. III. Results Figure 1 presents the dynamical history of the 2 lowest adiabatic eigenstates of the hydrated electron for a typical trajectory. At times before t=0, the electron resides in its nearly spherical s-like ground state, with the first p-like excited state lying -2.2 eV to higher energies. Solvent fluctuations modulate the energies of these states, and the strength of the coupling is readily manifest in the large changes in energy (nearly an eV on time scales of tens of femtoseconds). Previous work has established that size and shape fluctuations of the solvent environment have different effects on the quantum energy levels. 13,14 Upon promotion to the first excited state, the electron grows in size by a factor o f - 2 along the axial lobes of the p-like wavefunction, but remains unchanged in diameter in the other two dimensions, l0 The surrounding solvent cavity takes on a peanut shape to accommodate this change, and the net result is that the energy of the unoccupied nodeless ground state is raised while the energy of the occupied p-like state remains mostly unchanged. The electron eventually makes a non-adiabatic transition back to the ground state (which happens near t=200 fs for the trajectory shown, before the excited state equilibrium is attained). After the transition, the eigenenergy of the s-like state rapidly drops so the ground state equilibrium energy gap is quickly recovered, and
0-
'
~-l~
-100
0
100
Time
200
300
400
(fs)
Figure 1. Adiabaticeigenstatesof the hydrated electron for a typical trajectory. Solid and dashed lines denote the ground and first excited states, respectively. Diamondsmark the occupied state. the electronic density quickly becomes localized in one half of the peanut shaped cavity, creating a void in the solvent. As discussed elsewhere, 10 the behavior of the quantum energy levels upon excitation is in good agreement with previous simulations using a different model of the hydrated electron, 15 and the rapid re-establishment of equilibrium following radiationless relaxation is consistent with older, adiabatic calculations. 14 Since the hydrated electron experiences a substantial change in charge distribution upon quantum transition, undergoes a large increase in spatial extent upon photoexcitation, and experiences a corresponding collapse in size upon non-adiabatic relaxation, it serves as an outstanding quantum mechanical probe of the coupling between the mechanical and dielectric solvent responses. To better understand the behavior of the quantum eigenstates following photoexcitation and non-adiabatic relaxation, we have computed non-equilibrium ensemble averages of the quantum energy levels, shown in Figure 2. The left panel shows the response of the adiabatic eigenstates following photoexcitation (the "up" ensemble average); it is an ensemble average that only includes configurations in which the electron still occupies the excited state. Thus, the data at early times contain contributions from all 20 trajectories, but the statistics get poorer at later times as electrons make the radiationless transition to the ground state and are removed from the ensemble. The data clearly show that following photoexcitation, the excited state energy remains essentially unchanged while the ground state energy is raised on two time scales: a rapid increase which takes place in the first 30 fs, and a slower response which takes several hundred femtoseconds. At equilibrium, it is apparent that the gap has decreased from its initial value o f - 2 . 2 eV to 0.5-0.6 eV. The fight-hand portion of Figure 2 shows the change in the quantum energy levels following the non-adiabatic transition (the "down" ensemble average); in constructing this average, we have defined t=0 to be the point at which the non-adiabatic transition occurs for each trajectory. This is a fairly unusual kind of ensemble average: many of the initial configurations start after radiationless transition from the equilibrated excited state, but some initial configurations result from excited state trajectories in which the solvation response is not yet complete (such as the -200 fs point shown in Figure 1). Since all the excited trajectories undergo non-adiabatic relaxation at some point, the traces presented here are averaged over all twenty runs, resulting in good statistics over the entire time period displayed. After radiationless decay over an average gap size of -0.6 eV, solvent relaxation rapidly lowers the energy of the occupied ground state, with most of the response completed within 25 fs. There is evidence for slower relaxation of the ground state on longer time scales, but any slower component of the response plays a much smaller role than that following photoexcitation. Like the upwards transition, the first electronic excited state undergoes little change in energy in response to non-adiabatic
26
.
>
@
@
m
-2
0
I t I I I 300 600 900 1200 0 50 100 150 200 250 Time After Ph0toexcitation (fs) Time After Non-Adiabatic Transition (fs)
Figure 2. Non-equilibriumensemble averages of the lowest 2 adiabatic eigenstates of the hydrated electron. Solid line denotes the ground state, dashed line the first excited state. The left panel shows the response of the energy levels following promotion via photoexcitation; the right panel shows the response following the radiationless transition to the ground state. See text for details. relaxation. Within a few hundred fs of the non-adiabatic transition, the equilibrium structure of the hydrated electron is already re-established, with the gap enlarging to its original value o f - 2 . 2 eV. This rapid evolution to equilibrium once the ground state becomes occupied is in agreement with both the results of previous calculations 15 and the observed dependence of the different elements of the calculated ultrafast transient spectroscopy compared to experiment. 16 One interesting feature of the downwards ensemble average lies in the smoothness of the traces present in the right-hand plot of Figure 2. The energy levels of the individual trajectories (cf. Fig. 1) fluctuate by - 1 eV on a rapid time scale due to coupling with various modes of the solvent. The fact that these large fluctuations are washed out in the ensemble average following relaxation to the ground state indicates that there is no preferred phase of a particular solvent motion that drives the non-adiabatic transition. If, for example, particular configurations of water molecules characterized by being at the turning point of a collective librational mode were more favorable for non-adiabatic relaxation, then some of the oscillations observed in the individual trajectories would add coherently and hence be enhanced in the ensemble average. The lack of large amplitude oscillations in the average indicate that no special water configurations are responsible for driving the non-adiabatic dynamics. It is also interesting to note the small recurrence in dae response between 25 and 30 fs after the downward transition. This oscillation is most likely due to an underdamped librational mode of the solvent, and has been observed in previous aqueous molecular dynamics simulations. 17 IV. D i s c u s s i o n
The effects of mechanical and dielectric solvent fluctuations on the quantum energy gap of a solute are described by the equilibrium solvent response function: C(t) = (~SU(0) 8U(t))/((~SU) 2 ) where U(t) is the value of the quantum gap at time t and 5U(t) = U ( t ) - (U) represents the deviation of the gap from its average value. For small
27 perturbations, the regression of fluctuations due to a non-equilibrium perturbation should decay in the same manner as those present in equilibrium. Thus, in the limit of linear response, the non-equilibrium correlation function S(t) = (U(t) - U(oo))/(U(O) - O(oo)) (1) where the overbar denotes a non-equilibrium average quantity, should be identical to the equilibrium response, C(t). We note that the typical time-dependent Stokes shift experiment which investigates the solvent dielectric response or the transient holeburning experiment used to explore mechanical solvation measure the non-equilibrium solvent response function S(t). Figure 3 presents a comparison of the non-equilibrium solvent response functions, Eq (I), for both the photoexcitation ("up") and non-adiabatic ("down") transitions (cf. Fig. 2). The two traces are markedly different: the inertial component for the downwards transition is faster and accounts for a much larger total percentage of the total solvation response than that following photoexcitation. The solvent molecular motions underlying the upwards dynamics have been explored in detail in previous work, where it was also determined that the solvent response falls within the linear regime, i~ Unfortunately, the relatively small amount of time the electron spends in the excited state prevents the calculation of the equilibrium excited state solvent response function due to poor statistics, leaving the matter of linear response for the downwards S(t) unresolved. Whether the radiationless transition obeys linear response or not, it is clear that the upward and downwards solvation response behave very differently, due in part to the very different equilibrium solvation structures of the ground and excited state species. Interestingly, the downwards S(t), with its much larger inertial component, resembles the aqueous solvation response computed in other simulation studies, 17 and bears a striking similarity to that recently determined in experimental work based on a combination of depolarized Raman and optical Kerr effect data. 18 While the difference in the upwards and downwards solvent responses presented in Figure 3 is striking, this is not the first time that variations in solvation dynamics for the same solvent have been observed. Experimental studies have shown differences in solvation response for different probe molecules in the same solvent.2 This is a direct indication that probe molecules which have different excited state charge distributions and different mechanical interactions with the solvent produce differing relaxation dynamics. Computersimulations have also observed differing solvation dynamics for the forward and reverse transitions of the sudden appearance of charge, indicative of a solutedependent solvent response.17 Moreover, theoretical work has shown that dielectric solvation dynamics is sensitive to the shape of a solute,19 and that solute size is intimately connected to viscoelastic relaxation.2~ It is these effects which are manifest in the
\
0
- - -
100 Time (fs)
Down
200
Figure 3. Photoexcitation (up) and non-adiabatic relaxation (down) solvent responses (Eq. 1)
28 difference between the upwards and downwards solvent responses for the hydrated electron. The size, shape and charge distribution of the electron change continuously (on the time scale of the solvent response) following photoexcitation, creating a situation where the solvent dynamics polarize the solute which then alters the solvent dynamics. The solvent response is determined by both the dynamically changing charge distribution and the mechanical forces of the growing electron pushing on the first solvent shell during its initial expansion. Upon the non-adiabatic relaxation, the electronic density quickly localizes to a shape about that of the final equilibrium ground state. The solvent response to this rapid change in charge distribution is also affected mechanically by the presence of the large void recently occupied by one lobe of the excited state electron. These microscopic differences in the upwards and downwards transitions have profound implications for the coupling of viscoelastic and dielectric solvation dynamics and the nature of the solvent response. Previous work examining dielectric solvation has ascribed the initial relaxation to rotational motions of individual solvent molecules (i.e., polar solvent molecules inertially reorient to create a more favorable dipole interaction with the new charge distribution of the solute). 4,5 The rapid expansion of the hydrated electron upon photoexcitation, however, displaces predominantly translational modes of the solvent. Water molecules in the first solvation shell are simply pushed back by the Pauli repulsion (mechanical) forces as the electron expands. Thus, much of the initial relaxation of the solvent is the launching of an acoustic wave following this sudden expansion of the solute: the relaxation is predominantly viscoelastic in character. There are three major pieces of evidence pointing towards the strong translational nature of the initial response. First, time-dependent pair distribution functions show large solvent molecule displacements relative to the solute center of mass on the inertial time scale. 10 Second, computed ultrafast transient spectroscopic traces show ringing at frequencies corresponding to intermolecular H-bond stretching and bending motions. 16 Finally, deuteration changes the time scale of the initial relaxation by < 10%, a value in accord with motion involving translation of an entire water molecule whose mass has increased from 18 to 20 amu (the corresponding change in moment of inertia if the initial dynamics were predominantly rotational in nature would be 2 !/2, resulting in a --40% increase in the inertial time scale). This isotope effect is illustrated in Figure 4. This idea of predominantly displacing translational solvent modes upon excitation leads to an appealing microscopic picture for the coupling of the mechanical and dielectric solvent response. Upon photoexcitation, the free energy will be lowered by both solvent molecule translations, accommodating the new solute size and shape, and solvent molecule rotations, creating favorable electrostatic interactions with the new
08 ~
1_ ~ x
H20
0.6
0.4? 0.2 0
I ~,, 50 100 T i m e (fs)
150
Figure 4. Solvent response following hydrated electron photoexcitation (Eq. 1) for H20 and I)20
29 solute charge distribution. With a continuously expanding solute like the newly excited hydrated electron, the effectiveness of rotational motions of solvent molecules in the first shell will be hindered due to the overwhelming Pauli repulsion forces driving these molecules translationally. This leads to a kind of "dielectric frustration", where the electrostatic relaxation accomplished through solvent rotations is rate-limited by the overall translational rearrangement which accompanies the viscoelastic relaxation. This idea has support in recent experimental work examining the solvent response of the polar solvent propylene carbonate to photoexcitation of a completely non-polar solute, stetrazine. 21 In this work, the solvent response function near room temperature, which is well fit by a simple analytic theory incorporating only viscoelastic relaxation, 20 is virtually identical to that observed using the polar dye molecule coumarin 152 as a probe. 22 This opens up the possibility that in the latter work, the dye molecule undergoes a change in size and shape as well as dipole moment, and that the observed dielectric solvent response is actually rate-limited by the mechanical relaxation, producing the remarkable agreement between the two very different experiments. The relatively small amplitude of the inertial component of the upwards solvent response may also be the result of this mechanical-dielectric solvation coupling. While the motion of the first shell molecules is inertial in the sense that the solvent-solvent forces play little role in the dynamics, the pressure on these molecules due to the expanding solute hinders their rotational motion. Thus, the decreased effectiveness of these initial ballistic motions can also explain the differences in inertial component amplitude observed in Figure 3 (40%) and in previous aqueous simulation work 17 (8090%) where only the charge but not the size of the solute was varied. The downward solvent response for the hydrated electron also fits nicely into this picture. The effective size of the ground state electron in one lobe of the excited state peanut-shaped cavity does not significantly change upon non-adiabatic collapse. Molecules in the first solvation shell along the lobe into which the electron localizes do not need to significantly translate to accommodate the newly formed ground state electron. Molecules along the newly vacant lobe can freely translate into the void unhindered, providing rapid solvation. This picture is in accord with the agreement between the downward S(t) and that determined experimentally from the Raman-OKE data; the Raman-OKE experiments are not sensitive to relaxation by solvent translations, and show this same type of dominant rapid inertial response. 18 V. Conclusions In summary, we have used quantum molecular dynamics simulations of the hydrated electron to investigate the coupling between the mechanical and dielectric solvent responses. The solvation dynamics following photoexcitation shows a strong degree of translational character, because the increase in size of the hydrated electron provides mechanical pressure on the first solvation shell. This mechanical displacement hampers the effectiveness of solvent molecule rotations in providing relaxation, and results in a smaller amplitude of the inertial response. Following non-adiabatic relaxation, molecules in the first solvation shell are either displaced very little, or can freely translate through the newly created void in the solvent, providing for rapid and highly effective solvation. These results suggest that there is strong interplay between the dielectric and mechanical solvent responses, and that in certain cases, "dielectric frustration" can occur in which dielectric relaxation may be altered or even rate-limited by the viscoelastic response of the solvent.
Acknowledgments This work was supported by the National Science Foundation. BJS thanks Dave Vanden Bout for many stimulating discussions. BJS gratefully acknowledges the support of a National Science Foundation Postdoctoral Fellowship in Chemistry, and the allocation of computational resources from the San Diego Supercomputing Center.
References 1. see, e . g . J . T . Hynes in Ultrafast Dynamics of Chemical Systems, edited by J. D. Simon, p. 345 (Kluwer, Dordrecht, 1994). 2. M. Maroncelli, J. Mol. Liq. 57, 1 (1993). 3. M. Maroncelli, P. V. Kumar, A. Papazyan, M. L. Horng, S. J. Rosenthal and G. R. Fleming, in Ultrafast Reaction Dynamics and Solvent Effects, edited by Y. Gauduel and P. J. Rossky, AIP Conf. Proc. 298, p. 310 (AIP Press, New York, 1994); E. A. Carter and J. T. Hynes, J. Chem. Phys. 94, 5961 (1991). 4. S.J. Rosenthal, X. Xie, M Du and G. R. Fleming, J. Chem. Phys. 95, 4715 (1991); R. Jimenez, G. R. Fleming, P. V. Kumar and M. Maroncelli, Nature, 369, 471 (1994). 5. see, e . g . M . Maroncelli, J. Chem. Phys. 94, 2084 (1991); L. Perera and M. L. Berkowitz, J. Chem. Phys. 96, 3092 (1992); Ibid. 97, 5253 (1992). 6. R.M. Stratt and M. Cho, J. Chem. Phys. 100, 6700 (1994). 7. J.T. Fourkas and M. Berg, J. Chem. Phys. 98, 7773 (1993); J. T. Fourkas, A. Benigno and M. Berg, J. Chem. Phys. 99, 8852 (1993). 8. J.G. Saven andJ. L. Skinner, J. Chem. Phys. 99, 4391 (1993). 9. F. A. Webster, P. J. Rossky and R. A. Friesner, Comput. Phys. Commun. 63, 494 (1991); F. A. Webster, E. T. Wang, P. J. Rossky and R. A. Friesner, J. Chem. Phys. 100, 4835 (1994). 10. B.J. Schwartz and P. J. Rossky, J. Chem. Phys. 101, 6902 (1994); Ibid., p. 6917. 11. K. Toukan and A. Rahman, Phys. Rev. B 31, 2643 (1985). 12. J. Schnitker and P. J. Rossky, J. Chem. Phys. 86, 3462 (1987). 13. B.J. Schwartz and P. J. Rossky, Phys. Rev. Lett. 72, 3282 (1994). 14 J. Schnitker and P. J. Rossky, J. Phys. Chem. 92, 4277 (1988). 15. R.B. Barnett, U. Landman and A. Nitzan, J. Chem. Phys. 90, 4413 (1989). 16. B. J. Schwartz and P. J. Rossky, J. Phys. Chem. 98, 4489 (1994); Y. Kimuar, J. C. Alfano, P. K. Walhout and P. F. Barbara, J. Phys. Chem. 98, 3450 (1994). 17. see, e.g., M. Maroncelli and G. R. Fleming, J. Chem. Phys. 89, 5044 (1988). 18. E. W. Castner Jr., Y. J. Chang, Y. C. Chu and G. E. Walrafen, J. Chem. Phys., in press; Y. J. Chang and E. W. Castner Jr., J. Chem. Phys. 99, 7289 (1993). 19. see, e.g.E.W. Castner Jr., G. R. Fleming and B. Bacghi, Chem. Phys. Lett. 143, 270 (1988). 20. M. Berg, Chem. Phys. Lett. 228, 317 (1994); B. Bagchi, J. Chem. Phys. 100, 6658 (1994). 21. J. Ma, D. Vanden Bout and M. Berg, private communication. 22. M. A. Kahlow, W. Jarzeba, T. J. Kang and P. F. Barbara, J. Chem. Phys. 90, 151 (1989).
journal o f
~IOLECULAR
LIQUIDS ELSEVIER
Journ',d of Molecular Liquids, 65/66 (1995) 3 I--40
Fluctuating Charge Force Fields for Aqueous Solutions Steven W. Rick, Steven J. Stuart, Joel S. Bader, and B.J. B e r n e D e p a r t m e n t of C h e m i s t r y and Center for Biomolecular Simulation C o l u m b i a University, NY 10027
Abstract
A new molecular dynamics model in which the point charges on atomic sites are allowed to fluctuate in response to the environment has been developed in a previous work (J. Chem. Phys., 101:6151 (1994)). The model and its application to liquid water are briefly reviewed. Various properties of the model are calculated, with emphasis on the bonding characteristics. The water model is also used to investigate the aqueous solvation of formaldehyde.
1
Introduction
In simple molecular force fields the intramolecular electronic structure is often modeled by point charges fixed on well-defined sites in the molecular frame. The charges are constant and thus cannot respond to changing electrostatic fields which arise from movement of the atoms during the simulation. In reality, molecular electronic structure can be strongly influenced by the external environment. Charge induction effects are not pair-wise additive and improved models must go beyond pair potentials. A new simulation method in which the charges are responsive to environmental changes, presented in a recent paper[l], combines the electronegativity equalization (EE) method for determining atomic charges[2, 3, 4, 5, 6, 7] and the extended Lagrangian method for treating fictitious degrees of freedom as dynamical variables [8, 9, 10, 11]. This approach treats charges on the molecular sites as dynamical variables by introducing fictitious kinetic energy terms and selfenergy terms for these charges into the Lagrangian for the system, along with Lagrange constraints ensuring electroneutrality. In this extended Lagrangian approach the charges are propagated according to Newtonian mechanics in a similar way to the atomic degrees of freedom. An application of this model to liquid water and comparisons to wen-known water potentials are given in P~ef. [1]. In this paper we briefly review the fluctuating charge method, and describe its application to both liquid water and a dilute aqueous solution of formaldehyde.
2
Dynamical Fluctuating Charge Models
In an isolated atom the energy of creating a partial charge, QA, can be expanded to second order
E(QA) = ~~
lr0
,a2
+ -~AA~A,
0167-7322/95/$09.50 r 1995 Elsevier Science B.V. All rights reserved. SSDi 0167-7322 (95) 00842-X
(2.1)
32 where )~4 and J~4A are parameters dependent on the atom type. The parameter ~ is the Mulliken electronegativity of the isolated atom (per electronic charge lel) and J~4A is twice the atom's hardness. For a molecule, the polarization energy to create a set of partial charges is a sum of these atomic self-energies and intramolecular Coulomb interactions, Job(r), or E p~
1
Y~fr176 + "~Y~ Y]~ Qi~Qi/3Ja~(rio,iz)- E~p, cr
a
(2.2)
/3
rio,iZ is Iria - ri/31 and E~ p is the gas-phase energy of molecule i. The total energy for Nmoler molecules includes this polarization energy plus the usual Lennard-Jones and intermolecular
where
Coulomb terms,
The electronegativity per unit charge of atom A is given by XA = (OU/OQA). The charges, by the EE principle, are those for which the electronegativities are equal. This is equivalent to minimizing the energy subject to a charge neutrality constraint. Since the potential is quadratic in the charges, the minimization will lead to a set of coupled linear equations for the charge. The charges are not independent variables since there is a charge conservation constraint. In the following we constrain each molecule to be neutral, ~ a Qia = O. We treat the charges as independent and use the method of undetermined multipliers to enforce the constraint. T h e Lagrangian is then Nmolec N . . . .
Nmolec Natom
lrno§ + E i=1
c):l
E
i--1
Nmolec
1M'"O.~ - U({Q}, {r}) - E
~:1
i=1
Na.tom
~' E
Qi~,
(2.4)
o--1
where mo is the mass of atom c~ and Mio is a fictitious charge "mass", with units of energy • time 2/charge 2, and the ,~i are Lagrange multipliers. The nuclear degrees of freedom evolve according to Newton's equation and the set of charges evolve in time according to Mi.Qi,~
= _OU({Q}, {r}) _ )~i = -;~i,~ - ~ . OQi.
(2.5)
If the total charge of molecule i is a constant of the motion, then it can be shown that )q is the negative of the average electronegativity on molecule i. The force on the charge is simply the difference between the average electronegativity and the instantaneous electronegativity at that site. The charge mass, Mia, a fictitious quantity, should be chosen to be small enough to guarantee that the charges readjust very rapidly to changes in the nuclear degrees of freedom[l]. The Coulomb interaction, Ji/(r), for intramolecular pairs is taken to be the Coulomb overlap integral between Slater orbitals centered on each atomic site, with each orbital characterized by a principal quantum number, ni, and an exponent (i[6].
Table 1: Properties for potentials with the TIP4P geometry: the fixed-charge TIP4P and the flexible charge TIP4P-FQ models. Properties listed are the the gas-phase dipole moment, the dipole polarizabilities, c~ii (the y and z directions lie in the plane of the molecule, with the z-axis along the C2 axis), the energy of the dimer in its minimum energy configuration, the distance between oxygen atoms for the minimum dimer configuration, and properties of the liquid as indicated.
Gas-phase dipole moment (Debye)
~zz (h3) %~ (A 3) ~ (h 3)
TIP4P ~ 2.18
TIP4P- FQ 1.85
0 0 o
0.82
2.55 o
experimental 1.85c 1.468+0.003 d 1.5284-0.013 a 1.4154-0.013 d -5.44-0.7 9 2.98 e
Dimer energy (kcal/mole) -6.3 -4.5 Dimer O-O length (~) 2.75 2.92 Liquid state properties (7'=298 K, p=l.0 g/cm a) Energy (kcal/mole) -10.1 a -9.9 -9.9 a 0.0 b -0.16-1-0.03 0.0 Pressure (kbar) Dipole moment (Debye) 2.18 2.62 53+21 79-1-8 78~ ~0 1 1.5925=0.003 1.79g Diffusion constant(10 -9 m2/s) ' 3.6+0.2 ~ 1.9+0.1 2.30 h 1.44-0.2b 2.1-t-0.1 2.1 / r#M. (ps) 74-2b 84-2 8.274-0.02J To (ps)
a) Ref. [13], b) Ref. [14], c) Ref. [15], d) Ref. [16], e) Ref. [17], f) Ref. [181, g) Ref. [19], h)
l~ef. [20],i)Ref. [21],j)Ref. [22]
3
Results
P u r e w a t e r . The fluctuating-charge method (fluc-q) has been applied to two commonly used models of water[l], the simple point charge model (SPC)[12] and the 4-point transferable intermolecular potential model (TIP4P)[13]. Here we focus on the fluc-q version of the T I P 4 P model ( T I P 4 P - F Q ) . Systems of 256 TIP4P-FQ or TIP4P water molecules have been simulated in the NVE ensemble at an average temperature of 298 K and a density of 1 g/cm 3, using Ewald summation for the long-ranged interactions. For more detailed discussion of the model parametrization and simulation methodology, the reader is referred to our previous work[l]. The properties of the TIP4P and T I P 4 P - F Q models for the water monomer, water dimer, and liquid water are listed in Table 1. The T I P 4 P - F Q electrostatic parameters are chosen to give the correct gas phase dipole moment for the monomer. The dimer properties listed in Table 1 are the energy of the minimum energy configuration and the oxygen-oxygen distance of this configuration. Pair potentials, such as SPC and TIP4P, are parametrized to give the experimental liquid state energies and radial distribution functions and tend to overestimate the gas phase water dimer energy. The fluctuating charge potentials predict an oxygen-oxygen separation closer to the experimental value but underestimate the dimer energy.
34 We have calculated both static and dynamical properties of liquid water at a temperature 298 K and a density of 1 g/cm 3. The parameters for both fluc-q models are chosen to give a binding energy of-9.9 kcal/mol. This energy includes the self polarization energy, which is the difference between the self-energy in the liquid phase and the gas phase (see Eq. 2.2). The average self-polarization energy is 5.7 kcal/mol, which represents a large contribution to the total energy. Pair correlation functions give detailed information about the structure of the liquid. The TIP4P-FQ model gives pair correlation functions that are in good agreement with the neutron diffraction results of Soper and Phillips[23]. For details see Ref. [1]. The static dielectric constant, e0, is calculated from the fluctuations in M , the total dipole of the central simulation box, by (4~p~ eo = %0 + \ 3kT ]
( M ~ -- (___M)2] Nmolec J "
(3.1)
For the TIP4P-FQ model, a 1 ns simulation gave e0 = 79-/-8, in good agreement with experiment. This is consistent with earlier findings that models with a dipole moment of 2.6 D have a dielectric constant near 80[24, 14, 25]. The value obtained for the optical dielectric constant, coo, is 1.59, close to the experimental value of 1.79119], and is underestimated because the perpendicular polarizability, axx, is zero. The frequency dependent dielectric constant, e(w), for the TIP4P-FQ model is shown in Figure 1. The agreement with the experimental results[22, 26, 27] is very good. The close agreement in the low-frequency microwave range is due to the accurate T I P 4 P - F Q values of ~0 and the Debye relaxation time, rD (the exponential decay constant for the autocorrelation function of the total system dipole, M ) . The features at frequencies higher than 300 ps -1 are due to bond stretches and bends absent in the rigid geometry models used here. The highest frequency feature given by the TIP4P-FQ model is the librational mode, which shows a peak in e" at 130 ps -1. The experimental peak is at 90 ps-1127]. The feature at 25 ps -1 has been interpreted as a translational vibration of a water molecule in its cage of nearest neighbors[28], and is not present in the spectrum for non-polarizable water models such as TIP4P[18] or Matsuoka-Clementi-Yoshimine (MCY)[29]. As argued by Neumann, this translational motion will not change the system's dipole moment much for non-polarizable models, but for polarizable models, the translational motion will induce a change in the dipole moment[18]. Therefore this feature only exists in polarizable models. We have investigated the differences in bonding between the fluc-q and fixed-charge models by examining the distributions of pair energies, solvation energies, and residence times. In a molecular model with fixed charges, the pair energy Eij is easily defined as the sum of all pairwise interactions between molecules i and j. For many-body potentials, however, there is no unique definition of a pair energy; the sum of the pair energies Etj does not add up to the total system energy because the self-polarization energy is not included. There are several reasonable ways to distribute a molecule's polarization energy among the pairwise interactions in which it participates, each leading to a different definition of the pair energy. We have defined a normalized pair energy /~ij by including a portion of each participating molecule's polarization energy proportional to the strength of the pair energy,
E'i = E'.i +
Eij E~j AEyOl+ AZpOi. ~ k # i Eik ~kCj Ekj
(3.2)
This weighting reflects that the polarization energy varies as the magnitude of the induced dipole, and that large dipoles will be induced in pairs with strong interactions.
35 A distribution of pair energies is shown in Figure 2 for both TIP4P-FQ and TIP4P. The large peak centered near zero consists of the many distant, weakly interacting pairs in the liquid. The much smaller peak at low energies represents the strongly-bound close neighbors. A convenient definition of a hydrogen bond is any pair bound more strongly than the minimum in this distribution (-2.7 kcal/mol for TIP4P,-2.6 kcal/mol for TIP4P-FQ). Integrating up to this threshold gives 3.1 hydrogen bonds for both models. (Other authors have reported larger hydrogen bonding numbers, but used a different cutoff for the hydrogen bond energy[13].) The longer tail on the T I P 4 P - F Q distribution demonstrates the ability of fluc-q methods to allow for cooperativity in bonding; TIP4P water has a fixed lower limit of-6.2 kcal/mol on any pair interaction, while T I P 4 P - F Q waters have
100
10
~g'
1
t:
0.1
ir, il 0.01 0.01
-------L----
0.1
1
10
100
1000
w/ps -1 Figure 1: Real(top) and imaginary (bottom) parts of the frequency dependent dielectric constant for the TIP4P-FQ model (solid lines), compared to experiment (dotted lines).
36 no such restriction and can form much stronger bonds when other neighbors are oriented favorably. The average strength of a hydrogen bond for TIP4P-FQ waters is 4.38 kcal/mol, compared to 4.32 kcal/mol for TIP4P. Cooperativity can also hinder bonding, and will generally serve to broaden the bonding distribution. Although the high-energy (weakly bound) tail of the H-bond distribution is lost in the zero-energy peak, the broadening can also be seen in the total bonding energy between one molecule and the rest of the simulation cell (obtained by summing Eq. 3.2 over a single index). The distribution of these values is centered at -20 kcal/mot for both models (twice the solvation energy per molecule) but the widths differ. The full width at half-maximum is 8.2 kcal/mol for TIP4P, and 10.1 kcal/mol for TIP4P-FQ. Another way of examining the hydrogen bonding character of the liquid is to determine the first passage time of neighbors out of the first coordination shell, a sphere of radius equal to the first minimum in the oxygen-oxygen pair correlation function (3.3/~ for both T I P 4 P and TIP4P-FQ). A distribution of first-crossing times shows that the TIP4P-FQ molecules are more labile than TIP4P waters at times shorter than 0.3 ps, with a probability of escaping from the first shell increased by 8% over TIP4P waters. But at times between 0.5 and 4 ps, the TIP4P waters leave more readily, with an 18% higher escape probability. This suggests that the fluc-q waters are more fickle in forming hydrogen bonds, breaking away from unfavorable interactions more easily. But once a bond has been formed, it tends to last longer. After 4 ps, the number of molecules who have not left the first solvation shell decays exponentially, with the same decay constant for both models. A survival probability can also be defined by allowing for multiple recrossings[30], but this obscures the short-time behavior which illustrates an important difference between the two models.
i
I
!
i
1.4 1.2
if j
/
]
11
",,
,
',,
!
0.8
/ ',, 9 "", ....
II
I
I
iI ,,/
N(E)
0.6 0.4 0.2 J
0
-8
/I I
-6
!
-4
,
I
1
-2
0
2
4
E (kcal/mol)
Figure 2: Pair energy distributions for TIP4P (dashed line) and TIP4P-FQ (solid line).
37 In general, however, the flexible charge models have slower translational and rotational timescales than the fixed-charge models, primarily due to the stronger electrostatic interactions from the higher charges. The self-diffusion constant, D, is smaller than the fixed-charge models and closer to the experimental value (see Table 1). Rotational time constants for both molecular and system dipole reorientation (rNMR and rD) are also slower and in good agreement with experiment. F o r m a l d e h y d e . Fluctuating charge models for water can also provide insight regarding the structuraJ, energetic, and dynamic aspects of hydrogen bond formation between a water molecule and a solute. One expects that polarizable water molecules next to a solute molecule more polar than bulk water will have a dipole moment enhanced over the bulk value. We have studied the enhancement of the induced dipole on water molecules around a polar solute. As we will show below, the magnitude of the induced dipole on a water molecule next to a solute is strongly correlated with certain aspects of the solvation structure. The system we studied consisted of 209 TIP4P-FQ water molecules and one formaldehyde molecule in a cubic box 18.6/~ on a side. The formaldehyde solute was based on a model of Levy and coworkers [31, 32]. The rigid, planar formaldehyde molecule has a CO bond length of 1.184/~, a CH bond length of 1.093 ~, and an OCtt angle of 122.3 ~ The charges on 0, C, and tt are -0.577, 0.331, and 0.123 lel respectively; the Lennard-Jones a parameters are 2.85, 3.296, and 2.744 ~'; and the Lennard-Jones e parameters are 0.20, 0.12, and 0.01 kcal/mol. Standard combining rules were used to obtain the Lennard-Jones parameters for the interactions between formaldehyde sites and the oxygen site of TIP4P-FQ water. This model for formaldehyde yields a dipole moment of 3.9 D, which is 50% larger than the mean dipole moment of a TIP4P-FQ water molecule in bulk water. The results we present in Figure 3 are from a 20 ps run in the NVE ensemble with an average temperature of 300 K. In the top panel of Figure 3, the mean dipole moment of a water molecule is shown as a function of the distance between the water oxygen and the formaldehyde oxygen. Waters closer than 3 are seen to have a dipole moment enhanced over the bulk dipole moment by as much as 0.3 D. Past 4 ~, the water dipoles rapidly approach the value of 2.62 D characteristic of the bulk liquid. It is interesting to note the dip in the dipole moment of water molecules about 3/~ away from the formaldehyde molecule. Waters in this region are caught between the first and second solvation shell. These water molecules presumably lack the opportunity to form a strong hydrogen bond with either the formaldehyde or a neighboring water molecule. Consequently a smaller dipole moment is induced on these water molecules than the mean dipole induced on bulk water. The bottom panel of Figure 3 shows the strong angular correlations exhibited by water molecules between the first and second solvation shells. The cosine of the OHO angle formed by the formaldehyde oxygen and the H O of a water molecule is nearly 1 for water molecules closer than 3 /?k, indicative of a strong linear hydrogen bond. There is a precipitous drop at 3 /~, signaling the breaking of the hydrogen bond as the water molecule leaves the first solvation shell. Other angular correlations do not show nearly as sharp a jump between the first and second solvation shells. The dashed line, for instance, depicts the cosine of the angle made by the formaldehyde CO and a water oxygen. This angle tends to be more linear for waters closer to the formaldehyde, and angular correlation is lost slowly as the water moves away. The cosine of the angle between the symmetry axes of formaldehyde and water is shown in the dotted line. This angular correlation is also weak compared with the dramatic behavior of the OHO angle.
38
2.9
(,.,o>
I
I
I
I
I
2.8
(Debye) 2.7 2.6 2.5
2
0.75
0.5
(cos(0)) 0.25
i
i
i
i
i
i
i
2.5
3
3.5
4
4.5
5
5.5
I
I
I
I
I
I
~ "'""......
~
9
~
f
~
.'-..
QIIO '..'"'"....-......." - - . 7 - " = (
-0.25
f
~...
6
-
,, -.-.~.<---~ ......-. . . . . . . . " s"--,.s" " ~ ......
"
"'"
I
I
I
I
I
I
I
2.5
3
3.5
4
4.5
5
5.5
6
R (~,1 Figure 3: Dipole moments of TIP4P-FQ water molecules are shown with angular correlations with a formaldehyde solute. Top panel: The average dipole moment (p) of fluctuating charge water molecules is shown as a function of Roo, the distance from the water oxygen to the formaldehyde oxygen. B o t t o m panel: (cos(0)) for three different angles made between CH~O and H20 as a function of Roo. Solid line: O = O - HO angle for the H on 1~I20 closer to the O on CH70; Dashed line: 8 = CO - O angle; Dotted line: O = the angle between the C2v axes of CH~O (taken from C to O) and H20 (taken from the HH midpoint to O).
4
Conclusion
The fluctuating charge model, as applied to liquid water, was shown to be an effective method for treating polarization effects. The fluc-q method introduces no intermolecular interactions beyond those already present in fixed-charge models, and the propagation of the charges using extended Lagrangian methods increases the computational cost by only a modest amount (about 10%). In contrast, other methods which include polarizability to dipolar order introduce an additional interaction (the dipole-dipole interaction) which, together with solving for the induced dipole moments,
39 increases the CPU time by about a factor of 2 over fixed charge models[33, 34]. The electronic properties of the fluc-q water model are such that in the gas phase it gives the correct dipole moment and in the liquid phase the dielectric properties are well reproduced for a range of frequencies (see Figure 1). In addition, the fluc-q model gives good estimates for the liquid-state radial distribution functions and dynamical properties such as the diffusion constant and "rNMR (see Table 1). In comparison to fixed-charge models, average bonding properties appear to be no different under the fluctuating-charge model, but the charge flexibility and many-body character of the potential both help to broaden the distribution of bonding energies and encourage cooperative bonding behavior. In studies of formaldehyde solvation, we detect a strong correlation between the dipole induced on a water molecule and the presence of a linear hydrogen bond between the water and the solute. We are conducting ongoing studies of solvation dynamics to further characterize the relationship between electronic polarization, solvation structure, and solvation energetics. Other current work includes studies of the solvation of ions, using a Drude dispersion oscillator to model the ion polarizability, and of small organic molecules, examining the coupling of the molecular charge distribution to torsional bond rotation, to functional group substitution, and to the aqueous environment.
Acknowledgements This work was funded by a grant from the National Institutes of Health (GM 43340) and was done on the Thinking Machines CM-5 in the NIH Biotechnology Resource Center at Columbia University. SJS was supported by a National Defense Science and Engineering Graduate Fellowship (DAAL 03-91-G-0277).
References [1] S.W. Rick, S.J. Stuart, and B.J. Berne. J. Chem. Phys., 101:6141, (1994). [2] R.G. Parr and W. Yang. Density-Functional Theory o/Atoms and Molecules. Oxford University Press, Oxford, 1989. [3] L. Do~en-Mi~ovid, D. Jeremid, and N.L. Allinger. J. Am. Chem. Soc., 105:1716, (1983). [4] W.J. Mortier, K. Van Genechten, and J. Gasteiger. J. Am. Chem. Soc., 107:829, (1985). [5] K.T. No, J.A. Grant, and H.A. Scheraga. J. Phys. Chem., 94:4732, (1990). [6] A.K.R.app~ and W.A. Goddard. J. Phys. Chem., 95:3358, (1991). [7] C.A. Reynolds, J.W. Essex, and W. G. Richards. J. Am. Chem. Soc., 114:9075, (1993). [8] H.C. Andersen. J. Chem. Phys., 72:2384, (1980). [9] M. Parrinello and A. Rahman. Phys. Rev. Lett., 45:1196, (1980). [10] R. Car and M. Parrinello. Phys. Rev. Lett., 55:2471, (1985). [11] S. Nos6. Mol. Phys., 52:255, (1984). [12] H.J.C. Berendsen, J.P.M. Postma, W.F. von Gunsteren, and J. ttermans. Forces, page 331. edited by B. Pullman (/teidel, Dordrecht, Holland), 1981.
Interrnolecular
40 [13] W.L. Jorgensen, J. Chandrasekhar, J.D. Madura, R..W. Impey, and M.L. Klein. J. Chem. Phys., 79:926, 1983. [14] K. Watanabe and M.L. Klein. Chem. Phys., 131:157, (1989). [15] A.C. Shepard, Y. Beers, G.P. Klein, and L.S. Rothman. J. Chem. Phys., 59:2254, (1973). [16] W.F. Murphy. J. Phys. Chem., 67:5877, (1977). [17] J.A. Odutola and T.R. Dyke. J. Chem. Phys., 72:5062, (1980). [lS] M. Neumann. J. Chem. Phys., 85:1567, (1986). [19] A.D. Buckingham. Proc. R. Soc., 238:235, (1956). [20] K. Krynicki, C.D. Green and D,W. Sawyer. Discuss. Faraday Sot., 66:199, (1978). [21] J. Jonas, T. DeFries, and D.J. Wilber. J. Chem. Phys., 65:582, (1976). [22] U. Kaatze and V. Uhlendorf. Z. Phys. Chem. N.F., 126:151, (1981). [23] A.K. Soper and M.G. Phillips. Chem. Phys., 107:47, (1986). [24] S.L. Carnie and G.N. Patey. Mol. Phys., 47:1129, (1982). [25] M. Sprik. J. Chem. Phys., 95:6762, (1991). [26] M.N. Afsar and J.B. Hasted. J. Opt. Soc. Am., 67:902, (1977). [27] A.N. Rusk, D. Williams, and M.R.. Querry. J. Opt. Soc. Am., 61:895, (1971). [28] G.E. WaJrafen. Water-A Comprehensive Treatise, page Vol. 1. edited by F. Franks (Plenum, New York), 1972. [29] M. Neumann. J. Chem. Phys., 82:5663, (1985). [30] R. W. Impey, P.A. Madden, and I.R. McDonald. J. Chem. Phys., 87:5071, (1983). [31] J. T. Blair, K. Krogh-Jespersen, and R. M. Levy. J. Am. Chem. Soc., 111:6948, 1989. [32] R. M. Levy, D. B. Kitchen, J. T. Blair, and K. Krogh-Jespersen. J. Phys. Chem., 94:4470, 1990. [33] A. Wallqvist and B.J. Berne. J. Phys. Chem., 97:13841, (1993). [34] D. van Belle, M.P.G. Lippens, and S.J. Wodak. Mol. Phys., 77:239, (1992).
journal of
MOLECULAR
LIQUIDS ELSEVIER
Journal of MolecularLiquids, 65/66 (1995) 41-48
Soivation Dynamics of Dye Molecules in Polar Solvents Studied by Time Resolved Hole Burning Spectroscopy Katsura Nishiyama, Yasuaki Asano, Nobuhisa Hashimoto, and Tadashi Okada
Department of Chemistry, Faculty of Engineering Science and Research Center for Extreme Materials, Osaka University, Toyonaka, Osaka 560, Japan Abstract Tune resolved ground state hole spectra of cresyl violet in acetonitrile, methanol, and ethanol at room temperature have been measured in subpicosecond to picosecond time region. The time correlation function of the solvent relaxation expressed by the hole width was obtained. The main part of the correlation function decayed much slower compared with that of the reported correlation function observed in time dependent fluorescence Stokes shift. Some possible mechanisms are proposed for understanding of the time depencences of the spectral broadening under the condition with the distribution of the relaxation times in fluid solution based on the entropy term in the solvent orientation as well as the site dependent response of the solvent.
Introduction Absorption and fluorescence spectra of molecules in solution are broadened, in general, compared with those in gas phase because the energy of the molecule dissolved in solvent is fluctuating due to the orientational fluctuation of the surrounding solvent molecules. The fluctuational motion of the solvent plays an important role in chemical reactions in solution phase such as an electron transfer reaction, for example, where the energy fluctuation of the reactants caused by the solvent fluctuation is considered to be essential to realize the transition state of the reaction. In order to understand the dynamics of the solvent fluctuation, many experimental as well as theoretical efforts have been made intensively in the last decade. One of the most convenient methods to observe solvent reorganization relaxation processes within the excited state molecule is time resolved fluorescence spectroscopy. By using time resolved techniques a time dependent fluorescence peak shift, so called dynamic Stokes shift, has been dete~ed in nanosecond 1,2, picosecond3,4, and femtosecond5,6 time regions. Another method to observe solvent relaxation processes is time resolved absorption spectroscopy. This method is suitable for the observation of the ground state recovery of the solvent orientational distribution surrounding a solute molecule. We have performed picosecond time resolved absorption spectroscopy for organic dyes in alcoholic solution 7 and have shown the following results. The spectral shape of the difference spectrum before and after the excitation is expressed as the superposition of the absorption and fluorescence spectra detected under steady state condition when the solvent relaxation time is sufficiently short compared with the time resolution of the experimental equipment and the excited state lifetime. On the other hand, the spectrum in the viscous solvent at low temperature shows slightly sharp in initial and broadens its shape with time. Tune evolution of the ground state hole as well as fluorescence spectra initiated by a short pulse laser irradiation in solution has been conventionally explained in terms of the twodimensional configuration coordinate model by Kinoshita8. According to his theory, two adiabatic potential curves corresponding to the ground and excited states are assumed to have the same curvature but have the different potential minimum in the configuration coordinate, 0167-7322/95/$09.50 91995 ElsevierScience B.V. All rights reserved. SSDI0167-7322 (95) 00843-8
42
mg(Q,q)
- a Q 2 + bq 2,
to(Q,q) - a(Q
(1)
- 00) 2 4- b(q
- q0) 2 + e',
(2) where Q and q are the coordinates which give the slow and fast relaxation processes, respectively. Qo and q, denote the positions of their energy minima, a and b are the curvatures of the two adiabatic potential curves. In this model, three relaxation processes are taken into account: the relaxation in the ground and excited states, nonadiabatic one between the two states, and time evolutions of their populations are treated as stochastic processes along their potential surfaces. Tinae resolved hole burning spectra (transient absorption spectra) are thus expressed as
lv.s(w,,w2,t) ~162 -{[AI+[BI+[C]I+[DI
(3)
with
[A] - f d ~ ( o ~ ) ~ - ' ( a '
- o '4p . 2--1/2 ) exp[-Am 2 / G 2 ] e x p [ - { A O O e - 0 , : p = A o ) / G 2 } 2 / { G 2
-- (:Ts4p.2 / O'2}], [B ] .. e - r,f d tog( to ),r - , ( 0, 4 _ o,p,,, _ 4 _ 2.-,,2 ) exp[-Ato z/0,2]exp[-{At02 + 2bq 2
+2aQo2 (l - P . ) -
[C1-
0,s2p.Ao3 / cr2}2 /{ G2 - 0.4_ s P . 2 /o"2}],
-(1/Z'o)flodVfdeog(eo)x-'(a'o':p.2) -112 exp[-A(,02 / 0,2 ]exP[-{A~02 -
-u~p. A~ / a~} ~/{o ~ - ~'p
~/ a~}]cxp[-t/~o
+ 2a02(p.1 - Pro)
],
where g(~o) is a normalized spectral function of the exciting fight pulse centered at an angular frequency of a~1, and oJ2 is that of the probe light. A ~ o - o J - e , AoJ2 - ~ 0 2 - e , E " E'+ a Qo2 + bq 2, 0,2 w 0,2s +0,f2 with 0,2.4aQ2okT and 0,f-4bq2okT where k Boltzmann constant and T absolute temperature, z"0 is a population decay time from the excited state to the ground state. /9=-" p,,(t) is a normalized correlation function of the fluctuation and p,,,l = p,, (t - z). Here the same fluctuations are assumed for the ground and excited states. The terms [A] and [B] come from a hole created in the ground state population and the population created in the excited state by the light irradiation, respectively. In this expression the time resolved fluorescence spectrum can be expressed as only the term [B]. The term [C] corresponds to the additional population created by the nonadiabatic decay from the excited state to the ground state, i.e., hole filling effect. For an early time after the excitation, the term [C] does not contribute to the hole spectrum, because the amount of the relaxed population from the excited state is small. The term [C] is important only when the solvent relaxation time is comparable to the excited state lifetime. In addition to the three terms described above, the absorption spectra from the lowest excited state to the higher excited states (Sn*-'S1 spectra) are overlapped as the term [D] in eq (3). When the solvent relaxation is fast enough compared with the time resolution of the experimental equipment and the excited state lifetime, by putting p,, ( t ) - 0 for t > 0, the spectral shapes given by the terms [A]+[C] and [B] are reduced to
[A]+[C]-exp[-t/Vo]~r-eexp[-Aoge/0,e]exp[-Ao92/2 0,2 ], [ B ] - e x p [ - t / v 0 ] 0 , -2 exp[-Aw 2/G2]exp[-(Ao92 + 2aQ 2 + 2 b q 2 ) 2 / G 2 ] ,
(4) (5)
apart from the unimportant factor. The spectral shapes corresponding to the term [A]+[C] and [B] are coincident with those of the stationary absorption and fluorescence spectra, respectively. Then the hole spectrum is expected to be expressed by the sum of the steady state
43 absorption and fluorescence spectra. Under this condition, we can obtain the term [D] in eq (3), the Sn~--S1 absorption spectrum of the solute molecule, subtracting [A]+[C] and [B] from the hole spectrum. According to eq (3) the time dependences of the hole spectrum on the excitation energy are as follows. When we excite the dye solution at the center of the absorption band, a hole is created around the peak of the ground state population, while an excited state population created by the light irradiation shows a considerable deviation from the energy minimum of the adiabatic potential curve. The subsequent relaxation process causes a time dependent shift in position for the excited state population, while the ground state hole only broadens with time. Consequently, the fluorescence spectrum shows a time shift. The transient absorption spectrum also shifts toward the low energy side with broadening. However, this shift is caused mainly by the time dependent shift of the excited state population. Contrary to this, the case where the excitation is chosen at the tail of the low energy side of the absorption band is completely different. In this case, a hole is created at the tail of the ground state population and then shifts gradually to the peak of the initial population, while in the excited state the population is created almost at the bottom of the potential curve and hence does not show the time dependent shift. Therefore, the shift of the hole spectrum is caused mainly by the time dependent shift in the ground state population, and the shift in this case occurs toward the higher energy side. It is well known that the fluctuational behavior of the solvent molecules is characterized by the normalized time correlation function of the fluctuation, p,, (t), which is expressed as a following equation using observed time resolved fluorescence spectra,
p.(t)- v[(t)- ~1(oo), ~ (0) - r (~)
(6)
where ~;r(0), vr(t), and ~;i(o0) denote the peak energy of the fluorescence spectrum at t
=
0,
t, and oo, respectively. Fluorescence Stokes shift, i.e., ~5.(0)- vi(t), is usually explained as the dielectric dispersion of the surrounding medium and interaction of the solute dipole with the reaction field, in which the energy shift in the fluorescence spectrum is proportional to the square of the difference of the dipole moments between the ground and excited states of the solute molecule. On the other hand, the spectral width is linearly related with the difference of the dipole moment under the assumption included in eqs (1) and (2). Then p,, (t) is expressed using spectral widths as,
[,~(t)'- _,~(o~),~1~9m (t) ~ , V ' ~ 2
--. C~'(00) 2
(7)
where &(t) is the half width of the spectrum at time t. In this paper we investigate the time dependent ground state hole spectrum of cresyl violet in polar solvents by means of subpicosecond transient absorption spectroscopy. The time correlation function expressed by eq (7) showed large difference in time profiles compared with the reported one expressed by eq (6). Possible mechanisms will be discussed.
Experimental Time resolved hole burning spectra were measured by means of a femtosecond transient absorption spectrometer system. A second harmonics of a mode locked cw Nd 3+:YAG laser (Quantronix, 82MHz) was used for a pumping source. A synchronously pumped rhodamine 6G dye laser with a saturable absorber dye jet (DODCI/DQOCI) and dispersion compensating prisms in the cavity was used. The output of the dye laser (100fs fwhm, 600pJ/pulse) was
44 amplified up to ca. 400 p J/pulse by 4-stage dye amplifier excited by a regenerative YAG amplifier (Continuum) with 50Hz. The optical arrangement of the spectrometer is illustrated in Fig. 1. The amplified laser pulse was divided into two by a beam splitter; one was used as a pump pulse collimated to a 2mm sample cell, the other was focused into a quartz cell containing I)20 to produce a continuum probe pulse which was used for the absorption measurement employing the combination of a spectrograph and a multichannel photodiode array. A small part of the probe pulse was used as a reference employing another spectrograph/detector combination. The time resolution of this system was about 150fs. Dispersion effect of the probe pulse was corrected automatically using calibration table obtained from the measurements of a cross-correlation between pump pulse and continuum probe pulse or of a rise curve at each wavelength of a broad band Sn*--S 1 absorption spectrum such as chrysene and porphyrins. Cresyl violet (Exiton) was used as received. Spectrograde acetonitrile, methanol and ethanol were used without further purification. Two or three samples of each solvent with different dye concentration in the order of 10-5M were used for the measurements. No concentration effect on the time correlation function was observed. ,,
I
I~
H2
0
H2
cresyl violet
CPU
PC
^i.,
V, ~ j L P ~ ' f
s,.a.
\
M aS
M
I" O C ~ (DO)
]
"~:.-,.S..
,a f .......
~,
L .......
"
(c)
i
~'--'
E~L 7L
fromDyek'~. FIG. 1. The optical arrangement of the spectrometer. L:lens, BS:beam splitter, M:mirror, D:depolarizer, F:filter, PC:spectrograph, S:shutter, MCPD:multichannel photodiode array detector, CPU: computer,
Results
4OO
~) 6O0 wavelengthInm
700
FIG. 2. Steady state absoption and stimulated fluorescence spectra of cresyl violet in acetonitrile (a), methanol (b), and ethanol (c) at room temperature. Exciting wavelength is indicated by the arrow.
and Discussion
Normalized steady state absorption and stimulated fluorescence spectra of cresyl violet in polar solvents are shown in Fig. 2. Stimulated spectra were calculated from the fluorescence spectra divided by the square of the wavenumber. Exciting wavelengths in each solvent used for the measurements are also indicated in the figure by arrows. Mirror image relation between
45 absorption and fluorescence specWa was observed in each solvent and the amounts of Stokes shift were about 800em "1 in acetonitrile and methanol, and 700crn -1 in ethanol, indicating that the difference of the dipole moment of cresyl violet between the ground and fluorescent states is rather small, and that the equal curvature of their potential curves assumed in the model discussed in the previous chapter may be acceptable in this system. The exciting wavelengths used here correspond to the 0-0 transition of the molecule according to the literature9.
9
9
9
9
9
9
",...
(2 4ee
,~e
9 see
sse
~/
9 .;
O#
4) 6ee
6~
"tee
Toe
~e
o
0
.....
5
. . . .
,
10'
'
t i m e / ps
wavelength / rn
FIG. 3. Estimation of the Sn<--S1 absorption spectrum of cresyl violet in ethanol. 1: observed spectnnn at 166ps, 2: steady state absorption spectrum; 3: stimulated fluorescence spectrum; 4: superposition of absorption and stimulated fluorescence spectra; 5: the Sn<--S1 absorption spectrum.
FIG. 4. Time dependence of the absorbance of the Sn<-- S1 absorption at 500nm in methanol.
To obtain the Sn*--S1 absorption spectrum, the transient absorption spectrum (the difference of the absorbance with and without the excitation) detected at sufficiently long delay time after the excitation, when the solvent reorientation is considered to reach thermal equilibrium, was used. In Fig. 3, Sn*--S1 absorption spectra of cresyl violet in ethanol is indicated by a solid line (5) together with the transient spectra detected at 166ps after the laser excitation (curve 1), terms [A] and [B] in eq (3) at the steady states (curves 2 and 3, respectively), and [A]+[B] (curve 4). The contribution from the term [C] is negligible in this case, because the fluorescence life time of cresyl violet is in nanosecond time domain. The obtained absorbance of the Sn*--$1 spectrum rises with the response function of the detecting system and then decays with the fluorescence life time as indicated in Fig. 4. Transient absorption spectra in acetonitrile and ethanol are shown in Fig. 5. The spectra observed at shorter delay time than 100fs after the excitation are omitted here due to the contribution of the coherent coupling of the excitation and the probe light pulses. The spectra obtained in methanol were similar to those detected in ethanol. The peak shift of the spectra in acetonitrile could not be detected after the delay time longer than 100fs, while in the case of ethanol solution the red shift of a few nm in hundreds of femtosecond time region was observed, which was ascribed to fluorescence Stokes shift due to the excitation at the peak position of the absorption band as described previously. It should be noted here that the spectral broadening was observed until much longer time after the peak shift, although the time dependence of the spectral broadening and the peak shift should be consistent with each other under the assumption in eq (3). The ground state hole spectrum, the term [A] in eq (3), was obtained by subtracting the contributions from the term [B] and the Sn*'-S1 spectrum from the observed time resolved spectra shown in Fig. 5. The contribution from the Sn*---S1 absorption spectrum at each time
46 was corrected using the time dependence of its absorbance in each solvent. It was difficult in our present system to measure the time dependent fluorescence spectra with same time and energy resolutions as the transient absorption spectra. Therefore, we adopted the half width at the half maximum (hwhm) at the shorter wavelengths side of the spectrum without correction of the term [B] as a ground state hole width. The estimated error under this approximation was less than 8%.
(1)
(2)
O.2ps ._>.
0.4ps
0.2ps ~"
0.4ps
0 0 tin L_ 0 m ..I:3 m
I
400
I
I
I
500 600 700 w a v e l e n g t h / nm
I
800
( 400
I
I
I
500 600 700 w a v e l e n g t h / nm
I
800
FIG. 5. Time resolved hole burning spectra in acetonitrile (1) and ethanol (2) at room temperature. The time evolutions of the ground state hole widths in acetonitrile (1) and methanol (2) solutions are plotted in Fig. 6. The hole widths broadened up to 200-300cm "1 in the instrumental response time of the system in the all solvents used here. The band width of the fluorescence spectrum of cresyl violet in methanol/ethanol mixed glass solvent at 77K was also observed as about 300cm -1 hwhm. The rapid relaxation within the response time observed in the hole spectrum as well as the relaxation m 77K glass may correspond to the relaxation due to the librational or inertial component of the solvent which was reported in the studies of dynamic Stokes shift in femtosecond region 5,6. Fig. 7 shows the time correlation functions of solvent fluctuation in accordance with eq (7) using the values plotted in Fig. 6. In this calculation we used o,(0)=30cm -1 as the spectral width of the exciting laser pulse and 1430cm -1, 1360crn "1 and 1250crn -1 for ~oo) in acetonitrile, methanol, and ethanol, respectively. In Fig. 7 the reported correlation functions obtained by the dynamic Stokes shift measurements of LDS750 in acetonitrile5 and 1aminonaphthalene in methanol 6 in accordance with eq (6) are also plotted by dashed lines. According to the literatures 5,6, the correlation function decayed more than 80% in acetonitrile in the first 200fs and about 60% in methanol in the first 500fs. The peak shift of the absorption spectra in the present work was completed in the fin'st lps in methanol and ethanol solutions as indicated in Fig. 5 (2). However, the peak shift in acetonitrile solution could not be observed in the time resolution of our system. It is obvious from above results that the major part of the energy relaxation due to the fast response of the solvent dynamics is taken place in a few
47
hundreds of femtosecond in acetonitrile and lps in methanol. The spectral band shape, however, seems to be controlled by much slower solvent dynamics as indicated in Fig. 7. 9
I
'
i
1.5 ..(1) ,r", |
E
0
,
i
I
==_11= . . . . = _ = . = . . : ~(oo'
..............
,
(2)9 ===
"7,
E
1
i
!
,
i
9
i
9 ,,
_
- ..............................
1
-f
9
=_..: ~(=)
|
03
Q
03
o 'N"
""
%,
0.5
0.5
/ ,
-, 0-:ill 1 ~.,..
9
/
i
i
0
0
I
,
0
I
i
i
20 time / ps
t i m e / ps
'
410
'
FIG. 6. Hole widths of cresyl violet as a function of time in acetonitrile (1) and in methanol (2). 1
, ~ 0
~
"
!
9
o
.
1
.,
mOO0 .
.
.
~
(1)
~"
.
~'
(2)
% ~
r 0.5
o o
",..
"'-..,.
o
o
o
0
"-.
0
"""-.
0 0
o
"' 0
o
o
\
0
. . . . .
Ooo o
', ;
...
.
o
o
t
.
o ". . . . . . . .1. . . time / ps
.. 2
c
~
. . . .
~
,f
. . . . o.
2~
10
30
s0
time / ps
FIG. 7. Normalized correlation functions obtained from the analysis of the hole width in acetonitrile (1) and methanol (2). Dashed curves are reported P,, (t) of LDS750 in acetonitrile 5 and 1-aminonaphtalene in methanol 6, see text. In the previous paper 7 we have observed both the ground state hole shift and the hole broadening of rhodamine 640 in subnanosecond to nanosecond region in methanol/ethanol mixed solvent at 170K. It is also reported that the time constant of dynamic Stokes shift agreed with that of the broadening of the fluorescence band width in nanosecond time region in the systems of laser dyes in the low temperature viscous alcoholic solvent z. Therefore the two dimensional configuration coordinate model seems to work well in viscous alcoholic solvent at low temperature. In the fluid solvent at room temperature, however, the situation seems to be different. In the fluid solvents the fast and slow components are considered to be distributed in the same orders of magnitude in time scale, that is, the rotational relaxation times of the solvent molecule may distribute in tens of femtosecond (rotation of OH for example) to picoseconds
48 (molecular rotation) and the times of translational motions may distribute in subpicosecond to tens of picosecond. In the course of the energy relaxation mainly due to the fast component of the solvent response, some slower rotational and translational motions (corrective motion) surrounding the solute molecule may bring about the disturbance of the configulation which was distributed in the thermal equilibrium one before the laser excitation. This entropy term in the free energy of the system including the solvent orientational distribution may be considered to be a possible mechanism to explain the discrepancy between the energy and the spectral shape relaxation times observed in this work. It is also important to take into consideration the effect of the site dependent relaxation time. When a solute molecule of a larger size is dissolved into solvent molecules of a smaller size, the relaxation time around each atom of the solute molecule which has the different charge density, in general, may differ from each other due to the different solvent response to the local charge. The solvation dynamics depending on the solute charge as well as solvent has been studied by computer simulations of molecular dynamics6 and of the theory based on the site-site Smoluchowski-Vlasov equation]~ The results show remarkable specificity depending on the solvent and the local charge especially in the picosecond time region. In the case considered the site dependent correlation function, the total correlation functions in eqs (6) and (7) may be 1/2, respectively, where a, and b, approximated as Pm(t)- 2 a , p, and Pm(t)- {2 b~pj2} denote the factor of the contribution from the local correlation function, p~, around the atom i of the solute molecule. The contribution of this type of the site dependent relaxation times may be another possibility to explain the results in the present system. Acknowled2ments The authors are grateful to Prof. F. Hirata of Kyoto University, Prof. S. Kinoshita of Osaka University, and Dr. A. Yoshimori of Nagoya University for fruitful discussion on the relaxation processes of solution. The present work was partly supported by a Grant-in-Aid (06NP301 and 06239240) to T.O. from the Ministry of Education, Science, and Culture of Japan. References 1. 2. 3. 4. 5. 6. J. 7. 8. 9. 10.
N.G. Bakshiev, Y. T. Mazurenko, and I. V. Piterskaya, Opt. Spectrosc. 21,550 (1966). (a) S. Kinoshita, N. Nishi, and T. Kushida, Chem. Phys. Lett. 134,605 (1987). (b) S. Kinoshita and N. Nishi, J. Chem. Phys. 89, 6612 (1988) and references therein. V. Nagarajan, A. M. Brearley, T. -J. Kang, and P. F. Barbara, J. Chem. Phys. 86, 3183 (1987). E.W. Castner. Jr., M. Maroncelli, and G. R. Fleming, J. Chem. Phys. 86, 1090 (1987); M. Maroncelli and G. R. Fleming, J. Chem. Phys. 86, 6221 (1987). (a) S. J. Rosenthal, X. Xie, M. Du, G. R. Fleming, J. Chem. Phys. 95, 4715 (1991). (b) M. Cho, S. J. Rosenthal, N. F. Schemer, L. D. Ziegler, and G. R. Fleming, J. Chem. Phys. 96, 5033 (1992). M. Maroncelli, P. U. Kumar, A. Papazyan, M. L. Hong, S. J. Rosenthal, and G. R. Fleming, in "Ultrafast Reaction Dynamics and Solvent Effects," Eds. Y. Gauduel and P. Rossky, AlP Conference Proceedings, AlP Press, New York, 1994, 298, p. 310. (a) S. Kinoshita, H. ltoh, H. Murakami, H. Miyasaka, T. Okada, and N. Mataga, Chem. Phys. Lett. 166, 123 (1990). (b) H. Murakami, S. Kinoshita, Y. Hirata, T. Okada, and N. Mataga, J. Chem. Phys. 97, 7881 (1992). S. Kinoshita, J. Chem. Phys. 91, 5175 (1989). C . H . B . Cruz, R. L. Fork, W. H. Knox, and C. V. Shank, Chem. Phys. Lett. 132, 341(1986). F. Hirata, T. Munakata, F. Rained, and H. L. Friedman, in this proceedings.
journal of
~OLECULAR
LIQUIDS ELSEVIER
Journal of MolecularLiquids.65/66 ( !995) 49-57
Uitrafast Measurements of the Dynamics of Soivation in Polar and Non-Dipolar Solvents J. Gardecki, M. L. Horng, A. Papazyan, and M. Maroncelli Department of Chemistry The Pennsylvania State University University Park, PA 16802 ABSTRACT The dynamics of solvation of the probe solute coumarin 153 has been measured with -100 fs time resolution using the fluorescence upconversion technique. A wide variety of solvents, including polar aprotic solvents, hydrogen bonding solvents, and non-dipolar solvents have been examined. For all solvents of even moderate polarity (e0>5), the solvation dyna~cs observed follows the predictions simple models based on the solvent's bulk dielectric response. For a number of "non-dipolar" solvents such as dioxane and benzene, for which e o <5, we observe surprisingly large time-dependent shifts, which seem to reflect the solute interacting with the large quadrupole (and higher multipole) moments of the solvent molecules. These dynamics represent a solvation distinct from the dipolar solvation dynamics previously studied and they cannot be modeled in terms of any currently available theory.
INTRODUCTION An understanding of the time-dependence of solvation is a key ingredient in modern theories of chemical reaction in solution [1]. For this reason there has been a great deal of research focused on measuring and building predictive theories of this dynamics [2,3]. The main concern to date has been on polar solvation dynamics, where the interactions are strongest and are predicted to have large effects on electron and other charge-transfer reactions [1]. A few recent studies also begun to consider the dynamics of solvation in nonpolar solvents [4-6]. In either case, the phenomenon of interest is the time-dependent relaxation of the solvation energy subsequent to some step-function change in the solute's electronic structure. This dynamics is most often measured by monitoring the fluorescence of an appropriately chosen solute upon electronic excitation. (Other, less direct methods, are also under current development [7].)
0
Coumarin 153 0167-7322/95/$09.50 91995ElsevierScienceB.V. All rights reserved. SSD10167-7322 (95) 00844-6
50 In the present report we describe new time-resolved measurements of solvation dynmnics using the probe molecule coumarin 153 (hereafter referred to as C153) [8]. This weft-known solvation probe has been thoroughly characterized in a number of past studies [2,9]. Its main attribute which makes it an excellent solvation probe is that upon electronic excitation its dipole moment changes from a ground state value of - 6 D to a value of--15 D in S1 [10]. Furthermore, its $1 state is well separated from other excited singlet states and there is no indication of the occurrence of any excited-state reaction in most solvents [8,9]. Although the solvation dynamics of this molecule have been measured in a number of solvents previously [2,9], we have undertaken an extensive series of new measurements on more than 24 solvents. Motivation for this study comes from indications by computer simulation and theoretical considerations that earlier studies might have missed an important part of the dynamics due to limited time resolution [ 11]. Computer simulations first pointed to the presence of a prominent ultrafast component of the dynamics in small, highly polar solvents such as water [12], acetonitrile [13], and "methyl chloride" [14]. Recent experiments have confinmd the existence of this "inertial" component of the dynamics in a few important cases [ 15,16]. The present work reflects our attempts to more completely characterize such ultrafast dynamics in a broad range of solvents. Compared to earlier studies, we are aided in this pursuit by greatly improved temporal resolution (~ 100 fs FWHM instrument response), as well as having a method of accurately determining the position of the emission spectrum prior to solvent relaxation [ 17]. These two improvements allow us to make what we believe to be highly reliable measurements of the solvation dynamics of C153 in a range of solvents.
METHODS Emission decays were obtained from a fs Ti:sapphire laser/fluorescence upconversion spectrometer whose construction is reported elsewhere [8]. Here we only note that the overall temporal response used in these studies was between 112-125 fs as measured by the FWHM of the cross correlation between the pump and gate pulses. Decays (0-200 ps with a variable step size) were collected at a series of ten emission wavelengths (8 nm ban@ass) which were then used to reconstruct time-evolving emission spectra in the manner described in Refs. 8 and 9. From these spectra the solvation dynamics was extracted in the form of the spectral response function, s~(t)-
v(t)-v(oo) , v(0)-v(--)
(1)
where v(t) denotes the frequency of the peak of the emission specmun at time t (see Refs. 8 and 9 for details). All dynamics were measmed at room temperature, 23:1:2 C. DYNAMICS IN P O L A R SOLVENTS Figure 1 illustrates typical time-resolved emission spectra of C153 observed in the solvents acetonitrile, chloroform, and 1-pentanol. Also shown (dashed curves) arc the steady-state emission spectra (equal to the time infinity spectra in most cases) and the estimated time-zero spectra, the spectra expected prior to any solvent relaxation. (The latter were obtained through comparison of steady-state spectra in the solvent of interest and in a non-polar reference solvent as described in Ref. 17.) A few features of these spectra are noteworthy.
First, with the current time resolution, we are able to capture the full spectral evolution. That is, in nearly every solvent studied, the observed spectrum at t=0 agrees, to within roughly the accuracy of the time-zero method (--200 cm -i for C153), with the predicted time-zero specmml. Second, the time-evolution observed in all of the solvents studied involves a simple time-dependent shift with little change in shape or width. This behavior is indicative that we are indeed observing a continuous solvation process without interference from other excited state processes. The extent of the observed shift depends on solvent polarity. It ranges from --700 for weakly polar solvents such as chloroform to slightly over 2000 for strongly polar solvents such as acetonitrile. The magnitudes of these shifts can be reasonably correlated to the usual reaction field factors expected on the basis of dielectric continuum models [8]. Fmally, as illustrated by the three solvents portrayed in Fig. 1, the rate at which the emission specmun shifts with time varies over an impressive range as a function of solvent.
cm-i
cm-i
.8
,o~176
"
0
/// \\\ .i 16.
18. ~,
20.
~ "........
.2
ooo I
,ooo/ 22.
( 1 0 3 c m -1 )
24.
F'gure 1. Time-resolved spectra of C153 in acetonitfile, chloroform, and l-pentanol (connected points). Times in ps are indicated on the figure: the spectra evolve toward lower frequency with increasing time. The dashed curves are the steadystate emission spectra (red curve) and the spectrum expected prior to any solvent relaxation (blue curve: see Ref. 17).
0 I .05
"..........
1
-PeOH
""'-....
"~
M !':"::
.5
5.
50.
Time
")
\'\
500.
(ps)
F'~gure 2. Soivation response functions observed (solid curves) and calculated using the DMSAi theory (dashed curves; see text) for the solvents acetonitrile, methanol, chloroform, and 1-pentanol.
52 Figure 2 shows the spectral response functions (Sv (t), Eq. 1) derived from the spectra of Fig. 1. In order to adequately display data for these varied solvents, whose dynamics occur on very different time scales, we employ a logarithmic time axis. Such a representation is also useful because a number of solvents, especially the alcohols, show highly dispersive response functions. For example, one observes in methanol significant relaxation taking place over 3-4 decades in time. (Multiexponential fits to the methanol data yield roughly equal contributions from components with time constants of 0.2, 2, and 12 ps). Even in simple, non-associated solvents such as acetonitrile, one seldom observes Sv(t) functions that decay exponentially in time. Most often, biexponential fits are required to describe the observed relaxation. This biexponential behavior does not reflect any clear separation between fast inertial dynamics and slower diffusive dynamics in most solvents. Rather, the observed spectral shift usually appears to simply be a continuous non-exponential process. That is not to say that ultrafast inertial relaxation does not occur in many solvents, just that there is no clear time scale separation observed. Of the 18 polar solvents observed to date, a number of them do show prominent fast components that are probably inertial in origin. For example, in the solvents water [16], formamide, acetonitrile, acetone, dimethylformamide, dimethylsulfoxide, and nitromethane [8], we find that more than half of the solvation response involves components with time constants of <~00 fs. The dynamics observed in polar solvents can be readily modeled in terms of simple theories based on the bulk dielectric response (~(o~)) of the solvent. Although there are now a variety of more sophisticated theories currently available [3], we will only consider the "simple continuum" (SC) [18] and the dynamical MSA (DMSA) models [19]. The simple continuum theory assumes that the solute is a spherical cavity containing a point charge or point dipole. It interacts with the solvent only through the latter's bulk dielectric dispersion ~(r which is assumed to be known empirically. This limiting theory, which should be exact in the absence of specific interactions and in the limit of a solute of macroscopic size, forms the benchmark to which all other theories are compared. Previous studies have shown that predictions of the SC theory provide a useful guide to the dynamics observed in many real molecular systems [2,3]. However, the fact that the solvent is not a continuum fluid, but is in r e ~ t y composed of discrete molecules comparable in size to the solute, should cause deviations from the SC predictions. The DMSA theory corrects for some of the deficiencies of a continuum description by explicitly accounting for the fact that the electrical field generated by a molecular solute changes significantly on a length scale comparable to solvent molecule dimensions. This finite size effect causes a slowing of the solvation response compared to simple continuum predictions to an extent which depends on the solute/solvent size ratio [ 19]. The slow-down is much greater when one considers the response to a change in the solute's dipole moment compared to a change in its net charge (because of the difference in range of the electrical fields involved).
53
Figure 3. Solvation functions in dimethylsulfoxide. ,\
.8
"'-,\
9
9 "'\
b,
\N~x
~..6
--.
,
"':'~
"'b,
neglecting the unobserved uitrafust part of e(o~) (see tex0.
"',
"\
\
2 .82 2.0 MSi
3 3.~ 4.9 MSd
',,
'\, ,,
4 2.5 3.8 SC*
(ion), 3--DMSA(dipole),
El3.4 "\
The solid
curve ("0") is the obsen~ ~ and the dashed curves 1-4 are calculations using various models described in the text. The number of short dashes in the line pattern ctmeslxmd to the numbers wovided on inset tabulation of the characteristic times of these mslxmse functions: 1--SC (ion), 2ffiDMSA
0 .65 1 . 6 0 b s 1 .27 .80 SCi
" ',,
and 4ffiSC(ion)
.2 0
~
.05
,
,...~,- ~ . ~
.5 Time
5.
50.
(ps)
Figure 3 compares the experimentally observed solvation response of C153 in dimethylsulfoxide (#0, solid curve) to the predictions made by simple continuum and DMSA theories. The inset to the figure tabulates the 1/e times (t,,) and the average or integral correlation times (< %>) of the various response functions. These data show that the SC prediction for the response to a charge perturbation (#1) is too fast compared to the observed dynamics by a factor of approximately two. (The SC prediction for a dipole jump is sinfilar to that for a charge jump.) The DMSA prediction for a charge jump (#2; "DMSAi") is very close to the observed dynamics whereas the DMSA prediction for a change in solute dipole CDMSAd") is more than two-fold slower than the observed response. These observations concerning the performance of the various theories are quite general, as we will show presently. However, before doing so, we must make an important point regarding the use of dielectric data in such comparisons. The solvent response measured in time-resolved emission studies involves all solvent motion occurring on time scales slower than that of the electronic transition itself, i.e. slower than optical frequencies. Dielectric data, even those from recent experiments that, as in the case of DMSO [20], extend to the 100 GHz range, do not include all relevant nuclear motions of the solvent. For example, the dielectric data for DMSO used in Fig. 3 was reported to follow a Davidson-Cole distribution with parameters ao--46.40 ande..--4.16. The fact that some of the nuclear polarizability of the solvent is missed by this parameterization is indicated by the fact that at optical frequencies eop,--nv2=2.19. The difference e . . - nv2=1.97 reflects the contributions of nuclear responses in the FIR CPoley absorption") and IR regions (intramolecular vibrations) not captured in the dielectric data. Although this neglected part may seem small compared to the total polarizability of E0--46.40, the effect of these contributions on the computed solvation dynamics can be considerable, as illustrated by curve #4 in Fig. 3. This curve is a SC (ion) calculation which ignores such
54 contributions. The t~, and < ~ > times are slower than the same SC calculations that include these effects by factors of 9 and 5 respectively[ Although DMSO is an extreme example (in most solvents e . - no 2 is smaller than in DMSO), the difference is an important one, and proper comparisons must include the fast part of the nuclear response not directly observed in dielectric measurements. In the calculations described here we have treated this contribution as if it were a single relaxation m r m with a longitudinal time constant of 0.1 ps. This particular form of the time-dependence is arbitrary and is certainly an over simplification of the real dynamics. Luckily however, as long as the term describing the unobserved part of s is assumed reasonably rapid, its particular form and time scale do not appreciably affect the calculated values of t~, and < 1:>.
m 102
te
Figure 4. Comparison of ob~rv~ solvation times to predictions of the SC and DMSA theorics. The points show the comparison between the DMSA theory for an ionic solute with polar aprotic solvents denoted by circles and associated solvents by squares. For clarity,
j " ~ - - " .....--'" ........Q ...-'" .......
101 ._E~v i 0 ~ ~
. ~
..
the SC and DMSAd comparisons are only shown
E--
-o
I03
"~
I0 2 101
.j~ ~ '
I
t~'
L~ i 0 o
lO-t 10-1
10 ~
Observed
101 Time
102
103
as the linear fits to the log-log data with the SC predictions being the lower dashed line and the DMSAd the uPl~ dashed line. The solvents represented here are, in order of ~ i n g tie: acetonitrile, acetone, tetmhydrofuran, dimethylformamide, dimethylsulfoxide, propylene carbonate, dimethyl carbonate, chloroform, and benzonitrile (circles) and water [16], formamide, methanol, ethylene glycol, ethanol, l-prolmnol, l-peatanol, and l-decanol (squares).
(ps)
Figure 4 compares the characteristic times observed with C153 in a wide range of polar solvents to the predictions of the diele,ctric theories just described. As in Fig. 3, one finds good agreement between the times predicted by the DMSA theory for the response to a charge jump CDMSAi") and the observed dynamics. In contrast, the predictions of the simple continuum (ion) and DMSA calculations for a dipole change in the solute CDMSAd ~176 are r c s ~ v e l y too fast and too slow compared to experiment. The agreement between the observed dynamics and the DMSAi predictions also extends to the shapes of the response functions, as illustrated in the comparisons provided in Figs. 2 and 3. This is rather remarkable agreement, given that it extends over a wide variety of polar solvents, both associawxi and non-associated, whose solvation times span more than 3 orders of magnitude. It demonstrates the basic soundness of theories that link polar solvation dynamics to the solvent's dielectric response. However, it must be recognized that the DMSA theow is not a complete theory for the observed dynamics. If it were, one would expect to find agreement with predictions of the DMSA theory meant to model dipole jumps CDMSAd') rather than
55 with the ion version CDMSAi"), since the lowest-order multipole of the solute that changes upon electronic excitation is its dipole moment, not its charge. Indeed, calculations show that charge distributions of polyatomic molecules such as C153 and their change upon excitation are likely to be more quadrupolar in nature than dipolar [21]. Thus, one would expect that the observed dynamics should be even slower than that predicted by the DMSAd theory. The fact that the observed dynamics is in actuality faster is an indication that some aspect of the real system's dynamics is not properly captured by the DMSA theory. Simulations of lattice solvents [22] demonstrate that the DMSA theory does accurately predict the dynanfics of both charge and dipole jumps when (as assumed by the theory) the solvent dynamics involves only rotational motion of point dipole particles. What is neglected in the DMSA theory is: i) that solvent molecules translate in addition to rotate and ii) that the solute does not really see a nearby solvent molecule as a point dipole but rather as an extended charge distribution. The effect of translational motions, which contribute an additional relaxation channel for solvation via "polarization diffusion" [2,23] is clearly to speed up the response relative to the DMSA predictions. The effect of the extended charge distribution of solvent molecules is more subtle to gauge but, at least in some cases, this difference has also been shown to hasten the response [24]. We can summarize these results in the following way. The SC model provides useful first estimates for the solvation dynamics observed with a polar solute such as C153. However, due to its complete neglect of molecular aspects of solvation, it predicts response times that are uniformly too rapid. The DMSA approach, when properly applied (i.e. the DMSAd model), predicts too slow a solvation response. This latter model includes some important effects of solvent molecularity which serve to increase the response time compared to the SC predictions, while it neglects certain others that happen to act in the opposite direction. Thus, the agreement between the dynamics observed with the solute C153 and those pwxiicted by the "inappropriate" DMSAi theory should be regarded as partly fortuitous. Nevertheless, the surprising accuracy of the DMSAi predictions makes this simple model of considerable practical value. DYNAMICS IN NON-DIPOLAR SOLVENTS The success of dielectric models in describing the solvation of C153 in polar solvents is quite remarkable. Although, as just discussed, certain aspects of the dynamics are still not quantitatively described by theory, it is fair to say that the basic features of solvation dynamics in polar solvents are well understood. We would like to conclude by observing that the same cannot be claimed for the dynamics in non-dipolar solvents. Figure 5 shows that in the nominally "nonpolar" solvents dioxane and benzene, C153 undergoes time-dependent shifts that are in all respects analogous to the dynamics observed in weakly polar solvents such as chloroform [8]. However, theories of the sort described above, relying as they do on e(co), are inadequate to describe apparently similar solvation dynamics taking place in these latter solvents. Because they lack significant dipole moments, the solvation response predicted on the basis of their dielectric properties (i.e. by e0) is very small [6]. The fact that these and similar non-dipolar solvents we have examined do solvate C153 strongly is a consequence of the large quadrupole and higher-order electrostatic moments these molecules possess. At the short ranges relevant for molecular solvation, these higher moments (or alternatively the
56 charged groups or dipolar bonds) of the solvent molecules are as effective in solvating C153 as are the dipoles of weakly polar solvents. But the long-range dipolar correlations measured by the bulk property E(ca) are virtually absent in these solvents. Thus, to describe the dynamics observed in such solvents one does not have empirical information of the sort contained in e(o~) or an equivalent of the benchmark SC approach available for dipolar solvents. Furthermore, since the range of the interactions between the solvent and solute is shorter in such solvents, one expects that the details of the solvation structure and the effects of wanslational motion should be of much greater importance to the dynamics than in the dipolar solvent case. Thus, we anticipate that these types of solvents should provide a challenging testing ground for the future development of molecular models of solvation and solvation dynamics. l~gare5. Trine-resolvedspeclra of C153 in dioxane and benzene (connected points). The times shown are 0, 0.1, 0.5, 1, 2, and 10 ps from right to left. The dashed curves are the steady-state emission (red curve) and the speclrum expected prior to any solvent relaxation (blue curve; see Ref. 17).
i
3
0 16.
18.
Frequency
20.
22.
24.
(10 3 c m -I )
ACKNOWLEDGMENTS This research was supported by funds from the Office of Basic Energy Sciences of the US Department of Energy and by the Office of Naval Research. REFERENCES 1. See the reviews: J. T. Hynes, in Ultrafast Dynamics of Chemical Systems, I. D. Simon, Ed. (Klewer, Dordrecht, 1994), pp. 345-381; H. Heitele, Angew. Chem. Int. Ed. Engl. 32, 359 (1993). 2. Reviews of the experimental literature are provided in: M. Maroncelli, J. Mol. Liq. 57, 1 (1993); and P. F. Barbara and W. Jarzeba, Adv. Photochem. 15, 1 (1990); W. Jarzcba, G. C. Walker, A. E. Johnson, P. F. Barbara, Chem. Phys. 152, 57 (1991). 3. Reviews of theoretical and simulation studies can be found in Ref. 1 and B. Bagchi and A. Chandra, Adv. Chem. Phys. 80, 1 (1991); and F. O. Raineri, Y. Zhou, H. L. Friedman, Chem. Phys. 152, 201 (1991). Some of the most recent theoretical developments are described in: F. O. Raineri, H. Resat, B.-C. Perng, F. Hirata~ H. L. Friedman, J. Chem. Phys. 100, 1477 (1994); and B. M. Ladanyi and R. M. Stratt, J. Chem. Phys. (1995).
4. B. Bagchi, J. Chem. Phys. I00, 6658 (1994); Svaen and J. L. Skinner, J. Chem. Phys. 99, 4391 (1993) and luther work in progress. 5. M. Berg, "Comparison of a ViscoelasticTheory of Solvation Dynamics to Tune-Resolved Experiments in a Nonpolar Solution," Chem. Phys. Lett.,in press. 6 J.T. Fourkas, A. Benigno, M. Berg, J. Chem. Phys. 99, 8552 (1993); J. T. Fourkas and M. Berg, J. Chem. Phys. 98, 7773 (1993). 7. See, for example, E. T. J. Nibbering, D. A. Wiersma, K. Duppen, J. Chem. Phys. 183, 167 (1994); S. Y. Goldberg, E. Bart, A. Meltsin, B. D. Fainberg, D. Huppert, J. Chem. Phys. 183, 217 (1994); T. Joo, Y. Jia, G. R. Fleming, "Ultrafast Liquid Dynamics Studied by Thrid and Fifth Order Three Pulse Photon Echoes," preprint (1995). 8. A more complete account of these studies is cunently in preparation. (M. L. Homg, J. Gardecki, and M. Maroncelli, "Sub-Picosecond Measurements of the Solvation Dynmrdcs of Coumain 153," to be submitted to J. Chem. Phys.) 9. M. Maroncelli and G. R. Fleming, J. Chem. Phys. 86, 6221 (1987); M. Maroncelli and G. R. Fleming, J. Chem. Phys. 86, 3251 (1990). 10. P. K. McCarthy and G. J. Blanchard, J. Phys. Chem. 97, 12205 (1993); W. Baumann and Z. Nagy, Pure Appl. Chem. 65, 1729 (1993). 11. M. Maroncelli, P. V. Kumar, A. Papazyan, M. L. Homg, S. J. Rosenthal, G. R. Fleming, in Ultrafast Reaction Dynamics and Solvent Effects, Y. Gauduel and P. J. Rossky, Eds. (American Institute of Physics, New York, 1994), pp. 310-333. 12. M. Maroncelli and G. R. Fleming, J. Chem. Phys. 89, 5044 (1988); J. S. Bader and D. Chandler, Chem. Phys. Lett 157, 501 (1989). 13. M. Maroncelli, J. Chem. Phys. 94, 2084 (1991). 14. A. E. Carter and J. T. Hynes, J. Chem. Phys. 94, 5961 (1991). 15. S. J. Rosenthal, X. Xie, M. Du, G. R. Fleming, J. Chem. Phys. 95, 4715 (1991); S. J. Rosenthal, R. Jimenez, G. R. Fleming, P. V. Kumar, M. Maroncelli, J. Mol. Liq. 60, 25 (1994). 16. R. Jimenez, G. R. Fleming, P. V. Kumar, M. Maroncelli, Nature 369, 471 (1994). 17. R. S. Fee and M. Maroncelli, Chem. Phys. 183, 235 (1994). 18. B. Bagchi, D. W. Oxtoby, G. R. Fleming, Chem. Phys. 86, 257 (1984); G. van der Zwan and J. T. Hynes, J. Phys. Chem. 89, 4181 (1985). 19. I. Rips, J. Klafter, J. Jormer, J. Chem. Phys. 88, 3246 (1988); I. Rips, J. Klafter, J. Jormer, J. Chem. Phys. 89, 4288 (1988); A. L. Nichols HI and D. F. Calef, J. Chem. Phys. 89, 3783 (1988); see also M. Maroncelli and G. R. Fleming, J. Chem. Phys. 89, 875 (1988). 20. J. Barthel, IC Bachhuber, R. Buchner, J. B. Gill, and M. Kleebauer, Chem. Phys. Lett. 167, 62 (1990). 21. C. F. Chapman, R. S. Fee, and M. Maroncelli, "Measurements of the Solute Dependence of Solvation Dynamics in 1-Propanol: The Role of Specific Hydrogen Bonding Interactions," submitted to Chem. Phys. Lett. 22. H. X. Zhou, B. Bagchi, A. Papazyan, M. Maronceili, J. Chem. Phys. 97, 9311 (1992); A. Papazyan and M. Maroncelli, "Rotational Dielectric Friction and Dipole Solvation: Tests of Theory Based on Simulations of Simple Model Solutions," J. Chem. Phys., in press. 23. G. van der Zwan and J. T. Hynes, Physica 121A, 227 (1983). 24. F. O. Rained, Y. Zhou, H. L. Friedman, Chem. Phys. 152, 201 (1991).
This Page Intentionally Left Blank
journal of
~IOLECULAR
LIQUIDS ELSEVIER
Journal of MolecularLiquids,65/66 (1995) 59-64
Solvent and Nuclear Dynamic Effects on Intermolecular Electron Transfer Keitaro Yoshihara, 1, 2 Yutaka Nagasawa, 2 Arkadiy Yartsev, 1, + Alan E. Johnson,l,$ and Keisuke Tominaga 1 llnstitute for Molecular Science, Myodaiji, Okazaki 444, Japan 2The Graduate University for Advanced Studies, Myodaiji, Okazaki 444, Japan
Abstract An ultra_fast intermolecular electron transfer (ET) from electron donating solvent to an excited dye molecule was found. A temperature-dependent non-exponential time dependence was observed in aniline, and a temperature-independent single exponential process for Nile blue (160 fs) and oxazine 1 (260 fs) was observed in N,N-dimethylaniline. The solvation times of solvent anilines were obtained by dynamic Stokes shift measurements. The rate of ET in some systems was observed to be much greater than the solvation time of anilines. The dynamic behavior was simulated by the 2-dimensional potential energy surface for reaction, taking into account of the effects of both solvent reorientation and nuclear motion of reactants.
I. Introduction The universality of intermolecular electron transfer (ET) makes the mechanism one of the central questions to be solved in chemistry and biology. ET is strongly influenced by the nature of solvent and its dynamics in solution. Dynamical solvent effects on the course of reaction have recently studied both theoretically [1-5] and experimentally [6-15]. For ET with a rate comparable to the solvent fluctuation rate, the motion and structure of the solvent can determine the rate of ET and it becomes the "solvent-controlled" reaction [2,3, 6-9]. Most recently we have investigated ultrafast ET (as fast as ca. 1013 s-1) between excited dye and weakly polar electron-donating solvent molecules [10-14]. These systems have several interesting features. (1) The electron donor and acceptor are in contact and there is no translational diffusion to induce ET. (2) Non-equilibrium reaction caused by solvent dynamics can be observed. (3) Some systems have a rate of ET much faster than that of solvent relaxation. In this case the reaction is mainly due to the nuclear dynamics and is independent of solvent dynamics. (4) A clear substituent effect of electron acceptor molecules on the rates of ET is observed. The time scale of ET observed in these systems falls into the time scale of solvent relaxation and of nuclear motions and the reaction will be severely influenced by these dynamics. It is also the aim of the present study to investigate the role of intermolecular interaction, namely, solvent effects to the reaction as well as the effects of inter- and intramolecular dynamics. The oxazine dyes are Nile blue A perchlorate (NB) and oxazine 1 perchlorate (OX1), which are ionic dyes, and coumarin 102 is chosen as a fluorescent probe molecule in the dynamic Stokes shift measurements. The electron donating solvents are aniline (AN) and N,N-dimethylaniline (DMA).
II. Experimental Ultrafast fluorescence quenching dynamics were studied by the fluorescence-upconversion method with femtosecond mode-locked laser systems. For the studies of oxazine dyes, a synchronously pumped hybrid mode-locked dye laser with group velocity 0167-7322/95/509.50
91995 ElsevierScience B.V. All rights reserved.
SSDI 0167-7322 (95) 00910-8
60 compensation prisms was made to generate optical pulses at different wavelengths at 76 MHz. Using a combination of Rh-6G and DODCI/DQOCI and spatial filters between the second prism and end mirror, 70 fs pulses were generated at 615 nm. With a combination of sulforhodamine 101 and DQTCI generation of 50 fs pulses at 675 nm was achieved. For the studies of fluorescence dynamics of coumarins, we used the second-harmonic of a modelocked Ti: sapphire laser, at 395nm, to excite the sample. The remaining fundamental at 790nm was used to up-convert the fluorescence from the sample.
IH. Results and Discussion A. Solvent relaxation times of electron donating solvents W e observed ultrafastE T in the subpicosecond timescale which was thought to be much faster than the diffusive solvent relaxation process [I]. To confirm this the measurement of solvent relaxation time is carried out. The dynamic fluorescence Stokes shift is used to determine solvent relaxation times of the donor solvents ( D M A and AN). W h e n a probe dye molecule is photo-excited, the dipole moment increases instantaneously. The polarization of the surrounding solvent molecule responds to this change and startsto reorganize. Therefore, the energy relaxation process, caused by the solvent reorganization, shiftsthe fluorescence spectrum to longer wavelengths. To extract dynamical information of solvent relaxation process, the normalized spectral shiftcorrelation function C(O is measured [16],
C(t) = v(t)- v(..)
v(O)- v(..)
(1)
where v(t), v(,,), and v(O) are the fluorescence peak frequencies at times, t, ,0, and 0, respectively. For this method, a probe molecule which undergoes a large instantaneous change of dipole moment upon photo-excitation has to be chosen. Coumarin dyes are often used as probe molecules for this type of experiment [17-19]. Coumarin 102 (C102), which has a lifetime of 2.8 ns in DMA and 1.4 ns in AN, is the choice for this experiment. Figure 1 shows fluorescence decays of C102 observed at different wavelengths. At shorter wavelengths there is a fast decay and at longer wavelengths there is a concurrent rise. This implies that the fluorescence spectrum of C102 is shifting towards red with time. Time-resolved fluorescence spectra would illustrate this red shift. We have used the spectral reconstruction method to obtain a time-resolved fluorescence spectrum [13] When the fluorescence up-conversion method is used, the relative intensity between each wavelength becomes uncertain because the angle of the nonlinear crystal has to be tuned at each wavelength of observation. However, the intensity of the fluorescence l(/q,, t), at a given time t and wavelength ;I,, can be obtained from the normalized fitted decay series D(t ,;t) and intensity of the steady-state fluorescence 10()0;
l(/q;, t) = D(t,~) Io(~)
D(t, ~) dt
(2)
Figure 2 shows the reconstructed spectrum. The symbols correspond to the intensities of the fluorescence at the measured wavelength at a given time t. The lines are the least-square-fits to the points using a log-normal function. The log-normal function describes an asymmetric line shape and is often used to fit broad featureless absorption or fluorescence spectra. As expected from examinations of the individual fluorescence transients, the peak of the reconstructed spectrum shifts to red with time. In Figure 3, the spectral shift correlation functions C(t) are shown with the results of bi-exponential fitting. The observed solvent relaxation times are 7.9 ps (19 %) and 18.7 ps (81%) for DMA and 6.7 ps (81%) and 13.3 ps (19 %) for AN. These are much longer than the fluorescence lifetimes of NB, i.e., 0.1 ps
61
I (a)
497 nm
20 ps
Idl/// ~ll
(a)
I 16p~
Izr
;1111 o
'
(b)
~'
'~
. . . . .. . .
'
_ . ~ . , ~ ' ~ " *~ ~'~'~
~
.
.
~
.
~
a;
-
y
~o
.....
Time / ps
;o
.
.
.
.
.
.
.
.
!
500
....
16 ps
484 nm
9
o . . . . .
450
/ ' ~
....
497 nm ~
,,i .
400
~,
Figure 1. Fluorescence decays of C 102 (a) in DMA and (b) in AN, measured at various wavelengths.
/,P//
4~1
_'t,W/o~
1 ...... 500
5~}
I
6OO
Wavelength / n m
Figure 2. Reconstructed time-resolved fluorescence spectra of C 102 in (a) DMA and (b) AN. The dots are the data experimentally obtained, and the curves are the log-normal fits to the dots.
(95 %) and 2.5 ps (5 %) in DMA, and 0.4 ps (56 %) and 2.5 ps (44 %) in AN. Therefore, our previous prediction is confirmed; the ET occurs much faster than the diffusive solvation process. B. Temperature dependence of electron transfer The fluorescence of OX 1 in anilines is severely (ca. 103 times) quenched by ET, and, therefore, its dynamics reveal the ET dynamics. The temperature dependence of fluorescence decays of OX1 in anilines is shown in Figure 4. The ultrafast single exponential decay (280 fs) of OXI in DMA does not show any temperature dependence from 280 K (7 ~ to 373 K (100 ~ In AN, however, a clear temperature dependence was observed in its non-exponential decay. As the temperature increases from 283 K (10 ~ to 353 K (80 ~ the decay becomes faster. A tri-exponential fitting is performed at different temperatures. The first component (-430 fs) does not show temperature dependence, but the second component shows dependence which gives an activation energy of 1.0 kcal/mol. C. Effect of solvent dynamics on electron transfer and Sumi-Marcus two-dimensional model It is clear in the present system that ET occurs much faster than the diffusive solvation process. To explain exponential and non-exponential kinetics of ET, the idea of Sumi-Marcus two-dimensional reaction coordinate is used. In this treatment, instead of the usual one dimensional reaction coordinate (solvent coordinate), two coordinates are used, i.e., the solvent coordinate and the vibrational nuclear coordinate. A free energy surface is drawn in a two-dimensional plane spanned by the solvent coordinate, X, and the nuclear coordinate, q.
62 The Sumi-Marcus model treats both the solvent and intramolecular mode classically. However, the actual system should have not only the classical modes but also quantum mechanical high-frequency modes. Jortner and Bixon have developed an ET model which introduces the effect of the quantum mechanical high-frequency modes [4]. In this model, the reactant surface crosses not only with the vibrational ground state of the product but also with vibrationally excited states of the product. The reaction of oxazines in DMA is activationless, which means that the reaction is not in the inverted region but very close to it. We use a hybrid model of Sumi-Marcus and Jortner-Bixon developed by Walker et al. [15]. In this model, the intramolecular vibration is separated into the quantum mechanical highfrequency mode and classical low-frequency mode;
~p(X, t) = D ~ (9 + 1 dV(X) ] p(X, t)- ~, k(n, X) p(X, t) n ~t ~ [ 3t kBT dX
(3)
where p(X, t) is the population of the reactant at X and t, and k(n, X) is the reaction rate between the vibrational ground state of the reactant and vibrational state n of the product. The non-adiabatic formula is used for k(n, X) 9
(4)
k(n, X) =~ V~l2 (4ZCXlkBT)I/2exp( - AG*(n,X) ) kBT
,t
~
-:os
=o<
4
g
~
~ o., c
5
2
oo, 9
o.o-
,'o
_
20 '
6.7
30 '
,s(a
%1
40 '
I
~ - O.5
~'.s
'!I
o.o,, 6
o'.o
o'.s
/~
!,!,
,.o
,.s
2S3K
,,
(b)
'
i
1
0
2
Jo,,l ~
,
.i
I
6
Time / ps
oo-
~
i0
~
20
30
40
Time / ps
F i g u r e 3. The normalized spectral shift correlation function C(t) of C102 in (a) D M A and (b) AN. Results of bi-exponential fitting are also shown.
Figure 4. Temperature dependence of fluorescence decays of OXI (a) in DMA and (b) in AN.
63 where ,;Ll is the reorganization energy for the classical low-frequency mode. V'el is the electronic matrix element concerning the effect of Franck-Condon factor which can be written as
Ve~
= Sn
n!
(5)
exp(-S)Ve~
where the electron-vibrational coupling strength is S = Zh2/2. Zh is the reorganization energy for the high-frequency mode. AG*(n, X) is the activation energy which depends on the quantum number of the vibrational state n and the solvation coordinate X;
zlG*(n, X) = (As + AG - 2X;s + nhVh + ,~,/)2 / 4&t
(6)
For D, the time dependent diffusion coefficient D(t) is used [ 13];
D ( t ) = kBT
1
2A,s A(t)
dA(t)
(7)
dt
where
(8)
A(t) = al exp(- t / rl) + a2 exp(- t / z2) lr1, is the first and T2 is the second solvation time, and al + a2 = 1. This model contains m a n y parameters, some of them are difficult to obtain experimentally. But we try to fit the experimental data using as realistic parameters as possible. Firstly we estimate that the minima of the SO and S I free energy surfaces of OXI are located at the same point. This means that the electric dipole moment of OX1 does not change drastically upon photo-excitation. This approximation makes the initial distribution produced by the excitation an equilibrium one. The Stokes shift of OX 1 is very small, therefore, this approximation seems quite reasonable. Secondly, it is assumed that the reaction occurs only from the lowest vibrational state of the reactant. This approximation seems quite reasonable, since there is no excitation wavelength dependence. We used 0.13 eV for the solvent reorganization energy Zs, and 0.37 eV for the reorganization energy of the classical low-frequency mode ;tl. Solvation times of 7 ps (50%) and 15 ps (50%) which are close to the observed solvation times for AN and DMA are used. 0.13 eV and 1400 cm-1 are used for the reorganization energy 2h and the frequency Vh of the highfrequency mode, respectively.
1!~
(a)
II. 2
0.01
0
1
,:
6:
(b)
2.
.. O.lg z.
o..
6.(,
:
~
0.018-
5
Time I ps
Figure 5. Results of simulation using Sumi-Marcus Jortner-Bixon hybrid model. (a) AGo = -0.50 eV. (b) AGo = -0.33 eV, T = 273,293, 313, 333, 353, and 373 K.
64 The results of the simulations are shown in Figure 5. The normalized population of the reactant P(t) is plotted against time (ps) on a logarithmic scale. The energy gaps -AGo are 0.50 eV for Figure 5 (a) and 0.33 eV for (b). Temperature is changed in each set, from 273 K to 373 K The fluoresc6n6e decay of C102 in AN does not show any temperature dependence. Figure 5 (a) corresponds to the ET of OX1 in DMA, where there is no activation barrier. The free energy surface for the lowest vibrational state of the product crosses the bottom of the reactant surface. Such a situation occurs when -AGo = ,~l + 2s = 0.50 eV. The simulated reaction shown in Figure 5 (a) is almost exponential, which coincides with the experimental result of OX1 in DMA. In Figure 5 (b), -AGo is 0.17 eV smaller than that in (a). This difference corresponds to the difference in ionization potential between DMA and AN in condensed phase. The reaction becomes faster as the temperature increases and the change is rather large at the slower part of the reaction than the fast part.
Acknowledgment This work was supported in part by a Grant-in-Aid for Scientific Research on New Program (06NP0301) by the Ministry of Education, Science, and Culture of Japan.
References +Permanent address: Institute of Spectroscopy, Russian Academy of Science, 142092 Troitzk, Moscow Region, Russian Federation. $ Present address: Department of Chemistry, University of Rochester, Rochester, NY 14627, U.S.A. [1] [2]
[3] [4] [5] [6]
[7] [8] [9] [10] [11] [12] [13] [14]
[151 [16]
[17] [18] [19]
[20]
H.A. Kramers, Physica (Utrecht) 7, 284(1940). L.D. Zusmann, Chem. Phys. 49, 295(1980). I. Rips, J. Klafter, and J. Jortner, J. Chem. Phys. 94,8557(1990). J. Jortner and M. Bixon, J. Chem. Phys. 87, 167(1988). H. Sumi and R.A. Marcus, J. Chem. Phys. 84,4894(1986). E. M. Kosower and D. Huppert, Chem. Phys. Lett. 96,433(1983). M. A. Kalow, T. J. Kang, and P. F. Barbara, J. Phys. Chem. 91, 6452(1987). J. D. Simon and S.-G. Su, J. Phys. Chem. 92, 2395(1998). R. M. Neison, G. E. McManis, M. N. Golovin, and M. J. Weaver, J. Phys. Chem. 92, 3441(1998). T. Kobayashi, Y. Takagi, H. Kandori, K. Kemnitz and K. Yoshihara, Chem. Phys. Lett. 180,416(1991). H. Kandori, K. Kemnitz, and K. Yoshihara, J. Chem. Phys. 96, 8042(1992). A. Yartsev, Y. Nagasawa, A. Douhal, and K. Yoshihara, Chem. Phys. Lett. 207, 546(1993). Y. Nagasawa, A.P. Yartsev, K. Tominaga, A.E. Johnson, and K. Yoshihara, J. Chem. Phys. 101, 5717(1994). Y. Nagasawa, A. P. Yartsev, K. Tominaga, P. B. Bisht, A. E. Johnson, and K. Yoshihara, J. Phys. Chem., in press. G. C. Walker, E. Akesson, A. E. Johnson, N. E. Levinger, and P. F. Barbara, J. Phys. Chem. 96, 3728(1992). B. Bagchi, D. W. Oxtoby, and G. R. Fleming, Chem. Phys. 86, 257(1984). M. Maroncelli and G. R. Fleming, J. Chem. Phys. 86, 6221 (1987). M. A. Kahlow, T. J. Kang, and P. F. Barbara, J. Chem. Phys. 88, 2372(1988). M. A. Kahlow, W. Jarzeba, T. J. Kang, and P. F. Barbara, J. Chem. Phys. 90, 151(1989). R. A. Marcus, Discuss. Faraday Soc. 29, 21(1960).
journal of
~IOLECULAR
LIQUIDS ELSEVIER
Journal of MolecularLiquids, 65/66 (1995) 65-73
Theory on Rates of Solution Reactions Influenced by Slow Fluctuations in Viscous Solvents, and its Experimental Confirmation Hitoshi Sumi Institute of Materials Science, University of Tsukuba, Tsukuba, Ibaraki 305, Japan Abstract Recently much attention has been aroused on solution reactions whose rates decrease as the viscosity r / o f solvents increases. These reactions cannot be rationalized in the framework of the transition state theory. To describe them, two currents of theories have been developed by extending the Kramers theory. One was initiated by Grote and Hynes, while the other by Sumi and Marcus. Recent data on thermal Z/E isomerization of substituted azobenzenes and N-benzylideneanilines confirms the applicability of the latter for r/variation over 104 times under pressure.
I.
Introduction
The most traditional theory for chemical reaction rates is the transition state theory (TST) established in 1940's. It has recently been disclosed, however, that the TST cannot be applied to varieties of solution reactions. Examples can be found in biological enzymatic reactions ~, electron or proton transfer reactions2, atom-group transfer reactions 3, and isomerization reactions4. Study of solution reactions is one of the most traditionalas well as the most fundamental subjects in chemistry. The situation mentioned above means, nevertheless, that we have not yet established a general expression on rates of solution reactions. Accordingly, many discussions have been stimulated for investigating the unknown general expression. 5 In the TST, it is assumed a priori that fluctuations in the reactant state are so fast that the distribution of populations in the reactant state is always maintained in thermal equilibrium in the course of the reaction. Then, also the population of reactants in the transition state is always maintained in thermal equilibrium with those in the reactant state, and the rate constant can be calculated from the population at the transition state, which is determined only by the shape of the free-energy surface, without knowledge on dynamics of fluctuations for realizing the transition state. In this assumption, therefore, the rate constant should not depend on how fast fluctuations are in the reactant state. Molecular arrangements of the solute-solvent system in solution can take various conformations. They fluctuate from time to time due to shaking and/or damping by microscopic motions of solvent molecules. Since the speed of these conformational fluctuations decreases as the viscosity r / o f solvents increases, r/-' can be regarded as a measure of the speed. In the solution reactions mentioned earlier, the rate constant observed decreases as r/increases, that is, as r/-l decreases. This means that the rate constant of these solution reactions depends on the speed of conformational fluctuations in the solute-solvent system, and hence that these reactions are nonthermalized reactions located outside the framework of the TST. To be more exact, these reactions are controlled by slow speeds of these fluctuations. The first theory giving the 0-induced decrease of the rate constant is the Kramers theory6 presented as early as in 1940. He explicitly treated dynamical processes of fluctuations in the reactant state, not assuming a priori the thermal equilibrium distribution therein. His reaction scheme can be understood in Fig. 1 which shows, along a reaction coordinate X, a double-well potential W(X) composed of a reactant and a product well with a transition-state barrier between them. Reaction takes place as a result of diffusive Brownian motions of reactants surmounting 0167-7322/95/$09.50 91995 ElsevierScience B.V. All rights reserved. SSD! 0167-7322 (95) 00903-5
66
Figure 1. Potential curve W(X) for reaction in the Kramers scheme along the reaction coordinate X, and an example of reactive trajectories there. over the transition-statebarrier,as shown by a zigzag trajectory in Fig. I. In this situation, when the reactant population found in the interval from X to X 4 4 X at time t is written as P(X;t)dX, the distribution function P(X;O satisfiesthe Smoluchowski equation describing diffusion of reactants under the influence of a potential fielddue to W(X), as -~
t~X - ~ + ~ ksT
--
P( X ;t) ,
(1)
where D represents the diffusion constant and T the temperature. The diffusion constant D describes frictional effects arising from shaking and/or damping by microscopic motions of solvent molecules, and can be regarded as inversely proportional to the viscosity 7/of solvents, obeying the hydrodynamic Stokes-Einstein relation. When r/and hence D -I are sufficiently small in his theory, rapid diffusive motions make populations in the reactant well to be quickly recovered and maintained in the thermal-equilibrium form in the course of reaction, and the rate constant reduces to that expected from the TST. As 77 and hence D -t increase, however, the diffusive motions become slow, and the reaction becomes limited by their slow speed. In this non-TST regime, the rate constant given by his theory decreases in proportion to r/"t. Concerning the 7/dependence of the rate constant of solution reactions, most often investigated experimentally is photoinduccd E/Z (trans/cis)isomcrization of stilbencs.4 The isomcrization takes place by surmounting over a transition-statebarrier on the excited-state potential surface. The reaction is very fastwith a rate constant on the order of I010 s-' due to a small height (of about 15 kJ/mol) of the barrier.The rate constant observed in solvents kob s decreases as 7/increases. To be more exact, however, kob s often decreases more slowly than
r/-t,describable by a fractional-power dependence on 77-i as /Cobs o~ rl-a with 0 < o~ < I. Therefore, the 7/dependence of/Cot,s deviates from that expected from the Kramers theory. To understand the slow r/dependence of kob s, approximately two currents of theories have been developed by extending the Kramers theory. One was initiatedby Grote and Hynes 7 in 1980. They considered that frictionfeltby reactants should decrease when they surmount over the transition-statebarrier with a high speed. In thiscase, the rate constant in the non-TST regime should not be as small as expected from the Kramers theory, and hence its r/dependence should be milder than 7/"~expected from the Kramcrs theory. This idea of frequency (that is,s ~ ) dependent frictioncan be formulated by the generalized Langevin equation describing time (t) dependent stochastic motions of the position X(t) of a reactant along the reaction coordinate X in the field of the double-well potential W(X), as
d2X(t__._.___2)dW[X(t)] dt 2
=-
t ~ ( t - t' ) dX(t') d t ' + R(t) _ L** dX(t--------~ dr '
(2)
67
where R(t) represents random forces and ~ ( t - t ' ) (>0) represents the frictional memory function. The second term on the right-hand side of eq.(2) describes frictional forces whose strength at time t comes also from the magnitude of the reactant velocity dX(t)ldt' in the past at time t' (< t) through a nonvanishing tail of the frictional memory function ((z') at z= t - t' (> 0). Both R(t) and ((t) arise from a single origin of microscopic motions of solvent molecules interacting with reactants. Then, they must be related to each other. This relation is known as the fluctuation-dissipation theorem, and is written as
( R(t) R(t' )) = kBr ~(t - t' ) ,
(3)
where ( . - . ) represents the thermal average over microscopic motions of solvent molecules. Actual fitting of rate constants observed for stilbenes to that given by the Grote-Hynes theory was performed by several authors s. They all pointed out coincidentally that the fitting could be made only by adopting an unphysically small value for the downward curvature (the square of a frequency written as cot, later) of the potential W(X) at the top of the transition-state barrier. The Kramers theory corresponds to a limit of eq.(2) that the correlation time.between t and t' in eq.(3) is so short that ~ ( t - t' ) can be regarded as proportional to a delta function tJ(t- t3. In this case, the second term on the right-hand side of eq.(2) becomes proportional to dX(t)/dt, losing frictional memory effects, and the generalized Langevin equation (3) reduces to the Langevin equation, from which the Smoluchowski equation (1) can be derived. 9 The other current of theories for understanding the observed slow 7/dependence of the rate constant was initiated by Sumi and Marcus l~ in 1986. It corresponds to an extension of a theory by Agmon and Hopfield II for understanding rebinding kinetics of O 2 or CO molecules in hemoglobin. In this model, conformational fluctuations of intermolecular arrangement in the solute-solvent system take a reactant only midway to the transition-state barrier along the coordinate X, and final surmounting over the barrier is accomplished with the aid of much faster intramolecular vibrational fluctuations in the solute molecule, possible at each value of the coordinate X. Therefore, the reaction takes place as a result of sequential two steps which can be described in a two-dimensional potential surface shown in Fig.2. This surface is spanned by the coordinate X for conformational fluctuations in the solute-solvent system on the abscissa and by the other coordinate q for intramolecular vibrations in the solute molecule on the ordinate. Potential energies are shown by equi-energy contours in Fig.2. The part below the line C represents the reactant-state surface while the part above it represents the product-state surface. These two parts meet at the line C with the same energies, and hence the line C can be called the transition-state line. Energies on this line becomes minimum at the point S, which therefore represents the location of the transition state. Intramolecular vibrational fluctuations along the coordinate q on the ordinate are much faster than diffusive Brownian fluctuations of intermolecular arlatoa,ra surfotm
Reoztant tutfoce
Figure 2. Two-dimensional potential surface for reaction in the Sumi-Marcus scheme along the coordinate X for diffusive Brownian motions and the coordinate q for much faster intramolecular vibrational motions, and an example of reactive trajectories there.
68 rangements along the coordinate X. Then, the surmount over the transition-state barrier is possible with the aid of fast intramolecular vibrations at each fixed value of the slow coordinate X over the transition-state line C determined at this value of X. An example of reactive trajectories is shown in Fig.2. This shows that when the coordinate X fluctuating diffusively along the abscissa in the reactant surface gets an appreciable amplitude as described by a zigzag trajectory, fast intramolecular vibrational fluctuations along the ordinate let the reactant surmount over the transition-state line C as described by a vertical straight trajectory. After that, the system relaxes on the product surface, again by diffusive motions along the abscissa described by a zigzag trajectory there. Since the surmount over the transition-state line C takes place most favorably at X values around that of the transition-state point S, it is not necessary for the reactant to surmount over the barrier only by diffusive Brownian motions, at the point Xc in Fig.2 along the abscissa for q = 0. In the situation mentioned above, we can define a rate constant k(X) determined at each value of X for the surmount over the transition-state line C vertically along the coordinate q at this fixed value of X. Then, the distribution function P(X;t) for reactants with an intermolecular arrangement described by a coordinate value X at time t obeys, instead of the diffusion equation (1), a diffusion-reaction equation in the potential along the abscissa for q = 0, written as V(X), as
-0
Ot
P(X;t)
:oAFA
+ I c)X L ~)X k n T
dv(x)] P ( X ; t ) dX
k(x)P(X;t) .
(4)
The essential difference between eqs. (1) and (4) is the presence of the last term on the righthand side of eq.(4), and it describes loss of reactants by reactions during diffusion along X. The diffusion constant D is regarded as inversely proportional to the solvent viscosity r/as before, while k(X) is independent of r/, determined by intramolecular vibrational motions. It was shown by the present authod 2 that this Sumi-Marcus model gives as a rate constant k=l/(krsr -l+kf -l),
with
kf>O,
(5)
where kTSTrepresents the TST-expected rate constant independent of D (*~ r/-t), while kf(> 0) represents a part influenced by the speed of conformational fluctuations which become slower and slower as the solvent viscosity 7/increases. To be more exact, kf decreases with 7/as
kf ~ r / - a ,
with
0
(6)
This fractional-power dependence of kf on 7/"l can be obtained in the natural situation that k(X) has a nonvanishing width in X. When krs r << k / i n the low viscosity region, eq. (5) reduces to k = krs r, recovering the result of the TST. When kf << krs r in the high viscosity region, on the other hand, eq.(5) reduces to k = ky *~ r/-a, and the rate constant decreases more slowly than r/-l, as observed in photoisomerization of stilbenes in solvents 4. The rate constant of photoinduced isomerization of stilbenes in solvents, however, always decreases as r/increases. This seems to mean that the rate constant is so large because of a low height of the transition-state barrier that the reaction has already been controlled in solvents by a slow speed of conformational fluctuations in the solute-solvent system. This situation itself is very interesting, but is not most convenient from the standpoint of investigating the general expression for the rate constant of solution reactions. More adequate is a situation that the rate constant can be described by the TST in the small r/region, but decreases with r/in the large r/region, entering the non-TST regime. Such a situation was recently reported by Asano and his collaborators 13 with the pressure dependence of rate constants of thermal Z/E isomerization of substituted azobenzenes and N-benzylideneanilines. Their data enables us to determine which current of theories initiated by Grote and Hynes or by Sumi and Marcus is more appropriate.
69
Experimental Check of Theories by Thermal Isomerization in Solvents Although substituted azobenzenes and N-benzylideneanilines are similar in structure to stilbenes, the reaction investigated with these molecules was the recovering of the E form from the Z form obtained after the photoinduced E/Z isomerization, which was investigated for stilbenes. Differently from the E/Z photoisomerization of stilbenes, the Z/E reaction takes place slowly with a rate constant smaller than about 103 s -~ by surmounting over a high transition-state barrier (with height of about 50 kJ/mol) on the ground-state potential surface. 13The solvent viscosity 77 was changed over about 108 times by pressures. In the low-pressure region, the rate constant of this reaction showed a pressure dependence understandable in the framework of the TST. In the high-pressure region, on the other hand, it steeply decreased as the pressure increased, to be more exact, as 77 increased with the pressure. In this non-TST regime, the observed rate constant kobs showed a fractional-power dependence on 77"l, as observed for stilbenes in solvents. It was shown by a joint work of Asano and Sumi 14that the pressure (or, viscosity) dependence of kobs mentioned above could nicely be fitted to eqs.(5) and (6) derived from the SumiMarcus model. As concrete examples in molecules investigated experimentally in refs. 13 and 14, we take 4-(dimethylamino)-4'-nitroazobenzene (DNAB) and N-[4-(dimethylamino)benzylidene]4-nitroaniline (DBNA), since these molecules showed widest ranges of rate-constant variation with pressure. Solvents used were glycerol triacetate (GTA) which is polar, and Traction Fluid B (TFB) which is essentially nonpolar, being a commercial product composed mainly of 2,4dicyclohexyl-2-methylpentane. The structure of DNAB and DBNA is shown in Fig.3 together with the schematic profile of the ground (So)- and the excited (S 1)-state potential surfaces. The photoinduced reaction cycle of these molecules proceeds on these potential surfaces as follows: In the photoinduced E/Z isomerization process, denoted by 1 in Fig.3, a molecule with the stable form E on the S O potential surface is photoexcited to the S 1 surface, from which, surmounting over a small potential barrier, the molecule drops to an intermediate state at the lowest point of the S 1 surface. Then, the molecule makes a transition to the top of the S O surface, from which, along the S O surface, about half of it relax to the Z form and the remaining half relax to the original E form. In this process 1, surmounting over the potential barrier on the S 1 surface is rate limiting. Then, measuring a rise of population in the Z form, we can determine the rate constant of the barrier surmounting, as performed for stilbenes. Since the Z form is metastable on the S O surface, higher than the E form, it gradually transforms itself to the E form, surmounting over a potential barrier with height ,4 shown in Fig.3. This process is denoted by 2. Process 1 is very fast because of a low barrier on the S 1 surface. In stilbenes, the barrier height is about 15 kJ/mol and this process takes about 100 ps after photoexcitation, 4 as mentioned in Section 1. Process 2 is, on the other hand, slow because of a high barrier on the S O II.
S1
X- CH:DBNA~ X=N:DNAB
1
~O2 2
NMe2
. . . . .
Z
E
NO2
X=N NMo2
Figure 3. Molecular structure of DBNA and DNAB, and their photocycle composed of E/Z isomerization (process 1) and ZIE isomerization on the ground (So)- and the excited (S 1)-state potential surfaces.
70
surface. In both DNAB and DBNA, the barrier height is about 50 kJ/mol and this process takes at least about 1 ms at room temperature for DBNA. It is about 103 times slower for DNAB where polarity increase occurs at the transition state due to a reconstruction of electronic structures 15.t6 that NMe2 and NO2 in Fig.3 are converted into NMe2 + and NO2 in association with conversion of the X-N double bond into a single bond around which DNAB twists for reaction. Such an increase in polarity does not occur in isomerization of DNBA which proceeds via an inversion transition state, t5.16 Anyhow, in measurements with time resolution longer than I ns, we observe as if process 1 took place simultaneously after photoexcitation, followed slowly by process 2. In order for the reaction to take place with the mechanism in the Grote-Hynes theory as well as in the Kramers theory, the reactant must surmount over the transition-state barrier only by diffusional Brownian motions regulated by solvent fluctuations. In the two-step mechanism of the Sumi-Marcus model, on the other hand, surmounting over the transition-state barrier is accomplished as a result of sequential two steps. That is, the barrier is climbed ftrst by diffusional Brownian motions only up to intermediate heights, from which much faster intramolecular vibrational motions take the reactant to the transition state located at the top of the barrier. It is apparent, therefore, that differences between these two mechanisms manifest themselves most drastically when the transition-state barrier is much higher than the thermal energy kBT which is about 2.5 kJ/mol for T = 300 K. In this res~e~_t, isomerization of substituted azobenzenes and N-benzylideneanilines with a barrier height t3.t of about 50 kJ/mol on the S O surface is more interesting than photoisomerization of stilbenes with a barrier height 4 of only about 15 kJ/mol on the S 1 surface, if we can cast the former into solvent-fluctuation controlled under pressure. The observed pressure dependence of the rate constant kobs for thermal isomerization of DBNA in TFB and DNAB in GTA is shown respectively in parts (a) and (b) of Fig.4. Detailed experimental setup for the measurement was described in Refs. 13 and 14. The viscosity 77 of these solvents was determined also in Refs. 13 and 14 by best fitting of 77 values measured at several pressures P with a formula of log r/*, P since they met this formula in the whole P region used. The solvents remained transparent in this region without any indication of phase change. In Fig.4(a) for DBNA in TFB, ko/. is nearly independent of pressure when P < 100 MPa (= 1000 bar). This behavior can well be understood in the framework of the TST as a situation =5 that the volume of the reactant is almost unchanged between the reactant and the transition states. In Fig.4(b) for DNAB in GTA, on the other hand, kobs increases as P increases when P < 250 MPa (= 2500 bar). This behavior can also be understood with the TST as a situation of a negative activation volume 16that the transition state has a volume smaller than the reactant state. This difference between DBNA in TFB and DNAB in GTA in the activation volume in the TST regime arises 1000
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A
A
A
A
&
&
0
0
0
000000
~176176176176176
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A
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*
A
10
9 278 K O 288K & 298 K 9 308K
I
(a)
I 0
50
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A
6 0 9
o O 9
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4-" = 2-
o 0
kob=
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0
1
9 9
(sec")
20.1-
DBNA in TFB I
I
I
I
150
200
250
Pressure (MPa)
O
A O
& A
0 0
O
I-
4
300
0 (b)
0 9
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100
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9 A
kob,
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(sec")
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0 9
A &
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9 278 K O 288K A 298K 9 313 K i
i
I
200
400
600
I
8O0
Pressure (MPa)
Figure 4. Pressure dependence of the rate constant kobs of thermal Z/E isomerization of DBNA in TFB [part (a)] and DNAB in GTA [part (b)] at temperatures around 300 K.
71 1000=
§
2100~
§
h
4
2-
+
k~
o O 8 ~ o
+
o
4
0
DBNA I 50
in TFB
i 100
(El)
I 150
at
I 300
Pressure (MPa)
9
6
0
4)
DNAB in GTA at 2 9 8 K I 200
0.1 0
Co)
9
+
0
2-
I 250
9
1 ' e" 4-"
278 K i 200
kT~
o
0
0
S.1 4". 2-
os~
2-
h +
2-
ooooooe8~o o;tooooooo ~ ~ +
I0=
1
+
4"
0
s-I
§
100 7 .
+
4
i 400
i 600
I"' 800
Pressure (MPa)
Figure 5. Decomposition of kobs into kTS T and k./for DBNA at 278 K and DNAB at 298 K. from the difference in the static solute-solvent interaction, mentioned earlier, that the polarity at the transition state increases in a polar solvent in the latter, causing electrostriction there, while does not in a nonpolar solvent in the former. Fitting of the pressure dependence of kobs by the TST with an appropriate activation volume in the low pressure region was performed in Refs. 13 and 14. Then, analytically extrapolating the pressure dependence to the high-pressure region, we can obtain the TST-expected rate constant kTST in the entire pressure region. It is shown in Fig.5. In the low-pressure region, kobs is the same as kTST, but in the high-pressure region, kob s deviates downward from kTS T. This means that kob s can be expressed in the form of eq.(5). The kf derived by expressing kobs in the form of eq.(5) is also shown in Fig.5 together with kob s. The. pressure dependence of kf obtained above was converted to its dependence on the solvent viscosity 77 determined at each pressure. It is shown in Fig.6. We see therein that kf has indeed the 77 dependence of eq.(6) and it is satisfied for 77 variation extending over 104 times. We see thus that thermal isomerization of substituted azobenzenes and N-benzylideneanilines in solvents has rate constants which can well be rationalized in the framework of the Sumi-Marcus model which gives eqs.(5) and (6). More detailed discussions can be found in Ref. 14. Next, let us discuss whether the Grote-Hynes theory can be fitted to the rate constants kob s. In this thoery, kobslkTs T is related to the frequency (that is, the speed)/,t with which the reactant passes, by diffusive Brownian motions, through the transition-state-barrier region in Fig. 1, as
~oo~.
1000-
,-
4-
2:
2-
100-...
lO~
s.1 4! 2. 10-
s.1
,!
2:
1
(a)
103
1s" 4"
DBNA in TFB at 278 K ' ' '"'"1
10 4
'
,
,,
" ....I ' ' " ....I 10 s 10 6 10 7 Viscosity 11 (Pa s)
' '"'"1
' '
' '
'""1 10 a
4.. 2-
(b)
102
DNAB in GTA at 298 K '
' ' '""1
10 3
'
' " '""1
'
' ' '""1
10 4 10 s Viscosity 11 (Pa s)
'
' ' '""1
10 8
Figure 6. Viscosity dependence of kffor DBNA in TFB at 278 K and DNAB in GTA at 298 K.
72
kob,lkrsr
= l~lcob ,
and
/~lCOb = COb/[l~ +
g'(/~)],
(7)
where rob represents the imaginary frequency at the top of the barrier proportional to the square root of the downward curvature there and ~'(,u) represents the frequency dependent friction defined by the Fourier transform of the frictional memory function ~(t) in eq.(2), as
r
= SO ~(t)e-#tdt
We can derive the correlation time
(8)
"
~;sc of random forces in eq.(2) from (R(t)R(O))/(R(O) 2) by
~sc = ~o (R(t)R(O))/(R(O) 2) at .
(9)
Then the correlation function can be regarded as a function f(x) of a scaled variable x = t/'csc,as
/(tl~,~) = (R(t)R(O))I(R(O)2) = ~(t)/~(0),
with
/(0) = ~
(~0)
where the second equality is obtained from eq.(3). Equation (9) ensures ~o f(x)dx = I. W h e n
the Laplace transform of f(x) atthe Laplace variable 21,is written as F(/I,),eqs.(8)- (I0) give
F(/a~sc) - ~of(X)e-~'cXdx = ~(#)/~'(0).
