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Keviews in Computational Chemistry Volume 24 n 0 T H E W I L E Y BICENTENNIAL-KNOWLEDGE FOR GENERATIONS G a c h gen...

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Keviews in Computational Chemistry Volume 24 n

0

T H E W I L E Y BICENTENNIAL-KNOWLEDGE

FOR GENERATIONS

G a c h generation has its unique needs and aspirations. When Charles Wiley first opened his small printing shop in lower Manhattan in 1807, it was a generation of boundless potential searching for an identity. And we were there, helping to define a new American literary tradition. Over half a century later, in the midst of the Second Industrial Revolution, it was a generation focused an building the future. Once again, we were there, supplying the critical scientific, technical, and engineering knowledge that helped frame the world. Throughout the 20th Century, and into the new millennium, nations began to reach out beyond their own borders and a new international community was born. Wiley was there, expanding its operations around the world to enable a global exchange of ideas, opinions, and know-how. For 200 years, Wiley has been an integral part of each generation's journey, enabling the flow of information and understandingnecessary to meet their needs and fulfill their aspirations.Today, bold new technologies are changing the way we live and learn. Wiley will be there, providing you the must-have knowledge you need to imagine new worlds, new possibilities, and new opportunities. Generations come and go, but you can always count on Wiley to provide you the knowledge you need when and where you need it!

WILLIAM J. PESCE

PREelDENT AND CHIEF E X E C M M OFFICER

PETER BOOTH WILEV CHAIRMAN OF THE BOARD

Reviews in Computational Chemistry Volume 24 Edited by

Kenny B. Lipkowitz Thomas R. Cundari Editor Emeritus

Donald B. Boyd

BICmNtLNNlA L

BICCNTCNNILL

WILEY-VCH

Kenny B. Lipkowitz Department of Chemistry Howard University 525 College Street, N.W. Washington, D.C., 20059, U.S.A. [email protected] Thomas R. Cundari Department of Chemistry University of North Texas Box 305070, Denton, Texas 76203-5070, U.S.A. [email protected]

Donald B. Boyd Department of Chemistry and Chemical Biology Indiana University-Purdue University at Indianapolis 402 North Blackford Street Indianapolis, Indiana 46202-3274, U.S.A. boyd@chem .iupu i .edu

Copyright 0 2007 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., I I I River Street, Hoboken, NJ 07030, (201) 748-601 I , fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations br warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic format. For information about Wiley products, visit our web site at www.wiley.com. Wiley Bicentennial Logo: Richard J. Pacific0

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Printed in the United States of America. 1 0 9 8 7 6 5 4 3 2 1

Foreword Reviews in Conzputational Chemistry is a pedagogically driven review wries

covcring all itspccts of corriputatioiial clicmistry rclcvmt to tlic scicntific aiid engineering communities. Unlike traditional books and journals that are dedicated to reviewing a. given subject, this series also provides tutorials covering the topic at hand so that the reader is brought up to speed before the literature is reviewed. And, unlike tradit,ional textbooks that focus exclusively on delivering pedagogy, this series provides an overview of the extant literature: poi;iting oiit to t.hc r c d c r what, can and what cannot bc donc with a givcn computational method, illustrating for the practicing molecular modeler how one method compares with a rival technique to d d r e s s a given scientific of engineering proldeni, and highlighting for the novice computational chemist what to do: what not t,o do, and pitfalls to avoid. The success of this series lies in the fact that Reviews in Computational Chemistry is nontraditional; it is a hybrid species that, covers what, t,exthool<sand review journals accomplish indcpcndcntly of onc another. Texthoks, hv t.lieir very nat,iire, are limited in what, they can provide to tlic r c d c r in tcrrris of diversity of topic. and dcpth of covcragc. Thc covcragc of Revieuis in, Clornpi~tataonulChemisty is very wide in scope. Moreover, we publish in-dept.h, didactic c1ia.pters t,hat are niore highly focused on an important computational method than comparable chapters found in encyclopedias or in traditioiial textbooks. We want this series to be a learning experience for both the novice modeler who has a basic conipreliension of physical rharnsitry a? wall iL’i for seasoned professionals who may hc working in academia or industry aid ncctl t.o lcarii a IICW ~nctlioclquickly to solve a problem. Many topics covered in this series are sufficiently narrow in scope so that, coverage in a single, comprehensive chapter is possible. Some topics, however, are inore expansive in nature, cutting extraordinarily wide swaths through the scientific hndscape shared by researchers in disciplines as disparate as mathematics, biology, geology, pliysics, chemistry, and computer science. An example iiicludes the treatment of solvents where in the past we have covered V

vi

Foreword

individual topics likc coiit,itiuum solvation niodcls, corriputatioiial approachcs to lipophilicit.y, molecular models of water, cellular automata models of a e quous solution systems, and coriiputhig hydrophobicity among other related topics. Even broader in scope is the topic of quant,uin chemistry. We have published individual chapters in this book series covering t,echniques like semiempirical molecular orbital methods, density functional theory, postHartrcc-Fock methods, qiiarit,iirri Montc Carlo m c t h d s . hasis sots, t)asis sct, superposition errors, cffcctivc core potentials, relativistic cffcct,s, coup1c.d cluster methods, valence 11ond theory, ant1 molecular quantum similarity. Further att,esting to the breadth of quantum cheniist,ry are individual cliapters covering specific applications of those techniques focusing, for example, 011 calciilating the propcrtics of hydrogen bonds by ab initio methods, rleriving molecular electrostatic potentials aiid chemical reactivity, obtaining clcctron dcnsitics frorri qiianhim mcchaiiirs, c;llc:ilat,ing vibrational cirriilar tlitdiroisrti iiitonsitics, a1.1itiitio c.ottiputation of nuclcar rtiagnctic rcsonancc chemical shifts, quantum mechanical methods for predicting nonlincar optical properties, and many other topics. In this volume of Rewieurs in, Corri,puta.tional Chemistry, we deviate somewhat from the past. volumes by covering a. single topic: confined fluids. In this volurne Drs. Martin Schoc~riand Sabiiie H. L. Klapp highlight the theoretical undcrpinnings of this fiald of s t d y , and providc thr mat.hcmat.ica1 dcrivatiotis and cornputcr itii2)lciricntatioxisof t hose t Iicorics. Drs. Sdiocli aiicl Iilapp then compare the nunierical results of t,hose implementations with analytical solutions arid experimental data to illust,rate how one should treat confined fluids under various ooridit,ions. Understanding confined fluids arid predicting their complex behaviors is especially iniportant in the disciplines of science, engineering, and t,echnology. R.elevaiit areas of study that, quickly come to Inind inrhide t,hc hydrodynamiw and rlicology of groiind water in soils, tlic actioti of lubricants iii pistoil-drivcii itiotors or in autorriotivc trimsmissions, the chara.cteristics of compressed fluids used as refrigerants, and perhaps inore relevant to the traditioiial cheinisbry community, the behavior of confined fluids in chromatography, and, more receiitly, comprehending and manipulating the behaviors of fluids in t,he rnicrtr and nano-fluidic devices used for both synthesis aiid analysis. The twhavior aurl charact,arist,icsof confined fluids is rnorc romplcx than that of bulk liquids or of simple solvated systenis described in tlie chapters mentioned above. One must. consider t hc complexities of the interface between confining walls and the fluid along with confinernelit-iriduced phase transitions, critical points, the stratificat.ion of the fluid near the confining walls, tlie idea bhat confined fluids may sustain certain shear stress without exhibiting structural features normally associated with solid-like phases? and

Foreword

vii

other phcriorncna that sciciitists arid crigiriccrs alike iiccd to bc aware of to become proficient modelers. In t.his book, the authors put, all of these issues into perspective and assist the reader with many required niatheinatical derivations that are needed to corriprehcnd t.his teclriiiologically relevant and scientifically interesting area of research. Reviews in Computational Chemistry is highly rated and well received by tho sciontifir romniiiriity at. litrg(!; thc roasori for t h s c ac:cornplishmcnts rmts firmly on the slioulclcrs of the authors wliorri we liavc coritactcd to provide pedagogically driven reviews that have made this ongoing book series so popular. To those authors we are especially grateful. We are also glad to note that our publisher now makes our most r e cent volumes available in an onliiie form through Wiley InterScience. Please consult the Web (http://www.interxience.wiley.corn/onlinebooks) or contact [email protected] thc latcst inforrnat.ion. For raadcrs who apprcciatc tlic pcrmmxiw a i d tmvcriitm:c of bouiid books, thcsc will, of wurst:, coiltinue. .We thank the authors of this arid previous volumes for their excellent clmpters.

Kenny B. Lipkowitz, Washington Thomas R. Cundari: Denton December 2006

Preface In rlassifying condensed-matter syst,crns, it often proves iiscfiil to identify rclcvaiit lcrigt h scales if' oiic aims at a thcorctical understanding of their properties and behavior. For example, the notion of a mean-free path is a useful concept. in the kinetic theory of (dilute) gases. If sufficiently large, it tells us that we may t,reat the dynamic evolution of a gaseous syst,erri (and therefore its transport properties) as a result of isolated collision events involviiig no more than a pair of gas molecules. In solids the lattice constant poses another siich length scale t)v which different,cryst,allographic striictiires can he classified and distinguished coiiveniently. III tl(msc: g:i\.'it:s ant1 licliiids, w1iic.h wc sril)siinic: tinc1c:r t h : tc:rui ':fluids" hencefort,h, t,he correlation length niay be viewed as a key length scale in the above seiise bemuse it permits one t.0 classify phenomena in both the nonand near-critical regimes of the fluid. The correlation length is a quantitative indicator of density fluctuations in the fluid. It may thus be viewed as a inemure of the length over which fluid structures decay such that they may no longer be disthigiished from a raiidoni arrangmient, of fluid nioleciiles. Bc!cauisc of this iritcryrctatioii the corrclatioii lcngth is coiriparablc with thc range of a typical intermolecular interact.ion potential as long as one stays off t,he near-critical regime of t,he fluid where density fluctuations are rather minute; the correlatioii length niay, however, become macroscopic as one penetrates into the near-critical regime. From a theoretical perspective, the concept of a correlation length is the key ingredient in scaling theories that permit one t,o 1inderst.arrd crit,ic*alpht?norkienitin thernid many-partkle systciris quaritit,ativcly. Thc irlaa that a spacific propcrty of a thc:rinal syst,crn mav bc intimately linked to a characteristic length scale immediately raises the question: What, might happen if there is more than just one relevant lengt,li scale? Moreover: what might happen if different length scales governing different features of a thermal system are comparable in magnitude and what, will be the result of a competition between these different length scales! that is, a competition between physical features with which they are intertwined? To answer these ix

X

Preface

questions or, to be a bit Iriurc modcst, to shed somc light oii aspects touched upon by them, was at the core of our motivation to write t.his book. Although this rather broad exposition is interesting from a purely academic perspective, it. would be nice if systems exiskd in the “real” world where an answer to the above questions would help to better understand the behavior of such a system. The ultimate goal of such a venture would then he to cxplorc iicw t,cclinologics or t80optimiza certaiii procwsw so that in i rcsca.rcli of tliis k i d riiiglit bc useful to Inorc tlic w r y loiig r u ~ fundainciital applied scientific disciplines. Fortunately, an almost ideal realization of a situation meeting all these conditions and provisos does indeed exist in reality. The class of systems to which we are referring in this context are fluids confined by solid surfaces to’tiny volumes of nanoscopic dimensions, which we shall term “confined fluids” throughout this book. Confined fluids play an important role in a mrict#yof pr,zcticnl or applied contmt,s ranging from say, tlic Cjwclling of clay soils, which is iuigortarit in undcrstanding spec ts of frost heaving, to catalysis, where new nanoporous media are current.ly being explored as novel nanoreactors. In a confined fluid t8hecorrelation length remains R key length scale that determines t’heforniation of its equilibrium structures. We immediately limit the scope of our discussion to iioncriticul fluids where the correlation length is in the rango of nwomctcrs. However, tho dogme of confinement imposes a scco~idsurh lciigtli scale coniyarablc with that of the corrcltLtion lcngth. One may think, for example, of the width of a nanoporous medium, which can nowadays be produced rather routinely in a physical chemistry laboratory. As we shall demonstrate below, the presence of a solid surface causes the confined fluid to be inhoinogeneous; that is, its density depends on the position relative to the corifiniiig substrate. This inhomogeneity has profound repercussioris for thrrmophysical propertics of the confincd fluid. By xncaiis of rriodcrii litliographic a i d rclatcd tmhniqucs, it is also possible to impose on the confining substrate H. chemical or geometrical substructure of nanometer dimensions. These suhtructures impose yet a third length scale comparable with the two aforementioned ones, which will also have fundamental consequenccs for t,he confiiicd fluid. Perhaps the most fascinating of these consequences is that. the existence of thermodynamic phases may he triggcrd, which haw no counterpart in the hulk. Because of these intriguing and complex features of confined fluids, we feel that this text comes timely for several reasons. First,, research on confined fluids has been carried out for the past 15-20 years with increasing intensity so t,hat. it is still one of the in& active fields in soft condensed matter research. During this time, our understanding of many aspects of confined fluids has grown enormously. At the same time, experimeiit.al and

Preface

xi

theoretical “tools” have been dwigiicd arid applied to irivestigatc these systems so that the time seems right for a more comprehensive introduction to the field. Second, even though quite a few excellent reviews of actual scientific work in this field have already been published over the past years 11-51, there still does not exist any text aiming at a more pedagogic introduction from a statistical physics background. In particular this second aspect was oiir primary motivation of compiling t,hc mntm-id brlow. H m w r , looking back at rcscardi in this arca. it bccltliie ixiirriediately clear that a conscientious choice of techniques and applications had to be made to preserve the pedagogic impetus on the one hand and a sufficiently broad selection of illustrating material on the other hand. As the current text focuses on theoretical aspects, it goes without saying that mathematics is the appropriate language for such a discussion. In selecting the level of formal (i.c., mathcmatird) prascntation of thc a c c c s s n ~and relevant kky coriccpts, wc had in riiiritl a rcitdcr with knowlctlgc in dlgcbra mid calculus typical of a well-educated (physical) chemist. More specifically, we intend to address an audience ranging froin advanced stude1it.s in the physical sciences t o experienced scientists interested in beginning research in the field of nanocoilfined fluids. Unfortunately, the subject of this manuscript makes it inevitable to go quitc a bit beyond this lave1 at ccrtain points. In these wen& we sunimarizc thc mathiernatical tcchniqucs to an extent n c u w q to follow our line of arguments. To keep the main text legible, we deferred these discussions to various appendices where each appendix is related to a specific chapter of the main text. An example is a brief introduction to the theory of complex functions in Appendix B.2 for which excellent textbooks are, of course, available. Nevertheless the goal of keeping this manuscript self-contained as milch tls possihln promptcci 11s to incliidc this backgroimd matmid to somr Iriinirriurn extcrit. A guiding principle in selecting the topics to be discussed below was that a specific physical problem should be tackled by a combination of different techniques wherever feasible. For example, in Chapter 4, we employ meanfield theory to study the phase behavior of confined fluids. In Chapter 5 we revisit this issue but eniploy Monte Carlo simulations instead. The latter, bcing R first-principlcs numerical nict,hod, pcrinit, B rigoroils twt of prrdictions deduced from mean-field theory. This way we do not only emphasize the importance of applying niore than just one approach because of the complexity of the systems under study. Moreover, we can illustrate the mutual limitations and advantages of each one of them. Thereby we hope to provide a broader (and more useful) overview, which is particularly important from the educational or training aspect central to this work. At the same

xii

Preface

tiiiia we iiitcndcd to kccp this text sufficiently focused and tlicrcforc rriadc a deliberate choice from tlie oiibset. in 1imit.ing t,he discussion exclusively to equilibrium properties. From a theoretical perspective, therrriodynarriics is then the central t h e ory on which such a discussion iniist build. Below we use a formulation of thermodynamics usually applied to solid-like systems, because confined fliiids have it lot, i n cominnn with h l k solids in t h t , thcy arc highly inhnrriogcricous arid anisotropic. However, uiilikc il solid, a corifiricd fluid lacks any long-rangc spatial order. As wt: demonstratt: in Chapter 1, symnietry consideratioiis play an integral part, of the current. formulation of equilibrium thermodynamics with which the nonexpert. in the field will iiot necessarily be accnstonietl. The link to t,he molecular level of description is provided by statistical thcrniodynamics whcrc oiir foriis in Chapt,cr 2 will be on spccializcd statistical physical criscriiblcs d(:signctl spccifir:ally for capturing fcaturts t,liat iriakc confined fluids distinct among other soft condensed matter systems. 1% develop statistical t,hcrniodyriarriicsfrom a qiiaritum-mechanical foundation, which has at its core the existence of a discrete spectruni of energy eigenstates of the Hamiltmian operator. However, we quickly turn to thc classic limit of (quantum) statistical thermodynamics. The classic limit provides an adequate framcwork for the siihscqiicnt, rlisciission hccsiisc of the region of thcrrriodynamic st,atc s p a w in which most corifinctl fluids exist. We immediately apply the concepts of sta.tistica1 therniodynaniics in Chapter 3 to a class of systems that can be ha.ndled analytically, namely one-dimensional hard-rod fluids confined lietweeii hard walls. Despite its simplicity, this system exhibits key features of confined fluids as we shall denioiistrate by coinpa,ring the results obtained in Chapter 3 with those for moro rcalisti: systems in latcr chapt.crs. IIowcvcr, onc.-cliincnsiori~lcoiifiricd fluids with purely repulsive intcraclions can expected to be only of limited iiscfulness, especially if one is interested in phase transitions t#hatcaiinot occur in any one-dimeiisional system. In treating confined fluids in such a broader context, a key theoret,ical tool is the one usually referred to as “mean-field thory.” This powerfiil theory, by which the key problem of statistical thermodynamics, namely the compiit,a.t,ionof a part.it,ioii fiinction, bcconics t~ri~~t.id>l(>, is intmdiircd in Chapter 4 where we focris primarily on latt,ice models of confined pure fluids and their binary mixtures. In this chapter t,he emphasis is on features rendering confined fluids unique among other fluidic systems. One exarnple in this context is the solid-like response of a confined fluid to an applied shear strain despite the absence of any solid-like structure of the fluid phase. The niean-field theory developcd in Chapter 4 is, however, plagued by

Preface

xiii

several siiuplifying arid a priori uricontrollablc assuiriptions, the rriost prorninent one being the complete neglect. of intermolecular correlations and, in our case, the discretization of space by employing lattice models. To demonstrate the somewhat astonishing power and reliability of the mean-field treatment, we amend its discussion by presenting parallel Monte Carlo computer simulations in Chapter 5. As we emphasize in that chapter and as we already pointd nut above, Manta Carlo should hc vicwd cnscntially as a first prirrcipks approach frcc of any tdditioual assumptioil. Moutc Carlo simulations thus provide an ideal test ground for the mean-field results presented in Chapter 4. To make such a coinparison we selected applications of the Monte Carlo computer simulation inethod in Chapter 5. In all examples discussed in that chapter, fluid molecules interact with each other via short-range potentials decaying as rW6like dispersion interactions bctmcn polarizable molcculcs (7. bcing thc intmmdmular distancc) or evcri faster (as, for w r i p l c , in the nearcst-ncighbor lat$icc models also & cussed in Chapter 5 ) . There is, however, an increasing interest in modeling confined biological, electrochemical, or colloidal fluids where the dominant interactions are electrostatic in nature. As far as Coulonibic interactions b e tween charged molecules or molecules with a permanent dipole are concerned, these interactions are long-range. In these two cases, the relevant interaction potcntialx dway as r - l and r d 3 ,rcspwtiw1-y. In Chaptm 6 we discuss spccial techriiqucs to deal with such loug-rargc iutcractiorw hi corriputcr siniulatiom and prevent selected applications for confined dipolar fluids. Again we limit the discussion of simulation techniques in Chapter 6 to the Monte Carlo method as a key numerical technique to stay focused as much as possible. Molecular dynamics simulations, which are the other simulation technique one would iminediat,ely think of, are explicitly disregarded here hccausn t h y arc morc siiit,ablc to stiidy dynmiic rat,hcr than cquilibriiun propertics with which we arc coiiwrncd hcrc. hi a siirdar spirit. ofl-latticc density functional theory is also disregarded here because this is already a vast and flourishing field in its own right to which a separate such text should be devoted. For most of this book we consider cases of “ideal” confinement, that is, situations where the geometry of the confining substrates is simple. The most promincnt cxamplc is that of a slit-porc whcrc tho confining substrates are planar and parallel to one another. In Chapter 7 we focus on the o p posite extreme, that is, a fluid cozlfined to a randomly disordered porous matrix. Experimentally this situation is encountered in aerogels. The simultaneous presence of both confinement and (quenched) disorder representing the nearly-random silica network renders the treatment of such systems quite challenging from a theoretical perspective. In Chapter 7 we discuss one of the

xiv

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inost. powerful tccliiiiyucs t,o study fluids confined to raridorri porous rncdia, namely the so-called “replica integral equat,ions” which allow one to calciilate, at. an admittedly approximate level, both structural and thermodyiiamic properties of confined fluids. Given the amouiit of material covered by this manuscript, it, is apparent that over the years tnany people have contributed significantly to the hook w e arc now prtwcnt.ing. In particiilar, we t.hank oiir ciirrcnt and forrncr studcrits arid coworkers IIcmy Dock, TII~IIIW Grdin, Dirk Woywod, Sophie Sacquin-Mora, Carsten Spiiler, Ga.brie1M. Range, Fa.bien Porcheron, Jiirg R. Silbermann, Holger Bohlen, Jochen Sommerfeld?Matthias Gramzow! Madeleine Kittner, aiicl Vladiinir Froltsov. We are also grateful for many disciissic)ns, int,eractions, and most enjoyable aiid fruitful collaborations we enjoyed over the years with our colleagues Professors John H. Cushman (Purdue Univcrsity), Dennis .J. Dic?st,lcr(Univcasitv of Nebraska at Lincdn), Sic!gfricd Dictrich (hllax-Plarick-Institiit, fiir Mct,allfo~st:liui.Ig; and Uiiivcrsit5.t Stuttgart), Boll Evans (University of Bristol), Gerhard H. Findenegg (Technische Univmitat Berlin and North Carolina State University) , Frank Forstrnanii (Freie Uiiiversitat Berlin), Ala.in H. F u c h (Uiiiversit6 de ParisSud), Keith E. Gubbiiis (North Carolina State University), Eilrique D i e Herrera (1Jniversidad Authnoiiia Metropolitaria), Siegfried Hess (Technische Univcrsitiit Berlin), Christos N. Likns and lIart,miit Ii-iwcn (both at, HcinrichHciiic:-Uiiivcrsitat Dussclclorf): Pctcr A. Moiison (University of Mmwhiusett,s), and Gren Patey (Universit,y of British Columbia). Finally, we would like to thank our fa.iniliesand friends for their continued support. and patience. Withoiit their appreciation of the coritrairits on our time att the stage of writing and correcting the manuscript, this hook would never have come int,oexistence. One of us (S.H.L.K)would like to express her sinwrc gratitiiclc to D. Flkgncr for hrlp wticn(?vcrit, was n d c d . In addit,ion, M.S. wishes tootliarik G. ,4riiold. I<. BcIircms, E. Egorov, and M. Wahl for their friendship in difficult tirnos.

Sabiiie H. L. Klapp and Martin Schoen Berlin December 2006

Dimensionless units At this point, we would like to emphasize that throughout this book we are giving all quantities in dimensionless (i.e., “reduced”) units. However, these units will vary between lattice mid off-lattice models, pure fluids and binary mixtures, and systerris with short- and long-range interactions. To avoid specifying thc spccific diriicnsioiilcss quailtitics wc arc using a t any point in the text, which, in our opinion, would reduce the legibility of this text markedly, we compile the basic quantities in Table 1. The reader should note, however? that only basic quantities like lengths or energies are given explicitly; “derived” qiiantities like density or st,ress follow by siiitable combination of the basic units.

xv

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Dimensionless units

Table 1: Dimensionless (,i.e., “reduced”) units used for basic physical quantities in various parts of this book. Portion nf text Quantity In iinits of Chapter 3 length d stress/pressure kBT/d Section 4.2 length Crr energy Elf temperature Eff / k B Scc:t ioiis 4.3-4.5 lcngth e energy Eff temperature &ff / k B Sect ions 4.6-4.8 length e energy EAA temperature EAA/I;B Sections 5.3-5.7 length Of energy cff temperature Eff/kB Section 5.8 length cnorgy Elf t.eniperature Eff / k B Scctioii 6.3.3 lcr1gth c energy u s / p Sections 6.4.1 and 6.4.2 temperature &/kB dipole moment &3 Section 6.5 1engtsh U energv & Fig. 6.12 cncrgy , 3 / p Scctions 7.7.2 arid 7.7.3 Icr1gt.h 0 tciiipcrature E / ~ B dipole rnoment Fig. 7.4 length (I tmipc:r;it,iirc 1 1 2 1 ( li:Ra3)’ dipolc momcnt’

e

Jw

Op denotes the dipole rnorrient.

*Eloxen species.

Contents 1 Thermodynamics of confined phases 1 . 1 Intmdiictory rcmarks . . . . . . . . . . . . . . . . . . . . . . . 1.2 Deformation of macroscopic \, ociics . . . . . . . . . . . . . . . 1.2.1 Strain tensor . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Stress tensor . . . . . . . . . . . . . . . . . . . . . . . . 1.3 G i b h fundalnental equation . . . . . . . . . . . . . . . . . . . 1.3.1 Bulk fluids . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Slit-pore with unstructured substrate surfaces . . . . . 1.3.3 Slit-pore with structured substrate surfaces . . . . . . . 1.4 Equilibrium statcs and thermodynamic potantvials . . . . . . . 1.4.1 Coiiditions for t.hcrrnotlynamic cc4uilibriuni . . . . . . . 1.4.2 Thermodynamic potentials . . . . . . . . . . . . . . . . 1.5 Legeridre transformation . . . . . . . . . . . . . . . . . . . . . 1.6 Homogeneit.v of confined phases . . . . . . . . . . . . . . . . . 1.6.1 Mechanical expressions for the grand potential . . . . . 1.6.2 Gibbs-Duhem equations and syinnietry . . . . . . . . . 1.7 Phase trari~it~ior~s . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 5 3 5 10 12 14 15 16 16 19 21 23 23 26 27

2 Elements of statistical thermodynamics 2.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . 2.2 Concepts of quantum statistical thermodynamics . . . . . . . 2.2.1 The most probable distribution . . . . . . . . . . . . . 2.2.2 Justification of the most probable distrilmtion . . . . . 2.2.3 The Srhriidinp~-Hillapproarh . . . . . . . . . . . . . 2.3 Councction with t.hc.ririodyIiarnics . . . . . . . . . . . . . . . . 2.3.1 Determixiat.ion of Lagrangian multipliers . . . . . . . . 2.3.2 Statistical expression for the entropy . . . . . . . . . . 2.3.3 Statistical physical averages and therrnodynaniics . . . 2.3.4 FluctuatioiLs . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Equivalence of ensenibles . . . . . . . . . . . . . . . . . . . . . 2.5 The classic limit . . . . . . . . . . . . . . . . . . . . . . . . . .

33 33 36 36 41 46 49 49 51 54 56 58 59

xvii

CONTENTS

xviii

Syinnictry coiisitlcrations . . . . . . . . . . . . . . . . . Kirkwvood-Wigicr tlicory . . . . . . . . . . . . . . . . The canonical partition function in the classic limit. . . Laplace t.ransformation of probability densities . . . .

60 63 65 70

3 A first glimpse: One-dimensional hard-rod fluids 3.1 Intmdiictory remarks . . . . . . . . . . . . . . . . . . . . . . . 3.2 Pure hard-rod bilk fliiid . . . . . . . . . . . . . . . . . . . . . 3.2.1 Statist.i d t.hcrmorlynamics of hid-rod fluids . . . . . 3.2.2 Virial cquatioii of statc . . . . . . . . . . . . . . . . . . 3.2.3 Bulk isothermal compressibility . . . . . . . . . . . . . 3.2.4 Density distribution . . . . . . . . . . . . . . . . . . . 3.3 Hard rods confined between hard walls . . . . . . . . . . . . . 3.3.1 Aspects of sta.tist,ical therinodynarnics . . . . . . . . . 3.3.2 "Stratificat.ion" of confined one-dimensional fliiids . . .

73 $3

2.5.1 2.5.2 2.5.3 2.5.4

75

75 77 81 82 86 86 88

4 Mean-field theory 95 4.1 Iiitrodiictory rcrnarks . . . . . . . . . . . . . . . . . . . . . . . 95 4.2 Van der Wwls tlieory of adsorption . . . . . . . . . . . . . . . 97 4.2.1 Sorption experimenb . . . . . . . . . . . . . . . . . . . 98 4.2.2 An equation of state for pure confined fluids . . . . . . 100 4.2.3 Critical behavior and gas-liquid coexistence . . . . . . 107 4.2.4 Gas sorption in iiiesoscopic slit-pores . . . . . . . . . . 110 4.3 Lattice model of confined pure fliiids . . . . . . . . . . . . . . 115 4.3.1 The model systern . . . . . . . . . . . . . . . . . . . . 115 4.3.2 High-temperature expansion . . . . . . . . . . . . . . . 118 4.4 Thermodyliarnics of pure confined lattice fluids . . . . . . . . 121 4.4.1 The Rogoliubov variational theorem . . . . . . . . . . . 121 4.4.2 Mean-field approxirnat.ion . . . . . . . . . . . . . . . . 1.22 4.4.3 Thc limit. of vi~nisliinjit.cmprratiirc . . . . . . . . . . . 124 4.5 I ' l i a s c s Lchavior of pure lattice fluids . . . . . . . . . . . . . . . 128 4.5.1 Slorphologies in the limit. T = 0 . . . . . . . . . . . . . 128 4.5.2 Coexisting phases of lhe lattice fluid . . . . . . . . . . 132 4.5.3 The impact of shear strain . . . . . . . . . . . . . . . . 138 4.6 Binary mixt.iires on a 1at.t.ice . . . . . . . . . . . . . . . . . . . 146 146 4.6.1 Model systern . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Mean-field approximation . . . . . . . . . . . . . . . . 148 151 4.6.3 Equilibrium states . . . . . . . . . . . . . . . . . . . . 4.7 Phase behavior of biiiary lattice mixtures . . . . . . . . . . . . 153 4.7.1 Symmetric. binary bulk rnixtures . . . . . . . . . . . . . 153 4.7.2 Decoiiiposition of syinmetric binary mixtures . . . . . . 157

CONTENTS

4.8

Ncutron scatlcririg cxpcrimciits . . . . . . . . . . . . . . . . 4.8.1 Experimental details . . . . . . . . . . . . . . . . . . 4.8.2 Lat.lice model of water iBA mixtures . . . . . . . . . 4.8.3 Phase diagram . . . . . . . . . . . . . . . . . . . . . . 4.8.4 Concentration profiles and contrast factors . . . . . .

XiX

. 160 . 161 . 163 165 . 170

5 Confined fluids with short-range interactions 177 5.1 1iit.roduct.ory rcmarks . . . . . . . . . . . . . . . . . . . . . . . 177 5.2 Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . . 181 5.2.1 Importance sa.mpling . . . . . . . . . . . . . . . . . . . 181 5.2.2 The grand canonical ensemble . . . . . . . . . . . . . . 184 5.2.3 Corrections to the configurational energy . . . . . . . . 190 5.2.4 A mixed isostress isostrain ensernhle . . . . . . . . . . 193 5.3 Chemically homogeneous substrates . . . . . . . . . . . . . . . 197 5.3.1 Expcrimcnts with thc! siirfacc forccs apparatm . . . . . 197 5.3.2 Dcrjaguin’s approximatioil . . . . . . . . . . . . . . . . 199 . 5.3.3 Normal component of the stress tensor . . . . . . . . . 202 5.3.4 . Stratification of confined fluids . . . . . . . . . . . . . . 203 5.4 Chemically heterogeneous substrates . . . . . . . . . . . . . . 208 5.4.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . 209 5.4.2 Structure and phase t.ransitions . . . . . . . . . . . . . 213 5.5 Chemical patterns of low symmetry . . . . . . . . . . . . . . . 221 5.5.1 Nanopatterried model substrate . . . . . . . . . . . . . 221 5.5.2 Thcr&lyiiamic prrtiirbatim thcory . . . . . . . . . . 224 5.5.3 Coiriputdioii of 1110 grand potciibial . . . . . . . . . . . 228 5.6 Rheological properties of confined fluids . . . . . . . . . . . . . 237 5.6.1 The quasistatic approach . . . . . . . . . . . . . . . . . 238 5.6.2 Molecular expression for the shear stress . . . . . . . . 239 5.6.3 Fluid bridges exposed to a shear strain . . . . . . . . . 242 5.6.4 Thermodynamic st.ability . . . . . . . . . . . . . . . . . 248 5.6.5 Phase Iicliavior of shear-deformed confined fluids . . . 255 5.7 The Joule-Thomsoii effect . . . . . . . . . . . . . . . . . . . . 257 5.7.1 Expc~rimontalbac-kgroiirid ant1 iipplicat.ions . . . . . . . 257 5.7.2 Modcl systciii . . . . . . . . . . . . . . . . . . . . . . . 260 5.7.3 Tlierniotlyiiaiiiic consitlerations . . . . . . . . . . . . . 261 5.7.4 The limit of low densities . . . . . . . . . . . . . . . . . 264 5.7.5 Confined fluids at moderate densities . . . . . . . . . . 274 5.7.6 Exact t.reatnient oft.he Joule-Thomson coefficient . . . 277 5.7.7 Isostress isostrain ensemble MC sixnulatioris . . . . . . 281 5.7.8 Iiiversion temperature at low density . . . . . . . . . . 283 5.7.9 Density dependence of the inversion temperature . . . 287

xx

CONTENTS 5.8

Lat.lice h~lorltcCarlo siniulations . . . . . . . . . . . . . . . . . 291 5.8.1 5.8.2 5.8.3 5.8.4

Atlvaritagcs arid disaclvantagcs of lattice rriodcls . . . Grand canonical ensemble Monte Carlo simulat.ions . Thermodynamic integration for lattice fluids . . . . . Comparison with mcan-field density functional theory

. 291 . 293

. 295

. 296

6 Confined fluids with long-range interactions 301 6.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . 301 6.2 Three-dimensional Ewald summation . . . . . . . . . . . . . . 303 303 6.2.1 Ionic syst.ems . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 From point charges to point dipoles . . . . . . . . . . . 309 6.3 Ewald summation for confined fluids . . . . . . . . . . . . . . 312 6.3.1 Thc rigorous approach . . . . . . . . . . . . . . . . . . 313 6.3.2 Slah-adapt.cd Ewald summation . . . . . . . . . . . . . 315 6.3.3 R.cliahilitv of thc slahadaptcd Ewdd mcthod . . . . . 318 320 6.4 Irisulatiiig solid substrates . . . . . . . . . . . . . . . . . . . . 6.4.1 Dipolar iiitcractions aiitl 'norrnal stress . . . . . . . . . 320 6.4.2 Orientational order in confined dipolar fluids . . . . . . 325 6.5 Conducting solid substrates . . . . . . . . . . . . . . . . . . . 332 6.5.1 The boundary-value prohleni in elect.rostatics . . . . . 332 6.5.2 Image cha.rges in inetals . . . . . . . . . . . . . . . . . 333 6.5.3 Dipolar fluids . . . . . . . . . . . . . . . . . . . . . . . 336 6.5.4 Metallic substrates and ferroelect.ricity . . . . . . . . . 337

7 Statistical mechanics of disordered confined fluids 7.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . 7.2 Queiiched-annealed models . . . . . . . . . . . . . . . . . . . . 7.3 The int.roduct.ion of replicas . . . . . . . . . . . . . . . . . . 7.4 7.5

Clorrclation fiiiic.t,ionsand fliic.t.iiat.ions in tho disordrrcd fluid hit.cgra1 cquat ions . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Rcplica 0rnstcin-Zc.riiikc cyuatioiis . . . . . . . . . . 7.5.2 Closure rcdatioiisliips . . . . . . . . . . . . . . . . . . 7.6 Therinodynarnics of the replicat.ed fiuirl . . . . . . . . . . . . 7.7 Applica.tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.1 Modcl systems . . . . . . . . . . . . . . . . . . . . . . 7.7.2 Dipolar fluids in simple niatrices . . . . . . . . . . . . 7.7.3 Dipolar fluids in complex matrices . . . . . . . . . . .

341 341 343 . 34G . 348 353 . 353 . 356 . 358 361 361 . 363 . 366

A Mathematical aspects of equilibrium thermodynamics 369 A . l The trace of a matrix product. . . . . . . . . . . . . . . . . . . 369 369 A.2 Legeridrr transformation . . . . . . . . . . . . . . . . . . . . .

CONTENTS

xxi

A.3 Eulcr’s thcorcm . . . . . . . . . . . . . . . . . . . . . . . . . .

371

B Mathematical aspects of statistical thermodynamics

373 B . l Stirling’s approximation . . . . . . . . . . . . . . . . . . . . . 373 B.2 Elements of function theory . . . . . . . . . . . . . . . . . . . 374 B.2.1 The Cauchy-R.ieniann differential equations . . . . . . 374 B.2.2 The method of steepest descent . . . . . . . . . . . . . 376 B.2.3 GauB’s integral theorem in two dimensions . . . . . . . 379 B.2.4 Cauchv integrals and the Laurent series . . . . . . . . . 382 B.3 Lagrangian multipliers . . . . . . . . . . . . . . . . . . . . . . 386 B.4 Maxiinurn term method . . . . . . . . . . . . . . . . . . . . . 389 B.5 Basis sets and the canonical ensemble partition function . . . 389 B.5.1 Parseval’s equation . . . . . . . . . . . . . . . . . . . . 389 391 B.5.2 Proof of Eq . (2.41) . . . . . . . . . . . . . . . . . . . . B.6 The classic liniit of quantum statistics . . . . . . . . . . . . . 392 B.G.l The Dirac 6-fuiiction . . . . . . . . . . . . . . . . . . . 392 B.6.2 Proof of the completeness relation . . . . . . . . . . . . 395 B.6.3 Quantum correct.ions due to wave-function symmetry . 396 B.6.4 Quantum rorrectioiis to t8heHamiltonian function . . . 399

C Mathematical aspects of onodimensional hard-rod fluids 403 C.l Distance bat.wwn mirror imagas . . . . . . . . . . . . . . . . . 403 C.2 Integral transformations . . . . . . . . . . . . . . . . . . . . . 405 C.3 Galissican densitmydistrihiit.ion . . . . . . . . . . . . . . . . . . 409 C.3.1 Limiting behavior of t.ha probability dcnsity . . . . . . 409 C.3.2 h.loivrc.~Laplaccapproxiniation . . . . . . . . . . . . . . 411

D Mathematical aspects of mean-field theories D . 1 Van der Waals model for confined fluids . . . . . . . . . . . D . l . l Evaluation of the double integral in Eq . (4.21) . . . . D.1.2 Newton’s method . . . . . . . . . . . . . . . . . . . . . D .1.3 Ma.xwell’scoiistraint. . . . . . . . . . . . . . . . . . . D.2 Lattice models . . . . . . . . . . . . . . . . . . . . . . . . . . .

415 . 415 . 415 417 . 418 420 D.2.1 Nunierical sohition of Eq . (4.86) . . . . . . . . . . . . . 420 D.2.2 Binary inixtmrens . . . . . . . . . . . . . . . . . . . . . . 421

E Mathematical aspects of Monte Car10 simulations E.l Stochastic processes . . . . . . . . . . . . . . . E . l . l The Chapman-Kolmogoroff equation E.1.2 The Principle of Detailed Balance . . E.2 Chemically striped substrates . . . . . . . .

......... . . . . . . . . . . . . . . . . . . . . . . . . . . .

431 431 . 431 . 433 . 435

CONTENTS

xxii E.3 Molccular cxprcssioiis for strcsscs . . . . . . . E.3.1 Normal coniponent of strws tensor . . E.3.2 Shear stress . . . . . . . . . . . . . . . . E.3.3 Virial expression for the Inilk pressure

. . . . . . . . . 437 . . . . . . . . . 437 . . . . . . . . 442 . . . . . . . . . 444

F Mathematical aspects of Ewald summation 447 F. 1 Three-dimensional Coulombic systems . . . . . . . . . . . . . 447 F.l.l Energy contributions in Ewdcl forinillation . . . . . . . 447 F.1.2 Force alld stress tensor components . . . . . . . . . . . 457 . F.2 Threc-dirnellsional dipolar svstcrn . . . . . . . . . . . . . . . .461 461 F.2.1 Self-energy . . . . . . . . . . . . . . . . . . . . . . . . . 462 F.2.2 Form and t.orcpc . . . . . . . . . . . . . . . . . . . . . F.2.3 Stwss tawor . . . . . . . . . . . . . . . . . . . . . . . . 464 F.3 Slab geonietry . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 F.3.1 Rigorous expresions . . . . . . . . . . . . . . . . . . . 466 F.3.2 Force, torqiie, and stress in systems with slab geometry 473 475 F.3.3 Metallic substrat.es . . . . . . . . . . . . . . . . . . . . .

G Mathematical aspects of the replica formalism

479 G.l Replica expressions in the grand canonical ensemble . . . . . . 479 G.2 Derivation of Eq . (7.23) . . . . . . . . . . . . . . . . . . . . . 480 G.3 Molecular fluids . . . . . . . . . . . . . . . . . . . . . . . . . . 481 G.4 Proof of Eq . (7.33) . . . . . . . . . . . . . . . . . . . . . . . . 483 G.5 Niimcrical soliit.ion of intcgral cqiiations . . . . . . . . . . . . 484

Bibliography

491

Index

510

Contributors Martin Schoen, Stranski-Laboratoriuin fiir Physikalische iind Theoretisrhe Chemie, Sekretariat C7, Iiistitut fur Chemie, Fakultiit fur Mathernatik und Naturwissenschafteii, Twhnische Universitiit Berlin, Strafle des 17. Juni 115, 10623 Berlin, Germany (Elcct,ronic mail: martin.schoen9fluids.tu-berlin.de)

Sabine H. L. Klapp, Stranski-Laboratorium fur Physikalische uiid Theoret,ische Chemie, Sekretariat C7, Institut, fiir Chemie, Fakultat fur Mathematik iind Naturwissensrhafteri, Technische Universit,iit, Berlin, StraBe des 17. Juni 125, 10623 Berlin. Cermiuiy (Electronic rnail: [email protected])

xxiii

Contributors to previous volumes Volume 1 (1990) David Feller arid Ernest R. Davidson, Basis Scts for Ah Initio Molecular Orbital Calculations and Intermolecular Interactions. Jarpes J. P. Stewart, Semieriipirical Molecular Orbital Methods. Clifford E. Dykstra, Joseph D. Augspurger, Bernard Kirtman and David. J. Malik. Properties of Molecules by Direct Calculation. Ernest L. Plummer, The Application of Quantitative Design Strategies in Prsticidc Dwign. Peter C. Jurs, Chcuioinctrics aiitl Multivariiltc Aiialysis in Arialytical Cheinist ry. Yvonne C. Martin, Mark G. Bures, and Peter Willet, Searching Databases of ThrecDiriiensional Structures. Paul G. Mezey, Molecular Surfaces. Terry P. Lybrand, Computer Siinulatioii of Bioniolecular Systems Using Molccnlar Dynamics arid Rcc Eiicrgy l'crtnrbiition Mcthods. Donald B. Boyd, Aspects of Molccular hlodcling. Donald B. Boyd, Successes of Coniput er-Assisted Molecular Design. Ernest R. Davidson, Perspectives of Ah Initio Calculations. Volume 2 (1991) Andrew R. Leach, A Siirvcy nf hlcthods for Searching thc Conformatima1 Space of Small and Medium-Sized Molecules. John M. Troyer and Fred E. Cohen, Simplified Models for Understanding and Predicting Protciii Structure.

J. Phillip Bowen and Norman L. Allinger, Molecular Mechanics: The Art and Science of Parameterization.

Uri Dinur and Arnold T. Hagler, New Approaches to Empirical Force

xxvi

Contributors to previous volumes

Fields.

Steve Scheiner, Calculating t,he Properties of Hydrogen Bonds by Ah Initio Methods.

Donald E. Williams, Net Atomic Charge and Multipolc Models fort the Ah Initio Molecular Electric Potential. Peter Politzer and Jane S. Murray, Molecular Electrostatic Potentials and Chcmical Reactivity. Michael C. Zerner, Scrriicmyiricttl hlolr.culilr Orbital Mctliods. Lowell H. Hall and Lemont B. Kier, Thc IvIolecular Connectivit,y Chi Indexes and Kappa Shape Indexes in Structure-Property Modeling. I. B. Bersuker and A. S. Dimoglo, The Electron-Topological Approach to the QSAR Probleni. Donald B. Boyd, The Computathml Chemistry 1,iterature. Volume 3 (1992) Tamar Schlick, Optimization Methods’in Coinputational Chcmistry. Harold A. Scheraga. Predicting Three-Dimensional Structures of Oligupeptides.

Andrew E. Torda and Wilfred F. van Gunsteren. Molecular Modeling Using NMR Data. David F. V. Lewis, Cornpiitc:~-Assistcd Mctliods in thc Evaluatioii ol’ Chemical Toxicity.

Volume 4 (1993) Jerzy Cioslowski Ab Initio Cnlculatioiis on Large Molecules: Methodology and Applications. Michael L. McKee and Michael Page, Computing Reaction Pathways on Molecular Potcntial Eiic>rgySurfaces. Robert M. Whitnell and Kent R. Wilson, Computational Molecular Dynamics of Chemical Reaction iii Solution. Roger L. DeKock, Jeffry D. Madura, Frank Rioux and Joseph Casanova. Computational Chemistry in the Undergraduate Curricnlum. Volume 5 (1994) John D. Bolcer aiid Robert B. Hermann, The Developmeiit of Coniputatioiial Chcmistrv in the United States.

Contributors to previous volumes

xxvii

Rodney J. Bartlett arid John F. Stanton, Applications of Post-Hartrce Fock Met-hods: A Tutorial. Steven M. Bachrach, Population Analvsis and Electron Densities from Quantum hlechanics.

J e f i y D. Madura, Malcolm E. Davis, Michael K. Gilson, Rebecca C. Wade, Brock A. Luty and J. Andrew McCammon, Biological Applications of Elwtmst atsic. Cid(!llliLt,ions i t l d Brownian Dynamics Siinulatious. K. V. Damodaran and Kenneth M. Merz, Jr., Coinpiit;sr Simdation of Lipid Systems. Jeffrey M. Blaney and J. Scott Dixon, Distance Geonietry in Molecular Modelling. Lisa M. Balbes, S. Wayne Mascarella and Donald B. Boyd, A Perspective of Modrrn Mcthods in Computer-Aidcrl Drug Design.

Volume 6 (1995) Christopher J. Cramer and Donald G. Truhlar, Continuum Solvation Models: Classical and Quantum Mechanical Implementations. Clark R. Landis, Daniel M. Root and Thomas Cleveland, Molecular hlcchanics Forcc Ficld for Modcling Inorganic and Organonict,allic Compountls. Vassilios Galiatsatos, Computational Methods for Modeling Polymers: An Introduction. Rick A. Kendall, Robert J. Harrison, Rik J. Littlefield and Martyn F. Guest, High Performance Computing in Computational Chemistry: Methods and Machines. Donald B. Boyd, h.iolcc.nlar Modding Softwilr(! in USC:Piihlic,zt,ion Trcnds. Eiji &awa arid Kenny B. Lipkowitz, Appciidix: Publislied Forcc Field Parameters. Volume 7 (1996) Geoffrey M. Downs and Peter Willett, Similarity Searching in Databases of Chamical Striictiircs. Andrew C. Good and Jonathan S. Mason, Three-Dimensional Structure Database Searches. Jiali Gao, Methods and Applications of Combined Quantum hlechanical and Molecular Mechanical Potentials. Libero J. Bartolotti and Ken Flurchick, An Introduction to Density Functional Theory.

xxviii

Contributors to previous volumes

Alain St-Amant , Dc1isit.y Furiclioiial Mvlhods in Bioinolccular Modcling. Danya Yang and Arvi Rauk. The A Priori Calculation of Vihrat,ional Circular Dichroism Intensities. Donald B. Boyd , Appendix: Conipendiimi of Software for Molecular klodeling.

Volume 8 (1996) Zdenek Slanina, Shyi-Long Lee and Chin-hui Yu, Cornputa.tions in lleating Fullerenes and Carbon Aggregat,es. Gernot F'renking, Iris Antes, Marlis Bohme, Stefan Dapprich, Andreas W. Ehlers, Volker Jonas, Ariidt Neuhaus, Michael Otto, Ralf Stegmann, Achim Veldkamp, and Sergei F. Vyboishchikov, Psciidopot,rntial Calcnlat.ions of Transition Mctal Compoiinds: Scopc and Liiriitations. Thomas R. Cundari, Michael T. Benson, M. Leigh Lutz and Shaun 0. Sommerer, Effective Core h t e n t i a l Approaches to the Chemistry of the Heavier Elements. Jan Almlof and Odd Gropen, Relativistic Effects in Chemistry. Donald B. Chesnut, The Al:) Inito Chriputation of Nuclear Magnetic Resonaiicc Chcmical Shiclding.

Volume 9 (1996) James R. Damewood, Jr., Peptidt: hfiinetic Design with the Aid of Comput ational Chemistry. T. P. Straatsma, Free Energy by hilolecular Simulation. Robert J. Woods, Thc Application of hfolcciilar h.fodcling Tcchriiqiics too tlic Dctcrmiiiation of Oligosaccliaricic Solution Conforrtiat ions. Ingrid Pettersson and Tommy Liljefors, hfolccular Mechanics Calculated Conformational Energies of Organic hlolecules: A Comparison of Force Fields. Gustavo A. Arteca, Mobciilar Shape Descriptors.

Volume 10 (1997) Richard Judson, Genetic Algorit,hms and Their Use in Chemistry. Eric C. Martin, David C. Spellmeyer, Roger E. Critchlow, Jr. and Jeffrey M. Blaney, Does Combinatorial Chemistry Obviate ComputerAided Drug Design?

Robert Q. Topper Visualizing Molecular Phase Space: Nonstatistical

Contributors to previous volumes

XXiX

Effects in Reaction Dynamics. Ftaima Larter and Kenneth Showalter, Compiit.akiona1 Studies in Nonlinear Dynamics. Stephen J. Smith and Brian T.Sutcliffe, The Development of Coinputational Chemistry in t,he United Kingdom.

Volume 11 (1997) Mark A. Murcko, Recent Advances in Ligarid Design Methods. David E. Clark, Christopher W. Murray, and Jin Li, Current Issues in De Novo Molecular Design.

'lhdor I. Oprea and Chris L. Waller, Theorctical and Practical Aspects of Three-Dimensional Quantitative Structiire-Activity Relationships.

Giovanni Greco, Ettore Novellino, and Yvonne Connolly Martin, Approaches to ThrcxLDinicrisioiial Qiimit it atiw StructureActivity Relationships.

Pierre- Alain Carrupt, Bernard Testa, aiid Patrick Gaillard, Computatioiial Approaches to Lipophilicity: Methods and Applications.

Ganesan Ravishanker, Pascal Auffinger, David R. Langley, Bhyravabhotla Jayaram, Matthew A. Young and David L. Beveridge, Treatnirnt of Coiintrtrions in Coniputcr Simiilat,ions of DNA. Donald B. Boyd, Appciidix: Cornpcndiuin of Software arid Iiitcriict Tools for Computational Chemistry.

Volume 12 (1998) Hagai Meirovitch, Calculation of the Free Energy and the Entropy of Macromolcciilar Systcms by Cornpiitcr Simulation. rtiid T. P. Straatsma, Molccular Dyiiarnics with Gcrieral Holonomic Constraints and Application to Internal Coordinate Constraints. John C. Shelley and Daniel R. BGrard, Computer Siinulatioii of Water Physisorptioii at h.letmalWater Interfaces. Donald W. Brenner, Olga A. Shenderova and Denis A. Areshkin, Quantum-Rased Aiialytic Interatomic Forces and Materials Simulation. Henry A. Kurtz and Douglas S. Dudis, Qiiantmii hi(x-hariica1Mct hods for Predicting Nonlinear Optical Properties. Chung F. Wong, Tom Thacher and Herschel Rabitz, Sensitivity Analysis in Biornolcvular Simulation. Paul Verwer and Rank J. J. Leusen, Computer Simulation t o Predict Possible Crystal Polymorphs. Jean-Louis Rivail and Bernard Maigret , Computational Cheniistry in

Ramzi Kutteh

xxx

Contributors to previous volumes

Fra1ic.c: A Historical Survcy.

Volume 13 (1999) Thomas Bally and Weston Thatcher Borden, Calculations on OpenShell Molecules: A Beginners Guide. Neil R. Kestner and Jaime E. Combariza, Basis Sct, Siipcrposition Errors: Thcory arid Practice. James B. Anderson, Quniituni Monte Carlo: Atoms, Molecules, Clusters, Liquids. and Solids. Anders Wallqvist and Raymond D. Mountain, Molecular Models of Water: Derivation arid Description. James M. Briggs and Jan Antosiewicz, Simulation of pH-Dependent Propcrtics of Protrins IJsiiig Mcsosropic- Modcls. Harold E. Helson, Striicturc Diagram Gcncrat.ion.

Volume 14 (2000) Michelle Miller Franc1 arid Lisa Emily Chirlian, The Pluses and Minuses of Mapping Atomic Cliarges to Electrostatic Potentials.

T. Daniel Crawford and Henry F. Schaefer 111, An Introdiict,ion to Couplctl Clustcr Tlicory for Corriput,at,ioualChcrnists.

Bastiaan van de Graaf, Swie Lan Njo, and Konstantin S. Smirnov, Introduction to Zeolite Modeling.

Sarah. L. Price, Toward More .4ccurate Model Intermolecular Potentials for Organic Molecules.

Christopher J. Mundy, Sundaram Balasubramanian, Ken Bagchi, Mark E. Tuckerman, Glenn J. Martyna, and Michael L. Klein, Noucquilibriuni h4olc.cular Dyriariiics. Donald B. Boyd and Kenny B. Lipkowitz, History of the Gordon Research Conferences on Computational Chemistry. Mehran Jalaie and Kenny B. Lipkowitz, Appendix: Published Force Field parameters for I\llrtlec.ular Mechanics, Molecular Dynamics, and Monte Carlo Simulations.

Volume 15 (2000)

F. Matthias Bickelhaupt aiid Evert Jan Baerends, Kohii-Sham Den-

sity Functioiial Theory: Predicting and Understanding chemistry. Michael A. Robb, Marco Garavelli, Massimo Olivucci and Fernando Bernardi, A Computational Strategy for Organic Photochemistry.

Contributors to Drevious volumes

xxxi

Larry A. Curtiss, Paul C. Redferm arid David J. Frurip, Thcorctical Methods for Computing Enthalpies of Formation of Gaseous Compounds. Russel J. Boyd, The Development of Computational Chemistry in Canada. Volume 16 (2000) Richard A. Lewis, Stephen D. Pickett. and David E. Clark, Cornputer-Aidcd Molccdar Diversity Aiialvsis aiid Corribiiiatorial Library Design. Keith L. Peterson, Artificial Neural Networks and Their Use in Chemistry. Jorg-Rudiger Hill, Clive M. Freeman and Lalitha Subramanian, Use of Force Fields in Materials Modeling. M. Rami Reddy, Mark D. Erion and Atul Agarwal, Free Energy Calcnlations: IJsc wid Limitations in Predicting Ligarid Binding Affinitics.

Volume 17 (2001) Ingo Muegge and Matthias Rarey, Small Molecule Docking and Scaring.

Lutz P. Ehrlich and Rebecca C. Wade, Protein-Protein Docking. Christel M. Marian, Spin-Orbit Coupling in h~olaculcs. Lemont B. Kier, Chao-Kun Cheng arid Paul G. Seybold, Ccllular Automata Models of Aqueous Solution Systems. Kenny B. Lipkowitz and Donald B. Boyd, Appendix: Books Published on the Topics of Computatiorial Chemistry.

Volume 18 (2002) Geoff M. Downs arid John M. Barnard, Clustcring Methods aiid Tlieir Uses in Computational Chemistry. Hans-Joachim Bohm and Martin Stahl, The Use of Scoring Functions in Drug Discovery Applications. Steven W. Rick and Steven J. Stuart, Potentials and Algorithms for Incorporating Polarizability in Coniputer Simulations. Dmitry V. Matyushov and Gregory A. Voth, New Dcvclopmcnt,s in the Theoretical Description of Charge’llansfer Reactpionsin Condensed Phases. George R. F‘amini and Leland Y.Wilson, Linear Free Energy Relationships Using Quariturn Mechaiiical Dcscriptors. Sigrid D. Peyerimhoff, The Development of Coniputat,ional Chemistry in Germany. Donald B. Boyd and Kenny B. Lipkowitz, Appendix: Examination of

xxxii

Contributors to Drevious volumes

thc Eniploymoiit. Environincnt for Couiputatioiial Ckniistry.

Volume 19 (2003) Robert Q. Topper, David L. Freeman, Denise Bergin arid Keirnan R. LaMarche, Coniputatiorial Techniques and Strategies for hlontc Cnrlo Thcrrrlodynaniic Cidclll;~tfion,with Applications to Nanochistcrs.

David E. Smith and Anthony D. J. Haymet: Computing Hydrophobicity.

Lipeng Sun and William L. Hase, Born Oppenheimer Direct Dynamics Classical Trajectory Simulations.

Gene Lamm, The Poisson l3nltzmann Equation. Volume 20 (2004) Sason Shaik and Philippe C. Hiberty, Valcnce Bond Theory: Its History, Fundainentals and Applications A Primer. Nikita Matsunaga and Shiro Koseki, Modeling of Spin Forbidden Reactions. Stefan Grimme. Calr.nlation of thc Elcctronic Spwtxa of Largo ~lolcculcs. Raymond Kapral, Siiiiulatiiig Clicmical Waves aiid Pattcrns. Costel SQrbu and Horia Pop, Fiizzy Soft-Computing Methods and Their Applications in Chemistry. Sean Ekins and Peter Swaan, Development, of Computational Models for Enzymes, rllansporters, Channels and Receptors Relevant to ADhlE/Tox. Volume 21 (2005) Roberto Dovesi, Bartolomeo Civalleri, Roberto Orlando, Carla Roetti and Victor R. Saunders, Ah Initio QuatitsumSimulation in Solid State Chemistry.

Patrick Bultinck, Xavier GironCs arid Ramon CarbcSDorca. hlolecular Quantuni Siniilarity: Theory and Applications.

Jean-Loup Faulon, Donald P. Visco, Jr. and Diana Roe, Enlimorating Molecules.

David J. Livingstone and David W. Salt, Variable Selection-Spoilt for Choice.

Nathan A. Baker, Bioinolecular Applicatioiis of Poisson Boltzmami Methods.

Baltazar Aguda, Georghe Craciun atid Rengul Cetin-Atalay, Data

Contributors to previous volumes

xxxiii

Sources a d Corriputatiorial Approaches for Gciicrating Models of Gcnc Regulatory Networks.

Volume 22 (2006) Patrice Koehl. Protein Structure Classification. Emilio Esposito, Dror "obi and Jeffry Madura, Comparative: Protein Modcling. Joan-Emma Shea, Miriam F'riedel and Andrij Baumketner , Simulations of Protein Folding. Marco Saraniti, Shela Aboud and Robert Eisenberg, The Simulation of Ionic Charge Il-ansport in Biologiical Ion Channels: An hitroduction to Numerical Methods.

C. Matthew Sundling, Nagamani Sukumar, Hongmei Zhang, Curt Breneman arid Mark Embrechts, Wawl(:t,s in Clicmistry and Chernoinformatics.

Volume 23 (2007) Christian Ochsenfeld, Jorg Kussmann and Daniel Lambrecht, Linear Scaling in Quaiitimi Chemistxy. Spiridoula Matsika, Coiiical Iritcrscctioris in hllolccular Systciiis. Antonio Fernandea-Ramos, Benjamin Ellingson, Bruce C. Garrett

and Donald G. Truhlar, Variational Transition State Theory with Multidimensional Tumieling. Roland Faller. Coarse Grain Modeling of Polymers. Jeffrey Godden and Jurgen Bajorath, Analysis of Chemical Inform at'ion Coiitcnt wing Sha~inoriEntropy. Ovidiu Ivanciuc, Applications of Support Vector hlachincs ixi Cheniistry. Donald B. Boyd, How Coniputalional Chemistry Became Irriportant in the Pharmaceutical Industry.