(ll)
From Fig.5, we can get kobs/kTsT as a function of pressure. If it is describable by the Grote-Hynes theory, eq.(7) enables us to determine both # and ~'(#) at each kobs/kTsT and hence at each pressure P or viscosity r/, where rob should have a magnitude on the order of 10 B st
in eq.(7),s In eq.( l l) the zero-frequency friction ~'(0)can reasonably be regarded as given by the hydrodynamic Stokes-Einstein relation,proportional to 77.sThen we can determine F(l.t"rsc) at each 77 from eq.( I l). It is smaller than 10 "s at any 7"/orP where kobs/kTsT was determined from Fig.5, as explicitly shown in Ref. 17. The firstequality in eq.( l l) ensures that F(/I,)<< l can be obtained only for A >> I, and in this case F(;t) is nearly equal to I/:t. Since F(g "rsc)< I0 -s as mentioned above, we get F(l.t"rsc)= (Iz ~sc)l, and the second equality in eq.(l I) gives
~'sc= g-l ~'(0)/~'(g). Since all of/z, ((,u) and ~'(0) have been determined at each 77 or P, we can determine a ~:~cvalue which is required in order to describe the experimental data in Fig.4 by the Grote-Hynes theory. As explicitly shown in Ref. 17, ~:semust be as large as about 50 ns at 17 = 100 Pa. s at 308 K and about 5 ms at 77 = 5 x 107 Pa. s at 278 K for D B N A in ITB, and about 10 ns at 77 = 20 Pa. s at 313 K and about I ms at r/= 107 Pa. s at 278 K for D N A B in GTA. These requirements cannot be satisfiedin real systems, since "cscshould be at most on the order of I ps, as explained below: The average decay rate, l/ "csc,of the correlation function of eq.(3) should be on the order of the width of frequency distributionof solvent microscopic motions interacting with the reaction coordinate,isW a v e lengths of these motions must be at most on the order of the dimension, of several angstroms, of isomerizing moieties. It has been shown by inelastic neutron scattering of molecular liquidsI~ that even for a fixed value of wave length of this order, their frequencies have a width on the order of I T H z = I/(l ps). W e see thus that the experimental data in Fig.4 cannot be described by the Grote-Hynes theory. Discrepancies between the required and the reasonable values of "rscare so huge that the conclusion obtained above is unaltered
73 even if the Stokes-Einstein relation giving ~'(0) *~ 7/was violated so much, as shown in Ref. 17. The temperature dependence of kobs in the non-TST regime deserves also mentioning in relation to differences between the Grote-Hynes model and the Sumi-Marcus model. In the latter it corresponds to that of k f in the limit of this regime where kobs = ky in eq.(5). The thermal activation energy of kobs = kTST in the TST regime is given by the height of the lowest point S on the transition-state line C in Fig.2, while under the isoviscosity condition that of kf is given approximately by the height of the point on the abscissa with the same X value as point S. Then, in the Sumi-Marcus model the thermal activation energy of kobs under the isoviscosity condition should be considerably smaller in the non-TST regime than in the TST regime. In the Grote-Hynes model, on the other hand, it should be unchanged in both the TST and the non-TST regimes, since the reactant must surmount over a same barrier in any situation in Fig. 1. The r/dependence of k f was shown in Fig.6 only at 278 K for D B N A in TFB and at 298 K for DNAB in GTA. Points of k f vs. 71 at other temperatures were observed to line up approximately along the same straight line in Fig.6 or its extension for both DBNA in TFB and DNAB in GTA. ~4This fact means that the cr value in eq.(6) is nearly independent of temperature, and also that the isoviscosity activation energy of k f is nearly vanishing. In general, that of kot,s was considerably smaller in the nonTST regime at high viscosities than in the TST regime at low viscosities. This is another reason why thermal isomerization of these molecules cannot be described by the Grote-Hynes theory. The rate constant with the form of eq.(5) can be obtained only when reaction takes place as a result of sequential two (or more) steps. It is easy 14 to prove eq.(5) for a single intermediate state M, but in this case the o~ value in eq.(6) reduces to unity. It is when M is continuously distributed [described by a continuous variable X in eq.(4)] that a becomes smaller than unity. ~2 This form of eq.(5) has been used also by Troe 2~ for describing the pressure dependence of photoisomerization rates of stilbenes in solvents, without specifying sequential two steps underlying it.
Acknowledgments: This work was accomplished in collaboration with Professor T. Asano of Oita university. He is sincerely acknowledged for valuable discussions in preparing this paper. 1 2 3 4 5 6 7 8
9 10 11 12 13 14 15 16 17 18 19 20
For example, N.G. Goguadze, J.M. Hammerstad-Pedersen, D.E. Khoshtariya and J. Ulstrup, Eur. J. Biochem. 200, 423 (1991). As a review, M.J. Weaver and G.E. McManis HI, Acc. Chem. Res. 23, 294 (1990). For example, P.J. Steinbach et al., Biochem. 30, 3988 (1991). As a review, D.H. Waldeck, Chem. Res. 91, 415 (1991). For example, Ber. Bunsenges. Phys. Chem. 95, no.3 (1991), Rate Processes in Dissipative Systems. H.A. Kramers, Physica 7, 284 (1940). R.F. Grote and J.T. Hynes, J. Chem. Phys. 73, 2715 (1980), and 74, 4465 (1981). B. Bagchi and D.W. Oxtoby, J. Chem. Phys. 78, 2735 (1983); G. Rothenberger, D.K. Negus and R.M. Hochstrasser, J. Chem. Phys. 79, 5360 (1983); S.K. Kim and G.R. Fleming, J. Phys. Chem. 92, 2168 (1988); & N. Sivakumar, E.A. Hoburg and D.H. Waldeck, J. Chem. Phys. 90, 2305 (1989); see also, as a review article, B. Bagchi, Internat. Rev. Phys. Chem. 6, 1 (1987). N.G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland: Amsterdam, 1992). H. Sumi and R.A. Marcus, J. Chem. Phys. 84, 4894 (1986). N. Agmon and J.J. Hopfield, J. Chem. Phys. 78, 6947 (1983). H. Sumi, J. Phys. Chem. 95, 3334 (1991). K. Cosstick, T. Asano and N. Ohno, High Pressure Res. 11, 37 (1992). T. Anaso, H. Furuta and H. Sumi, J. Am. Chem. Soc. 116, 5545 (1994). T. Asano et al., J. Org. Chem. 58, 4418 (1993), and earlier papers cited therein. T. Asano and T. Okada, J. Org. Chem. 51, 4454 (1986), and earlier papers cited therein. H. Sumi and T. Asano, Chem. Phys. Lett. (1995), and J. Chem. Phys. 102, no.24 (1995), in press. R. Zwanzig, J. Stat. Phys. 9, 215 (1973); E. Cortes, B.J. West and K. Lindenberg, J. Chem. Phys. 82, 2708 (1985); & B.J. Gertner, K.R. Wilson and J.T. Hynes, J. Chem. Phys. 90, 3537 (1989). F.J. Bermejo, F. Batallan, J.L. Martinez, M. Garcia-Hemandez and E. Enciso, J. Phys. Condens. Matter 2, 6659 (1990), & E.G.D. Cohen, P. Westerhuijs and I.M. de Schepper, Phys. Rev. Lett. 59, 2872 (1987). J. Troe, Ber. Bunsenges. Phys. Chem. 95, 228 (1991), and earlier papers cited therein.
This Page Intentionally Left Blank
journal of MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids. 65/66 (1995) 75-83
D y n a m i c s of Brownian motion under a potential and theory on the rate of chemical reactions A k i o Morita
D e p a r t m e n t of Chemistry, College of Arts and Sciences, University of T o k y o , K o m a b a , M e g u r o - k u , T o k y o 153, J a p a n
Abstract We have introduced a two square-well potential system, solved the Smoluchowski equation, calculated exactly the concentration of the reactant, compared it to that resulting from the phenomenological rate equations and finally obtained expressions for rate constants for a first-order reversible reaction. It is found that the pre-exponential factors for both constants are just the diffusion coefficients divided by the square of the widths of potential wells. Our expressions for rate constants satisfy a relation between standard free energy and the equilibrium constant arising from chemical thermodynamics. We have also developed a general theory on the relaxation processes of approachment of a system to the equilib~Sum Boltzmann for an arbitrary potential with sufficiently deep well and the escape rate from the well for an asymptotically long time limit of the reaction, which enables us to calculate the rate constants for given potentials. Furthermore, we obtained the rate constant for a dissociation reaction of an ion-pair interacting through the Coulomb potential and found a maximum as the sum of the radii of the ion-pair increases.
I.
Introduction
In formulating the rate constant based a dynamical equation such as the Fokker-Planck-Kramers equation, the Smoluchowski equation and etc., we usually use the assumption of stationary state [1, 2, 3]. However, it is sometimes the case where dynamical information is suppressed by the assumption that the flux of Brownian particles must be uniform in space. It is thus sensible to calculate the rate constant by taking the dynamics explicitly. To this end, we must bear in our mind time scales of the dynamics. Imagine the diffusion of solute particles in solvent by regarding the position of a particle x as a reaction coordinate. When it diffuses on a fiat potential, the time scale we must be concerned with is of course that of diffusion, which is much longer than collision time between the solvent and solute molecules. Now, suppose it moves under a square-wen potential and its height is infinitively high, the main time scale will be roughly a relaxation time of the order of the square of the width of the potential bottom divided by the diffusion constant, which determines time approaching to an equilibrium state after colliding with the potential walls several times before settling into it. However, as lowering the potential height slightly, we see that the particles can now escape from the well after colliding with the potential walls much more times, which makes the time scale much longer to settle the motion. This is precisely why the exponential factor comes into play. The time scale in this region may be called a reaction time which is determined from the diffusion at potential bottom and the barrier height leading to the pre-exponential and exponential factors, respectively. We will be interested in tiffs long reaction time so that time scales representing molecular motion in small changes are trivial. In tiffs paper, we will devote ourselves to finding this reaction time based on the recent development on investigation of the dynamics of Brownian motion under a square-well potential [4]. Furthennore, we will develope a general theory for an arbitrary potential to obtain the reaction time. With this 0167-7322/95/$09.50 91995 Elsevier Science B.V. All rights reserved. SSD! 0167-7322 (95) 0(0)003
76
new method, we will be able to calculate the rate constants for given potentials. Also, this general theory enables us to estimate how fast a system approaches to the Boltzmann distribution in the reaction time-scale. We shall apply the theory to single minimum with a single maximum, double wells and the Coulomb potentials. Our aim of this study also includes how a potential without the curvatures affects the rate of reaction by remembering that they axe essential in Kramers' theory [1].
II.
Theory and discussion
r(x)
-------
We begin with the following Smoluchowski equation describing the Brownian motion under a potential V(x) for the probability density p(x, t):
L
c3p(x,t) Ot
0 [O,,p(x,t) = o--~ l
1
dV(x)
]
Ox " + kB----T d-----x---p(x't)
J
(1) which may be written after taking the Laplace transform with respect to t
dg(x,A) dx Figure 1" Parameters for the two square-weU potential where Ai = a i / k B T ,
dJ(x,A) dx
---- -eV(x)/kBT J(x, A)
(2)
= _A2e_V(x)/ksTg(x,A ) -t-~(x--xo)
(3)
where g(X,A) -- eV(z)/kaTp(x,A) and J(x, t) is the flux defined by
J(x,t)
-
1 dV(x) ] - [Op(x,t) l Ox + kB'--T d ~ p(x't)
Unless otherwise stated, we denote f(x,A) -"
f(x,t)e-~tdt
~0~176
with s = A2
that is, f(x,A) is the Laplace transform of f ( x , t ) with respect to t. Before treating a general case for an arbitrary potential V(x), we shall consider a specific and simple nmdel of a two square-well potential and obtain an exact solution to show how the reaction time may be found so that our approach will be clarified. It should be emphasized that in treating the Brownian motion with a discontinuous V(x) such as the two-square-well potential sketched in Figure 1., we must use Eqs. (2) and (3), otherwise we do not recover the Boltzmann distribution in the hmit of t ~ ~ . It follows immediately from Fig. 1 that the equihbrium Boltzmann distribution function, peq(X) is given by
peq(X) = [ H ( x ) - H ( x - L1)]e A' + [H(x - Ll) - H ( x - L2)] + [H(x - L2) - H ( x L(e Al + l + e A3)
L3)]e A3
77
where H ( x ) is the Heaviside step function, i. e. H ( x ) = 0 for x < 0 and H ( x ) = 1 for x > 0. Eqs. (2) and (3) readily leads to the following solution for a flat potential V ( x ) = - A k B T X(x,~)
(4)
= A ( x - a , A ) X ( a ) 4- H ( x - x o ) A ( x - x o , ~ ) B
where X ( x , )~) ---
g(x,~) J(x, )~)
'
A(x,)~) -
cosh()~x)
--)~eAsinh()~x)
'
B =
1
We can use this result for obtaining the solution for the present potential in Fig. 1 by introducing the boundary conditions JI(0,)Q = 0,
JI(LI,)~)=J2(L1,)Q,
J3(La,)Q = 0
J2(L2,)Q=J3(L2,)~),
(5)
and e-A1pl(L1,A)
= p2(L1,A),
p2(L2, A) = e - A s p 3 ( L 2 , ~ )
(6)
which enable us to write X l ( x , A) = A(x, A)X(0, ~) + H ( x - xo)A(x - zo, A)B
(0 < x <_ L1)
X2(x,)~) -- A ( x - L1,)t)X(L1,)~)
(L1 < x _< L2)
X3(x,)~) = A ( x -
(L2 < x <_ La)
L2,)QX(L2,)t)
where we have divided X(x,A) into three regions according to the divisions of potential in Fig. 1. Note in Eqs. (5) and (6) that although the fluxes are continuous, the densities are not just like the potential. Number of particles in the region 0 < x g L, N1 (t) can be obtained from
N~(~) = - ~ 1 {cosh[)~(L - xo)l - 1} + (1 + e A a ) c o s h ( $ L ) s i n h ( $ L ) s i n h [ A ( L -
xo)] + e A~ cosh[)~(x- xo)][cosh2(AL) + e As sinh2()~L)]
~2 [eA,+As sinh2()~L) + (1 + e A~ + eA3)cosh2($L)]
(7) The inverse Laplace transform of the above expression can be found by noting that there are simple poles at 1 s , = An2 = - ~-ff(n~r :t: sin-lq) 2 ( n = 0,1,2 .... ) (8) where i q=
14. e A~ 4- e Aa e A~+Aa 4. 1 4. e A~ 4- e As
We are particularly interested in so 1 so = -~--ff(sin -1 q)2 We see that A1 = A2 - 0 leads to q = vf3/2, which is equivalent to sin-lq = r / 3 that corresponds to the maximum value, and by increasing both A1 and A3, sin-lq approaches to zero asymptotically. When these are very large, it follows that sO ; - - -q2 ~
whereas
1 n27r2 sn = - L2
( n -- 1, 2, 3 .... )
78
Since the contribution from s, makes the relaxation much faster than that from so, for the large time scale the time-dependent behaviour of N1 (t) is mostly governed by two poles of s = 0 and s = so which are quite close together but not quite the same leading to e -As
1 k
Note that Nl(S) now is independent of the initial position of xo, which directly suggests that in this long time-limit the particles must hit the walls many times before escaping the wells, keeping almost quasi-equilibrium within the wells, thus forgetting the initial position. It is seen that since the small value of s corresponds to the long time behaviour, the direct substitution of s ~ 0 in Eq. (7) also leads to the same result in Eq. (9). This will become a useful point when we treat the general case. On using the phenomenological rate equation for the first-order reaction of the type A ~ B, that leads to the following concentration of A, x(s) z(s)
s+ k~ s(s 4- k 4- k')
=
and comparing it to the result in Eq.(9), we finally find k =
De-A~ L2
k t --
De-As L2
(10)
where k and k' are the rate constants for the forward and backward reactions, respectively. It explicitly shows that the pre-exponential factors are governed by diffusion processes in the potential wells and the exponential factors come into play due to rebouncing motion at the wells, viz the higher the barrier is, the more rebouncing is needed to escape from the wells. Furthermore, our expression in Eq.(10) satisfies the relation from chemical thermodynamics that is AG "9 = - R T log K where K = k / k ' is the equilibrium constant and G e is the standard free energy for tile reaction. This shows the difference in A s - A 1 determines the equilibrium state, whereas the barrier heights A 1 and As govern the dynamics of reaction. In conclusion, the dynamics of the square-well models can indeed take into account the phenomenological rate equations satisfactorily, which means we in fact have a rather microscopic dynamic model that goes quite well along the phenomenological blackbox relations. We will consider the irreversible first-order reaction with the square-well potential elsewhere [5], that also supports the same conclusion. Our method is different from that of the previous studies [6, 7, 8] investigating by means of the eigen-value problem. Now, we extend the above treatments to consider generally what is the reaction time after-all and to work out a method to obtain the reation times from arbitrary shapes of potentials. We write Eqs. (2) and (3) in the forms of the following integral equations: g(x,A) = g(a,A) - J(a,X)
e x p [ V ( x l ) / k B T ] d x l - H ( x - xo)
exp[V(xl)/kBT]dxl o
+ s
~X~X1exp
J(x,A) =-sg(a,A)
{[Y('xl)
-
(11)
V ( x 2 ) ] / k B T } g(x2, A)dxldx2
exp[-V(xl)/kBT]dxl
+ J(a,A) + H(x-
xo)
79
+ s
exp { [ - V ( x l ) +
V(x2)]/kBT} J(x2,)~)dxldx2
(12)
whose solutions can be expressed by g(x,A)
=
(l+sll-2>+s2ll-2+3-4>+...)g(a,A)-(ll>+sll-2+3> +s211 - 2 + 3 - 4 + 5 > + . . . ) J ( a , A) -
H(x - xo)(ll>o + sll - 2 + 3>0
+8211 -- 2 + 3 - - 4 + 5 > 0 + ...)
J(x,~)
=
-s(]-l>+s]-l+2-3>+s
(13)
2]-l+2-3+4-5>+...)g(a,A)+(l+s]-l+2>
+s2l - 1 + 2 - 3 + 4 > +...)J(a,)Q +
H(x - xo)(1 + s I - 1 + 2>0
+s2l - 1 + 2 - 3 + 4>0 + ...)
(14)
where we have employed bra ket notation through the following definition: [1 - 2 + 3... > .
.
.
JO"
J~
.
exp {[V(xl) - V(x2) + V(x3) - . . . ] / k B T }
dxldx2dx3...
(15)
and I...>o stands for the above integral on replacing the lower limit, a by x0. We are interested in the domain of reaction time, that is the barrier heights are sufficiently high in comparison with the thermal energy and the long time-behaviour of the dynamics. As a first general example, we shall consider the case where the potential heights at x = ul and x = u2 are infinitively high so t h a t
J(ul, A) =0
J(u~, )~) =0
These boundary conditions together with Eqs. (11) and (12) lead after taking the limit of s --, 0 to the following expression valid for the long time:
p(x,t.) _.exp[-V(x)/kBT]<_l> [ 1 - e x p ( - < _ l + 2 _ 3 >-<1 >
DE)]
(16)
where < ... > represents the integral in Eq.(15) with the upper and lower limits of us and ul, respectively. This shows explicitly the long time dynamical processes reaching the final Boltzmann equilibrium distribution. By separating the reaction coordinate into ul <_ x < u and u < x <_ u2 corresponding to the reactant and product regions, respectively, we find the rate constants, k =
D E~ exp[-V(x)/kBT]dx < -1+2-3
>
k ' - D f,~2 exp[-V(x)/kBT]dx
(17)
(18)
<-1+2-3> Whereas if we assume that the potential at ul is infinitively lfigh and that at u2 is infinitively low we can impose the conditions, J(ul,A) = 0 g(u2, A) = 0 which lead to
p(x,t) - exp[-V(x)/kBT] [< 1 > o - H ( x - xo)[l>o] exp [ - < 1 - 2 > D r ]
(19)
80
It is important to realize that p(x, t) takes the Boltzmann factor for x < x0 and does not lead to Boltzmann for x > xo before vanishing. However, when the initial position is near the potential minimum like the system we are concerned with, the pre-exponential factor in Eq. (19) hardly depends on x0 as far as the batTier is sufficiently high. If we apply Eq. (19) to an irreversible reaction, we find D <1-2>
k =
(20)
This expression is different from that of Kramers [1] in that we have taken into account the dynamics explicitly whereas Kramers assumed the stationary condition where the process is time-independent and obtained the following expression: D kc(~ <1> For three-dimensional motion under a central potential V(r) where r is the radial distance, the Smoluchowski equation corresponding to Eq. (1) can be expressed by
Op(r,t) 1~_.~ [OP~rt) 10V(r), ] Ot = "~ r2 + kBT -~r .p(r,t)
(21)
By defining functions, 1
P(r,t) -- eV(r)/ksTp(r,t) .8
Q(r,t) -- r2j(r,t)
.6
and taking the Laplace transform with respect to t, we can reduce Eq. (21) to
dP(r,s) dr
.4
dQ(r,s) dr
.2
e v(r)/keT ~Q(r,s) r2
--
(22)
-sr2e-V(r)/ksT p(r,s) -t- ~f(r -- r0)(23)
where 0 0
20
40
t
~0
Figure 2: Numerical solution of Eq. (1) with potential A (full curves) and theoretical results (marked curves) from Eq. (26)
j(r,t)
0p(r,t)
~
Or
-4
10V(r) p(r,t) kBT Or
It is extremely important to note here that we can easily shift from the one dimensional to three just by regarding
p(x,t) --. r2p(r,t),
J(x,t) --, r2j(r,t)
(24)
and
V(x) ~ V'(r) -'- V ( r ) - 2kBTln r
(25)
This states that we obtain the complete picture of the three-dimensional Brownian motion with the central potential from the one-dimensional one just on replacing V(x) by V*(r), i.e., for the
former motion, the extra potential -2kBTln r arising from an orientational Brownian motion nmst be considered. This is very much similar to the case where the apparent ove~-all potential for two or three dimensional motion in mechanics with a central force consists of the centrifugal mad the mechanical potentials. Now, we shall apply the above general formalism to three specific potentials, A. a system with single minimum and single maximum, B. double minimum system and C. the Coulomb attractive potential, given by the following expressions:
V(z) = b (x2 - 2x 3) V ( x ) = Vl x 4 + v 2 x 3 - 2Vl x 2
(A) 3v2x
-
(B)
V(r) = ---
(C)
r
respectively. Potential A has been set with the minimum at x -- 0 and with the maximum at x - 1. The potential barrier vo is given by 3b = kBTVo. Whereas, potential B has been set with the minima at x = - 1 and x - 1 and the differences in the minima, in the minimum at x -- - 1 and the maximum at xmax = -3v2/4Vl, and in the minimum at x = 1 and the maximum are Av = 4v2, AVl = Vl -- 2v2 + ?3max and Av2 = vl + 2v2 + vrna=, respectively, where ?3m~x= V(xmaz). 1
The number of the particles, N (t) for x <_ 0 as a function of time t has been calculated for potential A based on Eq. (19). Although the present theory cannot describe the accurate behavior of N(t) in the short time regime, on taking into account the rate constant in Eq. (20), we set N(t)=exp[
<1-2>t]
.8
.6
.4
(26)
with D = 1. The Smoluchowski equation with potential A has also been solved numerically and N(t) obtained from Eq.(26) and the numerical analyses o are shown in Fig. 2 where the full curves from o 20 40 60 the upper to the bottom are obtained numerit cally for V0=5, 4, 3, and 1, respectively, and the marked curves are from Eq. (26). It is seen that the agreement is remarkable especially for sys- Figure 3: Numerical solution of Eq. (1) with terns with high barrier tops. The pre-exponential potential A (full curves) and theoretical results factor in Eq. (19) has been also calculated nu- (marked curves) from Eq. (27) merically and compared with theoretical one. We confirmed the good agreement between them for relatively long time. Similar treatments have done for potential B that is a model for reversible first-order. The number of particles in the left well, N1 (t) is set gl(t) =
k' k k+k' +~exp
[
<-l>t] < -1 +2-
3>
(27)
82
with k and k ~in Eqs. (17) and (18), respectively. Nl(t) from Eq. (27) and the numerical calculation (full curves} are shown in Fig. 3 where the filled circles, open circles, and crosses are obtained from Eq. (27) for A v / k B T = 1, 2 and 3 with the fixed value of (Art + Av2 -- 2 v , n a z ) / k s T = 8 and open squares and triangles are ( A v l q- Av2 -- 2 v , , a ~ ) / k B T = l O and 6, for the fixed values of A v / k B T = 2 . Again, we see the agreement is excellent. Because we are now confident on the agreement of the dynamics between the present results and numerical ones, we re-consider the rate constant due to the Coulomb potential C. We see from Eq. (26) and potential C that there exists a maximum for V*(r) at r = rm~z = w / 2 k a T = rc/2 in which rc is the Onsager radius and an highly attractive region at a < r < rmaz. Because of the existence of this region, an ion pair produced at t - 0 is easily recombined. We seek the escaping rate of an ionized particle over the potential top analytically as tt functions of a and rmaz from Eq. (20). To this end, we have to calculate
-rc
4
< 1- 2 > =
~12drl
r22e'2dr2
(28)
where rc = w / k B T and R is a parameter finally to be put oo. On integrating the right hand side in Eq. (28) by parts and expressing the result in terms of a confluent hypergeometric function, we finally find the following approximate expression: D r c 2 =r.f..
k - ---~e
~
( f o r small a and R ---, cc)(29)
We have plotted this k as a function of a in Fig. 4 and it is interesting to note a maximum. Just for the sake of illustration, we show k including unrealistic values near a = 1 where obviously 0 .2 .4 .6 .8 1 Gr the present expression does not cover. When a is small, the potential is so deep that the main contribution to k arises from the usual exponenFigure 4: A theoretical prediction of k for po- tial factor in Eq. (29) which indicates the larger tential C with the initial position of an ion-pair a is, the faster the reaction proceeds. However. near a from Eq. (29) the barrier becomes shallow, the pre-exponential factor starts dominating the rate, where we note - 2 r c / a 3 is the second derivative of V(r) at r = a that leads to a frequency of an oscillator near at r = a. This frequency becomes small, when a is large, which results in the decrease in k. This is why k has a peak before reaching the potential maximum at r~a~. Euation (10) also indicates the pre-exponential factors are governed by tlus kind of frequency. It should be emphasized that in treating the Coulomb potential, our model in deriving Eq. (28) is different from that of Hong and Noolandi [7] who considered Brownian motion in a >_ r >_ oc from the beginning whereas we have introduced the absorbing boundary first at an finite distance, r = R and let it go to infinity in finding Eq. (29), because we know the integral in Eq. (28) does not differ considerably from that in large finite value of R. In other words, we were concerned with Brownian motion in confined space while Hong and Noolandi treated that in unconfined infinite space whose physics must be different. This is the very reason why the dyna~nics of our model has been expressed by a function of s only, whereas that of the latter needed v/~ in addition to s. For example, our case of p(r,t) decays exponentially as expressed by Eq. (19) whereas that of Hong and Noolandi decreases with e x p ( r c / ( r ~ ) .
83
I would like to record that results on irreversible and reversible reactions under square-well potentials are partly due to Mr. Satoshi Suzuki. References [I] H. A. Kramers, Physica 7, 284 (1940) [2] N. G. van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam (1992) [3] S. A. Rice, Di1~usion-Limited Reactions, Elsevier, Amsterdam (1985) [4] A. Morita,Phys. Rev. E, 4 9 , 3 6 9 7 (1994) [5] S. Suzuki and A. Morita to be published [6] E. Wong, Proc. Syrup. Appl. Math. 16, 264 (1964) [7] K. M. Hong and J. Noolandi, J. Chem. Phys.,68, 5163 (1978) [8] H. Risken, The Fokker-Planck Equation, Springer-Verlag, Berlin (1984)
This Page Intentionally Left Blank
journal of
MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 85-90
STRUCTURAL CHARACTERISTICS OF HYDROGEN-BONDED N E T W O R K S IN WATER AND ICE SYSTEMS
J C Dore and D M Blakey s
Laboratory, University of Kent, Canterbury, Kent, ~
7 N , UK.)
Abstract The molecular configurations of water molecules in the disordered condensed state are investigated by neutron diffraction. The results provide evidence for the interplay of hydrogen-bond goemetry with thermal disorder, leading to a sensitive response to external variables and local influences.
1)
Introduction
R is well known that liquid water exhibits a number of properties that are distinct from those of other molecular liquids.These characteristics are attributed specifically to the effects of a strong hydrogen-bonding interaction which leads to stmcaual ordering on a molecular scale. The tetrahedral nature of the interaction tends towards the creation of a space-filling network but the thermal disorder causes the static geometrical features to be modified. As a result, the water structure is extremely sensitive to the influence of thermophysical variables such as temperature and pressure and also to the presence of other molecules and ions, either in the form of solute entities in solution or due to the effects of a neighbouring interface. The importance of water in physical, chemical and biological sciences provides a compelling reason to study the microstrucmral and dynamical features in some detail. However, it becomes clear that the characteristics are rather subtle and although considerable advances have been made, particularly in the last decade, there are many fundamental questions that remain, at best, partially answered. The present paper provides a short critical review of current issues on the structural properties of pure water in the normal and under-cooled phases as studied by neutron diffraction incorporating information from studies of amorphous ice.
2)
Neutron diffraction techniques
a)
Structural characteristics at room temperature
Neutron diffraction measurements yield the liquid strucan~ factor Sin(Q) which contains information on the spatial correlations at an atomic scale. In the ease of water where the molecules comprise two atomic species, three independent measurements on H20/D20 mixtures of varying composition are required to determine the three pair correlation functions, goo(r), goD(r) and gDD(r) based on the assumption that hydrogen and deuterium occupy equivalent sites. The most frequently quoted results of Soper and Phillips [ 1] were based on reactor measurements with three sample compositions. Similar measurements for a different composition range have also been made by Garawi, Dote and Bellissent-Funel [2]but a detailed report has not been made of these datase~ as the five independent measurements revealed the possibility of minor differences in the hydrogen and deuterium characteristics. However, the data can be used to derive two of the partial functions and Fig 1 shows a comparison between the results of the two experiments. The consistency in the overall features is encouraging as the measurements are technically demanding but the shift in the peak position at low-r may have some significance in terms of detailed interpretation of the results. 0167-7322/95/509.50 91995 Elsevier Science B.V. All fights reserved. SSDi 0167-7322 (95) 00845-4
86
1.6
goo(r)
...
0.6
0.~
-0.2
1
2
3
4
S
(I
1
8
--0.~
1
2
3
4
5
6
r/A Fig 1: A comparison of gOD(0 and gDD(r) for water at 20oC from data of Garawi et al [2], (solid line) and Soper and Phillips [ 1] (dashed line). A full analysis of the systematic behaviour for the five d a t a s ~ has been delayed (by manpower resources) but seems to suggest that there may be small variations in the distribution of H and D atoms in the various mixtures. A further factor was also app.arent in the behaviour of the goo(r) curve where it was clear that the first peak at 2.8A was over-estimated in the Soper and Phillips data. The d~_a displayed in Fig 1 can be readily understood in terms of the local configurations of water molecules since the peak position are simply related to the distances of a hydrogen-bonded pair as shown in the insets. Most molecular dynamics studies using a range of interaction potentials are able to reproduce these featmes quite well on a qualitative basis. More recently, Soper and Finney [3] have used pulsed neutron techniques and obtained similar data for the goD(r) and gDD(r) functions but now have a goo(r) curve with a smaller peak height that is in beaer agreement with the results obtained from X-ray diffraction studies. The present situation is much beue~ than a decade ago when several groups had widely differing results [4] but a detailed examination of the s y s ~ c errors in the data treatment and a critical comparison of the various measurements has not yet been made. It therefore a p ~ that a definitive set of partial pair correlation functions has not yet been achieved for liquid water at room temperature. Although the present information is sufficiently accurate to give a clear indication of the inter-molecular features, it does not provide a sufficiently precise distribution to discriminate between the various forms proposed for the interaction potential. More recent approaches to the interpretation of either simulation or experimental results have now begun to emphasise the orientational characteristics rather than the site-site correlation [5]. Fig 2 shows a contour plot of the oxygen distribution function around a fixed water molecule by Svischev [6] using a SPC potential, and Soper has obtained a similar representation in a spherical harmonic (SHARM) analysis of neutron datao This relatively new approach should provide a greater insight into the way the basic tetrahedral order is disrupted by the thermal motion at 300 K.
7
r/,~
87
Fig 2: The orientational correlation of oxygen atoms around a water molecule shown as a contour plot from computer calculations of Svishev [6]. b)
Temperature variation studies
Temperature variation studies are based on a fast order difference technique, such that ADm(Q,AT) = Sm(Q,T2) - Sm(Q,T1) defines the change in the diffraction pattern for a temperature change AT = T2 - TI. Due to the density maximum (1 loc in D20), it is possible to choose pairs of temperatures either side of the maximum where the density is constant. This method defines the isochoric temperature derivative [7] which can be transformed to give the real-space function AdL(r,AT) where oo
AdL(r~T) = 41rrpA~(r) = 2 IQ ADm(Q,AT) sin Qr dQ 7r6 The results for two pairs of temperatures are shown in Fig 3 and reveal several interesting features. The two curves are in remarkable quantative agreement except at the low-r peak position corresponding the OD hydrogen-bond distance. It therefore seems that the local variation in the molecular correlations changes systematically across the density maximum. Furthermore, the re-arrangement is not restricted to the fast neighbour but extends to 8A, involving second and third-neighbour molecules. This result emphasises that the important characteristics of the structure are not simply dependent on adjacent molecules but are influenced by longer range effects arising from ring closure within the local network Of H-bonds. The significance of this observation becomes clearer at lower temperatures.
88
0.008
AT
0
----
-0.008
-
4
-
8
2 / 2 4 *C -o
e
se
*c
1;,
r/J, Fig 3: The isochoric temperature derivative for liquid water from Sufl et al [7] showing systematic variation over the density maximum. c)
The under-cooled liquid and amorphous ice
The structural correlations are strongly enhanced in the under-cooled state as the temperature is reduced towards the metastable limit of -40~ (for D20) and various thermophysical properties exhibit diverged behaviour [81. The exact nature of this anomaly is still the subject of some controversy. However, the diffraction pattern indicates that the structure is evolving towards that of amorphous ice which is characterised as a continuous random network of tetrahedral hydrogen-bonds [9]. Recent neutron measurements on amorphous ice [ 10] have re-inforced the earlier conjectures and shown that the structure is similar to that of hyper-quenehed glassy water produced by rapid cooling of micron-sized water droplets. It can now be realised that the CRN model for the disordered phase of ice is effectively the limiting structure of water at low temperatures. i
I
i
i
ii
i
i
]
2
4
6
8 r/A
10
12
14
16
Fig 4: The spatial function d(r) for a single single molecule form-factor fit and after removal of the cluster contribution for a-ice. The latest data has been analysed by a novel use of a cluster form-factor to fit the data in the high-Q region. The inter-molecular correlations within the tetrahedral network are evaluated up to a range of second nearest neighbour molecules. The transformation to a real-space representation in Fig 4, shows how well the local structure is represented by this approach and conf'ums that the four-bonded network gives excellent results.
89
d)
Argon/a-ice co-deposits
The presence of apolar solute molecules in liquid water is know to affect the local hydrogen-bond geometry and leads to the hydrophobic interaction. Isotopic techniques using natAr/36Ar difference measurements are feasible but difficult due to the low solubility of argon in water. Neutron results for this case have recently been given by Broadbent et al[ 111 using a pressure cell. d ~
i
,
,
,
|
i
!
2
4
6
8
10
|
,
12
14
I
g
2000
m
i
0
91A '
1.2
I
I
I
I
I
I
0.6
i5
......
o
-0.6
,
0
I
I
I
I
I
!
2
4
6
8 dA
10
12
,
I
14
16
Fig 5: a) Diffraction measurements for argon/a-ice co-deposit using Ar isotopes. b) The partial pair correlation function, dAr(r) obtained from the difference technique. An altemative approach to this question is to investigate the effect in the disordered solid phase where the spatial correlations are more pronounced and higher argon concentrations can be achieved. Fig 5a shows the two curves for a sample of codeposited amorphous ice with 8 atomic per cent of argon. A difference analysis yields a composite pair correlation function in the form 0.259gAro(r) + 0.595 gArD (r) + 0.146 gArAr(r)
90
which is shown in Fig 5b. The highly-structured curve shows that there is a wellformed cage of water molecules around the argon atoms which is very similar to the computer predictions for the liquid phase made by Guillot et al [12]. The plane of the water molecules is approximately tangentialto the Ar-O vector so that a localised clathrateg c o m c ~ is created. A doaflcd analysisof thisgeometry has yet to be made but further work is in progress. This type of measurement opens up many possibilities for further studies and is complementary to work currently being pursued by Soper, Finney and collaborators [ 131. 3)
Conclusions
The present work has shown that neutron diffractionhas an important role to play in the study of water structurein the normal and super-cooled liquid phases and also through the investigation of amorphous ice systems. The high sensitivity to minor permd3ations is seen to be due to the specificbchaviour of the hydrogen-bond geometry either through temperature change or proximity of solute particles. In the former case there is a systematic change resultingfrom a delicatebalance between the ordering effectsof the H-bond geometry and the disordering effectsof thermal motion. In the second case, the presence of the argon atoms effectivelycreates void volumes within the material and resultsin network modification such thatthe hydrogen-bond connectivity is preserved by forming cage-like structuresaround the apolar region.
The basic principles seem to be well established for the low t e m ~ regime although a full quantitative description has yet to be developed. The increased spatial correlations are important in providing clear indications of local structure but it remains to be seem whether this information can be extrapolated to predict the behaviour of water systems at ambient conditions. 4)
Acknowledgements
We would like to acknowledge many people who have contributed to this work over the last decade and would particularly like to mention Pierre Chieux (ILL) and Marie-Oaire Bellissent-Funel (LLB). It is also a great pleasure for one of us (JCD) to achknowledge the Yamada Foundation and the conference organisers for the invitation to the meeting and an enjoyable fast visit to Japan. References
1) 2) 3)
4)
5) 6) 7)
8) 9) lO) 11) 12) 13)
A K Soper and M G Phillips, Chem Phys 107, 47 (1985) M S Garawi, Thesis, University of Kent, (1987); M Garawi, J C Dore and M C Bellissent-Funel, in preparation. A K Soper and J L Finney, private communication J C Dore, p3 in Water Science Reviews' Vol 1, F Franks (ed), Cambridge (pub) 1985. 1994 A K Soper p97 in 'Hydrogen Bond Networks', NATO ASI Series Vo1435, I M Svischev and P G Kusalik, J Chem Phys 99, 3049 (1994). P A Egelstaff et al, Phys Rev Lett 47, 1722 (1981), M A M Sufi, Thesis, University of Kent, 1986; M A M Sufi and J C Bore, in preparation. A Angeil, Chap 1 in 'Water : a comprehensive treatise', Vol 7, F Franks (ed), Plenum (pub), 1982. (1984) M R Chowdhury, J C Dore and J T W enzel, J Non Cryst Solids, 53, 247 D M Blakey, p381 in 'Hydrogen Bond Networks', NATO ASI Series Vo1435, M C Bellissent-Funel and J C Dore (eds), Kluwer (pub), 1994. R D Broadbent and G W Neilson, J Chem Phys 100, 5743 (1994). B Guillot et al, J Chem Phys 95, 3643 (1991 ) A K Soper and J L Finney, Phys Rev Letts, 71, 4346 (1993).
journal of
MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65166 (1995) 91-98
S t r u c t u r e and D y n a m i c s of S u p e r c o o l e d and Glassy A q u e o u s I o n i c S o l u t i o n s Toshio Y a m a g u c h i , ~'* M o t o y u k i Y a m a g a m i , ~ H i s a n o b u W a k i t a , ~ a n d Alan K. Soper b ~ D e p a r t m e n t of C h e m i s t r y , Faculty of Science, F u k u o k a University, F u k u o k a 814-80, J a p a n bISIS Facility, R u t h e r f o r d A p p l e t o n L a b o r a t o r y , D i d c o t , Oxon, O X l l 0QX, U.K.
Abstract X-ray diffraction, neutron diffraction isotopic substitution(NDIS), and quasi-elastic neutron scattering(QENS) measurements have been made on concentrated aqueous solutions of lithium halides LiX (X=C1, Br, and I) in a temperature range covering the room-temperature liquid, the supercooled liquid, and the glassy state. X-ray radial distribution functions have revealed that the hydration shell of the halide ions is gradually structured with lowering temperature. Another interesting feature is the evolution of water-water interactions centered at 4.3 and 6.9 ~ independent of the halide ions, due probably to reinforced hydrogen bonds at low temperatures. The NDIS radial distribution functions related to Li + and C1- have demonstrated that the orientational correlation of bound water molecules for C1- is more strongly temperature-dependent than that for Li +. At low temperature the formation of a second hydration shell of Li + and C1- has been established. The QENS data have shown that the translational motion of water molecules in the LiC1 solution is more hindered with lowering temperature. The structural and dynamic characteristics of the supercooled and glassy solutions are discussed in connection with nucleation of ice, isotropic reorientational motion of water, and the partial recovery of hydrogen bonds in these solutions. I. I n t r o d u c t i o n The microscopic structure and dynamic properties of supercooled aqueous solutions have recently attracted much attention in various aspects such as nucleation of ice, crystallization of solutions, glass transition, hydrogen-bonded specificity, and ionic hydration. Lithium halides are very hygroscopic, and their concentrated solutions easily form glassy solutions at low temperatures. The physico-chemical properties of supercooled and glassy aqueous lithium halide solutions have been measured by various techniques such as thermal analysis (DTA and DSC), Rayleigh, Brillouin, and Raman scatterings) ,2 All of these data are, however, limited to qualitative features of the supercooled and glassy solutions. The structural information at an atomic level is essential for understanding the various properties of supercooled and glassy solutions. X-ray and neutron diffraction enables us to obtain direct structure information (bond distance and coordination number) of ionic solutions in terms of the radial distribution function. In the case of aqueous lithium halide solutions, X-ray diffraction data are dominated by halide-oxygen, halideoxygen, and oxygen-oxygen interactions. On the contrary, neutron isotopic substitution
0167-7322/95/$09.50 91995ElsevierScience B.V. All rightsreserved. 0167-7322(95)00896-9
SSDi
92
data with respect of Li and C1 give us the information related mostly to Li-O, Li-D, C1-O, and C1-D interactions. In addition to the static structure of the ionic solutions, their dynamic properties are also needed to explain various phenomena of the supercooled and glassy solutions. One of the available methods for this purpose is the quasi-elastic neutron scattering technique. In the present study, we have made X-ray diffraction, neutron diffraction with isotopic substitution, and quasi-elastic neutron scattering measurements on highly concentrated aqueous solutions of lithium halides in a wide temperature range from room temperature to below glass transition temperature, from which the microscopic behaviors of the static structure and dynamic properties of the solutions are revealed with lowering temperature. The results obtained are discussed in connection with ice nucleation, anisotropic motion of water, crystallization, and the partial recovery of hydrogen bonds. II. E x p e r i m e n t a l X - r a y Diffraction The sample solutions were prepared by dissolving anhydrous lithium halides into distilled water to the [H20]/[LiX] molar ratios of 5.22 (10.6 m, m=mol kg -1) and 12.1 (4.60 m, LiC1), 4.98 (11.2 m, LiSr), and 5.00 (11.1 m, LiI). X-ray scattering data were collected in reflection geometry with a 0 - 0 diffractometer (RIGAKU), except for the 5.22 m aqueous solution where a rapid liquid X-ray diffractometer with an imaging plate (Mac Science) was used in transmission geometry. Mo Ka radiation (A=0.7107 A), monochromatized by a LiF crystal for the former diffractometer and by a graphite crystal for the latter, was used for the measurements. For low temperature measurements, a cryostat described in ref. 3 was used for the 0 - 0 diffractometer, whereas cold N2 gas was used for the imaging-plate diffractometer. The scattering angle (20) covered for both diffractometer is 0,,,140 ~ corresponding to the scattering vector q=4rA -1 sin 0 of 0~16.8 A -1. The details of the X-ray measurements and the diffractometers have been described elsewhere. 4'5 The X-ray diffraction data were corrected for background, polarization, absorption, multiple scattering, and incoherent scattering with the usual procedures. 4'5 The structure functions i(q) were obtained by I(q)- Ex,f~(q) where I(q)is the observed X-ray intensities, xi and fi(q) are the mole fraction and the form factor of species i. The X-ray radial distribution function was obtained from eq. (1).
D(r) = 47rr2po+ (2r/r) f qi(q)M(q)sin(rq)dq
(I)
where p0(= [Zx,fi(O)]2/V) denotes the average scattering density of water and V is the stoichiometric volume (/~3) per halide ion. A modification function M(q) of the form [Y]xif2(O)/~xif2(q)] exp(-O.Olq 2) was used. Neutron Diffraction Sample solutions were prepared by dissolving into D20 ~ (a null mixture of 6Li and ZLi) and ZLi~tC1 for Li-enriched solutions, and ZLi3SC1 and ZLi3ZCl for C1enriched solutions. The sample solution was sealed into a Ti-Zr null alloy cell with inner diameter of 8 mm. Pulsed neutron diffraction measurements were made with HIT at KENS, Japan, and with SANDALS at ISIS, UK. Neutron scattering measurements were also made for background, an empty can, and a vanadium rod. The observed neutron
93 data were corrected for absorption, multiple scattering, and incoherent scattering, and scaled to absolute units by using the vanadium data as have previously been described. 6 The first-order difference function A.(q) between the two total structure factors for isotopic ions a and c~' is given by A,~(q) = A s . o ( q ) + Bs~D(q) + C s . ~ ( q ) + D s . , , ( q ) + E A = 2cacobo(b. - bt,,), B = 2C•CDbD(b. -- b~,) C = 2cac~b~(ba - b~,), D = %2(b~ - b~,),E = - ( A + B + C + D)
(2)
where sii(q) is the partial structure factor of atom pair i - j, c~ is the atomic fraction of atom i, and bi is its scattering length. The important parameter values are given in Table 1. The A a ( q ) is Fourier transformed to give the radial distribution function G~(r) represented by Ga(r) = Agao(r) + BgaD(r) + C g ~ ( r ) + Dg=~(r) + E
(3)
Table 1. Experimental data for the sample solutions" scattering lengths b (10 -12 cm), concentrations m (mol kg-Z), molar ratios R ( - [D20]/[LiC1]), densities p (g cm -a) at 298 K, and coefficients A , B , C and D (10 -27 cm -2) in Eq. (2) in the text. Solution ~ 7Lin~tC1 0Lir~tc1 7Linatc1 7Li35C1 'Li37C1
bLi -0.009 -0.222 0.011 -0.222 -0.222 -0.222
ba 0.958 0.958 0.958 0.958 1.163 0.345
m 4.96 4.98 9.32 9.41 9.61 9.42
R 10.0 10.0 5.36 5.30 5.20 5.30
p 1.20 1.20 1.27 1.27 1.27 1.28
A 2.40
B C D 5 . 5 1 0.'40 -0.05
4.41
1 0 . 1 1.36
-0.15
15.8
36.0
3.92
-1.15
Quasi-Elastic N e u t r o n S c a t t e r i n g The aqueous LiC1 solution with the composition of LiC1.6.0H20 was prepared in the same way as described for the X-ray samples. In the aqueous solution the proton (ZH) has a very large incoherent scattering cross section; the observed differential scattering cross section can be approximated to the incoherent dynamic structure factor through (d2aldFLdw) = (k/ko)[< b2 > - < b >2]Si"C(q,w)
(4)
Since water molecules are involved in translational, rotational, and vibrational motions, the incoherent dynamical structure factor may be described as a convolution of the three terms as
s~"~
~) = S'(q,~)
9S'(q,~)
9S~(q,~)
(5)
Here, the scattering law for the translational motion is expressed by the Lorentzian function as S t ( q , w ) = ( 1 / r ) [ D s q 2 / ( D ~ q 4 + w2)] (6) where DH is the translational diffusion coefficient of the proton.
The neutron quasi-elastic scattering measurements have been made from room temperature to glassy transition temperature on LAM-40 spectrometer at KENS. The details of the spectrometer and the data analysis have been described elsewhere/ III. R e s u l t s and D i s c u s s i o n
X - r a y Diffraction Figure l(a), (b), and (c) show the X-ray radial distribution functions (RDF) for the 10.6 m LiC1, 11.2 m LiBr, and 11.1 rn LiI aqueous solutions, respectively, at the various temperatures. The first prominent peak observed at 3.1 - 3.6 ~ in the RDFs corresponds mainly to the halide-water interactions due to the halide hydration. The contribution of the water-water interactions within the primary hydration shell of Li + also falls within this range. A characteristic feature of the RDFs with temperature is an appearance of a new peak centered at 4.3 /~. The position of the peak does not depend on the halide ions; the peak is gradually enhanced with lowering temperature. We have previously assigned this peak mainly to water-water interactions for the 11 m aqueous LiC1 solution, s The quantitative analysis has been made by a least-squares fitting procedure, and the important structural parameter values finally obtained are summarized in Table 2. The evolution of the 4.3 A peak with lowering temperature is clearly seen in Fig. 6
_
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10
r/A
Fig. 1. The X-ray radial distribution functions for the 11 m aqueous solutions of LiC1 (a), LiSr (b), and LiI (c) at the various temperatures. 2 (a) and (b) for the 5 m aqueous LiC1 solution. As is seen in Fig. 2(a) the solutions at 233 and 223 K contain hexagonal ice, together with supercooled liquid. The RDF at 223 K is dominated by the hexagonal ice, in which the peak appears at 2.8, 4.3, and 6.8 A, corresponding the tetrahedral network of ice. The continuous evolution of the 4.3 A peak from the supercooled to the ice-formed liquids suggests that at the low temperature the water molecules around C1- will be incorporated into the locally ice-like network locally involved in the solution. As seen in Fig. 2, this tendency is promoted in the solutions containing larger halide ions, probably because the surrounding water molecules are loosely bound to the large halide ion.
95
Table 2. Important structural parameter values obtained by least-squares fits for the aqueous lithium halide solutions at the various temperatures. T, r, b, and n axe the temperature, the interatomic distance, the temperature factor, and the number of interactions, respectively. The values in parentheses are their estimated errors. X - H20
T/K
r/h
LiCI
298 243 193
3.13(1) 3.13(1) 3.13(1)
LiBr
298
13s 3.13(~)
LiI
3.31(1)
26s
3.30(~)
243 298 268 243
3.31(1) 3.53(1)
3.54(1)
3.54(I) *Fixed.
b/lO-ah ~
10.2(5) ~'.S(3) 5.9(4) 5.8(4) 22.5(4) 20.0(4) 16.0(3) 20.0(3) 17.0(4) 17.0(3)
H20--.H20
n
r/h
6.1(5) 4.32(9) 6.4(5) 4.35(1) 6.5(5)4.30(1) 6.6(5)4.28(1) 5.7(5)4.35(3) 5.8(5)4.32(3) 5.6(5)4.33(3) 5.5(5)4.38(2) 5.6(5)4.40(5) 5.6(5) 4.42(5)
b/lO-ah 2
30* 30" 30* 30* 30* 30" 30* 30* 30* 30"
n
0.5(1) 4.0(2) 10.2(3) 10.2(3) 9.1(1) 10.4(1) 12.9(1) 11.5(2) 20.0(2) 22.0(3)
Neutron Diffraction with Isotopic Substitution Figure 3 shows the Li-related radial distribution functions GLi(r) for the 5 m and 9 m LiC1 solutions in D20 at the various temperatures. In the GLi(r) the predominant peaks observed at 2.0 and 2.6 /~, characterize the primary hydration shell of Li+; the former and latter peaks are ascribed respectively to the Li-O and Li-D interactions in the hydration shell. An interesting feature in GLi(r) is an appearance of the new peak centered at 4.2 /~ at 173 K for the 9 m solution and at 240 K for the 5 m solution. This peak should arise from the second hydration shell of Li +. This finding suggests that structure ordering is enhanced at such a low temperature probably through strengthened hydrogen bonds. 4 This conclusion is supported by the fact that the 4.2/~ peak appears at higher temperature for the water-rich solution. The interatomic distances and the hydration number of Li + were determined by a least-squares fitting procedure applied to the r2-weighed GLi(r). Characteristic Li-O and Li-D distances are 2.024-0.05/~ and 2.614-0.05/~ respectively, independent of temperature. The number of water molecules coordinated to Li + is 44-1 at the temperatures down to 213 K, but decreases to 34-1 at 173 K when the second hydration shell is established. The practical independence of Li-O and Li-D distances with temperature shows the strong orientational correlation of water molecules around Li+; it is most likely that an average orientation of the coordinated water molecules is such that the four atoms in a Li+-D20 unit is pyramidal. The strong orientational correlation of the bound water molecules rationalizes the anisotropic reorientational motion of the water molecules in the hydration shell found by nuclear magnetic relaxation data of supercooled LiC1 solutions. The evolution of the secondary hydration shell of Li + may be a hint of nucleation of ice, glass transition, and partial recovery of hydrogen bonds in the supercooled state. Figure 4 shows the Cl-related radial distribution functions Gel(r) for the 9 m LiC1 solution in D20 at the various temperatures. Since the glass transition temperature of the present solution is around 135 K, the values at 94 K corresponds to the glassy solution.
96
298 K
273 K
A
263 K ]
a
~3KI
A
243 K
]
8
10
263 K
~~" o
253 K
A
243 K
0
3
6
12
9
0
15
A
2
4
6 r/A
q / h -I
Fig. 2. The intensity curves (a) and the X-ray radial distribution functions (b) for the 5 m aqueous LiC1 solution at the various temperatures.
0.1
o.o3
-
'!
I
i
!
i
!
!
i
!
1
0
.~
o J
o J
E
-0.03 L0
i
I
2
I
I
4
I
i
6
i
i
8
I
10
-~
0
. . . . . . . . . 2
4
6
8
1
10
r/ ;~ Fig. 3. The Gu(r) function obtained for the 5 and 9 m aqueous solutions of LiCI in D20 at the various temperatures.
Fig. 4. The Gcl(r) function obtained for the 9 m aqueous solution of LiCl in D20 at the various temperatures.
97 The first peak at 2.29/~ is ascribed to the nearest neighbor C1-D interactions due to C1hydration. The second peak at 3.2/~, clearly defined at low temperatures, corresponds to the sum of C1-O and C1-D interactions within the hydration shell of C1-. A characteristic change in Gel(r) is that the first and second distinct peaks are gradually broadened with increasing temperature; this demonstrates a weakening orientational correlation of hydrogen bonds between C1- and the surrounding water molecules, probably due to large thermal motion of water molecules at high temperatures. From the least-squares fitting procedure applied to the 2.29 /~ peak, it has been found that the C1-D distances and the hydration number are 2.26:t=0.01 /~ and 5.8:t=0.1, respectively, at the temperatures investigated.
Quasi-Elastic Neutron Scattering Figure 5 shows the neutron quasi-elastic scattering spectra at q = 1.76/~-1 for the aqueous LiC1 solution at different temperatures. As is seen in Fig. 5, the peak width decreases with lowering temperature, indicating that the motion of water molecules in the solutions is gradually retarded with temperature. The spectra have been analysed by a curve fitting method using the following model function, S(q,~) = C[(A,,/~){F,,/(~ 2 +
r~)} +
(Ab/~){rb/(,., 2 +
r~)}]
(7)
where C is a constant, F~ (i=n and b) is the half-width at half-maximum (HWHM) of the narrow and broad Lorentzian components, respectively, and Ai is the fraction of each component. The HWHM values of the narrow component (b) are plotted against q2 in Fig. 6. The F,, values were analysed on the basis of a jump diffusion model:
Fn = DHq2/[1 + DHq2To]
(8)
where DH is the proton self-diffusion coefficient, and r0 is the life time of the proton oscillatory motion. The self-diffusion coefficient DH can be expressed by the mean jump length of proton L and r0 as D~ = < L ~ >/(6~o) (9) The obtained parameter values are summarized in Table 3. The self-diffusion coefficient of the proton decreases with lowering temperature, suggesting that the translational motion of water molecules is restricted at the low temperatures. This is probably because of the water-water interaction will be enhanced with decreasing temperature. The self-diffusion coefficients obtained from QENS are less than those determined by the 1H-NMR pulsedfield gradient method. A probable reason for the difference in DH is ascribed to different time scales observed by both methods (,-~200 #eV for LAM40).
Table 3. Parameter values for the jump diffusion model for the aqueous solution of LiC1.6.0H20 at various temperatures.
T/K LiCl 295 267
ro/ps L/A 7.1 17
DH/10-1~ I.I0 2.60 1.16 1.33
-1
D~_IMR/10-10m2s-] 5.35 (298 K) 1.74 (263 K)
98
o,
:7'
9
I/
""""
0.0/ -4.0-2.0
0.0
2:0
. . . . . . . . . . . . . . . . . . 206 K 4:0 6.0 8.0 16.0 ImeV
Fig. 5. The QENS spectra for the aqueous solution of LiCI.6.0H20 at different temperatures. The dots axe experimental values and the solid lines the fitted values by eq.(7).
0
I
o
5
o
3
0 4
5
0 206 K 6 7
a' I A"
Fig. 6. The Q2-dependenceof HWttM of the component r , for the solution of LiCI.6.0H20. The solid fines are the values calculated by the Fick law using the NMIt data in Table 3.
This work is partially supported by the Grant-in-Aid for Scientific Research (No.06453029) from the Ministry of Education, Science, and Culture of Japan. References 1 A. Elarby-Aouizerat, J.F. Jal, P. Chieux, J.M. Letoffe, P. Claudy, and J. Dupuy, J. Non-Cryst. Solids, 104, 203 (1988). 2 H. Kanno and J. Hiraishi, J. Phys. Chem., 87, 3664 (1983). 3 T. Takamuku, T. Yamaguchi, and H. Wakita, J. Phys. Chem., 95, 10098 (1991). 4 T. Yamaguchi, G. Johansson, B. Holmberg, M. Maeda, and H. Ohtaki, Acta Chem. Scand., A38, 437 (1984). 5 M. Ihara, T. Yaznaguchi, and H. Wakita, Adv. X-ray Anal., 25, 49 (1994). 6 M. Yamagami, T. Yamaguchi, and H. Wakita, J. Chem. Phys., 100, 3122 (1994). 7 K. Inoue, Y. Ishikawa, N. Watanabe, K. Kaji, Y. Kiyanagi, H. Iwasa, and M. Kohgi, Nucl. Instrum. Methods, A238, 401 (1984). 8 K. Yamanaka, M. Yamagami, T. Takamuku, T. Yamaguchi, and H. Wakita, J. Phys. Chem., 97, 10835 (1993).
journal of MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids. 65/66 (1995) 75-83
D y n a m i c s of Brownian motion under a potential and theory on the rate of chemical reactions A k i o Morita
D e p a r t m e n t of Chemistry, College of Arts and Sciences, University of T o k y o , K o m a b a , M e g u r o - k u , T o k y o 153, J a p a n
Abstract We have introduced a two square-well potential system, solved the Smoluchowski equation, calculated exactly the concentration of the reactant, compared it to t h a t resulting from the phenomenological rate equations and finally obtained expressions for rate constants for a first-order reversible reaction. It is found that the pre-exponential factors for both constants are just the diffusion coefficients divided by the square of the widths of potential wells. Our expressions for rate constants satisfy a relation between standard free energy and the equilibrium constant arising from chemical thermodynamics. We have also developed a general theory on the relaxation processes of approachment of a system to the equilib~Sum Boltzmann for an arbitrary potential with sufficiently deep well and the escape rate from the well for an asymptotically long time limit of the reaction, which enables us to calculate the rate constants for given potentials. Furthermore, we obtained the rate constant for a dissociation reaction of an ion-pair interacting through the Coulomb potential and found a maximum as the sum of the radii of the ion-pair increases.
I.
Introduction
In formulating the rate constant based a dynamical equation such as the Fokker-Planck-Kramers equation, the Smoluchowski equation and etc., we usually use the assumption of stationary state [1, 2, 3]. However, it is sometimes the case where dynamical information is suppressed by the assumption that the flux of Brownian particles must be uniform in space. It is thus sensible to calculate the rate constant by taking the dynamics explicitly. To this end, we must bear in our mind time scales of the dynamics. Imagine the diffusion of solute particles in solvent by regarding the position of a particle x as a reaction coordinate. When it diffuses on a fiat potential, the time scale we must be concerned with is of course that of diffusion, which is much longer than collision time between the solvent and solute molecules. Now, suppose it moves under a square-wen potential and its height is infinitively high, the main time scale will be roughly a relaxation time of the order of the square of the width of the potential bottom divided by the diffusion constant, which determines time approaching to an equilibrium state after colliding with the potential walls several times before settling into it. However, as lowering the potential height slightly, we see that the particles can now escape from the well after colliding with the potential walls much more times, which makes the time scale much longer to settle the motion. This is precisely why the exponential factor comes into play. The time scale in this region may be called a reaction time which is determined from the diffusion at potential bottom and the barrier height leading to the pre-exponential and exponential factors, respectively. We will be interested in tiffs long reaction time so that time scales representing molecular motion in small changes are trivial. In tiffs paper, we will devote ourselves to finding this reaction time based on the recent development on investigation of the dynamics of Brownian motion under a square-well potential [4]. Furthennore, we will develope a general theory for an arbitrary potential to obtain the reaction time. With this 0167-7322/95/$09.50 91995 Elsevier Science B.V. All rights reserved. SSD! 0167-7322 (95) 0(0)003
1t3o
there is little knowledge of the way in which they work at the microscopic level. For example, even the long known phenomenon of salting out in polymer solutions is far from being understood at the microscopic level. Of possibly even more importance are aqueous solutions of biological molecules. The majority of biochemical reactions take place in the aqueous phase and the conformation of the molecules, how they hydrate, and how their hydration is affected by the presence of salts or other molecules, is important in understanding basic biological functions. To date, the structure and behaviour of these molecules is generally inferred from the molecular structures of these molecules in the solid (crystal) phase, or indirectly through spectroscopic techniques (NMR, IR and UV etc.). Neutron techniques are now beginning to offer the possibilities of exploring these structures directly and unambiguously. In this paper we wish to consider how neutron scattering techniques can be used to study aqueous solutions containing large polymeric molecules. 2, Ext~erimental techniaues. _
NDIS techniques have been used for many years in the study of aqueous electrolyte solutions. Diffraction measurements for two liquids which differ only in the isotopic composition of one of the components reduce the total number of correlations observed in the data from N(N+I)/2 in the pure liquid to N (corresponding to correlations to the substituted atom) in the first order difference function. By careful analysis of the first order difference function details of the geometric arrangement of molecules around the substituted atom can be established. The supplementary technique of QENS allows a more detailed knowledge of the exchange times of the water in the hydration shells of these ions to be established. 3. Results and discussion. 3.1 Aqueous polyethylene oxide solutions.
Polyethylene oxide (PEO) is an uncharged polymer that has a high solubility in water, especially when compared to its analogues polymethylene oxide (PMO) and polypropylene oxide (PPO) which are essentially insoluble in water. PEO solutions
101
also show characteristic 'clouding' as the temperature is raised due to the formation of large micelles ha solution. The origins of the high PEO solubility and this behaviour have long been of interest to polymer chemists. It is generally believed that its high solubility is due a polymer structure that is commensurate with the underlying water structure, so that the ethereal oxygens on the polymer backbone are hydrogen bonded to the water molecules and the hydrophobic ethyl groups are essentially sitting in a water cage, without heavily perturbing the overall water structure. For the case of PMO and PPO the ethereal oxygens are too close or too distant to fit neatly into the water structure making it energetically untavourable for the polymer to dissolve. Despite this appealing idea no work has been carried out to investigate the water structure around PEO in solution. Figure 1 shows the AI(Q) obtained for the correlations around the (inexchangeable) hydrogen atoms on PEO using llH/21H substitution 9. . . . . .
l
- ~ ~"
iit i!t 0
./~--
I Figure 1. The first order difference I function AH(r) obtained for 1.88 I monomolal PEO ha 21H20. The , bottom (full line) is the uncorrected I difference f u n c t i o n . The top (full fine) is the difference after correction for the difference ha inelastic i scattering between 1l H and 2ill !(dashed line). I
.5 ;/
r
,
-
,
2
4
6
8
10
Q (A ~)
The corresponding AG(r) is shown ha figure 2. A clear peak is observed at 1.1 /~. which corresponds to the characteristic C-H bond distance and coordination number ha the polymer. It is clear that there is no strong structure at larger distances ha the first order difference. This is largely due to the fact that the polymeric hydrogen is not at any particular centre of symmetry ha relation to the hydrating water. However it is possible to reveal more of the hydration structure after the (known) intermolecular structure of the PEO molecule is subtracted from the difference function. A detailed analysis of the result after this subtraction,
1o2
shows that models in which the PEO is modelled into an unperturbed water lattice are not consistent w:,th the data. Further analysis shows that each monomer is hydrated by six waters; a result consistent with the hydration number deduced from cavity surface calculations performed on the polymer. Figure 2. The normalised AGH(r ) for 1.88 monomolal PEO in ZlH20. 2.S0 I
-~o
~.oo t
's~/ oo.[
o.so o.oo ~-~
~ I
.~ 2
J" 3
, 4
~ S
~ J)
QENS measurements have also been carried out on deuterated PEO solutions (2 molal) in ttH20 using the IN10 high energy resolution spectrometer at the ILL. The QENS spectrum is dominated by the translational motion of the protons in the water. After correction for multiple scattering and absorption the data was fitted to a single Lorentzian scattering law obeying Ficks diffusion law (WHM proportional to Q2) as observed for pure I~H20. This fit is clearly inadequate and demonstrates that the polymer is severely perturbing the dynamical behaviour of the water. An attempt to fit a double Lorentzian scattering law similar to the case for ionic solutions in slow exchange is also unsuccessful and indicates that either the hydrating water is exchanging on the timescale of 1 ns (the so caUed intermediate scattering regime) or the scattering law for the hydrating water is considerably more complex than the simple Lorentzian shape obtained from a free diffusion law and more like that of the scattering law of the polymer itself 10.
103 3.2 Aaueous oolvelectrolvte solutions and t~olvmer/ionic solutions, There are many interesting problems concerning the effects of polyelectrolytes and polymers in aqueous ionic solutions. For example, the microscopic origin of the behaviour of certain salts (in accordance with the Hofmeister series) in reducing the cloud point temperature of aqueous PEO is not understood. Originally, there was some speculation that the polymer was binding to the anions ( chlorine is the most effective anion) in these solutions and this was the origin of the salting out effect. An NDIS experiment 11 conclusively demonstrated that there was no modification of the CI- hydration sphere indicating that the origin of this phenomenon was more subtle than supposed. Recently studies of this type have been extended to other systems especially to questions concerning the binding of polymer to cations in solution. Figure 3 shows the first order difference functions for two systems, Ni-PSS 12 and Ca-Dextran 13 in which the hydration structure of a cation has been studied and compared with the corresponding structure in the pure salt solution. It is apparent that in these cases the Q- space data shows remarkably little change from the pure salt solutions for medium to large Q (2 < Q < ~20]k -1) ranges. However, small deviations (notably an enhanced scattering) are observed over the smaller Q region (0.3 < Q < ~2/~- 1) in these materials. This indicates that the close hydration structure of the ions in these solutions is not radically affected. The variations at small Q indicate a significant ordering on the medium range scale (310/~) suggesting that the polymer is binding to the ions over these length scales. The real space first order difference functions (AG(r)) of these solutions and that of Nickel adenosine 5' triphosphate (ATP) solution are shown in figure 4. The diagrams also show the AG(r)'s obtained from the pure aqueous solutions at the nearest available concentration (These have been renormalised to give the correct normalisation where appropriate). In all of these cases several points can be noted: i) The first peak in AG(r) corresponding to the coordination of oxygen around the ion remains unchanged.
1134
,io' o.:
,o,~.-
. . . . . . ,j ~-~,,.,~... " .
.
.
1 J~ . # . . . . . .
7.
q,
-
b
'Jr
~" . . . . . . . . . . . . .
Ji
!J
o..
.
i
............
t
,
.r
'\
~tbdl
,2a'::.,i ,
,I
;,
I '"
'l.
'"
~"
6
"
Im
io
i
~z
i 0
Figure 3. The first order difference function for a) 2~Ca(Q) fro pure 1.35 molal CaC12 solution, b) ACa(Q) for 1 molal CaC12 in 1 molal dextran solution, c) ANi(Q) for 2 molal NiC12 solution, d) ANi(Q) for 1.8 molal NiPSS..
Z
§
6
~
Figure 4. a ) A G c a ( r ) f o r the CaC12 solution (broken line) and CaCl2/Dextran solution (full line), b) A GNi(r) tbrthe NiC12 solution(broken line) and Ni-PSS solution (full line), c) AGNi(r) for 0.5 molal Ni-ATP solution (full line) and 0.4 molal NiC12 solution (dotted line).
105
ii) The second peak ( corresponding to the 21H peak in the pure aqueous solution) is reduced in intensity. iii) An extra peak ( or small shoulder in the case of Ni-PSS) appears in the region between the first and second hydration shell of the pure solutions. This is generally a relatively small feature and it is difficult to confirm that it is not an artefact of the Fourier transform. Such a strong peak has never been observed in any of the pure aqueous solution experiments. From these observations it is possible to conclude that the ion, in all cases, remains bound to the same number of oxygens as the pure solution. The reduced intensity of the second (2~H) peak is strong evidence that some water has been displaced from the hydration sphere and has been replaced by the polymer/polyelectrolyte directly binding to the ion. In all cases this leads to the reduction in the Z~H peak and predicts additional correlations further out due to the internal structure of the polymer/polyelectrolyte, for example, Ni-O-P correlations in Ni-ATP, Ni-O-S correlations in Ni-PSS and Ca-O-C correlations in Ca-dextran solutions. These correlations would all be consistent with the third observation above. Ctm'ently it is difficult to obtain quanttafive information concerning the precise nature of the eftect of the polymer on the ion hydration (especially in the important area of the second hydration sphere). In future, the higher neutron fluxes available and the use of selective hydrogen isotope substitution will lead to clearer information on the structure of these complexes. For example, using zero scattering water for the first order difference measurements will eliminate all the water contributions in the second peak of the difference function. This should lead to unmistakable evidence of the polymer binding to the ion and the ability to determine the conformation, and coordination number of the coordinating group. Selective labelling of the isotopes on the polymer chain should increase this precision even further.
1o6
Conclusions. The results presented here show that neutron scattering is an invaluable tool for studying the important area of ions binding to aqueous polymer solutions. Neutron difference methods allow direct evidence polymer-ion binding to be observed. Further developments in neutron scattering techniques and instrumentation will lead to increased precision in the future. Quantitative neutron quasielastic measurements provide an additional useful tool allowing studies of the modifications of the water dynamics to be studied. Acknowledgements. The authors would like to acknowledge the help and collaboration of the Leiden group, particularly T. Bieze, H. Tromp and J. Leyte. We would also like to thank the EPSRC for financial support. References 1. Enderby, J.E. and Neilson, G.W. (1981) Rep. Prog. Phys. 44, 593-653 2. Neilson, G.W. and Enderby,J.E. (1983) Proc. Roy. Soc. A. 390, 353-371 3. Turner,J., Soper, A.K. and Finney,J.L. (1992) Mol. Phys. 77, 411-429 4. Soper,A.K., Turner,J and Finney,J.L. (1992) Mol. Phys. 77, 431-437 5. PoweU,D.H. and Neilson,G.W. (1990) J. Phys. Condens. Matter 2, 5867-5876 6. Okan,S.E. and Salmon,P.S. (1994) J. Phys. Condens. Matter 6,3839-3848 7. Hewish, N.A., Enderby,J.E. and Howells,W.S. (1983) J. Phys.C: Solid State Phys. 16.1777,1791 8. Salmon,P.S. [ 1987) J. Phys. C: Solid State Phys. 20,1573-1587 9. Bieze,T.W.N., Barnes,A.C. ,Huige,C.J.M., Enderby,J.E. and Leyte,J.C. (1994) J. Phys.Chem. 98,6568-6576 10. Barnes,A.C., Bieze,T.W.N. Enderby,J.E. and Leyte,J.C.(1994) J. Phys. Chem. in press 11. Barnes,A.C., Enderby,J.E., Breen,J. and Leyte,J.C. (1987) Chem. Phys. Letts. 142,405-408 12. Tromp,R.H. and Neilson,G.W. (1994) J. Phys. Chem. in press 13. Vennix,A. and Barnes,A.C. to be published.
journal of
MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 107-114
Liquid-Liquid Phase Separations and Critical Behavior of Electrolyte Solutions Driven by Long-Range and Short-Range Interactions Hermann Weingiirtner Institut fiir Physikalische Chemie und Elektrochemie der Universitiit Karisruhe, D-76128 Karlsruhe, Germany and Woiffram Schriier, Institut t'fir Anorganische und Physikalische Chemie der Universitiit Bremen, D-28334 Bremen, Germany
Abstract The discovery of liquid-liquid phase separations in electrolyte solutions with critical points near room temperature enables the systematic study of the critical behavior of ionic fluids. Depending on the details of the molecular interactions mean-field-like criticality, Ising criticality or crossover from mean-field to Ising criticality is observed. Mean-field behavior occurs in systems which in the framework of a simple corresponding states model are fairly close to the theoretical critical point of the "restricted primitive model (RPM)" of equally-sized charged spheres in a dielectric continuum (so-called Coulombic phase separations). In these cases the phase transition is driven by the long-range Coulombic forces and may be modeled by a Bjerrum-type theory for ion pair association plus an additional term for the interaction between the ion pairs and free ions. The presence of additional short-range interactions shifts the phase transitions towards higher reduced temperatures and/or densities in the corresponding states plot. Two major examples are solvophobic unmixings in aqueous solutions of tetraalkylammonium salts and unmixings of salts with hydrogen-bonds between cations and anions (sticky ion pairs). Such systems exhibit Ising-like criticality. In the intermediate range crossover is observed. Possible scenarios for explaining the mean-field-Ising dichotomy are discussed, indicating a major role of the dielectric constant of the solvent in determining critical behavior.
Introduction Electrolyte solutions are of long-standing interest, and in many respects our understanding of their thermodynamics is in a mature state. The discovery of liquid-liquid phase equilibria in such systems has, however, introduced new features. ~3 Although already reported in 1903,4 and studied in more detail in 1963, 5 such phenomena have remained almost unnoticed. New impetus in the this field has now come from interest in the critical properties of ionic fluids. Experiments at high temperatures have indicated that, at least on a first study, ionic fluids appear to exhibit classical critical behavior, as opposed to the Ising-like criticality of uncharged fluids. 2 Recent experiments using liquid-liquid immiscibilities with critical points 0167-7322/95/$09.50 91995 ElsevierScience B.V. All rights reserved. SSDi 0167-7322 (95) 00898-5
108
(CPs) near room temperature have subsequently confirmed this behavior. ~-8 These form a major challenge to experiment and theory. 9 Of course, an ultimate way to treat this problem theoretically would lie in a renormalization group analysis, but successful analyses of this type have ever rested on crucial physical insights into the character of the phase transition. A general physico-chemical characterization of the systems and an understanding of the molecular forces driving the phase separation seems to be indispensable. The focus of our work is therefore on an understanding of the critical phenomena in relation to the molecular processes that control the phase separation.
Coulombic Phase Separations The simplest model for discussing electrolyte thermodynamics is the "restricted primitive model" (RPM) which describes the fluid as equally-sized, charged hard spheres of diameter a in a dielectric continuum with dielectric constant D. If we express the properties in terms of reduced variables, the basic energy scale is set by the binding energy ~ = q,qj/Da of an ion pair, where the qo. are the ionic charges. The length scale is given by the hard sphere diameter. Then, the reduced temperature and density scales are s 7" = k T/6 = kTDa/q,qj
and
C" = C d
(1)
where k is the Boltzmann constant and C the total number density of ions. This model possesses an instability regime with an upper CP near T" = 0.05 and 6" = 0.03, 2"9 substantially displaced from the CP of the Lennard-Jones (LJ) fluid (see Fig. 1).
T
"
210 , 5 --
solvophobic
0 . 2 --
0.1
- -
I
RPM sticky ions
0.01 7 0
I 0.1
! 0.2
oi,
oi,
01,
oi, C*
Fig. 1 Coexistence curves of the RPM and LJ fluids compared with experimental CPs of real systems. Systems according to Table 1" IV (e); V ( 9); VI (V); VII (A); VIII ( !"1 ). The circle A represents the regime of solvophobic unmixings. ( ~ ) denotes KI + SO2 not considered in Table 2.
io9
Via the law of corresponding states, the this coexistence curve is expected to match the general phase behavior of ionic fluids. ~'zl~ We are here interested in the case that a solvent forms a dielectric background with D > 1. In this case one expects the RPM coexistence curve to correspond to liquid-liquid immiscibilities of l:l-electrolytes in solvents of low dielectric constant (D < 8) and of higher valency types in water at high temperatures, where D is low. Owing to the low critical density, all phase separations should occur in the solvent-rich regime. Experimentally it turns out that, normally, at these conditions electrolytes are insoluble. A key step in searching for liquid-liquid phase equilibria has therefore been the use of low-melting salts to suppress crystallization. ~.3We have generally used alkylammonium salts which are chemically fairly stable and usually melt below 200 ~ If one mixes such electrolytes with low-D solvents one readily detects liquid-liquid phase separations. We denote these immiscibilities as Coulombic immiscibilities. Some representative systems are compiled in Table 1. Fig. 1 shows the location of the critical points in the corresponding-states plot.
T a b l e 1: Liquid-liquid immmiscibilities and critical behavior of electrolyte solutions
salt I
Pe4N Br
solvent water
type of unmixing
D at Tc
exponents effective criticality Ising
ref. 16
solvophobic
~50
13
~50
13,y,v
Ising
3
13
Ising
23
II
PrBu 3NI
water
solvophobic
III
Pe4N I
glycerol
solvophobic
IV
EtH3N NO 3 octanol
sticky ions
8.4
~/,v
Ising
24
V
EtMe3N I
chloroform
Coulombic
4.7
y,v
Ising
11
VI
22
BuaN Pie
dodecanol
Coulombic
4.6
y
crossover
VII Bu4N Pie
tridecanol
Coulombic
4.4
y,v
crossover ?
VIII HexEt3N HexEt3B
diphenylether Coulombic
3.7
~/
mean-field
7 6,8
As a typical example we show in Fig. 2 the coexistence curve of the system tetrabutylammonium picrate + tridecanol with an upper CP. ~The upper CP of the RPM does however not necessarily conform to an upper CP of real fluids. If D is a strong function of T, the coexistence curve inverts. Examples are found for 2:2-electrolytes and higher valency types like UO:SO4 or Na.~IPO 3 in water at high temperatures. ~ The system KI + SO: investigated in 1903 by Walden shows a lower CP as well. 4 We have recently reported on another system with a lower CP near room temperature, trimethyl-ethylammonium bromide dissolved in chloroform. ~ The coexistence curve is shown in Fig. 3. For the primary data and error estimates for these and all subsequently reported coexistence curves we refer to the original papers. Commercial chloroform contains ethanol for stabilization, and we have performed experiments under controlled ethanol content. Therefore, in principle, the system is a ternary one, and the minimum of the coexistence curve does not necessarily coincide with the CP. This is just what is seen from Fig. 3, where the CP determined by light scattering is displaced from the minimum of the coexistence curve.
I10
80-
T [oe)
35-
70-
T [ ~c]
Pic
6030-
504030-
25-
20100 0
I
0,2
I
I
0,4
0,6
I
0,8
I
1
20
0,02
, 0,04
, 0,06
, 0,1 x 1
Xl
Fig. 2: Coexistence curves of the systems tetrabutylammonium picrate + tridecanol and ethylammonium nitrate + oetanol, x ! is the mole fraction of the salt.
, 0,08
Fig.3: Coexistence curve of the system trimethyl-ethylammonium bromide + chloroform in the presence of 1% ethanol. The arrow indicates the CP.
Driving Forces for Coulombic Phase Separations Monte-Carlo calculations have established the approximate location o f the CP, but have provided little insight into the character o f these phase transitions. There seems however to be a simple, semi-quantitative way to account for these Coulombic phase separations. 9 For illustration we show in Fig. 4 the equivalent conductance A of the system tetrabutylammonium picrate + tridecanol as a function of the molar concentration C along an isotherm about 6 K above T c. A s2-
10.5-
0
0.2" 0.1" 0.05 0.02
0.0~
1r
i
i
C
Fig. 4: Equivalent conductance A (in S cm 2 mol~) of tetrabutylammonium picrate + tridecanol as a function of salt concentration C (in mol dm 3) about 6 K above Tc. The arrow indicates the CP.
111
The major feature is a rapid decrease of A at low salt concentrations, followed by a minimum and pronounced increase. At the CP there is a substantial conductance. To interpret this behavior, we first note that the Debye- Htickel (DH) theory itself predicts an instability regime at low T', but if compared with experiment C ~ is far too low. Taking account for ion association considerably improves thew results. In the presence of ion association, a higher salt concentration is needed to achieve the concentration of free ions to drive phase separation, i.e. C ~ is shifted to higher values. In particular, the Bjerrum model for ion pair association yields9 T~ = 1/16,
C* = 0.0452...,
K:ca = 1
(2)
in agreement with experiment, r is the usual Debye-HOckel parameter. This is just the picture resulting from the low-concentration branch of the conductance curve in Fig. 4, which yields an association constant K(Tc)= 3 x 109 dm3 mol! in agreement with the Bjerrum-prediction. 1 On the other hand, as indicated in Fig. 4, the CP is located in a regime of increasing conductances far above the minimum. Hence, another process will take part, which is associated with the coupling of these ion pairs with the remainder of the solution. 9 Traditionally, this coupling has been modeled by assuming ion triplets or higher associates which enforce the conductance to increase. More generally, the major effect may result from the coupling of ion pairs with the remainder of the screening free ions, regardless whether distinct species are formed or not. At this level, several scenarios have been discussed 9.1z~3and there exists a comparatively simple theory by Fisher and Levin ~3which matches the major features of the coexistence curve. "Sticky Ions" with Specific Interionic Interactions If in addition to the electrostatic interaction an additional "sticky" interionic interaction stabilizes the ion pair, C ~ is shifted towards higher values. The rationale is that the additional interaction gives rise to an increase of the association constant, so that again a higher overall salt concentration is needed to achieve the critical concentration of free ions. One illustrative example investigated by us is ethylammonium nitrate (EtNH3NO3) in o c t a n o l . 14 There is a comparatively strong hydrogen-bond between the cation and anion which shit, s the critical density towards a high value (Fig. 1) and results in a miscibility gap in the salt-rich regime. In fact, the association constant resulting from conductance measurements in this system is by more than one order of magnitude higher than predicted by the Bjerrum theory. We have recently extended the Fisher-Levin theory of ion-pair - ion coupling ~3 to include angular-averaged dipolar interactions between ion pairs, 15 which enables to describe a continuous transition from the ionic to the dipolar fluid. We obtain in the limit of complete association C~ = 0.13 and T~ = 0.167 which reflects this higher critical concentration.
Solvophobie Unmixings Another important class of liquid-liquid unmixings occur in aqueous solutions of higher homologues of the teraalkylammonium salts with large anions. 1'3'~6 Such miscibility gaps are found with many commonly used salts like tetrabutylammonium iodide at temperatures slightly above room temperature. These are generally closed gaps, the lower parts usually being suppressed by crystallization. ~Fig. 5 shows a particularly interesting example (propyl-tributyl-
112
ammonium iodide, PrBthNI), where a small closed gap is directly observed. 3 In a corresponding states diagram the CP is largely displaced from those of Coulombic immiscibilities (see Fig. 1). We have performed elsewhere a detailed analysis of these miscibility gaps in relation to the thermodynamic properties, m'~7'~8The latter exhibit the characteristic features known from aqueous solutions of hydrophobic molecules and attributed to hydrophobic cation-cation interactions. In addition, specific cation-anion pairing seems to play a role as well. It is known that pair association constants are comparatively small (typically 10 dm 3 mol") and increase with increasing cation and anion size, as opposed to expectations from electrostatic arguments. For anions one finds K to increase with the "lyotropic series" F < CI < B r < NO 3 < I < SCN < CIO; < picrate. This is just the series in which miscibility gaps increase. ~A simple picture is ~9that it is comparatively easy to desolvate a large anion which is the then stabilized in the low-D environment of between the alkyl groups of the cations. The interpretation is confirmed by HNC calculations of model systems characterized by Friedman-Gurney-type potentials. ~9We note that similar immiscibilities have been observed in other solvents of high cohesive energy density like ehtylene glycol, glycerol or formamide. ~ One may call this type of phase separation "hydrophobic u n m i x i n ~ 'i9 or "solvophobic unmixing". ! T~I
7!
65"
60"
-••.0.331
s5
wt. -9 0.446
T
i
0 35
!
0,40
0,48
i
i
0,50
0,55
Wl
Fig. 5: Upper and lower parts of the closed miscibility gap of propyl-tributylammonium iodide in water as an example for hydrophobie unmixing, wl is the weight fraction of the salt.
5.
Critical Behavior
The prime question is, how these various long-range Coulombic and short-range specific interactions affect the critical behavior. In general, liquid-gas and liquid-liquid coexistence curves in the asymptotic regime near the CP can be described by a power law of the form 2~ Ax
=
XI--X2
-"
B t 13 + . . .
(3)
where xl.2 characterizes the compositions of the coexisting phases (e.g. in mole fractions), B is a substance-specific amplitude and t - (7"- T ) / T c . In non-ionic fluids a "universal" critical exponent 13 = 0.325 is found, which corresponds to the figure predicted from the three-
ID
dimensional Ising model. Mean-field (or classical or van-der-Waals) theory predicts 13 = 1/2. However, often static or dynamic light scattering experiments are the method of choice for investigating critical phenomena. These yield the osmotic susceptibility Zr (e.g. derived from static scattering or turbidity measurements) which asymptotically diverges as ~ T = Xot -~
(4)
and the correlation length of the critical fluctuations (e.g. derived from dynamic scattering) which asymptotically diverges as ~ = ~ot -~
(5)
,/= 2v = 1 exactly in the mean-field model and 7 ----2v =__1.24 in the Ising model at the level of accuracy used here. Accurate work performed since the pioneering study of Sing and Pitzer 6 published in 1989 is summarized in Table 1, which lists the exponents investigated (13 via the shape of the coexistence curve, y via static light scattering or turbidity measurements and v via dynamic light scattering) and the universality class found. It is seen that three types of behavior are observable, i.e. sharp Ising behavior, sharp mean-field behavior and crossover from effective mean-field behavior far from Tc to asymptotic Ising behavior (although not complete in the cases under investigation). Such a crossover is expected on theoretical grounds, as far from T~ mean-field behavior should become accurate. The transition is characterized by the so-called Ginzburg-temperature to. 2~ In simple non-charged fluids to is obviously so large that meanfield exponents are never seen. Any observation of mean-field exponents may therefore either imply that criticality is truly mean-field, or that the asymptotic range is not reached in the experiments. General D i s c u s s i o n There are essentially two ways of looking at the data in Table 1. One is associated with the presence of short-range specific interactions in addition to the long-range Coulombic interactions. In these cases sharp Ising behavior is found. Two mechanisms have been detected until now: The dominance of solvophobic interactions shifts an apparent mean-field criticality to an Ising criticality. Similarly an enhancement of pair association by some stickiness between the ions leads to Ising behavior. Using the extension of the Fisher-Levin model to include dipolar interactions between ion pairs, we have found that in the limit of a completely dipolar fluid, t G is two orders of magnitude larger than in the free ion limit, so that an increasing Ising region is expected with increasing association. ~5 However, one may note that at least one observation does not fit to this simple picture: There is no reason for postulating any marked specific interactions in the system trimethyl-ethyl ammonium bromide + chloroform, and in fact, Fig. 1 shows that the RPM matches the experimental CP of this system surprisingly well. Nevertheless, we observe sharp Ising behavior. ~ One may however look at the data in Table 1 from a different point of view (see also ref. 22). It is seen that a transition from Ising to mean-field behavior is observed, if the dielectric constant decreases, the transition regime occuring somewhere near D = 4.5. Could the dielectric constant be the major parameter controlling criticality? In the latter case specific interactions play an indirect role: They shift Tc and Cc thus enabling phase separations to occur
114
at higher dielectric constants thus enabling phase separations under dielectric conditions which favor Ising-like criticality, but are otherwise incidental to the universality class. Recent experiments of Pitzer and Narayanan 22 give support to such an interpretation: In the system Bu4N Pic + dodecanol crossover from asymptotic Ising behavior to mean-field behavior is seen. If D of the solvent is increased by adding a second neutral substance, the Ising range increases. In fact, the role of dielectric phenomena in determining phase transitions and criticality of ionic fluids are widely recognized and subject to current investigations. 9"~2The relation between the dielectric constant and the crossover temperature needs to be investigated in the future. Acknowledgment: K.S. Pitzer and T. Narayanan are thanked for comments and sharing prepublication results.
References
.
10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
H. Weing~xtner, T. Merkel, U. Maurer, J.-P. Conzen, H. Glasbrenner and S. K~hammer, Ber. Bunsenges. Phys. Chem., 95, 1579 (1991). see e.g.K.S. Pitzer, Acc. Chem. Res., 23, 333 (1990). H. Weing~.rtner, M. Kleemeier, S. Wiegand and W. Schr6er, J. Stat. Phys., 78, 169 (1995). P. Walden and M. Centnerschwer, Z. Phys. Chem., 42, 432 (1903). H.L. Friedman, J. Phys. Chem., 66, 1595 (1963). R.R. Singh and K.S. Pitzer, J. Chem. Phys., 92, 6775 (1990). H. Weingiirtner, S. Wiegand and W. Schr6er, J. Chem. Phys., 96, 848 (1992). K.C. Zhang, M.E. Briggs, R.W. Gammon and J.M.H. Levelt Sengers, J. Chem. Phys., 97, 9692 (1992). see e.g.M. Fisher, J. Stat. Phys., 75, 1 (1994). H.L. Friedman and B. Larsen, J. Chem. Phys., 70, 92 (1979). S. Wiegand, M. Kleemeier, J.-M. SchrOder, W. Schr0er and H. Weingiirtner, Int. J. Thermophys., 15, 1045 (1994). G. Stell, J. Stat. Phys., in press. M.E. Fisher and Y. Levin, Phys. Rev. Lett., 71, 3826 (1993). H. Weingiirtner, T. Merkel, S. K~hammer, W. SchrOer and S. Wiegand, Ber. Bunsenges. Phys. Chem., 97 970 (1993). W. SchrOer, to be published. M.L. Japas and J.M.H. Levelt Sengers, J. Phys. Chem., 94, 5361 (1990). H. Weing~'tner, Ber. Bunsenges. Phys. Chem., 93, 1058 (1989). H. Weing~irtner and E. Steimle, J. Phys. Chem., 96, 2407 (1992). H. Xu, H.L. Friedman and F.O. Raineri, J. Solution Chem., 20, 739 (1990). see e.g.J.V. Sengers and J.M.H. Levelt Sengers, in "Progress in Liquid Physics", Chap. 4, ed. A Croxton, Wiley, New York, 1978. V.L. Ginzburg, Sov. Phys. Solid, 2, 1824 (1962). T. Narayanan and K.S. Pitzer, Phys. Rev. Lett., in press. T. Merkel and H. Weing~irtner, unpublished data. W. Schr6er, S. Wiegand and H. Weing~irtner, Ber. Bunsenges. Phys. Chem., 97, 975 (1993).
journal of ~IOLECULAR
LIQUIDS ELSEVIER
Journal of MolecularLiquids,65/66(1995) I 15-!22
Slow Relaxation in Pure and Binary Frozen-in Systems Hiroshi Suga Research Institute for Science and Technology, Kinki University, Kowakae, Higashi-osaka 577, Japan Abstract The glass transition phenomena that have been considered as one characteristic property of liquids are shown to be just one example of freezing transitions which must be of wide occurrence in condensed matters associated with freezing of any degrees of freedom in a system. Freezing transitions observed in some orientationally disordered crystals as revealed by their thermodynamic properties are reviewed. Adiabatic calorimeter used as an ultra-low frequency spectrometer has proven its usefulness in quantifying the relaxation rate and in determining the glass transition and residual entropy of non-crystalline solid as indicative of its non-ergodicity. Nature of relaxation processes observed in exotic non-crystalline solids produced by non-traditional methods is compared with those observed in liquid-quenched glasses. Similarity and dissimilarity in the relaxational processes between them are discussed.
I. Introduction All the classical liquids are believed in principle to realize their glassy states if their crystallizations are avoided in any ways. 1 The formation of glass is kinetic in origin and depends on the persistence of the metastable phase during a quenching process. This requires rapid passage of the crystallization regime in undercooled liquid to reach the low temperature domain where the driving force for the transformation is counterbalanced by the decreased molecular mobility. Similar situation is found in some of the orientationaUy disordered crystal when the transformation into a low-temperature ordered phase is bypassed. 2 The glass transition and residual entropy arising from freezing of the orientational degree of freedom of molecules are observed as in the case of the glass-forming liquids. Thus the glass transition is shown to be of wide occurrence in condensed matters irrespective of the three dimensional periodicity in spatial arrangements of the constituent molecules. The frozen-in disordered crystals were designated as "glassy crystals" There are many ways in producing non-crystalline solids other than the traditional liquid cooling. 3 Vapor deposition is particularly useful in the formation of non-crystalline solid from the most disordered state of aggregation of molecules onto a cold substrate. The process does not require any passage of sample through the "crystallization-dangerous region" in undercooled liquid. Kinetic energies of molecules in the vapor are extracted efficiently if the temperature of substrate is low enough and the condensation rate is sufficiently low. Novel exotic methods have been developed for the formation of non-crystalline solids. In these methods, the starting materials are crystalline solids? An excess amount of energy is impregnated into crystal and freeze it in an energized non-crystalline state. Obviously the method is just opposite from the traditional method in which the energies associated with disorder are removed rapidly. The excitation of energy is carried out either by bombardment of high-energy particles or by mechanical grinding. However, characterizations of these exotic non-crystalline solids are not sufficient compared to those of traditional glasses. It is the purpose of this article to examine similar and dissimilar nature of relaxational processes between the exotic and traditional non-crystalline solids as studied by calorimetric methods. 0167-7322/95/$09.50 91995ElsevierScienceB.V. All rightsreserved. SSD10167-7322 (95)(10846-2
116
II. Experimental Experimental methods for the study of non-crystalline solids are similar to those used for the study of phase transitions. 5 They are mainly aimed at resolving the dynamics of the freezing process. Dynamic nature is obtained from dielectric or elastic susceptibility measured over a wide range of frequency. These methods are supplemented by nuclear magnetic resonance technique, which provides information of local properties. Dynamic picture of motion in the time domain of 1 ms-.,10 s is obtained by these techniques. Adiabatic calorimeter can be used as an ultra-low frequency spectrometer owing to its high thermal stability and temperature resolution. It measures the enthalpy of a system as a function of temperature as well as time. Thus the method enables us to follow the enthalpy relaxation which may be exhibited by any frozen-in disordered systems. The time domain that can be covered by this method is in the range 100 s-1 Ms, depending on the quality of the calorimeter and patience of experimenter. The method is complementary with the dielectric spectroscopy. The principle of an adiabatic calorimeter as a relaxational spectrometer is simple. 6 If the temperature of a sample cell is constant with time under an adiabatic condition, the sample is in thermal equilibrium. If the temperature changes with time, the variation is due to irreversible process from a non-equilibrium to the equilibrium state. This can happen when a sample in frozen-in disordered state approaches the glass transition temperature Tg during warming from a low temperature. The non-equilibrium state has an excess amount of enthalpy referred to as the configurational enthalpy and tends to release the excess part when the relaxation time for the structural change crosses the experimental time scale. All the released energy is used to raise the sample temperature if the adiabatic condition is satisfied all the times. Therefore, the temperature versus time curve along the heat capacity data allows to calculate the rate of relaxation for the configurational enthalpy. Amount of correction for a natural heat leakage depends on the quality of the adiabatic control of the calorimeter. The temperature change due to the residual heat leakage is of the order of several mK h -1 in most of the temperature region. Integration of the temperature drift rate with respect to time gives rise to the total excess configurational enthalpy the frozen sample possessed initially. An adiabatic low-temperature calorimeter designed specifically for a vapor-deposited sample is depicted in Fig. 1. This calorimeter equips with a built-in cryorefrigerator which enables to keep the calorimetric cell at cryogenic temperatures for a long time.
Ill. Results and Discussion [1]
Relaxations in Crystalline Solids
One of the features observed in many glass-forming liquids is the non-linear nature of any relaxation processes that occur around and below Tg. The relaxation rate is found to depend on the sign of initial departure of actual sample from the equilibrium state. The relaxation rate is described well by the Kohlrausch-Williams-Watts' (KWW) empirical equation. 7 H(T) = H(O) exp[(-t/z)~],
(1)
where H(t) and H(0) are the enthalpy function at time t=t and t=0, respectively, t the relaxation time and 13 the fractional exponent. Obviously 13=1 corresponds to exponential decay. Generally the value 13is in the range 0.5-0.8 in most of the glass-forming liquids around Tg. One explanation for the deviation of 13from 1 is given by the distribution of the relaxation times. The wider the relaxation times, the smaller the 13value. This kind of non-exponential enthalpy relaxation can be found likely in some of the orientationally disordered crystals in their undercooled states, as exemplified typically by isocyanocyclohexane, s In the room temperature phase, each isocyanocyclohexane molecule reorients rapidly along several unique axes in an fcc unit cell, resulting in time- and spaceaveraged spherical symmetry of the molecule. The cubic phase is easily undercooled by avoiding the transformation into a low-temperature ordered phase. Finally the orientational
117
---A
V5
t
W
V3
~
"
II ,l'--i
I/
D
T4
G
~T3 AT2 ~TI
j K L
coldtrap
~
N
Figure 1 An adiabatic calorimeter for vapor-deposited sample. degree of freedom looses its mobility through a "glass transition" region. The heat capacity decreases discontinuously over a narrow temperature range and a finite amount of entropy remains down to the lowest temperature. The calorimetric results are summarized in Fig. 2. Since isocyanocyclohexane molecule has a large dipole moment, the relaxation rate can be observed dielectrically as well as calorimetrically. A special calorimetric cell with built-in alternately-arranged electrodes was developed in order to compare the relaxation times for the enthalpy and polarization relaxation around the main glass transition region Tg2. The results arc summarized in Table 1. Obviously the relaxations of both quantifies proceed faster at higher temperatures, as expected for a thermally activated process. It is interesting to note that the dielectric polarization relaxes faster than the enthalpy and the non-exponential parameter 13for the polarization is smaller than that for the enthalpy. It is clear that the polarization relaxation concerns only with the recovery of dipolar orientation toward the equilibrium state, while the enthalpy relaxation involves all the processes which accompany any enthalpy change of the crystal as a whole.
Z
isocyar,ocyclohexane
o
20O
,5o,
1
,~
~
Tfus "
!00
50
25o 0
KI
150
IOO
5o
~
5b
1,3o
1;o T
/
26o
2;0
K
Figure 2 Molar heat capacity and entropy of isocyanocyclohexane.
300
118
Table 1 The KWW parameters 13and x for the enthalpy (H) and polarization (P) relaxation in isocyanocyclohexane. T
TH
K
ks
123.04 125.37 126.01 126.82 126.84 127.66
/~H
71 8.3 7.5 4.7 4.6 2.5
fie
0.53 0.70 0.73 0.75 0.75 0.80
4.3 0.62 0.63 0.30 0.35 0.28
0.49 0.42 0.45 0.45 0.46 0.57
Clathrate hydrates 9 provide an interesting example of binary systems with two kinds of disorder. The host water lattice forms various kinds of Archimedes' polyhedra through the hydrogen bonding similar to many ice polymorphs, and the guest molecules are enclathrated in the polyhedral cages. The unit cell of the type II hydrates is composed of 16 pentagonal dodecahedral and 8 hexakaidecahedral cages formed by 136 water molecules. Most of the guest molecules in the type II hydrates are encaged only in the larger cages, so that the ideal stoichiometric composition of a guest molecule G is G-17H20. Both of the host and guest molecules have orientational disorder, being dynamic at high temperatures. For the clathrate hydrates encaging polar molecule like tetrahydrofuran (THF), there exist two kinds of dielectric relaxation. The high temperature one is similar in nature to that of ice and is ascribed to the freezing out of the water reorientational motion. The other occurs at cryogenic temperatures as low as 25 K at 1 ld-Iz and is due to freezing out of the THF reorientational motion inside the almost spherical cavity. The heat capacity of THF. 17H20 exhibits a glass transition at around 85 K which is similar to that of ice Ih observed around 100 K. Analysis of spontaneous temperature change of the sample due to the enthalpy relaxation observed around Tg gives characteristic time for the relaxation process as a function of temperature. The enthalpy relaxation times thus derived can be well correlated with the dielectric relaxation times in an Arrhenius plot. This justifies the assumption that the glass transition is associated with the freezing of the polarization of water dipole. Doping of a minute amount of KOH (1.8x10 -3 in water mole fraction) was found to accelerate the water reorientational motion and induced a first-order phase transition at 62 K T/K ~00
200
150
120
!
!
|
100
90
80
70
40
| ~
""
40
A
ut
v Ot
Pure
! BO.5 K
"-.
,"
0
2
41
6
9
1'2
0
-,
"'-..
50
"'--...
100
150
C'
T -t / k K-I
Figure 3 (Left) Calorimetric and dielectric relaxation time of pure and KOH-doped "ITqF hydrate crystal. (Right) A Cole-Cole plot of KOH-doped THF hydrate crystal.
119
The situation is quite similar to that of ice. ~~ A dielectric measurement on a KOH-doped THF showed that the relaxation time x for the reorientational motion was dramatically shortened by the dopant, possibly by creating a pair of the orientational defects proposed by Bjerrum. Not only the absolute value of x but also the activation energy for the process decreased by the dopant, as shown in Fig. 3. The value of x at 70 K is 10-8 times smaller than that for pure (undoped) sample. This is the reason why the ordering transition has escaped from observation for a pure sample by a kinetic reason appeared now at 62 K in the doped sample by a catalytic action of the dopant within a reasonable time. Also given in the figure is a Cole-Cole plot of the dielectric permittivity of the KOH-doped THF hydrate. The distribution of dielectric relaxation times is much wider in the doped sample than in the pure sample. [2] Relaxation in Non-crystalline Solids As has been described in the introduction, there are many ways in producing noncrystalline solids. These are summarized in Fig. 4. Chemical reaction sometimes produce
Vapor
Liquid i
I
I
Deposition
Rapid ~ooling
Non-crystalline Solids
[~ 1~"
ChemicalI reaction[
Grin~ding Low-temperature melting Bombardment of high-energy particles ,,w
[
Crystal Figure 4 Various methods for preparation of non-crystalline solids. amorphous solid. For example, it has been known that As 3+ ions in aqueous solution react with H2S gas to produce non-crystalline precipitates As2S3. In 1968, we have shown by thermoanalytical method that the precipitates exhibit a glass transition which is essentially the same as that of liquid quenched sample of orpiment (naturally occurred As2S3). 2 Likely, amorphous magnesium acetate Mg(CH3CO2)2 produced by dehydration of its tetrahydrate crystal exhibited a glass transition at around 470 K. Vapor deposited non-crystalline solids were characterized by a special thermal analysis apparatus to exhibit their glass transition phenomena. These observations are, however, merely of qualitative nature. Quantitative study by use of calorimetric method should be necessary for deeper understanding of these noncrystalline solids. A newly developed adiabatic calorimeter ~ for vapor-deposited sample was applied to several samples possessing good glass-forming ability in order to compare the relaxational behavior between the vapor-deposited and liquid-quenched non-crystalline solids. Since the condensation heat exceeds far the heat capacity of any sample, an extremely slow condensation rate is necessary for removal of the heat from the surface of the sample. Generally a non-crystalline sample of 1 g was produced during more than 30 hours' vapor deposition. One of the features shown by vapor-quenched (VQ) non-crystalline solids is their high configurational enthalpies compared to those of liquid-quenched (LQ) solids. ~2 Figure 5 draws
120
100
,
l
80
,
m
~
(//A
/" k
I
.t.~
,/,., 88
4o
. .I
I----I
I
J
I
I
I
I
1.4
YQ2
1.0
~
9
~: 0.6
"
0.4 20
I"
]'
-
0.2
o 00
50
TIK
100
150
50
GO
70
80 TIK
90
100
10
Figure 5 (Left) Molar heat capacity and (fight) configurational enthalpy of butyronitrile prepared by vapor-quenching (VQ) and liquid-quenching (LQ). the heat capacity and configumtional enthalpy of butyronitrile for its VQ and LQ samples. The heat capacity of crystalline phase is also included. Although the values of heat capacity and Tg coincide to each other, there is a large difference in the configurational enthalpy each noncrystalline solid possessed initially. The heat capacity was determined by extrapolating the spontaneously changing temperatures of the calorimetric cell to the mid-point of energizing period and thus correspond to iso-configurational heat capacity; i.e., vibrational heat capacity of the system that can responds rapidly to an energy input. Two kinds of the VQ samples were prepared. The one (VQ1) was deposited at 67 K, and the other (VQ2) at 40 K. It is clear that the lower the substrate temperature, the higher the configurational enthalpy. Here, the configurational enthalpy was referred operationally to that at Tg, at which the spontaneous temperature drifts change from being positive to being negative on warming. For the VQ2 sample, the excess frozen enthalpy amounts to 1.33 kJ mo1-1, compared with 0.08 kJ mo1-1 for the LQ sample, indicating the formation of a non-crystalline solid with extremely high fictive temperature. The enthalpy started to relax at much lower temperatures for the VQ samples than for the LQ sample. The rate of enthalpy relaxation observed for the VQ sample at three different temperatures showed highly non-exponential behavior. Still, the relaxation rate could be described well by the KWW equation. Thus the empirical equation turned out to be valid for non-crystalline solids not only near to but also far from the equilibrium state. The mechanical grinding is a novel technique for solid-state amorphization starting from crystalline sample and is quite popular in recent years in producing amorphous alloys of binary and ternary ~3 systems. The method is generally referred to as mechanical alloying and the grinding procedure is done either by ball-milling or rod-milling. The same thing can happen in non-metallic system. For example, tri-O-methyl-l]-cyclodextrin (permethylated-13-cyclodextrin) can be vitrified by a rod-milling. ~+ During the milling process, a pan of sample was taken out from a pot and examined by X-ray powder diffraction and differential scanning calorimeter. The temperatures of glass transition, crystallization and subsequent fusion were studied as a function of the grinding time. The Tg value decreased at the initial stage of grinding, then increased gradually and finally approached asymptotically the value for the LQ sample. For a fully amorphized sample, the heat capacity ~5 was measured as a function of temperature. Exothermic followed by endothermic spontaneous temperature changes associated with the enthalpy relaxation were observed around Tg. The results are summarized in Fig. 6, along with
121
50
100
150
200
250
300
350
2.5
i
i
,
,
,
Equilibrium
2.5
'
Liquid
2.0 4 o
/
/ !
t.5
]
E 1.5
"~
,..-,
"~ L.0
t.o
0.5
0.5
0
50
0
10
T/K
20
30
40
50
-0.5310
~
310
3~0
3~0
3;0
TIK
Figure 6 (Left) Molar heat capacity and (fight) configurational enthalpy of TMCD prepared by liquid-quenching (LQG) and grinding of crystal (GAS). those for a crystalline sample. Again, the vibrational part of the heat capacity and the Tg value coincided to each other for the ground amorphous solid (GAS) and liquid quenched glass (LQG) samples. However, the amount of configurational enthalpy obtained by integrating the spontaneous temperature drift rates with respect to time differed between the two kinds of noncrystalline solids. The enthalpy relaxation of the GAS sample likely started to occur at lower temperatures compared to the LQG sample. The molecule TMCD has a doughnut-like shape and is known to act as host molecules of the clathrate or inclusion compounds 19 to accommodate various guest molecules in the inner space. Benzoic acid (BA) is one of the guest species. Generally, the preparation of inclusion compound is carried out in solution. In this case, however, solid-state formation of the inclusion compound was found 20 to occur by shaking an equimolar physical mixture of TMCD and BA. About 75 % of the inclusion compound was formed by shaking of the mixture for about 180 min. The inclusion reaction was completed by grinding for additional 3 min with a vibrating mill. Further grinding of the mixture induced again the solid-state amorphization of the inclusion compound. X-ray diffraction pattern of a fully amorphized sample was similar to that of a liquid-quenched inclusion compound. A glass transition was observed for both noncrystalline solids in the temperature range between 331 and 337 K. The grinding-time dependence of the Tg value was about the same as that of pure TMCD. The non-crystalline inclusion compound crystallized into the stable phase via a metastable phase and fused at 447 K. A calorimetric study of the relaxational nature in the non-crystalline TMCD-BA inclusion compound is planning.
IV. Concluding Remarks 1) The observation of glass transition phenomena in some crystals requires a renewal of the concept of glass transition. 16 The glassy crystals are promizing aggregation of state of molecules in the sense that they simplifies the complex nature of non-crystalline solids arising from a coupling between the positional and orientational degrees of freedom of molecules. 2) The cases of ice and clathrate hydrates are particularly interest. 17 The hypothetical ordering transitions that have been concealed so far for a kinetic reason was induced by a minute amount of special dopant which relaxed severe constraints called "ice rules ''is imposed to the highly cooperative reorientational motion of water molecules in their hydrogen-bonded
122 networks. The dramatic acceleration effect of the dopant on the molecular mobility as a whole revealed a facet that a frozen-in disordered system would exhibit under the condition of thermal equilibrium. 3) The configurational enthalpy and its relaxation process in non-equilibrium solids depend strongly on the manner by which the solids were prepared. The amount of excess enthalpy can be changed continuously by modifying slightly the experimental conditions. The continuous hieralchical nature in energy of non-crystalline solids is in good contrast to the discontinuous hieralchy in energy of various modifications (polymorphs) of crystalline solids. 4) Among various techniques in producing non-crystalline solids, the vapor condensation is the most powerful method in extracting rapidly the energies of disordered system to form non-crystalline solids with extremely high fictive temperatures. The enthalpy relaxation starts to occur at temperatures far below Tg. The relaxation rate can be described likely by the KWW empirical equation, as in the cases of many liquids with moderate departure from the equilibrium state. The experimental works described here were carried out at Osaka University when the author had been in active service. The author would like to express his sincere thanks to Professor Emeritus Seki for his continuous encouragement, to many colleagues and students for their enthusiastic collaborations.
References 1 R. Zallen, The Physics of Amorphous Solids (Wiley Interscience, New York, 1983). 2 H. Suga and S. Seki, J. Non-cryst. Solids, 16, 1171(1974) : Faraday Discussion, No. 69 p 221 (Roy. Chem. Sot., 1980). 3 D.R. Secrist and J. D. Mackenzie, Modern Aspects of Vitreous State, edited by J. D. Mackenzie, Vol III, p 149 (Butterworths, London, 1960). 4 R.B. Schwarz and C. C. Koch, Appl. Phys. Lett., 12 146 (1986). 5 N.G. Parsonage and L. A. K. Staveley, Disorder in Solids (Oxford University Press, Oxford, 1978) 6 H. Suga and T. Matsuo, Pure & Appl. Chem., 61 1123 (1989). 7 S.A. Brawer, Relaxation in Viscous Liquids and Glasses (Am. Ceram. Soc., New York, 1985). 8 I. Kishimoto, J.-J. Pinvidic, T. Matsuo and H. Suga, Proc. Jpn. Acad., 67B 66 (1981). 9 H. Suga, T. Matsuo and O. Yamamuro, Supramolecular Chem., 1 221 (1993). 10 Y. Tajima, T. Matsuo and H. Suga, Nature, 297 810 (1982). 11 H. Hikawa, M. Oguni and H. Suga, J. Non-cryst. Solids, 101 90 (1988). 12 M. Oguni, H. Hikawa and H. Suga, Thermochim. Acta, 158 1143 (1990). 13 K.-J. Kim, K. Sumiyama and K. Suzuki, J. Non-cryst. Solids, 155 165 (1993). 14 I. Tsukushi, O. Yamamuro and H. Suga, J. Therm. Anal., 37 1359 (1991). 15 I. Tsukushi, O. Yamamuro and H. Suga, To be published. 16 H. Suga, Ann. N.Y. Acad. Sci., 484 248 (1986). 17 H. Suga, T. Matsuo and O. Yamamuro, Pure & Appl. Chem., 64 17 (1992). 18 J.D. Bernal and R. H. Fowler, J. Chem. Phys., 1 515 (1933). 19 J.L. Atwood, J. E. Davies and D. D, MacNicol (Eds), Inclusion Compounds, Vol III (Academic Press, London, 1984). 20 I. Tsukushi, O. Yamamuro and H. Suga, The rmoc him. Acta., 200, 711 (1992).
journal or
I~OLE~ULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 123-130
Slow Dynamics in Supercooled Fluids Y. Hiwatari, J. Matsui, T. Muranaka*, and T. Odagaki** Department of Physics, Kanazawa University, Kanazawa 920-11, Japan "General Education, Aichi Institute of Technology, Toyota 470-03, Japan "*Department of Physics, Kyushu University, Fukuoka 812, Japan Abstract Molecular dynamics simulation is a useful tool in general to investigate dynamical properties of condensed matters from a microscopic point of view. This is the case for highly supercooled liquids and the liquid-glass transition as well. However, we have to pay special attentions to the length of time as well as the time-mesh size and the system size of the simulations for the study of slow dymamics which governs the relaxation of such particular systems. In this paper, we study the slow dynamics of a highly supercooled fluid by calculating the general susceptibility x(q,w) with both two- and three-dimensional binary soft-sphere models and super-long-time molecular dynamics simulations. It is found that the Cole-Cole analysis clearly distinguishes between the a and fl relaxations of supercooled fluid states. A spurious sound-mode peak appearing in x"(q,w) (imaginary part of the general susceptibility) turns out to be located at oJ inversely proportional to the length of the simulation cell size and seems to be physically meaningless. This peak may be caused by the periodic boundary conditions used in the present simulation.
I. Introduction It is known that the density autocorrelation function or the intermediate scattering function F(q, w) in the supercooled fluid phase can well be described by a stretched exponential function of the form F(q,w) ~_ Aexp[-(t/to) ~] with positive constants A, to and fl(< 1) in a time region available in experiments and computer simulations. This particular relaxation is called a relaxation, and such a characteristic decay manifests itself in a slower decay of the dynamical structure factor S(q, w) ~_ w -~-1 and of the a peak of the general susceptibility X"(q,w) ~- w-n, as acompared to the usual Debye forms S(q,w) ~ w -2 and X"(q,w) ~- w -1. It has recently been reported from light scattering experiments that there are two distinct slow relaxations, a and ~ relaxations, in the supercooled fluid states of CKN. Similar slow relaxations have been observed in various systems. In our previous papers 1,2, we have shown that collective jump motions of atoms take place in highly supercooled fluid states, mainly contributing to the a relaxation, and therefore represents the molecular-level mechanisms. The main purpose of this paper is to study both a and fl relaxations from S(q,w) and X"(q,w) in a supercooled fluid by a super-long-time molecular dynamics (MD) simulation for a model fluid of binary soft-sphere mixtures. In particular, we focus on studying the type of each relaxation (Debye or non-Debye ?) and the molecular-level processes for the fl relaxation. In order to find chracteristic motions of atoms at a microscopic level and to reveal the 0167-7322/95/$09.50 91995 Elsevier Science B.V. All rights reserved. SSD! 0167-7322 (95) 00899-3
124 nature of dynamics most relevant to these relaxations, we have also carried out MD simulations for a 2-dimensional (2-d) soft-sphere model as well as for a 3-d model. In general, 2-d simulations are more advantageous to visualize the correlations of motions of atoms than 3-d simulations, and can remarkably help us find molecular-level solutions of the dynamics of our interest.
II. S o f t - s p h e r e m o d e l As a simple model for binary fluids, we consider in our MD simulation soft-sphere mixtures composed of N1 atoms of mass ml and diameter al and N2 atoms of mass m2 and diameter a2 in a volume V for a 3-d system or in an area S for a 2-d system; the atoms interact through the purely repulsive soft-sphere potentials: v,~a = e
,
(1)
where a, fl denote species indices, and e the energy unit. Here, we assume that a~a = (a,~ + h a ) / 2 . Using the notation of the effective one-component approximation, we introduce the effective coupling constant Felt, i.e., for the 3-d system r e ff - - n . ( T * ) - l / 4
(~ 3' \cr 1 /
3 _. ~a~,BXaXj30.3j3, Creff
(2)
and for the 2-d system
F~fr -" n ' ( T ' ) -1/6 (a~fr~ 2 \ (71 /
'
2
O'ef t ~--
E~
2 ~XaX~(Ta~.
(3)
Here, n" = N a 3 / V for the 3-d system and N a 2 / S for the 2-d system, denoting the reduced number density with the total number of atoms N(= N1 + N2). The reduced temperature T" equals k s T / e , and x l ( = N1/N) and z2(= 1 - z l ) are concentrations. Note that F4~ for the 3-d system and F~fr for the 2-d system are inversely proportional to T'; thus larger Feff'S corresponding to lower temperatures. In the present work we study binary soft-sphere mixtures with a core-size ratio a2/al 1.2 for the 3-d system and a2/al -- 1.4 for the 2-d system, respectively, a mass ratio m2/ml = 2.0 and an equimolar concentration (xl = 0.5) for both 3-d and 2-d systems. Using a constant-temperature MD technique and the peiodic boundary conditions, we have carried out MD simulations for the models. The pair potential, Eq. (1), was cut off over the distance r / a ~ - 3.5 and the number density was kept constant, i.e. n" = 0.8; the temperature was varied to achieve a desired Feff. The microscopic time scale was chosen to be =
E
,
(4)
which is nearly in the same order of magnitude as the Einstein period rE, namely the period of oscillations of atoms in an equivalent solid. This time scale T is of the order of 10 -13 sec, when we use the potential parameters suitable for argon liquid in Eq. (4). The equations of motion were solved numerically with a time mesh of the order of 100th of T, which precisely depends on the temperature of the system.
III. 3 - d i m e n s i o n a l results
1:2.~
We have carried out MD simulations for the 3-d binary soft-sphere model with N - 5 0 0 atoms in a cubic cell. First, we have simulated a liquid equilibrium at Fefr = 0.8; then with using the configuration at the final step of this run, the system was quenched down to Fefr = 1.50 (quenching process) followed by annealing MD simulation at this F~fr over ten million time steps. This ['eft is still lower than Fg (=1.58, the glass transition), but slightly higher than Fc (=1.45, kinetic transition) in the supercooled fluid phase. The Fourier and Laplace transformations, S(q,w) and G(q,w), of the intermediate scattering function F(q, t) are defined as:
S(q,w)
= ~1 f_~ F(q, t)ei~t dt,
(5)
oo
a(q,~o)
where
=
F(q,t) = (E i
/5
(6)
F ( q , t ) ~ - ~ ' at,
e iq'[ri(t+t~
-
ri(t0)]~
.
(7)
li,to
The averages are taken over the configuration of atoms {ri} and the initial time to. It is known that the general susceptibility x(q,w) is related to G(q,w) and S(q,w) as follows:
x'(q,w) x"(q,w)
= 1 + wa"(q,w), wG'(q,w), = ~ S ( q , ~),
(8) (9)
=
(~0)
where x(q,w) = x'(q,w) + iX"(q,w) and G(q,w) = G'(q,w) + iG"(q,w); prime means the real part, and double primes the imaginary part. In the present MD simulation, G(q,w) was calculated with the following equation: 3
G(q,w) = -~ Ej
dto eiq'r'(t~176
dt e -iq'ri(t)+iwt
,
(11)
i where < ... > means an average over the configuration of atoms (ri}, and T is the total time step of the simulation. x~(q,w) obtained from Eq. (11) is shown in Fig. 1. Here, the subscript s stands for the self part of the function, i.e. in the summation ~ j of Eq. (11) only the term j = i is taken into consideartion. The following two remarks should be noted. First the a (left) peak in x"(q,w) depends on T (the upper limit of the integration). Since the a relaxation involves the slowest relaxation process of the density fluctuations in the supercooled fluid, the value of the integration Eq. (11) in the low frequency region seems to depend significantly on the value of T. By careful considerations of x"(q,w) obtained by the present MD simulation (see Fig. 1), the c~ peak is clearly observed around log w = -3.2. We consider that the data below log w = -3.8, in particular the peak around log w -- -4.0, are not valid due to the finite T and do not have any physical meaning. Second, in Fig. 1, we observe the/3 peak around log w = 0.6 in the high frequency region. This peak is much clearer than the c~ peak. We note that the data in the high frequency region (log w > 1.2) monotonously increase as a function of log w resulting from the finite time increment (time mesh) used in the MD simulation and are therefore not correct. Figure 2 shows the Cole-Cole plots of xs(q,w). In this figure, as the frequency becomes higher, X's(q,w) moves from the right- to the left-hand side. As shown in Fig. 1, only the data without ! are reliable; the data marked by I are artificial caused by the finite T or the
126
finite time increment dt in the integration (11). From Fig. 2 we can conclude that the highfrequency data follows a half circle with a center on the abscissa, while the low-frequency data seem not to be fitted by a reasonable half circle with a center on the abscissa. This clearly indicates that the fl relaxation is regarded as being of a simple Debye type. This conclusion is consistent ~ith the decrement of x'(q,w) above the fl peak following w -1 (see Fig. 1 ).
0.0
9
-
9
1.0
,
-
f
-
.
f
3
A
tl v
!
f
3
f
"ID"
-1.0
v -=,
s o -2.0
-4.0
-2.0
Iog~o ~
0.0
2.0
0.0
.... o:o '
'
'
~'(q ,to)
1.o
(left) Fig. 1. The imaginary part of the general susceptibility (self part) , ~ ( q , ~ ) vs. w. The reduced wave number q'(= qL/27r) equals 8.0, where L is the side length of the cubic simulation cell. The circles represent the results of the numerical integration of Eq. (11); the solid symbols are for the lighter species, and the open symbols are for the heavier species. Unreliable data due to the finite numerical integration are marked by [ on their data points. The two peaks around log w ~ - 3 . 2 and 0.6 are the a- and ~-peaks, respectively. The solid lines indicate two power-law behaviors around the ~ minimum, represented by ,~(q,w) " w -0"47 and x~(q,w) " w 0"74 by the least-square fit. The dashed // line, x~(q, w) ~ w- 1 . 0 , is compatible to a typical Debye type. (right) Fig. 2. Cole-Cole plots of x,(q,w), i.e., X.;'(q,w) vs. ,~:',(q, w). The symbols are the same as in Fig. 1. At least two distinct relaxation modes are seen: the semi circle near the origin represents the fl relaxation, while that in the left-hand side represents the c~ relaxation. The latter seems to differ from a semi circle.
IV. 2-dimensional results Annealing MD simulation for the 2-d soft-sphere model with N=10000 atoms has been carried out over the time steps of 10007- at Fefr= 1.4 in the glass phase; the glass transition for the 2-d system is estimated to be between F g = l . 2 and 1.3. Figure 3 shows the displacements of a group of atoms in a portion of the whole system during the period of 47- in the annealed MD simulation. It is observed that the displacements of atoms are shorter than al, but substantially larger than the amplitude of thermal
127
vibrations, their directions being highly (nearly linearly) correlated with each other. In order to measure the degree of such a correlation between a referred atom (i-th atom) and the surrounding atoms within a cut-off distance r~, we introduce the correlated-motion coefficient CMi(r~, t): 4 N~
CMi(rc, t) = ~
A r i ( t ) . Arj(t)/Ar2(t)N~,
(12)
rij <~rc
where Ari(t) displacement from the i-th the i-th atom
is the displacement vector of the i-th atom and Ar2(t) the mean square for an elapsed time t, and Ni is the number of atoms within the distance rc atom at t = 0. The summation j is taken over the near-neighbor atoms of within the cut-off distance re, i.e., rij < re.
(Or)
0 position(5.59,16.77)
+5.0
Z~
=~ ~
: ~
~ ~
.
....
~-~
Y 0.0 ....
......... 4
......
.... ~
........
~,
,
..... ~ - - ~
.....
.......... ~ - ............ - ~ . . . . ~
.............
,
-5.0 -5.0
0.0
x
in of in of
+5.0
(o,)
Fig. 3. Movements of atoms around the position (5.59, 16.77), the center of this figure the present simulation cell with a side length of 111.80 in units of al during the period 47-. The circles represent the initial positions of atoms, and triangles the final positions the period of 47-. Both X- and Y-coordinates are the relative distances from the center this figure..
128 Figure 4 shows the space distribution of such correlations. It is clearly seen that there are large clusters of the dark points marked by the highest graduation in this figure, connected to each other in almost the whole simulation cell. For a shorter elapsed time, we have obtained a similar result, but dark and white points are almost randomly distributed. Note that the time scale used in Fig. 4 is about one order of magnitude longer than that of thermal vibrations. It will be shown below that such a time scale corresponds to the relaxation.
(o,) 50.0
9
Y 0.0
-50.0
-50.0
0.0
x
50.0
(o,)
Fig. 4. Spacial distributions of the strength of the correlations as calculated by the correlated motion coefficient C Mi(2.0, 5.0) for all atoms. The circle at the right-hand side of the figure denotes the size of the cut-off radius re, which is chosen here to be 2.0 al. All circles in different graduation and in different size represent the initial positions of all atoms of the simulation cell. The graduation is made according to the values of CM~, divided into five classes; the darker the graduation is, the larger the value of CMi is obtained. Both Xand Y-coordinates are in units of al.
129
The imaginary part of the generalied susceptibility (self part only) for the 2-d binary soft-sphere model at ['eft = 1.4 is shown in Fig. 5. The peak around co "-" 3 . 1 0 (in units of r -1) corresponds to the thermal vibrations (highest-frequency mode). The peak around co = 2 is the fl peak. This frequency is consistent with the time scale of the collective motions mentioned above. Therefore, these collective motions can be an origin of the /3 peak of x~(q,w). Analyzing the temperature dependence of CM~, we have found that the size of correlated domains becomes significantly larger as the temperature is lowered. In Fig. 5, there also appear largely scattered data in the lower-frequenc side of the/3 peak. Though physical meanings of these data have not yet been well understood, we think they are perhaps artificials. The reasons are followings: It has been found that the maximum among the data depends strongly on the system size of simulations and that the frequency of the maximum is inversely proportional to the length of the simulation cell and nearly equals the inverse of the time for the sound wave to propagate from one side to another of the simulation cell. This suggests that the maximum and the near-by data are seriously influenced by the periodic boundary conditions used in the simulations.
1 0.90.80.70.6-
........
!
........
'
''
.... I
I
|
I
II
o 0
)
0 0 4 0 ~
r~
o o
0.2-
0
0
0
oO00
|
00
~
Oo
|
~
0
0
o
0.1 -2
'
D
~ 0 , 5 -
3 EO.4=-_.. ~0.3-
I-
o 0
0
i , , .i|[
-1
0
i
i
i
i i,,,i
,
0 log,0( aJ )
,
s
, .,.,|
I
,
,
,
.
(1/~')
Fig. 5. Imaginary part of the generalized susceptibility (self part), xJ(q, co), for the lighter species at Fefr = 1.4 in 2-d system. The wave number is q = 27r in units of al, and in abscissa is in units of 7 -1 .
130 V. Conclusions
Main results of the present work are summarized as follows: (1) The a peak as well as the ~ peak in X"(q,w) spectrum was obtained. This is the first case to confirm the existence of of the a peak by MD simulation. (2) We have found that the fl relaxation is essentially of a Debye type. (3) We have obtained spacially collective motions in the time scale of the relaxation; the collective motions therefore are regarded as an origin of the ~ relaxation. (4) We have remarked that the periodic boundary conditions used in the simulations may cause an artificial peak in x"(q,w) and one therefore must be very careful. Finally, we comment on the difference between the self part and the full density autocorrelation function. The full density autocorreration function and the dynamical structure factor are experimentally measured, while in the present MD simulation only the self part was studied. However, the difference between both correlation functions (dynamical structure factors) is considered to be rather small except that additional modes associated with sound modes appear in the full density autocorrelation. We have previously computed the full density autocorrelation via MD simulations for the same model as the present one, and found that the general behavior of the a relaxation was little changed. General trends of the relaxation are nearly the same for both full correlation and self part. In addition, from a point of numerical calculations, the self part is more easily obtained than the full autocorrelation; the statistics of the data obtained from MD simulatons is much higher for the self part than for the full autocorrelation.
References
1 Y. Hiwatari, H. Miyagawa and T. Odagaki, Solid State Ionics, 47, 179 (1991) 2 Y. Hiwatari, J. Matsui, K. Uehara, T. Muranaka, H. Miyagawa, M. Takasu and T. Odagaki, P hysica A 204, 306 (1994). 3 J. Matsui, T. Odagaki and Y. Hiwatari, Phys. Rev. Letters, 73, 2452 (1994). 4 T. Muranaka and Y. Hiwatari, Phys. Rev. submitted.
joumaJ of"
MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 131-138
Density Functional Theory of Solution Dynamics Toyonori M u n a k a t a
Department of Applied Mathematics and Physics, K y o t o University, K y o t o 606 J a p a n
Abstract The density functional theory for classical(equilibrium) statistical mechanics is generalized to deal with various dynamical processes associated with density fluctuations in liquids and solutions. This is effected by deriving a Langevin-diffusion equation for the density field. As applications of our theory we consider density fluctuations in both supercooled liquids and molecular liquids and transport coefficients.
I. Introduction The density functional theory(DFT) 1-4 has become a useful method to study freezing transition from a quantitative standpoint.
The DFT is now generalized to deal with complex systems such as
molecular liquids s and liquid crystals 4. It is to be noted that the DFT is basically concerned with the (quasi)equilibrium density profile neq(r) and the corresponding free-energy Feq --- F[neq(r)], where
F[n(r)] denotes the free-energy functional of the system and plays the central role in the DFT. If we could introduce dynamics to the DFT and follow time evolution of the density field n(r,t), this might enable us to study some dynamic processes such as density fluctuations and transport coefficients. In this paper we present a time-dependent(TD) DFT together with some (preliminary) applications of the TD-DFT. In Section 2 the DFT is briefly reviewed and in Section 3 a Langevin-diffusion(L-D) equation for the density field n(r,t) is presented. In Section 4 we consider, as applications of the TD-DFT, (A) density fluctuations in liquids and solutions and (B)mass flow around a fixed particle to calculate transport coefficients of liquids. Section 5 contains some remarks and summary of this paper.
II. D e n s i t y F u n c t i o n a l T h e o r y ( D F T )
First let us consider a one-component liquid, whose density profile is denoted as n(r).
0167-7322J95/$09.50 9 1995 Elsevier Science B.V. All rights reserved. SSD/0167-7322 (95) 00847-0
From a
132 variational principle 2,3, we obtain an equation to determine the equilibrium density profile
$f[n]/6n(r)- p + r
neq(r) :
= 0,
(1)
where Celt(r) is the external field and p is the chemical potential. We may say that all the difficulties of a many-body problem are embedded in the free-energy functional methods heretofore 3. All the approaches divide (excess) part - ~ [ n ] , with
Fid exactly
F[n] into
F[n] and
with h the thermal wavelength [h 2/2rmkBT]
1/2. Standard
Taylor expansion around the uniform liquid state n(r) = =
Fid[n], and
an interaction
given by
Fid[n] = kBT f drn(r)[ln(n(n)h a) -
.[n]
there have been proposed many
an ideal gas part
(2)
1],
perturbational approaches employ a functional
nL, a
constant,
E(1/rn[)fdrl.-.fdrm6m~/6n(rl)...6n(rrn)lnL(n(rl)--nL)...(n(rm)--nL)
=_ (kBT)E(1/m!) f dr,.../drm
era(r1,'" ,rm)(n(rl)
which is usually truncated after m = 2. e In this case
F[n]'~Fid[n] -
Fin] takes
-
I'lL)...(n(bmirm)
-
I'lL) ,
(3)
the form
(kBT f dvlc,[n(v,)-nL]
+ (kBT/2)fdrlfdr2e2([r2--r~l)[n(rl)--nLl[n(r2)--nL]},
(4)
where use is made of the fact that for a uniform system cl is a constant and c2 is a function of Iv2 - r l I. It is not difficult to derive the relation between c2 and the radial distribution function g(r) =
h(r)+ 1, 2,3 (5)
h(r) = c2(r) + nL f dr'h(r')c~(Ir - r'l).
Equation (5) is the familiar Ornstein-Zernike relation and hereafter we denote the direct correlation function c2(r) as
c(r). Based on (1) and
(4) with (2), one can discuss freezing transition of one-component
liquids. 1-3 For an S-component mixture, which is specified by S density fields,
nj(r)(j = 1,... ,S),
we
have, instead of (4),
Fin] ~_ F~jFid[nj]- {kBtEif arlcl,j
[ n i ( r ) l - nj,L]
+ (kBT/2,Eij fdrl/dr2ci.j(lr2--rl,[ni(rl)--ni.L][nj(r2)--nj,L]},
(6,
where c i j ( r ) denotes the direct correlation function between the species i and j.7 For molecular liquids, the DFT is more complicated due to the constraints imposed by the fixed bond length. 5
III. Time-Dependent
Density Functional Theory (TD-DFT)
We now present a Langevin-diffusion(L-D) equation for the density field n ( r , t ) and the corresponding
133
Fokker-Planck(F-P) equation for the distribution functional f[n(r),t] and discuss general properties of
the TD-DFT. s,9 By combining a number-conservation law, the D F T and theory of Brownian motion, we
derived the following equations for the density n(r,t) and the m o m e n t u m density g(r,t) : cg(r,t)/Ot = - V . g ( r , t ) / m , cgg(r,t)/~ = - n ( r , t ) V 6 F / 6 n ( r , t ) -
(7)
r 0 g ( r , t) + y ( r , t ) .
(8)
where - V 6 F / 6 n ( r ; t ) represents a generalized force on a particle at r x~ and the random force f ( v , t ) satisfies the fluctuation-disspation (FD) theorem of the form
< f i ( r , t ) f j ( r ' , t ' ) > = 2 m k B T r o n ( r , t ) ~ ( r - r')6(t - t')6i~,
(9)
with i and j denoting the Cartesian components. Since we are interested in long time behavior, we employ an adiabatic approximation for (8), yielding g(r,t) = {-n(r,t)V6F/6n(r,t) + y(r,t)}/r0.
(10)
From (7) and (10) we finally obtain the L-D equation,
cgn(r,t)/Ot = - V . { - n ( r , t ) V 6 F / 6 n ( r , t ) + f ( r , t ) } / ( m r 0 ) = - 1 7 . { i s + J R } ,
(11)
with the FD theorem (9). Js and JR in (11) denote the systematic and the random current ,respectively. From the FD theorem (9) it is seen that the random current j a ( r ; t ) = f ( r ; t ) / r n r o is a multiplicative noise it and one must specify how one interprets the noise. Here for our purpose it is to be treated as an Ito type. Following a routine procedure n to derive a F-P equation from a Langevin equation, we see that the distribution functional f[n(r),t] evolves in time, with D - k B T / ( m r o ) , according to
Of~at= -/dr6n~J(f
),
J ( f ) = D { f l f V . n ( r ) V 6 F / 6 n ( r ) + V . n(r)V6 f /6n(r)} . When f is proportional to exp(-flF[n]), V . n ( r ) V 6 f / 6 n ( r ) =
V. n ( r ) { - f l f V 6 F / 6 n ( r ) }
(12) (13) =-/3fVn(r)V
6F/6n(r) and we confirm that the stationary solution is given by e x p ( - f l F ) . In other words, the L-D equation (11) actually samples,in a steady state, the density field n(r) according to the weight exp(-/~F). The multiplicativeness of the noise f ( r , t) can be interpreted based on a simple hopping diffusion model and can be related to the internal noise proposed by Mikhailov. t2 General properties of the T D - D F T are most concisely represented by the following two H-theorems the proofs of which are given elsewhere. 9 First we neglect the random current JR and consider the diffusion equation
On(r,t)/Ot = ~ D V . n(r,t)X76F/6n(r,t) = - V . ] s ( r , t ) .
(14)
First H-theorem : When the density field n ( r , t ) evolves in time according to (14), F[n] decreases in time according to dF/dt = - 0 3 D ) - t f dr { j s ( r , t ) } ~ / n ( r , t ) < 0 until ] s ( r , t ) vanishes and it holds that
6F/6n(r) = p, representing the variational condition (1) in the D F T to determine the equilibrium density
134
field for the case ~,=t(r) = 0. N o w we turn to the full L-D equation (I I).
Second H-theorem : W h e n the distribution functional fin,t]evolves in time according to the F-P equation (12) the generalized free-energy functional Fa[f] defined by
Fat/] = / DnFtnlf[n;t] + kBT / Dnftn]ln(ftn]),
(15)
decreases in time monotonically according to
dFa/dt =
-
/ Dn / dr {(kBT//Dn(v))x/2:is(r) - (DkBTn(r)//)l/2v(6//6n(r))} 2 < 0
(16)
until f[n, t] takes the form
/,,In] = const, exp[-F[nl/kBT].
(17)
We note that f Dn denotes the integration over the function space of n(v). When the integrand of (16) is zero, we have (17). Comparing the two theorems it is seen that the noise JR prevents the density field
n(r,t) from b~eing trapped in one of the local minima of the functional Fin]. la The results obtained above is readily generalized to mixtures.
V. A p p l i c a t i o n s of T D - D F T
In this section we apply T D - D F T developed in the previous section to study dynamic density fluctuations (A) and transport coefficients(B). In these studies the approximation (4) or (6) for the free-energy functional is employed since we have no reliable information on higher-order direct correlation functions cn(n > 3).
(A) Density fluctuations in liquids and solutions
Inserting (6) with (2) into (11) we obtain the following L-D equation :
cOn(r,t)/Ot = nV2n - DV. n(r,t)V f dr~c([r - r~l)[n(r',t) - nL]- V . JR
(18)
We note that the second term on the rhs of (18) describes a diffusion process, which is induced by the Vlasov field VF -
-kBTfdr'c(Ir-
r'[)[n(r',t)- nL]. Equation(18) with or without the random
current T M together with its generalization to two-component system 15 and to polar liquids 16'17 have been playing important roles and now generally called a Smoluchowski-Vlasov equation. As an application of the L-D equation (18) with the F-D theorem (9), we calculated the dynamic structure factor r of a simple liquid
G(q,t) = < n(q,t)n(-q,O) > / < n(q)n(-q) > .
(19)
135
with n(q,t)=_ f dr { n ( r , t ) - nL } e x p ( - - i q - r ) / N 1/2. Fourier-transformation of (18) and the FD theorem (9) yield
On(q, t)/Ot = -7(q)n(q,t) + Ek V(q, k)n(k,t)n(q - k,t) + r
(20)
> = - 2 D q . q'6(t - t') {6q+q,,0 + n(q + q',t)/NX/2},
(21)
where 7(q) = Dq2/s(q), with s(q) denoting the static structure factor, c(q) = 1 - 1/s(q), and V(q,k) =
Dc(k)q. k / N ~/2. If we neglect effects of both nonlinearity in(20) and the multiplicativeness of the noise, n(q,t) becomes a simple Ornstein -Uhlenbeck process ~1 and G(q,t) and the dynamic structure factor G"(q,w) = (1/2) f dtG(q,t)exp(iwt) are given by Go(q,t) = exp(-v(q)t) and Gg(q, w) = 7(q)/[v(q) 2 + w2],
(22)
respectively, where the subscript 0 on G means that we regard it as the zeroth aproximation to G.
In
calculating G(q,t) based on (20) and (21) we follow a nonlinear theory of fluctuations by Mori and Fujisaka. xr
Numerical calculation is performed for a hard sphere system characterized by a packing
fraction p = r~3nL/6, is Percus-Yevick approximation is used to supply the structural information. As p becomes large we observe narrowing of the central peak, reflecting slowing down of the density fluctuations due to nonlinear coupling in (20). At p = 0.53 a dynamic 'instability' occured where an effective diffusion constant crosses zero. As is well known the hard sphere system freezes at p ~_ 0.5 and the D F T in its various version has been applied to study the transition. 19 It is interesting to note that the interaction part of the free-energy(4) gives rise to the equilibrium transition on the one hand and it also gives rise to the instability through the nonlinear coupling in the L-D equation (20). Before leaving this section it is noted that if we take the free-energy functional (6) for an S-component mixture we have a set of L-D equations for S density fields ni(r,t)(i = 1,... ,S), which lead to
On,(r,t)/Ot=D, [ V 2 n , ( r , t ) - V
.n,(v,t)V { E j / d , ' c , ~ ( l r -
r'l)6nj(r', t)}]-
V . in,,
(23)
Furthermore if we take the free-energy functional as given by Chandler et hi. 5 for molecular(polyatomic) liquids, we obtain from (11) a L-D equation for each of the atomic species constituting the molecules, which is very similar to (20). After liniarization as done in (22), we obtain a linear diffusion equation, which was used by Hirata lr to investigate the dynamic structure factors of water.
(B) Some Transport Coefficients In this subsection we put a particle at an origin in a velocity field u ( r ) and study effects of flow on a stationary density profile n,t(r). When there is no flow u(r) = o, n,t(r) is obviously given by nLg(r), with g(r) a radial distribution function. Due to the flow u(r), this equilibrium distribution is distorted and from the distortion we can calculate some transport coefficients, like viscosity 7] and friction constant ~, as we show below. We consider a one-component system and neglect random current JR in (18) to obtain
8n(r,t)lbt = - V - j , ( r , t ) ,
(24)
136
j,(v,t) = DVn - a D ~ V t f dr'v~]] (Jr - r ' l ) , ( r ' , t ) + r where v,/l(r ) = -kBTc(r) and r
+ nu,
(25)
two-body interaction, represents effects of the particle fixed at
r = o. The last term on the rhs of (25) represents particle flow due to the velocity field, u ( r ) . We are interested in a stationary density profile n , t ( r ) around the fixed particle. First we consider the equilibrium solution n , q ( r ) when u = 0. In this case the particle flow j0 = o and from (25) we readily obtain ln[gCr)] __
ln[-,,C.)/-L] = -~ f dr'v,ss(l~ -
v'l)nL[g(r')
-
1] -- flr
(26)
where the boundary condition n,q(r) --* nL as r ---,oo is taken into account. From the relation (5) between h and c, we observe immediately that (26) is equivalent to the H N C equation to determine g(r). Thus our theory, when applied to anequilibrium situation, g!ves the H N C
result for g(r).z
Now let us turn to effects of the flow field u(r) on nat(r). We consider, to be concrete, shear flow
u(r) = 7yeffi where ez denotes a unit vector in the x-direction. In this case we assume a solution of the form n , i ( r ) - nLg(r)[1 4- w(r)7/D -t- o(7)]
(27)
where o(7)/~ --+ 0 as 7 ~ 0. Inserting (27) into (24) with i)n/Ot - O, we obtain
gV2w--nLgV2 i dr'c(Ir-r'l)g(r')w(r')+Vg.Vw-nLVg.V
i dr'c(Ir-r'l)g(r')w(,')
zyg'(r)/7. (28)
with use of Fourier transformation it is seen that (28) has a solution of the form w(r) = z y a ( r )
(29)
and a(r) satisfies a complicated integro-differential equation, which we do not write down here. The final step to obtain shear viscosity 71 is to calculate the xy component of the stress tensor axy, which is expressed, on the one hand, in terms of ,7 and 7 as =! =
'7")'
(30)
and also microscopically as T
ffxy = where r
(nLI2) ] drno,(r)xyr
(31)
- de~dr and we have neglected the kinetic contribution, which is very small at a liquid
density. Use of the solution (29) together with (30) and (31) gives
r] = (n2L/2D) / dr(xy)~ g(r)r
(32)
If we fix a particle in a uniform flow u ( r ) = uoe~, we can calculate the friction constant ( from a stationary force on a fixed particle by following similar lines of reasoning as above. Our preliminary results for shear viscosity r / o f a soft-core system with r
= e ( a / r ) 12 shows that
main contributions in the integrand in (32) come from the region r ~_ (V/N) ~/a (of course in the highdensity state) and that r/increases sharply as a function of p. =_ (elkBT)I/4(Nvra/V). Although detailed
137
analysis of 77 and ~ based on our TD-DFT needs some time to be completed, it seems that our approach is promising in view of the fact that we have at the moment nearly no (microscopic) theory for r/and (, especially for dense complex systems like water.
V. S o m e R e m a r k s a n d S u m m a r y
In this paper we gave a dynamic extension of the DFT, by deriving a L-D equation (11) with the /
fluctuation-dissipation theorem (9). We showed that the stochastic equation correctly samples the density field according to the probability exp {-/~F[n]},(17), based on the second H-theorem (16). At this point we note however that our TD-DFT is phenomenological and it is desirable to have a first-principle dynamics generalization of DFT. As applications of TD-DFT, we considered density fluctuations in liquids and transport coefficients. We are currently trying to solve the L-D equation in a real space-time to study slow dynamics in supercooled liquids. This study may be considered to be a dynamic counterpart of the work by Dasgupta and Ramaswamy is and as a similar attempt we mention the work by Lust and Valls 2~ We expect that our L-D equation has many fields of application. One example is dynamics in molecular liquids, as noted at the end of subsection IV(A), which is a very important field in connection with chemical reactions in solutions but, at the same time, is very complex to deal with from first principles.
References 1 R.Evans, Adv.Phys. 28 143(1979). 2 A.D.J.Haymet, Ann.Rev.Phys.Chem. 38 89(1987). 3 D.W.Oxtoby, In 'Liquid, Freezing, and the Glass Transition', (Eds. J.P.Hansen,D.Levesque and J.Zinn-Justin) (Elsevier:New York,1990) 4 Y.Singh, Phys.Rep. 207 351(1991). 5 D.Chandler, J.D.McCoy, and S.J.Singer, J.Chem.Phys. 85 5971;5977(1986). 6 T.V.Ramakrishnan, and M.Yussouff, Phys.Rev. B19 2775(1979). 7 J.P.Hansen, and I.R.McDonald.' Theory of Simple Liquid s'(Academic, New York, 1986). 8 T.Munakata, J.Phys.Soc.Jpn. 58 2434(1989). 9 T.Munakata, Phys.Rev.E(1994) to appear 10 T.R.Kirkpatrick and P.G.Wolynes, Phys.Rev. A35 3072(1987). 11 C.W.Gardiner ' Handbook od Stochastic Methods' (Springer, Berlin,1982) 12 A.S.Mikhailov, Phys.Rep. 84 307(1989).
13 C.Dasgupta and S.C.Ramaswamy, Physica A186 314(1992). 14 T.Munakata, J.Phys.Soc.Jpn. 43 1723(1977). 15 B.Bagchi, Physica 145 A 273(1987). 16 D.F.Calefand P.G.Wolynes, J.Chem.Phys. 78 4145(1983).
17 F.Hirata, J.Chem.Phys. 96 4619(1992). 18 H.Mori and H.Fujisaka, Prog.Theor.Phys. 49 764(1973). 19 T.Munakata, J.Phys.Soc.Jpn. 59 1299(1990). 20 A.D.J.ttaymet and D.W.Oxtoby, J.Chem.Phys. 84 1769(1986). 21 L.M.Lust and V.Valls, Phys.Rev.E 48 1787(1993).
journal of
~IOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 139-147
Dissociation and Solvation in W a t e r and A q u e o u s Solutions A.D.J. H a y m e t School of Chemistry, University of Sydney, N S W 2006 Australia Abstract Recently the properties of both charged and un-charged solutes in aqueous solution have been elucidated by computer simulations and approximate integral equation theories. While answering some questions, others are raised. In particular the structure of water next to a charged surface (or a very large charged solute) is a challenge. Approximate integral equation theories can address this challenge.
I.
Introduction
Water plays an important role in the chemistry and physics of bulk solutions and interfaces, including electrochemistry and macromolecules in solution. Usually the water is treated as a structureless, dielectric continuum, such as in the Debye-Hiickle approximation 1 for electrolytes, the Gouy-Chapman-Stern 2-4 (GCS) approximation for the electrical 'double layer' and the DLVO 5'6 approximation for colloids. Properties sensitive to the molecular nature of water cannot be determined by these theories. Recently we have published integral equation predictions for a flexible model of water next to a planar interface. T Experimental motivation for this work includes electrochemical experiments on ultra-pure (O2-free) water, s'9 surface EXAFS studies of the oxygenmetal distance for water at an electrode, 1~ and the tunnel junction device measurements of Porter and Zinn. 11 Vossen and Forstmann ~2 have published a related calculation using a different model of water and a different approximation for the bulk water bridge functions. Below we compare the input to the two calculations. First we review some results in bulk water and solutions of non-polar solutes.
II.
A m o d e l of water
We have studied via molecular dynamics computer simulation a number of properties of so-called CF1 central force model of water, 13'14 although our methods may be applied to any model of water with a unified Hamiltonian for intra- and inter-molecular degrees of freedom. The CF1 model is a slight variant of the central force (CF) model of Stillinger and Rahman, is which allows the water molecules to vibrate and even dissociate. 13,~6 Central force models view water as a 'molten salt' of two fractionally charged species, hydrogen and oxygen. MD simulations show that the static dielectric constant and pH of the CF1 model at 25~ and 1.000 g cm -a are in good agreement with experiment. Such properties 0167-7322/95/$09.50 91995 Elsevier Science B.V. All rights reserved. SSD! 0167-7322 (95) 00848-9
140
are important for models of water which are to be used in electrolyte theory and other solution phase studies. We have also determined a number of physical properties of the CF1 model using integral equation methods. We conclude from these studies that the CF1 model is a valuable addition to the family of classical models of water. The output from this calculation is used in theories of inhomogeneous solutions. 7 The revised central force (CF) potentials of Stillinger and Rahman 15 are defined by
uoo(r)
=
144.538 { 26758.2C1 - 0 . 2 5 e -4(r-3"4)2 - 0 25e -1"5(r-4"5)2 r
7.8.8591
UHH(r) -- 36.1345 ~18 r 1 + e 4~176 72.269 UOH(7")
:
r
,
"
'
-- 17e-762177('-1"45251)2
6.23403
10
r 9"19912
1 -l- e 4 0 ( r - I ' 0 5 )
4 -
I
-I- e 5"49305(r-2"2)
'
(:)
where the values of the constants are C1 = C2 = 1. The distances are measured i n / ~ and energies in kcal tool -1. At the temperature 25~ and a mass density of 1.000 g cm -3, this model predicts a pressure of 3.54 kbar, thousands of times greater than atmospheric pressure of approximately 1 bar. 1T'ls By changing the constants slightly to C1 - 0.9 and C2 - 1/1.025, it was found that the pressure could be reduced to a more acceptable value of 120 bar. 13 We call this model of water the CF1 model. For both CF1 and CF water, each hydrogen species is assigned a partial charge of qH -- 0.32998e where e is the proton charge, while the oxygen species each carry a charge of qo - -2qH to ensure electroneutrality of individual H20 molecules. This fractional charge should not be regarded as a defect of the model, 19 but instead as arising from the 'screening' induced by integration over polarisation terms, three- and higher-body potentials, etc., to obtain the effective pairwise additive potential energies displayed in Equation (1). III.
Hydrophobicity
In 1993 Blokzijl and Engberts 2~ published an important review article on hydrophobic effects. In 1992, Smith, Zhang and Haymet 21'22 showed that hydrophobic association of uncharged solutes in water is almost entirely an entropy effect. Haymet 2a went on to point out that experimental data show that methane "likes" water about as well as it "likes" any other solvent from the point of view of enthalpy. This view is underlined by the experimentally high solubility of water in cryogenic solvents such as liquid methane. Hydrophobicity in fact arises from the entropy penalty incurred by the water. The methane is not hydrophobic: the water is 'methophobic'. Very recently Guillot 24 has performed a beautiful series of simulations using faithful models of rare gases and water, and determined solubilities along the liquid-gas coexistence line. These new contributions revise substantially the traditional view of hydrophobicity.
141
IV.
Ionic
Association
and
Dissociation
The potential of mean force (PMF) for sodium chloride in water has been simulated by numerous investigators for many different models. Smith and Haymet is noted that to dissociate a model sodium ion and chloride ion at room temperature and pressure, two free energy barriers appear to have to be surmounted. The first barrier is associated with the formation of a 'shared' layer of water molecules between the ions, now constituting a so-called 'solvent separated ion pair', and the second barrier associated with the final unlinking of the solvations shells of the two ions. Very recently, Cui and Harris 25 have evaluated the P MF for sodium ion and chloride ion in a model of supercritical water. They decompose the PMF into enthalpy and entropy contributions, as did Smith and Haymet for uncharged solutes, 21 and find that, at least in supercritical water, the entropy effect is dominant, just as for uncharged solutes at room temperature and pressure. The dissociation of a single water molecule has been studied, for the CF1 model, by our group, la'14 By calculating the PMF for separating a single water molecule into two ionic fragments, an upper bound to the pH of the CF1 model was predicted to be 8.4. Future work will involve calculating the relative concentration of H(aq) , + H30~-a.~,( ~, ..., HgO+(aq) and related OH~-,q) species. V.
Properties
of bulk
water
from
an integral
equation
The results summarised here are for pure water at the temperature 25~ and the density 1.000 g cm -3, and are obtained by solving numerically the Ornstein-Zernike (OZ) equation for the pair correlation functions, using a closure that supplements the hypernetted chain (HNC) approximation with a bridge function. The bridge function is determined from computer simulations as described below. The numerical method for solving the OZ equation is described by Ichiye and Haymet 26'27 and by Duh and Haymet. 2s For a liquid mixture of u species at temperature T and volume V. The potential energy Uab(rl:) between particles of species a and b, separated by a distance rl: - Irl - r21, is spherically symmetric and pairwise additive. The (iirect correlation function C~b(r12) is defined by the OZ equation:
hab(r12) -- Cab(r12) + ~-~.ps /v dra cas(rla)hsb(r32) ,
(2)
s--1
where hab(rl2) = gab(rl2)- 1 is the total correlation function and ps is the density of species s. The pair potential is separated into short- and long-range contributions: U~b = u~b + l which leads to a decomposition of the direct and total correlation functions into Uab, ._. 8 8 Cab Cab ~ (~ab and 'hab = hab + qab, where the long range parts of Cab and hab a r e defined to be r = --~U~b and q~b = &~b + ~ = 1 P~ ( ~ * %b, where ~-1 = k T , k is the Boltzmann constant, and 9denotes the convolution integral as in Equation (2). The long-range part of the potential is chosen to be the Coulombic interaction and Cab(r) = --/3q~qb/47r~o~r, where q, is the charge on species a. We retain the symbol e to display the formal similarity
142
with standard theories of electrolytes. Of course, for the present case of CF1 water, E = 1. The function qab is the (negative, dimensionless) Debye-Hiickel (DH) potential of mean force, qab(r) = --~qaq_______~bexp (--~r)
(3)
4~rEo~r
where a is the Debye inverse length defined by ~2 = f l ( e o e ) - i ~ a paq~. For the CF1 model at 25~ and 1.000 g cm -a, a-1 _ 12.402/1,, so this is a DH "strongly interacting" case. To solve the OZ equation, we use the exact closure which may be derived directly from the partition function: hab -I- 1 = exp(--flUab + hab -- Cab ~- Bab) ,
(4)
where Bab is the bridge function. 29 In the HNC approximation, Bab = 0 for all distances. The bridge functions used in this work are discussed below. The above equation becomes C~b -- exp(--~U~b + Tab + qab + Sab) -- 1 -- Tab - qab ,
(5)
where Tab = h~b -- C~b, which is the renormalized closure. The OZ equation and equation 3
g~
2
1 0 2
go.
1 0
i_.
4 3
gHH
2 1 0
0
2
4
6
r/A
Figure 1" Pair correlation functions gab(r) from experimental data of Soper31(thick solid line), and for CF1 water (solid line) and the VF model (dashed line). (5) are solved iteratively to obtain Tab and C~b. Since (I)ab and qab are determined once the potential has been separated into short- and long-range contributions, Cab and hab may be calculated from their definitions. Further details of the integral equation method may be found in the references. 2a-2s For central force models of water, the bridge function is essential for accurate solutions of the integral equations. Thuraisingham and Friedman 3~ found that at room temperature the HNC [Bab(r) -- 0] approximation for the CF model was in very poor agreement with
143
$
COO
-50 -100
,
i
,
i
,
CSoH -50
s
CHH
......i.......i........i....... -50 8
Figure 2: Short-range direct correlation functions c~b for CF1 water (solid line) and the VF model (dashed line). "exact" results from computer simulations. In 1988 Ichiye and Haymet ~~ showed that hard sphere bridge functions can be used to improve the quality of the solutions. We have determined these bridge functions by a numerical method. 16 The pair correlation function gab(r) obtained by solving the OZ equation with these bridge functions is shown in Figure 1, together with the experimental data for water by Soper. al Our integral equation calculation agrees to within the thinkness of the line with MD simulation of the same Hamiltonian. Also shown are the pair correlation functions for bulk water predicted by Vossen and Forstmann12(VF). They use the original CF model of water, together with a bridge function based on the original hard sphere bridge functions of Ichiye and Haymet, 26 augmented with an empirical hydrogen-hydrogen bridge function. The agreement between experiment and the VF calculation is not as good as our calculation, due in part to the CF model of water, and in part to the much more approximate bridge function. The pressure of the bulk VF water, calculated from these pair correlation functions, is approximately -48,000 bar. A full range of properties (including the dielectric constant) arising from our calculation are available. 16 The dielectric constant of CF1 water has been determined by molecular dynamics simulation to be 69 • 11.13 These calculations can also be compared with the perturbation calculations on CF water by Holovko and collaborators, a2 The difference between the two calculation is displayed more dramatically in Figure 2, where the short-range direct correlation functions arising from the two calculations are plotted. They differ markedly. As show below, a convenient theory of the wall - solution interface uses moments of these functions as input into the calculation of the structure and properties of the inhomogeneous system. Not shown in Figures 1 and 2 are the predictions of the well-known HNC approximation, which yields unacceptable CF1 pair correlation functions.
14,4,
VI.
Theory
for the
Interface
To predict the normalized probability g~i(z) for finding an oxygen (i = O) or hydrogen (i = H) atom of a water molecule at a distance z perpendicular to the planar interface ('wall'), we use the 'singlet' theory version of the Ornstein-Zernike equation due to Henderson, Abraham and Barker a3-3S and Percus. 36 We have also used this level of theory to study soft-core primitive model electrolytes next to charged interfaces. 3r The input to the calculation is: (i) the structure of the bulk model water, obtained above, contained in the direct correlation functions cij(r) for all distinct pairs of species, ij = OO, OH and HH, separated by a distance r; (ii) the bulk water density pB and temperature T; (iii) the short-range potential energy between the wall and the oxygen and hydrogen atoms, r i = O, H; and (iv) the surface potential r The normalized density profile g~i(z) is related to the absolute density profile p~i(z) via pwi(z)/p~ - g~i(z) - [ 1 + h~i(z)] , (6) where h~i(z) is the total wall-species correlation function. At fixed surface potential r the total correlation function and the surface charge density a are obtained from the equations ln[1 + h~i(z)] = - f ~ r
+ 2r ~_, pj
~qir
and
j=O,H
zfO)(z) - f(i])(z)
Loo
o
=
E qJp~ dthw,(t),
-
(8)
j=O,H
where e - 4tee0, e0 is the permittivity of free space, and e = 1 is the 'relative permittivity'. The bulk quantities f[~')(z) - f ~ drr"ci~(r ) are the moments of the short-ranged bulk direct correlation functions, c~j(r) = cij(r) + ~qiqj/(er). The mean electrostatic potential, experienced by a test charge at a distance z from the surface, is obtained from r
= (4r/e) ~
j=O,H
qj p3B
L dt(z-t)hwj(t).
(9)
Differentiating ~p(z) with respect to z, and using the charge neutrality constraint of Equation (8), yields at z = 0 de(z)
dz
I
z=o
_
4__~_~~
e j=O,H
qJ pB
/o=
dt t hwj (t) _
4na ~
(10)
Thus the surface charge density is proportional to the negative slope of the mean electrostatic potential at z - 0. In fact, the initial shape of the mean electrostatic potential is necessarily linear with slope - 4 r a / e .
145
The potential energy between the charged, planar wall and the oxygen and hydrogen species may be divided into two parts: a short-range contribution r due to any specific interactions between the oxygen and hydrogen species with the interface (including softbetween the charged wall and core repulsions); and the Coulombic interactions r the partially charged oxygen and hydrogen species. The total potential energy u~i(z) , may be expressed as a sum of these two contributions u~i(z) = r + r where (I)~i(z) = - q i E z and E = 4 r a / e is the electric field strength at the wall. The Coulombic interactions are specified implicitly in Equation (7) through the value of the surface potential. The properties of our charged, planar interface are determined solely by the short-range potentials r and r and the surface potential Co. For our choice of the short-range interactions, at zero surface potential, by construction the wall shows no preference for either oxygen or hydrogen. The oxygen and hydrogen atoms of the water molecules may approach the interface equally. We observe that hydrogen prefers to be slightly closer to the wall than oxygen. Thus water molecules in the contact layer are oriented predominantly with both hydrogens directed towards the interface. The effect of the inhomogeneity created by the wall extends approximately 15/~ into the bulk of the liquid, equivalent to about four layers of water molecules, after which the water attains it bulk properties, as the normalized profiles approach unity asymptotically. An interesting feature of the normalized wall-oxygen profile is that in the region between approximately 2.4 and 3.7/~ from the wall there is very low probability for finding an oxygen atom. Similar behavior is not observed in gwn(Z). Therefore it is unlikely to find another oxygen atom near that of the one in the contact layer of water, due to both the favorable electrostatic interaction between hydrogen and oxygen and the unfavorable oxygen-oxygen interaction. Thus oxygen is largely excluded from this region. Outside the contact layer some structure continues in both profiles, but it is less well defined. The decaying oscillations in the oxygen and hydrogen profiles are approximately in phase outside the region within 5/~ from the wall. A useful way to see this fact is to plot the charge profile q(z) -- --qOP~[gwg(Z) -- gwO(Z)] shown in Figure 3. For this choice of wall-water potentials, the oxygen centres of the water molecules sit in layers at approximately 1.6 /~ and 4.3 /~ from the wall, with high probability of hydrogen species located at certain distances. As the surface charge on the wall is made increasingly negative, the positive end of the water molecule- the hydrogen - is attracted to the wall, and the oxygen end is repelled, as expected. The calculations performed to date suggest that (i) the details of the short-range wall-water potentials dominate the average orientation of the contact layer of water (in combination with the strong water-water and charged wall-water interactions); and (ii) beyond the first layer the structure is insensitive to these details. The layered solvent structure extends approximately 15/~ into the bulk of the liquid, representing about 4 layers of water molecules, a distance remarkably similar to that found in earlier simulations of an ice / water interface. 3s The potential of zero charge is calculated 7 to be -32 inV. The differential capacitance can be calculated, as discussed by Booth et al. 7 Our calculations are being extended to to incorporate the 'pair' theory. 34'39 Higher order correlation functions, such as the inhomogeneous pair correlation function g~ij(R; zl, z2)
146
q(z)
s
-10 -15
2
,
6_ Z
8
,0
/A
Figure 3" For the case of a wall with surface potentials: -1000,-750, -500, -250, 0, 250, 500, 600. 750 and 1000 mV the charge profile q(z) = --qOP~[g~g(Z) -- gwo(Z)] in units of 109 C cm -3 as a function of distance z from the wall. which appear in the pair theory, should provide a more detailed description of this region, particularly in terms of additional information on orientational correlations. Ions will be added to the water to yield a fully molecular description of the 'double layer:. A c k n o w l e d g e m e n t s : This research is supported by the Australian Research Council (ARC) (Grant No. A29530010). I thank Dr. D.-M. Duh, M. Booth and A. Eaton.
References 1p. Debye and E. Hfickel, Phys. Z. 24, 305 (1923). 2G. Gouy, J. Phys. (Paris) 9, 457 (1910). aD. L. Chapman, Phil. Mag. 25, 475 (1913). 40. Stern, Z. Elektro. Chem. 30, 508 (1924). 5B. Derjaguin and L. Landau, Acta Physiochem. 14, 633 (1941). 6E. J. W. Verwey and J. T. G. Overbeek, Theory of the Stability of L yophobic Colloids (Elsevier, Amsterdam, 1948). 7M. J. Booth. D.-M. Duh. and A. D. J. Havmer, J. Chem. Phys. 101. 7925 (1994). 8M. Heyrovsk~ and S. Vavricka, J. Electroanal. Chem. 353, 335 (1993). ~M. Hevrovsk~ and F. Pllcciarelli, J. Heyrovsk~ Memorial Congress on Polarography II. 65 (I 980). tow. Schmickler, D. Henderson, and O. R. Melroy, Chem. Phys. Letts. 216, 424 (1993). 11j. D. Porter and A. S. Zinn, J. "Phys. Chem. 97, 1190 (1993).
147
I2M. Vossen and F. Forstmann, Journal of Chemical Physics 101, 2379 (1994). I3A. Nyberg, D. Smith, L. Zhang, and A. Haymet, J. Chem. Phys. (1994), submitted. I4A. Nyberg and A. D. J. Haymet, in Structure and Reactivity in Aqueous Solution, edited by D. Trular and C. Kramer (American Chemical Society, New York, 1994). 15F. H. Stillinger and A. Rahman, J. Chem. Phys. 68(2), 666 (1978). 16D.-M. Duh, D. N. Perera, and A. D. J. Haymet, J. Chem. Phys. 102, 3736 (1995). IVT. A. Andrea, W. C. Swope, and H. C. Andersen, J. Chem. Phys. 79, 4576 (1983). ISD. E. Smith and A. D. J. Haymet, J. Chem. Phys. 96, 8450 (1992). tgj. W. Halley, J. R. Rustad, and A. Rahman, J. Chem. Phys 98, 4110 (1993). soW. Blokzijl and J. B. F. N. Engberts, Angew. Chem. Int. Ed. Engl. 32, 154 (1993). 21D. Smith, L. Zhang, and A. Haymet, J. Am. Chem. Soc. 114, 5875 (1992). 22D. Smith and A. Haymet, J. Chem. Phys. 98, 6445 (1993). 23A. Haymet, Annals of the New York Academy of Sciences 715, 146 (1994). 24B. Guillot, 12th International Conference on the properties of Water and Steam (1995). 25S. Cui and J. G. Harris, 12th International Conference on the properties of Water and Steam in press (1995). 26T. Ichiye and A. D. J. Haymet, J. Chem. Phys. 89, 4315 (1988). 27T. Ichiye and A. D. J. Haymet, J. Chem. Phys. 93, 8954 (1990). 2SD.-M. Duh and A. D. J. Haymet, J. Chem. Phys. 97, 7716 (1992). 29R. O. Watts, Statistical Mechanics (The Chemical Society, London, 1973), Vol. 1. 3~ A. Thuraisingham and H. L. Friedman, J. Chem. Phys. 78, 5772 (1983). 31A. K. Soper, Chem. Phys. 107, 61 (1986). 32A. D. Trokhymchuk, M. F. Holovko, E. Spohr, and K. Heinzinger, Mol. Phys 77, 903 (1992). 33D. Henderson, F. F. Abraham, and J. A. Barker, Mol. Phys. 31, 1291 (1976).
34D. Henderson and M. Plischke, J. Phys. Chem. 92, 7177 (1988). 35j. Quintana, D. Henderson, and A. D. J. Haymet, J. Chem. Phys. 98, 1486 (1993). 36j. K. Percus, J. Stat. Phys. 15, 1772 (1976). 37M. J. Booth, A. Eaton, and A. D. J. Haymet, J. Chem. Phys. 103(1), in press (1995). 38B. B. Laird and A. D. J. Haymet, Chemical Reviews 92, 1819 (1992). 39R. Kjellander and S. Mar~elja, Chem. Phys. Lett. 127, 402 (1986).
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journal o f
~OLECULAR
LIQUIDS ELSEVIER
Journal of MolecularLiquids,65166(1995) 149-155
Effect of Solvent, Temperature, and Pressure on Hydrogen Bonding and Reorientation of Water Molecules Masaru Nakahara and Chihiro Wakai Institute for Chemical Research, Kyoto University, Uji-city, Kyoto 611, Japan Abstract On the basis of the study of the solvent, temperature, and pressure effects, we show how the NMR rotational correlation times %21~for a heavy water molecule in neat liquid and organic solvents arc correlated with the strength of solute-solvent interactions, in particular, H bonds. At room temperature (30 "C), the correlation time is 2.1 ps in the random H-bond network in heavy water, whereas it is as small as 0.1 ps in such an apolar, hydrophobic solvent as carbon tetrachloride because of the absence of the H bonds between water molecules. Pressure distorts H bonds and accelerates the orientational motion of water molecules in neat liquid. Firm evidence is collected for the limitations of the Stokes-Einstein-Debye (SED) law in solution. I. I N T R O D U C T I O N Water is one of the most important H-bonding substances for the living. For a better understanding of reaction dynamics in aqueous systems, it becomes more important to investigate how the water dynamics are controlled by intermolecular interactions (density) as well as kinetic energies (temperature). Anomalous properties of water originate from attractive anisotropic interactions called H bonds, in sharp contrast to simple liquids whose equilibrium structure and transport properties are considered to be dominated by repulsive interactions; recall the van der Waals picture and the SED continuum model or its modifications through boundary conditions. There remains unsolved a nontrivial problem as to how the dynamics of water is controlled by the fluctuating H bonding structures. Here, we examine how the rotational correlation times for a water molecule in neat liquid and solutions are influenced by the H bonding between water molecules and between a solute water and organic solvent molecules. It is interesting to see how water responds to perturbations provided more or less selectively by the potential and momentum terms in the Hamiltonian through pressure, temperature, and solvent. In view of the anisotropic nature of H bonds, the rotational properties are of greater interest than those dominated by the radial distribution function. Nuclear magnetic resonance (NMR) spectroscopy has become one of the most powerful tools for studying the rotational dynamics as well as structure of water and aqueous solution under such extreme conditions as high and low temperatures and high pressures, t,2 Our newly developed multi-purpose NMR spectrometer has been successfully applied to the dynamical study of solutions under such extreme conditions as low and high temperatures, 3-5 high pressures, 6 and low concentration. 3-5,7 We have examined the role of H bonds in determining single-body water rotation dynamics in water and aqueous solution by changing or breaking H bonds through temperattm~, pressure, and organic solvents. II. E X P E R I M E N T A L 0167-7322/95/$09.50 91995ElsevierScienceB.V.All rightsreserved. SSDi 0167-7322(95) 0(}849-7
15o
To carry out high temperature experiment on water with a commercially available standard NMR probe (JEOL), we need to use a capillary in a normal glass sample tube (Wilmad) of 5-10 mm diameter. When a long glass sample tube with a normal size is used, the sample liquid contained vaporizes from a heated portion of liquid, condenses near the top of the tube, and falls down to the original liquid. The circulation prevents the efficient and homogeneous heating of the whole sample. A short capillary immersed in a temperature-transmitting medium (Demnum S-65, Daikin) can be employed at high (superheating) and low (supercooling) temperatures. In the case of dilute aqueous solution, the integrated capillary method5 is useful for avoiding the reduction of the signal intensity due to a low concentration. Glass capillaries can stand relatively high pressures; Yamada's method.l The high resolution can be attained by the capillary method at the expense of the sensitivity. We employed the high-sensitivity, high-resolution, high pressure NMR probe where a sample tube of 8 mm diameter can be introduced into a high pressure vessel made of a titanium alloy.2,6 The spin-lattice relaxation times T1 arc measured by the inversion-recovery method. By the capillary method, we have measured the spin-lattice relaxation times T 1 for heavy water (D20) over a wide range of temperature. The results are in good agreement with those given by Hindman and co-workers s,9 within the experimental uncertainties; our and their uncertainties arc +1% and +7%, r e s t i v e l y . The uncertainty of temperature is i-0.1 "C in the present work. IlL REDUCTION OF Tl TO ~2R The spin-lattice relaxation times T 1 due to the quadrupole interaction mechanism can be convened to the rotational correlation times X2R in a simple manner. In the extreme narrowing limit attained by rapid molecular rotational motions, the spin-lattice relaxation rate 1/T t for the 2I-1nucleus with the spin I--1 is expressed by T1-
2
1 +
X2R,
(1)
where e2qQ~ is the quadrupole coupling constant (QCC), ~ is the asymmetric factor for the electric field gradient, and x2a is the correlation time (the area of the time integral) of the time autocorrelation function of the second-order rotation of a bond or principal axis specified by q.6 In the application of Eq. (1) to water, assumptions are required for evaluating ~, QCC, and x2R. Now we can take QCC as 256 kHz and neglect the small ~ value according to a recent quantum mechanical and molecular dynamic study on liquid heavy water between 260 and 359 K. 10 In organic solvents, a gas-phase value of 308 kHz is used because of very weak H bonds between a water molecule and solvent molecules.4 Furthermore, we assume isotropic rotation Table 1. The rotational correlation times X2Rfor pure 1:)20 (99.8% D) T('C) -24 -22 -20 -18 -16 - 14 -12 -10 -5 0
x~8 (ps) 23.7 19.9 16.9 14.5 12.7 11.2 9.91 8.79 6.84 5.43
T('C) 1 5 10 15 20 25 30 35 40 45
X~l~(ps) 5.26 4.49 3.76 3.20 2.78 2.37 2.08 1.86 1.66 1.47
T('C) 50 55 60 65 70 75 80 100 105 110
x~8 (ps) 1.34 1.23 1.10 1.02 0.944 0.881 0.802 0.606 0.566 0.539
T('C) 115 120 125 130 135 140 145 150 155 160
x~8 (ps) 0.507 0.483 0.456 0.433 0.416 0.393 0.377 0.360 0.344 0.329
151
of a water molecule in the neat liquid and solution for the reasons described elsewhere. 6 In this case, the water rotational relaxation is described in terms of a single correlation time x2s. The value of ~2Rfor pure D20 has been determined over a wider temperature range than before, and the results are summarized in Table 1. IV. T E M P E R A T U R E EFFECT ON 't2R FOR
D20
The water molecule is characterized by the extremely small moments of inertia, the relatively small mass, and the large charge separation causing H bonds. The moments of inertia are so small that equilibrium angular velocities of water molecules distribute over a wide range according to the Maxwell-Boltzmann law. To what extent are the molecular characteristics reflected on the rotational correlation time? Is the H bonding between water molecules more impcmant than these molecular characteristics? To answer these questions, we plot the logarithmic values of x2s for pure D20 in Table 1 and those in the previous world on solutions in various organic solvents against inverse temperature in Figure 1. First, compare the X2Rvalues at 30 "C to see how the correlation time depends on the I
,,,
I
,,
I
,,,,
I
,,,
,
I m
l
41~ (CH3)2CO
j 2-1
A = ClIa
/
x C6H6 N
. f ~
. . P
cc14
1
Jo
~
,
-1
-2 I
I
2.5
3.0
i
3.5 1/T (10 -3 K -1)
Figure 1. Plots of In ~2R against 1/T for D20 in various solvents.
I
I
4.0
4.5
152
molecular environments of a water molecule. This is in the sequence, CC14 (0.10, 0.85) < C6I-I6 (0.22, 0.56) < CHC13 (0.23, 0.51) <
(2)
CH3CN (0.63, 0.32) < (CH3)2CO (0.83, 0.29) < D20 (2.1, 0.97),
where the x and y values in (x, y) denote the rotational correlation time (ps) and the solvent bulk viscosity (cP), respectively. The value of x2R (D20/D20) is much larger than that expected for the small-size molecule due to the presence of strong H (D) bonds. The very small value of 0.097 ps for D20/CC14 is attributed to the breakdown of the H bonds between water molecules due to the mixing with the hydrophobic solvent. Though the rotational correlation time for a solitary water molecule in the hydrophobic environment is small, it cannot be described by a free rotor: In fact, it has a small but non-zero activation energy barrier (--9 kJ mol-l). The rotational correlation time o f - 1 0 0 fs can be regarded as a lower limit in solution at room temperature in view of the exceptionally small moments of inertia for the water molecule. So far, we don't know whether the single-molecule orientation dynamics of water in the hydrophobic environment, which can exist locally as an active lubricating ~ i e s in membranes and protein solutions,3 is affected by quantum effect or not. The correlation time i n ~ s with increasing solute-solvent interactions but not with increasing solvent viscosity. Surprisingly, thus, the classical hydrodynamic concept is invalid. In fact, a linear relationship has been found between the correlation times and D or H chemical shifts which can be taken as a measure of the strength of the H-bonding interaction between water and solvent. In the solvent- and pressure-variable studies (see below), the SED model fails in predicting the rotational correlation time sequence. When we attempt to modify the hydrodynamic law by using a fractional power 0t and a constant A as
z2R = A(WT) cz,
(3)
the exponent becomes negative. This is a clear indication of the limitations of the continuum hydrodynamic model or its modified version which neglects the details of solute-solvent interactions in the short range. Second, note marked differences in the slope between the plots in Figure 1. The Arrhenius plots are linear for a solitary water molecule in the organic solvents, whereas the plot for neat liquid water is nonlinear as known for other dynamical properties in water. The Arrhenius activation energies at room temperature are in the sequence, CC14 (8.6) < C6H6 (9.8) _--CHCI3 (9.3) < CH3CN (10) < (CH3)2CO (11) < D20 (19), (4) where the values in the parentheses are the activation energies in kJ mo1-1 at room temperature. The sequence (2) agrees with the sequence (4), which indicates the important role of the solutesolvent interaction potential energies. V. PRESSURE EFFECT ON ~2R FOR
D20
Table 2. Pressure dependence of the rotational correlation time x2n for pure D 2 0 at different temperatm~es p(MPa) 0.1 I00 200 300
10 "C ~2R (ps) 11 (cP) 3.76 1.67 3.06 1.57 2.79 1.61 2.78 1.70
~2R (ps) 2.08 1.87 1.79 1.73
30 "C 11 (cP) 0.973 0.974 1.01 1.07
50
~2R (ps) i.34 1.24 1.19 1.17
"C 11 (cP) 0.652 0.676 0.710 0.753
153
As shown in Table 2, the rotational correlation time 't:2Rfor a heavy water in neat liquid decreases with increasing pressure in the pressure range studied at all temperatures. 6 This has been found by Jonas and co-workers. 11 At pressures lower than 100 MPa, the pressure effect on z2R and 11 (viscosity) are in parallel with a negative pressure coefficient, whereas at higher pressures, the correlation time continues to decrease despite the increase in viscosity with increasing pressure. It is to be noted that the rotational correlation time is much more sensitive to the H bonding state than the hydrodynamic property (11). The anomalous negative pressure coefficients for these dynamical properties are explained in terms of the pressure-induced bending or weakening of H bonds. The modified hydrodynamic expression, Eq. (3) accidentally appears to be valid at lower pressures with a positive value of ~ whereas it needs a negative value at higher pressures. The limitation of the continuum model is disclosed by the isothermal compression not at lower pressures but at higher pressures. The limitation of the
I
I
I
I
I
I
I
I
m
m
Q
I
in bulk in (0H3)200 in CHzCN in CHCI3
Z
inC6H 6
t,i
in CCI 4
& m
m
.,..,
..,.,
r~
& e,i m
m
0
=m,
m
I
0
I
1
I
2
i
3
I
4
I
5
I
6
I
7
n / T (10 .5 P/K) Figure 2. Plots of %2R for a heavy water molecule in different solvents against WT.
154
simple hydrodynamic model is unveiled by the mechanical rather than thermal response of the dynamical quantity; this is because the compression effect is associated exclusively with the intermolecular interaction potential effect in contrast to the thermal effect. Thus, pressure is unique and indispensable for testing the reliability of theoretical models or approximations. Recently, we have observed a similar inapplicability of the SED law to a much larger molecule (benzene) in water in a high pressure NMR relaxation measurement.l~ The initial compression does not decrease but increases the rotational correlation time for a benzene molecule in water at 30 "C. In other words, the viscosity exponent defined by Eq. (3) is negative at lower pressures. The exponent is turned over to a large positive at higher pressures. VL DEPENDENCE OF x2R ON TI/T
To test the reliability of the hydrodynamic variable W'I', we plots the rotational correlation times for a heavy water molecule in various solvents4 against ~/T over a wide range of temperature in Figure 2. The plots are approximately linear and expressed by ~
= ~2Ro + B(T~r),
(5)
where '~2R0 is the value in the limit of ~/T -+ 0. What is the most important is the splitting of the straight lines. The simple SED law predicts that the intercept ~2R0 is zero and that the slope value B, which is associated with the molecular volume characteristic of the solute, is independent of solvent (polarity). Neither of these are satisfied. The modified SED law, Eq. (5), is somewhat meaningful only in the sense that the slope is almost correctly in the sequence CCLs (0.022) < Cd-16 (0.11) = CHCI3 (0.12) < CH3CN (0.65) < D20 (0.66) < (CH3)2CO (1.2),
(6)
where the parenthesized values are the slopes in 10-1 ps p-I K. This corresponds to Eq. (4) except for water and acetone. Thus we can regard the slope B in Eq. (S) as an approximate measure of the strength of dynamical solute-solvent interactions. However, this modification of the SED law still causes such a problem that the intercept introduced for a fixed solute (water) is not common in the family of solvents studied. This indicates the limitations of the simple hydrodynamic scaling law despite the constancy of the slope; the constancy tells us that we are not allowed to change boundary conditions over the range of temperature studied here. The SED law has been modified into the two forms, and they are popular in the pressure- and temperature-variable studies, respectively, as mentioned in a previous paper. 6 Our recent pressure and temperature studyl3 on the rotational correlation time of a heavy water molecule in the organic solvents studied here indicates a serious shortcoming of Eq. (3). No molecular kinetic effect is taken into account in the hydrodynamic approach mentioned above, and one may expect that the kinetic collision effect plays a key role in controlling molecular rotational frictions under consideration. The Dcbye small-step diffusion model has been extended by Gordon 13 and developed later by others. 14.15 The rotational correlation times for infinitely dilute solutions of water in organic solvents have been calculated as a function of the collision frequency j3 using M- and J-diffusion models. 13 The predicted and observed values are summarized for comparison in Table 3. The extended diffusion models give considerably smaller values at atmospheric and high pressures. The pressure effect on the water rotational correlation times is not explained by the random isolated binary collision models; the predicted pressure effect is smaller, in particular in the apolar solvent. Thus, the extended diffusion theory, which originates from the isolated binary collision model in the gas phase, is inapplicable to the condensed phase because of the neglect of the intermolecular potential energy effect in the short range. For an interpretation of molecular rotations in
155
Table 3. Comparison of '~2R values calculated using extended-diffusion models (J- and Mdiffusion) with the values measured for D20 molecules in CH3CN, CHCI 3, and C6H6 at atmospheric (0.1 MPa) and high (300 MPa) pressures. II
Solvent CH3CN
CHC13
C6H6
II
p (MPa) 0.1 300
[~ (ps-1). 14.0 24.8
'172R(M) (fs) 90 133 (1.48) a
"I72R(J) (fs) 63 72 (1.13) a
"U2R(obs) (fs) 599 973 (1.62) j
0.1 300
9.2 13.1
73 88
62 63
227 324
(1.20) a
(1.02) a
(1.43) a
71 79
62 62
(1.11) a
(1.01) a
184 220 (1.20) a
0.1 90
8.7 10.7
aparenthesized values are the ratios of the ~
values to the atmospheric ones.
solutions in terms of attractive solute-solvent interactions, we hope molecular theories are developed. References
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
J. Jonas, Ed., High Pressure NMR (Springer-Verlag, Berlin, 1991). M. Nakahara, Rev. High Presure. Sci. Teehnol. (in Japanese) 1, 195 (1992). M. Nakahara and C. Wakai, Chem. Lett., 809 (1992). M. Nakahara and C. Wakai, J. Chem. Phys., 97, 4413 (1992). M. Nakahara, C. Wakai, and N. Matubayasi, J. Phys. Chem., 99 1377 (1995). C. Wakai and M. Nakahara, J. Chem. Phys., 100, 8347 (1994). M. Nakahara and K. Emi, J. Chem. Phys., 99, 5418 (1993). J.C. Hindman, A. J. Zielen, A. Svirmickas, and M. Wood, J. Chem. Phys., 54, 621 (1972). J.C. Hindman and A. Svirmickas, J. Phys. Chem., 77, 2487 (1973). R. Eggenberger, S. Gerber, H. Huber, D. Searles, and M. Welker, J. Chem. Phys., 97, 5898 (1992). J. Jonas, T. DeFries, and D. J. Wilbur, J. Chem. Phys., 65, 582 (1976). C. Wakai and M. Nakahara, in preparation. C. Wakai and M. Nakahara, J. Chem. Phys., (1995), in press. R.G. Gordon, J. Chem. Phys., 44, 1830 (1966). (a)R. E. D. McClung, J. Chem. Phys. 51, 3842 (1969); (b)R. E. D. McClung, ibid. 54, 3248 (1971); (c)R. E. D. McClung, ibid. 55, 3459 (1971); (d)R. E. D. McClung, ibid. 57, 5478 (1972). W.A. Steele, Adv. Chem. Phys. 34, 1 (1976).
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journal of
MOLECULAR
LIQUIDS ELSEVIER
Journal of MolecularLiquids,65/66 (1995) 157-168
Collective V i b r a t i o n a l D y n a m i c s of Simple Liquids Michael B u c h n e r a n d T h o m a s Dorfm/iller D e p a r t m e n t of C h e m i s t r y , U n i v e r s i t y of Bielefeld, 33615 Bielefeld, G e r m a n y Abstract
Instantaneous normal mode investigations of molecular liquids are presented for coupled translational and rotational coordinates as well as for vibrational coordinates. Illustrative examples are chosen so as to demonstrate the influence of structural and dynamical properties on the normal mode behaviour. It is demonstrated that among the structural properties characterising a liquid system the normal mode spectrum is uniquely well suited for being related to the dynamics. The theoretical derivation of the normal mode spectrum for molecular systems starting from configurational distributions is shown to be an ill-behaving problem. Normal mode spectral shapes for strongly infrared active vibrational modes are presented and correlated with transition dipole moment and liquid structure. Finally a perturbational procedure is proposed to quantify the motional narrowing effect of molecular dynamics on inhomogeneously broadened vibrational bands. I.
Introduction
Normal mode approaches enjoy growing interest for describing dynamical properties of liquids. 1'2 Many of their applications are found in numerical experiments where the eigenequation V(R) U(R)
=
U(R) A(R)
(1)
involving the matrix of second derivatives of potential energy V(R) is set up and solved for the set of eigenvalues As and eigenvectors IJ~. V(R) depends on the system configuration R and implicitly on time. Consequently in liquids the modes are momentary variables. These instantaneous normal modes show a number of useful properties. 9They provide a correct short-time description of dynamical processes. They therefore contrast to expansions around inherent structures 3 which are the configurations corresponding to local potential minima obtained by quenching instantaneous system configurations. The results of these two approaches show marked differences as has been demonstrated for the case of low frequency Raman spectra of water. 4's 0167-7322/95/$09.50
91995ElsevierScienceB.V.All fightsreserved.
SSDI 0167-7322(95)00905-!
158
9They can be used to model the long-time behaviour of a dynamical system. This can be done either by approximating the relevant correlation functions using only the stable portion of the frequency spectrum 6'7 or by using the unstable lobe of the spectrum for drawing (empirical) relationships to the diffusion constant, s-l~ 9They are collective coordinates and, therefore, provide a way to describe phenomena which may show high collectivity like solvation 11 or vibrational relaxation 12 processes. Collective velocity correlation functions, e.g., have been discussed as indicators for the onset of glass formation. 13'14 9For some simple systems normal mode spectra have been calculated theoretically by exploiting the isomorphism to a random walk 15 or by applying liquid state theory. 16 They may be regarded as a stepping stone between (structural) theory and experiment. 9Techniques of crystallography can be used to analyse and characterise the modes in terms of their wave vector lr and their spatial extension. 9In a wider perspective analogies have been pointed out with respect to the calculation of quantum time correlation functions, is In this contribution the concept of instantaneous normal modes is applied to three molecular liquid systems, carbon monoxide at 80 K and carbon disulphide at ambient temperature and two different densities. The systems were chosen in this way because pairs of them show similarities either in structural or in dynamical properties. The systems and their simulation are described in the following section. Subsequently two different types of molecular coordinates are used as input to normal mode calculations, external, i.e. translational and rotational coordinates, and internal, i.e. vibrational coordinates of strongly infrared active modes, respectively. The normal mode spectra are related quantitatively to molecular properties and to those of liquid structure and dynamics. Finally a synthesis of both calculations is attempted on qualitative grounds aiming at the treatment of vibrational dephasing effects. II.
S i m u l a t i o n a n d c h a r a c t e r i s a t i o n of t h e s y s t e m s i n v e s t i g a t e d
The configurations needed as input for the normal mode calculations were obtained by molecular dynamics simulation. For the purpose of the present investigation the simulation of rigid molecular models would be sufficient because only translational and rotational coordinates enter the normal mode analysis and the molecular vibrations are treated perturbationally within a frozen field approach. Nevertheless fully flexible molecules were assumed and a specialised algorithm for handling their high frequency motion was employed 1~ which allows time steps of 2 and 1 fs for CO and CS2, respectively. Comparing the results of vibrational normal mode analysis with the spectra obtained by molecular dynamics simulation allows assessing the effects of motional narrowing and anharmonicity. The anharmonic intramolecular 20 ' 21 as well as the intermolecular potentials for both CO and CS2 were taken from literature. The intermolecular interactions of CO are modelled by seven sites interacting isotropically. 22 Among the intermolecular potentials available for CS2 one including electrostatic interactions via partial chargcs 23
159
was preferred over simple and more common Lennard-Jones models 7'24,2s because electrostatics may be an efficient pathway for vibrational relaxation through dipolar or quadrupolar interactions. 26 Consequently, while there may be effective Lennard-Jones parameters for modelling translational and rotational dynamics as have been used in temperature dependent normal mode calculations for CS2, 7 the careful inclusion of electrostatic interactions appears to be indispensable when it comes to vibrational dynamics. CO possesses a small dipole moment of about 0.1 Debye (0.3336-10 -3~ Cm) pointing from carbon to oxygen. Since its vibrational mode is strongly active in the infrared a simple point charge model would be unable to reproduce its large transition dipole moment. In the present simulations an atomic charge model including charge fluxes was employed which was originally devised for describing gas phase infrared intensities. 27 It includes equilibrium charges z~ on sites a and their first-order fluxes Oz~/O~k upon elongation along a vibrational coordinate ~k. In these terms the change of dipole moment with ~k becomes
(2) under the condition of charge neutrality. Oza ~--~ -~k = 0
(3)
ot
If there are more charge parameters than conditions the extra constraint of minimal charge flux may be used by introducing Lagrange multipliers for conditions (2) and
(3).
(Ozo
" Min
(4)
Similar considerations apply for modelling dipole and quadrupole changes upon vibrational motion in CS2. In Table 1 the vibrational parameters entering equation (2) are presented. Table 2 summarises the systems simulated. In column three the experimental pressures corresponding to the simulated densities can be found. The simulated pressure coincides quite well with the experimental value in the case of CO but is as much as l kbar smaller for the CS2 systems, a fact which has not been reported. 23 Figure 1 shows some structural and dynamical properties of the systems under investigation. CO has a short bond length and a very small dipole moment. The directional forces on the molecular axis are, therefore, small. Accordingly, its intermolecular structure may be compared to that of an atomic liquid since it shows a simple and regular pattern in the radial distribution function g000(r) and only small orientational correlation as documented by the spherical harmonic expansion factor 2s g200(r). CS2 on the other hand is quite elongated and exhibits distinct features in its translational and orientational structure. The function g200(r) reveals that at distances below 4.5 /~ there is a relatively high particle density perpendicular to the bond vector of a reference molecule while above 4.5/~, there is a preferred positioning in the direction of the bond vector. As expected these structural features are more pronounced when the pressure is increased from 1 bar to 10 kbar.
160
molecule
Oz/O~
mode
[103 C m -1 kg -1/2]
Co
/]
9.331
2169.1
CS~
-0.847 -8.484 -6.328 3.164
672.6 /]2a,b
398.2
/]3
1558.8
o o o •
(C) (corn) (0) (C) (S)
(c) (s) (c) (S)
0r/0~ [1012 kg-1/2 ] -5.355 6z
0 4.017 ~z
0 -+-3.069 &z
(C)
ou/o~ [10-6Ckg-X/2] -0.98 ~
(com) (0)
(c) (S)
-3.88o ~,~ (c) 0.728 e~,u (S)
0.21 ~,y
-6.500 6, 1.220 ~
2.97 ~
(C) (S)
Table 1: Parameters entering equation (2) for the molecules and their electrostatic interaction sites (corn: centre-of-mass), k is the harmonic frequency, and 0~/0~ is the resulting change of dipole moment. The molecules are oriented along the z-axis. ~,,u,, are unit vectors along the axes of the molecular frame.
molecule CO CS2 CS2
temperature
pressure
[K]
[b~r]
80 293 293
1 1 10 4
density p [mole/l/ 28.56 16.60 21.88
N
Ibo,
256 256 256
[/~1 24.6 29.5 26.9
Table 2: Parameters of the systems investigated. The experimental pressure corresponding to the simulated thermodynamic state is specified. N is the number of particles simulated and lbo, is the length of the simulation box. Regarding the single-particle dynamical properties shown in Figure 1, the CS2 molecules in the high pressure system exhibit strongly bound motion as is revealed by the pronounced minimum in the angular velocity correlation function. On the other hand, the systems of CO and CS2 at ambient pressure show some resemblance in their single-particle dynamics. In both cases the reorientational correlation time is about 0.10 ps. The orientational correlation functions, however, reveal large differences in time scale. III. I n s t a n t a n e o u s n o r m a l m o d e s p e c t r a for t r a n s l a t i o n a l and r o t a t i o n a l coordinates For the normal mode analysis of translational and rotational coordinates of linear molecules6 the potential energy is expanded to second order around the configuration ON,r ri is the centre-of-mass position, Oi and R0 with R - {rl. O l , r Oi are the orientational variables of molecule i. In terms of the 5N-dimensional
161
2.5
,
j
1.00
9
2.0
'
I
,
~
0.75 0.50
1.5
I
((,Mi(O)'(.a)i(t))
((.~i(0)'~i(0))
-
0.25
1.0
0.00
0.5
......
.~ ....
-0.25 I
[
0.6
0.8
0.0 0.5
I
0.0 1.0
1
0.2
0.4
10
0.8 0.0
0.6 0.4
-0.5
0.2 -1.0
I
0.0 0
5
r [It]
10
15
0
-..
....... 4
I
I
4
6
8
2
t [ps]
10
F i g u r e 1" Structural (left column) and dynamical (right column) properties of the systems investigated. Upper left: centre-of-mass radial pair distribution function g000(r); lower left: spherical harmonic expansion coefficient g20o(r); upper right: angular velocity correlation function; lower right: orientational correlation function. Dotted lines: CO, 80 K, 1 bar; thin lines: CS2, 293 K, 1 bar; thick lines: CS2, 293 K, 10 kbar. mass-weighted configurational vector Z -
zi,
=
with components
{z1,...,ZN}
M1/2xi, M1/2yi, M1/2zi I1/20i, I1/2(sin0i0)r
( # = 1,2,3) (# = 4,5)
(5)
with molecular mass M and moment of inertia I, the Hamiltonian becomes H
=
1 1_~;.2~ + V(Ro) - F(Ro). (Zt - Zo) + ~ ( Z t -
Zo). V~*t(Ro) (9Z t - Zo)
(6)
with the mass-weighted force vector and dynamical matrix
[::i, =
OV Ozi,
Vi=,t~,-
and
02V Ozit~Ozj u
(7)
.~xt2 and the eigenvector V ~:~t is diagonalized numerically yielding the eigenvalues ~,,~ matrix U~t with entries ll ~t Using the eigenvalues the normalised density of states Dext(w) is defined.
D~(~)
=
/
~-g E ~ [~ - ~ ~=~(rto)] c~-'l
/
(8) Ro
162
svstem
CO, 1bar CS2, 1bar CS2, 10kbar
ft~s JU
frot JU
0.191 0.165 0.057
fext JU
0.083 0.104 0.041
0.274 0.270 0.098
L'~,S
.-=trs
=--:trs
I"gJU
-~sot
~-~uOt
47.0 55.6 82.1
21.3 18.0 20.4
67.6 62.1 90.1
32.5 23.3 24.9
=--~. ot /.---:trs
s - u•ot /rI~,----:tr u
1.44 1.12 1.10
1.53 1.30 1.22
~Js
/~.M$
Table 3: Moment analysis of the partial densities of states of external coordinates. f,, is the fraction of unstable modes and ~,/= the mean frequency for stable/unstable portions of the spectrum for the coordinates indicated, rot, trs, and ezt stand for rotational, translational, and external (rotational + translational) coordinates, respectively. Furthermore, for each eigenvector rotation p~ot can be singled out. N
o
U~ t
the contributions of translation
3
N
uo.,,~
=
,~.d
i=1 ~=1
= ~
p~s and
5
Z
uo.,~,
(9)
i=1 tt=4
Using these contributions partial densities of states may be defined. =
.ext g-~ ~ ~ [~- ~o (nol] P~,t r s / r o t (P.o)
~=1
(I0) R.o
It has been shown 6 that in terms of DtrS(w) and Dr~ short-time estimates for the single-particle translational and rotational time correlation functions can be formulated as plain Fourier transforms. For translational motion this estimate reproduces the second and fourth moment.
ksT f DtrS/rot(~)
(11)
By cumulant expansion 29 orientational correlation functions can be deduced from the exact or approximate correlation functions of the rotational velocity w~. (Pl [fii(0). fii(t)])
Pl(x)
,~ exp [ - ~1
l(,, + 1) ./o'(t - T/r
,,.,,(,//d~]
(12)
is the /th Legendre polynomial of its argument x. For the calculation of the normal mode spectra external and internal coordinates were assumed to be dynamically decoupled. Translational and rotational coordinates were extracted from the trajectories while all vibrational coordinates were set to zero. Dynamical matrices were set up for 50 configurations generated by molecular dynamics simulation. Long-range Coulombic interactions were treated using the Ewald summation technique. In Figure 2 the instantaneous normal mode spectra are depicted while in Table 3 some of their integral properties are compiled. The frequencies shown on the positive branch of the abscissa correspond to stable motion along principal directions of the potential surface with positive curvature. Following the usual convention the (imaginary) frequency distributions of the unstable
163
0.012
1
CS2, 293K, 1 bar -
0.010
1
CS2, 293 K, 10 kbar-
0.008 0.006 0.004 0.002 !
0.000
lOOi 0.012
I
0.010 -
I
CO, 80K, lbar
0.008 0.006 0.004 O.002 0.000
-
lOOi
o
lOO [cm-1]
'
200
A
o
lOO ~ [era-1]
200
F i g u r e 2" Instantaneous normal mode spectra. For simplicity the unstable lobes corresponding to imaginary frequencies are plotted along the negative branch of the abscissa. Dotted lines: rotational contribution D"~ thin lines: translational contribution Dt"~; thick lines: total spectra D ~=t.
modes are plotted along the negative branch of the abscissa. Along with the total spectra D e:~t the partial densities of states D t'~ and D T M are shown. Especially the stable lobes of D e*t for CO and CS2 as well as the total fraction of unstable modes f~,t are remarkably similar. Closer inspection, however, shows that the rotational contribution in CO is shifted to higher frequencies with respect to translation in CO as well as to rotation in CS2 (1 bar). This is certainly due to the small moment of inertia in CO which enters via the mass-weighted coordinates (equation (5)). Comparable observations regarding the effect of inertial factors have been made for other non-associated liquids in the case of noble gas mixtures. 3~ As expected the instantaneous normal mode spectrum for the high pressure sample of CS2 looks quite different with a strongly diminished portion of unstable modes. Furthermore, the stronger intermolecular interactions are reflected by an overall stretching of the partial spectra to higher real as well as imaginary frequencies (c.f. last two columns in Table 3). The choice of thermodynamic states for the low pressure systems of CO and CS2 and the resulting similarities in the density of states is, of course, fortuitous but corresponds to the similarity of velocity correlation functions in Figure 1. On the other hand, by comparing Figures 1 and 2 it is obvious that this similarity is not reflected by the distance dependent distribution functions. It has been stated before that the densities of states are structural properties but with uniquely good relation to dynamics. 31 This also sheds some light on the possibility of straight forwardly extending the theoretical derivation of mode spectra from the atomic case to that of simple molecular liquids, at least when using centre-of-mass and orientational distributions as input. Small changes in the radial distributions can cause large changes in the spectra and vice versa, resulting in an unstable process of transformation of structural information. In Figure 3 two normalised velocity correlation functions are shown together with
164
their normal mode approximations calculated using equation (11). Two extreme cases of predominantly unstable or stable motion were chosen. For both functions using only the stable modes gives a rather good approximation of the overall behaviour while including unstable modes introduces cosh-terms in equation (11) and diverging behaviour of the approximate correlation functions. The stable mode approximation, however, does not contain information about the diffusion coefficient, an integral content of velocity correlation functions. IV.
N o r m a l m o d e s for v i b r a t i o n a l c o o r d i n a t e s
Interactions between transition dipole moments of molecular vibrational coordinates are an important source of inhomogeneous line broadening. Coupled vibrations of strongly infrared active modes have been simulated in clusters, glasses, and in molten salts, a2-a4 The molecules exhibit efficient intermolecular energy transfer by resonance coupling. For polar liquids the frequency difference between the isotropic and anisotropic components of the Raman spectrum is a prominent effect as which can be understood as the result of a selection rule for collective coordinates. The transience of collective modes of vibrational coordinates can be directly observed by time resolved experiments as in picosecond holeburning, a6 Furthermore, vibrational energy relaxation has been discussed in terms of normal modes ar serving as a heat bath which makes up for small energy differences in stepwise relaxation processes. In the harmonic approximation the general form of the Hamiltonian for vibrational coordinates sci is H
=
1 9
1
{t~.~ + {t~.
vvib .t~
(13)
Oscillators on different molecules are coupled through their transition dipoles. V;~!b
1.00
0.75
OJc]
[ 1
\/ '.~
0.50 ~
i = j
T
.
with
#j
1
Tij
-
l
3
rijrij
1
(~,(0).v,(t)) (v, (0).v, (0))
_
CO, 80K, 1 bar
-
1
--
(14)
1
(w,(o).w,(t))
_
(w~(0).w~(0)) CS2, 293 K, 10 kbar -
0.25 0.00 : -0.25 0.0
"-"" . ....._~..._~...7..i_~. .... ~ . . ~ I
0.2
0.4
t[ps]
0.6
0.8
1.0
0.0
M YI 0.2
1
1
1
0.4
0.6
0.8
tips]
F i g u r e 3- Normalised single particle velocity correlation functions Left mode approximations ~#t"s(t)/zht"s(O) and ~ " ~ 1 7 6 mass velocity correlation function for CO, 80 K, 1 bar; right velocity correlation function for CS2, 293 K, 10 kbar. Thick dynamics result; dotted lines: normal mode result excluding thin lines: normal mode result including unstable modes.
1.0
and their normal panel: centre-ofpanel: rotational lines: Molecular unstable modes:
~Awvib, V/Awvib~/((Oo~):p)
system
~-~,,b [cm -1]
[cm -1]
(relative t o C())
CO, 1 bar
2169.11
1.86
1.000
CS2, 1 bar CS2, 10kbar
1558.78 1558.75
14.16 17.66
1.426 1.349
Table 4: Moment analysis of the vibrational density of states. The last column contains the reduced root mean squared frequency shift relative to its value for CO. I
I
0.1 x D~ib(~ ) CO, 80K, lbar
0.03
~!
DVib(5)
-
1600
1625
0.02
0.01 0.00 2165.0
2167.5
J
J~
2170.0 [cm-1]
2172.5
2175.0 1525
1550
1575 [cm-1]
F i g u r e 4: Normal mode spectra D'ib(~) for internal (vibrational) coordinates of samples of CO at 80 K (left) and CS2 at 293 K (right) and l bar (thin line) and 10 kbar (thick line), respectively. The dotted line indicates the harmonic frequency u0 (in wave numbers) of the undisturbed stretching mode. Here w0 is the fundamental frequency of an isolated oscillator, fii is the unit vector of molecule i along its vibrational coordinate which in the present case coincides with the molecular axis. The dipole tensor Tij depends on the molecular distance vector rij. As opposed to the molecular dynamics calculations the change of dipole moment with vibrational elongation is modelled by a single parameter 0#/0~ thereby ignoring changes of higher moments and electric anharmonicity. As in the case of external coordinates the eigenequation is solved for a set of eigenvalues w~rib 2 and eigenvectors ..~ll rib, this time for 4000 configurations. Analogous to equation (8) the vibrational densities of states DVib(w) are calculated for the cases of CO and CS2 at two different pressures, respectively. They are shown in Figure 4 while Table 4 contains information about their positions and widths. Since under the similarity transformation the trace of V~ib is preserved and because the relative band widths are small, the spectra are centred around the natural oscillator frequency w0. The widths of the distributions depend on the strength of coupling between two oscillators which, apart from factors of positional and orientational correlation, scale linearly with transition dipole (0#/0~) 2 and density p (equation (14)). CO is regarded as a reference system because of its simple translational and minor orientational structure. The last column in Table 4 expresses the influence of liquid structure on band width. All densities of states
166 1590 1588 1586 1584 1582 1580
0.0
2173.0
0.1
0.2 0.3 t [PSI
0.4
0.5
2172.8 2172.6 2172.4 2172.2 2172.0 0.0
0.1
0.2 O.3 t[ps]
0.4
0.5
F i g u r e 5: Variation of normal mode frequencies with time. The frequencies are specified in wavenumbers [cm-a]. Lower left panel: CO, 80 K, 1 bar; upper left panel: CS2, 293 K, 1 bar; upper right panel: CS2, 293 K, 10 kbar.
show a pronounced asymmetry with approximately triangular distributions stretched to higher frequencies.
V. Combining the approaches: broadened bands
Phase
relaxation in inhomogeneously
The band widths measured at normal pressure for CO in the isotropic Raman 3s and for CS2 in the infrared spectrum 39 are about 0.5 cm -1 and 20 cm -1, respectively. The CO band is narrower than expected by a factor of 4. Furthermore, the band of the v3 mode of CS2 is much more asymmetric than the CO vibrational band. A possible explanation comes to mind when looking at the orientational correlation functions in Figure 1. Because of the small moment of inertia the orientational correlation in CO drops at a much faster pace than in the CS2 systems in spite of the lower temperature. This rapid motion will quickly modulate the momentary coupling between pairs of oscillators, change the normal mode frequencies, and lead to motional narrowing of the corresponding band. Some qualitative insight might be gained if one tries to depict the time evolution of the normal mode frequencies as in Figure 5. In this Figure some representative ranges of frequencies are chosen from the vibrational densities of states in Figures 4. At each time step for which the eigenvalues were calculated (At = 4 fs for CO and At = 5 fs for CS2) the set of frequencies is represented by a column of bullets. This way an evolution of mode frequency emerges. The upper left panel for CS2 at low pressure is closest to the picture ideally expected for a low temperature crystalline solid, i.e. a laminar stream of undisturbed straight lines. Only now and then frequencies of different modes approach but avoid each other as is typical for an eigensystem (of vibrational modes) parametrically dependent on external parameters (the liquid configuration).
167
Small distances as in high pressure CS2 as well as rapid rotation as in CO can stir up this stream efficiently. How can the normal modes help us further to get some qualitative or even quantitative understanding of this kind of phase relaxation process? The flow of information might be illustrated using the depictions at hand. Explicit simulations or possibly structural information (Figure 1) and theoretical approaches enable us to calculate instantaneous normal mode spectra for translational and rotational coordinates (Figure 2). Approximate velocity correlation functions are derived (Figure 3) and using cumulant expansion (equation (12)) one arrives at orientational correlation functions. These can be used to investigate how the vibrational density of states (Figure 4) will be disturbed. To low order the shift of the vibrational eigenvalues can be expressed in terms of the perturbation of internal by external coordinates. [
vib~
~ 9
OVi~b Ij~t
IJvi.b "
--,J
~=t
Aq~:t ii,,ib
p = l
To get in touch with equation (14) the effect of external normal modes in terms of molecular coordinates. This way the effect of external inhomogeneously broadened bands can be evaluated. The contribution vs. rotational motion to motional narrowing in vibrational bands can
is reformulated coordinates on of translational be quantified.
Acknowledgement This work was supported by the Deutsche Forschungsgemeinschaft in the "Schwerpunkt" Schnelle molekulare Prozesse in Fliissigkeiten". M.B. wishes to thank Richard Stratt for numerous discussions and for sharing his results prior to publication.
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journal of MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 169-176
Mechanisms of Dephasing in Femtosecond Time-resolved Coherent Raman Scattering from Molecules in Liquids Y. Fujimura Department of Chemistry, Faculty of Science, Tohoku University, Sendal, 980-77, Japan Abstract The mechanisms of ultrafast dephasings appearing in the time-profile of coherent Raman scattering from molecules in liquids are examined. Two mechanisms are taken into account. One is associated with intermolecular interactions between molecules at different sites through heat bath modes. The other is associated with the interference between rovibrational Raman transitions. In the former mechanism, the time profile of the coherent Raman scattering is characterized by an exponential decay. This decay constant is called the intermolecular dephasing constant. For the latter mechanism, the time profile has an inhomogeneous nature and is characterized by a Gaussian form. The structure of the intermolecular dephasing constant is derived within the Markov approximation. We describe the relationship between the intermolecular dephasing constant and the intramolecular dephasing constant associated with spontaneous Raman transitions. Finally we analyze the ultrafast, sub-picosecond dephasings observed in coherent anti-Stokes Raman profiles of neat benzene liquid in terms of the rovibrational interference mechanism. I. Introduction In recent years, considerable attention has been giyen to ultrafast dynamics associated with molecular motions in liquids. Femtosecond time-resolved, nonlinear coherent Raman scattering spectroscopy is one method for clarifying the origin of such ultra_fast dynamics. 1-5 Ultrafast dynamics in liquids is reflected in the profiles of the time-resolved, nonlinear coherent Raman scattering spectroscopy. This method includes time-resolved coherent anti-Stokes Raman scattering(CARS), degenerate four-wave mixing, impulsive stimulated Raman scattering (ISRS),etc. When considering the molecular dephasing processes, differences in the creation of the molecular coherence between coherent and incoherent(spontaneous) Raman scattering should be noted. In the time-resolved, nonlinear, coherent scattering processes, each molecule in the liquid receives the photon wave vector transiently through spatial coherence of the photon field. Consequently, a transient intermolecular coherence is created. The transient macroscopic nonlinear polarization of the molecular ensemble is therefore given as the summation of the induced nonlinear polarization on each molecule. The output intensity of the coherent Raman scattering experiments is generally proportional to an ensemble average of the absolute square of the nonlinear polarization. 6,7 In the absence of long-range spatial correlation between molecules, the intensity can be expressed in terms of the absolute square of the ensemble-averaged nonlinear polarization. On the other hand, in incoherent Raman scattering processes, intramolecular rovibrational coherence of each molecule within the laser irradiation spot is created. The decay of rovibrational coherence is expressed by the ordinary intramolecular dephasing constant. This consists of the population decay constant and the pure dephasing constant. The former originates from an inelastic interaction between the molecule of interest and heat bath modes. The latter originates from an elastic interaction between the molecule and the heat bath modes. Thus, intermolecular phase dynamics from liquids is directly reflected in time-resolved, nonlinear, coherent Raman scattering profiles though the information content in both coherent and incoherent Raman scattering experiments is in principle the same. 8 It should be noted that the laser pump pulses create sets of intermolecular rovibrational coherence. For example, quantum beats appear in time profile of the CARS spectrum from a molecular mixture. 3, 9, 10 The beat frequency created is equal to the difference between frequencies of vibrational modes of molecules in the mixture. When many sets of intermolecular 0167-7322/95/$09.50
91995 Elsevier Science B.V. All rights reserved.
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170 rovibrational coherence are created, it can be expected that ultmfast dephasing will result from the interference between the sets of the rovibrational coherence. This phenomenon is one of the main origins of the inhomogeneous Gaussian component in the time profile. The time profile also has a photon polarization dependence. This polarization-dependent time profile has been observed in the sub-picosecond time regime in nonresonant CARS from benzene in liquid. 11 The present study uses a quantum statistical method to show how the time-profile of CARS from molecules in liquids is associated with the intermolecular dephasing mechanisms. Liouville space Feynman diagrams are used for the development of relevant transitions associated with pairs of molecules. 9-10 The structure of the intermolecular dephasing constant is clarified to identify the difference between the intermolecular dephasing constant and intramolecular dephasing constant in Sec.II. Section III presents the rovibrational interference mechanism. The sub-picosecond decay observed in CARS profile of neat benzene is explained in terms of this mechanism. II. Intensity of time-resolved C A R S Consider a time-resolved, electronically nonresonant CARS spectrum from a molecular liquid. In the CARS process, the laser pump pulses create a linear combination (that is the intermolecular rovibrational coherence) of Raman active rovibrational transitions between molecules at position rt and rm in the mixture. This stimulated Raman scattering process is carried out by two-coincident laser pulses(I, II) with central frequencies(wave vectors) mi(kI) and r By applying the third pulse with ml(km) to the liquid after time delay x, the time dependence of the intermolecular rovibrational coherence is detected through the measurement of the intensity of the scattered photon with ks.
The total Hamiltonian I2I is given as fl = I:t~ + fl~ + f l ~ ,
(l)
where I2IMBis the molecular Hamiltonian including the optically inactive heat bath modes, I2IRis the radiation field Hamiltonian, and I~IMR is the molecule-radiation-interaction Hamiltonian. Using the dipole approximation, and making use of the rotating-wave approximation, the interaction Hamiltonian is given as I21MR= Z
V e,
(2)
/=1 t,,,,q 9 ,v,,~ with
,
'
(-)
Ve = - 2.,2.., Mba Tba El b
exp(ikl "re)
a
, '9 (+) exp(-ikll - Z Z M~b Tcb E.ll c
r9e ) -
~~c M~cr Tdc •ll
exp(--ikiil ~ re).
b
(3)
g
The term M~tx denotes the matrix element of the electric-dipole moment between states ~t and ~, of the gth molecule. The operator ~
e
( -- I~t><~,l) is the transition operator from the ~, to the l.t
states. E denote the radiation field operators. The density operator for the total system ( ~ T )
is
satisfied with the equations of motion,
ih "~" [5T (t) = [V(t ),[5T ( t ) ] , (4) where the density operator with the overtilde denotes that it represents the density operator in the interaction picture. The dipole transition operator in the interaction picture is expressed as Q ( t ) = e x p I i ( I 2 I ~ + I21R)t/h] I2tMRexp[--i(121.~m+ I21p)t/h 1= f i g=l
Qe(t).
(5)
171 In the perturbative density-matrix method, the total density operator for the CARS process is given by expanding iST(t) in terms of the photon number (4) as (4) i OT (t)= { dt(8)} [V(tl), [V(t2) . . . . . IV(t7), [V(ts), OT(-~) .... ]]]1,
where { i d t ( 8 ' } = ( ~ ' ) S i d t i i d t 2 . . . .=.oo
-.oo
idt7idt
==.oo
8,
(6)
(7)
-.-oo
and OT (_oo) is the density operator for the total system at initial time t ---~-o.. This density operator is assumed to be given by
(8)
OT(__oo) = OM (-oo)O B (-oo) ~ R (...oo) , where 0M ("**) is the density operator of the molecules, 0B (--~
that of the heat bath and
(4) OR (.-o.) is that of the radiation field. The density operator for the scattering field 0s (t) is (4) derived from the total-system-density operator 0T (t) by tracing out the variables from both the molecules and the heat bath modes. The intensity of time-resolved CARS is defined in terms of the population of the scattered photons (W('r, Ak, Ws, t)) as 12 I(,, Ak) =
dadWs
Atdt~'~W(,, Ak, Ws, t), withAk = ki - kII + kiii - ks
(9)
The CARS intensity is expressed as I('t, Ak)--
ZZ Itm('l:)exp(iAk~ 9 (10) t=l m=l (t ~ m) After averaging Eq.(10) over the heat bath variables with the aid of the Liouville space Feynman diagram (shown in Fig. 1) within the factorization approximation, the main term for the CARS process is given as
oj" dt XXxX;X;XX;X a
b
c
d
a'
b'
c'
d'
At g m g g g m m m m X Pa Pa' Mdc Mcb Mba Ma'd' Md'c' Mc'b' Mb'a' t t ,,(-) (t2)EllI)(t3)) ,,(+ ,,(+ '(t4)Eli ,,(- )0.6)) X I _.,dt 1 " ' " I _oodt7 ( EIII ( EII ,, ((+) gm tm X ( EI )(ts) EI (t7)) Gaa:a,b,(t6-tT) Gba:a,b,(t5-t6) tm gm em em X Gba. a'c ,(t4-ts) Gca: a'c' (t3-t4) Gca: a' a' (t2-t3) Gda- a'd ,(tl-t2).
(11)
The term Pa denotes the population in the initial state of the molecule at site g. The term ..((El)(ti)
.. (+ EI )(tj))denotes ~ the incident radiation field correlation function.
The term
172 G ~ . es(ti-tj) represents the time-development matrix element of the intermolecular coherence between B ~ k and e ~ ~i from time ti to time ti in the presence of the molecule-heat bath interaction. This matrix element is d e f i n ~ by tm Gj~t: es(ti-tj)= [((I.tl~, X~I ~,m(ti-tj)I l.tl~, X~ ))]av. (12) where (3em(ti-tj), the propagator of the molecular pair whose constituents are located at sites g t~m
and m, is given as
gm
(3tm(ti-tj) = exp[-i f_.MB(ti-tj)],
trn
with I'~MB= [HMB, ]lb.
(13)
~al g site
t5 l
a
t4
~
t7
'"f
1.6
P
t
:
t
ti "la>
/ 7 m site
t2
.s/
1,3 t la'~'
Figure 1. A Liouville space Feynman diagram for CARS process. Each wavy line denotes the molecule-radiation field interaction. This diagram represents the time evolution of the intermolecular coherence between two molecules at g and m sites in the CARS process. Figure 1 shows the Liouville space Feynman diagram representing the time development of the intermolecular coherence between molecules at site g and m in the time-resolved CARS process. The two upper lines show the time development of intramolecular coherence of molecules at g. The two lower lines show the time development of intramolecular coherence of molecules at m, respectively. The upper and lower lines are connected through the incident laser fields(I, II, and III) indicated by wavy lines. In the ordinary pump-probe type time-resolved CARS experiments, the time duration from t3 to t4 corresponds to the pump-probe time x, and the intermolecular coherence time for pair molecules. In other words, it is the time dependence of the coherence between the Raman transition a <---~cat site g and a' <---~c'at site m which are detected in the CARS time profile.
Equation of motion for the intermolecular coherence The equation of motion for the intermolecular coherence between two Raman transitions gm i ~---}jand i' <---~j' at site g and m ( Gji:i'j ,(t) ) is" d tm ~m }'m gm dt Gji:i'j'(t)---[il~ji:i'J'+ Fji'i'j':ji'i'j'(t)] Gji:i'j'(t) + ZZZZ (pp 'qq' )m (ji'ij ')
tm gm Fj,,i,j,: pq. p,q,(t)l Gpq: p,q,(t),
(14)
173 tm
t
t
m
m
where tiSji:i'j' = tiSj-l~i -(l~]j'-1~Si' ) is the frequency difference between two Raman transitions. tm
The real part of l"ji.i,j,:ji.i,j,(t) represents the intermolecular dephasing rate between two transitions, i <--,j and i' <--r The imaginary part represents the time-dependent frequency shift of tm
the transition. The real part of the term l"ji.i'j': pq.p,q,(t) represents the magnitude of the transfer between two intermolecular coherences, i<--->j and i' <--~j', and p~-cq and p'+->q' from two molecules. Structure o f the intermolecular vibrational d e p h a s i n g constant Within the Mark~v approximation, the intermolecular dephasing constant associated with the coherence decay between i+-->j and i' <--->j' vibrational Raman transitions is expressed in terms of the population decay constant from each state and intermolecular pure dephasing constant as 13 tm 1 e e t g t~rn(d) r'ji, i,j,:ji.i,j,-~(r'ii:ii + Fjj:jj + Fi,i,:i,i , + I"j.j,:j,j,)+ I"ji.i,j':ji.i'j', (15)
e where l"ii:ii is the population decay constant from state i of the molecule at site ~, and trn(d)
F ji.i'j':ji.i'j' the intermolecular pure dephasing constant. The population decay constant is defined as t 2g (16) F i i : i i - h2 Z X X P i # 42 ~(~tJmfB, iis) 9 iB fs met and the intermolecular pure dephasing constant is defined as s 2~ r j i , i'j':ji,i'j' -- h2
i~B '~afB piBl(<j, iBI lqeBIj, fB>-')
- ( < j ' , iBI lqmBIj', f B > - < i ', iBI lqmBli ', fB>)P 5(C0iB.fB) 9 (17) Here lqeB(lqmB) represents the interaction Hamiltonian between the molecule at site e(m) and the heat bath mode. From the general structure of the intermolecular pure dephasing constant, the intermolecular dephasing constant for the transition i <--->jat e and i ,-->j at site m is expressed as ~m 1 t r'ji, ij:ji,ij -- 2 ( F i i : i i
~ m m + F j j . j j + Fii:ii + F j j : j j )
,
(18)
that is, there are no intramolecular and intermolecular pure dephasing terms. For Raman transitions i <--->jat site g and i <--->k(j ,:k)at site m, the dephasing constant is expressed as gm 1 e t m m gm(d) m ._ l"ji, ik'ji,ik -- 2(l"ii:ii + 1-'jj:jj + l"ii:ii + Fkk:kk) + l"ji, ik:ji.ik. (19) When the heat bath modes are different between sites g and m, these molecules are uncorrelated. In this case, the intermolecular dephasing constant consists of the sum of the intramolecular dephasing constants at the two sites, em e m Fji, ik: ji,ik = l"ji: ji + I ~ ki: ki. (20) When the molecules at different sites are correlated through heat bath modes, the intermolecular dephasing constant is not given by the sum of the intramolecular dephasing constants at the two sites. In this case it is given as em e m 1-'ji, ik:ji,ik = F j i : j i + I " k i : k i -
gm l'~ji, ik:ji,ik,
(21)
174 where I~ji,ik:ji,ik denotes the interference term between the two Raman transitions due to a common heat bath mode. At this poi.nt, a brief discussion about the site-dependence of the intermolecular dephasing constants is worthwhile. The population decay constant, and intramolecular dephasing constant are independent of the sites of the molecules in ordinary systems such as neat liquids. The magnitude of the intermolecular dephasing constant associated with i(--)j and i(---)k Raman transitions at different sites, on the other hand, do depend on the site when molecules are correlated through a common heat bath mode. The site-dependent intermolecular dephasing constant can be evaluated by using a multi-spherical-layer model. 9, l0 In this model the intermolecular dephasing constant is assumed to have the same magnitude within the same spherical layer. Under this assumption, the summation over sites ~ and m in Eq.(10) are carried out easily. 111. Intermolecular rovibratlonai interference m e c h a n i s m Okamoto and Yoshihara I l have reported a decay component of 0.39 ps in addition to a slow component of 2.4 ps in time-resolved CARS profiles of neat benzene at room temperature. This sub-pi~nd decay component is not attributed to the intermolecular vibrational dephasing effects discussed in the preceding section or to the effects of the pulsed lasers used. This is because the decay component has a Gaussian form and a photon polarization dependence. The slow component of 2.4 ps is identified with reorientational motions perpendicular to the C6 symmetry axis. Ultrafast intermolecular dynamics of benzene liquids has also been investigated by several groups using different nonlinear-coherent techniques. 14-18 The 0.39 ps dephasing time is explained in terms of interferences between rovibrational Raman transition amplitudes at different sites in this paper. To explain the origin of the 0.39 ps decay component on the basis of a rovibrational interference mechanism, we assume that the radius of the spot irradiated by the pump laser is much larger than an average correlation length between benzene molecules interacting through the heat bath modes. Further, the observation for the time-resolved CARS is performed in the direction of Ak=0. In this case, the molecules interact independently with the heat bath modes. The CARS intensity is proportional to the absolute square of the third-order nonlinear polarization, averaged over the heat bath mode and the distribution of rotational energy in the initial state. The rotational motion is treated quantum mechanically. For simplicity, rotational motions were treated within a free rotor approximation. This approximation is crude one in treating molecular librations in condensed phases and is only valid for high temperature limits in which the thermal energy is larger than an averaged librational bamer height. Photon polarization- dependence is taken into account by assuming that the geometry of benzene is that of a symmetric-top. Under the polarization condition [Z, X, Z, X] only anisotropic Raman transition contributes to the CARS process. The lime profile IZXZX(%)is given as 19 ZXZX
I
(x) oc I ~zxzx(%)12,
(22)
where
(2) X RJKN exp{-iB[N(N+ l)+2NJ]x}exp[-(i~fi+7 fi)x].
(23)
In Eq.(23), J and K are the rotational quantum numbers for the total angular momentum and the component projected to the molecular principal axis, ~ is the anisotropy of the Raman polarization tensor, pj is the thermal rotational distribution function in the initial state, and N specifies the selection rule of the rotational Raman transitions. ~(J)=0 if J<0 and ~(J)=l for all other values of J,
175
(2) R~r~N = [2(J+N)+ 1]
/
J
J +N
-K
K
1/2
c0il is the vibrational Raman transition frequency, and ~fi is the vibrational dephasing constant. Within the high temperature approximation, Eq.(22) is expressed as
I
('r -- ( - - - ~ ) 2 e x p [ - 2 7 f i l : ] {5+2exp(-8BkBTX2)}.
(24)
Here T denotes temperature, and ks the Boltzmann factor. This expression indicates that there are two decay components in the time-resolved CARS profile: one is Gaussian and the other is exponential. The Gaussian decay component originates from dephasings of the off-diagonal terms of the CARS intensity, Eq.(22).
,
2
.
:
_
o
C ,.-4 rr
-1
0.O
2.0
,~.o e.o Time/ps
a.o
1o.o
Figure 2. CARS time profile calculated under the polarization condition [Z, X, Z, X]. The Stokes and probe polarizations are perpendicular and parallel to the pump one, respectively. The figure inset represents the time profile observed. 1l Figure 2 shows the CARS profile calculated using Eq.(24). The parameter set used is 8 T ~ 2 = 0 . 4 0 with units of t~ called the mean polarizability, t.0fi=992cm -1, B--O.19Ocm -l, kBT=206cm -1, and ~/fi=3.0cm -1. The value of ~fi was determined from the spontaneous Raman spectra of neat benzene. 20 The figure inset shows the observed profile for comparison. Note the excellent correspondence between the calculated CARS profile and the observed one in the sub-picosecond regime. This implies that the sub-picosecond decay component is due to the quantum mechanical interference between the anisotropic rotational Raman transition amplitudes whose transition frequencies are located within the band width of the pump pulse. The physical origin of this interference is a coherent excitation of rotational Raman transitions of molecules by the pump pulse. In other words, a rotational wavepacket in molecular ensemble is created as a result of the coherent excitation, the dissipation of the rotational wavepacket is reflected in CARS time profiles. Classically, an initial angular phase relation between rotors created by the pump pulse disappears as time develops because each rotor has its own mean angular frequency. McMorrow and Lotshaw 17 and Lotshaw et al. 18 have recently given the most detailed analyses of benzene liquid by using femtosecond Fourier-transform of the optical-heterodyne detected optical Kerr effect. They have observed two relaxation constants of 80 fs and 0.45 ps in addition to that of 2.3 ps. The dephasing time of 0.39 ps discussed in this paper corresponds to that of
176 0.45 ps mentioned above. This relaxation component has been suggested that it is associated with a relaxation of overdamped local collective modes. 17 In conclusion, two types of intermolecular dephasing mechanisms are consistent with ultrashort, time-resolved CARS profiles of molecules in liquids. The mechanisms were discussed in terms of the intermolecular vibrational and rovibrational coherence created in molecules at different sites. The intermolecular vibrational coherence decay mechanism is due to molecular interactions through the heat bath modes. In the ordinary case in which molecules interact independently with the heat bath, the intermolecular vibrational dephasing constant associated with two Raman transitions i,-->j and i4-~k(kg:j) is equal to the sum of the intramolecular dephasing constants relevant to the two transitions. In a correlated system, the magnitude of the intermolecular dephasing constant is less than the sum of the individual intramolecular dephasing constants. In a two-level molecular system, the intermolecular dephasing constant is given by a sum of the populations decay constants of the two states. The other type of the dephasing mechanisms discussed is due to the interferences between rovibrational Raman transitions. The time profile of the CARS spectrum is characterized by a Gaussian form and by a photon polarization dependence. This phenomenon has been observed by Okamoto and Yoshihara 11 in neat benzene and is explained in terms of the rovibrational interference mechanism. Finally, it is interesting to clarify the origin of the rapid decay component of 80 fs of benzene liquid and a relation between collective modes and a creation of a long range librational (rovibrational) coherence, l a, 19
Acknowledgments The author expresses his thank to Professor Yoshihara for his fruitful comments. This work was supported by the Ministry of Education, Science and Culture, Japan(Grant No. 0664(041), and a Grant-in-aid for Scientific Research in the Priority Area of "Non-equilibrium Processes in Solution" (02245203).
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
S. Ruhman, B. Kohler, A.G. Joly, and K. A. Nelson, Chem. Phys. Lett., 141, 16 (198"0 D. McMorrow, W. T. Lotshaw, and G. A. Kenney Wallace, 1 EFF. J. Quantum Electron., 2 4 , 4 4 3 (1988). W. Zinth, R. Leonhardt, W. Holzaptdel, and W. Kaiser, I EFF. J. Quantum Electron., 24,455 (1987). K.A. Nelson and E. P. Ippen, Adv. Chem. Phys., 75, 1 (1988). S. Mukamel, Adv. Chem. Phys. 70, 165 (1988). N. Morita and T. Yajima, Phys. Rev., A30, 2525 (1984). S. Mukamel and E. Hanamura, Phys. Rev., A33, 1099 (1986). R.F. Loring and S. Mukamel, J. Chem. Phys., 83, 2116 (1985). M. Hayashi, Y. Nomura, and Y. Fujimura, J. Chem. Phys., 89, 32 (1988). M. Hayashi and Y. Fujimura, Chem. Phys., 140, 41 (1988). H. Okamoto and K. Yoshihara, J. Opt. Soc. Am., B7, 1702 (1990). M. Hayashi and Y. Fujimura, J. Opt. Soc. Am., 137, 1653 (1990). M. Hayashi, M. Sugawara, and Y. Fujimura, Phys. Rev., A43, 2416 (1991). T. Joo, M. A. Dugan and A. C. Albrecht, Chem. Phys. Lett., 177, 4(1991). T. Hattori and T. Kobayashi, J. Chem. Phys., 94, 3332 (1991). J. S. Friedman and C. Y. She, J. Chem. Phys., 99, 4960 (1993). D. McMorrow and W. T. Lotshaw, Chem. Phys. Lett., 2 0 1 , 3 6 9 (1993). W. T. Lotshaw, P. R. Staver, D. McMorrow, N. Thantu and J. S. Melinger, in Ultrafast Phenomena IX, Springer, p.91 (1994). M. Hayashi, Y. Fujimura, H. Okamoto, and K. Yoshihara, Chem. Phys. Lett., 196, 44 (1992). G.D. Patterson and J. E. Griffiths, J. Chem. Phys., 63, 2406 (1975).
journal of MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65166 (1995) 177-185
Ion Solvation a n d Ion Association Methods.
Studied
by I n f r a r e d
and
Microwave
Josef B a r t h e l Institut ffir Physikalische und T h e o r e t i s c h e C h e m i e d e r U n i v e r s i t g t R e g e n s burg, D - 93040 Regensburg, Germany. Abstract FTIR spectroscopy of electrolytesolutions has been employed to distinguishamong free solvent molecules, solvent molecules bound to ions, and ion aggregates themselves. Furthermore, solvation numbers and association constants have been calculated from quantitative absorption measurements. Microwave spectra confirm the information from FTIR investigations. High frequency permittivity data deduced from MW and IR measurements yield information on the dynamic processes in electrolyte solutions. I. I n t r o d u c t i o n The basic effects responsible for the properties of electrolyte solutions are ion solvation, ion association to ion pairs and higher ion aggregates with and without inclusion of solvent molecules. FTIR (Fourier transform infrared) and MW (microwave) spectra axe a valuable source of information on ion--solvent and ion-ion interactions and yield factual knowledge on the structure and dynamics of electrolyte solutions. The efficiency of these methods is exemplified for solvation in aprotic and protic solvents, hydrophobic solvation, association to charged and neutral ion aggregates, and stability of ion pairs.
II. I n f o r m a t i o n from F T I R s p e c t r a on ion solvation a n d association Solutions of alkali and alkaline earth metal perchlorates in acetonitrile (AN) are particularly suited to exemplify the investigations on ion solvation and ion association based on IR spectra. 1 - 5 The addition of alkali and alkaline earth metal perchlorates to AN causes in the region of the C-N stretching band of AN the appearance of a band produced by acetonitrile molecules in the solvation shells of the cations (fig. 1 for NaCl04 solutions); a band of the originally triply degenerate antisymmetric C1-O stretching vibration splits at increased salt concentration due to ion-aggregate formation (fig. 2, for a solution of LiCl04). 4 The bands (a) and (b) in fig. 2 are assigned to the free anions, band (1) is due to the vibration of a contact ion pair and band (2) to that of a solvent separated ion pair. Alkaline earth metal perchlorate solutions reveal in addition to the free anion bands (a) and (b) three bands due to: [M2+(CIOg)~I o, [M2+CIO;I + and [M2+(AN)CIO~I+,M 2+ = Mg 2+, Ca 2+, Ba 2+. 0167-7322/95/$09.50 91995 Elsevier Science B.V. All rights reserved. SSDi 0167-7322 (95) 00900-0
178
1.08 0.88
1.0 0.5 0 2200
2250
2300
2350
o.os
V cnl -!
1.25
"
5
1.0 0.75
4
0.50
3
0.25
0 1000
I
1
T
\k
2 1100
1050
1150
1200
0
0.25
v
Co
cm-I
tool d m - 3
0.50
Figure 1: IR spectrum of the C-N vibration of AN in AN solutions of NaCl04 (25~
c~ electrolyte concentration. Figure 2: IR spectrum of the CI-O vibration of the anion in AN solutions of LiCl04 (25~ For explanations see the text. Figure 3: Association constants KA(1) and Kc(2) of LiBr solutions in AN (25 ~ and comparison with those from other methods: A conductivity, O high frequency permittivity. Data analysis based on the absorption coefficients of the bands yields solvation numbers for Li+(4),Na+(4),Mg2+(6), Ca2+(6),Ba2+(6) and ClO~(O). Solvation numbers from IR measurements also are known for acetone solutions: Li+(4),Na+(4),Mg2+(6),6 , and for dimethyl sulfoxide solutions: Li+(4), Mg2+(6). 7 It is an interesting feature of the structure of AN solutions that the phase transitions of pure AN, liquid (1) ---, solid (s(a)) at -43, 8~ and solid (s (~)) ---, solid (s (~)) at -53, 6~ to the solid phases (s (~)) with glassy structure and (s (=)) with orthorhombic structure also take place for Li +, Na +, and K + solutions (solvation number 4), whereas the transition s(~) --~ s(~) does not occur for the alkaline earth metal perchlorates (solvation number 6) obviously caused by the incompatibility of the orthorhombic AN lattice with a solvation number of 6. 5
0.75
179
The association constants K~ of 1.1 electrolytes calculated with the help of free ion and ion-pair concentrations, c~ and cxp, deduced from the spectra show the usual concentration dependence relating K~ to the thermodynamic equilibrium constant Ks clp
YlP
]lIP
-
(y~
where y~ and y1~' are the mean activity coefficientsof the free ions and the ion pair. Fig. 3 shows the measured association constants Kc of LiBr solutions in A N (25~ The extrapolated value Ks -- lirn~o Kc -- 175 clrn3 mol -I 4 found in agreement with Ks from conductivity, 166 d m 3 tool-I 8, and M W measurements, 148 d m 3 mol-1. 9 Also shown is the concentration independent thermodynamic association constant Ks obtained from K~ (IR) according to eq. (1) with the help of activity coefficientsknown from vapor pressure measurements on LiBr/AN solutions.10 Also the association constants from FIR measurements on LiCI04 and NaCI04 in A N solutions, K.(LiCI04) = 25.4 dm 3 mo1-1 and K~(NaCI04) = 20.3 dm 3 mo1-1 are found in agreement with those from other methods of determination, conductivity: 23 dm 3 mo1-1 (LiCI04) 11, 21 dm 3 mo1-1 (NaCl04) 12 and MW: 31 dm s mol -~ (NaCl04). 9
HI. High frequency permittivity m e a s u r e m e n t s their solvents
on electrolyte solutions and
High frequency permittivity data are a rich and reliable source of information when the underlying absorption and dispersion spectra cover a sufficientlylarge frequency range. A survey on the method is given in refs. [13, 14]. In our laboratory spectra are now available in the frequency range from about 10 M H z to 90 G H z with the help of a time domain equipment (10 M H z to 4 G H Z ) 15 and a system of transmission lines (I to 90 GHz). 16 A n encompassing collectionof high frequency permittivity data of aqueous and nonaqueous electrolytesolutions and their solvents is given in ref. [17],parts 2 and 2 a. High frequency permittivity of electrolyte solutions is commonly represented with the help of the frequency dependent complex quantity 0(v)
#(.)
= ~(.) + i2~.~ ~ dr
(2)
the so-called generalized permittivity where the real part ~'(v) = ~'(v) represents the frequency dependence of permittivity (i. e. the dispersion curve) and r/"(v) -- ~"(v) + o'/2~rv~o is the loss function (i.. e. the absorption curve) composed of the dielectric loss e"(v) and the ohmic loss o'/2~rVeocaused by ionic conductance (~r, specific conductivity of the electrolyte solution at concentration c; co, permittivity of the vacuum). Eq. 2 neglects the small frequency dependence of conductivity and uses at all frequencies of the program the conductivity ~r measured at low frequencies (v < 10s Hz). For electrolyte solutions of the commonly used solvents the loss by conductance is the determining quantity at frequencies up to about 5 GHz, c. f. ,7" in fig. 5. Nonconducting systems are free of ohmic losses (or = 0), e. g. solvents, and r/'l(v) = ~"(v).
180
30
20
10
0.01
0.1
1
5 lO
~:"
0 10o
M
Figure 4: Dispersion (~') and absorption (e") spectra of AN (1 a, b) and PC (2 a, b) (25 ~ from time (open circles) and frequency (full circles) domain measurements
(25 oc). Figure 4 shows as examples of complete frequency coverage the dispersion and absorption curves of pure propylene carbonate (PC) and acetonitrile (AN). The maximum of g'(v) and the inflexion point of e'(v), situated at the same frequency, indicate the relaxation times r ~ (PC) = 43 ps and r ~ (AN) = 3.2 ps characterizing the response function Figr of orientational polarization P~r = E < fii > / V (fii, dipole moment of particle i; V, volume of the sample)
< P~'Ct)e.(o)> P~/(O)" >
F~" = < P~'(o)-"
f
oF;,. e x p ( - 2 r i v t ) d t
; eCt,) = [~" - eoo] ( - - - ~ )
(3a, b)
which completely determines ~(v) 13, see eq. (3 b); e" = e'(0) is the permittivity at low frequencies (v ~ o), eoo is the permittivity at frequency too (generally in the FIR region) at which polarization p~r is attenuated to zero by the incapability of the dipole molecules to follow the polarity changes of the external field. Aprotic solvents, such as PC or AN, generally reveal a single relaxation process, with a more or less broad relaxation time distribution hiding the particular rotational diffusion processes around the molecular axes. Dimethyl formamide (DMF) represents a case where two of the particular processes are separable on behalf of a sufficiently large difference in relaxation times .18 Protic solvents show two (1120, formamide FA) or three (alcohols ROH, N-methylformamide NMF) structural relaxation processes, each one with a characteristic relaxation time ri. 18,19 Eqs. 3 can be easily extended to this situation. 13,17 The dispersion and absorption curves of the pure solvents undergo drastic changes when an electrolyte is added, the most important being the superposition of conductivity shown in the absorption curve r/"(v) of fig. 5 a and in the Argand diagram 7?" = f(r of fig. 5 b. Reduction of r/"(v) to r is executed with the help of measured static conductivities a. Two relaxation processes are corroborated by two inflexion points of r two maxima
181
of ~"(v) and two regions in the Argand diagram r corresponding to the relaxation of AN molecules and the ion pairs IP of LiC104. The dispersion curve ~'(v) and the Argand diagram ~"(~') of pure acetonitrile are given for comparison (broken lines), cf. fig. 4. For the sake of clearness the absorption curve of pure AN is not shown; it would be very near to the boundary line of the shaded part AN of the absorption curve of the electrolyte solution. 50
4O g*
40 30 30 20 u 0
g' 10
20
10
0 0.1
0.5 1
5 10
50
0
~l~lv g~l
GHz
g
P
Figure 5: (a) Dispersion (e') and absorption (r/" and ~") curves and (b) Argand diagrams (r/" = f ( d ) and d' = f(~')) of a 0.389 M solution of NaCI04 in AN (25 ~ For explanations see the text.
IV. I n f o r m a t i o n on ion-solvent interactions from high f r e q u e n c y p e r m i t tivity s p e c t r a Free ions do not yield relaxation processes because of lacking proper dipole moments, but they orientate their neighbouring solvent molecules by their strong electrical fields against the external field, thus diminuishing the dispersion amplitude (~* -~oo) of the pure solvent to (~ool~(c) -~oo) where g,ol~(c) may be understood as the permittivity of the solvent in the electrolyte solution. The ion pairs in the solution, and also other polar aggregates capable to relax in the external field, contribute their dispersion amplitudes (eoot(C)- ~,o,,(c)) to the total dispersion amplitude (e,oi(C)- eoo), ~,ol(C) being the static permittivity of the electrolyte solution, see fig. 5. Hence the permittivity of the electrolyte solution e,o~ may be larger than that of the pure solvent when ion-association takes place, whereas the permittivity of the solvent e,ol, in the solution always decreases. The nonlinear decrease is commonly fitted to a polynomial of the type e,o1.(c) = e" - ~ c + ~ c " ; n = 1.5 or 2
(4)
182
Dipolar aprotic solvents such as AN and PC are much less affected than the liquids forruing hydrogen bonded chains, e. g. MeOH and NMF. 13 Values for 6, and ~/~ are reported in ref. 21. The dielectric depression, A~,o~, = ~~ - c*, is caused by three effects, a volume effect (r due to the dilution of the polar solvent by the apolar ions (~ ~, 2), irrotational bonding (IB) of solvent molecules to the ions caused by electrostatic interactions, and kinetic depolarization (KD) resulting from the movement of the ion against the direction into which the solvent dipoles are orientated. 22 (r and (IB) are equilibrium properties reflecting the structure of the solution and yielding solvation numbers, (KD) is dynamic in origin. Actually no unifying theory exists incorporating these three effects into the dielectric depression Ae~ in a consistent way and which can be applied to finite electrolyte concentrations. In our experience it is possible to analyse the experimental decrements 8, 13 (5a, b, c) The depolarization factor ~ is estimated with the help of the continum model of Hubbard and Onsager 22 for ion transport under stick or slip hydrodynamic boundary conditions. Recent MD simulations and theoretical considerations suggest that ~,ol~(c) might depend only on equilibrium properties, i. e. ~ = 0. 23 A decision about these contradictory theoretical statements is not possible with the restricted number of the existing experimental data. The separation of the measured effect into ionic contributions requires assumptions on reference ions depending on the nature of the solvent. 9, 24 For acetonitrile solutions, a attern of consistent solvation numbers is obtained at negligible kinetic depolarization for Li+(4),Na+(4),Br-(2),Bu4N+(O),I-(O),CI04(O)in agreement with those from FTIR measurements for Li +, Na + and CI04. Reasonable solvation numbers are found for Na + ions in FA and DMF both for ~ = 0(6) or ~ # 0(4). 25 Independent of the choice of ~, Na + ions in NMF show large solvation numbers (~ 10), just like in methanol solutions 26, 27, indicating that solvent association in chains produces ion solvation beyond the first solvation shell. The solvation numbers of aqueous solutions based on the assumption that the Cl- ion is the reference are in favour of ~ # 0 and slip boundary conditions. 24
~
Aqueous tetra~lkylammonium salts show a peculiar behaviour due to hydrophobic interaction of the cation with water. It is known that Me4N + ions (Me: methyl), as well as the larger alkali metal ions disorganize water structure in their immediate vicinity and yield negative temperature coefiicients of the Walden product, whereas Pr4N + ions and higher homologs behave just in the opposite manner. 28 Figure 6 shows how MW data reflect the "structure making" effect of Btt4NBr by the comparison of aqueous solutions of Et4NCl and an aqueous solution of Bu4NBr. 29 The Argand diagram of the 2.2 M solution of Et4NCl shows three relaxation processes typical for aqueous electrolyte solutions: (1)ion-pair relaxation (r(1) = rxp), (2)low frequency relaxation (r,(o21),,~ 8 ps) of water, (3) high frequency relaxation (r~o3)~~ 1 ps) of water, in contrast to that of the 2 M solution of B u 4 N B r where the relaxation process (2) splits up into two processes. Figure 7 shows the concentration dependence of the
183
~.(2.) and 'ool,,.,.(2b) corresponding relaxation times 'ool,,
50
30
,•eo
40
20
3O s., o 20 , q4
10
~,ee
10 !
10
20
30
p
8
3
40
'
8oll
=-
r
.
2
10
20
It
40
~.~. zo
0
1
2
3
e
tool
elm"s
Figure 6: Argand diagrams of aqueous solutions (25 ~ of (a) tetraethylammonium chloride (2.23 M Et4NCI) and (b) tetrabutylammonium bromide (2.00 M Bu4NBr). For explanations see the text. Figure 7: Frequency dependence of the low frequency relaxation time ',o1~'(2)of water in Et4NCl solutions (curve 3) and splitting of ',o1~-(2)in Bu4NBr solutions into .(2~) '~ol~ (curve ~.(~b) (curve 2), c. f. fig. 6. 1) and '.o.., Beginning at the relaxation time of 8.3 ps of pure water at electrolyte concentration zero the addition of Btt,NBr produces a relaxation process with the expected almost ~(2b) concentration-independent water relaxation time ',ol~ ~ 8.5 ps (bulk water), the other (2~) with a strongly increasing relaxation time r~ol~ indicating a water structure relaxing at much lower frequency, whereas Et4NCI exhibits no split of the water relaxation time .(2) ' solv"
184
V. I n f o r m a t i o n on ion cluster f o r m a t i o n f r o m high f r e q u e n c y p e r m i t t i vity s p e c t r a The ion-pair relaxation process, found at the low frequency side of the spectra is the consequence of the reorientation of ion pairs behaving like dipole molecules. However, the frequency window of ion-pair reorientation generally comprises the frequency range of the kinetic relaxation process of ion-pair formation and dissociation 14 k12 C+ + A-
~
IP
(6)
h, The overall time constant of the orientational relaxation process r ~ and the kinetic process r~i~'~ - [k21 + 2k12(c- cip)] -1 is given by the relation 1
rip
1
1
T~p i" ~IP
1
~r~P "~ ~21 "~" 2 ~12 (C
CIp )
(7)
permitting the determination of k,2 and [(r~)-'-t-k21]from the linear plot ri"~ vs. (c-cap). O n the other hand, the concentration dependent association constant Kc - kl2/k2x can be determined from the dispersion amplitude e~ ~,o~(c) of the ion-pair relaxation process, so that finally r~,k~2 and k21 are known. In ar~tonitrile LiBr (Ka =148 dm3/mol) and B u 4 N B r (Ka = 17 dm3/mol) form contact ion pairs. The association constants are in agreement with those determined with the help of other methods. However, M W measurements reveal a significant difference between the ion-pairs of Li+ and Bu4N + ions in A N due to their stabilities. For B u 4 N B r the rate constants kl2 = 8 9109 d m 3 tool-I s-l and k21 - 0.35 910 z s-l are in the range of diffusion controlled reactions, whereas no kinetic process can be observed for LiBr.9 Ion-pair formation data from high frequency permittivity measurements are also known for aqueous solutions of 1.1, 2.1 and 2.2 electrolytes 30, 31, M e O H 21, FA, N M F , D M F 21,25-and DMSO (dimethyl sulfoxide) 21 solutions. Charged ion pairs [M2+CIO~] + have been investigated by MW methods in aqueous solutions; CdCl2 forms the inner sphere complex [Cd(H20)sCl] + as can be concluded from the effective volume V~ of dipole rotation available from the analysis of the ion-pair relaxation times, eq. (7), in combination with the Stokes-Einstein-Debye equation 31
3 V~ OT ~'iP = ~ r /
(8)
The study of the effective volume of dipole rotation has proved to be a valuable tool for clearing up the structure of ionic aggregates in electrolyte solutions 9, 14,31,32, as well as the investigation of the effective dipole moments of the aggregates revealing the transition from solvent separated to contact ion pairs, e. g. NaCl04 solutions in AN at increasing electrolyte concentration, where LiBr reveals contact ion pairs down to the lowest measured concentrations. 9
18.5 References
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
18 19 20 21 22 23 24 25 26 27
28 29 30 31 32
J . F . Coetzee and W. It. Sharpe, J. Solution Chem. 1, 77 (1972). I.S. Perelygin and M. A. Klimchuk, Russ. J. Phys. Chem. 47, 1138 (1973). D.E. Irish and M. H. Brooker, in It. H. Clark and It. E. Hesters (eds.), Advances in Infraredand Raman Spectroscopy, Vol. 2, Heyden, London 1976, p. 212. J. Barthel and It. Deser, J. Solution Chem., in press. J. Barthel and It. Deser, J. Mol. Liq., in press. I.S. Perelygin and M. A. Klimchuk, Russ. J. Phys. Chem. 51, 838 (1977). J.P. Toth, G. Ritzhaupt and J. P. Devlin, J. Phys. Chem. 85, 1387 (1981). P.G. Glugla, J. H. Byon, and C. A. Eckert, J. Chem. Eng. Data 26, 80 (1981). J. Barthel, M. Kleebauer, and It. Buchner, J. Solution Chem., in press. L. Klein, Diplomarbeit, Regensburg 1988. F. Accascina, G. Pistoia, and S. Schiavo, Itic. Sci. 35, 560 (1966). G. Pistoia, A. M. Polcaro, and S. Schiavo,/tic. Sci. 37, 309 (1967). J. Barthel and 1t. Buclmer, Chem. Soc. Itev. 21,263 (1992). It. Buchner and J. Barthel, J. Mol. Liq., in press. J. Barthel, M. Mfinsterer, and It. Buchner, in preparation. J. Barthel, K. Bachhuber, It. Buchner, H. Hetzenauer, and M. Kleebauer, Ber. Bunsenges. Phys. Chem. 95, 853 (1991). J. Barthel, It. Buchner, and M. Mfinsterer, Electrolyte Data Collection, Parts 2 and 2 a, in: G. Kreysa (ed.) DECHEMA Chemistry Data Series Vol. XII, DECHEMA, will be published 1995. J. Barthel, K. Bachhuber, It. Buchner, J. B. Gill, and M. Kleebauer, Chem. Phys. Lett. 167, 62 (1990). J. Barthel, K. Bachhuber, It. Buchner, and H. Hetzenauer, Chem. Phys. Lett. 165, 369 (1990). J. Barthel, R. Buchner, K. Bachhuber, H. Hetzenauer, M. Kleebauer, and H. Oft maier, Pure Appl. Chem. 62, 2287 (1990). J. Barthel and R. Buchner, Pure Appl. Chem. 63, 1473 (1991). J.B. Hubbard, P. Colonomos, and P. G. Wolynes, J. Chem. Phys. 71, 2652 (1979). A. Chandra, D. Wei, and G. N. Patey, J. Chem. Phys. 98, 4959 (1993). J. Barthel, H. Hetzenauer, and R. Buchner, Ber. Bunsenges. Phys. Chem. 96, 988 (1992). J. Barthel, K. Bachhuber, and It. Buchner, Z. Naturwiss., in press. J. Barthel and It. Buchner, Pure Appl. Chem. 58, 1077 (1986). J.-P. Badiali, H. Cachet, and J.-C. Lestrade, J. Chim. Phys. Physico-Chim. Biol. 25, 1350 (1967). D.F. Evans and T. L. Broadwater, J. Phys. Chem. 72, 1037 (1968). J. Barthel, R. Buchner, M. Mfinsterer, and J. Stauber, 4th EurAsiaConference on Chemical Sciences, Kuala Lumpur 1994. It. Buchner, G. Hefter, and J. Barthel, J. Chem. Soc. Faraday Trans. 90, 2475 (1994). J. Barthel, H. Hetzenauer, and It. Buchner, Ber. Bunsenges. Phys. Chem. 96, 1424 (1992). J. Barthel and M. Kleebauer, J. Solution Chem. 20, 977 (i991).
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journal of
MOLECULAR
LIQUIDS ELSEVIER
Journal of MolecularLiquids.65/66(1995) 187-194
Dynamical Aspects of Water by Low-frequency Raman Scattering Yasunori Tominaga, Yan Wang, Aiko Fujiwara and Kohji Mizoguchi* Department of Physics, Ochanomizu University, Otsuka, Bunkyo.ku, Tokyo 112, Japan *Department of Applied Physics, Osaka University, Yamadaoka, Suita, Osaka 565, Japan Abstract Depolarized Raman spectra below 250 cm "1 in liquid water (H20, D 2 0 and H2180), aqueous electrolyte solutions (LiC1, NaCI, KC1, RbCI, MgCI2 and CaC12), D-glucose solution and ascorbic acid isomerism solutions (L-xylo and D-arabo) were investigated. The Raman spectra below 250 cm "1 in these liquids are interpreted with a superposition of two damped harmonic oscillators and one Debye type relaxation mode (liquid water) or one Cole-Cole type relaxation mode (aqueous solutions). (1) With increasing temperature the half width of half maximum of the central component gl in 1-120 has a small bend at about 20 *C. (2) For aqueous electrolyte solutions, with increasing salt concentration the ratio of relaxation time "r/~rwater shows just the same behavior as the ratio of viscosity 0/r/water. (3) In glucose solution the characteristic frequencies of two oscillator modes do not show any change. This means that the glucose can just embed in the dynamical structure of water. (4) The effect of L-xylo ascorbic acid (Vitamin C) on the dynamical structure of water is different from the effect of D-arabo ascorbic acid.
I. Introduction Water is one of the most familiar material in our life and is indispensable to all living things. In contrast to its apparently simple molecular structure, water shows many anomalous properties from both macroscopic and microscopic points of view. 1-2 However, the basic physical property of water, for example the dynamical structure of water, has not yet been fully clarified. To understand the dynamical aspect of water structure and its significant role in life, it is essential to clarify not only the dynamics of water molecules themselves but also the dynamics of water in the aqueous solutions. Raman spectroscopy has been usually employed to investigate the dynamical structure of water and aqueous solutions for a long time. In the high-frequency spectral region above 300 c m l , intramolecular vibration spectra of water molecule are measured and these spectra are analyzed and discussed by many researchers. 3-9 On the other hand in the low-frequency region intermolecular fluctuation bands which are due to the interaction between water molecules through the hydrogen-bond and/or the dipole-dipole interaction are observed. 9-19 In this low frequency region two broad bands around 190 cm -1 and 70 cm -1 have been observed and assigned to the restricted translational vibrations of the hydrogen bonds between water molecules based on a five-molecules cluster model. 9 The spectral profiles of these modes in aqueous solutions have been widely reported. 20-26 Recently it has been found that besides the above two broad bands one relaxation mode appears as a central component below 50 cm-1.10,17-19,27 This relaxation mode is due to the creation and annihilation process of hydrogen bond among water clusters. From the change of this spectral profile as well as the two broad bands in water and aqueous solutions we can obtain the dynamical aspect of water. 0167-7322/95/$09.50 91995ElsevierScienceB.V.All rightsreservgd. SSD! 0167-7322(95) 0085I-9
188 II. Experimental Raman scattering spectra were obtained by a four-slit double-grating spectrometer (Jobin-Yvon HG-2000M). The exciting light source was a NEC At-ion laser operating at 488 nm with a power from 100 to 300 mW. An apparent local heating due to the laser light was not observed. A right-angle-scattering geometry is always adopted in the present light-scattering experiments. The depolarized Raman spectra were measured with the configurations of X(VH)Y, where the XY plane is horizontal and X denotes the direction of incident light and Y denotes the direction of scattered light. The typical spectral resolution was 1.5 cm "1 - 2.0 cm "1 for the spectral region from -50 cm "1 to 50 cm "1 and 3.0 cm "1 for the spectral region from -250 cm "l to 250 cm "l. The scattered light signals were detected by a photo-multiplier in conjunction with photon-counting electronics. The samples of liquid water were 99.9% D20, natural H20 which was de-ionized and distilled and H2180 which was purchased from Isotec Inc., Japan. Solutes of LiCI, NaCI, KC1, RbCI, MgCI 2 and CaCI 2 arc extra-pure grade, which were purchased from Wake Pure Chemical Industries, Co. Ltd. The aqueous solution was prepared by dissolving solute into deionized distilled water. The concentration range of aqueous electrolyte solution is from 0.00 (water) to 0.11 molar ratio. The molar ratio is the mole number of solute against the mole number of water, which is written as nx/nwatcr in the present paper. The D-glucose was purchased from Wake Pure Chemical Industries Co. Ltd., Japan. Water we used as a solvent was de-ionized and distilled water. The concentrations of the D-glucose aqueous solutions arc from 0.005 to 0.04 molar ratio. The ascorbic acid aqueous solutions were prepared by dissolving L-xylo ascorbic acid or D-arabo ascorbic acid into the de-ionized and distilled water. The concentrations of ascorbic acid aqueous solutions are from 0.0025 to 0.03 molar ratio. The L-xylo ascorbic acid is a well-known vitamin C and the D-arabo ascorbic acid has very litde physiological activity although they arc isomeric with each other. All the samples were further purified by removing dust particles through 0.2 gm Millipore filter and sealed in fused silica cells or Pyrex cells. The sample cell was embedded in a specially designed home-made cryostat or furnace. The temperatures were measured with a chromcl-constantan thermocouple closely attached to a cell. The accuracy of the temperature control is within :1.-0.1 K. The thermocouples were prepared at Chemical Thermodynamics Laboratory, Osaka University. III. Results First we show in Fig.1 a high frequency (VV) and (VII) Raman spectral pattern of water at 295 K. The high frequency spectral pattern characterizes the internal molecular vibrations. Fig.1 shows raw spectra which are not calibrated by the spectrometer efficiency. However this spectral pattern has a typical characteristic of the tetrahedral-like C2v structure although the spectral line shapes arc very broad. In other words, between 1600 cm -1 and 4000 cm -1 essentially four kinds of molecular vibrations (el, v2, v3, v4) exist. This tetrahedral characteristic is consistent with the later discussion of the dynamical aspect of the water structure obtained by the low frequency Raman spectra.