Nanoconfined Fluids: Soft Matter Between Two and Three Dimensions Martin Schocn and Sabine H. L. Klapp St,rariski-Lsboratoriuiri fur Physikczlisclie und Theoretisclie Cheriiie Iristitut, fur Ckieinie Fakult,St fur Mat hernat ik uiid Nat urwissenschaft en Techriischc Universi tiit Berlin StraBe des 17. Juiii 135, 10623 Berlin? GERMANY

Reviews in Computational Chemistry Kenny B. Lipkowitz &Thomas R. Cundari Copyright 02007 by John Wiley & Sons, Inc

Chapter 1 Thermodynamics of confined phases 1.1

Introductory remarks

Our understanding of phenomena in the nonanirnated part of nature (and perhaps to a lesser extent evcii those in its animated part) is promoted by the four cornerstones of modern t,hcorrtical physics: classic mechanics, cliiantimi iricchariics, clcctrodynaiiiics, arid thcrmodyri~Kiics.Arriong these four fields, thermodynamic2 occupies a unique position in several respects. For example. its mathematical structure is by far the simplest and can be grasped by anyone with knowledge of clcrncntary calculus. Yet, most. students and at times even long-time practitioners find it, hard to apply its concepts to a given physical situation. The axiomatic basis of thcrrnodynamics is quite scarce coinparad with thc other threc thcorctical fields. .lust four clcr~icritaryarid coriiplctcly gciicral principles, taheso-called Laws of Thermodynanzics, are required to lay the axiomatic foundation. They are essentially deduced from everyday experience. Thus, thermodynamics is by far the most self-contained of the four theoretical fields. However, what appears as a particular strength and certainly a source of mathematical beauty also gives rise to perhaps the most serious shortcoming of thcrrnodynamics, namcly its almost total lack of any prrdictive power. This disadvantage is causrd by the fact that thermodynamics equips us only with general mathematical relations between its key quantities. It is virtually incapable of quantifying any of them without having to take recourse t o additional sources of information such as experimental data or (empirical) equations of state. Thus, there is a substantial price to pay for matheniatical rigor. self-contaimncrit. a i d beauty (i.e., struct>ural

1

2

Thermodynamics of confined phases

siniplicity) of tlicrinodyiiairiics its a ccritral theoretical cornerstone in thc modern physical sciences. The total lack of predictive power as far as properties of a specific physical system are concerned tmns out, to he caiised by the fact that thermal systems arc corriposcd of (a macroscopic piccc of) iriattcr whose properties dcpcnd 011 the interaction between the microscopic const,ituents (i.e., electrons, phonons, at,oms, or molecules) of which it is composed. Thermodynamics, on the other hand, has no concept. whatsoever of the underlying niicroscopic structure of the macroscopic world with which it is dealing; t,hat is, it knows nothing about interactions between microscopic constituents. To outhie the conceptual framework of this chapter, we think that, this latter aspect, can hardly be overemphasized because it has become a widely accepted but deplorable practice in some physical chemistry courses to i n t r e cluce stiident>sto thermodynamics hy taking recourse to inherently inolecular concepts that arc coiiiplctcly alicii to tlicrinodyiiiLlriics. Although this riot>iori is usually iriotivated tlitlactically (but ip our opinion utterly confused) , it must be regarded as ill-founded aiid conceptua,lly misleading. To establish a conceptmdly sonnd link lwt,ween molecules ;IScntities foriniiig a macroscopic piccc of niilttcr on t81ic!one liaiid and tlierriiodyliarnics on the other hand, one needs t.o resort to (quantum) statistical physics, which could not be established in its inodern form until after the advent of quantum mechanics in t,he early t,weiitieth century (see Chapter 2). In other words, an at,teinpt to base theriiiodyiiamics on molecular concepts like int,eracting molecules or, even worse, m01eciilur chaos, deliherat,ely ignore.. the fundamental character of the postulatory basis of thermodynamics among the laws of nature as wc know them today 16, 71. In light. of tlicsc conirnerits, tlieriiiodynamics appears as a typical physical t,heory of the nineteenth century, wit,h the engineers Watt,’ aiid Carnot2 being a.inong its “founding fakhers.” In the nineteenth century, it, was by no means iiiidispiit,ed whet.her entities like atoms or molecules really existed or whether they were merely a construction of the huniaii niind [8]. Although “the atom” was a well-established but piirdy philosophical cntity aroiind 1800 to which r a t h r bizarre properties were ascribed 1‘91, it did iiot, liavc prcciso Iiicaiiing iii a physics or chcmist.ry context. Meaning in the sense of a. sound concept. in the nat#uralsciences ‘.Ja.nies Watt: (1736-181 9) signilicant,ly improved the heat. engine developed by Thonias Newcomen. 21n 1824 Sadi Nicolas 1,Conwd Carnot (17961832) published a11 analysis of what became known as the “Carnot cycle” in his book entitled RPflezion .wr la Puzs.sance Motrim c h Feu el SIW leu Macliinas Pivpn:s ci Ilkrelopper Celtr: Puissance, where also he int.roduced the concept of a nonniolecular fluid, the t:crloric, as the working substance of heat. engines.

Deformation of macroscopic bodies

3

was @veil to it in t:xycrinicrits rcvicwcd by Lord I
1.2 Deformation of macroscopic bodies In the subsequeiit analysis of thernia.1 systems, a key issue will be a. transformation of energy bet,ween its various forms, namely heat and work. The latter term refers to energetic changes in the state of a thermal system on accoulit, of its interaction with the erivironnieiit through rneclianical means (i.c., pistons). In other words, t h cnvironmcnt irnposcs axtcrnal forcts on thc system to which it responds by deformation. In the context of this book, two types of forces are relevant: those changing tjhevolume of' the body (i.e., compressional/dila.tional forces) aad those changing its shape (i.e., shear forces). The purpose of this section is to develop a rigorous (macroscopic) description of the changes of the state of a rnacroscopic elastic body uiider these strains and to calcnlitte the associated stresses. Oiir treatment follows in spirit, the disciission in Cliaptcr 3.G of Arfkcri's book (1 11.

1.2.1

Strain tensor

Consider two mass e1ement.s of an elastic body in an unstrained reference state. One of them is located at a point. TO and the other one at a point ro b r ~ If. the body is strained, t,he first, one changes position according to

+

r = ro

+u

(TO)

+

(1.1)

whereas the other one is displaced simultaneously by u (ro &TO). Therefore, in the strained state, deforniat,ion of the body mily be described by the qi1antit.y d u = u (7.0 hrg) - u ( T " ) (1-2)

+

The dependence of the vector field u on the position of the iiiass elements in the rinst,raiiicd rcfcrancc state is a nacassary prcrcqiiisitc to dcscriha changes in t,he shape of the elastic body. N0t.e t,hat if u would be a vector (independent of the position of the mass elements), all mass elements would be displaced in the same direction and by the same amount Iul. Herice, we would be dealing with a rigid rather than an elitstic body incapable of changing its shape. In other words, the body would only be capable of moving in space as an undeformable entity.

4

Deformation of macroscopic bodies

If the two IIMSS clcIncrit,scoiddcrcd above arc scparatcd by ail iIifinitc+ in the unstrained reference state: it makes sense imally small distance (6~01 to approximate the displacement u (q) 6r0)by a Taylor series expansion of u ( T O ) accordiiig to

+

u (7-0 + 6To) = u ( T o ) +

c

O01

fl=

1

7 (ST" . V,)" u (T") 71.

(1.3)

where VL G d/&o,, a/ar,-,,). Retaining in this axpansioii only terms up to first order in 6r" (i.e., n = l)! we obtain from Eqs. (1.1) and (1.3) 6u = v,u (T") * 6T" (1.4)

In Eq. (1.4), the dyad

is a scrond-rank tcnsor in roniponcnt, notation, wlicrc r, and r d arc thc (Cartesian) cu-cornponents of the veclors T arid T O ,respectively. To proceed i l is convenient lo split tho tensor V,u into a symnietric (a) and antisymmetric part (77) ~riu

At this point it is instructive to analyze 77 in iiiore detail. To this end it, is useful to introduce a vector

such that [src Eqs. (1.4) and (1 .G)] where Ex, e7y, and e^, are unit vectors along t8he2-, 4-,and z-axis of the Cartesian coordinate system, respectively. Equation (1.8) has a lucid physical interpretation. It shows that the antisymnietric part of the displacement tensor V,.,,u( T O )describes a rotation about an instantaneous axis through the mass element a t r0 in the unstrained state in the direction of 6 by

5

Thermodynamics of confined Dhases

radians 1111. As we wish t.o focus on dcforriiatioris of t h clastic body and have already disregarded mere translations by Eq. (1.1)we shall henceforth ignore the antisymmetric part of the displacement t.ensor V,u ( T o ) . Instead we consider only its symnietrir part satisfying [see Eli. (1.6)] the symmetry relation fJ'rrlj = 0/h, va # P (1.9) where wc refer to u as the' strain tensor hencefdh. Let, us illustrate the above analysis by a specific example where Bro = brxoiZx

(1.10)

i.c., we considar two inass clemcnts saparatcd bv a small distance 6rxo along tlic x-axis in tlic uristrairicd refcram: state. Bccausc

bu has three nonzero components. Therefore, in the strained state, the two mash clcments are sc.pnratc.d hy br

= dro

+ 6~ = brxo

( flxi;)

(1.12)

so that the two mass elements, which were originally located at the same point in the 3-2 plane are now displaced also along the y- and z-axis. Therefore, we conclude that diagoiial elements of the matrix representing the strain tensor > 0) of the elastic bodv. dcscrilw comprtxsion (uucx< 0) or dilatation (o<,(~ whereas off-diagonal coniponents of u represent shear deformations.

1.2.2

Stress tensor

In the previous section, we focused on deforrnat.ions of elastic bodies. Clearly, such deforniations are the consequence of external forces acting on the body.3 These forces can be cast. in terms of a set of stresses (7,~)acting 011 the faces of an infinitesimally small parallelepiped, 7,$dAq: where A0 is the area of siich a face whose normal is pointing in the ~-dircction(scc Fig. 1.1). As wc would again like to disrcgartl I I I C ~ C translation of tlic para1k:lcpiped in space, certain symmetry relations between the stresses must hold, which are illustrated in Fig. 1.1. Moreover: we a ~ ~ u m c 3This argurnerit is based on the implicit but plausible ttssurnptiori of validity of the causality principle.

6

Deformation of macroscopic bodies

Figure 1.1: Skctdi of ii ymallclcpipcd (i.c., the. fluid 1arnc:lla) on which various stresses rap are acting. The quantity raodAp is the a-component of the force exerted on the P-directed face of the parallelepiped. Its direction is represented by the arrows. For the sake of clarity, only those stresses acting on the front faces are displayed.

1. The stress to be homogeneous throughout the body. 2. A stat,e of niechanical stability t,o exist.

3. Absence of external body forces (such '& gravity) or body torques (such as magnetic fields).

As in Section 1.2.1 we would also like to disregard rotcations of the parallelepiped as a result of thc: external forces acting on it. That is, we wish all torques about the three axes to vanish [ll]. Consider as a specific cxsmplc t.hc torqnc act,ing 011 thc parallclcpipcd about the z-axis. The norinal st,re!ses T,, do not contribute to this t-orque. Stresses T,, and T~ do not contribute because they point in the z-direction (see Fig. 1.2). Similarly, r,, and T ~ cannot , add t o the net torque because they are ba.lanced by stresses that, are equal in magnitude but point in the opposite direction on the bottom plane z = 0. This then leaves us with two remaining contributions to the net torque, namely ryx(dydz) dx and rxy(dzdz) dy.

Thermodynamics of confined Dhases

7

Figure 1.2: Side view of the parallelepiped sketched in Fig. 1.1.

These torques need to satisfy the equation

rYx (dydz) d z = rXy (d:rdr) d!/

(1.13)

if we wish the parallelepiped to stay a t rest, that is in the absence of rotation around the z-axis. Because in Eq. (1.13), dzdydz is arbitrary, ryx= T~ follows without further ado. The above argument, may he applied also to rotations around the x- and y-axis so that in general the stresses must satisfy the symmetry relation rafi zz ‘fiCk> va # [j (1.14) In other words, the set { r a p } can be represented by a symmetric matrix if we consider only external forces that cause deformations (i.e., conipres sion/dilatation or shear) of the uiistrained parallelepiped in its reference state. Notice also that, on account of our argumentation above, Eq. (1.14) is not a statcmont, about cquality of direction, hut mthcr onc about m,u.pitude of stresses. What remains to be shown is that the matrix {raa}is actually a representation of a second-rank tensor r to which we shall henceforth refer as the stress tensor. We need to demonstrate that the matrix representing T satisfies transformation properties under rotation of t.he coordinate system that constitute a second-rank tensor. To this end consider an infinitesimally

8

Deformation of macroscopic bodies

Figure 1.3: Sketch of an infinitesimal tetrahedron whose three faces coincide with the c-y, 2-2. and y-z planw of the original (unprimed) Cartesian coordinate system. The third slant face appears to be oriented such that the axis 2’ is norinal to t.he area of the slant face. The two remaining axes, y’ and z‘ lic?in the plane of the slant face but. are orthogonal to one another as well as to the d-axis. Hence, x’, p‘, and z’ define a Cart.esian coordinate system rotated with reqpect. to the original one.

siiiall tctrahcdron plottcd iii Fig. 1.3. Tlirce of its (triangular) faces coiiiciclc with the planes located at x = 0, y = 0, and z = 0, respectively, of the original (i.e., unprimed) coordinate system. The fourth face of area dA is slanted with respect to the original coordinate system. The fourth face rnay be employed to define a second (Cartesian) Coordinate system whose axis x’, say, points in a direction norinal to dA; the other two &es, 9’ and 2’: lie in the plane of arca dA siicli that any ono axis in thc primed coordinatc systeril is orthogonal to the other two just as any axis in t,he original coordinate syst,em is orthogonal to the other two. Clearly, bolh coordinate systems are rota.ted with respect to one another where the orientation of the primed coordinate system relative to the unprimed one may be expressed in terms of the matrix of direction cosines a, which is the cosine of the angle between any pair of axes! one in the prinwd and thc other one in the unpriiiicd coordinate system

Thermodynamics of confined phases

9

(see Sex-tioil 4.3 in 1111). From this description and the previous discussion in this section: it is clear that in the primed coordinate system forces acting on the slant face of the tetrahedron are given by r i j d A . We shall use Latin letters to refer to properties pertaining to the rotated coordinate system and Greek letters for the corresponding original coordinate system. Given the forces r;,dA we obtain t.hc corrcsponrling oiics r,,pu:;ddA in thc originid coordinatc system, whcrc njpd/l is t,hc area dA4of the slant face projected onto the plaric = 0. The force ra,3ajpd,4 p0int.s along t,he a-axis. Its component dong the 1:-axis of the rotated coordinate system is given by rapaiaajodA. Slimming these forces over cr gives the sum of the i-components of the three forces on the planes p = 0 in the i-direction. Finally, sumining over all three planes ,d = 0 the total force along the i-axis is given by (1.15)

and bccaiisc! dA is arhitrary

(1.16) which is the trarlsformatiori that a rnat,rix rcprcsciiting a tcilsor of scwond rank must. satify [l11. Finally, the reader should appreciate a significant difference between the way in which u and r were introduced. In Section 1.2.1 the strain tensor was defined on purely mathematical grounds, whereas the. conjugate stress tensor was introduced by purely physical reasoning (i.e., force balance). Howcvcr, both u and r arc clcfincd such that, nicrc transl;\t,ions or rotations of a rriacroscopic clast,ic body arc explicitly cxcluded. As far as u is coiiccrIicd, this is effected by introducing the displacement of mass elements u ( r )as a vector field and by defining u as the symmetric part of the displacement t8ensor V u ( T ) [see Eqs. (1.1) and (1.6)]; for t4hestress tensor T , the symmetry property stated in Eq. (1.14) serves to t4iminate rotations. Employing the concepts of stress and conjugate strain, and their proper mathcmat,ical fortnillation as sccond-rank tmsors, now cmablcs 11s to dcd with mechanical work in a general anisotropic piece of matter. One realization of such A. system are fluids in confinement t o which this book is devoted. However, at the core of our subsequent treatment are thermal properties of confined fluids. In other words, we need to understand the relation b e tween mechanical work represeiited by stxess-strain relationships and other forms of energy such as heat or cliemical work. This relation will be forrnally

Gibbs fundamental equation

10

established wit hiii t,hc Lramcwork of (cquilibriuin or phcrioriicriologic.al) thcrmodynamiw to which we shall turn now.

1.3 Gibbs fundamental equation Perhaps tlic most iiriport ant. concepts of tho axioniatic foundation of tllcrmodyiiamics are the ones referred to as the First arid Second Laws dealing with the internal energy U and the entropy S. They are essentially statements dealing wit.h energy conservation and the transformation of one form of energy (e.g., work) illto another one (e.g., heat). If .combined, the First and Second Laws give rise to the secalled Gihbs fundamental equation

dU = TdS + dM:

(1.17)

where T is the absolute temperature and dlV is an infinitesimal amount of work exchanged between t.he thermodynamic system and its environment. Following standard practice we refer to T as “absoliitd’tcmperat,iirc!bwause it is dcfiiicd such that any pliysical properties of the iiicasuriiig dcvicc (c.g., thermal expansivity of a thermometer substance) are complet,ely irrelevant. This reemphasizes our earlier argument. that t.hermodynamics as a physical theory ha5 no concept of matter. Incidentally, the only difficult and challenging step required in the analysis of any fhermodynaniic problem is to identify the system of interest,, its euvironment~and to specify the interaction bctwccri the two. The distinrthi 1wtwc:cn system atid en,vironm.ent is the key order priiiciplc of t1icniiodyiiamit:s. Mathematically speaking, both dU and d S are exact differentials; that is, they satisfy t.he equation

f

c

dX=0,

X=U,S

(1.18)

along any arbitrary closed path in the space spanned by the natuml variables (see Section 1.4.2) on which X depends (i.e., state space). Thermodynmnic functions X satisfying EQ. (1.18) are referred to as state functions because they depend only on initial and final states of the system but not on the path connecting the two in state space. On the contrary

{dlV = f ( C )

(1.19)

c

is the amount of work exchanged between the system and its environment and depends on how this exchange is beiiig effected. In other words, a state function I Y does not exist.

11

Thermodynamics of confined phases

So far nothing specific has bccii said about the iiaturc of the work exchanged bet,ween the system and its environment. In this book we shall restrict ourselves t,o two types of work. The first is chemical work

dWbem = pdN

(1.20)

in which inatter can be exchanged between the system and its environment. In Eq. (1.20). p is the chemical potential and N is the number of molecules accommodat,rd by the system. Thr somrwhat abstract, qiinntitzy11 < 0 may be viewed as the amount of work required t o add a new molecule to the system from some external reservoir of matter (i.e.l the e n v i r ~ n m e n t ~Hence, ).~ we are dealing with open systems in general. Note, that the use of the term molecule does not contradict our expository rernarks concerning molecular concepts in thermodynamic.. became nothing is being said about the (microscopic) properties of molecules at this point. .The socond typc of work to which we shall rcstrict oiir disciission is mechanical work. As we shall bee later, coilfincd pli&cs call bc cxposcd to two types of mechanical work, na,mely compression (dilation) and shear. In that regard, confined phases have a lot in common with bulk solids in that they are generally inhomogeneoils and anisotropic in one or more spa tin1 dimensions. Therefore, it seems sensible to cast the niechanical work term in terms of stress ( 7 )and dimeiisioriless strain tensors (u)introduced in Section 1.2 and suggestrd earlirr by Callen for a propcr trcatmcnt of thc thermodynamics of bulk solids [12], by writing (1.21) where V( is the volume of an undeformcd (uncompressed and unsheared) reference system. Thus, in its most general form suitable for this book, the Gihbs fundainental equation may be cast as

dU = TdS + p d N

+ bTr (Tdu)

(1.22)

which follows from Eqs. (1.17), (1.20), and (1.21). In Eq. (1.22), “TI-”r e p resents the trace of the product of the matrices representing T and du (see Appendix A . l ) . As we saw in Sections 1.2.1 and 1.2.2. T and du can be 41f d N < 0, dn’chrm system.

>

0; i.e., the snvironnient. does work on the thermodynamic

12

Gibbs fundamental equation

rcprmerited by syrnrnct,ric 3 x 3 niittriccs, riarricly

(1.23~~)

da

=

(

dun dux, duxz dnxy dmyy d n y z ) duxz day, durn

(1.2311)

As we also showed in Sections 1.2.1 and 1.2.2, both tensors have a lucid physical interpretation. Coinpoiients rap of the stress tensor r may be perceived as the P-compoiicnt of the force acting on an a-directed area; likewise, diagonal components of d a account for (infinitesinial) conipressiond/dilational strains acting oil t,lic systcmi, whcrcas off-diagonal coinponents represent various shear strains. By convention, ra0 < 0 if the force acting on the a-directed area point outward. In EQ. (1.23a), the factor 51 arises for off-diagonal eleiiierits of r to give the correct. magnitude of shear contributions to t8heniechanical work.

1.3.1 Bulk fluids Let us apply the above general formalism to two simple examples that are ccntral to this hook chaptrr. namcly that of a hulk fluid and a fluid confinrd to a slit-pore (see Sections 1.3.2 arid 1.3.3). In both cases, we take as llie reference system a rectangular prisin of volume 1'0 = sxosy~.sz~. where a bodyfixed coordinate system is employed such that the faces of the prism coincide with the planes I = fsXo/2. y = f s y o / 2 , a i d 2 = fsTB/2. If the instrained system is exposed to an iiifitiitesirnally sinall coinpressional or shear strain, Vo V = sxsysz. This implies that a mass element originally a t a point rg in t,hc iinstrained systcm ch
T

= Dro

(1.24)

where the transformation is effected by the matrix

(1.25)

13

Thermodynamics of confined phases

From Eq. (1.1) u'c' tlmn fiiid (1.26) such that from the definitioii of expression

t7

given in Eq. (1.6) we have the explicit

At this point it is important to realize that by definition bulk fluids are hornogcncoiis on ,account of thr ahscncc of any cxtcrnal fialds; that is, thc tlciisity of a bulk fluid is spatially coiistmit.. This iiriplics t,hat its propcrtics are txanslationally invariant in all t,hree spatial dimensions. Hence,

where the scalar Tb represents the cornpressioiial (dilational) stress exerted oil the hulk fluid (in any spatial direction) and 1is the unit tensor. Moreover, to preserve the isotropy of tha systcin [scc Eq. (1.28)] under compressional (dilahiial) stfrailis, these cannot be iiideyeriderit but inust satisfy (1.29) where q, is defined through t.he expression t 7=

similar to

7h

ob

-1 3

(1.30)

in Eq. (1.28). From Eqs. (1.21) and (1.29). we therefore have

which also defines the bulk pressure through the relation q, = -Pb. From Eqs. (1.17) arid (1.31). we finally arrive at the well-known form of the Gibbs fundanient,al equation for homogeneous, isotropic bulk fluids, namely

dU

=TdS

+ p d N - PbdV

(1.32)

Gibbs fundamental equation

14

1.3.2

Slit-pore with unstructured substrate surfaces

Tlie tliscussioii of bulk phwcs in the preccding scctioii is, of course, preserited solely for didactic reasons because it demonstrates that the somewhat niore involved formulatfion of mecha.iiical work in terrns of stresses and conjugate strains [see Eq. (1.21)] leads t.o well-known textbook results like Eq. (1.32). Hence, it should serve to help the reader gct familiar with the current treat.ment of mechanical work, which turns out t,o be particularly useful for conf i n d fliiids as wc shall dcmonstrat.r now. In this section we are considering one of the systems of key interest in this work, namely t,hat of a nanoscopic &-pore with chemically homogeneoils Substrate surfaces. For simplicity we disregard the atomic structure of the solid sutxtratc such that the external field exerted on the confined fluid from the siit)strat,c-surface. depc!nds o d y 0 1 1 t h clistaiicxr of it IlliLq, c:l~~iier~t In t,his case, the fluid confined t y the solid surfaces is anisotropic on account of the external potential represented by the pore walls. Assuming these walls to be parallel with the I y plane, the confined fluid is inhomogeneous along the z-diredion; that is, its local density depends on the position ( z ) with respect. to the confiiiing planar substrate surfaces. However, the fluid is honiogeneous arrow all planes parallel with the siibstmtc sllrfilces (at, different locations 2). Hmc:c:, in two t1irnc:nsioiis ( x ilii(1 u), isot,rop-yis prc:sc:rvc:d dt:spit1c confinement. In other words, compared with the previously discussed bulk, the confined phase has lower syminehy. However, note that like the bulk, the ciirrent.ly described confined phase cannot, be exposed to a shear strain because the external field depends only on the distance of a mass element from the substratte surfaccx The reduced syminctry is refie d by the relation

(1.33) which replaces Eq. (1.29) so that,

where A,-, = s,,-,s,,,-, is the (constant) area of the z-directed face of the unstraiiied reference system; likewise, A = A,-,q represents the area of the z-directed face in the strained system. Because of Eqs. (1.33) and (1.34), the

Thermodynamics of confined phases

15

where we employ the symmetry of the system to define

( 1.36a) ( 1.36b) and therefore have [see Eq. (1.22)]

dU = TdS + pdN

+ TISZodA+ 71Aodsz

(1.37)

Equation (1.37) is the Gibbs fuiidariieiital equation specialized to a fluid confined to a slit-pore with chemically homogeneous, (infinitesixnally) smooth substrate siirfaces.

1.3.3

Slit-pore with structured substrate surfaces

The next, slightly more complicated situdion concerns a fluid confined to a nanoscopic slit-pore by structured rather tliaii unstructured solid surfaces. For the tirile tieing, we shall restrict the discussion to cases in which the symmetry of' tlic cxt.criia1 field (rcprescntcd IJY t.hc substrabcs) prcscrvcs translational invariance of fluid properties in one spatial dimension. An example of such a situation is depicted in Fig. 5.7 (see Section 5.4.1) showing substrates endowed wit,h a. chemical struct.ure that is periodic in oiie direction (2)but, quasi-infinite (i.e., macroscopically' large) in t,he other onc (y). A new feature entering the picture is that fluids confined by substrates of the kind illustrated by the sketch in Fig. 5.7 can be sheared in addition to mcrc i:omprcssioii (dilatation). This is bc!c.ai.wctho rclativc: alignmcnt of the suhstmtes niatlters on account of the periodicity of their structure in the x-direction. 111 this case, t,hc transformation matrix assumes the form

so

that.

Equilibrium states and thermodynamic potentials

16 a i d thcrcforc [scc

Eq.(1.1)]

Thus, cr asslimes slightly more complicated forms, natnely

u=(

(SX

-

sxo)

0

/sxo

0

(sx

- sxo)

fl /sxo

0

0 SXO /szo

%o / sa, 0

(1.41)

(sz - szo) /szo

where 0 5 o 5 $ is a diinensionless parameter specifying the relative aligninent of the two substrates in the m-lirection. The substrates are perfectly dignctl for (b = 0, whrrcas thcir misdignmcmt, is maximimi for ( Y = $ bccaiisa of the periodicity of the chemical pattern. Note also that the symmetry of the coiifined fluid is once again reduced with respect to the case discussed in Section 1.3.2. This is reflected by the inequality Txx

# Tyy # Tzz

(1.42)

With Fqs. (1.41 ) tho spacialixcd Gihhs fiindamcntd aquation Isre Eq. (1.22)] can be written i-w

where Aao denotes the area of the a-directed fare of undeforined reference systein.

1.4 Equilibrium states and thermodynamic potentials 1.4.1

Conditions for thermodynamic equilibrium

To this point we have been concerned with various expressions for mechanical work depending on the symmetry of the thermodynamic system. A key issue in this regard was the unstrained reference system characterized by quantities like s d , A d , and 1%. By external agents (i.e., by exposing the system to mechanical or chemical work), the reference system may be transformed into a strained system that may or may not be in a statmeof thermodynamic

Thermodynamics of confined phases

17

Figure 1.4: Sketch of a fluid lamella (shaded area) whose faces in x-, v-, and t-direction can be moved independently in the direction of the double arrows. As an example we show states of the lamella differing in strain in the z-direction such that the right plot shows an expanded lamella (relative to the plot on the left side) of different width in that direction.

equilibrium. Thus, it is the purpose of this sect.ion to develop criteria that allow us to identify such equilibrium states. A key issue in establishing such criteria is to distinguish between the thermodynamic system and its environment and to specify the interaction mcchanisni between the two. In this case we s l i d take as the system a finite lamella of the (infinibely large) confined fluid, regarding the (infinitely large) remainder of the lamella as its environment. The faces separating the lamella from the environment can be moved as indicated in Fig. 1.4 by the double arrows. The lainella is therefore assumed to be coupled to its environment thermally, materially, and mechanically. Criteria for thermodynamic equilibrium of bhe lainella can be established by viewing it plus the environrncnt AS X Iisolated siipcrsyst,c:ni. That. is, if tonipcratiirt:, charnid potential! and strains are to be maintained const,ant by means of external agents (i.e.! virtual “pistons”), we wish t o find the appropriate thermodynamic function attaining a global minimum when the lamella is in a stat,e of thermodynamic equilibrium. Because the supersystem is isolated and rigid by supposition, its entropy miist, he constant in this eqiiilibriiim st,at,s and its internal energy miist be iiiiriirriurii according t o thc Second and First Laws, respectively. Hencc, an infinitesimal virtual transformation 6 that, would take the supersystem from this state must satisfy

6(U+U’) 2 0 6(S+S’) 2 0

(1.44a) (1.44b)

Equilibrium states and thermodynamic potentials

18

whcro uiiprimcd and prirncd quantitics rcfcr to thc lairiclla and its ciivironment, respectively, and the equal sign holds if the transformation brings back the system to the original equilibrium state. It is important- to note that Eqs. (1.44) constitute not-hing but. special versions of a general ext.rcmum principle applying to many different. problems and fields in physics. Examples are Gauss' principle of least constraint [ 131 or Hamilton's principle Irading to thc Lagrangian cqiiat,iow of motmionin c1;tssic: mwharrics [ 141. From Gibbs fundamental equation [see Eq. (1.22)], it follows that-

1 - --6N P bS = --bU T T

6s'

-

1

-6UIT'

P' --6N' TI

-

vo -Tr

T

-

(766)

v,l ( 7 ' 6 ~ ' ) -Tr T'

(1.45a)

(L.45b)

The supersystem is rnaterially closed such that

bN = -6N'.

(1.46)

and it is assumed rigid so that \/,at7

= -I/dba'

(1.47)

also holds. Because tohesupersystem is in a state of thermodynamic equilibrium, we have from Eqs. (1.44), (1.46), and (1.47) t8heexpression

As the transformations SU, S N , and ha arc independent, and arbitrary, Eq. (1.48) caii oiily bc satisficd if

(1.4%) (1.49b) (1.49c) hold simultaneously. These latter cxpressions constitute conditions of thermal [see Eq. (1.49a)], material [see Eq. (1.49b)]. and inechanical equilibrium [see Eq. (1.49c)l that inust be satisfied if the confined lamella is in therinodynamic equilil>rium.

19

Thermodynamics of confined phases

1.4.2

Thermodynamic potentials

In general, any transformation of the thermodynamic state of the confined lamella is associated with a variation of a characteristic function assuming a global niiniiniim if the laniella is in thermodyiiainic equilibrium. However, thorr arc various ways in which such R transformation may he cffcctcd in practice. HCIIC'C,tlic prc-cisc form of tlic charwtcristic furictiori rriay vary between different (experimental) situations. Below we slia.11 briefly discuss the characteristic functions that are key t o the analysis of confined fluids.

1.4.2.1

Closed system, fixed strains

Wc begin with a laniella oii wliirli iio comprcissiorial or shear stresses are acting: i.e., no mechanical work is exchanged between the lamella and its environment. Hence, ha = ha' = 0 , where 0 is the zero tensor. In addition, the lamella is separated from its environment by a fictitious, impermeable, diathermal membrane so that the number of molecules that constitute the lamella remains fixed; that is, 6 N = 6N' = 0. In other words, only heat is cxcliangcd hrtwcm t h lamella and its surroundings. Under thew conditions, we have froni Eqs. (1.44)?(1.45b), and (1.49a)

CU i + CiU'

2

iiU

+ T'o'S' -= (5 (U- T S ) G 6 3 2 0

(15 0 )

which defines the (Helmholtz) free energy F. Equation (1.50) statesl that nntfcr conditions of fixcd N and a,cqiiilihriimi statts of the lamclla arc cliaracterizecl by a global riiiriiiiiuiri of 3.

1.4.2.2

Open system, fixed strains

Consider now a situation where the lamella is still iiot subject to ariy mechanical work but is perriiitted to excliaiigc heat and matter with its eiivironmcnt. Thus, as in the previous exaniple we still have 6a = 6a' = 0 aiicl it follows froin Eqs. (1.44), (1.45b),and (1.49a) that

6U + 624'

= bU

+ T'6S' + p'6N'

= 6 (U- T S - p N )

652 2 0

(1.51)

whcrc 52 is the grant1 potential. Equation (1.51) trlls 11s t,hat, iindrr conditions of fixed compressional (dilational) and shear strains, fixed temperature, and fixed chemical potential, the equilibrium state of the confined lamella is characterized by a global niiriiinuni of R. Expeririiental situations where R is the relevant, thermodynamic potential are frequently encountered in the sorption of gases in nanoporous materials because the pore gas is coupled both thermally arid materially to a bulk reservoir (see Section 4.2.1).

20

Equilibrium states and thermodynamic potentials

Figure 1.5: Sketch of a fluid lamella (shaded area) sheared in the x-direction, where a shear strain of frr.sxo/2 is applied to the upper and lower substrate, respectively.

1.4.2.3

Closed system, nonvanishing strains

Last but not least we analyze a situation in which the laniella is still separated from its environment by a virtual impermeable, heat conducting membrane. However, this time we allow the confining substrates to move along the 2-direction thereby exert,ing a nonvanishing compressional strain of the enviroiirnciit ori the lamella. In addition, thc relative aligrirnerit of thc solid surfaces in the s-y plane may be alt,ered by external agents. For concreteness we assume the solid surfaces to be structured as shown in Fig. 5.7,which shows that only the alignment in the z-direction can be altered. This situation, depicted schematically in Fig. 1.5,is frequently encountered in experiments employing the so-called surface forces uppamtlls by which mechanical properties and the phase behavior of nanoscopic films can be investigated (see Section 5.3.1).Thus, thc strain teiisor can be expressed as

0 0 6uxz b = (0 0 0 ba,, 0 f5um

)

(1.52)

It is then easy to verify that the mechanical work term is [see Eq. (1.23a)l

TI'(T6U) = Txz6Uxz+ T,6U,

As we still liavc 611' = 6 N t

(1.53)

= 0, it is iinrricdiatcly clear from Eqs. (1.44),

(1.45b),and (1.53)that in this case

6U + 624'

6U + Tt6St + T : , ~ u ~+,TLSUL = 6 (U- TS - rx,aXz -T,o~) E& 2 0 =

(1.54)

Legendre transformation

21

whcrc G is a gc1it:ralizcd Gibbs potcritial. If tht: lamella is in a state of thermodynamic equilibrium for fixed N , T,T,,, and T ~ B , attains a global minimum. To arrive att the second line of Eq. (1.54), Eqs. (1.47), (1.49a), and (1.49~)have also been invoked. Note that, depending on the set. of strains controlled, several such Gibbs potentials exist.. They all have in common that, besides N and T , a subset of stresses is also controlled during a thcrmodynamic t.ransforination of t.hc confined lamclla. The above examples showed that. for a given set of variables, functions siidi a s U ,F ,a, or 4 may tw dofincd complying with t8hcgeneral cxtmmiim principle; that, is: fur a given sr-t of variables, these functions are rnininiurri if the system is in a state of thermodynamic equilibrium. Thus, by analogy with mechanica.1 systems, fiinctioiis such as U ,F,0, or G are frequently referred to as themodynam,ic potentials and their various sets of parameters are called natural zwiables of the functioii in question. According to the discussion in Section 1.3, the set of natural variables of the internal energy is {S,N , 6). .It needs to be emphasized at this point that one could, of course, express each thcrmodynamic pot.cntia1 in tcrms of differcnt sets of (nonnatural) variablts. The irriiricdiatc coriscqiic~i(:cis that, tlic..rinodyiiitrnit: potentials would not necessarily att.ain a minimum va.lue if the system is in a state of thermodynamic equilibrium. This point is important to realize because it implies that the set of natural variables of a given thermodynamic potent,ial is distinguished arid unique among other conceivable sets of variables.

1.5

Legendre transformation

Bwaiisc of thc disciission in Scrt,ion 1.3 thc qiicstion arises: Is thcrc R forr r i d way to switch bctwwn various tlicririodyriariiic potentials by changing the set of nat,ural variables? The answer to this question is particularly important, because controlling, for instance, the set { S, N,a} in a 1a.boratory experiment might be cumbersome at best, if not at all impossible, because it requires control of the entropy during a thermodynamic transformation. However, there is no direct way of measuring S directly nor is there any clcvicc to control it. Wit,hout, elahorating fiirt.hcr on this issiic, t,hc rcadcr will surely agree (at, least int,uitively) that it might at least, be hard t80think of a device that permits one to cont,rol S experimentally. Thus, one would like to transform variables such that. in our example S is no longer a natural variable but instead beconies a dependent function of another set of tliernic+ dynamic variables. The formal way of effecting such a change of variables is the Legendre trarisforrnation that we introduce formally in Appendix A.2.

22

Legendre transformation

Applyiiig tlic conccpls of Lcgciidrc trarisfoririatioli to thcrrnodynamic potentials, we realize that. Gihbs’ fundamental equation [see Eq. (1.22)] can be rewrit,ten as

hccnilsc dU is an cxart diffcrcntial. In Eq. (1.551,

by the matrix

V,U

can be reprcscritd

where we used shorthand notation “{ .} \en$’’to indicate that,, upoii performing the partial differentiation, the set of variables {S,N} is being held fixed together with all strain tensor elenients except for en(j. Elements of the matrix in Eq. (1.56) m a y bc idcntifid with thcrmodynamic forces acting on tlic: confi~icdlainclla oii accouiit of comprtssional (dilational) and slicar strains exerted on it. by its environment. The set of natural variablcs {S,A;, a} of U in Eq. (1.55) may thus be perceived as the set {xk} in Eq. (A.4)of Appendix A.2. Moreover, by comparison with Eq. (1.17). it is also clear that- the analogs of the {f:} in Eq. (A.4) are given by

(1.57a) (1.57b) (1.57~) In addition, we rcdizc from Eqs. (A.4) and (1.57) t,hat the grand pot,cntial defined in Eq. (1.51)can be ohtained its a Legendre t,ransforlri of U expressed as $2 ( T , / Ja) , = U ( S ,N , a ) - T S - /1.N (1.58) such that its exact. differential is given by

dR (T,p?a)= d (U- T S - p N ) = -SdT - Ndp + VtTr ( T d a )

(1.59)

Homogeneity of confined phases

23

whcrc Eqs. (1.55), (1.57a), and (1.5714 have also been cinploycd. Thcsc considerations can easily be extended to other kinds of thermodynamic potentials of interest. If the Legendre transformation is to be performed in order to replace a (sub)& of strains by conjugate strwses. one needs to realize that the operators “d” and “’I”’commute.

1.6

Homogeneity of confined phases

1.6.1 Mechanical expressions for the grand potential At an elementary level, one of the dogmas taught to almost every chemist is that in thcrniodynaniit.s only tliffi:rcnccs bct,wtx:ii t hcrmotfynarriic potcntials at various state points matter. This is essentially a consequence of the discussion in Section 1.3 where we emphasized that exact differentials exist for thermodynamic potentials siich as U ,S , T , G. or 52. These potentials therefore satisfy Eq. (1.113). However. one is frequently confronted with the problein of calculating absolute values of thermodynamic. potentials theoretically. An rxamplc is the dcterrninatiori of phmc qiiilihria, which is one of thc key issues in this book chapter. hi this coiitcxt a thwrcin associated with tsheSwiss niatheniatician Leonhard Euler5 is quite usehd. We elaborate on Euler’s theorem in Appendix A.3 where we also introduce the notion of horriogetieous functions of degree Ic.

1.6.1.1 Bulk phases Applying Euler’s ideas to the thermodynamic potentials introduced in Section 1.4.2, one realizes that homogeneous functions of degree 1 are of particular interest in the context of equilibrium thermodynamics [see F4. (A. lo)]. For cxamplc, considar thc grand pot,cnt.ial whosc cxact diffcrcntial is given by Eq. (1.59). For thc special rase of a honiogeiieous bulk phase, it follows that at constant T and 11 dR ( V ) = TbdV (1.SO) brxsuse of the special foriri of the mechanical work Lcrni [scc Eq. (1.31)]. OII account of the homogeneity of bulk phases, it is immediately clear that if we ‘kntiard Euler (1707-1783), Professor of Matherriatics in St. Petersburg (Russia). contributed in an outstanding way t a a variety of fields in niatlieinatics. His philosophical writ.ings had a great impact on the Gernian philosopher Irnmariuel Kaiit (1724-1804) who was Professor of Logic and Metaphysics in his lio~netowi~ Iiiinigsberg, which lie never left during his entire lifetime. With his book Krs‘tik der Reinen Vernunft, Kant laid one of the foundations of modern philosophy.

Homogeneity of confined phases

24

cliaiigc the volunic of our systcm by a factor X > 0 at coristarit T arid 11, the expression (1.61) 12 (T. p, AV) = AS1 (T, /I,, V ) must be satisfied. which tells us that R is a homogeneous function of degree 1 in V according to the definition given in Eq. (A.8). Ransforming variables in Eq. (1.60) according to V r/. = AV (A > 0) and rrplacing in Eq. (A.10), x, = V arid f = S2, wc' obtairi

-

(1.62) as a "mechanical" expression for the grand potential in terms of the bulk s t r w (prrssiiro). Equation (1.62) ran hr foiind in most, stmdard tcxt8son bulk phases. However, almost always it is not8 clearly stated that 51 still depends on T and p7 which is iinportant later on when we discuss the GibbsDuheni equation.

1.6.1.2 Slit-pore w i t h u n s t r u c t u r e d substrate surfaces A sonicwhat more coinpliratad sitiiiltion is cncoiintorcd for slit-porcs with (iiifixiitcsixnally) smooth (lioir~ogoiicoiis)sulstratcs. As wc cxplainctl iii Scction 1.3.2, the confined fluid is h.umqeneuw across each a-y plane located a t different positions L relative to the confining surfaces. Thus, from Eqs. (1.37) and (1.59) we find ~

Applying the honiogciieity argument, we realize that

52 (T, 11, X A , s,) = XR (T,p , A, s,) whcrc, of co~irsc,T , / I , , aiitl and (1.63). one obtains

s,

(1.64)

11wt1to bc hcltl (:onstant. Frorii Ei4s. (A.10)

As,nq = R

(T, p, A. s,)

(1.65)

as thr itnahg of Eq. (1.62). In Q. (1.65) it is important to rcalizc, that, despite s, being an extensive variable, 62 is riot a homogeneous funrtioii

of s, becaure of the inhomogeneous nature of the confined fluid in the Ldirection. The notion of extensivity of a thermodynamic function does not necessarilv imply that a thermodviiamic potential depending on this variable is homogeneous of degree 1 in that variable. In otther words, extensivity of a thermodynamic variable is a ncc(:~surycondition but riot suficient to

25

Thermodynamics of conflned phases

conclude that a tticrmodynaniic potcritial is a 1ioInogew:ous fuiiclion of degree 1 of that variable. Consequently. increasing s, by a factor of X does not cause nanoscopic slits to imbibe X times the amount, of matter it originally accommodated. We t,hus readily conclude that, for R to be a homogeneous function in any of its extensive variables, it, is necessary that the system is h,ornogeneous in a t least one spatial direction. In other words, system propertics niiist, lw translationully invcmnnt in ilt lcnst, on(’ spatial dinicnsion.

1.6.1.3

Slit-pore with structured substrate surfaces

Consider next the system depicted schematically in Fig. 5.7, namely a fluid confiiied to a slit-pore with cliemically striped walls. From Eqs. (1.59) and (1.43). wc obt,ain

dR = -SdT- Ndp+ AxOTXxd.~, + i / l , o ~ ~+AzO~zzd~z+AzO~xzd d~~ (C Y S ~ O (1.66) ) On account of the infinite length of the chemical stripes in the y-direction, system properties are traiislatiorially invariant in that direction. Hc~icc,J2 is a homogeneous function of degree 1 only in sy and therefore

0 ( T .p, sx, As,,

sz,

asxo) =

(T,p, Sx, s y , sz, asxo)

(1.67)

Eqimt.ion (A. 10) then gives .4yosy7yy

= f-2 (T. 11, sx, s y , $ 2 , a s x o )

(1.68)

as the desired mechanical expression for the grand potential in the current raw. Moreover, the reader riiay realize that, because of the periodicity of the substrate structure in the r-direction one may write

for 1 E N,which at first, glance may look as if R may also h pcrccivcd as a lioinogerieous functioii of degree 1 iii sx However, ou account of the structural periodicit!y of the substrates. 1is restricted to integer values unlike its counterpart X in Eq (1.67). In fact, cont-emplatingthe definition of partial derivatives. the relationship

(1.70) canriot be made arbitrarik small for arbitrary values of z. This is because, for integer values of X = A, AX = 6 1 cannot be made arbitrarily small and

Homogeneity of confined phases

26

therefore tlic riglit side of Eq. (1.70) docs riol exist. HCIICC,iii this casc the grand potential R T, 11, Xs,,-sy. s,! as.,> does not satisfy Eq. (-4.10)because

(

the partial derivative of R with respect to i s , is ill-defined. Thus, whcthcr or not R closcd mechanical cxprcssion for R cxists for a confined fluid (as well as for ariy other systeni exposed tto an external field) depends critica.lly on the nature of the external poteiiial. It must be such that the confiiied fluid is homogeneous in at least one spatial direction. In other words, the conclusion that R is a homogeneous function of degree 1 in one or inore of it,s extensive variables is inelucta.bly coiipled t o considerations of the svmnietry of the external potential representing the confining substrate.

1.6.2

Gibbs-Duhem equations and symmetry

Tlic cxistcncc of Inetfiaiiical cxprcssions for the grand potential introduces an additional equation for S2. Take as ail example Q. (1.62) whose exact, differential may be cast as dSl (T./ I , v ) = TbdV

+ VdTb

(1.71)

If this equation, which is valid for bulk systems. is combined with Eq. (1.59), we arrive at a secalled Gibbs-Duheni equation, namely 0 = S d T + Ndp + VdTb

(1.72)

which states that the t8hreeintensive variables T, p ? and q., cannot be varied independently but, are related to cadi ot,her through an (a yriori unknown) equation of state 71, = f (p.7'). This is standard textbook knowledge. However, as we shall deinonstrate shortly, Eq. (1.72) is by no means gcwral as far as confincd systoins (or aiiy systcin cxposcd to an external field) are concerned. Take as an example a fluid confined to a nlit-pore nith homogeneous (infinitesimally) smooth substrates. For this system, we derived an expression for 12 in Eq. (1.65). Differentiating it we obtain

dR (T,I / ,A, s,) = As,odTll

+ Tl1sddA

(1.73)

Conibinirig this latter expression with Eq. (1.63) yields yet another GibbsDuheni equation of the form

0 = SdT

+ Rrd//, + A.q,fidq1 - TIAOdRz

(1.74)

that tells us that an equation of state 1 = f (T,p, s,) exists in which one intensive variable (e.g., 11) can be expressed in ternis of the other two (T, p ) and one extensive variablc (s,) .

Phase transitions

27

If tlic cxtcrrial poteiitial is noricoiistaiit across thc z-.t/ planc but varics periodically along the x-axis,say. as the one describing the chemically striped substrate surfaces depicted in Fig. 5.7, the symmetry of the fluid is reduced even further. This causes the equation of state to depend on even more parameters as in the previously discwssed case. This can be realized from Eqs. (1.66) and (1.68), which permit us to derive yet another Gibbs-Duhern equation. namcly 0 = S d T t N d p - Ax"Txxdsx- AyO.sydTyy - AzOrz,dsZ - Ax07,,d

as,^) (1.75)

which suggests the existence of an equation of state of the form T~ = ryy(T?p, s,, s, as,") that now depends on the periodicity of the chemical stnictiire (i.e.,the widths of t,hc chemically distinct stripes in thc x-direction) through sx as well as on their lateral alignment (i.e., the shear strain as,") (see Fig. 1.5). Hence, symmetry considerations play an important r6le in the therniodynamics of confined fluids similar to bulk solids (see Chapter 13 in the book of Callen [12]). We shall return to this issue in Section 5.5 where the symmetry of the external potential representing a confining solid surface is such that the grand potcntial is not, a hornogcacoiis function of dcgrcc 1 in any of its cxtcnsive variables. The reason in this particular case is that the surface is decorakd with a chemical nanopattern of finite extcnt,, which together with the mere presence of the substrates, abolishes the translatima1 invariance of the local density in d l three spatial directions (i.e.>the homogeneity of the confined fluid). As a consequence, a Gibbs-Duhern equation does not exist, which precludes the existence of an equation of st,ate in the above sense as well.

1.7 Phase transitions Within the scope of this book, phase transitions play a prominent role. From a thermodynamic perspective, phase transitions can be discussed most, conveniently on the basis of the grand potential R introduced in Section 1.4. There is a twofold reason for this distinguished position of the grand among other thermodynamic potentials: 1. With regard to confined systems, one is often confronted with situations in which the confined phase is in thermodynamic equilibrium with a bulk reservoir with which it exchanges heat and matter. Under these conditions, it was shown in Section 1.4.2 [see Eq. (1.51)] that R is the relevant thermodynamic poteiitial to identify equilibriuiii states of the system of interest.

28

Phase transitions 2. As was dciiionstratcd in Section 1.6.1, a rricchaiiical expression for Sl can be derived in many cases of interest. This permits access to (absolute values of) R from stress tensor components that can be calculated6 or controlled experimentally. This also holds for other thermodynamic potentials where hr has been replaced by 11 as a thermodynamic state variable via a Legendre transformation (see Section 1.5).

The objcctivc then is l o identify stable pliascs in the context. of pliasc traiisitions on the basis of variations of 52. We will concentrate mostly on discontin-

uous (i.e., first-order.) phase transitions where in addition the participating phases will always be fluid (i.e., gas or liquid-like). In general, two phases cy and 13 undergo a discontinuous phase transition at some fixed temperature if their grand-potential density w (see below) satisfies the conditions (1.76a) (1.76b)

where 11'' denotes the chemical potential a t coexistence between phases i and j at. a given tempera.t.ure T. In other words, at a discontinuous phase transition, grand-potential density curves of different slopes intersect,. The following discussion is therefore devoted to an ixivestigation of conditions for the existence of such iritersect,ions and their relation t o measurable thermodynamic qiiantit.ies. The siiuplcst case that we shall bc discussing licrc in some detail is that 01 a fluid confined to a nanoscopic slit-pore with homogeneous (infinitesimally) smooth substrate surfaces. For this prototypical model, it was shown in Section 1.6.1 t,hat a mechanical expression for the grand potential exists. However, in what, follows, it is more convenient to fociis on the grand-potential density rather than on R itself. The former is defined through the relation (1.77) which has a niiniher of important, properties. For example, from Eqs. (1.63) and (1.77): it follows that. (1.78) wlicrc p is thc (r~ieari)density of the confincd fluid in tlie strained systcrn. Hence, for a given temperature and geomet,ry of the fluid lamella, w is a monotonically decreasing function of the chemical potential because p > 0. 'By, for example, statistical inechanical methods (me Chapter 2).

29

Thermodynamics of confined phases

Anothcr irnyorlant quantity in thc contcxt of cliscoxitiiiuous phasc transitions is the isothermal coinpressibility q .For this system, 611may be defined starting from the relevant Gibbs-Duhem equation [see Eq. (1.74)).which r e duces to N d p = -As,odTll (1.79) because T and s d are supposed to be constant. Under these conditions, both 11 and 711are solely fuiictions of N . Hence,

(1.80a) (1 3Ob) Substituting these expressions into Eq. (1.79) one obtains

- -A 2-s a -

N

(2)

'4Sd 1 - -T,N.s. N KII

(1.81)

because d N is arbitrary. This expression can be rearranged to give (1.82)

because both p and the isothermal transverse compressibility ~ 1 are 1 positive definite. Together, Qs. (1.78) and (1.82) d o w us to conclude that the function w (p) (for fixed T , A , and s,) is monotonic and concave; that is: w ( p ) satisfies the ineqnality

where cl0 and p1 are two arbitrary chemical potentials for which w ( p ) exists (scc hclow). The right, sidc of &. (1.83) rcprcsciits the secant, to w (p) between and p1. For fluid phases 2 and j differing in density, one realizes from Eqs. (1.78) and (1.82) that the associated curves w (p)will have different slopes and curvatures. Assuming pa < @. monotonicity of w ( p ) suggests that one and only one intersection p'j exists, which may be obtained as a solution of Eq. (1.7Ga) for each fixed value of T. In the thermodynamic limit. both curves (w'and

30

Phase transitions

T = const.

Figure 1.6: Schematic plot of thc grand-potential density w as a function of chemical potential p under isothermal conditions. The plot shows grand-potential density curva for a situatioii wherc thrw diffcreiit yliases i , j, and k are (meta)stable over certain ranges of p. Because of Eq. (1.78) their mean densities satisfy the inequality i? < < Fk. Notice that the concavity of the curves w ( p ) as predicted by Eq. (1.83) has been deliberatly ignored. Chemical potentials piJ and p pzj. However, as we shall demonstrate. .below, metastable states may exist for both phases such that the curves wLJexist above and below pij for a certain finite range of chemical potentials as indicated in Fig. 1.6. Thus, over a certain range of chemical potentials, the grand-potential density at fixcd tcmpcratnrc may tiirn oiit, to bc a rloiiblt:-valiicd fimction of p. The globally stable pliasc is then idciitificd its thc one satisfying

wa ( p i J T , ) = min 2 ( p i j ,T ) k

(1.84)

in addition to Eq. (1.76a) where we ta.citly assume that several phases axe conceivable. Notice also that

+

p' ( p * j . T )= pi (piJ!7') A p ( T )

(1.85)

where A p ( T ) 2 0 according to our definition of the relative magnitude of pi and @. Thus, at the intersection between we arid u j , the density changes

Thermodynamics of confined phases

31

discoiitminuously.This is charwtcristic: of a so-callcd “discoIitixi~ioiis”or firstorder phase transition. If, in the above equation, A p ( T ) = 0, then phases i and j are indistinguishable. This is indicative of a critical point ( T j . p : j ) . Because of the equality of pi and ~, the slope of w iand d’ is the same at the critical point [see EQ. (1.78)]. In other words, at p:’ and for T = q j , i ~ ”changes contiriiioiisly to id.Evcn though (*riti(:idph
(1.86)

t-n

where ,d is a critical exponent in standard notation [15], t E (T:j - T)/T:J, and the notation “0-” is used to indicate that t vanishes as T approaches from below. It, is thcn convcnicnt to introdiicc thc notion of a coexistancc linc 112 ( T ) as the set, of point,s {pi’,T}obtained as a solution of Eqs. (1.76a) and (1.84). We also note for later reference that one may encounter situations where a pair of coexistence lines intersects, that is; wliere ,j (pijk,

Tijk)

=& ’

(p7 T + ) = ),

( p k rJnjk)

(1.87)

is satisfied, thus defining a so-called triple point. (yijk,T*jk)at which three phases i, j, and k coexist. It therefore seeins sensible to int,roduce the notion of a phase diagram as the union of all coexistence lines (1.88) With Eq. (l.88), we conclude our discussion of phenomenological thermodynarnics of confined fluids. In Chapter 2, we shall turn to an interpretation of the various thermodynamic quantities introduced above in terms of interactions between the microscopic constituents forming the system at a molecular level of description (i.e., atoms and molecules).

Reviews in Computational Chemistry Kenny B. Lipkowitz &Thomas R. Cundari Copyright 02007 by John Wiley & Sons, Inc

Chapter 2 Elements of statistical thermodynamics 2.1

Introductory remarks

In Chaptcr 1 wo introduced tlicriiioclyiiairiicsas tho ccritral nr.c~croscopic physical theory that allows us to deal with bhermophysicaJ phenomena in confined fluids. However, as we mentioned a t the outset, thermodynamics as such does not permit. 11s t,o dra.w any quantit.at,iveconclusions about a specific physical system without taking recourse to additional sources of information such as experimental data or (empirical) equations of state based on these data. Instmd thcrmodynamics makzkcs rigorous stixt.omcnts ahout thc relotion, among its key qiiantitics such as t;cmipcrsturc, intcriial c!ncrgy, ciitrupy, hcat., mid work. It does not, permit one to calciilate any numbers for these quantities. -4s we pointed out in Chapter 1, t.he reason for this lack of predictive power is that thermodynamics as a theory of the macrciscopic world does not have any concept. of t,he underlving microscopic world goveriied by entities like electrons, atoms, or molecules, say. In fact, it was one of the great intellectua.1 cachievenientsof the nineteenth centiiry t,o realize that, for example, heat- as a t.1icnnodyiiamic quantity is irit iuiatcly couplcd to tlic rriot.ion of riioleculcs (see, for exa.inple, R.ef. 10). However: if one acceptasthe hypothesis that a.11 phenomena perceived by our physical senses are a result of the interaction and consequently the motion of microscopic constituents~we are imrnediately confront.ed with three central problems: 1. Any niacroscopic piece of matter is composed of an astronomically large number of molecules, which is usually of the order of loz3. How do we follow the spatio-temporal evolution of such a largc number of objects

33

34

Introductory remarks

given t,hc ovcrw1ic:lining aiiiourit of inforrriation? 2. Suppose we could, a t least in pn'nczple, solve the (classic or quantum mechanical) equations of riiotion for such a huge number of microscopic entities, we would still need t,o specify initial conditions of the systeiii, which, again, IS a task of ovcrwhrlniing complcxit,y in practice.

3. In Chapter 1 we saw t,hat only a rather small set of variables is required to completely specify the thermodynamic state of a system. How is the enormous reduction of information taking place as we go from a very large number of microscopic const.itucnts and thcir motion in space and t h i c to thc iiiarroscopic cquilibrimn t)chavior of nisttcr? Because of t.he substantial reductioii of information associated with the transition between the micro- and niitc:rnworltls, one may readily conclude that the ovcrwhclming amount, of informathi biiricd in the dct,ailcd description of thc spati* tcinporal cvolut,iori of ik niaiiylparticlc! syst,cIri rriilst bc largcly irrelevant for the t,hermodynamics of an equilbrium system. In fact, one may suspect that it is the on-average behavior of t+hemarly-particle system what matters for its macroscopic properties: which, in turn, iminediately suggests to employ statistical concepts. The extent to which statistical concepts enter t,he picture as we go from the micro- to the nia.croworld is not a t all at, our disposal. For example, quantum mechanics as we know it today teaches us t,hat it is impossible in princzple to obtain complete inforniation about a microscopic entity (i.e., the precise and simultaneous knowlcdgc of ail clectrori's location and rIIoIIic1ituIi1, say) at any instant, in time. On account. of Heisenberg's Uncertainty Principle, conjugate quantities like: for instance, position and momentum can only be known with a certain maximuni precision. Quantum mechanics therefore already deals with averages oiily (i.e., expect,ation values) when it comes to actual measurements. The key then is to somehow calculate the probability with which a specific quantum state contributes t.o the average values. As far its thermal system in thennodynamic equilibrium are concerned, this is the central problem addrcsscd l y statistical t,Iicrnioclyna,mics.Wc thcrcforc begin oiir disciission of some core elements of st,at,isticalthtx-modynaniics at, the quantum level but will eventually turn to the classic liinit. because the pheiiomena addressed by this book occur under conditions where a classic description turns out. to be adequate. We shall see this at, the end of this chapter in Section 2.5 where we introduce it quantitative criterion for the adequacy of such a classic description.

Elements of statistical thermodynamics

35

Two anlditiorial coriirncribs apply at, this point. First,, thcrc is a cona:ptual difference between the probabilistic element,in quantum and classic statistical physics. For instance, in quantum mecliaiiics, the outcome of a rneasurernent of properties even of a. single particle Cali be known in principle only with a certain probability. In classic mechanics, on the other hand, a probabilistic element is usuallv introduced for many-particle systems where we would in principle tw able to specify t,hc state of the systmn with ahsolnt,cccrt;ri;ity; however, in practice, this is not possible bccausc wc’ arc dcdiiig with too many degrees of freedom. R.ecourse t:o a probabilistic description within the framework of classic mechanics m&t therefore be regarded a nia.tter of mere convenience. The reader should apprecia.t.ethis less fundamental meaning of probabilistic concepts in classic as opposed to quantum mechanics. However, once this fundamental difference in the role played by probabilist.ic clcmcnts in qiiantiim and clas~irmechanics is arrcptcrt, thcrc is esscwtially rio operational t1iffi:rcncc iii tlic: way in whidi oiic would c:orriputc any property of a thermal many-particle svst0emregardless of whether the sysfem is treated at the quantum level. Oiir discussion below will reveal that in any case the key quantity required is the secalled partition function, which conveys information about the probability with which a particular (quantum or classic) microstate is realized. The second comment, applics t,o t,hc use of quant,iim statist#icalphysics iii t.his chapter. As is well kriowri thew arc two versions of quaiituin statistics, namely Bose-Einstein and Ferini-Dirac st.atistics depending on whether the spin of the quantum system is integer or half-integer. This difference, which manifests itself in symmetry properties of the wave function (see Section 2.5.2),is of importance if one wishes to calculate thermal properties of a specific many-particle quantum system. For most of the treatment here, t,hc diffcrcnt, statistics arc not an issuc?. What is important, however, is that, regardless of tlic spcxific vcrsiori of quaiitmii statistics, we arc always dealing with a discrete spectrum of eigenstates. The discreteness of this spectrum simplifies the formulation of statistical thermodynamics considerably as we shall see below. Classic systems do normally not have a discrete spect.rum of microstates. However, there are exceptions like the celebrated Ising model of a magnet. In the king niodel. one is dealing with a classic spin system where cach spin can attain one of two clisrrctc valiics (“spin up‘’ or “spin down’’). The Hamiltonian governing the interaction between t.hose spins is, however, purely classic. We mention the Ising model at this point because it is intimately linked to another model, which is central to the discussion in Chapter 4, namely t8hat referred to as “lattice fluid.’’ In the lattice fluid, molecules are not. permitted to niove continuously along classic trajectories but are restricted to discrete

36

Concepts of quantum statistical thermodynamics

positioris 011 S O I I I ~ lattice. hi this inodcl we may iiit.crprct the “spixi-up” configuration in the Ising model as a site on the lattice occupied by a fluid molecule, whereas the “spin-down” Configuration niay be interpreted as a11 empty lattice site. As one would anticipate, it is possible to translate between the “magnetic” language used for the Ising model and the one appropriate for the lattice fluid [lS].