189
Innnnlnnnnlnnnnlnllll qi[WATER (295K)I, If:' .
v,,vo
---"i.'
J .... J :
..........
:
!
~- . . . . . . .
~ :
: . . . . .
---8-
. . . .
-4----. ,
__i..._?_. -,..~ .... (vv)-.-~
~ li
" ~ /
-.4_~.... .
~
i
'I
i ~
"
-
.~L ....... i-~t---
v4
i
--
s
............. -/..4 .... -
-i/(vm
_
7
uuunlUnUUlUUnUlUUUu -200
2000 4000 RamanFrequency (cm"1)
0
0
200
RamanFrequency (cm"l)
Fig.l. High frequency Raman spectralpattern Fig.2L o w frequency reduced Raman spectrum in liquid water at 295 K. Four kinds of internal at 295 K. Solid curve is a best fitcurve and vibration modes from 1600 cm -l to 4000 cm -I. dottedcurves are components of three modes.
8
~-6
I.
ti.
,
I
i
I
,
I
i n
~5 >
~4 ~3 ~ 2 1 0
-200 0 200 Raman Frequency (cm"l) Fig.3 Reduced Raman spectra of various electrolytic solutions. (1):RbCI, (2):KCI (3):NaCI, (4):LiCI, (5):CAC12, (6):MgCI2.
0.00
0.04
0.08
0.12
n x / nw=er(molar ratio) Fig.4 Concentration dependence of the ratio of relaxation time (markers) and the ratio of viscosity (solid curves).
190 To clarify the spectral profile in the low frequency region, we reduced the Raman spectralintensityI(~') into the imaginary part of the complex dynamical susceptibilityX(V).27 The Z"(v) is given by
X"(~) = K (vi -~)--4[n(P)+ 1]-11(~),
(1)
where n(~')+l is Bose-Einstein thermal factor with n(~')=[exp(hcP/kT)-l] -l. The P is the Raman frequency shiftin cm -I,and the vi is the frequency of incident laser fight. The K is an instrumental constant. Fig. 2 shows the X"(V) spectrum of water at 295 K. Fig. 3 shows the typicalX"(V) spectra of various electrolytesolutions of 0.07 molar ratioat 303 K. The central components within :k20 cm-I vary with ion species. Since the reduced spectrum X"(v) clearly shows the low-frequency Raman modes, we introduced a simple model to analyze the spectral profile of Z"(v') for obtaining the quantitative information. The model is composed of two damped harmonic oscillatormodes and one Debye type relaxation mode (liquid water)27 or one Cole-Cole type relaxation mode (aqueous solution).28 Cole-Cole type relaxation is usually adopted in analyzing the dielectric relaxation. The formula of Cole-Cole type relaxationis represented as31
1
Z(to) = 1 + (,~)"'an;"/3'
(2)
where to = 2gc V and c = I/(2gcgl). The to is an angular frequency, c is the relaxation time, and the parameter fl (0<)0<1) represents the distribution of relaxation time. For fl=l, the equation (2) reduces to Dcbye type relaxation mode, and the gl corresponds to the spectralhalf width of the relaxation mode which is expressed by cm -l. For liquid water (H20, H2180 and D20) we adopt Debye type relaxation mode in analyzing the X"(v) spectrum, because the distribution of relaxation times can be neglected. Whereas, we found that it is better to introduce the parameter flfor aqueous solutions. The imaginary part of the susceptibilityof the damped harmonic oscillatormodes is
~,~,'(toj,Tj,to) =
toj27jto (toj2 _ to2 )2 .i. (07y j )2 '
(3)
where toj = 2~rcPj (j = 2,3) and ~ = 2~cgj (j = 2,3) are the characteristic frequencies and damping constants, respectively. The ~'j and gj are the characteristic frequencies and damping constants in cm -1 uniL Therefore the imaginary part of the total susceptibility is represented as Z"(P) = A 1Z~'(&,fl;V)+ A 2 Z"(v-2,g2;V)+
A 3 Z~'(v-3,g3; ~ ) ,
(4)
where A 1, A 2 and A 3 are the strength of each mode. Using Eq. (1) and (4), the reduced spectrum Z"(P) was fitted by a non-linear least squares method. We fitted each 5:250 cm -1 spectrum and +50 cm-1 spectrum separately and adjusted all the parameters consistently. The solid curves in Fig.2 and Fig. 3 are the best fitting curves. In Fig. 2 the spectral components of one relaxation mode and two damped oscillator modes arc also shown by dotted curves.
191
Table I shows the fitting parameters of H20, H2180 and D20 at 295 K. Table I Best fit parameters of liquid water at 295 K. H 2 0 (cm-1) gl v2 v3
H 2 1 8 0 (cm-1)
8.3 69.3 194.1
D20
7.4 64.8 184.9
(cm-1)
6.5 64.4 194.7
Fig.4 shows the concentration dependence of the z/r water and 0/r/water in various electrolytic aqueous solutions, where the z water is the relaxation time of pure water, and the r/ and r/water mean the viscosity of the aqueous solution and pure water, respectively. 28 The parameter fl of the electrolytic aqueous solution changes from f l - 0.9 to 0.7 for monovalent ions (LiC1, NaC1, KCI, RbCI) and from f1-0.9 to 0.5 for divalent ions (MgC12, CaC12) with increasing concentration. Solid lines represent the concentration dependence of the viscosity ratio of various electrolyte solutions. Fig.5 shows the concentration dependence of the characteristic frequencies of the two damped oscillators in the D-glucose aqueous solution. We can see that the characteristic frequencies are almost unchanged with concentration. 30 Fig.6 shows the concentration dependence of the damping constants in the ascorbic acid isomerism solutions. We can see that the change of the damping constant g2 in D-arabo ascorbic acid is more drastic than that in L-xylo ascorbic acid. 31 [
'
I
'
'
I.
350 ~§247 i........................... i. i'"........................... .... 4i...........i...... __,'--,300 i o Glucose ~ '~'~~ W ~a t e r9_ i i ~ ~ 200~'-.--............ ~ ) O~............ O ' ,...... O . ...~... O 0i0........
250" .-.
." ~
.... I,, ...... o L-xylo J D-arabo 9 water &
I,, i i !
!
A
-'E 200-: ................................! ........ i ..... e~
..6. 150 -,= ................................. ~100~50--1,............. 0 ~....~.0 ~~.......... ~0 ~.....~ ~
- a6
....if" 100
0
i'""; /
0.0
-
i
...... ; ...... ; . . . . . . ;I. . . . ? . . . . I . . . . . ; " .... ;l. . . . . . ;" I I I I
1.0
2.0
3.0
4.0
n x / n water ( xl 0 -2 m o l a r ratio) Fig.5 Concentration dependence of the characteristic frequencies of D-glucose solutions.
all
0.00
I
I l
0.01
I I
ill
0.02
I1"i
I
I I
0.03
n x/nwate r ( m o l a r ratio) Fig.6 Concentration dependence of the damping constant g2 of ascorbic acid isomerism.
192
IV. Discussion From Fig. 1 we propose that the water molecule has temporarily tetrahedral-like structure in a short time, because if the water has been constructed by a simple H20 (C2v) molecule there should be only three molecular vibration modes (Vl, v2, v3). In Fig. 1 we can see that between 1600 cm- I and 4000 cm-I more than three molecular vibrations. They can be classified into essentially four kinds molecular vibrations (vl, v2, v3, v4). Besides three or four vibrationcomponents in the Vl/V3 modes region there existsan extra broad mode at about 2200 c m -I. W e had betterto interpretthis spectralpattern as the molecular vibrationsof tetrahedrallike C2v symmetry which is composed by two O - H bonds and two O-..H hydrogen bonds in each oxygen. Although the conventionall explanation of 2200 c m -I mode is the combination m o d e 9 between the molecular vibration v2 and the lattice vibration VL, there is no direct experimental evidence. Rather the tetrahedral-likeC2v local structurecan produce the four molecular vibrationmodes (A I,A I, B I, B2) in the vl/v3 frequency region and three molecular vibration modes (A1, B I, B2) which arc bundled in the v4 frequency region. This lattermodes correspond to the broad 2200 cm'l mode. The above picture is consistent with the pcntamer model of liquid water which is stressed in the interpretationof the low-frequency R a m a n specwal pattern. According to the pentamer model of liquid water, the 190 cm-1 mode and the 70 cm-1 mode have been assigned to the stretching and bending vibrations between intermolecular hydrogen bonds, respectively 9. Later Walrafen et al. reassigned these modes to the longitudinal (dilation) type and the transverse (shear) type acoustic modes in the disordered system such as glass, respectively. 32 We think that these two assignments axe not essentially contradictory and it depends on how to take the coordinate system of the water molecule. Therefore in the present paper we refer to the former assignment. From Table I we can see that the characteristic frequencies of 190 cm -1 mode in H20 and D20 are almost the same but the characteristic frequency of H2180 is lower than that in H20 or D20. This result is the same as that of Nielsen's work. 33 This means that this mode is mainly due to the motion of oxygen atom because the mass of oxygen atom is the same between H20 and D20. This is consistent with the idea that the 190 cm -l mode is assigned to the stretching mode of the water pentamer because the stretching motion is parallel to the hydrogen bond. On the other hand the characteristic frequencies of the 70 cm -1 mode in H2180 and D 2 0 arc almost the same but the frequency of H20 is higher than that in H2180 or D20. This means that this mode is mainly duc to the motion of the whole water molecule because the total mass of water molecule is the same between H2180 and D20. This is also consistent with the idea that the 70 cm -1 mode is assigned to the bending mode between water molecules because the bending motion is perpendicular to the hydrogen bond. The g 1 -T plot in H20 changes linearly with temperature below about 293 K. The extrapolated temperature of the apparent divergence of the relaxation time would be 225 K. Whereas above about 303 K the g 1 -T plot in H20 deviates from the straight line which holds below 293 K. It has been shown in Fig. 4 that the relaxation time x of each electrolyte solution was always longer than that of water in both monovalent and divalent salt aqueous solutions. With increasing salt concentration, not only the relaxation time but also the viscosity r/of the solutions increases. 34 In Fig. 4 the ratio of viscosity U/r/water is shown as a function of concentration. From this figure we can see that the concentration dependence of the ratio X/Xwater has almost the same behavior with that of the ratio U/r/water. The viscosity is derived from the dynamical property of liquid. From a microscopic point of view, the molecules should be rearranged each other when flow occurs. The relaxation of structure is the process by which molecules of a system "flow" from a non equilibrium configuration to a new
193
equilibrium configuration. 35 Therefore the Raman relaxation time and the viscosity of aqueous solution are strongly correlated with each other and the ratio of viscosity against water exhibits almost the same concentration dependence. This means that the Raman relaxation time well characterized the dynamical structure of water from the dynamical point of view. In Fig. 5 we can see that the characteristic frequencies of D-glucose aqueous solution are almost independent as a function of concentration. In D-glucose aqueous solution, the predominant form is a chair form conformation because of the tetrahedral geometry of its saturated carbon atoms. 36 The chair form conformation of the six-member ring structure resembles a part of ice Ih structure. Since the tetrahedral structure is also one of the essential parts of ice structure, the tetrahedral configuration can constitute the crystal lattice structure of hexagonal ice. Franks 37 suggested that molecules of glucose are adjusted and harmonized with the water "lattice" which exists temporarily in liquid water. Thus because of the temporarily and locally existing tetrahedral structures, water can naturally accept glucose molecules and the dynamical structure of water is hardly disturbed by this kind of solutes. This picture can interpret why the characteristic frequencies scarcely varied with increasing concentration. In Fig.6 we can see that the damping constant g2 of the L-xylo and the D-arabo ascorbic acid solutions become different with increasing concentration. Since the damping constant characterizes the distribution of mode frequencies in the present system, the effect of the D-arabo ascorbic acid on water structure is larger than that of the L-xylo ascorbic acid.
V. Concluding Remarks The high frequency Raman spectral pattern of liquid water is better to be interpreted by a tetrahedral-like local structure than a simple H20 molecule. The low-frequency Raman specwa of water and aqueous solution are well fitted by a simple model which is a superposition of one relaxation mode and two damped oscillator modes. The dynamical aspect of liquid water and water in aqueous solutions can be characterized by the fitting analysis of these low-lying modes. The relaxation time in Raman frequency region is interpreted as an average life time of the tetrahedrally coordinate water cluster. In electrolyte solutions this relaxation time is strongly correlated with the viscosity of the solution. The characteristic frequencies of two damped oscillators in D-glucose aqueous solution are almost independent of concentration. This means that the molecular structure of glucose is just fitted with the dynamical structure of water cluster. As for the ascorbic acid solutions the effect on the dynamical structure of water is different between the isomerism. The effect of L-xylo ascorbic acid (Vitamin C) on the dynamical structure of water is less than that of D-arabo ascorbic acid which has little biological activity. It is interesting if this different effect on the dynamical structure of water may relate with a biological activity.
Vl. Acknowledgments This work is partly supported by a Grant-in-Aid (No.05452038) for Scientific Research from the Ministry of Education, Science and Culture. We also thank to Prof. Umpei Nagashima in Ochanomizu University for valuable and stimulated discussions. References 1 F. Franks, in Water, A Comprehensive Treatise, edited by F. Franks (Plenum, New York, 1972), Vol. 1, Chap. 1. 2 D. Eisenberg and W. Kauzmann, in The Structure and Properties of Water (Oxford University, Oxford, 1969). 1
194 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
27 28 29 30 31 32 33 34 35 36 37
W.F.Murphy, and H.J.Bemstein, J.Phys.Chem.,76,1147(1972). G.E.Walrafen, J.Chem.Phys.,36,1035(1962); ibid. 44, 1546(1966); ibid.47, 114(1967). G.E.Walrafen, "Water, A Comprehensive Treatise", vol 1 (edited by F.Franks, Plenum Press, 1972) 151. W.B.Monosmith, and G.E.Walrafen, J.Chem.Phys.,81, 609(1984). G.E.Walrafen, M.S.Hokmabadi, and W.-H.Yang, J.Chem.Phys.,85, 6964(1986). C.I.Ratcliffe, and D.E.Irish, J.Phys.Chem.,86, 4897(1982). G.E.Walrafen, J.Chem.Phys.,40,3249(1964) C.J.Montrose, J.A.Bucaro, J.Marshall-Coaldey, and T.A.Litovitz, J.Chem. Phys., 60, 5025(1974). S.Krishnamurthy, R.Bansil, and J.Wiafc-Akenten, J.Chem.Phys.,79, 5863(1983). W.Danninger, and G.Zundel, J.Chem.Phys.74, 2769(1981). G.E.Walrafen, M.R.Fisher, M.S.Hokmabadi, and W.-H.Yang, J.Chem.Phys., 85, 6970(1986). G.E.Walrafen, M.S.Hokmabadi,W.-H.Yang,Y.C.Chu, and B.Monosmith, J.Phys. Chem.,93,2909(1989). F.Aliotta, C.Vasi, G.Maisano, D.Majolino, F.Mallamace, and P.Migliardo, J.Chcm.Phys.,84, 4731 (1986). J.L.Rousset, E.Duval, and A.Boukenter, J.Chem.Phys.,92, 2150(1990). V.Mazzacurati, and P.Benassi, Chem.Phys.,112, 147(1987). V.Mazzacurati, M.A.Ricci, G.Ruocco, and M.Sampoli, Chem.Phys.Lett., 159,383(1989). V.Mazzacurati, A.Nucara, M.A.Ricci, G.Ruocco, and G.Signorelli, J.Chem. Phy s.,93,77 67 (1990). G.E. Walrefen, J. Chem. Phys. 36, 1035 (1962). P. Terpstra, D. Combes, and A. Zwick, J. Chem. Phys. 92, 65 (1990). David W. James and Raimund Irmer, J. Raman Spectrosc. 14, 184 (1983). Murray H. Brooker and Michel Pcrrot, J. Chem. Phys. 74, 2795 (1981). G.E. Walrafen, J. Chem. Phys. 44, 3726 (1966). O.F. Nielsen and P.-A. Lund, Chem. Phys. Lett. 78, 626 (1981). O.F. Nielsen and P.-A. Lund and S. B. Petersen, J. Am. Chem. Soc. 104, 1991 (1982). K. Mizoguchi, Y. Hori, and Y. Tominaga, J. Chem. Phys. 97, 1961 (1992). Y.Wang and Y.Tominaga, J.Chem. Phys. I01,3453(1994). Kenneth S. Cole and Robert H. Cole, J. Chem. Phys. 9, 341 (194 I). Y.Wang and Y.Tominaga, J.Chem.Phys.,100, 2407 (1994) Y.Wang and Y.Tominaga, J.Phys.Soc.Japan 62,4198(1993). G.E. Walrafen, J. Phys. Chem. 94, 2237 (1990). O.F.Nielsen,Chem. Phys. Letters 60, 515(1979). Chemical Table edited by Chemical Society of Japan, (Maruzenn, 1984), No. 2, p.42 (in Japanese). C.M. Davis, Jr., and J Jarzynski, in Water, A Comprehensive Treatise, edited by F. Franks (Plenum, New York, 1973), Vol. 1, Chap. 12, p.443. Lubert Stryer, Biochemistry (W. H. Freeman and Company, New York, 1988). F. Franks, in Water, A Comprehensive Treatise, edited by F. Franks (Plenum, New York, 1972), Vol. 2, Chap. 1, p.36.
journal of MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids. 65/66 (1995) 195-204
ULTRASONIC RELAXATION STUDY OF FAST EXCHANGE PROCESSES IN MIXED MICELLE SYSTEMS OF ALCOHOLDECYL ONIUM BROMIDE-WATER R.E. VerraU* Department of Chemistry University of Saskatchewan Saskatoon, SK S7N 0W0, Canada D.J. Jobe Research Chemistry Branch Whiteshell Laboratories, AECL Research Pinawa, MB ROE 1L0, Canada and E. Aicart Departamento de Quimica Fisica Facultad de Ciencias Quimicas Universidad Complutense 28040 Madrid, Spain
Summary Recent studies of the fast exchange processes of micellar components between mixed micelles of alcohol-surfactant and the aqueous bulk phase are reviewed and discussed in the light of some further work. Ultrasonic absorption measurements (0.6 < f < 210 MHz) of aqueous solutions of decyltrimethylammonium bromide (DTAB) and short to medium hydrocarbon chain length alcohols show two relaxation frequencies. The lower frequency is assigned to the exchange of the monomer surfactant and the higher frequency to the exchange of alcohol between the micellar and bulk phases. Conductometric and time resolved fluorescence measurements have enabled complementary information to be obtained regarding the mean aggregation number of the surfactant and, indirectly, of the alcohol in the aggregates. The ultrasonic data were analyzed according to the theories of Aniansson and of Hall to obtain estimates of the backward rate constants and volume change for the exchange processes of surfactant and alcohol between the micellar and bulk phases and of the polydispersity of the mixed micelles. The Hall model gives larger values of the kinetic parameters. Generally, the results of the analyses for the surfactant show that its exit rate from the mixed micelle and the mean aggregation number and the polydispersity of the micelle decrease as the alcohol chain length increases. Reasons for the observed influence of the alcohol on the kinetic and thermodynamic parameters are presented.
Introduction Chemical relaxation methods are widely used to study the kinetics of fast chemical reactions in solution. In particular, ultrasonic absorption techniques have been used to investigate fast exchange processes with relaxation times in the range 0.3 Its to 0.3 ns. The longer end of this time range has become accessible in recent years through the use of cylindrical resonator methods [1] which have lowered the frequency range covered by traditional pulse echo methods. Recently, we have shown [2,3] that, if the surfactant has an appropriate hydrocarbon chain length, then the relaxation processes associated with the rapid exchange of both the 0167-7322/95/$09.50
91995 ElsevierScience B.V. All rights reserved.
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196 alcohol (A) and the surfactant (S) between mixed micelles of surfactant-alcohol (SnAa) can be investigated using ultrasonic relaxation techniques. The two relaxation processes possible are shown in eq 1.
k~ A + SnAa_ 1 <
k i> SnA a (
k/
) Sn_
1Aa + S
(1)
where k + and k i- are the forward and backward rate constants, respectively, for the exchange of surfac'tant witfi the mixed miceUes and k~ and k~ are the forward and backward rate constants, respectively, for the exchange process of the gdcohol with the mixed micelles. While it is generally difficult to accurately resolve two ultrasonic relaxations separated by a small factor, in the systems considered in this paper the separation of the relaxation times Xl and x2 was sufficient to allow the two corresponding relaxation frequencies, frl and fr2, to be estimated by fitting the ultrasonic absorption data to the expression
1+-~-_2 +B
o r / f 2 = ~ Ar i / i=l
\
(2)
ri j
where ot is the ultrasonic absorption coefficient, f is the frequency of the ultrasonic wave, Ari !s the relaxation amplitude associated with relaxation process i, B is the background ultrasonic absorption due to any processes occurring above the frequency range of interest, and fri is the ultrasonic relaxation frequency for the relaxation process where 1/1: i = 2n-fri
(3)
The ultrasonic data were interpreted using the Aniansson model [4] and the phenomenological model of Hall [5]. The former is strictly applicable to nonionic micelles. However, for ionic surfactants having a low degree of micelle ionization (13), the use of this model to give a detailed statistical and analytical analysis of the relaxation times is not likely to lead to significant errors in any derived parameters. On the other hand, when alcohols are added to surfactants, their inclusion in the head group region of the micelles tends to decrease the surface charge density of the micelle and can dramatically increase the value of 13. Under such circumstances the acceptability of using the Aniansson relations to interpret the experimental relaxation data may be questioned. Equations (4) and (5) are the relations derived by Aniansson for mixed micelles and they relate the relaxation times to kinetic constants and other parameters as follows: 1 / 1:1 = ki" / c2[1 + t~2Cm / cmc - t~A O'S 1/1:2 = k 2 / I:12[1 +tI2Cm / C~]
(5)
where t~A 2 and tJS 2 a r e the variances in the mean alcohol aggregation number (~) and the mean surfactant aggregation number (,~), ,respectively, GAS is a measure of the correlation (or covariance) between tJAZ and tJSZ , C At is the concentration of alcohol not micellized, cmc is the critical micelle concentration in the presence of the alcohol, and Cm is the concentration of micelles, given by
197 Cm = (Cs - cmc)/ff
(6)
where CS is the total surfactant concentration. More recently, using a model that accounts for the effect of the ionic character of the surfactant on the rate constant for the surfactant exchange process, Hall [5] derived the following expressions corresponding to the surfactant and the alcohol processes: 1/1:1= k l / C r 2 + k l ( 1 / c m c + (1-13)2/(cmc+[~CSm)+2T')Cm
(7)
~tmax2 = [ ~ V 2 / 2RT]I;2k2CAm / KS = [r,21V22 / 2RT]CfCsm / IcsC A
(8)
where I/max2 is the maximum absorption per wavelength for the alcohol relaxation process, KS is the adiabatic compressibility for the solution, CAm is the concentration of alcohol solub'flized in the micelle, given by CA - CA~ where CA is the total alcohol concentration, CS m is the concentration of surfactant in the miceUe, given by CS - cmc where CS is the total surfactant concentration, ~' = -0.587/cmcl/2(1 + cmclT2)2, and R and T have their usual meaning. When 13--- 1 and ideality is assumed eq 7 becomes identical to eq. 5, which is the expression derived by Aniansson for the exchange of a nonionic surfactant with its micelle [4] in the absence of alcohol. It is evident that in order to obtain information about the rate constants and variances using these theories, estimates of experimental parameters such as ~, a-, CA f, KS and [3 must be obtained. The alcohol components were chosen to reflect three different types of miscibility behaviour in water at 25oc; (i) an alcohol that is widely miscible, 1-propanol (Pr); (ii) alcohols that have relatively narrow miscibility ranges, 1-butanol (Bu) and 1-pentanol (Pc); and (iii) an alcohol that has a relatively long equivalent hydrocarbon chain length but, because it contains an ethylene oxide unit, is completely miscible, 1-butoxyethanol (BE). The surfactant used in this study is decyltrimethylammonium bromide (DTAB). It was chosen because it has the longest hydrocarbon chain length for which the fast monomer exchange between the miceUe and bulk phases can be monitored by ultrasonic techniques. The results of this study show that ultrasonic absorption techniques, when used in conjunction with other aggregate properties, provide important kinetic information about the partitioning process of alcohols between ionic micelles and the bulk phase. The effect of hydrocarbon chain length of the alcohol on the exit rates of monomer surfactant and alcohol components from the mixed miceUes, aggregate size distribution, and the electrostatic stabilization of the head group region of the micelles will be considered. As well, previously unre.ported data for water- 1-butoxyethanol-DTAB will be used to extend conclusions based on previous work [2,3].
Experimental The experimental techniques used to obtain the properties necessary to derive the kinetic constants of interest from the ultrasonic relaxation times have been previously described in detail [2,3]. Briefly, the degree of micelle ionization (13) and the binding constant (Ka) of an alcohol to mixed micelles were obtained from specific conductivity measurements as a function of surfactant concentration at various fixed alcohol compositions. The binding constant was determined from the slopes of the curves above the cmc, as proposed by Abu-Hamdiyyah et al
198 [6]. The ratio of the slope above to the slope below the cmc provided an estimate of 13 [7]. The binding constant was also used to estimate the concentration of alcohol in the bulk phase. Time resolved fluorescence quenching of dimethylnaphthalene by cetylpyridinium bromide solubilized in the micelles was used to obtain estimates of the mean aggregation number of the surfactant (n-) in the mixed micelles. High concentrations of short hydrocarbon chain length alcohols are known to break down micelles. However, in this work the analysis of the ultrasonic relaxation data to obtain kinetic information was restricted to alcohol and surfactant concentrations where the mean aggregation number of the surfactant did not show a significant decrease [2,3].
Results and Discussion Figure 1 shows typical results of edf 2 plotted versus frequency for solutions where the surfactant concentration is fixed (0.12 m) and the concentration of alcohol (Pr) is varied. Figure 2 shows similar results for systems containing a fixed concentration of butanol (0.695 M) and various concentrations of surfactant. p s z cm -1)
a/fZx
10000
1017 (Np s z cm -1)
o.ooo
lOOO,
ooooo 0.012 M DTAB eeeee 0.020 M DTAB AA/W^ 0.040 M DTAB DTAB
1ooo
lOO
1oo
,,,!
,
1
,
,
,,,,,!
lO
,
,
,
,|,,,!
loo
,
,
f (MHz) Figure 1. Plot of odf2 versus frequency at 25~ for solutions containing 0.12 m DTAB and various concentrations of Pr: (o) 0.00 m, (I"l) 0.20 m, (z~) 0.49 m, (X) 1.60 m and (*) 2.23 m. Reprinted with permission from J. Phys. Chem. 1992. 96, 2348. Copyright 1992 American Chemical Society.
l0
,,,,!
,
1
,
,
,
,,,,!
lo
,
,
,
,
,,,,1
ioo
,
f (MHz) Figure 2. Plot of (z/f2 versus frequency at 25~ for solutions containing a fixed concentration of Bu (0.695 M) and various concentrations of DTAB. Reprinted with permission from J. Phys. Chem. 1992 96, 6811. Copyright 1992 American Chemical Society.
Best fit parameters of the ultrasonic data to eq 2 for systems containing propanol, butanol and pentanol have been previously published [2,3]. Subsequently, studies were extended to include ultrasonic absorption measurements of ternary systems of DTAB-BE-water. Values of
199 the best fit parameters Ai, fri and B, as well as Z 2, that result from fitting the ultrasonic absorption data of these systems to eq 2 are shown in Table I. In all of the ternary systems Table I: Best-Fit Values of the Ultrasonic Data for BE A c c o r d i n g to Equation 2 A2x 1017 CA CS A1 x 1017 frl (Np s2 cm4) (molk~"1) (molkg -1) (Nps2cm -1) MHz 0.000 0.12 1113 • 108 1.4 + 0.1 16 • 2 0.100 0.12 1332 • 27 2.2 • 0.1 22 • 4 0.200 0.12 1525 • 29 3.0 • 0.1 138 • 26 0.400 0.12 1850 • 44 3.4 • 0.2 200 • 38 0.600 0.12 1521 • 34 6.2 • 0.5 0.800 0.12 1167 • 82 8.8 + 1.1 332 • 86 0.932 0.12 1186• 75 13.2+ 1.3 213 • 76 243 • 76 0.800 0.050 2262 • 48 6.4 + 0.5 0.800 0.030 2636 + 54 5.7 • 0.4 229 • 59 171 + 26 0.800 0.010 2025 + 31 ' 4 . 7 + 0.3 140 + 20 0.800 0.005 1183 • 28 4.2 + 0.3
1oooo e//fz x 1017 (Np s z cm-') ooooo 0.80 m BE
ooooo 0.60 m BE ZXZXZX,',A0.40 m BE ..... 0.20 m BE 9o 0 . 0 0 m BE
1000
100
|0
,,!
,
1
,
,
, , , , , |
,
10
,
,
,
....
!
,
9
100
f (MHz) Figure 3. Plot of ot/f2 versus frequency at 25~ for solutions containing 0.12 m DTAB and various concentrations of BE.
fr2 MHz 86 • 25 70 • 17 34 • 5 41 • 6 39 • 6 54 • 12 43 • 6 38 • 7 39 + 4 39 + 5
u m s"1 1511.0 1515.8 1520.2 1527.2 1531.7 i534.3 1533.7 1541.3 1545.0 1551.1 1554.1
Bx 1017 (Np s2 cm 4) 25 + 1 22 • 2 24 • 1 26 • 1 27 • 2 28 • 2 28 • 2 28 • 2 28 • 2 28 • 2 28 • 1
X2 0.25 1.0 0.90 0.71 0.93 1.1 1.5 1.5 1.7 2.1 1.2
studied, two relaxation processes were required to minimize Z 2, and the residual difference between the experimental and calculated values of o~/f2 were found to be randomly distributed in every case. Figure 3 shows the results of o~/fz plotted versus frequency for solutions of D T A B - B E water. The lines drawn through the data p o ~ t s in Figures 1-3 are the fitted values of ct/fz according to eq 2 using best fit values reported previously [2,3], and in Table I for the case of the ternary system containing BE. Some of the calculated relaxation frequencies reported are outside the experimental frequency range studied and therefore are expected to have a higher uncertainty associated with them. A single relaxation was observed for the binary water-surfactant system and the results obtained are in good agreement with those previously reported [8]. A relatively weak sound absorption (ca. 1-4 MHz) is apparent in alcohol-water systems for the more concentrated solutions of Bu and Pc.
The amplitude increases with hydrocarbon chain length of the alcohols but it is still much smaller than the experimental sound absorption for the ternary systems in this frequency range. It probably originates from self-associative equilibria of the alcohol molecules and/or equilibria in alcohol-water aggregates. Because of its relatively weak magnitude, it was neglected in the analysis of data for these systems. The concentrations of BE in the ternary systems studied in this work were maintained below 0.90 m. This was done in order to minimize the contribution of sound absorption due to the chemical relaxation of self-aggregates that form at higher concentrations.
200 For a constant concentration of surfactant, the amplitude of the higher frequency relaxation process increases with increasing alcohol concentration. This is the basis for assigning it to the alcohol exchange process since its magnitude would be expected to be proportional to the amount of alcohol partitioned in the micelle. The lower relaxation frequency is assigned to the exchange of a surfactant monomer between the mixed micelle and the bulk phase. It was assumed that the relaxation associated with the formation/break down of the mixed micelles occurs at a much lower frequency, outside of the range investigated in this work. Although the amplitude of the lower relaxation process is less affected by the presence of the alcohol, which is consistent with its assignment as the surfactant exchange process, the magnitude of its relaxation frequency increases almost linearly with alcohol concentration for all alcohols studied. The analysis of the relaxation data according to the Aniansson model to obtain estimates of k i- is not straightforward in all cases. Whereas the third term within the brackets of eq 4 can be~ shown to be < 1 for systems containing propanol, and therefore can be ignored, this is not the case for systems containing butanol, pentanol and 1-butoxyethanol. For most of the conditions examined in the latter cases, c~Cm/C~, >> 1 and eq 4 simplifies to yield 1 / x 1 = k i" / C~S 2 + ki-[C m / cmc - CfA / (CmO4S)]
(9)
where the value of OAS must be known in order to obtain k~ from the Xl data. concentration ranges studied, the statistical value of C~ASwas used ~AS = ~
(n - fi)(a- a)/N
For the
(10)
where fi and ~ are the average values of n and a, respectively, and N is the total number of samples. Equations 5 and 9 were used to analyze the ultrasonic data. Only experimental data which complied with c~2Cm/Cf >> 1 were used As a result, only solutions where the alcohol concentrauon was vaned and surfactant concentrauon was kept constant were used. Plots of 1/1;1 vs Cm/cmc were, for the most part, linear and the deviations from linearity probably reflect the crude method used to obtain an approximate value of ~AS. The values of the kinetic parameters and statistical results are shown in Table II. 9
.A
*
.
Table II. Kinetic Parameters for Aqueous Solutions of D T A B Alcohol Systems Derived from the Aniansson Model.
and D T A B
+
9,.i 1010s oS AV1, AV2, Kc kis 108 k+b~I "1 k2"1010 ~A . . cm3mol-I cm3tool -1 mol-ldm3 noned 4.5 3.0 10 4.4 n-C3H7OH 2.2 1.4 -+0.1 2.4 4.9:t:2 12:t3 4.2 _+0.3 29"2-_2 3.5:k0.3 0.49 n-C4H9OH 1.7 1.1 s 5.6 3.5:t:1 15:1:8 3.1_+0.4 30-+9 2.9+0.2 1.6 n-C5H11OH 1.5 0.95+0.3 8.3 1.7"1-0.4 17-1-102.5-t-0.5 24-1-4 2.4-+0.5 4.9 CaHo(EO)OH 5.1 3.1 -+0.7 2.5 2.2_+0.6 16"3:10 !3_+5 4.3-+0.8 1.2 Reprinted with permission from J. Phys. Chem. 1992, 96, 6811. Copyright f992 American Chemical Society. a Calculated using the cmc in the absence of alcohol where k'~ = k [/cmc. b Calculated from the relationship Ka = k~/k~, c Obtained from conductivitymeasurements, d Obtained using the data of ref. 8. alcohol
k~'a"1109s M "1
l
Values of the average volume change associated with the exchange of both surfactant (AV1) and the alcohol (AV2) can be calculated, according to the Aniansson model, from the following expressions
201 l.tmaxl = (Alfrl u) / 2 = [xAV? / 2RT]F(cmc) / [K:s(I+F)]
(11)
A 2 = 1.18x10 -7 AV 2 C m k~ 1;2
(12)
and
where ~tmaxl is the absorption maximum per wavelength, u is the speed of sound, and 1" = t~22/n)(Cm/cmc) 9 The slopes of graphs of ~tmaxl vs F(cmc)/[r,S(1 + 1")] and A2 vs Cm k -2 x 22 . were analyzed to obtmn estimates of AV1 and AV2, respectively, the results of which are shown in Table II. The volume changes associated with the surfactant are large and comparable for all systems. The rather large values of the exchange volume of DTAB with the mixed micelle, compared to the value of 4.4 cm 3 mo1-1 for this process in a regular mieelle, is puzzling. However, estimates of this property from the data using the model developed by Hall (see below) also show large values. We have speculated [2] that small fragments of alcohol and surfactant may break off from the micelle. A similar conclusion was drawn from the results of time correlated fluorescence measurements of pyrene in mixed alcohol-surfactant micelle systems [9]. On balance, further studies are required to confirm the exchange volume data for the surfactant and the tentative explanation for its magnitude. The volume changes associated with the exchange of the alcohol are much smaller and decrease with increasing chain length of the alcohol. This is contrary to the thermodynamic estimate of AV2 which increases with increasing alcohol chain length [ 10, 11]. Table III shows the values for the kinetic and statistical parameters obtained by analyzing the data for all ternary systems according to the phenomenological model of Hall [5]. A comparison of the values of the various parameters (cf. Table II and Table 111) obtained from analyses employing these two models show, generally, that larger values are obtained with the Hall model; however, the trends in the derived parameters are similar. The values of the rate constants for the alcohol exchange process in the Hall model were obtained by extrapolating CAf --->0 to avoid any concentration dependence of k~. Using the extrapolated value of k~, it was shown [5] that k~ is given by k ~ = n ~ k 2 (Cr~ / C f C~ ) Table III.
(13)
Kinetic P a r a m e t e r s for A q u e o u s Solutions of D T A B and DTAB + Alcohol Systems Derived f r o m the Hall Model.
alcohol
k~,a 109 ki ' 108 k+ b 1010 k~, ib8 t~S AV2, Kc M-1 s-1 s-1 ~-1 s-1 s-1 cm3 mol-1 mol-1dm3 noned 3.8 2.3 10 n-C3H7OH 3.6 2.2 + 0.5 1.6 80 + 6 6.4 + 3 4.9 + 1 0.49 n-C4H9OH 2.9 1.8 + 0.5 0.41 6.5 + 1 5.2 + 2 4.5 + 1 1.6 n-CSHllOH 2.4 1.5 + 0.4 1.6 9.2 + 1 4.0 + 1 3.3+ 1 4.5 C4H9(EO)OH 6.2 3.8 + 0.5 4.4 10 + 5 5.2 + 1 1.2 Reprinted with permission from J. Phys. Chem. 1992, 96, 6811. Copyright 1992 American Chemical Society. a Calculated using the cmc in the absence of alcohol where k~ = k~/cmc, b Calculated using eq. 13. c Obtained by extrapolating to the limit C~ --> 0. d Obtained using the data of ref. 8. where n ~ is the aggregation number of DTAB in the absence of alcohol. In the Aniansson model k~ = K k~. The difference in the values of k~ obtained from the two models is greater for the shorter chain alcohols. There is good agreement in the values of k~ and k i-
202 obtained from analyses of the data using the two models. The term in the Hall model which accounts for the ionic character of the surfactant narrows the range of values for k~" and k ifor the various alcohol-DTAB systems, yielding values closer to those obtained in the absence of any alcohol. From eq 8 one can obtain estimates of AV2 from the slope of a plot of I.tmax2 vs CAf C ~/~SCA. The values of the exchange volume of an alcohol between the mixed micelle and bulk phase are larger using the Hall model. However, both models predict the value of AV2 decreases with increasing chain length of the alcohol. The model developed by Hall does not enable a value of AV1 to be estimated although this parameter has been related [5] to the quantity (~gVI~A2 as follows
(~v~ / ~)A2 = av~~ + av2 (ck c~ / cAc~)
(14)
where AVy is the volume change for the surfactant in the absence of any alcohol. At high alcohol and surfactant concentrations, (0V 1/0~)A2 plateaus at a value of ca. 30 cm 3 mo1-1, similar to the value of AV 1 derived from use of the Aniansson model. If, as mentioned above, this large value were thought to be the volume change associated with the exchange of a fragment, then given that the exchange volume of an alcohol is ca. 5 cm3 mo1-1, the stoichiometric ratio of alcohol to surfactant in such a fragment would be ca. 5:1. As previously mentioned, this interpretation is tentative in the light of our more broadly based understanding of micelles. The data in Tables II and HI show that the exit rate of the monomer surfactant decreases in a linear manner with increasing chain length of the n-alcohol in the mixed micelle. It is also evident from the rate constant data in these tables that the value of k i" for the DTAB-BE system does not readily fit into the above trend. In fact k i- has a value near to that for the exit rate of a monomer surfactant from regular micelles of DTAB. At the same time the magnitude of the exit rate of BE from the mixed miceUes is between the values for Bu and Pc. To explain these results it is important to recall that micelle aggregates exist by virtue of a delicate balance of intermolecular forces. On the one hand, there are the stabilizing contributions arising from the presence of counterions near the head groups and the hydrophobic interactions in the micelle interior. On the other hand, there is the destabilizing, repulsive forces that result from the charged head groups of the ionic surfactant monomers when they are in proximity to one another in the micelle. The degree of solvation of the counterions and of the head groups may also play a role in the balance between stabilizing and destabilizing forces in this outer region of the micelles. The decrease in the magnitude of kiand k~ in mixed micelles containing the n-alcohols, as the hydrocarbon chain length of the alcohol increases, can be explained by increased hydrophobic interactions. There does not appear to be a significant change in the balance of intermolecular forces in the head group region when the hydrophilic moiety of the n-alcohol replaces water. However, in the case of BE it appears that there is a change in the balance of intermolecular forces in this region of the mixed micelles. The data in Table III indicate that the magnitude of k~ for BE is approximately in the range of values for Bu and Pc. This suggests that the butyl segment of BE is embedded in the mixed micelle. However, the magnitude of the exit rate constant for DTAB from mixed
203 micelles containing BE is greater than for mixed micelles containing normal alcohols and for regular micelles of DTAB. We have shown [ 12] that the surface charge density of DTAB-BE mixed micelles is generally greater than that of DTAB-n-alcohol systems. As a result, the increase in repulsive forces in the head group region in the presence of BE appears to cause a more rapid exit rate of the surfactant. It is of interest to speculate about the role of BE in the head group region of the mixed micelle. If the CH2-CH2-O- segment of BE locates on the surface of the micelle or just outside this surface, then it must displace solvent (H20) molecules. As a consequence, solvent of a higher dielectric constant is replaced by solvent of a lower dielectric constant which is less effective in decreasing the repulsive force between head group charges. Also, if ion-solvent dipole interactions play some role, then relatively stronger ion-solvent dipole interactions in the presence of water are replaced by weaker-ion EO dipole interactions. Of the twenty five sterically possible conformers of the EO segment, a large majority of them have small or no dipole moments [ 13]. It is also probable that, when the butyl group of BE enters the micelle, these relatively nonpolar conformations are favoured. This qualitative picture provides an explanation for the different behaviour of BE compared to the n-alcohol series. It is also consistent with a report [14] that the Gibbs energy of transfer of a homologous series of alcohols having the general formula, C4(EO)xOH where 0 ___X ~_ 4, from the bulk to the cationic micellar phase is independent of the number of EO segments in the alcohol. If we exclude the results for BE and assume for the other alcohols that the forward rate constant k~ is diffusion controlled and approximately constant, the decrease in the exit rate of the surfactant represents a measure of the Gibbs energy of stabilization per CH2 group of the alcohol. From AGs = RT din (ki-)/dn c, AGs = -450 J mo1-1 per CH2. A similar calculation using data derived from the Hall model gives an estimate of AGs = -220 J mol-I per CH2. The values of (~A and t~s reported in Table II for the n-alcohols show that the addition of alcohols of increasing hydrocarbon chain length to DTAB micelles decreases the value of aS and increases the value of OA. The increase in (~A probably reflects increased binding of the alcohol in the mixed micelle with increasing alcohol hydrocarbon chain length. This appears to lead to greater polydispersity of alcohols in the mixed micelle, i.e., more of the mixed micelles contain a broader range of alcohols. From the perspective of the surfactant it would appear that there is a reduced polydispersity as the surfactant aggregation number decreases [ 11]. The interesting behaviour observed for aggregate systems of surfactant and BE requires a broader investigation. Systematic thermodynamic and ultrasonic absorption studies of systems where the number of EO units in the alcohol is increased are currently in progress. The information obtained should be of benefit in extending the commercial application of these surfactant-cosurfactant systems.
Acknowledgement We thank the Natural Sciences and Engineering Research Council of Canada for financial support.
204 References
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
Eggers, F.; Funck, Th. J. Acoust. Soc. Am. 1975, 57, 331. Aicart, E.; Jobe, D.J.; Skalski, B.; VerraU, R.E.J. Phys. Chem. 1992, 96, 2348. Jobe, D.J.; Verrall, R.E.; Skalski, B.; Aicart, E. J. Phys. Chem. 1992, 96, 6811. Aniansson, G.E.A. In Techniques and Applications of Fast Reactions in Solution; Gettins, W.J., Wyn-Jones, E., Eds.; Reidel; Holland, 1979; p. 249. Hall, D.G.J. Chem. Soc. Faraday Trans. 2 1981, 77, 1973. Abu-Hamdiyyah, M.; Kumari, K. J. Phys. Chem. 1990, 94, 2518. Evans, C.H.J. Chem. Soc. 1956, 579. Kato, S.; Nomura, H.; Honda, H.; Zielinski, R.; Ikeda, S. J. Phys. Chem. 1988, 92, 2305. Malliaris, A.; Lang, J.; Sturm, J.; Zana, R. J. Phys. Chem. 1987, 91, 1475. De Lisi, R.; Milioto, S.; Triolo, R. J. Solution Chem. 1988, 17, 673. Desnoyers, J.E.; H~tu, D.; Perron, G. J. Solution Chem. 1983, 12, 427. Jobr D.J.; Verrall, R.E.; Skalski, B.D. Langmuir, 1993, 9, 2814. Karlstrtm, G. J. Phys., Chem. 1985, 89, 4962; Karlstrtm, G.; Carlsson, A.; Lindman, B. J. Phys. Chem. 1990, 94, 5005. Marangoni, D.G.; Kwak, LC.T. Langmuir 1991, 7, 2083.
journal of
MOLECULAR
LIQUIDS ELSEVIER
Journal of MolecularLiquids.65/66(1995) 205-212
Organometallic Free Radicals in Solution James H. Espenson Ames Laboratory and Department of Chemistry, Iowa State University, Ames Iowa 50011, USA Abstract Laser flash photolysis of [CpM(CO)3] 2 (M = W, Mo, and Cr) provides a convenient source of CpM(CO)3 ~ an organometallic free radical with 17 valence electrons. It is a transient and highly reactive spedes. D e p e n d i n g on the circumstances and the other reagents present, the radical will dimerize, u n d e r g o halogen and h y d r o g e n atom abstraction reactions, and electron transfer reactions. With t e t r a m e t h y l - p h e n y l e n e d i a m i n e , there is a cyclic process of electron transfer steps, the net result of which is the catalyzed disproportionation of the metal radical. I. Introduction Organometallic compounds of the transition metals with carbonyl and other ~ - a d d ligands are generally constrained to 18 electron configurations, if they are to be stable. Compounds with only 17e are the subject of this paper. Because such species are one electron short of a closed shell, they react like free radicals. The 17e compounds are generally transient entities, except for those cases w h e r e the steric bulk of some of the ligands p r e v e n t s their coupling. The transient entities are typified by the c o m p o u n d s that are the subject of this study, (rl5-C5H5)M(CO)3 . (M = Cr, Mo, W) To achieve stability, these reactive entities may u n d e r g o any n u m b e r of reactions. The purpose here is to enumerate what these reactions are and the mechanisms by which they occur. Moreover, an a t t e m p t is m a d e to show how the organometallic free radicals are analogous to organic radicals, which are in general more familiar. The 17e organometallic species are isolobal with alkyl radicals which are 7e species, also one electron shy of a closed shell. II. Reactions of an OrganometaUic Free Radical To outline the processes the reactive o r g a n o m e t a l l i c radical m a y undergo, we consider the following: (a) dimerization, so as to regenerate the stable dimer from which they were formed by photohomolysis; (b) halogen abstraction from alkyl halides; (c) halogen abstraction from transition metal halides; (d) O H g r o u p abstraction from peroxides; (e) h y d r o g e n atom 0167-7322/95/$09.50 91995ElsevierScienceB.V. All fightsreserved. 0167-7322(95) 00853-5
SSD!
206 abstraction from certain metal hydrides; (f) addition of 30 2 and subsequent reaction; (g) ligand-catalyzed disproportionation; (h) electron-transfer reduction; (i) electron-transfer oxidation, before or after ligand addition; and (j) electron-transfer induced disproportionation. These reactions are outlined in Scheme I. The occurrences and mechanisms of these transformations will be dealt with here.
Scheme I CpW(CO) 3- + CpW(CO)3PPh3 + ~ h CpW(CO)3 L+ ~
(
3P
(a~,, ~ o ~cpw(co)3
(i) CpW(CO)3-
1
~
30 CpW(CO)3OO'~f
[CpW(CO)3]2
\ (h)
OC /
CO
i CO
(b) ~-C p W (CO)3X+ R~ RX
[ ~ ) ~ " (ZpW(CO)3X,M"
(eC~pM ~(e (CO)3H ROOH ~
M = Rh(dmg~2PPh 3) CpW(CO)3OH + RO ~
CpW(CO)3H
III. Experimental Procedures The chemistry of the CpM(CO)3 ~ radicals occurs on the microsecond time scale, and flash photolysis with optical detection is the technique most generally applicable. The radical is created by the photohomolysis of the stable dimer, [CpM(CO)3] 2, available commercially for M = Mo and W. The excitation sources used were Nd-YAG (~'exc 532 nm) and flashlamp-pumped dye lasers (~'exc 490-525 nm). When shorter wavelengths are used, the desired process in accompanied by increasing amounts of CO loss; this is in general undesirable because it introduces other transients into the system. The methods of generation of the radical and of analysis of different types of kinetic data have been referred to. ~-6 Optical detection of species was made, primarily of the parent dimer itself or of certain products referred to later.
20? IV.
Results
Radical recombination. The results p e r t a i n i n g to m a n y of the reactions in S c h e m e I will n o w be presented, starting with (a). W h e n a solution of the metal dimer is subjected to a laser flash, -25% of the d i m e r is dissociated; in a typical experiment a 20 ~M solution of [CpM(CO)3] 2 (M = Mo, W) yields about 25 ~M of the dimer. A slightly different p r o c e d u r e was u s e d for M = Cr, but with comparable results. 7 The recombination of the radicals occurs o v e r about 300 I~s a n d follows s e c o n d - o r d e r kinetics. A typical e x p e r i m e n t for the m o l y b d e n u m radical and the fit to s e c o n d - o r d e r kinetics is s h o w n in Figure 1. The rate constants (k/109 L mol d s -1) in acetonitrile at 23 ~ are: Cr: 0.27, Mo: 2.16, and W: 4.7. In other organic solvents the rate constants are c o m p a r a b l e to these, reflecting the relatively small differences in viscosity. In a q u e o u s solution (CsH4CO2-)Mo(CO)~" has k/109 L mol -I s "I = 3.0.
Absorbance
i
0
0
1 10-4 2 104 Time/s
Figmre 1. The results of a laser flash photolysis experiment, showing the combination reaction of CpMo(CO)3", as monitored by the recovery of the dimer at its 384 run maximum. The smooth curve through the points shows the fit to second-order kinetics. On closer inspection, the c o m b i n a t i o n rate constants are a b o u t 1 / 4 of the estimated diffusion-controlled rate constant. For acetonitrile, for example, kdc .- 2.9 x 1010 L mol "1 s -1 from the y o n S m o l u c h o w s k i e q u a t i o n 8 with a diffusion coefficient from a modified version of the Stokes-Einstein relation, D - . kT/4xTlr. 9,10 O w i n g to the restriction to singlet state recombination, an experimental rate constant 1 / 4 of kdc is quite reasonable. O n the other hand, for these h e a v y metals, the spin restriction m a y not apply, in which case one w o u l d a r g u e that the geometrical a n d o r i e n t a t i o n a l r e q u i r e m e n t s of these large species c o u l d well give r e c o m b i n a t i o n rates s o m e w h a t b e l o w the theoretical m a x i m u m .
208
Halogen atom abstraction. The metal radicals react with alkyl and aralkyl halides (RX, X = C1, Br, I) to yield CpM(CO)3X and R ~ The metal halide p r o d u c t is a species k n o w n i n d e p e n d e n t l y , and its f o r m a t i o n in these reactions was verified by IR. The organic free radicals w e r e not detected directly, but were inferred from the final organic products determined by GC; these products were consistent with those k n o w n to be f o r m e d when R ~ undergoes dimerization and disproportionation reactions. A large n u m b e r of rate constants for organic halides have been reported. 2 A small selection will be presented here. First, the trend with the group R, and then with X, using data for CpW(CO)3": Alkyl group trends
I XUrl: .22 •
[ Bu': 6.0 •
Halide trends AUyl, CH2CHCH2-X Trichloromethyl, C13C-X
k/L mol "1 s "1
Pr': 3.0 •
! Pr":
•
[M,, < 6 •
I
k/L mol "1 s -1 X = I: 1.22 x 108 X = Br: 5.3 x 108
X = Br: 5.4 x 104 X = CI: 2.92 x 104
These results suggest that the transition state features an incipient free radical (note the trends with R), and that the metal atom has begun to make a b o n d to the halogen of appreciable strength. The t r e n d with halogen substitution is a particularly pronounced one, since the trends in BDE(R-X) and BDE(M-X) are in the opposite direction (that is, the low-valent metal center is a soft acid). H a l o g e n atom abstraction occurs also with m e t a l halides, both ( N H 3 ) s C o l I I - x 2+ and X-RhIII(dmgH)2PPh3 having been studied. For the latter, kcl = 9.2 x 107 and kBr = 1.57 x 108 L mo1-1 s -1 for the reactions of C p W ( C O ) 3 ~ The i m m e d i a t e p r o d u c t of the r h o d i u m reaction is the i n d e p e n d e n t l y - k n o w n r h o d i u m radical, RhIl(dmgH)2PPh3 ~ a 17e species. It in turn dimerizes to yield the stable dimer, [Rh(dmgH)2PPh3] 2. Both of these p r o c e s s e s can be m o n i t o r e d d i r e c t l y , the b u i l d u p a n d decay of RhII(dmgH)2PPh3 ~ at 580 nm and the formation of the r h o d i u m dimer at 452 nm. The mechanistic discussion concerning the alkyl halides is pertinent here as well; the rhodium system allowed the direct detection of the radical product, which had only been inferred in the organic reactions.
Hydrogen atom transfer also occurs, as represented in one instance by the reaction of CpW(CO)3 ~ with CpMo(CO)3-H, a process driven by the higher bond energy of the tungsten hydride. 11 The atom transfer process is
209 not to be confused with proton transfer; the reaction of CpMo(CO) 3- with CpMo(CO)3-H is also known.
Reduction of the metal radicals. The anionic complexes CpM(CO) 3- are well k n o w n species; they are stable entities with 18 valence electrons. The standard reduction potential for the CpMo(CO)3~ - couple is -0.08 V vs SSCE. 12 The m o l y b d e n u m radical is thus a mild oxidizing agent; with suitable electron donors it can be reduced to the anion. For example, the radical oxidizes Fe(T15-CsMes)2 with a rate constant of 2.2 x 108 L mo1-1 s "1 in acetonitri]e at 23 ~ 6 Application of the Marcus equation for electron transfer affords the electron exchange rate of the molybdenum radical/anion couple. The value is kee = 3 x 107 L mo1-1 s-1. The high value argues that very little nuclear reorganization is needed to add an electron to the SOMO of the 17e radical. Oxidation of the metal radicals. The cationic d e r i v a t i v e s of these o r g a n o m e t a l l i c c o m p o u n d s contain an a d d i t i o n a l ligand. For example, CpM(CO)3PPh3 + and CpM(CO)3NCCH3 + are readily obtained; the latter can be g e n e r a t e d electrochemically in acetonitrile. The C p M o ( C O ) 3 N C C H 3 +CpMo(CO)3 ~ couple has a standard electrode potential of-0.50 V vs SSCE. 12, which shows that the metal radical is a strong reducing agent. For example, Fe(TIS-CsHs)2 + reacts with CpW(CO)3 ~ with a rate constant of 1.9 x 107 L mol d s "1 in acetonitrile at 23 ~ 6 As large as this this value is, it corresponds to an electron exchange rate between CpMo(CO)3NCCH3 + and CpM(CO)3 ~ of only -10 -12 L mol d s -1. The small value signals a large inner-shell reorganization; the two species differ by one coordinated molecule of solvent. Ligand-catalyzed disproportionation; the role of 19e radicals. There is now considerable evidence for the association of a 17e radical and Lewis bases, such as phosphines, pyridine, and even acetonitrile. The resulting species has the formula CpM(CO)3 L~ and is often referred to as a 19e radical. It is probably just that, but it might instead represent a species with a "slipped" Cp ring, T!3CsH 5. In any event, the 19e species is a much stronger electron donor than C p M ( C O ) 3 L itself. The complex CpW(CO)3PPh3 ~ which has a formation constant of 6 L mol d, reacts at a diffusion-controlled rate with CpW(CO)3 ~ The products are those of the pure electron transfer steps that occur without solvent or ligand redistribution and without W - W b o n d formation. The occurrence of electron transfer alone is the feature that m a k e s the reaction so efficient. The diagram in Scheme II details the various reactions
210 and it presents the electron count for the various species. The electron count is the guide to reactivity.
S c h e m e II:
A t o m Transfer vs Electron Transfer: Electron C o u n t s +L
CpMo(CO)3 ~
~
~-~
C p M o ( C O ) 3 L ~ FeCp2+ ~
CpMo(CO)3 L+
p2§ CpMo(CO)sX + R ~ CpMo(CO)3 + CpMo(CO)3NCCH3 +
Radical disproportionation induced by electron transfer. Several interesting and interrelated p h e n o m e n a occur when N,N,N',N'-tetramethyl1,4-phenylenediamine (TMPD) is a d d e d to the solution of [CpM(CO)3] 2 prior to the laser flash. As expectedfrom the electrode potential of TMPD ~ 0.16 V vs SSCE in acetonitrile, the first event is the rapid growth of the intense absorption band of the amine radical cation centered at 613 nm (e = 1.2 x 104 L mol "1 cm-l). This absorption then fades fairly rapidly. The fading was quite u n e x p e c t e d , since TMPD ~ n o r m a l l y persists indefinitely. This phenomenon is illustrated in Figure 2. Indeed, when independently-prepared T M P D ~ was injected into the cuvette after the laser flash its blue color persisted indefinitely. Clearly, the disappearance of TMPD ~ signals that some intermediate in the system is causing its destruction. Tests based on the kinetics and on the addition of CpMo(CO) 3- and CpMo(CO)3NCCH3 + to the reaction mixture and to samples of TMPD and TMPD ~ have revealed the sequence of chemical events. 4 The reaction occurs by a catalytic cycle of electron transfer events. The net result is the disproportionation of CpMo(CO)3 ~ Scheme III shows the catalytic cycle.
211
0.08
9
,
,
,
,
[Mo'l/gM a 40.5 b 28.4 c 9.76 d 4.43
0.06 ~ a ~ }b\\ 0.04 J \ \ 0.02 0.00 I 010 o
I
I
I
I
1 104
2 10-4
i
3 104
Time/s
Figure 2. Experimental absorbance changes following [TMPD~ at 613 nm in a series of experiments with different initial concentrations of [CpMo(CO)3]2. In each, [TMPD]0 --- 5.00 mM. (Mo~ = CpMo(CO)3"). The fitting to the model in Scheme III is shown by smooth curves.
S c h e m e III
CpMo(CO)3 +
CpMo(CO)3 ~/
CpMo(CO)3 ~
\
~
~, CpMo(CO) 3-
The t h e r m o c h e m i s t r y of this system, discussed intermittently in w h a t has c o m e before, can be s u m m a r i z e d here. D i s p r o p o r t i o n a t i o n is less favorable t h e r m o d y n a m i c a l l y than recombination; the e q u i l i b r i u m constant is 10 7 as c o m p a r e d to 1016. Nonetheless, disproportionation can be m a d e to be the more i m p o r t a n t reaction, even the nearly exclusive one, by virtue of the c o n t r o l e x e r c i s e d by t h e c o n c e n t r a t i o n of T M P D . T h e p r o d u c t s of disproportionation, M o - a n d M o - A N +, do n o t react r e a d i l y . The disproportionation products are " t r a p p e d " as such. In other words, the metal radicals are c a p t u r e d into an electron transfer cycle w h i c h p r e v e n t s their falling to the most stable products, the dimer. The energetics of this system are depicted in Figure 3.
212 Electron-Transfer Catalysis to the Disfavored Product 1:
~
.... 7--Yi
/
F
AG~= 32.1kJ
....... \
- '
\
,
Figure 3. The reaction coordinate diagram comparing the dimerization (left) and electron transfer (right) processes. Note that the ionic products of disproportionation, the products of photolysis in the presence of TMPD, are very slow to yield dimer. One should note that the light energy provided by the initial laser flash has thus been "trapped" in part in the form of the chemical energy of the ionic pair relative to the dimer. They lie in a free energy well above the thermodynamic product, the Mo 2 dimer, but do not rapidly revert to it.
Acknowledgment. This work was supported by the U. S. Department of Energy, Office of Basic Energy Sciences, Chemical Sciences Division under contract W-7405-Eng-82. The author is grateful to many individuals for their assistance: J. Balla, A. Bakac, W-J. Chen, O. Pestovsky, S. Scott, T-J. Won, Q. Yao, and Z. Zhu, and to the Yamada Foundation for sponsoring the conference for which this paper was presented. References (1) Scott, S. L.; Espenson, J. H. Organomet. 1993,12, 4077. (2) Scott, S. L.; Espenson, J. H.; Zhu, Z. J. Am. Chem. Soc. 1993,115, 1789. (3) Scott, S. L.; Bakac, A.; Espenson, J. H. Organomet. 1993, 12, 1044. (4) Balla, J.; Bakac, A.; Espenson, J. H. Organomet. 1994, 13, 1073-1074. (5) Zhu, Z.; Espenson, J. H. Organomet. 1994, 13, 1893-1989. (6) Scott, S. L.; Espenson, J. H.; Chen, W.-J. Organomet. 1993, 12, 4077. (7) Yao, Q.; Bakac, A.; Espenson, J. H. Organomet. 1993, 12, 2010. (8) vonSmoluchowski, M. Z. Phys. Chem. 1917, 92, 129. (9) Edward, J. T. J. Chem. Educ. 1970, 47, 261-270. (10) Beckwith, A. L. J.; Bowry, V. W.; Ingold, K. U. J. Am. Chem. Soc. 1992, 114, 4983-4992. (11) Pestovsky, O.; Espenson, J. H. unpublished results (12) Pugh, R. J.; Meyer, T. J. J. Am. Chem. Soc. 1992, 114, 3784.
joumal of
MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 2 ! 3-219
The Structure of Metal Complexes of Small Models of Glycoproteins in Aqueous Solution
K._l~_urg~, L. Nagy and B. Gyurcsik
Department of Inorganic and Analytical Chemistry, A. J6zsef University, Szeged, Hungary
Abstract The combination of potentiometric equilibrium measurements and NMR, EPR, CD, EXAFS and M6ssbauer structural studies of metal complexes of small models of the moiety of glycoproteins which connects the protein and carbohydrate parts of the macromoleeule permitted determination of the composition and structure of the coordination sphere of the metal in equilibrium species in solution. Introduction Natural and synthetic compounds with biological activity exert their biological effects only in solution. Simple relationships between the solid-state structures of such compounds and their biological activities can not be expected, since dissolution of the substances in water may change their structure and composition dramatically. For kinetieally labile metal complexes, the presence of several species in equilibdurn with one another in the same solution makes the picture even more complicated. This is why the equilibrium study of complex formation in biologically active systems has to be complemented by structural investigations. To model the moiety of glycoproteins which connects the protein and carbohydrate parts of the molecule, amino acid - carbohydrate adducts (e.g. N-D-gluconylamino acids, 1 thiazolidine-4-carboxylic acid derivatives of carbohydrates, 2"4 D-fructose amino acid derivates, 5 etc.) have been synthetized. The protonation and metal ion (Cu2+, Ni 2+, Co2+, Zn2+ and Et2Sn2+) coordination equilibria of these model compounds have been studied by means of potentiometric equilibrium measurements. 1-5 CD, EPR, NMR, EXAFS and M6ssbauer investigations (the latter in frozen solutions) have been used to determine the structure and symmetry of the coordination sphere of the complexes.l'5, 6 Some typical examples are shown below. Copper(II) complexes of N-D-gluconylamino acids N-D-Gluconylglycine, a pseudopeptide derivative of D-glucono-5-1actone and glycine, was prepared according to ref. 7. The equilibrium constants of its copper coordination, determined according to ref. 1 (Table 1), were used to calculate the concentration distribution curves in Fig. 1, which show the compositions and protonation states of the different species in solutions of different pH. 0167-7322/95/$09.50 9199.5 Elsevier Science B.V. All rights reserved. SSD1 0167-7322 (95) 00854-3
214
100.
.,=.,.=.,,..=
M
MLH_3 ML
MU-L2
03
,
2
i
4
i
6
8
J
10
\
pH Figure 1. Concentration distribution curves of the system coppcKII) - N=D-gluconylglycinc at a metal to ligand ratio of 1:8, [Cu 2+] = 5.0x10 .3 mol rim-3.
13C.NMR relaxation measurements in solution of pH ~ 8.6 suggested the participation of carboxylate, deprotonated amide and deprotonated alcoholic OH groups in the chelate ring of the copper(II) complex formed in the physiological pH range. Since the complex in which this ligand is coordinated only by carboxylate-O and amideN atoms would not be optically active, CD spectroscopy was used to confirm the coordination of the alcoholic OH group on C(2) of the sugar moiety to copper(II), resulting in the formation of a second chelate ring, beside that between the carboxylate-O and amide-N donor atoms. Similar extra stabilization was observed in complexes of carbohydrate derivatives of thiazolidine-4carboxylic acid.2, 3 This effect was shown to depend on the conformation of the sugar moiety. EPR spectra recorded for solutions of different pH were simulated on the basis of the assumption of coordination spheres with different compositions and structures. These measurements indicated the coordination of one N donor atom to copper(II) at pH > 6 in MLH.2, and dimerization of the complex between pH 6 and 10 (M2L2H. 4, M2L2H_3). The EPR spectra also reflected the decomposition of the dimeric species at pH > 10, i.e. the formation of MLH. 3. In general, the amino acid residues of these organic ligands dominate as the primary binding site in metal ion coordination. Therefore, variation in this part may result in complexes with completely different stabilities and structures. Among the amino acids with weakly or noncoordinating side-chains, oc- and 13-amino acid-containing ligands are expected to show the largest differences in chemical behaviour. This appears, for instance, in the ligand excess needed to keep copper(II) ions in solution. However, in spite of the increased distance between the
215
carboxylate and other donor groups in comparison with the a-amino acid derivatives according to both CD and EPR investigations, the species MLH_ 1 and MLH. 2 proved to have analogous structures in the copper(II) complexes of tz- and 13-amino acid-containing ligands. The only difference in the two systems is the higher pK 2 of MLH. 2 (Table 1).
Table 1. Overall formation constants (log 13) for the complexes and pK values for deprotonation processes in the systems copper(II) - N-D-glueonylamino acid (T = 298 K, I = 0.1 mol dm -3, NaCIO4)
Composition HL CuL CuL2.' . CuLH. 1 CuLH. 2 . CuI~.H. 3 Cu2L2H-3 Cu2L2H-4
'
,,,
iii
.
pK1 pK2 _ p8 3
N-])-gluconyl-13-al~e 4.24(1) 2.84(6) 4.87(10) . -2.86(8) -9.45(10) . -20.54(1 O) -8.70(7) -16.78(16) 5.70 " 6'.5'9 .
.
.
.
11.09
N-D-gluc0nylglycine* 3.39 1.94 3.51 -3.82 ._-...... -9.63 ... '-20.0 ] -10.20 -16.63 5.76 5.81 10.38
i
i i
ill
9
i
9
Standard deviations in parenthesis * from ref. 1
EXAFS measurements 6 provided information on the coordination number of the copper, and on the Cu-O/Nequ, Cu-Oax bond lengths and Cu.-.C and Cu...Cu nonbonding distances for the N-D-glueonylglycine complexes in aqueous solution at two different metal:ligand ratios (Table 2). The data distinguished between equatorial Cu-O/Neq u and axial Cu-Oax bond lengths. The latter was found to be significantly shorter than that expected for coordinated water molecules, indicating that the axial donor O atoms are coordinated more strongly than water. NMR and CD measurements indicated that these are OH O atoms of the ligand. 1 Change of the metal:ligand concentration ratio from 1:10 to 1:100 did not cause significant changes in the CuO,Nequ, Cu-Oax and C...C distances, but increase of the ligand excess prevented dimerization (Cu...Cu interaction) in the complex.
216
Table 2. Structure parameters of the copper(II) complexes of N-D-gluconylglycine in aqueous
solution
Metal:ligand ratio 1"10
Coordination number
4 1
4
1"100
Atomic distances
pm
Cu-O/Nequ Cu-Oax Cu...C Cu...Cu Cu-O/Nequ Cu-Oax Cu...C
190 215 270 298 193 218 272
The structures of the species suggested on the basis of the complex investigations outlined above are given in Fig. 2. o
o
0
c ----~ O'l,C),=
0 ....
H.~
:
c~o"
. .OH,
\
(H,C)~--- N"
~ O"
OH
:/
c ---~, CH
!\
o~
!\ [~R,J~.,II
0
Y \ ?"-"
o
/
~c),,
c--o'\ Cu"' o:,,
~.~/\ //
....HO" ! ~ O (c~. ,, _. cu~..... / i ~cx.i " o~c o
o
\
O.~c),~-- N- /
:
~Or --"
o.~C~ \
o~c
\\ o
o
~t~ Figure 2. Structures of the species formed in the equilibrium system in the physiological and basic pH range
217
Diethyitin(IV) complexes of NoD-gluconylamino acids A similar series of complex investigations was performed on diethyltin([V) complexes of N-D-gluconylamino acid ligands. 8 The potentiometrie equilibrium measurements led to the concentration distribution curves in Fig. 3.
100 A
~" o .w
~.
eo ~,
6o
i_
"i
~
,,o
o
ffl
3
4
5
6
8
9
10
pH
Figure 3. Concentration distribution curves of the system diethyltin(IV) - N-D-gluconylglycine at a metal to ligand ratio of 1:10, [Et2Sn2+ ] = 1.0xl0 -1 mol dm-3. 13C-NMR investigations revealed which donor atoms participate in the complexation process. As a result of the very slow ligand exchange in comparison with the NMR time scale, the 13C signals of the bound and the free ligands were observed separately in the 13C-NMR spectrum of the complex MLH_ 2 of diethyltin(IV)-N-D-gluconylglycine. The significant shifts of the -CH2_, -COO- and -CONH- C signals of the ligand (the latter two were distinguished via the proton-coupled 13C-NMR spectra) suggested coordination of the carboxylate group and the deprotonated peptide-N. Shifts observed in the C signals of the polyhydroxyalkyl chain indicated that the alcoholic OH groups are also involved in the coordination. The shift of one of latter signals is significantly larger than that of the others, suggesting the stronger interaction of this specific (and therefore presumably deprotonated) alcoholic OH group, assigned to C(2), with the metal ion. The 13C-NMR spectnun of the N-D-gluconyl-13-alanine complex, soluble in water at pH ~ 10, reflected a different structure. The differences in the chemical shifts of the ligand C atoms in the presence and absence of diethyltin(IV) are small, but the signals of the carboxylate and three other C atoms (C(3), C(4) and C(5)) which carry alcoholic OH groups are broadened, which is characteristic of complexes with an intermediate ligand exchange rate. The above
218
observations suggested coordination of the carboxylate and deprotonated alcoholic OH groups to the organotin(IV) cation. Neither a noticeable shift nor line broadening was observed for the amide and methylene C signals, excluding deprotonation and consequently coordination of the amide-N. Mtssbauer study of quick-frozen solutions of the diethyltin(IV)-N-D-gluconylglycine complexes permitted determination of the geometry of the species in solutions of different pH. The Mtssbauer spectra recorded for the system at pH ~ 4 indicated the presence of two overlapping doublets (Fig. 4), in good agreement with the equilibrium analysis, due to the presence of species ML and MLH. 1 in the same solution. A comparison of the experimental quadrupole splitting (QS) values with those calculated on the basis of the partial quadrupole splitting (PQS) concept 9"10 revealed that the complex ML contains a pentacoordinated, trigonal bipyramidal central tin(IV) atom, while MLH. 1 contains a hexacoordinated, distorted octahedral one. The Mtssbauer spectra recorded for systems at pH between 6 and 9 reflected the presence of only one species, which was shown by equilibrium measurements to be MLH_2. PQS treatment of the data indicated a penta-coordinated central tin(IV) atom in latter complex.
..... .._..---~-_.. ~..
~ -.'..-.... ,\9":..'~ '~ ',. ', I'.'.
:..-- ~::---
.'. i.
I .... ~ . . . . .
-0
-4
..
~]1
'"J
.~,
\~
-q
0
"..,
,
4~
~.[
_0 v ( ~ / s )
Figure 4. 119Ti n Mtssbauer spectrum of the system diethyltin(IV) - N-D-gluconylglycine at a metal to ligand ratio of 1"10, [Et2Sn2+ ] = 0.1 mol dm "3, pH ~ 4, at liquid nitrogen temperature Conclusions
The combined application of equilibrium and structural studies in solution led to the following main conclusions on the studied systems: 1. Equilibrium investigations permitted determination of the compositions and concentrations of successively formed metal complex species in solution. The concentration
219 distribution curves of the species in solution permitted the assignment of independent experimental structural data to the single species in solution. 2. NMR spectroscopy could be used to determine which donor atoms of the polyfunctional ambidentate ligands participate in the coordination processes. 3. The symmetry of the coordination sphere of the complexes could be determined by M6ssbauer measurements on quick-frozen solutions; even the differences in structure of coordination isomers in different protonation states could be demonstrated. 4. The effects exerted by the configuration of the ligand on the complexation process could be determined by means of CD studies.
Acknowledgement The present work was supported by the Hungarian Research Foundation (Grants OTKA 84/1991 and BO11045)
References .
2. 3. ~
10.
B. Gyuresik, T. Gajda, L. Nagy, and K. Burger, d. Chem. Soc. Dalton Trans., 1992, 2787 T. Gajda, L. Nagy, and K. Burger, J.. Chem. Soc Dalton Trans., 1990, 3155 T. Gajda, L. Nagy, N. Rozlosnik, L. Korecz, and K. Burger, d. Chem. Soc. Dalton Trans., 1992, 475 N. Buzz, B. Gyurcsik, L. Nagy, Ying-xia Zhang, L. Koreez, and K. Burger, lnorg. Chim. Acta, 218, 65 (1994) B. Gyurcsik, T. Gajda, L. Nagy, K. Burger, A. Rockenbauer, and L. Korecz, lnorg. Chim. Acta, 214, 57 (1993) L. Nagy, T. Yamaguchi, T. Mitsunaga, B. Gyurcsik and H. Wakita, to be published F. Schneider and H. U. Geyer, Hoppe Seyler's Z Physiol Chem., 330, 182, 189 (1963) B. Gyurcsik, N,. Buzz, T. Gajda, L. :Nagy, E. Kuzmann, A. V6rtes, and K. Burger: Z Naturforschung, to be published G. M. Bancroft: Mi~ssbauer Spectroscopy: An Introduction for Inorganic Chemists and Geochemists, McGraw-Hill Book Company, U.K. (1973) G. M. Bancroft, V. G. Kumar Das, T. K. Sham, and M. G. Clark, J. Chem. Soc. Dalton Trans., 1976, 643
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journal of MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 221-228
Kinetic Evidence for Short-Lived Intermediates in Metailoporphyrin Formation Masaaki Tabata Department of Chemistry, Faculty of Science and Engineering, Saga University Honjo-machi, Saga, 840 Japan Abstract The rate of metalloporphyrin formation is accelerated by (1) large metal ions like mercury(II) and cadmium(II), (2) reducing agents like hydroxylamine, and (3) amino acids. The detailed reaction mechanism was studied from kinetic and spectral methods for the incorporation of metal ion into 5,10,15,20-tetralds(4-sulfonatophenyl)porphyrin (H2tpps 4-) or 5,10,15,20-tetrakis(1-methylpyridinium-4-yl)porphyrin (H4tmpyp4+). The main focus is on the intermediates formed in the reaction of metalloporphyrin formation, i.e. heterodinuclear metalloporphyrin, mixed-valence metaUoporphyrin, and molecular complex with amino acid. Cd(tpps) 4- is 7900 times as reactive as H2tpps 4- in the incorporation of manganese(II). The coordination of cadmium(II) deforms the porphyrin ring favorably for the attack of mediumsized metal ions and a heterodinuclear metalloporphyrin was postulated as an intermediat~ ~The heterodinuclear metalloporphyrin was detected kinetically in the reaction of zinc(II) or copper(II) with homodinuclear mercury(II) porphyrin complex of Hg2tpps2-. The reaction showed a biphasic kinetic behavior of the following steps: Hg2(tpps) 2- + M 2+ --, Hg(tpps)M 2+ Hg 2+ - M(tpps) 4" + 2Hg 2+, where M 2+ denotes Cu2+ and Zn 2+. The rate constant for the second-step reaction was characteristic of a unimolecular reaction: the rate constant was independent of copper(II), zinc(II), and mercury(II) concentrations, and also of hydrogen ion concentration at high pH. An extended X-ray absorption fine structure (EXAFS) measurements revealed the deformed structure of Hg2(tpps) 2- for the metal-substitution reaction of Hg2(tpps) 2- with Cu 2+. The rate enhancement by 104 in the presence of reducing agent was attributed to the formation of Cu(1) (r = 96 pro). The Cu(I) behaves similarly to the large metal ions. Amino acids enhanced the formation of Zn(tpps) 4" and formed a molecular complex as an intermediate. The rate constant was linearly correlated to the hydrophobicity scale of the side chain of the amino acid. 18-Crown-6 stabilized the metalloporphyrin towards the ligandsubstitution or acid-dissociation reaction of the metalloporphyrin by the formation of a sandwich-metalloporphyrin [M(tpps)18C6] 4-. The intermediate was confirmed by 1H NMR and spectrophotometry.
I Introduction A number of studies on metalloporphyrin formation have been reported over 20 years, as a result of the important role of metalloporphyrins in biological systems. The general mechanism for the metalation of porphyrins was first ]?l'Oposed by Hambright and Chock, 1 and has been reviewed from time to time by Hambright,Z Lavallee3,4 Schneider,5 and Tanaka. 6 The rate of metalloporphyrin formation is several orders of magnitude slower than that of the 0167-7322/95/$09.50 91995 ElsevierScience B.V. All rights reserved. SSDi 0167-7322 (95) 0090 !-9
222
complex formation of open-chain ligands, 7 and the over all mechanism is not straight forward in some cases. 8 "The great difference in their reaction rate is presumably due to a less flexible porphyrin ring system and a less accessible lone-pair electron on the porphyrin for the incoming metal ions. In order to accelerate the slow metalation, several methods have been proposed:9 (1) use of substitution reaction of cadmium(ll) or mercury(II) porphyrin, (2) use of N-substituted porphyrins at the pyrrole nitrogen, (3) addition of aromatic heterocyclic bases such as pyridine and imidazole, (4) introduction of functional groups to bind metal ions in the vicinity of the porphyrin nucleus (e. g. tetracarboxylic acid "picket-fence" porphyrins) and (5) use of reducing agents such as hydroxylamine and ascorbic acid in copper(lI) incorporation. The present paper describes the mechanism of porphyrin metalation in the presence of catalyst with special emphasis on the reaction intermediates including heterodinuclear metalloporphyrin, mixed-valence metalloporphyrin, and sandwich-metalloporphyrin with 18crown-6. II
Catalytic Effect of Mercury(ll), Cadmium(ll), and Lead(ll) in the Metailoporphyrin Formation I 0-12
The rate of incorporation of manganese(II) into H2tpps 4- is very slow, and a little spectral change was observed even at 60 minutes after the o initiation of the reaction 25 *C. The reaction proceeded only 25 % after two days. However, in the presence of ~e mercury(II) or cadmium(ll) rapid spectral ~' change occured: the absorbance at 413 "nm decreases and the absorbance at 467 nm increases. 11,12 I n the metalloporphyrin formation catalyzed by metal z ion, the ionic radius(r) of the metal ion acting as catalyst is important: large metal I v --J l-t ions such as Cd 2+ (r = 95 pm), Cu+(r = 0~ pH 96 pm), Pb 2+ (r = 118 pm), and Hg 2+ (r =114 pm) give rise to appreciable catalytic effect, while medium-sized metal Figure 2. A typical biphasic kinetic run for the ions like Cu 2+ (r = 73 pro) and Zn 2+ (r = reactionof Hg2(tpps)2- with Zn2+. 75 pm) exert hardly the catalytic effect. In Fig. 1 the observed rate constant is plotted versus pH. 10 The rate of formation of Mn(tpps) 4- in the absence of cadmium(ll) is independent of pH, while in the presence of cadmium(II) the rate depends on pH. Cadmium(lI) accelerates the rate of Mn(tpps) 4formation at pH higher than 6. The following mechanism was proposed for the formation of Mn(tpps) 4- in the presence of cadmium(II). 10 _
,
r
--
_
6
.
(hI
.
-
.a
-
,-
7
8
9
kl Mn 2+ + H2tpps 4-
- 9Mn(tpps)4- + 2H +
(1)
Cd(tpps) 4- + 2H +
(2)
k2 Cd 2+ + H2tpps 4k_2
223 k3 Cd(tpps) 4- + Mn 2+ ---" Mn(tpps) 4- + Cd 2+
(3)
At low pH and/or low concentration of manganese(II), the rate law is rewritten as kCd =
k2k3k-2-1[Mn2+][Cd2+][H+]-2
(4)
On the other hand at higher pH, the rate equation is simplified to kCd = k2[Cd 2+] (5) The rate constants at pH 7.8 to 8.7 are independent of [Mn 2+] over the concentration range from I x 10-4 to 10-3 tool dm -3 as expected from eq.5. An independent experiment revealed that manganese in the product is in the trivalent state and that the rate of oxidation of Mn(II)(tpps) 4- to Mn(III)(tpps) 3- by dissolved oxygen is rapid compared to the formation of Mn(II)(tpps) 4-. The Mn(tpps) 4- formation in the presence of cadmium(II) proceeds through Cd(tpps) 4-Because of large ionic radius of cadmium(II) (r=- 95 pm), the coordination of cadmium will give a highly distorted porphyrin core which favors the attack of porphyrin nucleus by manganese(II) from underneath. Cd(tpps) 4- is 7900 times as reactive as H2tpps 4- in the incorporation of manganese(II). Cadmium(II) liberated after the incorporation of manganese(II) combines again with the free base porphyrin as a catalyst. At pH > 7.8, the formation of Cd(tpps) 4- is rate-determining step (eq.2): the intermediate involves only cadmium(II), and it is of course mononuclear. At low pH and/or lower concentration of manganese(II) eq.3 is the rate-determining step and the rate law given by eq. 4. The formation of Mn(tpps) 4- proceeds through a heterodinuclear intermediate involving cadmium(II) and manganese(II).
III
Heterodinuclear
Metalloporphyrin
as
Intermediatesl
3
Heterodinuclear metallopor~hyrin, postulated in the metalloporphyrin formation catalyzed by mercury(II) or cadmium(II) 1U-12 was detected kinetically in the reaction of zinc(II) or copper(II) with homodinuclear mercury(II) porphyrin complex of Hg2tpps 2-- The following biphasic reaction was observed: Hg2(tpps) 2- + M 2+ --~ Hg(tpps)M2- + Hg 2+ ~ M(tpps)2- + 2Hg2+ (6) where M 2+ = Zn 2+ or Cu 2+Dinuclear mercury(II) porphyrin forms quantitatively in large excess of mercury(II) at pH 6.15 The first-step reaction in eq. 6 completes in a few seconds and it is followed by the second-step reaction with a half-life time of a few minutes (Fig. 2). 13 The rate constant for the first-step reaction increases with increasing c6ncentration of copper(II) or zinc(II) and reaches a plateau. The rate constant also decreases with increasing mercury(II) concentration (Fig. 3a). These observations lead to the following rate law:
0.2
0.1 r~ .< 0.0 0
5
10 tls
15
20
Figure 2. A typical biphasic kinetic run for the reaction of Hg2(tpps)2- with Zn2+
224
(7)
ko f = (k lk2[M2+][OH-])(k, l [Hg(OH)+] + k2[M2+]) -1
The plot of (k0f)" 1 vs. [M2+] - 1 gives a straight line with an intercept. The rate constants for the second step reaction are independent of the concentrations of mercury(II), copper(lI), and "Hg zinc(II) (Fig. 3b). 13 These results indicate; the unimolecular reaction, i.e. the self-dissociation [kz. MV of the intermediate. The kinetic results of the two-step reaction are consistent with the reaction mechanism shown in scheme 1. Hg The reaction paths for k l , k-l, and k2 , , ,,N k3 correspond to the first-step reaction, and the Hg(oa)" ~ ~ / 1 ~ ) ----------reaction paths for k3 and k4 are involved in the ~t...~N :x . / U~/uN second-step reaction. The rate constants are \ H / summarized in Table 1. T he homodinuclear mercury(ll) k, M = znz'. cuz" porphyrin dissociates to a 1:1 mercury(II) Scheme 1. Reaction mechanism for the reaction porphyrin which reacts with z i n c ( l l ) or of Hg2(tpps)2- with Zn 2+ or Cu2+. c o p p e r ( I I ) to f o r m the c o r r e s p o n d i n g heterodinuclear metalloporphyrin. The final step is the dissociation of mercury(II) from the 103[Zn=~ll t o o l d r n " corresponding heterodinuclear metalloo I 2 3 4 porphyrin. Interestingly k l, k3, and k4 values for zinc(ll) are a l m o s t the s a m e as for copper(II), while the rate k-2/k-1 for copper(ll) "7 m z is about 80 times as high as for zinc(lI). These findings are consistent with the p r o p o s e d "~ reaction mechanism. The reaction paths for k 1, t k3, and k4 involve only the dissociation of mercury(II) and thus they are independent of the 0 nature of incoming metal ions. On the other hand the reaction path for k2 depends on b characteristics of incoming metal ions, because "7 :2 "-1 it involves coordination of these metal ions to form heterodinuclear metalloporphyrins. " o The r e a c t i o n m e c h a n i s m was also ,,,-~ / confirmed by a series of absorption spectra o recorded on a rapid-scan spectrophotometer. The c h a n g e in a b s o r b a n c e at 5 6 4 n m 0 (absorption m a x i m u m o f Hg2(tpps) 2-) is o ~ ~. ~ ~ ',o smaller than for the second-step reaction. This Io'(znZ~lmol dm 3 finding points to one mercury(II) ion remaining Figure 3. Dependence of rate constants for the in the intermediate. The increase in absorbance first-step (a)and the second step (b) reactions, a, at 551 nm during the first-step reaction suggests CHg = 1.00 x 10-5; 5.02 x 10"4, b, 5.02 x 10-5; the coordination of zinc(II) to form the 9.58 x l0 -5 tool dm -3. hetrodinuclear metalloporphyrin. 9
,
,
i -
v
,,
- -
~
9
9
~
0
i
225 Table 1. The rate constants for the reaction of Hg2(tpps) 2- with Cu 2+ and Zn 2+ at 25 *C, and I = 0.1 (NAN03).
Metal ion
k21k-1
10-8kl mol- ldm -3 s- 1-
Cu 2+
1.95 • 0.05
Zn 2+
1.56 + 0.06
102k3 s- 1
10-3k4 tool- 1din3 s- 1
1.00 • 0.03
2.70 + 0.06
(9.6 • 0.2) x 10 -2 0.92 • 0.02
1.75 + 0.03
7.1 • 0.3
IV EXAFS Study of Reaction of Hg2(tpps)2- with Copper(ll)lS,16 Kinetic studies have p r o p o s e d evidence for the formation of heterodinuclear intermediate. In order to study the structure of the intermediate in solution as r e l a t e d to the r e a c t i o n mechanism, extended X-ray absorption fine structure (EXAFS) measurements have been u n d e r t a k e n for the metal-substitution reaction of mercury(lI) porphyrin complex with copper(II) in an acetate buffer (pH =
+
?
5.6). Cu 2+ + Hg2(tpps) 2- -~ Cu(tpps) 4- + 2Hg 2+
(8) Mercury(II) is readily replaced by copper(II) ~on at rate faster than direct reaction of copper(II) with the free base porphyrin H2tpps 4-. Nine solutions were prepared and EXAFS was measured at both Hg LIII and Cu K edges. The structure of acetatomercury(II) is tetrahedml and the Hg-O (CH3COO-) bond length of 219 pm is shorter and the Hg-O (H20) bond 262 pm is longer as compared to that (241 pm) of Hg(H20)6 ion.
o
? ,o S
S c h e m e 2. Reaction scheme of the metalsubstitution reaction of Hg2(tpps)2- with
in an acetate buffer. 1, free base porphyrin; 2, mercury(II) acetato complex; 3, homodinuclear mercury(II)porphyrin;4, cotver(II) acetato complex; 5, heterodinuclear intermediate; 6, mereury(II) acetato complex; 7, copper(II)porphyrin.
o 4
o
I o
6
7
226 Mercury(II) ion in Hg2(tpps) 2- is bound to two nitrogen atoms and one acetate ion. The Hg-N and Hg-O bond lengths are 222 and 265 pm, respectively. Copper(II) acetato complexes exhibit a distorted octahedral structure: the axial and equatorial Cu-O bond lengths are 225 and 197 pm respectively. The bond lengths are the same as in the Cu(HO2) 6 ion. Mixing of Hg2(tpps) 2- with copper(ll) ion gives rise to the acetatomercury(II) as well as the copper(II) porphyrin complex of which structure was four-coordinated square planar with the Cu-N bond length of 200 pm. The possible structural change during the reaction of Hg2(tpps) 2- with Cu(ll) is postulated as in Scheme 2. The tetrahedral mercury(II) acetato complex reacts with H2tpps4- to form the deformed homodinuclear mercury(II) porphyrin complex, where two mercury(II) ions bind to pyrrole nitrogens from upper and lower sides and one acetate anion coordinates to each mercury(II). Pyrrole rings are twisted alternatively up and down, like "saddle" porphyrin skeleton, and deviated about 30* from the mean porphyrin plane. The nonbonded Hg Hg distance was estimated to be 310 pm. The deformed structure favors the attack of another metal ion from underneath. Very recently, Ohtaki, et al. 17 have directly determined, by using a stopped-flow EXAFS, structure of the deformed porphyrin. IV
Mixed-Valence
Cu(ll)-Cu(1)
Porphyrin
as
an
Intermediatel8
0 " Reducing agen.ts such as hydroxylamine and ascorbic acid accelerate the formation of copper(lI) porphyrin: in the presence of hydroxyl-1/ amine, formation rate of Cu(lI) porphyrin is about 104 times as high as _ f inits absence (Fig. 4). 18 Some authors ~'1tt o have assumed the effect of these reducing ~,.~-2agents to be attributed to high reactivity _o of the aqua copper(I) ion as compared to g the copper(ll) ion. The copper(ll) ion has o two types of water molecules (axial and -3 / , / / equatorial), and the formation-rate constant of copper(II) porphyrin has been correlated to the dissociation rate of the axial water molecule (k~lO9 s-l),l-4,19 -4 //o which is close to the diffusion-controlled rate. Therefore the rate enhancement by a factor of 104 in the presence of -4 -2 reducing agent cannot be attributable to t og(ccJ3/mot d~ 3) higher reactivity of the copper(1), Figure 4, Dependenceof the formation of Cufl(nnpyp)4+ because the water molecule bound to on the concemratioo of coppe~]:]) in the absence( 9and precopper(I) may dissociate rapidly but only seine(C)) of metallic copper at pH 3.50. by a factor of 10 compared to copper(II) at largest. The ionic radius of Cu+(96 pm) is significantly larger than that of Cu2+(73 pm): sitting on the porphyrin nucleus, Cu + distorts it favorably for the attack by Cu 2+ from underneath. Thereby copper(I) behaves similarly to large metal ions such as mercury(II), cadmium(ll), and lead(lI).10,11,12 We postulate, as intermediate, mixed-valence copper porphyrin, [Cu(II)(tmpyp)Cu(I)], in which copper(I) corresponds to mercury(lI) and copper(II) corresponds to zinc(II) in Zn(tpps)Hg 2-. 13
/
227 From the above experimental results, it is anticipated that copper(I) can also catalyze the incorporation of other medium-sized metal ions, such as manganese(II). As expected copper(I) as low as 10-7 mol dm -3 favored the incorporation of manganese(II) into H2tmpyp ~ . 1 8
V M o l e c u l a r C o m p l e x e s as I n t e r m e d i a t e s 2 0 - 2 2 The incorporation of zinc(II) into H2tpps 4- is accelerated in the presence of some amino acids. 20 The reaction sequence is described as follows: K ZnA + + H2tpps 4" ~ I
ZnA+(H2tpps ~ ) I "~ Zn(tpps) 4" + A- + 2H +
(9)
(10)
ZaA + (HA = amino acid) rapidly associates with the free base porphyrin to form a molecular complex (I) in which ZnA + is weakly bound to the porphyrin plane. Then the water molecule coordinated zinc(II) in I dissociates, and zinc(II) is incorporated into the porphyrin core. Thus the over-all rate constant of formation of zinc(II) porphyrin in the presence of an amino acid is given by k Z n A = Kk
(11) [
9
,
T•r• ,
.
where K is th formation constant of the 2 molecular complex and k is the rate r)p constant of substitution of a water r~ Phe molecule on ZnA + with one of the basic "6 nitrogens in the porphyrin. Figure 5 shows that the rate of incorporation of r~ zinc(II) into H2tpps 4- is linearly related ~,, ! to the hydrophobicity scale of the amino acids which is the free energy of solution "~ of amino acid in kJ mo1-1. Among the 2 studied amino acids L-tryptophan exhibits the largest catalytic effect on the metallo0 porphyrin formation. .gtlkJ mol "110 The formation on the molecular complex of porphyrin or metalloporphyrin with amino acids has been inves- Figure S. A correlationbetweenthe formationrate constant tigated by thermodynamic and 1H NMR of Zn(tpps)4- and the hydrohobicityscale of the amino acid. methods.21 1H NMR experiments on the porphyrin and amino acids system indicate directly the interaction between the side-chain of amino acid and the Po4_rphyrinp l a n ~ / h e up-field s~lft has been observed for L-tryptophan in the presence of H2tpps , H2tmpyp or Zn(tmpyp) at pD 9.0. This is attributable to the hydrophobic interaction between the side-chain of amino acid and the porphyrin plane. Furthermore, in the Zn(II) porphyrin system, the chemical shift of [3-proton of L-Trp separates to two signals due to the axial ligation of the amino group to Zn(II). The ligand-substitution reaction of M(tpps) 4- ( M 2+ = Zn2+, Cd2+, pb2+) with N,N'1,2-ethanediylbis[(N-carboxymethyl)glycine] (H4edta) and the acid-dissociation reaction of the metalloporphyrins are inhibited by the presence of 18-Crown (18C6). 22 The rate suppression by 18C6 was explained by the formation of [M(tpps)(18C6)] as a precursor complex. The
228 intermediate hinders the attack of Hedta3- or H + to the metal in the metalloporphyrins. The formation constants of [M(tpps)(18C6)] are 102-5, 10. 2-7, and 101-8 for zinc(II), cadmium, and lead(II), respectively. The intermediate is formed by the interaction of the metalloporphyrin plane with 18C6 ring, and the interaction was confirmed by 1H NMR and absorption spectra. The 1H NMR signal of 18C6 shifted to a higher field and the change in absorption spectra was larger for Cd(tpps) 4- than for Zn(tpps) 4- and Pb(tpps) 4-.
VI Summary The kinetic studies on metaUoporphyrin formation demonstrate the importance of intermediates such as heterodinuclear metalloporphyrin, mixed-valence metaUoporphyrin, and molecular complex. The important features of catalytic incorporation of metal ions is the deformation of porphyrin ring by coordination of large metal ions and the hydrophobic interaction with amino acid and aromatic bases. While, strong interaction of metaUoporphyrin with 18-crown-6 stabilizes the metalloporphyrin and prevents the attack of Hedta3- or H+ to the metal ion in the metalloporphyrin. Another interesting aspect is high reactivity of copper(I) porphyrin. The deformed structure of Hg2(tpps)4-was revealed by EXAFS-.
Acknowledgments We acknowledge with thanks the supports by the Grant-in aid for Scientific Research (No. 04215218, 05453071) from the Ministry of Education, Science and Culture of Japan.
References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
P. Hambright, P. B. Chock, J. Am. Chem. Soc., 96 3123 (1974). P. Hambright, in Porphyrins andMetalloporphyrins, edited by K. M. Smith, p. 232-278 (Elsevier, Amsterdam, 1975). D.K. Lavallee, Coord. Chem., 6 1, 55 (1985). D.K. Lavallee, Comments Inorg. Chem., 155 (1986). W. Schneider, Struc. Bonding(Berlin), 23, 123 (1975). M. Tanaka, Pure. Appl. Chem., 55, 151(1983). D.W. Margerum, G. R. Cayley, in Coordination Chemistry, edited by A. E. Martell, p 1-194 (American Chemical Society, 1978). S. Funahashi, Y. Yamaguchi, M. Tanaka, Bull. Chem. Soc. Jpn., 5 7, 204 (1984). M. Tabata, M. Tanaka, Trends Anal. Chem., 1 0,126 (1991). M. Tabata, M. Tanaka, J. Chem Socc., Dalton Trans. 1955 (1983). M. Tabata, M. Tanaka, Mikrochim. Acta, 149 (1982). M. Tabata, M. Tanaka, Anal. Lett., 1 3, 427 (1980). M. Tabata, W. Miyata, Chem. Lett., 785 (1991). L. R. Robinson, P. Hambright, Inorg. Chem., 3 1,652 (1992). M. Tabata, K. Ozutumi, BuU. Chem. Soc. Jpn., 65, 1438(1992). M. Tabata, K. Ozutsumi, Bull. Chem. Soc. Jpn., 6 7, 1608 (1994). H. Ohtaki, S. Funahashi, Y. Inada,, M. Tabata, K. Ozutsumi, K. Nakajima, J. Chem. Soc., Chem. Commun., 1023 (1994). M. Tabata, M. Babasaki, Inorg. Chem., 3 1, 5268 (1992). S. E. Hay, P. Hambright, Inorg. Chem., 23, 4777 (1984). M. Tabata, M. Tanaka, Inorg. Chem., 2 7, 203 (1988). M. Tabata, A. Shimada, Bull. Chem. Soc. Jpn., 63, 672 (1990). M. Tabata, K. Suenaga, Inorg. Chem., 3 3, 5503, (1994).
journal of ~IOLF~ULAR
LIQUIDS Journalof MolecularLiquids,65/66 (1995) 229-236
ELSEVIER
Structure and dynamics of solvated ions: new tendencies of research Tamds Radnai Central Research Institute for Chemistry, Hungarian Academy of Sciences, Budapest, Pusztaszeri dt 59-67, H-1025, Hungary Abstract Some recent developments in the research of the structure and dynamics of solvated ions are discussed. The solvate structure of lithium ion in dimethyl formamide and preliminary results on the structure of sodium chloride aqueous solutions under high pressures are presented to demonstrate the capabilities of the traditional X-ray diffraction method at new conditions. Perspectives of solution chemistry studies by combined methods as e.g. diffraction results with reverse Monte Carlo simulations, are also shown. I. Introduction In the last decades the overwhelming majority of the experimental studies of the structure and dynamics of ions targeted the aqueous solutions and the water itself, and less attention has been paid to other solvents and non-aqueous solutions. Theoreticians and "traditional" (e.g. IR, Raman, NMR) showed more interest to non-aqueous systems, but the predominance of aqueous solutions is still well reflected in the reviews [ 1]. A split between scientists involved in structural studies and those dealing with the dynamics can be observed, too. Information obtained about the same solutions was only seldom treated as complementary to each other [2]. Researchers applying a specific method often neglected the results of a related method. The appearance and fast development of computer simulation methods seemed to bridge the gaps between the different fields, but there are only a few studies when a simulation really aimed at a complete study of a system. The last several years brought substantial changes to the field. New systems are studied by traditional methods, and new methods have been also born. More complex studies are reported and often various methods are used to target the same subject. The attention of an increasing number of researchers seems to turn to non-standard (often extreme) physicochemical conditions. Complete series of systems are more often studied in order to find relations rather than to describe simply the properties of the system. Rarely though, new concepts seem to appear as the sign of better understanding of the chemistry of solutions. This paper does not intend to be a review; rather comments and examples are given for some of the recent progress. The related literature is not searched exhaustively and the selection is rather arbitrary. Preliminary results of two new studies by the XD method are presented in order to demonstrate the capabilities of the method at new conditions. The reverse Monte Carlo (RMC) technique is also discussed in more detail to show a new perspective in the structural modelling of solutions. II. New tendencies in structural studies Traditional methods o f investigation - n e w methods o f interpretation
The XD, ND as well as molecular dynamics (MD) and Monte Carlo (MC) computer simulation methods are considered well-established since a great number of investigation has been performed with them. With rare exceptions, experiments on liquids at normal conditions are considered as routine laboratory work by now. Similarly, even commercially available computer software helps the researchers to run simple simulations. However, the interpretation of the results is complicated and therefore a continuous improvement of the interpretation techniques is required. The XD experimental structure functions are often 0167-7322/95/$09.50
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230 interpreted by various geometrical models which extend to the first neighbors of a central atom only, or the second neighbors are also comprised [1,3]. Another method is the separation of some contributions of interest from the total structure- or pair distribution function. For this purpose several XD and ND experiments are carried out with a wellplanned strategy and differences of structure functions are calculated. The isotopic substitution and the subsequent first- or second-order difference method is well established in the ND practice [4]. With its application the local order around the ions can be studied separately. More recently, attempts have also been made to isolate the structure of the bulk solvent from the ion-water and ion-ion distributions, and thus the effect of ions on the solvent structure can be observed. For instance, it was concluded from ND experiments on LiC1 aqueous solutions that the lithium ion does not influence the bulk water structure at 1 molal concentration, but about 70% of H-bonds are broken when the concentration is 10 molal [5]. Instead of isotopic substitution, the isomorphic replacement can be used by the XD method. Here two solutions of equal concentration with the same solvent but with different ions are measured, and the difference of the structure functions is calculated. If the ions are selected carefully, i.e., they have nearly the same ionic radii, ionic charges and similar chemical properties, the isostructural contributions cancel out and the differences in the ionic solvate structure can be analyzed. The method proved to be successful for solutions involving erbium or holmium ions with an isomorphic pair of yttrium. Thus the XD studies showed that the erbium(m) ions form inner complexes with nitrate ions [6], and in high concentration, with chloride, bromide and perchlorate ions [7] in its aqueous solutions. The investigation was later extended to dimethyl sulfoxide solutions in which the difference in complex formation between chloride and nitrate ions proved to be less [8]. The effect could be explained by the fact that the solvating power of the DMSO for chloride and perchlorate ions is lower then that of the water. XD studies with isomorphic replacement in combination with ND studies with isotopic substitution have been reported for nickel(l/) and magnesium(l/) ions [9] as well as sodium(I) and silver(I) ions [10] in aqueous solutions and the cation-water and cation-cation partial pair distribution functions were separated experimentally.
New experimental techniques The appearance of new experimental techniques is connected with, but not exclusively, the establishment of big research centers where high power synchrotron sources and/or neutron generators are available. A fairly large number of liquid structure studies has been performed by now with the help of the EXAFS and XANES spectroscopy [11], especially since laboratory-scale instruments could also been constructed. Their performance is very good in determination of coordination numbers and ion-solvent distances for several suitable ions in the solution, and the applicable concentration is much lower than for the XD or ND method. This is an advantage in non-aqueous solutions, where the solubility is generally more limited. On the other hand, the more quantitative analysis of the ordering in the solutions is not possible without the help of other structural methods. White X-ray sources, energy dispersive techniques, soft x-ray sources offer new and alternative tools for structural studies. Small angle neutron scattering (SANS) is also in development. It offers another tool for experimental separation of contributions, ascribed to the cation's solvate shell, from the total pair distribution function especially for large size ions [12]. Structural properties around hydrophobic solutes in water can also be studied, as it was done for aqueous solutions of tetramethyl urea [13]. Quasi-elastic neutron scattering (QENS) is a promising technique for the determination of some dynamic properties in the solutions at the picosecond timescale, in contrast to the NMR spectroscopy, which works at a scale of milliseconds. Indeed, the self-diffusion coefficient D § could be determined for a series of cations [14], a quantity, which can not be accessed directly from experiments. A few computer simulation studies also succeeded in determination of separate values of selfdiffusion coefficients for cations, anions and bulk water [ 15].
23!
Solutions at non-standard conditions Most of the structural and dynamic studies in solutions have been carded out at ambient temperature and atmospheric pressure or not far from it. An increasing number of papers is devoted to supercooled [16-19] and glassy [ 18,20,21] water and solutions as well as to studies of water [22] and of aqueous solutions [23,24] at high temperature and/or pressure. Computer simulation methods are very flexible and are suitable for various studies at almost any thermodynamic conditions, and therefore a new strategy of the research can be established easily: simulations may predict some properties of the solutions at conditions which can be later verified when the appropriate experimental conditions become available. New range of solutes and solvents Studies on structure and dynamics of liquids have recently been extended to solvate structure of ions in non-aqueous solutions, and to the structure of complexes with relatively complicated ligands. We can also handle special problems like hydrophobic solvation is. Diffraction studies have been performed on new solvents as e.g. trifluoroethanol [25] and tetramethyl urea [26], and on solvent mixtures [27-30]. More recently the preferential solvation of ions has been subjected by an XD investigation in MgC12-water-methanol ternary systems [31 ], and the solvation structure around the cations proved to undergo the change of solvent molecules proportionally to the relative concentration of the two solvents. Extension of the research to new fields of solution chemistry A significant step to the combination of our knowledge about the static structure in liquids and their kinetic behavior has recently been made by application of an in-laboratory stopped flow EXAFS experiment [32]. This is an EXAFS spectrometer operated in the dispersive mode and a stopped-flow unit positioned along the x-ray path. Since results of very short time measurements can be accumulated in this way, with the appropriate selection of the system structural studies of reaction intermediates can be determined, which has not been possible before. Results are reported [33] on a partial structural change around the Cu(II) ion of a reaction intermediate at the formation of a Cu(ID porphyrin complex in the metal substitution reaction of the Hg(II) porhyrin complex with the Cu(II) ion in an aqueous acetate buffer solution. The measurements showed that the Cu-N distance in the reaction intermediate are elongated by about 0.04A in comparison with the final product. Combination of investigative methods and information The separate pieces of information are now often placed into a complex environment by the application of more than one method of investigation. Typical examples are: combination of diffraction experiments of deverse kinds; diffraction experiments with computer simulations and/or with thermodynamic and spectroscopic studies. The further continuation of the above tendencies can hopefully lead us to new and unified concepts of structure and dynamics in solutions. III. Solvation of lithium ion in N,N-dimethyl formamide Lithium(I).is one of the most puzzling ion from structural point of view. Due to its small radius, its surface charge is high and therefore its ability to form stable and well ordered solvate shells is comparable with the transition metal ions. The direct application of both the XD and ND methods to the solvate structure is difficult owing to the special scattering properties of the ion. As a consequence, the structural parameters describing the hydration structure of lithium(I) ion in aqueous solutions, although subjected to intensive studies, show a contradictory picture [1,34]. The average coordination number n varies between 4 and 6, and the average value of Li-O distance shows an unusually large variation from 1.94 to 2.25A. The effect can be explained partly by experimental errors, but a hypothesis has been also formed that the lithium ion tends to have a value of n=6 in its dilute solutions while n decreases at high concentrations [34]. The distance between the ion and the neighboring water molecules changes with the coordination number, the smaller is the latter, the smaller is the
232
distance. It is worth noting that lithium is able to form stable complexes in the solid phase with coordination numbers from 4 to 6 and with different geometries, and the lithium-binding atom distances correlate with n as it was found in aqueous solutions [35]. Only two structural studies have been reported for lithium solvates in non-aqueous solutions. In an XD study of concentrated formamide solution of LiC1 [36] the lithium ion was found to be solvated by 5.4 formamide molecules in average and the Li-O distance was reported to be 2.24./k, in keeping with the values found in diluted aqueous solutions. More recently, a detailed study has been performed by combination of the ND method with a series of theoretical methods on an 0.6 mol dm "3 LiBr solution in acetonitrile [37]. The lithium ion was found to be tetrahedrally coordinated by three solvent molecules and one bromide ion. The Li-N distance resulted in 2.05 ]k. From these data it is suspected that the molecules of the solvate structure of lithium ion might be largely effected by the solvent molecules. Since the solubility of some lithium salts is relatively high in N,N-dimethyl formamide (DMF), concentrated solutions can also be examined. In a previous study the solvate structure of lithium has been described in an 1.5 mol dm "3 LiNCS solution in DMF [38]. A new XD measurement has been carded out for a LiCI solution of the same concentration. Table 1 zhows the structural parameters for the lithium solvates in both solutions. The structural parameters were determined by a leastsquares fitting method (LSQ). After the subtraction of the contributions ascribed to the intramolecular structure of the DMF molecules and to the assumed structure around the anions from the total structure function of the solution, the resulted difference curve was approximated by calculated model curves. The result is shown in Figure 1. From the geometrical model described above, an average coordination number of 6 around a lithium ion was resulted for both solutions. In the LiNCS solution the cation is fully solvated, while in the LiC1 solution 5 DMF molecules and 1 chloride ion are located in the neighborhood of a cation. The Li-O distance values agree within the limits of errors. The contact ion-pair formation seems to be plausible because the solution of LiC1 was already near to saturation and thus the probability of the contact ion-pair formation is more enhanced. Table 1. Structural parameters for solvation shell of lithium ion in DMF as determined by XD measurements and least-squares fitting procedure: average distances rij, root mean square deviations l U, and coordination numbers nij. Solution of LiC1 (left) and of LiNCS (fight).
i,j Li-O O-O C..O Li..C Li..N O..O O..N Li..C Li..C1 CI..O
rii/A 1.97 2.74 a 2.99 2.79 4.00 3.94 a 3.63 "~ 4.56 2.40 3.25 a
Li(dmf)~C1 lii/A 0.07 0.12 0.18 0.14 0.18 0.20 0.20 0.30 0.15 0.20
nil 5 10a 10a 5~ 5~ 3~ 10a 10a 1 2a
.
i,j Li-O O-O C..O Li..C Li..N O..O O..N Li..C
rii/A 1.95 2.76 a 2.95 2.80 4.00 3.96 a 3.50 a 4.60
Li(dmO~§ lii/A 0.07 0.20 0.24 0.20 0.24 0.24 0.24 0.31
nil 6 12a 12a 6a 6a 3a 12a 12a
233 Figure 1. Experimental difference pair-distribution functions g(r) (circles) for solvation shell of lithium ion in a LiCl solution in DMF. The curve corresponding to the octahedral coordination, resulted in the best fit, is drawn by solid lines. Contribution calculated with an assumption of tetrahedral coordination is shown by dots. 0.6
I
I
I
I
I
I
I
I
I
I
I
I
00
0.4
0.2 ~
!
0
2
I
4 r/A
6
8
IV. Structure of an aqueous solution of NaCI at high pressure One of the most exciting questions in solution chemistry is concerned with the effect of the pressure on the solvation shell of ions. From conductance data it was suggested that the pressure breaks up the structure in the bulk water and also in the local water structure near the ions. Opposite opinions also exist according to which the increase in pressure leads to an enhancement of the close hydration. ME) simulation studies on a concentrated aqueous NaC1 solution [23] 3howed almost no changes in the hydrate structures of the ions even at 10 kbar pressure. On the other hand, the H-bonded network of the water molecules is distorted in a similar way as it was found in pure water [40]. Because of extreme technical difficulties, only a few attempts were to determine the structure of pure solvents at elevated pressures and only one ND experiment is reported on aqueous solutions of a LiC1 and a NiC12 aqueous solution [24], leading to contradictory results. No XD diffraction studies were reported. Preliminary results of an XD investigation of 2 molal aqueous NaC1 solution are given here. The pressure changed from 1 to 2000 bars. The measurements were performed with an XD diffractometer equipped with a specially designed high pressure unit. The solution was kept in a cylindrical cell made of beryllium. The details of the experimental technique and the data elaboration are reported elsewhere [22]. Figure 2 shows the total pair-distribution function of the NaC1 solution at different pressures. The radial distributions are broken because of experimental errors and spurious ripples are observable. Nevertheless, the average structural parameters listed in Table 2 seems reasonable. The structural parameters were determined by a LSQ fit to the main peak of the g(r) curve. The starting values of the parameters were taken from a previous study [3] of the same solution at room temperature. The parameters describing the hydration of the chloride ions were kept fixed and the calculated contribution was subtracted from the total curve. The remaining part was analyzed in terms of two contributions, namely, the Na-O and O-O. The results show that the average value of water-water first neighbor distances is contracted, but all other parameters do not change due to compression. The contraction ratio is 0.02 A/kbar, similar to what was found in
234 pure water [22]. However, this tentative results have to be confirmed by more systematic experiments and with improved data elaboration and interpretation technique. Table 2. Estimated structural parameters for sodium and chloride ions and for the first neighbor water molecules in 2 mol dm 3 aqueous solution of sodium chloride at room temperature and 1 to 2000 bar pressure, as derived from X-ray diffraction measurements: average distances rij and coordination numbers nij. pressure/b
rNa~)
ar
nNa-O
rci-o
1a 2.42 4v 3.14 1 2.42 5.5 3.14 c 1000 2.43 5.4 3.14 c 2000 2.42 5.7 3.14 c aRef. [3] bRegular tetrahedral structure of the ionic hydrate was eFixed at values taken from ref. [3].
ncl-O
roo
noo
6c 6e 6e 6e
2.92 2.91 2.88 2.87
4o 4.4 4.7 4.8
assumed.
Figure 2. Experimental pair-distribution functions g(r) for 2 mol dm "3 aqueous solutions of sodium chloride at room temperature and at 1 bar (lower curve) and 1000 bar (upper curve) pressures. 2
/
I
I
I
I
I
i
I
1
/
t--
I
2000 bar
1~1 . . . . . . . . . . .~. . . . . . . . . . . . . "
0L .......
/A
~
.
1000 bar
.
.
-
. .
.
.
.
.
.
-
.
1 bar
-
0 0
2
4
r/A
6
8
i0
V. Reverse M o n t e Carlo simulation of molecular liquids
In spite of the great success of the computer simulation methods in the determination of the microscopic properties of the solutions, the capacity of the traditional MD and MC simulations is always limited by the choice of the suitable potential functions to describe the interatomic interactions. The potentials are most often checked by comparison of the structural properties calculated from the simulation with those determined experimentally. The reverse Monte Carlo (RMC) method, developed by McGreevy and Pusztai [41 ] does not rely upon knowledge of any interaction potential, instead it generates a large set of atomic configurations on the condition that the difference between the experimental and calculated structure functions (or pair-distribution functions) should be minimum. The same structural
235
properties can than be calculated and analyzed as from any traditional simulation. Since no supplementary information on the studied.system is required, the RMC method seems to be a promising substitute for the average geometrical models to describe the static structure in the liquid. The RMC method has been frequently applied to atomic liquids and amorphous materials or two-component melts. There is only one attempt reported when ionic solution has been examined by the RMC method [42]. More recently an RMC simulation of liquid water has been performed [43] and the results were compared to those obtained from an MD simulation. A more distorted geometry of hydrogen bonds was found by the RMC study than in the MD simulation, and a significant fraction of the nearest neighbor molecules were observed to be distributed around a central one randomly rather than forming a tetrahedral network. The first RMC study on a heteroatomic, linear, molecular liquid was performed on liquid acetonitrile [44]. The molecule was described by three sites and deviation from linearity was also allowed. Intramolecular structure, partial pair correlation functions, angular distributions and the orientation of the molecules together with density profiles characteristic to spatial distributions of the neighboring molecules were discussed. One of the most interesting result is that the average coordination number of the first neighbors was found to be about 5-6, but the nearest two or three molecules seem to form a "subshell". This is demonstrated in Figure 3, where the N-CH3 partial pair distribution function is shown, together with the contributions of the f'trst n neighbors. The changes in the peak shapes suggest that the neighbors additional to the first two or three molecules are located at somewhat larger distances. A strong preference for antiparallel orientation of nearest neighbors and slight preferences for other configurations have been found, in qualitative agreement with previous results. However, these preferences decay rapidly after the first two or three neighbors. It can be seen that the RMC method is a more useful modelling tool to analyze the diffraction experiments than the construction of average geometrical models. Since several experimental data sets can be simultaneously input in the RMC procedure, ND and XD studies can be used in combination with it. The further development of the technique will hopefully help us to better understanding of the structures of aqueous and non-aqueous solutions as well. Figure 4. Partial pair-distribution functions gij(r) as computed from an RMC procedure in liquid acetonitrile for N-CH3 pairs (stars) and contributions from the nearest n neighbors of a central molecule (full lines). 3.00
gU 2.50
2.00
-
-
1.50 1.00
1
-
0.50 0.00
2.00
4.00
6.00
8.00
10.00
12.00
236
References 1 H. Ohtaki, and T. Radnai, Chem. Rev., 93, 1157 (1993), and references quoted therein. 2 H.L. Friedman, Chim. Scripta, 25, 42 (1985). 3 G. PAlink,fts, T. Radnai, and F. Hajdu, Z. Naturforsch. 35a, 107 (1980). 4 G.W. Neilson, Z. Naturforsch. 46a, 100 (1990), and references quoted therein. 5 R.H. Tromp, G. W. Neilson, and A. K. Soper, J. Chem. Phys. 96, 8460 (1992). 6 H. Yokoyama and G. Johansson, Acta Chem. Scand. 44, 567 (1990). 7 G. Johansson and H. Yokoyama, lnorg. Chem. 29, 2460 (1990). 8 G. Johansson, H. Yokoyama, and H. Ohtaki, J. Solution Chem. 20, 859 (1991). 9 N.T. Skipper, G. W. Neilson, and S. C. Cummings, J. Phys. Condens. Matter, 1, 3489 (1989). 10 N.T. Skipper and O. W. Neilson, J. Phys. Condens. Matter, 1, 4141 (1989). 11 See e.g. X-Ray Absorption: Principles, Applications, Techniques of EXAFS, SEXAFS and XANES, ed. D. C. Koningsberger and R. Prins, Wiley: New York (1988). 12 W. Kunz and P. Turq, J. Phys. Condens. Matter, 2, SA151 (1990). 13 V. Yu. Bezzabotnov, L. Cser, T. Grass, G. Jancs6, and Yu. M. Ostanevich, J. Phys. Chem. 96, 976 (1992). 14 N.A. Hewish, J. E. Enderby, and W. S. Howells, J. Phys. C: Solid State Phys. 16, 1777 (1983). 15 K. Heinzinger, G. P~linkfis, In: Interactions of Water in Ionic and Nonionic Hydrates, Springer: Berlin (1987). 16 R. Corban and M. D. Zeidler, Ber. Bunsenges. Phys. Chem. 96, (1992). 17 J.C. Dore, J. Mol. Struct. 237, 221 (1990). 18 T. Takamuku, T. Yamaguchi, andH. Wakita, J. Phys. Chem. 95, 10098 (1991). 19 T. Takamuku, K. Yoshikai, T. Yamaguchi, and H. Wakita, 7_, Naturforsch. 47a, 841 (1992). 20 M . C . Bellisent-Funel, L. Bosio, A. Hallbrucker, E. Mayer, and R. Sridi-Dorbez, J. Chem. Phys. 97, 1282 (1992). 21 K. Yamanaka, M. Yamagami, T. Takamuku, T. Yamaguchi, and H. Wakita, J. Phys. Chem. 97, 10835 (1993). 22 T. Radnai, and H. Ohtaki, in preparation. 23 G. Jancs6, K. Heinzinger, and T. Radnai, Chem. Phys. Lett. 110, 196 (1984). 24 G.W. Neilson, Chem. Phys. Lett. 68, 247 (1979). 25 T. Radnai, S. Ishiguro, and H. Ohtaki, J. Solution Chem. 18, 771 (1984). 26 T. Radnai, H. Ohtaki, Z. Naturforsch. 47a, 1003 (1992). 27 T. Radnai, S. Itoh, and H. Ohtaki, Bull Chem. Soc. Jpn. 61, 3845 (1988). 28 T. Radnai, S. Ishiguro, and H. Ohtaki, Chem. Phys. Lett. 159, 532 (1989). 29 G. PAlinkAs, I. Bak6, P. Bopp, and K. Heinzinger, Mol. Phys. 73, 897 (1991). 30 G. PAlinkfis, and I. Bak6, Z Naturforsch. 46a, 95 (1991). 31 T. Radnai, I. Bak6, and G. P~ilink~s, Acta Chim. Hung. in press. 32 Y. Inada, S. Funahashi, and H. Ohtaki, Rev. Sci. Instr. 65, 18 (1994). 33 H. Ohtaki, Pure Appl. Chem. 65, 2589 (1994). 34 Y. Marcus, Chem. Rev. 88, 1475 (1988). 35 U. Olsher, R. M. Izatt, J. S. Bradshaw, and N. K. Dalley, Chem. Rev. 91, 137 (1991). 36 H. Ohtaki and H. Wada, J. Solution Chem. 14, 209 (1985). 37 W. Kunz, J. Barthel, L. Klein, T. Cartailler, P. Turq, and B. Reindl, J. Solution Chem. 20, 875 (1991). 38 T. Radnai, S. Ishiguro, and H. Ohtaki, Bull. Chem. Soc. Jpn. 65, 1445 (1992). 40 G. P~ilinkfis, P. Bopp, G. Jancs6, and K. Heinzinger, Z. Naturforsch. 39a, 179 (1984). 41 R.L. McGreevy and L. Pusztai, Mol. Simul. 1, 359 (1988). 42 M.A. Howe, J. Phys. Condens. Matter, 2, 741 (1990). 43 P. Jedlovszky, I. Bak6, and G. P~ilinkfis, Chem. Phys. Lett. 221, 183 (1994). 44 T. Radnai, and P. Jedlovszky, J. Phys. Chem. 98, 5994 (1994).
"journal of /~OLECULAR
LIQUIDS
ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 237-244
The Role of Solvent in Inorganic Reaction Mechanisms, as Elucidated by High Pressure Studies Thomas W. Swaddle Department of Chemistry, The University of Calgary Calgary, Alberta, Canada T2N IN4 Abstract
Pressure effects on the rates of simple inorganic reactions in solution act primarily on intermolecular interactions, particularly solvation. They afford the possibility of "tuning" solvent properties without altering the solvent chemically. Outer-sphere self-exchange electron transfer reactions of transition metal complexes provide an opportunity to test this idea quantitatively, through an adaptation of Marcus-Hush theory. The theory is only moderately successful for organic solvents, as the solvent "seen" by the reactant behaves as if it is less compressible than bulk solvent. Furthermore, for multiply-charged ions or for solvents of low relative permittivity, the calculations become unstable even without the likely complication of ion pairing. For aqueous systems, the theory is successful except where one or more of the following complications applies: (a) changes in total spin multiplicity accompany the electron transfer process in complexes with flexible ligand systems; (b) cation-mediated pathways emerge in anion-anion self-exchange reactions; (c) the reactants are so small that the assumption that the solvent is a continuous dielectric fails; or (d) an inner-sphere sphere mechanism is operative. I. Introduction
Rigid solute molecules may generally be expected to be intrinsically one to three orders of magnitude less compressible than the bulk solvent. 1 This may be understood intuitively by comparing the compressibilities ~ of solids such as diamond (which may be regarded as infinitely extended molecules) with those of typical solvents such as methanol, and reflects the availability of free volume in liquids. Some solvent compressibility is lost when ionic solutes are solvated; for example, it has been estimated 2 that the molar volume of solvating water in the outer coordination spheres of cationic cobalt(HI) ammine complexes in aqueous solution is, on the average, some 2.5 cm3 mol"1 less than that of bulk solvent (18.07 cm3 tool"] at 298 K), corresponding to an effective average electrostrictive pressure of about 500 MPa outside the first coordination sphere. Thus, it transpires that the effect of pressure on the kinetics and equilibria of ionic reactions in solution reflects primarily changes in intermolecular interactions rather than intramolecular restructuring of the reacting solute(s). Again with reference to the 0167-7322/95/$09.50 91995 Elsevier Science B.V. All fights reserved. SSD! 0167-7322 (95) 00855-i
238 cobalt(IH) ammine example, the volume of activation A//r (= -RT(O In k/OP)T, where k is the rate constant and P is the pressure) for reaction 1 Co(NH3)sX ~
+ H20 ~
Co(NH3)sOH~ 3§ + X ~"
(1)
varies widely from +1 to -18 cm3 mol"l at the atmospheric-pressure limit as z is increased from 0 (X -- H20) to -2 (X = SO42"), and is markedly pressure dependent for anionic X. These results may be interpreted to mean that bond making~reaking contributes only about +1 cm3 mol"l to AF~ in the series, the balance (and the pressure dependence of A~) being due to solvation of the emerging 3+ and z- ions with concomitant loss of both volume and compressibility of part of the solvent - - an estimated eight water molecules per Co, in the case of the sulfato-complex. 3 A particular advantage in the use of pressure to probe solvational effects is that pressure can "tune" the characteristic bulk properties of the solvent without changing it chemically. The use of mixed solvents to vary the dielectric constant, for example, will likely be confounded by preferential solvation by one of the solvents, while the system will be chemically different if solutions in different neat solvents are considered. Pressure studies, however, have their own limitations. Technical complications 4 limit the pressure range normally used in chemical kinetic studies to about 0.1-200 MPa, and this produces only small changes in such solvent parameters. Furthermore, one needs to have available a realistic theory that relates changes in solvent parameters to their anticipated effects on reaction rates and equilibria, if pressure is to be used as a reliable probe of solvation phenomena. Our recent efforts in Calgary have been directed towards establishing and testing such a theoretical framework for suitable model reactions. The simplest of such reactions are solvent exchange on metal ions (representing substitution processes) and the outer-sphere electron transfer reactions of transition metal complexes (representing redox). For the latter, the theories of Marcus s and Hush 6 have provided a starting point for a theory of pressure effects on reaction rates. ~'7
IL Solvent exchange kinetics Pressure effects on the kinetics of solvent exchange on metal cations have been described at length elsewhere, 4.s and will not be considered in detail here. Suffice it to say that it is possible, using an empirical relation that predicts the molar volumes of aqueous cations, to predict the volume changes that would accompany the gain or loss of one solvent ligand and so to set limits upon AFr for associative (A) and dissociative (D) water exchange (about -13 and +13 cm3 tool"l, respectively). 4's In fact, A ~ values for all water exchange reactions of metal 2+ and 3+ ions studied to date have fallen between these limits (Ti(H20)63§ is close to the associative limitS), implying that the water exchange process is appropriately described as interchange (I,, I~). In particular, the D limit is not approached at all closely, as might be expected in view of the high abundance of water molecules surrounding the metal ion, with the negative ends of their dipoles pointing inwards on a time average. At present, there are insufficient molar volume data for metal ions in nonaqueous solvents to permit construction of empirical relationships like that established for aqueous solutions, and so this approach has so far been limited to solutions in water.
239
IH. Theory of pressure effects on outer-sphere self-exchange reaction rates In the absence of ion pairing and rate limitation by solvent dynamics, the volume of activation AIA for adiabatic outer-sphere electron transfer in couples of the type ML~ez§ can, in principle, be calculated as in equation 2 from an adaptation of MarcusHush theory. ~'7 In equation 2, the subscripts refer respectively to volume contributions from internal (primarily M-L bond length) and solvent reorganization that are prerequisites for electron transfer, medium (Debye-Htickel) effects, the Coulombic work of bringing the reactants together, and the formation of the precursor complex. AV~ = AVIR~:+ AVSR~ + AVDH~+ AVCOUL:~+ AVPRi/C~
(2)
Of these contributions, AVeR* is calculated to be between 0 and 1 cm 3 mol "~ for rigid complexes ~ (a value of +0.6 cm3 mol "] is adopted for the complexes discussed in this paper), AVpREC~ is typically between +1 and +2 cm3 mol'~,7 and, for practical values of the ionic strength I, A VDH~ and A Vco~ ~ will be of opposite sign but similar magnitude unless the relative permittivity D of the solvent is very low" AVDH:l: =
{R~ZIZ2CII/2/(1
-I- Ba/lf2)2}[(O
In D/OP)T(3 + 2 B a I la) - 13)]
AVcouL $ = (NxZ~Z2e214m~oa)(aD'~laP)T
(3)
(4)
If we assume the reactants to be hard spheres of roughly equal radii r and electron transfer to occur with highest probability at an M-M separation a that varies with compression of the solvent (isothermal compressibility 13), we have for solvent reorganization AVsR ~ = (NAe2/1B~F,o)[(r '' -- o"l){0(n "2 - D ' I ) / a P } T
- (n "2 - D ' l ) ~ / 3 o "]
(5)
In equations 3-5, a is the anion-cation close-approach distant, B and C are the DebyeHt~ckel parameters at 0.1 MPa, n is the refractive index of the solvent, and the other symbols have the usual SI meanings. Thus, theory predicts that AV~ can be equated approximately with AVsR~. Since the pressure dependences of both n and D are closely linked to that of the density p of the solvent, i.e., to [3, it follows that AVt is determined in large part by the compressibility of the solvent. Sample calculations for model 3+/2+ couples are given in Table 1. As Table 1 shows, however, the compressibilities of liquids, particularly organic solvents, decrease markedly with increasing pressure, and so the calculated AVsR~ and indeed all the parameters in Table 1 decrease in absolute magnitude over the typical experimental pressure range, especially at low pressures. The result is that A V~ itself can be expected to depend markedly upon pressure, but the effect is less significant for water than for nonaqueous solvents. The implication is that, given the usual experimental uncertainties, one can meaning~lly discuss pressure effects on electron transfer kinetics in aqueous systems over the 0-200 MPa range in terms of either an average value of AV~ over that range (but this raises the question of how the theoretical values of AVt are to be
240 Table 1. Calculated contributions to AF ~for self-exchange of a t~ical couple Ml~3+a+.' Solvent water
P /MPa 0 100 200
[3 AVSR:~ /104o Pa "l /cm3 mol "l 4.5 -7.5 3.6 -6.1 3.0 -5.1
~VDHt /cm 3 mol "1 2.9 2.4 1.9
A VCOUL~: /cm 3 mol "l -3.4 -2.8 -2.2
A V~(calc) /cm 3 mol"l -6.3 -5.1 -4.1
methanol
0 100 200
11.6 5.7 3.9
-18.3 -9.1 -6.2
20.6 10.5 7.0
-20.7 -10.8 -7.3
-14.9 -7.6 -5.2
acetonitrile
0 100 200
11.1 5.8 4.0
-17.8 -9.3 -6.4
16.1 9.3 6.5
-16.3 -9.7 -6.9
-14.7 -7.9 -5.5
0
12.6
-18.4
49.7
-47.6
-12.5
100
6.1
-9.3
19.5
- 18.3
-6.4
acetone
200 4.2 -6.4 12.1 I9= 0.1 mol kg "~, r = 500 pm, o = 1.0 nm, a = 800 pro, T = 25~
-11.3 -4.4 AFrot = 0.6 cm 3 mol"~.
averaged, since their pressure dependence is nonlinear 7) or its value at mid-range (I00 MPa). In practice, experimental values of In k often appear to be linear functions of P within the error limits, although non-linearity does seem to exist in some cases where I AFt I is unusually large 9"ll and direct comparison with theory is then possible. Wherland ]2 has reviewed the experimental results on pressure effects on the kinetics of nonaqueous electron transfer reactions. Table 1 gives theoretical values of ~VDH~ and AYcouL~, and, although few 3+/2+ reactions have been studied in nonaqueous media to date (largely because of solubility and redox stability problems), the point emerges that these opposing contributions can become so very large that the calculation of A Fr becomes numerically unstable. Even for 0/- or +/0 couples, for which the DebyeHfickel and electrostatic work terms should in principle be nonexistent, one suspects that second-order ionic strength and Coulombic effects could contribute significantly and possibly unequally to AF r Furthermore, ion pairing of the reactants (at least one of which has to be ionic) with the counterions that are inevitably present will become increasingly evident as D decreases. If the paired ion(s) are more reactive than the free reactant ions, the contribution AVw ~ of ion pair formation to AVr will be positive, since, as Fuoss theory shows, ~ increasing pressure acts through increasing D to disfavor ion pairing. Conversely, if paired reactant ions are less reactive in electron transfer than the parent species, A Ywt will be negative m Wherland m! has shown that this is more generally the case, although the assumption made for convenience of calculation in some of our publications, ]~ that ion pairs are inactive in electron transfer, is excessive. Of course, the value of AVn,~ will approximate to a constant only if ion pairs constitute a very small or a very large fraction, respectively, of the total reactant in these two cases.
241
Even if ionic strength, Coulombic, and ion pairing effects can be ignored, however, Table 1 shows that AVsR~ (and hence A/~ itself, for 0/- and +/0 couples) in nonaqueous solvents is predicted to be strongly negative at low pressures but to assume values not very different from those for aqueous systems as P increases. In reality, such sharp pressure dependence of AV* (i.e., nonlinearity of the In k vs. P plot) for electron transfer reactions has not been observed to date. For example, for the Ru(hfac)3 ~ couple (hfac- = 1,1,1,5,5,5-hexafluoropentane-2,4-dionate), the In k vs. P plot is linear to within the experimental uncertainty from 0-200 MPa, with AVt = -5.8, -5.5, and -6 cm3 mol"~ in perdeuterated methanol, acetonitrile, and acetone, respectively (cf. -7.3, -7.5, and -7.6 cm 3 mor l, respectively, calculated for 100 MPa as in Table 1 m no measurements are available for water, in which gu(hfac)3 ~ is insoluble), only for methanol is there a hint of nonlinearity, though still less than predicted by equation 5.13 The agreement between theoretical and experimental A ~ values is reasonably close at 100 MPa, and might be better at higher pressures. The implication is that the solvent, as "seen" by the reacting complexes, is more compressed and so less compressible than bulk solvent by virtue of its solvational action (cs the hydration of aqueous ammines as discussed above); as one goes to higher pressures, the errors introduced by equating solvating solvent with bulk solvent
diminish in magnitude.
IV. Electron transfer in aqueous solution
If local loss of solvent compressibility near solute molecules is indeed a major cause of failure of the theory of AVt for electron transfer in organic solvents, then the better record of success of theory in aqueous systems (Table 2) is understandable, as bulk water is less compressible than organic solvents at low pressures, the gap narrowing as P increases (cs Table 1). Aqueous systems listed in Table 2 for which the adiabatic Marcus-Hush-based theory predicts AVt with acceptable accuracy include the hexaaquairon(III/II) couple (which is a benchmark of theoretical analysis), (1,10-phenanthroline)iron(IIIM) and bis(trithiacyclononane)cobalt(IHM) (in which couples both the reduced and oxidized partners are low-spin in their ground states), and cobalt(RIM) complexes in which the metal ions are encapsulated in the rigid cages sepulchrate, diaminosarcophagine, and diprotonated diaminosarcophagine. In contrast, the tris(ethylenediamine)cobalt(UI/R) and tris(1,10-phenanthroline)cobalt(III/II) exchanges give AV~ values that are 10-15 cm3 mol"~ more negative, and are several orders of magnitude slower, than predicted. (The self-exchange reaction of the ethylenedinitrilotetracetatocobalt(III/II) complexes is also of this class, but the numerical analysis of AV~ is complicated by the fact that, in the Co n partner, the EDTA 4" ligand is only quinquedentate, the pendant carboxylate arm being protonated and the sixth coordination site on the Co n being occupied by a water molecule. ~s) This can be understood in one of two ways, both of which relate to the fact that the cobalt(II) is high spin (quartet state) and the cobalt(IL0 is low spin (singlet) in the ground state, so that electron transfer must be accompanied by major multiplicity change and is nominally "spin forbidden". One possibility is that electron transfer is preceded by a cobalt(H) quartetdoublet equilibrium with a volume change of-10 to -15 cm3 mol"~, indeed, a volume
242
Table 2. Volumes of activation at 100 MPa for self-exchange in ML~(z+~)/z+ couples in aqueous media. 9 A/~(expt)/cm 3 mol "l A//C(calc)/cm3 mor t gel. Couple I/mol kg "l T/~ -10.4 11 Fe(H20)63+/'2+ 0.5 2.0 -11.1 • 0.4 b Fe(phen)33+/2+ 0.3 3.0 -2.2 + 0.1 -2.5 c 14 Co([9]aneS3)23+t2§ 0.1 25.0 -4.8 + 0.2 -5.3 15 Co(sep) 3+t2+ 0.2 25.0 -6.4 + 0.2 -6.4 15 Co(diamsar) 3+t2+ 0.1 25.0 -10.5 • 0.6 -7.3 16 Co(diamsarH2) 5+/4+ 0.63 25.0 -9.4 • 0.9 -9.2 16 Co(en)33+/2+ 0.5 65.0 -15.5 • 0.8 b -5.3 10 Co(phen)33+/2+ 0.1 (NOr) 25.0 - 16.0 • 0.7 - 1.7 17 0. I(CI') 25.0 -17.6 • 0.7 -2.2 17 "Co(EDTA) /2'' 0.5 85.0 -3.2 • 0.3 -18 -9 c 9 MnO4"t2. 1.1 45.0 - 17 • 2 b Na+,MnO4"rz" 1.1 45.0 +3 • 1 -6 ~a 9 K+,MnO4"t2" 1.1 45.0 -1 • 1 -6 ~'d 9 (K+)3,Fe(CN)e3"/4" 0.5 25.0 +22 • 2 -5 e'd 19 Fe(I-I20)5OH2+/Felq 2+ 0.5 2.0 +0.8 • 0.9 -11 11 9AVnt: taken as 0.6 cm 3 mol "l except as indicated, b A ~ shows possible pressure dependence, c AVn~:taken as 0. d Includes an estimate of the effect of ion pairing.
change of this magnitude has been reported for the bis(terpyridine)cobalt(II) spin equilibrium, 2~ but there is evidence2~ that dechelation of the high-spin Co(terpyh 2+ is also involved, and furthermore the low-spin cobalt(n) state appears to be energetically inaccessible for Co(phen)3 2+ and especially Co(enh 2§ The other possibility is that electron transfer in the Co(phen)3 3+~+ and Co(enh 3+r2+ couples is nonadiabatic; we have tended to favor this explanation because the contribution of nonadiabaticity to AV~ can be calculated in terms of the distance scaling factor a, and the required values of a (16-19 cm~) fall within the range expected for these couples. This raises the question, however, as to why the Co(sep) 3+r2+, Co(diamsar) 3+~+, and Co(diamsarH2) s+/4+ couples, in which the Co n partners are also high-spin, should conform to expectations based on adiabatic Marcus-Hush theory. The answer seems to be that the encapsulating ligands, while permitting some lengthening of the Co-N bonds when Co m is reduced to Co n, preclude the angular deformations that presumably occur following spinstate change with three independent bidentate ligands. Qualitative arguments to explain the impact on A/~ might include the opening up of spaces between the bidentate ligands to additional solvation on going from high-spin to low-spin Co n, but such explanations are not readily quantified. Significantly, Wherland ]z has identified flexibility of certain ligand systems as a cause of anomalies in AVt for reactions of the Mn(CNR)6 2§ series in nonaqueous solvents. While the role of the counterions in cation-cation electron transfer seems to be confined to Debye-Htickel-type interactions, anion-anion self-exchange can be strongly catalyzed by the cations, so that cation-dependent paths are evident in addition to
243
exchange of the "bare" anions for the aqueous manganate-permanganate couple, 9 while for the ferri/ferrocyanide reaction the only detectable pathway at practical ion concentrations involves three K + ions. ~9 Such cation-dependent pathways are characterized by A/~ values that are much more strongly positive than theory permits, even when the contributions of ion pairing are allowed for using Fuoss theory. 7 The A/~ values for the MnO4r cation-dependent paths need to be judged with reference to the anomalously strongly negative A/~ for the cation-independent path. Part of the negative discrepancy for the latter may be due to the non-adiabaticity invoked by Dolin, Dogonadze and German, 22 but an additional factor is likely to be the unusually small size of the reactant anions, which means that the solvent can no longer be treated as a continuous dielectric. We have attempted to improve the continuous-dielectric model by treating the encounter complex as an eUipsoidal entity rather than as the usual two spherical conductors, 9 but Matyushov23 has tackled the general problem of solvent "graininess" in electron transfer theory directly. Unfortunately, the resulting expressions for AVsg~ are unwieldy and give positive values for model reactions in a range of nonaqueous solvents, contrary to the predictions of equation 5 (to which the Matyushov expressions should reduce when the reactant dimensions are much larger than those of the solvent molecules) and to all experimental results for outer-sphere electron transfer reactions other than some counterion-mediated processes. These positive values of the predicted AVsx~ arise because the pressure dependence of the expression that replaces the Pekar factor ((n "2 - D "~) in equation 5) is overwhelmed by that of solvent density reorganization, contrary to intuitive expectations, and it may be that the promising Matyushov approach simply requires numerical reevaluation. Finally, the apparent failure of Marcus-Hush-based theory of AV~ in electron transfer between Fe(H20)sOH 2§ and Fe(H,O)6 '~+ is quantitatively attributable to the operation of a hydroxo-bridged (inner-sphere) mechanism, 11 and indeed AV* can serve to distinguish between outer- and inner-sphere processes in appropriate cases such as this.
Acknowledgement I thank my coworkers, listed in the references, for their contributions, and the Natural Sciences and Engineering Research Council of Canada for financial support of our work.
References I. 2. 3. 4. 5.
D.g. Stranks, Pure Appl. Chem., 38, 303 (1974). T.W. Swaddle, Adv. Inorg. Bioinorg. Mech. 2, 95 (1983). W.E. Jones, L. g. Carey and T. W. Swaddle, Can. J. Chem. 50, 2739 (1972). T.W. Swaddle, Can. J. Phys., in press. g. A. Marcus, J. Chem. Phys., 24, 966, 979 (1956); Discuss. Faraday Soc., 29, 21 (1960); Faraday Disc. Chem. Soc., 74, 7 (1982). 6. N.S. Hush, Trans. Faraday Soc., 57, 557 (1961). 7. T.W. Swaddle, Inorg. Chem., 29, 5017 (1990).
244 8. A.E. Merbach, Pure Appl. Chem. 59, 161 (1987), and in High Pressure Chemistry and Biochemistry, edited by R. van Eldik and J. Jonas, p. 311 (Reidel, Dordrecht, 1987). 9. L. Spiccia and T. W. Swaddle, Inorg. Chem., 26, 2265 (1987). 10. W. H. Jolley, D. 1L Stranks and T. W. Swaddle, Inorg. Chem., 29, 385 (1990). 11. W. H. Joiley, D. R. Stranks and T. W. Swaddle, Inorg. Chem., 29, 1948 (1990). 12. S. Wherland, Coord. Chem. Rev. 123, 169 (1993). 13. H. Doine and T. W. Swaddle, Inorg. Chem., 27, 665 (1988). 14. H. Doine and T. W. Swaddle, Can. J. Chem. 66, 2763 (1988). 15. H. Doine and T. W. Swaddle, Inorg. Chem., 30, 1858 (1991). 16. R. D. Shalders and T. W. Swaddle, forthcoming publication. 17. M. R. Grace and T. W. Swaddle, Inorg. Chem., 32, 5597 (1993). 18. W. H. Jolley, D. R. Stranks and T. W. Swaddle, Inorg. Chem., 31, 507 (1992). 19. H. Takagi and T. W. Swaddle, Inorg. Chem. 31, 4669 (1992). 20. R. A. Binstead and J. K. Beattie, Inorg. Chem. 25, 1481 (1986). 21. 1L D. Shalders, H. Takagi and T. W. Swaddle, unpublished work. 22. S. P. Dolin, R. 1L Dogonadze and E. D. German, J. Chem. Sor Faraday Trans. I, 73, 648 (1977). 23. D. V. Matyushov, Chem. Phys. 174, 199 (1993); D. V. Matyushov and R. Schmid, J. Phys. Chem. 98, 5152 (1994).
journal of
MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 245-252
EXAFS of Bromide Ion in Solvents and at Air/Solution Interface Iwao Watanabe Department of Chemistry, Faculty of Science, Osaka University, Toyonaka, Osaka 560, Japan. Abstract Solvation structures of Br- ion dissolved in 23 solvents have been studied at the Br Kedge using extended X-ray absorption fine structure (EXAFS). The results are summarized as follows. 1) Coordination bond lengths r are 0.31-0.32 nm for B r - - - O in protic solvents and 0.35-0.36 ran for B r - - - C in aprotic solvents. 2) The r values for water and formamide are larger than those for other protic solvents, possibly due to the strong solvent-solvent interactions of these two solvents. 3) Coordination numbers N widely differ, ranging between 3-6, from solvent to solvent. The value of N for alcohol linearly correlates with the MayerGutmann's acceptor number A N. 4) In aprotic solvents containing two methyl groups per molecule, Br- is found to contact with the two methyl groups, thus the number of primary solvent molecules must be equal to half the N value. 5) EXAFS amplitude is found to correlate with A N of the solvent. To study the surface structure, a new EXAFS technique, total-reflection total-conversion-electron,yield EXAFS, has been developed and applied. The new technique utilizes polarized X-rays from synchrotron radiation, providing not only information on the solvation structures of ions at the aqueous solution surface but also on the preferential orientation of coordination bond at the solution surface.
I. Introduction Solvation structures of anions in water are difficult to determine due to their weak interactions with the solvent. Various coordination distances r and coordination numbers N have been reported for bromide ion in water. X-my diffraction(XRD) technique predicts an r value ranging between 3.12 and 3.43 ,~, and N values from 6 to 8-9.4.1 Other techniques, such as nmr, neutron diffraction, and ir spectroscopy gave different values, suggesting that the results depend to large degree on the techniques used. The hydration number of Br- obtained by nmr is zero, which is an extreme case. 2 The interactions are even weaker in organic solvents, thus very little is known about the geometry of anion solvation in organic solvents. Computer-simulation calculations describe the solvation structure of Br- as ceaseless motion of solvent molecules distributed around the anion at equilibrium r and N. 3 Diffraction techniques can provide the distribution functions for these values, though they require rather high concentration samples. Concentrated salt solution may have different solvation structures from that of dilute ones, not only because of the smaller number of free solvent molecules but also the higher possibility of ion-pair formation especially in organic solvents. 4 Extended X-ray absorption fine structure (EXAFS) is a useful technique to provide the local structure of solutions. It is able to determine the r and N values at relatively dilute concentrations and in organic solvents as well. The solvation structures of Br- in various solvents as determined by EXAFS 5 is discussed in the present paper. Moreover, the EXAFS technique has been applied to the aqueous solution surface. At the surface the solvation 0167-7322/95/$09.50 91995 Elsevier Science B.V. All rights reserved. SSDI 0167-7322 (95) 00856-X
246 structure of Br- may not be the same as that of the bulk. Some of the molecular dynamics calculations predict that halide anions in water tend to float on the surface of clusters consisting of water molecules rather than within water. 6 This effect may cause a dissimilar solvation structure to that of the bulk. In addition, if the anion is segregated at the surface by surfactants such as large alkylammonium cations, the anion density at the surface should be high and its environment differ from the bulk. This is a preliminary report of the first experimental study of the solution surface by the EXAFS technique. This technique provides us information on the gas/liquid interface, the structure of l.angmuir films, and the effect of the interface on chemical reactions. II. E x p e r i m e n t a l
Br K-edge absorption measurements in transmission mode were performed using a 1 crn or a 5 mm thick cell at BL-6B and -10B of the Photon Factory (National Laboratory for High Energy Physics, Tsukuba). The spectra were taken at the concentrations of 0.1 and 0.05 mol dm -3 at room temperature. The concentration dependence of the spectra was not observed at this concentration range. The surface study was performed at BL-7C of the same facility by the total-reflection XAFS(TR-XAFS) method. The experiments were done at 1 mrad grazing incidence angle, since at the Br K-edge energy, the critical angle for the total-reflection is about 2 mrad at the water surface. In order to reduce the X-ray absorption by air and water vapor, the solution 0 /'~ n H20(1) surface was covered by helium gas. The X-ray absorption was determined by detecting the totalconversion-electron current. When o X./_Wv ,,j-- I the Br- ions at the surface absorb X-rays, they emit Auger electrons of 10 keV energy. The helium is, then, ionized by Auger electron to A Pk fo~amide 17) create many He + and e- couples amplifying the signal. An electrode t-butyl alcohol (12) placed above the solution surface then collects the conversion X./ v V "I electrons. This is a simple and 01~ m A ~ acet~ (14) sensitive technique to measure the
LX'A'A'
LV V ,,,v ,,
' ' I
/\ ,.,,.,, V V
~
%0
X-ray absorption by surface species. By reversing the polarity of the electrode, the He + current can be measured. In both cases (l~sitive and negative polarity) similar results were obtained. The solution was cooled to around I*C in order to reduce the vaporization of solution and the temperature was kept constant during the experiments within 0.1*C.
V
0/~ O~
%/
V
A
v
'
2
,,j
--
--'-I
~-di/~me~ylf~~ ~_~Lide(I
L/
~
v
rN f~ X../ N.J v " '
~ - -
I
I
4
.,~
v
_ I
kl A -~
pyridine 119 -.-,~. , ---I
I
6
I
I
8
Figure 1. The Br K-edge EXAFS k3 Z (k) spectra of Br- ion in several solvents. Taken from ref.5.
247 III. Bromide Ion in A q u e o u s and Organic Solvents
The EXAFS spectra for Br- display weak oscillations, indicating either a small coordination number or a large disorder of the coordination bond lengths due to the weak Br-solvent interactions. Some of the spectra display strong multi-electron excitation structures7 and this structure had to be removed before the EXAFS analysis. The EXAFS osciUations 5 for several typical solutions are given in Fig. 1. In all cases the oscillations for aprotie solutions are weaker compared to aqueous or protic solutions. Most of the spectra could be analyzed by using a one-shell model, but it is clear that the data for acetic acid (4) and tbutyl alcohol (12) can not be adequately analyzed by the one-shell model. The EXAFS oscillation amplitudes are quite different from sample to sample and seem to be correlated with the anion-solvent interaction strength. Here, the EXAFS amplitudes are evaluated with the peak heights h in the Fourier transforms plotted against the anion solvation strengths represented in terms of the Mayer-Gutmann's accepter number AN .8 It is quite interesting to find a linear correlation between them as shown in Fig. 2. In most EXAFS studies it is not considered that the amplitude alone directly reflects a physical or a chemical property of the sample, except for the correlation of the EXAFS amplitudes for transition metal aqua complexes with their ligand exchange reaction rates. 9 Since the Fourier transform calculation gives rather consistent results compared to the curve fitting calculation, it could be suggested that the present h value can be used as another solvent parameter to characterize electron pair acceptors or hydrogen bond donors. The two-shell solvation structure found for acetic acid solution (4) and its difference in behavior in Fig. 2 must indicate that the solvation structure of Br- in acetic acid is quite different from the others. The EXAFS data were analyzed to obtain the r and N values by using a one-shell curve fitting method except for acetic acid and 0.3-' ' ' ' ' ' i , i .. ~ = , t-butyl alcohol solutions. The results are plotted against the A N values in Figs. 3 and 4. The r , values pertain to the distances of m 9 B r - - - O for protic solvents and Br1(1 - - C for aprotie solvents. The == 0 . 2 9 d coordination distances for alcohols are fairly constant irrespective of ~ 08 their wide spread in A N value. Thus, the quite different solvent I properties, e.g., from hexafluoropropanol (1) to butyl alcohol (11), 0.1 ~ ~ 2 1 3 are not reflected in their anionsolvent interatomic distances. This -~.~, . ,., , . , . , finding contradicts the generally 20 40 60 observed fact that stronger interacAN tions lead to shorter distances. There is another interesting finding Figure 2. Correlation of the peak height h of the in Fig. 3. The r value for water (2) Fourier-transform spectrum for Br- ion with the is clearly larger than those for acceptor number A N of the solvent: (O) protic and alcohols. This is also difficult to (O) aprotic solvents. Taken from ref. 5. understand because water solvates i
248 ions far better than alcohols and it is smaller than alcohols in size, thus the geometrical constraints to solvate the anion must be less in water. Another solvent which has an unreasonably larger r value is formamide (7). The suggested mechanism for the large r values for these two solvents is that the second solvation sphere molecules or perhaps more distant solvent molecules pull the primary solvent molecules outward. The position of the primary solvents is determined by two opposing attractions, one from Br- and the other from outer shells. It is well known that water and formamide are the most strongly ordered, thus the interactions between the primary and the secondary solvation shells must also be stronger. Obviously alcohols have weaker solventsolvent interactions and arc able to come closer to Br- than water or formamide. The r values for aprotic solvents arc larger than those for protic solvents and similar to that found for tetraalkylammonium bromide crystal.10 The N values for protic solvents in Fig. 4 are referred to the generally accepted value of six for Br- in water. The values for aprotic solvents are determined by using tetrapropylammonium bromide crystal as a reference material, for which the XRD result is known. 10 The N values are distributed over a wide range and do not seem to correlate with A N value. However, careful inspection gives us the following conclusions. Firstly, the N values for alcohols are linearly correlated with A N values. The solvents, water (2), acedc acid (4) and formamide (7) are exceptions again. Mayer has
I
3.61~ 01; 22
'
I
'
I
'
I
F;~0190 15 L- 0 ~ 6 18 14
,.<
O13
3.4
3.2 -
12 e~
0
sos
11 ~
3
'
'
2O
'
!
"s
4O AN
,
4ee2
!
-
"3
,
!
,
9
I 6O i
Figure 3. Correlation of the coordination distance r for Br- ion with the aeeeptor number A N of the solvent: (Q) protic and (O) aprotic solvents. Taken from ref. 5.
'
-
I
'
2~O17
I
'
O15 O19 O ~
I
'
I
0'
4
I
e2
. ~ ;
1
3
g o ,
I
20
/
~/l~le 9 i
,
l
40 AN
,
I
~
I
60
Figure 4. Correlation of the coordination number N for Br- ion with the acceptor number A N of the solvent: (O) protic and (O) aprotic solvents. Taken from ref. 5.
249
Table 1. EXAFS Curve Fitting Results for Br- in solvents and the Mayer-Gutmann acceptor number A N. solvent
r/nm
N
(1) (2) (3) (4)
0.311 0.319 0.312 0.314 0.379 0.314 0.315 0.326 0.314 0.312 0.313 0.312 0.318 0.366 0.343 0.348 0.353 0.350 0.361 0.349 0.352 0.356 0.355 0.357 0.356
4.7 6a 4.4 6.4 3.2 3.3 3.7 4.9 3.1 2.5 2.9 2.7 2.6 7.8 4.3 4.7 5.6 3.1 6.3 4.6 5.1 6.3 2.9 5.8 5.4
1,1,1,3,3,3-hexafluoro-2-pmpanol water 2,2,2-tdfluoroethanol acetic acid, b 1st shell 2rid shell (5) ethylene glycol (6) methanol (7) formamide (8) ethanol (9) isopropyl alcohol (10) propanol (11) butyl alcohol (12) t-butyl alcohol, c 1st shell 2nd shell (13) nitromethane (14) acetonitfile (15) dimethyl sulfoxide (16) propylene carbonate (17) ~.,V-dimethylformamide (18) nitrobenzene (19) pyddine (20) N,N-dimethylacetamide (21) N-methyl-2-pyrrolidinone (22) acetone (23) ethyl methyl ketone
AN 61.6 54.8 53.3 52.9 42.8 41.5 39.8 37.1 33.6 31.0 30.0 27.1 20.5 19.3 19.3 18.3 16.0 14.8 14.2 13.6 13.3 12.5
aThr bromide" ion in water is used as a reference for the solutions of pmtic solvents and its value is assumed to be six. bThe results of a two-shall model. The N' value ratio for the 1st and the 2nd shell is fixed at 2:1. CThe results of a two-shell model. The N value ratio for the 1st and the 2nd shell is fixed at 1:3. originally intended that the acceptor number should pertain to the interaction between the negative site on an indicator molecule, triethylphosphine oxide, and a single coordinating solvent molecule. Later, however, he found that the primary solvation numbers at the site are around two or three in alcohols and water by ir spectroscopy. 11 The present result indicates further that the A N values for alcohols are strongly governed by the solvation number. Secondly, the larger molecules like N,N-dimethylacetamide (20) or N,N-dimethylformamide (17) have larger coordination numbers compared to those for smaller molecules such as nitromethane (13) and acetonitdle (14). This difference in coordination number may be explained if we consider that each solvent directs two methyl groups to a Br- ion. Thus for dimethylfomamide (17) the N value is 6.3 whereas for nitromethane (13) it drops to 4.3. The other solvents having two methyl groups, dimethyl sulfoxide (15) and acetone (22), also have large N values. The large N values for aromatic solvents, nitrobenzene (18) and pyddine (19),
250
could be explained if the Br- is solvated by facing the ring of the solvent molecule. As linear small solvents like acetonitrile (14) or nitromethane (13) have relatively large N values, these solvents must be interacting with Br- through ion-dipole interactions, orienting the methyl group toward it.
0.5
at
the
With the knowledge of the solvation structures in the bulk phase, we now look at the structure of the solution surface. The surface sensitivity of the present TR-XAFS technique is demonstrated in Fig. 5. In the figure are shown two near edge spectra from the aqueous solution surfaces of 0.1 mmol dm -3 stearyltrimethylammonium bromide (STAB) and 0.1 mol dm -3 potassium bromide. In spite of 1/1000 times smaller concentration, the STAB solution gives a signal four times stronger than the KBr solution. The spectrum for the KBr solution is similar in shape to that obtained by the transmission method, 5 although the STAB spectrum is different. The EXAFS oscillation spectra are shown in Fig. 6, one from STAB at the surface by TR-XAFS and the other for Br- in the bulk by transmission. The TRXAFS signal from 0.1 tool dln -3 KBr solution was weak and noisy because of its small density at the surface, therefore, the bulk transmission spectrum for Br- is used in Fig. 6. It is obvious that the oscillation pattern for STAB is complex, i.e. the second positive peak has a shoulder. This spectrum clearly indicates that the Br- ion segregated at the surface by STA cations must have at least two
I
I
I
i
I
I
i
I
I
i
e ~ / . , ~ m M STAB =
A
IV. Bromide Ion Air/Solution Interface
I
~_
--
0.4 03
0.1 M KBr
0.'2
l
I
I
I
13450
I
I
I
i
,
13500
I
I
E~ eV
Figure 5. The Br K-edge absorption spectra of Br- at aqueous solution surfaces by the TRXAFS method, l; total-conversion-electron current and Io; the gas ionization current of X-ray intensity monitor. "1
I
I
1
I
I
I
I
I
I
!
I
r
%0
-1 V |
I
2
I
4 k / A -1
6
8
Figure 6. The Br K-edge EXAFS k3 X(k) spectra of Br- ion; the surface of 0.1 mmol dm -3 STAB by the TR-XAFS (solid line) and the bulk of 0.1 mol dm -3 KBr by the transmission (dotted).
251 Table 2. EXAFS Curve Fitting Results for Br- and Rb +. solute/method i
....
BrSTAB/TR-XAFS, 1st shell 2nd shell TBAB/TR-XAFS LiBr/TR-XAFS KBr/transmission Rb + RbCI+SDS(2 mmol dm -3) /TR-XAFS RbcIfrR-XAFS RbCl/transmission
,
c/mol dm -3
r/nm
N
10--4 0.5 0.5 0.1
0.312 0.522 0.318 0.323 0.323
9.6 2.6 6.4 5.5 6.0 a
0.0283 0.0209 0.0188 0.0145 0.0160
5x10 -3
0.288
8.3
0.0225
0.1 0.1
0.288 0.288
7.7 6.0 a
0.0220 0.0199
cT/nm
9
STAB; stearyltrimethylammonium bromide, TBAB; tetrabutylammonium bromide, TR-XAFS; total-reflection total-conversion-electron XAFS, SDS; sodium dodecylsulfate, aThe reference for coordination number, c; concentration, r; coordination distance, N; coordination number, and ~, the Debye-WaUer-like factor. distinct coordination shells. Some of the curve fitting calculation results are given in Table 2. The first shell distance of 0.31 nm found for the STAB solution surface is similar to that of the bulk, thus pertains to the B r - - - O distance. Our interpretation of the r value of 0.52 nm for the second shell is illustrated in Fig. 7. Since the second shell does not appear in the spectrum for the bulk solution, we believe this peak does not correspond to the second hydration water. The geometry shown in Fig. 7 with the two distances of 0.31 and 0.52 nm gives an angle of about 110" between the O - - - B r and the O - - - C bonds, in accordance with the tetrahedral hydrogen bonding model. V. Preferential Orientation at the Surface The surface provides an asymmetrical field. Two kinds of coordination bonds of the ion at the surface are possible, one in parallel with surface and the other perpendicular to it. Each will have different structures. The present technique using the horizontally polarized X-ray radiation from synchrotron is sensitive to the coordinating atoms sitting on the horizontal plane of the X-ray absorbing atom. The N value for Br- from an LiBr solution is less than six, while the value from the STAB solution is very large, more than nine. These results indicate that for normal Br- the possibility of finding the coordinating water molecules at horizontal positions is less at the surface than in the bulk., while when the STA cation at the surface attracts Br- anions, the anions are surrounded by water molecules horizontally as illustrated in Fig. 7. The TR-XAFS measurements were also performed with rubidium cations which have the same electronic configuration as that of Br- and the K-edge energies are also similar to each other. The measurements were performed on two solution surfaces, one at 0.1 mol dm -3 and the other at 5 mmol dm -3. To the lower concentration sample, sodium dodecylsulfate (SDS) was added, which accumulates the cation at the surface. These samples and conditions were chosen in order to check the conclusions given above. The results are included in Table 2.
252 The effective N value for Rb + without surfactant is clearly larger than its average value in the bulk. This indicates that the R b - -O bond prefers the parallel orientation to the solution surface in contrast to the case of Br-. The surfactant of SDS enhances this orientation preference as STA does to Br-.
qHz
gas solution
H,C
1st shell
"H /
|
~;H2
,"
H-O-. H § ... ~I("C H, H"~ CH3 H
Acknowledgements
This work has been Figure 7. The suggested model for the solvation structure of Brperformed under the segregated at the air/solution interface by surface active cations. approval of the Photon Factory Program Advisory Committee (Proposal Nos.91-179 and 92G182). A part of this work was supported by Grant-in-Aid for Scientific Research No.05453057 from the Ministry of Education, Science and Culture.
References 1 H. Ohtaki and T. Radnai, Chem. Rev., 93, 1157 (1993); Y. Marcus, Chem. Rev., 88, 1475 (1988). 2 J. F. Hinton and E. S. Amis, Chem. Rev., 71, 627 (1971). 3 L X. Dang and B. C. Garrett, J. Chem. Phys., 99, 2972 (1993); S. -B. Zhu and G. W. Robinson, J. Chem. Phys., 97, 4336 (1992); L. Perera and M. L. Berkowitz, J. Chem. Phys., 96, 8288 (1992); J. W. Caldwell and P. A. Kollman, J. Phys. Chem., 96, 8249 (1992). 4 M. KreII, M. C. R. Symons and I. Barthel,J. Chem. Soc., Faraday Trans. 1, 3419 (1987); R. L. Buckson and S. G. Smith, 1. Phys. Chem., 68, 1875 (1964). 5 H. Tanida, H. Sakane and I. Watanabe, J. Chem. Soc., Dalton Trans., 2321 (1994). 6 L. S. Sremaniak, L. Perem and M. L. Bcrkowitz, Chem. Phys. Lett.,218, 377 (1994); L. Perem and M. L. Berkowitz, J. Chem. Phys., 99, 4222 (1993); W. L. Jorgensen and D. L. Severance, J. Chem. Phys., 99, 4233 (1993). 7 H. Tanida, H. Sakane, I. Watanabe and Y. Yokoyama, Chem. Lett., 1647 (1993). 8 V. Gutmann, The Donor-Acccptor Approach to Molecular Interactions,Plenum, New York, 1798, p.29. 9 T. Miyanaga, I. Watanabc and S. Ikeda, Chem. Lctt., 1073 (1988); T. K. Sham, J. B. Hastings and M. L. Pcrlman, Chem. Phys. Lett.,83, 391 (1981); T. IC Sham, Acc. Chem.
Res., 19, 99 (i986).
10 A. Zalkin, Acta Crystallogr.,10, 557 (1957). 11 U. Mayer, H. Hoffmann and R. Kellner, Monatsh. Chem., 119, 1223 (1988).
joumaJof I~OLECUI~R
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 253-260
Redox Properties of Oxo-Bridged Di- and Tri-nuclear Complexes of Heavy Transition Elements: Stability of Mixed-Valence State. Yoichi Sasaki Department of Chemistry, Faculty of Science, Hokkaido University, Sapporo 060, Japan Abstract Redox properties of some oxo-bridged di- and tri-nuclear complexes of the second and third transition elements, Re, Ru, Os, and Mo, are discussed with special attention to stability of the mixed-valence states.
These complexes display multi-step one-electron transfer
processes and give fairly stable mixed-valence oxidation state. Stability of the mixed-valence states is discussed in terms of qualitative molecular orbital schemes based on d~ (metal) - p~ (bridged oxygen) interactions.
I. Introduction Oxide bridge is common to di- and poly-nuclear transition metal complexes. 1 In the diand tri-nuclear complexes of second and third-transition elements, metal-to-metal interaction through oxide bridge is more significant compared with that in the complexes of first-transition elements. Thus dx-p~ molecular orbital description is often applied for interpreting the electronic structure of the di- and tri-nuclear complexes of groups 17 - 19 second and third transition elements. Redox chemistry of these di- and tri-nuclear complexes are versatile as a result of such stronger metal-metal interaction as well as of kinetic stability of various oxidation states involved. Mixed-valence state is very often observed as a stable state. In this paper we wish to discuss on some oxo-bridged di- and tri-nuclear complexes, which were studied recently in our laboratory, from view points of stability of the mixed-valence state. For the complexes of heavy transition elements of group 16 or earlier, metal-to-metal direct bonding interaction is more common rather than the indirect interaction through the dx-p~ overlap. A doubly oxide-bridged dinuclear molybdenum(V) complex with a direct single Mo-Mo bond is also considered for comparison. The complexes discussed here are as follows. Dinuclear complexes with a single oxide bridge: (a) Re(HI) dimers, [Re20t-O)(CI)2(L)4] 2+ (L = 1,10phenanthroline (phen) (1), 2,2'-bipyridine (bpy) (2)) 2 and [Re2(lx-O)(Cl)2(tpa)2] 2+ (tpa = tris(2-pyridylmethyl)amine) (3). 3 (b) dinuclear Os(IV) complex, [Os2(lx-O)(CI)5(CH3COO)0167-7322/95/$09.50 91995 Elsevier Science B.V. All rights reserved. SSDi~167-7322 (95) 00857-8
254 (py)4]2+ (py = pyridine)(4).4 (c) DinuclcarRu(III) compelxes with a (I~-O)(W-CH3COO)2 core, [Ru2(~-O)(~-CH3COO)2(bpy)2(L)4]2+ (L = py (S), 1-mcthylimidazol, (Meim) (6)). 5 (d) Dinuclear Ru(III)complex with a (~-alkoxo)(w-CH3COO)2 core, [Ru2(~t-dhpta-O)(~CH3COO)2]" (dhptaH5 = 1,3-diamine-2-hydroxyl~ntan, tcwaa~dc acid) (7). 6 (e) Dinuclear Mo(V) complexes with double oxide bridges, [Mo202(W-O)2(W-t~n-N,N')] 2+ (tl:~n = tctra(2pyridylmethyl)ethylenediamine) (8). 7 (f) Trinuclear Ru(III) complex with tt3-O bridge, [Ru3(tt3-O)0t-CH3COO)6(py)3] + (9). 8 Electronic structures of all the complexes except for 7 and 8 are considered on thebasis of d~-l~ molecular orbital interaction without direct metal-
Cl
Cl
CI
Cl
Cl
N
CI--~Os-- O-- Os-- O2CCH3
(1,2)
(3)
(4)
.CHs
/~N" Ru I~
tRu I ' N~ J
.CH3
"~'0 I~
CH3
O
o
(~)
p y ~ / " / ~ ' ~ IPY "Ru ~Ru "O H3C ~
H3c
(s)
O"1
CI'13
(s,6)
/~. n ..o. II . ~ ,
tl
Ru
I
py
CH
3
(9)
Figure 1. Schematic drawing of the basic structures of the di- and tri-nuclear complexes discussed in this paper.
255 to-metal bond. Such interaction isnot effective in 7, which has a bridging alkoxide of the binucleating ligand dhpta(5-) in addition to two acetate bridges. The complex 8 has a single Mo-Mo bond. Basic structural units of all the complexes are schematically given in Figure 1.
II. R e d o x Properties
Redox properties of all the complexes except 7 were studied in aeetonitrile. The complex 7 was studied in aqueous phosphate buffer solution (pH, 6.88). While the complexes 1 - 3 are isolated in trivalent (III,III) states, and 4 in tetravalent (IV,IV) and one-electron reduced mixed-valence (III,IV) states, their cyclic voltammograms all display two one-electron reversible redox processes between the oxidation states (III,III) and (IV,IV) in acetonitrile.2"4 The mixed-valence diosmium(III,1V)complex has been isolated and its structure has been determined by the X-ray structural analysis. 4 Three dinuelear Ru(III) complexes are reduced with a quasi-reversible one-electron and irreversible one-electron processes. The oxo-centered trinuclear mthenium(III) complex displays four one-electron reversible processes from (II,II,III) to (III,IV,IV). Complexes with three different oxidation states (II,III,III), (III,III,III), and (III,III,IV) have been isolated. 9 The X-ray structural determination have been made for the (III,III,III) and (III,III,IV) complexes. 10 The triphenylphosphine derivative in the (II, III,III) state was also characterized by a single crystal X-ray sturactural determination. 11 Electronic absorption spectra of all the mixed-valence states have been obtained by using speetroelectrochemical technique where they have not been isolated. Redox potentials relevant to the mixed-valence states discussed in this paper are given in Table 1.
III. Comproportionation Constants
From the data of redox potentials given in Table 1, we can calculate the comproportionation constants of the mixed-valence state defined as follows. Kcom = [M2(n,n + 1)]2/[M2(n,n)][M2(n+ 1 ,n+ 1)] (1) where n and n+l denote the oxidation number of the metal ions. The Kcom values were calculated from the separation of the two redox potentials (AE) by using the equation 2, and given in the Table 1. Kcom = exp(FAE/RT) (2) The Kcom values are taken as a measure of the thermodynamic stability of the mixed-valence state against disproportionation. All the mixed-valence complexes have fairly large Kcom values, indicating their high stability. Enormous stability of the mixed-valence
256
Table 1. Comproportionation constants and associated redox potentials of some mixed-valence complexes of second and third transition elements, a complex mixed-valence oxidation state (n,n+l) formalnetcharge in parenthesis 1 (III,IV) (+3) 2 (Ill,IV) (+3) 3 (Ill,W) (+3) 4 (III,IV)(+I) 5 (II,III) (+1) 6 (II,III)(+l) 7 (II, III) (-2) 8 (IV,V) (+1) 9 (II,III,III) (0)
El/2 or Eap (Ecp) /V b
(n,n+l) / (n+l~n+l) +1.01 +0.97 +0.90 +0.03 -0.39 -0.55 -0.34 -1.39 -0.32
Kcom
reference
(n,n+l) / (n~n) +0.31 +0.28 +0.23 -1.15 -1.09 -1.16 -0.94 -1.86 -1.59
6.8 4.6 2.1 8.8 6.8 2.0 1.4 6.2 2.9
x x x x x x x x x
1011 1011 1011 1019 1011 1010 1010 107 1021
2 2 3 4 5 5 6 7 8
9 (III~III~IV) (+2) +0.74 +1.67 7.7 x 1015 8 a Solvent is acetonitrile except for 7 where the solvent is aqueous phosphate buffer (pH, 6.88). b V vs Ag/Ag+ for I - 6, 8 and 9, and vs Ag/AgCl for 7. Although the potentials were measured with different reference electrodes, direct comparison of the Kcom values is valid since they are the function of only the difference in two relevant potentials.
(II, III,III) state of the oxo-centered trinuclear ruthenium complex 9 and the (III,IV) state of the oxo-bridged diosmium complex 4 is noted.
IV. Discussion Table 1 clearly shows high thermodynamic stability of all the mixed-valence states involved. Since the solvational change during rcdox processes would affect the rcdox potentials, Kcom values which is the function of the difference in two successive redox potentials would also be affected by solvational change. Electronic structure of the mixedvalence states would be more important, however, in determining Kcom values (vide infra). Electronic structures of all the complexes except 7 have been interpreted in terms of molecular orbital consideration. Simplified qualitative molecular orbital schemes are given in Figure 2.
257 ."
es~t~
.."
\
\
/ .t.~ooo
I
~
t
\
d~
9
L ooo~
,,
\
"
(M)
\
~
\
\
(M)
\
",,__.~176176176 p~
x%
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~~ ~176
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(o)
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,
9
t
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dn /o"- .. . . . . t
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;
"-
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,0" do
(M) '"
"
o~ ~176 dn 3
do 9
\
~t t
":
(o)
(iii)
(iv)
Figure 2. Simplified qualitative molecular orbital description of the ~-type interaction of oxidebridged di- and tri-nuclear complexes (only the relevant orbitals are shown). (i) Linear oxobridged dinuclear complexes. (ii) Bent oxo-bridged dinuclear complexes (with additional carboxylate bridges). (iii) Oxo-centered trinuclear complexes. (iv) o- and ~-type interaction of edge-shared dinuclear complexes.
Stability of the mixed-valence states are discussed on the basis of these scheme.
Possible
influence fzom solvational change is also commented briefly. (1) Influence of Solvation.
The mixed-valence states in Table 1 have various net
charge from-2 to +3. The solvational change on reduction or oxidation would be different depending on the net charge of the mixed-valence species. It was pointed out, however, that the solvation would give less significant influence on the redox potentials than expected. 12 The complexes in Table I are di- and tri-nuclcar complexes in which net charge is less influential as it may be delocalized over cluster core. The biggest Kcom value in Table 1 is observed for the mixed-valence complex of net charge zero. This species would gain significant solvational
258
stabilization on redox interconversion and thus Kcom would be small if no other factors were concerned. It is thus reasonably assumed that the net charge of the mixed-valence species is not important in determining Kcom values. (2) Dinuclear Complexes with a IAnear Oxide Bridge, 1 - 4. The molecular orbital scheme applicable to the dinuclear complexes 1 - 4 is given in Figure 2(i). If the z-axis is taken along the direction of metal-O-metal bond, two virtually equivalent interactions, dxz-Pxdxz and dyz-py-dyz, give two bonding, two essentially non-bonding and two anti-bonding orbitals. 13 For the dirhenium(III,IV) complexes, three electrons occupy two 'non-bonding' orbitals. In the case of the diosmium(III,IV) complex, one electron enters into 'degenerated' anti-bonding orbitals. In both cases, however, two non- or anti-bonding orbitals may not be degenerated due to different ligands in the x-y plane or due to Jalm-Teller distortion. The structural comparison of the diosmium complexes in the (IV,IV) and (III,IV) states does not provide unambiguous evidence for the Jahn-Teller distortion.'~Nevertheless, non-equivalence of dxz and dyz orbitals are clear and the splitting of the two anti-bonding orbitals would be an important factor for stabilization of the mixed-valence diosmium state. Because of the existence of another empty anti-bonding orbital, kinetic stability of the mixed-valence state is not significantly lowered. For the dirhenium complexes, unpaired electron occupies the nonbonding orbital whose splitting would be less influential to the stability. Thus the Kcom values of the mixed-valence states are smaller. (3) Dinuclear Ruthenium Complexes with Bent Oxide Bridge, 5 and 6. The two dx orbitals considered for the linear pL-oxocomplexes are now non-equivalent in the bent Ix-oxo complexes. Only one set of the dx-px-cl~ interaction is now significant. 14 In the scheme (ii) in Figure 2, two other dx orbitals which are omitted in the scheme may be in the energy level close to the dx-px non-bonding orbital. In the mixed-valence state, an additional electron occupies the anti-bonding orbital. The redox processes (II,III)/(III,III) and (II,II)/(II,III) are not reversible. This irreversibility is compared with higher reversibility of the diosmium complexes where two anti-bonding orbitals are relevant. The Kcom values of the mixedvalence states of 5 and 6 are in similar magnitude to that of 7 which has no dx-px stabilization (vide infra). Thus the dx-px interaction of 5 and 6 does not appear to stabilize the mixedvalence state. (4) Dinuclear Ruthenium Complex with a Binucleating Ligand, 7. The dX-l~ molecular orbital description is not applicable to this complex, since bridging alkoxide does not appear to operate as x-electron mediator. While all the complexes, I - 6, are essentially diamagnetic as expected by the electron-pairing in the molecular orbitals, the complex 7 is paramagnetic.6 It is known that on protonation at the oxide bridge the p-oxo-di-~-acetate diruthenium complexes becomes paramagnetic.15 Antiferromagnetic coupling of the two ruthenium ions would operate in these complexes. Electronic absorption spectrum of the mixed-valence (lI,III) state fIom 7 shows intervalence band. 6 Stabilization by the exchange interaction would thus explain the relatively high Kcom value.
259 (5) Oxo-Centered Trinuclear Ruthenium Complex, 9. Electronic structure of the oxo-centered trinuclear ruthenium complex 9 has been explained previously in terms of the dxpx interaction between the metal dx and central oxygen px orbitals (Figure 2(iii)) 8 ' 9, 11 One bonding, one anti-bonding and two essentially non-bonding orbitals are resulted by the interaction. Remarkable stability of the (II,III,III) state is explained by this molecular orbital scheme because in this oxidation state all the molecular orbitals except one anti-bonding orbital are just occupied. It is considered in this molecular orbital scheme that additional electron is fully delocalized over the three ruthenium ions. Analogous trinuclear ruthenium complexes with different terminal ligands (at the positions of pyridine ligands in 7), also show similar high stability of the mixed oxidation state (II,III,III). 16"18 The comproportionation constant of the (III,III, IV) state is 7.7 x 1015. This value is also significantly big, although it is less than that of the (II,III,III) state. Two other dar orbitals have similar energy to the 'non-bonding' orbitals, and it is not certain which orbital is involved. (6) Dimolybdenum Complex with a Single Mo-Mo Bond, 8. The molecular orbital scheme given in Figure 2(iv) is based on the qualitative calculation reported earlier for the [Mo204(R-cysteinato)2] 2" ion which has the same structural core. 19 It has been suggested subsequently that the electronic structure of the Mo204-complexes are not significantly affected by the type of non-oxo ligands. 7' 20 According to the scheme, additional electron must occupy the az-bonding orbital, and the metal-metal interaction would become stronger. Stability of the mixed-valence state (IV,V) is rather small as compared with other complexes with metal-metal indirect interaction. It is concluded that the stabilization on ~z half-bond formation is not particularly favorable as compared with a ~ single bond of the (IV,IV) species. In this paper, we demonstrated that oxide bridged di- and tri-nuclear complexes of the second and third transition elements show thermodynamically very stable mixed-valence states against disproportionation. Contribution of the d3Z-l~ molecular orbital formation is responsible for the high stability.
Acknowledgments. The author is grateful to Dr. K. Umakoshi, Department of Chemistry, Hokkaido University, for valuable discussions. This work is supported by a Grant-in-Aid for Scientific Research (No. 05225201) on Priority Area of "New Developement of Organic Electrochemistry" and a Grant-in-Aid for Scientific Research (No. 05403008) from the Ministry of Education, Science, and Culture, Japan. A research grant from Mitsubishi Foundation is also gratefully acknowledged.
260 References
1 2 3 4 5 6
7 8 9 10 11 12 13 14 15 16 17 18 19 20
F.A. Cotton and G. Wilkinson, Advanced Inorganic Chemistry, 5th ed. (Wiley Interscience, 1988). T. Takahira, K. Umakoshi, and Y. Sasaki, Chem. Lett., 2315 (1994). T. Takahira, K. Umakoshi, and Y. Sasaki, unpublished. Y. Imbe, K. Umakoshi, C. Matsunami, and Y. Sasaki, Inorg. Chem., in press. T. Fukumoto, A. Kikuchi, K. Umakoshi, and Y. Sasaki, unpublished. T. Tanase, M. Kato, Y. Yamada, K. Tanaka, K. Lee, Y. Sugihara, A. Ichimura, I. Kinoshita, M. Haga, Y. Sasaki, Y. Yamamoto, T. Nagano, and S. Yano, Chem. Lett., 1853 (1994). R. Hazama, K. Umakoshi, A. Ichirnura, S. Ikari, Y. Sasaki, and T. Ito, Bull. Chem. Soc. Jpn., in press. Y. Sasaki, A. Tokiwa, and T. Ito, J. Am. Chem. Soc., 109, 6341 (1987). J.A. Baumann, D. J. Salmon, S. T. Wilson, T. J. Meyer, and W. E. Hatfield, Inorg. Chem., 17, 3342 (1978). O. Almog, A. Bino, and D. Garf'mkel-Shweky, Inorg. Chim. Acta, 213, 99 (1993). F.A. Cotton and J. G, Norman Jr., Inorg. Claim. Acta, 6, 411 (1972). A.A. Vlcek, Coord. Chem. Rev., 43, 39 (1982). J.D. Dunitz and L. E. Orgel, J. Chem. Soc., 2594 (1953). Y. Sasaki, M. Suzuki, A. Nagasawa, A. Tokiwa, M. Ebihara, T. u C. Kabuto, T. Ochi, and T. Ito, Inorg. Chem., 30, 4903 (1991). P. Neubold, K. Wieghardt, B. Nuber, and J. Weiss, Inorg. Chem., 28, 459 (1989). H.E. Toma and C. J. Cunha, Inorg. Chim. Acta, 154, 63 (1988). H.E. Toma and C. J. Cunha, Can. J. Chem., 67, 1632 (1989). M. Abe, Y. Sasaki, T. Yamaguchi, and T. Ito, Bull. Chem. Soc. Jpn., 65, 1585 (1992). D.H. Brown, P. G. Perkins, and J. J. Stewart, J. Chem. Soc., Dalton Trans., 1105 (1972). K.Z. Suzuki, Y. Sasaki, S. Ooi, and K. Saito, Bull. Chem. Soc. Jpn., 53, 1288 (1980).
journal o f
MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65166 (1995) 261-264
Study of the Reduction of CoONH3)4(pzCO2)2. by Fe(CN) 4- in Binary Aqueous Mixtures: An Interpretation of Solvent Effects Based on Spectroscopic Data Pilar P~rez-Tejeda', J~in Benko t, Maria Luisa Moyli and Francisco S,Anchez Department of Physical Chemistry, Seville University, C/Profesor Garcia Gonz~ilez s/n, 41012 Seviila,SPAIN 1Department of Physical Chemistry, Comenius University, Bratislava, SLOVAKIA Abstract The reduction of [Co(NH3)4(pzCOg] 2+(pzCO2=pyrazincarboxylato) by [Fe(CN)6]4- has been studied in several water-methanol mixtures. This study, in addition to the spectroscopic data obtained for the metal-to-metal charge transfer band of the ion-pair [Co~H3)4(pzCO2)]2+-[Ru(CN)6] 4- and of the binuclcar complex [(NH3)sCONCRu(CN)5 ]- in the same mixtures, supports the idea that the classical (continuun) model of solvent, with a linear response of the polarization, does not give correct predictions for the free energy of solvent reorganization in solvent mixtures. I. Introduction A particular challenge in electron transfer studies has always been the determination of the quantities relevant to the reaction, e. g. the free energy change AG~ the solvent and intramolecular reorganization energies, k, and k~, respectively, ..etc. Approximate estimations of these parameters at different levels of sophistication can be used for the calculations of electron transfer rates, which can be compared to experimental rates. An alternative approach to the independent estimation of medium and intermolecular parameters, which govern electron transfer, rests on the analogy between radiative and radiationless processes in condensed media ~. The purpose of this work is to use the second approach in order to explain solvent effects on the title reaction in binary aqueous mixtures. The selected reaction is especially suitable since working in excess of the acceptor, the equilibrium constant for the precursor complex formation and the rate constant for the electron transfer step can be separately obtained. Therefore, one can discuss solvent effects on the p u r e electron transfer rate constant. H. Experimental Materials.- All materials were commercial with the exception of the complex [Co(NH3)4(pzCOz)](CIO4) 2 which was prepared and recrystallized according to the method described in ref. 2. The water-cosolvent mixtures were made up by weight. Kinetics.- Kinetic runs were carried out in the thermostated cell compartment of a Hitachi 150-20 UV-visible spectrophotometer. The cell temperature was kept at 298.2-,-0.1 K. The kinetics were followed at 420 nm. The cobalt(llI) complex concentration was always 5 10-4 tool dm -3. All runs employed an excess of ferrocyanide. Na(H2edta ) had to be added to prevent precipitation of Co3[Fe(CN)6]2. The pH was kept constant at 5.7 by using the appropriate acetic/acetate buffer. The ionic strength was fixed at 0.5 mol dm -a by adding the adequate quantity of sodium perchlorate to the working medium in each case. Spectra.- All the spectra were obtained in a Hitachi t50-20 UV-visible spectrophometer at 298.2 K. HI. Results and Discussion Table 1 gives the values of the observed (pseudo-first order) rate constants, l~s. In 0167-7322/95/$09.50
91995 Elsevier Science B.V. All rights reserved.
SSDI 0167-7322 (95) 00817-9
262 this Table are also listed the electron transfer rate constants, k~,, obtained from the plot of k ~ - ' against [[Fe(Cr%]'-]-' 3. Table 1.- Pseudo-first order rate constants, l q J s -t, for the reaction [Co(NH3)4(pzCO2)] 2§ + [Fe(CN)6] 4- in various water-methanol mixtures at 298.2 K. Ionic strength=0.5 tool dm -3 . 102 k , J s - '
D, %MethanoP
78.5 0.00
76 5.68
[[Fe(CN)6]4-]/mol dm -3 0.0025 1.4 0.0034 . . . . 0.0050 2.2 0.0075 2.9 0.0100 3.6 0.0200 4.7
1.5 . . 2.7 3.7 4.1 6.6
102 k~/s -I
13.1
7.2
74 10.20
.
.
2.1 . 3.8 5.0 6.4 9.6
70 18.37
.
20.0
.
.
3.8 . 6.6 8.5 10.3 14.6
66 26.53
5.8 .
9.6 12.4 15.1 18.3
6.2 7.7 10.5 13.4 -----
26.5
32.9
.
24.0
64 34.64
"Percentage by weight; D, is the static dielectric constant at 298.2 K The spectroscopic data corresponding to the metal-to-metal charge transfer bands of the binuclear species [(NH3)sCoNCRu(CN)5 ]- and of the ion-pair [Co(NH3)4(pzCOg]2+[Ru(CN)d 4- are given in Table 2. In this Table E ~ = h v ~ is the energy corresponding to the absorption maximum, and Av~, the full width at half-height. From the values of Eop and &vu2 in Table 2 and by using the Hush equations 4 the reorganization free energy, k, and the free energy ~:hange, AG~ corresponding to the electron transfer processes were obtained. The values of these parameters are also included in Table 2. According to current theories of electron transfer processes 3, the rate constant of an electron transfer reaction can be formulated as:
k.r=K.10 . exp (-AG*/RT)
with
AG ~
( k + A G o') 2 -H~ 4k
(i)
Here •a is the electronic transmission coefficient, ~, is the nuclear frequency factor, AG' is the Dee energy of activation for the electron transfer reaction, k is the reorganization energy and AG~ is the free energy change accompanying the reaction. In our case H~,--5 kJ mo1-1 (see ref. 5). This value assures the adiabatic character of the process. According to this ~:~,=1. The nuclear frequency factor depends in our case only on the internal vibrational modes. Assuming that there is no coupling between internal and external (solvent) modes, ~, can be considered constant when changing the medium. In this case a value of 10 *4 s -~ seems reasonable for this parameter 6. So we will assume that solvent effects on the process considered are a reflection of the changes in AG" when the solvent changes. The free energy AG~ is expected to decrease with increasing alcohol content. This is because the electron transfer process implies a decrease of the absolute values of the charges of both, the donor and the acceptor. On the other hand, taking into account that the
263 Table 2.- Spectroscopic data corresponding to the metal-to-metal charge transfer bands. T-298.2 K. D,
Eo~k.l mol q
[(NHOsCoNCRu(CN)~78.5 320 76 316 74 323 70 332 66 325 64 324 [Co(NH3)4(pzCOOl~-[gu(CN)~ ~ 78.5 310 76 308 74 308 70 310 66 315 64 316
Avtry'crn-l
6959 7315 7575 8094 8614 8874 7380 7430 7780 8060 8220 8400
k/k.l mol q
AG~
253 279 299 342 387 411
67 37 24 -10 -62 -86
284 288 316 339 352 368
26 20 -8 -29 -37 -52
mol -~
factor (1/D,-1/Dop) (D, and Dop are the static and optical dielectric constants, respectively) decreases as the alcohol content increases, the reorganization energy is also expected to decrease as the amount of alcohol increases. So, these two parameters, according to the predictions of the continuum model of the solvent, will make the free energy of activation of the reaction smaller when the proportion of alcohol in the reaction medium increases. This is, indeed, in agreement with the observed trend in the rate constants, k~,, given in Table 1. Therefore, the dielectric continuum model explains, apparently, at least qualitatively, the observed variations in k~. However, consideration of Table 2 revels that the s u ~ of this simple model is only casual: k increases when the content of alcohol increases. At first, this result is rather unexpected. In order to check in some way the validity of this, the solvent effects on the metal-to-metal charge transfer band of the binuclear complex [(NH3)sCoNCRu(Cb05 ]- was studied in the same water-methanol media. A good linear correlation observed between the X values corresponding to the above binuclear species and to the ion-pair implies, in the authors" opinion, that the same factors are relevant in the configuration of the X values in the binuclear complex and in the ion-pair. This point constitutes one of the main results of the work: simole continuum models of the solvent cannot be applied to calculate solvent reorganization ener~,ies, even oualitativelv, in structurally complicated media such as solvent mixtures, where the phenomena associated tO preferential solvation can make the macroscopic dielectric solvent parameters meaningless. Moreover, from the data in Table 2 it is possible to estimate the free energy of activation of the reaction studied. In order to do this we must assume, above all, that the optical electron transfer, to which the band correspond, is the same as the thermal electron transfer, to which the ka values correspond. However, the data in Table 3 cannot be used directly for estimating the activation free energy of the thermal electron transfer reaction. The first obvious reason comes from the fact that the band corresponds to the [Co(NH3 )4(pzCO2) ]2+_[Ru( CN)6]4- ion-pair instead of to the [Co(NH3)4(pzCO2) ]2+_[Fe(CN)6] 4one. On the other hand, the ionic strength present in the working media is different in thermal and optical experiments and it is necessary to correct the data from its effects. Finally, in the
264 case of the optical electron transfer ~ contains, in addition to the ~ and go terms contributing to the thermal process, an additional term g,, 6. However, for the thermal electron transfer, only the k o and g t contributions are relevant. Besides, the value of AG~ given in Table 2 must be corrected from the difference in the redox potentials of the iron and ruthenium cyanide complexes (0.5 VV). From data in Table 2, once they are corrected as indicated above, the free energy of the thermal electron transfer has been calculated by using eXl. I and the H~b in ref. 5. A value of H,6=3.68 kJ mol -~, that is about 75% of the H,b value obtained from the band, was used in this eq. This difference arises because the optical and thermal electron transfers are produced at different points along the reaction coordinate 7. The calculated values of AG* as well as the experimental AGmvalues (obtained from the rate constants with v,= 1014 S-2) are also given in Table 3. It can be seen that both series of values follow the same trend. Indeed, the order of magnitude is the same for both series of data. However, there is not a quantitative agreement. These differences can be related to an overestimation of the ~. values from the metal-to-metal charge transfer band. In other words, ~. is different for optical and thermal electron transfers due to a nonlinear response of the medium to electric fields. In this c,mse the slope of the free energy surface would increase more rapidly than in the case of a parabolic surface, far from the minima given a ~,>;k~. If this explanation is accepted we must conclude that the linear restmnse of the medium does not hold in solvent mixtures. .
Table 3.- Activation free energies estimated for the optical electron transfer within the [Co~H3)4(pzCO2)]2§ *" ion-pair (see the text). T=298.2 K.
D, AGm,~/kJmol -a AG~/l~Jmol -~
78.5 58.8 87.6
76 57.4 86.1
74 51.9 85.0
70 49.2 84.6
66 49.3 84.3
64 47.4 83.8
IV. Acknowledgements The authors wish to thank Professor Sumi, Tsukuba University (Japan) and Professor Balahura, New Guelph University (Canada) for helpful suggestions concerning the interpretation of the results. The authors also wish to ackalowledge the financial support from the Consejeria de Educaci6n y Ciencia de la Junta de Andalucia (D.G.C.Y.T. PB92-0677). V. References and Notes 1 J. Cortes, H. Heitele and J. Jortner, 3. Phys. Chem. 98, 2527 (1994). 2 J.M. Malin, D. A. Ryan and T. V. O'Halloran, J. Am. Chem. Soc. 100, 2097 (1978). 3 F. Sfinchez-Burgos, M. L. Moy~i and M. Galfin, Prog. React. lO'net. 19, 1 (1994) and references therein. 4 N.S. Hush, Electrochim. Acta. 13, 1005 (1968). 5 In fact the reported value corresponds to Co(NH3)4(pzCO2)2+/Ru(CN)6't- ion-pair. But a similar value has been obtained for similar ion-pairs with Fe(CN)64- instead of Ru(CN)64-. See for example A. J. Miralles, A. P. Szecsy and A.Haim, lnorg. Chem. 21, 697 (1982). 6 B.S. Brunschwig, S. Ehrenson and N. Sutin, J. Phys. Chem. 90 (1986) 3657. 7 Standard Potentials in Aqueous Solutions, A. J. Bard, R. Parsons and J. Jordan eds., IUPAC, New York, 1985 chap. 14. 8 M . H . Chou, C. Creutz and N. Sutin, Inorg. Chem. 31, 2318 (1992).
journal of
NIOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65166 (1995) 265-268
Ion Pair of Tetraalkylammonium Picrates Kiyoshi Sawada and Fumie Chigira Department of Chemistry, Faculty of Science, Niigata University, Niigata 950-21, Japan Abstract The ion-pair formation equilibria of tetraalkylammonium ion with picrate ion (C6H2(NO2)30-) in 1,2-dichloroethane were studied at 25.0 ~ by means of conductivity measurement. The cation studied were tlie symmetric (TAAn+; (CnH2a+I)4N+, n = 1-8) and asymmetric (TOMA§ (CsH17)3(CH3)N§ CTMA; (C16H33)(CH3)3N+, ) tetraalkylammonium ions, Q§ Limiting molar conductivities of salts, A o, and ion-pair formation constants, Kip were obtained by the analysis of the conductometric data with Fuoss-Justice Equation. Stokes radius, r~, of the tetraalkylammonium ions was estimated from the limiting molar conductivity. The closest approach, a, between the cation and anion of the ion pair was estimated from the ion-pair formation constant. The value of r s estimated for asymmetric Q+ is comparable to that of symmetric Q+ having a corresponding number of carbon atoms, whereas the closest approach of the asymmetric Q+ is much smaller than that predicted from the number of carbon atoms. The structures of ions and ion pairs were discussed.
Introduction Tetraalkylammonium salts are the most common electrolytes for the non-aqueous solvent systems and high molecular weight ammoniums have been used for the ion-pair extractant. The thermodynamic properties of the salts such as partition equilibria, solubility, ion-pair formation etc. have been studied in a variety of solvents. The detailed equilibria or structure of ion pair, however, are not fully elucidated. We studied the partition equilibria and structures of the ion pair of cationic complex of polyether compounds with picrate ion and reported that the dissociation of ion pair in the organic phase plays an important role for the partition equilibria. 1-4 In the present paper, the ion-pair formation in 1,2-dichloroethane was studied by using the one of the most simple system of electrolyte, tetraalkylammonium picrate. The effect of the chain length of the alkyl group was studied for the symmetric and asymmetric tetraalkylammonium ions. Experimental The solvent 1,2-dichloroethane was washed twice with deionized water. Tetraalkylammonium bromide Q+Br- was converted to picrate salt Q+pic- by solvent extraction method. A portion of 1,2-dichloroethane saturated with water (40 ml, Cmo = 0.13 mol dm -3) was titrated with 10-3 mol dm -3 tetraalkylammonium picrate by a piston burette in a thermostated vessel (25.0 ~ Conductivity was measured by Toa-denpa Model CG-511C. The cell constant of the electrode was calibrated by KCI aqueous solution and obtained as 0.0920 cm -1. A portion of titrant was dried and dissolved in an alkaline aqueous solution. Then the concentration of picrate ion was determined by spectrophotometry. 0167-7322/95/$09.50 91995 Elsevier Science B.V. All fights reserved. SSDi 0167-7322 (95) 00879-9
266 Results and Discussion The molar conductivity experimentally obtained, Aobsd, is plotted as a function of square root of the concentration of tetraalkylammonium picrate (Q§ V'c. The results of tetrahexylammonium (TAA6+), trioctylmethylammonium (TOMA +) and cetyltrimethylammonium (CTMA § ions are depicted in Fig. 1. Similar results were obtained for other symmetric tetraalkylammonium runs (TAAn*; (CnH2n+I)4N+, n = 1-8). The ion-pair formation of tetraalkylammonium picrate, Q+pic-, is written by
Q* + pic- ~
(1)
Q*'pic-
The ion-pair formation constants, Kip, is given by [Q*-pic-] Kip = [Q+][pic-]
(2)
The conductivity data was analyzed by Fuoss-Justice equation: 5 At,
= A0 - S(cot) u2 + Ecot log(cot) +
JlCot -
(3)
J2(cot) 3/2
where At: is the molar conductivity under the assumption of the complete dissociation of electrolyte. Thus, the degree of dissociation is given by ot = Aobaa/A F and the i o n - p a i r formation constants is Kip = ( l - a ) / ot2cy2. The activity coefficient, y, of 1 19 electrolyte was 50 evaluated by the extended D e b y e - H ~ c k e l -r equation, log y = - A z 2 V'l / (1 + Ba "r A is the limiting molar conductivity of Q*Pic-. The values of parameters of limiting molar conduc,.q tivity, A o, and the ion-pair formation constant, Ki, were evaluated from the data Act,. shown ~ 4t) inr~ig. 1 by the successive approximation with i~ a help of a computer. The value of the closest approach of the ions, a, in the ion pair was = estimated by comparing the value of K l with 8 the prediction of the Bjerrum theory, g' The -~ II r 30 results of A o, K i and a are listed in Table. The value of the m~'lar conductivity calculated by these constants listed in Table are shown in Fig. 1, solid lines. The experimental results show good fitting with the calculated curves. Structure of the ion pair of tetraalkyl20 0 0.5 1. 1.5 ammonium picrate was calculated by molecular mechanics (MM2) with a help of CAShe R ~/c/ 10-2 mollr2 dm-3/2 system. The estimated structure of ion pair of tetraoctylammonium picrate is shown in Fig. 2. Similar structures were obtained for other te- Fig. 1. Plot of molar conductivity as a functraalkylammoniums. The linear chain of alkyl tion of square root of concentration groups spread tetrahedrally from central nitroof tetraalkylammonium picrate. gen atom (left hand moiety of Fig. 2). The A : TAA6+pic phenolate oxygen of picrate ion (right. hand 9 TOMA*pic9 moiety of Fig. 2) is in contact with central part " CTMA+pic of ammonium ion.
",. "-,.
\
267
Table Limiting molar conductivity, ion-pair formation constant and ion size parameters. Cation TAA1 § TAA2 § TAA3 § TAA4 § TAA5 § TAA6 § TAA7 § TAA8 + TOMA § CTMA §
Ao/S cm2mo1-1
log
74.52 69.98 64.08 59.45 54.74 53.19 51.13 49.96 52.77 54.88
Kip
4.29 3.58 3.50 3.43 3.41 3.39 3.37 3.36 3.74 4.11
•A
rsA
rvdwA
4.4 5.8 6.0 6.3 6.3 6.4 6.5 6.6 5.4 4.6
2.43 2.72 3.21 3.73 4.48 4.79 5.35 5.62 4.89 4.45
2.86 3.41 3.82 4.12 4.42 4.69 4.92 5.11 4.74 4.45
:ructure of ion pair of tetraaikylammonium Lcrate calculated by MM2. Left hand moiety: Tetraoctylammonium cation. Right hand moiety: Picrate anion.
Closest approach The closest approach, a, reflecting the distance between the cationic charge of the alkylammonium and anionic charge of the phenolate, are plotted in Fig. 3 as a function of the total number of the carbon atom of the tetraalkylammonium ion. The closest approach, a, of tetramethylammonium (CH.).N § (AT.. = 4) is much smaller than other Q+ and the increment of a of Q§ by the increase m the number N c is very small for other Q§ These results support the structure of the ion pair shown in Fig. 2. That is, the closest approach might scarcely be affected by the increase in the number of the carbon atom except for tetramethylammonium ion. In the case of the CTMA + (N c = 19), only one alkyl group is long chain and others three alkyl groups are methyl groups. Consequently, it is quite reasonable that the value of a of CTMA + is much smaller than that of TAA + having a comparable number of carbon atoms and close to the value of the tetramethylammonium ion. The TOMA + ion which have one methyl group shows also relatively small value of a compared with Q+ having a comparable number of carbon atoms. 9
.
.~
q
.
t.,
.
268
Stokes
radius
The van der Waals radius, r V ..W , . was e s t i m a t e d from the partial molar volume by regarding the Q+ cation as spherical and is plotted in Fig. 3. By the assumption that the Stokes radius of Q+ of low m o l e c u l a r weight is almost the same as r V ..~N , the limiting molar conduc9 . . . tlvlty of t~ose cations m 1,2-dichloroethane were estimated. Thus the limiting molar conductivity of picrate anion was estimated as 35 S cm 2 mo1-1. By using this value, the limiting molar conductivities of other Q+ cations were evaluated. So called Stokes radius, r.., which corresponds to the radius of sphere having the same limiting molar conductivity as that of Q§ was estimated by using the Stokes equation 9Stokes radii, r~, of the tetraalkylammonium ions thus obtained are plotted in Fig. 3 as a9 function of N... The . t., van der Waals radms_ r V GJ.W _is also plotted in Fig. 3. The value of r.. Increases almost l i n e a r l y with the number of carbon atoms of Q+, N c. Although the increment of r V(I.. decr~.ases by the in. crease m the n u d g e r of carbon atoms, the value of r s still increase linearly. These findings support the structure of ion pair shown in Fig. 2. That is, the alkyl group of Q+ is u n f o l d e d and linearly spread outward. In contrast to the results of closest approach, r s, of asymmetric Q§ does not so differ fr6m that of the symmetric Q+ having a corresponding number of carbon atoms. These results indicate that the asymmetric Q§ also has the similar structure as the symmetric Q§
O
o< ~ 6 oo
-
~5
-
o
O
._= O I::1 O 0
4
-
O
~ 3r t/) 4.) o2
-
r..) 1
0 0
-
4
8
12
1
2
24
2
32
Number of carbon atoms Fig. 3. Plot of closest approach and ionic radii of tetraalkylammonium. O "a, Closest approach of ions A : rs, Stokes radius of cation * : r 9, van der Waals radius of cation Filled'~r~arks are asymmetric cations.
References
1 Y. Kikuchi, N. Takahashi, T. Suzuki and K. Sawada, Anal. Chim. Acta, 256, 311(1992). 2 Y. Kikuchi, Y. Nojima, H. Kita, T. Suzuki and K. Sawada, Bull. Chem. Soc. Jpn. 65, 1506(1992). 3 Y. Kikuchi, T. Suzuki and K. Sawada, Anal. Chim. Acta, 264, 65(1992). 4 Y. Kikuchi, M. Kubota, T. Suzuki and K. Sawada, Bull Chem. Soc. Jpn. 67, 2111(1994). 5 E. Renard and J. C. Justice, J. Solution Chem., 3, 633(1974) 6 R.A. Robinson and R. H. Stokes, "Electrolyte Solutions", Butterworths, London (1959)
journal of /91OLECUI~ R
LIQUIDS Journal of MolecularLiquids,65/66 (1995) 269-272
ELSEVIER
Ion-Pairing Effects on the $9Co Electric Field Gradients Relaxations of Tripositive Cobalt(Ill) Complex lons 1
in the NMR
Masayasu lida,* Toshie Nakamori, Yuri Mizuno, and Yuiehi Masuda ? Department of Chemistry, Faculty of Science, Nara Women's University, Nara 630, Japan 1"Department of Chemistry, Faculty of Science, Ochanomizu University, Tokyo 112, Japan Abstract 59Co NMR relaxation rates for the [Co(chxn)3] 3§ ( c h x n - trans-(1RC2R)-l,2diaminocyclohexane) and [Co(en)3] 3§ (en m ethylenediamine) ions were measured in the presence of sulfate or oxalate ions. IH relaxation rates for the chxn ligands in the same systems were also measured. The ratios of the 59Co relaxation rates to the 1H ones indicate that the 59Co electric field gradient of [Co(chxn)3] 3+ is appreciably decreased by the formation of the 1:1 ion pair with the sulfate ion. The difference of the concentration dependence of the 59(]0 NMR relaxations between the sulfate and the oxalate ions can be attributed to the faster change in the configuration of the 1:1 ion pair of the oxalate ion.
I. Introduction NMR relaxation method is a very powerful tool for the study of ionic interactions. 2 Many nuclei studied are with spin quantum number I > 1/2 and their relaxations are governed by quadrupole interaction between the nuclear electric quadrupole moment and the electric field gradient (efg) at the nuclear site. Under extreme narrowing condition, the longitudinal relaxation rate (RI) for the 59Co nuclei(l=7/2) can be expressed as
R1 = 2-~e2qQ !2Tc
(1) 491 h I where xe is the correlation time of the fluctuation of principal axis component of the efg tensor, and eq and eQ are the efg along the principal axis at the nuclear site and the quadrupole moment of the quadrupolar nucleus, respectively. As the 59(]0 NMR relaxation is dominated by the nuclear electric quadrupole relaxation mechanism and as the tripositive cobalt(Ill) complex ions strongly interact with anions, 59Co NMR relaxation is useful to see the effect of the outersphere complex on the relaxation rate. Previous studies for the various [Co(en)3] 3§ salts revealedthat the ion pairing with the sulfate ion significantly reduces the 59(]0 relaxation rate, while those with the succinate and tartrate ions slightly increase the relaxation rate. 3'4 However, the effect of the ion pairing on the 59Co efg did not explicitly appear in the [C~en)3] 3§ system. In the present paper, we report 59Co relaxation studies for the [Co(chxn)3] 3§ and [Co(en)3] 3+ in the presence of excess sulfate or oxalate ions. The chxn complex has 0167-7322/95/$09.50 91995ElsevierScienceB.V. All rights reserved. SSD! 0167-7322 (95) 00818-7
270
hydrophilic amino protons in the direction of the C3 axis and has bulky and hydrophobic cyclohexane-moiety in the direction perpendicular to the C3 axis. We can thus expect larger selectivity in the ion pairing with the anions than the en complex.
II. Experimental Section The complexes of A-[Co(chxn)3]CI3-SH20 and [Co(en)3]CI3.3H20 were prepared by the standard method. 5"6 The sodium sulfate and the potassium oxalate used were guaranteed reagents of Wako Pure Chemical Industries, Ltd. The 59C0 and IH NMR spectra were measured with a JEOL GX-270 FT NMR spectrometer operating at 64.1 and 270.1 MHz, respectively. The temperature of the sample solution was controlled at 27~q3.5 *C. The solvents used were I-t20 or D20(99%, for a comparison with the 1H NMR data) in the 59Co NMR measurement and D20(99%) in the IH one. The concentration of the cobalt(III) complex was adjusted to 5x10 "3 tool dm "3 in all cases. The longitudinal relaxation time(Tl) was determined by using the usual inversion-recovery method and the relaxation rate(R1) was obtained as the reciprocal T 1. For the I H relaxation measurements the solution was bubbled with argon gas for 10 min before each NMR measurement to remove oxygen gas from the solution. The chemical shift was measured from that for a 5x10 "3 mol dm "3 cobalt(Ill) complex solution containing no added salts. III. Results and Discussion The changes in the 59Co relaxation rates for [Co(chxn)3] 3+ with addition of sodium sulfate and potassium oxalate are shown in Fig. l(a). The result i n the presence of the sulfate ion is unusual since the relaxation rates of ions generally increase with increasing the salt concentration as seen for the oxalate ions. 2"4 On the contrary to the relaxation rates, the chemical shift changes in Fig. l(b) show that the sulfate and oxalate ions have similar effect. As the 59Co 400. ,',u,,,u,,,u,,,u,,,
10. ' ' ' u ' ' ' u ' ' ' u ' , , n , , ,
.9
=
35~.
9 9
300:
v~ 2 5 0 nr-"-
200
De
9
9
9
: .10 ~
9
o
D
:~ E -20 Q. ib " ~ -30 ~ 9 ~ o 9
oxalate sulfate o
15(> ~ : oo
o
-4(>
0.2
0.4
-60 0.6
0.8
1
' ." "
~
o
"
9
'''n'''n'''n'''=''' 0
c (Added Salt) / mol dm 3 (a)
oxalate sulfate
-50-
10(2" ' ' ' n ' ' ' n ' ' ' n ' ' ' n ' ' ' 0
o
oOo 9 0 0
0
o
9
0.2
0.4
(I_
,i
-
0.6
0.8
c (Added Salt) / mol dm "3 (b)
Fig. 1. Dependences of the 59Co longitudinal relaxation rates (a) and the chemical shifts (b) on the concentrations of the potassium oxalate or the sodium sulfate.
271
chemical shift generally changes to upfield with increasing the electrostatic interaction, 7 the present result indicates a gradual increase in the ionic interactions. Although the presence of 1:1 ion pairs is usually assumed for the analysis of the experimentally obtained NMR parameters, significant mounts of multiple ion-aggregates are present for the multivalent cations at higher anion ~ ' ' 1 ' ' ' 1 ' ' ' 1 ' ' ' 1 ' ' " concentrations. 1 Reasonable explanations have been 240 performed for the concentration dependences of the relaxation rates and the chemical shifts by assuming the ion-association model. 3"4'8 In order tOtheaSSignthe change in theofrelaxationthe ion "~ 2 0 0 j.~'dc rates to respective steps ~_ aSsociations, we calculated the concentration cC fraction for each species, [Co], [Co * A], and 160"1# oxalate [A * Co * A], by using the association o sulfate constants for the sulfate ion 9 and by considering the change in the ionic strength. It 120"~ was certain that the concentration of the added T ,,, I,.= I , , . I , , , I . . = = anions at the minimum of the relaxation rate 0.2 0.4 0.6 0.8 corresponds to the point where the population 0 of the major ion pairs switch from the 1:1 c (Added Salt) / mol dm 3 species to the 1:2 one. Fig. 2. 59C0 relaxation rates for [Co(en)3]3* The unusual relaxation rate for the [Co(chxn)313+SO42" ion pair is distinct when in the presence of the sulfate or oxalate ions. we compare the relaxation rates with the case of [Co(en)3]3+(Fig. 2) where both the sulfate and oxalate ions have very similar effects. It seems significant to estimate the effect of the ion pairing of [Co(chxn)3] 3+ on the parameters 1-4"1 "' ' I ' ' ' I ' ' ' I ' ' ' I ' ' of eq(efg) or x C in Eq. (1). The rotational correlation time of the complex ion can be .-. 1.2 estimated by the 1H NMR relaxations of the chxn ligands since the proton relaxation is mostly dominated by the magnetic dipolar oxalate interaction with intramolecular protons of the ligands. If the anisotropy of the rotational "" I o sulfate motion of the complex ion induced by the ion-9 O pairing is not appreciable, 10 the proton O relaxation rate(therotationalcorrelationtime of the dipolarinteractingaxis in the complex ion) 0.6 ~ o ~ is approximately proportionalto the correlation time for the main axis component of the efg at 0 . 4 1 , , , I , I , , , I , = I _ 59Co (the rotationalcorrelationtime of the C 3 0 0.2 0.4 0.6 0.8 1 axis of the complex ion). Thus the relative c (Added Salt) / tool drn "3 change in x C in Eq. (I) with the anion concentration can be estimated by the proton Fig. 3. The ratios of the 59Co relaxation rates relaxationtimes. W e then took the ratioof the to the tH ones for [Co(chxn)3]3§ depending 59Co relaxation rate to the IH one which is averaged for the 2 and 3 protons of the on the concentrations of the added salts. chxn ligand; the ratio can be regarded as
o
9
t
1
~~0.8
t
eee I
9
272 relative change in the effective efg caused by the ion pairing. The result is depicted in Fig. 3. The trend still shows similar profiles as shown in the 59Co relaxation rates and is roughly reflected by the change in the 59Co efg. It is clear that the 59Co efg is appreciably decreased by the 1:1 ion pairing with the sulfate ion and that it is increased by the further associations with this anion. The appreciable increase in the effective 59Co efg by the oxalate ion is also characteristic of the [Co(chxn)a] 3+ ions. For the consistency of the distinct features appeared in the [Co(chxn)3] 3+ system, a reasonable explanation is possible on the basis of Eq. (1) with considering the difference in the dynamic behavior of the ion pairs b e t w e e n [Co(chxn)3]3+SO42- and [Co(chxn)313+C2042". The sulfate ion should be located on the C3 axis of the [Co(chxn)3] 3+ ion in the ion pair and its life time is much longer than the rotational correlation time(xc). 3 In this case the C3 axis component of efg can be expressed as: eqMA = eqM+ e ~ (2) where eqMA is the effective efg for the ion pair, eqM the inherent efg for the metal complex ion, and eqA the efg caused by the anion. If the eqA is very effective on the eqM value in the chxn complex, the eqMA value is appreciably smaller than the eqM value. For the ion pair with the oxalate ion, on the other hand, the configuration of the complex ion and the anion is assumed to be quite flexible and the life time of the configuration (the results of the chemical shift measurements indicate that the oxalate ion interacts with the complex ion to a similar extent in average as the sulfate ion does) is much shorter than the rotational correlation time of the unpaired complex ion. In this case there is a slight correlation between the fluctuation of the field gradients of the unpaired complex ion and that of the anionpaired complex ion. Thus the 59Co relaxation rate can be expressed as: R I = Kt(eqM)2XM + (eqA)2XA] (3) where K corresponds to (2x2/49)(eQ/h) 2 in Eq. (1), xi is the correlation time of the fluctuation of the efg of each species. Equation (3) indicates an increase in the R 1 with increasing the field gradient caused by the counter anion irrespective of its sign. The increase in the effective efg by the formation of the triple ions for both the sulfate and the oxalate ions can also be explained from the same model. References 1 A preliminary report on this work has been published in Chem. Lett., 1994, 1433. 2 H.G. Hertz, Water, A Comprehensive Treatise, ed by F. Franks, Plenum Press, New York(1973), Vol. 3, Chap. 7. 3 Y. Masuda and H. Yamatera, J. Phys. Chem., 92, 2067(1988). 4 Y. Masuda and H. Yamatera, J. Phys. Chem., 87, 5339(1983). 5 S.E. Harnung, B. S. Serensen, I. Greaser, H. Maegaard, U. Pfenninger, and C. E. Schaffer, Inorg. Chem., 15, 219_3(1976). 6 J.B. Work, Inorg. Synth., Vol II, ed by W. C. Fernelius, McGraw-Hill, London, 1946, p. 221. 7 M. lida, Y. Miyagawa, and S. Kohri, Bull. Chem. Soc. Jpn., 66, 2398(1993). 8 A. Delville, P. Laszlo, and A. Stockis, J. Am. Chem. Soc. 103, 5991(1981). 9 M. Iida, M. Iwaki, Y. Matsuno, and H. Yokoyama, Bull. Chem. Soc. Jpn., 63, 993(]990). 10 As the conformation of the chxn ligand is rigid in the complex ion, the change in the IH-IH vectors of the ligand by the ion-pair formation would be negligible in the present discussion.
joumaJ of
I~IOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 273-276
Concentration Dependence of s 9Co Relaxation Rates of Tris(acetylacetonato)cobait(III) in Some Organic Solvents Mitsuhiro Kanakubo, Haruko Ikeuehl, and Gen P. Sat5 Department of Chemistry, Faculty of Science and Technology, Sophia University, 7-1 Kioicho, Chiyoda-ku, Tokyo 102 Abstract The 59Co longitudinal relaxation rates of tris(acetylacetonato)cobalt(III) in dichrolomethane and benzene were measured over the concentration range between 20 and 110 mol m-3 at several temperatures. The dependence of the relaxation rate on the temperature and the complex concentration is primarily attributable to the change in the viscosity of solution. The values of eQqlh were calculated from the concentration dependence of the 59Co relaxation rate by usmg the Debye-Stokes-Einstein equation and the Einstein relationship between molar volume and viscosity B coefficient. I. Introduction Tris(acetylacetonato)cobalt(III), [Co(acac)3], is suitable for the study of the complexsolvent interaction because it contains cobalt-59, one of the most sensitive nuclei for the NMR measurement. This diamagnetic complex is soluble in many common organic solvents, and its molecule is bulky and nearly spherical. We have studied the concentration dependence of the 59120 relaxation rate of [Co(acac)3] in acetonitrile, and concluded that the relaxation is attributable to the rotational reorientation of the complex and the macroscopic viscosity of the solution is the dominant factor in this dependence. 1 Busse 2a determined the 59Co relaxation rates of [Co(acac)3] in several solvents by using the dual spin probe technique. Busse and Abbott 2b conclude that the relaxation rate for cobalt in such an ostensibly symmetrical complex can be explained by small dynamic distortion from ideal symmetry that fluctuate owing to rotational reorientation in C6D6, C D s C O C ~ , and diglyme. If the relaxation is through the rotational reorientation, the relaxation rate, lIT1, is given by 1 3 ~ 9- 2 1 + 3 r2 "-TT= 1 0 / 2 ( 2 / - 1) ( 1 + ' ~ - ) (
eOq ~, )2rr
(1)
where I = 7/2, % is the rotational correlation time, and eQq/h is the quadrupole coupling constant The principal axis of efg coincides with the C3 axis of the complex. The asymmetry parameter, ~r will be practically zero because of the highly symmetrical arrangement of ligating oxygen atoms.
II. Experimental Commercial products of [Co(acac)3] were purified through column chromatography as previously described. 1 Reagent-grade CH2C12 obtained from Wako Pure Chemical Industries, Ltd., was purified through fractional distillation from 1:'205. The distillate was dried with Molecular Sieves 4A for more than 12 hours and then distilled again under a reduced pressure. 0167-7322/95/$09.50 91995 Elsevier Science B.V. All fights reserved. SSD10167-7322 (95) 00819-5
274
Spectroscopic-grade C6H6 purchased from Dojindo Laboratories was dried with Molecular Sieves 4A and distilled under a reduced pressure. Dichloromethane-d2 (99.95% up D) obtained from Aldrich Chemical Co., Inc., and benzene-d6 (99.6% up D) from Wako Pure Chemical Industries, Ltd., were used as received. The molalities of the samples are listed in Table 1. The amount concentrations of the complex solutions were calculated at each temperature from the molalities and the partial molar volumes of the components at infinite dilution. For the CH2C12 3 and C6H6 4 solutions, all the relevant partial molar volumes are available. For the C6D6 solutions, 5 the partial molar volume of the solute was assumed to be the same as that in C6H6. For the CD2CI 2 solutions, the partial molar volume of the solvent was replaced by that of CH2C12 and the partial molar volume of the solute by that in CH2CI2. The inversion recovery method was used for measuring the relaxation time. The instruments and the procedure were described previously. 1
Table 1. The molalides of [Co(acac)3] of the sample solutions (mmol kw 1) Sample Solvent number CH2CI2 CD2CI2 C6H6 C6D6 1 15.3 20.5 22.8 29.6 2 30.6 39.1 43.1 55.7 3 46.2 56.7 68.4 86.5 4 62.0 74.9 89.2 118.3 $ 77.6 112.7
280
!
260
"7
279.2 K
289.0 K
240
"7,
298.8 K
220
200
303.7 K
180
,
0
III. R e s u l t s a n d D i s c u s s i o n
The longitudinal relaxation rates are presented in Tables 2 and 3. At each temperature, 1/T1 was a linear function of the amount concentration c of the complex (e.g., Fig. 1),
i
40
, 80
120
c ~ n ~ l rrga
Fig. 1 I / T l as a function of c (solvent, C I-L2C12)
Table 2. The 59Co Lon~tudinal Relaxation Rates of [Co(acac)~] in CHTCI7 andCI~Cl~ a) (sd) Sample CH7C12 CD2C12 number 279.2 289.0 298.8 3(13.7 279.2 289.0 1 254(4) 231(1) 209(2) 196(1) 261(2) 235(2) 2 259(3) 231(1) 210(1) 1970) 265(2) 238(1) 3 262(3) 235(2) 213(1) 200(3) 270(1) 242(2) 4 264(2) 237(2) 214(1) 203(2) 272(1) 243(1) $ 268(2) 239(2) 217(2) 205(3) a) Each figure is the average of more than 4 runs. The 95% confidence limit is given m parentheses.
293.9
223(2) 226(1) 230(1) 230(1)
Table 3. The 59Co Longitudinal Relaxation Rates of [Co(acac)3 ] in C6H6 and C6D6 a) (s-l)
Sample
C6H6
r
number 284.1 289.0 298.8 308.6 284.1 298.8 1 366(2) 337(2) 289(2) 253(1) 388(2) 307(1) 2 373(2) 345(2) 294(1) 256(2) 395(2) 310(1) 3 381(2) 351(2) 298(2) 262(1) 406(3) 318(2) 4 386(2) 355(3) 303(2) 263(2) 420(2) 328(2) 5 392(2) 361(2) 307(1) 267(2) a) Each figure is the average of more than 4 rims. The 95% confidence limit is given in parentheses.
308.6
265(3) 270(2) 275(2) 282(3)
275 Table 4. IITI | and B' for [Co(-~--~-)3]--CH2C12and-CD2Cl2 systems.a) CH2C12 T/ K (1/TI| / sq B' / mol"t dm3 B b) / mol-t dm3 279.2 2510) 0.67(0.15) 0.77 289.0 227(2) 0.51(0.08) 0.72 293.9 298.8 2060) 0.49(0.07) 0.67 303.7 193(2) 0.65(0.13) 0.65 a) The 95% confidence limit is given in parentheses. b) Values of B are obtained from Ref.9 by linear interpolation. Table 5. llTl**and B' for [ C o ( a c a c ) 3 ] ~ md -C-O)6 systems a)
c~
T /K
B,I tool-ldm 3
(IITIco)/ s-I
284.1 360(2) 0.91(0.06) 289.0 332(2) 0.91(0.09) 298.8 2860) 0.78(0.06) 308.6 249(1) 0.75~0.07) a) The 95% confidence limit is given in parentheses.
( I +B' c)
258(1)
232(|) 22o0)
B' 1 moi '-Ia m 3
2980) 259~I)
0.89(0.05) 0.83(0.05)
375(2)
,
CD~I2 B' I mol"1dm3
0~0.0s~ 0.51(0.04)
0.48(0.05)
c~
(I/TI**)I S-l
which is expressed by 1 1 "-~1= T1|
(I/TI,**)/ sq
1.04(0.04)
(2)
with 1//'1'* being the relaxation rate at infinite dilution, and B' a coefficient dependent on the solvent and the temperature. These values are given in Tables 4 and 5. Since the rotation of the complex molecule will be isotropic as in the cases of several tris(didenate)cobalt(III) complexes, 6 and the molecule can be regarded as a sphere, we assume that the rotational correlation time is expressed by the Debye-Stokes-Einstein equationT, 8
~=FY-
Vrn
9
(3)
Here, Vr the volume of the spherical molecule, r I is the viscosity of the solution, k is the Boltzrnarm constant, and T is the thermodynamic temperature. This equation is based on the theory of rotational Brownian motion of rigid spheres in a viscous fluid. Viscosities of the [Co(acac)3]--CH2Cl2 9 and - - C H 3 C N 10 systems at a given temperature increase linearly with the complex concentration for c < ca. 100 tool m-3: r/= r/0 ( 1 + B c )
,
(4)
where r/0 is the viscosity of pure solvent. Substituting Eqs. (3) and (4) into Eq. (1), and identifying the resultant expression with
Eq. (2), we see B'= B
and
3~;2 2 / + 3 e Q q )2 Vr r/o 1,1| = 1--0-/2 ( 2 / - 1) ( . . k T "
(5)
1
(6)
In the case of the acetomtrile solutions, 1 B' agreed well with B. As seen in Table 4, the B' values for the dichloromethane solutions roughly agree with the corresponding B values,
276
but the agreement is not as good as in the case of acetonitrile. Such direct comparison is impossible for the benzene solutions, owing to the lack of the viscosity data. An experimental valueof 111"1"*gives the product(eQqlh )2Vr with Eq. (5). The value of Vr may be estimated by resorting to Einstein's relation as a first approximation Vr = B / 2.5, (7) into which we substitute B' for B, ignoring the difference between them. Thus we have
eQq_4{2.5 h
-
10 P ( 2 1 - 1) 3x z
2 I+
3
k T r/0
1 } B' TI |
(8) "
The values of the right-hand side of Eq. (8) are included in Table 6 as Y along with 11o. In acetonitrile the Y value was 4.8 MHz at 298.8 K (calculated from the data of ref. 1). Busse 2a reported a value of 4.2 MHz for eQq/h by measuring the relaxation times of 59Co and 13C of this complex in C6D6 at 298 IC We see that the Y values for C6H6 and C6D6 are close to Busse's value. The Y value in dichloromethane was appreciably higher than those in the other solvents (eQq/h = 4.5 MHz for CDsCOCD3 and diglyme2a). This discrepancy seems to indicate the failure of Eq. (3) and/or contribution of some relaxation process(es) other than rotational quadrupolar mechanism possibly owing to specific interactions such as observed in chloroform. 12 Table 6. Values of Yfor [Co(acac)3]-CH2CI2, ~ , and -C6D6 systems c.~c12 c~ TIK r/oa) I mPa s YIMHz rlob) I mPa s YIMHz 279.2 0.4902 4.8 284.1 289.0 298.8 303.7 308.6
0.4481 0.4059 0.3849
5.5 5.8 5.0
0.7456 0.6921 0.6006
4.0
0.5259
,
c~ rlob) I mPa s
YIMHz
4.0
0.~
3.7
4.4
0.~
4.1
4.6
0.5586
4.3
a) Values of r/0 are obatainedfrom Ref.9 by linear interpolation. b) Valuesof 1'/0are obatainedfrom Ref.11 Reference
1 M. Kanakubo, H. Ikeuchi, and G. P. Sat6, J. Magn. Reson., Ser. A, in Press. 2 (a) S. C. Busse, Ph.D. Thesis, Montana State University(1986). (b) S. C. Busse and E. H. Abbott, lnorg. Chem., 28, 488(1989). 3 S. Okuno, private commumcation. 4 S. Sato, private communication. 5 J.A. Dixon and W. Schiessler, J. Am. Chem. Soc., 7 6, 2197(1954). 6 Yuichi Masuda and Hideo Yamatera, J. Phys. Chem. 9 2, 2067(1988). 7 A. Einstein, Ann. Phys. Leipzig, 1 9, 371 (1906). 8 P. Debye, "Polar Molecule," traslated by T. Nakamura and K. Sato, Kodansha, Tokyo (1976), p.98. 9 R. Sato, private communication. 10 H. Ikeuchi, M. Takano, Y. Kimata, H. Matsuoka, and G. P. Sat6, Abstracts of the Chemical Society of Japan, 59th National Meeting. Kanagawa, Abstract 11=328(1990). 11 J.A. Dixon and W. Schiessler, J. Phys. Chem., 5 8, 430(1954). 12 J. F. Steinbach and J. H. Burns, J. Am. Chem. Soc., 8 0 , 1839(1958).
journal of
~IOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65166 (1995) 277-280
Mechanistic Study of Oxidation Reactions of Hydroquinone, Catechoi, and L-Ascorbic Acid by Dicyanobis(l,10-phenanthroline)iron(lll) in Dimethyl Sulfoxide Hideo D. Takagi, Nobuyoshi Kagayama, Mitsuru Matsumoto, Toshiyasu Tarumi, and Shigenobu Funahashi Laboratory of Analytical Chemistry, Faculty of Scienece, Nagoya University, Chikusa, Nagoya 464-01, Japan Abstract Reactions of hydroquinone, catechol, and L-ascorbic acid with dicyanobis(1,10-phenanthroline)iron(M) were studied in dimethyl sulfoxide (DMSO). Application of the Marcus theory to the reactions of catechol and hydroquinone provided the electron exchange rate constant for the Fe(III/II) couple in DMSO. The self-exchange rate constant for the ascorbic acid/radical couple was estimated for the first time in DMSO. The one electron-oxidation process of L-ascorbic acid in an aprotic solvents such as DMSO may be completely different from that in aqueous solutions.
I. Introduction Reactions of ascorbic acid have been intensively studied in relation to the behavior of this familiar compound in biological systems. 1,2 Most of the studies treated ascorbic acid as an simple outersphere reducing reagent, until Creutz published an article concerning the complexity of the reaction pathways of ascorbic acid and related radicals. 3 The authors recently demonstrated that the oxidation reactions of ascorbic acid in acidic aqueous solutions are not adiabatic from the volume analysis of the reactions. 4 As the non-adiabaticity of the ascorbate reactions implies the involvement of the proton dissociation and/or ring-closure processes at the rate-determining step, it is expected that the reactions of ascorbic acid in dipolar aprotic solvents such as dimethyl sulfoxide (DMSO) are adiabatic. 5 The electron exchange rate constant of the iron(IN) complex in DMSO was estimated from the cross reactions with hydroquinone and catechol, which was compared with the rate constant obtained electrochemically. The mechanism of the ascorbic acid oxidation reaction in DMSO is discussed based on the Marcus theory.
II. Experimental DMSO (Wako Pure Chemicals Inc.) was distilled twice from 4A molecular sieves(Wako) under reduced pressure. Dicyanobis(1,10-phenanthroline)iron(II) [Fe(CN)2(phen)2] was synthesized by mixing 0.03 mol of phen and 0.01 mol of ammonium iron(II) sulfate hexahydrate in 400 cm 3 of water, followed by the addition of KCN (0.15 mol). The resulting crude crystals were then dissolved in 30 cm 3 of concentrated sulfuric acid followed by the addition of ldm 3 of water. Dicyanobis(1,10-phenanthroline)iron(III) nitrate was obtained by the oxidation of corresponding iron(II) complex with concentrated nitric acid. The perchlorate salt was obtained by the addition of sodium perchlorate to the nitrate solution. Analytical grade hydroquinone, catechol, and L-ascorbic acid (Wako) were used without further purification. A Shimadzu UV-265FW spectrophotometer was used for the determination of stoichiometry of the reactions. Kinetic measurements were carried out with excess amount of reducing reagents by using a Unisoku RA-401 stopped-flow apparatus. Electrochemical measurements were carried out by a BAS-100B Electrochemical Analyser. 0167-7322/951509.50 91995 Elsevier Science B.V. All rights reserved. SSD! 0167-7322 (95) 00827-6
278
III. R e s u l t s The stoichiometry of the reactions was determined by the spectrophotometric titrations. It was observed that each mole of reductants reduced two equivalent moles of [FeIII(CN)2_ (phen)2] +. The overall reaction was, therefore, expressed by eq. 1 for each reaction: 2[FelII(CN)2(phen)2] + + H2A 2[FeII(CN)2(phen)2] + quinone or dehydroascorbic acid
(1)
where H2A represents hydroquinone, catechol, and ascorbic acid. In dipolar aprotic solvents such as DMSO, proton dissociation from H2A is negligible. The dependences of conditional first-order rate constants on the concentrations of reductants at various temperatures are shown in Fig. 1 A, B, and C for the reactions of hydroquinone, catechol and ascorbic acid, respectively. As the reactivity of the radicals formed as intermediates is expected to be rather high, consecutive one-electron transfer reactions are suggested as follows: 6 k [Felll(CN)2(phen)2] + + H2A
~
[Fell(CN)2(phen)2]
+
H2A .+
(2)
fast [Felll(CN)2(phen)2] + + H2A .+ [Fen(CN)2(phen)2] + quinone or dehydroascorbic acid
(3)
Therefore, kobs was given by eq. 4: /Cobs = 2k[H2A]
(4)
The rate constants, k, and the corresponding activation parameters are summarized in Table 1.
d
40 4 3.
I,,
A
d
"7, 30
B
$0,
40.
c
2.
0
b
i i ~ s o 103[hydroquinonel/moi dm-3
C
d
30.
,,~ o 20
.
~
o
.
.
. s ~o ~s 103[catechol]/moi dm "3
2o.
01}
1 i ~ ~ 103[ascorbic actd}/mol dm-3
F i g u r e 1. Dependence of the conditional first-order rate constants on the concentrations
of reductants (A: hydroquinone; B" catechol; C" L-ascorbic acid) at various temperaures (a: 293K; b: 298K; c: 303K; d: 308K). An attempt was made to obtain the electron self-exchange rate constant for the [Fe(CN)2(phen)2] +/0 couple by an electrochemical method. Real space Laplace analysis6, 7 was used for the chronoamperometric response of 7.05 x 10 -4 M (M = mol dm -3) solutions of the iron(Ill) complex with 0.1 M tetra(n-butyl)ammonium perchlorate as supporting electrolyte. A glassy carbon working electrode with 3 mm diameter was used for the measurements. The ButlerVolmer plot 7 gave an excellent straight line, and the electrochemical self-exchange rate constant was obained to be 1.2 x 10-2 cm s-1 from the In k value at the zero over-potential. From the
279 Table 1. Activation Parameters for the Oxidation Reactions of Various Reductants by Dicyanobis(1,10-phenanthroline)iron(III) in DMSO
Reductants Hydroquinone Catechol Ascorbic acid
k/dm3mol-ls-1 (298K) 1.41 x 102 7.1 3.7 x 103
AH*/kJ mo1-1 46 :t: 1 55:1:1 60 + 3
AS*/J mol-lK -1 -49 + 1 -45 :!: 1 23 + 11
classical Marcus theory, 8 the relationship between electrochemical, kel, and homogeneous, kex, exchange rate constants is derived as follows: kex ] kex0 = ( kel ] kelO)2
(5)
kex0 and kel 0 are the diffusion-controlled rate constants for homogeneous and heterogeneous reactions, respectively, which can be roughly estimated from the following equations. 9 kex0 = 8rdgRNA,
kel0 = 2D/~R
(6)
where D, R, and NA are the average diffusion coefficient of the exchange couple, the distance of the closest approach between the reactants, and the Avogadro number, respectively. Setting the R value to 1200 pm, kex0 and kel 0 are 3.2 x 109 M-ls -1 and 13 cm s -I, respectively, by using 2.1 x 10-6 cm 2 s-1 as D, which was also estimated from the electrochemical measurements. Finally, the homogeneous self-exchange rate constant, kex, was estimated to be 2.6 x 103 M-Is 1 from eq. 5. The redox potential of [Fe(CN)2(phen)zj ~-.~+/0 couple was obtained in DMSO as 0.466 V vs NHE, assuming the redox potential for the ferrocene/ferricinium couple being 0.400 V vs NHE. 10 IV. Discussion Catechol and hydroquinone are well known outersphere reducing reagents for the reactions with inert metal complexes. 11 The self-exchange rate constant of 2 x 106 M - l s -1 for H2Q+./H2Q (H2Q represents catechol and hydroquinone) couple was obtained from the direct measurements using pulse radiolysis. 10 The electron self-exchange rate constant for the [Fe(CN)2(phen)2]+/~ couple was calculated from the cross reactions with catechol and hydroquinone from the Marcus theory. As the effect of electrostriction is minimal for +1/0 redox couples, 10 the redox potentials reported for the aqueous reactions at pH 0 (1.04 V and 1.12 V for hydroquinone and catechol, respectively) were used without corrections for the discussions below. 12 Ignoring the work terms, the self-exchange rate constant for the [Fe(CN)2(phen)2] +/0 couple was calculated to be (2.8 + 2.5) x 107 M-ls -1, which may be compared to the self-exchange rate constants for the tris(1,10-phenanthroline)iron(III/II) and hexacyanoferrate(III/II) couples. 13,14 Electrochemically obtained heterogeneous rate constant, however, was much smaller than that obtained for the homogeneous reactions. The difference may be attributed to the fact that the electron exchange rate of the [Fe(CN)2(phen)2]+/0 couple is fast enough to exceed the application limit of eq. 5. It was suggested that the electrochemical rate constants reach a diffusion-controlled limit significantly below the expected value, i.e. at nearly 1 cm s-1.15 If this applies, the corresponding homogeneous rate constant may be 1 x 106 M-ls-1, which is close to that estimated from the cross relation. The value of 2.8 x 107 M-ls -1 will be used here as the upper limit of the exchange rate constant for the [Fe(CN)2(phen)2] +/0 couple.
28o The complexity in the oxidation reactions of ascorbic acid originates from the uncertainty whether the redox potential reported to date corresponds to the formation of the open-chain radical or not. 3 Considering Scheme 1, the redox potential of the ascorbic acid/radical couple may be expressed by eq. 7 with the use of the equilibrium quotient, K, for the formation of the bicyclic radical from the open-chain ascorbic acid radical: Scheme 1 .~~
O 140
E 0 = E' - (RTIF) In K
o
o
+ H+
14o
(7)
where E' and E0denote the redox potentials between ascorbic acid/open-chain radical and between ascorbic acid/bicyclic radical, respectively. If the oxidation reaction of ascorbic acid involves the ring closure l~rocess within the radical, the redox potential between ascorbic acid/open-chain radical, E , may be higher than the reported value (E 0 = 0.99V in aqueous solution), taking into account the K value larger than unity. The electron exchange rate constant estimated by the Marcus cross relation, therefore, may have been underestimated as pointed out by Sisley and Jordan. 5 In dipolar aprotic solvents such as" DMSO, the dissociation of a proton following the ring closure within the ascorbic acid radical may not be a preferred process. Therefore, the redox potential so far reported (0.99 V) may be taken as a lower limit. Indeed, the cyclic voltammogram of ascorbic acid in DMSO showed an anodic p e a k at 0.64 V vs ferrocene/ferricinium couple, which corresponds to 1.04 V vs N.H.E. 1 0 Using 0.99 V as the redox potential, a value of 1 x 109 M-1 s-1 was calculated as a lower limit of the self-exchange rate constant for the ascorbic acid/open-chain radical couple from the Marcus cross relation. The estimated exchange rate constant here is much larger than the one reported to date for the aqueous reactions. In conclusion, the oxidation reaction of ascorbic acid in DMSO proceeds through a simple adiabatic process, while non-adiabaticity including the ring closure process is suggested in aqueous solutions. 4 References 1 K. Kustin and D. L. Toppen, Inorg. Chem., 12, 1404(1973). 2 E. Pelizzetti, E. Mentasti and E. Pramauro, Inorg. Chem., 15, 2898(1976). 3 C. Creutz, Inorg. Chem., 20, 4452(1981). 4 N. Kagayama, M. Sekiguchi, Y. Inada, H. D. Takagi and S. Funahashi, Inorg. Chem., 33, 1881(1994). 5 M.J. Sisley and R. B. Jordan, Inorg. Chem., 31, 2137(1992). 6 E.D. Moorhead, V. E. Dauyotis and M. M. Stephens, Anal. Chim. Acta, 162, 161(1984). 7 A.J. Bard and L. R. Faulkner, E l e c t r o c h e m i c a l M e t h o d s (John Wiley & Sons, Inc.,1980) 8 R.A. Marcus, Electrochim. Acta, 13, 995(1968). 9 R.A. Marcus, J. Chem. Phys., 43, 679(1965). l0 R. R. Gagne, C. K. Koval and T. J. Smith, J. Amer. Chem. Soc., 101, 4571(1979). 11 J. W. Herbert and D. H. Macartney, J. Chem. Soc., Dalton Trans., 1986, 1931. 12 S. Steenken and P. Neta, J. Phys. Chem., 83, 1134(1979). 13 H. Doine and T. W. Swaddle, Inorg. Chem., 31, 4669(1992). 14 H. Doine and T. W. Swaddle, Can. J. Chem., 66, 2763(1988). 15 T. Saji, T. Yamada and S. Aoyagi, J. Electroanal. Chem., 61, 147(1975).
joumaJof
~OLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65166 (I 995) 281-284
Molecular Orientation in Fluids near Solid-Liquid Interface as Studied by the Density Functional Method Yasuhiro Teramoto
and Koichiro Nakanishi
Division of Molecular Engineering and Department of Industrial Chemistry Kyoto University, Sakyo-ku, Kyoto 606-01, Japan Abstract T h e density functional method as applied by Tarazona to deal with classical fluids has been used to calculate the orientation of triatomic molecular fluids near the solid-liquid interface. T h e results give valuable suggestions about the effect of molecular shape on the orientation of
real molecules such as liquid crystals near the solid-fluid interface. I. I n t r o d u c t i o n
The density functional(DF) method is one of the most promising tools for the calculation of structure and orientation in heterogeneous fluids near phase boundaries.
Among many
proposals for the application of DF method, that by Tarazona 1,2 has been frequently used in the studies of interfaces and phase transitions in hard sphere systems. The purpose of the present study is to apply the Tarazona method to the fluid systems of polyatomic molecules, thereby eliminating some disadvantages inherent to his method. We formulate a weighting function by the intermolecular potential alone 3'4. Another improvement is with the use of the equation of state. A volume-corrected hard-sphere equation of state is used as the equation of state for polyatomic molecular fluids. In the present study, the DF method is used to clarify the molecular orientation in fluids near two-dimensionM solid-liquid interface. Our molecular model is the triatomic molecule shown in Figure 1. We expect that the results of the present study will give valuable information about the effect of molecular shape on the orientation of real molecules such as liquid crystals neaz the solid-fluid interface. II. M e t h o d
of Calculation
The present method is based on that proposed by Tarazona.
Since it is well described
elsewhere 5'6 for two-dimensional system, we will not repeat it here. An ultra-simplified account will be given here for three-dimensional system. We start from the formulation of the grant potential f / o f the system, 0167-7322./95/$09.50 9 1995 Elsevier Science B.V. All righet reserved. SSD!0167-7322 (95) 00916-7
"S
>
Figure 1: The model of a triatomic molecule and the orientation vector: Bond angle(left), atomic diameters(center) and orientation vector(right).
/,
f~[p]
F[p(z, 0)] - J dzdOp(z, 0)[# - r
0)]
(1)
where z is the distance between the center of gravity of the molecule of interest and the interface, 0 is the angle between the molecular orientation vector and the perpendicular vector to the interface as shown in Figure 1, # the chemical potential, and r the potential function between molecule and interface. F is the Helmholtz free energy. Combination of Eq.(1) with 6~
,--: pu
=
(2)
o
leads to
6F~
mp(~, o) = -/3~:(z,O) +/3[# 6s
6/,(z,o) =
:[~(z)]
~/~(z') 6p(~, 0)
- r
O)]- ln[A2A,]
+ J ez'dO'p(z', o'):, ~ ( z ' )
= 2~:l do' W(Iz - z'l, o', o) .1
(3)
(4)
(5)
and the density p(z, O) can be obtained by the self-consistent solution of Eq.(3). III. M o n t e Carlo S i m u l a t i o n Comparison is attempted between the results of density distribution calculated by the DF method and those from Monte Carlo(MC) simulation. The present MC simulations have been
283
performed with usual Metropolis scheme. The basic cell is of two dimension where the cell length is Lz and Ly, the solid surface is in the z direction and the periodic boundaries are considered in the y direction. The basic cell contains N molecules. N is 100 in the present calculation. L, is fixed to be 15a and Lv is adjusted so that Ly = N / L , p where p is the number density. The solid surface is assumed to be a hard sphere. The MC sampling has been performed 3 x l0 s steps for equilibration and further 2x 106 steps for averaging. IV. R e s u l t s a n d D i s c u s s i o n
Results have been obtained for the density distribution, intermolecular excluded volume effect at the interface, the relation between molecular orientation and bulk density, and the relation between molecular orientation and diameter. Only the first one will be discussed here in detail. As a typical example, calculation has been made for density p(z,8) in the condition that bulk density p=0.5, d/a=0.2 and the bond angle a=90 ~ In Figure 2 are shown the results ot the present DF method(left) and Monte Carlo simulation(right).
p(x,o)
0.3
0.25
0.5
o.~
.~
o~
0.30.2
. . . .
O. 05
o.~
o z z
o
o
0
0
Figure 2: Comparison of the DF results with the Monte Carlo simulation: DF(left) ant MC(right). It is seen from the figure that the DF results give a smooth density distribution. Althoug~ the first peak of the density calculated by the DF method is higher than that obtained b] the MC method, the density p vs. tilt angle 0 relation shows qualitative agreement for earl approach as seen in Figure 3(upper). As an application of this kind of calculation, we refer to the case of liquid crystals. Okan( et al. T have discussed the role of excluded volume effect on the molecular orientation in th~ interface involving liquid crystals. They used spherocylinder systems, but considered only limited case of orientation.
,.,
0.4
-9 -
/
'....
. . . . "_ _ s - % ~[: : ,-,,
:
0.2
Ii
/. ]
(b~ ~
p
i
1
',
ts 0.2
0.5
0.5
| .......
0
0.6
=
is ~ . |
100
0
0
30
60
90
<> (~ (,)
0.)
(')
Figure 3: The molecular density p(solid line) and the length h(dotted line) between the center of the mass of molecule and the surface as a function of the tilt angle from the DF method: Molecule with a = 90~ and molecule with cr = 180~ Molecular density from MC calculation(broken line) is also given for comparison. V. Conclusion
We have successfully used the classical density functional method to investigate the behavior of triatomic molecules near interface. Although perfect agreement with a Monte Carlo simulation cannot be achieved, we have obtained interesting and important information on the surface-molecule excluded volume effect. Acknowledgments
We thank Prof. X. C. Zeng, University of Nebraska, for critical reading of the manuscript and helpful advice. References
[1] Tarazona, P., 1984, Mol. Phys., 52, 81. [2] Tarazona, P., 1985, Phys. Rev. A31, 2672. [3] Poniewierski, A. and Holyst, R., 1990, Phys. Rev. A41, 6871. [4] Sokolowski, S., 1991, J. Chem. Phys., 95, 7513. [5] Curtin, W.A. and Ashcroft, N.W., 1985, Phys. Rev. A32, 2909. [6] Evans, R., 1979, Adv. Phys., 28, 143. [7] Okano, K., 1983, Jpn. J. Appl. Phys., 22, L343.
journal of
MOLECULAR
LIQUIDS ELSEVIER
Journal of MolecularLiquids, 65/66 (1995) 285-288
The Stability and Dynamics of Clathrate Hideki
Hydrates
Tanaka
Division of Polymer Chemistry, Kyoto University, Sakyo, Kyoto
Graduate School 606-01 J a p a n
of
Engineering
Abstract The t h e r m o d y n a m i c
stability of a clathrate hydrate encaging nonspherical ethane molecule has been investigated by examining the free energy of cage occupancy. A generalized van der Waals and Platteeuw theory is extended in order to treat rotational motions of guest molecules in the clathrate hydrate cages.
I.
Introduction
Gas hydrate consists of guest and the host water molecules which form a hydrogen bonded network. The clathrate hydrate is stable only when guest molecules exist in the cages of the hydrate. The thermodynamic stability of the clathrate hydrates has been explained by van der Waals and Platteeuw (vdWP). 1 With some empirical parameters, this theory is applicable over a wide temperature r a n g e . 2,3 However, its application sometimes predicts an incorrect phase behavior for large guest species. We generalized the original vdWP theory for the hydrate encaging a comparable size of a molecule with the larger c a g e ? ,5 It was revealed that this generalization is very important to account for the discrepancy between the prediction from the original vdWP theory and the experimental result. In the present study, we examine the magnitude of the anharmonic contribution from the host water molecules and also the anharmonic free energy arising from the rotational degrees of freedom of guest molecule by Monte Carlo (MC) simulation with the Gaussian statistics. T h e o r y a n d method The water-water intermolecular interaction is described by the TIP4P potential. 6 The ethane molecule consists of two interaction sites, each of which interacts with each other via Lennard-Jones (LJ) potential. 7 The reference of ethane molecule is spherical and is of LJ type interaction with size and energy parameters of 4.18 A and 1.72 kJ/mol, s The LJ parameters for methyl group of ethane are 3.78 A a n d 0.866 kJ/mol. For the water-guest interaction, the Lorentz-Berthelot rule is assumed. The interaction potentials for all pairs of molecules are truncated smoothly at 8.655 A. 9 A generalization of the vdWP theory was made to deal with the coupling of the host m o l e c u l a r motions with those of guests. The original theory and its generalization are detailed elsewhere. 4's We obtain the potential energy minimum structure of clathrate hydrates encaging spherical guest molecules. The potential energy is expanded to a polynomial of the displacement from the equilibrium position. The expansion is truncated at the quadratic order for the small displacement. This truncation is not appropriate for the hydrate of a nonspherical guest molecule because it rotates with a low energy barrier and the anharmonic part of the potential is not negligible. II.
0167-7322/95/509.50 91995 ElsevierScience B.V. All fights reserved. SSDi 0167-7322 (95) 00906-X
286 The anharmonic contribution to the f r e e energy is evaluated by a t h e r m o d y n a m i c integration method with a reference system of harmonic oscillators. This free energy difference between the real and the reference system A - Ao is given by A - Ao = - k T In < exp[-13(O - Oo)] >o,
(1)
where 9and Oo are the real and reference system potential, respectively and the average < >o is taken over the reference harmonic oscillators. Since the potential of the harmonic oscillator system having the potential minimum value Uo is written by
Oo = Uo + ~ o~ q ?/2,
(2)
the probability for the system (ice or empty hydrate) to have a set of normal mode coordinates q =(q~,q2 . . . . q e N . 3 ) is given by P ( q ) = l-li (13to12/2n) l tz exp(-13 ~i2 q ?/2).
(3)
This method provides a much more efficient sampling way for a harmonic system than the usual Metropolis scheme. ~~ This is because the distribution of the amplitudes is the Gaussian and each mode is independent of other modes; the generated configurations have no correlations. In the case of occupation of nonspherical ethane molecules, the reference system is chosen to be the hydrate of spherical guest molecules. The orientations of the guest ethane molecules in the real system are assigned randomly.
discussion I I I . Results a n d The guest molecular motions are examined by performing a molecular dynamics (MD) simulation. The linear and angular velocity correlation functions of guest ethane are depicted in Figure 1 together with their power spectra. The spectra of the translational motions have distinct peaks which shift to lower frequency region with decreasing the temperature. The rotational spectra have peaks at about 0 cm -l. There are small peaks in non-zero frequency region indicating that the rotation of ethane is somewhat hindered. The rotational motions have been studied for mostly polar and some of apolar molecules. 3 Those suggest that guests molecules rotate rather freely, which agrees with our observation. The anharmonic free energy is evaluated for empty hydrate and cubic ice (ice Ic). The calculated free energy due to the anharmonic potential energy surface is given in Table 1. The anharmonic contribution to the free energy of empty hydrate is the same as ice, which is as large as -0.61 kJ per mole of water. The free energy differences between the real and the reference clathrate hydrates are also given in Table 1. The anharmonic free energy change from the spherical guests with harmonic approximation to the nonspherical guests is -0.25 kJ /mol. Thus, we can calculate the total free energy change upon accommodation of nonspherical guest molecules. The dissociation pressure Pd at 273.15 K is obtained from the intersection between the chemical potential curve and the horizontal line corresponding to the difference in chemical potential between ice and empty hydrate ~i-gt~ which is calculated to be - 0.72 kJ/mol, using the previous calculation and the anharmonic free energy obtained in the present study. The chemical potential differences between occupied and empty hydrates are plotted in Figure 2 for the nonspherical (harmonic + anharmonic terms) and the spherical (harmonic term) guest molecules together with that calculated from the original vdWP theory.
287
1
0.5 .5
0 -0.5 0
250
500
0
time/2.5fs
50 100 150 frequency/cm -1
Figure 1, The translational (thin) and rotational (heavy) velocity autocorrelation functions and power spectra at 273.15 K(solid line), 223.15 K(dashed line) and 173.15 K(dash-dot line). Left (time correlations), right (power spectra).
-0.2
I~~I
~-0.4 9
I
I
I
"
~
~-o.6 b \?, -1
I
F_
0
.
.
.
.
'
.
~ '
2
-.............. '
-''--
4
p/O. 1 MPa
1
6
Figure 2. Dissociation pressure of ethane hydrate at 273.15 K. Solid, dashed, and dash-dot lines show the dissociation pressures for the nonspherical guest evaluated by anharmonic and harmonic free energy, for spherical guest molecule evaluated by only harmonic free energy, and for spherical guest according to the original vdWP theory, respectively. The horizontal lines show the chemical potential difference between ice and empty hydrate, dashed; harmonic+anharmonic, dotted; harmonic.
288
Table 1. Free energy due to the anharmonic contributions. Free energy is in kJ/mol. The anharmonic free energy of the guest molecule is denoted by A (kJ per mole of guest). The reference systems for ice, empty hydrate and the hydrate encaging spherical guest molecules are corresponding harmonic oscillators. The reference system for the hydrate encaging nonspherical guest molecules is the hydrate occupied by spherical guest molecules. free energy ice empty occupied A
-0.50 -0.61 -0.25 +2.53
mode energy 6.80 6.80 6.35
The experimental dissociation pressure P a is 0.53 MPa z which is very close to our present result, p d=0.50 MPa, but is different,' from the previous harmonic oscillator a p p r o x i m a t i o n , pd=0.24 MPa, and the original vdWP theory pd=0.16 MPa. The occupation number of the cage per unit lattice is ranging from5.5 to 5.6 in those methods. There is some variation in experimental results, 5.6 to 6.0. tt Although the present method is approximate one and the chemical potential difference between ice and empty hydrate does not agree perfectly with the experimental observation, this provides a way to evaluate the free energy of the cage occupation taking account of nonspherical nature of guest molecules. IV. Concluding Remarks In the present study, a generalized vdWP theory is further extended in order to treat rotational motions in the clathrate hydrate cages. The vibrational free energy of both guest and host molecules is divided into harmonic and anharmonic contributions. The anharmonic free energy associated with the rotational degrees of freedom of the guest molecules is evaluated as a perturbation from the spherical guest. The anharmonic term is found to be essential in determining the free energy of the hindered rotation for the guests. It is revealed that thermodynamic properties according to the present method enable us to evaluate the phase behavior of the clathrate hydrate more accurately. References
1 2
J . H . van der Waals and J. C. Platteeuw, Adv. Chem. Phys., 2, 1 (1959). E. D. Sloan, Clathrate Hydrates of Natural Gases, (Marcel Dekker, New York, 1990). 3 D. W. Davidson, W a t e r - A Comprehensive Treatise, Vol.2, edited by F. Franks, (Plenum, New York, 1973). 4 H. Tanaka and K. Kiyohara, J. Chem. Phys. 98, 4086 (1993). 5 H. Tanaka and K. Kiyohara, J. Chem. Phys. 98, 8110 (1993). 6 W . L . Jorgensen, J. Chandrasekhar, J. D. Madura, R. W. lmpey, and M. L. Klein, J. Chem. Phys. 79, 926 (1983). 7 W . L . Jorgensen, J. D. Madura, C. S. Swenson, J. Am. Chem. Soc. 106, 6638 (1984). 8 J . O . Hirshfelder, C. F. Curtiss, and R, B. Bird, Molecular Theory of Gases and Liquids, (Wiley, New York, 1954). 9 I. Ohmine, H. Tanaka, and P. G. Wolynes, J. Chem. Phys. 89, 5852 (1988) 10 A. Pohorille, L. R. Pratt, R. A. LaViolette, M. A. Wilson, and R. D. MacElroy, J. Chem. Phys. 87, 6070 (1987). 11 Y . P . Handa, J. Chem. Thermodyn, 18, 915 (1986).
journal of MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 289-292
Deviation of the TST rate constant through the nonlinear couplings Masataka Nagaoka*, Naoto Yoshida a),b), and Tokio Yamabe b) Institute for Fundamental Chemistry, 34-4, Takano-Nishihiraki-cho, Sakyo-ku, Kyoto 606, Japan a)Daicel Chemical Industries, LTD., 1239, Shinzaike, Aboshi-ku, Himeji, Hyogo 671-12, Japan b)Department of Molecular Engineering, Kyoto University, Kyoto 606, Japan Abstract On the assumption of the external bath equilibrium, a set of simultaneous generalized Langevin equations (GLE) for a microscopic Hamiltonian is derived, whose potential function includes cubic (i.e., nonlinear) coupling terms, which are linear in the internal coordinates but quadratic in the external bath coordinates. Furthermore, on the GLE treatment, a closed expression of time-dependent friction coefficient and the rate constant in the Grote~Hynes theory (GHT) axe derived microscopically, reflecting the reactant and solvent structures. Comparing the rate constant of GHT with that of the multi-dimensional transition-state theory (TST) for the whole solution system, we conclude that these rate expressions are different each other and the difference is due to dynamic effects via the nonlinear couplings among the reaction coordinate and the internal and external normal coordinates.