2.2 2.2.1

Concepts of quantum statistical thermodynamics The most probable distribution

For the sake of concreteness of the following developments, we consider a fluid coilfined to a slit-pore such that t,he solid surfaces representing the pore walls are pla.nar, parallel to one another, arid perpendicular to the z-axis of a Cartesian coordinate system. The separation bet,ween the pore walls will bc dcnot,crl s,. In addition, thc two solid snrfaccs can he manipulatcd by exteriial agerit s rioririal to the fiuid-solid iiiterface such that s, may be altered. Eventually, these planar surfaces will conie to rest a t some equilibrium separation s,. As we shall see later in Section 5.3.1,the situation just described is akin t,o laboratory experiments in which the rheology of confined fluids is investigated by means of the so-called surface forces apparatus (SFA). Coiisidcr iiow x i astroIiomicillly largc mmibor of N - 1 identical virtual replicas’ of the original slit-pore. Each replica is capable of exchanging heat with the slit-pore through diathermal walls to maintain the confined fluid a t constant T. In a.ddition, niatter niay be exchanged between the slit-pore and its environiiient formed by the replicas. As the confining substrates may move on account of external manipulation. the slit-pore may also exchange (normal) compressional stress with the replicas. Together the N systems are forming a supcrsystoin (i.c., a stmatisticalphysical ensemble) that is assu~ncd to be closed in the thermodynamic sense, meaning that the supersystem is insulated against it,s own surromidings such that it has fixed energy E , a fixed number of molecules N , and fixed volume V. Moreover, we ilSSuine that, for the slit-pore (and therefore for all its N - 1 idnnt,ical replicas) wc can somrhow solvc the timc.-indcpcmdcnt (i.c., ’Use of the term “replica” for a virtiial system in Statistical physical erisembles should not be confused with the same t.errri used in conjunction with integral equation theories in Chapter 7.

Elements of statistical thermodynamics

37

stationary) N-puticlc Scliriidirigcr equation

where

c

i-3 = " 2 9+ r? (V;N , sz) i=l

2nr

is the Hamiltonian operator. For simplicity we assume our fluid to be composed of particles of mass rn that have only translational degrees of freedom.* In J3q. (2.2)! $, and F, are operators pertaining to momentum and position of a particle, respectively, and (FN;AT, sz) is the configurational-energy operator that can be forinally split into an intrinsic (i.e., fluid-fluid) and ~)) an external (i.e., fluid-siibstrate) contribution. In Eq. (2.1), l $ j ( ~ , ~are the quantum mechanical wave functions in the well-known representationindependent Dirac notation. The reader should realize that this treatment of the fluid-substrate interaction assiimcs that we may perccivc the siibstratc as ail external field supcririiposed onto thc iiitcrrnolccular inlcrxtioris. This is possiblc if wc disregard thermodynamic equilibrium between the fluid molecules and the substrat,e. In other words, the substrates are not thermally coupled to the confined fluid. Such a coupling may be incorporated but would make the statistical physical analysis much more involved than is needed for our current purposes. We use the not~ation E j ( N , s z ) to indicate that the spectrum of energy cigcnstatas dopands { Ej(N,szl} on the mimbcr of molrciilcs A' accommodated by the slit-pore and the actual substrate separation sZ, which niay, of course; vary such that a t any instarit in t,imt?n . i ( ~ , ~ systems ,) of t,he ensemble are in a quantum state characterized by E ~ ( N , ~n, )j (, ~ t , ~in; )a quantum state charaderized by E ~ ( N , , and ~ ; ) so on. As the spectrum of energy eigenstates obtained by solving Eq. (2.1) is discrete, we may order them such that

Henceforth, we shall tacitly ilSsume the set of orthoriormal functions t o be complete. Therefore, we know froin Eq. (2.1) the complete spectrim of cnergy eigenstatrs { Ej(N,sz)} (i.c., energy cigenvaliies of the N particlo Hamiltoriiaii). Wc r1ot.c iri passiilg that this is, of course, a formidable if not, entirely inipossible task in practice if one is dealing with a many-particle

l$J(~,sz))

'The treatment can be extended to rionspherical particles (see,for example, Chapters 6. 7, and Ref. 17).

38

Concepts of quantum statistical thermodynamics

systcrii of intcractiiig inicroscopic coiisti tucnts because of the inany dcgrccs of freedom one needs to consider explicitly. One of the simplest. cases for which a solution of Eq. (2.1) turns out to be possible (and with which each chemist should be familiar) is the part,icle in a box: which we shall briefly discuss to get a feel for the number of quantum states accessible to a macroscopic piece of rnatt+erunder conditions relevant to the current discussion. Considcr a cubic box of sidc Iciigth L containing N nonintcracting particlcs of ~iiasstn,. At, thc boundaries of the box, that is, at, plaiicsy -- 0, I,, y = 0, L, and z = 0: L, the part,iclw are exposed to a potential U = U,, = 00, which keeps them inside the volume V = 1J3 of the box. By solving Eq. (2.1) for this model, the energy eigenstates of each particle are given by A

(2.4) For sufficiently large valucs of the quantum nurnbers k,, ky, and k,, these are distributed quasi-continuously on one octant of R sphere of radius R Hence, the number of quantum states with an energy less than or equal to E maybe calculatcd from

Jm!.

where the factor of f arises because the quantuin numbers are positive semidcfinitr such that only onc octant of thc sphcre needs t o he considcrctl. Frorii Eq. (2.5), wc niay mtirriatc the iiunibcr of quantuin statw within a small energy interval AE from the expression

Ad,(E,AE)

# ( E + AE) - d ( E )

+

whcrr wc aswmcd AEIE << 1 such that, we can oxpand (1 A E / E ) 3 / 2 in a Taylor seriev truncated after the first-order term. To estimate A+ we take as an example a particle having thermal energy E = j k ~ Ta t room teniperature T = 300K, where k~ is Boltzmann’s constant. The particle has a mass rn = 10-25kg typical of a raregas atom like Ar. Assuming I, = O.lm and a typical thermal “noise” of AE = 0.01E, we obtain A$ = 0 for each particlc in our box. As we are coiisidering a macroscopic piece of matter.

39

Elements of statistical thermodynamics

N = 0 (loz3).Thus, our critirc systciii lias of Ihc order of 0 (lo"') acccssiblc quantum states that, are compatible with the specified conditions. This is indeed an astronomically large number of states, which becomes even larger by many more orders of magnitude if we also include interactions between all these particles as we should for more realistic situations. Notice also that, if we t,ake our system as a quantum mechanical model for an ideal gas, t,hc thermal stat,(!of t h gas woiild be complct,oly spcrificd hy the density a i d tcmpcraturc of t lie gas (which would ycrinit us t o corr1put.c its pressure). Hence, in going from the microscopic to the macroscopic level of description, an enormous reduction of informatioii from 0 ( clown to only 2 degrees of freedom 11% taken place. Let us. now return to our original situation, namely that we can solve Eq. (2.1) for the systems of interest in this book and that we can arrange cncrgy ciganstatm according t,o t,hc inrqualit,g given in Eq. (2.3). For tho subscquciit. dcvcloprnciitb,it will turn out to bc uscfiil to ixitroducc: tlic co~icept of a distribution of quantum stmalesthrough the vector n = In) = ( ~ X ( N , n~ 2~ () ~, . ~ . .. .), ) , where n j ( N . s , ) is the occupation number of quantum state j for all those replicas having the same number of molecules N and substrate separation s,. Thus, we can interpret n as a quantity that tells us how the available quantuni states are distributed over the ensemble of replicas. Thcrcforc, wc shall honccforth rcfor to thc vector n as a "distributioii" of quaiituxn states. Bcca.usc tlic supcrsystcm is tlicrrriodynarnically closed against its own environment, the distribution n is subject, to four constraint,s, namely q 1

(n)

N-

7Lj(N.s,)

=0

(2.7a)

N,s.-j

v2

(n)

f

-

nj(N,sz)Ej(N,s,) =

0

(2.7b)

lV,sE-+j

v.3 (n)

N-

C

nj(N,s,)Iv

=0

(2.7~)

N.r.-+j

p4 (n)

sz

-

nj(N,s.)% =

0

(2.7d)

N,s.+j

J3quation (2.7a) oxpressas t.hc fact that. tho iiiimbcr of systems forming thc ensemble is fixed and finite but may he iiiade arbitrarily large at the end of our calculation. Equation (2.7b) states that the energy of the isolated supersystem is fixed as well as its total volume V = N2s,s,S, [see Eq. (2.7d)j and total number of molecules N [see Eq. (2.7c)l. The expression for V implicitly assumes all systems of the ensemble to be exposed to the same compressional straiiis proportional to s, and sY as well as shear strain proportional to as,,-,.

40

Concepts of quantum statistical thermodynamics

In Eqs. (2.7) wc arc using surii

“CC C N

sz

“

C

N.z.dj

. . .” its shorthand notation for thc triplc

. . .” whcrc tlic arrow is used to emphasize that. within the

j(N?s.)

sct of qiiant,iiiii states { } depends on tlic actual mirrihcr of part,iclcs N and the specific substrate separa.t,ion s,, which may differ between the individual systems of t,he enseniblc. Let 11s now assume N to be finite but overwhelmiiigly large. Because we are imposing only four constraints on the construction of n,consisting ) themselves of an ovcrwhclmiiigly large number of elements n j ( ~ , ~as=well,, it. seems reasonable to oxpoct, many dist.rihntions {n}to exist, all of which arc coriiplyirig with Eqs. (2.7). Frorri thc prccrdirig discussion, it should bc obvious that at, a.ny instant it is the specific quantum state occupied by one of t,he (many-particle) replica systems of the ensemble that, makes it distiiict and distinguishable from any other. In other words, the replicas can only be discriminated on an energetic basis. However, it is conceivable that, two or inore replicas occupy the same quantum state and are thus indistinguishable. Tticrc arc snvc!ral siich groups in which r ~ q i , ~rcplicas ,) arc diaractcrixcd by the same eiiergy E i ( N , s z ) , ??j(Nf,.;) liaviiig ciicrgy E ~ ( N I ,and ~ ; ) SO 011 illid SO forth. Hence, a spccific distribution {n}can be realized in a number of

different ways by assigning n , i ( ~ , ~randomly .) chosen systems of the enseinble to quantum state E ~ ( N , ~n j,()N:I , s ; ) to quantuiIi state E j ( N I , s ; f ) and so forth. Hence, if we pick any one systein of the ensemble at random and ask for the probahility to find it on average in a quantam state Ej(j(~,~,) accommodating N molwulcs at, a subst.ratc scparatiori s,, this probability is givw by

(4

where E ~ ( N , . ~is= )the number of times a quantum state j is realized on average with N molcrulm prcscnt and siibstratC scparathi s,. Tho sums in Ekl. (2.9) extend over all a priori equally probable distributions. We assume these distributions to be equally likely because they all coinport with a state of the isolated supersystem characterized by fixed N , V. and &. The assumption of equal likelihood of all distributions {n}coinplying with & = constant is known as the Principle of Equal A Priori Probability. It is part of the postulatory basis of st.at,istical thermodynamics.

41

Elements of statistical thermodynamics

As wc poiritcd out: N can bc rriadc arbitrarily largc arid rriay, in fact, become infinite. Suppose now that. in the limit N -+ m a distribution nf exists such that W (n*) overwhelms all other values { W (n)}. An immediate

consequence of this assuiription is that, in the sums on the right-hand side of Eq. (2.9), all summands except those involving n* become negligible so that Eq. (2.9) may be replaced bv a single term, namely

(2.10)

2.2.2

Justification of the most probable distribution

The existerice of a most probable distribution n* with properties reflected by Eq. (2.10) is by no mcaiis ot)vinits, howcvcr, or cv(m plnidblc dcspitc thc fact that N is at, our disposal and may therefore be increased beyond limits. A formal way of proving the existerice of n* was suggested by Darwin and Fowler [18]. Their method is based on function-theoretical arguments that are' mathematically a bit, involved. The advantage of Darwin and Fowler's argument is that it is mathematically rigorous and completely general. We have therefore decided to include it. in this hook. Moreover, we were prompted t,o do so bccawc thc Darwin-Fowlcr approach (:an hardly hc foiind clsrwhcrc in the literahire (191. Because of the somewhat, more formal charact,er of Darwin and Fowler's analysis, however, less interested readers may skip this portion of the current section. Nonetheless, we feel that these readers should at, least' be equipped with an argument suggesting that the assumption of a inost probable distribnt'ion in t,he smse of Section 2.2.1 is jiLstified and physically scasihle. However, unlike thc Darwin-Fowler approach, this alternative reasoning, which is based 011 a Taylor expansion of In I V (n), arnounls only to a plausibility a.rgument and is by no means rigorous.

2.2.2.1

Taylor expansion of Eq. (2.8)

U'c bcgiii by rewriting Eq.

In

w (n)

N

(2.8) as

N In N - N -

C

N.s,+j

NlnN -

n $ ( N , s z ) 111 r i j ( N , s ; )

+

C

nj(N,s.)

N.s,-+j

(2.11) N,s.-rj

where we employed Eq. (2.7a) and Stirling's approxiniation [see Eq. (B.7)]. Expanding In W (n)in a Taylor series around the most probable distribution,

ConceDts of auantum statistical thermodynamics

42 w(: obtain

where we retain in the expansion only terms up t.o second order in the devi&ion of n from n*. The prime attached to the snmination sign emphasizes that the summation is to be carried oiit siibject to the const-raint,sposed by Eqs. (2.7). 1Jndcr thcsc conditions, thc first,-ordor tcrm vanishcs bccausc it, is the necessary condition for the exisknce of the most probable distribution. Assuming In* - nI to be sufficiently sinall, higher-order t,erins may be neglect.ed in Eq. (2.12). We niay then rewrite Eq. (2.12) as

=

{ M;'(n*), nn =+nn** 0,

if if

(2.13)

-

This follows by noting that, for n = n*,the exponential fact,ors are exactly one, whereas thcy are always less than one otherwise. However, because 00 the product the nuinber of such factors is astrononiically large as N in Eq. (2.13) goes to zero regardlms of how srriall the deviation between n aid n* bccornts. In othcr words, cornyarctl with 14,' (n*),W ( n )M 0 for all other distributions { n } \n*, which is the plausibility argument jurtifying the subsequentodevelopment detailed in Section 2.2.3. 2.2.2.2

Darwin-Fowler analysis of the most probable distribution

of microstates

To justify the concept of existence of a most probable distribution of microstates overwlielming all other possible distributions [i.e., all other distributions consistent with the constraints posed by Eqs. (2.7)] a treatment originally due t o Darwin and Fowler may bc cinployed3 [IS]. It starts by defining tlw zcro of tlic ericrgy scdc and energy unit such that the discrete energy spectrum { E 3 ( ~ . Rcan L ) }bc represented hy a set of nonnegalive 'This section may be skipped by less interested readers.

Elements of statistical thermodynamics

43

integers { E}. Let u s thcii dcfiric an auxiliary luunct.ion

Q(t)

C z E t ( E )=

XZ”CW(TZ)

E=O

E=O

{n}

where 2 E C is coinplex [see Eq. (R.8b)l and the prime attached to the second summation sign sipifics that this snnimation has to he carricd out, such that, for each cricrgy G , tlic constraint in Eq. (2.7a) is satisfied. Keeping this in mind Eq. (2.14) can be recaqt as

One immediately realizes that the previous expression can be rewritten more compartJy iising the iniiltinoniial expansion [ 171 r

where

(2.17)

One member of the set of (nonnegative integer) energies {E} is the fixed energy E of the ensemble [see J3q. (2.7b)l. Therefore, by definition, t ( E ) is the coefficient associated with zE in the expansion of Q in terriis of a power serics in z in Eq. (2.14). As the c*omplexqimnt,it,yt mav h o m e zero, Q ( z )/ z E + l has a siiigularit,y for this wluc of z. Because of our definition of the zero and unit of the energv scale, E 2 0. To avoid an undet,ermined expression of the form Oo, we consider zE+l rather than t Ehenceforth and in a Laurerit series around 20 = 0 [see Appendix B.2.4, expand Q ( z ) J3q. (B.40)] according to

(2.18)

44

Concepts of quantum statistical thermodynamics

whcrc thc { n k } arc unknown cocficicd.s of tlic Laurcrit cxparisioii. TWIIIS in the Imt expression may be rearranged slightly to give m

(2.19) k=-m

Corriyariiig tlic tcriii for k: = -1 in tlic Laurcnt expalision with our startiiig expression for Q ( z ) in Eq. (2.14), i l is dear that a-1 = t ( E ) because (1-1 is the expansion coefficient associated with 8,This coefficient, the so-called residue [see Eqs. (B.33), (B.41)], mav be obtained froni (2.20) whcrc C is soma yct, to he spccifirtl closc?dpath in the coinplax planc anclosing tlie singularity of the iritcgraiid at z = 0. Let. u s now define lhe complex a.uxi1iar-j functiori [see Eqs. (B.8)] *f ( z ) = hiq ( 2 ) which permits us to rewrite Eq. (2.20) as

E+1 ~

N

f

111z

t ( E ) = - d: exp [NJ ( z ) ] 2ni c

(2.21)

(2.22)

where we used l3q. (2.16). The integral in Eq. (2.22) is identical with the oiie in Eq. (B.17) and can therefore be cvaluated using the method of steepest descent detailed in Appendix B.2.2. It requires f ( z ) to assuine a inaxinium at some 2 = a.Expanding f ( 2 ) in a Taylor scirics and rafaining t,crms to sccoiid ardor only, it, is shown in Appendix B.2.2 that Eq. (2.22) can be rewritten as 1 (2.23) lim In1 ( E ) = -- lirn {In [2nNf” (TO)] Nf (20))N N f (Q) NdQO 2 N-ca whcrc .cg = RczO is t,hc ran1 part, of a. Thcwc cxprtwions permit, 11s now to calculate the mean - occupation number ? r j ( ~ , s ,of) quantum state j (N, s,) and its variance n23(N,s,) - ?t;(N,.qs). To proceed we introduce

+

45

Elements of statistical thermodynamics

whcrc { < j ( N , s z ) } is a set of arbitrary parariiet,crs that, will be sct. equal to at the end of t,he calculakion. From the definition in Eq. (2.9), it follows that

hi Eq. (2.25) we crnyloycd Eq. (2.23) alld tllr fact that

[SCC

Eq. (2.21)]

(2.26)

In a similar fashion we calculate

(2.27)

46

Concepts of quantum statistical thermodynamics

In Eq. (2.27), wc can rcwritc

(2.28) Hence, from Eqs. (2.27) and (2.28). we realize that

where the term in hrackebs is constant because ~ T ~ ( N a ,~N J . Because of this proportionali tv, we may conclude that

which proves the existence of a. most probable distribution n* = Ti with vanishing variance in the limit N 4 00 so that indeed n* overwhelms all other distrihiitions of quantum states consist,ent wit,h the constraints [see Eqs. (2.7)] imposed on the (astronomically large nurnber of) virtual systems of the ensemble.

2.2.3

The Schrodinger-Hill approach

The problein we are now confronted with consists of determining the most prohahlc distrihiit,ioii n* discusscyl in tho prccding Scctions 2.2.1 and 2.2.2.

This approach was first suggested by Schrodinger [20] and later extended and reformulated by Hill [21]. We already noted that, mat.hematical1y speaking, n* is that distribution rriaximizing W (n)subject. to the constraints specified in Eqs. (2.7). This, n* can be determined using the method of Lagrangian multipliers detailed in Appendix B.3. However, we apply this technique t o Eq. (2.11)rat.her than directly to W ( n ) .This is possible because both IY (n)

Elements of statistical thermodynamics

47

and 111W (n) arc irionotonic functions ol' N, whith permits us to prcfcr one over the other on the grounds of mere computational convenience. By analogy with the trcatnient, dcvtlopcd in Appcndix B.3 wc begin by introducing F(n)= l n b V ( n ) + A . c p ( n ) (2.31) where [see Eq. (2.7)] (2.32) arid in the current situation = ( X I , x2,x3, A,) is a four-dirneiisiona1vector of undeterminecl Lagrangian multipliers. From Eq. (2.1l), it is easy t o verify that (2.33) because N is constant (but. can be made arbitrarily large). Moreover, it follows from Eqs. (2.7) that a matrix cp' (n) can be formed whose elements in each row are given by (2.34a) (2.3413) (2.34~) (2.344 Hence, using Eqs. (2.33) and (2.34) in Eq. (B.47), we readily obtain njt(N,,)= exp(-X1 - 1)exp ( - X ~ E ~ ( N , . ~exp(-X3N)exp(--XqsZ) .))

(2.35)

after a trivial rcitrraiigcincnt of terms. Wc call imiricdiatcly clirninatc one of the four Lagmngian multipliers by summing both sides of the previous expression over j,N:and s,, which leads to the more compact expression (2.36)

48

Concepts of quantum statistical thermodynamics

whcrc

N

*a

is the partition function of a so-called grund mixed isostress isostmin ensenibled and (2.38) Q ( N s,) = extxl, ( - X 2 E J ( N . Y z ) )

c J

is the canonical parbition function, which of course, still depends on the substrate separation and the number of particles. It is termed “canonical” bccaiisc of the distinfliishcrl rolc playcd by this cnscmblc, as we shall dcmonstratc below (scc also Scctioii 4.1). Aii alternative formiilation of Eq. (2.38)is obtaiiied 1)y realizing that , with the aid of Schriidiiiger’s equation, one may write [see Eq. (2.1)]

=

Tr [rxp

(-41

(2.39)

(

-1

where the expression in [. . .] is a matrix element of the operator exp - X 2 H defined via its MacImirin-series expansion on the second line of Eq. (2.39). Thc Q r a wopcrat,ion is dofincd in Eq. (A. I). Providcd on(’ can expand an arhitrary complete-set of orthoriormal functions 9, in terms of the eigenfunctions { d j ( ~ , ~of~I7) }according to (2.40) j=O

‘The term gnnd f 7 i U f d isostrws isostmirr.ensemble is used to indicate that the slit-pore is materially coupled to its eiivironrrient and that its thermodynamic state depends on t,he control of a wt of stresses aid strains.

Connection with thermodynamics

49

whcrc we charigcd the riotation ;j (Ar, R,) -+ ,j tcmporarily for the sake of clarity. We emphasize that the functions {$*} are not eigenfunctions of fi. One can then show that

which expresses the fact that, in order to calculate the canonical partition function Q, one does not need to know the complete set of energy eigenfunctions. In fact, an,g complete set of orthonormal functions permits one to calcii1at.c Q. A proof of FA. (2.41) bawd on Parscval’s eqiiation [see Eq. (B.65)] is presented in Appendix B.5.2.

2.3

Connection with thermodynamics

From the previous discussion it should be apparent that, in the limit of an iwtronomically large mimbrr of systcms, cxprcssiotis siich as t h i oncs given in Eqs. (2.37) or (2.38) for x or Q, rmpcctively: arc exact.. The corrcct,Iiess of this statemeill, is, of course, intimately linked to the existence of a most probable dist,ribution n* (see Sectmion2.2.2). However, as expressions for both parttitmion fiinct.ions have been derived quantum mechanically, they do not. perinit us per se to calculate equilibrium properties of thermal systems because quantum (as well as classic) mechanics as such does iiot have any concept of krnperature. However, aii inspection of Eqs. (2.37) and (2.38) rcvcals that h t l i part,it.ion fiinctions still contain oiic or more yet- to-bedelcrmined Lagrangian multipliers. These need to be calculated in a way that the resulting expressions become consistent with thermodyimmics as it was introduced in Chapter 1.

2.3.1

Determination of Lagrangian multipliers

We begin by noticing from Eqs. (2.10) and (2.36) t,hat the probability of observing (on average) our slit-pore in quantum state j with N particles and substrat.e separation s, is giveii by

Conscqiiantly the avcragc! cncrgy of the slit,-porc (ix., it,s tncan cncrgy) can be written as (2.43)

50

Connection with thermodynamics

such that its exact. differential riiity bc cast as

[E j ( N , s r ) d p j ( N , s , )+ p j ( N , s . ) d E j ( N , . s . ) ]

d ( E )=

(2.44)

N,s,-j

As we pointed out in Sect,ion 2.2.1, all systems in the enseinhle are subject t,o t,hc samc fixcd comprcssional sttrainsproportional to s, and s,; in addition, t h y arc all exposod to the SXIIC fixatl shear st.raiii mx0.Hcntrc, thc cncrgy eigf3nVallleS E ~ ( N ,=. ~E j~( N), s % )(sx, sy. QS,") so tha,t,

(2.45) and from Eq. (2.42)

Inserting now Eqs. (2.45) and (2.46) into Eq. (2.44),we obtain

(2.47) where, of course, (. . .) sbands for

C

N.sl+j

,

. . p , ( ~ , , ~ ,To ) . arrive at Eq. (2.47),

we invoked the Principle of Probability Conservation, which is

This follows because (2.49)

51

Elements of statistical thermodynamics aiid thcrcforca

d

)

dpj(N,a.) = 0

Pj(N,s,) = (N,sK&~

(2.50)

N,s.+j

We now postulate that ( E ) = U. Hence, comparing Eq. (2.47) with the thermodynamic expression given in Fx.(1.43), we readily conclnde that,

(2.51a) (2.5111) Moreover, we identify

(2.52a) (2.52h) (2.524 so that we are still left with multiplier.

2.3.2

A2

as tlie remaining undetermined Lagrangian

Statistical expression for the entropy

Cornparing Eqs. (2.47) axid (1.43), one iiiay spcculatc that physical mcariing can be assigned to A2 by relating it to the entropy S , i.e., by making the 1 identification (2.53) wlicrc

-

PJ(N,sz)lllpj(N,Sz)

(2.54)

N,s'+J

Moreover, as pointed out in Section 1.3. d S is an exact. differential such that the factor 1/ (AzT) in Eq. (2.53) mnst be an integrating factor because dw cannot be expected to be an exact differential per se. Hence, we can write

4 (711) du! G d f

(11,)

(2.55)

where dfmust be an exact differential because of Eq. (2.53). Thus, we may integrate dS along an arbitrarily chosen path in thermodynamic state space to obtain (see Section 1.3) (2.56) = f (10) c

s

+

52

Connection with thermodynamics

whcrc c is an iiitcgratioii coilstant.. Two addit,ional p r o p c r t k of S t,iirn oiit to be important for tho ciirrent discussion. 1. As wt: saw iii Scxtion 1.3, S is a state furictioii such that. /dS = S(CJ -S(Ct,)

i7

(2.57)

N,,a,) and c‘t, - (&,, A$,, o b ) dcnotc thc start and whcrc (& - (Ua, end point of some thermodynamic transformation, respectively, along an open path C in thermodynamic state space. Without loss of generality, we caii thercforc set c = 0 in Eq. (2.56). 2. S is additive in the sense that, if we were giveii two systems A and B both in the same thcrrnodynamic equilibrium state, S A ~ =B SA SB. This prompts us t,o roncliide from Eq- (2.56) that

+

should also hold. Let A arid B be system pertaining to our cr~scmblc.We rriay thcn ask what is the prohabilit4y of finding A in a quantum state characterized by E ~ ( N A , whereas ~ ; ) , B is simultaneously in a quantmn stake represented by an energy eigcrivalue E,(,B,:,‘? As we tacitly assumed the interact,ions between systems in the ensemble to be very weak, so that the spectrum of energy eigenstates is essentially that of an isolated system, we conclude that the answcr to thc qiitstioii is givcw 1)y

so that, from Eq. (2.54), we niay writc

A

B

53

Elements of statistical thermodynamics

whcrc the suinniation cxtcnds over all quantum states accessible to systems A and B, respectively. In other words, becawe ol Eqs. (2.58) and (2.60),our function f must satisfy the relation

Differentiating both sides of the previous expression with respect. to WB, respectively, we obtain

df ( U J A+ wig) 3 ( I U A + W ~ R ) d ( u ? ~+ ~ L ‘ B ) ihl’~ df (WA + ~ U B 8 ) ( W A + WB) d ( U ~ A+ ~ g ) (htl~

-

+ + WB)

df ( 7 1 1 ~ d (WA

d f (WA

d ( 7 1 ’ ~ 4-

11%)

- df --

wB) -

(”’A)

WA

and

(2.62a)

dWA

d f (WB)

7143)

dW&j

(2.62b)

Equations (2.62) result, because

3 (WA

+ WB)

- 3 (WA -

aWA

+ WB)

3WB

=1

(2.63)

or, altcriiativcly, because fuA aiid iiig arc iiiutually indcpciidciit of oilc ailother. This is a direct consequence of our initial assumption of a negligibly weak interaction between various systems of tlie cmsemble such that, in each system the spectrum of energy eigenstates is identically the same. Equations (2.62) caii onlv be satisfied simultaiieously if and onlv if

(2.64) where k~ is some constant, henceforth referred to as Boltzrri.cmn’sconstant. It is then immediately clear from Eq. (2.55) t,hat the integrating factor is given bv

(2.65) where the far right side follows by comparing the right side of Eq. (2.53) with the left side of Eq. (2.55). Equation (2.65) determines the remaining Lagrangian mnltip1ic.r A2 siich that, from Eqs. (1.33) and (2.47) we have

s = -kB

P j ( N , s z )l ~ ~ ~ j ( N , s z )

(2.66)

N.n.4)

as the expression for the entropy in the grand mixed ismtress isostxain ensemble. Similar expressions may be derived in other statistical phvsical ensembles

PI.

54

Connection with thermodynamics

Rcplacirig iii Eq. (2.66) p,(N,sz) through thc cxprcssion givcii in Eq.(2.10), another interesting expression may he derived. We readily obtain

Noting also that Ssup = NS is t,he ent.ropy of the isolated supersystem we may write from the previous expressiori &up

= -kB

~Z;(,V:~,) 111 n;(N.*.)

-N

In N

J

=

-kB In IV (n*) (2.68)

wlicrc Eqs. (2.7a) arid (2.11) havc also bcc~iciriploycd. As ciicrgy E [sex: Eq. (2.7b)], substrate separation S, [see Eq. (2.7d)], and the number of molecules N [see Eq. (2.7c)l of the isolated supersystem are fixed, we cam interpret W (n*) as the number of quantum states accessible to the supersystem R ( N ,V ,I )subject to the constraints spelled out in Eqs. (2.7). Hence, we rnay write (2.69) Ssup = -kB In fl ( N ,V :E )

which is nothing but thc wc:ll-kiiowii cxpressioii for bhe entropy iii the inicromay be interpreted as the thermocanonical In Eq. (2.60), Ssup dynamic potential associated with the microcanonical ensemble.

2.3.3

Statistical physical averages and thermodynamics

Inserting now Eqs. (2.51a), (2.51b), and (2.65) into the expression for the partition fuiictioii in Ey. (2.37),wc cvcutudly arrive a t

(2.7Ob) =

pP(-)a

(2.70~)

5An expression or I ha rorm of Eq. (2.69) am first propcmed by the Austrian physicist Ludwig Boltixrnan~i(1844-1906) within the framework of classic physics. To honor Boltzmann’s contributions to statistical physics, Eq. (2.69) is engraved on his tombstone at the Zentralfriedhof in Vienna (Austria).

Elements of statistical thermodynamics

55

whcra wc dcfinc the partition function iu tho graiid canonical crlscriiblc*3 in Eq. (2.70~)for laber reference. I

Another important relatioil can be derived from the expression for the entropy in the grand mixed isostress isostrain ensemble. Substituting in Eq. (2.G6), p j ( ~ , ~via , ) Eq. '(2.42) together with Eqs. (2.51n), (2.5lb), and (2.65) permits us to write

where the second line follows from the equivalence between statistical and therniodynaniir expressions in tho tliormotlyiiami(. liniit. Elpation (2.71) also defines the thermodynamic potential of the grand mixed isostress isostrain ensemble. Taking the exact differential of 9 , we realize that as

9 = CP (T,p, sx, .sy, T,,, crs,~) depends on a mixed set of stress ( rzh)and strains (s,, sy, crsxo) as natural parameters. Therefore, the ensemble characterized by the partition function y has been termed the grand mixed isostress isostraiii enseinble in Section 2.2.3. Note that, in the derivation of &. (2.72), Ed.(1.43) has also hccn cmployd

A comparison between Eqs. (2.71) and (2.72) reveals that st,atistical averages are related to first-order derivatives of the relevant thermodynamic potential. For example, frorn Eqs. (2.70a) and (2.71), we have

=

-(N)

(2.73)

Another example is the average substrate separation that the system will attain if exposed to a fixed external (normal) stress. From Eqs. (2.70a) and

56

Connection with thermodynamics

(2.71), orie iriirricdiatcly sees by a similar tokcri t,hat

(2.74)

2.3.4

Fluctuations

By virtue of their nature: therrnal averages like (N) [see Eq.(2.73)] or (s,) [see Eq. (2.74)]are assocint,edwith a certain variance of their instantaneous values N or s,. Thc variance is associatcd with sccond-order partial dcrivativcs of the relevant thermodynamic potential(s). For example, from Eq. (2.73), we find

(%>,,\,,

=

a( N ) -(.i; t 1\11 ;)

(2.75)

where UN is the variance of (N). Noting from Eq. (2.75) that. @ O( ( N ) O( a;! it. follows t,hat.

In other words, as we let the system size grow, fluctuations about the average value of a thermal quantity become negligible. However, there are exceptions.

57

Elements of statistical thermodynamics

For cxar~iplc,iicar a critical poilit, or at.a spiriodal (i.c., a t thc stability limit of a fluid phase in the metastable regime), the above is no longer true. However, it should also be not,ed tha.t, a.t a discontinuous phase transition (i.e.: before entering thc riietastable regime), Eq. (2.76) still holds. To arrive a t the far right side of Eq. (2.76), we employed the addztiwzty of a. That is, if we prepare two identical equilibrium systems characterized by @A and @B, then the composite system has twice the number of particles than any one of t,hc two separate systcnis and @A”B = @ A @B = 2@. It is iiriportarit to rcdizc that. additiwitz~in this seusc docs riot ncccssarily imply eztensivity in the sense of Section 1%: where we showed that thermodynamic potentials inay be hoinogeneous functions of degree one in only a few of their extensive variables depending or1 the symmetry of the external field represented by the (structured) substrates. Expressions similar to t,he one given in Eq. (2.75) result for other thermal averages rzs well. In most cases, a relation like Eq. (2.76) also holds, which is csycntially the rrason why one ineasiircs “sharp” valucs of t,hcrmal propcrtics in macroscopic systciris (cxcopt.for statistical crrors) . A result pcculiar to confined fluids is, however, obtained by considering thermal fluctauatioiisof s, about (s,,). By manipulations precisely parallel to thosc leading to the expression for a~ in Eq. (2.75), one caii show that

+

Consider now the relative variance usz/(sz). Clearly, it does not decrease if wc make the system larger in thc 2- or y-directions hiit, will only decrease as w e approach the bulk limit by lcttiiig s, approach irifiriit*y.IIowcvcr, tllis inevitably changes the physical nature of the confined fluid. Notice lhat the same problem would be encountered for the ratio ON/ (N) in cases where the external potential prevents the thermodynamic potential to be a homogeneous function of degree one in either sx or sy and these two quantities would be increased by noriinteger fsctors. These examples show that for confined fluids one has to be cautious t,o approach the thermodynamic limit, propcrly. Thcsc considerations also have an irnmcdiatc pract,iral coiiscqucncc for t.hc experimental det,ermination of (5,) (in, say, the SFA experiment; see Section 5.3.1) because t,hese average values would inevitably be plagued by a certain systematic error on account, of t,hermal fluctuations. In other words, there will be a distribution of substrate separations s, about the average value that is characteristic of the physical nature of the confined fluid. Therefore, this error cannot. bc: reduced by any more sophisticated device

Equivalence of ensembles

58

but, is characteristic of the cxpcririicrital silualion. Howcvcr, in practice thc error may still be small enough to be inconsequential from a purely practical perspective.

2.4

Equivalence of ensembles

So far our discussion has focused on the grand mixed isostress isostrain ensenihle, which was devised to mimic a t a microscopic level operating condition.. eiicounterd in the SFA. However, this eiiseinble is by no means distinguished among other conceivable ensembles. In other words, it was defined merely out of convenience. The close relation between the grand riiivcd isost>rcssisostraiii cnscinblc and tlic rnorc coiivcritional carionical or grand canonical eiiseinbles is already suggested by Eqs. (2.70). However, one can demonstrate that-the grand iiiixed isostxess isost,rain ensemble is not, only linked to others but also forinally equivalent to the other two. The demonst,rat,ionof equivalence depart,s from t,he observation that, partition fnnctions in m y statistical physical ekenible comply with the general form given in l3q. (B.52) with individual sumrnands as given in Eq. (B.54). Thiis, according toothc rliscwssion in Appcndix B.4, wc may approximatc thc grand mixed isostrcss isostrairi ciisciiiblc partition fuiiction y, in Eq. (2.70a) by the maximum terms in the sums appearing on t;he right side of that equation Q,

= -kBTln y

21

-pN*

- T,,A,~s: - lcuTln

Q ( N * ,T ,R,, sy.s:,

o s , ~ ) (2.78)

where N* and s i denote values of N and s,, respectively, corresponding to the niaxiniurn term in tlic suiiis in Eq. (2.70a). The previous cxyrcssion niay be recast wilh lhe aid of Eq. (2.71) as

F ( N * ,T,sX, syrs;. m,~)= -kBTln & ( N * , T, sx, sy. st,ass)

(2.79)

where the definition of the (Helmholt,z) free energy given in Eq. (1.50) has also been employed. Thcse cxprejsions show that the canonical ensemble for fixcd N = N' and s, =- .5: is cqiiivalcnt to tho grand mixad isostrcss isostrain cwsrrriblo. which is solcly a ~oriscquc~icc I)otli of the irifiiiitcly largc iiunibcr of terms contributing to the partition function x in J3q. (2.70a) and of the functional form of these terms [cf., Eq. (B.54)]. Likewise, starting from Eq. (2.70~)arid employing the rriaxinium term approximation (see Section B.4). we niay write

The classic limit

59

which together with Eq.(2.71) aiid tlic definition of tlic gritiid potcnt.ial ~ ~ V C I I in Q. (1.51)yields

after some straightforward a1gebra.k manipulations. This expression states that the grand mixed isostress isostraiii ensemble is equivalent8to the grand canonical oiie at fixed s, = sr. Similar considerations can be employed to demonstrate equivalence of any two (physicdly sensible) statistical physical cnscmblcs. I n closing, we note that fluct.uatioiis in statistical physics arise in two scparatc! contexts. As we saw in Scction 2.2.2, fliictnations around thc most probable distribution of quantum states are completely suppressed on account of the astronomically large number of systems of which an ensemble is composed in the thermodyriariiic limit. However, within this inost p r o b able distribution, themrtal fluctuations arise. Their magnitude can be quite large depending on the specific thermodynamic conditions, as we pointed out above.

2.5

The classic limit

Until now, our formulation of statistical thermodynamics has been based on quantum mechanics. This is reflected by the definition of the canonical ensemble partition function Q3which turns out, to be linked to matrix eleinents of t'he Harniltonian operat,or H in Eq. (2.39). However: the svstems treated below exist in a region of thermodynamic state space where bhe aact qiiant,iim mechanical treatomentomay be abandoned in favor of a classic description. Thc trarisit,ion from qiiantmn to classic st.atist,ics was workcd out by Kirkwood [22, 231 and Wigner [24] and is rarely discussed in standard t,exts on statistical physics. For the sake of completeness, self-containment, arid as background informatioil for the iiiterestcd readers we suinriiarize t.he keg consideratioils in this section. In essence: the classic approximation will enable us t,o replace sunis over discrete quantum stattes in the various partition functioils [see, for example, k s . (2.70)] by integrals over the so-called phase space spanned by the 6 N coordinates rN and nioinerita p"' that fully specify a classic microstate of a systc~~ ini which tlic N particlw liavc orll traislational degrees of freedom. T ) (the transThus, (rN) = ( T I , T Z , . . . ,T N ) and (p") = ( P I , & , .. . , p ~are poses of) 3N-dimensional vectors. We refer to = r N @ p Nas a point in (classic) phase space and to r N as a configurat,iori of particles. h

4?

r

60

The classic limit

At thc corc of t he following dcvclopnicnt arc syrriirict ry propcrtics of thc wave function. In quantiina mechanics, one is concerned with two general types of particles, namely, Bosom (e.g., photon, a-meson, and 4He) and Ferniions (e.g., electron, proton, iieiitron, and 3He). Fermions are characterized by half-integer spin whereas for Bosoiis the. spin is integer. This difference has an immediate consequence for the wave function describing the state of a niany-particlc Fcrmioii as opposcd to a Boson systcni. If wc arc dcaling wit-h identical yarticlcs, thcn t,hc wavc furictiori nwst be aitisymmctric with respect l o the exchange of any two particles as far as a fermionic system is concerned; for a bosoiiic system, the wave function remains symmetric during such a permutation. This is a consequence of Pauli’s Principle. which can only be fully appreciated witliiri the framework of quantum electrodyuainics.

2.5.1

Symmetry considerations

Consider a systein of N particles whose qiiantiim state can be described by a wavc functioii 1/: (rlr~ 2 . ,. . , TN), whc:rc ri is the location of particle i in space. Here and below we deliberately ignore spin coordinates. This seems justified because iiiclndirig those a.dditiona1 degrees of freedom would not affect, our final conclusions concerning conditions that render adequate the desired classic treatment of microscopic systenis, with which we shall be concerned almost exclusivelyG[ 191. Let us then define. an operator P, that serves to permute pairs of coordinates ~i and r,. If this operator is applied to t,he wave function, we may thus write 6

(k1)l9r1%/!!n,

( T h f 2,.

. . ,T N ) = i j t n

l- I

..

( T I ?~ 2 . :

.

rN)

(2.82)

whcrc. P, is t,he total riuriiber of pair-wise exchanges into wliidi tlie perrnutation can be decomposed arid the set { s $ ~is} t,akeii to be energy eigenfunctions [see Fq. (2.1)]. However, here and below we shall be working in the canoiiical ensemble for convenienc*e. Hence, N and .s, are asiimed constant such that we niay replace tlic label j ( N . sz) on the wave function (and its associated energy eigenvalues) by an int,eger m, say (which is independent of both N and s,). In Eq. (2.82)’ the prefactor (j.e.’ the parity of the wave function) is +1 for a system of Bosons because of their sym.met~icwave function. In other words, regardless of whethta is even or odd, the sign of the wave function remains unaltered. For a system of Fermions, however, the prefactor is -1. The wave function changes sign if P, is odd so that the wave function

PrI

I- I

6The only t-xception is the coiifiiicd ideal qiiantiim gas disciissrd in Section 5.7.4.3.

Elements of statistical thermodynamics

61

of a system of Fermions is untisym.metric with rcspcct to the pcrrnutation of a sanyle pair of particles. These symmetry properties give rise to different types of statistics associated with the names BoseEinstein and Fermi-Dirac [17]. Moreover, in a system of N particles, N! different permutations are possible so that froin Eq. (2.82)

(2.83) follows. To proceed, we now cxpand the wave function iu terms of cigenfunctions exp (ikN. r N )of the momentiini operabor, $m

( v N )= / a , , , ( k N )exp ( i k N. r N )dkN

where t8heN-dimensional vector k N whose elerrieiits arc is the momentum of particle i and h = h/27r. That,

(2.84)

ki= p,/h,where pi

N

exp (ikN . r N )= n e x p (ikj . ~

j

)

(2.85)

j=l

is an eigenfuriction of the inorriciiturii operator space represent&ion lqi considaring

its

can easily be verified in

which also shows that pj is the eigrnvalue of the momentum operator 6,. Mathematically, Eq. (2.84) constitutes a Fourier inversion of the wave function from momentum (k)to real ( T ) space. The corresponding Fourier transformation [i.e., the transformation conjugate to Eq. (2.84)] is then given by n,

/

(kN) = (2r)3N.

( r N cxp ) (--ikN. r N )d r N

(2.87)

Thiis, inserting t,hr cxpansion givan in Eq. (2.84) into trhc right side of Eq. (2.83), we ol)tttiii 1

li,, ( r N = ) -/ a e r n( k N )6 ( k N T,

m.

~dkN )

(2.88)

The classic limit

62 whcrc

and the periniitat,ion operator ’Pk acts on elements of thc vector k”. The: second line of Eq. (2.89) follows hccmise any pcrmut-ation of a pair of clcmerits of the vecLor P” is equivalent. t.o a permutation of the corresponding elements in the vector k N as far as the scalar product k” . r” is concerned. Note that the funct,ion 8 (k”, P”) introduced in Eqs. (2.88) and (2.89) is an eigenfunction of the permutation operator to an eigenvalue of fl. Eventually, the current considerations will serve t,o express the partition fuiiction Q as a (niultidirnensional) integral over configuration r N and 1110mcntiiin space p ” . It, is thcrcforc ncccssary to invcst,igat,cthc cffcrt of Pk on integrals of the general Lype

-

=

I F (..., k ~ - l l k ~ t l k ~ b. +. . dl ~k .~.,.d) k : , - , ~ d k. .~. +(2.90~) ,

=

I F (. . . , kk-,, kil,k k + l l ..). . . .dkk , , d k ~ ! d k ~ . . +. ,(2.90d)

where we exchanged variables k,t-l ++ k, [see Eqs. (2.90a) and (2.90b)], changed variables according to k, -, k: (i 5 11 - 2 A i 2 n 2), kn 1 + k;, illid kn + k:L--l.Finally, tlic ordcr of intcgrdion is cliaiiged bctwccrl Eqs. (2.90~)and (2.90d), thus bringing us hark to the original expression in Eq. (2.90a), which leads us to conclude that permutations of the arguments of the function F are ine1t:vaiit for the integral I. Hencc, the operations detailed in Eqs. (2.90) prompt us to write

+

(2.91)

Elements of statistical thermodynamics

63

Applying thcsc: considcrations to tlic riglit sidc of Eq. (2.88), wc obtairi

-

1

!V! ,

b,n (kN)0 (kN,r”) dkN

(2.92)

Notice that. 0 (kN.r N )is int,rodiiced as the complete set, of permutations

(-1

caused by the action of the set of perinutat,ion operators pk

on the function

{- 1

exp (ikN . r N ) [see Eq. 2.85)]. Hence, if any one of thc members of Pr; acts

on 0 (kN,rN), it is impossible by definition to generate new permutations t,hat are not already accounted for by the original 6 (kN.rN). However, the may change the parity of 0 (k”,@” such action of a specific operator

&

that the factor (H)l’kl appears as the eigenvalue of the operator @k acting r ” ) . Coefficients b, (kN)in Eq. (2.92) are then on its eigenfunction 0 (kN, given by

-

1

qm(rN)0’ (kN,r N ) drN ( 2 4 3 ~,’

(2.93)

where the asherisk denotes the complex conjugate and Eqs. (2.87) and (2.89) have also been used.

2.5.2

Kirkwood-Wigner theory

We now realixc from Eqs. (2.39) and (2.65) that, Q = Tr [exp ( - f i / k ~ T ) = ]

j(N.3.)

( q j lexp (-fi/k~T)I $j)

(2.94)

in the representation-independent Dirac ( “bra”-“ket” ) notattion. As we already emphasized in Section 2.2.3 (see also Appendix B.5.2), the trace is independent of the choice of a specific basis set. However, the subsequent discussion will benefit from turning imriitdiately to a specific representation

64

The classic limit

in which we cxprcss the set {lbj} iii the basis of cigcnfunctions of thc pcrinutation operator 0 (k”,r”) [see Eq. (2.92)]. Moreover, writing the Hainiltonian operator defined in Eq. (2.2) as

(2.95) ill spacc rcprcscntation, we iiiay rccast. Eq. (2.94) ruorc explicitly as

(2.96) In Eq. (2.95), we USE: the shorthand Ai Vi * Vi [gi = ( A / i ) Vi,VT (il/axi.a/&, a/&,) in Cartcsiari coordinatcs] t.o simplify the notation. The secoiid line of Eq. (2.96) follows by iiiscrting Eq. (2.93) arid rioticiiig that the t.wo integrat,ions over spatial coordinates have to be independent. In Eq. (2.96), we also used t,he fact that the Hamiltoniaii operator as defined in Eq. (2.95) acts only on particle coordinates and not on their momenta. Became the { Q j } form a complete orthoiiormal basis, the term in brackets can be replaced through the celebrated completeness relation

c m

[d,ft, (7.N)?,/,”,(r”)] = b (r” - 4”)

(2.97)

where 6 denotes Dirac’s &function. A proof of Eq. (2.97) is deferred to Appendix B.6.2. The integration over r” in Eq. (2.96) can then be carried out in closed form to yield

(2.98)

Elements of statistical thermodynamics

65

5~

where tile second and tliird lilies follow froiri EY. (2.8'3). 111 ~ q (z.w), . arid Fl are permutation operat.ors acting on elements of k" (see Section 2.5.1). To proceed, let 11s define the inverse permutat4ion operakor P k ' - l , which reverses the permutation cffected bv @L (see Section 2.5.1). We rnay then start with an expression like the one given in Eq. (2.90d), change the order of integration [see Eq. (2.90c)l: and change variables to arrive at Eq. (2.90b). Finally, let P k l - ' act, on the integral in Eq. (2.90b) to recover t,he original expression given in Eq.(2.90a). Therefore, like its counterpart. ?;, the inverse operator &' does not, affect integrals like the one given in Eq. (2.98). Thus, wc may niiiltiply both t,crms in brnckcts in Eq. (2.98) by &I-' to obtain h

h

'

Q =

1 -1 N ! (2743N

c A

//

(k1)W

xTexp ( i k N .r " ) drNdpN

&

-

I (

(4". T") cxp

p&xp

-k:T)

(2.99)

-

-// where = P k 1-1 P~ and 'i= ?jk'-'?L. 111 Eq. (2.99), 7 is the unit operator. We also used the fact that C C . . . N! C because each of the two @; F; 5 k independent sums generates N! perrnutations.

2.5.3

The canonical partition function in the classic limit

Turning now to the classic limit,, we assume that €or sufficiently high temper a tures N

n

(2.100) where H is the Haniiltonian function of classic mechanics. This is the sirnplest approximation that will turn out to give rise to correction terms arising solely from symmetry properties of the wave function. For a more refined trcatmcnt., thc intcrcstcd rosdor is rcfcrccd to Appondix B.6.4 whcrr wc show that &. (2.100) is tho lowwt-ordcr approximatiori in our scinic:lassic trcatrnent of the quantum mechanical Hamiltonian operator. However, for current purposes J2q. (2.100) will be sufficient (see the discussion iii Section 5.7.4.3). R.eplacing in Eq. (2.99) the Haniiltonian operator by its classic analog and noting t.liat dk" = ti-"dp" ( h = h/27r, h is Planck's constant), we can

The classic limit

66 rewritc Eq. (2.99) as

[-

]

H (rN.pN) kBT

drNdkN

(2.101)

wherc! we split the suiri ovcr permutations Pk such that the identity perniutation Pk = 1 is treated sepa.rately, thus giving rise to the first summand in J3q. (2.101). Note that in front. of the second summand of Eq. (2.101) the operator likt‘‘ instcad of ‘b(kl)l’k177 arises whcrc the “+” sign rcfcrs to Bosons and the ”-” sign to Feriiiions. This t~ccoinmpossible by realizing that the sum over permutations in Eq. (2.99) generates N! terms. The integer N! is inevitably an even number regardless of N . If we treat the identity permutation separately, only N! - 1 terms are generated by the sum over permutations in Eq. (2.101), which turns out. to be an odd number independent of N. Therefore, syminetry properties of the wave fiinction dictate that in a fcrinionic systam thc fact.or in front, of the second term in Eq. (2.101) ttiust always be tiegative, wliereas a positive coritriLutioI1 results for a syst.et11 of Bosons. One may derive a more explicit expression for the second summand in Eq. (2.101) by realizing that the slim over all permutations (except, the ideiitity permutation) can be rerraiiged such that terms resulting from a permutation of pairs of particle momenta, triplets, quadruplets, and so on are grouped together. Each of these suiiis contains products of two, three, four, and so on factors of t,hc gcnernl form h

A

A

where (2.103) is the thermal de Broglie wavelength, r;, = J r i j= ( IT, - rjpil is the distance between particles i and j , and “ + I ’ and ”-” refer to Bosons and Fermions, respectively.

67

Elements of statistical thermodvnamics

To rationalize Eq. (2.102), cousidcr as an cxarriplc an operator Pk that exchanges the momenta of molecules 1; and j. From Eq. (2.101), one realizes that h

n N

Fk

exp (-ikN r N ) =

Fk

exp ( - i l ~ lT. I )

1=1

-

exp (-ikj

n

*

T , ) exp (-ikj

. vJ)

N

x

exp(-ikl

.rl)

(2.104)

/=l#i,j

It is t,hcn clear t,hat,, in Eq. (2.101), N - 2 of the N fnrtms in thc intcgrcmd survive, namely

becsiisc

n N

cxp (-&

. T , ) r’xp (ikl . T I ) = 1

(2.106)

l=lfa,j

Thus, the surviving t,erms have precisely the form of the complex exponential fiinction in the integrand of Eq. (2.102) if we replace the wave vector of a molecule by its momentum. The only difference between Eq. (2.105) and Eq. (2.102) is that for general permutations of three or inore wave vectors the indices on momentum and on the distance vector may be different. The general form of t,hc resiilting terms remains unalt,crcd as thc rcxlcr may vcrify: which is a bit tedious Init straightforward. The Gaussiaii iii the integmncl of Eq. (2.102) represents the kinetic part of the Hainiltonian fuiiction because of [see Eq. (2.100)]

The right side of Eq. (2.102) is then obtained as detailed in Appendix B.6.3 [see Eq. (B.103)]. Focusing from now 0x1 oiily on pair-wise permutations, it is immediately clear from the previous discussion that, for each permutation k, ++ kj,N - 2

The classic limit

68 iritcgrals of thc form

Jh3

J J/p2cxp 2n r

do

(--)

dpsinqdpdh' == 1

A3

0 0 0

(2.108)

arise because the permut,ation does not affect, N-2 wave vectors and therefore Eq. (2.10G) holds for those wave vectors. Hence, as far as the momentum integration is concerned in Ec4. (2.101), one is left only with the kinetic part of the Hamiltonian function as the integraiid. Realizing that nioirientum space is isotropic and homogeneous for a system in therriiodynaniic equilibrium, one may convcnicntly IISC sphrical coordinatw, which gives rise to a product of N - 2 t m n s of thc form prcswtcd in Eq. (2.108). In addilion, a factor

f-1 exp 146

(-%)

exp

(-2)

=k 1 p exp

(2.109)

appears, which is caused by thc pair-wise permutation as discussed above. Hence, we can recast Eq. (2.101) as

(2.110) whcrc

Q,l

is thc "c'lassic'' partitioil function givcii by

(2.112) is thc configuration intcgral. From Eq. (2.110), i t is clcar that Q N &cl in the limit r k [ / A -+ 00. For most fluids, A = U(lO-'a) or smaller where a is the "diameter" of a spherical fluid molecule. As r k l = U (a):the second term in Q. (2.110) vanishes to a good approximatioil provided m and T are not too small [see Eq. (2.103)]. However, it is important to realize that this does not hold for ideal gases. Because there is no interaction between thc molecules of an ideal gas, the

69

Elements of statistical thermodynamics

Boltzrriarin factor cxp [-(I (r’)/ k ~ T docs l iiot prcvciit molcculcs ~ I I apI proaching one another closely. In fact,, because in an ideal gas I/ (r.”)= 0 configurations are conceivable in which two or more molecules occupy the same point in space. This, in turn, implies that in an ideal gas the s e p aration between any pair of molecules may and will become inuch smaller than the thermal de Broglie wavelength regardless of T and rn. In other words, qiiantmn cffr:ct.s arc muximized in a scniic-lassic idcal gas compared with riorlideal fluids. In Eq. (2.110), wc dclibcratdy ignorcd nddit,ional tcrms arising from thc permutation of triplets of momentma,quarlruplets, and so on because we will be interest,ed mostly in situations in which rij >> A. Because these higherorder terms involve sums over products of three and inore factors of the form exp ( - r r ; / A * ) , their contribution to the seniiclassic correction to vanishes rapidly. We note in passing that a simple graphical method can be devised to derive explicit forins for the contributions from triplet,, quadruplet, and so on pcrniiitations. A detailed discussion of this t,cchniquc is, however, beyond the scope of this chapter. It is instruct.ivc to suiiixiiarixc thi: above itiialysis, which is it bit irivolvcd at certain points, in a more qualitative manner. The argument is based on the well-known fact that, in quantum mei-hanics one may associate a wave lengt,h

h A=(2.113) lPl with a free particle of mass rn and momentum p . Using [see Eq. (2.100)) (2.114) we realize that (2.115)

where Eq. (2.103) has also been used. In Eq. (2.114) the far right side is obtained by invoking the equipartition theorem, which states that, each of the (three) translational degrees of freedom of the particle contributes an ainoiint of k ~ T / 2to the total kinetic energy &in. The important point about Eq. (2.115) is that apparently the thermal de Rroglie wavelength A is a measure of the size of the quantum mechanical wave packet X associated with the (free) particle. Hence, one may argue that a classic description is adequate whenever the mean distance between the particles M l/@ ( p density) is larger than the size of the wave packet, i.e., larger than A. Typical examples of fluids where, on the contrary, quantum corrections are important

-

The classic limit

70

arc Hz or (liquid) Hc. In both cascs, thc h i t 1 inolcculcs liavc a small rriass tn and exist, a t quite low temppratures T. Therefore, in the classic limit, tlierrnal averages in the graiid mixed isostress isostrairi enseinble may be cast as / d r N O ( r NN. ; sz) p ( r NN: , sz)

(0)=

(2.116)

Nsz

where 0 ( r NN, : sz) is a microscopic analog of the macroscopic thermal average (0).For example, taking 0 ( r NN, ; s.) = U (@" N , sz), (0)= ( U ) would be the configurational contribution to the internal energy, which is to sav that U = $ N ~ B T ((1).In Eq. (2.116)

+

Y ( r NN; , sz)

=

1 N!A3Nycl exp

[g][ exp

Tzz A 7Ds'z

kET

]

exp

[- u

( r N ; N . sz)

kBT

1

(2.117) is the probability dcrisity in thc grand mixed isostress isostrain ensemble replacing its quantum statistical counterpart pjjnrsz in the classic limit where

is the classic analog of the quantum statistical partition function defined in Eq. (2.37) and 2 (N, sz) is the configuration integral already introduced in Eq. (2.112).

2.5.4

Laplace transformation of probability densities

Equations (2.116)-(2.2 18) can he rewritten in a slightly different, way, which permits to derive a general relation between partition functions in various mixed isostress isostrain m i a ~ m b l ~ Notice, ~. for cxaniplc. that we may define

x

where

J drNO( r NN; ,

s.)

cxp

[-

IJ ( r NN; , s,

~ B T

'1

(2.119)

71

Elements of statistical thermodynamics

is thc partition function of‘ t hc g r a d caiioiiicd enscrnblc in tho classic limit.. Noticing that in the classic limit s, is continuous on t,he interval [0,00], we may thus rewrite Eq. (2.116) as

replacing in Eqs. (2.116) aid (2.118), C . .. 8.

--+

Jds, . . .. Comparing thc 8.

previous expression with the Laplace transform of a fimclion j (t), namely M

(2.122) we notice that except for the prefactor in Eq. (2.121) both expressions are formally equivalent if we make the identifications 1 = s,

(2.1234 (2.123b)

where s is positive semidefinite because r,, 5 0 on account of mechanical stability (see Section 1.3). If the variance of the distribution of (0(s,)) around its maxinium (0(s:)) vanishes so that one may replace (0(s,)) in Ey. (2.121) by (0(s,)) 6 (sz- s:), the integration in Eq. (2.121) may be carried out and one obtains (2.124) wherc Eq. (2.80) has also lxen eriiployed. Tlic equivalence between (0(s;)) and (O(r,,)) may be interpreted as a reformulation of the equivaleiice between statistical physical enst?rnl>lrsdernonst8ratedin Section 2.4.

Reviews in Computational Chemistry Kenny B. Lipkowitz &Thomas R. Cundari Copyright 02007 by John Wiley & Sons, Inc

Chapter 3

A first glimpse: One-dimensional hard-rod fluids 3.1

Introductory remarks

In Chapter 2, we saw t,hat the configuration integral is t8hekey quantity to be calculated if one seeks to comput,e thermal properties of classical (confined) fluids. However, it is immediately apparent that this is a formidable tsaskbccausc it, rqiiiras a cdciilat,ion of Z , which tiirns oiit, to involvc a 3 N diiiicrisiorial i~itcgrationof a horrcndously coniplcx intcgraid, nanicly thc Boltzmann factor exp [ - C l ( T ~l )k ~ T [see ] Eq. (2.112)]. To evaluat!e 2 we either need additional simplifying assumptions (such as, for example, meanfield approximations to be introduced in Chapter 4) or numerical approaches [such asLs, for instance, h,lont,e Carlo computer siniulations (see Chapters 5 and 6 ) : or integral-equation techniques (see Chapter 7)]. There is an alt,ernatiw, however. It consists of employing a sufficient.ly simple model for which the configuration integral can be computed analytically without having t o take recourse to additional simplifying assumptions. The iinniediate disadvantage of such models, on the one hand: is a certain unavoidable lack of realism as far as experimental systems are concerned; they may therefore seem to be of little or no iise to the practitioner. On the other hand, if iiot ovcrsirnplificd, thew iriodcls soriictiriies permit a surprisingly deep insight into the fundamental physics governing in a qualitatively similar fashion a more complex model or even experimental systems. Based on this notion and reemphasizing the pedagogical impetus behind this work, we find it instnictiw to begin ;A deeper disciission of thermal

73

74

A first dimme: One-dimensional hard-rod fluids

properties of coiifiiicd fiuids by considoring oiic‘ of tho sirriplest nontrivial models still capable of embracing the basic physics cha.racteristic of these systems. This model coiisists of one-dimensional rods of length d without internal (e.g., spin-like) degrees of freedom, that is “niolecules” that cannot orient themselves as, for example, in tlie one-dimensional Isiiig model. However, in the model considered below, niolecules are not, restricted to discrete sitcs on a onc-dirncnsional lattkc: hiit, mav irinvp continiioiisly in spwc. In ddition, a pair of rods is riot itllowcd to overlap 011 accouiit of “liar#! rcpulsive interactions, t,hat, is, we arc dealing wilh a om-dimensional model fluid whose properties are completely determined by entropic effects. We defer a more detailed discussion of this latter issue to Section 3.2.1 where we consider statistical thermodynamical aspects of our model system. As we shall also see below: the confined hard-rod fluid exerts a stress on thc confining “siirf;~ccs”that dcci~ysto tsha (ncgat.ivc) h l k prcssiirc! as t,hc distaiica t)c:twccm the siirfaccs iIicrcasta. Moroovcr, tlic strtxs oscillatcs as a function of substrate separation with a period roughly equal to thc rod length. The oscillations may be iuterpret,ed jts fiiigerpririts of a confinernentinduced structure (i.e.? inhomogeneity) of the ha.rd-rod fluid. As we shall demonstrate later in Section 5.3.4,this structure really is a stratification of the confined fluid. That is, in confinement, fluid molecules tend to arrange thcir centers of mass in individual layers. Stratificat,ion is perhaps thc most proniincnt, structural feature causcd by solid surfaccs that arc separated by a distance compara.ble with the range of intermolecular interaction potentials. However, we should also emphasize at t.he outset of this chapter that the confined fluid is not suitable for the study of yet. another feature of central importance to us, namely confinement-induced phase transitions. This is because one can rigorously prove t,hat, in general, one-diniensional systems cannot undergo discontiniloils phasa changos [ 161. However, this apparcnt, lack of rcalisrn is outweighed by the arialyticity of thc current niodel systcrn and i t.s capability l o reproduce otlicr important fea,turesof more sophisticated models or even experimental systems siifficienbly realistically as we pointed out above. Our analysis in this chapter is based upon the original work by Vanderlick et al. (25j and has in part, bccn adopted from t,he book of Davis 1261.

75

Pure hard-rod bulk fluid

3.2 3.2.1

Pure hard-rod bulk fluid Statistical thermodynamics of hard-rod fluids

Let us begin by introducing a systeni of one-dimensional rods of length d whcrr t h intcnction hctwccn a pair of rods is rlcscribcd l y thc intcrmolcciilar potential

That is to say the potential just prevents any pair of rods from interpenetrating. For convenience, we treat the fluid as a thermodynamically open system such that its equilibrium properties are determined by t,he grand potential [cf., Eqs. (1.32) and (1.51)]

where the last term expresses the niechanical work exchanged between the oncdimcnsional fluid of "voliimc" I, and itti surroundings. On accoiint of tlic diriicrisiori ol' our systciri the bulk strcss has dirnciisioris of cncrgy per unit length rather t,han unit voiunle as in a corresponding threedimensional system. The connection to the microscopic level of description is then provided by the standard relation [cf., Eq. (2.81)]

where Zcl is defined as in Eq. (2.120) replacing, however, A3N by AN because of the dimension of the system aiid because our molecules have only translational degrees of freedom. For the same reason. the configuration integral is given hcrc by

where the configurational potential energy is given by [see Eq. (3.1)] N

N

(3.5) Thus, it is apparent, froin Eq. (3.5)that I/ depends on the hard-rod configuration z N only through interinolecular distances {qj}. Thus, we can apply the

76

A first KlimDse: One-dimensional hard-rod fluids

aiialysis dcvclopcd it1 Appendix C.2 arid rcwritc thc corifiguration integral as [see Eq. (C.24))

Because of the form of the intermolecular interaction potential introduced in F4.(3.1)we realize that the Roltzmann factor in the integrand of Eq. (3.6) can be zero if any pair of hard rods overlap; it will be equal to one, however, if this is riot the case. Hence, we cilri rtwljust the integration limits in Q. (3.6) to restrict the range of integration to those regions in which the Boltzmann factor does not vanish aiid rewrite Eq. (3.6) as

J

J

t4v- d

(L-d)/2

Zld =

N!

[- L+(2N - l ) 4 / 2

d z ~

13 -d

dzN-l . . .

[-L+(2N-3)4/2

dz2

(-L+3d)/2

J

22-d

dzl

(- L + d ) / 2

(3.7) At this point., it is corivenient to introduce a transformation of variables zi

4

Tj = f i

+ -21 [I, - (22 - 1)d.J

so that we can rewrite Eq. (3.7)as

=

( L - Nd)N

(3.9)

Equat,ion (3.9)permits us to verify that properties of our model system are completely determined by entropy. This becomes apparent by considering the st,atist.ic:alcxprtssion for tlic iritcrnal crierby, riairicly

nJ N

= -1N ~ R T

2

+1 2ld .

n=l

L/2

-L/2

dzJ/ ( z N exp )

u kN> (-r)

77

Pure hard-rod bulk fluid

which shows that U consists of only a kimtic coiitributiori. Equation (3.10) is consistent. with the equipart,it,iontheorem axsigning a kinetic energy of kBT/2 to each of the degrees of freedom of the N molecules. In any permissible configuration: no pair of hard rods is permitted to overlap on account, of the illfinitely hard repulsion between both rods [see b.(3.1)]. Therefore, the configurational potential energy ( U ) vanishes in Eq. (3.10). Hence? it follows from Eq. (15 0) t,hatt 1 F = -NkRT - T S (3.11) 2 is completely determined by entropy S apart from temperature, which a p pears to be a trivial scaling variable. It. also follows from Eqs. (2.120) and (3.9) t h t ,

-

=d =

such t,hat,we obtain

c m

1

N=O

= -knT(-)

exp

[fl] ( L- Nd)N kBT

M

I, - M d

(3.12)

(3.13)

froin Eqs. (3.2) and (3.3) for the bulk equation of state (i.e., the negative bulk pressure S,).

3.2.2

Virial equation of state

Assuming that the density of thr hard-rod fluid is below the density of a closepacked configuration, that is N d / L < 1, we may expand

in a power series, which may be reinserted into Eq. (3.13) to give

78

A first alimDse: One-dimensional hard-rod fluids

In Eq. (3.15) we iritrodiicc>the cquation of state of thc ideal gas of hard rods

as

T:'

=-kBp

(3.16)

where the mean density is given by

(3.17) (3.18) is the Ic-th virial coefficient of thc hard-rod gas. Because the rricriders of lhe set {&} are all positive sernidefinite, it is clear that Tb is a monotonically decreasing and continuous funct,ion of the density for all ijd < 1. Not unexpectedly, the hard-rod gas cannot undergo any phase tramitions at any density as we already pointed out in Section 3.1. The expression for the virial coefficients can also be obtained in a different fmhion (sw also Scc. 5.7.4). In t h s cam. the dcrivatioii dcparth from

-

=,I

= exp

(-&) (-g) = exp

(3.19)

where the last equality follows because fl is a homogeneous function of degree om! in L (scc Saction 1.6.1). Expanding t,hr cxponant,id fiinction, wc may rewrite thc previous cxpressiori as k

m

( L - ,vd)N -N N! I

k=O

N=O

(3.20)

where the far right side follows directly from Ecl. (3.12) arid the definition of thc activity 2 ex11( p / k ~ T/A) (3.21) Following Rowlirisori [27] arid McQuarric arid Rowlinson [28], we write as ai unsutz m

(3.22) j=l

Inserting Eq. (3.22) into Eq. (3.20), we obtain

z

= 1 -L

(biz +

L2 + b3,'3 + . . .) + 2

+ 2b,hz3 + . . .>

I;? -- ( b y + . . .) + 0 (2) 6

= l+(L-d)z+

( L - 2d)2 2

2

r t

( L -3d)3 6

+ 0 (z4)

i3

(3.23)

Pure hard-rod bulk fluid

79

Coiriyaririg in this cxprcssiori coefficients of cyual powcr ill z, it. follows aftcr straightforward but, somewhat tedious algebraic manipulations that

d bl = - - 1

(3.24a)

L

(3.24h) (3.24~) However: following the discussion at the beginning of this section, we wish to express 7 b in a power series in p rather than one in terms of the activity. To accomplish this we notice from Eq. (3.20) that (3.25)

where we also used Eqs. (3.20) and (3.22). We now make another ansat: expressing (3.26) z = alp azp2 (13j?3 O ($)

+

+

+

which wc inscrt. into the far right sidc of Eq. (3.25) to yicld

-

+

+

+

p = -61 ( U I ~ u.2p2 a 3 3 . . .) -2b.2 (a'fp2 + 2u,u27? + . . .) -3b3 (a;$ . . .) 0 ($)

+

+

(3.27)

Eqiiating in this cxprrssion terms of cqiial power in i j on hot,h sides, we obtain

(3.28a) (3.28b) (3.28~) Inserting Eqs. (3.28) into Eq. (3.26) allows us to reexpress the activity in terms of the set of the original expansion coefficients {bj}. Using the resulting expression for z and irisertirig it into Eq. (3.22) eventually gives us

A first nlimme: Onedimensional hard-rod fluids

80

thc dcsircd cxparisioii of q.,in tornis of

d 2-3dlL - 2 (1 - d / i , ) 2 -

H

powcr scries iii 7,

(3.30a)

( 2 - 3d/ L)2 _ -d2 9 - 26d/L (3.30b) 3 (1 - d / 1 5 ) ~ (1 Comparing these expressions with the equivalent ones in Eq. (3.18), it is apparcnt that, thc Iattcr arc indcpcntlcnt of thc systcm s i x , whcrcas t,hc foririer still depend on tlie ratio d / L . This is l~ecauseEy. (3.18) was ohtaiiied directly from the equation of state, tha,t, is, from Eq. (3.13), which involves a summation over all particle nurnbers. In ot,ht:r words, we took the thermodynamic limit, NIL = const, N , I , 4 00 prior thoexpanding 7 b in a power series in p. The coefficients ( L ) and ( L ) , on the other hand, were obtained from the first few tcrms of the expansion in Eq. (3.23). In other words: starting frorii J3q. (XU),we arrive at. the filial exprcssioris in Eys. (3.30) without taking the t,IierrIio~yIiarIii~ liiiiit ariywhere during the ent,ire derivation. However, we recover I32 and B3 given by Ekl. (3.18) by noticiiig that for the bulk I, can bc made arbitrary large so tha.t, d / L << 1 and thereforc -

d2

(3.31a)

.Id2 - 3d2 = d2

(3.31b)

which are in complete agreenient with the previously obtained results presented in Eq. (3.18).

Pure hard-rod bulk fluid

3.2.3

81

Bulk isothermal compressibility

It is also instructive to consider fluctuations in the one-dimensional hardrod fluid. Focusing on density fluctuations one realizes from Eqs. (1.81)and (2.75) that the isothermal compressibility is a quantitative measure of such fluctuntions. For t hc onr-dimansiond fluid ronsidcrcd in this scction, wc may dcfi11c

(3.32) by analogy with Eq. (1.81). Identifying in this expression N with the average number of fluid molecules ( N ) , we may employ the virial equation of state to obtain [see Eqs. (3.15) and (3.18)]

Alternatively we may use the definition [see Eq. (1.81)]

(3.34) for the isothermal compressibility of the one-dimensional hard-rod bulk fluid. From Eqs. (3.2); (3.3), and (3.12), we find that

Remembering that 3 also depends on p , we may differentiate t.he previous expression one Inore time to obtain

(3.36) so that we filially arrive at,

L

[

Kh=kgr

]

(N2) - (N)2

(N)2

(3.37)

82

A first glimpse: One-dimensional hard-rod fluids

whcrc Eq. (3.34) has also bccn used. Coriiyaririg Eq. (3.37) with its couritcrpart that is valid for a three-dimensional system [see, for example, Eq. (5.78)], it turns out that the two expressions are identical except for the volume V . which, in the three-dimensional system, replaces the variable L in Eq. (3.37). Equations (3.32), (3.33), and (3.37) may be combined to give

(3.38) where we also used the definition of the variance of the mean particle number int,rodiic-cidin Eq. (2.75). Bccaiwc the physically scnsihlc density range of the hard-rod fluid curnplies with the inequality

O
(3.39)

it follows from Eq. (3.38) that

(3.40) That, is to say, density flnctiiat,ions (rciative to t,hc mcan dcnsity) vanish as oiic approaclics tlic limits of cithcr variisliing tlcrisity (i.c., tho hard-rot1 idcal gas) or the density of a close-packed hard-rod fluid (ix., jjd = 1). Notice that at jjd = density fluctuations assume a niaximuni of

(3.41) but, rcrnairi finite over the entire dcrisity range dcfiiied by tlic iricquality in Eq. (3.39). Hence, density fluctuatioiis do not diverge t.0 infinity, which implies the absence of a critical point.

3.2.4

Density distribution

Despite the absence of capillary condensation, the onedimensional hard-rod fluid is still so useful because we havc an analytic expression for its partition function [see FK~. (3.12)] that permits us to derive closed expressions for any tnhcrmophysicalpropcrtv o f irit,crcst. Onc siich qiiantity t,hat is closcly r e I a t d to the isothermal compressibility discussed in the preceding section is the particlenumber distribution P (N), which one may also employ to compute thernioinechanical properties [see, for example, Eqs. (3.65) and (3.68)]. Moreover, in a three-dimensioiial system P ( N ) is useful to investigate the system-size depeiidence of density fluctuations as we shall demonstrate in Section 5.4.2 [see Eq. (5.8O)l.

83

Pure hard-rod bulk fluid

As wc dcrnonstratc in Appcndix (3.3: P ( N ) sliould lw Gaussian in thc thermodynamic limit,. In Appendix C.3.1, we present an argument showing that P (N) should always approach the Gaussian limit regardless of the specific form of the partition function (see also Section 5.4.2). Moreover, even if we do riot, know the partition function, the discussion in Appendix C.3.2 gives us the Gaussian limit of P (N) by applying the Moivre-Laplace theorem to thc Barnoiilli distrihiition charactmizing n gcnrral nicasiirrmcnt procrrjs [SCC Eq. (C.37)]. Hc-IICC,as wc arc givcii an aiialytic exprwiori, it S C ~ I I I S wortliwhile to apply the analysis detailed in Appendix C.3 to the one-dimensional hardrod fluid. Our starting point is the probability to find N hard rods in a (one-dimensional) bulk system of “volunie” L given by [see Eq. (3.12)j

-

1 =exp

-

d

(-)k”T pRr

( L - iVd)N N!AN

where t>hesecond line follows with the aid of Eq. (3.9) ancl the last line is a direct consequence of Eq. (3.19). Rewriting in Eq. (3.42), N! according to Eq. (B.6). one realizes that P (N) may be recast. formally as

(3.43)

is a monotonically decreasing function of N becaiisc p < 0 and N d / L < 1.

(3.45) on the other hand, increases monotonically with N because the current analysis is based on the explicit, assumption of validity of the classical limit, characterized by the inequality

d A

->1 according t o our discussion in Section 2.53.

(3.46)

84

A first glimpse: One-dimensional hard-rod fluids

The quarititics I$ ( N ) arid I$ (N) arc introduced only to split P (N) into monotonically incrPnsiry and decrpminy contributions, respectively, such (N) P* (N) must have a maximum at some value Ry that the product P~J which we seek t,o determine iiow. That the ext,remum of Pfi (N) Pu (N) rriust be a maxinium follows from the fact that both P,-, (N) and Pu (N) are positive semidefinite over the entire physically seiisible parameter range. Clcarlv, thc nccossary conditioii for t+hcmistmcc of this maximum may be stated as d P ( N )/d N = 0. However, bccausc tlic fuiictions P (N) a i d lnP(1V) share the same monotony, it turns out. to be more convenient to determine the inaximum of In P ( N ) rather than that of P (N) itself. Ft-oin Eq. (3.42),we obtaiii

PN + N l n ( L - N d ) In P ( N ) = - 1nZ + -

kBT

- N In N

+ N - -21 In (27~N)- N In A

(3.47)

which we may differentiate with respect to N. Solving theii

(3.48)

gives us an expression for the rhemical potential as a function of

-

-

N 1Vd 1 1- -i = -I In .A ~ B T I,-Nd L-Fd 2N hiscrting this crliiatim into Eq. (3.47)>wc obtain P

(3.49)

- = 111

111 P

(N) =

- 1115

m,namely

+ N 111 ( L - Nd) - N hi N + N - -21 111 ( 2 r N )

(3.50)

which satisfies Eq. (3.48)as it inustoarid as the reader may verify for himself. For the second-order derivative, we obtain 2d

I, - Nd -

Nd2

( L- ~

1 +. , N (1, - Ed)2 2N

--

L’

From Eq. (3.51)one can also verify that in general

d

- 1N)

~

(3.51)

85

Pure hard-rod bulk fluid

in a straightforward fashioii that. turns out to be algebraically a bit tedious. Hence, if we consider these expressions in the thermodynamic limit, it follows that the leading term in Eq. (3.52) is

(3.53)

lIlP(N)

=

Oo

lIIP(F)+Ctc=l

=

1 d”lnl’(N)l

dN”

TL!

1 L2 lIiP(T) - 277 ( L -

N=~J

(N

m)2

-q2

(3.54)

or, equivalently, if we take the antilogarithm of the previous expression we

The previous expression should be compared with J3q. (C.29a) where we emphasize that, Imlikc Eq. (3.55),Eq.(C.29a) was dcrivcd wit,hoiit cmploying a syccific form of thc canonical ciisciiiblc partition function &. Moreover! thc discussion in Appcridix C.3.1 - rcvcals that for a Gaussian distribution like the one given in Eq. (3.55),M = (N) [see Eq. (C.32)]. Hence, we can rewrite the argument of the exponential function in Eq. (3.55) [see also Eq. (3.38)]

L2

I

N (I,- iVd)2 - pd (1 -

-$= - 1

L

d

(3.56)

using t.hc definitions of arid ON given in Eqs. (3.17) and (2.75), respcctivcly. Hence, we stx: from Eqs. (3.55) aiid (3.56) t,liat

(3.57)

86

A flrst glimpse: One-dimensional hard-rod fluids

aiid dctcrrninc: P ((N)) such tJitlt, P ( N ) is propcrly normalized (see Appendix C.3.1). This approach eventually yields

(3.58) according to the arguincrits giver1 in Appendix C.3.1. In closing this saction the roatlcr should a1so apprecintc the fact, that. P (N), as it, may be determined from Eq. (3.50), does not equal P ((N)) = l / & a ~ , which we obtain from the normalization condition. This is because in reaching Eq. (3.58) we took the thermodynamic limit, and triincated the Taylor expansion of P (N) after the quadratic term in Eq. (3.54).

3.3 3.3.1

Hard rods confined between hard walls Aspects of statistical thermodynamics

The analysis of the virial expmsion of the bulk stress in the preceding section showcd that thr! systmi-size dependonce of t.hc birial cocfficicnts in Eqs. (3.30) was an artifact 1)cc:ansc: tho tlicrniodyniLinic.:limit. was riot takcn propcrly iii deriving those expressions. In other words, the ratio d / L does not. have any physical meaiiirig as far as the bulk fluid is coiicerncd. Turning our attention now to a hard-rod fluid confined between hard walls! this situation changes because now the system boundaries become physically significant in that they define the space of a one-dirnensional pore accommodating the fluid niolmmles. To emphasize this we rcplacc L by the distance between the pore walls s,. The evaluation of the configuratioii iritegral proceeds in identically t,he same fashion as in Secbion 3.2.1 so that we ohtain the equivalent expression (3.59) Zla = (s, - Ard)N from the Analysis in Appendix C.2. However, wc now havc t.o amend this axpression by thc condition that Nd must. riot exceed s, for Eq. (3.59) to be rneanirigfiil because the yore is completely filled if N d = s,. To implement, this additiona.1 constraint into our statistical thermodynamic: treatment, we replace the grand canonical partition function derived in Eq. (3.12) for the (infinitely large) bulk fluid by c

c do

=

N=O

(s, -

N d ) N8 (5, - N d )

(3.60)

87

Hard rods confined between hard walls

whcrc

(3.61) is thc Heaviside function. The link to therinodynamics is provided by Eq. (3.3), where, however. the (exact differential of the) grand potential is now given by

di2 (T,p, L ) = - S d T

-

N d p + TLdS,

(3.62)

and TIis the stress exerted by the fluid on the confining substrates (i.e., the pore walls) [cf., Eq. (1.63)]. Hence;

l.)\sz

kBT N (s, - N d ) N z N 6) ( s , - N d ) - - T -C ~ Z - N d N! N=O kBT 00 (s, - N c ~z N) ~ Nl 6 (s, - N d ) W

c.-

I

Y

(3.63)

N=O

where the activity z was defined in Eq. (3.21) and we used the fact that

(3.64) and the Dirac &“function” is defined in Eq. (B.75). From that definition, we conclude that the second surnniand in the above expression does not rontribiitc to TI so that wr may rcwritc it as W

(3.65)

where

(3.66) is the stress exerted particles and

011

the substrat,es by a confined fluid accommodating N z N (s,

-

P (N;s,)= y

I

- N d )N 8 (s, N!

-

Nd)

(3.67)

is the prohahilitmyof finding N particles in a pore of width s,. At this point it seems worthwhile to point out that P ( N ; s , ) ,unlike its bulk counterpart P ( N ) . docs not comply with a Gaussian dist,ribution like

88

A first glimpse: One-dimensional hard-rod fluids

the o ~ i cbivcIi in Eqs. (3.55) or (3.58), say. The rcasoIi is that we cannot take 9, to infinity because it. represents the degree of confinement. Hence, a variation of s, inevitably changes the physical nature of the confined fluid, whereas the properties of the hulk fluid must 71ut depend on a corresponding variation of L. The cutoff represented by the Heaviside function in Eq. (3.67) prevents P ( N ;5,) from becoming Gaussian except in the bulk limit where s, + 00. In otlicr words, for thr confi11c:d hard-rod fliiid, thc thormorlynamica limit docs riot exist in tlic scwc of thc second liru of Eq. (3.54).Thcrcforc, the one-dimensional confintd hard-rod fluid must be considered a somewhat pathological model. Another quantity of interest is the mean pore density = ( N ) /sz. From Eqs. (3.62) and (3.60), we find

CG

(3.68)

3.3.2

“Stratification” of confined one-dimensional fluids

On t h basis of thc pravioiis thcord ical trcatmcntt of onc-diniansional fluids, in both the bulk aiid thc coiifiricd stiltc, we now discuss sonic key features of these systems. Specifically, we shall consider the confined fluid to be thermally and inaterially coupled to the (infinitely large) hulk so that in thermodynamic equilibrium both systems arc maintained at the same chemical potential p and tcmperatiire T. However, in the absence of any attractive interactions between either fluid molerules or between a fluid inolecule and the hard siibstratt-, t,he latter tiecomes a more or less t.rivial parameter that does riot affcct thcrnial propcrtics of tliv l i d - r o d fluid. Bccausc of Eqs. (I .SO) and (2.79). we have

(3.G9b) where we used Stirling’s approximation [see Eq. (B.7)]. With the definit,ion of the bulk stress given in Eq. (3.13), Eq. (3.69b) can be rearranged to give

(3.70)

89

Hard rods confined between hard walls

1

0.8

2-

W

n

0.4

0.2

0

0

1

2

3

4

5

6

7

8

Figure 3.1: Probability density P (N;8 , ) as a function of pore “width” -9,. Curves are plotted for N = 1 ( - ) 1 N = 2 (- . -.), and N = 3 (... ).

for the activity [see Eq. (3.21)] of both the bulk and the confined fluid because both are assumed to be in thermodynamic equilibrium. Fixing the hiilk dcnsity to a siifficicntly high flidd tlcnsity Fd = 0.75, we calciilatc a corresponding bulk strcss ~ , d / k , T = -3 (i.c., a bulk prcssurc P,d/k,T = 3) from the equation of state given in E‘q. (3.13) using also the definition of the mean density [see h.(3.14)]. With these numbers, we calculate a value of zd = 60.26 for the activit,y, which w e shall use in the calculat8ionspresented in this chapter. In addition, we fix t.he pore “width” to a nanoscopic range of 1 5 s,/d 5 10, which is small enough to illustrate confinement. effects as wcll as tha onset of ordinary hulk hohavior.

Wr begin with a bricf discussion of I’ ( N ;s,)! which rcprcscnts tho probability of finding N molecules in a pore of “width s,. Plots in Fig. 3.1 show that this quant,ity is zero as long as s, is not large enough to accommodate N inolecules as one would have guessed. If .s, exceeds this threshold, P (N; s,) increases quite rapidly until it, assumes a niaximum at some characteristic value of s, at which the pore is just becorning large enough to accommodate N 1 molecules. Because of the competition with larger pore occupancies,

+

A flrst glimpse: One-dimensional hard-rod fluids

90

12

10

a

6

4

2 0 -2 -4

0

2

4

6

8

10

4

6

8

10

1

0.8 tQ

U

0.6 0.4

0.2 0

0

2

s,d-l Flgurc 3.2: (a) AH Fig. 3.1, but

but for the mean pore density

c@ = 0.75.

fur

tho &joining

(h) A s (a), the bulk density

prcssiiro f ( s z ) .

0. The horizonal line demarcates

91

Hard rods confined between hard walls

p (N; s,) dccrcwm if s, cxcccds this threshold urilil it vaiiishcs for a sufIiciently large pore width. If s, is not, too large (i.e., for typical nanoscopic pore widths), E is completely determined by just a few of t,he probability

distributions { p (N; s,)}’~. Hence, for nanoscopic one-dimensional pores, we are in a position to calculate the secalled disjoining pressure defined as [cf., Eq. (5.57)]

1(SZ)

f -71 ( S Z ) -

1.b

(3.71)

which is a measure of the excess pressure exert4edby the confined fluid on the substrates. For the current system, we calculate 71 from Eq. (3.65). Clearly, as the distance between the substrates becomes n~~ocroscopic in magnitude, that is. in the limit (3.72) lirn f (s,) = 0 Yg-CJ

the inipact of corifiiierrieiit, diniinishrs wider thwc miditions and therefore fluid properties become indistinguishable from those of the corresponding bulk systeni with which it, is in thermodynamic equilibrium. If, on the other hand, for wifficiontly smdl s,, j’(Y,) < 0 tho niochanical stat(: of t h fluid is such that it tcrirls to pull the corifiiiirig substrat,cs together, whcrcas if .f (s,) > 0. khe tendency is to push t.he substrat;es a.part4.As we shall explain later in Section 5.3.1, f (s,) is in principle accessible in experiments employing t,he surface forces upparatus (SFA). Plots in Fig. 3.2 show that in general f (s,) is a rionmonotonic function of s, tacillating with a period that is slightly larger than the rod length d. These osc:illatoionsrcflect, the inhornogcneoiis st,riic:t,iire of t.he confined fluid. In fact, as wc shall scc Lclow in Section 5.3.4,oscillations iii t,he disjoining pressure are fi1igerprint.sof sLrat.ificat.ionof three-dimensional confined fluids, which is the tendency of fluid molecules to forin individual layers parallel with the confining subst.rate. This structural interpretation is somewhat indirect, however, unless one correlates it, with variations in the local density. We shall establish this correlation later in Section 5.3.4. For our cnrrent- purposes, it. siiffires to concliirle that, the confined fluid is apparent>ly higlily inhomogcncous if t h c c1c;grcc of confincincnt is sufficiently largc (ix.: if s, is sufficiently small). This notion is supported by plots of the mean pore density in Fig. 3.2(b), which we calculated from Eq. (3.68). The plot in Fig. 3.2(b) indicates that, like f (sz), i j oscillates as a function of the pore “width” s, where we notice again t.hat

1. The period of the oscillations corresponds roughly to the rod length d

2. This period increases with s,.

92

A first rrlimDse: One-dimensional hard-rod fluids

0.7405

0.740

0.7475 -

0.7465

I

60

70

80

90

100

110

I

120

Figure 3.3: Mean pore density as function of substrate separation sz. The full = 0.75. line is a plot of tlie right side of J3q. (3.74) where The increase of the period of oscillations visible in t,he plot in Fig. 3.2(b) rcflccts that for largcr porc widths thc! hard rods can pack morc comfnrt.ably because of t,he larger space available to them. Comparing plots in Figs. 3.2(a) and 3.2(h) ravcals that, minima in S (s,) arid coiIicidc, as far as thc substrate scparatioii is conccrncd, at, wliich thcy occur. At tlitae values of s,, a confined fluid of n, “layers” appears to be strained minimally in the sense that, the pore space is not large enough to accommodate n. 1 such layers. Hence, over the associated range of substrate separations, the iiiean pore density decreases until .s, eventually becomes large enough to ;Icconimodate n + 1 layers. A s s, increases, “layering” b e corrim lcss and lcss distinct, as indicatcd t y thc damping of tho oscill~tions in tlie plots of Figs. 3.2(a) and 3.2(1>),which is a direct consequence of the diminishing influence of the substrate surfaces. However: we notice that, another subtle confinement effect prevails up to the largest substrate separations considered in Fig. 3.2(h). As the plot clearly shows, t.he mean pore density approaches the bulk density from below. This approach is rather weak with increasing substrate separation so that one expects this pheiiomenoii to prevail up to substrate separations exceeding the largest one considered in the plot in Fig. :3.2(b) by more than an order

+

Hard rods conflned between hard walls

93

of iiiaguitudc. This slow dccay of‘ thc ~iicaridcnsity can be ratiorializcd as follows. Consider a hulk system at, some density &,. Confining this system to a pore with hard walls is equivalent to putting the bulk fluid between two imrnobile hard rods whose centers are separated by a distance s,. Hence, the space accessible to the fluid is smaller by some excluded volume on account. of the presence of the two “wall particles.” The rnagnitude of the excluded voliimc is d / 2 for tach pair of wall-fluid particlw so that, s;a = s, -

Id -22

(3.73)

(3.74) where we assign half of the total excluded volume to each particle (i.e., wall and fluid) of the interacting pair. A plot of 7 in Fig. 3.3 shows that the simple excluded-volume argumeiit presented above is capable of explaining the slow decay of the mean pore density toward its bulk value. In other words, the slow dac*ayis nothing Init, iitrivial c:ffwt that c.oiil(l cdiiiiinat,cd 1b-y properly rescaliiig the pore volume according to Eq. (9.74).

Reviews in Computational Chemistry Kenny B. Lipkowitz &Thomas R. Cundari Copyright 02007 by John Wiley & Sons, Inc

Chapter 4

Mean-field theory 4.1 Introductory remarks

In Chapter 2, we developed statistical thermodynamics as the central theory that cnahlcs 11s in principle to calciilittr thcrmophysicsl propertics of macroscopic c:orifinc:cl fluids. A kcy fcaturc! of statistitral t,hcrrriod?rriariiicsis mi enormous reduction of information that takes place as one goes from the microscopic world of electrons, photons, atoms, or molecules to the macroscopic world at which one performs measurements of thermophysical properties of interest. This iiiformatioii rednction is effected by statistical concepts such as the most probable distribution of quantum states (see Section 2.2.1). By int,rodidng tho notion of varioiis sttatistical physical ciisembles in Section 2.2.1, wc saw that wc can rnakc tlic quaiit.mil incdiariical trcatirierit consisknt with several constmint,s imposed at the macroscopic level of description. That way we obtain a n understanding of a thernia.1 system at the microscopic: level; that is, we can interpret thermodynarnic properties in terms of the int,eract,ion between the microscopic coiist,ituents forming a macroscopic system. Thc not,iori of an cnscinhlo was first siiggrxtcd by Gihhs’ in a rrrnarkably insightful manner. In the preface of his book Elementary Principles in Statistical Mechanics Developed with Special Reference to the Rational Foundation of Th,errnodynanii.csGibbs writes [29]: “We consider especially enserriblcs of systems in which the index (or logarithm) of proba.bility of phase is a linear function of the energy. This distributioii on account of its unique importance in the theory of statistical equilibrium, I have ventured to call ‘Josiali Willard Gibb (1839-1W3),professor of Inatherriatical physics at Yale University and one of the “founding fathers” of statist,ical mechanics and vector calculus.

95

96

Introductory remarks

cunonicd, arid tlic divisor of the cncrgy! tlic modulus ol’ the distribution. The moduli of distributions have properties analogous to temperature . . ..”* In his writings Gibbs based statistical thermodynamics on entirely classical concepts when, for exaniple, he writes about therniodynaiiiics as pertaining to the “department of rational mechanics” [29]. Nevertheless he knew that classical physics was not entirely adequate. In fact, Gibbs expresses a dccp iindcrstanding of t,hc st,at,iisof statist,icaI nice-hanics of his era in writing:

“III thc prcscnt. statc of sciciice it S C ~ I I I Shardly possiblc to frame a dynamic theory of molecular action which shall embrace the phenomena. of thermodynamics, of radiation, and of the electrical manifestatioiis which accompany the unioii of atoms. Yet any theory is obviously inadeqiiate which does not take account, of all these phenomena.” Being a contemporary of the nineteenth century, Gibbs could obviously not have had any conccpt. of qiiantiini machanics and its role in laying 8 sound foundatioii of moclcrii statistical t1iormodyrianiit:s. hi this iiiodcrri formulation, t8heclassic Gihhsian version of statistical thermodynamics does! however, emerge as a limiting case as our discussion in Section 2.5 reveals. Howwor, rogardlcss of whctticr we bas(: our trc:atmc:nt on classical or qua,ntimi statistics, the development of statistical thermodynamics in C h a p ter 2 shows that the partition function is a key ingredient of the t,heory. This is because we may deduce from it explicit expressions for the thermophysical properties of equilibrium systems that may be of interest. At its core (and irrespective of the specific ensemble cmployrd) , the partition function is detcrmincd by t.hc Doltzinann factor exp [-U ( r N ) where the tot.al coiifigurational potcutid (xicrgy I/ ( r N turris ) out. to bc it horrendously COIIIplex function of the configuration r N on account of the interaction between the microscopic constituents. Bccausc of theso irit4crilctioiwccrtain spatial arrarigcrrients of thc microscopic constituents will turn out to be more likely than others. This is immediately apparent from a. purely energetic perspective because it will be more likely to find a pair of atoms or molecules at separations from one another corresponding to the niiniinum of the interactioii potential rather than at very short distances where the partial overlap of their electron clouds gives rise to mora or less st,rong rcpiilsion. On xcoiint of tha intmactions, particla

2See,for example, Eqs. (2.46), (2.51), and (2.65)of this work.

Van der Waals theory of adsorption

97

positions appear to bc cor~elatedarid one would ncctl to know tlic corrclations in configuration space to eventually evaluate the configuration integral [see Eq. (2.112)] (and with it the classical partition function &). From the discussion up to this point the reader will surely appreciate that a rigorous, first-principles calculation of the partition function for a macroscopic system is generally precluded eveii in the classical limit with the exception of rather simple models of limited usefulness (see Chapter 3). However, the problem of calculating the partition function (or the configuration integral) in closed form bccomcs tractdde if we introduce as a kcy assumption that corrclatioris bctwceii rriolecules arc entirely xiegligiblc. In effect, cadi niolcculc is then cxpos~dto a 7 1 ~ ~ field 7 1 exerted on it by all other molecules and external fields such as confining substrate surfaces. Hence, the same mean-field can represent a large number of different configurations which we no longer have to worry about explicitly. The introduction of a nienn-field approximation reduces the problem of calculating the configuration integral in Eq. (2.1 12) greatly because the coinplex N-dimensional integral then factorizes into singlcparticlc contributions that. arc obviously far easier to handle computationally. This retilarkable reduction of thc computational problem is particularly important in the case of confined fluids as we shall demonstrate in this chapter.

4.2

Van der Waals theory of adsorption

Correlations in confined fluids essentially originate from two soiirccs. On account of fluid-fluid interactions one would imrnediately anticipate shortrange order t.o exist in confined fluids similar to the bulk. This short-range order manifests itself in, say. pair (or highrr-order) correlation functions [30]. However, because of the external potential representing the confining sub strates, the fluid in their vicinity is highly inhomogeneous (see Section 5.3.4 for a comprehensive discussion). This iiihoniogeneity niay also be viewed as a manifestation of cormlat,ions in t2hcflnid phase. One may, for cxampl(>,regard t h (planar) siibstrat,c(s) of a slit-pore as thc surface of a sphcrical particle of irifiiiitc radius. Tlie confincd fluid plus the substrates may then bc perccived as a binary mixlure in which macroscopically large (i.e.. colloidal) particles (i.e., the siihstrates) are immersed in a “sea” of small solvent molecules. The local density of the confined fluid niay then be interpreted as the mixture (A-B) pair Correlation function r e p resenting correlations of solvent niolecules (A) caused by the presence of the soliit,c (B). As we shall demonstrate in this section, a simple mean-field theory of

98

Van der Waals theory of adsorption

coiifi~icdfiuids itiay be dcvelopctl bascd on thc assumption that both typm of the aforementioned correlations (i.e., fluid-fluid and fluid-substrate) can be disregarded altogether. As a result one obtains an analytic equation of state of the van der Waals type for the confined fluid that permits one to understand some very basic features of sorption experiments. As an illustrative exaniple, wc discuss below the volumetric determination of the phase hrhavior of a piirc’ fliiid confiiicd to a niwoporons silica glass ( s ~ c ,for cxample, Fig. 4.1) carried out by Tlioiiirrics a i d Firidwegg [31]. If one is interested only in properties of the pore phase as a whole, such as the excess adsorption and the phase behavior, and not in properties that depend explicitly on local density, or on interrriolecular correlations, then it may be sufficient to neglect entirely variations in the local densit-y. It is in this spirit that we present a simple model for the adsorbed phase that yields closed expressions for the free eiiergy and for the equation of state. The iriodel is a dircct cxterisiori of \wi dcr Wat1.1~’iiiodcl for thc bulk fluid. For simplicity we adopt the slit-pore gcorriet ry, although the significant conclusions of the study are not altered for pores of other shapes. As we shall demonstrate below, some features of the Thomnies Findenegg experiment [31] can indeed be understood in terins of a simple van der Waals equation of state.

4.2.1

Sorption experiments

Using a volumetric t.edinique, Thommes and Findeiiegg [31] have measured the excess coverage r of SFe in cont,rolled pore glasses (CPG,see Fig. 4.1) as a function of T along subcritical isoclioric paths in bulk SFe. The experimental apparatus, fully described iii Ref. 31! consists of a reference cell filled with pure SFG and a sorption cell containing the adsorbent in thermodynaniic cqiiilihriiim with hiilk SFGgas at, a givcn initial tcmpcratiirc Ti of thc! fluid in both cells. The pressure P in the reference cell and the pressure difference A P between sorption and rcfcreiice cell are measured. The density of (pure) SFG at, is calculated from P via an equat,ion of state. A t the beginniiig of an experimental scan, the reference-cell volume is adjusted such that A P ( T , ) = 0; that is, the theriiiodynamic state of SFG is the sanie in both cells. The temperature is then lowered from r, to a ncw tcmpcratiirc Ti+, = 7: - AT. at, which Ai’ f 0 hccaiisc morc SFc is adsorbed. The volume of the sorption cell is t,hen adjusted t o reestablish the original condition AP = 0 at the new temperature Ti+l. The change in the excess coverage is given by A r o( pAV, where AV is the change in the volume of the sorption cell between r, and T,+, . Measurements are repeated by lowering the temperature in a stepwise fashion uiitil the bulk coexistence temperature Txb of SF6 for the given isochore is reached and the gas in the

Mean-field theory

99

Figure 4.1: The spongelike structure of a typical sample of controlled-pore glass used in sorption experiments. The silica matrix in lighter gray surrounds the mesopores appearing in darker gray. rcfercricc cell bcgins to condeilsc. By I I I C ~ I I Sof a high-pressure niicrobalancc technique [32],the absolute value of I' (T,) is determined in an independent, experiment so that r ( T ) can be calculated from AI' for each temperature T, L T ITxb. Froin a theoretical perspective, these experiments are particularly appealing for two reasons. First. CPG is characterized by a very narrow poresize distribution. As pointed out in R,ef. 31, 80% of all pores have a diameter within 5% of thc avcrngc radiiis of thc (approximatcly) cylindrical porcs. If connections bet,ween individual pores are disregarded, the phase behavior of the adsorbate should therefore closely resemble that of an adsorbate in a single pore (see Section 4.2.4). Second, the CPG employed by Thommes and Findenegg [31] is mesoporous, as reflected by the nominal average pore radii of 24 nm (CPG-240) and 35 nm (CGP-350). As these values are large compared with the range of fluid-substrate intermolecular forces, the inho-

100

Van der Waals theory of adsorption

iriogcricous rcgiori of the porc fluid is imich sinallcr than thc horriogcneous region. Therefore the shape of the pores should not matter greatly. This notion is corrobrated by the fact that the structure of CPG is largely bicontinuous having a nearly vanishing inean curvature. The characteristic pore widths of CPG, on the other hand, are still small enough such that confinement effects can be expected to prevail to a significant extent.

In the meantime it. has also become feasible to synthesize other mesoporous inaterials that differ from CPG in tha,tXhey consist of individual, disconnected cylindrical pores. These so-called SBA-15 or MCM-41 silica pores can be synthesized using a technical-grade triblock copolymer as the structure directing template in aequous H2S04 solution and tetraethyl orthosilirate as the silica soiirce [33, 341. After calcination [35], one obtains a regular array of iiidividud cylindrical pores as illustratod by the tmrisrriissiori electron micr0gTaph.s (TEMs) shown in Fig. 4.2.

A key result of the sorption experiments conducted by Thommes and Findenegg concerns the pore condensation ljne Txp( p b ) > Txb( p b ) at which porc condcnsation occurs along a suhcritical isochoric path Ph/Pch < 1 in the bulk (I& arid /&b arc thc dcxisity of this isochorc arid thc bulk critical density, respectively). Experimentally, T,, (pt,) is directly inferred from the temperature dependence of r (T), which changes discontinuously a t T,,(yb) (see Ref. 31 for details). The pore condensation line ends at the pore critical temperature Tcp (rigorously defined oiily in the ideal single slit-pore case) [31]. Because of confinement TcDis shifted to lower values with decreasing pore size. If, on the othcr hand, the porc becomes large, Tcp--+ (the bulk critical teinpcraturc) aiid T,, -, T,, ( w c Fig. ci of Ref. 31).

Tc..

4.2.2

An equation of state for pure confined fluids

The Thomines Findenegg experiment [31] can be analyzed theoretically via an equation of state for the pore fluid, which can be calculated from the (Helmholtz) frcc energy of thc porc fliiid 3, givcn formally by Eq. (2.79). From a molecular perspective, 3 is linked to t,he configuration integral via Eqs. (2.111) and (2.112) assuming that the experiment is carried out. in a temperature regime where the classical treatment is adequate according to the discussion in Section 2.5 [seeEqs. (2.110) and (2.103)]. Moreover, Eq. (2.112) iniplies that molecules possess only translational degrees of freedom, which seems justified for SFG given its molecular structure.

Mean-field theory

101

Figure 4.2: Transmission electron micrographs of mesoscopic SBA-15 silica pores. Upper: along the pore axes; lower: perpendicular to the pore axes (361.

Van der Waals theory of adsorption

102 4.2.2.1

Perturbation theory

As we see from En. (2.112); the key quantity is the configurational energy U (rN)Iwhich we henceforth separate into the potential energy of an unperturbed (reference) system, Uo (r"'), and its perturbation represented by rl1 ( r N ) We . may then rrwritc Q. (2.112) as

where the angular brackets signify the ensemble average over the iinpertLiirhed probability distributioii 2;' cxp [-prr" ( T " ' ) ] . hi E.1. (4.1)

is the configuration integral for the reference system. Assuming the pertiirbation to bc siifficicntly small over thc tmpcraturc range of intcrrst, wc approximate the ensemble average in l3q. (4.1) by

For sufficiently high temperatures (i.e.l for sufficiently low values of l/T), we may truncate the expansion in Eq. (4.3)after the linear term. Coinbining then Eqs. (2.79), (2.111), (4.1), and (4.3) yields 1

F z F o + F 1 =--In~ B T zo

=

where we used the fact that .r ([!& /kBT expanded in a MacLaiirin series3 to give In (1 - x) = --z

+ (W"

(4.4)

<< 1 such that In (1 - z)can be

+ o (2) -p

(Ul)o

(4.5)

Henceforth, we consider a Lennard-Jones(l2,6) (LJ) fluid between two plane parallel solid substrates. The basic setup of our model is schematically depicted in Fig. 4.3. We awime the fluid-substmte interaction to be a pair-wise additive surn of L.1 poteiitials. As Lhc rcfcrcricc systcrrl wc take a hard-sphere fluid (diameter of)between hard-sphere substrates (diameter a8). Moreover, we "sinear" the hard spheres of the surface layer of each SThat is, a Taylor scries around x = 0.

Mean-field theory

103

Figure 4.3: Side view of model slit-pore showing wall atoms (white) wid fluid

molecules (gray).

substrate uniformly over the plane of the layer to obtain a hard wall that is iilfinitr!sirnally sniooth in traiisvcrsc (z,y) dircctions. The srncarcd fluitl-wall interaction thus depends only on the distance of the fluid molecule from the wall. The potential energy of the reference system can then be expressed as 1

uo = 5

N t=l

N If1

uk (l.J +

c e(4 N

G I

(4.6)

where the fluid-fluid (ff) contribution is given by

and that of the fluid-wibstmtc! by

In Q. (4.6),rij z Iri - rjl is the distance between a pair of hard spheres located at ri and r j , uf,= (uf g8)/2 is the distance between a pore molecule and a substrate atom in contact? and sz0 is the distance between the walls

+

Van der Waals theory of adsorption

104

(scc Fig. 4.3). We approximate t hc c:onfiguration intsgral of tlic rcfcrcncc system by where ZF)is the effective single-moleciile configuration integral. Notice that

Eq. (4.9) neglects correlations between molecules in the reference system and rnav therefore be considered a mean-field approxirnation in itself. We take 2,( 1'1 to be eqiial to the voliinie accessible to m y given riioleciile, as dictated 1)y &IS.( 4.6)-( 4.8): (4.10) 2p = A ( S d - 2UfS)- N b whcrr A is the awn of t,hc wall and

274

bG-

(4.11)

3

is (half) the volunic excluded to onc rriolcculc bv another. Hciic~,we have for the reference free energy =

~0

N

[ A (SZO - 20fs) - Nb1

(4.12)

The perturbation is similarly given by -

N

N

N

(4.13) where (4.14) and 4 s ( G ) = - 2TpsEfsu; 3d

[(g( +

s;co(Tfs - 2%)31

1

Of,

< 2,

S d

- UfS

(4.15) In Eq. (4.15):ps is the areal density of the solid substrate. The fluid-fluid perturbation is just the attractive term of the LJ pair potential. Likewise, the (original) fluid-suhstxatc pcrtiirhation is tho L.J attraction -lief, (qs/r) 6 . To be consistent with the smooth-wall approximation to the reference pot.ential. we average t,he fluid-substrate attractions over the (2,y) positions of the substrate atonis in the ylarics in which they lie. Wc suppose each substrate to coinprise an infinite half-space of atomic plaiies, separated successively by distance d. Approximating the sun1 over these planes by the Euler-MacLaurin formula (371 yields the expression in Eq. (4.15).

105

Mean-field theory 4.2.2.2

Mean-field approximation

Based on Eqs. (4.13)-(4.15) we have

where po(1) (r1)and po(2) (

T ~ , T are, ~ )

respectively, the local deilsity and the

pair distribution function in the reference system (211, which are related to onc anothcr through t,ho pair corrclation fiiiic-thn by Pf)

(T l , 7-2)

= P I ) (T1) P!)

( T 2 ) 9 (Tl. r2)

(4.17)

Equation (4.16) may be rationalized by recalling that

where we use the definition of the Dirac 6-function (see Appendix B.6.1) and the angular brackets denote an average in the canonical ensemble. Because we are seeking an equation of state at the mean-field level, we ignore intermolecular (fluid-fluid) correlations and set (4.19) Moreover, neglecting fluid-substra,te correlations, we take the fluid to be homogeneous throllghout t,he pore volume; that, is, we approxirnat,e the local density by

(4.20)

It is well established [38] that a fluid confined to a slit-pore is stratified (i.e., the fluid molecules order themselves in strata parallel with the substrates; see Section 5.3.4 for a more detailed discussion). Stratification diminishes rapidly with increasing distance froin the substrates, because of the decay of the fluid-substrate interaction; beyond a few molecular diameters cf,the fluid is essent,ially homogeneous, as computer sirnulatioris have repeatedly

Van der Wads theory of adsorption

106

dcrnorwtratcd (39-461. Wc tlicrcforc cxpcvt E l . (4.20) to be rcasonablc for mesoscopic pores ( S , o 2 lows). From Eq. (4.16) we then obtain

(4.21) The onedimensional integral defining Q (c) in Er4. (4.21) can be readily performed to yield

(4.22) where the energy scale is set by

(4.23)

<

Because of constrairits eriforced by the hard interactions, and G s z ~ / u f ~ . the evaluation of the double integral defining a,, (t)is algebraically more demanding. 1t.s cvaliiat,ion is thcrrforc clcfcrred to Appendix D. 1.l. Thp final rcsnlt of this c:alc:iilat,ion is

where Q, refers to the honiogeneous bulk fluid, which can readily be obtained by transforming the variables in the second term of Eq. (4.21) according to {PI ~ 2 ) { T I ~ 1 2 and ) reexpressing the t.hree-dimensional int<egralover ~ 1 in spherical polar coordinates. This yields

A comparison of Eqs. (4.24) and (4.25) shows that np (<) 5 U b , where the equality applies in the limit, ( = 00. The inequality is a direct, consequence of the weakened attractive field "experieiiced" by a molecule in confinement due to the presence of the substrates. which may, of course, compensate this loss of attractivity through the term Q (<) /kBT, that is, by virtue of its

2

107

Mean-field theory

chemical naturc (ix., thc constarit e,) at least- as far its the cquatiori of state is concerned as we shall see shortly. Inserting now Eqs. (4.12) and (1.21) into Eq. (4.4) gives

F

A (s," - 206) - h'b

- - kBT ="In

NA3N

+ I + - - 8 (0 kBT .4kBT (SzO - afs) (4.26)

For a fluid confined bctwccrr sniooth walls, tlic exact cliffcrcrrtial of thc frcc energy can be expressed as

where we used the definitioii of F given in Eq. (1.50) as well as thc expression for the (exact differantid of the) internal cncrgv of a. fluid confined between structureless, planar substrates surfaces displayed in Eq. (1.37). From Eqs. (4.26) and (4.27), we obtain the niean-field equation of state of the confined fluid as

where we also iisctf Eq. (4.20). Equation (4.28) tiirns out to Iw indcpcnclont to the perturbatmion.In Ey.(4.28) of the fluid-substrate contribution N Q we introduce the transverse pressure 91. In the limit + 00, np (6) = ah and Eq. (4.28) rediices to the well-known van der Waals equation of state for the bulk fluid

(0

<

(4.29)

4.2.3

Critical behavior and gas-liquid coexistence

It is well known that the van der Waals equation of state is qualitatively correct in the sense that it, is capable of predicting gas-liquid phase equilibria as well as critical phenomena. Mathematically speaking this is because the van der Wads equation may br pcrccived as a thirtl-order polynomial in p regardless of whcthcr wc consider the hiilk or a confined fluid. A s for the bulk. the location of the critical point of the confined fluid is determined by the conditions

Van der Waals theory of adsorption

108 which togcther with ECq. (4.28) yiclds

(4.31a) (4.31b) 1

f’cp

s,)

= - -- Pcb

(4.31

cu)

where Tcb, P&, and prh denote critical valucs. In the inequalities given ill Eqs. (4.31), the equal sign pertains to the limit, E = 00 [see Eq. (4.24))The shift of Tcp(6) to a valiie lower than Tcb is well estddished by niore sopliistic.;ttc:d t,lic!orim as well as c!x~)c~rirri~!iit~lly. Introducing “reduced” variables through the relations

(4.32a) (4.32b) p - - f)

(4.32~)

PCP

it. is possiblo t,o rc.writc Eq. (4.28) as

-

p-= - -8?$

3-p

32

-

(4.33)

Expanding P in a Taylor series around the critical point (i.e., around T = p = 1) and retaining terms up to t,hird-order derivatives, the resulting (reduced) cqiiation of state can hc cast, as 3 (4.34) P z1 1) [ 4 + 6 ( p - 1 ) + 3 ( p - 1)2] + -(p- 1 ) 3 + . . .

-

+ (T-

2

Consider now coexisting gas and liquid phases a t densities & and p i , respectively. If wc- t,akc t,hcsr dc!nsitit:s t,o hc siifficicntly dose to t h rritkal dcnsit4y, wc riiay writc

&

= 1

+t

(4.35a) (4.3%)

i;%XP = 1 - c

where c << 1 is a sriiall nu~iiber.We may then recast Eq. (4.34) for gas arid liquid phases in coexistence as 3 (4.36a) 4 + 6 ~ . + 3 ~ ~-8] 2 33 (4.361-3) 4 - 6 c + 3 c 2 ] -;c. 2

+

Mean-fleld theory

109

-

Howcvcr, if' gas and liquid arc in cocxist,oricc qy previous expression can be rearranged to give

by dcfiriition a i d the

where t (<) is defined as in Eq. (1.86) from which we conclude that the critical exponent p = for the order parameter Ap irrespective of (. In other words, in the irririicdiate vicinity of the critical point, the coexistence curve for the van der Waals fluid is symmetric.. but its location does, of course, depend 011 thc dcgrcc of confincmcnt,, that is, on t. To illustratc t h : I arigc- of tmnpcrdturcs owx wliich thv powcr law in Eq. (4.37) is valid, we plot the order parameter p!,, - ptt, as a function of T/T,b in Fig. 4.4. The coniparisoii sliows that the critical behavior as predicted by Eq. (4.37) prevails for temperatures that are about 10% lower than the (bulk) critical temperature. However, as one would have guessed, the scaling law docs not hold for 111uc:h lower ternperaturcs as one can also scc from Fig. 4.4. 0.4

0.3

?F I

0.2

-
0.0

0.6

0.7

0.8

0.9

1.0

Figure 4.4: Plot. of the order parameter p: - & as a function of T/T,, calculated from the pore coexistence curve plott.ed in Fig. 4.5 (-) and from the power law (- . -) [SCX &. (4.37)).

A somewhat, niore detailed discussion of fluid phase behavior becomes

Van der Waals theory of adsorption

110

possible or1 the basis of the full cocxistcncc c u r v e shown in Fig. 4.5 for both the bulk van der Waals fluid and one confined to a slit-pore where s, = 50. Figure 4.5 serves to illustrate t,he impact of spatid confinement on the phase diagram of a “simple!‘ fluid (i.e., a fluid in which molecules possess only translational degrees of freedom). These curves are obtaiiied by a numerical solution of Eq. (4.43) detailed in Appendix D.1.3. As is evident from the plots in Fig. 4.5, corifincniont, caiiscs a dcprwsion of t,hc critkal tompcratiira, whereas tlic critical dcrisiby rcriiaiiis uiialtcrcd. Thcac featurcs are consistent with the analytic expressions for Tcpand pcp given in Eq. (4.31). However, the reader should note that. the la,t.ter feature is not correct with respect. to corresponding experimental observations where the critical density is usually shifted to higher values and the coexistence curve of the confined fluid turns out to be narrower with reqpect to its hulk counterpart [31]. This r c f l c h thc fact that, with regard t,o nican dcnsitics, gas- and liquid-likc confiiictl phascs arc I I I ( . ) ~ Cdikc than iii thc bulk. Thc abscricc of a shift in critical density in the theoretical curves is caused by the fact that wit,hin t,he context. of the current perturbational approach the density dependence of the free energy remains the same in both confined and bulk fluids [see, for example? Eq. (4.26)], which shows that corifinenient effects arc solely restrictcd to the dcnsity-independent van der Wads parameter up (<). Howcwr, on thc positivc sido, wc arc now cqiiippd with equations of statc for both t,hc confiricd fluid [SCC Eq. (4.28)] arid its bulk counterpart [see Eq. (4.29)]. Together these equtions of state enable us to revisit the Thornmes Findenegg experiment at mean-field level.

4.2.4

Gas sorption in mesoscopic slit-pores

From a theoretical perspective, the Thommes Findenegg experiment 1311 can be represeiited by the equation /lh

(T,A) - ~p (T, pp)

0,

= CoIEt.

T

--t

T,+b

(4.38)

whcrc. T,, is t h bulk liquitl-gas c.ocxist,cmcc:tcinpcraturc. To dcrivc cxprcssions for pb and /ip, we differentiat<eEq. (4.26) [see Eq. (4.27)]

froin which

111

Mean-field theory 1.0 0.9

0.8

+3 \

I-

0.7 0.6

0.5 0.4

0.0

0.5

1.5

1.0

2.0

2.5

3.0

P/Pcb

Figure 4.5: Phase diagram of bulk (-) fluid (- . - .).

and confined (s, = 50) van der Waals

follows. Equation (4.40) is obtained from Eq. (4.39) using the fact that [see Eqs. (4.22), (4.24)] lim 9(<)= 0

(4.41~1)

lini up(() =

(4.4 11))

€--

t-+m

ah

Combining now Eqs. (4.38), (4.39), and (4.40), we get

(4.42)

As Eq. (4.42) cannot be solved explicitly for pp in ternis of pb and T, we must havc rccoursc to a nimicrical mcthod. which is dctailcd bclow. In the remainder of this section, we expres all quant,ities in tAhecustomary dimensionless (ix., “reduced”) units, where length is given in units of of and energy in units of ~ t f . In addition: for simplicity we set of = a, and d = a,/& assuming the (100) configuration of the face-centered cubic lattice for substrate atoms. Then in dimensionless units we have a b = 8 ~ / 3and p, = 1.

Van der Waals theory of adsorption

112

To solvc Ey. (4.42) for subcritical bulk isochorcs in thc terripcraturc rarigc Tcb 2 T 2 Txb, we need to determine r x b first by solving [see Eq. (4.29)]

(4.43) which has three real roots pt,l (T), yb2 (T), and (T) for T 5 are given analytically by the (Cardanic) formulas [37]

Tcb.

These

(4.45) As only the gas density &, 3 Pb is fixed, and the density of coexisting liqiiid and the coexistence temperature are unknown, we must solve Eqs. (4.43) and (4.45) siinultaiieously for Txband p:. We accomplish this numerically by a procedure also detailed in Appendix D. 1.3. Equation (4.42) can iiow be solved under experimentally relevant conditions [31]! that is, for bulk isochoric: paths (1% = const) and T + T i . Again we dcfcr B dctailcd description of thc numerical proccdurc to Appendix D.1.3. Once the numerical soliltmion has been found, we are in a position to calculate the excess coverage for the t,hermodynamically sta.ble pore phase wiu

r (T!Pb) =

(% - 2la w p )

(pp - m,)

(4.46)

which is the primary experimeiit,al quantity [31]. Remlts are plott.ed in Figs. 4.6-4.7 for various values of Efs and sz = 50 corresponding closely to the expeririiental pore width of CPG-240 where we assume up = 0.47 nm for SFs (471. If the walls are purely repulsive (&fs = 0), ( T .pb) < 0 (see Fig. 4.6). This situation corresponds-to “drying” because p,, < pb regardless of T and

r

Ph.

Mean-field theory -0.3

-0.5 -Oe4

113 1

I

I

-0.8

-0.9

'

-1.0 0.001

I 0.1

0.01

Figure 4.6: Excess coverage r (T,m,) as a functioii of t.emperature T for fluid ~ for representative bulk isochore f i / P & = confined by hard substrate ( E =~0) and 0.640. (a), -9, = 50; ( o ) ,Sz = lo2; (*), Sz = lo3; (0); S, = lo4; (-), + 00 limit..

<

If, 011 the other hand, the subst,rates are sufficiently attractive, one notices from the plots in Fig. 4.7 that r (T,yb) may eit.her vary continuously or discontinuously depending on whether the (bulk) isochoric path is suparor subcritical, respect,ively, with regard to the critical point of the conf i n d fliiid. Hcncc, discontinuities in tho plots in Fig. 4.7 indicatc capillary coiidciisatioii (cwporatioii) in the iiiodcl porc prior to coiideiisation in the bulk, which would, of course, occur a t bulk gas-liquid coexistence, i.e., a t (T- T x b ) / T x b = 0. Notmicealso that, at. temperatures higher than that corresponding to c a p illary condonsation: r (7'.p b ) incrmscs with docrcasing T . This is indicstivc: of a regime where one would observe growth of a wetting film, which at a mean-field level, manifests itself as an increase in overall density of an otherwise homogeneous low-density phase adsorbed all across the slit pore. For temperatures lower t,hari that a t which capillary coiidensat,ion sets in! I? ( p b ) turns out to be nearly indepeiident of T the lower T becomes. This reflects that, when sufficiently close to bulk gas-liquid coexistence, the pore is com-

Van der Waals theory of adsorption

114

lo

c

n

F-

W

L

1 -

L

0.001

0.01

0.1

Figure 4.7: As in Fig. 4.6, but for a fluid confined by strongly attractive substrate (Efs = 1.0)?Sz = 50, and bulk isochores h / p & = 0.545 (+), pb/pch = 0.595 (*), &/P& = 0.645 (o),pb/pcb = 0.675 (m), &//I& = 0.730 (o),and &/p& = 0.780 ( 0 ) .Solid lilies are intended to guide the eye.

pletely filled with a (homogeneous) liquid-like phase resisting compression.

An ,dclit,ional fcatiire oiic notices from Fig. 4.7 ronccrns thc rhangc (4.47)

a t the disrontinuity where p:,, and Pg,, are liquid and gas densities in the porr at cocxistcnrc. Thc plots in Fig. 4.7 clcarly show that A r (7') bccoincli larger when the density of the suhcritical isochoric path in the bulk is lower. The magnitude of AI? (2') is a direcl nieasure of lhe difference in density of coexisting pliases in the pore, which apparently are becoming inore alike the greater the proximity of the bulk isochore is t o its own critical point. This is another feature also prewnt in the parallel experiments of Thommes and Findenegg [31].