I. Introduction
Recently, the understanding of chemical reaction dynamics has been enriched by extensive experimental and theoretical studies. 1-4 Grote and Hynes (GH)5, 6 generalized Kramers' treatment 7 and formulated the dynamic effect on the rate processes in solution, regarding solvent molecules as a surrounding medium in thermal equilibrium. The generalized Langevin equations (GLE) with respect to the mass-weighted coordinates of the reactive and nonreactive modes are the starting point for GH's treatment. GH solved a set of the simultaneous GLEs in terms of the reactive mode and derived a closed GLE of the reactive mode. Owing to the long time behavior of the solution of the closed GLE, so-called, GroteHynes equation (GHE) was derived. The major result was a simple expression for the escape rate:5,6
korl= K:. kT~ST= ~1... kT~T S~ f21
(I)
where ~ s, is the absolute value of the imaginary normal-mode frequency at a transition state (TS) on the free energy surface and Z.I'is the reactive frequency defined by the GHE and k~rsr is the rate constant on the free energy surface with an abbreviation FE. If the time-dependent friction coefficient ~u(0 > 0 for all t, then Z.T< f~s,, so that the motion in the vicinity of the barrier should slow down. Therefore, GH concluded that the rate constant becomes smaller than k~r. 5,6 According to the molecular dynamics simulations, 8,9 the frictional effect is quite large and solvent molecular motions play an important role in the reaction dynamics. However, since GH started with the GLEs that include phenomenological friction coefficients, microscopic connection between the coefficients and molecular interaction remained unclear. From the chemical point of view, such a theory is more desirable that the influence due to the differences in microscopic interaction of real solute and solvent molecules should be 0167-7322/95/$09.50 91995 Elsevier Science B.V. All rights reserved. SSD! 0167-7322 (95) 00880-2
290
describable. In order to elucidate the influence, van der Zwan and Hynes treated the microscopic model of chemical reactions, where a one-dimensional chemical reaction system couples bilinearly with a one-dimensional bath mode, and Pollak also treated the onedimensional chemical reaction system whose coordinate is coupled bilinearly with each of a large number of harmonic oscillators that constitute the heat bath. They obtained a very interesting and profound result: 10-12 the GH expression for the rate of escape over the barrier is just the continuum limit of the usual gas phase harmonic TST expression. Moreover, we have treated the multi-dimensional chemical reaction system in order to consider the difference of the internal bath mode frequencies at a stable state (SS) and "IS and to evaluate the indirect friction coefficient, which Pollak did not take into account. It was confirmed that the GH rate constant coincides with the multi-dimensional TST for the whole solution system.13,14 It means that the multi-dimensional TST for the whole solution system can contain the dynamic effect treated by GH, assuming the bilinear couplings between the internal and external bath modes. Therefore, in order to deal with more realistic interaction than the bilinear couplings and to treat the dynamic effect that cannot be described by the multi-dimensional TST, we should treat not only bilinear couplings but also nonlinear couplings. In this paper, we will treat the nonlinear couplings, which are linear in the system coordinate but quadratic in the external heat bath coordinates, and show that the GHT is not equivalent to the multi-dimensional TST for the whole solution system if the potential function contains the nonlinear couplings. In See. II, we introduce the microscopic Hamiltonian (IIA), evaluate the GH rate expression (liB), and, in See. IIC, we compare it with the multidimensional TST rate for the whole solution system. Finally, the main points are summarized in See. III. II. Theory A. Microscopic Hamiitonian for reacting system in solution Our starting point is the microscopic Hamiltonian for the chemical reacting system in solution. After the transformation of the coordinates, it can be expressed within the third order approximation in the neighborhood of any equilibrium geometry as follows: 13.14 H= V0+
89
+ 2]..pBTpB + l q S T AS2qs + 89 +
AB2qB
(2)
1 sr~3) .,B)
9 q SBB~'I where V 0 is the potential energy at an equilibrium geometry of the whole solution system. Furthermore, qS and qS express vectorially two sets of normal coordinates of the internal and external bath modes each with the degrees of freedom NS and N B, respectively, and pS and p8 are the corresponding conjugate momenta. As and AB are the diagonal matrices whose elements consist of normal-mode frequencies. The reaction coordiante for the whole solution system corresponds to the component q~. Then, the last term of r.h.s, of Eq.(2) denotes the nonlinear interaction term between the internal and external modes, and can be rewritten as follows: (3) (qa) =~-q s l q sTv SBB
9
=
~ ~ t:SBB,,S,,B,,B r i k l ~li ~lk ~ll
(3)
where Es~B are the cubic coupling coefficients. B. GHE and GH rate expression Cortes et al. obtained the GLE for the one-dimensional system whose coordinate is coupled not only bilinearly but also nonlinearly with each of a number of harmonic oscillators that constitute the external heat bath; i.e., the couplings are linear in the system coordinate but arbitrarily nonlinear in the heat bath coordinates.15,16 We apply the result to the chemical reaction system, which is described by the Hamilonian as Eq.(2). As a result, we obtain
291 perturbatively a set of the simultaneous GLEs on the free energy surface as follows: 13,14 Pl
~js(t) + (As2- Z(0))q s(t) + I^" dxZ(x)ils(t-x) = R(t)
(4)
dU
where R(t) is the random force vector along the internal modes by the external bath and Z(t) denotes the friction coefficient matrix, whose elements are the friction coefficient ~j(0 : NB ~
t:SBB~SBB
"ikl "jkl cos ABt cos 9 Atat
;ij(t) = 1 ~ -~
k l#k
(5)
B2. B2
Ak" A 1
where ~ denotes 1/kBT. By diagonalizing As2 - Z(0) by a unitary matirx U, we obatin the diagonal matrix ta s2 whose diagonal elements are the normal mode frequencies on the free energy surface and a new set of internal normal modes. After averaging the transformed GLEs over the external bath equilibrium distribution, firstly, the GLEs are solved to find the mean internal bath normal coordinate ~u (i = 2 ..... N S) in terms of the mean reactive normal coordinate ~u. 5,6,13,14 Here - mark indicates the average by the external bath equilibrium distribution. By substituting the solutions into the averaged GLE with respect to the reaction coordinate ~v, one obtains finally the following averaged equation of motion: -s#2 _--StJ0 ~'~lu0(t) " t2t ql (t) +
~lU(t-X)~llU0(1;)dx =0
a s..= ~- f~s2
(6)
where ~s,, is the absolute value of the normal-mode frequency of ~tr at the TS on the free energy surface and the superscript 0 indicates that the solution is obtained under the initial .--su condition; ~s~(0) = 0 and q i (0) = 0 (i = 2 ..... N S ), and ~(t) is the time-dependent effective friction coefficient.5,6,13,14 According to Eq.(6), the rate constant expression in solution is obtained as Eq.(1) and the reactive frequency ki' is defined by the GHE as follows: / s~ 2 ~.t =
~:
^U
(7)
:m
where ~(~.t) is the Laplace transform of ~(t).
C. Comparison with TST rate constant for the whole solution system The TST rate constant for the whole solution system is defined as follows: 1,2,17 f ~ ' " "f.**"dpSdqSdpBdqBexp{-I3H(TS) }~5(qS)cls0(cls) kTST = (8(qS)~ISO(~IS)~ =
(8) f-**".... I-**"dpSdqSdpBdqBexp{-lBH(SS)}
where iS(q~)and 0(tis) are the delta and step functions, respectively. Perturbative integration of Eq.(8) with respect to all coordinates with the microscopic Hamiltonian at SS and TS leads to the next formula: kTST _ ASl(TS) i 1 + 1 ~_~ E "+ikl ....._]~. | e-13AV0 --
271:
i A/s(TS) /
4~ i
. B2.AlB2 AS(Ts)2 k l* k Ak
Ais (SS) 2
292 As(TS) 2r:
"ikt (9) I~SBB2 B2 B2 S 2 ' Ak At AI(TS) where AV0 is the difference between the potential energy at the SS and TS and AS(ssfrs) are the internal mode frequencies at the s s f r s on the potential energy surface. The ratio of the microscopic GH rate expression to that of the TST for the whole solution system becomes Na Na t::SBB2 - S~2 B2 B2 kTST
413
As(SS) 1 N~ k Nl~k As(TS) 413
. B2, B2
flA 1 + Where A s..is the absolute value of A~s(TS). Eq.(10) means that G H rate expression deviates from the T S T one by the second term of r.h.s,and k ~ is always smaller than not only k ~ but also kvs-r. The result is contrasted clearly with both Pollak's and the previous ours, where the bilinear couplings were considered. Furthermore, the second term of r.h.s, in Eq.(10), representing the deviation from the T S T one, mainly originates from the Laplace transform of the frictioncoefficient. Therefore, the deviation is due to the dynamic effect via the nonlinear couplings. As is distinct from the treatments of bilincar couplings, the friction cocfficicnt depends on temperature and hence the deviation becomes larger with temperature increasing. Ill. S u m m a r y In this paper, startingwith a microscopic Hamiltonian, whose potential has a cubic form and the internal degrcc of freedom is multi-dimensional, we have obtained the microscopic friction coefficient and the averaged GLE of the reactive mode. Noting the long time behavior of the reactive mode motion, we have derived the GHE. Furthermore, comparing with the TST rate constant for the whole solution system, we have proved that the GH rate expression deviates from that of TST for the whole solution system and ktm is always smaller than both ka~-r and kn-r, because of the dynamic effect via the nonlinear couplings. With temperature increasing, kGn deviates from krsr more largely by the temperature dependence of the friction coefficient. References
1 J.T.Hynes, in Theory of Chemical Reaction Dynamics, edited by M.Baer (CRC, Boca Raton, 1985), Vol. IV. 2 P.H~inggi, P.Talkner and M.Borkovec, Rev.Mod.Phys., 62, 251 (1990). 3 A.Warshel, Computer Modeling of Chemical Reactions in Enzymes and Solutions, (John Wiley & Sons, New York, 1991). 4 W.L.Jorgensen, Adv.Chem.Phys., 70, 469 (1988). 5 R.F.Grote and J.T.Hynes, J.Chem.Phys., 73, 2715 (1980). 6 R.F.Grote and J.T.Hynes, J.Chem.Phys., 74, 4465 (1981). 7 H.A.Kramers, Physica, 7, 284(1940). 8 B.J.Gertner, K.R.Wilson, and J.T.Hynes, J.Chem.Phys., 90, 3537 (1989). 9 S.Hayashi, K.Ando, and S.Kato, J.Phys.Chem., 99, 955 (1995). 10 G.van der Zwan and J.T.Hynes, J.Chem.Chys., 78, 4174 (1983). 11 E.Pollak, J.Chem.Phys., 85, 865 (1986). 12 E.Pollak, Chem.Phys.Lett. 127, 197 (1986). 13 M.Nagaoka, Y.Okuno, N.Yoshida, and T.Yamabe, Int.J.Quantum Chem.,51,519(1994). 14 M.Nagaoka, N.Yoshida, and T.Yamabe, (to be submitted). 15 E.Cortds, B.J.West and K.Lindenberg, J.Chem.Phys., 82, 2708 (1985). 16 K.Lindenberg and B.J.West, The Nonequilibrium Statistical Mechanics of Open and Closed Systems, (VCH, New York, 1990). 17 H.Eyring, J.Chem.Phys., 3, 107 (1934); E.Wigner, ibid, 5, 720 (1937); S.Glasstone, K.J.Laidler and H.Eyring, The Theory of Rate Processes, (McGraw-Hill, New York, 1941).
journal of MOLECULAR
LIQUIDS ELSEVIER
Spatial
Journalof MolecularLiquids,65/66(1995)293-296
Correlations
in Reaction-Diffusion
Systems
in N o n e q u i l i b r i u m Conditions K a z u h i k o Seki I a n d K a z u o K i t a h a r a 2 1 T h e o r e t i c a l C h e m i s t r y Lab., N a t i o n a l I n s t i t u t e of M a t e r i a l s a n d C h e m i c a l R e s e a r c h , T u k u b a s h i H i g a s h i 1-1, I b a r a k i , 305, J a p a n 2 D e p a r t m e n t of A p p l i e d Physics, F a c u l t y of Science, T o k y o I n s t i t u t e of Technology, T o k y o 152, J a p a n
Abstract The steady state spatial correlations in reaction-diffusion systems involving many reversible chemical reactions are examined. It has been already discussed that the spatial correlations are related to the breaking of detailed balance in chemical kinetics for both one species and for two species reversible reactions. Here, we focus our attention on how the spatial correlations of concentration fluctuations in a macroscopically homogeneous systems approach to the instability point. The spatial correlations depend strongly on the stability of systems for two species reactions compared to one species reactions. I.
Introduction
It has been pointed out that the nonequilibrium spatial correlations appear in the system when the reactions are coupled with diffusion of chemical species 1. The appearance of spatial correlations is directly related to the breaking of the detailed balance in chemical kinetics, as it was pointed out for the Schl6gl model by Nicohs and Malek Mansour 2 and later studied in the case of an arbitrary set of reversible reactions by our previous paper 3-4. The microscopic simulations of these correlations were performed with various techniquesS-6; reasonable agreement was found with the analytical expression for the wide range of the length scale. In this note, the spatial correlations in a nonequilibrium steady state involving two variable constituents X and Y are analytically examined in details. Compared to the one-vaxiable reaction diffusion systems, the fluctuations are shown to be significantly enhanced as going to an instability point. The result is in consistent with the divergent form factor found by Lemarchand and Nicolis for a specific model 7. We start from the multivariate master equation for the probability distribution of number of particles in local elements. The local concentration of a species X in a volume element at the position ~' is denoted by x(v-") and its statistical average is denoted by xs, which is assumed to be independent of the position due to the assumption of a homogeneous steady state. Then, the covariance of the fluctuations 0167-7322/95/$09.50 91995ElsevierScienceB.V.All rightsreserved. 0167-7322(95) 00820-9
SSD!
294
~x(~ _---x(r') - x o is obtained in the form Co 1 e-~1r162'1 (,Sx(r-"),Sx(g ')) = x o 6 ( g - ~' ') + 8~rD I F - F -'--~[ '
(1)
where D is the diffusion coefficient. Nicolis and Malek Mansour showed that, in SchlSgl's model, when the detailed balance of the chemical kinetics is broken C, becomes nonvanishing 2. In our previous paper, we generalized the argument of Nicolis and Malek Mansour to an arbitrary set of reversible reactions for one species and show that Co is indeed related to the net flows of reversible chemical reactions 3. The theory is extended to two-species systems 4. In this note, we comment that the spatial correlation amplitude is not just given by 6'8 but is related to the stability of the kinetics for two-species systems. II.
Two-species system
In the previous paper, the derivation of differential equations for spatial correlations is given for the case of two species, X and Y whose concentrations axe denoted by x and y, respectively 4. In this note, we report only the solution. Let us consider the reversible chemical reaction of the scheme,
k~ Aj + n i X + m j Y
(1)
= (nj + Pi)X + (mj + q j ) Y + Bj. k_j
The net rate of the reaction j is defined by
(2)
Jj(x, y) ~ ajx~Jy"~j - bjxnJ+niy'~+q~, where aj and bj axe constants. Introducing quantities f(x,y)
-
g(x,y)
=
(3)
~'~p.igj(x,y), i ~-~qjJj(x,y), J
(4)
the stability condition can be written in terms of Kxx =
,Kxv = $
,Kvx = $
,Kvv = $
,
(5)
$
where the suffix s stands for the homogeneous steady state. The constituents, X and Y, may diffuse with diffusion coefficients, D x and Dr. When the system is stable under the condition, T =_ K x x + K v v < O, A -- K x x K v v - K x y K y x A = (DxKvv - DvKxx) 2 + 4DxDvKxvKvx
> O, < O,
(6) (7)
295
the solution for
pxx(e, ~") = ( , ~ x ( ~ , ( e ' ) ) - x,,5(e- e ')
(8)
is obtained as
pxx(e,~') = 1 Dy Cxx(F + x/~) 2 + 4 D x K x v C x v ( r + v ~ ) + 4D x~ K ~2y C y y e(i.n_~ )~ 8rrv/'A 2Dx (Dy - Dx)r + (Dx + D y ) v ~ +complex conjugate
r
Kxy (Dx + Dy)(KxyCyy -- K y x C x x ) + 2FCxy -r 87rr F 2 + K x y K y x ( D x + Dy) 2
(9)
where
F = DyKxx - DxKyy,
"/1--212(KxxKyy--I'(xyKyxI 2~ .y~ =
( K x x K y y - K x y K y x 1/~ 2 ~, -~x D 7 )
(10)
+
DxKyy + DyKxx , DxDy DxKyy + DyKxx Dx Dy
(11) (12)
(13) and
Cxx
= ~ ( 2 . ~ + pj)p~j~(~o,y.),
(14)
cxy
= ~ ( , . j p j + ,~jq~ + p~qj)Jj(~o, ~~ J = ~(2m~ + qj)q~j~(~o,y.).
(15)
Cxx
(16)
J
CXX~ Cxy aJ:Id Cyy are source terms for the long range correlations, which are related to the chemical reaction flows Jj(x,y). These relations were pointed out previously 4. In this note, we remark on the other point. The first two terms tends to diverge as the system approaches to the instability point, A = 0, because the denominators include vZ'A. The spatial correlations depend strongly on the stability of the systems. Such a strong dependence is not found for the one-species reaction systems. By putting K x y = 0 Eq. (9) reduces to the result of the one-species reaction systems, Eq. (1), where D = Dx and C~ = Cxx. We see that the in the denominator of Eq.(9) cancels out in Eq. (1). In the case of two-species reaction systems with Kxy, ~ in the denominator of Eq. (9) in general remains. For a specific model of two species reaction systems, Lemarchand and Nicolis found that the spatial correlations diverge in the vicinity of the critical state z. Although
296
our result suggests such a divergence, Eq. (9) is derived under the conditions that the system is far from the instability point and is in an uniform steady state on average. The more sophisticated theory is needed to analyze fluctuations around the instability point. The point in this note is that the spatial correlation amplitude is sensitive to the stability of the kinetics for two species in a distinct way from one species. Finally, we comment on the exponents in Eq. (9). The first two terms decrease with oscillations as the distance, r, increases. 3'1 characterizes the wave number of oscillations and selects the typical wave number of spatial pattern at the instability point. 72 and the exponent of the third term are related to the stabihty of kinetics. As long as the system is stable, these two exponents remain finite. References 1. N. G, van Kampen, "Stochastic Processes in Physics and Chemistry", North Holland, Amsterdam. (1981), Chap. XII and references therein. Y. Kuramoto, Prog. Theor. Phys. 52, 711 (1974). 2. G. Nicolis and M. Malek Mansour, Phys. Rev. A 29, 2845 (1984). 3. K. Kitahara, K. Seki and S. Suzuki, J. Phys. Soc. Jpn. 59, 2309 (1990). 4. K. Kitahara, K. Seki and S. Suzuki, in Proceedings of the Emil-Warburg Symposium, "Dynamical processes in condensed molecular systems", eds, A. Blumen, J. Klafter and D. Haarer, World Scientific(1990). 5. F. Baras, J. E. Pearson and M. Malek Mansour, J. Chem. Phys., 93, 5747 (1990). 6. J. Gorecki, K. Kitahara, K. Yoshikawa and I. Hanazaki, Physica A(in press, 1994). 7. H. Lemarchand and G. Nicolis, Physica 82 A, 521 (1976).
joumal of I~'IOLECULA R
LIQUIDS ELSEVIER
Journalof MolecularLiquids,65/66 (1995) 297-300
Nonlinear effects of soivation dynamics Akira Yoshimori
Department of Physics, Faculty of Science, Nagoya University, Nagoya 464-01, Japan Abstract The nonlinear Smoluchowski-Vlasov equation is calculated to investigate nonlinear effects on solvation dynamics. While a linear response has been assumed for free energy in equilibrium solvent, the equation includes dynamical nonlinear terms. The solvent density function is expanded in terms of spherical harmonics for orientation of solvent molecules, and then only terms for s and 1, and m=0 are taken. The calculated results agree qualitatively with that obtained by many molecular dynamics simulations. In the long-term region, solvent relaxation for a change from a neutral solute to a charged one is slower than that obtained by the linearized equation. Further, in the model, the nonlinear terms lessen effects of acceleration by the translational diffusion on solvent relaxation. I. Introduction Many molecular dynamics simulations have shown similar nonlinear effects on solvation dynamics for a change from a neutral solute to a ion or a large-dipole molecule ta. In long-time regions, the simulated relaxation was slower than that of the equilibrium time correlation function (tcf) for a neutral solute. In addition, the tcf relaxed faster for a neutral solute than for a charged solute. On the other hand, in some simulations, ''z solvent relaxed more slowly for a change from a neutral solute to a charged one than for a change from a charged one to a neutral one. All the above relaxation should be exactly the same in the linear response theory. The purpose of the present study is to investigate physical reasons of the nonlinear effects on solvation dynamics in the simulations, using a simple model. Few authors have studied nonlinear effects on solvation dynamics theoretically "=,though such effects were obtained in molecular dynamics simulations. Thus, the physical reasons of the effects have not been established yet. II. Model The basic equation in the present model is
(l) Here, p(r,to, t) is a density function of solvent with the position r and orientation to of the molecules, DR and Dr are the rotational and translational diffusion coefficients and tSF/rp is the functional derivative of free energy F in inhomogeneous-equilibrium solvent. When free energy F is given by the ideal gas expression plnp, eq. (1) reduces to an usual diffusion 0167-7322/95/$09.50 91995 ElsevierScience B.V. All fights reserved. SSD! 0167-7322 (95) 00881-0
298
equation. Equation (1) is called the nonlinear Smoluchowski-Vlasov equation 5 or the molecular hydrodynamics equation in the diffusive and Markovian limit (generalized Smoluchowski equation) 6. If we assume a linear response for the functional derivative of the free energy in the basic equation, then we have4 81:: = r 8p with
f + 4_~--~p(r,o~,t) - [ dr'do~'C(r-r',o~,o~')Sp(r,co,t) po J
8p(r,co,t) = p(r,co,t)
p0 4Jr"
(2)
(3)
Here, P0 is constant number density; C(r-r',co,co') is the two-particle direct correlation function, given by the mean spherical approximation. The first term O(r,co) in eq. (2) denotes the interaction between solute and solvent molecules, assumed to be the following dipole-charge interaction in the present: ~(r,eo) =
-~r.o~),
r> a
(4) 0
r
with r=-[rl, where Ix and ze are the dipole moment of solvent molecules and the charge of a solute molecule, kBT is thermal energy and a is the sum of solvent and solute molecule radii. In addition, the density function in eq. (3) is expanded in terms of the spherical harmonics 8p(r,co,t) = ~ alm(r,t)YI,m(CO), I,m
(5)
where the z axis is parallel to r in YI,m(CO). For simplicity, the present calculation includes only the harmonics for I--0 and 1 and m--0. Here, ao0(r,t) shows the number density of solvent and al0(r,t) shows the polarization. The expanded density function was numerically calculated under some conditions. Boundary conditions for r=a are 0/0r( 18F/SpYI m(CO)dco)=0. For r=-rmax, al m(r,t)=0, where rmax is the length of the model domain. In addition, since the number density of solvent molecules should be positive, we have P0 + aoo(r,t) > 0. 4~
(6)
The numerical method was the simple Eular discretization. Time step was less than 0.01 (2DR)-'l, the space interval was 0.05 R where R was the diameter of solvent molecules, and the model domain length rmax was 6 R. Note that 13~1 had the dimension of time though the dimension of I3~l is [time][radian] 2, because radians do not have dimensions. Even if the step or interval is small or rmax is large, calculation gives qualitatively similar results.
299
III. Results Numerical calculation of the model gave the interaction between solvent and solute molecules and the normalized function x(t) = f 6p(r, ta, t ) ~ r , to)drdta,
S(t) =
x(t)-x(o0)
.
(7)
x(0)-x(0o) Here the employed parameters were Ix = 2.0Debye, p = 3.3x 1022 cm -3 , Dr = 0.05 (2DRR2), T = 300 K, a = 1.0 R. The calculated results agreed qualitatively with that by molecular dynamics simulations (Fig. 1). In the long-time region, solvent relaxation for a change in a solute charge from 0 to e (z=0--, 1) was slower than that obtained by the linearized equation. In addition, relaxation for z=0--, 1 was slower than that for z= 1---0. On the other hand, in the short-time region, solvent relaxation for z = 0 ~ 1 was similar to that by the linearized equation.
10 ~
z = 0---r 1 z=l-90 linearized eq 10"2
(~
015" ........
Time
1:0
"
(2Dn) - l
'
11.5
. . . . . . .
2.0
Figure 1. Normalized functions S(t) in solvation dynamics calculated by the nonlinear Smoluchowski-Vlasov equation for a change in a solute charge from 0 to e (z=0--- 1) and from e to 0 (z= 1 --,0) (solid lines), and by the linearized equation (dashed line). Calculation with the variation of translational diffusion coefficients showed that the nonlinear effects lessened effects of translational diffusion (Fig. 2). In the linearized equation, solvent relaxed quickly for a large translational diffusion coefficient. That is, a value of S(t) at t=(2DR)l decreased with the increase in the translational diffusion coefficient for the linearized equation. For calculation with the nonlinear Smoluchowski-Vlasov equation (1), however, a value of S(t) at t=(2DR) 1 increased slightly with the translational diffusion coefficient. Thus, the ratio of a nonlinear value to a linear value increased with the translational diffusion coefficient.
300
10 3
J
102
ratio .........----
101 o0~ (L_ 13100 L_
~
z = 0 ---~ 1
0
10 -I
10 -2 9
o
,
10 . I 10 .4
o
o.~s
o.~o
o.~s
0.20
Translational diffusion coefficient D T / 2 D n R 2 Figure 2. Values of S(t) at t=(2DR)"t for some translational diffusion coefficients. Circles show values by the nonlinear equation for z=0--* 1, diamonds show values by the linearized equation, triangles show ratios between them. IV. Discussion The calculated results by the present model agreed qualitatively with that by molecular dynamics simulations. However, it is not clear whether nonlinear results of the simulations can be explained only by the present model. The present model does not include all nonlinear effects in the simulations. In particular, the nonlinear effects in the solvent free energy can be important in the solvation dynamics. Equilibrium values of the number density a00(r,t) can depend on a charge distribution of a solute, though the linear-response free energy does not include the effects. The equilibrium number density for a neutral solute approaches to the homogenous constant density P0. The number density near a charged solute, however, can be larger than that of bulk solvent. The present model does not include such effects. Such cffects of the nonlincar frce energy can change a role of translational diffusion from that in the present model. The present model showed small nonlinear effects for small translational diffusion coefficients. In the model of nonlinear free energy, however, small translation diffusion has slowing effects on relaxation of solvent. This is because nonlinear free energy requires translational relaxation of the number density. The expansion of 8p(r,ta, t) by spherical harmonics has an instability. If the condition (6) is removed, the calculation cannot be stabilized. The reason for this instability is neglect of the higher-order harmonics in eq. (5). The neglect can give negative values of the density function. From time evolution of the free energy F, we can find that the density function in eq. (l) should be positive if a value of F always decreases. The positive density function needs the condition (6).
References 1 M. Maroncelli and G. R. Fleming, J. Chem. Phys., 89, 5044(1988). 2 M. Maroncelli, J. Chem. Phys., 94, 2094(1991). 3 For example, E. A. Carter and J. T. Hynes, J. Chem. Phys., 94, 5961 (1991). 4 A. Yoshimori, Chem., Phys., Letters, 184, 76(1991). 5 D.F. Calef and P. G. Wolynes, J. Chem. Phys., 78, 4145(1983). 6 A. Chandra and B. Bagchi, J. Chem. Phys., 91, 1929(1989).
journal of
MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 301-304
Solvation Dynamics Studied by Uitrafast Transient Hole Burning Jangseok Ma,* David Vanden Bout + and Mark Berg
Department of Chemistry and Biochemistry, University of South Carolina, Columbia, SC 29208, USA Abstract The time-resolved solvation of s-tetrazine in propylene carbonate is studied by ultrafast transient hole burning. In agreement with mode-coupling theory, the temperature dependence of the average relaxation time follows a power law in which the critical temperature and exponent are the same as in other relaxation experiments. Our recent theory for solvation by mechanical relaxation provides a unified and quantitative explanation of both the subpicosecond phonon-induced relaxation and the slower structural relaxation. 1. Introduction Because of the fundamental importance of solvent-solute interactions in chemical reactions, the dynamics of solvation have been widely studied. However, most studies have focused on systems where charge redistribution within the solute is the dominant effect of changing the electronic state.[1,2] Recently, Fourkas, Benigno and Berg studied the solvation dynamics of a nonpolar solute in a nonpolar solvent, where charge redistribution plays a minor role.[3,4] These studies showed two distinct dynamic components: a subpicosecond, viscosity independent relaxation driven by phonon-like processes, and a slower, viscosity dependent structural relaxation. These results have been explained quantitatively by a theory of solvation based on mechanical relaxation of the solvent in response to changes in the molecular size of the solute on excitation.J6] Here, we present results on the solvation of a nonpolar solute, s-tetrazine, by a polar solvent, propylene carbonate over the temperature range 300-160 K. In this system, comparisons to several theoretical approaches to solvation are possible. 2. Transient Hole Burning Experiments We used transient hole burning to measure the solvation dynamics (Fig. 1).[3-5] These experiments yield both a time-dependent Stokes shift and a time-dependent width, both of which are related to the solvation dynamics. Here we consider the time-dependent Stokes shift. The time-dependent widths will be discussed elsewhere.J9] Detailed measurements of the s-tetrazine gas-phase spectrum were made. With these data, measurement of the absolute Stokes shift S(t) is possible. Because the Stokes shift is zero in the absence of solvent nuclear dynamics, the magnitude of the Stokes shift at the earliest times represents the amount of relaxation within the experimental time resolution. The steady-state absorption and fluorescence spectra were also measured to provide an independent value of the equilibrium Stokes shift S,~ With this data, the absolute solvation response function R(t)=l
S(t)
( 1 )
s(~)
* Current address: Institute for Molecular Science, Myodaiji, Okazaki, 444 Japan + Current address: Department of Chemistry, University of Minnesota, Minneapolis, MN 55455, USA 0167-7322/95/$09.50
91995 Elsevier Science B.V. All rights reserved.
SSD! 0167-7322 (95) 00821-7
302 was calculated without any normalizing factors. Measurements were made at 13 temperatures between 298 and 160 K, spanning a viscosity range of 2.6 cP to 3.2x1011 cP. Comparisons to four different theoretical approaches are made. 3. Results A. Mode-Coupling Theory Mode-coupling theories have recently shown great promise for explaining the microscopic origins of the glass transition.J10,11] These theories predict that the dynamics are dominated by the existence of a critical temperature Tc distincdy above the glass transition temperature. Above Tc, the primary (or) structural relaxation time is predicted to have a power-law temperature dependence, x = ot(T-Tc)-3'. Tc and 3' should be the same for all relaxation processes in the same solvenL In Fig. 1, the average solvation time measured by transient hole burning is compared to Du, et al.'s measurements of relaxation by depolarized light scattering.[12] These experiments measure relaxation on very different length scales; solvation on a molecular length scale, and light scattering on the scale of the wavelength of light. Both sets of data fit with the same exponent of 3'=2.74 and extrapolate to nearly the same value of Tc, in agreement with mode-coupling theory.
.6
9
-
Z ~
O.4
-
~- o.2 0
. 0 ~ 200 250
300
Temperoture (K)
B. Mechanical Relaxation Theory Berg suggested that a change in the solute size on excitation could produce a significant solvent response.[6,8] He treated the solute as a spherical cavity of radius r c and the solvent as a viscoelastic continuum with time-dependent compression and shear moduli K(t) and G(t). For a spherically symmetric change in cavity size, two time scales are relevant,
350
Fig. 1 A fit of solvation times to a power law temperature dependence. Depolarized light scattering (DPLS) times [12] fit to the same power and intercept, in agreement with modecoupling theory. Neutron scattering (NS) times [13] match solvation times, supporting Bagchi's solvation theory.
~
l:ph "-
f
K~ +
G**
'l;s= <1"1_ _
9
4 1
G(t) dt
(2)
where p is the solvent density, and 1] is the viscosity. In the normal case, where ts >> tph, the response function simplifies to R ( t ) = g+4G**g { ex[{tx -t 1;ph] ( COS [(X'~/~tph] - g1 sin [OtTXtph] ) }
p(s)
A
4G~ +g-~G.
I' I-
dt'L-)
I - --~ -- ^
]+
4p(s) g
(t') (3)
303
Koo+~oo
IX=
and
g+4G**
T2 = 2IX - 1
2.0
The frequency dependent modulus is defined by the Laplace transform
A
It(S) = L[G(t)](s) =
I
e-stG(t) dt
(4)
and K** and G**are the infinite frequency moduli. This model predicts that the solvation will have two distinct components. The first term of Eq. 3 is a subpicosecond response determined entirely by the high frequency elastic moduli. The moduli, and therefore tph, are only weakly temperature dependent. This term represents the creation of vibrations (phonons) of the instantaneous structure of the liquid. The second term is a slower, temperature dependent response caused by reorganization of the instantaneous structure. Figure 2 shows fits of this model to the dynamic Stokes shift of s-tetrazine in propylene carbonate. A stretched exponential form for G(0 has been used, and the temperature dependence of G.,, has been estimated from light scattering data.[ 11] The details of the phonon-component are not observable with current time-resolution, but the predictions of the magnitude of the early phonon component can be meaningfully compared to the experiment. A critical test of the reasonableness of these fits is the ability of the fit parameters to correctly predict the solvent's viscosity. The viscosity is well predicted by the fits 9The values of G** are a factor of 2 larger than expected from light scattering experiments. This discrepancy could be explained by a deviation from a spherical change in size, or by a local value of Go. which is larger than the bulk value.
.~
1.5
~.o 5 o.5 0.0~-2
-1
0
1
2
3
Log Deloy Time (ps)
Fig. 2 A fit of a viscoelastic model of solvation (solid) to transient hole burning data (points). Temperatures top to bottom: 160K to 298K
C. Bagchi's Theory Bagchi has presented a molecular theory for solvation of spherical nonpolar solutes.J7] It relates the solvation response function to the interaction potential and the dynamics structure factor. Complete k-vector dependent neutron scattering experiments can determine the dynamics structure factor, but such measurements are not widely available. Relaxation times from neutron scattering at a single k-vector are available for propylene carbonate.J13,14] Assuming this k-vector is representative of the length scale of the interaction potential, Bagchi's theory predicts the solvation times and the neutron scattering times would be the same. Figure 1 shows that this prediction is true.
D. Dielectric Relaxation Theory The solvation dynamics of Coumarin 153 in propylene carbonate has been studied by Barbara and coworkers by femtosecond fluorescence spectroscopy.J15] This solute has a large charge redistribution in the excited state, and solvation is dominated by the dielectric response of the solvent to the new change distribution. We
304 compared the solvation times of the Coumarin to our measurements of the solvation times of s-tetrazine to test whether dielectric solvation is active in our case. At room temperature, the solvation response functions of Coumarin 153 and s-tetrazine are almost identical.(Fig. 3) As the temperature is lowered (267 K and 257 K), the s-tetrazine response function becomes distinctly slower than the Coumarin response. From this comparison, we conclude that dielectric solvation is relatively unimportant for this nonpolar solute, even though the solvent is strongly polar. 4. Conclusions These experiments have shown that the slower component of solvation is linked to the overall structural dynamics of the liquid, and that mode coupling theory predicts many of the overall features of these dynamics. Dielectric solvation, which is the most widely studied solvation mechanism, does not play a major role for this nonpolar solute. Two theories of nonpolar solvation give better agreement with the data. Bagchi's theory is more rigorously derived, but our model permits a more detailed and rigorous comparison with experiment.
298 K
0.75
c
0
"~ (,3 C
0.25
~
0.00
~) 03 C 0
I .oo
CL 03
257K
0.75
0.50
0.0
ee
e
0.5
1 0
1.5
~.0
z5
Log Delay Time (ps)
Fig. 3 A comparison of the solvation of stetrazine (points) and Coumarin 153 (solid) [2] in propylene carbonate. Different solvation mechanisms appear to operate for different solutes.
5. References 1. M. Maroncelli, J. Mol. Liq. 57, 1 (1993). 2. P. F. Barbara and W. Jarzeba, Adv. Photochem. 15, 1 (1990). 3. J. T. Fourkas and M. Berg, J. Chem. Phys. 9 8 , 7 7 7 3 (1993). 4. J. T. Fourkas, A. Benigno and M. Berg, J. Chem. Phys. 9 9 , 8 5 5 2 (1993). 5. H. Murakami, S. Kinoshita, Y. Hirata, T. Okada and N. Mataga, J. Chem. Phys. 97, 7881 (1992). 6. M. Berg, Chem. Phys. Lett. 2 2 8 , 317 (1994). 7. B. Bagchi, J. Chem. Phys. 100 , 6658 (1994). 8. M. Berg, in preparation. 9. J. Ma, D. Vanden Bout and M. Berg, in preparation. 10. W. GOtze, in Liquids, Freezing and the Glass Transition, edited by J.P. Hansen, D. Levesque, and J. Zinn-Justin (North-Holland, Amsterdam, 1990), p. 287. 11. W. G6tze and L. Sj6gen, Rep. Prog. Phys. 55, 241 (1992). 1 2 . W . M . Du, G. Li, H. Z. Cummins, M. Fuchs, J. Toulouse and L. A. Knauss, Phys. Rev. E. 4 9 , 2 1 9 2 (1994). 13 L. B6rjesson and W. S. Howells, J. NonCryst. Solids. 1 3 1 - 1 3 3 , 5 3 (1991). 14 L. B6rjesson, M. Elmwonh, L. M. Torell, Chem. Phys. 149, 209 (1990). 15. W. Jaezeba, G. C. Walker, A. E. Johnson and P. F. Barbara, Chem. Phys. 152, 57 (1991).
journal of MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 305-308
Hydrogen-Bonded Structure in Aqueous Sulfuric Acid Solutions Yasuo Kameda, Kiyohiko Hosoya, Shuji Sakamoto, Hirohito Suzuki, Takeshi Usuki and Osamu Uemura Department of Chemistry, Faculty of Science, Yamagata University, Yamagata 990, Japan
Abstract
X-ray and time-of-flight (TOF) neutron diffraction and Raman spectroscopic measurements have been carded out at 25~ on concentrated aqueous sulfuric acid solutions, (H2SO4)x(H20)I_x, x = 0, 0.25, 0.5, 0.75 and 0.86, in order to investigate the hydrogen-bonded intermolecular structure in the solutions. The intramolecular parameters for regular tetrahedral SO 4 unit were determined from the least squares fit to observed X-ray and neutron interference terms in the range of Q > 8 A -1. The nearest neighbor intermolecular hydrogen-bonded distances, ro... O = 2.7 ,~ and ro... D = 1.7 ,A,, respectively determined from X-ray and neutron intermolecular distribution functions, exhibited both ---0.2 ~ shorter values than those reported for pure liquid water. The isotropic Raman spectra of O-H stretching region for solutions of higher H2SO 4 content (x _-->0.5) indicated an extremely broadened band centered at 3030 cm -1 , which corresponds to ---300 cm -1 lower frequency shift compared with that for pure liquid H20. These results suggest the existence of strong hydrogen-bonded intermolecular interaction in these solutions.
Introduction The information conceming the structural feature of the intermolecular hydrogen-bond in the liquid state is indispensable for understanding of the proton transfer reaction which plays a fundamental role in various chemical and biological processes. 1 The sulfuric acid - water system has been of particular interest due to extremely fast proton transfer property which has been frequently found by Raman 2 and quasielastic neutron scattering 3 studies. Structural investigations on the water-sulfuric acid system have been carded out using X-ray and neutron diffraction techniques. 4-6 Caminiti proposed the eightcoordinated hydration model for SO42- in 3.76 tool% aqueous sulfuric acid solution from X-ray diffraction data. 4 Andreani et al obtained the information concerning the hydrogen-bonded structure in this system from observed X-ray and neutron total distribution functions. 5,6 The combination of X-ray and neutron data has a significant advantage in determining the intermolecular O-H correlation in the system. However, to obtain more definite information on the intermolecular hydrogen-bonded structure, it is necessary to separate the intra- and intermolecular contributions from observed total interference term, which was not applied in the previous works because of the limited Q-range measured (Qsl0 ,~,-1)5,6. In the present paper we describe the results of X-ray and TOF neutron diffraction and Raman scattering measurements for aqueous sulfuric acid solutions with extended composition range in order to obtain the information both on the intermolecular hydrogen-bonded structure and on the intramolecular structural parameters of SO 4 unit in the solutions.
Experimental
X-ray diffraction measurements for sample solutions with natural isotopic composition, (H2SO4)x(H20)1_ x, x=0, 0.25, 0.5, 0.75 and 0.86, were carded out at 25~ in the reflection geometry using a 0-0 X-ray diffractometer manufactured by Rigaku Co. The Mo k(x radiation (k = 0.7107 ,~) was employed. The scattered intensity was counted at an interval of 0.5 ~ over the range of 3 < 20 < 150 ~ corresponding 0.5 < Q < 17.1/~,-1 (Q = 4 :t sin0 / X). The TOF neutron diffraction measurement for the sample solution sealed in a cylindrical quartz cell (8 mm in inner diameter and 0.4 mm in thickness) was 0167-7322/95/$09.50 r 1995 Elsevier Sciencc B.V. All rights rcscrvcd. SSDi 0167-7322 (95) 00882-9
306 performed at 25~
using the HIT spectrometer 7 installed at KENS, Tsukuba, Japan.
Deuterated
samples i> 99% D) were used to avoid large incoherent scattering from the hydrogen atom.
The
scattering data from the forward angle detectors located at 20 = 14, 25, 32 and 44 ~ were used to minimize the inelasticity effect. The data correction and the least squares fitting procedures for X-ray and neutron diffraction data are almost identical to those described previously.8, 9 Raman spectra were observed at 25~
in the frequency region of 1500 - 4500 cm -1 using a JASCO NR-1100 spectrometer
with a 514.5 nm line of NEC GLG-3200 Ar + laser operated at 200 mW. The details of the experimental procedures and data corrections are described elsewhere. 10
Results and Discussion Intramolecular structure of H2SO4 and HSO 4Figure l a shows the observed X-ray total interference term, Qix(Q ).
The least squares fitting analysis was applied to the observed interference term in the range of Q ~ 8 A -1 in order to determine the intramolecular structure of H2SO 4 and HSO 4- in the aqueous solution. The calculated intramolecular interference term was evaluated by the sum of intramolecular contributions from H2SO4, HSO4- , SO42- , H30+ and H20 molecules, icalc(Q) = AiH2so,i(Q ) + BiHSO4_+ Ciso, i2_ + DiH30+ + EiH~(Q),
(1)
where, the coefficients A - E were referred from a paper of Robertson et al. 11 In the fitting procedure, the intramolecular parameters for H30+ and H20 molecules were fixed to the values found in our previous work. 9 The tetrahedral geometry of isolated SO42- (rSo=1.477J,) 12 was assumed. The intramolecular bond lengths, r(S-O) and r(S-O(H)), and the root mean square amplitude, I(S-O), within H2SO 4 and HSO 4- molecules, were treated as the independent parameters. The bond angles, / O - S - O and Z. (H)O-S-O(H) in H2SO 4, and Z_O-S-O(H) in HSO4-, were also refined independently. The molecular structure of H2SO 4 and HSO 4- reported in the gaseous 13 and in the crystalline states,14,15 is characterized by two distinct S-O bond lengths, r(S-O(H)) = 1.54 - 1.57 A and r(S-O) = 1.40 - 1.43 J,. In the present analysis, the fitting procedure was carried out using the model function involving two different intramolecular S - O distances, r(S-O) and r(S-O(H)), however, the optimized solution of the 150 ,,
3
100~' \.
(HiS0i)x(H'O)i-x
-
~~
- i
o ~ .k..It,j./ " - ~ ~ { < ~
o
.,oI~.~L_...." ~
v ' ~
"
A
8 O//~_= 12
16
20
I
.~
-1
'
I
i
I
'
I
x:o.,,
ii . ~
0.75
.-. .....
i ...,.... .. 9 0
'
(HzSO')x(HzO'-x
. . . . .
o
-50 0
''
2
2
.,ob z. r/A
6
L
1 8
A
10
Fig. 1. a) Dots: The observed X-ray total interference term, Qix(Q ), for aqueous sulfuric acid solutions. Solid lines: The calculated intramolecular contribution, b) The observed intermolecular total distribution function, gxinter(r).
3 0 7
o5~
,~ /
s o ~.. -/ ~..-...
04
0[-
)
?,,/ V
o~o
~
o2s
.
o - . . . .o 1 s
o9 0
z.
8
Q/j_,
12
16
20
0
2
~
~ 4
r/A
b 6
8
10
Fig. 2. a) Dots: The observed neutron total interference term, iN(Q), for aqueous sulfuric acid solutions. Solid lines: The calculated intramolecular contribution, b) The observed intermolecular total distribution function, gNinter(r). least squares fit always indicated that the two S-O bonds are identical in the bond length (rso = 1.48 _ 0.01 ,&) with the root mean square amplitude of Iso = 0.05 _+ 0.01 ,& in these solutions. The same analytical procedure was applied to the observed neutron interference term (Fig. 2a). Additional independent parameters, r(O-D), I(O-D) a n d / - S - O - D , were also refined in the fitting procedures for the neutron data. The result again exhibited composition independent single S - O distance of rso = 1.50 _+ 0.01 /~ and r.m.s, amplitude of ISO = 0.04 _ 0.01 A, which are consistent with those obtained from the present X-ray data. These results suggest the regular tetrahedral geometry of SO 4 structural unit in concentrated aqueous sulfuric acid solutions.
Intermolecular hydrogen-bonded structure Figures l b and 2b represent the X-ray and neutron intermolecular distribution functions for aqueous sulfuric acid solutions, respectively. In the gxinter(r) function, the nearest neighbor (hydrogen-bonded) !
!
!
i
I
i
i
i
f
!
(H2SOI.)x(H20)f-x .:. '.-;.. ~; .
-"9
~-
. . . . . .
--"
[:':',;,
1500
"~',-,
"
"
3000 v/cm-i
x=0.86
"
"
"
0"7
......:
i
/.500 1500
3000 u / c m -I
/.,500
Fig. 3. Composition dependence of the isotropic Raman spectrum in the O-H stretching region for aqueous sulfuric acid solutions. -__
308 O..-O interaction can be identified as an incompletely resolved peak at r = 2.7 A. This intermolecular distance is ,--0.2 .~, shorter than that for pure water. A well defined first peak at r = 1.7 ,~, observed in the present gNinter(r) functions can be ascribed to the hydrogen-bonded intermolecular O .--D interaction. This intermolecular O..-D distance is again considerably shorter than that reported for pure liquid water (ro... D = 1.85 ~.).16,17 These short intermolecular hydrogen-bonded distances indicate the existence of strong hydrogen-bonds in these solutions. The composition dependence of the gNinter(r) and gxinter(r) functions seems to be very small for solutions of higher sulfuric acid content above 50 mol%, suggesting the similarity of the hydrogen-bonded liquid structure among these solutions. This similarity can also be found in the isotropic Raman spectra of O-H stretching region shown in Fig. 3. Except for 25 tool% H2SO4 solution, an extremely broad (fwhm = 640_40 cm -1) O-H stretching band at 3030__.20 cm-1 with a small shoulder at 2420_20 cm -1 was observed. According to the vOH - ro... O correlation curve proposed by Wall and Homig, 18 the present value vOH = 3030 cm -1 leads to a very short hydrogenbonded O " - O distance of 2.72 A. The value is in excellent agreement with that determined from the present gxinter(r) function. The peak components at 3300 and 3500 cm -1 observed in 25 mol% solution may be attributed to the O-H stretching modes of remaining H20 molecules in the solution. Acknowledgment
The authors would like to thank Profs. M. Misawa, T. Fukunaga and T. Yamaguchi for their help during the course of the neutron diffraction measurement. All of the calculations were performed using the ACOS $3600 computer at the Computing Center of Yamagata University. References
1. T. Bountis, ed., "Proton Transfer in Hydrogen-Bonded Systems", Plenum Press, New York (1992). 2. 3. 4. 5.
H. Chen and D. E. Irish, J. Phys. Chem., 75, 2672 (1971). D. Cavagnat and J. C. Lass~gues, J. Phys.: Condens. Matter, 2, SA189 (1990). R. Caminiti, Chem. Phys. Lett., 96, 390 (1983). C. Andreani, C. Petrillo and F. Sacchetti, Mol. Phys., 58, 299 (1986).
6. C. Andreani and C. Petrillo, Mol. Phys., 62, 765 (1987). 7. N. Watanabe, T. Fukunaga, T. Shinohe, K. Yamada and T. Mizoguchi, "Proc. 4th International Collaboration on Advanced Neutron Sources (ICANS-IV ), KEK, Tsukuba", eds. by Y. Ishikawa et al, p. 539 (1981). 8. Y. Kameda, R.Takahashi, T. Usuki and O. Uemura, Bull. Chem. Soc. Jpn., 67, 956 (1994). 9. Y. Kameda and O. Uemura, Bull. Chem. Soc. Jpn., 65, 2021 (1992). 10. Y. Kameda, H. Ebata and O. Uemura, Bull. Chem. Soc. Jpn., 67,929 (1994). 11. E. B. Robertson and H. B. Dunford, J. Am. Chem. Soc., 86, 5080 (1964). 12. J.-O. Lundgren, ,~. Kvick, M. Karppinen, R. Liminga and S. C. Abrahams, J. Chem. Phys., 80, 423 (1984). 13. R. L. Kuczkowski, R. D. Suenram and F. J. Lovas, J. Am. Chem. Soc., 103, 2561 (1981). 14. C. Pascard-Billy, Acta CrysL, 18, 829 (1965). 15. I. Taesler and I. Olovsson, Acta CrysL, B24, 299 (1968). 16. A. K. Soper and M. G. Phillips, Chem. Phys., 107, 47 (1986). 17. J. C. Dore, J. Mol. StrucL, 250, 193 (1991). 18. T. T. Wall and D. F. Homig, J. Chem. Phys., 43, 2079 (1965).
journal of MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 309--312
Raman Spectroscopic Study of Rotational and Vibrational Relaxation of CF3H in the Supercriticai State
Susumu Okazaki, Nobuyuki Terauchi, and Isao Okada
Department of Electronic Chemistry, Tokyo Institute of Technology Nagatsuta, Midori-ku, Yokohama 226, Japan
Abstract Instrumentation of polarized and depolarized Raman scattering measurement for supercritical fluids at high pressures up to 20 MPa along its isotherm of moderate temperatures has been established. Preliminary measurements have been performed for supercritical CF3H to examine density dependence of rotational and vibrational relaxation of the molecule in the fluid. Reorientation of the molecule changed from free-rotor-like motion to liquid-like diffusional one via intermediate specific one with increasing density. Strong density dependence was found for vibrational relaxation time of the supercritical fluid reflecting the collision frequency or inhomogeneity in the fluid.
I. Introduction Supercritical fluid continuously changes its density along isotherms from gas-like one to liquid-like one. This presents a variety of environments to the molecules in the fluid. With respect to the rotational and vibrational relaxation of the component molecule, this characteristic property is very interesting in that collision frequency in a dynamical sense or inhomogcneity in a static sense can be widely controlled as a function of pressure keeping the kinetic cnergy unchanged. 1,2 Thus, polarized and depolarized Raman spectroscopy for this fluid may lead us to a deeper understanding of the fundamental process of the rotational and vibrational relaxation of molecules in condensed phase. In the present study, an instrument has been established for polarized and depolariz~ Raman scattering measurement of supcrcritical fluid along its isotherm to realize the investigations stated above. Preliminary results for supcrcritical CF3H are given to draw a schematic picture of the dynamics.
II. Experimental A sample cell was designed to accomodate supercritical fluids at high pressures up to 20 MPa and at moderate temperatures using stainless steel and fused silica glass, as schematically shown in Fig. 1. The fluid was prepared by compressing the low pressure sample gas from the tube using a piston compressor outside the cell. The cell and the attached part for high pressure lines were immersed in an air thermostat. The pressure and temperature were 0167-7322/95/$09.50 c: 1995 Elsevier Science B.V. All rights reserved. SSDI 0167-7322 (95) 00828-4
310
to spectrometer
~ : ~
i
incident light
/~ ..j~~~~
-"i'~.!:-~;.~.L...-~ --;.::..:.:.-.-- N
windows seals
I
"i
t
Fig. 1 Pressure-resistant sample cell for Raman scattering measurement of supercritical fluids.
monitored during the measurement by a semiconduaor strain gage(Druck PTX 520) and a platinum resistance thermometer, respectively. Arrangement of the spectrometer and data analysis of the observed spectra are the same as those previously employed.3 The measurements were done for the v 1 stretching mode of CF3H molecule in the supercridcal fluid
0.8 0.6 E cnO.4 0
(3.
0.2 ..i--P/MPa Fig. 2 Measured points(closed circle) in the PpT-diagram. isotherm at 313 K.
The solid line represents the
311
.,.., 0 t-"
..Q
--~0 2.0 MPa
3200
1
3000
v/cm-1
2800
Fig. 3 Depolarized(VII) Raman spectra for the v, stretching mode of CF3H in the supercritical fluid.
at 313 K and at 2.0 MPa, 5.0 MPa, and 7.0 MPa. The critical constants of this material are Tc=299.3 K, Pc=4.77 MPa, and pr g cm3. The density region covered in the present study was, thus, the lower one along the isotherm, as shown in Fig. 2.
III. R e s u l t s and D i s c u s s i o n The observed VII spectra are presented in Fig. 3. Although sufficient statistics have not yet been obtained for quantitative evaluation of the rotational autocorrelation function, it is enough to give a qualitative discussion of the relevant dynamics of the molecule. In this sense, one of the most interesting aspects is the shape of the spectra. The spectrum of the fluid at 7.0 MPa is ordinary Gaussian-Lorentzian, representing the almost diffusional reorientation which is generally found in liquids 9On the other hand, for the fluid at 2.0 MPa, the spectrum clearly shows side peaks on both sides. The Fourier transformation of the spectrum gave a rotational autocorrelation function similar to that of the free rotor, although a difference between them is certainly found. It is of a triangular shape for the fluid at 5.0 MPa, showing existence of an intermediate relaxation process between those found in the gaseous and liquid states. Among
312
1"01
9
"~ O.8 0.6 o 0"~:0
015
o oo
110 115 P/g cm -3
2.0
Fig. 4 Vibrational relaxation time x v as a function of densit~ 13 for the v, stretching mode of CF3 H in the supereritical fluid(closed circle) and in the liquid-~(open circle).
the models, the J-extended model4, where rotational libration with large amplitudes is taken into account, may be most suitable for describing this behavior at least qualitatively. The linewidth of VH spectrum for the fluid at 7.0 MPa is as broad as about 54 cm"(FWHH), from which the rotational relaxation time is calculated to be as short as about 0.3 ps. This may be based upon the fact that the density of the fluid is so low that the molecule can reorientate with small perturbation; the collision rate is still low at this density compared with that in the liquid. V~rational relaxation time % evaluated from the linewidth of the isotropic spectrum is plotted in Fig. 4 together with that of the liquid.5 The relaxation time for the supercritical fluid is larger than that in the liquid. The % much depends upon density in the supcrcritical fluid, whereas it does not in the liquid, showing a good contrast between them. The difference in the behavior can be understood in terms of the collision rate of the molecules and inhomogeneity with respect to the environment of the oscillators. Detailed experiments with sufficient statistics to obtain quantitative evaluation of the autocorrelation functions are in progress.
References
1 2 3 4 5
D. Ben-Amotz, F. Laplant, D. Shea, J. Gardecki, and D. List, ACS Symp. Ser., 488, 18(1992). S.M. Howdle and V. N. Bagratashvili, Chem. Phys. Lett. 214,215(1993). S. Okazaki, N. Ohtori, and I. Okada, J. Chem. Phys., 91, 5587(1989). R.G. Gordon, J. Chem. Phys., 44, 1830(1966). J. DeZwaan, D. W. Hess, and C. S. Johnson, Jr., J. Chem. Phys., 63,422(1975).
journal of MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 313-316
Anomalous Conformational Behavior of Short Poly(oxyethylene) Chains in Water:. An FT-IR Spectroscopic Study Hiroatsu Matsuura and Tatsuya Sagawa Department of Chemistry, Faculty of Science, Hiroshima University, Kagamiyama, Higashi-Hiroshima 739, Japan Abstract Conformational behavior of short poly(oxyethylene) (POE) chains in water has been studied by FT-IR spectroscopy. The spectra of aqueous solutions of CH3(OCH2CH2)mOCH 3 with m = 2--6 were measured for various concentrations by using a horizontal ATR accessory. The observed spectra show that, with increasing water fraction, the population of the gauche conformation about the OC-CO bond increases, but the direction of the conformational preference is reversed for lower concentrations. The observed maximum of the gauche population is interpreted as a consequence of anomalous enhancement of the gauche population in a specific concentration region. This effect is more significant for shorter POE chains. These observations suggest that there exist some specific interactions, which highly favor the gauche conformation of the O-CH2-CH2-O segment, for limited composition regions of the binary systems of short POE chains and water.
I. Introduction Poly(oxyethylene) (POE) (-OCH2CH2-) m is an unusual polyether with practically tmlimited solubility in water, unlike other structurally related polymers. 1 At elevated temperatures, however, the isotropic aqueous solution of POE separates into two phases. 2 The mechanism of the water solubility of POE and the phase behavior has attracted much attention of many investigators. Various mechanistic models have in fact been proposed to account for these phenomena: 3 a water structure model, 4 a hydrogen bond model, 5 and a conformational model. 6 The conformation of POE in aqueous solution has been studied by a number of experimental and theoretical methods. Among those studies, Raman spectroscopy, which provides direct information of relative conformational populations, has shown that the POE chain in water greatly favors the gauche conformation about the OC-CO bond. 7 Raman spectroscopy further indicated that the population of the gauche conformation increases with an increase in water 0167-7322/95/$09.50 ~c) 1995 Elsevier Science B.V. All rights reserved. SSDI 0167-7322 (95) 00883-7
314
fraction of the binary system. In a recent Raman spectroscopic study, we have performed spectral measurements for more diluted aqueous solutions of short POE compounds, s In the present work, we have investigated the conformational behavior of POE chains in water by FT-IR spectroscopy, which is another vibrational spectroscopy that gives important conformational information, to gain a further insight into the structural aspect of the POE-water system. The POE compounds studied are CH3(OCH2CH2)mOCH 3 (C1EmC 1) with m = 2--6.
IL Experimental C1E2C1, C1E3C1, and C1E4C 1 were commercially supplied, while C1E5C 1 and C IE6C 1 were prepared by the Williamson ether synthesis. The infrared spectra of aqueous solutions of these compounds were measured for various concentrations at room temperature by using a horizontal ATR accessory with a ZnSe prism. The lowest concentration studied was a mole fraction of approximately 0.005. The spectra were recorded on a JASCO FT/IR-7300 spectrometer. Each of the spectra was obtained by the coaddition of 200 scans at a resolution of 4 tin--1.
HI. Results and Discussion III.1. Spectral Analysis For studying the conformational behavior of the POE chain in water by infrared spectroscopy, we focused on two distinct marker bands at about 1330 and 1350 cm -1, which are assigned to the CH 2 wagging mode of the O-CH2-CH2-O segment in the trails and gauche conformations, respectively, as established by normal coordinate analysis.9, lo The infrared spectra of aqueous solutions of C1E2C1 in the 1500-1200 cm-1 region are illustrated in Figure 1. It is apparent that the absorbance of the 1330 cm-1 band (the trans band) is greatly diminished relative to that of the 1350 cm -1 band (the gauche band) with an increase in water fraction. In Figure 2, the absorbance ratio of the gauche band to the trans band, Agauche/Atram, is plotted against a mole fraction of C1EmC1. A remarkable observation is that, with increasing water fraction, the population of the gauche conformation about the OC-CO bond increases, but the direction of the conformational preference is reversed for lower concentrations. The mole fractions of C1EmC 1 giving maximum gauche population are 0.0165, 0.027, 0.032, 0.0275, and 0.025 for m = 2, 3, 4, 5, and 6, respectively, with uncertainties of +0.002. These values agree, within the estimated experimental errors, with those obtained from the Raman marker bands, i.e. 0.019, 0.030, and 0.030 for m = 2, 3, and 4, respectively, lo III.2. Stabilization of the Gauche Conformation of POE Chains in Water The increase of the population of the gauche conformation about the OC-CO bond with increasing water fraction can be explained for a considerable part by
315 the dielectric effect, which implies that the conformers with larger dipole moment are more stabilized in the medium with higher dielectric constant. 11 According to the calculation using the atomistic force field, the value of the dipole moment for the O-CH2--CH2--O segment in the gauche conformation is generally larger than that for the same segment in the trans conformation, although the value depends significantly on the conformation about the CO-CC and CC-OC bonds. 12 The second possible factor of the stabilization of the gauche conformation is specific hydrogen bonding between the ether oxygens of the POE chain and water. The importance of the hydrogen bond is suggested by molecular dynamics and Monte Carlo simulations. 13,14 It is shown that the gauche conformation of the O-CH 2CH2-O segment is more favorable than the trans conformation for water molecules to form hydrogen-bonded bridges between the adjacent ether oxygens. 25
(a)
2o
~ ~
15 o
I
i
I
I
10
|
(b)
20
-- 15
o
c.o
~
I
I
l
I
l
o
10
l 0
= 10
0
I
I
o 15 f (d) < ~o 10 o
1
10
1500
1400 1300 Wavenumber / cm-1
1200
Figure 1. FT-IR spectra of aqueous solutions of C 1E2C 1 for mole fractions of (a) 0.202, (b) 0.057, (c) 0.014, and (d) 0.005. The conformational marker bands are indicated by arrows.
~
0.0
o
o
,.
,
011 0:2 0:3 Mole fraction of C 1EmC 1
0.4
Figure 2 Absorbance ratio, Agauehe/ Atrans, "for aqueous solutions of C 1EmC1 as a function of mole fraction; (a) C1E2C1, (b) C1E3C1, (c) C1E4C1, (d) C1E5C 1, and (e) C1E6C 1.
316
Figure 2 indicates that the observed maximum of the gauche population is most likely a consequence of anomalous enhancement of the gauche population in a specific concentration region. It is also shown that this enhancement is more significant for shorter C1EmC 1 compounds. These observations suggest that there exist some specific interactions, which highly favor the gauche conformation of the O-CH2-CH2-O segment, for limited composition regions of the binary system of short POE chains and water. According to our preliminary results, the gauche population for C 1EmC1 (m = 2--4) in methanol increases and that for the same compounds in carbon tetrachloride decreases steadily with increasing solvent fraction without exhibiting ma~mum or minimum. 15 These experimental findings further support the importance of the specific interactions involved in the POE-water system. The mechanism of the anomalous enhancement of the gauche population for short POE chains in water is not clear at present. Possible effects relevant to the specific interactions may be associated with the conformational adaptation of the POE chain to the water structure and/or the amphiphilic behavior of the POE chain in water. A clue to the elucidation of the anomalous conformational behavior probably lies in the observation of the more significant effect for shorter chains and in the particular composition of the binary system at the point of the maximum gauche population. For a more comprehensible understanding of the structural aspect of the POE chain in water, more extensive experimental work and sophisticated theoretical treatment are necessary. References
1 F. E. Bailey, Jr. and J. V. Koleske, Poly(ethylene oxide) (Academic Press, New York, 1976). 2 S. Saeki, N. Kuwahara, M. Nakata, and M. Kaneko, Polymer, 17, 685 (1976). 3 G. KarlstrOm and B. Lindman, in Organized Solutions: Surfactants in Science and Technology, edited by S. E. Friberg and B. IAndman, pp 49--66 (Marcel Dekker, New York, 1992). 4 R. Kjellander and E. Florin, J. Chem. Soc., Faraday Trans. 1, 77, 2053 (1981). 5 R. E. Goldstein, J. Chem. Phys., 80, 5340 (1984). 6 G. Karlstr6m, J. Phys. Chem., 89, 4962 (1985). 7 H. Matsuura and K. Fukuhara, J. Mol. Struct., 126, 251 (1985). 8 H. Matsuura, S. Masatoki, M. Takamura, K. Kamogawa, and T. Kitagawa, to be published. 9 H. Matsuura and K. Fukuhara, J. Polym. Sci., Part B, 24, 1383 (1986). 10 H. Matsuura, K. Fukuhara, and H. Tamaoki, J. Mol. Struct., 156, 293 (1987). 11 R. J. Abraham, L. Cavalli, and K. G. R. Pachler, Mol. Phys., 11,471 (1966). 12 G. D. Smith, R. L. Jaffe, and D. Y. Yoon, J. Am. Chem. Soc., 117, 530 (1995). 13 M. Depner, B. L. Schttrmann, and F. Auriemma, Mol. Phys., 74, 715 (1991). 14 Y. C. Kong, D. Nicholson, N. G. Parsonage, and L. Thompson, J. Chem. Soc., Faraday Trans., 90, 2375 (1994). 15 H. Matsuura, unpublished results.
journal of MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 317-320
Time-Resolved Resonance Raman Study of the Recombination Dynamics Photodissociated Carbon Monoxide to Sperm Whale Myogiobin and Its Mutants Satoru N a k a s h i m i t , Teizo Kitagawa*, John S. O l s o n * * Department of Chemistry, Faculty of Engineering Science, Osaka University, Toyonaka 560, Japan *Institute for Molecular Science, Okazaki National Research Institute, Okazaki 444, Japan **Department of Biochemistry and Cell Biology, Rice University, Houston, Texas 77251, USA
Abstract Time-resolved resonance Raman (TR 3) spectroscopy was applied to study the recombination process of photodissociated carbon monoxide with recombinant sperm whale myoglobin (Mb) and its several mutants. The measurements were carried out, with various delay times (zxt=-20 ns - 1 ms) following photolysis for several mutants in which dital or E7His (His64) was replaced with a aliphatic or aromatic residue. The Fe-CO stretching bands (vFe-CO) showed that the environments around CO in the steady state as well as the kinetics of the CO rebinding process are very different among mutants and these differences seemed to be correlated with the hydrophobicity of the replaced residues. In spite of these differences for each mutant, a common intermediate species was observed at almost the same time region. In this intermediate CO was not bound to the iron ion yet, but the heme state was different from the initial deoxy form. Since this intermediate is common to all the mutants, the difference of the distal residues did not play an important role. The intermediate may it reflect the global dynamics of the protein, but correspond to so-called "open form" of Mb in which the His-64 swings out and a pathway for the ligand to enter the active site in general.
Introduction Mb is one of the highly characterized proteins and is considered to be a good model molecule for the analysis of the dynamics of the structural changes in general proteins. In attempts to elucidate the role of the protein dynamics toward their function, the recombination of photodissociated CO of carbonmonoxy myoglobin (COMb) has been extensively investigated over times that range from femtoseconds to seconds at various temperatures. 1-3 In x-ray crystallographic analysis of COMb it has been noted that there is no pathway for the migration of CO between the buried binding site and solvent. 4 Therefore, rapid rearrangements of the protein structures should be accompanied by recombination of CO. The transient absorption study on the relaxation process of photolysis product of COMb revealed that the recombination of the photodissociated CO was nonexponential. Fe-CO stretching band that appears in Raman spectra, is a good maker for this distal pocket environment. We pursued the time profile of this band to monitor the dynamics and found an intermediate species common to all the mutants. We will discuss the existence of the intermediate which correspond to "open form". 0167-7322/95/$09.50 'c' 1995 Elsevier Science B.V. All rights reserved. SSDi 0167-7322 (95) 00822-5
318
Experimental Wild type Mb (WT) and several mutants, where distal Histidine was replaced to aliphatic residues (His64Gly, His64Ala, His64Leu) and was replaced aromatic residues (His64Tyr, I-lis64Trp), double mutants (His64TyrLeu29Phe, His64TrpVa168Phe) and Leu29 mutant (Leu29Phe), were examined, s (Their abbreviations are H64G, H64A, H64L, H64Y, H64W, H64YL29F, H64WV68F, and L29F, respectively) Sperm whale Mb was isolated from stored meat and mutant sperm whale Mbs were expressed in E.coli and purified. Mbs solution (50mM Tris-HCl buffer at pH=8.0) were reduced by small amount of dithionite in the CO atmosphere and used in air-tight cell. Nanosecond pump/probe time-resolved resonance Raman experiments were carried out with two Nd:YAG lasers (t~t=7 ns) which provide the pump- (532 nm, 4 mJ) and probe pulses (416 nm, 100 gJ), a triple polychromator equipped with an intensified photodiode array, and a slow spinning cell (25rpm). For every sample, spectra were taken for delay times of At=-20, 0, 20, 100 and 500 ns, 1, 10, 100 and 500 ~ts, and 1ms. 6 To measure all these spectra in the same condition, spectra for each delay time were recorded quasi-simultaneously. The controller of the detector, the pulse generator and mechanical shutter were connected and managed by a personal computer. One spectrum was accumulated for 8 see. in one of the conditions and stored in the memory of the computer. Then the conditions were changed one after the other, and in each case it accumulated the spectra for 8 sec., which were recorded into the corresponding memories. After measuring all these conditions, it went back to the fast condition and repeated this sequence again, where each specman was added onto the corresponding spectrum on the memories. In all it repeated this procedure for 500 times, which was equal to about 12 h for whole accumulation time and to about 70 mills, for each spectrum.
Result and Discussion In the steady state, the VFr band appeared in a frequency range from 490 to 526 cm1. The VFe.COand the vco frequencies for each mutant observed by RR and IR measurements, respectively, are listed in table I. They are classified into three groups, A0, A1 and A3. This classification was originally based on the CO stretching 7 band, but is known that the VFc.CO frequency have a negative linear correlation with the vco frequency. H64G, H64A, H64L, H64W, H64Y and H64WV68F belong to A0, and the wild type to A1 and H64YL29F and L29F to A3 type. These results mean that the replacement of distal residues changes the heme environments and that there are at least three major types in the local field at CO in COMb. Table I. The VFe.CO, bFe-C-Oand VFe-COband of the mutant COMbs. (cm d) 9
VFe.CO
i
Wild Type
H64G
H64A
H64L
H64W
H64Y
H64W V68F
H64Y L29F
L29F
508
496
494
490
494
491
490
519
526
~Fe-C-O
579
-
570
571
550
550
-
579
582
VFe-CO
1945
1965
1966
1965
1969
] 966
-
-
1932
The results of TR 3 measurements are shown in Figure 1-4 for wild type with 12C160, wild type with 13C180, H64A and L29F, respectively. The CO rebinding kineti~ for each
319
mutant is correlated with the frequency of the VFe-COband. The lower the frequency of the VFe. CO band is, the faster the kinetics of CO rebinding becomes. We have reported 6 the linear correlation between the frequencies of the VFe-CO band and the hydropathy indeices of the replaced residues. It is empirically evident that the vre.co frequency becomes lower as the environment around CO becomes more hydrophobic. Therefore the rebinding kinetics is also affected by the hydrophobicity of the replaced distal residues; i.e. when the distal residue favors the presence of solvent water near itself, the rebinding kinetics becomes slow and v.v. In the case of the wild type, which belongs to A1 type, the equih'brium Fe-CO stretching band was identified in the spectrum for t~t=-20 ns (508 cm -1 for 12C]60 and 497 cm -1 for 13C180) and this band was absent in the spectrum for At=0 ns. A new broad band around 497 cm 1 started to appear from t~t=20 ns, which did not show the isotope shift. The VFe.CO band which reappeared from zxt=10 ~ts, was restored by 65% at 1 ms. Although in other two types of mutants the rebinding kinetics were very different from this A1 type, the intermediate band at 497 cm "1 was observed in almost the same time region and didn't show an isotope shift. It provides the evidence for the presence of a transient species, in which CO migrates into the heine pocket but not binds to the heine.
'qldType
m 410
"i
3441 ]'Wild Type
497
9
,.c,.o.o ,
i~
i '-,one__.., 20ns
i
700
-=
600
"L
3~s
J ill
~
V I=~ V ~
'
]
SO0
400
Raman Shift/cm "~
I! 700
,
1 600
I SO0
..
! 400
Raman Shift/cm-1
,
!
Figure 1. TR 3 spectra of wild type 12C]6OMb. Figure 2. TR 3 spectra of wild type 13C18OMb.
320
In spite of the such difference of the heme environments in the resting state for each mutants, there is a common intermediate in the rebinding process of photodissociated COMb. Since this intermediate appears at the initial stage of rebinding, it is considered that this is the precursor of the COMb form. In the time region where this intermediate appears, there is no band to show isotope shift so that CO does not bind to the heme in this stage but affects to the porphyrin to change its state slightly. As this intermediate is common to all the mutants, it is considered that the appearance of the intermediate reflects the (global) dynamics of the whole protein, but not the local dynamics. Therefore we propose that this intermediate can be so caged "open form".
H64A -20ns
I
r,,,
~
[ I L29F 379 t
526 I 497
581
lOOns llOOns
500ns
1lxs 1ixs
1opts
i I 700
I I I 600 500 400 Raman Shift/cm-1
Figure 3. TR 3 spectra of H64A COMb.
,4
700
600 500 400 Raman Shift/cm-1
Figure 4. TR 3 spectra of L29F COMb
References
1 Martin, J. L.; Migus, A.; Poyart, C.; Lecarpentier, Y.; Astier, R.; Antonetti, A. Proc. Natl. Acad. Sci. USA 1983, 80, 173-177. 2 Frauenfelder, H.; Slingar, S. G.; Wolyness, P. G. Science 1991, 254, 1598-1603. 3 Jackson, T. A.; Lim, M.; A.anfinrud, P. Chem. Phys. 1994, 180, 131-140. 4 Kuriyan, J.; Wilz, S.; Karplus, M.; Petsko, G. A.J. Mol. Biol. 1986, 192, 133-154. 5 Springer, B. A.; Sligar, S. G.; Olson, J. S.; Phillips, G. N., Jr.. Chem. Rev. 1994, 94, 699-714. 6 Sakan, Y.; Ogura, T.; Kitagawa, T." Fraunfclter, F. A.; Mattera, R." lkeda-Saito, M. Biochemisty 1993, 32, 5815-5824. 7 Li, T.; Quillin, M. L." Jr., G. N. P." Olson, J. S. Biochemistry 1994, 33, 1433-1446.
journal of
MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 321-324
Solvent Viscosity Dependence of Bimolecular Reaction Rate Constant of the Excited 9-Cyanoanthracene Quenched by 1,3-Cyciohexadiene Minoru Kitazawa, Tetsuro Yabe, Yoshinori Hirata, and Tadashi Okada Department of Chemistry, Faculty of Engineering Science, Osaka University, Toyonaka, Osaka 560, Japan Abstract The photocyclization reaction rate constant at the encounter in nonpolar solvents has been evaluated for the excited 9-cyanoanthracene-l,3-cyclohexadiene system by analyzing the transient effect in the fluorescence decay curves. The reaction rate decreased with increase of the solvent viscosity, while the increase of the reaction probability in the encounter complex was observed. The reaction dynamics was discussed in relation to the solute orientational motion in the solvent cage.
Introduction The bimolecular reactions in solution are considered to take place in the solvent cage by approaching with mutual diffusion. However, there has rarely been experimental studies to clarify the reaction dynamics between reactants in the solvent cage. The excited 9-cyanoanthracene (CNA) -1,3-cyclohexadiene (CHD) system is considered to be one of the best combinations to study the reaction in the solvent cage. CNA reacts in the excited singlet state with CHD to give the [4+4] cycloadduct with high quantum yield in solution as shown in Scheme 1. The produced [4+4] cycloaddut photodissociates very rapidly to the excited singlet state of CNA and ground state CHD by irradiation of UV lights as indicated in Scheme 21. After the photodissociation of the adduct, the geminate diffusive motion will start from some restricted conformation, probably face-to-face, and reproduce the adduct or escape from the cavity. We can observe this processes by measuring the time dependencies of CNA fluorescence after irradiation of UV laser pulse. Actually, we have observed non-exponential decay curves in some solvents. On the other hand, the fluorescence quenching reaction of CNA with CHD gives us the informations of the dynamics in the encounter pair of the system. In this report, we estimate reaction rate constant in the encounter pair as a function of the solvent viscosity by measuring the fluorescence decay curves with picosecond laser excitation and analyzing the transient effect in the fluorescence quenching process.
9
9 CN
400nm Emission Ex.
I
lMe mbna n
CN
Scheme 1. Bimolecular fluorescence quenching reaction.
~ CN
Scheme 2. Fluorescence quenching by a geminate recombination.
0167-7322/95/$09.50 r 1995 Elsevier Science B.V. All rights reserved. SSD! 0167-7322 (95) 00823-3
322
Experimental Fluorescence decay curves were measured by means of a time correlated single photon counting method using a mode locked Ti:sapphire laser (Spectra Physics, Tsunami) as an exciting light source. The repetition frequency of the excitation pulse was modulated by electro-optical modulator to be 1.6-8 MHz. CNA was excited by the second harmonics of the laser (400nm) under the condition of the polarization at the magic angle. The response time of the apparatus was 35ps fwhm. SAI~ program at the Osaka University Computation Center was used for the nonlinear least-squares analysis of the observed decay curves. All measurements were made at 20"c. Liquid paraffin and decalins were purified by column chromatography on activated silica gel and alumina. Viscosity of liquid paraffin was measured by Ostwald viscometer. Spectrograde n-hexane and cyclohexane were used without further purification. All solutions for the measurements were deoxygenated by flushing with nitrogen gas.
Analysis We use the time dependent "rate constant", k(t), obtained by solving diffusion equation under appropriate boundary conditions 2-4. l _ 1 k(t) (1 + tCS e x p ( x E ) e r f c ( x ) ) N ' (1) with
exp(-A2)dA
erfc(x)-
x .... '
, R
ko
where kR is the reaction rate constants, kD=4~tRDN' the diffusion rate constant, D the sum of the reactants' diffusion coefficients, R the specified encounter distance and N' the Avogadro number in milliliter. When x>> 1, exp(x2)erfc(x) converges to 1 exp(x2)erfc(x) -
and k(t) is given by 1
1+
(2)
The time dependent fluorescence intensity is given by l(t)-
I o e x p ( - a t - bt v2)
where a - zo ~ + 4~rR'DN'[CHD]
b- ~O)"~R'~N'[CnO]
(3) (4)
(5)
R' - k s R / ( k n + k s ) (6) "to is the fluorescence life time of CNA without CHD, and [CHD] the concentration of CHD. By means of eq.(3), we have simulated the observed fluorescence decay curves of CNA in the presence of CHD. In the course of the simulation, the time walk between the response function of the system and the fluorescence decay curve was decided to minimize the chi-square parameter. A small change of the time walk leads to a large difference in the fitting parameter b at the low concentration of CHD. R' and D values were obtained from the linear relation of a and b vs. [CHD] with the aid of eqs.(4) and (5).
Results and discussion Figure 1 shows the non-exponential decay curves of CNA and simulated results in n-hexane and liquid paraffin including 0.100M and 0.508M CHD, respectively. Fitting results seem to be quite well, which indicates that the conventional formula of eq.(3) may be acceptable at the observed time scale. The quencher concentration dependencies of a
323
(a) I
I
I
I
I
(b) a = 6.331x108
a = 1.886x108
b = 1.393x103
b = 9.320x103
--..,.
..
I
I
I
I
I
I
lOOch/div. (20.35ps/ch)
I
lOOch/div. (20.24ps/ch)
Figure 1. The fluorescence decay curve (dotted line) and its simulation with eq.(3) (solid line). (a): [CHD] - 0.100M in n-hexane, x 2 = 1.111, DW (Durbin-Watson parameter)- 1.662 (b): [CHD] - 0.508M in liquid paraffin, x 2 = 1.394, DW = 1.742
3.0xl0g ].[. i I I 2.5 ~o n-Hexane (n a:l
I /o
~ t t
2.0~-&trans'Decali~t 1.5
12x10 3
'o0
1.0
8 4
0.5
o.o k'r'-
J
i
0.0
0.1
0.2
~
~
-H
0.3 0.4
!
0.5
0.6
[CHD]/M
0 0.0
0.1
0.2
0.3
0.4
0.5 0.6
[CHD]/M
Figure 2. The parameter a and b as functions of [CHD] obtained by simulation with eq.(3) for some solvents indicated in the panels. Table 1. R' and D values of CNA-CHD system in different solvents obtained by analysis of fluorescence decay curves. Calculated D values by using empirical equations, D~I., and Stokes-Einstein equation, DSE, are also listed. solvents n-Hexane Cyclohexane trans -Decalin cis-Decalin liquid paraffin
R,/10-Scrn 1.44 2.28 2.65 2.31 5.67
DFl0-Scm-2s-1 4.77 2.04 1.07 0.939 0.0506
DCal./10-Scm-2s-1 6.53" 2.09* 1.24 * 0.778* 0.0935**
DSE,/10-5cm2s-1 4.73 1.53 0.706 0.445 0.00953
* From 9 Wilke-Chang correlation, D~215 10"8(0MB)I/2T/TIBVA0-6 ,where D~ "1 is mutual diffusion coefficient of solute A in solvent B, MB /g-mol "] is molecular weight of solvent B, riB/cp is viscosity of solvent B, VA/cm3"mol "j is molar volume of solute A at its normal boiling temperature, and r is association factor of solvent B. ** 9From Hayduk and Minhas correlation, D~ -0-71 where E-(10.2/VA)-0.791.
324 and b in same solvents are indicated in Figure 2. The lifetime of CNA without CHD in each solution agreed very well with the value obtained from the intercept at [Q]-0 of a vs. [CHD] plot. R' and D values are summarized in Table 1 with calculated D values using empirical equations, Wilke-Chang and Hayduk-Minhas correlations 5. We have also estimated D values by using Stokes-Einstein equation, DSE in Table 1. The mutual diffusion constant obtained from Stokes-Einstein equation shows a large difference with the values obtained from the measurements of fluorescence quenching reaction and also empirical equations especially in viscous solvents. By using the values of R' and D, we estimated the reaction rate constant, kR, and reaction probability in collision complex, rp, when we assumed the van der Waals radii of CNA and CHD estimated from space-fitting molecular model to be 5.0 A and 2.0 ,~,, respectively, that is R=7.0 ,A, in eq.(6).The obtained rp, kD and kR values are listed in Table 2. Table 2. rp, kD and kR values obtained from R' and D assuming R=7.0/~,. solvents n-Hexane Cyclohexane trans-Decalin c/s-Decalin liquid paraffin
rp 0.206 0.326 0.379 0.330 0.810
kD/109s "IM1 25.3 10.8 5.66 4.97 0.268
kR/109s "1M'I 6.56 5.22 3.45 2.45 1.14
It should be noted here that kR decreases with decrease of kD while rp increases. In the fluid solvents such as n-hexane, the kD value is much larger than the value of kR and rp is relatively small, whereas rp is larger and kD is small compared to kR in liquid paraffin. These results may be explained probably as follows. The conformational orientation between the excited CNA and CHD should be restricted very much to produce a photocycloadduct in the collision complex indicated in the scheme 1. In the fluid solvents like hexane, the rotational relaxation times of the solute molecules are rather fast compared to the reaction rate, which increases the escape probability of the reactants from the solvent cavity due to the large value of kD. On the other hand, the transit time in the reactive conformation, probably symmetrical face to face, may be longer in the liquid paraffin. This means that the observed kR may be expressed as a function of the mutual rotational relaxation time of reactants and the real reaction rate in the face-to-face conformation. In this sense, it is very important to make precise time-dependent measurements in the course of geminate recombination reaction indicated in Scheme 2, because the initial conformation after photodecomposition of cycloadduct is considered to be close to the face-to-face conformation. The studies on the geminate processes of the system in solution by the time resolved spectroscopy are now progress in our laboratory.
Acknowledgments This work was partly supported by a Grant-in-Aid (06NP301 and 06453025) to T.O. from the Ministry of Education, Science, and Culture of Japan.
References 1 2 3 4
T. Okada, K. Kida, and N. Mataga, Chem. Phys. Lett., 88, 157 (1982). F . C . Collins and G. E. Kimball, J. Colloid Sci., 4,425 (1949). R. M. Noyes, Progr. React. Kinetics, 1, 129 (1961). S. Nishikawa, T. Asahi, T. Okada, N. Mataga, and T. Kakitani, Chem. Phys. Lett., 185, 237 (1991). 5 R.C. Reid, J. M. Prausnitz and B. E. Poling, The Properties of Gases & Liquids, Fourth Edition p597 (McGraw-Hill Book Company, New York).
journal o f
MOLECULAR
LIQUIDS
Journal of Molecular l.iquids. 65/66 (1995) 325-328
ELSEVIER
Extremely Slow Reorientation Dynamics of Molecular Tracers in Glassy Polymers Qui Tran-Cong, Hirotaka Kanato and Shinya C h i k a k i Department of Polymer Science and Engineering Kyoto Institute of Technology Matsugasaki, Sakyo-ku, Kyoto 606, Japan
Abstract A n e w experimental method based on the polarization-selective photochromic reactions is proposed to monitor extremely slow reorientation dynamics of molecular tracers in glassy polymer matrix. The correlations between the local relaxation processes of polymers and the reorientation dynamics of the tracers with different sizes are found from the experimental results obtained by this method. I. Introduction Reorientation dynamics of molecular tracers in polymers is not only important for the understanding of slow relaxation phenomena in glassy polymers 1, but plays also a critical role in practical problems such as molecular design of nonlinear optical materials with long-term stability based on dyes/polymers complexes 2. We s h o w here the reorientation dynamics of molecular tracers in glassy polymers obtained by the annealing-after-irradiation method described below. These experimental results are compared to the local relaxation processes of glassy polymers obtained by the already established measurement techniques such as dielectric relaxation and solid state NMR. Finally, the molecular interpretation of the relaxation of free-volume distribution in polymers will be discussed. II. Experimental Section A) S a m p l e s . The tracer molecules are 9-hydroxymethyl-10-{(naphthylmethoxy) methyl} anthracene 4 (HNMA, Fig. l-a)and tetraethyl {3.3}(1,4) naphthaleno (9,10) anthracenophane-2,2,15,15-tetracarboxylate5 ( C y c I o p h a n e, F i g. 1 - b ) . T h e intramolecular photodimerization of these molecules is induced by exciting the anthracene moieties with linearly polarized light at 365nm. The polymer matrix is poly(methylmethacrylate)(PMMA, Mw = 1 . 4 4 x 105, Mw /Mn = 1 . 7 ) . T h e microstructures of this polymer are 58.9% syndiotactic, 35.4 % heterotactic and 5.7% isotactic) as observed b~ 1H NMR. The tracers were doped in these PMMA films at the concentration ca.10"~M by using solvent casting method. R : CH=OH
R : COOC2Hs
/ C
0
H
hv
R
hv
~
~
-"
>
~hv',A
h e , A ~
Hg.l(a) Chemical structure of HNMA.
Fig.l(b) Chemical structure of Cyclophane.
B) Data A n a l y s i s . Irradiation was performed with a high pressure mercury lamp (250W, Ushio Electrics Co.). The two dichroic absorbance components of HNMA and Cyclophane at 393.5nm and 402nm were monitored at room temperature after irradiating the sample over different time intervals with linearly polarized light at 365nm. The reorientation dynamics of these molecular tracers were obtained from the analysis of the time-dependent induction efficiency rl0) which is defined as4: r/(t) = 100. { O D • (t) - OD,, (t)}/ O D o (1) 0167-7322/95/$09.50 r 1995 Elsevier Science B.V. All rights reserved. SSDI 0167-7322 (95) 00824-1
326 where OD_t_ (t) and OD II (t) are the absorbance measured in the directions perpendicular and parallel to the polarization of the exciting light, respectively and ODo is the initial absorbance. Experimental data were fitted to an approprmte model function by using a non-linear least square regression program. C) Results
and Discussion.
Upon irradiation with linearly polarized light, the induction efficiency aq(t) increases and reaches a maximum which strongly depends on the experimental temperatures 4. This behavior can be explained as the result of the competitions between the reorientational relaxation and the selective photodimerization of the molecular tracers6. The independent contribution from the reorientational relaxation to rl(t) can be extracted by using the annealing-after-irradiation described hereafter. At first, PMMA films doped with these tracers were irradiated until rl(t) reaches its maximum magnitude rlmax. Subsequently, irradiation was stopped and the sample was annealed at seven temperatures ranging from 40"C to 100"C in the dark. Due to the reorientation of the tracer, rlmax decays with annealing time. This relaxation process can be analyzed by fitting the experimental results to an appropriate decay function. Fig. 2 shows some typical decay curves of the rlmax of H N M A obtained by the annealing-after-irradiation. The change of+lrnax w i t h annealing time can be well expressed by the following function: +7(t) = (+7(0) - +7** ) exp(- kr t ) + +7** (2) where rl(0) is the induction efficiency before annealing, r100 is the equilibrium value of rl(t) obtained at long time of annealing and kr is the reorientational relaxation rate of HNMA. On the other hand, as shown in Fig. 3, the decay curves of the Tlmax obtained for Cyclophane are well fitted to the following equation: +7(t) - A (F1 e -•" + F2 e -k2, ) + +7** where A = (11(0) - rl|
(3)
) is the pre-exponential factor expressing the contribution of the
restricted reorientational process of cyclophane to the decay of +l(t) and kl, k2 are the decay rates of Vl(t) with the corresponding fractions F1 and F2. ~ t
i
i
1~4mA
i
i
Lr
I" "/.Oo - +ct
~
+ + + o
io
wZ /,.
cyclol~hllml i
,
+ 1so Annemiinl T i m e (rain)
i-ig.2 Annealing time dependence of the i.rlducllon efficiency rl0) of H N M A in P M M A at different temperatures.
T-rrz '~ ! o
so
i
!
i slo
Annealing Time (rain)
,]+
l ~,oo
Fig.3 Annealing tame dependence of the indu~ion efficiency rl(t) of Cyclophane m P M M A at dif~rent temperatures.
The solid curves depicted in Figs.2 and 3 are obtained by the fitting using Eqs. (2) and (3), r e s p e c t i v e l y . For both H N M A a n d Cyclophane, the presence of the temperature-dependent base line rloo i m p l i e s that the tracers undergo restricted reorientation in the glassy state of PMMA. A measure of this restriction can be expressed by the quantity +loo/rl(0) whose temperature dependence is shown in Fig.4.
327
The restrictions imposed on Cyclophane are stronger than on HNMA over the whole range of the experimental temperatures, suggesting the size effects on the rotation dynamics of molecular tracers in polymeric glass. The details of this restriction can be clearly seen in Fig.5. Here, the reorientation of Cyclophane and HNMA is assumed to follow the wobbling-in-cone model which has been proposed to analyze the rotational diffusion of fluorescent probes in lipid bilayers 7. Upon approaching the glass transition temperature (Tg = 110"C ) of PMMA, the cone angle 0max which is a measure of the free volumes surrounding the tracers, increases with increasing temperature and is almost 70* at 100*C as seen in Fig.5 . On the other hand, the temperature dependence of the reorientational relaxation rates kr of HNMA and kl, k2 of Cyclophane follows the Arrhenius behavior within the temperature range of this
1.0
|
,!
|
!
l 14NMA
Cyclophane I
]
0.8
0.8
~40d
I
0.4-
0.2-
'o
4;
o.o
0
8 lOO Temperature ('C) Fig.4Temperature dependence of 11,,/riO).
40
6
lo
100
]
'*' 9
HNI~I~ I PMMA
eye.bane
-""
,~
o.1
..................
"ck l o -~
"-
"" "" " A . . , ~ ~ . "-..
l O .2
1.?0
.
i
a.*o
,
I
lo*
.5
4"
o.os
I
SO
Temperature ( ' C ) Fig.5Temperature dependence of Omn.
|
:.oo
,
I
,
i
3.10
.
i
3.20
I / T xlO3 OK'I) Fig.6 Temperature dependence of reorientational rate constants of H N M A in P M M A . (m):lS-relaxation (dielectric relaxation); (----) : y-relaxation ( C P / M A S 1 3 C N M R ) ; For comparison , the relaxation frequencies of these data have been shifted by an appropriate amount along the ordinate.
10-~.70
,
I 2.80
,
I 2.90
A
I
3,00
1 / T x l 0 3 0K'~) Fig 7 Temperature dependence of reorientational rate constants of cyclophane in PMM.A.(--):lS-relaxation (dielectric relaxation); For comparison, the relaxation frequencies of these data have been shifted by an appropriate amount along the ordinate.
328 work. It was found that the activation energy Ea of kr of HNMA is 7.8kcal/mol and is close to the T-relaxation process of PMMA (rotation of the ot-CH3 groups) which has been directly observed by CP/MAS 13C N M R as depicted in Fig. 6 8. F o r Cyclophane, the values Ea of kl and k2 are in the range 16.9kcal/mol - 22.8kcal/mol. These values are close to the activation energy of the 15-process of PMMA (rotation of the -COOCH3 groups) observed previously by dielectric relaxation as illustrated in Fig. 7 9. The above results suggest that the difference in the "best fit" model functions (2) and (3) used in the curve fitting reflects the existence of a free-volume distribution in the sample. As shown in Figs. 6 and 7, the reorientational relaxation of molecular tracers with different sizes is affected by the local relaxation of different segments. In other word, the rearrangements of the free-volume distribution in the glassy state of polymers are driven by the local relaxation of polymer chains. Finally, the equilibrium value rloo probably corresponds to the reorientational relaxation processes of HNMA and Cyclophane in the regions with the free-volumes smaller than their sizes. It is of great interest to compare these results with the temperature dependence of the free-volumes data obtained by positron annihilation techniques. These experiments are currently in progress. In summary, we have demonstrated that extremely slow reorientation dynamics of molecular tracers in glassy polymers can be observed by the annealing-after-irradiation method proposed in this work. The correlations between the rotational motions of the tracers and the local relaxation of polymer matrix can provide an insight into the molecular interpretation of the relaxation of free-volume distribution in glassy polymers. Furthermore, the experimental results obtained here might give some useful guiding principles for the design of nonlinear optical polymers with long-term stability.
Acknowledgements. Q. T.-C. gratefully acknowledges the research fundings from the Ministry of Education, Science and Culture, Japan (Grant-in-Aid No. 06239238), and the Ogasawara Foundation for the Promotion of Science and Engineering. H. K. thanks Nippon Paint Co. for the scholarship.
References 1) See, for example, K.L. Ngai in "Non-Debye Relaxation in Condensed Matter" (World Scientific, Singapore, 1987). 2) M. Eich, B. Reck, D.Y. Yoon, C.G. Wilson and G.C. Bjorklund J. A p p i . P h y s . 66, 3241 (1989). 3) Q. Tran-Cong "Polarization-induced Photochromic Reactions m Polymer Solids " in "The Polymeric Materials Encyclopedia, Synthesis, Properties and Applications", CRC Press, to appear. 4) Q. Tmn-Cong, T. Kumazawa, O. Yano and T. Soen M a c r o m o l e c u l e s 23, 3002 (1990). 5) D. H. Hua, B. Dantoing, P.D. Robinson, Q. Tran-Cong, C.Y. Meyers Acta C r y s t . C 50 , 1090 (1994). 6) Q. Tran-Cong, N. Togoh, A. Miyake and T. Soen M a c r o m o l e c u l e s 25, 6568 (1992). 7) K. Kinosita, Jr., S. Kawato and A. Ikegami B i o p h y s . J. 20, 289 (1977). 8) F. Horii, Y. Chen, M. Nakagawa, B. G a b r v s a n d R. Kitamaru Bull. I n s t . Chem. Res. Kyoto U n i v e r s i t y 66, 317 (1088). 9) J.L.GomezRibelles, R. DiazCalleja J. P o l y m . Sci. P h y s . E d . 23, 1297 (1985).
journal of MOLECULAR
LIQUIDS ELSEVIER
Journal of MolecularLiquids,65166(1995) 329-332
Dynamics near a Liquid Surface: Mechanisms of Evaporation and Condensation K. Y a s u o k a , M . M a t s u m o t o , and Y. K a t a o k a " D e p a r t m e n t of A p p l i e d P h y s i c s , N a g o y a U n i v e r s i t y , F u r o , C h i k u s a - k u , N a g o y a 464-01, Japan *Department of Materials Chemistry, Hosei U n i v e r s i t y , K a j i n o 3-7-2, K o g a n e i 184, J a p a n
Abstract The rates of evaporation and condensation under the vapor-liquid equilibrium condition are investigated with a molecular dynamics computer simulation for argon at 80K, methanol at 300 K, and water at 400 K. The molecular reflection at the surface, which is related to the condensation coefficient a, is divided into two parts, self reflection and molecular exchange. There is no significant difference among the three substances concerning the ratio of self reflection to collision. The total ratio of reflection is estimated as a ~ 0.8 for argon, 0.2 for methanol, and 0.4 for water. The ratio of molecular exchange is much larger for methanol and water than for argon, which suggests that the conventional assumption of condensation as a unimolecular process fails for associating fluids. I. Introduction Evaporation and condensation are fundamental and important in various fields of science and engineering. Although a number of experimental and theoretical studies have been made on evaporation and condensation rates for decades, there still remain many problems. Although the collision rate is easily estimated with the kinetic theory of gas dynamics, all incident molecules are not necessarily caught on the liquid surface, and thus the condensation coefficient a, which is the ratio of the observed condensation rate to the ideal condensation one, becomes important. It is usually assumed that c~ is very close to unity (complete condensation) for many fluids, 1 but there has been controversy over a of associating fluids (e.g., water and alcohols). This arises from the experimental difficulty in estimating the absolute rate of condensation or evaporation under the vapor-liquid equilibrium condition. A transition state theory has been applied to calculate a under the assumption that the condensation process is a unimolecular chemical reaction. 2 It predicts a ~ 1 (complete capture) for simple fluids such as argon. In the case of associating fluids, however, much smaller a is expected due to the potential barrier ca.used by the rotational restriction of liquid molecules. Since the experimental results of c~ widely scatter, 1 we still have to check with great care the validity of assumptions made in the theoretical treatments. Computer simulations with molecllla,r dynanaics (~iD) technique are very suitable to investigate these evaporation-condensation phenomena at a. molecular level. Thus, we have carried out MD simulations for three systems, argon (a typical simple fluid) a,t T = 80 K and two associating fluids, methanol at T - 300 [( and water at T = 400 K. In this paper, we briefly describe the results of our simulations and data. analyses, and discuss the differences among the three from the viewpoint of dynamics a,t the surface. 0167-7322/95/$09.50 (c) 1995ElsevierScience B.V. All rightsreserved. 0 !67-7322 (95) 00892-6
SSDI
330
i_/
Fig. 1 A snapshot of the simulation (Argon at 80 K).
II. Simulation Technique We adopted a microcanonical ensemble MD method. The Lennard-Jones potential was used for the molecular interaction in the argon, OPLS potentials for methanol and water. We make a liquid slab with thickness of about 10 molecules a.t the center of the rectangular unit cell with periodic bo~indary conditions for all three dimentions. Figure 1 is a snapshot of a typical molecular configuration. Both sides of the slab are free liquid surfaces, on which molecules can evaporate and condense. The cell size along the surface normal is typically 100 A. and the surface area is 50 ]k x50 ~. The number of molecules is 1200 for argon, 864 for methanol, and 1024 for water. Other technical details are described elsewhere, a,4 III. Results and Discussion After equilibrating the system, we a ccllmulated the configurations for 997 ps (argon), 300ps (methanol) or 375 ps (water), from which we study dynamic properties concerning evaporation-condensation processes. Self condensation In most of studies so far (experimental as well as theoretical) on the condensation coefficient c~, the condensation process l~as I)een considered as a unimolecular chemical reaction, which means that the evaluation o t ' a is equivalent to estimating how many vapor molecules are reflected after colliding wit.h the liquid surface. In order to avoid confusion, let C~s,if be the condensation coefficient from this point of view, since we show later that the assumption of the unimolecular reaction is llot always correct. In principle, we can evaluate c~sr by direct counting of collision and reflection events from the molecular trajectory data; for both ca.ses, we observed sufficient number (about 200) of collision events. To estimate o'.~lf quantitatively, we have developed an autocorrelation function method. 5 The result is CYself ~ 90 % and there is no qualitative difference among various types of fluids, which suggests that the condensation as a unimolecular chemical reaction is a barrierless ~process. Calculation of the local chemical potential also supports this barrierless pictlire.
Total c o n d e n s a t i o n In visualizing tile nlolecl,lar configllrations, we sometimes observed that vapor molecules colliding with tile liquid s~lrfacc apparently drove other molecules out of the liquid. This kind of correlation bet,weell the conr tltlx and the ex,aporation flux, or ':molecular exchange," lnay llave a significailt effect oil t,lle rate of condensation. Following
331 Table 1 Comparison of condensation behaviors estimated from the simulation data. Argon (T=80K)
Methanol (T=300K)
Water (T=400K)
Self (t asr Exchange Total (1 a)
0.12 0.08 0.20
0.14 0.66 4- 0.10 0.80 4- 0.10
0.05 0.55 4- 0.05 0.60 4- 0.05
Condensation coefficient a
0.80
0.20 4- 0.10
0.40 4- 0.05
Ratio of reflection
1.5
15"-5 Water
E
o
T=400
--" "
1.0
o
K
m o
E
bonding non-bonding
e~
0.5
8
5-6
o.o bonding 9
>,
"" -0.02 r ._o ...., .__ -0.04 v 0
1]
0z
~ o.oo 8
El
L i q u i d I _ Surface region ,
,
10
z
n o n - b o n d i n g I~
9 ,
(h)
_ I
-I-
20
/
Vapor .
]
-I
30
Fig. 2 Surface of liquid water at T = 4 0 0 K. Density profile and number of neighboring molecules (top) and the kinetic energy correlation (bottom). a general framework of chemical reaction theory, we have developed a memory function method to estimate the phenomena. T The total condensation coefficient a is calculated by using the correlation of colliding and evaporating fluxes with a memory function. The estimated value is _~ 0.80 for argon, ~_ 0.20 + 0.1 for methanol, and ~ 0.40 + 0.05 for water. Comparing a with asar, we can also evaluate the ratio of the molecule exchange. The results are summarized in Table I. The ratio of self reflection and molecular exchange is very similar for argon. However, in the case of associating fluids, the ratio of the exchange is several times larger than that of the self reflection. This implies that the condensation cannot be regarded as a unimolecular process, especially in the case of associating fluids. Heat transfer by hydrogen bonds A key point to understand the above difference between simple fluids and associating fluids is the mechanism of molecular heat transfer accompanying tile evaporation and condensation at the surface. If the energy is not transferred (or dissipated) fast enough, the hitting molecule cannot lose the extra energy to condense itself, and will finally re-
332 evaporate, which is the self-reflection. On the other hand, the surface molecules receive the extra energy from the hitting molecule, and if the amount of the received energy is too much, the surface molecule may get evaporated in exchange of the condensation of the hitting molecule, which is the molecular exchange. Since it is difficult to evaluate the precise local heat conductivity from the simulation data, we instead calculate the kinetic energy correlation between neighboring pairs; if a large amount of heat is transfered between a pair of neighboring molecules, there should be a strong correlation between the kinetic energy fluctuations of each molecule. A part of the results is shown in Fig. 2, where the kinetic energy correlations are shown for hydrogen-bonding pairs and non-bonding pairs in the case of water. The correlation is several times stronger for bonding pairs than non-bonding ones; a similar result is obtained for methanol. In the case of argon, the correlation is similar to that between non-bonding pairs of associating fluids. From these evidences, we can easily imagine what happens in the condensation process of associating fluids: The latent heat released in the instance of condensation is rather large, but is transferred efficiently by the strong hydrogen bonds. However, since the number of bonding pairs is only 1 or 2, the neighbor becomes easily evaporated by the transferred heat, and thus the molecular exchange is often observed. Further analyses are now under way. Summary Using a molecular dynamics computer simulation, we have investigated the evaporation-condensation dynamics for argon, methanol, and water. The ratio of condensed molecules to surface-colliding ones, a,~jl, is about 90~7c for all substances, although surface properties of methanol and water are very different from those of argon. However, the total condensation coefficient c~ estimated with flux correlation of evaporation and condensation is much smaller for methanol and water than for argon. The difference between a and a,~ll is caused by molecular exchange at the surface. In the case of methanol and water, the molecular exchange is a dominant factor to determine the condensation-evaporation rate. A rough sketch for the difference among the species is given from the viewpoint of heat transfer near the liquid surface.
Acknowledgments Thanks are also due to the Conlputer Center, Institute for Molecular Science, for the use of their colnputer facilities. This work has been financially supported in part by Grant, s in Aid for Scientific Research (No. 0523901.5) from tile Ministry of Education, Science alid Culture, Japan
References 1 2 3 4 5 6 7
H.K. Cammenga, CurT~ent Topics in ~laterials Science, ed. bv E. Kaldis (NorthHolland, Amsterdam, 1980), \1ol. 5, and references therein. S. Fujikawa and M. Maerefat, JSME II, 33,634 (1990). K. Yasuoka, M. Matsumoto, and Y. I(ataoka, 3. Chem. Phys., 101, 7904 (1994). M. Matsumoto, K. Yasuoka, and Y. I,~a.taol~a, J. C]lem. Pllvs., 101, 7912 (1994). M. Matsumoto and Y. Kataoka., Nlol. Silnlllatiotl. 12, 211 (199,1). K. Yasuoka, M. Matsumoto. ;ttld Y. I(ataoka, l$~lll. (',lleln. Soc..Jpn., 67, 859 (1994). M. l~ia.tsumoto, I,/. Yasuol~a, a llcl Y. I{ataoka, "l'llerln. Sci. Eng., 2, 6,1 (1994).
journal of
MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 333-336
The Effect of Ionic Atmosphere on the Tracer Diffusion of Miceiles Toshihiro Tominaga and Masayuki Nishinaka Department of Applied Chemistry, Okayama University of Science, 1-1 Ridai-cho, Okayama 700, Japan Abstract By use of the Taylor dispersion method, diffusion coefficients for pyrene solubilized in micelles of octadecyltrimethylammonium chloride (C 18TAC) and tetradecyltrimethylammonium bromide (C14TAB) have been measured in aqueous NaCI and NaBr solutions, respectively, at 35 ~ These values can be regarded as tracer diffusion coefficients for the micelles because essentially all pyrene molecules are solubilized in the micelles. In the range 0.24 < ~r <1, where x "1 is the Debye screening length and a is the radius of the micelle, diffusion coefficients at the critical micelle concentration (cmc), Dcmc, decrease with With further decrease in ~ca from 0.24 to 0.18, the decreasing ionic strength, i.e., ~r Dcmc value does not decrease. This fact suggests that the drag of ionic atmosphere reaches maximum when ~a becomes ca. 0.2.
I. Introduction We have recently reported tracer diffusion coefficients for micelles of decyl-, dodecyl-, tetradecyl-, and hexadecyl-trimethylammonium bromide (C10TAB, C 12TAB, C 14TAB, and C16TAB, respectively) in aqueous NaBr solutions. 1 In the range 0.3 <~r a < 1.0, where x -1 is the Debye screening length and a is the radius of the micelle, the tracer diffusion coefficients decreased with decreasing ionic strength, i . e . , x a . Some studies show that selfdiffusion coefficients of latex particles decrease with decreasing ionic strength. 2, 3 Schurnacher and van de Ven 4 have reportexi that diffusion coefficients of highly dilute solutions of gold sol (20nm) and latex particles (40 rim) show minima at ~r - 1. They have explained their results by using a theory for the ,sedimentation velocity. 5, 6 This theory predicts that the sedimentation velocity has a minimum at aa~ - 1 when the ~"potential is constant while it has a minimum at la~ - 0.25 when the particle charge is constant. In this study we have measured tracer diffusion coefficients of octadecyltrimethylammonium chloride (CIsTAC) micelles in water and aqueous NaCI solutons at 35 oC. Due to its low critical micelle concentration (cmc), the smallest ~r a value is as low as 0.18. We have also determined tracer diffusion coefficients for C14TAB micelles at 35 ~
II. Experimental C18TAC (Tokyo Kasei, >97 %) was dried in vacuum, was dissolved in hot ethanol under nitrogen atmosphere, and was recrystallized by adding acetone and cooling. C 14TAB (Tokyo Kasei, guaranteed reagent), NaCI, and NaBr (both Nacalai Tesque, guaranteed reagent) were used as received. Pyrene (Wako Pure Chemical Ind.) was purified by sublimation. Water was distilled twice. Diffusion coefficients were measured by the Taylor dispersion method. 7 Other details have been described elsewhere. 1
III. Results and Discussion 0167-7322/95/$09.50 "C)1995 ElsevierScience B.V. All rights reserved. SSD! 0167-7322 (95) 00825-X
334
Figures 1 and 2 show diffusion coefficients of pyrene solubilized in C 18TAC and C 14TAB micelles, respectively, at 35 oC. Because essentially all pyrene molecules are solubilized in the micelles, the diffusion coefficients can be interpreted as tracer diffusion coefficients of the micelles. Diffusion coefficients decrease with increasing concentration of the micelles, and increase with increasing concentration of the salts added. Diffusion coefficients at cme's, Dcmc, were obtained by extrapolation, and are listed in Table 1
0.9
E
in water
a
0.0003MNaCI
zx 0.001MNaCI
r O4
o
v
0.002MNaCI
i
o
0.005MNaCI
~ 0.7
,~ 0.01MNaCI
0
0.8
o
*
0.6 0
0.01
0.02
0.02MNaCI
0.03
018TAC / M Figure 1. Tracer diffusion coefficients of CI8TAC micelles in aqueous NaCl solutions at 35 oC. Cmc's are very close to zero in this abscissa scale. 1.2
r
1.1
i
o
in water
8
0.005M NaBr
O4
E
O
zx 0.01M NaBr
i
o
0.9
-
(
v
0.02M NaBr
o
0.1M NaBr
<~ 0.15M NaBr 0.8 0
0.05
0.1
C 14TAB / M Figure 2. Tracer diffusion coefficients of CI4TAB micelles in aqueous NaBr solutions at 35 ~ Arrows indicate cmc's.
335
Table 1. Tracer diffusion coefficients and Stokes radii at the cmc in aqueous NaCI or NaBr solutions at 35 o c [NaX] / mM a
cmc / mM
0 0.3 1 2 5 10 20
0.29 0.20 0.10 0.07 0.03 0.02 0.01
0 5 10 20 100 150
3.8 3.5 2.0 1.3 0.3 0.2
Dcmc / 10 "10 m 2 s -1 C18TAC
a emc / nm
0.799 0.797 0.839 0.867 0.923 0.955 0.955
3.92 3.93 3.74 3.62 3.40 3.29 3.29
1.043 1.100 1.128 1.156 1.132 1.138
3.01 2.85 2.78 2.71 2.77 2.76
C14TAB
a NaCI for C 18TAC and NaBr for C14TAB.
together with cmc values and Stokes radii, acmc, calculated from acmc = k T 1 6~ r/D cmc
(1)
where k is the Boltzmann constant, T the absolute temperature, and r/the viscosity of the medium. Booth 5 derived an expression for the sedimentation velocity. It is expressed as a power series in either the particle charge Q eft or the ~ potential
U = Uo[ 1 - c Q eli 2] = U 0 [1 - d ( e UkBT )2]
(2)
where U 0 is the sedimentation velocity for uncharged particles, and c and d are products of the scaled drag coefficient for small ions and complicated functions of r a . Recently, Ohshima eta/. 6 derived more general expressions for the sedimentation velocity. For small charge density case, they have given an analytical expression as a power series in ~S as in eq. 2. Using a relationship between ~ and Q eft their results can also be expressed as a power series in Q eft
Both of these theories predict minumum value of U at Ka =1 when ~" is
constant, and at Ka = 0.25 when Q eft is constant. In Figure 3, Dcmc values are plotted as a function of ra. In all cases including our previous results for CI4TAB at 25 ~ and C16TAB at 35 ~ values decrease with
336
i
a
1.1
E e 0
Figure 3. Tracer diffusion coefficients at the cmc as a function of
~ o 0.9 E u a
0.7
. . . .
I
. . . .
1
I
2
. . . .
I
,
,
ru.. O, C14TAB at 35 ~ El, C16TAB at 35 ~ (ref. 1); A, C18TAC at 35~ O, C14TAB at 25 ~ (ref. 1).
3
=:8
decreasing value of ra when the latter is less than one. In other words, the condition of constant ~" potentilal does not hold, but the condition of constant particle charge holds much better in the present systems. In the case of CIsTAC, the smallest two ma values are 0.18 and 0.24 corresponding to 0 and 0.3 mM NaCl, respectively. It is quite interesting to find that the Dcmc valuse does not decrease when ~ca decreases from 0.24 to 0.18. 8 This fact suggests that the drag of ionic atmosphere is the largest when its thickness is 4 to 5 times larger than the radius of the micelle. Although this was predicted by the theories, 5, 6 as far as we know, this is the first experimental observation for micellar systems. The authors thank Professor M. Kodama for letting us use the vacuum line.
References 1 2 3 4
5 6 7
8
T. Tominaga and M. Nishinaka, J. Chem. Soc., Faraday Trans., 89, 3459 (1993). T. Okubo, J. Phys. Chem., 93, 4352 (1989). S. Gorti, L. Plank and B. R. Ware, J. Chem. Phys., 8 1 , 9 0 9 (1984). G. A. Schumacher and T. G. M. van de Ven, Faraday Discuss. Chem. Soc., 83, 75 (1987). F. Booth, J. Chem. Phys., 22, 1956 (1954). H. Ohshima, T. W. Healy, L. R. White and R. W. O'Brien, J. Chem. Soc., Faraday Trans. 2, 80, 1299 (1984). G. I. Taylor, Proc. Roy. Soc., A219, 186 (1953); A225, 473 (1954); A. Alizadeh, C. A. Nieto de Castro and W. A. Wakeham, Int. J. Thermophys., 1,243 (1980); H . J . V . Tyrrell and K. R. Harris, Diffusion in Liquids. A Theoretical and Experimental Study, Butterworths, London, 1984, p. 193. Note that small ion drag coefficient is larger when g'a--O. 18 than when g'a---0.24 because in the former case small ions are 0.29 mM C 18TAC and in the latter case they are mixtures of 0.20 mM C 18TAC and 0.30 mM NaCI.
journal of
MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids. 65/66 (1995) 337-340
Effect of Addition of Surfactant on the Miceile-Monomer Exchange Process Tatsuro Matsuoka, Tetsuya Shibata, Shinobu Koda and Hiroyasu Nomura Department of Chemical Engineering, School of Engineering, Nagoya University, Nagoya, 464-01, Japan Abstract Effects of addition of n-tetradecyltrimethylammonium bromide (Ct~TAB) on the micellemonomer exchange processes of n-decyltrimethylammonium bromide (C~0TAB) were investigated by the ultrasonic relaxation method. The relaxation frequency increased and the relaxation strength decreased with increasing amount of C~4TAB added. The dependence of the relaxation frequency on the amount of C~4TAB added was in fair agreement with the relaxation theory of Annianson for mixed micelle. I. Introduction The micelle-monomer exchange process has been extensively investigated by many investigators using several chemical relaxation techniques. Using the wide-band ultrasonic relaxation technique, we have investigated alkyltrimethylammonium bromides derivatives which are different in the alkyl chain length by the factor two and discussed the effect of alkyl chain length on the kinetic constant and volume change at the miceUe-monomer exchange reaction. ~ In this study, we investigate how addition of a surfactant affects the micellemonomer exchange process.
H. Experimental a) Materials n-decyl- and n-tetradecyltrimethylammonium bromides ( abbreviated as C10TAB, and C14TAB, respectively ) were purchased from Tokyo Kasei Co., Ltd. They were used without further purification after drying in vacuo at room temperature for at least 48 hours. The critical micellar concentration for C t0TAB and C 14TAB are 6.63 • 10.2 and 3.73 • 10"s M. The concentration of the sample solutions prepared is summarized in Table I. b) Measurements Ultrasonic absorption measurements were carried out over the frequency range from 0.2 to 65 MHz. Three different techniques were used depending on the frequency ranges; from 0.3 to 1.7 MHz, we used the piano-concave resonator method coupled with the optical detection techniques, from 2 to 7 MHz range we used the cylindrical resonator method and from 15 MHz to 65 MHz we used the pulse method. The details of the experimental system are described elsewhere. 2~
IlI. Results and Discussion Figure 1 shows the results of the ultrasonic absorption measurements. The data is expressed as log ( c t / f 2 ) vs. log f, where f represents the measurement frequency and a, the absorption coefficient, Solid lines in Fig. 1 represent the curve fitted to the single relaxation 0167-7322/95/$09.50 (e) 1995 Elsevier Science B.V. All rights reserved. SSDI 0167-7322 (95) 00893-4
338
Table L Sample concentration and ultrasomc relaxation parameters
C,,,TAB
C,4TAB
/M
t 10~' M
f,/MHz
/.4,
(). 122
0
1.000
1.000
1.55
O. 130
() 122
1.12
0.991
0 982
1.60
0 137
O. 122
1.86
() 985
0.971
1.65
O. 134
(1 122
2.60
() 979
0.960
1.72
O. 133
(}.122
4.09
0.968
0.939
1.80
O. 130
O.122
7.44
0 943
0.898
1.90
0.125
0.122
14.88
0.891
0.825
2.00
O. 115
spectra expressed by the following equation. a
A
f"
1+(/,,/,)
(])
.,-B
z
where f~ is the relaxation frequency, A the relaxation amplitude and B the contribution due to absorption from any other processes that may be occurring at higher frequencies beyond our frequency range. In Fig. 1. the concentration dependence o f the relaxation spectra is not clear. Thus we replot the data in a form of log f vs. absorption per wavelength after subtracting the contribution of B, as shown in Fig. 2.
For the case o f single relaxation described by the
equation (2), the peak frequency and the peak height represent the relaxation frequency, f~, and the relaxation strength,/.~, respectively
= In Fig. 2, the decrease in relaxation strength and the increase in relaxation frequency with increasing concentration of C~4TAB are clear. ........
IOOO
9
!
........
,
oo
.......
The values fi and/a,, are also summarized in
!
I
o
% 0
.
.ell
o.,
"q,
too
lO
0.1
1
10
100
o
1 o.1
Frequency
/ MHz
Figure 1 Ultrasomc absorption as a funcuon of frequenc3' for aqueous solution of C~,,TAB added by C~4TAB. Concentration of C~4TAB added are: open circles: 4 09• 10.3 M: closed circles: 2.60• 10; M; open triangle 1.49• !0: M
1
lO
lOO
F r e q u e n c y / MHz
Figure 2 Ultrasonic absorption per wave length a function of frequenc) for aqueous solution of CI()TAB added bv C,4TAB. Concentration of CiaTAB added are: open circles: 1.86x10 3 M; Closed circles: 4 09• IO3M. Open triangle: 1.49• 10z M.
339 Table I. We now discuss the dependence of the relaxation frequency on the concentration of C~+TAB along the line of the theory of Annianson 3) for the relaxation process of mixed micelle. In this argument, the micellar solution consists of two kinds of surfactants. Under the condition that the dissociation rate constant of surfactant I is much larger than that of surfactant 2, the reciprocal of the relaxation time of the f a s t p r o c e s s is written in the following way. r I
= kz._ l
+
(3)
,
where r, represents the relaxation time,
k,,_~ the mean dissociation rate constant, or, 2 the
variance of the size distribution, c~ the monomer concentration, n the number concentration of the mixed micelle. The index i means surfactant i. obtained in the following way.
C - ,C _-~7
n= ~
The number
concentration can be
(4)
//7
where ~ is the mean aggregation number of the mixed micelle and C the total concentration of the surfactants. To obtain n, we have to estimate the monomer concentration. Similarly as proposed by Clint 4), we assume that the rnicelles behave as ideal mixture. In this assumption, the relation of the monomer concentration to the critical micellar c o n c e n t r a t i o n is written in the following way,
c? =
xc.,.,
(5)
where x is the mole fraction of surfactant 1 in the mixed micelle and CM, the critical micellar concentration of surfactant i. Consideration of the mass balance gives the relation between x and fl, the mole fraction of surfactant 1 in the total mixed solute, 2.4
~(c - 4" - q ) = ~ " - ~ . From equations equation
(5)-(7),
,.,
we
((~, -c~,.,)x(x-t)
obtain
the
(8)
C
~
2.1
-~
1.
~"
1.8 1.7
Solution of equation (8) gives the parameter x, and hence the monomer concentration from equations (5) and (6). The dissociation rate constant of Ct0TAB (3.0 x l0 s s"t) is much larger ( 3.2 x l06 s"t ).
-
(7)
than that of C~+TAB Therefore we can treat
C t0TAB as surfactant 1. On the assumption
~
1.6 1 . 5 I_.
0
.
.
.
.
.
.
.
0.005
.
.
0.01
+
0.015
C o n c e n t r a t i o n o f C14TAB
0.02
/ M
Figure 3 Comparison of the theoreucal estzmation (solid line) with the experimental data (open squares.)
340 of the constant value of the aggregation number at any concentration we calculated the relaxation frequency as a function of the concentration of C x4TAB. The comparison of the theoretical estimation with the experimental data is shown in Fig. 3, in which solid line represents the theoretical estimation discussed above. In this estimation, we used the value in the literature, '1 k~._~= 3.0 • l0 s s"~ 9cr x2= 10 and ~ = 38. The theoretical line is in good agreement with the experimental data except for the concentration of C~4TAB equal to 1.488 • 10-2M. The assumption of constant value of the mean aggregation number is not always reasonable especially at larger C x~TAB concentration. At C~4TAB concentration of 1.488 • 10-2 M, l-x amounts to about 20% so that the size of the mixed micelle must increase. Hence the mean aggregation number also increases. This effect may suppress the increasing tendency of the relaxation frequency.
References 1 2 3 4
S. Kato, H. Nomura, H. Honda, R. Zielinski and S. Ikeda, Bull. Chem. Soc. Jpn., 92, 2305, (1988) S. Kato, H. Nomura, R. Zielinski and S. Ikeda, Bull. Chem. Soc. Jpn., 59, 707, (1986) E . A . G . Aniansson, in Technzques and Applicatzotts" of Fast Reactions in Solution edited by W J. Getting and E. Wyn-Jones, p. 249-258, (D.Reidel Publishing Co., 1979) J.H. Clint, J Chem. Soc. Faraday Trans., 71, 3320, (1975)
journal of
MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 341-344
Dissipative Structure in Aqueous Polymer Solutions H. M a e d a , S. Sasaki, T. M a s h i t a , K. N a k a m u r a , H. O j i m a , T. Gotoh, T. F u k u d a and M. Y a m a n a k a D e p a r t m e n t of Chemistry, Faculty of Science, K y u s h u U n i v e r s i t y , H a k o z a k i 6-10-1, H i g a s h i - k u , F u k u o k a 812, J a p a n Abstract Study on the rapid transport of a polymer in dextran solutions, first observed by Preston et al., is extended into two directions. They are (1) enhancement effect on the transport rate of polyvinylpyrrolidone (PVP) by the addition of a simple salt, and (2) extension to the transport of linear polyelectrolytes. The enhancement effect was observed on the structured flow as well as on the transport rate. The enhancement effect was correlated with the densities of the solutions in the lower compartment of the diffusion cell. The correlation was improved when the rate was corrected for the differences in viscosities. We have found that effects of charges on the polymers favor the rapid transport of polyacrylates (PA) and sodium hyaluronate. Counterion condensation was manifested in the transport rate of PA. Transport rates of several salts of PA in the absence of added salt increased linearly with their partial specific volumes in water.
1. Introduction Rapid transport of polyvinylpyrrolidone (PVP) in dextran solutions has been observed by Preston et al. 1 This rapid transport was ascribed to the occurrence of a finger - like structured flow and was characterized by the linear dependence of the transported amount on time t rather than on t1/2. The molecular mechanism of the phenomenon is not yet fully elucidated. Two aspects of our study related to this phenomenon will be presented. 2. Effects of added simple salts on the transport rate of PVP. 2, 3 Rapid transport of polyvinylpyrrolidone (PVP) in dextran solutions was found to be further accelerated when a simple salt was added to both upper and lower compartments of a diffusion cell at the same concentration. The enhancement effect increased with the salt concentration in all the examined cases: CsCI, NaBr and NaSCN. The transport rates v(10-7m s -1) of PVP (Mn = 3.2 x 105) in the presence of 2M salts + 80 kg m "3 dextran (Mn = 4 x 104) at 25"C were: (4.2~0.6) x 10 (CsCI), 9.5+_0.3 (NaBr), 9.2-+0.3 (NaI), 7.6+_0.2 (NaNO3) , 5.5+_0.5 (NaCl), 5.0+_0.2 (NaSCN), 2.6_+0.2 (LiCI) and 2.8___0.3 (no salt). The transport rates at 2 M were in the following order: CsCI >> NaC1 > LiCI -- No salt : NaBr -- NaI > NaNO3 > NaCI > N a S C N > No salt. In the preliminary report, we have shown that the enhancement effect of the salts cannot be ascribed to their effect on the coil dimension of PVP. 2 Intrinsic viscosity of PVP in aqueous solutions considerably decreased in the presence of 2 M CsCI. In the presence of 80 kg m "3 dextran, however, the coil dimension of PVP was decreased so much as not to be further influenced by the addition of CsCI. The enhancement effect was correlated with density of the solutions. Solutions of higher densities showed greater acceleration effect, as shown in Table 1. The correlation 0167-7322/95/$09.50 :C 1995 Elsevier Science B.V. All rights reserved. SSDI 0167-7322 (95) 00826-8
342
was further improved when the transport rates were multiplied by the viscosity. The effect was not + 20 k~; m "3 PVP + 2M salt correlated with the osmotic coefficients or lyotropic numbers of the salt density / 103 k[[ m "3 v / 10 -7 m s -1 salt solutions (in the absence of the polymers). The effect depended on no salt 1.030 2.8 :_4).3 the viscosity but correlation between LiCI a) 1.069 2.6 :d).2 them is not simple. Finger like flow NaSCN 1.101 5.0 *-0.2 patterns (a dissipative structure) were NaCI 1.107 5.5 _+0.5 observed under a microscope. As NaNO3 1.136 7.6 *-0.2 compared with the case of no added Nal 1.250 9.2 -,-0.3 salt, followings were observed (a) NaBr 1.185 9.5 *.0.3 fingers were wide, (b) the number of fingers was reduced and (c) the time CsCI a) 1.275 42 ---6 for the fingers to appear was short. a) PVP concentration" 10 kg m -3 Critical PVP concentrations required for the formation of structured flow in the presence of 2 M salt were determined by examining the presence or absence of the structured flow pattern. They are in the following order : CsCI < NaCI < LiCI. The order is well correlated with the enhancement effect of these salts. It is pertinent to consider separately the enhancement effect of salt on two steps: the initiation step (onset of the flow) and the structured flow. The transport rates are related to the properties of the final structured flow and are contributed from the effects on both steps. The effect on the initiation step is clearly noticed since the critical PVP concentrations for the occurrence of the structured flow depended on the kind of salt. Effects of a salt on the cross diffusion constants of the two polymer components will be examined on both excluded volume and frictional effect. The effect on the excluded volume interaction between the two polymer components is expected to be small. This expectation is partly supported by the result 2 that coil dimension of PVP was not influenced by the addition of a salt at 2 M in the cases of three salts LiCI, NaCI and CsCI, while these salts showed quite diverse effects on the transport rates of PVP. Since viscosities vary with the kind and the concentration of salt, frictional coefficients are influenced by the presence of a salt. In this respect cross diffusion constants may be affected by salt through a change in viscosity of the medium. We have shown that the salt effect is well correlated with the densities of the solutions. Accordingly the essential part of the salt effect is expected to operate in the steps after density inversion occurs. The enhanced transport rates are expected to be related to the growth speed of fingers. We anticipate the friction in the boundary layers between upgoing and down-going fingers affects the velocity in the centers of both fingers. For thin fingers the velocities will be small. Relationship between the density of the solution and the characteristics of the fingers (such as their thickness, growth rate) is to be clarified. Table 1. Transport rates of PVP and densities of solutions containing 80 kg m 3 dextran
3. Rapid transport of polyelectrolytes in dextran solutions. We have observed rapid transport of polyacrylate (PA) and hyaluronate (HA) in dextran solution matrix, which will be the first extension to linear polyelectrolytes. The whole transport process did not follow the flow regime(linear in t) nor diffusion(linear in t 1/2) but a combination of the two, when ionic strengths were not high enough. In the media of low ionic strengths, diffusion of linear polyelectrolytes is very rapid due to the effect of counterion diffusion. 4 We sometimes observed structured flows under a situation that transport rate followed diffusion law. This behavior was more clearly observed on HA than PA. Effect of charges favored the rapid transport of both polyclcctrolytcs, since (a)
343
I
4
I
I
i
n r'l D 2
O
!t3-0
8
DD 0e
r,,)
1 ; I I I 4 6 8 time/10 4 s
0 0
2
Figure 1. Effect of the counterion species on the transport rate of polyacrylates in dextran solutions without added salt. NH4+(Q),Li+([[]),Na+(O) and Cs+(Z:X).