Lattice model of confined pure fluids

4.3

115

Lattice model of confined pure fluids

In t.hc prcccdirig sortion, we derived a mean-ficlrl cqiiat,ion of statc for confined fluids bassccl 011 the assuiiiptions of 1. Irrclcwaricc of fluid-fluid corrclatioils

2. Homogeneity of the confined fliiid Within the approach dcvclopcd in the prcvioiis section, ncit.hcr of these assurnptioiis (:MIbe replaced cwily by a morc realistic onc. Howcver, it t,urris out, that, if one abaiidons the continuous model fluid in favor of a discrete rnodel in which molecules are restricted to positions on a rigid lattice, the second of the above assumptions is no longer necessary to derive mi analytic expression for the partition fuiiction of the fluid. Latt,ice niodels are of great iniportance in a variet.y of contexts in statistical physics (see, for example, Ref. 48). The charm and usefulness of lattice models lies primarily in their discrete nature arid their syrnmetry. The discreteness rcdiires grcatly the niimbcr of configurations t,hat>nwd to be corisidcrcd iii evaluating the partit,ioii function. Their irilicrcrit syxiirrietry, if combined with short-range interaction potentials, may be exploited to derive a fairly sinall set of equat.ions describing the location of minima of the relevant thermodynamic potential that can be solved numerically by standard techniques. A key ingedient of this approach is that we deliberately neglect interinolecular correlations as before in Section 4.2.2. The current section is t,hcrcforc devoted t80discwssiiig thcsc fcat,iircs in sorno detail and illustrate the power of lattice rriodcls within thc scope of this book.

4.3.1

The model system

Here we consider a lattice model of a "simple" pure confined fluid, that is, a fluid composed of inolecules having only translational degrees of freedom.

The posit,ions of t,hesscmolcciilc=.,arc rcstrict*cdt o N - 71,71.,,7~, sitrs of a siniplc cubic lattice of lattice coiistaut E. Each site on the lattice can be occupied by one molecule at, most which accounts for the infinitely repulsive hard core of each molecule. In addition to repulsion, pair-wise additive attractive interactions between the molecules exist. They are modeled according to squarewell potentials where EE is the depth of the attractive well whose width equals t . In addition, the flnid is confined between two planar solid surfaces (slit- . pore) exerting aii external field on the fluid niolecules. Specifically, these solid

116

Lattice model of confined pure fluids

surfiiccs arc decorated with altmiatirig strongly (width G)arid weakly attractive stripes (width it,,,) that. mimic different chemical materials. Over the last decade, researchers have made substantial progress in controlling surface heterogeneities even down to the nanometer length scale. It is now possible to imprint specific geometric or energetic patterns on a surface experimentally at such a sinall length scale [4!+57]. The chemicallv striped substrate is t,hc most complex rnodrl siibst,rst,c that, w e wish t,o coiisidrr in this section. It incorporates siinplcr sulxtratc rriodels such as tlic purely repulsive one ( E l s = Erw = 0) or the chemically hornogcneous substrate (&fs = ~ hQ,= 0) as special cases. The latter are also of interest as reference systems. For the chemically striped substrate, the external field representing the composite solid inat,erial can be cast as Zj

> n,

where superscripts “[l]!’ and “[2]” refer to lower aiid upper substrates, respectively, and cy = Anx/% is a (discrete) diiiiensionless parameter specifying the misalignment of the upper relative to the lower substrate surface in the xclircction. Clcarly, if ck = 0, the two surfaces arc pcrfcctly aligned, whereas rnisaligiiiient is iiiaxirnurii for a = $ on account of the peridicit8ythat we in thc! :I:-direction. By rnisaligning the? towosiibstxatc siirfacr!q, assiimc for the confined latt,ice fluid can be exposed to a shear strain represented by.a strain tensor u = ( 0 0 0x2

@yl

.)

ax2

(4.49)

0

0

In Eqs. (4.48),we model the fluid-substrate interaction according to a squarewell potential; .qsand Efn represent the depths of the attractive wells associated with strongly and weakly adsorbing portions of the solid s u h strat,cs, rcrspwtivcly. H(mcc! oiir ciirrcnt rnocicl is very similar t o t,hc one depictrd in Fig. 5.7 iii the iwxt chapter. I3ecirusc of Eqs. (4.48)the latticcl fluid at any site i in the 2-2 plane is exposed t,o a total external field

a+ -= 4) ( . X i , q) = d11(Xi,

. I )

+ (9121 (Xi,8)

(4.50)

Mean-field theory

117

The Harriiltoriiari function of the latticc fluid iriay then be written as

H ( s N )= -%

*

CC N 44

sisj

i = l j#i

+ C @is* = C h (sj) N

N

i=l

i= 1

(4.51)

whcrc h, (si)is the singleparticlc Harriiltorliaii function that wc introduce for future reference in Section 5.8.2.In Ell. (4.51), si

= s (Xi,Z l ) = { 0,1,

lat,tice site occupied lattice site empty

(4.52)

is the occupation number; sV = ( s 1 , s2.. . . , S N ) is a specific occupationnumber pattern, which is a lattice-fluid configuration (and therefore the discrete analog of r N iritroduccd in Section 2.5), aiid u (i.) is th: riuiribcr of nearesbneighbor sites of site i. Depending on whether this site is located at the solid substrate zi = 1:n,or elsewhere on the lattice, the number of nearest. neighbors is 5 or 6,respectivcly, because of the simplecubic symmetry of our lattice. The confined lattice fluid is coupled to an (infinitely large) external reservoir of heat and matter. Hence, from an equilibrium perspective, the grand potciitid is the rclcvarit thcrrriodyriariiic potcutid [see Eq. ( 1.58)] whcrc N is equivalent to the nurnber of occupied lat,tice sites. The exact differential of R is given in Eq. (1.59), and the link to the microscopic level is provided by Eq. (2.81) where we need to replace the continuous variables s, by their discrete analogs nu (a= x, y, or 2). Hence, the discrete analog of %. (2.120) may be cast as

where we dropped f,he subscript, “cl” of S to simplify the notation and define N i=l

Becailsc of Eqs. (2.81) arid (4.53) wc liavc R((T,p,u)= -kBTlnZ = -kBTln

exp

(4.55)

as the exact expression for the grand pot,ential of the 1att.ice fluid where we note in passing that the dependence of R on the strain tensor u arises through the external potential @; [see Eqs. (4.48)-(4.50)].

118

4.3.2

Lattice model of confined pure fluids

High-temperature expansion

In the limit, of high temperatures one may derive approximat,e expressions for thermodynariiic properties of the lattice fluid introduced in Section 4.3.1. Here we are particularly interested iii the energy. Our interest, in this quantity is motivat,cd 1 y a 1at.w romparison hctwccn rcisiilts obtainod for t,hc meanfield 1at.ticc fluid to bc discussed iii Scvtion 4.5 aiicl corrc!spoiiding Monte Carlo simulations to be presented in Section 5.8. As we shall describe in Section 5.8.3,in these simulations, one has accesi to the grand potential as the quantity of key interest. only through therniodynainic integration along suitably chosen paths in thermodynamic state space. In this regard it will turn out to be iriiportant t.o consider the limit, of high temperatures as a reference siich that an analytic* axprcssion for thc energy in t.his limit, is highly desirable. To simplify its derivation we begin by transforming variables according to (4.56) s; 4si G 2Sl - 1

sucli tliat. the occupat,ioii-nuint)cr vcctor SN is rcplacctl by a spiii vcctor i P where the elements Zi = fl (“spin down” si = -1; “spin up” si = +1). Replacing in Eq. (4.51)occupation numbers by spins permits us to rewrite the Hamiltonian function in so-called “niagrietic” language as

-

-

N i l l j#i

where

.I

=

Eli -

4

i= 1

(4.58)

is the coupling constant,.

(4.59) is a (local) magnetic field

and EO is a trivial constant that. only shifts the zero of the energy scale. Except, for thc local magnctic field /,, Eq. (4.57) rcprcsmts t h r Hamiltonian functioii of tlic: wcll-known Isiiig niagiic:t. From Q. (4.53)we obtain the corresponding partition function of the Isirig magnet by replacing by its counterpart HI given in Eq. (4.57),where,

fi

119

Mean-field theory

however , tlic sumination over occupat,ioxi-xiiiiribcrpattcrns has to be rcplaccd by a summation over spin patterns (configurations). We may cast the expectation value of HI by differentiating Eq. (4.53)for the Ising magnet according to

' d

(HI) = kBTL-lnE:l

dT

= Z;'

Hlexp

I+)

(a

In thc limit, of wifficiontly high tcmprratmc! cvaluatod approximatcly iisirig the cxpansion

+ 0)

(-5) kBT

(4.61)

this cxprwsion may bc

(4.62)

exp(-z) = 1 - :r + 0 (2) such t.hat El in the denominator may be apyroxiniated by

where L is the total number of configurations. One realizes fairly easily that, the term proportional to 1/T niust vanish. To see this notice that the expression in brackets has a fixed value for each i. Now, when summing over all configirations, there will bc! onc othcr ronfigiration wherc all 11(2) neighbors of spin i haw thc smic value. but S, lias chitiigcd its sign. Therefore each of the ternis in brackets arises twice when the s u m over all configurations is performed: oiice with a positive and once with a negative sign so that all these contributions cancel each other in a pair-wise fashion. According to our initial astsiiinption of sufficiently high temperatures, we neglect terms of higher than linear order in 1/T. By a similar token we have in the numerator of Eq. (4.61) 1 C iIIexp (-,LIH,) = C H,- C H: + o (T-*)

W}

(+I

lcRT (+I

N

1

-kBT

C H;

(*'I

(4.64) which follows bccausc we wish t o retain oiily the lvadirig term in thc tamporatiire expansion and because

(4.65)

Lattice model of confined pure fluids

120

for reasons just cxplairicd. HCIICC,we iriay writo rrioxx: cxplicitly as

because of the symmetry between spin-up and spin-down states that we exploited already in Eq. (4.63). From Eqs. (4.61). (4.63), (4.64), and (4.66) we finally obtain

(4.67) Transforming back to the original lattice-fluid parameters may writc thc last. cxprassion as

Elf,

p, and

@i,

we

where we used the fact that %if (nz - 2) and nXny2sites have 6 and 5 newest neighbors, respectively. Siniilarly3the external potential Qivanishes a t n.,rzY (n,- 2) sites; at the remaining nSny2and nwnY2 sites, it assumes a valiia of -Efs and -Elw, rcspcctivcly [scc Eq. (4.48)]. Together, Eqs. (4.60) and (4.68) provide a means to calculate H in the limit T 00. At siifficieiitly high chemical pot,entials one may also expect all latticc sitcs to ba fiilly occiipicd siich that, si = (si) = 1 so that, in the liiriit of sufficiciitly high tcinpc:ratures oric has access to tlic int,crnal oiicrgy of the lattice fluid through the expression ---$

(->

where H is given in Eq. (4.51) mid Eqs. ( 4 . 5 4 ~ and ) (4.57) have also been used. Equation (4.69) serves as a suitable starting point for the calculation of the grand potential via thennodynamic integration (see Section 5.8.3).

121

Thermodynamics of pure confined lattice fluids

4.4 4.4.1

Thermodynamics of pure confined lattice fluids The Bogoliubov variational theorem

As wc arc nltimatcly intrrcstcd in thc phwc behavior of the confincd lattice fluid at arbitrary ternperatures we are seeking solutions of Eq. (1.76a),where, of course, w" = N-'R" on account of the discreteness of the lattice. Because of Eqs. (2.81) and (4.53) this amounts to finding the chemical potential pap corresponding t o the intersection between w" and w 4 a t a given temperature T as we explained in Section 1.7. To identify those configurations we take recourse to tliermodynainic perturbation theory [58]. Lct us assurric that we can split thc Ilarr~iltor~ia~~ fuIictiori iiito two contributions accordiiig to

H

(8%

A) = Ho (8")

+ AH1 (&)

(4.70)

wli& H" and HI arc the Harnilonims goveriiirig sorric rcfcrcncc systcin ( s u b script 0) and a perturbation (subscript l ) , respectively; 0 5 A 5 1 is a dimensionless coupling parameter that permits us t o switch continuously between the reference system and the system of interest. Because of Eqs. (2.81), (4.53). and (4.70) we may also write

(4.71) Assuming that t.he perturbat,ion is not. t.oolarge we may expand R (T,p; A) in a hlacLaurin series in terms of the coupling parameter A, t>hatis, as a Taylor expansion in A around A = 0 (i.e., the reference syst,eni). Retaining in this expansion terms up t o first order, we obtain

52 (T,p; A) = R (T,p; 0 )

"b:IA="

+A -

+

P2)

(4.72)

From Eq. (4.71) it is irrirncdiatcly appitrcnt that

(4.73)

122

Thermodynamics of pure confined lattice fluids

which is the perturbation Hariiiltoriian avcragcd over coilfigurations rcyrescnting t,he unper-turbml system. The reader should notice the similarity between these expressions and t,he ones derived in Section 4.2.2.1. Thus, set.ting A = 1 in Eq. (4.72) and using the definition of H I given in Eq. (4.70), we may write

R (T.y;1) = R (T,11; 0) + ( H - Ho)A,o

+0

(A2)

(4.74)

As oiie ran verify by differentiating Eq. (4.73) with respect to A, (4.75) which is a direct. consequence of the concavity of the exponential function arising in the grand canonical erisernble part,ition function [see Eq. (4.53)] 1591. Eqnatioii (4.74) may then be recast, as f-l(T.p; 1)

I R ( T ,p; 0) + ( H - Ho)A=o

(4.76)

which constittut,c!sthc cclchratd Bogoliiihov incqnality [59, 601. It shall scrvc as a basis for the treatment in Section 4.4.2.

4.4.2

Mean- field approximation

Equation (4.76) stmates that. the grand polential calculated by thermodynamic perturbatioii theory is an iipper hound for its “true” value. Hence, one wishes to minimize the deviation between R (T,p; 1) and R (T,p; 0) ( I 1 - H O ) ~ , ~ ) . In this case, we take ns an ansatz

+

(4.77) where Q i is an a priori unknown external reference potential (i.e., the mean field) to be determined by minimizing the right side of Ekq. (4.76) with respect to the set {a;} 1581. Putting together Eqs. (4.51), and (4.77) we obtain [see Eq. (4.7G)l

Mean-field theorv

123

Moreovcr, wc liavc

= fi{l+exP[-s]}

(4.79)

in Eq. (4.53) by HO (.") [see Eq. (4.77)] which follows by replacing H (.") and noting that si = 0 , l is discrete and double-valued. To determine the yet unspecified external potential { Q i } in the ansatz [see Eq. (4.77)], we realize

is the nccessary condition dctcrmining t,hc opt,imaI external field. Becaiw Eq. (4.80) must hold for arbitrary infiriitcsirrial chaiigcs dQa, wc find froni Eqs. (4.71), (4.76), (4.78), and (4.79) (4.81)

where pi is the mean occupat,ion number (equivalent to the dimensionless lord donsity in units of t 3 ) . Bcc.ausc: of thv cnsc~r~il~lc avc~agctaken in Eq. (4.81), pi is continuous on the interval [O, 11 unlike s, itself (see above). Equation (4.81) can be rearranged such that (4.82)

which shows that Q, can be interpreted as an eflective local chemical potential that also reflects the symmetry inherent. in the lattice model, that is, Qi = p for pi = If p, > f , Q i < p and vice versa. Because of this interpretation, Eq. (4.77) constitutes a mean-field approximation, i.e., Ho (&) in Eq. (4.77) disregards all intermoleciilar correlat,ions. Using Eqs. (4.78), (4.79), auld (4.82), we finally oLt,ain

i.

"<"[pN]

=

N

k B ~ x [ p ~ l tn (pl -~p i ) l n ( l - - p i ) ] i= I

Af

(4.83) i=l j # i

i= 1

124

Thermodynamics of pure confined lattice fluids

froiri Bogoliubov’s iricquslity [scc Q. (4.76)1, which dcfincs thc grarid potential of the lattice fluid at mean-field level where # {PI, h,. . . ,p ~ } To . arrive att Eq. (4.83) we also assumed that. [see Eq. (4.78))

=

(4.84)

Clearly, Eq. (4.84) ignores correlations in the occupation-number patterns (i.e., the configurations of the 1at.tice fluid). This (independent) assumption is. however, required to be consistent with the mean-field ansatziii Eq. (4.77). Finally, the best estimate of 52 is found by minimizing thc functional 52 [p”] with respect, to f l for fixed values of T and p; that is, we reqiiire the functional derivative [26, 301 to satisfy (4.85)

in the sense of Callen (sea above) 1581. Fquations (4.83) and (4.85) eventually lead to the coupled set of Euler-Lagrange equations

kBT In Pi - ER 1 - Pi

44

pJ

+

- / I = 0,

i = 1, . . . ,

(4.86)

32’

which we solve numerically according to a recipe detailed in Appendix D.2.1.

4.4.3 The limit of vanishing temperature 4.4.3.1

Bulk lattice fluid

In thc limit of vanishirig tciiipcraturc, tlic rncan-ficld trcatrricnt of the currcrit model becomes exact. To see this we begin by examining the bulk system, which is obtained as a special casci of Eq.(4.86) for {@,} = 0. In the absence of an external field, all elements of the vector p” assume the same value p and each site on the siinple cubic latt,ice has v ( i ) = 6 nearest neighbors. Hence, Q. (4.86) can be simplified to (4.87)

which lias a solutiori Tc, $ arid I/& -= -3 dcfiriirig tlic critical poiiit of thc bulk lattice gas in the customary dimensionless units (distance in units of P, e n e r u in units of E R , temperature in units of € E l k s ) . In the spirit of our discussion in Section 1.7, the set of points (4.88)

125

Mean-field theory

defines llie cocxistcricc line of thc bulk lattice fluid aloiig which gascous and liquid phases coexist. In dimensionless units, Eq. (4.87) can then be recast as 1 p=-(Tx-p) 6

(4.89)

where x is defined through the expression

(4.90)

mid --oo 5 :c 5 00 because 11 E [O, 11. Thus, plotting t-hc right sidcs of Eqs. (4.89) and (4.90) along p, ( T ) as functions of x, we obtain three intersections located at -zo, 0, and zo corresponding to densities 1 - ~ ( a o )f , and p (zo), respectively. Evaluating the second derivative [see Eq. (4.83)]

Hmce, R has a maximum at, p = $ corrcsponding t,o an iinstable thcrmodynamic stmate.The two remaining solutions must t,herefore correspond to minima of R, and in light of the symmetry inherent in the expression in Eq. (4.83) for px (T), it is clear that f2 [p (x~)]and R [ p ( 1 - so)] are the equal values of the grand potential of coexisting “gas” and “liquid” phases. Furthennore, it is clear from Eq. (4.90) that in the limit T = 0 (i.e., 20 = 00)

liiii p ( T o ) = lini 11 - cxp (x0)l-l = 1

.q)+m

IUdOo

(4.92)

and thcrcforc 1 - p (.7:o) = 0. By a similar analysis: we ohtdn for the limiting stable solutions of Eq. (4.89) (T = 0) for p > pc arid p < pc, the respective values p = 1 (liquid) and p = 0 (gas). We conclude that,, regardless of the value of the chemical potential, the only stable solutions of Eq. (4.89) are these two, respectively corresponding to the completely filled and completely empty lattice. This conclusion, reached on the basis of our mean-field approach, is confirmed by thc: ezact cxprcssion for R . From Eqs. (2.81), (4.51), and (4.53) wo may write

126

Thermodynamics of pure confined lattice fluids

whcrc thc prirnc is attwhcd to the sunimatioii sign to indicate that iii the 0, the summation the ground-state configuration sf is oniitted. As T logarithmic term in Eq. (4.93) vanishes, leaving ---$

N

=

-N

2=1

+

( ~ E R s , ~ p.90)

Nw,

(4.94)

where the second line follows for the special case of a bulk lattice fluid (@, = 0 ) in which SO is discrete and bivaliied according to the definition given Eq. (4.52). In &. (4.94) we also define the grand-potential density of the bulk lattice fluid w,. The expression on the second line of Eq. (4.94) is therefore equivalent to the mmn-field one [see Eq. (4.83)] in the limit T = 0 wherr p = so as WP rcaoncd abovc. 4.4.3.2

Lattice fluid confined to a chemically heterogeneous slit-pore

Thc abovc considerations may IJC extciidcd

t o the situation of primary interest here: The fluid is constrained in one dimension ( z ) by plane-parallel substrates that are chemically decorated with weakly and strongly adsorbing stripes. These stripes alternate periodically in the r-direction, so that the external potential depends only on zl and z, [see Eq. (4.52), Fig. 4.81. Thus, for a given value of z,and 2, the occupation numbers do not vary with the y-coordinat,r of thc lattice sit,c. That is, by symmetry all dcnsitics along lincs parallel with the y-axis are equivalent. Thus. using Ey. (4.86) we can write for a particular site i

where the factor of 2 comes from the two neighbors in the y-direction and { p 3 } are the C (i) = 3.4 densities at t,hc nearrrc?st,-neighborsites of site i in thc T-z planc. Using thc dcfiriition of .7: givcri in Eq (4.8'3), we can rewrite Eq. (4.95) in dimensionless variables as p. - 1

'-2

( T r - p1eff)

(4.96)

where 1 ~ may : ~ be interprcted as ail eflectivc chemical potential idil

(4.97) j=1

127

Mean-field theory

Figure 4.8: Schematic of the model. Dashed lines demarcate four types of energetically distinct regions (slabs): inner (subscript i) and outer slabs (subscript 0) in weak (superscript w) and strong (superscript s) (honiogeneous) modules. A molecule in the inner region (black circle) interacts with its six nearest neighbors (the four in the 2--2 plane are depicted as white circles; the two in the 9-direction above and below the z z plane are not shown). Sites at which molecules are = -qs,see Eq. (4.48)]are indisubjected to the strongly attractive substrate [@i cated by dark gray squares; those at which niolecules are subjected to the weakly attmrtive substrate [az= - E l s , see Eq.(4.48)]are denoted by light gray squares. acting on site i in the 5-2 plane. Because &. (4.96) assumes the same functional form as Eq. (4.89) for the bulk lattice fluid, the same reasoning may be applied to allow us to conclude that in the limit T = 0, the only stable solutions of Eq. (4.96) are pt = 0 , l . irrespective of the value of p:R. Therefore, a t T = 0, all of the sites aloiig lines parallel with the y-axis are cit,hcr cmpty or fillcd. Thc stablt. soliltion of thc ovcrall problcm is t,hc set, { p z } rriiiiirriizing thc furictioiid

0 [PI=

N

N

-f C C p t p j + C (asa=1

~ ( t )

]#a

pa

(4.98)

r=l

where all pt are either 0 or 1. Again, this conclusion is exactly in accord with (the first line of) Eq. (4.94). The specific pattern of p minimizing n ( p ] in

128

Phase behavior of pure lattice fluids

Eq. (4.98) [or: cquivalcritly, tlic exact cxyrcssioii ill tlic first, line of Eq. (4.94)l constitutes the “morphologies” of the confined lattice fluid atLT = 0.

4.5

Phase behavior of pure lattice fluids

Because of the relative complexity of the external field [see Eqs. (4.48)], it is instructive to enumerate possible morphologies in the limit of vanishing tempcraturc. This is particiilarly so hccausc in the limit T = 0 phasc equilibria can be dctcriniricd analytically as our discussion in Section 4.4.3 already indicated. Our recipe for identifying possible morphologies is based on a modular approach in which we construct a hierarchy of increasingly complex modules sequentially from simpler ones, starting from the bulk [61]. Any module, which gives rise to R set of morphologies { M’} consists of a juxtaposition of one or niore of the previous (simpler) modules. The grand yotciitial of a givcii iiiorpliology within a iiiorc complcx module can be expressed as a sum of grand potentials of t,he simpler ones, plus corrections accounting for the breaking of “bonds1’between nearest neighbors in the simpler modules and the formation of new bonds acroxs the interfaces between modules that make np the composite (more complex) module. The sum of all these contributions determines the nuniber and eventually the stability of thc possildc rnorphologics.

4.5.1

Morphologies in the limit T = 0

4.5.1.1

Bulk lattice fluid

In the simplest case, @ = 0 and all sites are equivalent. Thus, as we saw in Section 4.4.3 [see Eq. (4.94)].a closed exqxession for the grand potential may he derived where the ocriipiition nnmhcr so = 0 , 1 is discrete and doublethat valuccf. From Eqs. (1.764 aiid (4.94), we coiicluc1~~

where wa = Ra/N1 w i = 0 (so = 0) is the grand-potential density of the gasmas and wi = - 3 -~ p ~(so ~= 1) that, of thc liquid phase in the bulk. Frorm Ey. (4.99),/ifh = - 3 f t is readily deduced. Thus, for 11 < /& gas is the thermodynamically stable bulk phase, whereas for p > &, this is true for the liquid phase.

Mean-field theory 4.5.1.2

129

Confinement by “hard” repulsive substrates

The next slightly more complicated sitmuationis one in which the lattice fluid is confined in the z-direction by two planar hard substrates represented by

(4.100) According to our modular approach, the lattice fluid confined by ‘‘harfl serves to repulsive .substrates may be viewed as a bulk system, in which introduce “surfaces.” We can then express the grand-potential density as @h (%)

= wb -k Awh (nz)

(4.101)

where wb pertains again to the bulk module defined in Eq. (4.94) and the correction Awh (%) accounts for the interactions that are inissing for molecules located in the surface planos z = 1, TI,. Becaiisc! cwh nearest-neighbor interaction coritributcs - E R S ~ / ~to tlic corifiguratkmal ciicrgy of thc origi;inal bulk module, t,here are nxr%molecules in each surface. The total correction then becbines 7i,ny~fls$ and we can rewrite the previous expression more explicitly as Eff s; (4.102) u h (71,) = Ldb nz However, as SO = 0,1, 110 new morphologies arise. The only effect of confinement, by hard, repulsive substrates is an upward shift in the chemical potential a t gas liquid coexistence. By solving the analog of &. (4.99) we obtain &-I = 1 cff -3+(4.103)

+

712

which shows that the shift vanishes in the limit n, 00 where we recover the chemical potential at gas liquid coexistence in the bulk. The upward shift, in p$ effected by hard, repulsive walls relative to the bulk value may be interpreted as “drying.” This term refers to the fact that, a larger rheniicd potential is neodod to initiate condensation of the confined gas relative to its bulk counterpart,. This is because the effect of the substrates represented by Eq. (4.100) is to create an energetically less favorable situation by reducing the number of nearest-neighbor attractions from six (bulk) t,o five in the surface planes of the confined fluid. ---$

4.5.1.3

Confinement by chemically homogeneous substrates

The previously discussed confinement scenario becomes slightly more complex if we allow the substrates to attract molecules in addition t o just repelling them. We focus 011 a chemically homogeneous substrate surface first.

Phase behavior of pure lattice fluids

130

That is, wx*takc in Eq. (4.48) R, = 71,

(N

-- 0) such that

z, < 1 and z%> r ~ , zi = 1 and 2, = n,, 0, 2 1 . ; 5 n z - 1

00,

-Efs,

(4.104)

The possible morphologies of the lattice fluid can thus be determined by sandwiching a hard-substratc riiodulc of “voluiIic” 7i,7iy ( 7 ~ , - 2) sitcs bc! tween two hard-substrate inodules of nxny(n, = 1) sites. Using the modular principle in this case, we may express the grand-potential density as “Jtiom

= Whl

+ 2Wh2 -k AWhorn

(4.105)

where Whl

(4.106a) (4.1 0th)

where we use the notation soc to indicate that the ground-state occupation numbers may aSsunie the values 0 and 1 uniformly but independently in both rnodidm (i = 1,2). Thr tcrni proport+ionalto E$ in Eq. (4.106h) arises because of at,tractive fluid substrate inleractioris represented by Eq. (4.104). As the expression for Wh (n,) in J3q. (4.102) already accounts for the breaking of “bonds” to create free surfaces, the correction Adhornin Eq. (4.105) is due solely to the formation of bonds across the two interfaces and is therefore 2 / n ~ all this together we may rewrite Eq. (4.105). given by - 2 ~ f f s ~ ~ s ~Putting &!

Equation (4.107) illust.rates the hierarchical charact.er of our modular a p proach. For example, if the substrates were purely repulsive, Efs = 0 so that Eq. (4.104) has to be replaced by Eq. (4.100). However, t,his also implies Wb] = Wb2 and sol = so2 = SO in E l.(4.107) so that the second and fourth terms vanish and we are left with the expression for iJh given in Eq. (4.202), which rcdiiccs fiirthcr to W b rcprcscnting thc hilk in thc limit riz -+ 00. As 301 and so2 can assume values 0 and 1 independently, four different morphologies arise for the cheniically homogeneous module. These can be identified by sets of occupation numbers M = {sol, 502). For example, Mg = (0,O) corresponds t.0 gas, Mg = {0,1} to a monolayer film adsorbed on each substrate. The grand-potential densities w&, associated with each of the four morphologies M” can be deterinined in closed form froin Eq. (4.107).

Mean-field theory 4.5.1.4

131

Confinement by chemically heterogeneous substrates

Consider now the situation of ultimate interest, namely a lattice fluid confined between substrates endowed with weakly and strongly attractive stripes that alternate in the z-direction. In this case, the external potential is given by Eqs. (4.48) where we shall focus for simplicity on the case in which thr substratrs arc prrfrctly aligncd, i.c., cy = 0 in Eqs. (4.48). Followirig again the riiodiilar coiatruction principlc, wc’ c‘mi criwncratc possiblc morphologies by juxtaposing (in the z-direction) two modules corresponding to the previous, simpler one: the lattice fluid confined between homogeneow attractive substrates. Thus, we caii write the grand-potential density as Whet

1

= - (%W;;bnl 4-

n X

%d&,m) 4- AWhet

(4.108)

wlicrc w:oL arc given hy

(4.109) by analogy with Eq. (4.107). The correction in Eq. (4.108) can be derived as follows. We must first create surfaces by breaking bonds between nearest neighbors in the homoge IICOUS rnodiilcs across a plmic plaric parallel with the y-z plaie. This prosw 8 W sw s w cess increases wllet liy the amounts EE [(n,- 2) sdl s& 2sd2 s& ] /n,n,. We must then join the strong and weak homogeneous niodules by forming bonds across the interfaces. This joining decreases the grand-potential density by ~ f [(n, i - 2 ) s&stl 2s;;(,st2]/nxn,. Hence,

+

+

(4.110) An immediate consequence of the lower symmetry of the current system compared with the lattice fluid confined between chemically homogeneous substrates is a larger number of possible morphologies for the former. Inspection of F ~ s (4.107)-(4.110) . reveals that the grand-potential density is dctcrmincxl by thc sat M - { s;, szl,sl;;, sz1}wharc each occupation niimbar can independently assume the value 0 or 1. Thus: 16 different. morphologies are possible an principle. This fairly large number can be reduced substantially on physical grounds by taking into account the relative magnitudes of E R , ~ f s and , qw.For example: if both ~k and Efw are small compared with E E , the morphology characterized by M = { O , O , 1 , l ) is physically not sensible because it refers to a situation where energetically less favorable sites in the

Phase behavior of pure lattice fluids

132

iminicdiatc vicinity of tlic substrates arc occupied, wh:rcas (71, - 2) cncrgctically more favorable “inner” sites remain empty. By similar considerations, most of the remaining morphologies can be ruled out, without the necessity of calculating their grand potentials. Thc relevant reniaining morphologies and their associated grand poteiit.ials are compiled in Table 4.1.

4.5.2

Coexisting phases of the lattice fluid

4.5.2.1

The limit T = 0

The remaining morphologies are those where the lattice is coinpletely empty (i.e., “gas”) or filled (i.e., “liquid”) with molecules. In addition, we have a “bridge” morphology in which fliiid spans thc gap bct,wccn the strongly attractive parts of thc solid substratm lcaviiig criipty tliosc portioris of thc! lattice that are corit,rolled by t,he weak part of the substrate. Other r e l c vant morphologies in the current context. are those listed in Table 4.1 as “droplet ” and “vesicle,” respectively. The former morphology refers to a situation in which a monolayer is adsorbed by the strongly attractive part of the substrates leaving the entire rerna.iiider of the lattice empty. The latter morphology dcscrihos in a smsc t,ha op1,osit.c situathn in which t,hc lattirc is corriplctcly fillcd wi tli niolcc~iilcscxccpt for a riionolaycr ricxt, to tlic weakly adsorbing part of the substrate. At T = O! w” is a linear finiction of p regardless of the nature of a specific morphologies as Table 4.1 shows. This implies that the equality in Eq. (1.82) holds where, however? on account of the symmetry of the external field, 611 has to be replaced by K~~ defined in Eqs. (5.75) and (5.76). Because the Table 4.1: Possiblc lattice fluid Inorphologies M” arid associated grand potential

Ra ( p ) for T = 0.

a g= (g) droplet (d) monolayer (in) bridge (h)

vcsiclc (v) liquid (1)

M”

{O,O?0,O)

{O,O, 0,l) { 1 , 1,0,O) {0,1,0,1)

{I, 1,0,1) {1,1. 1 , l )

n/n, (See Ref. 61)

+

0

+

-n, (v - 2 2p) 2 - 2 % ~ ~ not stable at T = 0 -srhn, 1 (v 2p) n, + n, - 2 % ~ ~

+

+

f (2wS- rtXnz)(v t- 211) t 2 -t ?L, - 2rwrs - i ~ n (v , + 2p) + n, - 2 7 ~ b2 7 2

~ ~ ~

133

Mean-Beld theory

-3.05

-3

P

-2.95

Figure 4.9: Plots of n/n, as a function of p in mean-field approximation for various temperatures ;uid rnorpholo@cyjof the lattice fluid idcritificd in Tatde 4.1. Data are shown for .qs = 1, .qs = 0, n, = 20, n, = ns = 10. (a) T = 0, (b) T = 0.6, (c) T = 0.9, (d) T = 1.2.

Phase behavior of pure lattice fluids

134

-8

-12

-16

-20 -3.05

-3

-3.05

-3

P

-2.95

-28

-32

-36

P

Figure 4.9: Continued.

-2.95

Mean-field theory

135

(4.111) does riot vanish in general (exccpt for the “gas” niorphology, see Table 4.1) we conclude that in general K I I must vanish instead. Because the fluid at T = 0 is in one of its ground states, the t*ransformation {sz} + { p a } [see Eq. (4.84)] is bijective [62]. This is because sI is doublevalued and assumes one or the other of the two values in every single configuration depending on the actual morphology, that is, R, = ( s t ) = pz a t T = 0. The uniqueness of the t,raiisformation rcflccts the cquivalcncc bctwwn thc exact expression for tlic exact [scc Eq. (4.55)l .. and t h : incau-field cxprcssion for tlic grand potential [see Eq. (4.83)). Yet. another way of looking at Q. (1.82) is to conclude that at T = 0 density fluctuations are completely suppressed. To realize this, consider

(4.112) where Eqs. (4.54a) and (4.55) have also been employed. Hence, the equality in Eq. (1.82) implies t1ia.t

r=l

J=1

a=1

j=1

a=1

must hold a t T = 0 where Si, is the Kronecker symbol. hi other words, at T = 0 there are no correlations between molecules. Wc arc thus in a position to calciilatta thc sct of chemical pot,cntials { p a p } at wliich pliascs n and /3 c-ocxist a t T = 0 from Eq. (1.764 and cntrics for $2 in Table 4.1.

4.5.2.2

Nonvanishing temperatures

In t8hclimit, of nonvanishing t,cmpcraturcs, simplc analytic forms for thc w”’s, such as the ones compiled in Tabla 4.1: do not exist. Hence we need to resort to a numerical scheme to solve the Euler-Lagrange equations [see Eq. (4.86)]. This can be accomplished by an approach detailed in Appendix D.2.1 that starts from the set of (exact) morphologies { M u } compiled in Table 4.1 a t T = 0 as starting solutions for a temperature T 2 0. Once convergence has been attained, the algorithm yields new morphologies for this sufficiently

136

Phase behavior of Dure lattice fluids

-3.0

P :c'

-3.2 -

xe g

-3.4 (a)

-3.6

\

db

-3.2

-3.6 I 0

0.5

1

T

1.5

Figure 4.10 Phase diagrams p, ( T )for a lattice fluid using a mean-field approximation (full lines) and from GCEMC ( x ) . Pairs of phases a and p indicated in the figure coexist, along coexistence lines pFJ(T)(see Table 4.1). For T > 0 R monolayer film (m) arises as a thcrmodvnamirally atablc phase for n, = 20, = % = 10, nz = 10. ( R ) ~g = 1.0: ~ f = s 0.0 (b) Els = 1.5, = 0.0 (c)

Efn

= Efw = 1.5.

Mean-field theory

137

-3.0 1

'

-3.6 0

0.5 Figure 4.10: Continued.

small but nonvanishing temperature. These morphologies are then taken as new starting solutions a t a slightly higher temperature T ST. Thus, by varying p a t each fixed T, we obt,ain plots of S2" ( p , T ) similar to the ones plotted in Figs. 4.9(b)-4.9(d). The difference between these latter plots and those presented in Fig. 4.9(a) is that they can no longer be represented by st,raight,lines hiit are increasingly convex t,he higher T becomes bwaiise of thc inequality Eq. (1.82). According to tho discussioii in Scct.ion 1.7, wc arc XIOW dcaling with coexistence lines rather than isolat.ed points p:? along which phases a and p coexist. The corresponding phase diagrani is defined in Eq. (1.88) as the union of all coexistence lines. Figure 4.10 shows examples for f i (T) for a number of different systems. The simplest situation is the one depicted in Fig. 4.10(c) where Efs: = Efw = I .5, which is a relatively strongly attractive but rhcmically homogeneous solid siibstratc. In this cilw wc obscrvc! first-ordcr phase txansitions along t,he coexistsencecurve p y (T) at. fairly low chemical potentials. Along pErn(T) gaseous phases coexist with adsorbed monolayer films. If one increases p along lines of constant temperature, a second line of first-order phase trailsitions is encountered. That is to say, we have capillary condensation at ( T )for temperatures T 5 T,"' E 1.452 which is smaller than the bulk gas liquid critical ternperature Tcb = on account of confine-

+

9

Phase behavior of Dure lattice fluids

138

incrit. Simultaricously, tlic chemical potcritial is depressed iii the coiifiricd system rela.tive to the biilk IRcaiise of the presence of the subst,rates. Plots in Figs. 4.10(a) and 4.10(b), on the other hand, pertain to cases in which the substrate is chemically heterngeneow. To illustrate the impact of this chemical heterogeneity, we fix Ef. = 0 mid vary &fs. In other words, the “weak” portion of the substrate is purely repulsive and supports drying rather than wetting. Generally speaking, when comparing Figs. 4.10(a) and 4.10(b) with Fig. 4.10(c), we not,icc that, the d~cr~ii(;id hcrhcogcncit,yapparcritly givcs risc to a more coiiiplcx phase diagram. For cxamplc, plots in Fig. 4lO(a) reveal coexistence lines betwven various phases and the fluid bridge, as, for (T), pf’’ (T), and &’ (5”). The existence of the fliiid bridge example, is a direct consequence of the chemical structure of the underlying solid substrate that serves to “imprint” its OWII structure on the fluid adjacent to it. Thus, the fluid bridge, while being a generic thermodynamic phase: has no coiintmpart in the hnlk. The uniqncncss of spccial morphologies iriducxxl by (iiario)pattcriicd substratcu; has bccii the focus of scvcral studies over the past- fcw years [I, 61, 63-83]. It is now an accepted fact, that these morphologies have the status of legit,iniate thermodynamic phases like the ordinary gaseous, liquid, or solid phases in the bnlk. Moreover, a comparison between Figs. 4.10(a) and 4.10(b) shows that, as the “strong” portion of the substrate becomes more attractive, layering transitions become more pronounced. For instance, the gas droplet coexistence line p t (7’)hecomcq detached from the remainder of px (T),thereby caiising the onc~phascregion of droplet. phasts to iiicrcasc in size substantially.

4.5.3

The impact of shear strain

As wv poiiitod oiit, in Scc*t,ion4.3.1:tho c.onfiiic~1fliiid (’an l x (:xposcct too a iioiivariisliirig sliear straiii by iiiisaligiiing the two dieniically striped surfaces. Misalignment is specified quantitatively in terms of the parameter a in Q. (4.48a). On account of the discrete nature of our model, (Y can only be varied discretely in increments of Aa = 1 / ~ This . section is devoted to a discussion of both structure arid phase behavior of a confined lattice fliiid exposed to a shear strain. 4.5.3.1

Substrates in registry

We begin with the simplest situation in which the substrates are in registry, that is, a = 0 in Eq. (4.48a). Applying the numerical procedure detailed in Appendix D.2.1 permits us to calculate the local density p ( z : z ) as a solution of Eq. (4.86). Because of the discrct,e nature of the our model,

Mean-field theory

139

p (x,z ) is tlcfiried only at lattice sites. However, to visualize p (2,z ) , it provcs convenient to interpolate between neighboring sites. Figure 4.1l ( a ) shows the typical structure of a bridge phase, namely a high(er) density over the strongly attractive. portions of the substrate alternating in the d i r e c t i o n with a low(er)-density regime over the weakly attractive ones. In the zdirection, high(er)- and low(er)-density portions of the fluid span the entire spare betwcc~nthe siihstratcs with comparably little variation of p (c,z) along cuts 3: = const. Uiidcr suitable thcrinodyiainic conditioiis, a bridge phase may condense and form a liquid-like phase [see Fig. 411(d) for a typical liquid-like phase]. Alternatively, a bridge morphology may evaporate leaving behind a gas-like phase [see Fig. 4.11(c) for a typical gas-like phase]. The bridge phase is unique in the sense that it has no couriterpart in the bulk because its structure is sort of ''imprinted" 011 the fluid by the chemical striictiirc of the confining slihst,rilteaq.The Importance of confinementtfor the cxistciicc of bridgo p1iast.s is illustratctl by plots of phasc diagrams for various degrees of coihiement. in Fig. 4.12. The horizontal line in Fig. 4.12(a) represents the bulk phase diagram7 which we include for comparison. Thermodynamic states p < kb(T)= -3 and p > pu,b = -3 pertain to the one-phase region of bulk liquid and gas, respectively (T 5 Tcb = More subtle effects are observed if the lattice fluid is confined by solid siibstratcs as plota in Fig. 4.12(a) show. For sufficiently large n,, chemical decoratioii of tlic substrat c docs iiot mattcr but, confiiicment cffccts prevail. For example, for n, = 15, the critical point is shifted to lower T','and p:' compared with bulk Tcb = and p& = -3. Moreover, p:' (T) is 110 longer parallel with the temperature axis as in the bulk. If n, decreases, a bifurcation appears at T = Ttr.Oiily (inhomogeneous) liquid- and gas-like phases coexist along the line pf' ( T )(T < Ttr). At T = T,, the lat,tcr two arc in thcrmodvnamir cyiiilihriiirn with a bridge phase. For T > T,,, t8hccocxistcncc curve consists of two brdiichw. The upper one, p:' ( T ) ,can be interpreted as a line of first-order phase transitions involving liquid-like and bridge phases whereas the lower one, pEb ( T ) ,corresponds to bridge and gas-like phases. respectively. Both branches terminate a t their respective critical points { p:', qbl} and { /ifb. Tfb}. The entire coexistence (T) of the lattice fluid is formed by p$ (T), pEb ( T ) ,p:' ( T ) ,and the curve point {h,,T&}. biorcovrr, we vrrifird niinicrically t,hat,

4).

4

(4.114) where ApiJ is the average-density difference between coexisting phases i and &, 2 N within numerical accuracy for our three-dimensional lattice fluid niodel, indicating that the

j. For the critical exponents we obtain

4

140

Phase behavior of pure lattice fluids

P 7

1

P 0

Figure 4.11: Locd density p ( z , z ) for confined lattice fluid at, T = 1.0, p = -3.03658. Substrates are characterized by n, = 14, nz = 7, n, = 8, n. = 6, q. = 0.4, and qs= 1.4. (a) bridge morphology (a= 0), (b) bridge phase (a= &), ( c ) gaslike phase ((I = (d) liquidlike phase (a = Plots in (c) and (d) correspond to coexisting phases. Two periods of p ( s , z ) in the z-direction are shown because of lattice periodicity.

i),

3).

Mean-field theory

141

1

P 7

0

Figure 4.11: Continued.

142

Phase behavior of pure lattice fluids

rnca11-fic1d character is prrscrvcd at, both critical poiiits (see Sectiori 4.2.3). However, unlike for the vart der Waals fluid discussed above, an analytic determination of the critical exponents is much more demanding here because a simple equation of state like the one given in Eq. (4.28) does not exist for the current. model. Comparing in Fig. 4.12(a) coexistence curves for n, = 8 and 9, i t is evident that t.hc triple point is lowered the Inore scverc the confincrncnt,becomes, that is, the srriallcr n, is. Simultaneously, ,121 iricrcascs, -whereas ptb decreases such that the one-phase region for bridge phaseu widens. Because of these complex variations of px (T) with n,, it is conceivable that for a fixed thermodynamic' state { p ,T} the confined phase is gas-like initially if n, is sufficiently large. Upon lowering n,,this gas-like phase may condense to a bridge and eventually to a liquidlike phase at even smaller nb. This is illustrated in Fig. 4.12(b) for a spccific thcrniodynamic state clctcrniincd by T = 1.325 and p = -3.0235. Froni the plot it is claar that for n, 2 10 the corifincd fluid is gas-likc bccausc its thermodynamic state lies below all branches of p, (T). As the substrate separation decreases, however, one notices from the plot corresponding t o n, = 9 that the same thermodynamic state now pertains to the one-phase regime of liquid-like phases. That is to sav, it falls above all branches of p, (T). Thus, in going from n, = 10 to nz = 9, the confined lattice fluid undmwcnt R first-ordcr phase transition from a gas- to a, liquid-liltc phase. For an cwii srridlcr substratc scparation n, = 8, oiic sees from Fig. 4.12(a) that the triple point. has shift.ed to rather small {br,T,r} and that the onephase region of bridge phases has widened considerably. Thus, as can be seen from the parallel Fig. 4.12(b): the thermodynamic state eventually bclongs to the one-phase region of bridge phases where it reniains for all smaller n,. Hence, as one decreases the substrate separation from n, = 9 to nz = 8, an originally liquid-lilw phaw is t,ransformcd into a bridge phase rhring a firsborder phase trailsition. 4.5.3.2

Substrates out of registry

The preceding section clearly illustrates the complex phase behavior one can expect, if fluids are confined between chemically decorated substrate surfaces. Three different length scales, which are present, in our model, are priinarily responsible for this complexity. hi addition t o the one corresponding to the range of interactions between lattice-fluid molecules (i.e., l ) ,another length scale refers to confincmcnt, (i.c., 71%) and is alrcarly present, if thc substrates arc cGrinically honiogciic:ous. It causr.s

1. A critical-point shift t o lower { p$, Tf'}(Q, = Efw > ER) compared with the bulk {pcb?T c b } .

143

Mean-field theory

T -3.015

-3.02

-3.025

-3.03

1.3

T

1.35

1.4

Figure 4.12: (a) Phase diagrams p x ( T ) for various confined lattice gases as functions of substrate separation n, indicated in figure (a= 0: n, = 14, n, = 8, E f . = 0.3, Els = 1.4; (---) &(T),(- - -) pfb(T), and (----) ,e'(T).(b) Enhancement of the shaded region in the plot of panel (a). In the plots of both panels ( 0 )reprwerits a (fixed) tlimiiodyniunic statc of the coilfiiicd fluid.

Phase behavior of rmre lattice fluids

144 2.

/i$'

( T )to form an arigle larger than

o with the temperaturc

axis.

Tlie third leiigtli scale, intmduced by cheiiiical decoratioii of the substrate, is set by n, (or, equivalently, n, - ns),exceeding P by almost an order of magnitiide for the various coexistence c:urves plotted in Fig. 4.12. Corusequmces of this third leiigth scale are 1. Existence of bridge phases as a new thermodynamic phase. 2. Two independent critical points { pEb,Tfb} and { p;', Tp'}. Figure 4.12 already showed that the precise form of /L,(2') is caused by an interplav of these different length scales. To further elucidate this interplay, it s e e m interesting to expose the Iatticc fluid t o a shcar strain h-y varying fr [sw F4. (4.48a)I. Comparing the plots in Fig. 4.11(a) and Fig. 4.11(b) illustrates the effect of a shear strain on the structure of a typical bridge phase. However, depending on the therriiodynainic state, a bridge phase will sustain only a maximum shear strain and will then eventmilly be either "torn apart" and undergo a firstorder phase. transition to a gas-like phase [see Fig. 4.11(c)] or condense and form a liquid-like phase [see Fig. 4.11 (d)]. Corresponding coexistence curves px(T) plottctl iii Fig. 4.13 show that upon incrcasiiig (r. from its initial value of zero causes the triple point to shift to higher Tt, and hr.Simultaneously, the one-phase region of bridge phases shrinks. The one-pliase regime of bridge phases may: however, vanish completely for some Q < amaxdepending on substrate separation (i.e., .n2),chemical corrugation (i.e., ,%/nx),or strength of interaction with the chemically different parts of the substrate (ix., Ers, ~ f , ) . Notice that for the special case amRx = (i.e., n, even) the one-phrrse region of bridge pliascs must. vanisti in ~Aicliiiiit (I! = om= for spiirictry rcwo~is[sex Qs. (4.48)). In addition, Fig. 4.13 shows that, critical t>emperaturesT$"and T!" depend only weakly on the shear strain unlike p!' and p!b such that the critical points are essentially shifted upward and downward, respectively, as Q increases. Consider now a specific isotherm T = 1.25 in Fig. 4.13, intersecting with different branches of the (same) coexistence curve px (T) a t different cliemical pot,cntials. According t,o thc dcfinition of p, (T),cach intcrscctiori corresponds to a pair of (separately) coexisting phases. For example, at p!b (I")21 -3.053 and a =-0, a gas-like phase coexists with a (more dilute) bridge phase, whereas a (denser) bridge phase coexists with a liquid-like phase for p!' (T)N -3.029. Because the one-phase region of bridge phases shrinks --+ 0 the with Q (see Fig. 4.13), the '?dist,ance" A b (T)zz lp$b(T) - pt' larger cr liecomes, that is, with increasing shear strain. From the plot in

(")I

145

Mean-field theory

Fig. 4.13, it is clear that a shcar strain exists such that Als, = 0 , that is:

T 5 T,, (an,).For this and larger shear strains only a single intersection re-

mains, corresponding to coexisting gas- and liquid-like phases (see Fig. 4.13).

Y

5

Figure 4.13: As Fig. 4.12, but for various shear strains cr indicated in figure (n, = 14, n, = 6, n,, = 7, qs= 1.6, q, = 0.4). Intersectioiis between isotherm 'E (vertical solid line, see text) and coexistencecurve branches represent coexisting Phases. P) PZh(T), (v) P : " n 0 = Or ( 0 ) Pfh('E), (0) P:*(n a = +;(W

pg'

(T),cr = $.

Before returning to the issue of shear deformation of fluid bridges in a broader context. in Section 5.6, we emphasize the mere fact that a fluid phase in Confinement is capable of sustaining a nonvanishiiig shear strain. This is yct nnothcr fcatiirc of confinctl fluids that mnkcs thcin distinct from other, more conventional, soft matter systems. As we will show below in Section 5.6, the fluid bridge "responds" to a shear strain in a fashion qualitatively similar to a bulk solid in terms of its rheological properties while maintaining a fluid-like structure. In retrospect it is this mixed fluid and solid-like nature of soft condensed matter in confinement which rnakes its thermodynamic treatment developed in Chapter 1 particularly insightful and appropriate.

Binary mixtures on a lattice

146

4.6

Binary mixtures on a lattice Model system

4.6.1

We now extend the previous discussion of pure confined lattice fluids to binary (A-B) mixtures on a simple cubic lattice of n/ = n z sites, whose lattice constant. is again C. Wc dcviatt from our prcvioiw notation (i.c., N = n.xnynz)bccausc we coriccritratc: on dicmically hornogcricous substratw where n = ~ n located y in a plane at some fixed dista.nce from the substrate, which are energetically equivalent. Moreover, our subsequent, development will benefit notationally by replacing henceforth ~hby just z. The position of a fluid molecule on this latt,ice is then specified by a pair of integers (k,l),where 1 5 k: 5 n labels the position in ail x-y plane and 1 5 1 5 z determines the position of that plane along the z-axis. A specific sitc niay bc occupied either by a molcculc of spocics A or B, or it may be altogcthcr crnpty. Hcrico, this riiodcl accouiits for rriixcd arid dcrnixcd liquid phases as well as for gaqeous ones. To describe individual configurations on the lattice, we introduce a mat,rix of occupation numbers s with elements

sk.1

=

{

+1, site occupied by niolecules of component A 0, empty site -1, sitc occupicd by iriolcculcs of coriipoxient B

(4.115 )

For a given configuratioii s, the total number of sites occupied by molecules of species A or B is given by

(4.117)

is the total number of occupied sites in a given configuration s (i.e., for a given occupation-number pattern). Equations (4.116) account for the fact that ( S k J f 1)s k , l must not contribute to the sums if a site ( I c , 1 ) is empty or occupied by a molecule of type B in h.(4.116a) nor must this term contribute to the sum in Eq. (4.116b) if the specific site is empty or occupied by a molecule of type A.

Mean-field t heorv

147

Morcovcr, it is straightforward to show that tlic total ~iumbcrof riiolcculcs of species A at, either substrate is given by NAW (s) =

c

1 "

5

[(I + S k J ) Sk,l

+ ( 1 + s k , z ) sk,z]

(4.118)

k=l

which follows from considerations similar t80the ones leading to Eq. (4.116a). Thus, the total number of molecules of species B at the substrate is given by

(4.119) Similarly, one can work out expressions for the number NAA( NBB)ofAA (B-B)pairs, which are directdy connected sites, both of which are occupied

by a molecule of species A (B). These somewhat more involved expressions are given by

(4.120b) m= 1

where the summation over m extends over the 4 nearest neighbors G ( k ) of lattice site k in the x-y plane. A slightly more complicated expression obtains for the number of A-B (nearest-neighbor) pairs, namely

nc= 1

I

(4.121)

Binary mixtures on a lattice

148

Because of tlic irifiriitc repulsion “ f d t” by fluid ~nolcculcsat vanishing distance from the siibstrate surface. we amend Eqs. (4.120) and (4.121) hy the boundary conditions *3k,O = %J+I = 0, Qk (4.122) The Hatniltonian function governing our system can then be cast as

H (s)

=

[ A r(s) ~+ ~ x ~ l l r(s)] , ~ + EABNAB (s) +Es [NAW (s) iXRNBW (s)J

E

-p

[ h r A (s) -k NB

(s)]

(4.123)

where for convenience = /&A =

(4.124)

and (4.1254 (4.125b) k‘B

EBB

EAA

(4.125~) (4.1254

In Eqs. (4.125), E determines the depth of the attractive well (ie., the attraction strength) of the A-A potential function. Likewise, E~ describes the attraction of a molecule of species A by the solid substrate. Parametcr XB will hcncctfort,h hc refcrrcd to as t,he “asymmetry” of the model mixture, where XB > 1 characterizes a binary mixture in which the formattion of B-B pairs is energetically favored, whereas for XB < 1, this is the case for A-A pairs. For the special case X B = 1 the asymmetric mixture degenerates to the symmetric case previously studied in Refs. [MI and [85]. In addition, we define the “selectivity” of the solid surfaces by specifying xx in Eq. (4.125d) in a fashion similar t o XB in Eq. (4.125~).Hence, the paramotcr space of our rriotlcl is s p a t i i d by the sct,{&, EAR^ E ~ ,yg, , xS}.

4.6.2

Mean-field approximation

As we are again int.erest>edin determining the phase behavior of the binary mixture in confinement and near solid interfaces, we are essentially confronted with the same problem already discussed in Section 4.5, namely finding niinima of the grand potential for a given set of thermodynamic (T,p ) and model parameters [see Eys. (4.125)]. To obtain expressions for w that are tractable, at least numerically, we resort again to a mean-field approximation. That

Mean-field theory

149

is, we wish to rcplacc H in Eq. (4.123) by its incan-ficld arialog Hmf. Be cause we are dealing wit.h a binary mixture,.applying the approach taken in Section 4.4.2 is somewhat tedious. This is because in a binary mixture we do not only need to consider its density but. also the composition to specify its physical nature wit,hout ambiguity. In the language of Section 4.4.2, we would thus have to involve a. second field [besides {Qi}, see Eq. (4.77)], which would randcr rathcr involvcd thc approach takcn in that swtion. Alterriatively, wc assumre that, within cadi plaric I parallel t,o the solid sulxstra,tes the occupat,ion number at each lattice site can be replaced by an avemge occupation number for the entire plane. On account of the symmetrybreaking nature of the solid substrate, tliese average occupat,ion numbers will generally vary between planes; that is, they will change with 1. Hence, we introduce the total local density (4.126) and the local "miscibility"

in.1

(4.127) as convenient alternative order parameters a t the mean-field level. In the thcrrnodyxiaruic limit TI. -+ 00. pl (in units of P ) is tliiiicxisiorilcss arid continuous on the interval [O, 11, which implies that ml is coiitiiiuous and dimensionless as well but on the interval [-1,1]. Mathematically speaking, the meari-field assumption consists of m a p ping the m x z occupation-number matrix s onto the z-dimensional vectors nA= (nf,n$,. . . ,n9) and nB = (n?,@. . . . ,n:) where nf is the total number of molecules of species z on lattice plane 1 regadless of their specific arrangement. HC~ICC, we icpldcc I f ( 5 ) by its mcaii-field arialog If,f (nA,nB) where we note in pwsing that the trailsformation s + n A nH , is not bijective in general (see below). To derive the mean-field arialog of Eq. (4.53) for the current model we rewrite it more explicitly as 1 s1.1=-1

1

1 sm,r=-l

s2,1=-l

1

1

Binarv mixtures on a lattice

150

Hencc, at. the mcaii-field lcvcl, we may replaco tlic 7rt x 2 above according to

Z

-+

Z,,,f

=

ncc

SUIIIS

in parentheses

8 (nA,nB)cxp [ - ~ H , , f(nA,n")]

(4.129)

where the conibinatorial factor

fi ( np np ) ( nf;nP ) fi ( n", ) ( ; )

(nA*nB) = I=

71

1

+

= I=1

(4.130) represents thc a priori possible c.onfigiirat,ions corresponding t,o the same value of Hmf,that is, thc degeneracy of a particular microstate characterized by vectors nA and n". In the thcrmodvncmic limit#(i.c., as rr + cm)it is convenient to replace nf/n. thc discrete variables nf by tlicir (quasi-) continuous counterparts pi : so that the double sums can be replaced by double integrals,

where tlic z-dimicilsional vectors pA aid pB arc dcfiried aiialogously to nA and nB,respectively. Changing variables pf.py pl,rnl via Eqs. (4.126) and (4.127) in t.his last expression permits us to eventually cast Eq. (4.129) as --$

where w (p,m; T,p ) defines an energy hyperplane in the multidimensional spncc spanned by the set of local order paramctms { p , m } for givcn valucs of T and 11. The function w ( p .m;T,p ) may have many extreriia in p-m-space. The

Mean-field theorv

151

iiccc-ssary conditions for tlicsc cxtrcma to exist riiay be stat.cd as

where explicit expressions for the functions h$ and h!j are given in Eqs. (D.32). Equations (4.132) may have several solutions LY = 1,.. . , i. It is then sensible to introduce the notion of a phase M a through the set of 2%elements

M" = { p " . m n }

(4.133)

where pa and ma are not only simultaneous solutions of Eqs. (4.132) but also mznima of w (p,m;T ,p ) . A t this point, it is important to realize that in t,hc thrrmodynamic limit, (i.c. as n 4 00) thc global minimiim p*, m' of the function w will cornpletely deteriiiine t,he integral in Fkl. (4131). In the 00, this permits us to rewrite Eq. (4.131) as limit m ---$

12,f In Z,f w ( T . p ) = -= -

N

( N ,TI11) ON

-

- 1118 (p*,m * )

DN

+ Hmf(Np * ,m*)

(4.134) where p* and m* represent the "configuration" a t the absolute minimum of the grand-potential density w (2': 1-1). which is the therrnodynamically stable phase (i.e ., morphology) M*, whereas all other i- 1 phases are only mrtastable (except, for points of phase cocxistmcc, sci*Soction 1.7).

4.6.3

Equilibrium states

4.6.3.1

The limit of vanishing temperature

Let us now briefly discuss the special case in which the transformation

s k , l --.,

p1. rnl is bijective. From the definition of pl and ml in Eqs. (4.126) and (4.127), it is iminediatcly clcar that this can only be tho cwc if all matrix clerneiits in the mbh row of s are equal awiming one of the three values given in Eq. (4.115). This then implies that pl = 0 , l is discrete and doublevalued.

In other words, across any given lattice plane 1. all sites must be empt-y or occupied by nioleciiles of one or the other species so that pl = pf = 1 or pi = pp = 1, respectively. To discriminate between these cases, Eq. (4.127) givcs rril = 1 if pi = lo;" = 1 , whrrcas v n l =. - 1 if pi = lf = 1. Thiis,

ni = A

1%

=

11,

{ II,

(4.135a) (4.135b)

152

Binary mixtures on a lattice

implying 8 = 1 from Eq. (4.MI), which is rnathcmatically cquivalcrit to saying that the transformatbn Sk,l 4 pl, ml is bijective. If this is so we conclude from Q. (4.131) that

(4.136) This l a t h expression is idmtical t+oEq. (4.134) in the limit, T = 0 replacing, howeVcr, in Eq. (4.136), Ma by M'. Tlius, in tliissezisc

(4.137) is a conscqucncc of the fact that a t T -= 0 tlic incan-field treatrricrit bccoincs exact (i.e.l the transformation s k , J 4 p1,rnl becomes bijective) where the subscript " 0 was introduced to emphasize the limit T = 0. Equation (4.136) is iniportant because H,f (M") can be calculated aiialytically for our current model.

4.6.3.2 Nonvanishing temperatures For T > 0 we are concerned with solutions of Eqs. (4.132). To find these it. is convenient to introduce the (transpose of the) 2z-dimensional vector

zT = ( P I , ~ , l ,f.. l

r

n

l

~

~: Pl z l,

7%) ~

~

~

(4.138)

which perillits us to rcwritc Eqs. (4.132) as f

f (4=

1); h), ?rlolpi 7111,~ 2 ~ 7 1 ~ 2 ) h:: (Po, m,o,P11 m 1 1 P 2 9 m 2 )

=! o

i

(4.139)

hf ( 1 ) 2 - 1 1 m , E - 1 ~ ~ 3 , 7 ~ ~ 1 ~ z + l l 7 ~ , z + l ) hh ( P Z - 1 , m2-1,P z , m 1 P t + l , mz+d

Suppose a solution zo of Eq. (4.139) cxists for a given tempcrature TOand chemical polential h. We are then seeking B solution 5 for slight,ly different thermodynamic conditions

(4.1404 (4.140b)

Phase behavior of binary lattice mixtures

153

where bT aid S p are sufficiently stria11 so that we may exparid Eq. (4.130) in a. Taylor serie3 around zo

f (2)= f ( 2 0 )+ (VfT)I=,

*

(Z

- zo)

+ 0 (1s- 201~) 0

(4.141)

rebaining only the linear term. In Eq. (4.141), the a-dimensional vector V = (a/apl) a/am, . . . ,d/ap,, d/&u2). Introducing the functioiial matrix D through the dyad V f (z), that is )

where Eq. (4.139) has also been used and the elements of D can easily be computed with the aid of Eqs. (D.32). However, we may employ synimetry properties of D t o simplify the numerical treatment. These symmetry properties are summarized in Appendix D.2.2.3.

4.7

Phase behavior of binary lattice mixtures

We begin with the simplest case of a confined binary mixture, which is a symmetric mixture coilfined between chemically homogeneous, nonselective planar substrates (slit-pore) . The grand-potential density governing the equilibrium properties of such a mixture is given by Eq. (D.29) for the special raw XB = xS = 1 and EAW = E ~ .These cqiiilihriiim stat.es are ohtailled in principle by again solving Eqs. (4.132), wlicw, liowcvcr, l i t a i d h$ arc ~iow given by Eqs. (D.33) rather than by Eqs. (D.32). Except for this difference, we may, however, compute the phase diagram according to the algorithm detailed in Appendix D.2.2.3.

4.7.1

Symmetric binary bulk mixtures

We begin the discussion with bulk mixtures, which shall serve as a reference for confined binary mixtures to be discussed below in Section 4.7.2. For a more comprehensive disrussioti of the phase behavior of general bulk

154

Phase behavior of binary lattice mixtures

rnixturcs, the intcrestcd rcadcr is rcfcrrcd to Ref. 86. Cliaractcristic pliasc diagrams are displayed in Fig. 4.14 for selected values of &An. To realize a binary bulk inixcure, we choose EW = 0, t = 1 in Eqs. (D.33) and replace the hard-substrate boundary conditions po = pz+l = 0 by periodic boundary conditions po = pz+l = p1 to account for the syminetry of the bulk mixture. Results plott.ed in Fig. 4.14 for various values of EAB illuwtrate generic types of phase diagrams defined in J2q. (1.88). Bulk phase diagrams have also been discussed earlier by Wilding et al. [87];-These authors studied the phase behavior of a contitmow square-well binary bulk mixture by means of Monte Car10 siniulations arid a niean-field approach. For EAB = 0.40, plots in Fig. 4.14(a) show that for tcmpcratmas T ;L 1.32 only gas aiid chmixad liquid cocxist dorig il liiic of first-ordcr pfillsc transitions. This line tmds at a tricritical point located a t pt,ri2: -1.75 and Tt,i ‘v 1.32. For teniperaturw exceeding Ttri:gas and dernixed liquid coexist along the so-ca.lled X-line [i.e., a line of critical points indicated by the t,hiii solid line iii Fig. 4.14(a)]. This type of phase diagram reseinhles the one showii by Wilding et al. in their Fig. 1(c) [87].4 For higher EAB = 0.5, the phase diagram differs qua.litatively from the previous one. This can be seen from Fig. 4.14(b) where a bifurcation appears (i.e., at a triple point) for ‘Y -2.25 and Ttr 21 1.075 at which a gas phase coexists simultaneously with both a mixed a.nd a deniixed fluid phase. Conseqiieiit,ly a crit,ical point, exists (p,b N -2.25, Tch ‘v 1.15) at which the line of first-ortlcr trmsitions bctwccn rnixctl liquid and gas stattcs cnds. The line of first-order transitions involving mixed and demixed liquid states ends at a higher temperature and chemical potential of ptri 21 -2.00 and Gri 2: 1.18, and the X-line is shifted toward lower temperatures as one can see from the plot in Fig. 4.14(b). This type of phase diagram coniports with the one shown in Fig. l ( b ) of Wilding et a]. (871. A further slight increase of E A R to 0.56 does not cause the phase diagram to change qualitatively but quantitatively from the previously discussed case. This can be seen iii Fig. 4.14(c) where for E A B = 0.56 the triple point is shifted to a lower temperature and chemical potential compared with EAB = 0.50. Likewisc, the linc of first-order t,ransit,ions hdween gas and mixed liquid appears a t lowcr chemical pot critial but. is soruewhat lorigcr because the critical point, is elevated to a higher Tcb ‘v 1.18. The opposite is true for the coexistence between mixed and demixed liquid pha.sea as one can see from Figs. 4.14(6) and 4.14(c). Eventually, as E A R bccoriics sufficiently largc, first.-order trailsitions b e ‘As was shown recently in Ref. 86 the classificationscheme proposed by Wilding et al. [87] is incomplete.

155

Mean-fleld theory

c1 -2.0 -

D

G 0.6

0.8

I .o

T

1.2

1.4

Figure 4.14: Bulk phase diagrams p , ( T ) [see &. (1.88)] where G, M, and D refer to oncphase regions of gaseous, mixed liquid, and demixed liquid phases, respectively. Pairs ofneighboring phases coexist for state points represented by solid lines where thick and thin lines refer to first- and second-order phase transitions, respectively. (a) EAB = 0.40, (b) EAB = 0.50. (c) EAB = 0.56, (d) CAB = 0.70.

156

Phase behavior of binary lattice mixtures

-1.5

P -2.0

-2.5

t

0.6

G 0.8

1.o

1.2

1.4

0.8

1.o

1.2

1.4

T

-1.5

CL -2.0

-2.5

0.6

T

Figure 4.14: Continued.

Mean-field theory

157

tween iriixcd arid demixcd liquid pliascs disappear as the plot in Fig. 4.14(d) shows. For E = 0.70 the A-line intersects a line of first-order phase transitions at a critical end point peep N -2.55, Tcep21 0.84 because the nature of the participating phases along the A-line differs from those involved in the first-order transitions for T < Tccpor T > Tmp.This type of phase diagram resembles the one plotted in Fig. l(a) in the paper of Wilding et al. [87]. In the h i i t EAB = 1.0, the symrnetxic binary mixture degenerates to a pure flttid. In this case T,, + 0 and the A-line becomes formally indistinguishable from the paxis (and therefore physically meaningless). The remaining coexistence line 1.lxb = -3 = p c b (i..., the phase diagram) involving gas (G) and liquid phases (L) becomes parallel with the T-axis and ends at the critical point where Tcb = $ as expected for the bulk lattice gas (161 [see?for example, Fig. 4.12(a)].

4.7.2

Decomposition of symmetric binary mixtures

If we now confine the binary mixture to a slit-pore of nanoscopic dimension, we may, in fact, change the tapolgy of the phase diagram. For example, by varying the degree of confinemento (i.e., z in our current notation), it turns out to be possible to switch between various types of p h a diagrams ~ with profoiind conscqiiences for liquid liquid and gas liquid phnw cqiiilihria. This phenomenon may have practical implications for the decomposition of mixture.. of immiscible liquids in nanoporous matrices. Consider as an exa,mple the case EAR = 0.5 for which the bulk phase diagram is plotted in Fig. 4.15(a). It consists of a line of first-order phase transitions involving gaseous and demked liquid states for T 5 1.08. At Tt, N 1.08, the phase diagram bifurcates into a line of first-order phase transitions between gzscoiis aiid mixed liquid stmatesending at the critical poiiit pc = -2.25, T, 3 1.13, arid a liric of first-ordcr transitions irivolviiig mixed and demixed liquid states. The latter ends a t the tricritical point /Lt.,i ‘V -2.04 and Tt,i N 1.16. If this binary mixture is now confined to a relatively wide slit-pore, the phase diagram remains of the same type, but the plot referring to t = 12 in Fig. 4.15(a) clearly shows the confinement-induced downward shift of coexistence lines and the displacement of characteristic (ix., triple, critical, and t(ricritica1)points tiiscrisscd in the prcccrling saction. However, if the degree of confinement becomes more severe [see plot for z = 6 in Fig. 4.15(a)], the topology of the phase diagram changes. In other words, by going from t = 12 to t = 6, the mixed liquid vanishes as a thermodynamically stable phase, whereas the entire phase diagram is further shifted to lower chemical poteiitials. This latter trend persists if the pore width is reduced even more with 110 further change in the topology of the

Phase behavior of binary lattice mixtures

158

phasc: diagram. A classification of mixtures with respect to the topology of their phase diagram has been presented by van Konynenburg and Scott [88]. More recently, Woywod and Schoen have also studied the topography of phase diagrams of binary bulk mixtures [86]. This latter study was inspired by the geometrical approach to equilibrium thermodynamics discussed in' the book by Wightman [8Y]. In their st,ndy, Woywod and Schocn prcscrita an argiincnt which precludes tlic cxistciice of tricrit.ica1poiiits iri birtary Jnixturcs in general (86). This is a consequence of a purely geometrical argument based on an analysis of the number of ways in which coexistence surfaces can be joined in the' (Euclidian) space of the three therniodyiiainic fields T , j i G ( / L A p ~/2,) a i d A p 3 ( p -~CLR)/2 specifying the thermodynamic state of a binary fluid mixture. However, Woywod arid Sclioen show that, by the same token, tricritical points may axist in casas, wharc the mixt,iirc: posswsc?s sonic spccial symnictry. If one then fixes the thermodynamic state such that the bulk mixture is a gas [represented by in the inset in Fig. 4.15.(a)], confinement to a relatively wide pore ( i e , z = 12) may first cause capillary condensation to a mixed liquid mixture analogous to ordinary capillary condensation in pure fluids. If the fluid is confined t.0 a narrower pore ( z = 6), however, decomposition into A-rich and B-rich liquid phases is triggered by confincmcnt upon condcnsation. Thus, by choosing an appropriate porc widtli, oric can cithcr prornotc condensation of a gas to a mixed liquid phase or, alternatively, initiate liquid liquid phase separation in the porous matrix where both processes are solely confinement-driven because the pore walls are nonselective for molecules of either species in our present model. This process is further illustrated by the plots in Fig. 4.15(b) where the mean dansit,y 7 of t,hcr~iodynaniicallystable confiricd phases is plottcd as a function of z (i.c.: the porc width). Thrcc diffcreiit, brtliiclies arc discernible. For small z < 8, i j is relatively high indicating that the pore is filled with liquid. A corresponding plot of the local densities of a representa.tive phase for z = 5 shows that this liquid consists locally of A- (or R)rich, high-density fluid (because the two cannot be distinguished in a symmetric mixture). Hence, for z < 8, we observe (local) decomposition of liquid mixtures. Along an int.crmcdintc branch of porc widths, t,hat is, for 8 < t < 16, 7 is somewhat smaller thaii for the t,ightest pores ( z < 8). An inspection of a prototypical plot -ofthe local densities for t = 12 reveals that the confined phase now consists of a locally equimolar mixture. Hence, for intermediate pore sizes, t,he confined phase is a mixed liquid. is.stil1 smaller t,han along the two previously disFinally, for z > 16, i~ cussed branches. The local density of a representative stat.e for t = 20 now

+

*

159

Mean-field theory

-2.1

CL -2.2

-2.3

-2.4

1.05

1.10

T

1.15

1.20

Figure 4.15: (a) As Fig. 4.14,where the inset is an enhanced representation of that part of the phase diagrams bounded by the box with a fixed thermodynamic state represented by (b) Mean pore density j j as a function of pore width z, where stability h i t s between pairs of phases are demarcated by vertical lines; also shown are histograms of local density of representative phases where shading of the bars refers to p t and p:, respectively.

*.

160

Neutron scattering experiments

clearly shows that. a coiiiparativcly low-density Muid exists a t the ccntcr of the pore. As either substrate is approached, the density increases, indicating that this mixture wets the substrates. However, as expected for such a "gas" state. the fluid is composed locally of an equirnolar mixture similar to states along the intermediate branch 8 < :< 16. The change of j j between a pair of branches is discontinuous at characteristic pore widths where first-order transitions occur between these phases. The confinement-induced change in topology of .the phase diagram may have important rcporrnssioiis for t,hc dacompositsionof I)inary mixtures in sorption expcriinents where o m may eiivisiori pore condensation in nanoscopic solid matrices leading either to a mixed or deniixed liquid such that the physical nature of the coilfined phase depends solely on the pore width.

4.8

Neutron scattering experiments

The discussion in the preceding section already showed that binary mixtures may decompose on acconnt of t h prcscncc of a nanoscopic poroils matrix. In general, biiiary liquid mixtures separate into two phases of different compositions below a critical solution point. If such mixtures are imbibed by a mesoporous ma.trix, phase separation cannot occur on a niacroscopic length scale. Microphase separation of the system may lead to metastable local geometries of the two phases in the pore, depending on the relative amounts and strength of interaction of the two components with the surface [4]. The structure of m i c r o p h ~ s o s c p a r ~iiiixturcs tc~ in porous matcxials is of iiitercst in a variety of different fields, ranging from liquid chromatography or microfiltration and relat.ed membrane processes to techniques by which liquids may be extracted fkom porous materials (for a review see R d . 90). A porous medium affects a liquid mixt,ure not only by mere confinement to volumes of nanoscopic dimensions [91] but, also by the energetic preference of the solid substrate for niolwules of one of the components of the mixture [92, 931. This selectivity causes an enrichment of the component in the proximit)y of t,hc porc walls. For siifiricntJy wida porcs, tha decay length of t-he rcsultiiig concentratioii profile corresponds t o the correlation length [c of concentrat,ion fiuctuations [94]. In narrow pores, on the ot$herhand, when the mean pore width D is less than &, concentratmionprofiles near the pore walls overlap, thereby causing enhanced adsorption. Because the correlation length
Mean-field theory

161

preferred corriponeiit. is t,hc major corriponcrit 195-97). Iiicipicnt wcltirig of the pore walls manifested by multilayer adsorption is expected to occur in the one-phase region close to the liquid liquid coexistence curve in that region of thermodynamic state space where t,he energetically favored component is the minor component of the mixture [9&100]. To test some of the structural implications of these predictions, smallangle neutron scattering (SANS) experiments were carried out on the binary mixt,urc iso-biityric acid (iB.4, comporicnt, A) -1- DzO (tmmponcnt. W) in a controlled-pore glass (CPG-10) of about 10 nm mean pore width (see Fig. 4.1). In this section, some results of the experimental work are compared with the predict,ioris of a theoretical model. We eriiploy the lattice model for an asymmetric binary mixture confined to a slit-pore wit,h selective walls that we already introduced in Section 4.6.1. By adjusting model parameters properly according t o criteria spelled out in Sections 4.8.2 and 4.8.3, we are able to rcprcscnt, a binary mixture of tJic aniphipliilc iBA and DzO studied in the parallel SANS experiments to be described in some detail in the subsequept Section 4.8.1. Unlike in the preceding sections of this chapter dealing with mixtures, we are not able to focus 011 the entire phase diagram of the current. mixture. The relative crudeness of our model, on the one hand, and the complexity of the current mixture, on the other hand, do not permit us to do so. Rather, we shall be focusing on liquid liquid phase equilibria hcriccforth. This narrower scope of our discussioii below is rriotiwtcd arid justified by the conditions under which the parallel SANS experiments have been carried out.

4.8.1

Experimental details

Small-imglc neutron scat,tering was used to study t,hc tclmpcratnrodcpcndent mesoscale structure of the confined liquid iBA+D,O mixture. As the CPG-10 materials are not monoliths but consist of granules (mesh size ca. 100 nm), the sample was compressed to cylindrical pellets of about 1 mm thickness and 12 mm diameter. The pellets were soaked with a mixture at a temperature well above the phase separation temperature to prevent demixing during the filling process. A 10% excess of liquid relative to the pore space of the sample was added in order to fill coiiipletcly the pore s p x c ruid tlic iiiterstitial spmc among the granules of the pellet. The SANS experiments were made in a temperature range from 10°C to 70°C after an equilibration time of at least 45 minutes. The scattering data were analyzed by a function I ( q ) similar to that proposed by Formisano and Teixeira [101, 1021, which is coinposed of additive contributions accounting for the scattering of the silica matrix and an ad-

162

Neutron scattering experiments

sorbcd film, as wcll as contributions due to corimitration fluctuatioris in thc confined liquid mixture and-the microphase separation in the pores at low temperatures; that is

The first term on the right-hand side of Eq. (4.M)describes the coherent superposition of the scattering by the silica matrix a.nd an adsorbed film' caused by preferential adsorption at the walls. Here, s G ( 9 ) represents the' structure factor arising from the quasi-periodic structure of the pore network in CPG-10,and & ( q ) is a structure factor representing the adsorbed layer at. the poro walls. As in t.hc work of Formisano and Tcixoira [101, 1021, tho= structurc factors arc rc:prcscnt,cd by scalilig furictioris proposed by F'urukawa [103], which exhibit a correlation peak characterizing t,he periodicity of the matrix and adsorbed layer. The two structure factor terms are weighted by contrastf factors, kG and kF, resulting from the difference in scattering-length density between the silica matrix and the adsorbed layer and between the layer and the core fluid, respectively (see Fig. 4.16). The temperature and composition depcndenco of tho contrast parameters arc of central importgame to the tlicwrctical analysis oT t hc SANS cxpcriiiicnts in the current section. The term Ip/q4 in Eq. (4.144) accounts for the Porod scattering [104], which is caiwad hy thc grariiilar st,riictiiro of tho CPG-10 and thc rrsiilting contrast, between the matrix and the liquid in the interstitial space. The term ZBG xcount8sfor the noncoherent scattering background resulting from the nondeuterated organic liquid. At high temperatures (i.e., in the onephase region of the liquid mixture) the scattering of the pore liquid is described by an Ornstein-Zernike term, Z(9) oc (1 q2(6)-': with a correlation length &! characterizing the composition fliict,tiations. At temperatures well below the bulk-phase separation temperature of the pore liquid, the leading contribution to the scattering of thc liqiiid is oxpcctd to arise from domains of t-ha two liquid p h a t s : which havc sliarp iriterfaccs. This contribution, rcprcsentcd by tlic theory of Debye et al. [105], has the form I ( 9 ) o( (1 + q2&-2, where & is a correlation length representing the mean separation of the scattering objects. Analysis of t,he SANS data [I061 reveals that the Ornstein-Zernike term can be neglected-at low temperatures, whereas the Debye term can be neglected at high temperatures. Details of the data analysis based on the expression given in Eq. (4.144) are presented in Ref. 106. The mean-field model described in Section -4.6.2cannot account for fluctuation-induced effects. In the current work, we focus on the effects caused by confinement. and

+

Mean-field theory

163

Figure 4.16: Sketch of the irregular silica network, an adsorbed layer at the pore wall, and the liquid in the core of the pore (cf. Fig. 4.1). The contrast parameters arise from differerires in the scattering leiigth density between the ) between the core fluid and the adsorbed layer inatrix and adsorbed layer ( k ~and ( k ~ )respectively. ,

by tlic iiiteraction of thc coinpoiicrits with tlic pore walls. In that coiitcxt, fluctuation-induced effects of the sort just described are believed to be of minor importance.

4.8.2

Lattice model of water iBA mixtures

To gain an understanding of the experimental findings, we adopt a lattice model of a binary fluid mixture similar to the one introduced in Section 4.6.1. As in Section 4.6.1, we consider a simple cubic lattice with lattice constant P. However?unlike in Section 4.6.1, we now assume molecules t o occupy the cubic rclls of voiuma t3 forrnd by the siirrountling 1attir.c sitm rather than occupying ttic sites thcrnsclves. This approach allows us to accourit for the differenl sizes of water and iBA molecules (see below). Each crll may bc ocmpird by cithcr D 2 0 (W) or iBA (A), or i t may bc empty. The attractive interactions between the moleculw are accounted for by a square-well potential where the width of the attractive well is P as before, which is to say that only molecules locat,ed in nearest-neighbor cells on the lattice interact directly with each other. The depths of the attract,ive wells are E W , EWA, and EAA for water water, water iBA, and iBA iBA interactions, respectively. Because iBA molecules are about five times larger than water

164

Neutron scattering experiments

~riolcculm,the cells occupied by w&cr coiitain niort: parallel molcculm than those occupied by iBA. Thus, the strength of the attraction between two water-occupied cells is greater than that bet,ween two iBA-occupied cells. By the same token we expect that the interaction parameters decrease in the order ERW > EWA >> EAA. In this analysis, we deliberately set EAA 0 to reduce the number of model parameters. In thc modol of the porc, t,hc rriixtmc is coiifincd batwccn two planar. honiogcnous substrates pcrpcndiculw to the z-axis. Thus the two substrata are at Z = 0 and 2 = z + 1, where x is the nurnber of lattice layers of the mixture parallel with the x-y plane. The width of the slit-pore is d . Molecules do not occupy lattice cells at Z = 0 and 2 = t + 1, which reflects the hard-core repulsion of the substrate. In the experimental system the water molecules are favored by the pore wall. This preferent,ial interaction wit8hthe siibstratc is modalad by a potmtial [107] (4.145)

Equilibrium states of the water-iBA model mixture are chara.cterized by minima of the grand-potential density

1

(4.146)

where the mean-field iiitrinsic free energy is defined in Eqs. (D.30) and (D.31). Unlika in our previous discussion of t,hc ronfincd binary mixtiire, we now abandon the restriction stated in Eq. (4.124) w indicated in Eq. (4.146) where the mean chemical poteiitial (4.147)

arises. According to its definition, ji couples to the mean density of the water-iBA mixture pw + FA.where

- -

(4.148)

On the other hand, the incremental chemical potential in Eq. (4.146) defined as 1 (4.149) APE + - P A )

Mean-field theory

165

+

coritrols thc corripositioii 7jw FA of tlic mixturc. To obtain the density profiles of the confined binary mixture, which are of primary concern in this study, we minimize w(p, rn; T, p, Ap) according to the recipe described in Appendix D.2.2.3 for giveii values of PI AIL,T, E W , EWA, -I, and E S . Only the latter two of these parameters describe the confinement effect due to the substrates. However, in the experimental systems, the pore geometry of CPG is more or less that of cylinders rather than of slits as in tlie theoretical model. Thus, the siirfacc?-t,o-vohinic:ratio uS/up is larger for a cylindrical porc gmnictry. To take this into account, we introducc the coiicept. of a hydraulic pore radius TI, G 2as/.up following the suggestion of Rother et al. [108]. Whereas for cylindrical pores f h is just the radius of the cylinders, for slit-pores Th is taken to be twice the pore width; that is, Th = 2zC. Setting the lattice constant equal to the meaii distance of molecules of type W and A, and C = 1 nni, we find for the nominal pore size l'h. = 6.8 nin of CPG-10-75 pores [108]. and thcrcforc, the overall number of layers corresponds to z = r h / l = 7. Additionally, all ticnsitics p . arc mcasurcd in w i t s of F 3 , whcrcas kuT, 11, Ap, EWA, and ES are in units of EMW.

4.8.3

Phase diagram

4.8.3.1

Bulk Mixture

As explained in Sectioii 4.8.2, our bulk inodel depends on four adjustable parameters, namely T ?p, AIL,and EWA. Among these the mean chemical potential 11. determines mainly gas-liquid phase coexistcncc. Bwaiisc the bulk cxperirxiciital ixiixturc is always in tlie liquid state, p should bc larger than -2.0 (in our dinlensionless units) so that we are dealing with a (bulk) liquid state in the niodel calculations for all considered temperatures. The incremental chemical potential Ap, on the other haiid, is the thermodynamic variable conjugate to the coinposition cast here in tcrms of jjw - j j A for convenience [see Q. 4.1461. In accord with the experimental conditions we adjiist Ap such that. thc model mixturc is at the liquid-liqiiid cocxistcncc for a given temperature below t.he critical point. Therefore, Ail is fixed by this coexistence between water-rich and iBA-rich liquid phases. To link the dimensionless model temperature T t.0 that of the experimental study, we use the criterion tha.t, the critical solution temperature of the experimental system, Y P , is to be matched by that of our model mixture, T,. Moreover, we introduce a temperature offset , to^, because: this lattice fluid model is less suitable at low temperatures. Because this model is based on a meaii-field treatment and does riot account for the difference in inolecular size

166

Neutron scattering experiments

of tlic two coIIipoIicrits, the calculated cocxistcxicc curve: docv riot represent the phase behavior of the real system over an extended temperature range, especially at low temperatures where packing effects play an important role. This deficiency of the model is compensated by introducing a temperature shift as an empirical parameter by which the coexistence curve of the model can be tuned to that of the real system. Thus, the (dimensionless) model temperatiire T is tmnsforrncd into the model temperature Tmdin K by --Tlnd - T (4.150) - - (yp - to^) To, TC

+

In the experiment, mass fractions rather than volume fractions are used. The conversion from the volume densities p" and pA into the mass fractions wsand W A is effected by (4.151)

with [cf. Eq. (4.127)] m.

jjw - F A

(4.152)

pw tpA and q = 0.850 is the ratio of the mass densities of iBA and D20 at 30°C. The model bulk parameters Tow,p, and EWA are adjusted to the experimental bulk phase diagram in the following way. Figure 4.17 shows the cxpcrimcntal and calciilat,crl cocxistencc curves for thc adjusted paramctcrs to^ = 261 K (-12"C), / r = -1.05, arid EWA = 0.30. The resulting inoclcl critical temperature, Tc,is 0.5 K higher than the experimental value reported by Gansen and Woermann [log] (T;e"P = 318.19 K).5 Close to the critical point the coexistence curve is expected to conform to a power law AUJA oc (T, - T ) f i

(4.153)

where /3 is a critical exponent [cf. Eq. (1.86)]. Our calculation verifies the usual mean-field critical exponent 9 = (see Section 4.2.3 for comparison), whereas Gamen and Wocrmmn [log] found N 0.330 as expected for a real mixturc. The critical point of tlic systmi is located on thc water-rich side, reflecting the fact that the strength of the attraction between water molecules exceeds that between iBA molecules. We note that the influences of ji and EWA on the bulk phase diagram in Fig. 4.17 are essentially independent. Changes in jZ lead to a horizontal

4

5Thevalue of Y P reported by Gansen and Woermann [lo91may be too high on account of impurities in the sample [110], which is, however, inconsequential for this analysis.

Mean-field theory

167

40

10

0.2

0.4

0.6

0.8

1

Figure 4.17: Liquid-liquid bulk coexistence curve of iBA+D20 (A+W) in the WA-Tprojection. The shaded region marks the region of liquid-liquid phase coexistence. S, changes in shift of the whole coexistence curve along the Z O A - ~ ~ ~whereas modify the curvature. The shaded area in Fig. 4.17 marks the liquid liquid coexistence at which the bulk system separates into a water-rich and an iBA-rich phase.

EWA

4.8.3.2

Confined system

Now we consider the water+iBA mixture confined in pores. As outlined in Section 4.8.2, we use the concept of the hydraulic radius to compare the r e siilts for thc controllcd-pore glass matrrial (CPG-75, Th = 6.8 nm) with those for a model slit-pore of width approximately 3.5 nni. Taking the adjusted parameter set from bulk, that is, To* = 261 K, F = -1.05, and EWA = 0.30, we have the only remaining model parameter ts,which measures the preferential strength of attraction of D2O for the substrates [see Eq. (4.145)]. The parallel experiments of Rother et al. (1081 suggest that ES should be larger than 1.5 but less than 5 (in our current units) to avoid unrealistic strong

168

Neutron scattering experiments

u

0

Figure 4.18: Phase diagram of the model mixture in the WA-Tprojection for the bulk systeni (-) and rnnfined in the slit-pore (---) with ES = 3.0. Shaded regions are phase coexistences of the confined system. Dots ( 0 ) indicate critical points. Paths I aid I1 display ternperaturc qucnclicr at two fixed compositions (mean mass fractions WA = 0.25 and WA = 0.54,respectively). adsorption cffccts. The model coexistence curves for our choice of ES = 3.0 are shown in Fig. 4.18. The shaded areas indicate regions of phase coexistence of the confind syst,cm. The rcmarkablc change of the phme diagram relativc to t#hatof the bulk system is caused by the strong confinement t,ogether with the strong selectivity of the pore for water. As expected, the critical temperature of the pore fluid is shifted downward. The critical composition has moved toward the water-rich side because of the selective character of the substrates. In addition to the liquid liquid coexistence curve, the confined fluid exhibits two further, smaller phase coexistence regions at larger W A and lower T . The coexisting phases represent water-rich films of a thickness corresponding to one or two layers, which are distinguishable only at lower temperatures. The existence of such first-order layering transitions may be overestimated by our lattice model on a homogeneous surface and enforced unrealistically

169

Meamfield theorv

&;=1.5

50 -

-

40

20

10

0

0.2

0.4

0.6

0.8

1

Figure 4.19: Phase diagrams of the niixture in the T-WAprojection for the model slit-pore for different, water substrate interaction strengths E S . The bulk coexistence curve is represented by the thin solid line. Dots ( 0 ) indicate critical points.

by the geometry of the iinderlying (simple-cubic) lat,tire. Siich tmmsitions arc, in fact, wrakcncd or abscnt, in continuous models or in inodcl systems with rough or heterogenous walls [ 111, 1121. As shown in Fig. 4.18, confineinent of the liquid mixture leads to a strong dcprwsion of the critical tcmpcratiirc T,, in ;t<.c.ordance with carliar stiidics [84,91, 113, 1141. This critical yoiiit shift caii bc attributcd to tllc cutoff of the range of concentration fluctuations when the correlation length & becomes equal to the pore width. Confinement also leads to a shift of the critical point t o a more watcr-rich cornyosition (i.e., in the direction of the component, which is preferred by the walls). This finding is consistent with results of earlier studies (93, 1151 and can be rationalized by the fact that thc pore liquid is highly inhomogcncous. Figure 4.19 shows the influence of the substrate attraction strength ES. As can be seen, the larger the selectivity of the substrates, the lower the critical temperature and the smaller the critical iBA-concentration. W A . Also, we

170

Neutron scattering experiments

observe a wcakcriirig of thc filrri phase trarisitioiis for strongly adsorbiiig walls as almost all water molecules stick to the substmtes. But for all E S values, these film phases are distinguishable only a t temperatures well below the corresponding critical solution point.

4.8.4

Concentration profiles and contrast factors

We now focus on the behavior of the mixture as afunction of T at constant overall composition in order to compare the predictions of the model with the results of the experimental study. Specifically, we consider the "trajectory" paths I arid I1 indicatcd in Fig. 4.18, which corraspoiid to the mass kCwt,ions llJ~ choscri ill tht: t:xpcrinicntal study. Schcin~nclv t al. [ 1061 showed that along these trajeclories the Porod parameter Ip (see Sec ion 4.8.1) does not change with temperat.ure, whicli indicates that the m e w composition of the mixture in the pore stays constant#for all temperatures. We adopt the followiiig prescription to coinpare the measured contrast factors, kG and kF (see Section 4.8.1), with. those predicted by our model. Tha dcnsit,ies, and &,, of thc first layer at, the pore wall and the mean dciisitics, p& awlp&,,, , irisiclc the rcmaiiiirig porv spa(:(: arc givcii by

F

pg

(4.154a) (4.154b) whcrc o = A, W and wv set, again z = 7. Taking the valiics of scattering length densities d [116] in units of 10" cmV2,d, = 6.37, d A = 0.537, and dsioz = 3.64. for D20,iBA, and SiOz (silica matrix), respectively, we may define the contrast paranieters in iinits of 10" cm-2 for a given model phase bY kG

k~

G

(dl, - &iozl = IS.S7mW, 4- 0.537plaY A - 3.641

E

I L r e

=

16.37 ( P Z e -) :A

-

+ 0.537 (Pkre - k

y )

I

(4.155a) (4.155b)

To achieve the stable phase of constant mean concentration, W A ,for fixed temperature T, we have to distinguish two scenarios. If the state is in the o n e phase region (outside the shaded regiorls in Fig. 4.18), then the differential chemical potential A p couples naturally to and behaves monotonically with the coiicentration W A . Therefore, Ap has to be varied until the chosen value of U!A is reached. On the other hand, for states in the phase coexistence

Mean-field theory

171

region, the rriixturc separates iiito two stable pliascs [i.o., (1) itlid (2)] with Also, AILis fixed by t,his mean composition such that w!) < WA < I$). coexistence at the fixed T. The constant mean concentration, W A , is then realized such that the condition

(4.156) is in&, wlicrc z is tlic riiolc frwtioii of pliasc (1) and (1 - x) that of piiasr (2). The cont-rast,parameters of the t,wo coexistent, phases are assumed to be additive such that the mean contmst parameters are given by

(1 - X ) k G(2)

(4.157a)

rk,(1) + ( 1 - ~ ) k (2) p

(4.157b)

kG

= X k G(1)

k~

=

With the above prescriptions we can compute the density profiles of all stable phases along paths I and I1 in Fig. 4.18, respectively. Froni these density profiles, we calciilatc the contrast parameters using Eqs. (4.155) and, if rimessary, Eqs. (4.156) and (4.157) t.o compare them with the values extracted froni the experiments. 4.8.4.1

Path I

(WA

= 0.25)

Here we consider the behavior of tlic coiifimd wat-cr-richliquid rriixturc: (mcari mass fraction of amphiphile W A = 0.25) on the trajectory I in Fig. 4.18. S A M measurement.s of the intensitmyZ(q) [10G] show that Z(q) is independent of temperature in the experimcntal range (10 - 70°C). In tcrms of Eq. (4.144), this implies that within this temperature range the two contrast parameters: kc. and k ~ reinain , constant, which is surprising in view of the fact that the mixtnrc iindergws R phase separation in this tcmpcratnrr: rnngc. Consider now the results of tlic rriodcl calculations. As shown for path I in Fig. 4.18, the confined mixture separates iiito two phases (shaded region) at temperatures below T 21 30°C. Volume density profiles for a 3 . 5 m slitpore are shown in Fig. 4.20 for four different temperatures. It, is seen that water is preferentially adsorbed at the pore walls. Figure 4.20(a) and (b) show the situation in the singlephase region at 7'= 70°C and 50"C, whereas thc graphs in Fig. 4.20(cl),(c2)and (dl),(d2) illiistrat3cthc dansity profilm for the. two- hase region at 25°C and 10°C. The mean mais fraction of 3hase (1) [i.e., w i'; see Fig. 4.20(dl)] is lower and that of phase (2) [i.e., u$; see Fig. 4.20(d2)] is larger than W A = 0.25, which gives the portions z of phase (1) and (1 - F) of phase (2) across all pores according to Eq. (4.156). If the system exhibits a phase separation, almost the entire aniount of water in the iBA-rich phase is adsorbed by the walls and may not contribute

172

Neutron scattering experiments

P

Figure 4.20: Volume density profiles for constarit mean mass fraction of W A = 0.25 for temperatures given in the graphs. (a) and (b) show stable single phases for T = 70°C and T = 50°C, respwtively; (cl),(c2) and (dl),(d2) show coexisting phases at the same temperature, respectively. Dark bars represent DzO, and bright bars iBA amounts.

to the phase coexistence [see, e g , Fig 4.20(dl)]. Thus, the iBA-enriched liquid inside the core region, which is confined by the water-rich layers, needs more water for pliase separation. This explanation is coilsisteiit with the result that the shift of the critical composition becomes more pronounced for strongcr matrix sclcrtivity, as shown in Fig. 4.19. In addition, prcfcrcncc for the minor component, by the walls leads t o surfaceinduced phase transitions [94, 971. Figure 4.18 displays film phases of one and two water-rich layers at low temperatures on the iBA-rich side in the phase diagram. Because the lattice fluid model may overemphasizes these layering transitions, we believe that they are irrelevant in this cont,ext.

In the model the two coexisting phases are well separated and the influcncc of the inbrfacc htwccn thcm is not included. Therefore, quwtions of the morphology of the phwe-separated liquid in the pore space are beyond the scope of this model. For the experimental system, it is believed that the two coexistent, phases form a domain structure on a length scale of the pore size [lOG], with the domains of the iBA-rich phase [Figs. 4.20(cl) or 4.20(dl)] located mostly in rather wide pores or pore junctions of the network [9S] and the water-rich domains [Figs. 4.20(c2) or 4.20(d2)] in narrow

Mean-Beld theory

173

porc rcgioiis. Such a distribution of thc doniaiiis is suggested by Kohonen and Christenson [117] who showed that the phase in which the preferred component is the majority component is found in regions of smaller pore widths. The sandwich-like structure of the iBA-rich phase as in Fig. 4.20(dl) for slits suggests a tube-like structure of that phase in cylindrical pores and in CPG-10. Such shapes have been found in earlier studies [95-971. The SANS maasiirc!ment,s [106] on t2hcmixtnrc! of constant. mean rnws fraction of WA = 0.25 showcd that the contrast factors kG and k~ reiriain constant in the temperature range of 10 - 70°C. A constant k~ means that composition of the first layer adsorbed at the walls is unchanged in all phases in this temperature range, which can be seen in the model density profiles in Fig. 4.20. This also implies the constant contrast between this layer and the composition of the core region because the total mean mass fraction stays constant at UIA = 0.25.

T I OC Figure 4.21: Model contrnst parameters kG and kF calculated with Eqs. (4.155) and (4.157) as functions of temperature for W A = 0.25. The shaded area indicates

the region where phase separation occurs.

Contrast parameters kG and k~ computed from Eqs. (4.155), (4.156), and (4.157) for various temperatures are displayed in Fig. 4.21. The shaded

174

Neutron scattering experiments

z /nm Figure 4.22: As Fig. 4.20 but for W A = 0.54. Here, we have no phase transitions. region again denotes the two-phase coexistence below T N 30°C. It is seen that neither of the two parameters exhibits a pronounced temperature dependence. This result can be rationalized by the fact t
Path I1

( W A = 0.54)

U'e now consider the behavior of the confined iBA-rich liquid mixture (mean mass fraction of iBA W A = 0.54) along path I1 in Fig. 4.18. Density profiles from the model calculation for four different temperature are shown in Fig. 4.22. At, t,his composition (mean maw fraction W A = 0.54) the mixture does not undergo phase separation (see path I1 in Fig. 4.19). As water is the minor component of the mixture and is strongly preferred by the substrate, almost all water accumulates in the first and secoiid layer at low temperatures (T 30°C), whereas the core liquid is almost depleted of water [see Figs. 4.22(c) and 4.22(d)].. For this reasoii the contrast factor k~ attains a high and nearly constant value at low temperatures, as can be seen in

s

Mean-field theory

175

Fig. 4.23(b). F'urthcrinorc, thc contrast yararrictcr k~ in Fig. 4.23(a) sliows a similar behavior in this temperature range. In t8hislow temperature regime (T 5 30°C) the layer next to the substrate is composed almost completely of water [see Figs. 4.23(c) and 4.23(d)] and there is a sharp interface from this water-rich layer bo the iBA-enriched core region of the pore. Thus, the surface-directed structure of such a phase rcvc:al.s a sharp layering reminiscent, of the tiibc-likc morphology in cylindrical pores (O5-(37]. Wc agaiii cniphasizc that, the thermodynamic statc of the syslem here is away from two-phase coexistence. These surface layers have a thickness of about half of the pore width [sm Figs. 4.23(c) and 4.23(d)]. Iu this temperature range the SANS nieasurements suggest that, the Debye term Corresponding t o sharp structures of size & is now dominant [106]. These structures have a size of about 6s = 3 - 4 nm, which is the half of the nominal pore size of 6.8 rim [108]. This, our riiodcl calciilations rcproducc thc oxpcrimcntal findings. At higher temperatures, the density profiles exhibit more gTadual concentration changes from t,he pore wdls into the core region. As can be seen in Fig. 4.22(a) and 4.22(b), the layer core interface becomes more diffuse corresponding to water enrichment of the core region. Thus, the two contrast fact,ors arc decreasing with increasing temperature (see Fig. 4.23). Because tho cxpcrimcntally dctcrrnincd contrast factors arc uncertain to within a tcrni~cralurt~iride~c~i~ciit, scdc factor, thcy arc plottcd in Fig. 4.23 iii abitrary units adjusted so that the values at. T = 10°C coincide with t,he corresponding values predicted by the model. Therefore, alt,hough we cannot compare the predicted and experimental magnatudes of k~ and kF, we can see from Fig. 4.23 that the dependeiice of k~ and k~ on temperature for the niodel agrees qualitatively with that of the experiment. Comparing Figs. 4.23(n) and 4.23(b), we find good qurilit,at,ivcagrccmcnt bctwccri the rriodcl arid cxpcrirrieiital codrast factors iii thc inodcl mid in the experimental study. The experirnental curves of Fig. 4.23(b) also show that no phase transition occurs within this temperature range. As water is in the minor component. ( W A = 0.54),a first-order transition would exhibit a discontinuous change in the composition of the adsorbed layer corresporiding to a discontinuity in both, kc: and k ~ which , is not apparent in Fig. 4.23(b). Howrvcr, t,hc graph of kG shows a proiioiinccd adsorption of water by the pore walls even for high temperat,ures. The plot of k~ [see Fig. 4.23(b)] indicates a decreasing contrast bet,ween the layer and the core region, because the sharp interfaces become diffuse.

176

Neutron scattering experiments

1 -

(b)

'

T / OC Figure 4.23: Contrast parameters (a) kc and (b) k~ as functions of temperature obtained frorn the [nodel calculations with Eq. (4.155)(-) and from the scattering measurements (0)at WA = 0.54. The experimental data are scaled arbitrary as described in the text.

Reviews in Computational Chemistry Kenny B. Lipkowitz &Thomas R. Cundari Copyright 02007 by John Wiley & Sons, Inc

Chapter 5 Confined fluids with short-range interactions 5.1

Introductory remarks

As we mentioned a t the beginning of Chapters 3 and 4, the key problem in statistical physics of equilibrium systems is to somehow get a handle on ralriilating thr: partition function Q [scc Eqs. (2.38), (2.111)] or, equivalently, thc coiifiguratiori intcgal Z lscc Eq. (2.112)1. This problcrri arises bccausc in general the microscopic constituents forming the system of interest are not independent but rather correlated in their spatial arrangements because of iiiterniolccular interactions. This, in turn, inakes it. impossible in general to factorize the multidimeiisiorial integral in Eq. (2.112) into a nuinber of simpler ones that are tractable individually. Such a factorization is only possible if thc systcm is simplc cnongh. Ail vxamplc is thc on(.-diinmsional hardrod fluid coiifiried betwcwi hard substratcs that we discussed in Chapter 3. Unfortunately, this model is virtually useless if one wishes to address one of the key issues of this text, namely that of phase transitions in confined fluids. Therefore, we tackled the problem of evaluating the partition function of a three-dimensional many-particle system of interacting constituents by a different strategy in Chapter 4. Tn this chapter we introduced various versions of mcan-ficld thmry. In rsscnc'c, and rcgardlcss of whether one is dcaling with lattice or off-lattice models, all these approaches share the complete neglect of intermolecular correlations. Examples, are given in Eqs. (4.16)-(4.19) for the van der Waals treatment, Eqs. (4.77) arid (4.84) for the onecomponent lattice fluid, or h s . (4.129), (4.130), and (4.131) for the binary lattice-fluid mixture. Despite the differences in the precise form of the inean-field approximation, it always reduces the original problem of solving the overwhelmingly 177

178

Introductory remarks

coriiplcx iiitcgral 2 to the extent that the rciritliriiiig cxyrcssioris can be liaridled analytically, whicli is a great achievement indeed. Invoking a.dditiona1approximations like a high-temperature expansion of the free energy [see Eqs. (4.3), (4.4),or (4.63)] or the Bogoliubov variational theorem [seeEq. (4.76)] combined with a restriction t,o nearest-neighbor interactions eventually leads to closed expressions for the relevant thermodynamic potentid throiigh which cqiiilibriiiin propartics of many-particle systems ran be calculated [sce Eqs. ( 4 2 4 , (4.83), or (4.134}.. Unfortunately, despite its surprising power in predicting properties of many- particle systems, the meanfield approximation is crude a.nd breaks down in particular in the near-critical regime. However, it is noteworthy that this breakdown is quantitative rather than qualitative. This is reflected by the fact that in mean-field theories order parameters usually exhibit a power-law dependence on the relevant t h e m e dynamic fiald whcrc only the valiia of the (critical) exponent governing the I ~ with riiorc sopliisticatd thcorics or powcr law is iisiially W ~ O I c:oriiparwl experimental data (see Sectmion4.2.3). If one wishes to abandon the mcan-field approximation in dealing with complex and realistic syst,ems (with respect to parallel experiments) , there is essentially only one alternative t o solve the key problem in statistical physics of equilibrium systems. This alternat.ive route is offered by c:ornputer sirnulations whmc onc aims at, raprascnthg niimaricallv the cvolution of a manyparticle system in pliasc (i.c. nioriicnturn arid configuration) space by cxplicitly considering the interactions between the microscopic constituents of such a system. Because evaluating the int,eractions between the constit,uents is nuniericdly demanding, simulation systems are usually microscopic in size: accommodating between, say, of the order of lo2 to about lo6 molecules given the current capacit,y of computers. Pcrhnps thc: biggest advantage nwociatcd with the advent, of c:ompiitcr sirnulatioils chririg tlic 1950s a i d 1960s is that they should bc regarded ils first-principles methods: Apart from adopting a specific int.eraction pot,ential, no other a.pproximations are involved. In part,icular, intermolecular correlations, which were completely neglected in the mean-field theories, are exactly accounted for, at least in principle, with the exception of the near-critical regime where special techniques need to be invoked t,o deal with the problem of (qiiasi-) macroscopic rorrclnt,ioii lcngths in inicrosropic simulation systcms [ 118, 1191. However, the theory of critical phenomena is not of central interest here so that we defer the interested reader to the vast literature in this field [15, 120, 1211. Another .great advant,age of computer simulations is that they offer the possibility to directly visualize the spatietemporal evolution of a system in addition to being anienable to a niathematical analysis in terms of

179

Confined fluids with short-range potentials

rigorously defiucd statistical physical ciititics like local dciisitics or various other correlation functions. In fact, the graphical analysis of either individual configurations generated in a computer simulations (i.e., “snapshots”) or video sequences often gives one an idca of how to define a meaningful quantity in a mathematical rigorous way to analyze the results of such a siniulation in terms of well-defined statistical physical quantities. Ordered stmctiircs with many d c f d s that can bc analyzed in terms of suitably dcfined bond-orientat,iori correlation fuiictions provide mi cxaiiiplc [ 1221. The choice of a proper bond-orientation correlat,ion function is usually dictated by the symmetry inherent. in the ordered structure, which can be determined best by directly looking a t individual snapshots from the simulation. Another example illustrating the suitability of such a graphical analysis will be discussed below in Section 6.4.2. From a more general perspective, computer simulations can be grouped into two different classes. Monte Carlo (MC) methods aim at generating a sequence of configurations in a specific statistical physical cnscmblc according to the relevant probabilit>y density p (T”’;X) [see, for example, J3q. (2.117)] governing the distribution of these configurations in configuration space where X is a vector consisting of the implicitly fixed variables on which p depends [for example, N and s, in Eq. (2.117)]; in molecular dynamics (MD) simulations, on the other hand, one solves numerically the equation of motion of a many-particle system and obtains the full information about the temporal evoliitim of the many-particle system in phase spacc. If the equilibrium syst,em is ergodic [123-1251, that, is, if

(0)= 1 0 (rN)p ( r NX; ) d r N = lim t-a!

t

0 [r” ( t ) ]d t

(5.1)

n

both M C and MD can be expected t o yield identically the same results for equilibrium properties (O).’ We briefly note that, glasses are an important class of systems for which Eq. (5.1) frequently turns out t,o be invalid2 even for quite long periods of observation 1. Glasses are characterized by complex free-energy landscapes with rather dcap “valleys” that. cannot easily be surmoiintd given the low c : ~ system ~cc bwoines thcrixial energy t$ypical of a glaxs. As a c o ~ ~ s c ~ ~ uthc locked into a rather narrow region of (classic) phase space r (see Section 2.5) ‘We explicitly disregard the presence of inaccuracies associated with, for example, inadequate saiiipliiig of cotitigurtttiori space. ‘Of course, Eq. (5.1) would always be valid if one could let the observation time t bernme st,rirtly infinit,e, which is generally precliided i r ~pmctice.

180

Introductory remarks

for a lorig tirric until, eventually, a fluctualioii arises pcrriiitting thc system to escape ancl to explore a wider range of I?. In Eq. (5.1) thc scqiicncc of points { T ( ~1 ) ) rcprcscnts a trajectory in configuration space obtained as a solution of the (classic) equation of motion and 1 denotes time. Moreover, we tacit,ly assumed that the microscopic quantity 0 (rN) whose average (0)we wish t o calculate depends only on the positions r N of the N molecules of the system but not on their monienta PN. In this work we shall exclusively concentrate on the MC method for two main reasons:

1. Compared with MD, MC is conceptually closer to the Gibbsian version of equilibrium statistical physics on which we essentially based the analysis in Chapter 2. 2. M C can be more easily adapted to specific statistical physical ensembles. The second point is of particular importance if one wishes to study phase equilibria where one needs to have access to a thermodynamic potential. This is rather cumbersome in MD so that it is generally not advisable to employ MD to investigate phase equilibria. Consequently, MC has been much more widely used in numerical studies of equilibrium properties of confined systcnis. The advaritagc of MD is that it, ycririits one t o calculate trarisport properties in equilibrium systems such as diffusion constants. Investigat?ion of transport phenomena is hampered in MC where one (at least in most cases) does not know on what time scale the system of interest evolves. The only serious disadvantage of both simulation techniques ( M C and MD) is that they will always be limited t o systems that are extremely tiny on a macroscopic scale because of storage limitations and lack of computational speed. Because of the small sizes of simulation systems, special techniques have been devised to save computer tiine and to avoid domination of surface cfferts in the minute saniplcs o ~ i cis siinulating. Thesc techniques ran be iniplemented in a more or less straightforward manner as long as the interaction poteiitials between the constituents of the system decay sufficiently rapidly. Special precaution is, however, required, if this proviso is not met, that. is, in systems where the interaction potentials decay rather slowly with intermolecular distance. Two prominent examples pertaining to this latter class are Coulomb interactions between charged molecules or interactions between molecules with a permanent dipole. In these cases, highly sophisticated techniques have been devised to obtain reliable results in computer simulations

Monte Carlo simulations

181

dcspitc tlic srriallricss of thc sariiplcs uiiclcr study. Bccausc of tlic great irriportance of confined fluids with siich “long-range” interaction potentials, we defer a discussion of these techniques to the subsequent Chapter 6. Accordingly. this chapter will be devoted to confined fluids with short-range iiiteraction potentials where the aforenienticned limitation of system sizes poses a far less serious probletn.

5.2 5.2.1

Monte Carlo simulations Importance sampling

Because MC is a numerical technique to calculate multidimensional integrals like the one over configuration space in Eq. (5.1), we begin by discretizing configuration space and rewrite the integral as M

( 0 )= lim

M-.w

C o (r:) p

(T:;

X)

m= 1

where M denotes the tiumber of “points” r: in configuration space.3 The basic idea then is to soinehow generate a sequence of “points” in configurat,ion space { r : } . However, because of the high dimension of configuration space one should not generate the points {r:} according to some regular array hcxaiisc. one can dcmonstrate that, thc ovcrwhclming niimher of points on this array will lie on tlic surfacc of this rrrultidirneiwiotial spac~cc~.Hence, in this case one focuses on a narrow and perhaps largely irrelevant region of configuration space, which consequently might be sampled rather poorly. To appreciate this latter point, consider the trajectory of a single molecule in space. Let us discretize this trajectory such that we represent the trajectory of the molecule by a succession of regularly spaced points. Thus,these points may be viewed as the p3 nodes of a cubic lattice. Clearly, in each spatial dimension, 2 oiit of the t,ot,al mimbcr of tho p nodcs in that, direction lie on the surface of tlic cube. Exteiidiug tliese coiisidcratioiis to N instead of just a single molecule, it is immediately clear that we need to replace the original cube by an N-dimensional hypercube such that. in cadi dimension the fraction p - 2 / p represents the ratio of nodes not on the surface of the hypercube relative to the total number of nodes. To estimate this fraction 3For a more comprehensive introduction to Monte Carlo simulations we refer the interested reader to the excellent text by Landau and Binder [126]. In Ref. 126 the authors discuss many applications of the Monte Carlo technique beyond the scope of the present book.

Monte Car10 simulations

182

for t h . oritirc: hypercubo, wc‘ tlicrcforc ricud to consider

Moreover, to represent the continuous trajectory of each molecule by a succession of regularly spaced nodes, the lattice constant of the hyperciibe should be sufficiently small; that is, p should be large enough so that L 2/p << 1. In this case, we can approxiriiate the logarithm in Eq. (5.3) through In ( 1 - z) = -z 0 (x2)CT --z such that

+

which vanishcs for nach fixcd and finit,(: valiic of p in thc limit N --+ 00. Iri othcr words, iii tlic lirnit, of a sufficiciitly largc: iiurribcr of molec.ul(:s, tho sampling of configuration space is severely biased because almost, all configurations { ~ g turn } out to be represented by nodes on the surface of the hypercube. As a result configuration space may be sampled highly inadequately, which may cause the sum in Eq. (5.2) to converge very slowly toward the correct (0); that is, A1 needs to be very large. This argument has been adoptcd .from t-hc hook of Rindcr and Hccrmann [127]. Tiic problem Lcr.omcs cvcri riiorc scvcrc in view of our discussion iii Scction 2.2.2.2, which showed that in a sufficiently large system the relevant region of (quanturn mechanical) state space contributing t,o statistical thermodynamical averages is very srriall [see, for example, Eq. (2.3O)I. Tkanslating this observation into the language of classic statistical thermodynamics that we are using here, we conclude that as N becomes large the region in configiiration spacc coritribiiting to thc siim in Eq. (5.2) dcclincx as well. This. liowcvcr, rriakcs it cvcii harder to sariiplc configuration spacc adcquatcly by using a regularly spaced hiwyperlatticein configuration space on account, of the above considerations. Because of this additional effect, M in J3q. (5.2) must be made. even larger than was already necessary due tVothe biased distribution of riodes on the hypercube discussed above. It is therefore obvious that, a numerical solution of Eq. (5.1) based on a rcgiilar distribiition of nodcs in configration spacc is qiiitc inallicient, if not. completely prohibit.ive. A much better idea would be toogenerate the “points” { t :} at rundom, which guarantees that configuration space will be sampled without any bias provided hl in Q. (5.2) can be made large enough. One may then calculate 0 (r:) at these points, multiply each value by its associated probability (density), and estiriiate the left side of Eq. (5.2) by summing up all these values.

183

Confined fluids with short-range potentials

However. there is a twofold problcrn with this iiaive approach. First, because the amount of computer time is inevitably finite, we need to limit the summation in Eq. (5.2) to some maximum value M = lllm,. It turns = 0 ( lo5- lo7) is sufficient from a practical out that in most, cases A!=, perspective so that this problem can be surmounted. The second and by far more serious problem with an application of Eq. (5.2) involves the prohdility dcnsity p (r:; X) which is a priori iinX) dcpcnds on known. Ail irispcctiori of Eq. (2.117) reveals that p the (classic) partition funct,ion, which involves the configuration integral as Eq. (2.118) shows. However, tlie partition fuiiction itself is unknown such that the probability with which 0 (r:) needs t o be sampled a t points {T:} remains undetermined. Alternatively, one could eiivision generating points { T: } according to thcir amportance dctcrmincd 11-yp (r:: X).In this cam, it woiild be possible to roplarc Eq. (5.2) by a sirnplor one, namcly

(rt;

where the prirne has heen attached to remind the reader that in the sequence { cach configuration is rcalixcd according to its corrwt probability dcnsity. At this point OIIC may woiidcr what oxic might have gairied by Eq. (5.5). On the surface it seems much more straightforward to generate the random which have to satisfy the additional sequence of points rather than constraint of comyatibilit,y with some a priori unknown probability density. The key t o appreciating tlie great improvement represented by Eq. (5.5) is to realize that in order to generate numerically the sequence of configurations {fit>it tiirns oiit, that only thc: relatitre prohabilit,y mattcrs with which any two “ncigliboring” nicrnbcrs 9-2mid rc++l in tlic scqueiicc { T:;} occur. This is a direct consequence of the Principle of Detailed Balance, which we introduce in Appendix E.1.2 as a stationary solution of the Chapman-Kolmogoroff equation. Based on &. (E.20) we niay design an algorithm compatible with the importancesampling concept. We shall illustrate this below for two separate caws that arc particularly rclcvant, to confincd fluids. Thc first of t h a c concerns the grand canonical ensemble, which mimics sit,iiationv encountered in sorption experiments [31, 128-1321. The second set, of experiments concerns measurements employing the so-called surface forces apparatw (SFA), which permits one to deduce in an indirect way information about, the local structure of confined fluids [133-1371. The algorithms we shall be introducing below are adapted versions of the classic Metropolis algorithm proposed in

rr}

{rr},

184

Monte Cario simulations

1953 to sirriulatc propcrtics of a two-dirncxisiolial fluid of hard disks [l3% 1401. By this algorithm one can generat,e a numerical representation of a so-called Markov process. The sequence of configurations generatsed by the Metropolis algorithm is therefore frequently termed a “hlarkov chain” (see Appendix E. 1).

5.2.2

The grand canonical ensemble

By analogy with Eqs. (2.117) arid (2.118), i t follows that the probability density in the grand canonical ensemble is given by p

(P;N)

=

1 N !A3”’I-,:

exp

[””1

kB T

[

exp -

I/ (FN;N kBT

’1

(5.6)

where the grand canonical ensemble partition fnnction for a classic system is given in Eq. (2.120). For convenience, we introduce in Eq. (5.6) “reduced” coordinates through the transformation

such that the simulation cell is a unit cube rather than the original parallelepiped with side lerigth s , ~ ,where LY = x, y, or z. From the form of p (F”’; N),one may guess iiit,iiitively that the getncration of a Markov chain of confibqrations (see Appendix E. 1.1) should iiivolvc two types of proccsscs, namely

1. A random disp1ac:cmcnt of ~nolccul~’~, that, is

-+

Fr,

2. A random change of the number of molecules accommodated by the system, that is, N,-] + N , both of which we may associate separately with the abstract random process yn-l yn introduced in Appeiidix E.l. It is then possible to identify with F‘, in Eq. (E.20), p (FN;N) from Eq. (5.6). As we realize from the discussion in Appendix E.1.2, the quantity that matters for the transition between gn- 1 +-+ gn is the transition probability Il introduced in Eq. (E.20). Thus. we rcalizc that a ralrulation of I1 docs not raquirc knowlcdgr of thc grand canonical partition function which turns out to cancel between numerator and denominator in %. (E.20). As we also point. out in Appendix E.1.2, the Chapman-Kolmogoroff equation is derived under the assumption of small changes in the random processes represented by y. Hence, the Metropolis algorithm proceeds in two consecutive steps, namely. -+

185

Confined fluids with short-range potentials 1. Pick ~ n o l c ~ u7.l carid replace i t according to

-

~

-

+

i = ,~ j ~, ~ - l6,

(1 - 2 0 .

1: = 1 , . . . ,h‘

(5.8)

where lT = ( 1 , 1 , l), 6, is the side length of a small cube centered on T , , ~ -(i~siially ~ 1 - 10% of the “diameter” of a molmule), and ( is a vector whosc thrce componcnts arc (psciido-) random niinibcrs uniformly distributed on the interval [0,1]. Thcre are two options to realize t,his. The first option is to pick inolerules consecutively according to their storage location in the computer’s memory. The second one consists of picking a molecule at mndorn. In practice, however, it turns out that both approaches lead to the same results provided the Markov chain generated is sufficiently long.

-

2. Change the number of molecules according to

N , = Nn-lf 1

(5.9)

where the “decision” t o add or remove one molecule hau to be drawn at random with equal probability for both options (addition and removal) t o avoid biasing the creation or deletion frequency in the long run, that is, for a large number of creatioii/destruction attempts. Henceforth, the sequence of N displacement attempts followed by Nn-l creation/destruction attempts in grand canonical erlseinhle MC (GCEMC) simulations will be referred to as a ‘.GCEblC cycle.’’ For step 1 of this cycle, we notice that Nn-l = IY,= N remains constant between members n - 1 and n in the Markov chain. Hence, we compute [see Eq. (E.20)]

(5.10) where FE and Fr-, are consecutive configurations distinguished by the location of molecule i [see Eq. ( 5 . 8 ) ] .The transition probability for step l of our GCEMC cycle is the11 givexi l y

ll, = niin [l.exp ( - A U n - 1 + n / k ~ T ) ]

(5.11)

Hence, if the energy decreases in the course of a displacement of a molecule, that is: if AUn.-l+,, 5 0, the displacement. attempt will immediately be

186

Monte Car10 simulations

acccptcd. If on t,hc other hand, AUn-,4,t > 0, tlic displacclrient attempt is not immediately rejected despite the increase in potential energy. Instead another random number ( E [0,1] will be picked. A decision about the outcome of the displacement process will then be made 011 the basis of

<

> exp [-AUn-~-+lt/k.~Tl exp [-A(llz-l+n/k~T] 5 [

--

accept displacement, reject displacement

(5.12a) (5.12b)

Equations (5.11) and (5.12) form the core of the Metropolis algorithm in its classic form [138], which is for the canonical ensemble where, by definition, N = constant. At this point it may seem a bit, difficult to immediately see why Q s . (5.12) work in practice, that is, to see why Eqs. (5.12) will generat.e a distribution of configurations t,hat comply with the partition fiinction of a gvcn s t a t i s t m i d physical ensemble. Let us therefore elaborate on a simple intuitive argument from which the validity of Eqs. (5.12) emerges. Consider a specific configuration r N such that displacing a single molecule gives rise to an increase in total configuratioiial potential energy corresponding to exp [-AUn-l-.n/kgT] = 0.1, say. It is conceivable that there are numerous different configurat,ions { rN } all being characterized by identically thn samc valiic of cxp [-Af/n-l-.n,/kBTI upon particle di~placemcnt.~ Moreover, we inay assume to have access to all these configurations in parallel, that. is, a t the same time. If we then pick a random number distributed uniformly on the interval [O?11 and compare t,his number with the quantity exp[-AUn-l,n/kBT] = 0.1 to reach a decision about whether t o accept each individual displacement, it is clear that in 90% of the cases the decision will be to reject the displacement, whereas in 10% the displacement will bc! acccptcd according t,o Eqs. (5.12). However, whether we have. access to the configurations in parallel or se quentially is irrelevant, which permits us to conclude that, for a suflciently long and eryodic Markov chain, displacenients will be accepted on menage with the correct probability dictated by the principles of statistical physics (i.e., the probability density of a given statistical physical ensemble). Noticn that,, on ;wcwant, of displaing sorric of the N molccnlcs during step 1, a niolecule may cvcntually eiid up outside the unit cube (i-e., the simulation cell) in which it was pla.ced originally. The most convenient way of preventing this from happening is to apply periodic boundary conditions at the faces of the unit cube. That is, one surrounds the simulation cell with other unit cubes accomodatiiig precisely the same configuration as the simulation cell. 41n fact, an irnfinilr!nrimbcr of siich mnfie;lirat,iorwis cnnceivrrblc?.