12 10
(Me)4N
O
)+N*
10
transport rate decreased as NaCI concentration increased for both PA and HA and (b) transport rate increased as the charge density c~ of PA increased but it remained nearly constant when o~ exceeded about 0.5. Thus, counterion condensation phenomenon affects the transport rate in the present study. Details of these results will be presented elsewhere. In this report, we will confine ourselves on the effects of counterion species on the transport of polyacrylates in dextran solutions without added salt. Effects of counterion species on the transport rate of PA are shown in Fig. I for alkali metal counterions and NH4+. O, A and CO represent, respectively, the transported amount, the cross section area and the initial PA concentration. In this type of experiment, the upper solutions contained only dextran and changes of degree of ionization may occur due to selfprotonation of PA as a result of dilution when polyions move across the boundary into the upper compartment. The rates (10 -7 ms -1) were in the following order:
8 E ~'
6
~"
4
(Pr)4N+(9.8-+ 0.5) > (Me)4N+(9.3-+ 0.6) > Li + (2.5_+0.5)> (Bu)4 N+ (2.2~ 0.3) > Na+(1.5+_0.2) > (NH4+)(0.75_+0.11) > Cs+(0.4_0.1).
-
_
Nay
The order concerning alkali metal ions is the reverse of that found for the enhancement effect by added salts on 0 0.2 0.4 0.6 0.8 1 1.2 the transport rate of PVP. 2, 3 The partial s p e c i f i c v o l u m e / mi g-1 rates of tetrapropyl homolog (Pr)4 N+ PA and of tetramethyl homologs Figure 2. Effect of the partial s p e c i f i c (Me)4N+PA are large, while that of volume in water on the transport rate tetrabutyl homolog (Bu)4 N+ PA is of v a r i o u s p o l y a c r y l a t e s ( o p e n circles) comparable with those of alkali metal in d e x t r a n s o l u t i o n s . cations. This may be ascribed to the Solid s y m b o l s s t a n d for the c a s e s of three o t h e r p o l y m e r s . increased viscosity of (Bu)4N + PA solutions. In Fig. 2, the transport rates of several polymers in dextran solutions are plotted against their partial specific volumes in water (not in dextran solutions). A fair correlation is found between the two properties. Cs+O
I
ONattA L llP~tAm t
344 Data on several species of polyacrylates fall on a line except for the data on (Bu)4N+ PA. The results on other polymers including polyacrylamide (PAAm) don't fall on the same line. The difference may be related to different nature of hydrations of different polymers. 4.
Discussion Comper et al. have proposed a mechanism in terms of the hydrostatic instability as a cause for the occurrence of the structured flow in the matrix-gradient systems consisting of polymers.5, 6 Density inversion which triggers the instability is reached mainly as a result of large cross-diffusional flux of the matrix component caused by the diffusion of the gradient component. 510 Significant extent of cross diffusion is considered to arise mainly from the excluded volume effect and the hydrodynamic frictional effect. 7 As to crossdiffusion between polymeric solutes, entanglement effect should be taken into account in addition to excluded volume effect and frictional one. The hydrostatic model is to be applied to the step of the onset of the flow or to the transition from diffusion regime to flow regime. The total phenomenon undoubtedly consists of many steps. The flow grows to form flow structure and stabilization and growth of the structure need other mechanisms. In addition to the hydrostatic model, two other mechanisms have been examined: 6, 8 osmotic model and the hydrodynamic model. As to the osmotic model, Comper and Preston have questioned its validity as a probable mechanism leading to density inversion. 6 According to their argument, the main diffusion constant of PVP should equal to the cross diffusion constant of dextran if the osmotic model is valid. This expected relation was not found experimentally. Swelling of ionic gels will be expected to be related with the initial event of the present phenomenon. Swelling of ionic gels occurs as a result of osmotic flow of solvent when gel contacts with solvent. This swelling often takes place nonuniformly and gel surfaces moving into solvent are wavy. To examine the hydrodynamic instability, fluxes of the two polymer components are described with equations consisting of terms including the velocity of the solution in addition to those representing diffusive motion 9 The fluid velocity is given by the NavierStokes equation where kinematic viscosity and hence density of the solution plays a role. In this respect, the present salt enhancement effect might be related to the hydrodynamic instability through its effect on the density and viscosity. References
1
B.N. Preston, T. C. Laurent, W. D. Comper, G. J. Checkley,
Nature, 287, 499 (1980). H. Maeda, T. Mashita and S. Sasaki, Chemistry Lett., 635 (1991). H. Maeda, T. Mashita, H. Ojima, S. Sasaki, Y. Tomita, T. Fukuda, M. Yamanaka, J. Phys. Chem., in press. 4 M. Nagasawa, H. Fujita, J. Am. Chem. Soc., 86, 3005 (1964). 5 W . D . Comper, D. M. MacDonald, B. N. Preston, J. Phys. Chem., 8 8 , 6031 (1984). 6 W . D . Comper, R. P. W. Williams, G.J. Checkley, B. N. Preston, J. Phys. Chem., 9 1 , 9 9 3 (1987). 7 W . D . Comper, Van Damme, M. - P. I., G. J. Checkley, B.N. Preston, J. Phys. Chem., 89, 128 (1985). 8 W . D . Comper, G. J. Checkley, B. N. Preston, J. Phys. Chem., 8 9 , 1551 (1985). 9 T.J. McDougall, J. S. Turner, Nature. , 2 9 9 , 812 (1982). 10 T.J. McDougall, .L Fluid Mech., 126, 379 (1983) 2 3
journal of
MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 345-348
N o n - L i n e a r E f f e c t s in the F r e e E n e r g y P r o f i l e o f E l e c t r o n T r a n s f e r Reaction:
An Integral Equation Approach Song-Ho Chong and Fumio Hirata Department o f Chemistry, Faculty of Science, Kyoto University, Kyoto 606-01, Japan
Abstract The free energy profile for the electron transfer reaction in a polar solvent is examined based on the extended reference interaction site method (ex-RISM) applying it to a simple model of a charge separation reaction which was previously studied by Carter and Hynes with molecular dynamics simulations. Due to the non-linear nature of the hypernetted chain (HNC) closure to solve the RISM equation, our method can shed light on the non-linearity of the free energy profiles, and we discuss these problems based on the obtained free energy profile.
L Introduction Electron transfer (ET) reaction in a polar solvent has been one of the central issues in physical chemistry and biophysics. In the presence of an immersed solute, a polar solvent around the solute continuously fluctuates due to the translational and orientational motion of its constituent molecules. The concept of a free energy profile governing these fluctuations as a function of solvent coordinates plays a central role in the theory of ET. Recently, we developed a molecular theory for obtaining the free energy profiles of ET reactions based on ex-RISM? and obtained quantitatively good results in terms of agreement with the simulation data. Recently, validity of the two assumptions regarding the free energy profile, parabolic and solute independence of the force constants, has been examined by several authors. These assumptions have been predicted from the continuum dielectric models and commonly adopted in many of the early works. Kakitani et aL 2 discussed the nonlinearity of solvation related to ET in polar solvents, and Carter and Hynes 3 performed molecular dynamics simulations of the charge separation (CS) and the charge recombination (CR) reactions to observe such non-linear effects. More recently, Ando et al. 4 discussed these problems and they observed no such non-linear effects. Due to the non-linear nature of the hypernetted chain (HNC) closure to solve the RISM equation, our method can shed light on the non-linearity of the free energy profiles. In section III, we apply our method, which is outlined briefly in section II, to the CS reaction which was previously studied by Carter and Hynes 3, and discuss the problems mentioned above based on the obtained free energy profiles. II. Theory Here we briefly outline the method for obtaining the free energy profiles of ET reactions based on ex-RISM, 5 an integral equation theory developed for molecular liquids. (For more detailed description of the method, see Ref. 1.) We denote the charge distribution of the solute ions, the donor (A) and the acceptor (B) ions, at the reactant state as {e^, eB }. The distance between A and B is fixed. Let z denote a fraction of the electronic charge transferred from A to B. We define a reaction coordinate as,
AE = e( VB - VA) , 0167-7322/95/$09.50 (~ 1995 Elsevier Science B.V. All rights reserved. SSDi 0167-7322 (95) 00917-5
(I)
346 where e is the unit charge and VA(VB) denotes electric potential that the solvent molecules produce at the redox site A(B). In the spirit of Marcus theory,6 we characterize the non-equilibrium solvent polarizations at the intermediate states of the reaction as the equilibrium polarizations corresponding to the hypothetical charge distributions of the solute ions, {eA+ze, en-ze}. Then, specific values of AE can be calculated with those charge distributions from the following equation,
(2)
"CA = 4~rp~,J r2cb"g'Aa(r)e..........._..~, $
where gU"A, is the radial distribution function for the sites A and s, where s refers to the sites of the solvent, and e, denotes the charge of the site s. The equation for VB is given by replacing A in Eq. (2) with B. We denote the value of AE corresponding to the charge distribution {e^+ze, es-ze} as AEz. Then, the expression for the reactant free energy profile is, 1 ZW'(aE,)
- ~,
- ~ o § ~'~-,,
(3)
where AF, is the equilibrium solvation free energy of the solute ions of the charge distribution {e^+ze, es-ze}, and AFo is for z --- 0. The solvation free energy AFz for the solute ions with the charge distribution {eA+ze, eErze} can be calculated with ex-RISM by the following equation, 7
~Al,tso," =4~rp~,ir:Zdrrl h,,,a :Z(r)_c~,m(r)_ 1 h~,aa(r)c~, (r)l, ~.,
L2
-i
(4)
where hu"a,, and cu*,,~ are the solute(u)-solvent(v) total and direct correlation functions between the site a of the solute and the site s of the solvent, respectively, p and 13 are the bulk density of the solvent and the inverse temperature, respectively. The product free energy profile can be also obtained from the similar procedure, and it can be proved that the reactant and product free energy profiles satisfy the following well known equation, ~a,s FR(AI~'~)- FP(A~-) -" AE- ,
(S)
where F R and F P denote the reactant and the product free energy profiles, respectively. HI. Results and Discussion
In this section, we apply our method described in the previous section to the CS reaction which was previously studied by Carter and Hynes, 3 and briefly discuss the results. The specific reaction we analyze here is the CS reaction,
A~ ~ ._+A§
-l"
for a solute pair AB, with A=B, in diatomic solvent molecules which have structure similar to that of CHsCI if CHs group is regarded as an united atom. The solute at the reactant state is a neutral pair (NP) AB, and that at Ire product state is an ion pair (IP) A§
-l, and separation between A and B is fixed at 3.0 A,. The solvent molecules have a fixed
atomic separation of 2 A, with partial charges such that the dipole moment is 2.4 D. The number density and the
347
60 t--"l ,.-w
o 40
L--.,
20
o
0
-20 -40
-1~0
0
20
:
:
:
4'0
(50
8'0
100
D e l t a E [kcal/mol]
Fig. 1" The diabatic free energy profiles for NP and IP states, o: ex-RISM results. dashed line: 2nd order fitting profiles, solid line: 4th order fitting profiles.
Table 1: The force constants for the NP and IP free energy profiles. force constants a for NP
IP
2nd order fitting
0.018
0.018
4th order fitting
0.018
0.016
(cf. Carter and Hynes) b
0.017
0.028
a Force constants (in [kcal/mol] d) are calculated from the second derivative coefficients at the minimum points of the profiles. b The result of Ref. 3.
348 temperature of the solvent are 0.012 molecules A -3 and 250 K respectively. All the sites of the solute and the solvent interact via Coulomb and Lennard-Jones (LJ) interactions. The LJ parameters a=3.5 A and s K are identical for all the sites of both the solute and the solvent molecules. The diabatic free energy profiles obtained by our method is plotted in Fig. 1. By symmetry of the system we consider, the NP profde is symmetrical with regard to the minimum point of the profile. Our resultant prof'des do not agree well with those of Carter and Hynes. 3 Now we discuss the problems related to the parabolic free energy profiles and the solute independence of force constants. As is obvious from Eq. (5), the force constants of NP and IP profiles have to be the same if the profiles are exactly parabolic. But it is possible that the free energy profiles are non-parabolic, and that this causes the difference in the force constants. To elucidate these problems, we first attempted to fit the NP profile by parabola and then calculated the IP profile using Eq. (5). From the plotted profiles in Fig. 1, we can see that the IP profile can not be well approximated by the curve so obtained. We therefore attempted to fit the NP profile by 4th order polynomials, and then calculated the IP profile using Eq. (5), which is also plotted in Fig. 1. It is seen from Fig. 1 that 4th polynomial fitting curves give better results. The force constants of two prof'des are summarized in Table 1. The resultant force constants are in contrast to the data of Carter and Hynes; they reported that the force constant for the IP profile is 1.6 times larger than that for the NP prof'de. The motivation that Carter and Hynes performed the simulation is to check the validity of the hypothesis proposed by
Kakitani et al. 2 which says that dielectric saturation of the first solvation shell causes the curvature of the free energy profile for IP to be much larger than that for NP. Although the results of Carter and Hynes indicate that the difference in the force constants between IP and NP is not so large as had been assumed by Kakitani et al., it seems that 1.6 times difference in the force constants between IP and NP profiles is still large. More recently, King and Warshel9 performed more detailed analysis of the system that Carter and Hynes had studied. The results of King and Warshel are quantitatively in good agreement with ours. King and Warshel concluded, from the their results that the force constants for IP and NP are nearly identical, that the saturation effect in the first solvation shell seems to be almost completely compensated by underpolarization of the next solvation shells, thus the dielectric saturation effect does not alter the force constants of the free energy profiles appreciably. Another feature one should note from Table 1 is that our results show that the force constant for the NP profile is slightly larger than that for IP. This trend in force constants in case that the solvent is water is indicated by Fonseca et aL to Investigation of the free energy profiles of ET reactions in aqueous solutions tx and the study to elucidate why the force constants of this system show the trend presented in Table 1 are currently underway in our group. References
1 S. Chong, S. Miura, G, Basu and F. Hiram, J. Phys. Chem. 99, 10526, (1995). S. Chong and F. Hiram, J. Mol. S/re. (in press). 2 A. Yoshimori, T. Kakitani, Y. Enomoto and N. Mataga, J. Phys. Chem. 93, 8316, (1989). 3 E.A. Carter and J. T. Hynes, J. Phys. Chem. 93, 2184, (1989). 4
K. Ando and S. Kate, J. Chem. Phys. 92, 7241, (1990).
5 F. Hirata, P. J. Rossky and B. M. Pettitt, J. Chem. Phys. 78, 4133, (1983). 6 R.A. Marcus and N. Sutin, Biochim. Biophys. Acta 811,265, (1987) and references cited therein. 7 D.A. Zichi and P. J. Rossky, J. Chem. Phys. 84, 1712, (1986). 8 M. Tachiya, J. Phys. Chem. 93, 7050, (1989). 9 G. King and A. Warshel, J. Chem. Phys., 93, 8682 (1990). 10 T. Fonseca, B. M. Ladanyi and J. T. Hynes, J. Phys. Chem., 96, 4085 (1992). 11 S. Chong and F. Hiram, (in preparation).
journal of MOLECULAR
LIQUIDS Journal of MolecularLiquids,65/66 (1995) 349-352
ELSEVIER
Induction Time of Precipitation of Calcium Carbonate Lars-Fride Olsson Gambro AB, Technical Research, P.O.Box 10101, S-220 10 Lund, Sweden. Abstract A study of the life-time, or "induction time" Tin d of the metastable state of supersaturated aqueous solutions containing calcium and hydrogen carbonate ions is presented. The measurements have been performed at room temperature, in the physiological pH range 7.2 to 7.5 and with hydrogen carbonate in large excess over calcium. The method is useful for the study of physiological solutions, where control over pH is crucial, pH is kept constant within __ 0.01 unit by means of the buffer system carbonic acid/hydrogen carbonate, and by controlled addition of carbon dioxide if pH tends to increase. The Gibbs free energy, zxG, of the supersaturated solutions has been calculated and correlated with the induction time. As expected the induction time is shorter the greater the supersaturation is. I. Introduction In supersaturated solutions the formation of a second phase, e.g. precipitation of a solid substance, will start some time after the creation of the solution 1. The time from mixing to the first sign of a precipitate is known as the "induction time", T~,.d. During Ti,d there is spontaneous formation of precipitation nuclei 1'2'3. This process is the precursor to crystal growth and as such a knowledge of the induction time is useful where unwanted precipitation may occur, e.g. in industrial and biological crystallization or in processes where hard water is used. A special case, which is connected to both industrial and biological processes, is represented by a dialysis solution. This water solution, which is physiological with respect to sodium chloride, has a pH around 7.4 and is supersaturated with respect to hydrogen carbonate and calcium ions. Dialysis solutions are useful as "simulated" body fluids in chemical research concerning reactions in the body fluids, e.g. transport processes across membranes and precipitation reactions. The macroscopic properties of the supersaturated solution are seemingly unchanged during the induction time. It is reasonable to assume that the initial supersaturation, given by its free energy value, zxG, is an essential parameter for the calculation of Tin d. In the study reported here, the time Tin d has been measured. Since the reaction describing precipitation most appropriately can be written as
Ca 2+ + HCO 3- --) CaCO3(s ) + H +
(1)
the change in the Ca 2+ concentration at time t = Tin d is measured with an ion selective calcium electrode. During the experiment pH is kept constant by means of the buffer system COJHCO3-. However, a small decrease of pH can often be seen at the moment when reaction (1) starts. 0167-7322/95/$09.50 91995 ElsevierScience B.V. All rights reserved. (95) 00829-2
SSDi 0167-7322
350
I. Experimental section Apparatus. The measurements were performed in a 100 mL thermostated glass beaker with a cover and a magnetic stirring bar. The stirring rate can be varied between 300 and 600 rpm without affecting the results. The Ca 2+ concentration is measured by an ion specific electrode (Orion, model no. 9320) and a double junction reference electrode (Orion, model no 9002; 3 M KCl(aq) as the fill solution), pH of the solution is measured by an Orion Ross electrode (model no. 8102). The electrodes are connected to an automatic measuring system (Fluke | Hydra), which measures the two e.m.f, every 20 seconds, pH is kept constant by means of the buffer system carbonic acid/hydrogen carbonate. Since there is, however, an equilibrium between carbonic acid and gaseous carbon dioxide, CO2(g), and a small leakage of CO2(g ) occurs, the amount of carbonic acid tends to decrease and thus pH tends to increase. To suppress this, the pH electrode is connected to the Fluke | Hydra Alarm System, which operates a valve for the addition of CO2(g), as soon as pH exceeds a prescribed value. CO2(g ) is added via a Pasteur pipette, which ends a few mm below the surface of the solution. It has been possible to keep pH constant within • 0.01 units. Chemicals and solutions. Aqueous salt solutions were prepared from water purified through reversed osmosis (conductivity 5 to l0 ~tS/cm at 25 ~ The following salts were used: calcium chloride tetrahydrate, CaC1E4H20 (Merck Suprapur), sodium hydrogen carbonate, NaHCO 3 (Merck p.a.) and sodium chloride, NaCl (Merck Suprapur) as supporting electrolyte. Carbon dioxide, CO2(g ) was of analytical grade. The experiments have been performed at 20, 25 and 30 ~ (• and in the concentration ranges 1.96 to 4.67 mM (CaCl2) and 49.5 to 66.7 mM (NaHCO3). NaCl is added to a total ionic strength between 61.1 and 80.7 mM. The range of the induction time is 5 to 394 minutes. The pH electrode was calibrated before and after each experiment in a Standard solution (Merck buffer solution) at pH 7, and once a day the slope was checked by calibrating at pH 7 and 4. Similarly, the slope of the e.m.f, of the Ca electrode was checked once a week. Both electrodes show the theoretical (Nernst) slope. Measurements. A supersaturated solution is created in the following way: a thermostated solution (approximate volume 50 mL) of NaCl(aq) and NaHCO 3 (aq), pH --8.2 is made and the automatic measuring system (Fluke | Hydra) starts. CO2(g ) is slowly added until pH is about 0.1 unit above that at which the experiment is intended to be performed. A few mL of 100 mM CaCIE4H20(aq) is added via a pipette, pH decreases slightly and the e.m.f, of the Ca electrode changes. This marks the beginning of the measurement period. After some experience it is possible to attain the expected pH and maintain it there. As long as the time is less than T~,d the e.m.f, of the Ca electrode is constant, c.f. Figure 1. When reaction (1) starts, the e.m.f, of the Ca electrode drops, pH decreases slightly and the solution turns slightly dim. All these signs marks the end of the metastable period. The induction time can thus be calculated from three observations. In most cases they agree rather well. When the calcium concentration is low, the change of the e.m.f, is slight and it can be hard to evaluate the induction time. Calculations. The Gibb's free energy of the supersaturated solutions is given by AG = RTIn(Q/K) = RTIn(~o/Q,p ) = -RTlnS, where K is the equilibrium constant and Q the activity ratio according to reaction (l). K,p is the solubility product for CaCO3(s). CaCO3(s) can exist in at least three crystal structures. It has been assumed that it is the most soluble form, known as vaterite, which is formed initially 4. Q,p is the corresponding product in the
351 supersaturated state and S the degree of supersaturation, S = Qsp/K~,p. p I ~ and PKaz for carbonic acid, p I ~ for the ionic product of water, the solubility of carbon dioxide in water, the stability constant for the formation of Ca(HCO3) § and the solubility product, K~p of CaCO3(s,vaterite) have been taken from IUPAC Stability Constants Database 5. The concentration of free ions, [Ca 2.] and [HCO3-], has been calculated from the total concentrations and the measured pH. The activity coefficients have been calculated from the Debye-Hiickel theory, with hydration radii according to Kielland 6. Ill. Results and discussion.
A typical measurement of e.m.f, vs. time is displayed in Figure l a. In Figure l b, Tind has been plotted versus AG. T~llmin 100
e.m.f./V 0.02
0.01 0.00 -0.01 -0.02 -0.03 -0.04
6O
4O
1
+
I
I
1
I
I
I
44+ +
9
-0.05 O-
,~,UJ1 r
J~r,~
20
I~
[
t~te/s 1000
"I~
3000
4000
+
+
01.3 t'.4 115 z'.6 ti7
-aG/RT tls xi9 2
+ +
2.x
Figure 1. a) e.m.f, of calcium (upper curve) and pH electrode, b) induction time versus AG/RT, at 25 ~ The expected corl;e!ation between ~G and Ti, d is confi..rmed, i.e. the greater the supersaturation is, the shorter is the induction time. Reasonably, Ti, a goes to zero in the limit where LxG goes to infinity. Conversely, in a saturated solution, zxG is equal to zero and Tind is of course infinitely long. SOhnel and Mullin l, Garside 2 and recently Barlow and Haymet 3 have discussed the molecular interpretation of induction times from the standpoint of classical nucleation theory. Crystal nuclei with a critical size must be formed before the new solid phase is visible. According to the model there exists a free energy barrier, AG* to the formation of the crystal nuclei. AG* is proportional to (lnS) 2, where S is the supersaturation ratio. The Gibb's free energy, AG of the supersaturated solution is equal to -RTlnS (R=gas constant; T=temperature). The induction time is a function of AG* and thus AG according to the following equation 1'2 ln(Tina/min) = B/(T3(AG/RT) z) - lnA
(2)
where A and B are constants ~'2. A plot according to equation (2) of all the data is given in Figure 2, together with a few data from ref. 7, Table 1. These 7 data refer to the mixing of equimolar concentrations (about 1 mM) of sodium carbonate and calcium chloride, at pH 10.5 and 25 ~ It is obvious that equation (2) gives a reasonable description of all the measured values.
352
l n ( T Imm) 7
I
I
I
I
I
I
O O +
o ~8§ o
+ -1-
o 9 0
~
x - 10=/['I4(aG/RT) 2] I
I
I
I
I
I
0.5
1
1.5
2
2.5
3
3.5
Figure 2. Plot according to equation (2). 'zx' denotes values at 20, 'o' at 25 and '+' at 30 Points denoted " . " are from ref. 7, Table 1 (see also text above for details).
In natural chemical systems a stable state of moderate or low supersaturation can exist for a fairly long time, even on the geological time scale s. In such systems, calcium carbonate plays an important role. The measurements reported in this work, indicate that the classical nucleation theory can correctly predict the the trend in the induction time for a known change in the Gibb's free energy of supersaturation. Equation (2) has the correct behaviour at small supersaturation: when AG is zero, T~nd will be infinitely long. It seems unlikely, however, that (2) can correctly represent data at very large supersaturation, since T~,d will approach a finite value (I/A) when AG is very large. Classical nucleation theory is, however, based on the assumption that macroscopic thermodynamic properties are applicable to particles of molecular dimensions 1'2'3. This may be a too drastic simplification. The spread of the data is mainly due to the stochastic nature of the process, and not to experimental errors, cf. Barlow and Haymet 3 and references cited therein.
References 1 0 . S/Shnel and J.W. Mullin, J.Colloid Interface Sci., 123, 43 (1988). 2 J. Garside, in "Biological Mineralization and Demineralization", ed. G. Nancollas. Springer Verlag, Berlin 1982. 3 T.W. Barlow and A.D.J. Haymet, Rev. Sci. Instrum., accepted for publication. 4 J.J.R. Frausto da Silva and R.J.P. Williams, "The biological Chemistry of the Elements", Oxford University Press, Oxford 1993. 5 IUPAC SC-Database ver. 1.1. Compiled by L.D. Pettit and H.K.J. Powell, University of Leeds, Leeds 1994. 6 J. Kielland, Amer.Chem.Soc., 59, 1937, 1675. 7 O. S6hnel and J.W. Mullin, J.Crystal Growth, 60, 1982, 239. 8 W. Stumm and J.J. Morgan, "Aquatic Chemistry", Wiley, New York, 1981, 2:nd ed.
journal of MOLECULAR
LIQUmS ELSEVIER
Journal of Molecular Liquids, 65166 (1995) 353-356
Photophysical Properties of Viologen-linked N-Alkyiporphyrins and Their Zinc(ll) Complexes in Solutions Keiichi Tsukahara,* Naoko Sawai, Satoko Hamada, Takashi Nakazawa, and Ryoichi Nakagakit Department of Chemistry, Faculty of Science, Nara Women's University, Nara City, Nara 630, Japan ?Faculty of Pharmaceutical Sciences, Kanazawa University, Kanazawa City, Ishikawa 920, Japan Abstract The fluorescence properties of viologen-linked N-alkylporphyrins and their zinc(II) complexes having a tri-or hexamethylene chain are discussed on the basis of the orientation between the viologen moiety and the porphyrin plane. The fluorescence decay of the porphyrins with a trimethylene chain in acetonitrile solutions has two components corresponding to the two conformers, closed and perpendicular ones, the latter being slower than the former. The hexamethylene-bridged dyad has one major component in the fluorescence decay.
I. Introduction The orientation between electron-donor and acceptor must be one of the important factors determining electron-transfer (ET) reaction rates. An electronic coupling between the donor and acceptor orbitals with a n character must be weak in a perpendicular orientation for porphyrin systems and the ET rate is expected to be slower than that for the face-to-face interaction or the edge-to-edge cofacial one. We have recently synthesized a new type of viologen-linked Nalkylporphyrins and their zinc(II) complexes. 1 N-Monoalkylation of porphyrins results in distortion of the porphyrin plane by an sp 3 hybridization of one of the pyrroleninic nitrogens; the alkyl group is located perpendicularly on the porphyrin plane. 2 The use of N-alkylporphyrin makes therefore the realization of the fLxed orientation easy; in the dyad system (1 and 2) the viologen moiety is forced to be located perpendicularly on the porphyrin plane by not only the conformationally characteristic N-alkyl bond but also the steric hindrance between N-alkyl group and relatively bulky four phenyl groups at meso positions.
to
b
T:f/2
ct
o- . -
2'
"-
3'
3
2
n =3 n =6
6
[HtppprMVI(PF6)2 [Zn(tppprMV)C1](PF6)2 [HtpphxMV](PF6)2 [Zn(tpphxMV)Cl] (PF6)2
(1) (2) 0 ) (4)
Figure 1. Structures of the dyad systems.
0167-7322/95/$09.50 91995 Elsevier Science B.V. All rights reserved. SSDI 0167-7322 (95) 00894-2
354 In this work we report the fluorescence properties of the dyad systems (1 - 4) on the basis of the orientation between the viologen moiety and the porphyrin plane. II. Experimental Dyads (1 - 4) were synthesized by the method described previously. 1 Fluorescence spectra were measured in degassed acetonitrile solutions at 25 *C with a Hitachi 850 spectrofluorometer. The excitation wavelength was the Soret maximum. Fluorescence lifetimes were measured at 25 *C using a Horiba NAES-500 ns-fluorometer interfaced to an NEC PC-9801 RX personal computer. The excitation light below 420 nm was cut off with a glass filter. The fluorescence was detected by a single-photon counting system and analyzed as the sum of two exponential components after deconvolution of the instrument response function. 1H NMR spectra were recorded on a JEOL JNM GX-270 NMR spectrometer. IIl. Results and Discussion
Table I shows the 1H NMR spectral data for I - 4 in acetonitrile,d3 at 30 *C. Proton signals were assigned by 2D 1H-1H phase-sensitive COSY and NOESY spectra. Table 1. 1H NMR spectral data for N-alkylporphyrins and their zinc(II) complexes (1 - 4) in acetonitrile-d3 at 30 *C
6/ppm (J/Hz) assignment pyrrole C 2,6-bpy pyrrole B,Da pyrrole B,Db o-phenyl I o-phenyl II 3,5-bpy m,p-phenyl I m,p-phenyl II pyrrole A 3',5'-bpy 2',6'-bpy bpy-CH3 a-cn2 I~-CH2 N-H "r 6-CH2 e-CH2 oJ-CH2
1
8.83 (s) 8.76 (d, 7.3) 8.59 (d, 4.4) 8.35 (d, 5.1) 8.32 (br) 8.12 (br) 7.99 (d, 6.6) 7.88-7.94 (m) 7.76-7.84 (m) 7.55 (s) 7.41 (d, 7.3) 6.90 (d, 6.6) 4.40 (s) 2.32 (t, 6.6) -0.66--0.57 (m) -2.30 (br) -4.64--4.55 (m)
2
8.87 (s) 8.77 (d, 7.3) 8.93 (d, 5.1) 8.77 (d, 4.4) 8.65, 8.22 (br) 8.10 (br) 8.08 (d, 6.6) 7.798.02 (m) 8.38 (s) 7.83 (d, 6.6) 7.58 (d, 6.6) 4.37 (s) 2.36 (t, 7.3) -0.26--0.15 (m) -4.63--4.57 (m)
3
8.77 (s) 8.73 (d, 6.6) 8.59 (d, 4.4) 8.38 (d, 5.1) 8.55, 8.28 (br) 8.18 (br) 8.16 (d, 6.6) 7.79-7.88 (m) 7.72-7.83 (m) 7.40 (s) 8.07 (d, 7.3) 8.35 (d, 7.3) 4.32 (s) 3.90 (t, 7.7) 1.05-1.14 (m) -2.06 (br) -0.07-0.09 (m) -0.66- -0.53 (m) -1.62- -1.54 (m) -4.57- -4.49 (m)
4
8.34 (s) 8.80 (d, 6.6) 8.97 (d, 5.1) 8.89 (d, 4.4) 8.62, 8.25 (br) 8.09 (br) 8.24 (d, 6.6) 7.837.91 (m) 8.28 (s) 8.16 (d, 7.3) 8.44 (d, 7.3) 4.37 (s) 3.98 (t, 7.7) 1.08-1.20 (m) 0.05-0.17 (m) -0.83--0.71 (m) -0.92- -0.86 (m) -4.69- -4.63 (m)
355
N-CH2 proton signals in the porphyrin plane appeared at a high field due to the ring-current effect of the porphyrins and did not exhibit a triplet signal, but a multiplet one. This strongly suggests a restriction of the free rotation of N-CH2-CH2 bonds. Broad and complex Jcouplings of o-phenyl proton signals indicates the inequivalence of these protons by a steric hindrance for the rotation around the porphyrin-phenyl bond. In the trimethylene bridged dyads (1 and 2), the proton signals for the bipyridinium ring (2,6-, 3,5-, 3',5'-, and 2',6'-bpy) appeared from the upfield in order the proton close to porphyrin, indicating that the viologen moiety is located perpendicularly on the porphyrin plane. On the other hand, the signals of bipyridinium ring protons in the hexamethylene bridged dyads (3 and 4) appeared differently from those for I and 2; the order of the signals from the upfield is 2,6-, 2',6'-, 3,5-, and 3',5'-bpy, which is similar to that for free bipyridinium ions. This suggests that the viologen moiety in 3 or 4 is free from the ring current of porphyrin or is located in the face-to-face orientation. The molecular models of 2 and 4 by molecular mechanics calculations are shown in Figure 2. The optimized structures for 2 and 4 are quite reasonable judging from the NMR spectra.
Figure 2. Optimized structures of 2 (left) and 4 (fight) calculated by a CAChe system. Table 2. Fluorescence data for N-alkylporphyrins and their zinc(II) complexes in degassed acetonitrile at 25 *C
compound H2tpp [Zn(tpp~] HtppPr ~ [Zn(tppPr)C1] b 1 2 3 4
Z.ex/nm 419 421 429 436 429 437 429 436
xl/ns (yield/%) "t2/ns (yield/%)
kem/nm 650, 713 602, 654 687, 745sh 668,732 687, 745sh 670, 740sh 668,740sh 668, 740sh
aIn methylcyclohexane. Ref. 5. bRef. 6.
0.38 0.080 0.014 0.011 0.0004 0.0002 0.0020 0.0023
13.6 2.7 0.8 1.6 0.4 0.9 0.4 0.4
(100)a (100) a (47) (95) (80) (75) (90) (92)
4.9 5.9 6.9 2.0 3.5 3.5
(53) (5) (20) (25) (10) (8)
356 Although a number of studies of the photophysics and photochemistry of non-N-substituted porphyrins and metalloporphyrins have been reported, there are very few studies of Nalkylporphyrins; Lavallee et al. have reported fluorescence spectra of N-methylporphyrins and their zinc(II) complexes.3, 4 There is no report on the lifetime and its correlation with the structure. Fluorescence data for N-alkylporphyrins and their zinc(II) complexes in degassed acetonitrile are shown in Table 2. Red-shifts in the emission maxima and a considerable decrease in the fluorescence quantum yield were observed in N-alkylporphyrin and Zn(lI) complexes compared with non-Nsubstituted porphyrins. This fact can be explained by the noncoplanarity of the porphyrin plane and a weaker ligand field than that of a typical planar porphyrins. 2 The fluorescence lifetimes of N-alkylporphyrins listed in Table 2 consist of two components, although those of H2tpp and [Zn(tpp)] have only one component. It is strongly suggested that the excited singlet state has two conformers, because the alkyl chain is still flexible. The fluorescence quantum yields and the lifetimes of the viologen-linked dyads are smaller than those of HtppPr and [Zn(tppPr)Cl] which have only a propyl group, suggesting that the excited singlet state of the N-alkylporphyrin unit is quenched by the linked viologen via an intramolecular electron transfer.
hv [Zn(tppp,'NeW+)CW]'+ _-"-'--" "[Zn(tppprN~W+)C[F+ (1) [Zn(tpp+'prMV+')CIF+ The viologen-linked dyads have also two components. The fast component (Xl) might be a closed form, where the viologen moiety is close to the porphyrin plane; the bipyridinium ring is able to wag, although the free rotation around the N-CH2 bond is restricted. The other slow component (x2) must be a perpendicular form. The slowness of the fluorescence decay of the perpendicular form might arise from a weak coupling between the porphyrin n-donor orbital and the viologen n-acceptor orbital. In the case of the hexamethylene bridged dyads, the perpendicular orientation is relaxed and the viologen moiety is located on the porphyrin plane, as is shown in Figure 2. The fluorescence lifetimes of 3 and 4 are almost one component and become shorter than those for I and 2. It is strongly suggested that the only closed conformer exists in 3 and 4 and that the intramolecular ET quenching occurs effectively in the closed conformer. The fluorescence quantum yields for 1 and 2 are smaller than those for 3 and 4, suggesting that the intersystem crossing to the excite triplet state for the former is faster than that for the latter. The photochemical study of these dyads is interesting and is in progress. References
1 2 3 4 5 6
K. Tsukahara, N. Sawai, S. Hamada, and T. Nakazawa, Proceedings of the 29th International Conference on Coordination Chemistry, Lausanne, P712 (1992). D.K. Lavallee, The Chemistry and Biochemistry of N-Substituted Porphyrins (VCH Publishers, Weinheim, 1987) D.K. Lavallee and M. Bain-Ackerman, Bioinorg. Chem., 9,311 (1978). D.K. Lavallee, T. J. McDonough, Jr., and L. Cioffi, Applied Spectroscopy, 3 6 , 4 3 0 (1986). A. Harriman, J, Chem, Soc., Faraday Trans. 1, 76, 1978 (1980). K. Tsukahara, N. Sawai, S. Hamada, R. Nakagaki, and M. Itoh, unpublished results.
journal of MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 ( i 995) 357-360
Conductometric Study of Hydrophobic Properties of Tris(1,10-phenanthroline) and Tris(2,2'-bipyridine) Complexes. Haruhiko Yokoyama, Kazuteru Shinozaki, Shin Hattori, Fumiyo Miyazaki, and Masuo Goto Department of Chemistry, Yokohama City University, Seto, Kanazawa-ku, Yokohama 236, Japan Abstract Conductivity measurements have been made for aqueous solutions of chlorides and perchlorates of [Fe(phen)3]2+, [Co(phen)3]3+, and [Co(bpy)3] 3+ (phen=l, 10-phenanthroline; bpy=2,2'-bipyridine) at temperatures from 0 to 50~ The temperature coetiicients of the Walden product for the complex ions, having positive signs and increasing in the order [Co(bpy)3] 3+ ~ [Co(phen)3]3+ < [Fe(phen)3]2+, revealed these complex ions to be hydrophobic structure-makers of which quality is enhanced in this order. The ion association of the perchlorates was significantly stronger than those of the chlorides and was strengthened with increasing ionic charge. The excess free energy in the ion association with perchlorate ion was attributed to the partial loss of the hydrophobic hydration water molecules. The entropy and enthalpy changes are also discussed with respect to the hydrophobic interactions. I. Introduction The [Fe(phen)3]2+ complex dissolved in water is known to behave as a hydrophobic ion, as demonstrated by the ion association with o- and m-benzenedisulfonate and 2,6- and 2,7naphthalenedisulfonate ions, which was investigated by the conductivity method. 1 Similar hydrophobic properties were also observed for trivalent [Co(phen)3]3+ and [Co(bpy)3]3+ complexes. 2 In the present study, the temperature dependence of molar conductivities of [Fe(phen)3]2+, [Co(phen)3]3+, and [Co(bpy)3] 3+ as well as the ion association with chloride and perchlorate ions were examined in order to elucidate the effect of the ionic charge on the hydrophobic interactions. II. Experimetal Tris(1,10-phenanthroline)iron( II ) salts were prepared according to the usual manner. Chloride of [Co(phen)3]3+ was synthesized by refluxing a water-ethanol mixture containing pentaamminechlorocobalt(l]I) chloride and 1,10-phenanthroline and by precipitating the salt with an ethanol-acetone mixture; the perchlorate was precipitated with sodium perchlorate from the chloride solution. The salts of [Co(bpy)3] 3+ were prepared in a similar manner. All the salts were recrystallized and then air-dried; their purities were confirmed by the silver titration for the chloride ions, by the visible-ultraviolet spectrum measurement, and by the analysis for the number of water of crystallization, 3 of which values were 7.0, 4.7, and 4.0 for the chlorides of [Fe(phen)3]2+, [Co(phen)3]3+, and [Co(bpy)3]3+, respectively, and 0.6, 1.5, and 3.3 for the corresponding perchlorates. Conductivity measurements were made by use of a Fuso 360 linear bridge conductometer at temperatures of 5~ from 0 to 50~ Further details of the experimental procedure 0167-7322/95/$09.50 9 1995 Elsevier Science B.V. All rights reserved. SSDI 0167-7322 (95) 00830-6
358 have been described previously. 3 Concentrations of the solutions are 2 • 10"4 to 2 • 10-3 mol dm-3. Ill. Results Observed molar conductivities were analyzed by assuming the ion association (ion-pair formation) between the complex ions and the counter ions in the same manner as described previously.3 The closest distances of approach of ions (a) in the Robinson-Stokes conductivity equation and in the Debye-HOckel equation were taken as 6.8 and 7.3 A for chlorides and perchlorates of the tris(phen) complexes; 6.6 and 7.1 A for those of the tris(bpy) complex, respectively, using the effective ionic radii (let') of the complex ions, shown in Table 1, and those of CI" (1.81A) and CIO 4" (2.30,/k). 3 The values of ref were estimated from the ionic partial molar volumes (Vi ~176 by use of Glueckauf' s equation.3, 4 Limiting molar conductivities of the complex ions (2 ~176 obtained were expressed as a function of temperature (t in ~ by the equation: 2 ~176
= 2 ~176176 + a (t- 25) + b (t- 25) 2 + c (t- 25) 3,
(1)
where the parameter values (averaged) are summarized in Table 2. Stokes radii (rs) and temperature coefficients of the Walden product (din( ~ oo r/o)/dT) are given in Table 1. The ion-association constants (KA) are shown as logarithmic values in Fig. 1 and were expressed by the equation:3 (2)
log KA = P (t- t m)2 + log KA(m),
where the parameter values are summarized in Table 3. Entropy, enthalpy, and heat capacity changes ( AS*, A H ~ and A Cp 0) of the ion association are given in Table 4.
Table 1. Partial Molar Volume (V i~176Effective Ionic Radius (ref), Stokes Radius (rs), and Temperature Coefficient of Walden Product (din( ;t oo 77o)/dT) at 25~ Complex ion
Vi oo
r_c,.
rs
cm3mol"1
A
,~,
rs/ref
din( ;t oo r/o)/dT K -1
[Fe(phen)3]2+
399--- 1
4.95
5.24
1.06
0.0021 •
[Co(phen)3]3+
374+--2
4.96
5.13
1.03
0.0012-.+0.0001
[Co(bpy)3] 3+
333 ---2
4.78
4.90
1.03
0.0011 - 0.0001
Table 2. Parameter Values of Eq. 1 Complex ion
2 ~176176 S cm2mol "1
a S cm2mol'lK "1
b • 103 S cm2mol'lK "2
c • 105 S cm2mol'lK "3
[Fe(phen)3 ]2+
35.1
0.874
4.8
-3.4
[Co(phen)3]3+ [Co(bpy)3]3+
53.9 56.4
1.291 1.344
6.4 6.8
-3.0 -2.4
359 Table 3. Parameter Values ofEq. 2 tm oC
log KAfm)
-1.17-+0.16
39-+2
1.580---0.001
1.79-+0.45
33 -+2
1.467-+0.001
Salt [Fe(phen)3](CIO4) 2
p • 105 K-2
[Co(phen)3]Cl 3 [Co(phen)3](CIO4) 3
1.55 -+0.14
81 -+5
2.073 -+0.005
[Co(bpy)3]Cl 3
3.43 -+ 0.10
24 --- 1
1.495 -+ 0.001
[Co(bpy)3 ](CIO4) 3
0.94 --+0.22
79 -+ 12
2.023 -+ 0.006
Table 4. Thermodynamic Parameters of the Ion Association at 25 ~ Salt [Fe(phen)3]Cl 2
~aa ~
AS ~
31-1 ~
~C,. ~
dm3mol "1
KA
kJ mol "l
J K'lmol "1
kJ mol -I
J K-lmol -l
5 --+2
-4.0 --- 1.0 -36-+5
[Fe(phen)3](CIO4) 2
38---3
-9.0-+ 0.2
32.1 -+0.5
0.5 -+0.2
[Co(phen)3]Cl 3
29-+2
-8.3-+0.2
26.4-+ 1.4
-0.5-+0.4
132-+2
-12.1 -+0.1
30.7-+0.5
-2.9-+0.2
33---4
31 -+ 2
-8.5 -+ 0.2
29.1 -+ 0.3
0.1 -+ 0.1
118 -+ 4
112-+3
-11.7-+0.1
33.5-+0.7
-1.7-+0.2
20_.+7
[Co(phen)3](C104) 3 [Co(bpy)3]Cl 3 [Co(bpy)3](CIO4) 3
57-+ 15
IV. Discussion All the complex ions studied are regarded as hydrophobic structure-makers, since the din( 3. ~176 values are positive, similarly to the tetraalkylammonium ions. 5 The hydrophobicity of the trivalent [Co(phen)3]3+ ion is inferior to that of the divalent [Fe(phen)3]2+ ion since the din(3, oo 7?o)/dT value of the former complex is smaller than that of the latter complex. This suggests that the hydrophobic atmosphere around the complex ion is weakened with increasing ionic charge, correlated to the reduction of the partial molar volume (Vioo) which is explicable as the increase in the electrostriction by Glueckauf's equation. 4 The ion-association constants for the perchlorates are significantly larger than those for the chlorides, contrary to the relation observed for hydrophilic complexes such as hexaammine- or tris(ethylenediamine)cobalt(l]I) complexes.3, 6 The ion association of the hydrophobic complex ions with perchlorate ion has a larger advantage due to the weak hydration of the anion compared with chloride ion. Moreover, the perchlorate ion, being a structure-breaker with a relatively large volume, may partly expel some hydration water molecules and weaken the hydrophobic hydrogen-bonding of the adjacent water molecules. This effect may produce some excess free energy to stabilize the ion association, because the positive free energy due to the hydrophobic hydration is reduced. The ion-association constants of the perchlorate of [Co(phen)3]3+ are larger than those of [Co(bpy)3]3+, while the K A values of their chlorides are similar to each other. This indicates that the [Co(phen)3]3+ ion is somewhat more hydrophobic although the difference in din( 3, oo ~7o)/dT is slight.
360 The K A values for the chlorides are a little larger than the prediction of the ion-association theory7 as depicted in Fig. 1. The difference in A G ~ the perchlorates of [Co(phen)3]3+ and [Fe(phen)312+ (ca. -3 kJ tool"l) is not so much different from that between the chlorides (ca. -4 kJ mol'|). This means that the excess free energy produced by the partial loss of the hydrophobic hydration water molecules is additive to that due to the long-range electrostatic interaction although the excess free energy is slightly reduced with increasing charge of the complex ion. The experimental values of A H ~for the 2.5 perchlorates are smaller than those for the chlorides as well as the theoretical ones:3, 7 AH~ and 3.7 ld mol"1 ( a = 7 A ; 25~ for 2:1 and 3:1 electrolytes, respectively. Therefore, the excess free energies for the per2.0 chlorates against the chlorides are apparently based on relatively large AS ~ small A H ~ values. Negative or small AH~ be explain",c
AH~ the perchlorates become smaller than those for the chlorides or the theoretical values, 7 actually as observed for A H ~ For the naphthalenedisulfonates of [Fe(phen)3]2+, both observed values ofA S ~ AH ~ significantly smaller than the theoretical ones. 1,7 The negative A Cp ~ the perchlorate of n 2+, aso 1 observed for the naphtha[Fe(p he)3] lenedisulfonates, l indicates the hydrophobic effect to be dominant on the ion association.
JI..--It---ll.--a
6
I0
2'0 3'0 4'0 50 t/'c Fig. 1. log K A values for chlorides (open symbols) and perchlorates (solid symbols) of" [Co(phen)-~]3+(C), O), with Eq. 2; dotted and dashed lines are theoretical curves (a = 7 ,~,) for 2:1 and 3:1 electrolytes, respectively.7
References 1 H. Yokoyama, Y. Koyama, and Y. Masuda, Chem. Lett., 1988, 1453. 2 M. Iida, M. Iwaki, Y. Matsuno, and H. Yokoyama, Bull. Chem. Soc. Jpn., 63, 993 (1990). 3 H. Yokoyama and H. Kon, d. Phys. Chem., 95, 8956 (1991). 4 H. Yokoyama, M. Mochida, and Y. Koyama, Bull. Chem. Soc. Jpn., 61, 3445 (1988). 5 R. L. Kay, Ionic Transport in Water andMixed Aqueous ,Solvents, in Water, edited by F. Franks, Vol. 3, Chap. 4 (Plenum, New York, 1973). 6 H. Yokoyama, T. Ohta, and M. Iida, Bull. Chem. Soc. Jpn., 65, 2901 (1992). 7 H. Yokoyama and H. Yamatera, Bull. Chem. Soc. Jpn., 48, 1770 (1975).
journal of ~OLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65166 (1995) 36 i-364
Structure of the Monohalogenocadmium(II) Complexes in
N,N- Dime th yi form amide Kazuhiko Ozutsumi Department of Chemistry, University of Tsukuba, Tsukuba 305, Japan Abstract The structure of the monohalogenocadmium(II) complexes, [CdCl] +, [CdBr] +, and [CdI] +, in N,N-dimethylformamide (DMF) has been studied by the X-ray diffraction method at 23 ~ revealing the six-coordinate octahedral structure of the complexes. The Cd-O bond length within [CdX(dmf)5] § is 239(3) pm for X = CI, 238(1) pm for X = Br, and 241(2) pm for X = I. The Cd-O bond lengths within the three complexes are practically the same within experimental uncertainties but are significantly longer than that (230 pm) within the solvated [Cd(dmf)6] 2§ ion in DMF. The Cd-X bond length within [CdX(dmf)5] + is 259(3) pm for X = C1, 265(1) pm for X = Br, and 279(1) pm for X = I.
I. Introduction The complexation of cadmium(H) with halide ions has extensively been studied in water as well as in nonaqueous media, and the formation of mononuclear [CdX] § [CdX2], [CdX3]-, and [CdX4] 2- (X = C1, Br, I) complexes has been elucidated. 1-3 The logKl values for the formation of [CdX] § increase in the order C1 < Br < I in water, while the values decrease in the sequence C1 > Br > I in N,N-dimethylformamide (DMF) and dimethyl sulfoxide (DMSO). This suggests that the cadmium(U) ion behaves as a hard ion in DMF and DMSO. With manganese(II), cobalt(U), and nickel(II), which are relatively hard acceptors, the logK1 values for the formation of [MX] § (M = Mn, Co, Ni) decrease in the sequence CI > Br in DMF but are smaller than those of [CdX]+. 4 Also, the decreasing extent of the logK1 values for [MX] § (M = Mn, Co, Ni) with increasing the ionic radii of halide ions is greater than that of [CdX] § These facts reflect a soft character of the cadmium(II) ion in DMF. Thus, the cadmium(H) ion has hard and soft characters to a similar extent. In the present study the structure of [CdX] § in DMF was determined by the X-ray diffraction method in order to elucidate hard and soft characters of the cadmium(H) ion. II. Experimental All chemicals used were of reagent grade. Cadmium(II) perchlorate DMF solvate was prepared and DMF was purified as described elsewhere. 3 Cadmium(II) chloride, bromide, and iodide were used without further purification and dried in a vacuum oven at 100 ~ Four sample solutions were prepared for X-ray diffraction measurements. Solutions A, B, and C were prepared by dissolving cadmium(H) perchlorate DMF solvate along with CdC12, CdBr2, and CdI2, respectively, in DMF. The [CdX] § complex exists as a main species in these solutions. Solution D is a cadmium(II) perchlorate DMF solution and contains the solvated [Cd(dmf)6] 2§ ion. 5 The composition of the sample solutions is summarized in Table 1. All solutions were prepared and treated in a dry box under nitrogen atmosphere. X-Ray diffraction data were obtained on a 0-0 type diffractometer (RINT 1100S, Rigaku) at 23 ~ An Mo tube (~ = 71.07 pm) was used as the X-ray source. The accessible range of scattering angle 20 was 2 - 140 ~ Times required to accumulate 120000 counts were recorded at 0167-7322/95/$09.50 (C) 1995 Elsevier Science B.V. All rights reserved. SSD! 0167-7322 (95) 0083 !-4
362 each angle of 20 and the measurement was repeated three times over the whole angle range. The method of X-ray diffraction measurements and data treatments were described in a previous paper. 5 Calculations were performed by using a program KURVLR. 6 Table 1. The composition (mol dm-3), stoichiometric volume V per cadmium atom, and density p of the sample solutions
Cd 2§ XCIO4DMF V I 109 pm 3 p / g cm -3
A [CdCI] +
B [CdB r] §
C [CdI] +
D Cd 2+
0.8280 0.3384 1.318 12.38 2.006 1.141
1.029 1.028 1.030 12.30 1.614 1.199
0.9455 0.9488 0.9422 12.20 1.756 1.212
0.9424 1.885 12.28 1.762 1.191
III. Results and Discussion The reduced intensities i(s) multiplied by s for solutions A to D are shown in Fig. 1. Figure 2 depicts the differential radial distribution curves, D ( r ) - 4nr2po, where po represents the average scattering density in the stoichiometric volume V per cadmium atom. In the ( D ( r ) - 4m'2po) curve for solution D involving the [Cd(dmf)6] 2§ ion, 5 two large peaks are observed at 140 and 230 pm and two small humps at 325 and 450 pm. The first peak at 140 pm is ascribed to the C-H, C-N, and C=O bonds within DMF and the C1-O bonds within CIO4-. The second peak at 230 pm is due to the Cd-O bonds within the [Cd(dmf)6] 2§ ion. The nonbonding C.-.C, C..-O, and N...O interactions within DMF and the O...O contacts within CIO4" also contribute to the peak. The third and fourth peaks at 325 and 450 pm are mainly attributed to the nonbonding Cd-.-C (DMF) and Cd-..N (DMF) interactions, respectively. The peaks in the ( D ( r ) - 4nr2po) curves for solutions A, B, and C are assigned similar to solution D, though the second peak for solutions A and B shifts to 235 and 250 pm, respectively, and that for solution C is split into two peaks, consistent with the formation of [CdX] § In order to extract peaks solely consisting of the intramolecular interactions within [CdX] § the peak shapes due to the intramolecular interactions of DMF and CIO4- should be subtracted from the observed radial distribution function. The structure parameters for DMF and C104have been reported and the parameter values were quoted from the literatures. 7,8 Subtraction of the calculated peak shapes due to the intramolecular structure of DMF and CIO4- from the original curve (solid line) gives the residual curve (dashed line). A peak appears at 235, 255, and 275 pm in the residual curves of solutions A, B, and C, respectively. According to the formation constants of the cadmium(H) halogeno complexes, 3 [Cd(dmf)6] 2§ [CdX] § [CdX2], and [CdX3]- coexist in solutions A, B, and C. Since the relative amounts of [CdX2] and [CdX3] do not exceed 10 %, the three solutions were treated to contain [Cd(dmf)6] 2+ and [CdX] § The relative amount of [Cd(dmf)6] 2§ was evaluated to be 60, 20, and 30 % in solutions B, C, and D, respectively. In the course of the analysis of the ( D ( r ) - 4nr2p0) curves, the [Cd(dmf)6] 2§ ion coexisting in the solutions was taken into consideration by subtracting the theoretical peaks resulting from [Cd(dmf)6] 2§ The area under the peak appearing at 235 pm in the residual curve of solution A is larger than that due to six Cd-O bonds of solution D and corresponds to one Cd-C1 and five Cd-O bonds. Also, the peak in the residual curves of solutions B and C is well reproduced by considering one Cd-X and five Cd-O bonds. Thus, [CdX] § in DMF has a six-coordinate octahedral structure. Further subtraction of the peaks due to the O.--O, O...X, Cd-..C, and Cd-..N interactions based on the octahedral [CdX(dmf)5] §
363
structure leads to smooth background curves (dotted lines) in Fig. 2, with no distinct peaks over the range r < 450 pm.
0 \k J ,\
-lfi 16
,.
T
B: [CdBr ]+
~
:ji
Q C1
-16 !~
... 9
....
%
C" [Cd I ] +
:. v
-
-
0
o
L
-
k.
-16 16 o
":" i i
o
D : C d 2+ ,,_... . . . .
!
............
-,:,'
l
0
4
, , ,
8 12 s/10 -2 pm-1
16
1
Figure 1. The reduced intensities i(s) multiplied by s for sample solutions A to D. The observed si(s) values are shown by dots and calculated ones by solid lines.
-24.
0
2
-"
9 4.
6
r/1 0 2 pm
8
2+ 0
Figure 2. The (D(r) - 4m'2po) curves for the sample solutions. Dashed lines represent residual curves after subtraction of the peak shapes for the intramolecular interactions within DMF molecules and perchlorate ions. Dotted lines represent residual curves after subtraction of the peaks for the Cd-O, Cd-X, O--.O, O...X, Cd---C, and Cd--.N interactions within the complexes.
The structure parameters of [CdX(dmf)5] + were finally determined by a least-squares calculation applied to the high-angle region of si(s) values (s > 4 x 10-2 pm-[). 5 In the course of the refinement, the structure parameters for DMF and CIO4- were fixed at the literature values. 7,8 The interatomic distances r and temperature factors b for the interactions of the cadmium(II) ion with X- and DMF in the first coordination sphere were independently optimized. Based on the octahedral [CdX(dmf)5] § structure, the frequency factors n of all interactions were kept constant and the interligand O..-O and O..-X distances were calculated from the optimized Cd-O and Cd-X bond lengths. The temperature factors of the O..-O and O-..X interactions were evaluated as the sum of those of the Cd-O and Cd-X bonds. The solid lines calculated by using the optimized parameter values in Table 2 reproduce well the experimental values over the range s > 4 x 10-2 pm -l as shown in Fig. 1. The Cd-O bond length within [CdX(dmf)5] § is 239(3) pm for X = C1, 238(1) pm for X = Br, and 241(2) pm for X = I. The Cd-O bond lengths within the three complexes are practically the same within experimental uncertainties but are significantly longer than that (230 pm) within [Cd(dmf)6] 2+. The Cd-X bond length within [CdX(dmf)5] + is 259(3) pm for X =
364 CI, 265(1) pm for X = Br, and 279(1) pm for X = I. The ionic radius of halide ion is 167 pm for X = C1, 182 pm for X = Br, and 206 pm for X = 1.9 As far as the electrostatic interaction between the cadmium(H) and halide ions is concerned, the Cd-Br bond length is expected to be longer by 15 pm than the Cd-CI one, and the Cd-I bond length is 24 pm longer than the Cd-Br one. The difference in the observed Cd-X bond lengths is 6 pm for X = Br and CI, and 14 pm for X = I and Br. These values are appreciably smaller than those estimated on the basis of the ionic radii. Moreover, the difference between the observed Cd-X bond length and the ionic radius of halide ion is 92, 83, and 73 pm for X = CI, Br, and I, respectively. Thus, the covalent character in the Cd-X bond increases in the order Cd-C1 < Cd-Br < Cd-I. Table 2. Structure parameters for the cadmium(H) complexes in N,N-dimethylformamide a Interaction Cd-O
Cd-X
Parameter r / pm b / 1 0 pm 2 n r /pm b / 1 0 pm 2 n
O-'-O
O..-X
r c/pm b d/10 pm 2 n
r */pm b f/IO p m 2
n Cd...C
r/pm b / 1 0 pm 2 r/
Cd-..N
rlpm b / 1 0 pm 2 n
[CdCI] +
[CdBr] +
[CdI] +
Cd 2+
239 (3) 6 (2) 5b 259 (3) 12 (5) 1b 338 12
238 ( 1) 9 (2) 5b 265 (1) 6 (2) 1b 337 19
241 (2) 9 (2) 5b 279 (1) 2 (1) 1b 341 19
230 (1) 9 (1) 6b -
8 b
12 b
4b 312(2) 9 (2)
321 (1) 17 (2)
5 b
5 b
4b 311 (4) 17 (3)
431 (23) 72 (37) 5b
428 (3) 26 (5) 5b
443 (4) 23 (5) 5b
435 (2) 23 (4) 6b
8 b
352 18
4b 319(11) 28 (12)
8 b
356 15
369 II
5 b
-
325 19 -
6 b
a Estimated standard deviations are given in parentheses. b The values were kept constant during the calculation. c r(O--.O) = 21/2r(Cd-O). d b(O--.O) = 2b(Cd-O). e r(O-..X) = {r(Cd-O) 2 + r(Cd-X) 2 }t/2. f b(O...X) - b(Cd-O) + b(Cd-X). References
1 E. H6gfeldt, Stability Constants of Metal-lon Complexes, Part A (Inorganic Ligands); IUPAC Chemical Data Series 21 (Pergamon Press, Oxford, England, 1982). 2 S. Ahrland, Pure Appl. Chem., 51, 2019 (1979). 3 S. Ishiguro, K. Ozutsumi, M. Miyauchi, and H. Ohtaki, Inorg. Chem., 28, 3258 (1989). 4 K. Ozutsumi and S. Ishiguro, J. Chem. Soc., Faraday Trans., 86, 271 (1990). 5 K. Ozutsumi, T. Takamuku, S. Ishiguro, and H. Ohtaki, Bull. Chem. Soc. Japan, 62, 1875 (1989). 6 G. Johansson and M. Sandstr0m, Chem. Scr., 4, 195 (1973). 7 H. Ohtaki, S. Itoh, T. Yamaguchi, S. Ishiguro, and B. M. Rode, Bull. Chem. Soc. Japan, 56, 3406 (1983). 8 H. Ohtaki and M. Maeda, Bull. Chem. Soc. Japan, 47, 2197 (1974). 9 R. D. Shannon, Acta Crystallogr., Sect. A, 32, 751 (1976).
journal of
MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 365-368
Effect of Pressure on the Viscosity at 25 *C
of Aqueous Cesium Chloride Solution
T. Nakai, S. Sawamura, and Y. Taniguchi Department of Chemistry, Faculty of Science and Engineering, Ritsumeikan University, 1916 Noji-cho, Kusatsu, Shiga 525-77,
Japan
Abstract Viscosity of aqueous cesium chloride (CsCI) solution was measured in the range of 0.1-5.0 mol kg -1 and 0.1-375 MPa at 25 *C. The Jones-Dole B coefficient of CsCI was obtained from the concentration dependence of the viscosity. It is negative not only at atmospheric pressure but also at high pressure, having a maximum against pressure at about 160 MPa. Similar maximum of the B was observed for aqueous sodium chloride (NaCI) solution. The similarity is discussed in terms of the water structure and dielectric friction theory.
Introduction Concentration dependence of the viscosity of electrolyte solution has been empirically expressed by the Jones-Dole equation 1 (eq.(1)),where 7/and r/0 are the viscosities of solution (1) r//r/0 = 1+ A~c+ Bc and solvent, respectively. A and B are coefficients, and c is the molar concentration. The B, informing us of ion-solvent interactions, depends on not only the sort of solute and solvent but also external conditions such as temperature and pressure. It is well known that the B for aqueous solution strongly depends on temperature because of the structural change of water by heat.2,3 Our purpose is to elucidate the correlation between the B and water structure from the viewpoint of the pressure effect. Some abnormal behaviors of water under high pressure have been observed in dynamical properties at room (or lower) temperature, such as the shareviscosity minimum, activation-energy minimum of viscous flow, 4 and spin-lattice relaxationtime maximum of proton NMR. 5 These phenomena suggest that the structural change of water by pressure is an important factor for the B as well as the change by temperature. In an earlier study we measured the high pressure viscosity of aqueous NaCI solution and estimated the B value. 6 At 25 ~ the B increases with increasing pressure initially and then decreases after having a maximum at about 100 MPa. In the present paper we estimate the pressure dependence of the B for aqueous CsCI solution and compare it with that for aqueous NaC1 solution.
Experimental Section Cesium chloride (Nacalai Tesque Co.) was recrystallized from distilled water. Sample Sample solutions were prepared by weight at the concentration of 0.100-5.000 (• mol kg -1. The molality was converted into the molarity (c) using the density of aqueous CsC1 0167-7322/95/$09.50 91995 Elsevier Science B.V. All rights reserved. SSD! 0167-7322 (95) 00832-2
366 solution at high pressure. 7 The solution were passed through membrane filter with a pore size O. 1 tam to remove dust. Viscosity measurement Viscosity was measured using a rolling-boll viscometer up to375+0.5 MPa with 25 MPa increment. The accuracy was estimatedas =el %. Detailsof the viscometer were described previously. 6
Results and Discussion High Pressure Viscosity
Figure 1 shows the concentration dependence of the viscosity of aqueous CsCI solution. At 0.1 MPa our value agrees well with the literature values. 8,9 Reduction of the viscosity with increasing concentration has been ascribed to the breaking effect of electrolyte on the water structure. Fig. 1 shows that the effect does not disappear up to 375 MPa, suggesting that the structure-breaking effect of CsCI in water still existed at high pressure.
111
375
300 1.0 200 100
0.9
0.1
MPa 0.8 l 0
I
1
I
2
I
I
3
c / mol dm
4 -3
Figure 1. Concentration dependence of the viscosity of aqueous cesium chloride solutions at 25 ~ and several pressures. O ' T h i s work, i1' Ref. 8, A" Ref. 9.
367
.|ones-Dole B coefficient The B coefficient was obtained by fitting the data to the extended Jones-Dole equation 10 (eq.(2)) using the least square method. The A value was (2) r//r/o = 1+ A~/c + B e + Dc 2 theoretically calculated from the method by Falkenhagen et a111 using the high-pressure data, i.e., the limiting molar conductances of ions 12, the viscosity 13, and the static dielectric constant 14 as tO be 0.0049 mo1-1/2dm3/2 at 0.1 MPa and 0.0047 mol -1/2dm3/2 at 200 MPa. Because the difference between these values is negligible, we used 0.0049 mol-1/2dm3/2 as the A values at all pressures. Figure 2 shows the pressure dependence of the B of CsCI in water. That of NaCI is also plotted for comparison. The B of CsCI, being negative at all pressures, increases and then decreases with increasing pressure having a maximum at about 160 MPa. Similar maximum of the B against pressure is observed for NaCI at about 100 MPa. The increment of the B for CsCI from 0.1 MPa to the maximum pressure is larger than that for NaCI. And the maximum pressure for CsCI is larger than that for NaCI. The maximum in Fig. 2 may be caused by two opposite effects. One is a loosening
NaC1
0.10
o
0.05 _ I I
0.00 CsC1
-0.05
I
0
1
200
!
400
P / MPa Figure 2. Pressure dependence of Jones-Dole B coefficients of cesium chloride and sodium chloride in water at 25 *C.
368 of bulk-water structure by pressure, causing a lowering of water-viscosity. Because the structure of solvation water around ion is strongly fixed by the electrostatic force and is much less sensitive to pressure than that of bulk water, the breaking-effect of CsC1 and NaCI on the water structure becomes relatively small at high pressure. And the B apparently increases with increasing pressure because of a loosening of the bulk-water structure. The other effect is the dielectric friction between an ion and water. It can be estimated from high-pressure properties of water, i.e., the viscosity, 13 the static dielectric constant, 14 and the dielectric relaxation-time 15 using Ibuki and Nakahara's method 16 based on Hubbard-Onsager dielectric friction theory. The effect decreases the B value with increasing pressure and the degree is almost the same for both CsCl and NaCl. These two effects are essential for pressure effect of the B in aqueous electrolyte solution and the observation in Fig. 2 is a balance of them. The larger increment of the B of CsCl up to the maximum pressure than that of NaCl may be ascribed to the fact that the former effect for CsCl is stronger than that for NaC1 because the breaking-effect of CsCl on the water structure is larger at atmospheric pressure.
Acknowledgment This work was supported by a Ritsumeikan Academic Grant for Specific Research A.
References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
G. Jones and M. Dole, J. Am. Chem. Soc., 51,2951(1929). M. Kaminsky, Discuss. Faraday Soc., 24, 171 (1957). D . J . P . Out and J. M. Los, J. Sol. Chem., 9, 19(1980). R. A. Home and D. S. Johnson, J. Phys. Chem., 70, 2182(1966). J. Jonas, T. DeFeries, and D. J. Wiblbur, J. Chem. Phys., 65,582(1976).; T. DeFeries and J. Jonas, J. Chem. Phys., 66,896(1977). S. Sawamura, Y. Yoshimura, K. Kitamura, and Y. Taniguchi, J. Phys. Chem., 96, 5526(1992). T. Nakai, S. Sawamura, and Y. Taniguchi, to be submitted. P. A. Lyons and J. F. Riley, J. Am. Chem. Soc.,76, 5216(1954). T. Isono, Rikagaku-Kenkyusho Houkoku, 61,53(1985). G. Jones andT, Talley, J. Am. Chem. Soc., 55, 4124(1933). H. Falkenhagenand E. L. Vernon, Philos. Mag., 14,537(1932). M. Nakahara, T. T/Jr/sk, N. Takisawa, and J. Osugi, J. Chem. Phys., 76, 5145(1982). InternationalAssociation for the Properties of Steam: The lAPS Formulation1985for the Viscosity of Ordinary Water Substance, 1985. K. R. Srinivasan and R. L. Kay, J. Chem. Phys., 60, 3645(1974). R. Pottel, E. Asselborn, R. Eck, and V. Tresp, Ber. Bunsenges. Phys. Chem., 93, 676(1989). K. Ibuki and M. Nakahara, J. Chem. Phys., 85,7312(1986).
journal of
MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 369--372
Melting Behaviour in Hexagonal CeCls and Monoclinic ErCi3 Crystals Yasuhiko lwadate, Norikazu Okako, Yasuyuki Koyama, Hirokazu Kubo, and Kazuko Fukushima
Department of Materials Science, Faculty of Engineering, Chiba University, Yayoi-cho 1-33, Inage-ku, Chiba 263, Japan Abstract Precise analyses of liquid structures of pure CeCI3 and ErCI3 were made by consulting the bond lengths in crystals and the volume changes on melting, allowing for octahedral geometry around the lanthanide atoms and the corner-sharing linkage of octahedra in molten CeCl3 and the edge-sharing combination of octahedra in molten ErCl3.
I. Introduction The liquid structure of pure lanthanide trichlorides as well as their mixtures with alkali metal chlorides has so far been analyzed by X-ray diffraction. 1 High temperature Raman spectroscopy has been applied to rare earth chlorides. 2 Lanthanide atoms in the melts were found to be surrounded by six chloride atoms, that is, octahedrally coordinated. In the previous work, 1 the existence of dioctahedra was also deduced from the analysis of interatomic distances. Recently, a novel melting mechanism and liquid structure of YCl3 have been proposed by Saboungi et al. 3 and structural correlations in molten trivalent metal chlorides have been estimated as functions of the metal ion size across the range from LaCl3 to AICl3 using a charged soft-sphere model and the hypernetted chain approximation. 4 The main purpose of the present work is to investigate the short range structure of molten CeCl3 and ErCI 3 and to consider the melting behaviours in terms of the volume changes on melting, the metal-metal correlation lengths, and the difference in linkage of octahedra.
H. Short Range Structure in Solid and Liquid, and Volume Changes on Melting The details of sample preparation, molar volume measurement, and X-ray diffraction analyses have been reported in the preliminary works on molten CeCl3 and ErCl3 .5'6 CeCl3 is known to be hexagonal with a bimolecular unit, 7 in which each Ce atom is surrounded by nine Cl atoms. On the other hand, ErCl3 forms a monoclinic crystal, 8 the unit cell of which contains four molecules. The number of the nearest Cl atom around Er atom is six. As shown in Fig. 1, the first peak centered at 0.28 to 0.29nm appeared sharply in the correlation function G(r) for molten CeCI3, and the broad peak existed in the range of 0.46 to 0.6nm upon which the shoulder-like peak overlapped in the neighborhood of 0.4nm. According to the ionic radius list by Shannon, 9 the first peak was assignable to the Ce-Cl pair since the ionic radii of Ce3"(VI) and CI-(V[) were 0.101 and 0.181 nm, respectively, and the 0167-7322/95/$09.50 91995 Elsevier Science B.V. All rights reserved. SSDI 0167-7322 (95) 00833-0
370
----r--r--T
l-i
I
1'--T"----'r--""l--~
1 __.___.+____q_
---+---k molten E r C I 3 _
0
0.5
1
r/nm Figure 1. Correlation functions G(r)'s for molten CeCl 3 and ErCI3. summation was 0.282nm close to the peak position. By integrating the corresponding peak in the radial distribution function D(r), the number of CI atom around Ce atom was roughly estimated at about 6. In contrast, Er atom was invariably coordinated by some six CI atoms also in the melt, as was observed in the crystal. If this situation stands reasonable, the large variation of coordination number might provoke the corresponding change in volume on melting. The molar volume changes on melting are listed in Table 1, where Tmp, Vs, and V m refer to the melting point, the molar volumes of solid and melt, respectively. Table 1. Molar volume changes on melting System Trap/K V, / cm3mol -~ V , / cm3mo1-1 IO(~m-VJV/' / %
LaCI 3 1150 63.90 76.30 19.1
CeCI 3 1095 62.42 76.06 21.9
PrCI 3 1059 61.40 74.30 21.0
NdCI 3 SmCI 3 GdCI 3 DyCI 3 HoCI 3 1029 935 875 928 993 60.47 59.23 58.03 74.38 73.02 73.65 73.73 73.34 74.62 74.00 21.8 24.5 26.4 0.3 1.3
ErCI 3 YCI 3 1046 987 72.11 74.83 75.65 75.17 4.9 0.5
The molar volume of molten CeCI 3 at the melting point was estimated by extrapolation of the empirical equation reported in the literature, ~~and that of molten ErCI3 was taken from the previous work. 6 The volume changes on melting could roughly be classified into two groups, the hexagonal group expanding by about 20% or more on melting as well as the monoclinic group less expanding by 5% at most. Therefore, the nearest neighbour ordering in molten ErCI3 appears to be similar to that in the solid state. By a least-squares fitting the nearest neighbour distance, r(Ce-Cl), was estimated at 0.284nm and the number of C1 atom around Ce atom was 5.6, virtually equal to 6. Moreover, the interatomic distances of CI-CI and Ce-Ce pairs were computed to be 0.405nm and 0.521nm, respectively. The corresponding pair distances in solid were calculated to be 0.292nm (0.303nm) for Ce-CI,
371
0.335nm (0.341nm, 0.350nm, and so on) for CI-CI, and 0.431nm (0.481nm) for Ce-Ce, in which the distances in the parentheses refer to the second neighbour of the corresponding pair, the third one, and so on hereafter. The ratio of r(CI-CI)/r(Ce-CI) obtained for the melt was 1.43, being approximately equal to ~ characteristic of octahedral configuration. As for molten ErCI3, r(Er-Cl), r(CI-CI), and r(Er-Er) were evaluated at 0.263nm, 0.375nm, and 0.405nm, respectively. The corresponding distances in solid were 0.254nm (0.261nm, 0.267nm), 0.342nm (0.343nm, 0.365nm, and so on), and 0.391nm (0.393nm), respectively. The number of CI atom around Er atom in the melt was 5.8. It was, anyway, confirmed from the above results that the octahedral configuration of CI atom around lanthanide atom existed in molten CeCI3 and ErCl 3. The short range ordering in CeCI 3 melt was quite different from that in solid, especially in coordination, but not for molten ErCI3.
HI. Intermediate Range Structure Assuming that the Ce atoms are homogeneously distributed in the melt, the mean interatomic distance, Ce-Ce, is calculated from molar volume 5 to be 0.504nm. As mentioned before, the most probable interatomic distance for Ce-Ce pair was 0.521nm, being larger than the above value and indicating that there is some special configuration between cerium hexachloride units (octahedra). When two octahedra approached each other as closely as possible, the shortest Ce-Ce distance was estimated at 0.604nm by considering the steric hindrance of CI ligands and the equality of intra-octahedral and inter-octahedral C1-CI distances (impossible to be separated from each other in XRD analyses). It is, therefore, not the real arrangement that the melt structure in the intermediate range is composed of discrete octahedra. Next, it was tried to consider the model that two octahedra shared two CI atoms and formed, for example, Ce2Cllo-type dioctahedron, being called an edge-sharing model. Due to the short range XRD analysis and the Raman spectroscopy, 5 the angle of C I - C e - C I and the distances of Ce-CI and CI-CI were 90 ~ , 0.284nm and 0.405nm, respectively. The Ce-Ce distance should be, thus, equal to 0.405nm but the observed value was 0.521nm, indicating that the edge-sharing model was not good for the CeCI 3 melt. As the third model, we adopted a comer-sharing model that two octahedra shared one C1 atom and formed a Ce2Clla-type dioctahedron. When the Ce-C1-Ce configuration was linear, the interatomic distance between Ce-Ce was the same as the summation of respective ionic radii, being 0.564nm according to Shannon. 9 This value was indeed greater than 0.521nm but if the dioctahedron was bent and twisted in such a manner that the C e - C I - C e angle was about 133 ~ , the corner-sharing model became valid in a sense. Moreover, clusters of octahedra described as a linear "polymerization" in the form CenCl~sn~l) did not satisfy the stoichiometry of CeCI 3 on the whole since the value of CI/Ce ratio only approached to 5 as much as the linear clustering proceeded. Therefore, any clustering of octahedra on the basis of corner-sharing other than linear "polymerization" seemed to be needed in order to reduce the CI/Ce ratio (satisfy the stoichiometry). Additionally, it should be stressed in understanding the melt structure that the bondings between Ce and CI atoms were not rigid, that is, broken and linked from time to time. As for molten ErC13, the edge-sharing and the corner-sharing octahedral models were
372 taken into account. When undistorted octahedra form an edge-sharing configuration, the Er-Er pair distance would be about 0.373nm at most. This value is, however, smaller than the experimental value. The rigid octahedral edge-sharing model is thus not suitable for the intermediate range structure of molten ErCI3, but a model assuming flexible octahedra with variable Er-CI and CI-CI bond lengths could be valid. This supports the suggestion by Saboungi et al. 3 that molten ErCI 3 may have a structure similar to YCI3 (see Table 1). Next, a linear arrangement of octahedra was considered on the basis of corner-sharing. In this model, the Er-Er pair distance became 0.528nm, being a too large value. If the arrangement was not linear, i.e. the angle of Er-CI-Er was nearly equal to 100" and two octahedra were twisted with each other, this structure might exist. But comer-sharing is thought to be less probable than edge-sharing because of the steric hindrance between CI atoms, although corner-sharing has been expected from numerical calculations on Raman frequencies. 11 Since the Er-Er distance has good correlation between solid and liquid and the volume increase on melting well corresponds to the elongation of Er-Er distance as discussed in the preceding section, it is concluded that the melt structure of ErCI3 is very similar to solid. .
IV. Conclusions The melting behaviour in hexagonal CeCI3 and monoclinic ErCI3 were summarized as follows; (1) the nearest ncighbour chloride coordination number of Ce atom changed from 9 in the crystal to 6 in the melt and that of Er atom was unchanged to be 6 in both crystal and melt, in any event allowing for octahedral geometry around the lanthanide atoms; (2) thcre appears to exist the comer-sharing linkage of octahedra in molten CeCI3 and the edgesharing combination of octahedra in moltcn ErCl3. References 1 J. Mochinaga, Y. Iwadate and K. Fukushima, Mater. Sci. Forum, 73-75, 147 (1991). 2 G.N. Papatheodorou, J. Chem. Phys., 66, 2893 (1977). 3 M.-L. Saboungi, D. L. Price, C. Scamehorn and M. P. Tosi, Europhys.Lett., 15, 283 (1991). 4 H. Tatlipinar, Z. Akdeniz, G. Pastore, and M. P. Tosi, J. Phys. Condens. Matter, 4, 8933 (1992). 5 J. Mochinaga, M. Ikeda, K. Igarashi, K. Fukushima, and Y. Iwadate, J. Alloys and Compounds, 193, 36 (1993). 6 Y. Iwadate, T. Iida, K. Fukushima, J. Mochinaga, and M. Gaune-Escard, Z. Naturforsch., 493, 811 (1994). 7 R. W. G. Wyckoff, Crystal Structures, Vol.2, 2nd Ed., 1964, pp.77-78 (Intersciencr Pub., New York). 8 ibid. (pp.55-57). 9 R.D. Shannon, Acta Cryst., A32, 751 (1976). 10 G. J. Janz, F. W. Dampier, G. R. Lakshminarayanan, P. K. Lorenz, and R. P. T. Tomkins, Molten Salts, Vol. 1, p. 53 (National Bureau of Standards, Washington, D. C.). 11 A. Erb61iikbas, Z. Akdeniz, and M. P. Tosi, ll Nuovo Cimento, 14, 87 (1992).
journal of
/~OLECULAR
LIQUIDS ELSEVIER
Journal of MolecularLiquids. 65/66 (1995) 373-376
Polarization Effect on the Static and Dynamic Properties of Molten Lithium Iodide Sumiko Itoh Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, U.K. Paul A. Madden PCL, Oxford University, South Parka Road, Oxford OX13QZ, U.K.
Abstract We present how to treat the polarization effect on the static and dynamic properties in molten lithium iodide (LiI). Iodide anion has the biggest polarizability among all the halogen anions and lithium cation has the smallest polarizability among all the alkaline metal cations. The mass ratio of I to Li is 18.3 and the ion size ratio is 3.6, so w e expect the most drastic characteristic motion of ions is observed. The softness of the iodide ion was examined by modifying the repulsive term in the Born-MayerHuggins type potential function in the previous work 1. In the present work we consider the polarizability of iodide ion with the dipole rod method in which the dipole rod is put at the center of mass and we solve the Euler-Lagrange equation. This method is one type of Car-Parrinello method.
I. Introduction Our interest is how to treat polarizability of ion in the molecular dynamics calculation. There have been paid a lot of attention up to n o w and three methods to include the effect of the polarizability are considered; a shell model method, a modified potential method, a dipole rod method. In the shell model, the electron cloud is treated as a spherical shell connected to the center of mass of ion with a certain spring 2. In this method the deformation of the electron cloud is not reproduced, since the electron cloud is fixed as the spheric shape, although the real polarizability is brought from the deviation of the electron cloud from the spheric shape. The modified potential function method is the implicit way to include the polarizability in the calculation by modifying the potential function based on the physical properties in solid state so that it reproduces the pair distribution function obtained by the X-ray and neutron diffraction measurements in the liquid state. The dipole rod method is the explicit way to put a dipole rod at the center of mass of ion so as to reproduce the deviation of the electron cloud from the spheric shape. The aims of this work are to compare the modified potential method to the dipole rod method and we want to make clear which method can express the physical properties more correctly. 0167-7322/95/$09.50 91995ElsevierScience B.V. All rights reserved. 0167-7322 (95) 00834-9
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374 II. M e t h o d
In this method the motion of the atom and the m o v e m e n t of the electron cloud have been treated independently, via an application of the CarParrinello (CP) method 3. The self-consistent values of the induced multipoles at each time step are calculated from the values at the previous time step. The applicability of this method was successfully shown by Sprik and Klein in the calculations of pure water system and the solvation system of the chloride ion, in which the permanent and induced dipoles were considered 4. The polarizability is treated as an embellishment of a rigid-ion model. The induced dipole is expressed as a set of variable charges at certain fixed separation from the ionic center, via a rod of fixed length with a pair of variable charges of the equal magnitude q with opposite sign on its ends. The rod length is taken sufficiently short that the higher-order multipole effects are negligible. So the three degrees of freedom are required to represent the dipole and the two polar angles are necessary to determine the rod orientation. The total kinetic energy is expressed as follows;
T= 89
miv~+89 i~=l r
N N ites . 2 Z Z qi~' 2 qi=l y = l
{1)
w h e r e i represents the ion and i~, the variable charge ~,on ion i, N the total number of ions in the system and Nsites the number of variable charge sites per ion. In the case that the only dipole moment is treated, Nsites is two. The total potential energy is expresses as follows; U=Z
N
N Z
N ~,Qi V C (ri)+
i=lj=i+l u(rij )+
89 i=1
891~ ~vC(riy)qiy+lk q Ny~ y2 q2, i = Iy = 1
where ri represents the center of ion i. The first term type potential function
i = 17 = 1
u(rij ) is
u(rij ) = Bq exp(-aijrij ) - Cq i r6ij- Dij I r~.
(2)
the Born-Mayer
(3)
The second term is the Coulombic term expressed as follows;
v C(r i)=
N 2 q~ Qj y ( y~ ~ + m ) j~:i 7=lrij7 rij
(4)
where Qi is the formal charge on ion i and V C(ri ) is the total Coulomb potential due to formal and variable charges on other ions. The equation of motion is derived from the Lagrangian mechanics ( L = T - U ). The rotational motion is derived from the rotational equation of motion for a rigid rod 5.
375
(5)
~,/o, = ~-, 2
vi = _1 y. diy x VriyU y=l where
,
(6)
diy is the vector from rod-end to rod-end on ion i and vl is the torque on iy
rod i. For the charge variation, the Eular-Lagrange equation for charge becomes as follows;
d ~L ~L =o dt O~iy OqiT, mqitiy = - kqqiy - V(ri)d
y=7,2
(7)
(8)
In the case of the pure Coulombic induced dipoles, the total Coulomb potential can be written by the Taylor expantion, so
mqq i= -kqq i + E C( r i) "di/2
(9)
EC(ri)= (-vVC(ri)) is the electric field at the center of the ion and di is the vector joining the rod ends. The induced dipole #= qidi is where
expressed as
If di2/2kq is chosen to be equal to a, the ion polarizability, the induced dipole is represented by (eEC(ri) and, as can be seen by substituting it into Eq. (2), the induction energy is-la(EC(ri))
2. The equations of motion for
the ion positions are
dOL dt Ok. l
OL...= o
Or.
(ll)
l
mii;i=--VriU
(a2)
That is
N
miJ:i=-Z, [ Vriu(rij)+QiEC(ri)+PiV~iEC(ri)], j~
(13)
where the potential has been Taylor expanded about the center of ion i and
376
l,t= qidi 9 In the calculation we use the Nos6-Hoover thermostats 6. The Eq. (13) becomes as follows; N (14)
mii:i=- Z, [ - - Vriu(rij) +QiEC(ri)--~li VriEC(ri)]-(transmiri,
y~
where (trans is the translational friction term. The Eqs. (6) and (9) become
2 zi = _ 1 ~, diy x V y=l
riy
U - (rot I qro (15)
and
mq "qi - - qqi + EC ). /2 - G : , q :7,
(16)
respectively. ~rot and (chg are the friction paremeters of the rotational motion and the change in the variable charges, respectively. These parameters are updated as (17) where fl represents the translational, rotational or variable charge motion, Tkin,fl is the kinetic temperature of the ]3 system, Tfl the target temperature, and z/t the characteristic thermostat relaxation time. By this method we seek the appropriate parameters of the mass and charge for the dipole rod of iodide ion to perform the molecular dynamics calculation and we evaluate the polarizability effect on the static and dynamic properties on molten LiI.
References 1 S. Itoh, M. Konagai and K. Takahashi, Z. Naturforsch., 46a, 155 (1991). 2 M. J. Sangster and M. Dixon, Adv. Phys., 23, 247 (1976). 3 R. Car and M. Parrinello, Phys. Rev. Lett., 55, 2471 (1985); D. K. Remler and P. A. Madden, Mol. Phys., 70, 921 (1990). 4 M. Sprik and M. L. Klein, J. Chem. Phys., 89, 7556 (1981) M. Sprik, J. Phys. Chem., 95, 2283 (1991); M. Sprik, Computer Simulation in Chemical Physics, edited by M. P. Allen and D. J. Tildesley (Berlin, Kluwer Academic Publishers) p. 211, 1993. 5 Classical Mechanics, by H. Goldstein (Reading, MA: Addison-Wesley) 1980. 6 S. Nos6, Mol. Phys., 52, 255 (1984); S. NosG J. Chem. Phys., 81, 511 (1984); W. G. Hoover, Phys. Rev., A31, 1695 (1985).
journal of
I~IOLECU LA R
LIQUIDS ELSEVIER
Journal of MolecularLiquids, 65/66 (1995) 377-380
Evaluation of Thermodynamic Q u a n t i t i e s of F l u i d M i x t u r e s by t h e P Y Equation Fumio Kawaizumi, Y a s u h a r u A m a k a s u , Masaharu Ohba 1) Hiroyasu N o m u r a School of E n g i n e e r i n g , N a g o y a U n i v e r s i t y . 1 ) K a w a i j u k u Educational
Institution Abstract The enthalpy, internal energy and their excess quantities of the Lennard-Jones binary mixture have been determined using the PY approximation. The values obtained are in good agreement with the results of MC calculation. The enthalpy and isobaric heat capacity are calculated using the extended expression of the thermodynamic quantities in terms of pair correlation functions.
I. Introduction Experimental determination of excess molar quantities such as excess molar enthalpy H E and excess molar volume VE is very important for the discussion of solution properties of binary liquids. Recently, calculation of these thermodynamic quantities becomes possible by computer simulation of molecular dynamics (MD) and Monte Carlo (MC) methods. On the other hand, the integral equation theory has played an essential role in the statistical thermodynamics of solution. The simulation and the integral equation theory may be complementary but the integral equation theory has the great advantage over simulation that it is computationally easier to handle and it permits us to estimate the differential thermodynamic quantities. We have calculated enthalpy, internal energy, excess molar enthalpy, and excess molar internal energy based on the integral equation theory. Validity of its use has been confirmed by the comparison of our results with those of MC calculation. Then, we have calculated the differential thermodynamic quantities of the isobaric heat capacity
Cp and the excess isobaric molar heat capacity, CpE.
lI. Formalism for Calculation and Model S e t t i n g Differentiation of the Fourier transformed Ornstein-Zcrnike equation with respect to T(temperature) gives the following expression. 0167-7322/95/$09.50 ~'~ 1995 ElsevierScience B.V. All rights reserved.
SSD!0167-7322 (95) 00907-8
378
h- =p c
p
+~
+~
(1)
p
Hereafter, the expressions of h, c and D are matrices. In Equation (1) the notation fab denotes the derivatives of quantity f with respect to a under constant value of b. The integral equation of the number density derivatives of the radial distribution function is obtained in a similar way. The differential form of the closure relation is obtained in a similar way. In what follows a rather extended description is given for calculation of the heat capacity. Detailed procedure for calculation of these thermodynamic quantifies will be published elsewhere. 1) The functions
gqr,a and gq'akr,,a'
arc calculated from
differentiation of the Fourier transformed Ornstein-Zernike equation and the differential form of the PY-approximation. The superscript p k attached to gq means Z~k. Expansibility, at,, is calculated from the following equation.
1
ap = - f
,(,
P--~~Pj~Y~(,"
(2)
+f q
Then, the derivatives of the radial distribution function with respect to T at constant P,N is calculated from the following equations.
g q ~,, = g qp
ct e P, g q r ~,
g~q T,x
and
Pk g~i] T:'
where x are the composition. Finally, the hcat capacity is calculated from the following equation. The heat capacity consists of three terms.
___1§247
C~/ N,-
{,
}
2 p , ,=, J=, T A(Cp)
T B(Cp)
~ C(Cp)
A(Cp) : contribution from the structural change with temperature B(Cp) : contribution from thermal expansion of the system as a whole C(Cp) : contribution from external work Hereafter the contribution from kinetic energy is discarded in
Cp and H .
As a first stage of this work, the comparison of our method with the result of MC calculation has been done for the system of binary monoatomic system. The set of parameters used is the same as used by Nakanishi et al. 2) component 1 : e 11/k=84.71 K
~r11=3.405 • 10-10m
379 component 2 : e 22/k=169.42 K
o'22=3.405 • 10-10m
Here, k is the Boltzmann constant. The Lorentz-Berthelot rule has been adopted for the parameters working between different species of the mixture and the values are thus e 12/k=119.8 K and or 12=3.405 •
10-10m, which correspond to the interaction
parameters of neat argon. The critical temperature of the Lennard-Jones monoatomic fluids 3) evaluated by the integral equation theory is 1.321 multiplied by ( e/k ) , which is equal to 111.9 K and 223.8 K for the component 1 and 2, respectively.
The
temperature set in our present calculation is higher than any of these critical temperature. Thus the fluids discussed in this work are always at supercritical state. HI. Results and Discussion
A comparison of our evaluation of thermodynamic quantity with the result of MC 2) is given in Fig.1 for the case of
pressure dependence of enthalpy.
The values
calculated from the integral equation theory are in good agreement with those from MC. As the pressure becomes higher, the integral equation theory gives values slightly smaller than those from MC. However, the values by these two different methods are in good agreement as a whole. As an example, the excess molar enthalpy is shown in Fig.2 together with the result obtained by MC2). The values from integral equation theory are slightly larger than those from MC, but the general tendency is compatible with MC. Pressure dependence of cnthalpy is shown in Fig. 3 together with its components U and P V . The enthalpy increases linearly with pressure at high pressure region and decreases rapidly with pressure at low pressure. The former behavior is duc to the contribution of PV, and the latter is to the internal energy. Variation of isobaric heat
12.5
.''''1
....
I ....
1 ....
I ....
I ....
i ....
400
I ....
10 ,7,
~
. . . . .
,
9
.
.
,
.
,
.
,
........
I
,
,
.
.
.
I
,
,
,
35O
7.5
""
300
~
250
,--, 200
5 2.5
lOO 50
0 -2.5
,J,I
0
....
I ....
1000
I ....
I ....
2000 P/atm
I ....
i ....
3000
0
!,,,=
40(
Fig.1 Pressuredependence of H at T=250K r-h xl=0.15, /k: xl=0.5 , O: xt=0.85 (PY)
I : Xl=0.15, &: xl=0.5, O" xi=0.85 (MC)
1
0
.
0.2
.
.
.
0.4
0.6
,
,
0.8
Xl
Fig.2 Excessmolar Hat T=300K C):1000 atm (PY), Q:1000 atm (MC) l-']:4000 arm (PY), 1:4000 atm (MC)
1
380
25
10 8
20
6
2
~
10
o -4 ....
-6 o
, ....
, ....
lOOO
.......
2ooo P/atm
O
,....1
3000
4000
Fig.3 Pressure dependence of H and its components at T=300K and x i--0.5
9 H, Q: U, ,I~: t'V
O
1000
2000
3000
4000
P/atm Fig.4 Pressure dependence of the Cp and its components at T=3OOK and x 1=0.5 O: Ct,, Q: A(Ce), A: B(Ct,), I1: C(Ce)
capacity with pressure is shown in Fig.4, which illustrates the respective contributions to the heat capacity. The appearance of maximum at pressures around 200-300 atm originates from the approach to the critical state of component 2. Although not illustrated in this paper, this appearance of maximum in the relation between Cp and pressure is more remarkable for the pure component 2 even at 300K. Our analysis of Cp demonstrates clearly that the most significant contribution to Cp arises from the term C(Cp). But as for the variation of Cp, the contribution of B(Cp) is predominant at low pressure region, while at high pressure, the terms C(Cp) and B(Cp) cancel out. Finally, the contribution from the structural change by temperature A(Cp) increases slightly with pressure, b~,t this contribution is weak and it doesn't affect the general tendency of the pressure dependence of heat capacity over the pressure ranges investigated. The authors wish to thank Profs. K. Nakanishi and I. Fujihara for giving them the detailed information of MC data. References 1
M. Ohba and It. Nomura, to be submitted.
2
K. Nakanishi and I. Fujihara,
3
F. Kawaizumi, H. Nomura, and M. Ohba, J. Phys. Chem.,
Netu Sokutei 20, 2 (1993) 98, 13050 (1994)
journal of MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 38 !-384
Solvation Free Energy of Protein Reproduced by The Combination of The Extended Scaled Particle Theory and The Poisson-Boltzmann Equation
Masayuki Irisa, Takuya Takahashi*, Fumio Hirata +, and Toshio Yanagida Yanagida bio-motron project, ERATO JRDC, Senba.Higashi, Mino, Osaka 562, Japan *Nagayama protein array project, ERATO JRDC, Tokodai, Tsukuba 300-26, Japan +Department of Chemistry, Faculty of Science, Kyoto University, Kyoto 606, Japan Abstract A hybrid approach of the extended scaled particle theory (SPT) and the PoissonBoltzmann (PB) equation for the solvation free energy of non-polar and polar solutes has been proposed by us. This new method is applied for the hydration free energy of the protein, avian pancreatic polypeptide (36 residues). The contributions form the cavity formation and the attractive interaction between the solute and the solvent to the solvation free energy compensate each other. The electrostatic contribution is much larger than other terms m this hydration free energy, because hydrophilic residues are ionized in water. This work is the first step toward further applications of our new method to free energy difference calculation appeared in the stability analysis of protein.
Introduction Increasing attention has been paid to the theoretical determination of the solvation free energy from a variety of fields including the stability analysis of protein. Solvation free energies of molecules, especially large ones, are very difficult to calculate based on the molecular simulation due to the overwhelmingly large configurational space. This uncomfortable situation urged us to develop a new method extended from the scaled particle theory (SPT) 1 for the determination of the solvation free energy where the solute can take an arbitrary shape 2. We have further developed a new approach in which the extended SPT and Poisson-Boltzmann (PB) equations work complementarily in order to analyze the solvation free energy of polar solutes. The applicability of the hybrid method has been demonstrated by taking a series of the normal alcohols as an example 3. In this paper, we present the first attempt to calculate the hydration free energy of protein based on SPT coupled with the PB equation.
Method Here, we give a brief sketch of the hybrid approach. The detailed explanations of the theory has been presented in preceding two articles.2, 3 Some information related to the application for protein is also provided later. We employed the Pierotti's recipe 1 to calculate the solvation free energy, A~~ The dissolution process of the solute is decoupled into three steps by a thought experiment: (I) creating a cavity in solvent, which has a right size as well as a shape to accommodate the solute molecule; gc, (II) establishing the Lennard-Jones type interaction between the solute molecule and solvent; gI. (III) coupling 0167-7322/95/$09.50 9!_995 Elsevier Science B.V. All rights reserved. SSD! 0167-7322 (95) 00835-7
382 m
the electrostatic interaction between the solute molecule and the solvent molecules; gB. Then the solvation free energy can be written as a sum of the three terms: Ag~ (1)
Extended Scaled Particle Theory We consider the rigid body system consisting of one solute molecule with an arbitrary shape and solvent molecules with a spherical shape. The solute molecule is modeled as fused hard spheres with different radii and the solvent molecules are modeled as hard spheres with the same radius. This represents aqueous solution of organic solute at infinite dilution. The reversible work for creating the cavity with an arbitrary shape can be written as follows 4 1
gc=kB~-ln(l-pVc(~))+pf~
F
0Vc(~.) (~,)OZ, d~,]
(2)
where VC(~) is the excluded volume of the scaling solute, I"(~) is the distribution function of solvent centers at the surface of the cavity. The first term in equ. (2) is the work required to dissolve a material point as a seed of a solute molecule into pure solvent. The second term represents the work required to scale up the solute molecule from ~--0 to ~= 1. The meaning of the second term is rather obvious since it is a product of the pressure of solvent exerted on the surface of the cavity, kBTpl"(~ ), and the infinitesimal increment of the volume of the cavity. The difficulty appeared in the calculation of equ. (2) is that we must have analytical expressions for excluded volume, V c ( ~ ) , and their first and second derivatives. Because in this work solvent molecules are regarded as single spheres, the excluded volume of the solute molecule is exactly the same as volume of fused spheres, each of which radius is a sum of radii of a solute atom and a solvent molecule (Figure 1). The analytical formulation of the volume of the fused spheres with unequal radii made by one of us, is explained in the elsewhere 5.
.......................
~s
~!i~i.~i~=:
. . . .
'iiii~:!i;i!:!iii!ii~ ii~i!ii!i;: ~:#i~i!C~~jii!i#:iiiZ
Figure 1 A schematic drawing of protein and water. The regions of the protein and the excluded volume of it are represented as A and A+B, respectively. The regions of solute (low dielectric media) and solvent (high dielectric media) are represented as A and B+C, respectively. Hydrogens bound to heteroatoms are treated specially in the manner explained in the text.
383 Lennard-Jones interaction of the solute molecule with the solvent. To estimate the contribution from the Lennard-Jones interaction gI, we also utilized Pierotti's formula 6. Because the solvent entropy change caused by the establishment of the Lennard-Jones type interaction is thought to be small, gI is approximately equal to the
corresponding enthalpy, h I, due to the attractive force between solute and solvent. Then h I is estimated as an integrated value of the Lennard-Jones type interactions between the solute and solvent molecules. The overall solute-solvent interaction is a superposition of interactions of individual solute atoms with solvent corrected for the exclusion of solvent molecules due to other atoms in the solute. The calculation was performed numerically using equally spaced 3-d grids. Electrostatic interaction between the solute molecule and the solvent To evaluate the electrostatic part (Born energy) of the free energy change, gB, we numerically solve the Poisson-Boltzmann equation of the solute-solvent system 7-8 using a finite difference method. The system is divided into two regions: solute region and solvent region (Figure 1). In this case, the dielectric constants of the solute molecule and water are 1 and 78.4, respectively. The boundary between these dielectric media is just the surface of the solute molecule except hydrogen atoms bound to heteroatoms. For hydrogens bound to heteroatoms, which have partial charges, we used the dielectric boundary with radius of 1.0/~ in order to avoid a situation in which the hydrogen charge comes too close to water 9. In the numerical calculation of the electrostatic interaction, the grid size is 0.5 A and the Debye screening constant K: is set to 0.1. We treated the both terminals and hydrophilic residues of protein as ionized both in water and in vacuum. m
Energy parameter and structure The OPLS 10 parameters are employed for the united atoms and partial charges in protein. The parameters for water are those determined by Pierotti 11. The crystallographic structure of avian pancreatic polypeptide (1PPT) 12 is taken from protein data bank in BNL. For electrostatic calculation, we added 67 hydrogen atoms which bound to heteroatoms to crystallographic structure data which includes 301 heavy atoms. In this structure, N- and Cterminals are positively and negatively ionized respectively, and 3 residues (arginine) are positively ionized and 5 residues (asparagine and glutamine) are negatively ionized. Net charge of this structure is -2 electron unit.
Results and Discussion The accessible surface area and excluded volume of this protein are 3336.8 A,2 and 8524.1/~ 3, respectively. However this protein has two helicies (Figure 2), most of which heavy atoms are accessible to water molecules (only three heavy atoms are not accessible). The calculated values of the solvation free energy for this protein are 239.8 kcal/mol, -142.5 kcal/mol,-620.6kcal/mol and-523.3 kcal/mol for g--c, g~, g--Band Ags~ respectively. In the hydrophobic part of the solvation free energy of this protein, g c and gI compensate each other. In the solvation free energy of the protein, the Born energy is much larger than other terms, because hydrophilic residues are ionized in water. The Born energy calculated here is the preliminary one, so more investigation is needed. We have presented the first application of the newly developed method for calculating the solvation free energy to protein, which is based on the extended scaled particle theory and the Poisson-Boltzmann equation. Although the results are still preliminary, it demonstrates a possibility of obtaining the quantity theoretically, which is difficult even for the modern
384 simulation technique using a computer with high performance. The most time consuming part in the calculation is the numerical integration of attractive forces between the solute and the solvent. A faster calculation of this part may be possible by using the analytical formulation of the excluded volume calculation. The Born energy appears dominant in the solvation free energy of the protein. However, the results are not conclusive at this moment because the calculation is somewhat sensitive to the boundary condition and the electrolytes are not included. Nonetheless, the method proposed here is sensible for some applications including thermal stability of protein, in which the temperature dependence of hydrophobic term is important. In order to apply the empirical method using accessible surface area of the solute to such a problem, the experimental parameter sets at individual temperatures are needed. It will be also a good application of our method to investigate the conformational equilibrium of protein in aqueous solutions, because our method can treat the arbitrary shape of the molecule. That type of application includes the stability of a complex of proteins (e.g. motor proteins, myosin and actin).
C-terminal
N-terminal
Figure 2 A secondary structure of avian pancreatic polypeptide.
References 1 R.A. Pierotti, Chem. Rev. 76, 717 (1965) 2 M. Irisa, K. Nagayama, and F. Hiram, Chem. Phys. Letters 207,430 (1993) 3 M. Irisa, T. Takahashi, K. Nagayama, and F. Hiram, to be submitted. 4 T. Boubl~, Mol. Phys. 27, 1415 (1974) 5 M. Irisa, to be submitted. 6 R.A. Pierotti, J. Phys. Chem. 67, 1840 (1963) 7 H. Nakamura and S. Nishida, J. Phys. Soc. Jpn. 56, 1609 (1987) 8 T. Takahashi, H. Nakamura, and A. Wada, Biopolymers 32, 897 (1992) 9 W.C. Still, A. Tempczyk, R. C. Hawley, and T. Hendrickson, J.Am.Chem.Soc. 112, 6127 (1990) 10 R.A. Pierotti, J. Phys. Chem. 69, 281 (1965) 11 W.L.Jorgensen and J. T-Rives, J.Am.Chem.Soc. 110, 1657 (1988) 12 T.L.Blundell,J.E.Pitts, I.J.Tickle, S.P.Wood, C.-W.Wu, Proc.Nat.Acad.Sci.USA 78, 4175 (1981)
journal of
MOLECULAR
LIQUIDS ELSEVIER
Journal of MolecularLiquids,65/66 (1995) 385-388
Transport Properties of a Spherical Molecule in Solvent with Position-Dependent Viscosity Kazuyasu Ibuki and Masakatsu Ueno Department of Molecular Science and Technology, Faculty of Engineering, Doshisha University, Tanabe, Kyoto 610-03, Japan Abstract The rotational friction coefficient of a spherical molecule in solution is calculated applying the Navier-Stokes equation for a continuum solvent with a position-dependent viscosity as a model of "microscopic viscosity." The rotational friction coefficient decreases with decreasing surface viscosity. The results are compared with the translational friction and viscosity B coefficients which are previously obtained from the same model. The B coefficient is most sensitive to a local viscosity change. The Gierer-Wirtz model overestimates the effect of the "microscopic viscosity" on the translational friction coefficient comparing with the present results.
I. Introduction In the study of the transport properties of electrolyte and non-electrolyte solutions, "microscopic viscosity" is one of the most important ideas to explain the limitations of the uniform continuum model, such as the Stokes-Einstein law for the diffusion constant and the electrical conductivity, 1 the Debye-Stokes law for the rotational relaxation time, 2 and the Einstein law for the viscosity B coefficient. 1 The term "microscopic viscosity" has long been used to express a local viscosity which is significantly lower than the bulk one. The origin of the local fluidity is attributed to the short-range solvent structure effects such as the breaking effect of halide ions in aqueous solutions. 3 Since a low-viscosity region is surrounded by a bulk-viscosity region, "microscopic viscosity" can be naturally modeled by a position-dependent viscosity. In order to investigate the effect of "microscopic viscosity" on the translational friction coefficient (T and the viscosity B coefficient quantitatively, we have recently extended the continuum model to incorporate the position-dependent solvent properties starting from a general form of the Navier-Stokes equation for solvent motion. 4 In the present paper, we apply our extended continuum model to the calculation of the rotational friction coefficient (R Among many molecular theories so far proposed to calculate (R, a theory by Gierer and Wirtz ~ is well-known in relation to the concept of"microscopic viscosity." In their theory, the solvent flow is divided into successive layers each with the thickness equal to the diameter of the solvent molecule. Such an artificial treatment is not satisfactory in both molecular and continuum senses and we ought to improve the model. Before turning to molecular theories, however, it is worthwhile to investigate (R by the extended continuum model since, in solution chemistry, the concept of "microscopic viscosity" is an intuitive extension of the primitive continuum model based on the uniform solvent properties. Experimentally, it is well-known that there are strong correlations between the transport 0167-7322/95/$09.50 91995ElsevierScienceB.V.All rightsreserved. SSDi 0167-7322(95) 00908-6
386 properties of different modes, i.e., the translational, rotational, and shear modes. For a better understanding of the validity of the concept of "microscopic viscosity," we examine the correlations using the extended continuum model. II. T h e o r e t i c a l
Here we explain in brief the extended continuum model 4 and its application to the calculation of (R We start from the Navier-Stokes equation with a position-dependent viscosity rl(r) for a slow steady flow of an incompressible fluid given as 6 0 = - Vp(r) + [V q (r)]. o (r) + q(r)V2v(r) (1) Here, v(r) is the velocity field at position r, p(r) the pressure field, and o(r) the rate-of-strain tensor defined as the symmetric part of the velocity gradient tensor. In the calculation below, rl(r) is assumed to be spherically symmetric around a solute. v(r) around a rotating sphere can be expressed in the form v(r)
- ~ r ) f~ • r
(2)
where D is the angular velocity of the sphere, f(r) is a scalar function which expresses the decay of the angular velocity t~(r) of the solvent flow. Here we assume that t~(r) = t) at the surface of the sphere, i.e., the stick boundary condition. At a long distance where the viscosity is virtually the bulk one 7~0,f(r) has the same form as that for the Stokes law with a positionindependent viscosity.
fir) = S R 3 / r 3 (3) where R is the radius of the rotating sphere and S a constant. For the Stokes law, S - 1. Putting Eq. (2) into Eq. ( 1) and the definition of o(r), we get the following equation: d y + ( 4 + l d r l ) df _- 0 (4) dr2
~r
11 d r ] d r
Note that Vp(r) in Eq. ( 1) vanishes everywhere by the symmetry of the problem. We numerically solve Eq. (4) by a method slmdar to the cases of (T and B with a condition that the long-distance form of./(r) at large r is given by Eq. (3). Stokes' theorem tells us that the friction exerted on the rotating sphere can be calculated from the long-distance behavior of.](r). With S in Eq. (3) obtained by the numerical solution of Eq. (4), (R is given by ~R - 8 n q 0 R 3 S (5) We have already discussed in detail the appropriate functional forms of the positiondependent "microscopic viscosity" in our previous paper. 4 Here we employ the following two functions: First, we assume that the magnitude of the viscosity reduction around an ion is proportional to the strength of the structure breaking effect arising from the ion-dipole interaction. This leads to the following function: q(r) = q0(l - a R 3 / r 3 ) (6) For a non-ionic solute, it is plausible to attribute the viscosity reduction to more shorter ranged interactions such as the dipole-dipole interaction. This suggests (7) q(r) = lq0(l - a R 6 / r 6) In Eqs. (6) and (7), the parameter a, which reflects the viscosity reduction at the solute surface from the bulk, is introduced as a variable to examine the correlations between transport properties of different modes. We employ two functions for the position dependent viscosity, instead of one, in order to show that the validity of our results does not depend on the particular choice of the functional form. 4
9
9
387
HI. R e s u l t s
and Discussion
Figure 1 shows the calculated rotational friction coefficient as a function of the parameter a in the range of 0 _ (R > (T ~' (b) 7 B becomes even negative for some cases. 19 The first result agrees with what solution 6 chemists expect for the effect of the "microscopic viscosity." The second result tells us that the sensitivity of the friction coefficients on a local viscosity change largely depends on ~o, 17 the mode of solvent motions. The shear mode (the viscosity B coefficient) is the most sensitive of the three. It is to be noted that these results do not depend on the particular I I choice of the functional form of the position(c) dependent viscosity as expected. o We can find corresponding experimental results 7 in relation to the limitations of the 7 uniform continuum model. The halide ions have negative B coefficients in water while the Einstein laws predicts only positive B value. % 0 However, the electrical conductivities for the same ions are at most 20% larger than the prediction of the Stokes-Einstein law with the crystallographic radius for the slip boundary I I I -1 condition. 0.0 0.2 0.4 Gurney 3 considered that the negative B coefficient is an indication of the local loosening Figure 1. Surface viscosiw dependences of the solvent structure in the vicinity of the of (a) rotational and (b) translational solute molecule and can be used as a measure friction coefficients and (c) viscosit3' B for the structure breaking effect. The present coefficient. 6 and 7 indicate the results quantitative analysis of the effect of the local with Eqs. (6) and (7). respectively. viscosity change supports this idea.
e~
-
oo
e~
388
20 (a)
6 J
/
/
-"
Aw I
10
I
(b) "7 C)
j
0
6 -1
0
I
I
10
20
30
~ /R ~rlo
Figure 2. Correlations (a) bet~een the translational and the rotational friction coefficients and (b) between the viscosity B and the rotational friction coefficients. GW indicates the Gierer-Wirtz theory and other symbols have the same meanings as Figure I.
In Figure 2 (a), we plot (a" against ~R for the two functions of the "microscopic viscosity." The correlation curves for Eqs. (6) and (7) are roughly identical; the correlation between (T and (R is not so sensitive to the functional form. From these correlation curves, we can estimate the significance of the effect of non-uniform solvent properties on (R" Such effects are usually concealed by large effects of the solute shape and the surface boundary condition which is often treated as an adjustable parameter. In Figure 2 (a), the result of the GiererWirtz theory is also depicted. The correlation predicted by the Gierer-Wirtz theory is much different from our results; it overestimates the effect on (a- In fact, Gierer and Wirtz did not define the "microscopic viscosity." Hence we cannot directly compare their result with the present one. However, their method to divide the solvent into concentric spherical layers is quite unrealistic for the translational motion in which no spherical flow of the solvent occurs. Such an unphysical assumption may be reflected in the unusual correlation which significantly differs from the present result based on physically realistic approximations within the continuum model. In Figure 2 (b), the correlation between B and (R is examined for each functional form of the "microscopic viscosity." Two different correlation curves suggest that the viscosity B coefficient is sensitive to the functional forms of the local viscosity change. This corresponds to the sensitivity of B on the parameter a as pointed out above.
References
1 2 3 4 5 6 7
A. Einstein, lnvesttgattous ou the Theory of the Browman Motion (Dover, New York, 1956). P. Debye, Polar Molecule (Dover, New York, 1929). R.W. Gurney, lomc Processes m Sohttion (Dover, New York, 1962). K. Ibuki and M. Ueno, Bull. ('hem. Soc..]pu., in press. A. Gierer and K. Wirtz, Z. Naturforsch. 8a, 532 (1953). L.D. Landau and E. M. Lifshitz, FlutdMechamcs (Pergamon, London, 1959). R.A. Robinson and R. H. Stokes, Electrolyte Soluttous, 2nd ed. (Butterworths, London, 1959)
joumal of MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 389-392
Liquid Dynamics Studied by Higher Order Non-Linear Optical Spectroscopy Keisuke Tominaga, Gary P. Keogh,* and Keitaro Yoshihara Institute for Molecular Science, Myodaiji, Okazaki, 444 Japan Abstract Temporally two-dimensional non-resonant Raman spectroscopies, which are based on higher order susceptibilities, have been developed to distinguish homogeneous and inhomogeneous contributions to vibrational dephasing. The five-pulse correlation experiment was performed to study low frequency intermolecular vibrational dephasing of liquid CS2, and the Raman echo experiment was conducted to investigate the C-N stretching of benzonitrile. In the both cases it is found that homogeneous dephasing is dominant.
I. Introduction Vibrational lineshape is a rich source of structural and dynamical information about liquids since the dephasing of vibrational modes is strongly affected by perturbation of the environments of the oscillator. 1,2 In order to discuss the microscopic dynamics it is very important to distinguish homogeneous and inhomogeneous contributions to vibrational dephasing. The homogeneous contribution results from rapid perturbation to the oscillator and the inhomogeneous contribution from difference of the environment around the molecule of interest. However, it is well known that frequency- or time-domain spectroscopies which are based on the third-order non-linearity are incapable of distinguishing the homogeneous and inhomogeneous contributions unambiguously. 3 On the other hand, echo-type experiments, which make use of temporal two-dimensionality, have been proved to be a powerful technique to determine relative importance of inhomogeneity of dephasing process as demonstrated in magnetic resonance. 4 In this paper we report vibrational echo-type experiments with ultrafast lasers on liquids in order to distinguish homogeneous and inhomogeneous contributions to the vibrational linewidths. Two different experiments were done, namely the five-pulse correlation experiment 5 was performed to study dephasing of low frequency intermolecular vibrational modes, and the Raman echo experiment 6 was conducted for high frequency intramolecular vibrational modes. ks
II. Five-Pulse Correlation Experiments; Low Frequency Vibrational Dephasing The dephasing processes of low frequency modes have been observed by femtosecond optical Kerr and impulsive stimulated light scattering (ISS) experiments using pulses with durations that are short compared to the period of vibrational oscillation. 7 For example, the third order response from CS2 has been extensively studied by femtosecond optical Kerr and ISS experiments. 7- I0 The ISS signal from 0167-7322/95/$09.50
--
Fig. 1 Pulse configuration for a fifth-order experiment. The system interacts with two pairs of pulses at different time t = 0 and Atl. The last pulse at Atl + At2, kp, is the probe that generates the signal, ks.
91995 Elsevier Science B.V. All rights reserved.
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~tl
390 CS2 shows an underdamped motion, which was analyzed with a distribution of the o s c i l l a t i o n f r e q u e n c y due to an inhomogeneous broadening. 7-10 However, inherent ambiguity remains in determining the relative importance of inhomogeneity. Recently Tanimura and Mukamel have proposed a fifth order nonlinear spectroscopy that is c a p a b l e of distinguishing homogeneous and inhomogeneous contributions to low frequency vibrational modes. 11 As shown in Fig. I, a medium interacts with two pairs of short pulses with the same wavelength at t = 0 and Atl, and the vibrational coherence is probed by monitoring scattered light induced by a fifth pulse at t = Atl + At2. According to their theory, an echo-like signal is observed at around At2 = At I in the inhomogeneous limit. In the purely homogeneously broadened case, on the other hand, the time dependence of the fifth order signal is almost identical to that of the third order response when either Atl or At2 is scanned with the other time-variable fixed. We have observed the fifth order non-linear optical response of liquid CS2 by non-resonant six-wave mixing with five different short pulses. In Fig. 2 the fifth order signal from CS2 is illustrated as a function of At2 with different fixed At l. 5 The scattered light is identified as a fifth order signal by its disappearance when any of the five pulses is blocked, its fifth-order dependence on the input laser power, and its appearance at the correct phase-matching angle expressed as, ks = kp + (k2 - k 2 ' ) - ( k l - k l ' ) ,
(1)
/
A ~-x
1000"-~
:1
Observed Intensityof 1.0
1.0
.
,
,
,
~176176
,
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,
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/
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ttt,t,L4[ll l !
it.
.
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Fig. 2 Solid lines; Fifth order response from CS2 as a function of At2 with different Atl. Atl is (a) 0.33 ps, (b) 0.50 ps, (c) 0.67 ps, and (d) 1.00 ps. Broken lines; the ISS signal from CS2. The relative intensities of the fifth order signal are shown together with those of the ISS signal at corresponding time.
where ks and kp are the wave vectors of the fifth order response signals and probe pulse and ki and ki' are the wave vectors of the i-th pair of pulses. The dotted lines are the ISS signals from CS2, which are identical to reported ones. 7-10 In the ISS signal the instantaneous response at around t = 0 is due to a purely electronic contribution. The following component with rise and decay is due to an underdamped nuclear motion, and the slowest single exponential part is assigned to the decay of an anisotropic distribution of molecular orientations. 8-10 In the fifth order signals
7 3
391
I(5), scattered light at At2 = -Atl is mainly due to a third order response. It is notable that the variation of I(5) as a function of At2 is identical to the time profile of the ISS signal within experimental error. In the figure the relative intensity of I ( 5 ) a t maximum is shown together with that of the ISS signal at the corresponding delay time. The change of I(5) as a function of Atl is also quite similar to that of the ISS signal. All the experimental results are consistent with the prediction of Tanimura and Mukamel for the purely homogeneous case, and a static inhomogeneous contribution is minor.
IIl. Raman Echo Experiments; High Frequency Vibrational Dephasing The Raman echo is a phenomenon based on a seventh order non-linear process. 3 This technique is capable of distinguishing the homogeneous and inhomogeneous contributions to the high frequency vibrational dephasing, 3,12,~3 which cannot be differentiated by third order nonlinear spectroscopy such as timeresolved coherent anti-Stokes Raman scattering. A pulse scheme of the Raman echo is shown in Fig. 3(a). Two short pulses for the excitation have different wavelengths whose difference matches the vibrational frequency of interest. The second pair of pulses acts on the medium twice, corresponding to a "180" degree rephasing pulse" in the magnetic resonance. In the Raman echo experiments the phasematching condition can be expressed as,3 ks = kp + 2(k2 - k2') - ( k l - k l ' ) .
(2) In Fig. 3(b) Raman echo signals of the C--N stretching of benzonitrile 6 are
(a)
k
k 2f
!l
kll
At1
kl p
ke
At2
(b~
"~i
0
1 At 2 / ps
Fig. 3 (a) Energy and time evolution diagram of the Raman echo. The solid and broken arrows represents the ket- and braside dipole transition. The wavy arrow is the signal. (b) Raman echo signal intensities for the C - N stretching of benzonitrile as a
function of At2 with fixed Atl of 0 ps (1) shown as a function of At2 with fixed At l and 2 ps (2). The broken curve indicates the of 0 ps and 2 ps. A strong signal at around exponential decay with time constant of 0.6 At2 = 0 ps is due to an artifact of the ps. The dotted curves are calculated decay coherence resulting from a seventh-order profiles by eq. (3). See text in detail. non-resonant response. It is found that the echo signals show exponential decay profiles with time constants o f - 0 . 6 ps for both Atl = 0 and 2 ps within experimental uncertainty.
392 If the inhomogeneous distribution in Raman transition frequency is assumed to be a Gaussian with a characteristic width (i.e. standard deviation of the transition frequencies), 5, the intensity of the Raman echo signal I(Atl,At2) is given by the following equation, 3 I(Atl ,At2) = exp[-2(Atl +At2)/T2]exp[-62(At 1-At2) 2]
(3)
where T2 is the homogeneous dephasing time constant. If the inhomogeneous broadening has a significant contribution to the dephasing, the decay of the signal should be retarded further as Atl increases, or in an extremely inhomogeneous broadening case, the signal should show a rise-and-decay profile. In Fig. 3(b), the calculated decay profiles are also indicated for (i) T2 1.22 ps and ~5- 0.92 cm-l, and (ii) T2 - 1.33 ps and ~i - 1.41 cm-1. From a comparison of the observed decay profile with the calculated one, the characteristic width of inhomogeneous broadening, 5, is estimated to be less than 1 cm" 1. In conclusion, the C - N stretching Raman band of benzonitrile is almost completely homogeneously broadened.
Acknowledgments We thank Dr. R. Inaba, Prof. K. A. Nelson, and Prof. M. Tasumi for their help to the Raman echo experiments, and Dr. Y. Naitoh and Prof. T.J. Kang for their contribution in constructing the laser system.
References * Permanent address; Department of Chemistry, Imperial College of Science, Technology, and Medicine, London, SW7 2AY UK. 1 D.W. Oxtoby, Adv. Chem. Phys. 40, 1 (1979). 2 J. Jonas, Acc. Chem. Res. 17, 74 (1984). 3 R.F. Loring and S. Mukamel, J. Chem. Phys. 83, 2116 (1985). 4 R.R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Clarendon, Oxford, 1987). 5 K. Tominaga, Y. Naitoh, T. J. Kang, and K. Yoshihara, to be published in Ultrafast Phenomena IX, edited by G. Mourou, A. H. Zewail, W. H. Knox, and P. F. Barbara (Springer-Verlag, Berlin, 1994). 6 R. Inaba, K. Tominaga, M. Tasumi, K. A. Nelson, and K. Yoshihara, Chem. Phys. Lett. 211, 183 (1993). 7 Y.-X. Yan, L.-T. Cheng, and K. A. Nelson, In Advances in Non-linear Spectroscopy; edited by R. J. H. Clark and R. E. Hester (Wiley, New York, 1987) p.299. 8 S. Ruhman, B. Kohler, A. G. Joly, and K. A. Nelson, IEEE J. Quantum. Electron. 24, 470 (1988). 9 D. McMorrow, W. T. Lotshaw, and G. A. Kenney-Wallace, IEEE J. Quantum. Electron. 24, 443 (1988). 10 B. Kohler and K. A. Nelson, J. Phys. Chem. 96, 6532 (1992) 11 Y. Tanimura and S. Mukamel, J. Chem. Phys. 99, 9496 (1993). 12 D. Vanden Bout, L. J. Muller, and M. Berg, Phys. Rev. Lett. 67, 3700 (1991). 13 L.J. Muller, D. Vanden Bout, and M. Berg, J. Chem. Phys. 99, 810 (1993).
joumal of MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 393-396
Temperature Dependence of Singlet Excitation Energy Migration in Liquid Benzene as Revealed by Picosecond Laser Photolysis Hiroshi Miyasaka 1 and Noboru Mataga 2 ~Department of Polymer Science and Engineering, Kyoto Institute of Technology, Matsugasaki, Kyoto 606, Japan 2Institute for Laser Technology, Utsubo-Hommachi 1 - 8 - 4 , Nishiku, Osaka 550, Japan Abstract Temperature dependence of the excitation energy migration in liquid benzene was studied by monitoring the S1-S 1 annihilation process induced by excitation with an intense picosecond (ps) 266 nm laser pulse. In the temperature region above c a . 20~ the energy migration rate constant decreased with temperature lowering ( which means that the energy migration is assisted by some thermally activated molecular motions), while, in the temperature region below ca.20~ down to the freezing point (5.5~ the rate constant increased with temperature lowering. Such an increase of the rate constant with temperature lowering was not observed for the cyclohexane solution of benzene ( 50 vol %), where the usual Arrhenius type temperature dependence was demonstrated. These results suggest that a kind of short range excitonic state dependent upon the periodicity of liquid structure plays an important role in the energy migration in neat benzene at low temperature region.
Introduction Several mechanisms have been proposed for the singlet excitation energy transfer processes in aromatic molecular liquids. Voltz predicted the multipole-multipole interaction for the excitation energy transfer from the solvent toluene to the solute molecule 1, although the equilibrium between the excited monomer and the excimer was neglected in this theory. Contrary to this, Birks and Conte proposed that the rapid formation and dissociation of the excimer provided the energy migration in aromatic molecular liquids such as benzene and toluene 2. Ohno and Kato suggested 3 that the exciton diffusion, which was in narrower region than in the crystal, might take part in the energy migration processes in the excited liquid naphthalene derivatives. For the elucidation of the energy migration mechanism in aromatic liquids, the investigation for pure benzene is meaningful since the liquid structure of neat benzene was reported to be very similar to that in the crystal 4 by X-ray diffraction and the electronic absorption band arising from the crystalline field in Sn~-S 0 transition was observed 5. In this study, we measured the concentration and the temperature dependences of the excitation energy migration rate in liquid benzene by directly observing the S1-S 1 annihilation process induced by the excitation with an intense picosecond 266nm laser pulse. On the basis of these results, we will discuss the energy migration process in liquid benzene from two viewpoints; the molecular translational motions and the exciton diffusion dependent upon the periodic liquid structure in short range. 0167-7322/95/$09.50
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394
Experimental A microcomputer-controlled picosecond laser photolysis system with a repetitive mode-locked Nd3+:YAG laser was used to measure transient absorption spectra. The details of this system have been reported elsewhere 6. The samples were excited with a single 266nm pulse with ca.20ps fwhm and 0.5-1mJ output power. Cyclohexane (Wako Spectrosol) was used without further purification. Benzene (Merck Uvasol) was purified by passing through a column of activated silica gel (Wakogel 200) and some samples were further dried by using activated molecular sieves (Wako 3A)in vacuo so as to remove small trace of water. The experimental results were, however, unaffected by this procedure. Samples were deaerated by repeated freeze-pump-thaw cycles or by irrigating with nitrogen gas stream. Temperature control system was constructed with a refrigerated circulation unit (Neslab RTE-8), by which we controlled the temperature of the sample from 5~ to 70~ with the accuracy of __.0.2~
Results and Discussion Figure 1 shows transient absorption spectra of pure liquid benzene excited with a picosecond 266nm laser pulse at 22~ These broad spectra with an absorption maximum around 505nm were ascribed to the excimer of benzene 7,8,9. AS shown in this figure, this absorption signal decays in a few ns region, while the fluorescence lifetime of benzene excimer is 27ns 7. Actually, no spectral evolution nor decrease in absorbance was observed when excited with two-photon absorption of 355nm picosecond laser pulse at this temperature 9. The decay profile is plotted in Fig.2, demonstrating the linear relation between the reciprocal absorbance and the time after excitation. Further, the slope was unaffected by the excitation intensity, indicating that this decay profile is due to the second-order kinetics. A simple calculation of the absorbance of liquid benzene at 266nm in the ground state gives the value of more than 100 in a lcm optical cell. Accordingly, the one-photon absorption of an intense 266nm picosecond pulse produces dense population of the excited molecules on the surface of the sample cell, which induces effective deactivation due to S1-S 1 annihilation.
o.8~,
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Figure 1. Transient absorption spectra of liquid benzene excited with a picosecond 266nm laser light.