Confined fluids with short-range potentials

187

If a m o l ~ ~ uoriginally lc Lclorigirig t,o the sirnulat,ion ccll c'rosscs tlic bouudary bet,ween it and a neighboring unit, cell, another molecule from the opposite unit cell will simultaneously enter the simulation cell, thereby guaranteeing a constant number of molecules in the simulation cell. In practice, one needs to store only molecules in the simulation cell. The periodic boundary condit,ions are accounted for by replacing

where sign (6i) is a function returning the sign of its argument and 6i = yi. % for a bulk system, whereas Gi = Zi: F, is for a slit-pore where the fluid suhstratc rcpiilsion at short distancc hctwcrcn a molcculc and tha substratme scrvcs t o coiistraiii the z-coordinate of a fluid molcculc to thc simulation cell. In addition we assume in Eq. (5.13) that, t.he origin of the coordinate system is located at t8hecenter of the simulation cell such that Zi, yi, z, E However, because of periodic boundary conditions, one needs to make sure that, as far as short-range interaction poteiit,ials are concerned, a molecule in the simulation cell interacts only with another particle in the simulation cell or one of its periodic images depending on which is closest. This so-called minimum zm.uge cxmrention can easily lie implemented through the equations -

-

I

- - [-it+$I.

where 6i = Zi,Gi, Zj for a bulk system, whereas Cri = ;Ci,.G, is for a slit-pore as before [see Eq. (5.13)]. If one contemplates Eqs. (5.13) and (5.14). one realizes that, Eq. (5.13) adds and subtracts a box length (or fl in reduced units) depending on whether the molecule has left, the sirnulat,ion cell in the -a-and +a-direction, thereby restoring it by a periodic image. Similarly, Eq. (5.14) serves to ensure that a molecule interacts either with another one in the central cell or one of it,s imagcs dcpcnding on which onc is closcr. In CFWS whcrc! molcculc3 i. and j are separated by more than half the relevant side length of the simulation cell, a periodic image of molecule j will be considered rather than molecule j itself. Thus, in effect the int.eraction between molecules is truncated a t some cut-off distance, which is determined by the dimensions of the simulation cell and its geometry. Although this is not critical for systems governed by short-range interaction potentials, the truncation is the source of considerable

Monte Car10 simulations

188

difficulty in siriiulatiiig syst,eiiiswith loiig-range intcractious. We shall return to this issue in greater detail in Chapter 6. In the second steb of our adapted hletropolis algorithm, we change the number of molecules in the simulation cell by f l . More specifically, we either attempt to create a new molecule at a randomly chosen position in the simulation cell, that is

(5.15) or an existing molecule will be deleted. Both processes need to be carried out with equal probability to avoid biasing the generation of a (numerical representation of) Markov chain of configurations in favor of one or the other process. To determine the trailsition probability in this case, it turns out to be convenient. to introduce the auxiliary quantity

A3 V

c1 B---ln-

kBT

(5.16)

following the origind proposal by Adains [41] where A is the thermal de Rroglie wavelength defined in F4. (2.103). From Eqs. (2.79), (2.111), and (2.112) it follows that the chemical pot.entia1 of the ideal gas is given by

"I

(TN)

= 01

(5.17) where we used also the thermodynamic definition of the free energy [see Eq. (1.50)] as well as the Gibbs fundamental equation in its most general form [see Eq. (1.22)]. Introducing the excess chemical potential via

p

-

- $d

(5.18)

and using Eq. (5.17), it turns out that, Eq. (5.16) may be recast as

(5.19) which has a somewhat more trnrisparent physical interpretation than the original (but numerically more useful) expression given in Eq. (5.16). The decision of whether the attempt to create or destroy a molecule is accepted will again be based on a trarlsitiori probability defined analogously to the one in Eq. (5.11). It depends on the ratio

(5.20)

Confined fluids with short-range potentials

whcrc the argument of thc pscudo-Boltzrniuni factor in

189

a.(5.20) is given by (5.21)

where thc upper sign rofcrs to addition iuid thc: lower orie to rcrrioval of one fluid molecule, respectively. In Eq. (5.21) the meaning of N is that of cither the number of molecules after adding a new one to the system or prior to the removal of ail already existing one. Similarly, U, denotes the configurational energy of the molecule to be added or to be removed from the simulation cell. The creation/destructiori attempt is then realized based on the transition probability = inin [l,exp

(.+)I

(5.22)

In case crc?ation or tlcstruction is iinfavorahlc, that is, if rk < 0 thc attempt will not Lc rcjcct8ctliirirricdiatcly but, rcaliacd ac:cording to a riiodificd Met,ropolis criterion [see Eqs. (5.12)], that is

-

exp ( r k ) > [ accept creation/destruction exp (r+) 5 [ -+ reject creation/destruction

(5.23a) (5.23b)

<

where again E [0, 11 is a pscudw-andom niimhcr. Because stcp 2 of the adapted Metropolis dgorithni for GCEMC siniulations involves a change in density by f l / V between members n - 1 and n of the Markov chain, some care has to be taken in computing U* if the interaction potential is short-range, that is, if it, decays sufficiently rapidly but does not go to zero at any finite separation between molecules. An example is t8heLennard-Jones (12.6) (LJ) potential defined by (5.24) which is frequent.ly employed to model t,he interactions between spherical molecules of “diameter” o that art! separated by a distance rij = Iri - rjl. The strength of repulsive (proportional to r;”) and attractive interactions (proportional to r,j6)is scaled by E > 0, which determines the depth of the attractive well. The attract.ive part of the LJ potential represents dispersive (or van der Wads) intemctions arising from induced dipole moments generated by fluctuations in t,he electronic charge distributions of two interacting particles. Because u (rjj) decays rather quickly, it is convenient t o employ some cut-off parameter and compute (1, only for those molecules located inside

190

Monte Car10 simulations

F

sorric subdomain of tlic cntirc syst,em volurnc ccntcrcd on tlic mo~cculc to he created or removed. Assuming pairwise additivity of intermolecular interactions represehed by u ( r t j ) .we may write

(5.2513) j=I#zCF

where AUc is a correction due t80longer-range attraction that is neglected by limiting the sums in Eqs. (5.25) to interactions within the cut-off solid of volume Q. Explicit analytic expressions for ACT, are derived below in Section 5.2.3 for a slit-pore: which is the most important confined geometry in the context- of this book.

5.2.3

Corrections to the configurational energy

For a coiifiried fluid

A(/, = AIJc,~+ AUC,k

(5.26)

where AU,,K and represent, cnt-off corrections due to fluid fluid and fliiid substrata intcractions. Thc latter arisc in caws whcrc the substrate itself is composed of individual atoms arranged according to some solid structure and interacting with a fluid molecule via a LJ (12,6) potential, say [see Eq. (5.24)). Formal expressions for both corrections to the potential energy can he derived by noting that [cf., Eq. (4.16)]

v\i;

P

v

(5.2713)

whcrc N, dcnotcs thc nrinihar of solid atoms of which the confining solid substrate consists and p(') ( T I ) and p(2) (rl,r 2 ) are one- and two-particle densities in configurations 71 - 1 and n, respectively whose standard definition can be found in textbooks on statistical mechanics (see, for example, Eqs. (4.18) and, for more details, Ref. 17). These corrections differ in step 2 of the Metropolis algorithm adapted for GCEMC because the density differs by f l / V on account of addition/deletion attempts of one molecule as

191

Confined fluids with short-range potentials

poiritcd out, in Scctiori 5.2.2. I n thc first stcp of this algorithm, N is the same before and after displacement of one molecule so that both expressions in h s . (5.27) vanish identically. Notice that the integration over the coordinates of molecule 1 extends over the volume of the cut-off solid, whereas the iiitegration over coordinates of molecule 2 is restricted to the surrounding volume V\? in Eq. (5.27~1). It is customary to relate p(') (rl)and p(2) (vl: r2)to the pair correlation ) Eq. (4.17). To proceed we introduce two key assump function g (TI, ~ 2 via t-ions,narnoly 1. A mcan-field approximation by as.surning [cf.

FKA.(4.19)]

2. We assume the confined fluid to be homogeneous represented by a constant density p [see Eq. (4.20)]. [Jnder these conditions we can rewrite Eq. (5.27a) as

-

where we used t = 22 - -tl and associate with V a cut-off cylinder of radius rc and height s, in the z-direction. In additioii we use cyliiidrical coordinates (p, z ) for the integration over positioiis outside the cut-off cylinder. In Eq. (5.29), .4,0 is the area of the 2-directed face of the urideformed lamella introduced in Sertioii 1.3. Assuming now the radius r, to be sufficiently large, we may approximate the relevant interaction potentials by their attractive contributions only [see Eq. (5.24)],that is -4e,ra6 (5.30) Il.ff ( p , z) N (p2

+ z2)3

192

Monte Car10 simulations

Wc insert Eq. (5.30) inlo J2q. (5.29),carry out thc iiitcgratiori over p, aid obtain

(F)' / /

8./2-21

s./2

U,,n = -mffu6Aa

dzl

-s&

dr

1

(1.:

-sz/2-2,

+ 22)'

(5.31)

The remaining two integrations can be carried out with the help of tabulated integrals [ 14 11. One finally arrives at ?TEffU6 N' UC,B = --

Vl.2

):(

(5.32)

srctan

Now: as configurations 71.- 1 and n in the second step of the GCEhlC-adapted Metropolis algorithm differ in N , = N,%-I f 1, we obtain from the previous expression

(5.33) where A' denotes either the l i m b e r of molecules before removal or after add ition. In a bulk fluid, similar considerations may be used to derive an expression for ciit-off corrections to tha confignrational cncrgy. Using in this caw a cutoff sphere ratlicr tliari a cylinder gives rise to

(5.34) where N has the same meaniiig as above. Onc! may also rlcrivc a closad exprrssion for the corrcction Uc,fsusing thc liornogericity approxiiliation p ( ' ) ( T I ) M p togct,hcr with Eqs. (5.2711)and (5.30) to obtain

J

s.o/a

uc.fs

= -4xlvsEW -

V

dZ2

-s,o/2

froin which Kfs =

--27r&&U6Ndvs[arctan Vr:

1 [rc"+ (z2 - S & / 2 ) 2 J 2

(z)1+

I-,"

+ SZO

(5.35)

(5.36)

follows after performing the remaining integration over q . Assumiiig as before that N , = N,-1 f 1, we yield from the previow expression

Confined fluids with short-range potentials

193

which pcrriiits us to cstirriatc A[/, in Eq.(5.26)using Eqs. (5.33)arid (5.37). In Rcf. 46 it, was shown that the homog(mcit,yasslimption [sec Q. (4.20)] is very reliable in estimating AU,. However, care has to be taken with respect to the mean-field approximatioil [see Ell. (5.28).As was shown by Wilding and Schoen, this assumption is prone to break down if the thermodynamic state of the fluid is in the near-critical regime because intermolecular correlations become long raiige and the assumption of g ( T I , r z ) being unity outside the cut-off solid is invalicl [142]. Hence, Ekp (5.33)and (5.37)[and therefore AUCin Eq. (5.26)]inav lead to crroncoiis rcwilts if cmplovcd uncritically in GCEMC simulations. To avoid these complications, it is advisablc in most cases to replace the infinitely long-range LJ(12,6) potential in Q. (5.24)by

(5.38) where the explicitly short-range interaction pot,ential USR ( r i j ) is defined by

which vanishes continuously together with its first derivative at the cut-off radius r, and is equal to zero for all larger intermolecular separations rij > rc. The advantage is that in this case AU, = 0 because U S R vanishes identically everywhere outside P. Clearly, the same "trick" can also be applied t o other continuous interaction potentials provided they decay with intermolecular distancc 7 f&cr than rT3.A decay slower than (and cqiial to) T - ~is typical for electrostatic potentials such as the dipolc dipole potential. hi this case longer range interactions are importaiit and must not be eliminated in the spirit of Eq. (5.39).In these latter cases, special techniques have been devised to treat the longer range interactioiis properlv. We defer a discussion of these methods to the subsequent Chapter 6.

5.2.4

A mixed isostress isostrain ensemble

GCEMC was introduced as a way to compute thermal properties of a s y s tern in contact with an infiniLely large reservoir of heat and matter in Section 5.2.2. We shall now turn our attention to a sit,uation where the t'hermodynamic system may exchange coinpressional (dilational) work with its surroundings. To siniplify the treatment below, we shall assume there exists no longer any coupling to a reservoir of matter for the time being. However,

194

Monte Car10 simulations

at the cud of this scctiori, wc shall describe how S U Cari : ~ additiorial couplirig can he reiniplemented. As we saw in Section 1.3, the prototypical lamella representing the confined fluid from a purely therrnodynainic perspective may be deformed in a number of ways. For example, the most general expression for tlie exact differential of the internal energy in Ey. (1.43)shows that in the context of t h c w cwmprcwional strains (proportional to thc current analysis thcrr s,, sy, s,) arid oric sliear strairi (proportioiid to ns,o) actirig 011 the larnclla. To mimic a real experimental situation encountered. for instance, in the SFA. one may fix a subset, of stresses and strains and study thermal properties of the laniella under these conditions. For one such example, we present the parallel statistical physical analysis culriiinating in an expression for the gmnd mixed isostress isostrain partition function y in Eqs. (2.70a)[or the classic analog in J3q. (2.118)]. Herr wc shall consider n slightly siinplrr version of riiixctl isostrcss isostrairi ciiscnibl~r;charactcrizcd by N = wrist. Suppose we set N = .Ar* so that the sum on N in Eq. (2.70b)can be ) Section 2.4),where represented by its maxiniiim term exp ( p N * . / k ~ 2 ’ (cf.

Taking the logarithm of this expression and using

&. (2.71),we realize that

(5.41)

@ = - k ~ T l n x =- k B T l n T l - p N * which pcrriiits us to iritroducc a gcricralized Gibbs potcritial Q,

+ p N * = U - TS -

(5.42)

Tzz.4&Sz

as the thermodynamic potential of tlie current m i x d isostress isostrain ensemble. Moreover, replacing on the right side of &. (2.118)the sum over N by it,s maximiim term, onc realizes that the factor cxp (pN*/kBT) canccls in J3q. (2.117)so that we rnay writ$‘ p

(7.”;

sz) =

1 cxp N*!h3” Ycl,l

[1-

~ZzA7.0%

kBT

[u

exp -

( T y

s,)

~ B T

]

(5.43)

for the probability density in the mixed isostxess isostrain ensemble where

(5.44) 50bviously, the factor N*!h3” would also cancel in Eq. (5.43). However, we shall leave it alone for purely formal reasons to preserve p (r” ;sz) as a probability density.

195

Conflned fluids with short-range potentials

is the associated partition function in th: classic limit. To case the riotatiorial burden henceforth, we shall drop the superscript ”*” as well as the subscript, “c1”. Under these premises, t,he analog of Eq. (2.116) may now he cast as

(0) = 8%

1

dr”0 (r”;s,) p (r”;s,)

whcrc we ixitzodiiccd unit-cube coordinates [see fi.(5.7)] replacing, however, sa, by its (variable) analog s,. From the last line of Eq. (5.45), we also realize that the weighting factor for the microscopic quantity 0 (FN;s,) now becomes (5.46) p’ (P; -5.) = s,Np (P; S”) rathcr than p (FN;5,) in t,hc original coordinntc system. Thcrcforc, applying thc Importance Sampling concept [ s a &. (5.5)] to wtiinatc (0) in a niived isostress isostrain M C (MIEMC) simulation, microstabes need to be generated according to the scaled quantity p’ (FN;R,)rather than p (F”; s,) itself. Hence, by analogy with Eqs. (5.10) arid (5.21), t,he generatmion of the Markov chaiii will be determined by the ratio (5.47) where frorii Eqs. (5.43) and (5.46) 7’, r

-

AU - T,,A,&~ kBT

+ Nln

(“>

*%vn-I

(5.48)

In Eq. (5.48) we also used

AU

q l - l ; S2.n-1) U ( F t ; ~ z , n ) L’- ( -N

Asz

sz,n

- Sqn-1

In this case, we realize the random process yn-l S,,n

= -%,n.-l

-+yn

+ 62 (1 - 2 0

(5.49a) (5.4913)

[see Appendix E.11 via (5.50)

196

Monte Carlo simulations

by arialoby with Eq. (5.8). With this process w e associate a transition probability [see Eq. (E.20)] (5.51) Il3 E min 11,exp ( T , ) ]

As before, wc realize this by imrncdiatcly acccpting .my changc in substrate separation if T , 2 0: if, ou the other lia~id,T, < 0,

exp (rz) > exp ( r z ) 5

< <

--

accept change in suhstrate separation reject. change in substrate separation

(5.52a) (5.52b)

according to a modified Metropolis criterion where again $, denotes a pseudorandom number distributed uniformly on the interval [0,1]. In practice, we generate n new configurat~ionin two steps. Step 1 is identical with step 1 of the Metropolis algorithm adapted for GCEMC, namely a random displacement of molecules governed by a transition probability [sw Swtiori 5.2.2. Eq. (5.11)J. In stcp 2, tiic s1ilxdzat,cscparation is changed according to Eq. (5.50) so that

Because in the step 2 of the current MIEMC algorithm all N z-coordinates are changed at once, steps 1 arid 2 are carried out with a frequency N : 1. Some care must also to be taken if a potential cut-off is employed. Then one has to makc sure that, after rcscalirig particle c:oordiriatc%according to Eq. (5.53) the same subset of molecules employed in calculating U (Ft-l; s,,~-I) is also considered in calculating U ( F t ;s,,.) . Last but not lcast wc c~nplislsizc:that, wc may amend both steps of the MIEMC algorithm by step 2 of the GCEMC algorithm; thaL is, the number of molecules accommodated by the slit-pore may fluctuate as well. In this case, all bhree steps are realized with a frequency of N : 1 : N on account of the computational effort as-ociated with generating a new configuration in all three steps. In this lat,ter case, the distribution of microstates in configirat,ion space complics with thc: probability density given in Eq. (2.117). Regardless of whether a fixed number of molecules is used in a MIEMC sirnulntion, energy corrections need to be added t>oAU if a long-range interaction potential is employed such as tlie one in Eq. (5.24). These corrections can be worked out from Eqs. (5.33)and (5.37). However, we emphasize that for reasons pointed out in Section 5.2.2 it is advantageous to employ finiterange interaction potentials snrh as the one iritroduced in Eq. (5.39).

Chemically homogeneous substrates

5.3

197

Chemically homogeneous substrates

Employing the MC simulation technique introduced in the previous section we now turn t o a detailed discussion of thermophysical properties of confined fluids. In particular: we intend to illustrate the intimate relation between these properties and unique structural features caused by the competition between various length scale.. pertinent to specific confinenient scenarios. Thew stijdics arc largely motivated by parallel cxpcrimcntal work cmploving the SFA. Therefore, we bcgin with a concise description of some key aspects of SFA experiment.s.

5.3.1

Experiments with the surface forces apparatus

The main purpose of the SFA is to measure the forces exert,ed by a thin fluid film on a solid substrate with nearly molecular precision (1431. In the SFA, a thin film is confined between the surfaces of two macroscopic cylinders arranged such that their axes are at a right angle [143]. In an alternative setup, the fluid is confined between the surface of a macroscopic sphere and a planar suhstratt [ 1441. Howcvcr, crossed-cylinder and sphcrc-plane configurations can bc mapped onto oiic ariotlicr by diffcrcritial-gmmctrical argmncnts ( 1451. The surface of each macroscopic object, is covered by a thin mica sheet with a silver backing, which permits one to measure the separatioii h. between t,he surfaces by optical interferonietry ( 1431. The radii of the curved surfaces in either setup are macroscopic as we mentioned so that they may be taken ns approxiniately parallel on a molecular length scale around the point of minimum distance h between the opposite bodies (i.e., thc two cylinders or the sphere and thc plane). In addition, they arc locally planar, bccaiwc mica c'ui be prepared with atomic snioothness over molecularly large areas. This set,up is then immersed in a bulk reservoir of the same fluid of which the confined film consists. Thus, a.t thermodynamic equilibrium, T and p are the same in both subsysteins (i.e., bulk reservoir and confined fluid). By applying an external force in the dircction normal t o both substrate surfaces, the thickness of the filin can be altered by either expelling inolecules from it or imbibing thcm from thc reservoir iiiitil t,hcrmodynamic equilibrium is reestablished, that is, until the force exerted by the film on the surfaces equals the applied external normal force at the same T and p. Plotting this force per radius R, FIR, as a function of h yields a damped oscillatory curve in most cases. This is illustrated by plots in Fig. 5.1 where typical curves are shown for several fluids consisting of branched and unbranched hydrocarbons [ 1461. As one can see, both the period and the amplitude of oscillations depend on

Chemically homogeneous substrates

198

thc: details of tho molecular arcMtcctnrc of tiic fluid rriolcculcs.

31

a1

4-

a1 81 11

3n

E

z

\

E

2-

a \

1-

5

0

U

u,

1

z9

bl

Tetradme, Hexadecans

-1-

2-Methyloctadecane

-2-

-3

-

-4l-

I

0

I l l

1

2

I

I

I

3 4 5 Distance, D (nm)

I

6

I.

7

Figure 5.1: Force-diutarice FIR curves measured in the SFA for various hydre carbon fluids (from Ref. 146). In this plot, D corresponds to h in Fig. 5.2.

In another mode of opcration of the SFA, a confincd fluid can bc rxpowd to a shear strain by attaching a movable stage to the upper substrate via a spring characterized by its spring constant A: [ 147-1511 and moving this stage at some constant velociw in, say, the :c-direction parallel to the film substrate interface. Experimentally it is observed that the upper wall first “sticks” to the film, as it were, because the upper substrate remains stationary. From the known spring constant and the measured elongation of the spring, the shear

Confined fluids with short-range potentials

199

stress. sustained by the filrii car1 be determined. Bcyorid so~iiccritical shear strain (i.e., a t the so-called “yield point”) corresponding to the maximum shear stress sustained by the film, the shear stress declines abruptly and the upper substrate “slips” across the surface of the confined film. If the stage moves a t a sufficiently low velocitv, the movable substrate eventually comes to rest again until the critical shear stress is once again attained so t,hat the stick-slip cyc:la repeats i t.self pcriodically. The stick-slip cycle, observed for all types of compounds ranging from long-cliain (e.g., hexadecane) to spheroidal [e.g., octanietliyltetracyclosiloxane (OMCTS)] hydrocarbons [136], has been attributed by Gee et al. [146] and la.ter on by Iilein arid Kumacheva [l50, 1511 to solidification of the confined fluid. This suggests that the atomic struct.ure of the walls induces the formation of a solid-like film whan tha siibstratcs arc properly registered a i d that this film “Int:lts” when the substrates are iiiovcd out of’ thc correc‘t registry. As was first demonstcratedin Ref. 457 such films niay, in fact, form between commensurate subst>ratesurfaces on account, of a template effect iinposed on the film. However, noting that. the stick-slip pheiiomenon is quite general, in that it is observed in every liquid investigated regardless of whether its solid structure is commensurate with that of the confining siihstrat.as, Graiick [ 1361 hiLq arguccl that mcra confinement niay so slow mcc:hanical relaxation of thc filrri that flow xilust, bc activatcd on a tirric scale comparable with that of the experiment. This more general mechanism does not necessarily involve solid films. In the discussion below, we shall therefore concentrate on this latter, more interesting and spectacular scenario in which confined fluids sustain a nonvanishing shear stress without attaining a highly ordered solid-like structure.

5.3.2

Derjaguin’s approximation

To make contact with the SFA experiment, one has to realize that the confining surfaces are only locally parallel. Because of the macrnwopic ciirvatilre of tlic substrate surfaces, the stress cxcrtcd by the fluid on t,hcsc curved substrate* becomes a local quantity varying with the vertical distance s, (z,y) between the substrate surfaces (see Fig. 5.2). As the sphereplane arrange inent (see Section 5.3.1) is immersed in bulk fluid at some pressure p b (T,p ) , the total force exerted on the sphere by the film in the t-direction can be expressed as (1521

200

Chemically homogeneous substrates

Figure 5.2: Side view of the geometry in which a fluid film (not shown) is confined between n sphere of macroscopic radius R and a planar substrate surface. The shortest distance between two points locatcul on the surface of the sphere and the substrate is denoted by h.

which must be regarded as an effective rather than n typical intermolecular force because it depends on the thermodynamic state through T and 1.1. This solvation, or depletion, force plays a vital role in the context of binary mixturcs of colloidal particles of differcut sizc~j[153, 1541. To evaluate tlic iIitcgral in Q. (5.54), it, is convcnicrit to transform from Cartesian to cylindrical coordinates dz dy + detJ dpd& to obtain 2n

o

R

R

o

= 27i T d s z

0

(R

5.

+ h) f (s,)

(5.55)

h

where the determinant of the Jacobian matrix is detJ = p. For pedagogic reasons we restrict the current discussion to fluids interacting with chemically homogeneous substrates where the fluid substrate interaction is modeled according to &. (5.71), and dropped the arguments T and 1.1 to simplify

Confined fluids with short-range potentials

iiotation. In Eq. (5.55) wc uscd s,

=

(SLY:

20 1

Fig. 5.2)

h + H - J r n

(5.56s) (5.56b)

which follows from elementary geometrical considerations. In Eq. (5.55) we also introduced the disjoining pressure [cf., Eq. (3.71)] (5.57) which may be interpreted as the excess pressure exerted by the confined fluid on the substrate surfaces. This interpretation readily follows froin Eqs. (1.60) and (1.63), which permit us to write

where flb is the grand potential of t,he bulk fluid and Eq. (1.31) has also been employed. In Eq.(5.58) we also used the fact that V = Aos,, which follows from the isotropy of bulk phases (see Section 1.3.1). Equation (5.58) then permits us to define the excess grand potential Rex= 52 - R b of the coilfined fluid. In Eq. (5.55), F (h) still dcpcnds on thc curvatiirc of thc substrate siirfaces through R. Experimentally, one is typically concerned with measuring F (h) / R rather than the solvation force itself [143] because, for macroscopically curved swfaces, this ratio is independent of R and therefore is independent of the specific experimental setup. This can be ratiorialized by realizing that f (ss) must vanish on a molecular length scale because this quantity is nonzero only over a range of substrate separations comparable with the range of the fluid sithstrate interaction potential, which is orders of magnitude smaller than R. We may therefore t.ake the upper integration limit in Eq. (5.55) to infinity, which gives 00

1

7.

/anex\ (5.59)

because flexvanishes in the limit. s, -+ 00 according to its definition in Eq. (5.58). In Eq. (5.59) we int.roduce the grand potential per unit area, w y (h) of a fluid confined between two planar substrate surfaces separated by a distance h. Equation (5.59) is the celebrated Derjaguin approximation [see Eq. (6) in Ref. 145).

Chemically homogeneous substrates

202

It was poinlcd out. by Giitzclmarin ct al. that thc: Dcrjabain approximation is exact in the limitaf a macroscopic sphere, which is the only case of interest here [155]. A rigorous proof can be found in the Appendix of Ref. 156. A similar “Derjaguin approximation” for shear forces exerted on curved substrate surfaces has been proposed by Klein and Kumacheva [150]. Equation (5.59) is a key expression because it links the quantity F (h) / R that can t x mcasiirod in an SFA oxpcrimcnt, dirwtly to thc local strms T,, available fromi MC siniulatioils. Moreover, it is intatsting to note that from Eq. (5.59) we obtain 1 d F ( h ) dwex(h) --= 2ndh. R d11

d

00

=-J dszf(s2) = - f ( h ) dh

(5.60)

h

which shows that a derivative of the experimental data is directly related to the stress exerted locally on the niacroscopically curved surfaces at the point (O,O, s, = h ) .

5.3.3

Normal component of the stress tensor

To dcrivc a molccular oxprwsion for thr strcsts tansor compoiicnt T,, which is the basic quantity if one wishes to compute pseudo-experimental data F ( h )/ R , we start from the thcrrnodynamic expression for the exact differential given in Eq. (1.59), where the strain tensor cr is given in J3q. (1.41). F’rorn these two expressions, it follows that

where we wed Eq: (2.81) and the fact that durn= ds,/sd. The last expre.. sian in &. (5.61) niay be recast as (5.62)

which follows frorn Eqs. (5.61), (2.112), arid (2.120). As wc dcmonstratc in Appendix E.3 (5.63) Trn = 7,.FF -k Tzz FS where [see Eqs. (E.33) and (E.40a)l (5.64)

203

Confined fluids with short-range potentials

a ~ i dT:’ is defined in Eq. (E.40b). As deriioiistrutcd in Appendix E.3.1.2, ari alternative route to calculate T,, is provided by the so-called force expression given in Eq.(E.46). Together, virial and force routes provide a check on internal consistency of the simulations, an assessment that is highly recommended in practice. The consistency check is possible because of the different functional forms of the molecular expressions for T~~in Eqs. (E.40) and (E.46). Morcovcr, as t h systcrn is not siipptwcd t,o movc in spare, t,hc total force om tlic slit-pore must vaiiisli ox1 c~vcragc.That is to say [scc Q. (E.47)]

(5.65) Thus, the syrnnictry of the force cxprcssiori provides another useful dicck on the simulations. The a.ccuracy to be expected is illustrated by entries in Table 5.1. Table 5.1: Normal component of the (microscopic) stress tensor fZzfrom virial [see Eqs. (5.63), (5.64), and (E.40b)l and force [see JQ.(E.46)) expressions for /. = i -11.50 a d Efs = 1.00.

s, r.., [Eq.(E.46)] 1.90 -2.251 2.10 0.020 2.20 0.341 2.30 0.379 2.50 0.227 2.70 -0.043 3.00 -0.183 -0.040 3.80 4.50 0.052 0.015 5.00 10.00 -0.026

5.3.4

T=

(Eq. (5.63)] -2.261 0.021 0.339 0.385 0.232 -0.056 -0.177 -0.037 0.055 0.020 -0.028

:T

-0.112 -0.140 -0.138 -0.136 -0.140 -0.195 -0.271 -0.187 -0.111 -0.120 -0.048

7 2

-2.149 0.161 0.477 0.521 0.372 0.139 0.093 0.150 0.170 0.141 0.020

Stratification of confined fluids

To illustrate the relation between microscopic structure and experimentally accessible information, we focus on the computation of pseudo-experimental solvation-force curves F (h) / R [see Eqs. (5.57), (5.59), (5.63), and (E.46)] as they would be determined in SFA experiments. However, here these curves are computed from computer simulation data for rm and f b where fb iS

204

Chemically homogeneous substrates

I

I

I

I

4

5

6

8

6

cn“

n W

rc

0 -2

-4 2

3

7

Figure 5.3: Excess pressure f(s,) [see Eq. (5.57)] ( 0 , - . -) and the solvation force F ( I t ) / R ( ---) as a functions of s, and h, respectively.

calciilated from h s . (E.70), (E.73), and (E.74) in a separate simulation of a bulk fluid maintained at t,he same T and p. R,esults are correlated with the mic:roscopic:stmc.t,iirc:of a t,hiii filrii corifinc:d 1)ctwc:c:ii plan(: parallel substrates separated by a distaiice s, = h. Again we focus on “simple” fluids, which serve as a suita.ble model for the approximately spherical OMCTS molecules bet.ween mica surfaces, which is perhaps the most thoroughly investigated system in SFA experiments [143, 14GI. Because OMCTS is chemically inert and electrically neutral, the influence of charges on the mica surfaces may safely be ignored (see Chapter G for a discussion of electrostatic interactions in c:orifiiic.:d fliiitls). Plots of f (3,) and F (h.)/ R versus s, and h, respectively, are shown in Fig. 5.3. The oscillnt,ory decay of both quantities is a direct consequence of the oscillatory dependence of r,, on s,, which has also been investigated by integral equations of varying degree of sophistication [157-1611. As can be seen in Fig. 5.3, zeros of f (s,) correspond t,o successive extreina of F (h) / R

Codned fluids with short-range Dotentids

205

A

4

3 n

...

k#

2 5 2 Q

1

-0.5

-0.4

-0.3

-0.2 -0.1

0

0.1

0.2

0.3

0.4

0.5

Figure 5.4: Local density p (2) as a function of substrate separation s,; s, = 2.60 (m),8 , = 2.80 ( O ) , and s, = 3.00 ( 0 ) .

because of EQ.(5.60). In actual SFA experiments, the only portions of the F (h.)/ R curve generally accessible are those where the inequality

(5.66) holds because ueX increases upon compression of the film [see l3q. (5.60)]. Alternatively, one may employ colloidal probe atomic force microscopy (AFM) to measure force distance curves such as the ones plotted in Fig. 5.1 [162]. The important difference between SFA and colloidal probe AFM expcrimcnts is that, in thc latkcr tho entire forcci distancc ciirvc is x c m i b l c rather than only that portion satisfying Eq. (5.66) [163, 1641. In Ref. 164 a comparison is presented between theoretical and experimental data for confined poly-electrolyte systems. In any case, structural changes accompanying the variation of F (h) / R are rather obscure regardless of the experimental technique. These changes can be inferred more directly from Figs. 5.4-5.6 where plots of the local

206

Chemically homogeneous substrates

3

2 n

J: .^ N

W

Q

1

-0.5

-0.4

-0.3 -0.2 -0.1

0

0.1

0.2

0.3

0.4

0.5

Figure 5.5: As Fig. 5.4, but for sz = 3.20 (a),s, = 3.40 (0). and s, = 3.55 (0).

density (5.67)

arc' presented. In Eq. (5.67): N (2) is the niimhc.r of fliiid molccdcs whosr writer of mass is located in a prisrii of diiricrisions s d x syo x S,, where 6, is typically of the order of - 10-'a for a LJ(12,6) fluid [see &. (5.24)]. In general, p ( z ) -, 0 as 1.1 -, 4 2 because of the increasing repulsion of fluid molecules by the substrates. Maxima in p (2) reflect stratification, which is the arrangement of fluid molecules in individual layers parallel with the solid substrates. Recause of the layered structure of the confined fluid, neighboring maxims in plots of p ( z ) arc scparatcd by minima that. rcflcct a rcdiiccd probability of finding the center of mass of fluid molecules in this region. Oscillations in p ( z ) are damped as one moves away from the substrates because of diminishing fluid substrate interaction. In other words, if the slitpore is sufficiently nide, stratification is pronounced only in the vicinity of the substrate surfaces such that the inhomogeneity of the fluid persists only over distances roughly comparable with the range of intermolecular forces.

207

Confined fluids with short-range potentials

4

I

I

1

I

3

n

$

2

.C

N

v

P

1

-0.5

-0.4

-0.3 -0.2 -0.1

0

0.1

0.2

0.3

0.4

0.5

Figure 5.6: As Fig. 5.4, but for s, = 3.20 (W), s, = 3.40 ( O ) , and sz = 3.55 ( 0 ) .

Because of Eq. (5.60), experimentally accessible portions of the pseudcexprrimcntal data can be related t,o t,hc local st+rcwat thc point#(0, 0, s, = he) of minimum distance between the surfaces of the macroscopic sphere and the planar substrate (see Fig. 5.2). By correlating the local stress T~ (h) with the confined fluid’s local striicture at ( O , O , h) \la p ( z ) , one can establish a direct correspondence between pseudo-experimental data [i.e., F (h) / R ] and the local microscopic structure of the confined fluid. Plots of a sequence of local densities p (2) in Figs. 5.4-5.6 over the range 2.60 5 h 5 4.00 illustrate this correlation. In an actual SFA experiment 2.59 5 h 5 3.06 and 3.53 5 h 5 4.00 are accessible portions of the solmtionforce curvc, whcrcw 3.06 < h < 3.53 dcmarcatw thc inacccujsiblc range of distances because here the inequality stated in Eq. (5.GG) is violated. Plots in Figs. 5.4 and 5.5 show that in the experimentally accessible regions the film consists locally of two and three molecular strata, respectively. For h = 2.60, the film is locally compressed because F (h) > 0 whereas it is stretched for h = 3.00 because here F (h) < 0. Under compression the film appears to be Im stratified. as is reflected by smaller heights of less well-separated

208

Chemically heterogeneous substrates

peaks of p ( z ) cornpared with the two other curves iu Fig. 5.4.For h = 2.80, F (h) 2 0, and T,, (9, = h) hw almost irssumed its minimum d u e , indicating

that for this particular value of h film molecules are locally accommodated most satisfactorily between the surfaces of the macroscopic sphere and the planar substrate. It is therefore not surprising that peaks in p ( : ) are taller for h = 2.80 compared with the two neighboring values of h (see Fig. 5.4). In the next accessible region 3.53 5 h 5 4.00,the film consists of three molecular strata for which the most pronoundstructure is observed for h N 3.80, corrwponding to a point, at which F ( h )/If ncarly vanisha (scc Fig. 5.6). As before in Fig. 5.4this is reflected by the peak height of the coutact strata (i.e., those layers being closest, to the substrate surfaces), whereas inner portions of the film remain largely unaffected by the change in pore width. Plots of p ( z ) in the experimentally inaccessible regime of pore widths in Fig. 5.5 show that here the film undergoes a local reorganization characterized by the vanishing (appearance) of a whole layer of fluid niolecules. The raorganixatsionis padiial, at onc can s w in t8hcplot of p (2) for it = 3.4 whcrc two shoulders appear at z/s, N f0.1. Stratification, as illustrated hy thc plots in Figs. 5.4-5.G) is due to constrain& on t,hc packiiig of molwulm ncxt to the suLWt.ratc surfacc: arid k there fore largely determined by the repulsive part of the intermolecular potential [38].Stratification is observed even in the complete absence of intermolecular attractions, such a$in the case of a hard-sphere fluid confined between planar hard walls [165-1671.For this system Evans et al. [168]demonstrated that, as a consequence of the damped oscillatory character of the local density in the vicinity of the walls, T= is itsclf a d a m p d oscillatory function of s,, if s, is of tlie order of a few niolmular diameters, which is confirmed by the plot in Fig. 5.3.

5.4

Chemically heterogeneous substrates

In tha pravioiw smtion w amployd GCEMC simiilations to ilhwtratte the clo(~'rclatiori bctwecn thcrnioyliysical propertics [i.e., F (h.)/ R or T ~ mid ] the microscopic structure of the confined fluid [i.e., p (z)]. The characteristic damped oscillatory dependence of F (h) / R on h observed in both SFA experiments [135,1691 and computer simulations [5,39, 44, 170))is a direct consequence of the interplay between two relevant length scales, namely the range of fluid fluid interactions and the degree of confinement represented by h or s,. GCEMC simulations at fixed T, p, and sz [39,42-45] and in a grand

Confined fluids with short-range potentials

209

iriixed isostrcss isostrain crlscrnblc [ 17&172] denionstrate that t,hc fhid p i l a up in layers parallel with the walls and that, in coincidence with the oscillatioris in T,,, whole layers of fluid abruptly enter the pore. This stratification, due to constraiiit,s on the packing of riiolecules against the rigid planar walls, thus accounts for the oscillatory dependence of T,, on s, (381. GCEMC simulations [173, 2741 of a monoatomic film between walls comprising like atoms fixed in tha configination of t,hc facc-ccntcrcd cubic (fcc) (100) planc show that if the walls arc in the right rcgistry they can induce frcming of a 111o1ecularly thin film. The frozen film resists shearing (i.e., the walls stick) until a critical shear strain is surpassed, whereupon t,he film melts and the walls slip past one another. This effect has been invoked to explain stick-slip lateral movenient observed by the SFA [146, 148). In this section we shall focus on the behavior of confined phases exposed to a shear strain. However, nnlike in these earlier studies, the confined phase will not be solid-like but will remain fluidic, which, in our opinion, makes the rheology of confined phases even more fascinating. It will turn out that. the fluid’s capability t,a resist a shear strain can be linked to a third length scale competing with the other t,wo mentioned above. This t,hird length scale can be identified with some inherent, structure of the solid surfaces themselves. The inherent structure could be geometrical in nature such as a sequence of narioscopic grooves rendering the confining substrates nonplanar. It could also be chemical in nat,ure like some sort, of imprinted structure by which the wdtability of the solid surfaces varics locally. We shall focus on the swond situation in this aiid Scctioils 5.5 aid 5.6, whcrc in the lattcr section we address the rheology of confined fluids. In all three sections, the third length scale entering our discussioii can be associated with the dimensions of the chemical patt,ern witahwhich the confining siibstrates are endowed.

5.4.1

The model

5.4.1.1

Continuum description of the substrate potential

For simplicity we employ a model system sketched in Fig. 5.7. It consists of a film coniposed of spherically symmetric molecules that is sandwiched b e tween thc surfaccs of t,wo solid substrates. Thc siibstmtc siirfaccs are planar, parallel, and separated by a distance s, along the z-axis of the coordinate system. They are semi-infinite in the z-direction, occupying the half spaces s,/2 5 z 5 00 and --x 5 z 5 - 4 2 , and infinite in the z- and y-directions. Each substrate comprises alternating slabs of two types: strongly adsorbing and weakly adsorbing. The “strong” and “weak” slabs have widths d, and &, respectively, in the 2-direction and are infinite in the y-direction. The

210

Chemically heterogeneous substrates

Figure 5.7: Schcmc of R simpla fluid ronfincd hy a rhcrnicdly hctmogcnwiis model pore. Fluid molecules (gray spheres) are spherically symmetric. Each substrate consists of a sequence of crystallograhic planes separated by a distance & along the z-axis. The surface planes of the two opposite substrates are separated by a distance s,. Periodic boundary conditions are imposed in the x- and y-directions (see Section 5.2.2).

+

system is thus periodic in the x-direction of period d, d, such that its p r o p erties are translationally invariant in the pdirection. In practice we take the systcrii to hc a finite picca of thc film, imposing pcriodic boiindary conditions [140] (sec Sectioii 5.2.2) on the plariw :1: = f . 4 2 arid = f.9,/2. The substrates are in registry meaning that slabs of the same type are axytl-y opposite anch ot.hcr. Substrata at.oms arc assumed to be of the snmc i'diameter'' ( 0 )and to occupy the sites of the fcc lattice [the substrate surfaces are taken to he (100) plyies] having lattice constant C: which is taken t.o be the same for bot
Confined fluids with short-range potentials

21 1

pair of film rriolcculcs E : ~m [i.c., ii,R ( r ) ] . Thc iiaiioscale heterogeneity of the substrate is characterized by E = &fs [i.e., ufs( r ) ] for the interaction of a film molecule with a substrate atom in the strong (central) slab, and by E = Efw [i.e., ufw( r ) ] for the int#eractionof a film molecule with a substrate atom in either of the two weak (outer) slabs (see Fig. 5.2). We take qs2 E R and Efw << EB (see Section 5.4.2 for specific values). To avoid complicationis result irig frorii a detailed description of the atomic structure. of the heterogeneous substrate surfaces, we eniploy a continuum representation of the fluid substrate potential. Details of its derivation can be found in Appendix E.2. Combining the expressions in Eqs. (E.21), (E.24). (E.26), (E.27), and (E.29), we obtain for the potential energy of a film molecule in the contiiiuuni representation of the substrate k (= 1.2)

(5.68) where the sign on z is chosen according to the convention + ct k = 1 and - c-$ k = 2 (see Fig. 5.2). Before discussing the implementation of h.(5.68) in thc GCEMC simulatioii, we cornincrit briefly on the propcrtics of thc whole fluid substrate pot,ential CP = @['I+ d21that follow stxictly from considerations of symmetry. When the t-coordinate of the fluid molecule is reflected through the mirror plane z = 0, - z in the arguments z" = 4 2 d 6 e f z of d l 1 changes to +z in the arguments of @I2] and vice versa. That is, CPl1l(x? -z) + (z, z ) and vice versa. The sum @ is therefore invariant under reflection in the z = 0 plane. Likcwisc, CP is invariant iindcr rcflcction in the .r-plane, although the proof involves more subtle interconversions. For example, under the transformation z + -z, the first term in braces (for m)is converted into the second term in braces (for -m).Likewise, the third term in Eq. (5.68) is converted into the fourth term. Of course, as the potential is periodic in z, of period sx, we need t.o represent the whole continuum fluid substrate potential in only one quadrant (sav, 0 5 2 5 sx/2, 0 5 z 5 4 2 ) of the z-z plane.

+

212

Chemically heterogeneous substrates

5.4.1.2

Computation of film-wall contribution t o configurational energy

As we demonstrated in Section 5.2.2, the generation of a Markov chain of configurations in GCEMC simulations is governed by a change in configurational cncrgv associated with hot11 psrtklc displaccmcnt and cmntion/dcst.riiction attcmpls. For this systcrn t he configurational energy M can be written as .

N

2

N

N

where w = U S R is given in Eq. (5.39),dk]in Eq. (5.68)? and rij = ,lri - r j ( is the tlist.ancc: t)ctwccn thc centers of filrri riiolcculcs i and j located at ri and r j , respectively. Equation (5.69) also defines the film-film and filmsubstrate contributions (IFFarid I/FS to U. To implement, the expression for @Ik] in Eq. (5.68), we truncate the infinite summations according to 00 Mi Em=-, Cz,=o C,"==_, Cm,=", where in.tcgcrs A4 and M'arc sufficiently large to yield @Ik] with a prescribed precision. For a system size of s, 2 10 and a lattice spacing of & = 1.0, we find that M = 2 and A T = 50 are large enough to give @Ik] to a precision of 0.3% regardless of the position of a film molccnlc with rmpnrt, to thc suhstratc. Ho~vevcr,M aid ikl' arc still too large to crnploy tlic truncated version of Eq. (5.68) directly in each GCEMC step. Tests show that for hi = 2 and M I = 50 t8heevaluation of @Ik] for a single film molecule requires approximately the same amount of computer t h e as thr computation of [IFF for N = 100, so that a GCEniiC sixnulation of a typical length of lo5or lo6 cycles (see Section 5.2.2) would be prohibitively expensive. Iiistead of computing @Ik] diiring cavh stcp of the GCEMC sirmilat,ion, wc adoptcd t,hc following proccxlurc. Prior to tiic siiiiulatiori wc' coiiiputcd * I k ] by the truncated vcrsion of Eq. (5.68) and stored it a.t the nodes of a square grid {zk, zk}k=l,...,K in the quadrant 0 5 z 5 4 2 , 0 5 z 5 4 2 . During the simulation the value of @Ik] at the t,he act,ual (instantaiieous) position (xi,z.i) of film molecule i (which does not nec.cessarily coincide with any node) is obtained by bilinear interpolation [ 1751 among the values of @Ik] at the four nearest-neighbor nodrs of (xi,z i ) . We tested the inberpolation scheme for a special case in which @Ik] can be readily evalurtted during each GCEMC step. The substrate consists of a single, chemically homogeneous plane for which we set Efs = Efw = E R = E . Thus, the discrete sum on m and piecewise integrations over the strips are replaced bv a single integration on z' from -m to 00. The summation 011 m' also reduces to a single term n2' = 0. IJnder these conditioris Eq. (E.21)

-

213

Confined fluids with short-range potentials

can bc rcwrittcn as

in cylindrical coordinates. so that the integrations on $ and p ran be carried out in closed forin to yield

where (+ H k = 1, - t-) k = 2). Employing a mesh of 6, = 6, = 0.025, corrc,.ipontling to K = 7.6 x lo3 (sx = 10, s, -= 1.9) for thc srnallcst suh stratc scparatioii aid K = 5.0 x 10' (sx = 10, sz = 12.50) for the largest; we compare (see Table 5.2) results obtained using the inlerpolation scheme [Eq. (5.68)] with those based on direct evaluation of the potential given by Eq.'(5.71). It is noteworthy that the agreement is good, even for fluid fluid (FF) and fluid substrate (FS) coiitributions to the normal component of the stress tensor T ~ , which . are particularly sensitive to numerical inaccuracies in c a w thc magnitiidc of thwse contril,utions is small. Table 5.2: Comparison of intcrpolation [truncated version of E<1. (S.SS)] and dircct rvdiintion [Eq. (5.71)] of fluid sitbstrxtc potential energy for various prop erties of a fluid confined by substrates consisting of single, chemically homogeneous pliuies. Entries, give11 in diniensionlerjsuiiits defiried in Tdde 1, refer to simulations based on either direct evaluation (D) or interpolation (I). s,

-/A

- (UFF/N) n I

- (['FS/N)

D

I

-TzF

D

-2

I D 1 3.00 9.56 3.676 3.668 2.920 2.912 1.09 1.10 0.21 0.20 2.70 9.46 3.086 3.088 3.021 3.013 2.45 2.44 3.16 3.20 2.20 9.26 2.103 2.113 4.860 4.858 0.42 0.42 -1.53 -1.54

5.4.2

Structure and phase transitions

In Section 5.3.4, we demonstrated that fluids confined to nanoscopic volunies are highly inhomogeneous in that. they forni molecular strata. The most direct way of realizing this was through plots of the local density (see

214

Chemically heterogeneous substrates

Figure 5.8: Local density p (2,t) a function of position in the x-z plane for qs= 1.25, 4 = 4.0 [seeEq. (5.68)); (a) s, = 7.2, (11) s, = 7.5? .9, = 12, qw= (c) S, = 8.2.

Confined fluids with short-range potentials

215

Figs. 5.4-5.6) based on thc definition of p (2) dcfixicd in h.(5.67). However, because in the current model, the fluid substrate interaction potential is a function of z and z [see &. (5.68)], the local density must also depend on both (Cartesian) coordinates, which we redefine by writing (5.72)

In Eq.(5.67), N (2,z ) is the number of fluid molecules in a given configuration that are located in a .sqiiarr prism of dimcnsions rSx x .sfl x 6, ccntcrcd on R point (t, 2 ) . Three characteristic examples for the local density are plotted in Fig. 5.8 for different substrabe separations. Because of the symmetry of @Il p (t, z ) must be symrnetric about the 2 = 0 and z = 0 planes (see Fig. 5.8). Similar to the plot of p (2) in Figs. 5.4-5.6, peaks in its two-dimensional counterpart represent positions of molecular strata. For s, = 7.2 a stratified fluid bridges the gap between the strongly attractive central portions of the opposite substrates [i.c., for 1x1 2.0, scc Fig. 5.8(a)]. Bcrausc of tlic decay of the fluid substrate interaction potential, stratitification diminishes as z increases along lines of constant x. Stratification is absent over the weakly attractive portion of the substrate [see Fig. 4.13)]. Here an inhomogeneous low-density fluid exists, as indicat-edby the relatively low value of p (5,z ) and its weak dependence on 5 and z for 1x1 2 4.0. By analogy with the mean-field data p1ott.d in Figs. 4.11(a) and 4.11(b), we refer t,o sitiiations akin to the oxic dcpicted in Fig. 5.(l(a) as "bridge" phasic: A stratified high-density fluid stabilized by the attractive part of the substrate plus a surrounding gas over the outer two, weakly attractive portions of the substrate material. For larger sz = 7.5 [see Fig. 5.8(b)ll the structure of the fluid changes significantly. Over the strongly attractive portion of the substrate, the fluid remains stratified. However, the low-density portion has given way to an illhomogeneous high-density fluid over the weakly attractive part of the s u b strate. Consequently, the interface between higher- and lower-density portions of the confined fluid visible in the plot of p ( z , 2) in Fig. 5.8(a) has disappcarcd, and mi 110 loiigcr be w r i in Fig. 5.8(b). As the weak portions of the substrate are essentially repulsive, p (2, z) decreases for Is12 4.0 from the center of the fluid ( z = 0) toward the substrate (121 -+ 4 2 ) . If the distance between the substrates is increw4 even further, another structural change occurs in the fluid. It is illustrated by the plot of p (5,z ) for s, = 8.2 in Fig. 5.8(c), where the fluid bridge disappeared and only two strata of fluid molecules “cling” to the strongly attractive portion of the substrate. For example, for1.1 5 3.0 and t = 0, the density is rather low and decreases monotonically toward the center of the confined fluid located at z = 0. The

216

Chemically heterogeneous substrates

0.6 0.5 1Q

0.4

0.3

'

'

'

0.2 . 0.1 .

0

2

4

8

6 sz

10

12

14

Figure 5.9: Mean density 7 [see Eq. 5.74)) as a function of substrate separation s,. Data we plotted for various degrees of chemical corrugation of the substrate (A) [MX Eq. (5.73)]. Cr = ;$ (o),C, = 4 (a), Cr = 4 (o), C, =

The sequence of plots in Figs. 5.8(a)-5.8(c) illustrates the peculiar phase transition from a fluid bridge to a liquidlike phase and eventually to a thin adsorbod film with inrrcatsing s,. That. thc morphologies p l o t t d in those figures do, in fact, have the status of legitimate thermodynamic phases is borne out by the corresponding meaii-field lattice density functional calculations discussed in detail in Section 4.5.2. In particular the plots presented in Figs. 4.12 illustrate the sequence of phase transitions discussed in the preceding section. As can be seen froni Figs. 4.12, the change from a11 initial bridge phase to a liquid-like and eventually gas-like confined fluid is caused

Conflned fluids with short-range potentials

217

by e shift of the gas bridge liquid triple point to higher tcrnpcritturcs arid an associated narrowing of the onephase region of the bridge phases. -4t this point it. seems worthwhile to investigate in some depth the impact of chemical corrugation on this phase behavior by varying (5.73)

which is the relative width of the strongly adsorbing stripe on the substratme surfaces. Instead of visualizing the associated phase changes by plots of the local density as before. we shall focus below on the mean density related to the local density via (5.74) -S,Q/~

-8d.o/2

Plots of versus s, are shown in Fig. 4.15 for various degrees of chemical corrugation of the substrates. Sta,rting with the largest degree of corrugation c, = $, we notice from Fig. 5.9 that j5 oscillates for s, 5 6.0 with a period of approximately one molecular “diameter.” A similar behavior of 7 is found for c, = and which can bc intcrprotcd m a, fingcrprint of stratification [see, for cxar~iplc,Fig. S.S(a)], that is, the change iri the n u d m of rnolecular strata accommodated bet,ween the substrates with vanishing s, [38, 1721. However: in the limit s, 4 00 the confined fluid becomes increasingly bulk-like on account of the vanishing iiifluence of fluid substrate interactions. Because the bulk phase a t the current values of T = 1.0 and p = -11.5 turns out. to be a gas, one intuitively expects at, least one phase transition from a dcnscr confincd fliiid at, small s, to a lowc?r-donsit,yfliiid at somc cliarxteristic largcr substrate scperatioii. This transition, kriowri as capillary condensation/evaporation (see also Section 4.2), is, in fact, observed for c, = around s, N 11.0. For c, = capillarv condensation shifts to a smaller value s, N 11.0, which is reasonable in view of the reduced net strength of attractive fluid substrate interactions compared with the previously discussed case. The shift of capillarv condonsation to lowcr siibstratc soparation also pcrsists for 4 c, = 12 and c, = but it is much less pronounced, as Fig. 5.9 reveals. However, comparing only the latter two values of .cr,chemical corrugation of the substrate seems to be only of marginal importance for the location of the discont,inuity (that is, the location of the phase transition) in the plot of p. For c, = this second discoiitinuity vanishes in favor of a rather steep increase of j5 over the range 5.5 5 s, 5 6.5. A corresponding plot of the

&

4

&,

A,

&:

&

218

Chemically heterogeneous substrates

isotlicririal coiriprwsibility K ~ , . in Fig. 5.10 has a tall, cusylilw peak iii thc same range of substrate separations. The derivation of ‘cys parallels precisely the one for ~ 1 in 1 Section 1.6.2 where, however, here (5.75) From Eqs. (1.66) and the associated Gibbs Duhem Eq. (1.68) it follows that (5.76) wlicrc tho analysis follows prcc.iscly thc oiic lwtwccn Eqs. (1.7‘3) aid (1.t12) and we used V = AYosy. However, here the grand-potential density is defined in a slightly different fashion CIS [see Eq. (1.77)] (S.77) Using the statistical expression for SZ given in Eq. (2.81) [using the (classic) expression for Z given in Eq. (2.120)], we ran apply precisely the same derivatioii as thc om prcscntcd iii Section 2.3.4 to obtain %Y

=

v

koT

(N2)-(N)2

(W2

(5.78)

In the vicinity of the cusplike peak in the plot of ‘cyu for c, = 4 in Fig. 5.10, t,he isothermal coinpressibility depends on the size of the simulation system. This is illustrated by the plot of the density distribution P ( p ) plott,ed in Fig. 5.11 as a function of sy. To aiialyze the data plotted in Fig. 5.11, we have nornializrd P ( p ) nnmcrically surh that (5.7‘3) where Ap = p - p in which p and are the instantaneous density (of a given configuration of the Markov chain) and the mean density of the confined fluid, respectively. For a sufficiently small density fluctuations, P ( p ) should be approximately Gaussian as denioiistrated in Appendix (2.3. We can therefore use Eq. (C.36) as a basis for a finitesize scaling analysis of lcyy obtained from GCEMC siriiulations in which the computational cell is inevitably finite in

Confined fluids with short-ranne potentials

219

100

50

10

1

0

2

4

6

8

sz

10

12

14

Figure 5.10: As in Fig. 5.9, but for the isothermal compressibility ‘5.y: c, = (0) and c, = ( 0 ) .

6

4

size. To this end we notice from Eq.(5.78) that the variance of P ( p ) is given bY (5.80)

Thus, we may construct P ( p ) as a histogram during a GCEMC siniulation and calculate up by fitting Eq. (C.36) to the discrete data points that constitute this histogram. We niay t,hen plot a,,as a function of 1/&, which should give us a straight line if the simulation cell is large enough in the y-direction. From thc slopc of this linear relationship, we (*ant,hcn also cstiirratc K,,,, via Eq. (5.80). An additiorid scaling rclatiorisliiy is obtairicd for the height of the densily histogram. Because of Eq. (C.36)we also realize that the height of the peak in the plot of P ( p ) should scale with f i because the normalization constant in Eq. (C.36) is inversely proport+ionalto u,,in the Gaussian limit (i.e., in the limit sy -+ 00). The system-size dependence of density fluctuations is illustrated by the plot of P ( p ) in Fig. 5.11. If sy = 10, for example, the density distribution is by no means Gaussian but rather bimodal, indicating that the system “oscillates” between higher- and lower-density states. As sy increases, this bimodal

220

Chemically heterogeneous substrates

iiaturc of I’ ( p ) gradually dcclirics such t,hilt for sy 2 50, I’ ( p ) turns out to he Gaussian with the required dependence of peak height and standard deviation on sy. Data for fiYy obtained from the finite-size analysis just described are plotted in Fig. 5.10. However: as we pointed out in Ref. 176, a similar finite-size dependence was not observed for any other quantity computed in the present context. For t,he system sizes employed here, these findings arc complctdy in accord with i1 rriorc roccnt, stwly o f systcm-sizc affects in computcr sirnulatioris [ 1771.

20

15

10

5

0 -0.20 -0.15 -0.10 - 0 . 0 5

0.00

P-P

0.05

0.10

0.15

0.20

Figure 5.11: Normalized density distribution P ( p ) as a function of the deviation of the instankneous density p = N / V from the average densit.y = (N) / V for various values of s y : ( x ) sy = 10, ( 0 )sy = 30, (+) sy = 50, (A) .9y = 65, and ( 0 ) sy = 100. Solid liiiw represent tlic fit. of a Gaussian tauP ( p ) .

The observed syst,em-size dependence clearly indicates that the correlation length associated with density fluctuatioiis in the confined fluid exceeds the dimensions of the siniulatioii cell [178]. This is indicative of a nearcritical therinodynamic state of the confined fluid. Because of the density of the participat,ing phases in this near-critical region we conclude that the critical point is the one at which fluid bridge and liquid-like phases become in-

Chemical patterns of low symmetry

22 1

dist.inguishable. As the variation of 7 with s, is coritiiiuous (see Fig. 5.9), tlir thermodynamic state of t,he confined fluid is still slightly supercritical with respect to the critical point where the phase diagram is qualitatively similar to the mean-field phase diagram plotted in Fig. 4.12 for n, = 8 - 10. As can be seen from that figure. a triangular-shaped region exists correspondiiig to the onephase region of fluid bridges. On either side of the cusplike peak, ‘5y decays rapidly for c, = to rather small values typical of dense LJ (12,6) fluids (see Fig. 5.10). For s, 5 5.5, tcYY oscillates with a period of about one molecular diameter reflecting stmt,ifiration in the scnsc of thc discussion in Section 5.3.4. In addition to its cusp-like mtiximuni K~ also ctiaxiges discontinuously during a first-order For all s, 2 8.3 the magnitude of tcYy phase transition at s, = 8.3. (cr = corresponds to that of a typical LJ (12,6) gas in accord with the corresponding plot of 7 in Fig. 4.15. Stratification-induced oscillations of tcYy can also be seeii for c, = $ and small s,. However, in this case, tcYy remains rather small up to the substrate separation where its discontinuous change again signals a first-order phasc transition t,o a low-density phasc (s, N 11.0). Thc smallcr value of K~~ (c, = compared with c, = indicates the presence of a denser fluid. This seems sensible because the net attraction of fluid molecules by 4 the substrate is larger for c, = than for c, = E.

&

2).