395
The Sl-S 1 annihilation rate constant was determined in the following manner. First, the concentration of the excimer was obtained by dividing the observed absorbance by its extinction coefficient and by the effective cell length where the SI*--S0 absorbance at 266nm was 1. In addition, in pure liquid benzene as well as in solution, there exists rapid equilibrium between monomer and excimer, of which time constants of association and dissociation are in the order of a few ps 1~ Hence, the sum of the monomer and excimer concentrations obtained by the equilibrium constant at each temperature 11 was used as the concentration of the excited singlet species for the analysis. Although this assumption may affect to some extent the accuracy of the obtained rate constant, the error of this estimation would not depend upon the temperature. 1014
'
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'
O t~,~
'
'
-'1-._4
Figure 2. Inverse of transient absorbance vs. delay time relation of benzene excimer, excited at 266nm and observed at 505nm in neat liquid state. High excitation intensity for closed circles and low for open circles.
O ......... ........ -""
..... o" ...... L_
0
00 .13
5
.o.....~
00
.... "O'" ....
2 Time / ns
4
Figure 3 shows the temperature dependence of the S1-S 1 annihilation rate constants for pure liquid benzene (a) and benzene (50 vol%) in cyclohexane (b). In both cases, the rate constant decreases consistently with the temperature, but an increase at low temperature region was clearly observed in the case of pure liquid benzene. It is worth noting that the similar temperature dependence as in Fig.3 (a)was also observed for the energy transfer process from excited benzene to the solute biphenyl in benzene solution 12, where the rise of Sn*--S1 absorption of the guest molecule was monitored. Accordingly, it could be concluded that the rate constant obtained by the present method corresponds to the energy migration processes in liquid benzene. Moreover, these results suggest that some intermolecular interactions responsible for the energy migration processes peculiar to the pure benzene arise as the temperature decreases.
11.8--
i
I
i
11.6
i
I
i
i
9 , , ,
_
9
J
I 3
~
11.l~
I
i
I
I
J
i
I
i
I
I
I
i
I
11.4
i
-
11.2
i
(a) -
9
-
::
i ~ t ~ 3.2 3.4 3.6
_
11.2 11 1
3
3.2
3.4
I
3.6
1/K (xl 0 3) Figure 3. Temperature dependence of rate constant of S1-S 1 annihilation.(a) neat benzene and (b) benzene(5.6M) in cyclohexane.
396
The activation energy estimated for the energy migration rate constant in the normal Arrhenius type temperature dependence region was 3.5-4.5 kcal/mol, which was in the order of the activation energy for the molecular diffusion 13 and for the dissociation of the excimer into monomer excited state 14. From these results, it may be concluded that the energy migration processes assisted by some molecular motions is dominant at higher temperature region. On the contrary, another mechanism may be responsible for the energy migration process in pure benzene at the temperature region below ca.20-25~ As mentioned in the introductory section, the liquid structure of benzene at 25~ which was very similar to that in solid was demonstrated by means of X-ray diffraction 4. Moreover, an exciton state in liquid benzene was predicted on the basis of the similarity of the Sn~--S0 absorption spectrum in liquid phase to that in solid crystalS; K series on SI~---S0 transition induced in the crystalline field was also observed in liquid benzene and this series became more apparent and sharper with temperature lowering. From these results, it can be deduced that the structure of liquid benzene in a small domain is similar to that of the solid. Integrating above arguments with the present results on the temperature and the concentration dependences of the Sx-S 1 annihilation rate constants, it is suggested that the increase of the rate constant in pure liquid benzene below ca.20-25~ is attributable to an exciton state arising from the periodicity of the local liquid structure. As the temperature increases from the melting point to the room temperature, the thermal motion of liquid leads to an decrease in the effective exciton size (reaction radius ). The negative slopes observed in Fig.3 (a) might be related with the destruction of this periodicity. Hence the efficiency of this pathway decreases with increase of temperature. In the high temperature region, other mechanisms, such as Birks mechanism and/or the molecular diffusion, would play a dominant role in the energy migration processes. On the other hand, for the diluted solution, it is predicted that some molecular motions in the excitation migration process is dominant over the temperature region as examined here. More detailed studies on these problems relating to the effective size of the domain with crystal-like structure are now going on, results of which will be published shortly.
References R.Voltz, Rad.Res.Rev., 1(1968)301. J.B.Birks and J.C.Conte, Proc.Roy.Soc.London, Ser.A,303 (1968)85. T.Ohno and S.Kato, Bull.Chem.Soc.Jpn., 54(1981)1517. A.H.Narten, J.Chem.Phys., 48(1968)1630. T.Inagaki, J.Chem.Phys., 57(1972)2526. H.Masuhara, N.Ikeda, H.Miyasaka, and N.Mataga, J. Spectrosc.Soc.Jpn.,31(1982)19; H.Miyasaka, H.Masuhara, and N.Mataga, Laser Chem.l(1983)357. J.T.Richards and J.K.Thomas, Chem.Phys.Letters, 5(1970)527. N.Nakashima, M.Sumitani, I.Ohmine, and K.Yoshihara, J.Chem.Phys., 72(1980)2226. 8. H.Miyasaka, H.Masuhara, and N.Mataga, J.Phys.Chem., 89(1985) 1631. 9. J.B.Birks, C.L.Braga, and M.D.Lumb, Proc.Roy.Soc.Ser.A., 283(1965)83. 10. R.B.Cundall and D.A.Robinson, J.C.S.Faraday Trans. II, 68(1972)1133. 11. H.Miyasaka, F.Ikejiri, and N.Mataga, to be published. 12. R.E.Rathbun and A.L.Babb, J.Phys.Chem., 65(1961)1072. 13. F.Hirayama and S.Lipsky, J.Chem.Phys., 51(1969)1939. 14. .
journal of I~IOLECULA R
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 397--400
Solvent Effect on Ultra-fast Electron Transfer in Mixed-Valence Biferrocene Monocation. A. Masuda, Y. Masuda*, and Y. Fukuda Department of Chemistry, Faculty of Science, Ochanomizu University, B u n k y o - k u , T o k y o 112, Japan Abstract The thermal-induced intramolecular electron transfer rates of mixed-valence biferrocene monocation (Fe(II),Fe(III)) were determined in seven solvents and at various temperatures by the proton paramagnetic relaxation measurements. The rate constants of pico-second order were obtained at 298 K and the frequency factors showed a solvent dependence. The effect of solvent friction on the barrier crossing in the reaction trajectory was examined in the strong adiabatic regime. I. Introduction The solvent effect on electron transfer (ET) reaction has been traditionally treated by Marcus theory, where static and dielectric continuum properties of solvent are taken into account within the transition state theory (TST). 1 In this decade, however, obvious effect of the dynamical properties of solvent on the ET reactions, whose rates exceed 1011s -1, has been demonstrated by the direct observations of photo-induced ET processes as well as by the elucidation of the ET rates within the precursor complexes (i.e., the donor-acceptor encounter pairs geometrically suitable for the electron transfer) in intermolecular ET reactions. 1,2 These experimental studies, on the other hand, lead progress in the theoretical approach for the dynamical solvent effect on the ET reactions. 1,2 The examinations of these theories are, however, made difficult sometimes by the following factors other than the solvent dynamical effect; 1 e.g., in photo-induced ET reaction, the higher excited vibrational states as an escape route of the reaction contribute to accelerate the rate, 3 whereas, in intermolecular ET reaction, there exists some ambiguity to estimate the formation constant and the geometry (the donor-acceptor configuration) of the precursor complex, nevertheless these parameters are required for the determination of the intrinsic ET rate within the precursor complex from the observable overall reaction rate. 4 Thermal-induced intramolecular ET reaction is flee from above the additional problems when one examines the dynamical solvent effect on the rate. Nevertheless, no quantitative and systematic data for the rate constant exceeding 1011 s -1 have been reported because of experimental difficulties. 1 In the present study, the rate constants of the thermal-induced intramolecular ET reaction of pico-second order were first determined for mixed-valence biferrocene monocation (Fe(II),Fe(III)) in seven solvents at various temperatures by the NMR relaxation method. The procedure of the determination of the rate constants from the measured 1H spin-lattice relaxation times (T1) is shortly mentioned and the effect of dynamical property of solvent on the ET rate is discussed. II. Experimental Hexafluorophosphorous salt of biferrocene monocation, [(rlS-CsHs)Fe(II)(rl 5C5H4)2Fe(III)(rlS-C5Hs)] PF6 (bfc +" PF6") was obtained by oxidation of biferrocene (bfc), 0167-7322/95/$09.50 91995 Elsevier Science B.V. All rights reserved. SSDI 0167-7322 (95) 0091 I-6
398 which was synthesized by the literature method, 5 by adding stoichiometrical amount of AgPF6 under dry Ar atmosphere. The sample solutions of various solvents for the NMR measurements were degassed by twice freeze-pump cycles and were sealed. The 1H NMR line width and the spin-lattice relaxation time, T1, which was determined by inversion-recovery method, were recorded by JEOL GX and GSX-270 NMR at 270MHz at various temperatures.
III. Results and D i s c u s s i o n The proton relaxation of the Determination of ET rate c o n s t a n t s . 6 cyclopentadienyl (cp) ring of the present mixed-valence complex is caused by the magnetic dipolar interaction with the electron spin on the Fe(III) ion of the ferricenium unit (S=1/2). 6 Then the proton spin-lattice relaxation rate of bfc +, l/T1, is represented by: 7 (l/T1) = (1/10) y2 g2132 r-6[j(t.Ol)+7J(o~S)] J(co) = "re/(1 + o021;2c)
(1) (2).
Where, y, g, and 13are the gyromagnetic ratio of the proton, the electronic g factor, and the Bohr magneton, respectively. The distance between the proton and the electron spin, r, is given by the crystallographic data, 8 and the g value is determined by the glassy state ESR measurements, col, c0s, and J(co) respectively indicates the proton and the electron spin Larmor frequency, and the spectral density of the fluctuation of the magnetic interaction. In the present system, the fluctuation of the local magnetic field at the proton caused by the electron spin is originated by the electron spin relaxation, the molecular reorientation, and the electron transfer between the two Fe atoms. Then, the correlation time, zc, is given by: "rc"1= ket + "rR-1 + T l e "1
(3)
where ket and Tie indicate the rate constant of the electron transfer and the electron spin relaxation time in bfc +, respectively. XR is an effective rotational correlation time relating to the cp ring rotation and the reorientation of the whole bfc + ion. The xc"1 value calculated by eqs. (1) and (2) was 1.3x1012 s 1 in acetone at 298K. The I:R"1 value was estimated to be 5.0x1010 s 1 at 298K in acetone from the proton T1 values measured for bfc. The electron spin relaxation time, Tle, was estimated as follows. The proton paramagnetic relaxation of ferricenium (Fe(llI)) ion, fc +, is caused by a similar way with that for bfc + and the correlation time for the proton relaxation of fc + is represented by eq.(3) without the ket term. Then the Tle -1 value for fc + can be determined from eqs. (1) and (2) with the estimated "rR value for fc + and is given as 2x1011 s -1. The ESR band widths in the glassy solutions indicate longer T2e for bfc + than fc +. This leads a relation, Tle-l(bfc +) <. T2e'l(bfc +) < T2e'l(fc +) - 2x1011 s "1. Thus we concluded that ~:c-1 for bfc + (1.3 xl012 s -1) was much higher than Tle "1 as well as "rR"1 for bfc +, that is, the Xc-1 value was approximately equal to ket. (See eq. (3)). Then, the ket values can be determined from the observed T1 in various solvents and temperatures. 2 The obtained rate constants, ket, are shown in Figure 1. The values in most solvents exceeded 1012 s "1 at room temperature.
Dynamical solvent effect on ket. The solvent dynamical effect can be examined by the so called frequency factor, Vn, in the usual TST like format, ket = K:elVn exp (-AG*/kBT)
(4).
Where kB, AG*, and ~el represent the Boltzmann constant, the flee energy barrier for the ET process, and the transmission coefficient, respectively. In the present system, the electronic coupling between the reactant and the product is sufficiently strong (see bellow) so that ~Cel*- 1. AG* is represented by the electronic coupling matrix element, H 12, and the reorganization free
399
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9. .
j . .
9
i
,
''
'
,
'-14.1
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~AI
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,
l
~
,
,
I
MeOH 9 0
i
9 AC
e., O
-13.9
m
EtOH 9
0
~
--11.5
9
Bm ' 2.5
'
'
' 3.5
'
'
'
--13.7
13.5
' '
4.5 K/1000 T
9
C rl
11.0
/
o
5.5
Figure 1. Themperature dependences of observed ket in various solvents. o : acetone(AC), o : acetonitrile(AN), ~ : dichrolomethane(DCM), I1: MeOH, 9 : EtOH, A: nitromethane(NM), A : nitrobenzene(NB).
9.2
10.2 11.2 log(v(cai))
12.2
Figure 2. Plot of observed frequency factors versus those calculated according to Hynes Model. (298K) Open circles indicate the calculated values taking account of the multiple relaxation of alcohols.
energy, k, of the inner and outer contributions respectively denoted by Z.is and ~Kos,1 i.e., AG* = k/4 - H12, (~J4 = Z.is/4 + kos/4 ). The Z. and H12 values can be determined from the band maximum frequencies and widths of the IT bands in various solvents, 9 and Z.is is calculated from the Fe-cp distances 8 and the force constants for the symmetrical stretching modes of the ferrocene and ferricenium units in bfc+. 10 The representative AG*, ~.is, ~os, and H12 values in acetone were, respectively, 10.8, 7.1, 60.6, and 6.1 kJ/mol. The much larger Los values compared with kis indicate that the solvent reorganization mostly dominates the barrier of the present ET reaction. The Vn values in eq. (4) in various solvents were obtained by using the ket values and the AG* values determined above. A variety in the Vn values in the different solvents indicates that the solvent dynamics contributes the ET process. In the stable-state picture, the solvent dynamics affects both of the dynamics within the potential well of the reactant and over the barrier top along the reaction coordinate. 11 For the present system the electronic coupling, H12, is sufficiently large (H12>>kBT), i.e., the roundness of the barrier top is enough to be affected by dielectric solvent friction when the reaction trajectory paths through the barrier so that this process becomes rate determinant. 12,13 In such case the rate constant, ket, can be calculated according to Hynes treatment utilizing a generalized Langevin equation (GLG) in the strong adiabatic regime. 12,13 Figure 2 shows a comparison of the observed Vn with those calculated regarding solvent as a Debye continuum, which are roughly proportional to -[L-1 for the present system. The plot shows a distribution closely around a slope of unity indicated by the solid line in Figure 2 except for the alcohols and nitrobenzene. Such anomaly in alcohols is also reported for other chemical processes and time-dependent fluorescence stokes shifts and is attributed to their non-Debye multiple relaxation behavior; 14,15 the shorter relaxation components, which are assigned to local motions such as the OH group reorientation, contribute the friction for the barrier crossing rather than the slower main relaxation component, which corresponds to the longitudinal dielectric relaxation time, ZL, when one regards the solvent as a Debye dielectric medium. If one takes account of the multiple relaxation of the alcohols, 12 the theoretical ket (or Vn) values increase and approach to the trend of the other solvents. (See open circles in Figure 2.)
40o The deviation of nitrobenzene from the solid line (the slope = 1) in Figure 2 is probably attributed to the frequency dependent dielectric friction for the reaction dynamics around the barrier top, i.e., the much slower dielectric fluctuation of nitrobenzene ( ' r L - 6 ps at 298K) compared with the ET rate hardly works as friction for the barrier crossing. In such case, the fi'iction is shows "~:L-ct (ct
Conclusion The 1012 s -! order rate constants for the thermal-induced (ground state) intramolecular electron transfer rates of the mixed-valence biferrocene monocation were first elucidated in various solvents by the 1H_NMR relaxation measurements. The obtained solvent dependent frequency factors indicated significant contribution of the solvent dielectric friction on the barrier crossing. An existence of the faster processes compared with the ET rate such as the internal vibration as an escape route of the reaction dynamics along the solvent coordination was also proposed in some extent. Acknowledgment This work was partially supported by Grant-in-Aid for Scientific Researches, No. 06640717 for the Ministry of Education, Science and Culture.
References 1 2
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
J.D. Simon, Ace. Chem. Res., 21,128(1988); M. J. Weaver and G. E.McMais, Ace. J. Chem.Res., 23,294 (1990); H. Heitele, Angew. Chem. Int. Ed. Engl., 32,359 (1993). P.F. Barbara and W. Jarzeba, Adv. Photochem. 15, 1 (1990); Dynamics andMechanizms of Photoinduced Electron Transfer and Related Phenomena, ed. N.Mataga, T. Okada, and H. Masuhara, (Elsevier, Amsterdam, 1992); "Ultrafast Reaction Dynamics and Solvent Effects", Y. Gauduel and P. J. Rossky Eds., AIP Conference Proceedings 298 (1994). T.J. Kang, W. Jarzeba, P. F. Barbara, and T. Fonseca, Chem. Phys., 149, 81 (1990); K. Tominaga, G. C. Walker, T. J. Kang, P. F. Barbara, and T. Fonseca, J. Phys. Chem., 95, 10485(1991); K. Tominaga, D. A. V. Kliner, A. E. Johnson, N. E. Levnger, and P. F. Barbara, J. Chem. Phys., 98, 1228 (1993). G.E. McMains, R. M. Nielson, A. Gochev, and M. J. Weaver, J. Am. Chem. Soc., 111, 5533 (1989); G. Grampp and W. Jaenicke, Ber. Bunzenges. Phys. Chem., 95, 9094(1991). H. Shechter and J. F. Helling, J. Org. Chem., 26, 1034 (1961). The procedure to determin the ket values is given in elsewhere in detail. T.J. Swift, "NMR of Paramagnetic Molecules", G. N. La Mar, W. Dew. Horrocks, Jr., R. H. Holm, ed., Academic Press New York, 1973, chap. 2. T. Y. Dong, D. N. Hendrickson, K. lwai, M. J. Cohn, S. J. Geib, A. L. Rheigold, H. Sano, I. Motoyama, and S. Nakashima, J. Am. Chem. Soc., 107, 7996 (1985). G.E. McManis, A. Gochev, R. M. Neilson, and M. J. Weaver, J. Phys. Chem., 93, 7733(1989). B. F. Gachter and J. A. Koningstein, J. Chem. Phys., 62, 4628 (1975). G. Van der Zwan and J. T. Hynes, J. Phys. Chem., 89, 4181 (1985). J. T. Hynes, J. Phys. Chem., 90, 3701 (1986). G. E. McManis and M. J. Weaver, J. Chem. Phys., 90,912 (1989). M. J. Weaver, G. E. McManis, W. Jarzeba, and P. F. Barbara, J. Phys. Chem., 94, 1715 (1990); ibid, J. Phys. Chem., 90, 6563 (1986). M. Maroncelli, J. Mol. Liq., 57, 1 (1993). H. Smni and R. A. Marcus, J. Chem. Phys., 84, 4247 (1986); ibid. 84, 4896 (1986). D. K. Phelps and M. J. Weaver, J. Phys. Chem., 96, 7187 (1992).
journal of MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 40 i-404
Translational Diffusion of Transient Radicals Studied by the Transient Grating Method Masahide Terazima, Koichi Okamoto, Noboru Hirota Department of Chemistry, Faculty of Science Kyoto University, Kyoto, 606, Japan Abstract Diffusion coefficients (D) of various radicals created by the photoinduced hydrogen abstraction reactions from alcohols (ethanol and 2-propanol) as well as those of the parent molecules are measured by using the transient grating (TG) method. Dependence of D on the viscosity, molecular size, and temperature are investigated, and the results are interpreted in terms of microscopic aggregation of the radicals with solvents or solutes.
1. Introduction Since translational diffusion process is sensitive to the microscopic structure in the solution, understanding the diffusion provides an important insight into the structure as well as the intermolecular interaction. Therefore, dynamics of molecules in solution have been one of the main topics in physical chemistry for a long time. 1 Recently we have studied the diffusion process of transient radicals in solution by the TG method aiming to understand the microscopic structure around the chemically active molecules. This kind of study will be also important in a view of chemical reaction because movement of radicals plays an essential role in the reactions. Here we present anomalous diffusion of the radicals created by the photoinduced hydrogen abstraction reaction. The origin of the anomality is discussed based on the measurments of the solvent, solute size, and temperature dependences.
2. Method Interference between two coherent beams produces a sinusoidal pattern (grating) in a sample solution. At the bright region of the grating, reactants are excited to the excited states, and they abstract hydrogen atoms from the alcoholic solvents. Spatial modulation of the chemical species is probed by another cw laser (He-Ne laser). The diffracted signal, ITG(t), induced by grating with wavevector ~ decays with the diffusion process and D can be calculated from the decay rate constant, k, with D=k/q2. Assumptions made for deriving this relation are discussed in more detail in Refs.2 and 3.
3. Results First we show the method of the measurement by taking an example of benzoquinone (BQ). According to the photochemical investigations of BQ, the triplet state BQ abstracts hydrogen from alcohol to form transient species of benzosemiquinone radical (BQH) and ,2-hydroxypropyl radical (HPr). 4 Typical time dependence of the TG signal after the photoexcitation of BQ in 2-propanol is depicted in Fig.1. The initial strong spike-like signal is due to the contribution of the thermal grating. The decay of the signal is determined by the thermal diffusion. After it decays to the baseline once, another slowly developing signal appears. Since this signal develops almost 2 orders of magnitude slower than that of the thermal grating, the time profile must reflect the molecular diffusion in the solution. The 0167-7322/95/$09.50 91995 Elsevier Science B.V. All rights reserved. SSDI 0167-7322 (9.5) 00885-3
402 square root of the TG signal is expressed well by a sum of two exponential functions with different signs of the pre-exponential factors ITG 1/2 =as exp(-kst)+arexp(-kft) (1) where subscripts s and f stand for the slow and fast components, respectively. The dip between the thermal grating and the population grating implies that the absolute signs and magnitude of the pre-exponential factors in eq.(2) are as > 0 > af and lasl>lafl Assignment of the chemical species in the TG signal is made based on the following facts. (a) All of these species have their absorption bands at shorter wavelengths than the light of the HeNe laser. A theoretical consideration on the refractive index associated with the absorption bands states that the slow and fast components originate from the radical species (or the reaction products) and the parent molecules, respectively. (b) The slow component decays faster under the air saturated condition. (c)BQH is revealed to be the only stable radical during the TG measurement from the EPR and transient absorption spectrum. On the basis of these considerations, we conclude that the slower signal is due to the radical and the faster one comes from the parent molecule. Figure 2 shows the q dependence of ks, and kf. The each slope of the plot gives D of the each chemical species. Similar arguments also hold for the other chemical species and also in ethanol. The determined D of the radicals (Dr) and parent molecules (Dp) are plotted against 1/r (r; radius of the molecules) in Fig.3. It should be noted that Dr are generally smaller than Dp. This is not an apparent result because the difference in the molecular volume between the parents and their radicals is just one hydrogen atom, which is negligible compared to the whole molecular volume. The relation, Dr < Dp, holds for all the radicals regardless of the nature of the solute. O
OH [~
OH +
O
H3C/~CH3
+
OH HaC~CH3
O,
O
O
2 ol
E
9
O
[]
O
0.0
01S
~.0
1.S
2.0
2.5
[]
f"l
3.0
t/ms
0
I
2
2
q /l.~m
3
-2
Fig.1 (left) Time profile of the TG signal after the photoexcitation of benzoquinone in 2propanol (dotted line) and the best fitted curve with cq.(1) (solid line). Fig.2 (right) Plots of the observed decay rate constants (k) of BQH (squares) and of BQ (circles) vs. square of the wavcvcctor,q 2. 4. Discussions 4.1. Comparison of D with the calculated D
403 The Stokes-Einstein equation, which describes D of a diffusing molecule in a solution with viscosity rl, is given by DSE = kT/fvlr (2) where r denotes the radius of a spherical molecule and f is a factor for correction of the deviation from the spherical shape and of the boundary condition. The determined Dr agrees well with this DSE. On the other hand, we found that a semiempirical modification of eq.(2) proposed by Spernol and Wirtz 5 reproduces Dp very well. In the following, we discuss the origin of the difference in D between the radicals and the parent molecules. 4.2. Molecular size effect and a radical diffusion model Fig.3 shows that Dp almost constantly decreases with increasing molecular size. On the other hand, the Dr plot first decreases with decreasing 1/r until 0.3X10-10m-1, and then it increases gradually ('inverted' region,). In terms of the friction, this tendency can be treated as follows. The friction coefficient of a species i (i=p or r), fi, is defined by fi=kT/Di . (3) An excess friction, Af, of the radical gained by the conversion from the parent molecule to the corresponding radical is given by Af=fr-fp 9 (4) When this quantity is calculated for each solvent and plotted against l/r, it apparently shows a decreasing trend with increasing r. The trend in the excess friction of the solutes studied herein may be accounted for by the following model. First we consider that the radical is diffusing accompanied with several solvent (or other solute) molecules and the effective molecular volume of the radical is defined by including the attached solvent molecules around the radical. Next we assume that an unpaired electron has the capability of increasing the effective molecular volume only by a certain amount V. Then we obtain a simple relation between Af and r as, Af ~ 1/r . (5) This relation can explain the observed 1/r dependence of Af qualitatively. In this paper, the excess friction is treated as an apparent excess volume of the radical. However, since the radicals cannot be tightly bounded with solvent (or solute) molecules, a description that the radicals feel a larger friction from the surrounding molecules during the movement by the intermolecular interaction may be more adequate. 4.3 Solvent effect Since the radical concentration is very low ( 10-5 M) in the measurement, we might neglect a possible radical-radical interaction. The radicals should be surrounded by the solvent molecules in our observing time scale. Thus, an attractive intermolecular interaction between the radical and solvents (or solute) is necessary for increasing the apparent volume. One possibility for such an interaction is a hydrogen bonding interaction.The effect is examined by measuring D of benzophenone kethyl radical (BPK) in non-polar solvents. Producing BPK in nonreactive solvents, we add hydrogen donors. We find that the values of D are inversely proportional to the viscosity of the solvents which is predicted from the SE relation (Fig.4). Therefore, we believe that the hydrogen bonding is not the origin of the interaction. 4.4 Temperature dependence To examine the adequacy of the proposed diffusion model, Dp and Dr are measured at various temperatures. The temperature dependence of both D can be well expressed by the Arrhenius relation with activation energies close to that of the viscosity of the solution. The similar ED of Dr and Dp indicates that both diffusions are governed by the molecular motion, not by chemical reactions. (If a chemical reaction is involved in the decay of the
404
TG signal, the activation energy of Dr should be totally different from that of Dp because of the activation energy of the reaction.) Further, we find that the small but non-negligible difference in ED between the radical and the parent molecules can be explained by the concept of the larger volume of the radical, which supports the our radical diffusion model. Conclusion" Ds of various transient radicals are measured by the TG method. We found that Dr are 2,-, 4 times smaller than Dp. The discrepancy is interpreted in terms of an attractive intermolecular interaction between the radical and solvents (or solute), which may create microscopic clusters in solution. 2.2
-
9 i
I
a)
2.0 1.8
9 ,
9 |
9 i
9 i
*
i OH
1.6 Tu~ N
|
9
o
E m 'o
o
1.2
.~ o. tm
OH
4
1.0 0.8 0.6
i
.,...
9 i , t . i . I , i , i , I , 1 0.0 U. 8 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34
03
r "t 1 I 0 " l ~ rn "l
0.9
'
E
O
[]
0.6
El
~
t
0
s
b)
0.8 0.7
,
3
1.0
79
i
o
0.4. 0.2
N
mcAcH,
9
1.4
8
0
[]
E "9 'o "-
J
0.5 0.4
~
[]
0.3
.
0
z 1
i
1
~-I
0.2
i
2 /
cp-1
1
3
4
0.1 0.0
9
i
,
i
.
i
.
I
,
i
,
i
i
|
.
1
U.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 r "l
/
10
"l~
rn-~
Fig.3 (lcft) Ds of the parent molecules (a) and thosc of the transient radicals (b) in 2propanol (closed circles and squares) and in ethanol (open circles and squares) against the reciprocal of the radii (r-1) of the molecules. The solid lines represent DSW in (a) and DSE in (b). Fig.4 (right) Plots of Dp (circles) and Dr (squares) for benzophenone in various solvents against viscosity. Solid line represents DSE. References 1 (a)E.L.Cussler, "Diffusion", Cambridge University Press, Cambridge,(1984). 2 M.Terazima, N.Hirota, J.Chem.Phys.,95,6490(1991). 3 M.Terazima, K.Okamoto,N.Hirota, J.Phys.Chem.,97,13387(1993).; M.Te razi ma,K. O kamoto, N. H irota,J. Phys. C hem.,97,5188( 1993 ). 4 J.B.Pedersen,C.E.M.Parbo,L.T.Muus, J.Chem.Phys.,63,2398(1975). 5 A.Spernol,K.Wirtz, Z.Natureforsch,89,522(1953).
journal of MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 ( i 995) 405--408
Intra and I n t e r m o l e c u l a r E n e r g y T r a n s f e r in Thermal Dissociation in the Gas and Liquid N 2 0 4 Toshiko Kato, Soichi Hayashi,* and Katsunosuke Machida**
Seibo Jogakuin Jr. College, Fukakusa, Fushimi-ku, Kyoto, 612, Japan *Institute for Chemical Research, Kyoto University,Uji, Kyoto-fu, 611, Japan **Kyoto Pharmaceutical University, Yamashinaku, Kyoto, 607, Japan Abstract Molecular dynamics simulations were performed for the dissociation and association (D/A) reactions N204=2NO2 in the gas and liquid phases. The transition state of the D/A reactions was identified as the phase space surface ET=0, where ET is the sum of the potential and kinetic energies of the interfragment motion. The dissociation reaction in liquid N204 is found to occur by two steps: first, a reactant is activated by intermolecular energy transfer from solvent to reactant, and second, the dissociation occurs by energy transfer from intrafragment vibrational and rotational modes to interfragment translational mode. The association reaction proceeds by two steps: interfragment translational energy of an encounter NO2 pair is transferred to intrafragment modes and then the associated N204 is deactivated by intermolecular energy transfer from product to solvent.
I. Introduction Thermal D/A reaction N204=2NO2 is interesting in that it is an equilibrium reaction at room temperatures and the dissociating fragments have internal structure beyond that of a diatomic. By studying the molecular dynamics of this system, we shall encounter many problems which will arise in other no-barrier dissociation reactions into polyatomic molecules. We shall focus here on the roles of intermolecular energy transfer between the reactant and solvent, and intrareactant energy transfers between the vibrational(V), rotational(R), and interfragment translational(T) modes.
II. Molecular dynamics simulation The classical dynamics simulation was initialized from a distribution of all variables in the equilibrium MD simulation of liquid N204,1 except the NN distance of a selected N204 was compressed to some value R ex so as to give the NN mode enough energy to dissociate. After replacing the harmonic intramolecular potential of the N204 by a sum of an intermolecular potential between NO2 molecules and harmonic intramolecular potential for NO2, 2 the dynamics were calculated both forward and backward in time, for a time period of 20 ps with a time step At=1/2048 ps. We used the Morse+LJ potential for the NO2-NO2 interaction, which is a sum of a Morse potential between N atoms and atom-atom Lennard-Jones potentials for the N-O and O-O pairs (Table 1). From each trajectory of a selected reactant NO2 pair in the gas phase (isolated reactant) 0167-7322/95/$09.50 91995 Elsevier Science B.V. All rights reserved. SSDI 0167-7322 (95) 00886-I
406
Table 1. Atom-atom potential parameters for the NO2-NO2 interaction. Interaction
Type
N-N
Morse
(~-1)
re(A)
a
1.782
2.4987
m (10 -18 J ~6)
De (10 "18 J) 0.0945
n (10 "18 J
N-O
Lennard-Jones
0.5978
115.14
O-O
Lennard-Jones
1.4177
646.30
0.2
---
0
'2
0.2
A12)
~
Etotal
,e
9 ~
o
_
r,~,,,,
_
-
-
~
. . . . . . .
Ev
0.2 0 0.2
/
+
+
/
0
ET
8 .-,
6
2
E
F
I
I
I
0
10
20
t(ps)
Figure 1. An example of the time dependent energy partitioning and RGG(t) for a DAD trajectory in the gas phase. The ordinates are for Etotal, EV, ER, ET, and RGG from top to bottom.
407
0.2
~-----
0 ,., 0.2
.
.
.
.
.
.
Etota !
.
t..
0.2 0 _ 0.2
~
Ea
-
~
ET
6
4 2
13
It
_1
i
t
t
!
0
1
2
3
4
t(ps)
Figure 2. An example of the time dependent energy partitioning and RGG(t) for a ADA trajectory in liquid N204. The ordinates are for Etotal, EV, ER, ET, and RGG from top to bottom.
and in 124 N204 molecules, we stored as a function of time: (1) RGG the distance between the centers of masses of two NO2 molecules, which is the reaction coordinate, (2) ET the sum of the potential and kinetic energies of interfragment motion, (3) ER the sum of the rotational kinetic energies of two NO2, (4) EV the sum of the intrafragment vibrational energies of two NO2, and (5) Etotal the total energy of the reactant. III. R e a c t i o n m e c h a n i s m s and s o l v e n t e f f e c t
By investigating all the trajectories (about 1500 MD) initiated from states activated to various energies of each N204 in 125 N204, characteristics of energy transfers among intrareactant V, R, and T modes and solvent modes(S) were studied in detail. Energy ET, which is defined to be the sum of the potential and kinetic energies of interfragment motion, is found to play a key role in the D/A dynamics in both the gas and liquid phases; a reactant pair is associated (A state) when ET<0 and the pair is dissociated (D state) when ET>0. The transition state to the D/A reactions is hence defined by ET=0 in the phase space.
408 In the gas phase where total energy is conserved, energy transfer between the reactive mode T and intrafragment V and R modes gives a decisive role in determining the reaction fate. Figure 1 shows an example of the gas phase dynamics; the fragments associate at E via T-V energy transfer (ET changes sign from positive to negative), then energy redistribution occurs among V, R, and T modes in the associated state, and finally dissociate at F via R-T energy transfer. The life time of N204 for this DAD type trajectory is 13.3 ps. It should be noted that energy transfer between intrafragment V and R modes and reactive T mode occurs usually at the inner turning point on the interfragment potential, and in very few cases R-T energy transfer occurs near the outer turning point. In liquid N204, total energy of reactant is not conserved because of intermolecular interaction. The relaxation of Etotal occurs in two steps; rapid relaxation at 30-70 fs comes mainly from the decay of translational kinetic energy of fragments because of collisions with surrounding molecules, and gradual decay in the time range 0-1 ps comes mainly from potential energy relaxation. 2 Figure 2 shows an example of liquid phase dynamics. The fragments dissociate at G via S-T and R-T energy transfer, they diffuse for about 1.8 ps, and associate again at H via T-S and T-R energy transfer. If we compare the gas and liquid phase dynamics initiated from the same condition of a reactant, the following behavior is observed. The dynamics in both phases are essentially the same for about 0---30 fs. After this period, kinetic energy of interfragment motion gained by the excitation remains constant, leading to dissociation in the gas phase, while in the liquid phase, the fragments lose their kinetic energy quickly because of collisions with solvent molecules and remain trapped in a local cage (the cage effect). The effects of intrafragment modes on the D/A reactions are similar in both the gas and liquid phases. The V-T and/or R-T energy transfer is found to excite the reactive T mode or gives rise to dissociation, and T-V and/or T-R energy transfer often causes association. In addition to these mechanisms, the D/A reaction is found to occur by an energy transfer between reactant relative translational mode and solvent modes. The reaction rate is determined by the flux which cross the transition state surface ET=0, and this flux arises essentially by energy transfers between the reactive T mode and V, R, and S modes of reactant and solvent. From the gas phase MD of a reactant NO2 pair initiated from various R ~ values, the lifetime x of the A state was obtained as a function of the reactant's total energy. It is found that
x~ ~ at low Etotal and x ~ 0 at high Etotal, and this is because a reactant is always associated at low energy and dissociates directly at high energy. However, the lifetime x of a reactant is found not to be a monotonically decreasing function of Etotal, but shows a striking variation within a narrow energy range. The lifetime, and hence the dissociation rate, of N204 is found to be highly dependent on the specific state of the reactant. References 1 T. Kat6, S. Hayashi, M. Oobatake, and K. Machida, J. Chem. Phys. 77, 177 (1992). 2 T. Kat6, S. Hayashi, M. Oobatake, and K. Machida, J. Chem. Phys. 100, 2777 (1994).
journal of
MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65166 (1995) 409--412
Photoelectron Emission Spectroscopy Applied to Bromide Ion in Solvents Hajime Tanida and lwao Watanabe Department of Chemistry, Faculty of Science, Osaka University, Toyonaka, Osaka 560, Japan. Abstract The solution phase ionization potentials of Br- in 16 solvents have been determined by photoelectron emission spectroscopic technique. The values obtained as the threshold energy E t for Br- in various solvents are found to be correlated well with the Mayer-Gutmann acceptor number of solvent. The reorganization energy A G r of solvent after the photoionization of Br- has been obtained from the E t value. The A G r values are well reproduced by using a simple model which incorporates the dipole-dipole repulsion and the hydrogen-bond formation in the first solvation layer. The solvation structures of Brdetermined by EXAFS are used for the A G r calculation.
I. Introduction The vertical ionization potential for a solvated chemical species can be the measure of its reactivity in the solution phase, especially for a single electron transfer reaction. It has been reported that the ionization potentials of anions in solution are correlated with the kinetic parameter for nucleophilic substitution reaction. 1 This implies that an important aspect of the activation process of the reaction is a single electron transfer from anion to substrate. The ionization potential for solvated species has been available as the threshold energy E t by photoelectron emission spectroscopy for solution (PEES). a This spectroscopic technique is able to provide the E t values of almost any solvated species, such as organic, inorganic, cations, anions and neutral molecules in aqueous and nonaqueous solutions. When a solvated anion Br-(solv)is irradiated at the solution surface by a vacuumultraviolet light, it is photoionized if the photon energy hv exceeds the minimum energy hvrnin, or Et, necessary for the following reaction, Br-(solv) ~
*Br~
+ e-(vac)
hvmin=Er
(1)
The E t value reflects the stabilization energy of the negative charge by surrounding solvent molecules. Thus its variation by solvation can be used as one of the solvent parameters. Since reaction 1 is nothing but a one-electron oxidation reaction, E t can also be called optical oxidation potential. 3 The standard oxidation potential value is often difficult to be determined because many redox reactions are not reversible. Therefore, the E t value should be a good alternative as a measure of redox reactivity for which the equilibrium redox potential is not known. In this study, the E t values are measured by PEES for Br- in 16 solvents 4 and the reorganization energies A G r pertaining to the relaxation process after the photoelectron emission are obtained from the E t values. The product *Br~ in reaction 1 still preserves the solvation structure of Br-. Thus the study of A G r provides the structural and energetic 0167-7322/95/$09.50 9 1995 Elsevier Science B.V. All rights reserved. SSD! 0167-7322 (95) 00887-X
410 information for Br- solvation. The A G r values are also predicted by using a simple model. In order to discuss the value for Br-, its solvation distance and solvation number obtained by extended X-ray absorption fine structure (EXAFS) 5 are taken into account.
Table 1. Threshold energies E t and reorganization energies AGrexp for Br- (in eV) 4 and the Mayer-Gutmann acceptor number AN. 6 No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Solvent
N-Methyl-2-pyrrolidinone Pyridine N,N-Dimethylacetamide Dimethyl sulfoxide N,N-Dimethylformamide Tetramethylene sulfone Nitrobenzene Propylene carbonate Acetonitrile Ethylene glycol Nitromethane Formamide Propyl alcohol Ethyl alcohol Methyl alcohol Water
Et
-AGr exp
,.l N
6.27 6.55 6.27 6.46 6.46 6.62 6.48 6.90 7.11 7.16 7.16 7.32 7.36 7.32 7.61 7.79
0.15 0.27 0.23 0.24 0.34 0.48 0.28 0.71 0.93 0.73 0.96 0.93 1.09 1.01 1.23 1.29
13.3 14.2 13.6 19.3 16.0 13.3 14.8 18.3 19.3 42.8 20.5 39.8 31.0 37.1 41.3 54.8
II. Results The PEE spectra were obtained with the apparatus in this laboratory. 4 The E t values measured with 16 solvents are summarized in Table 1. It is found that the E t value correlates with the Mayer-Gutmann acceptor number AN6. This correlation indicates that the electronic energy states of solvated anions are mainly stabilized by the Lewis acid-base interaction. The absorption of a photon produces e-(vac) and leaves a neutral atom *Br~ which has the same solvent cage as that of Br-(solv). Thus the solvent molecules must reorient and form the most stable solvation structure for Br ~ According to the energy diagram in Figure 1, the reorganization energy AG r can be determined as A Gr=EA-Et- A Gs+ AGn, 't I - - B (solv)+ewhere E A is the electron affinity of Br, 3.14 eV, and AG s and AG n the solvation A Gr energies of Br- and Br ~ respectively. We Br~ +e- 4, I~ Br~176 + ehere adopt the approximation of the null solvation energy of Br ~ i.e., AGn=0.
0
|
~Gn
Br-(vac)
|
l
i
A Gs
~'
.o0
/
~
o
E (vs.V.L)
Br-(solv)
r
r Figure 1. Schematic representation of the energies for the process, Br- ---" Br~ + e-.
III. Discussion We estimate the A G r energy by using the following simple model. 4 (1) The anion as a hard sphere is coordinated by solvent molecules which are also hard spheres with a radius r s and a dipole moment # at the center. The solvents are assumed to be arranged, for example, tetrahedrally or octahedrally with their dipoles oriented toward the anion. (2) During the
411
Table 2. Theoretical reorganization energies AGrcal, dipole-dipole repulsion energies AGdd, and hydro~;en-bond formation ener~;ies A Grm in eV with solvent parameters. No. 1 2 3 4 5 7 8 9 10 11 12 13 14 15 16
- AGrcal - A Gdd - AGHB 0.15 0.22 0.26 0.20 0.25 0.29 0.41 0.73 0.84 0.60 1.02 0.88 0.73 1.19 1.34
0.07 0.13 0.07 0.07 0.06 0.24 0.38 0.67 0.05 0.56 0.36 0.28 0.10 0.49 0.73
0.09 0.09 0.19 0.13 0.19 0.05 0.03 0.06 0.79 0.04 0.66 0.61 0.64 0.70 0.61
/t 4.09 2.37 3.71 4.06 4.24 4.00 4.94 3.53 2.31 3.56 3.37 3.06 1.66 2.87 1.82
6h2/Cal cm -3 rs/~ 12.25 8.41 25 25 30.25 4 4 9 161.29 6.25 86.49 72.25 90.25 118.81 428.49
2.72 2.56 2.68 2.46 2.52 2.77 2.61 2.22 2.26 2.24 2.02 2.50 2.30 2.04 1.55
r/,~ a
R/A,
3.55 3.52 3.56 3.53 3.61 3.49 3.50 3.48 3.14 3.43 3.26 3.13 3.14 3.15 3.19
5.20 5.01 5.18 4.92 5.07 5.19 3.50 3.48 4.44 3.43 4.27 2.65 2.66 2.67 2.71
Na 2.9 5.1 3.2 2.8 3.2 4.6 3.1 4.7 3.3 4.3 4.9 2.9 3.1 3.7 6.0
c 0.55 2.86 0.72 0.52 0.73 2.07 0.67 2.27 0.85 1.77 2.47 0.55 0.67 1.13 4.44
a Ref. 5.
reorganization process, only the orientation of solvent dipoles is rearranged so that the unfavorable dipole-dipole interactions in *Br~ disappear and the new bonds between solvents represented by the hydrogen-bond are created. The dipole-dipole repulsion energy A Gdd is calculated from a sum of all the 101 repulsion energies between dipoles as follows,
AGdd=-Clt2/R 3 eV debye -2 ~3,
(2)
where c is the constant calculated from the dipole configuration. The value of c can be calculated easily for each solvation number N of 2, 3, 4, 6, or 8 with symmetrical arrangement and is plotted against N in Figure 2. Since the EXAFS study 5 for Br- in various solvents indicated that the average solvation number is widely different from solvent to solvent, the value of c for a given N value is determined by using the relation in Figure 2. R is the distance between Br- and a dipole, which is estimated using the coordination distance r from
5
2
I
I 4
I N
I 6
I
"18
Figure 2. The values of the constant c in equation 2 for dipole-dipole repulsion energy A Gdd as a function of N.
412 the EXAFS results as R=rB,_c-rc_H+rs for aprotic solvents. For protic solvents the dipole must be at the site of OH group and the OH groups come very close to each other. Therefore the R values are evaluated to be equal to R=rar_o-ro_n/2 accordingly. The hydrogen-bond formation energy AGm3 is estimated from the following equation,
A GIIB = -6h2(4zt/3)rs 3 xN eV ca1-1 cm 3 A -3,
(3)
.'''
>
1
el (..1
'l
_
;66 <
I~//
100/ "
<11 0 . 5
I'
....
/
_
/
;,0
/
/
/
/
/
O13
/
-
90 014 o 11
/
_
-
_
o
4 2
/ , , , , I , , , , I , , i where 6h2 is the hydrogen-bond compo0.5 1 nent of the solubility parameter by - A GreXp / e V Barton 7 and corresponds to the cohesive energy per unit volume of solvent originating only from the hydrogen-bond Figure 3. Comparison of the experimental formation. The volume used for estiA GreXp and theoretical A GrCal for Br-. mating AGrm is limited to the solvent shell taken to be the product of N and the molar volume of solvent. The values for 13-16 are multiplied by 1.5, since the hydrogen-bond network may be formed with the second solvation sphere. The calculated values AGrcal (=AG~+AGm3) are listed in Table 2 and compared with the experimental ones in Figure 3. The agreement is surprisingly good in spite of the very simple model used. As a conclusion the electronic energy data by the PEES technique can be used to discuss the solvation structure around anion. References
1 E. Buncell, S. S. Shaik, I.-M. Um, and S. Wolfe, J. Am. Chem. Soc., 110, 1275(1988); S. S. Shaik, J. Org. Chem., 52, 1563(1987); S. S. Shaik, E. Dusy, and A. Bartuv, J. Phys. Chem., 94, 6574(1990). 2 P. Delahay, Acc. Chem. Res., 15, 40 (1982); P. Delahay in Electron Spectroscopy: Theory, Techniques and Applications, edited by C. R. Brundle, and A. D. Barker, vol. 5 p123 (Academic Press, London, 1984); I. Watanabe, J. B. Flanagan, and P. Delahay, J. Chem. Phys., 73, 2057(1980). 3 I. Watanabe, Anal. Sci., 10, 329(1994). 4 I. Watanabe, H. Tanida, K. Maya, S. lkeda, and Y. Yokoyama, Bull. Chem. Soc. Jpn., 67, 39(1994). 5 H. Tanida, H. Sakane, and I. Watanabe, J. Chem. Soc., Dalton Trans., 2321(1994). 6 U. Mayer, V. Gutmann, and W. Gerger, Monatsch. Chem., 106, 1235(1975); V. Gutmann, The Donor-Acceptor Approach to Molecular Interactions (Plenum Press, New York, 1978). 7 A. F. M. Barton, Chem. Rev., 75, 731(1975).
journal of MOLECULAR LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 413-416
Direct C o m p a r i s o n b e t w e e n F e m t o s e c o n d O p t i c a l K e r r Effect and HighR e s o l u t i o n Light S c a t t e r i n g M e a s u r e m e n t s Y.Kai, S.Kinoshita, M . Y a m a g u c h i 1 a n d T.Yagi I
Department of Physics, Faculty of Science, Osaka University, Toyonaka, Osaka 560, Japan 1Research I n s t i t u t e for Electronic Science, H o k k a i d o University, S a p p o r o 060, Japan Abstract We have performed optically heterodyne-detected optical Kerr effect measurement for transparent liquids with ultrashort light pulses. In addition, the depolarized low-frequency light scattering measurement has been performed by means of a double monochromator and a high-resolution Sandercock-type tandem Fabry-Perot interferometer. The frequency response functions obtained from the both data have been directly compared. They agree perfectly for a wide frequency range. This result is the first experimental evidence for the equivalence between the time- and frequency-domain measurements.
1. Introduction The dynamical behavior of molecular liquids is directly observed via femtosecond optical Kerr effect (OKE) measurement in the time domain, x~s The signal intensity, SOKE(r), obtained by the measurement using an optically heterodyne-detected (OHD) technique (mixing the OKE signal with a local oscillator) can be expressed in the form 4'6
/? dtE~o(t - r)Ep~ob~(t -- r)Q(t),
SOKE(T) CX
(1)
o~
where 9 , d t ' R ( t - t')Ep~mp(t ) E p ~ p ( t ' )
Q(t) o( oo
is the material excitation. In these expressions, R(t) is the impulse response function, and Eprobe(t), Epu~p(t) and ELo(t) represent the electric field components of the applied probe and pump beams, and the local oscillator introduced for the OHD, respectively. Since Eprobe(t), Ep~mp(t) and ELo(t) are derived from a single laser pulse, Eq.(1) can be expressed in the useful form 3'4 SOKE(r) =
dtG(2)(r - t)R(t),
(2)
oo
where G ( 2 ) ( T ) is the intensitv autocorrelation function [IAF) of the input laser pulse and r is the time delay between the two pulses. The OHD-OKE signal is expressed bv the convolution integral and thus the response function is obtained without the comp[exitv of analyzing the signal, while it is quadratic in the (homodyne) OKE measurement. By 0167-7322/95/$09.50 91995 Elsevier Science B.V. All fights reserved. SSDI 0167-7322 (95) 00888-8
414
means of the Fourier transform of Eq.(2), the frequency response function, /~(Aw), can be calculated as R(Aw) = SOKE(A~o)/(~ (2)(Aw). (3) We note that in this expression two of the three quantities are experimentally determined. On the other hand, the depolarized low-frequency light scattering (LS) measurement under stationary excitation in a right-angled configuration, offers the information on the fluctuation in the frequency domain. The frequency spectrum obtained by this measurement can be represented as 6
SLs(Aw) (x w3{nT(Aw)+ l}Im/~(Aw),
(4)
where a factor denoted a s nT(AW)-+-1 is a Bose factor with the phonon o.ccupation number of nT(Aw) and w is the angular frequency of the emitted photo,l. By referring to Eqs.(3) and (4), the imaginary part of the Fourier transform of the OHD-OKE response divided by the Fourier transform of the IAF of the laser pulse is directly comparable to the LS spectrum divided by the Bose factor in addition to w3. Because the dynamical behavior and the fluctuation ought to be correlated with each other through the fluctuation-dissipation theorem, this comparison can verify that the information provided by the two experiments are identical. To our knowledge, however, the direct comparison has not been made yet. In this report, we will describe the details of our experiments and thier results on this problem. 2. E x p e r i m e n t a l The OHD-OKE measurement is performed in a pump-probe configuration using a cw mode-locked Ti:sapphire laser. The average power, the duration o t t h e pulse and the repetition rate are 500mW. 80fs and 80MHz, respectively. Both the probe and pump beams are first directed to the computer-controlled optical delays and then focused into a sample cell of 1 mm pathlength containing a transparent liquid such as nitrobenzene, chlorooenzene, toluene, benzene, CS2, acetonitrile and CC14. The probe beam is polarized at 45 ~ to the incident laser beam by a prism polarizer, passed through a quarter-wave plate which is inserted between the polarizer and a focusing lens, and analyzed bv the second polarizer with the crossed polarization direction to the first polarizer. The/4uarter-wave plate is slightly rotated by ,-~ 1~ in order to mix the OKE signal with a local oscillator. The signal is detected by an avalanche photodiode and amplified by a lock-in amplifier which is referenced to the sum of the different chopping frequencies for the pump and probe beams. This procedure takes away the effect of unwanted scattering of the pump or probe beam in the sample cell and markedly improves the signal-to-noise ratio. To analyze the OKE transient, it is necessary to measure the IAF of the incident laser pulse. To do this, we put a KDP crystal i n p l a c e of the sample and measured the intensitv of noncollinear second harmonic signal. On calculating the Fourier transform of the signal, special care should be taken to determine accurateIv the time origin for the OKE measurement. For this purpose, we have measured the IAI: very carefully and have paid a particular attention to the mechanical stability of the optical system between the OKE and IAF measurements. The depolarized low-frequency LS measurement is performed under stationary excitation by means of a double monochromator for a wide frequency range and a Sandercocktype tandem Fabry-Perot interferometer T for a high-resolution study. The same samples as above are measured using 1 cm glass cell. The LS is measured under a depolarized condition in a right-angled configuration. 3. R e s u l t s and D i s c u s s i o n
The temporal profile of the OHD-OKE measurement on nitrobenzene is shown in Fig. la. The insert shows the short-time behavior of the OKE transient. The solid line
415
1
o
'
'
6
'
'
1'o
-,"4
6
6
'
'
'
'
lbo
i
2oo
. . . .
delay time
-2'oo'
/ps
'
'
6
'
'
'26o
wave number/cm-1
Figure 1 a) The temporal profile of OHD-OKE on nitrobenzene. In the insert are shown the short-time behavior of the OKE transient (solid line) and the intensity autocorrelation function of the laser pulse (dashed line) b) The light scattering spectrum of nitrobenzene measured by the double monochromator and the tandem interferometer (insert).
II
tX..A ., .~, I
0 '
t
i ,
I 10
I
q
0
100 200 wave number/cm-I
Figure 2 The imaginary part of frequency response function, Im/~(Aw). obtained by the OHD-OKE (dots) and the light scattering measurements by the double monochromator (dashed line). In the insert is shown Im/~(Aw) obtained by the tandem interferomator (solid line). A small peak appearing at 3.3 cm -1 (shown as a symbol of ,) is the ghost due to the tandem interferometer.
416
indicates the OHD-OKE response that consists of an instantaneous response, which is the electronic contribution to the O KE at time origin, a fast decaying relaxation with a time constant of ,--0.3 ps and a slower one with that of ~30 ps. ~ The dashed line indicates the IAF of the laser pulse. The depolarized LS spectrum on the same sample obtained by the double monochromator are shown in Fig. lb. The insert shows a spectrum obtained by the tandem interferometer. It consists of a very strong peak around the center and the weak shoulders at both sides. The former is called a central mode and comes from the diffusive reorientation process, while the latter is called a low-frequency phonon mode and comes from the librational motion of the molecule around its mean configuration. The imaginary part of the Fourier transform of the OHD-OKE response divided by the Fourier transform of the IAF of the laser pulse is shown as dots in Fig. 2. A sharp peak around the origin which corresponds to the slowly decaying relaxation is caused by the central mode, while a broad band extending ~250 cm -1 which corresponds to the fast decaying relaxation is caused by the low-frequency phonon mode. The small hump around 176 cm -1 is due to the intramolecular vibrational mode s as shown in the insert of Fig. la. The LS spectrum is divided by the Bose factor in addition to ~a. The result of the above calculations is shown as the solid line and the dashed line in Fig. 2. Obviously, the time-domain data acquired by means of the OHD-OKE measurement agrees completely with the frequency-domain data obtained by means of the depolarized LS measurements for a wide frequency range from 0.1 cm -1 to 250 cm -1. 4. C o n c l u s i o n s We have measured the time domain OHD-OKE response and the frequency domain depolarized LS spectra on the same sample, and have compared the data. The relationship between these measurements through the fluctuation-dissipation theorem is for the first time verified experimentally. This experiment has a meaning of a great significance. In the study of the fluctuation, these measurements play a complemental role over a wide time or frequency range. Especially, the time domain OHD-OKE measurement has suitability for the measurements which are hardly carried out in the frequency domain, such as the relaxation phenomena having a longrelaxation time or the resonance condition, for instance. The frequency resolution in OHD-OKE depends upon the length of the optical delay. Therefore, in principle, high-resolution spectra can be obtained at any time when the mechanically steady scanning of the optical delay and stable ultrashort laser pulses are attained. The work along this course is now in progress.
References 1 W.T.Lotshaw, D.McMorrow. C.Kalpouzos and G.A.Kenny-Wallace, Chem.Phys.Lett. 136, 323 (1987)
2 S.Ruhmam, L.R.Williams, A.G.Joly, B.Kohler and K.A.Nelson, J.Phys.Chem. 91, 2237 (1987) 3 D.McMorrow, W.T.Lotshaw and G.A.Kenney-Wallace, IEEE J.Quantum Electron. QE-24, 443 (i 988) 4 D.McMorrowand W.T.Lotshaw, J.Phys.Chem. 95, i0395 (1991) ; Chem.Phys.Lett. 174, 85 (1990) 5 Y.J.Chang and E.W.Castner,Jr., J. Chem.Phys. 99, i13 (1993) ; J.Chem.Phys. 99, 7289 (1993) 6 R.W.Hellwarth, Prog.Quantum Electron. 5, I (1977) 7 R.Mock. B.Hillebrands and R.Sandercock, J.Phys.E:Sci.lnstrum. 20, 656 (1987) 8 J.H.S.Green, W.Kynaston and A.S.Lindsey, Spectrochim. Acta 17, 486 (1961)
journal of
MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 417--420
Vibrational Cooling Process in Solution Probed by Picosecond Time-Resolved Raman Spectroscopy. Analysis of the Cooling Kinetics. Koichi Iwata and Hiro-o Hamaguchi Molecular Spectroscopy Laboratory, The Kanagawa Academy of Science and Technology (KAST), KSP East 301, 3-2-1 Sakato, Kawasaki 213, Japan
Abstract The vibrational cooling process in the first excited singlet ($I) state of trans-stilbene in hexane was observed at room temperature with picosecond pump-probe time-resolved Raman spectroscopy. From the time-dependent change of the peak position of the 1180 cm 1 band, the cooling kinetics of the solute stilbene was clarified. The cooling kinetics was well explained by a simple model in which the vibrational cooling rate of the solute molecule is controlled by the heat conduction in the surrounding solvent. The simulated cooling curves for two different initial heat distributions (8-function and box function) are compared with the observed values. I. Introduction When a molecule is photoexcited in solution with vibrational excess energy, the energy is first redistributed in the vibrational manifold. As the molecule relaxes to its vibrational ground state, it dissipates the excess energy to the surrounding solvent. By observing this vibrational cooling, we can monitor the energy dissipation process from the solute to the solvent, which plays an important role in chemical reactions in solution. There have been active studies to characterize this process mainly by using absorption or fluorescence spectroscopies in the visible region. ~'2 Recently, we constructed a transform-limited picosecond time-resolved Raman spectrometer which makes it possible to observe the time-dependent change of transient molecules with the maximum possible time and frequency resolutions. 3'4 Because the vibrational cooling process occurs in about 10 ps and because we can directly monitor vibrational levels with the time-resolved Raman spectroscopy, it is expected to be a suitable method to monitor the vibrational cooling process in solution. In this paper, we present the time-dependent .positional change of the 1180 cm ~ band of trans-stilbene in the first excited singlet (S t) state m a hexane solution, which represents the vibrational cooling of the solute stilbene. We simulate this cooling process with a simple model in which the vibrational cooling of a solute is controlled by the heat conduction in the solvent and compare the simulated decay kinetics with the observed values. II. Experimental The time-resolved Raman spectra were measured with a picosecond time-resolved Raman spectrometer which employs a standard pump-probe technique. The details of the spectrometer have been published elsewhere. 3'4 The followings are concise description of the apparatus: The output from a synchronously pumped mode-locked dye laser is amplified with the output from a cw Nd:YAG regenerative amplifier. The second harmonic (294 nm, 2 kHz, 1-2 roW) of the amplified light (588 nm, 3.2 ps, 2 kHz, 15 mW) was used as a 0167-7322/95/$09.50 91995 Elsevier Science B.V. All rights reserved. SSDI 0167-7322 (95) 00889-6
418 pumping light while the remaining visible was used for the probe. The probe light power was reduced to 0.1 mW to avoid the effect of the probe light electric field on the Raman band shapes. 5 The spectral width of the probe light was measured to be 3.5 cm t. The collected Raman signal was dispersed with a single monochromator (Instruments SA, HR320) and detected by a CCD detector (Princeton Instruments LN/CCD-1024 TKB). A narrow band filter (Kaiser Optical Systems, Holographic SuperNotch Filter) was used to reject the elastically scattered light. ElL Results The time-resolved Raman spectra of the S t state of trans-stilbene were measured in a 3x10 3 mol dm 3 hexane solution (Figure l(a)). The 1160 to 1200 cm "~region is expanded in Figure 1(b). From the figure, it is clear that the position and the width of the 1180 cm ~ band change as the time delay increases. The Raman bands in Figure 1(b) are well fitted to Lorentzian functions regardless of the time. From this band shape analysis, it is possible to determine the precise peak position. The obtained peak positions are plotted against the corresponding time delays in Figure 2. The peak shifts caused by the pump light electric field~ have been compensated. The position changes by about 2 cm l in 50 picoseconds after the photoexcitation. The observed kinetics agrees to that for the 1570 cm t band of the S~ trans-stilbene that was measured simultaneously with the 1180 cm ~ band. It has been reported that the position of the 1570 cm ~ band changes linearly as the temperature increases 6 and that the position for t=O, where the change starts, depends on the amount of the excess energy given by the pump light." We also observed that the anti-Stokes/Stokes intensity ratio for the 285 cm ~ band of S t trans-stilbene decays with the same time constant as that for the positional change of the 1570 cm ~ band. s From these observations, it is most probable that the observed time dependence of the peak position of the 1180 cm ~ band represents the temperature change, that is, the vibrational cooling process of the S t trans-stilbene.9.t~
0 1 2 3 5 7 10 15 20 30 40 50 70 10( 1600
1400 1200 Raman Shift / c m l
p$ ps ps ps ps ps ps ps ps ps ps ps ps ps 1200
1180 Raman Shift / cm ~
1160
Figure 1 Time-resolved Raman spectra of the 3x10 3 mol dm 3 solution of S t trans-stilbene in hexane (a). The 1160 to 1200 cm ~ region is expanded in (b). Time delays are shown in the center. The position and the width of the 1180 cm ~ band is changed depending on the time delay.
419
IV. D i s c u s s i o n
The temperature change caused by the heat conduction in a macroscopic media can be described by a diffusion equation, ~U/~t = (~./cp) AU = ~: AU,
(1)
where U is temperature, 2Lis thermal conductivity, c is heat capacity, p is density, and ~: =cp is thermal diffusivity. The general solution for Equation (1) is O(x, y, z, t) = (4rtK t) ~ f~I f(~, 11, ~) exp (-[(x-~)2+(y-Ti)z+(z-~)2]/4k: t) d~drld~,
(2)
f(x, y, z) = U(x, y, z, 0)
(3)
where
represents the spacial distribution of the heat at t---0. Based on the solution (2), we estimate the temperature change at the point x=y=z=0 for two different cases. First, if we assume that the initial heat distribution is described by the ~i-function, (4)
f(x, y, z) = ~i(x) 5(y) ~5(z), then the solution (2) can be integrated to U(x, y, z, t) = ( 4 ~ t) ~
exp (-[x2+yZ+z2]/4K: t).
(5)
At x=y=z=0, U(0, 0, 0, t) = (4nK: t)~ .
--
1180.5
,o,,s oS'S~
-
E e-
(6)
1180.0 -
~0
..... ,5-function
oQ .
1179.5 -
~
1179.0 -
~
.... box function ......... exponential
J] i
1178.5 ( 0
i
I
20
i
40
I
60
I
80
100
Time / ps
Figure 2 Time dependence of the peak position of the 1180 cm 1 Raman band of S 1 trans-stilbene. The observed values are shown with circles. Best fits with the model functions derived from the initial heat distributions of the k-function (dashed curve) and the box function (solid curve) are shown. Best fit with the single exponential (dotted curve) is also shown in the figure.
420 In the second case, if the initial heat is localized in a cube of 2r o x 2r o x 2ro, that is, if f ( x , y , z ) = 1 ( x , y , z < r o) 0 (x > r o or y > r o or z > ro),
(7)
then the temperature at x=y=z=0 is U(0, 0, 0, t) = {erf( r0/(4K: t) ~n )}3.
(8)
By using the functions (6) and (8) as model functions, we tried to explain the observed time dependence of the solute temperature. The best fits for the both cases are shown in Figure 2. The thermal diffusivity for hexane, 8.24 x 104 m 2 s~, was used in the model functions. When fitting with the model function (6), the time region before 7 ps was excluded from the fitting because of the unrealistically steep change of the function (6) in this time range. The time 0 was also excluded from the fitting for the function (8). The fit with the equation (6) is poor especially at earlier time delays, while the model function (8) gives a good fit when r 0 = 1.6 nm. The other fitting parameters for the function (8) are the value at t=** (offset) and the amplitude of the change. It is interesting to note that the fit with the exponential decay is also quite good, showing that the time constant based on a single exponential decay scheme is a good measure to characterize the kinetics of the vibrational cooling. It is not naturally expected that the vibrational cooling process in a picosecond time range is so well explained by using the function (8) which is based on the thermal properties of the surrounding solvent, and not the character of the solute itself. This fact suggests that the vibrational cooling process of a solute molecule is controlled by the heat conduction in the solvent. In this scheme, the energy flow at the interface between the solute and the nearest solvents, or solvents in the fast layer, is expected to be much faster than the flow at the solvent-solvent interface. The latter serves as the rate determining step. The r0 value of 1.6 nm also implies that the excess energy spreads rapidly to a volume which is larger than the dimension of trans-stilbene, again suggesting that there is an efficient energy transfer between the solute and the fast layer solvent molecules. The observation that the cooling processes for three different vibrational modes of S~ trans-stilbene, 285 cm 1, 1180 cm ~, and 1570 cm 1 are indistinguishable means that the excess energy is well distributed among vibrational degrees of freedom and that this intramolecular redistribution process is much faster than the observed vibrational cooling time scale. Recently we also observed that the vibrational cooling rates in different solutions have a good correlation with the thermal diffusivities of the solvents. These results will be published soon. References 1 T. Elsesser and W. Kaiser, Ann. Rev. Phys. Chem., 42, 83(1991). 2 Y. Hirata and T. Okada, Chem. Phys. Lett., 187, 203(1991). 3 K. Iwata and H. Hamaguchi, Chem. Phys. Lett., 196, 462(1992). 4 K. Iwata, S. Yamaguchi, and H. Hamaguchi, Rev. Sci. Instrum., 64, 2140(1993). 5 K. Iwata and H. Hamaguchi, J. Raman Spectrosc., 25, 615(1994). 6 R.E. Hester, P. Matousek, J. N. Moore, A. W. Parker, W. T. Toner, and M. Towrie, Chem. Phys. Lett., 208, 471 (1993). 7 W.L. Weaver, L. A. Huston, K. Iwata, and T. L. Gustafson, J. Phys. Chem., 96, 8956 (1992). 8 K. Iwata and H. Hamaguchi, to be published. 9 H. Hamaguchi and K. lwata, Chem. Phys. Lett., 208, 465(1993). 10 J. Qian, S. L. Schultz, G. R. Bradburn, and J. M. Jean, J. Phys. Chem., 97, 10638(1993).
journal of MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65166 ( i 995) 42 i - 4 2 4
Rotational Relaxation of Rod Like Molecules: Diphenylacetylene in Various Solvents. Yoshinori Hirata, Yasuhiko Kanemoto, Tadashi Okada, and Tateo Nomoto* Department of Chemistry, Faculty of Engineering Science and Research Center for Extreme Materials, Osaka University, Toyonaka 560, Japan and tDepartment of Chemistry, Faculty of Education, Mie University, Tsu, Mie 514, Japan Abstract By using fluorescence anisotropy decay and polarized transient absorption spectrum measurement techniques with a picosecond time resolution, we have studied the rotational relaxation of 4-[[4-(dimethylamino)phenyl]ethynyl]-benzonitrile in various solvents. Our aim is to clarify the solvent-solute interactions revealed as a solvent dependence of the rotational relaxation time of solute molecules. Among the solvents we used, ethanol shows characteristic behavior compared with other polar solvents. The rotational relaxation time in ethanol seem to be close to that calculated on the basis of the slip boundary condition, while in other polar solvents, the stick boundary condition can reproduce the experimental value. The results suggests that the strong hydrogen bonding between the solvent molecules allows the solute to rotate more freely within the solvent cage than in other polar solvents. I. Introduction Rotational diffusion of molecules in liquids should be quite important to establish the microscopic description of chemical reaction dynamics in the solution phase. Sometimes, geminate process was suggested to be governed by rotational diffusion rather than a translational diffusion. It may play an important role in the radical dimer formation process on the geminate encounter of p-aminophenylthiyl radical pair. 1 Reorientation relaxation of a solute molecule is usually measured by monitoring either the time dependent fluorescence depolarization or the transient absorption of a polarized light pulse. Rotational diffusion of dye molecules in various solvents has been investigated extensively. Although many of them aimed for the elucidation of solvent-solute interactions including hydrogen bonding, there are ambiguities which come from the electronic relaxation and geometrical change of solute molecules. In order to simplify the situation and extract the information about the rotational diffusion of solute molecules, it is necessary to reduce the flexibility and complexity of the molecular structure, which cause the change of the direction of the transition dipole moment within the molecular frame. Considering these factors, we have chosen 4-[[4-(dimethylamino)phenyl]ethynyl]-
H3C/
I benzonitrile as a probe molecule. This molecule has a rigid rod-like structure and the transition moment between the ground and fluorescence states (Sx state in nonpolar 0167-7322/95/$09.50 91995 Elsevier Science B.V. All fights reserved. SSDi 0167-7322 (95) 00890-X
422
solvents and charge separated state in polar solvents) are parallel to the long molecular axis. Although the electronic relaxation and twisting of the phenyl rings which occur on the intramolecular charge separation may give some difficulty, such characteristics should allow us a simplified view of the rotational relaxation phenomena. Boundary condition of the rotational motion in the solution phase should reflect the solvent-solute interactions. The solvents used in this work is found to be devided into two rOUpS. One is slip and the other is stick. In ethanol, the rotational relaxation time was und to be shorter than those estimated for aprotic solvents with a similar viscosity. The viscosity dependence of the rotational relaxation time in various solvents will be discussed. II. Experimental The rotational reorientation times of the sample in several solvents at room temperature were measured by picosecond time--resolved fluorescence and absorption depolarization spectroscopy. Details of our experimental setups were described elsewhere? For the time-correlated single photon counting measurement of which the response time is about 40 ps, the sample solution was excited with a second harmonics of a femtosecond Ti:sapphire laser (370 nm) and the fluorescence polarized parallel and perpendicular to the direction of the excitation pulse polarization as well as the magic angle one were monitored. The second harmonics of the rhodamine-640 dye laser (313 nm; 10 ps FWHM) was used to measure the polarized transient absorption spectra. The synthesis of the sample is given elsewhere. 3 All the solvents of spectro-grade were used without further purification.
III. Results and Discussion As shown in Figure 1, fluorescence spectrum of I shows a significant solvent effect. In nonpolar solvents, the fluorescence maximum is observed around 370 nm and the absorption and fluorescence spectra show mirror image symmetry. On the other hand, in polar solvents the mirror image symmetry is broken and the broad fluorescence band of which the maximum appears in the long wavelength region is observed. The band maximum shifts to the red with increasing solvent polarity. Measuring picosecond time resolved absorption spectra of I in polar solvents, we observed the spectral change due to the intramolecular charge separation. In polar solvents such as diethylether ether and acetonitrile, immediately after the excitation, the S, ,--$1 absorption band is observed, while the spectrum ascribed to the charge separated state is dominant in the delay time region longer than 10 ps after the excitation laser pulse. The major part of the observed fluorescence in polar solvents seems to be from the charge separated state of I. Because of the short lifetime of $1 in polar solvents, rotational diffusion of $1 should not be significant. It was suggested that the ground and $1 states of I are almost planar and in the charge separated state, two phenyl rings are perpendicular to each other. The geometrical change on the charge separation may not affect the rotational diffusion of the charge separated state significantly because the molecular volume and the ratio of the long/short molecular axis are essentially the same for the $1 and charge separated states. The time dependence of the fluorescence polarized parallel and perpendicular to the pump polarization direction were fitted simultaneously to
I(t) = (I(O)/3)exp(-t/r!)l + 2r(O)exp(-t/ror )
(1)
I(t) = (I(O)/3)exp(-t/r t)l - v(O)exp(-t/ror).
(2)
423
By using nonlinear least squares fitting, the orientation relaxation time (rot), fluorescence lifetime (rf), and the initial value of anisotropy (r(0)) were obtained. The typical results of the fitting to the time dependence of polarized fluorescence of I in n-hexane is shown in Fig. 1. In order to confirm the validity of the data analysis, the fluorescence lifetime was also measured with the magic angle arrangement. 3_. 8
1 ,..,
7.")("
4 "~""~
' ' '
A4i "i,...\ , -",,"Y,.",. \ -~
,
t
i ", ' ', .'. \1;./
.,
i./
400
',..
"-;x \
~,,,,,,,,,,..
500 600 ~,Vavelength [ nm ]
",,(,,:..,.,,..,,:., 700
F i g u r e 1. Fluorescence spectra of I in cyclohexane (6),trans-decalin(8), diethylether (1), ethanol (7), acetone (3), and acetonitrile (4).
"~ 0
I
i
200 ~00
I
0 200 ~00 600 t [ chonnel ]
F i g u r e 2. Time dependence of the polarized fluorescence of I in n-hexane (1 channel = 2.54 ps). The residuals are also shown in the figure.
Except for ethanol solution, a good fitting was obtained for the fluorescence decay curves. In ethanol, even the fluorescence measured at the magic angle shows a complex decay, which should be due to the dynamic Stokes shift. Although we got a rather poor fitting to the biexponential decay function in the short delay times, the estimated fluorescence decay times obtained from the measurements at the magic angle were about 10 and 220 ps. From the fitting of the polarized fluorescence in the long delay times, the rotational relaxation time was obtained to be 124 ps. Reliable value for r(0) was not obtained for this case. The measured rotational reorientation times, initial value of the fluorescence anisotropy, and fluorescence lifetimes in several solvents are listed in Table I. The initial value of anisotropy is very close to 0.4, which suggests that the directions of the transition dipole of the absorption and the fluorescence are the same and the depolarization not due to the reorientation of the molecule can be ignored. Figure 3 shows the plot of the rotational reorientation time versus the solvent viscosity. The straight lines show the reorientation times calculated as rot = C . f~tick " V r # / k B T ,
(3)
where V is molecular volume of solute, 77 is solvent viscosity, and kB is the Boltzmann constant, fstick depends on the molecular shape and is calculated and tabulated by Hu and Zwanzit~.4 The C value is the ratio of the friction coefficient with slip to the friction coefficient ot stick. The probe molecule has been modeled as a prolate ellipsoid with the molecular volume and the ratio of about 70 A 3. Acetone (3) and N,N-dimethylformamide (5) are on the solid line, which was calculated for stick. On the other hand, several aliphatic hydrocarbon solvents (2, 6, 8) and ethanol (7) seem to be
424
on the broken line. The ratio of the slopes of the broken and solid lines is about 0.78, which is slightly smaller than the calculated value from the molecular shape. It is known that the reorientation relaxation times of uncharged molecules are rather close to those estimated under slip boundary condition. The nonpolar solution of I seems to be the case. On the other hand, the stick boundary condition seems to give a better prediction in polar aprotic solvents. This should be due to the strong solvat]on of the large dipole moment produced by the intramolecular charge separation. The orientation relaxation time of I in ethanol was found to be very close to that predicted for slip boundary condition. Important point of our results is that the rotational diffusion in ethanol is faster than that expected for the stick boundary condition. Our results may suggest that the strong hydrogen bonding between the solvent molecules allows the solute to rotate more freely within the solvent cage than in other polar solvents. Table I. Excited-state rotational reorientation times, fluorescence lifetimes (ps), and initial values of the fluorescence anisotropy of I in several solvents at room temperature.
1 2 3 4 5 6 7 8
to,. 43 38 63 49 130 102 124 226
Solvent (viscosity(cP)) diethylether ether (0.24) n-hexane (0.32) acetone (0.32) acetonitrile (0.38) N,N-dimethylformamide (0.92) cyclohexane (0.94) ethanol (1.19) trans-decalin (2.13)
200
1"/ 1880 600 1230 750 850 670 224 760
-
7"(0) 0.39 0.39 0.37 0.43 0.38 0.39 0.37
t tt
5o
loo
.og
-
~
13
Oo
1
Visc~sity [(:P ] Figure 3. Viscosity dependence of the reorientation relaxation time of I.
Acknowledgement: The present work was partly supported by Monbusho International Scientific Research Program (05044055) to T. O.
References I. Y.Hirata, Y. Niga, and T. Okada, Chem. Phys. Lett. 221,283 (1994) 2.Y. Hirata, T. Okada, and T. Nomoto, J. Phys. Chem., 97, 9677(1993). 3. T. Nomoto, F. Tanaka, and N. Suzuki, Chem. Express 6, 319 (1991) 4. C. M. Hu. and R. Zwanzig, J. Chem. Phys. 60, 4354 (1974)
journal of
MOLECUlaR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 425--428
Ground State Recovery of Phenol Blue in Fluids near the Gas-Liquid Critical Density Yoshifumi Kimura and Noboru Hirota
Department of Chemistry, Faculty of Science, Kyoto University, Kyoto 606-01, Japan
Abstract We have made a sub-picosecond pump-probe transient absorption study on Phenol Blue at 308 K in trifluoromethane near the gas-liquid critical density and at the high density (about twice of the critical density) where small amount of 2,2,2-trifluoroethanol is added as c ~ solvent. The ground state hole created by the pump pulse is mostly broadened by about 1 ps, and we could not detect any meaningful density effect on this process.
I. Introduction Recently it has been demonstrated by Barbara and co-workers that the ground state recovery of a solvatochromic dye such as betaine-30 gives us a lot of information on the energy dissipation process from the excited state in liquids. They found an important role of a high frequency intramolecular vibration of the ground state on the energy dissipation process from the excited state, and also found a "hot" ground state configuration after the energy transfer from the excited state molecule to the solvent, which is observable as a red-shifted absorption spectrum. 1-5 In the present work, we have tried to measure such an energy dissipation process with a function of solvent density. In considering the problem of the energy transfer, the solvent density plays an important role, since the collision frequency and the thermal diffusivity, which are dominant factors in the energy transfer process, are strongly dependent on the density. 6 By changing the solvent density, we hope to observe such an effect on the energy dissipation rate from the excited state and on the dynamics of the "hot" ground state. We have also another interest in the fluid near the critical density. Near the critical density, there is a developed local structure around the solute molecule, 7 which is also spectroscopically observable. 8 If the relaxation time of the local structure is slow enough, we could see some density effect on the ground state hole broadening process just after the excitation, since this process is related to the relaxation of inhomogeneity. For these purposes we have studied the solvatochromic dye of Phenol Blue (PB, Scheme 1) in trifluoromethane (CF3H) near the critical density. Although betaine-30 is more preferable for the comparison with liquid solvent, it was impossible due to the extremely low solubility of betaine-30 to CF3H. PB is more soluble in CF3H, and has absorption around 560 nm which is accessible by our laser system. Unfortunately the photophysics of PB has not been studied until now even in liquid solution. Therefore we have also studied the ground state recovery in several liquid solvents. According to the theoretical calculation, 9 the charge 0167-7322/95/$09.50 fc3 1995 Elsevier Science B.V. All rights reserved. SSDi 0 i 67-7322 (95) 00836-5
426
O
O-
N~'~N(CH3)2 N~+N(CH~2
Scheme 1. Structure of PB.
a
b
separated state (b) is more stable in the excited state, and non-polar state (a) is more stable in the ground state. The ground state recovery corresponds to the charge recombination process, not to the charge separation process like betaine-30. II. Experimental The ground state recovery was measured by a sub-picosecond pulsed dye laser. A train of frequency doubled pulses from a modelocked Nd:YAG laser (Coherent: Antares 76) synchronously pumped a dye laser (Coherent: Satori) which produced 200 fs pulses centered at 640 nm at 76 MHz. A small fraction of the output of the Nd: YAG laser was used to seed a Nd:YAG regenerative amplifier (Continuum: RGA60). The output from the dye laser was amplified in three flowing dye cells (Continuum: PTA60) by the doubled output of RGA60. The final output (Ca. 300 ~tJ at 50 Hz) was introduced to the standard pump-probe experimental setup. 10 A white continuum generated by focusing the laser pulse on water was used as the probe light, and the transient absorption spectrum was measured by the optical multichannel analyzer. The polarization of the pump light was rotated to the magic angle to that of the probe light to eliminate the effect of molecular rotation. All measurements were performed at 3138 K. A quartz cell with the path length of 1 mm was used for the liquid solution. For the measurement under high pressure conditions, an optical high pressure cell with the path length of 12 mm was used. 11 To obtain enough concentration of PB in the fluid near the critical density, we added a small amount (Ca. 1xl0 -1 mol dm -3) of 2,2,2-trifluoroethanol (TFE) as co-solvent. We have measured at two different pressures of 5.7 MPa (pr4).8) and 18.0 MPa (pr,~l.9), where Pr is the reduced density by the critical density of solvent. The density of CF3H was calculated by the empirical equation of state 12. We neglected the effect of the co-solvent on the PVT relation of solvent. III. Results and Discussions Figure 1 shows the transient absorption spectrum of PB in methanol after the excitation at 640 nm. In the figure we set time zero as the rise of the pump pulse. The ground state hole created by the pump pulse is broadened mostly by 1 ps. Near the absorption maximum of the ground state, more than half of the bleach signal observed at 1 ps is recovered until 2 ps, and the residual bleach is diminished with a decay constant of about 10 ps. The hole broadening
427
0.1
,
,
0
.~
,
o
\ ~ .
-0.2
.
,
0.1
" ~ =..
f/
o
oot
a
~o -
,
0
~
~
~
I
o
I
9 0.2~, 0.4 I ~
9
i
'~
9 o.6.
,
,
a
~
%
&io~
--
o
~, o
I
I
I
I
550
600
650
700
750
9
800
500
1.0 pe
9 ~.o.
o:.o.
-OD
I
,
o
~o
-0.2
I : ._oo. 0~
[: 500
,
_
-OD N
I
I
I
I
I
550
600
650
700
750
Wavelength (nm)
800
Wavelength (nm)
Figure 1. Transient absoprtion spectrum of PB in methanol at 308 K after the excitation at 640 nm. The difference of the absorbance with pump pulse from that without pump pulse is plotted. Solid lines denote the equilibrium absorption spectra, of which the sign is inverted and the peak value is rescaled to that of the latest time data in each figure.
0.05
=
I
=
=
0.05
,
=
=
i
~~176 9 t
o
X
-0.05
o
-0.1
,,
~
,
9
r 0.2 ps 0.4 ps 0.8 ps
o o
"
=,~
o
0
=,,~
,,
-
0.05
",,
-0.1
~o~,~
-
0.8 lOS
450
9 r~
~Y~,
2.0 ps 3.0 pS
~,o o ~176 ~'~'
~ - O D q
-OOeq
-0.15 400
o.
,m,
i
I
I
I
I
500
550
600
650
Wavelength(nrn)
0.15 700
400
I
I
I
I
I
450
500
550
600
650
700
W a v e l e n g t h (nm)
Figure 2. Transient absoprtion spectrum of PB in CF3H with small amout (Ca. l x l 0 -1 mol dm -3) of TEE at 308 K and 5.7 MPa after the excitation at 640 nm. The legend is the same for figure 1. process after 2 ps was not well resolved within our signal to noise ratio. There appears an absorption signal around 750 nm with a decay rate of about 1 ps. The absorption around 750 nm, red-shifted to the ground state, is considered to be due to the non-solvated excited state, not to the "hot" ground state, since absorption due to S0"-'$2 is observed around 340 nm in equilibrium and the sum of the energy of 640 nm and 750 nm corresponds to 340 nm. This fact suggests a possibility that the growth of absorption due to some new "state" from the non-solvated excited state, probably the solvated excited state, contributes partially to the fast component of the bleach recovery near the absorption maximum
428 of the ground state absorption; i.e., in this time range we observe the complex spectrum dynamics related to the solvation and the relaxation of local inhomogeneity. Figure 2 shows the transient absorption spectrum of PB in CF3H at 5.7 MPa. The pattern of the transient spectrum is almost the same as those in methanol. The hole broadening occurs mostly within 0.8 ps, and the bleach is recovered with two different time constants in a similar manner as in the liquid solvents, although the recovery after 2 ps is much slower in the fluid near the critical density (about 40 ps). It is also to be noted that the bleach signal after 2 ps is narrow banded in comparison with the equilibrium absorption. This can be interpreted by the overlapping of the excited state absorption, and/or, the small inhomogeneity remained after 2 ps due to the long time density fluctuation. In the earlier events by 2 ps we could not see any meaningful difference between the medium- and high-density fluids within our time resolution and signal quality. This suggests that the density effect on the solute environment which determines the fast process, the solvation and the relaxation of the inhomogeneity of solvent in this time range, is small under the density region studied. The elucidation of the nature of the long time component of the bleach recovery such as the solvent effects, the density effect and the temperature effect, is now under work.
Acknowledgment This work is supported by the Research Grand-in-Aid from the Ministry of Education, Science and Culture (No. 0674~.A.4). References 1 E. Akesson, G. C. Walker, and P. F. Barbara, J. Chem. Phys., 95, 4188 ( 1991). 2 N. Levinger, A. E. Johnson, G. C. Walker, and P. F. Barabara, Chem. Phys. Lett., 196, 159(1991). 3 E. Akesson, A. E. Johnson, N. E. Levinger, G. C. Walker, T. P. DuBruil, and P. F. Barbara, J. Chem. Phys., 96, 7859 (1992). 4 G.C. Walker, E./~kesson, A. E. Johnson, N. E. Levinger, and P. F. Barbara, J. Phys. Chem., 96, 3728 (1992). 5 A.E. Johnson, N. E. Levinger, W. Jarzeba, R. E. Schlief, D. A. V. Kliner, and P. F. Barbara, Chem. Phys., 176, 555 (1993). See, e.g., A. B. Bhatia, UltrasonicAbsorption (Dover, New York, 1967). Y. Kimura and Y. Yoshimura, Mol. Phys., 72, 279 (1991). See, e.g., O. Kajimoto, M. Futakami, T. Kobayashi, and K. Y amasaki, J. Phys. Chem., 92 1347 (1988). V. Luzhkov and A. Warshel, J. Am. Chem. Soc., 113, 4491 (1991). 9 G. R. Fleming, Chemical Application of Ultrafast Spectroscopy (Oxford University 10 Press, New York, 1986). 11 Y. Kimura, Thesis, Kyoto University 1992. 12 R.G. Rubio, J. A. Zolloweg, and W. B. Streett, Ber. Bunsenges. Phys. Chem., 93, 791 (1989).
journal of
MOLECUlaR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 429-432
Dynamics of Two-Component Liquids near the Glass Transition Yutaka Kaneko and Jiirgen Bosse Department of Applied Mathematics and Physics, Faculty of Engineering, Kyoto University, Kyoto 606, Japan and Institut fiir Theoretische Physik, Freie Universit~it Berlin, Arnimallee 14, 14195 Berlin, Germany Abstract Dynamics of a supercooled binary hard-sphere liquid is investigated within a modecoupling theory. For a disparate-size mixture, a glass is formed at a critical packing fraction rib by only the big particles, while the small particles diffuse through the voids of the glassy structure. Diffusion of the small particles is blocked in a localization transition at rL4 (> riB). Near the glass transition (r/B), we find a change of the diffusion mechanism for the small particles which we discuss from a dynamical point of view.
I. Introduction The glass transition of simple one-component liquids has been extensively studied within a mode-coupling theory supplying us with many predictions concerning the relaxation dynamics in a supercooled state. 1 Recently, this theory has been extended to multi-component systems with the aim of studying such interesting phenomena as interdiffusion and ionic conductivity. 2'~ The long-time limits of density correlation functions have been determined for disparate-size binary liquids, which allowed to identify the localization-delocalization transition of the small particles within the glassy structure of the big particles. 2'4 In this paper, we study the dynamical properties of a binary hardsphere mixture near the glass transition by solving the full space- and time-dependent mode-coupling equations for this system. The appearance of the delocalized phase of the small particles is confirmed by investigating the diffusion constants of both particle species. We discuss the change in the diffusion mechanism of the small particles from liquid-like diffusion to slow diffusion in a (static) random potential in terms of the incoherent density-relaxation function and the frequency-dependent diffusivity. II. Theory For a system of classical particles, the time evolution of the tagged particle correlation function
r
~)=< exp[-iq. (r0(,) - r0(0))] >
0167-7322/95/$09.50 ( 91995 Elsevier Science B.V. All rights reserved. SSDI 0167-7322 (95) 00837-3
(1)
430
is expressed by the generalized oscillator equation (v~ = kBT/m,) r176
+ q~v~ r
dr' K ~
with the initial conditions r 0) = 1 and r the mode-coupling approximation reads
t') r
= 0
(2)
0) = 0. The relaxation kernel K,(q, t) in
K,(q, t) MCA kBTv~
v, ~ ~ ''c'(')c'''(')r162 k
(3)
U'
where Coo,(q) is the direct correlation function and ~ - [ q - k[. Equation (2) is formally solved by the Fourier-Laplace transformation as
r176
-1
=
2 ~
r176
q v, z--
= 0).
(4)
z + K~
The frequency-dependent diffusion constant D,(w) is determined from the incoherent relaxation kernel as 2
Do(w) = Im
w + K~
v~
= O,w + iO)"
(5)
In order to solve the self-consistent equations (3) and (4), knowledge of the coherent density-correlation functions (I),~ t) is required as an input. We calculated (I)~ t) from the (2 x 2)-matrix equation
r
= - { z I - ( z I + g(q,z)) -t . g~2(q)}- t . (I)(q, t = 0),
(6)
where I denotes a unit matrix, and (I)(q, t = 0) and f~2(q) are determined from the partial structure factors So,,(q). The relaxation-kernel matrix elements g,,,(q, t) reduce to
g,~
t) MCA (kBT)~ 2 )
~
+
V3"rt'~
vo ~ 5-.:~{k2Oo,(k)C,,,,(ic)Ou,(k,t)aoo,(lq _ kl,t) k
U'
k z ( q - k,)Ca(k)Cv~
k[)r162
kl,t)}
(7)
with k~ = k . q/q in the mode-coupling approximation. III. R e s u l t s
The theory described above is applied to the binary hard-sphere system with the size ratio al/a2 = 0.2 and the concentration of small particles cl = 0.5. Figure 1 shows the diffusion constants D, of small (s = 1) and big (s = 2) particles versus the total packing fraction 77. The dotted lines show the power-law fit D = D01o - )7[~ with 7=1.31 and 2.36 for small and big particles, respectively. The diffusion constant of the big particles is found to vanish at T/B=0.52, which is close to the liquid-glass transition point (77=0.516) of a one-component hard-sphere system, s while D1 becomes zero at T/A =0.53 (> r/B). This means that for r/B < T/ < T/A there exists a new phase (delocalized phase) with mobile small particles diffusing through the voids of a glassy structure formed by the immobile big particles.
43 ! -1 ..............
-3
o.
......
o9.
9 .o.~%
_
9 9 ,(~ 9
9 ....
.,~149
9 ,.
ta0 o
-5
9
-
"(~..
9
#
C>
9
Fig.1
_
'
-
(>
-7 -9
.
I
I
0.4
0.44
.,,
1
0.48
1.
. 9
0.52
Diffusion constant D , ( w ) at ~ / ~ 0 = 1 0 - ' ~ i t h ~0 = for s = l (o) and s=2 (<>).
~/~:sT/(miai)
In Fig.2 we display the incoherent density-relaxation spectra r w) = Imr + i0) as a function of w in a logarithmic scale 9It is interesting to note that there is an additional quasi-elastic peak in r at small w (W/Wo < 10 -s) near the glass transition. This additional low-frequency intensity, which has not been detected for a one-component system, grows further as 77 approaches T/B. Figure 3 shows the frequency-dependent diffusion constants Da(w) for the same values of 77 as in Fig.2. We note that a little bump appears in Dl(w) in the same frequency range as that of the additional quasi-elastic peak in r for corresponding 77. This bump in Dl(w)is also found for c~=O.1 and 0.9. 6 For the big particles, on the other hand, no additional structure at small w is observed in r and D2(w) for all 77 of our calculations. The appearance of the bump is considered to indicate the change in the diffusion mechanism of the small particles. As the glass transition 77 = r/s is approached from the liquid side, the motion of the big particles becomes very slow compared to that of the small particles. Note that D~. is about 10 s times smaller than D~. Near 77=rjB, the big particles are almost frozen, producing an 'almost static' random potential which the small particles will experience when diffusing through the glassy matrix. Therefore, the diffusion mechanism of the small particles is expected to change from liquid-like diffusion to that of a particle moving in a 'static' random potential when passing the transition point ~BVI. S u m m a r y In this paper, we studied the glass transition and the localization-delocalization transition in a disparate-size hard-sphere mixture from a dynamical viewpoint 9 For Cl = 0.5, the existence of the delocalized phase of the small particles is confirmed by investigating the frequency-dependent diffusion constant. Near the glass transition, we and Dl(w) at small w, which sugfound an additional quasi-elastic structure in r gests that the diffusion mechanism of the small particles would change from the liquid-like diffusion to a slow diffusion in a random potential.
432
1
0.518
v
-'=
0
r/--0.395
o
r/:0.395
0-
-,%,
,,--t
-2- (b)
(a)
-2
J
I
-6
1
I
-1.5
2
-8
J
I
J
I
-6
-4
-2
0
logxo(W/Wo)
Incoherent density-relaxation spectra r for s = l (a) and s=2 (b) at q = 7.047a~-1 (the position of the first peak in $22(q)). 7/is chosen as r] = rIB - 1/2" with n = 3, 4 , . . . , 9.
Fig.2
:3
-4
I
-4 -2 loglo(W/Wo)
\
I
I
1
I
[
.... I
I
-
.
I
I
I
I
l
[ J
-2
-1
0
1
T}=0.395
-2 -2.5 -7
-3
-8
-6 Fig.3
-5
-4 -3 -2 -1 loglo(W/Wo)
0
1
-6
-5
-4
-3
loglo(W/Wo)
Frequency-dependent diifusivity D,(w)for s = l (a) and s=2 (b).
This work is supported by the Deutsche Forschungsgemeinschaf~, Sonderforschungsbere-
ich 337. References 1. W. GStze in Liquids, Freezing, and the Glass Transition, eds. D. Levesque, J. P. Hansen, and J. Zinn-Justin p287 ( North Holland, Amsterdam, 1990 ). 2. J. Bosse and J. S. Thakur, Phys. Rev. Lett. 59, 998 (1987). 3. J. Bosse and M. Henel, Ber. Bunsenges. Phys. Chem. 95, 1007 (1991). 4. J. S. Thakur and J. Bosse, Phys. Rev. A 43, 4378, 4388 (1991). 5. U. Bengtzelius, W. GStze, and A. Sj61ander, J. Phys. A 17, 5915 (1984). 6. J. Bosse and Y. Kaneko, in preparation.
journal of MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 433-436
Relaxation Characteristics of Quantum Filtered (DQf) NMR
Intracellular
Na + as Measured by Double
Yoshiteru Seo and Masataka Murakami Department of Molecular Physiology, Sciences, Okazaki 444, Japan
National
Institute
for Physiological
Abstract We have measured the transverse relaxation rate constants depending on statistic field (Bo) and the diffusion coefficient of 23Na+ in an agar gel as a model of the intracellular Na +. We have estimated a binding site correlation time (x c) and a quadrupole coupling constant (e2qQ/h) on assumption of the discrete-exchange model. Transverse relaxation constants of "free Na+" (Sfree) in the agar gel were estimated from the self-diffusion coefficient of 23Na+ in the agar gel using the observed relationship between the self-diffusion coefficient and the transverse relaxation rate constant of Na+ in glycerol solution. As a result of minimization (r2 = 0.98), the fraction of the 23Na+ under the slow-motion condition (Pb for 1% agar/1 M NaC1/ water gel) was calculated as 9-10 -4. The values of e2qQ/h and the correlation time were 1.0 MHz and 16 ns, respectively, and the calculated value of e2qQ/h (1.0 MHz) lies in the reasonable range (0.2 M H z - 2 MHz). I. Introduction The progress of the multi-quantum nuclear magnetic resonance spectroscopy gave new information about the physical condition of electrolytes in the cells. In the perfused rat salivary gland, we have reported double-quantum filtered resonances due to intracellular 23Na+ signals. The resonance consists of sharp and broad Lorentzian components (the homogeneous biexponential (type c) spectrum). In order to discuss the relaxation mechanism of the intracellular 23Na+, we have measured the transverse (T1) and the longitudinal ('1"2)relaxation rate constants of the intracellular 23Na+ signal from the perfused rat mandibular salivary gland, and have measured their dependencies on Bo, temperature and contentl. The mandibular salivary glands (ca. 0.25 g) were isolated and perfused arterially with a modified Krebs solution. 23Na NMR spectra were collected at 8.4, 4.7 and 2.34 T by using MSL-100, AM-200wb and AMX-360wb, respectively. The T2 double-quantum filter and T1 double-quantum filter were used. Results are summarized in brief: 1) The transverse relaxation rate constants showed a slight Bo dependency, which promises a poor fitting by the single population model. 2) When temperature was decreased (37 - 5~ the ratio of the relaxation rate constants (o~2 = T2s/T2f) and 1/T2f itself increased, and the value of 1/T2s was almost constant. 3) Even when the intracellular Na+ content increased, the relaxation rate constants were almost constant. Results of Bo dependency were applied to the two models: 1) Static model: We assumed a homogeneous population of Na § in the intracellular fluid, 2) Exchange model: We assumed an exchange between Na+ under the slow-motion condition and the Na+ in the extreme narrowing region. Results obtained in this study can be fitted with the two-site exchange 0167-7322/95/$09.50 91995 Elsevier Science B.V. All rights reserved. SSD! 0167-7322 (95) 00891-8
434 model, which postulates the presence of a rapid exchange between 2 populations of Na + ions: a small fraction of Na + ions "trapped" with high-molecular-mass-solutes ("bound Na+"), and a major part moving freely in the cytosol ("free Na+"). One unknown parameter in the exchange model is the relaxation rate constant of the "free Na +'' (Sfree). Measurement of the molecular diffusion coefficient of the intracellular Na+ could give the answer, since this value would mainly reflect the movement of "free Na+". In this paper, we have measured Bo dependencies of the transverse relaxation rate constants and the diffusion coefficient of 23Na+ in an agar gel model. We have estimated a binding site correlation time (Xc) and a quadrupole coupling constant (e2qQ/h) on assumption of the discrete-exchange model.
II. Double-Quantum Filter NMR The relaxation rate constants of 23Na+ in agar gel (1, 2, 3, 4% agar and 1 M NaC1 in water) were measured at 8.45 T and 2.34 T at 25~ 23Na NMR spectra were collected using an AMX-360wb (8.45 T) and an MSL-100 (2.34 T) spectrometer with broad-band probes (10 mm in diameter). The T 2 double-quantum filter (d-90~ - 180~176176 was used. The peak heights of Fourier transformed spectra with various creation time (x) values were used for determining the transverse relaxation rate constants. Figure 1 shows flip angle dependency (0) of the signal passed by the T 2 doublequantum filter (d-90~ - 180~ Since the signal virtually disappeared at the flip angle of 54.7 ~ the observed signal from the agar gel rise from the T32 component and the contribution from the T22 component is negligibly small. When the quadrupolar coupling of 23Na+ is not averaged completely, the T22 component are not suppressed by the doublequantum filter 2. 3. Thus, we can use Na+ in agar gel as a model system for the discreteexchange model.
~>" 3.0
...,, c-
.c: 2.0 e..,
._~ 1.0 o3 0.0
,
0
,
,
,
30
,
,
,
,
,
,
'
60
,
90 e
'
'
,
i
120
'
'
'
i
150
,
,
'
,
,
180
(degree)
Figure 1. Flip angle dependency of double-quantum filtered signal from Na ion in agar gel at 8.45 T and 24~ 0 is the flip angle of the 3rd and 4th pulses.
III. Pulsed Field Gradient NMR The self-diffusion coefficient of 23Na ion in the agar gel was measured at 2.34 T using a micro-imaging unit with a home-built 23Na coil (15 mm in diameter). The conventional pulsed field gradient spin-echo sequence (d-90~ - 180~ was used. Typical values used are a pulsed gradient field strengths of 5.4 mT/cm, 90 ~ pulse of 18 Its, 180 ~ pulse of 36 Its, A (pulse distance = t 1 + 8 + t2) of 40 ms and 5 (width of pulsed gradient) was varied from 0.3 ms to 4 ms 4.
435
IV. Discrete-exchange model We have applied the discrete-exchange model 5 to these data. An exchange between Na ion under a particular slow-motion condition and in the extreme narrowing limit is assumed. Transverse relaxation time and diffusion coefficient are written as follows: 1/T2i = Sfree-(1 - Pb) + si'Pb (1) D = Dfree'(1 - Pb) + Db'Pb (2) Where, 1/T2i and D are the observed values of 1/T2for 1/T2s and diffusion coefficient, respectively, si and D b are the transverse relaxation rate constants (s 1 or s2) and the diffusion coefficient of Na ion under the slow-motion condition, respectively, s 2 and sl correspond to the 11/2><-1/21 coherence and the I-1/2><-3/21, 13/2><1/21 coherences, respectively. Pb is the fraction of the Na ion under the slow-motion condition (1> Pb _>0), which is proportional to the fraction of agar. Sfreeand Dfree are the relaxation rate constant and the diffusion coefficient of Z3Na+ under the extreme narrowing condition, respectively. The observed relaxation time is time-averaged value of Sfree and Sl. The fraction of Na ion under the slow-motion condition is expected small, but the transverse relaxation rate constants (s i) are expected larger than Sfree by 10 to 100 times. Therefore, we applied this equation to the observed relaxation rate constants to estimate the 2 compartments. One problem is accurate knowledge of the value of Sfree for Z3Na+ in the agar gel. V. Estimation of Sfree As shown in Eq. 2, the observed diffusion coefficient should be time-averaged value of Dfree and D b. However, the observed value is virtually equal to Dfree since values of D b and Pb are so small compared with values of Dfree and (1 - Pb)- Therefore, observed diffusion coefficient of Na ion in the agar gel mainly represents the motion of Na ion in the extreme narrowing region. Since the relaxation rate constant of 23Na+ depends strongly on the viscosity of the solution, we measured the self-diffusion coefficient of 23Na+ in glycerol/water solution, and found a relationship between the self-diffusion coefficient and the transverse relaxation rate constant on viscosity [Fig. 2-a]. Then, we have estimated the Sfree value of 23Na ion in the agar gel from the self-diffusion coefficient of 23Na ion in the agar gel using the observed relationship [Fig. 2-b].
35
35
a)
30
3O
25
25 -,q,
20 15 0.7
b)
0.8
0.9
1.0
"-....
20 1.1
15 1.2 0.7
0.8
0.9
1.0
1.1
1.2
Diffusion Coefficient (10-5 cm2/s) Figure 2. Relationships of transverse relaxation rate constants (1]T2) and diffusion coefficient of Z3Na ion in glycerol/water solution (a), and estimated 1/T2 value for 23Na ion in the agar gel from the diffusion coefficient of Na ion (b).
436
VI. Results of minimization The result of the minimization is shown in Fig. 3 (Relaxation rate constants (s 1 and s2) of 23Na+ in agar gel at 25~ Solid lines represent result of fitting the data obtained at 2.34 T (x) and at 8.45 T (o). The fraction of the 23Na ion under the slow-motion condition (Pb for 1% agar/1 M NaCI/water gel) was calculated as 9.10 -n. The values of e2qQ/h and the correlation time were 1.0 MHz and 16 n, respectively. The observed data fit this model well (r2 = 0.98), and the calculated value of e2qQ/h (1.0 MHz) lies in the reasonable range (0.2 MHz - 2 MHz). These results support the exchange model. Thus, we will proceed to analyze the relaxation of the intracellular 23Na+ using the discrete-exchange model. v
to
2.4
c...
2.2-
to t-.0 o
2.0-
.-., 9
1.8--
L_ tO
x
n~"
1.60
1.4-
G~ 1.2
o
0
..-0 I
1.0
i
1
i
2.0
i
3.0
I
i
4.0
1 03-Pb
Figure 3. The transverse relaxation rate constants of 23Na ion in agar gel at 2.34 T (x) and at 8.45 T (o) as a function of the fraction of Na ion. Solid lines are results of fitting data to the discrete- exchange model. We have applied 23Na double-quantum filter NMR and Z3Na pulsed field gradient NMR to analyze the relaxation of Na ion in agar gel, which is a model for the intracellular Na+ to test the discrete-exchange model. From diffusion coefficient of Na+ in agar gel, we estimated transverse relaxation rate constant of 23Na+ in extreme narrowing region. The transverse relaxation rate constants obtained by the double-quantum filter NMR (2.34 T and 8.45 T) fit well the exchange model, and we estimated a correlation time (16 ns) and a quadrupole coupling constant (1 MHz) for Z3Na+ under the slow-motion condition.
Acknowledgements
We thank O. Ichikawa, M. Takagi, and H. Ohkawara for their technical assistance, and Professor C. S. Springer Jr., Dr. J.-H. Lee, Dr. S. Wimperis and Dr. H. Shinar for stimulating discussions.
References Y. Seo, M. Murakami and H. Watari, Biochim. Biophys. Acta, 1034, 142 (1990) G. Jaccard, S. Wimperis and G. Bodenhausen, J. Chem. Phys., 85, 6282 (1986) U. Eliav, S. Shinar and G. Navon, J. Magn. Reson., 98, 223 (1992) J. K~irger, H. Pfeifer and W. Heink, Adv. in Magn. Reson., 12, 1 (1988) T. E. Bull, J. Magn. Reson., 8, 334 (1972)
journal of
MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids 65/66 (1995) 437-439
Author Index Aicart, E., 195 Amakasu, Y., 377 Asano, Y., 41 Bader, J.S., 31 Barnes, A.C., 99 Barthel, J., 177 Benko, J., 261 Berg, M., 301 Berne, B.J., 31 Blakey, D.M., 85 Bosse, J., 429 Bout, D.V., 301 Buchner, M., 157 Burger, K., 213 Chigira, F., 265 Chikaki, S., 325 Chong, S.-H., 345 Dore, J.C., 85 Dorfmiiller, T., 157 Enderby, J.E., 99 Espenson, J.H., 205 Friedman, H.L., 7, 15 Fujimura, Y., 169 Fujiwara, A., 187 Fukuda, T., 341 Fukuda, Y., 397 Fukushima, K., 369 Funahashi, S., 277 Gardecki, J., 49 Goto, M., 357 Gotoh, T., 341 Gyuresik, B., 213
Hamada, S., 353 Hamaguchi, H., 417 Hashimoto, N., 41 Hattori, S., 357 Hayashi, S., 405 Haymet, A.D.J., 139 Hirata, F., 15,345,381 Hirata, Y., 321, 421 Hirota, N., 401,425 Hiwatari, Y., 123 Horng, M.L., 49 Hosoya, K., 305 Ibuki, K., 385 Iida, M., 269 Ikeuchi, H., 273 Irisa, M., 381 Itoh, S., 373 Iwadate, Y., 369 Iwata, K., 417 Jobe, D.J., 195 Johnson, A.E., 59 Kagayama, N., 277 Kai, Y., 413 Kameda, Y., 305 Kanakubo, M., 273 Kanato, H., 325 Kaneko, Y., 429 Kanemoto, Y., 421 Kataoka, Y., 329 Kato, T., 405 Kawaizumi, F., 377 Keogh, G.P., 389 Kimura, Y., 425 Kinoshita, S., 413 Kitagawa, T., 317
438
Author Index/Journal of Molecular Liquids 65/66 (1995) 437-439
Kitahara, K., 293 Kitazawa, M., 321 Koda, S., 337 Koyama, Y., 369 Kubo, H., 369 Ma, J., 301 Machida, K., 405 Madden, P.A., 373 Maeda, H., 341 Maroncelli, M., 49 Mashita, T., 341 Masuda, A., 397 Masuda, Y., 269, 397 Mataga, N., 393 Matsui, J., 123 Matsumoto, M., 277, 329 Matsuoka, T., 337 Matsuura, H., 313 Miyasaka, H., 393 Miyazaki, F., 357 Mizoguchi, K., 187 Mizuno, Y., 269 Morita, A., 75 Moya, M.L., 261 Munakata, T., 15, 131 Murakami, M., 433 Muranaka, T., 123 Nagaoka, M., 289 Naga~sawa, Y., 59 Nagy, L., 213 Nakagaki, R., 353 Nakahara, M., 149 Nakai, T., 365 Nakamori, T., 269 Nakamura, K., 341 Nakanishi, K., 281 Nakashima, S., 317 Nakazawa, T., 353 Neilson, G.W., 99 Newton, M.D., 7 Nishinaka, M., 333 Nishiyama, K., 41 Nomoto, T., 421 Nomura, H., 337, 377 Oclagaki, T., 123 Ohba, M., 377 Ohtaki, H., 1
Ojima, H., 341 Okada, I., 309 Okada, T., 41, 321,421 Okako, N., 369 Okamoto, K., 401 Okazaki, S., 309 Olson, J.S., 317 Olsson, L.-F., 349 Ozutsumi, K., 361 Papazyan, A., 49 Perez-Tejeda, P., 261 Pemg, B.-C., 7 Radnai, T., 229 Raineri, F., 15 Raineri, F.O., 7 Rick, S.W., 31 Rossky, P.J., 23 Sagawa, T., 313 Sakamoto, S., 305 Sanchez, F., 261 Sasaki, S., 341 Sasaki, Y., 253 Sat6, G.P., 273 Sawada, K., 265 Sawai, N., 353 Sawamura, S., 365 Schr'oer, W., 107 Schwartz, B.J., 23 Seki, K., 293 Seo, Y., 433 Shibata, T., 337 Shinozaki, K., 357 Soper, A.K., 91 Stuart, S.J., 31 Suga, H., 115 Sumi, H., 65 Suzuki, H., 305 Swaddle, T.W., 237 Tabata, M., 221 Takagi, H.D., 277 Takahashi, T., 381 Tanaka, H., 285 Tanida, H., 409 Taniguchi, Y., 365 Tarumi, T., 277 Teramoto, Y., 281
Author Index/Journal of Motecular Liquids 65/66 (1995) 437-439 Terauchi, N., 309 Terazima, M., 401 Tominaga, K., 59, 389 Tominaga, T., 333 Tominaga, Y., 187 Tran-Cong, Q., 325 Tsukahara, K., 353 Uemura, O., 305 Ueno, M., 385 Usuki, T., 305 Verrall, R.E., 195 Wakai, C., 149 Wakita, H., 91 Wang, Y., 187
Watanabe, I., 245,409 Weingfirtner, H., 107 Yabe, T., 321 Yagi, T., 413 Yamabe, T., 289 Yamagami, M., 91 Yamaguchi, M., 413 Yamaguchi, T., 91 Yamanaka, M., 341 Yanagida, T., 381 Yartsev, A., 59 Yasuoka, K., 329 Yokoyama, H., 357 Yoshida, N., 289 Yoshihara, K., 59, 389 Yoshimori, A., 297
439
This Page Intentionally Left Blank
journal of
MOLECULAR
LIQUIDS ELSEVIER
Journal of MolecularLiquids, 65/66 (1995) 441--446
Subject Index absorption curve, 179 acetone solutions, 178 acetonitrile(AN), 13, 18, 177, 180 adiabatic calorimeter, 116 air/solution interface, 250 alkyl radicals, 205 alkylammonium salts, 109 amorphous ice, 85, 87, 89 anharmonic free energy, 286 approximate correlation function, 161 aprotic dipolar liquid, 18, 19 aqueous CsCI solution, 365 aqueous electrolyte solutions, 187 aqueous solution of NaCI at high pressure, 233 aqueous solutions, 9 I, 229 aqueous solvation, 31 aqueous sulfuric acid solutions, 305 argon, 329 argon/a-ice co-deposit, 88 Argand diagram, 181,182 ascorbic acid, 187, 277 association constant, 177, 179, 184 average distances, 232 avian pancreatic polypeptide, 381 azobenzene, 65 benzene, 13, 273 N-benzylideneanilines, 65 biferrocene monocation, 397 bimolecular reaction rate, 321 binary soft-sphere mixtures, 121 bipyridine, 357 bi-exponential decay, 19, 20 Bom-Mayer-Huggins, 373 bromide ion, 230, 245 Brownian motion, 75 butanol, 199 1-butoxyethanol, 199 calcium, 349 Car-Parrinello method, 373 carbonmonoxy myoglobin, 317 catalytic effect, 222 CeCI 3,369 central component, 187 central force (CF) model, 139
CD, 213 CF3H, 313 charged ion pairs, 184 chloride ion, 230 classical (continuum) model of solvent, 261 clathrate, 89, 118, 185 closest approach, 265 cluster form-factor, 87 [Co(NH3)4(pzCO2)] 2+, 261 cobalt (III) ammine complexes, 237 coherent anti-Stokes Raman profiles, 169 Cole-Cole plots, 125 Cole-Cole type relaxation mode, 187 collective jump motions of atoms, 123 collective variables, 15 collectivity, 157 59Co longitudinal relaxation rates, 273 combination reaction, 207 complexes of N-D-gluconylamino acids, 213 computer simulation, 329 condensation coefficent, 329 conducti v ity, 265, 357 configurational enthalpy, 120 conformational behavior, 313 conformational fluctuations, 65 contact ion pair(s), 177, 184 continuous random network, 87 continuum model, 385 coordination distance, 245 coordination numbers, 232, 245 correlated atomic diffusion model, 15 correlated motion coefficient, 127 correlation time, 433 corresponding state, 109 Coulombic phase separations, 108 cresyl violet, 41 critical behavior, 107 critical density, 425 crossover, 107 crystalline solids, 115 crystallization, 115 Cu(II) porphyrin complex, 23 1 9-cyanoanthracene- I, 3-cyclohexadine, 321 Debye type relaxation mode, 187
442 density autocorrelation function, 123 density correlation, 16 density fluctuations, 131 density functional (method), 16, 281 density functional theory, 131 density maximum, 86 density of state, 162 dependence of relaxation rate on the temperature, and the complex concentration, 273 dephasing, 131 dephasing constant, 169 detailed balance, 293 di- and tri-nuclear complexes, 253 diabatic free energy profiles, 8 dichrolomethane, 273 dielectric constant, 114 dielectric depression, 182 difference pair-distribution function, 233 diffusion coefficients, 333 diffusion constant, 17 diffusion controlled reactions, 184 diffusion mechanism, 429 diffusion-controlled rate constant, 207 diffusive Browinan motions, 65 diffusive motions, 65 N, N-dimethylformamide, 180, 229, 361 dimethyle suifoxide solutions, 227, 230, 178 dipolar and non-dipolar (polar) solvents, 7 dipole-dipole interaction, 409 direct correlation, 16 discrete-exchange model, 435 dispersion amplitude, 181 dispersion curve, 179 disproportionation, 205 dissipative structure, 341 dissociation pressure, 286 DLVO approximation, 139 double-quantum filter NMR, 434 dropping effect, 118 dye, 59 dynamic effect, 289 dynamic process in electrolyte solutions, 177 dynamic variables, 15 dynamical nonlinear terms, 297 dynamical structure factor, 15, 16 dynamical structure of water, 187 dynamics, 91
dynamics of solvation, 49 d~ - p~ interactions, 253 effective dipole moments, 184 effective ionic radius, 358 effective volume of dipole rotation, 184 electric field gradients, 269 electrolyte solutions, 107 electron donors, 209 electron exchange rate, 209 electron transfer, 59 electron transfer reaction(s), 7, 205, 237, 345 electrostriction, 359 encounter pair, 321 energetics, 211 energy transfer, 405 enthalpy change of the ion association, 358 enthalpy relaxation, 120 entropy change of the ion association, 358 EPR, 213 erbium (III) ions, 230 evaporation and condensation, 329 EXAFS, 230, 245, 213 EXAFS oscillation amplitudes, 247 excess free energy, 357 excimer, 394 excitation energy migration, 393 excition diffusion, 393 [Fe(CN) 6]4-, 261 femtosecond, 59 femtosecond time-resolved coherent Raman scattering, 169 finger-like structure flow, 341 first excited singlet, 417 first- order difference method, 230 fluctuation dissipation theorem, 18, 414 fluorescence quenching, 321 fractional exponent, 116 free energy profiles, 7, 11 free-volume, 325 friction kemel, 16 frozen-in systems, 115 FT-IR spectroscopy, 177, 313 functional derivative, 16 general susceptibiity, 123 generalized Langevin equation, 15, 289
443 generalized permittivity, 179 glass transition, 115, 126, 429 glassy, 91 glassy crystals, 115 glassy polymers, 325 glassy water and solutions, 231 D-glucose solution, 187 Gouy-Chapman-Stern (GCS) approximation, 139 Grote-Hynes theory, 65, 289 ground state recovery, 425 halogen abstraction, 205 hard-sphere liquid, 429 heat capacity, 118 heat capacity change of the ion association, 358 heat conduction, 417 heat transfer, 331 heavy water, 149 heterodinuclear metalloporphyrin, 221 hexagonal ErCl3, 369 high frequency permittivity, 177, 179 high pressure, 365 homogeneous solvent, 18 homogeneous and inhomogeneous contributions, 389 homogenisation, 158 1H relaxation, 269 iH spin-lattice relaxation, 397 hydrated electron, 23 hydration, 91,357 hydration entropy and enthalpy, 360 hydration free energy, 381 hydrogen abstraction reaction, 401 hydrogen atom abstraction, 205 hydrogen bond, 18, 91,149 hydrogen carbonate, 349 hydrogen-bond connectivity, 89 hydrogen-bond formalisim, 409 hydrogen-bonded intermolecular structure, 305 hydrogen-bonded pair, 86 hydrophobic, 177, 357 hydrophobic complex, 359 hydrophobic interaction(s), 182, 357 hydrophobic structure-makers, 357 hyper-quenced glassy water, 87
inorganic reaction mechanism, 237 Instantaneous normal modes, 157 integral equation method, 15 integral equation theory, 345, 377 Integral motion, 20 interaction site model, 7 lntermolecular, 169 mtermolecular dephasing, 169 mtermolecular interactions, 7, 169 lntramolecular electron transfer, 397 intramolecular photodimerization, 325 ion association, 177, 357 ion solvation, 177 ion-aggregate formation, 177 ion-association theory, 360 ion-pair formation, 265, 358 Ion-pair relaxation, 184 ion-pairing effects, 269 ionic atmosphere, 333 ionic products, 212 lrrotational bonding, 182 lsochoric temperature derivative, 86, 87 Isolobal, 205 isomorphic replacement, 230 isotropic substitution, 99 Jones-Dole B coefficient, 365, 385 kinetic depolarization, 182 kinetic relaxation process, 184 Kohlrausch-Williams-Watts' equation, 116 Kramers theory, 65 Langevin-diffusion equation, 131 laser flash photolysis, 205 Lennard-Jones binary mixture, 377 LiCI solution in DMF, ~ light scattering, 113, 413 limiting molar conductivity, 265 linear response, 19, 261 LiNCS solution in DMF, ~ liquid, 91,169 liquid acetonitrile, 235 liquid benzene, 393 liquid crystal, 281 liquid surface, 330 liquid-liquid phase separation, 107 lithium halides, 91 lithium iodide (LiI), 373 lithium ion, 229
444 local relaxation processes, 325 localization transition, 429 long time limit of the reaction, 75 low-frequency Raman scattering, 187 Marcus equation, 209 Marcus-Hush theory, 237 Markov limit, 16 MD simulation, 17, 229, 286, 329, 405 mean-field-like criticality, 107 mechanical grinding, 120 mechanical relaxation theory, 301 melting behavior, 369 metal radical, 205 metal-to metal charge transfer, 261 metalloporphyrin, 221 metastable phase, 115 methanol, 18, 329 methyl chlorides, 18 MgCIE-water-methanol ternary systems, 231 micelle, 333 microscopic Hamiltonian, 289 microscopic viscosity, 385 mixed micelles, 196 mixed-valence, 253,397 mode coupling theory, 301 mode-coupling theory, 429 molecular liquids, 131 molecular theory of solvation dynamics, 7 molybdenum radical, 209 momentum density, 15 monoclinic ErCl3, 369 monohalogenocadmium (II) complexes, 361 Monte Carlo (MC) computer simulation, 229 Mossbauer structual study, 213 multi-exponential behavior, 17 multi-exponential decay, 17 MW spectra, 177 23Na NMR, 433 N204, 405 naphthalenedisul fonates, 360 neutron diffraction, 85, 99 neutron scattering, 91 neutron scattering, 99 new renormalized linear response development, 7
nickel (II) and magnesium (II) ions, 230 nitrate ions, 230 NMR, 149, 213 NMR relaxations, 20, 269 non-adiabatic relaxation, 23 non-aqueous solutions, 229 non-crystalline solids, 115 non-exponential relaxation, 116 nonlinear coupling terms, 289 nonlinear Smoluchowski-Vlascov equation, 297 normal mode spectra, 158 octadecyltrimethylammonium chloride, 333 open form, 317 optical Kerr effect, 413 organic free radicals. 208 organic halides, 208 organometallic free radical, 205 orientational structure, 158 Ornstein-Zernike equation, 378 oxidation, 277 oxide bridge, 253 pair correlation function(s), 85, 86, 88 partial molar volumes, 358 partial pair-distribution function, 235 partial specific volumes, 341 pentanone, 199 perchlorate (ion), 230, 357 phenanthroline, 357 phenol blue, 425 photoelectron emission spectroscopy, 409 photoexecitation, 15, 18, 23 photohomolysis, 205 photophysical properties, 353 physiological solutions, 349 picosecond laser photolysis, 393 picosecond time-resolved Raman spectroscopy, 417 POE-water system, 313 Poisson-Boltzmann equation, 381 polar and non-dipolar solvents, 7 polar liquid, 15, 18 polarizability, 373 polarizable, 31 polarization charge density, 15 poly (oxyethylene) (POE), 313 polypeptide, 381 porphylin, 221
445 potential height, 81 power spectra, 286 precipitation nuclei, 349 pressure effects on reaction, 237 propanol, 199 propylene carbonate (PC), 180, 301 protein, 381 protein dynamics, 317 proton transfer, 209 pulsed field gradient NMR, 434 PY approximation, 377 quadrupole coupling constant, 433 quantum molecular dynamics, 23 quasi-elastic neutron scattering (QENS), 99, 230 radical, 401 Raman echo experiment, 389 Raman spectra, 305 Raman spectroscopy, 187, 309 rapid transport of a polymer, 341 rate constants, 196 rate of chemical reactions, 81 recombination, 211 recombination process, 317 redox properties, 253 relaxation, 115 relaxation mode, 187 relaxation of inhomogeneity, 425 relaxation process, 180 relaxation time, 116, 180 relaxation time distribution, 180 relaxation times in fluid solution, 41 renormalized solute-solvent interactions, 8 reorganization energy, 7, 409 reorientation dynamics, 325 residuasl entropy, 115 response function, 180 reverse Monte Carlo simulation, 229, 234 rhodium radical, 208 RISM, 10, 15, 16 root mean square deviations, 232 rotational and vibrational relaxation, 309 rotational correlation times, 149 rotational diffusion, 421 rotational friction coefficient, 385 rundom force, 16 S 1-S 1 annihilation, 394
scaled particle theory, 381 scattering, 169 Schlogl model, 293 second order difference method, 230 self-diffusion coefficient, 433 self-exchange reaction rate, 239 short range structure, 369 short-lived intermediates, 221 silver ion, 230 single-particle dynamics, 159 singlet excitation energy transfer, 393 site dependent relaxation time, 48 site-number density, 15, 18 site-site Smoluchowsky-Vlascov (SSSV) equation, 15, 19, 21 small angle neutron scattering (SANS), 230 small model of gilcoproteins, 213 Smoluchowski equation, 75 sodium (I) ion, 230 sodium chloride, 229 solid-fluid interface, 281 solid-state amorphization, 120 solution reactions, 65 solvate structure, 229 solvation, 23, 177 solvation dynamics, 10, 15, 297, 301 solvation free energy, 7, 381 solvation numbers, 177 solvation numbers, 178 solvation of lithium ion in N, N-dimethyl formamide, 231 solvation structure, 245 solvation time, 59 solvation time correlation function (STCF), 18, 19, 20 solvent dynamical effect, 397 solvent exchange, 238 solvent orientational distribution, 48 solvent pressure effect, 149 solvent relaxation, 41 solvent separated ion pair, 177 solvent viscosity, 65 solvent-solute interaction, 421 solvophobic unmixings, 111 spatial correlations, 293 SPC model, 17, 19 stability of ion pairs, 177 stearyltrimethylammonium bromide, 250 sticky ions, 111 trans-stilbene, 417
446 Stokes radius, 265, 358 Stokes-Einstein-Debye law, 149 stopped flow EXAFS experiment, 231 stretched exponential function, 123 structual parameters, 232 structure, 91,361 structure and dynamics of solvated ions, 229 structure of coordination of isomers, 219 structure-breaker, 359 structure-makers, 367 Sumi and Marcus, 65 supercooled, 91, 23 1 supercooled liquids, 131 supercooled fluid phase, 1 ~ supercritical fluid, 309 supersaturated aqueous solutions, 349 surfactant-alcohol, 196 temperature coefficients of the Walden product, 358 temperature pressure effect, 149 temperature dependence of molar conductivities, 357 tetraalkylammonium, 265 n-tetradecyltrimethylammonium bromide, 333,337 tetramethyl urea, 23 1 s-tetrazine, 301 theory of soivation dynamics, 7 thermal dissociation, 405 thermochemistry, 211 thermodynamic derivatives, 377 time correlation function of solvent fluctuation, 46 time evolution of the ground state hole, 41 time resolved absorption spectroscopy, 41 time resolved fluorescence spectroscopy, 41 time resolved hole burning, 41 time resolved resonance Raman, 317 time resolved Stokes shift, 18 time-of-flight (TOF) neutron diffraction, 305 TIPS model, 19 tracer diffusion, 333 transient, 205 transient effect in the fluorescene decay curves, 321 transient grating, 40 1
transient hole burning, 301 transition metal complexes, ~_37 transition-state theory, 289 translational diffusion, 401 translational friction coefficient, 385 transport coefficents, 131 transport of linear polyelectrolytes, 341 transport properties, 385 transverse relaxation rate constants, 433 trajectory, 17 triatomic molecule, 281 trifluoroethanol, 231 trifluoromethane, 425 tris(1, 10-phenanthroline)ion(ll), 357 tri-exponential decay, 19 tungsten, 208 ultrasonic absorption, 195 ultrasonic relaxation method, 337 under-cooled liquids, 87 van der Waals and Platteeuw theory, 285 vapor deposition, 119 variances, 197 velocity correlation function, 157 vertical energy gap, 7, 9 vibrational coiling, 417 vibrational lineshape, 389 viologen-linked N-alkylporphyrins, 353 viscosity, 187, 365 viscosity correlation functions, 286 volume change, 369 volume of activation, 238 XANES, 230 X-ray, 305 X-ray diffraction, 91,229, 361 a-peak, 1~ a - relaxation, 10_3 /3 - relaxation, 123