4)

&

8

5.5

Chemical patterns of low symmetry

5.5.1

Nanopatterned model substrate

The situation studied in the previous section becomes more complicated if t9hcchcniiral pattern with which thc siilmtrat8csiirfacc is daroratcd is of finite cxtciit.. For cxi~~~iple, tlic iriodel substrates cmyloyed in Section 5.4 were endowed with chemically dis!.inct, stripes that. are infinitely long in one spatial direction. As a result, the potential function, describing the fluid substrate interaction turned out to be independent of the y-coordinates of fluid molecules [see Eq. (5.68)). As a consequence, properties of the confined fluid are translationally invariant in this direction, which has important repercussions for thc invcstigation of tha phasc hchavior of fliiids confincd bctwccn such (chemically) heterogeneous surfaces. For example, if t,ranslational invariance of fluid properties is not preserved on account of the symmetry of the heterogeneity on the substrate, we demonstrated in Section 1.6 that a Gibbs Duhem equation may not exist, which, in turn, precludes the existence of a “mechanical” expression for the grand potential, which is the key quantity on which an investigation of phase behavior is based.

222

Chemical patterns of low symmetry

Precisely this latter situation ariscs if the coiifiriirig solid surface is cndowed with a chemical pattern that is both nanoscopic in size and finite in extent. Such chernical patterns may be created by lithographic methods [ 1791. Atomic beams have been employed to produce hexagonal nanostructures [180]. Other met.hods capable of creating chemically nanostructured substrate surfaces involve microphase separation in diblock copolymer films [181] or the me of form mic:roscopy t,o locally oxidizc silicon snrfaces (1821. Ariotlicr cxarriplc is tlic irripriiitirig of ring patterns over thc surface 1531. In this situation, thc pat,tern is charwlerized by an innctr radius R,,, and an outer radius Lt forming an a.iinulus of constant width in which the surface displays preferent.ia1 interaction for the fluid. Those surfaces may be prepared by rnicrocoiitact printing using alkanethiols on gold 153, 541. In this method an elastomer stamp is used to deposit molecules on surfaces. Thc stamp is first, “inkcd” with a solution of alkancthiol inolcciilw and t.hcn prossc:tl onto a gold surfitcc: wliicli rcsults in wcll-tlcfincd liydropliilic s i t m The remaining bare gold surface is then made hydrophobic by dipping it into mother alkanethiol. A liquid is then subsequeiitly adsorbed onto the surface in a closed cell. The experiments are conducted by cooling down or heating up the system, thereby changing the voluriie of liquid formed on the surface [s4j. Expcrimci~t~s performed on thew ring-pattmncd substrates show that, wficri thc volume iricrcascs, tlic liquid gconictry foriiicd over thc pattern undergoes a tra.nsition from a ring morphology (where the liquid forms a homogeneous covering over the pattern) to a bulge niorpliology where a droplet of liquid is formed over one side of the pat,t,ern. This bulge geometry “breaks” the symmetry imposed by the pattern. As the volume of liquid increases, the bulge progressively spreads over the nonwetting disk inside the pattern and This finally a spherical cap is formcd over t,hc whole disk of radius bulgc geometry ctlli also be obscrvcd 011 surfaces pattcrricd with microirictric stripes [73]. If one aims at understanding this rather peculiar phase behavior from a inicroscopic perspective, oiie again needs to know the relevant, ther~iiodynamic potential. However, one is immediately confronted with a complication because a “mechanical” expression for thermodynamic potentials cannot be dcrivcd due to the low symmetry of thc confined fluid (see also discussion in Section 1.6.2). Therefore, a different means of calculating these potentials must be devised. This alternative computational t.eclinique will be based on a perturbational approach to which the current section is devoted. Specifically, we consider the sihiation depicted schematically in Fig. 5.12: a “siniple” fluid confined between two planar substrate surfaces composed of like atoms where the fluid fluid interaction is described by Q. (5.39) for

223

Confined fluids with short-range potentials

Figure 5.12: Schematic representation of a fluid (black spheres) confined between two plaiiar substrat.esdecorated with a circular region of radius R that attracts the fluid molwnles. Ontsidt. the circiilar region fluid-snbstmte interaction is purely repulsive.

the LJ (12,6) potential. If the substrates are free of any nanoscopic chemical patterns the fluid substrate interaction is described by Eq. (5.71), which we split into attractive and repulsive contribut.ions according to

(5.81a) 4

27r€fsn.*a ( z *:J2) such bhat

dk1( z ; Sz)

=

+

cpg (2) - w (). ‘Fatt

(5.81b)

(5.82)

employing again the convention c-) k = 1 and - H k = 2. In Eqs. (5.81): Efs sets the energy scale of fluid substrate interactions. For simplicity we take

224

Chemical patterns of low symmetry

n~cr’ = 1, -cfs : E K , and trcat the substrates as scirii-iiifinitc solids. Thc fluid substrate abtraction is long range, that is qatt(2) oc zP3 [see Eq. (5.81b)J.

w

To iiiodcl substrate surfaces with irnpriiited chcniical riariopattcriis (scc Fig. 5.12), we modify J3q. (5.82) according to

where T denotes the (vector) posit,ion of a fluid molecule and t,he ”switching” function 1

is introduced as a continuous representation of the Heaviside function (i.e., the Fermi function [17];see Fig. 5.13) such that @Ik] ( T ) describes the interaction between a fluid moleciilc and an infinit,esimally smooth, repidsive solid surface endowed with an attractive circular area of (fixed) radius R (and infinite height in the fz-directions) centered a t (O,O, fs,/2). In Eq. (5.84), IE 2 0 is a measure of “softness” with which the attractive part of @Ik] ( T ) is turned off as a fluid molecule moves away from the center of the circular area, that is, froni the 2-y plane. In other words, the range which s ( T , y; R ) varies between 0 of in-plane distances arid 1 is dctcrmincd by 6.

5.5.2

Thermodynamic perturbat ion theory

To proceed wc‘ cniploy a perturbational approach that enables us to calculate (the absolute value of) t,he grand-potential density. The key idea in any perturbation theory is to somehow link properties of the system of interest to those of a reference system whose properties can readily be calculated. One then needs to “pertiirb the reference system in a controllable manlier so that one switches continuously from the known reference system to the system of iritercst awuning that one can obtain system propcrtic. at any instant of‘ the applied perturbation. Lct, us apply this gencral philosophy to the grand potential for thc systcrn depicted schematically in Fig. 5.12. Clearly, Eq. (1.66) represents the (exact. differential of the) grand potential for the situation shown in Fig. 5.12. Because the fluid siibstratc potential in Eq. (5.83) depends on the (vector) position r of a fluid molecule rather than a subset of Cartesian coordinates, it is immediately clear that we are dealing with a fluid that is inhomogeneous in all three spatial directions. Accordiiig to our discussion in Section 1.6.1,

225

Confined fluids with short-range Dotentids

1

0.8 0.6

S 0.4

0.2 0

4.2

4.4

4.6

4.8

r

5

5.2

5.4

5.6

Figure 5.13: Switching function s (z, y; R, K ) [see Eq. (5.84)] as a function of distance T = d m from the center of the coordinate system: (m) K = 1, (0) K = 5, (- -) K = 10.

this low s.ymmctry of thc fluid prccludc5 translational invariancc of its p r o p ertics in any dircctioii. Corlscyuciitly a closcd “rricc:hariical” cxprcssiori for fl is not obtained because there is no Gibhs Duhem equation pertinent to the system depicted in Fig. 5.2. However, we also not,icr from Eq. (5.83) that in the a1wm-e of the switrhing function, the rcrriairiiiig fluid substrate potential would dcpciid only on the z-coordinate of a fluid molecule. In this case, fluid properties would be translationally invariant in the r- and y-directions. Hence, in this case, the exact differential of tha grand potential would be given by Eq. (1.63), where this higher symmetry of the confined fluid lias already been exploited. As a consequence we obtain a closed expression for the grand potential [see Eq. (1.65)J in terms of the transvcrsc stress T I I as wc show in Section 1.6.1. The discussion a t the begiiining of this sectioii therefore suggests taking the fluid confined between undecorated surfaces as a reference system. For the reference system, we may derive a molecular expression for -ryy= rxx= 711, whcrc (5.85)

Chemical patterns of low symmetry

226

following the derivation for these stress tciisor clciricrits as outlintul iii A y pendix E.3.3 because in the reference syst,em the fluid substrate interaction potential does not depend on either the 2- or the z-coordinate of a fluid molecule. Equation (5.85) can easily be evaluated in a GCEMC simulation so that w = ql can also be obtained for the reference system. Following the general philosophy of perturbation theory, we replace the fluid hiihstmte potential for thr systmi of intcrcst hy

dkl( r ;A)

=

(z)

- A&{

(r)

(5.86)

PI ( z ) where X E [O? 11 is a perturbation parameter aid the contributions prep and @$i(r)to the fluid substrate potential are introduced in Eq. (5.83). Clrarly, if X = 0, we arc dealing with the nnpertiirhcd reference systmi in arid rioiiwcttablc: due which thc solid substratts arc clicinically 1ioiiiog~~1icou.s to the absence of attractive contributions to the fluid substrate potential. For X = 1, on the other hand, we are concerned with the situation of interest shown in Fig. 5.12. Because X is a..surned to be continuoils, we can continuously switch between the reference system and the system of interest by increasing X from 0 to its maximum value of 1. What is thc cflcct of introdncing the pertiirhation par(ameter X from a rriolccular pcrspcxtivc'! Tlic reader will irnnicdiatcly realize that tlic configurational energy

becomes a function of X as well as

This is apparent froiri Eq. (2.120), tlic dcfiiiitioii of IT ( r NA) ; in Eq. (5.87), and that of the configuration integral in &. (2.112). From Eqs. (2.112), (5.88), and (5.87) it is also evident, that, for fixed values of the set of natural ~ is solely a function of the perturbation variables {T,p, sx, s y , sz, L Y S , ~ } R paranieter so that we may write

(5.89)

227

Confined fluids with short-range potentials

Iiitroducirig (5.90) k=1 i = l

we may rewrite the previous expression as (5.91)

where the right side can be obtained as an ensemble average in a GCEhlC simulation but depends on X because E (A) as well as CJ ( r N ;A) both are functions of t8hatparauictcr. If wc iiitcgratc this lattcr cxprcssion, wv finally arrive at. x 0

according to the foregoing discussion of the limit X = 0. This approach, iisiially rcfcrrcd to as “X-expansion,” has I.)ccncniplovcd frcqiicritly as a siiitable starting point in thermodynamic perturbation theories [ 1831 (see also Ref. 30). In &. (5.92) we a,ssiinie that the confined fluid lamella. is not strained during the process of “switching on” the circular pattern on the substrate so that V, = sxOsyOs~. From the above derivation it is also clear that w is still a function of the set. {TIp , sx, sy,s, O S , ~ } regardless of the value of A. This fact may be cxploitcd to dovisc two additional roiit,cs on which w can bc calciilatcd from GCEMC data. For cxarriple, diffcrcritiatirig thc cxprcssioii for 12 (A) with respect bo p one finds [cf. Eq. (2.71)] 112

Ill

where F ( p ) = (N ( p , ) )/V0. In addition we find from Eq. (1.66) that (5.94)

Defining w = R/A,Os, = Q / V ,we may formally integrate the last expression to obtain

+

sZ2w (sZ2) = s Z I w( s , ~ )

.%I

ds,

r,,

(Y,)

,

TIp , s,. sy,A, cwX0 = const (5.95)

228

Chemical patterns of low symmetry

which can bc cvaluatcd using rnolccular cxprcssions for thc strcss tensor derived in Section 5.3.3. Because of the specific form of the fluid srihstrat,e potential, we have

so that the stress tensor element

(5.97) where 7:: and

7EF

are given in Eqs. (E.33) and (E.40a): respectively, and (5.98)

and an altcrnativc "force" cxprcssion follows froin Eqs. (5.96) and Eq. (E.46).

5.5.3

Computation of the grand potential

5.5.3.1

Chemically homogeneous substrate

As a test of Eq. (5.92) we apply t,he procedure outlined in Section 5.5.2 to a fluid confined to a. slit-pore with infinitesimally smooth, chemically horn& geneoiis substrate surfaces. These can be realized by replacing in Eq. (5.83) s (z, y; R, K ) G 1 so that, cPatt PI (zi) in Eq. (5.87) is replaced by patt Ikl ( z i ) . Ac-

cording to the discussion in Section 5.5.2 we may t,heii also replace Eq. (5.92) bY ?I

x

(4= 711 (0) - vi ./dA%%:)A! 1

0

(5.99)

such that A q (A) = 1 (A) - q (0) can be calculated either directly froin Eq. (5.85) or by numerically integrating Plots of A711(A) are shown in Fig. 5.14 for representative gas ( p = -7.8) and liquid phases ( p = -7.0) at a temperature T = 0.65 and s d = 12. Alt.hongh low, this tcmprraturc is still c:xpcctcd to cxccod thc triplepoint temperature in the bulk according to Fan and Monson [99]. The plots show that A711(A) is a monotonically decreasing function of X regardless of the thermodynamic state considered. For the gas, the dependence on X is largest in the vicinity of X N 1 [see Fig. 5.14(a)]. One also notices that t,he statistical uncertainty is somewhat larger for liquid than for gas states. This is not siirprisirig in view of typical liquid and gas densitnieslisted in Table 5.3

229

Confined fluids with short-range potentials

0 -0.002 -0.004 -0.006 -0.008 -0.01 -0.012

0

-0.1

-

-0.15

-

0.2

0.6

0.4

0.8

1

0.8

1

-0.2 -0.25

0

0.2

0.4

x

0.6

Figure 5.14: The incremental shear stress AT,, as a function of the perturbation parameter A. (A) from Eq. (5.85), (-) from Eq. (5.99); (a) gas, (b) liquid (see text).

230

Chemical patterns of low symmetry

n b l e 5.3: Acceptance ratio and mean density for selected gas and liquid states of a fluid confined hctwwn homogcnmiis substrates ( s w Fig. 5.14).

Stat(* liquid liquid gas ga~

A Acc.cpt.ancc ratio P 0.0 3.790 . 6.667. lo-' 1.0 2.010 . lop4 7.567. lo-' 0.0 3.836. lo-' 7.000. 1.0 1.690 . lo-' 3.400. lo-'

and the associated acceptance prohabilities. Acceptance probabilities for gamms states cxcccd those for liqiiid-like states hy approxirnatcly two ordcrs of rnagiiitutlc. For thc S ~ I I I Ciiumbcr of GCEMC c*yclcsoric tliercforc expects configuration space to be sampled more efficiently for gas than for liquid states. Howevcir, it is noteworthy that even for the highest-density state listed in Tahle 5.3 the acceptance probability exceeds, by more than a factor of two, the threshold value of approximately at which point the current GCEMC algorithm becomes prohibitively inefficient.

5.5.3.2

Chemically heterogeneous substrate

Consider now the chemically decorated substrates where the fluid substrate interaction potential is given by Eqs. (5.81)-(5.84). Over the temperature range 0.60 5 T 5 0.75 to which this stiidy is rcstric-tcd, the confinctl fluid may form three distirictly different rnorphologies characterized by the local density defined here as (5.100) where N (r,z ) is the number of fluid molcculty in an annulus of width br and thickness dz located at a distance r = and t from the center of the coordinate system and the substrate surface, respectively. Because the system is symmetric with respwt to the plane z = 0, we enhance the statistical accuracy of the histogram representing p (r,z ) [see Eq. (5.100)) by averaging over spatially cqiiivalcnt points in thc iippcr (z > 0) and lowcr ( z < 0) half of our system From the plots in Fig. 5.15(a), it is evident that for T = 0 75 and p = -8.36 a small portion of fluid is adsorbed by the circular arca on each suhstmte; the remainder of the system volume is occupied by low-density gas. The portion of fluid in the immediate vicinity of the substrates is stratified as indicated by the noii-monotonic decay of p ( T , z ) with decreasing IzI along lines of constant I'. Morphologies similar t o

d

w

Confined fluids with short-range potentials

231

the oric depicted in Fig. 5.15(a) arc thcrcforc rofcrrcd t.0 as “gas.” Notice also that the statistical error increase9 substantially as T -, 0 because the area of the ring segment [and therefore N ( T , z ) ] goes to zero in that limit. If, on t,he other hand, p = -8.30, the entire volume is occupied by fluid at a much higher density compared with t,he gas morphology [see Fig. 5.15(c)]. The plot in Fig. 5.15(c) shows that p ( T , z ) decays more or less monotonically as IzI -, s,/2 and T > R (ix., outside thc circnlar attractive! region). This is characteristic of repulsive substrates, which arc riot wet by the confined fluid. The morphology illustrakd by the plot in Fig. 5.15(c) is representative of what we shall call “liquid.” From the plot in Fig. 5.15(c), it is furthermore evident that the liquid is not only stratified in the direction perpendicular to the substrate (i.e., along the z-axis and lines T = const), but also in the contact layer (i.e., the one closest to either substrate) as one moves out of the attrartivc! circular rcgion in radial tlircction, which is with increasing distance T from tho ccntcr of that rogioii. Thc: separation t.)ctwccn successivc: maxima in p ( r , 2 ) as T increases in the contact layer is approximately 1. It also seerns worthwhile poiiiting out that, these radial oscillations of the local density in the contact layer become more pronounced with increasing I’ as one approaches I’ = R, which is the boundary separating the circular attractive from the purely repulsive part of the substrat,e. Thus, the fluid is more ordered along the rircumfercncc compared with portions controlled by “inricr” parts of tlic circular area. Near the cciitcr of tlic circular at.tractivc regon (i.e., for T N 0), fluid order has nearly vanished as reflected by p ( T , z ) that is nearly independelit, of I’ in this regime of the contact, layer. However, in the ,--direction (along lines of constant r 5 R), the separation between successive rnaxinia of p ( T , 2) is also approximately 1, indicating stratification as it would be expected in fluids confined among planar, chemically homogencoiis, and attractivc solid siihstratcs (see, for example, h f . 5). This packing efFcct, is due solcly to tlic geometry of the chc111ical decoration of the substrate and has not, been observed for other geometries, which is for, say, alternating striplike domains composed of different solid materials (see Figs. 5.8). For intermediate chemical potentials, the confined fluid may condense only partly in a subvolume V = { ( T , z ) 10 5 T 5 R7- s , / 2 < 2 < s,/2} “controlled” approximately by the circdar attrart,ive area with which the sub strates are decorated. This is illustrated by the plot of p ( r ,z) in Fig. 5.15(b) where high(er)-density fluid is spanning the gap hetween the circular attractive regions on the opposite substrates. Therefore, this morphology will be referred to as “bridge.” Notice that in the contact layers the fluid is stratified in radial directions similar to the liquid [see Fig. 5.15(c)]. However, here the radial ordering of fluid molecules is less pronounced compared with the liquid

-

232

Chemical patterns of low symmetry

Figure 5.15: Local density p ( r ,z ) as function of pcxsition relative to substrate plane ( z ) and distance T from center of attractive circular nanopattern (see Fig. 5.12): (a) gas ( p = -8.36), (b) bridge ( p = -8.33). ( c ) liquid ( p = -8.30). In all cases T = 0.75, R = 5, s, = 12, and .sz = 20.

233

Confined fluids with short-range potentials

(c)

Figure 5.15: Continued.

state. FoI gas aiid liquid states, w ( p ) can be calculated in GCEMC siinulatious by thermodynamic integration employing a s . (5.92)and (5.93)for fixed T and s, = 12. In general, w is a monotonically decreasing function of p [see (1.78)].Because of Eq. (1.78)and Fig. 5.15,we expect different slopes for w ( p ) depending on the morphology in question (e.g., gas, bridge, or liquid), that is, fluids characterized by distinctly different local densities (and thcrafora manifrstly diffcrcnt j 7 s ) . For sufficiently low p1 = -7.60 (T = 0.63) and X = 0, a gas forms between the purely repulsive substrate surfaces. This chemical potential is sufficiently low to guarantee that as X -, 1 the original gas is not subject to any discontinuous phase transition [sm Fig. 5.16(a)]. For A = 1 the interaction between fluid molecules and the decorated substrate has been fully “switched on.” Thermodynamic integration then proceeds by raising the chemical potential and employing Eq. (5.93)along the remainder of tlic. intcgration path its twfore. Oiily points aloiig this lattar path are show~iiri Fig. 5.16(a). For a corresponding liquid state one begins with X = 0 and a sufficiently high p1 = -7.15 such that a liquid morphology is stable even though the substrates are not wet (because they are purely repulsive) [see Eq. (5.87)]. Once X = 1 the chemical potential is now lowered and w is again calculated by

a.

Chemical patterns of low symmetry

234

0

-1.5

-7.40

-7.46

-7.44

-1.42

-7.4

-7.12

-1.1

-7.68

P

0.02

0.01

0

-0.01,

0 0.02

0.03

-0.04

0.05 -1 E

-1.16

-1.14

P Figure 5.16: Grand-potential density w as function of chemical potential p for gas (O),bridge (A), and liquid niorphologies (0): (a) T = 0.63, (b) T = 0.67, (c) T = 0.75.Solid lines are fits to simulation data int,ended to guide the eye.

235

Confined fluids with short-range potentials 0.01

I

I

4.014.05 I

-8.30

-8.36

-8.34

-8.32

-8.3

Y

-0.28

Figure 5.16: Continued.

thermodynamic integration employing ECq. (5.93). Again only points along the second integration path are shown in Fig. 5.1G(a). Plots in Fig. 5.16 also indicate that, over certain ranges of 11, w ( p ) is a multivalued function where the lowest value of w obviously corresponds to tlie thcrinodyn~nicallystablc morphology (i.c., pliasc): thc othcrs arc o~ily metastable. Metastability ends (i.e., the confined fluid becomes unstable) if the inequality in Eq. (1.82) can no longer be satisfied. The reader should realize that in general inetastability in MC simulations is an artifact caused by the limited system size and insufficient length of the Markov chain (i.e., the finite computer time availahle) [184]. Metastability would not be observed in an infinite system where the evolution of the system could be followed iiidcfinitcly In othttr words, Irictastability vuiishcs iii thc therrnodyntunic limit. Here the situation is slightly more delicate. Because the nanopatterns are finite in extent by definition, there is no way of increasing the system size Bithout altering the physical conditions of the confined fluid. Hence, in a sense, metastability here is “real” and associated with the (physically meaningful) small size of the fluid bridge. However. this also raises the question of whether the morphologies triggered by finite-size chemical patterns should

Chemical patterns of low symmetry

236

bc regarded as thcrrnodynaniir pliascs in t hc strict. scnsc. Nevertheless, thcsc morphologies are characterized by distinctly different. grand-potential densities as plots in Fig. 5.16 clearly show. Therefore, the notion of a "thermod p a m i c phase" does not seem to be totally nonsensical even in light of the above remarks. Suppose now two morphologies n and d exist with associated grandpotential densities wn and d.Under isothermal conditions (i.e., with T , srr,syrand s, fixcd) and according t o the nhovc. rationalc J3q. (1.7621) mav liitve a solution / L ~ PEE itno at which tlic morphologies cy and @ corrcsporid to coexisting phases. If a third morphology y exist,s, one may have three solutions pop,pp7,and /iTn from equations analogous to Eq. (1.76a) involving pairs of these morphologies. Suppose the mean densities associated with the morphologies satisfy the inequality

pa < ;ii' < $

(5.101)

irrespective of IL. Bccausc of Eq. (1.82) this iIiiplics

Donoting by dki' tho vahie of tho grand-potential dnnsity at p"", thrcc different scenarios are discernible as one can verify geometrically: 1,

>( p B p ~ P< p 7 w51j

> woY ,p

In this case only morphologies N and p comport with thermodynamically coexisting phases at. pno z PEA'; at, pro and pnY morphologies y: Y and N , 7 are only metastable. 2. w?b = w"b = w"' p ? ~ j = lL4' =p The three intersections coincide a t the given temperature thereby defining a triple point {rGT,/it,.} at which all tlircc ~riorpliologicsarc thcrmodynarnically stable.

3. w'fl < wQ4 < way p P

>

>

This describes a situation in which two pairs of separately coexisting morphologies are thermodynamically stable phases, namely 7 and ,f3 at pTP E p:' and a and y at p q pp'; at pnfi morphologies a and p are only metastable.

Rheological properties of confined fluids

237

According to this logic, plots of w versus 11 iii Fig. 5.l(i(a) iiidicatc that for T = 0.63 gas morphologies are thermodynamically stable over the range -m < p 5 -7.46, whereas liquid morphologies are thermodynamically stable over the range -7.46 5 p 5 p:, keeping in mind t.he general possibility of solidification of the confined phase a t a sufficiently high chemical potential pt where liquid and solid phases may coexist. At the intersection p$ N -7.46 gas and liryiid morphologies cocxist. Figirc 5.16(a), amcndctl by a parallol ralr.ulat.i.ionof p ( r ,z ) , also shows that bridge inorpliologics form as riietastable phases over the range of chemical potentials where this morphology may exist subject t,o thermodynamic consistency as spelled out in Eq. (1.82). Apparently, this situation resembles scenario 1 above. Ca.1culatingLJ ( p ) for bridge morphologies is significantly more demanding in terms of the thermodvnaniic integration procedure. To avoid a discontinuous phase transition during the initial stage where the si1bstrat.e at.traction is “turned on” [i.e., as X -, 1, see Eq. (5.92)], one needs to start from small valIICS s, = 3 - 3.5, which am!too small for any tliscontiniioiis transition t,o occiir 11851. Thus, as one increases X from 0 to 1, riiorc arid more molecules p d ually assemble in t,he vicinity of t8heincreasingly attractive circular regions between the substrates. Once X = 1, an addit,ional integration must be done to carry the substrate separa.t,ion to the desired value s, = 12. Along this path w is calculated via Eq. (5.95). Because r,, depends non-monotonically on sz, as the plot in Fig. 5.3 shows! this curve needs to be known with high resolution for Eq. (5.95) to provida sufficimtly ,accurate rcsiilts. Hcrc we calculatc r,, (s,) in stcps of As, = 0.1. Oiicc t h : substrate scparatioii has reached s, = 12, the remaining integration proceeds as discussed above for liquid and gas.

5.6

Rheological properties of confined fluids

In the preceding two sections we deniollstmted that confined fluids are highly inhomogeneous on account of the external field represented by the confining substrates. This is bccause the external field adds a new relevant length scale to the system competing with the characteristic length of fluid fluid intcrmolcciilar int,cractions. As a rcsiilt confincd fluids appcar gencrally to be stratified, at least to some extent, which manifests itself in a characteris tic oscillatory dependence of the solvation force with respect to a variation of substrate separation. If the substrates themselves are structured either chemically, as in the example discussed in some detail in Section 5.4, or geometrically the external field may depend on more than just one (Cartesian) coordinate. In these cases, confinement may give rise to new thermodynamic

238

Rheological properties of confined fluids

phases tliat have 1 1 0 couiit.crpart in thc bulk. An cxainplc is the fluid bridgc to which we devoted considerable attention in Section 5.4.2. An equally remarkable feature to which we shall turn now is the fact that confined fluids may sustain a certain shear stress without exhibiting structural features normally pertaining to solid-like phases; that is, they do not necessarily assume any long-range periodic order. We tacitly assumed this from thc very baginning of this hook iii our dcvalopmcnt of a tharmodynamir dcscriptioii of coiifiiicd fluids, wliicli closcly r ~ ~ o n $that h ~ appropriate for solid-like bulk phases (see Section 1) [12]. In addition we pointed out in Section 5.3.1 that the shear deformation can be measured experimentally in one mode of operation of the SFA. Hence, this section will be devoted to an analysis of these experiments in the framework of various coniputer siniulation approaches.

5.6.1

The quasistatic approach

Many att,empts have been made to elucidate detaiLs of the behavior of confined fluids under shear using theory. The approaches can be grouped into two different categories, which may be labeled “dynamical” [ 186-1921 and “qiiasistatic” [122, 171-173, 193-1953. In the dynainical approaches a stationary nonequilibriuin state is created either by applying an external driving force [ 1861 or by cxplicitly moving a siibst,rat.c [ 187, 189-1921 in noncqiiilih rium molecular dynamics (NEMD) siniulations in order to miinic dynamical aspects of a corresponding SFA experiment directly on a molecular scale. However, t,he relationship between NEMD siniulations [187, 189-1921 and SFA experiments remains eliuive for a nmnber of reasons. First, to describe the motion of the substrate on a physical time scale, an eqiiatioii of motion needs to be solved that, inevitably involves the siibstrate maw. However, thcrc: arc no physical criteria on which tha choicc of a specific value for this mass could be based. Second, even though the substrate is a macroscopic object in the SFA experiment, its mass cannot be t.oo much larger than the mass of a film molecule in the NEMD simulations because ot,herwise the wall would reniain at rest, on the time scale on which film molecules move. In fact, the ratio of the mass of a single film inolecule to that of t,he elitire wall is somatinies as small as 1/8 [191, 1921 so that, one can expcct relaxation plieiiornciia in tlie filiii to depend swisibly (and thercfore unphysically from aii experiment,al perspective) on this arbitrarily selected wall mass [170]. Third, t,he speed at which the walls are slid in the SFA experiment is typically of the order of lo-’ - lO-’Aps- [136] so that under realistic conditions the walls remain practically stationary on a tvpical length and time scale of molecular relaxation processes.

*

239

Confined fluids with short-range Dotentids

To avoid these problems aid in vicw of thc characteristic low shear rates in the actual SFA experiments, we employ it "qiiaistatir" or reversible approach in which the thermodynamic state of the film passes through a succession of equilibrium states (see Section 3.3 in Ref. 196), each being distinguished by a different (average) lateral alignment, of the walls [122, 170-173: 1931. Equilibrium properties of the film can be computed within the framework of M C sirniilations tfcsigricd to captm-c: kcy cliarilc.t,orist,i(~sof a corrcqwnding SFA cxperiincrit to a maxiIIiuin dcgrcc.

5.6.2

Molecular expression for the shear stress

Because of the chemical decoration of each substrate, a confined fluid can be exposed to a shear strain by riiisaligriing the substrates in the +x-direction according to

(5.103) where a &/s, is a dimensionless number and 6, is the magnihde of the relative displacement of the substrates with respect to each other where { N 10 5 a 5 f } may vary continuously between its limits. In this range a = 0 refers to substrates in registry. whereas a = if the substrates are out of registry. A kcy qiiantitativc nicmirr of the rcsist,ance of any confined fliiid to an external shear strain is the shear strcss rxz.As bcforc for the coinpressional stress T,, it. follows from Eqs. (E.54) and (E.57) that. a "virial" expression for the shear stress can be derived here, too (see Appendix E.3.2.1). It may be cast as (5.104) 7x2 = rx2 FF + rx, FF

4

Alternat.ively, we may derive a "force" expression for r,, following the derivation presented in Appendix E.3.2.2 for the stress tensor component T ~ , . It follows if we combine Eqs. (E.62) .and (E.63) with Eq. (E.49) from which we obtain

FJ

rxz= -(5.105) 2AXO As before [see Eq. (5.65)] mechanical stability of the cntirc system reqiiires

(Flf]) = - (Ff])

(5.106)

which serves as a useful internal check on consistency of the GCEMC simulation together with Eqs. (5.104) and (5.105). Another interesting quantity is the shear modulus c u where we use Voigt's notation ([12], see p. 14 in R.ef. 196). We reemphasize the appropriateness of

Rheological properties of confined fluids

240

this notation, origiiially tlcviscd for a thcririodyiiaxnic: description of solids, for the current case in which one is dealing with fluid phases that are nevertheless distinct from solids by their lack of any long-range order. The shear niodulus is defined by

A inicroscopic definition of

c4$

can be derived directly from Eq. (E.49); that

is.

(5.108) Focusing for convenience on the ‘:force’’ treatment it is a simp2 matter to vvrify frorn Ey. (E.62) that

...

x cxp

(--> I:uT

...

(5.109)

where the integration limits (represented synibolically by “. . . ’ I ) are the same as in Eq. (E.62) and wr employ thc ixrgimcnt, for differentiating Z presented atwvc [see Ekp. (E.58)-(E.62)]. Thus, with thc aid of Qs. (E.62), (E.63), (5.107), and (5.109) it follows from Eq. (5.108) that c44

=

I keT A20

(5.110)

which shows that the shear modulus depends on fluctuations of the fluid substrate force and on the curvature of the fluid substrate contribution to the configuratioiial potential energy with respect to the shear strain.

Conflned fluids with short-range potentials

24 1

Figure 5.17: As Fig. 5.13, where the fluid bridge is unsheared in part (a) but exposed to a shear strain in part (b). To enhance the clarity of the presentation two periods in the x-direction are shown.

242

Rheological properties of confined fluids

5.6.3

Fluid bridges exposed to a shear strain

As a first illustratioii we consider the model discussed in Section 1.3.3,namely a fluid of “simple” molecules confined between chemically striped solid surfaces (see Fig. 5.2). As before iii Section 5.4 we treat the coiifiiied fluid as a thermodynaniically open system. Hence. equilibrium states correspond to minima of the grand potential R introduced in Eqs. (1.66) and (1.67). The fluid fluid interaction is described by the ixitermolecular potential uff ( r ) introduced in Ey. (5.38) where the tssociated shifted-force potential is introduced in Eq. (5.39). The fluid siibstrate interaction is described by @Ik] (5,z ) in the continuum representation [see Eq. (5.68)], where T replaces 5 because of t.hc rnisalig~i~ncnt of tho substratcs rclativc: to oach other [scc Eq. (5.103)]. Frorri a morphological perspective, the confined flnid can exist either as a tthiii inhoinogoncous film [sm Fig. 5.8(a)], a high-dcnsity inhornogclncous liquid phasic [scc Fig. 5.8(b)], or a h i t 1 bridgc [see Fig. 5 . 8 ( c ) ] .As was already evident from the mean-field calculations daqcribed in Section 4.5.3, the bridge morphology is distinct from the other two in that it can be deformed in a direction parallel to t-he solid substratgs wit,hout breaking apart instantaneously [see Figs. 4.11(a) and 4.11 (b)]. Likewise, in the current GCEMC simulations the fluid bridge may be deformed by applying a shew strain in tho x-dircction [see Eq. 5.103)]as plots of thc lord density p (x,z ) in Fig. 5.17 clearly show. As a quantitative measure of the extent to which a confined phase is capable of resist.ing a shear dcforrnatioii, w(: iritroducc iii Scctiori 5.6.2 tlic slicar stress T ~ For ~ a . fluid bridge a typical shear-strew curve T,, (cys,o) is plotted in Fig. 5.18. Regardless of t,he thermodyminic state mid the thickness (i.e., s,) of a bridge phase, a typical stress curve exhibits the following fec\tures: 1. For vanishing shcar strain (i.o., roasons. 2.

(YS,~

= O),

T,,

(0)

3

0 for symmctry

(ctsdm)depends linearly on the shear strain as,o in t,he limit a -+ 0. That is to say, the response of the bridge phase to small shear strains follows Hooke’s law. T,,

3. For larger shear strains, negative deviations from Hooke’s law are observed, eventually leading to R yield point (cryd, 7 : : ) defined by the constitutive equation

(5.111)

243

Confined fluids with short-range potentials

or, alterxiatively

[SCC

Eq. (5.110)]1 cqq (QYdS,O)

0

1

=0

2

(5.112)

3

4

5

Figure 5.18: Typical stress curve T,, ( L Y S ~ O )for a monolayer bridge phase and cr = &. Solid line is a least-squares fit of a polyiiomid to the (discrete) MC data points ( 0 ) intended to guide the eye.

As far as the current model is concernedl the degree of chemical corrugation of the substrate has significant consequences for t8heyield-point location (cryd, 7::). Plots of stress curves for various values of c, are shown in Fig. 5.19(a). For monolayer bridge phases arid fixed s, = 10 one can see from Fig. 5.10(a) that both T,'," and ddarr smnllwt for thc- smallrst c, = For c, < only gas phases are thermodynamically stable because the strongly attractive portion of the substrate is too small to support formation of denser (bridge) phases. As c, increases both T,'," and aYd increase until they reach ' ) (2.740,0.169) for c, = For larger their maximum values ( a Y d s x , 7 ~ ~ M c, > the plots in Fig. 5.19(a) show that both 7,; and cryd decrease again : ) x (1.550,0.069) for c, = which is the largest substrate until (aYds,. 7

&.

6

5.

6,

244

Rheological properties of conflned fluids

&

corrugation for which bridge phascs wcrc observed. For c, > only tlicrmodynamically stable liquid phases formed in the simulations, incapable of sustaining a shear strain. One also notices from Fig. 5.19(a) t,hat stress curves for c, = and apparently do not, cover the entire range of shear strains. In these cases the strongly attractive port.ion of the substrates is too narrow to stabilize thc dcnscr portion of a bridgc phaso rcgardlcw of t,hc applied shear strain. At sorne strain threshold cycsxo;thc bridge p l i w is sirriply Yarn apart” arid undergoes a first-order phase transition to an inhomogeneous film. This film, by virtue of it,s niicroscopic structure [see Fig. 5.8(c)], is incapable of sustaining a shear stress. Thus, a t a,sxo, T,, drops to zero discontinuously such that T,, = 0 for all {a lo, 5 (Y 5 f }. For thc sake of clarity we do not plot this part of the stress curves in Fig. 5.19(a). Dwpitc this non-monotonic variation of thc yidd-point location with cT it, turns out that withiii tlic t,lioory of c.orrtapontling s t a t e [ 1971 it is fcasiblc to renormalize stress curves such t1ia.t all data point,s fall onto a unique master curve. Renormalization is effected by iiitr.oducing dimensionless variables T,, G T,, ((YsxO; cr)/T:: (cT)and li (q).Normalization by dd and T,?: is consistent with the t,heory of corresponding states because it was pointed out in [198] that the yield point niay be perceived as a shear critical point analogous to t,hc liquid gas crit,ical point in p r c hoinogcncous fluids. If ttic silriulatioii data plotted in Fig. 5.N(a) arc rciiornializorl accordiiig to this recipe, they can indeed he represent>edby a. mastser curve as the plot in Fig. 5.19(b) shows. The remarkable insensitivity of Cx(Zsxo) to variatioiis of c, can be r a t i e iialized as follows. Because of the Hookenn regime in the limit as, + 0, c44 should he approximately constant aiid positive in this limit. A typical plot in Fig. 5.20 confirms this notion. Howcvcr, hccaiwc of Eq. (5.112) onc cxpccts c44 to dccliric frorri its Hookcaii valuc as (rs, + d d s X o also iri agrcenicxit with Fig. 5.20. Furthermore, as Fig. 5.20 shows that. the variation of c44 with as, is not too strong over the range { cr 10 5 N 5 nYd}, it seems sensible to expand cj4 in a power series according to

6

&, 2,

-

where we refer to the far right side as the small-strain approximation in the current context (cf., Section 1.2.1). Notice that the set of coefficients {uk} refer to the mstmined bridge phase (i.e., a = 0). A inolecular expression for G c44 (0) is given in &. (5.110). In the small-strain approximation u2 accounts for deviations froin Hoolcean behavior and may therefore be

245

Confined fluids with short-range potentials

0.15

0.1

8 0.05

0

0

1

2

3

4

5

0.2

0.4

0.6

0.8

1

1 0.8

0.6

k?,.

0.4

0.2 0

Figure 5.19: (a) Strew curve T , ~(as,)for various chemical corriigations c, = & (*), (0). Solid lines are intended to guide (+), (x), (O), (A), (o), the eye. (b) rteduced stress curve Fz, (Gsxo) [see Ey. (5.116)]where symbols are referring to data plotted in (a). The solid line is a representation of Eq. (5.117).

&

&

&

R.heologica1 properties of conflned fluids

246

-0.2

i

0

1

I

1

2

3

QSxO Figure 5.20: Shear modulus c44 as a function of shear strain as&. (0):MC simulations in grand mixed isostress isostrain ensemble; (-): representation of smallstrain approximation cj4 (as,")= CQ + a2 ((YS~O)~ [seeEqs. (5.113) and (5.114)]. interpreted as a measure of plasticity of the unsheared confined film. Moreover for a = 0 symmetry requires a2k-1 -= 0 (k = l , . . . ,00). However, we note in passing that these coefficients do not vanish a priori for a # 0. F'rom Eqs. (5.107) and (5.113) wa obtain thc (shaar strrss) cqiiation of statc

based on the small-strain approximation. In principle, a()and a2 are determined by ordinate and initial curvature of the function c u (os,~) (a -, 0) (see Fig. 5.20). The latter is extremely difficult to extract given the typical accuracy with which the shear n~oduluscan be calculated in our MC simulations (see. Fig. 5.20). However, an accurate estimate is possible based on

247

Confined fluids with short-ranne potentials

Table 5.4: Comparison of shear modulus c44 from iiiolecular expression and yieldpoint. location (see text).

Eq. (5.115a) Ey. (5.110)

2

2 ! !

$

2.113 2.075 3.057 2.069 3.044

1.350 2.499 2.588 2.743 2.412

0.075 0.161 0.101 0.169 0.095

0.084 0.096 0.058 0.092 0.059

0.079 0.088 0.060 0.101 0.066

Eq. (5.111), which, together with Eq. (5.114), leads to (5.115a)

in terms of yield st,ress and strain. Thwr latter quantities ran be det,ermined with high precision from Eqs. (5.105), (5.104), mid plots siriiilar to thc oncs shown in Figs. 5.18, 5.19(a). The validity of Eq. (5.115a) is illustrated by Table 5.4 where we compare it with the shear modulus obtaiiied directly from the molecular expression Eq. (5.110) for a selection of unsheared bridge phases. Inserting now Eqs. (5.115a) and (5.115h) into the equation of state Eq. (5.114) (in the small-strain approximation) together with the transformations

(5.116a) (5.116h) permits us to recast Eq. (5.114) as

(5.117)

It is furthermore noteworthy that universality of stress curves, as defined here, is riot restricted to monolayer fluids. Plots of F,, versus lis, in

Rheological properties of confined fluids

248

Fig. 5.21(b) show that. siiiiulatiori data for triono-, bi-, arid trilaycr bridgc phase2 can also be mapped onto the master curve Eq. (5.117) according to the treatment detailed in this section. Again, the stress curves in Fig. 5.21(a) end at, some QS,, because the bridgc phases evaporate if they are strained bevond this limit.

5.6.4

Thermodynamic stability

From a fundamental point of view, bridge phases coniprisirig different nunibers of moleciilar strata may be viewed as different thermodynamic phases. This intcrprctation is evident if one consirlcrs the thcrmodynamic pot,cntial @ tiofiricxl in Eq. (2.71). Togcthcr with Eq. (1.43), wc' obtain d@ = -SdT - N d p

+ AxOrxxdsx+ Ayor,dsy

- .4xos,dr,,

+ A d r x Z d(cys,~)

(5.118) Applying the arguments detailed in Section (1.6) and in view of the fact, that the fluid substrate potential does not, depend on y: we immediately conclude from Eq. (5.118) that

(5.119) Rccausc fluid hridgcs of diffcrcnt length in the z-direction will gcncrally he characterized by diffbreiit T~~~it is clcar tliat tlicsc bridges will have different values of Cp and must therefore be considered as legitimate thermodynamic phases. This, however , causes a complication because the thermodynamic state is uniquely specified by the set { T , p .s,, sy:rZZ, crs,~}. Because bridge phases of variable leiigth in the r-direction may generally be compatible with the same fixed set of thermodynamic state variables, we apparently have a midtiplicity of phascs cicspit,c the fact that the thrmodgnamic state is uuiqucly syccificd. IIowcvc'r, from $11 equilibrium perspective, only the Inorphology corresponding toothe global minimum of d is a thermodynamically stable phase; the others must be metastable. Fortunately, only a small, finite number of possible morphologies can exist under the current therinodynamic constraints. This can be understood by considering the (normal) compressional stress T ~ plotted , as a function of snbst,ratc separation s, in Fig. 5.22(a). Data plotted in Fig. 5.22 wcrc o h tainecl in GCEhlC simulations in which a thermodynamic state is specified by a choice of natural variables similar to the ones determining 9,replacing, however, T,, by its conjugate variable s, [cf. Eqs. (1.66) aiid (5.118)]. Again, the plot in Fig. 5.22(a) shows that rzz is a damped oscillatory function of s,. As we saw in Section 5.3.4, these oscillations are fingerprints of stratification, which corresponds to the formation of new fluid layers as

Confined fluids with short-range potentials

249

0.15 0.1 X

I?

0.05

01

1

0

1

2

3

4

5

0.2

0.4

0.6

0.8

1

1

0.8

12

0.6

0.4

0.2 0

0

Figure 5.21: (a) As Fig. 5.19(a), but for mone (0),bi- (A, and trilayer (+) morphologies and c, = (b) As Fig. 5.19(b) but for data points plotted in (a).

6.

Rheological properties of confined fluids

250

the substrate scparatioii incrcascs at constant T arid p . Darripirig can bc ascribed to the decreasing influence of the fluid substrate potential, which becomes negligible if s, exceeds soiiie critical value s,,~. For s, 1 s , , ~ onc expects a honiogeiieous region to exist in the confined fluid. The homogeiieous region is centered halfway between both substrates, increases in size with s,, and its local density (which is independent of position) equals that of H rorrcsponding bulk phasr for tho same T and p . As a rrsnlt lim r,,

#.--too

(-9,)

= -Pt,

(5.120)

where Pt, ( p , T )z 0.03 is the bulk pressure. In other words, because stra,tification diminishes with increasing sz, oscillations in ru.( s z ) also vanish eventually [168]. Therefore, the plot in Fig. 5.22(a) shows that, under the current conditions, and for s, 2 6.0, stratification becomes subdominant. In tlic grand iriixcd isostrcss isostraiii cris(:rriblc, rriorphologics coilsistent with the set {T,p , sx, sy,rm,QS,~,} of state variables can now be identified wit,h intersections between the oscillatory curve T= (s,) and the isobar r,, = const 5 0. However, only intersectioiis for which dr,,/ds, 2 0 correspond to (tliermodpiiamicallv or meta-) stable states as pointed out in Section 5.3.4 [see Eq. (5.66)]: intersectioiis for which dr,/ds, < 0 pertain t+oiinstablc states: which cannot be realized in MC sirniilat~ioiisin the paiid rriixcd iswtrcss isostraiii mscmblc. Thc t~l~crinodyiiai~iically stable rnorphology corresponds to the intersection having the smallest d, ( T = ~ 0) according to Eq. (5.119). Based on this rationale, an inspection of Fig. 5.22 shows thatj the thermodynamically stable, iinstrained morphology (cy = 0.0) is a monolayer film with s, N 2.1 (r,, = 0.0). If confiiied films are progressively sheared, a parallel analysis of plots in Fig. 5.23 a i d 5.24 shows that the miiiirniim of q3 for s, N 2.1 bccomw shallowcr. whcrcas anothcr minimum arouiid R, N 3.1, corrcsponding to a bilaycr film, bccoines dccpcr with increasing shear strain. Eventually the depth of the lakter minimum exceeds that associated wit,h the monolayer film so that a bilayer film becomes the thermodynamically stable morpholdgv. Thus, a shear strain exists such that d, is the same for mono- arid hilayer films. At this shear strain both morphologies may therefore be viewed as coexisting phases in the classic sense (scc Scctiori 1.7). To obtain a more concise pictaure of thermodynamic stability of different film morphologies, we plot q5 as a function of QS,O in Fig. 5.25 for the same system analyzed in Figs. 5.22-5.24. In a sequence of MC simulations in the grand niixed isostress isostrain ensemble, we calculate 4 directly from Eq. (5.119) using the molecular expression for T~~ [see ECq. (5.85)], which does not contain aiiy fluid substrate trontribution between the fluid substrate

251

Confined fluids with short-range potentials

2

3

5

6

4

5

6

S Z

-0.4

-8-

4

-0.6 -0.8 2

3

sz

Figure 5.22: (a) Normal compressional stress T~~(see Appendix E.3 for molecular expressions) as a function of substrate separation from GCEMC simulations (0) ( m X o = 0.0). Solid line are intended to guide the eye. (b) As (a) but for 4 [see Eq. 5.1191. Intersections between the latter and the vertical lines demarcate (meta- or therrnodynamically) stable states in the grand mixed isostress isostrain ensemble for T,, = 0.0 (see text).

interaction potential as it does not depend on the y-position of fluid moleculcs. An alternative expression for r$(asd) can be obt,ained by integrating Eq. (5.118)

4 (os,o)

=

4 (0) +

J

OSXO

d (os,o)I

T,,

[(asxo)']

n

N

a0

a2 4 (0) + -2- (ctsxo)2 + (CIS,")

12

4

, fixed T ,LL.s,, .sy. G~ (5.121)

where the second line is based 011 the small strain approximation introduced in &. (5.113). Full lines in Fig. 5.25 are representatioiis of Eq. (5.121). Solid lines plotted in Fig. 5.25(a) are therefore obtained without further adjusting % arid a2; d(0) is taken from MC sirnulatioiis for unstrained bridge

Rheolonical DroDerties of confined fluids

252

0.5

c

N

0 -0.5 2

3

5

6

4

5

6

SZ

-0.4 -8-

4

-0.5 -0.6

2

3 SZ

Figure 5.23: As Fig. 5.22, but for

aSxO

= 2.25.

phases. The excellent agreement bet.ween q5 (ct.s,O) from the MC simulat.ions in the grand mixed isostress isost.raiii ensemble and the small strain approximation in Eq. (5.121) highlights once more the validity of the latter for all {a la 5 dd}. However, the plot in Fig. 5.25(a) also shows that the small strain approximation is doomcd to fail for siifficicntly lnrgc shear st.rains in accord with onc''s cxpectatioii.

From the plots in Fig. 5.25(a) one notices t1ia.t 4 is lowest for a. monolayer bridge phase over the range 0.0 5 crs,O 5 2.2, indicating that, the monolayer is the tliermodynainically stable morphology in this regime. Figure 5.25(a) also shows that intersections between the curves exist at which 4 for a pair of differcnt, bridge morphologics assiinics the same vahic. Thus, at thc corresponding valiiw r r s x O , t,hese different, morphologies coexist, so that, the intersections can be ascribed to first-order phase transit,ions between bridge phases comprising different, numbers of molecular strata. Although there is no obtious relationship linking QSxO a t coexistence bet,ween mono- and bilayer morphologies cy-Vdsko, we notice that for all cases investigated a monolayer film is the thermodyiiairiical~ystable morphology for all { a Jcr cryd }

<

Confined fluids with short-range potentials

253

0.5

c

N

0 -0.5

2

3

5

6

4

5

6

sz

-0.4 -8-

4

-0.5 -0.6

2

3 SZ

Figure 5.24: As Fig. 5.22,but for a s X o = 2.50.

so that, up t,o the yield point, plots in Fig. 5.19 apparently pertain to thermodynamically stable phases. Thicker filrris arc t,hcrforc tlicrrriodyriarriically stable only if the shear strain exceeds the yield strain. For example, plots in Fig. 5.25(a) for c, = $ show t,hat Q for a bilayer bridge phase is lower than for the corresponding monolayer bridge phase over the range 2.3 5 as,n 5 5.0 where the bilayer bridge phase is the thermodynamically stable one according to the above discussion. An additional trilayer bridge phase was observed for c, = ils plots in Fig. 5.25(b) show. For rr thc hilayer is t,hcrmod~nrnicnlly stable over the range 2.4 5 ms,O 5 3.3, whereas the t,rilayer film seems to be thermodynamically stable over the range 3.3 5 as,^ 5 4.0 where all three curves end. However, for the trilayer morphology the statistical error of 4(as,o) is already quite large because T~ is small (see Figs. 5.22-.-5.24). For a S x n 2: 4.0 bridge phases become unst,able mid the system undergoes a first-order phase transition and evaporates.

&

6

254

Rheological properties of conflned fluids

-8

-8

Figure 5.25: (a) q!~HS a fimct.ion of shear strain a s x o for mono- (O), bi- (A), and trilayer (+) morphologies calculated in grand mixed isostress isostrain ensemble MC simulations [see Eqs. (5.119) and (5.85)] for c,. = Solid lines are calculated from l3q. (5.121). (b) As (a), but for c, =

&.

6.

Confined fluids with short-range potentials

5.6.5

255

Phase behavior of shear-deformed confined fluids

A confined fluid may undergo phase transitions among thin gaseous films, liquid-like, and bridge phases similar to those observed for the confined lattice fluid in Section 4.5.3. To demonstrate the close correspondence bet,weeii the two models as far as the phase behavior is concerned, we calculate the average overall density defined in Eq.(5.74) for various substrate separations s,. A plot. of p in Fig. 5.9 for c,. = cxhibits t,wo discont,iniiitics. By a parallel analysis of p ( : I ; , z ) in Fig. 5.8, thc oiic around s, N 8.2 turns out to correspond to a first-order phase transition involving gas- and liquidlike phases, whereas the one at s, N 7.5 refers to a transition between a liquid-like phase and a bridge phase (upon reducing sz). Therefore, the sequence of phase transitions in Fig. 5.9 resembles precisely the scenario observed for the lattice fluid in Fig. 4.12(b). However, depending on the prccisc chcniical striictmc of these siirfaccs, diffcrcnt. phasc transitions arc yossiblc (see Fig. 5.9), wliicli c w i also bc cxplaincd qualitativdy withiii the framework of the niean-field lattice fluid. Oscillations of p in Fig. 5.9 over the'range 2 5 s, 5 6 reflect stratification of tlie confined fluid as decrihed above. To make direct, cont,act with t.hc mean-field cnlciilatioiis of the related discrete iiiodel in Section 4.5.3, we erriploy the grarid carioriical ratlier thaii the mixed grand isostmss isostrain ensemble used in the preceding section. However, investigations of phase transit,ions by GCEMC simulations are frequently plagued by metastability, that is, the existence of a sequence of corresponding only to a local minimum of t,he confiwrations { grand-potential density w where A1 can be quite substantial. In other words, the "lifetime" of a metastable therrnodyriariiic state can be large compared with the time over which the microscopic evolution of the system caii be pursued on account of limited computational speed. The origin of metastability is lack of ergodicity in the immediate vicinity of a first-order phase transition that ariscs on account of the iiiicroscoyicdly small systciiis cinploycd in cornputer simulations [ 1731. As we pointed out above, rnetastability is manifest as hysteresis in a sorp tion isotherm (like the one plotted in Fig. 5.9). Metastability involves a range of finitc width As, around t8hct,riic transition point, over which for t,hc same T and p, (s2) is a double-valued function. To distinguish the metastable from the bhermodynaniically (i.e.: globallg) st.able phase one needs to compare w for the two states pertaining to different brandies of tlie sorption isotherm at the same p and s, [see Eq. (1.68)]. The branch having lowest w is the globally stable phase; tlie other oiie is only metastable. In Fig. 5.9 we plot only data for therrnodynarnically stable phases identified according to this

6

4>k=,,....hf

256

Rheological properties of confined fluids

rationalc. Because of the siniilarity between the lattice fluid calculations and the hlC siniulations for the continuous 111oclc1,it SCCIIIS instructivc: to study the phase behavior in the latter if the confined fluid is exposed to a shear strain. This may be done quantitatively bv calculating j5 as a function of p and as,. For sufficiently low p., one expects a gas-like phase to exist along a subcritical isotherm (see Fig. 4.13) defined w the set of points (T = const)

T = { ( p , T ) lptr < p < rnin (p:'.

p!') Ttr< T

< min (TESb,TF) } (5.122)

At an intersection between T and p:' (T), the gas-like phase will undergo a spontancois transformation to a hridgc phase. In a corrosponding plot, of F ( p ) , one should see a discoritiiiuous juniy to a higher density. Eventually, another intersection between T and p:' (T) exists and a second discontinuo i jump ~ to an evcn higher valiie of p ( p ) should be visible. Both of these transitions are indeed observed in Fig. 5.26 for as, = 0, p 21 -8.40, and p N -7.98, respectively. Notice that, in Fig.. 5.26, pz' for as,^ = 0.0 exceeds its bulk counterpart p x b ; that is, for p:', the corresponding bulk phase is liquid. This can hc rationalized by noting that the low(cr)-density part of a bridge phase is prcdoininautsly involved in tliis sccoiid t.ransitior1. Recall also that, this part, of a bridge phase is stabilized by the weak portions of both (perfectslyaligned) substrates characterized by Efw << 5 ~ Hence, . the second first-order txansition is inhibited rather than induced by the substrates (with respect to the bulk) because of the dominating repulsive interaction of a fluid molecule with the weak part of the substrate. If a shear strain is applied, the region of overlap of the weak substrate parts in the s-direction shrinks [see Eq. 5.1031 such that a fluid molecule locat-cd at, {:c J 4 / 2 5 I:c( 5 s,/2, u s , = 0.0) is exposed to a strongcr nat, attractive fluid substrate interaction. Consequently, OIIC cxpccts an associatcd shift of p:' to lower values. Tlic plot in Fig. 5.26 confirms lhe expectation. In addition, Fig. 4.13 shows that. the one phase region shrinks because Tt, shifts to higher temperatures and because the slope of the coexistence lines does not change much. The plot in Fig. 4.13 therefore suggests that for a > 0 the two discontinuities in p ( p ) approach each other so that the branch of p ( p ) belonging to Iwidge phnscs lwcomcs narrower with incrcvwing w , O . This cffect is indeed visible in Fig. 5.26where the width of the int8ermadiat.e-density branch of j5 ( p ) (corresponding to t,hermodynaniically stable bridge phases) diminishes from [Apt 21 0.42 (as,,, = 0.0) to lApl 21 0.14 (asxo = 7.5). Finally, if the shear strain is large enough, the lattice fluid results in Fig. 4.13 suggest t.hat for a given temperature T', Ttr(as,)> T' for sufficiently large shcar strains (sec thc curvc for cr = f in Fig. 4.13). Hence, under these

The Joule-Thomson effect

0.8

I

I

-8.5

-8.4

257 I

1

-8.2

-8.1

1

0.7 0.6 0.5 IQ Om4

0.3 0.2

0.1

0

-8.3

P

-8

-7.9

Figure 5.26: Sorption isotherms 7( p ) from GCEMC simulations: (0):a s x = 0.0; (a): a s x = 2.5; (a):a s x = 5.0; (m): as, = 7.5; (A): a s x = 10.0. Also shown are corresponding bulk data ( 0 ) . Results were obtained for T = 0.7, .sx = 20.0, d, = 10.0, and s, = 8.0.

circumstances, one would expect ;ii(p) to exhibit just a single discontinuity referring to a phase transition between gaseous film- and liquid-like phases. The plot in Fig. 5.26 for as,^ = 10 konfirins this notion.

5.7 5.7.1

The Joule-Thomson effect Experimental background and applications

Aftcr ilhistrating thc rathcr fascinating strwtiiral and rhrological propertics of confined fluids we conclude our discussion of MC simulations of continuous model systems (i.e., models in which fluid molecules move dong continuous trajectories in space) with yet another example of the unique behavior of confined fluids. For pedagogic reasons we selected a topic that is standard in physical chemistry textbooks [26, 199-2031 as far as bulk fluids are concerned, namely the Joule-Thomson effect.

258

The Joule-Thomson effect

The JouleThoriison cffccb refers to a phcnomciiori observed if a gas in a vessel 1 a t temperature TI and pressure PI expands slowly through a valve or porous plug into another vessel 2 where its pressure P2 < PI. During this expansion a temperature change AT = T, - T I is otxmved, which can be positive, negative, or vanish altogether depeiiding on the precise experimental conditions. This phenomenon is referred to as the .Joule-Thomson effect and was origirially rc-portetl by .loiilc m d Thomson (latm tit led Lord Kclvin) 12041. During the expansion the gas does not exchange heat with its environinent. However, it exchanges work because of the expansion against the rio1izm-o prcx;siira 1’2. It is than a sirnplc mat,t,cr t,o dcmonstrate that the gas expands isenthalpically [26, 199-2031. This makes it coiivmient to d i e cuss the Joule-Thoinson process quantitatively in terms of a JouleThornson coefficient

(5.123) where ‘H is the enthalpy. Froin Eq. (5.123) it is clear that during an isenthalpic expansion (df < 0) 6 > 0 if the gas is cooled and 6 < 0 if it is heated instead. The fact t,hat the gas ran be cooled during a .Joule-Thomson expansion is of grcat tcchiiological rclcvaiicc in appliad fields like cryogciiics and in particular in the liquefaction of gases [200]. Frorn a fundamental perspective the JoubThomson effect is important becaiwc it, can be linkcd dirc?ct,ly to the nature of intermolecular forces betwcwi gas molecules [205]. Cotisidcr, for cxa~i~plc, a classic ideal gas as tlic simplest case in which molecrileq do not interact by definition. For this model it is simple to show t,hat as a consequence of the absence of any intermolecular interactions a .JoulcThomsori effect does not exist, that is, 6 = 0 [200, 2011. If, on the other hand, the ideal gas is treated quantum mechanically, it can be demonstrated [206] that a Joule-Thomson effect exists (b # 0) despite the lack of intcrniolcciilar interactions. The origiii of the rionvaiiishing Joule-Thomson effect is the effective repulsive (Fermions) and attractive (Bosom) potential exerted on the gas molecules, which arises from the different. ways in which quantum states can be o w u p i d in systems obcving Fcrmi-Dirac and Bozic-Einst,cin statistics, respectively (17). In other words. t.he effective fields are a consequence of whether Pauli’s antisymmet,ry principle, which is relativistic in nature [207], is applicable. Thus? a weakly degenerate Ferrni gas will always heat up (6 < 0): whereas a weakly degenerate Bose gas will cool down (6 > 0) during a Joule-Thornsoii expansion. These conclusions remain valid even if the ideal qiiaiit,uin gas is treated relativistically, which is required to understand

Confined fluids with short-range potentials

259

certain aspwts of stellar rnattcr [208?20'3). We shall rcturii t o thcse issues in Section 5.7.4.3 where we consider the Joule-Thomson effect, in confined ideal quantum gases as a first application. Beyond the (classic or quantum) ideal-gas level, molecules in a gas are subject to a.tkractive and repulsive iiit,erniolecular interactions. Thus, intuitively, one expects a real gas to show both a positive and a negative .TouloThornson cffcct depending on the t hormodynaniic conditions. I n other words, the s i p of b dcpcrids csscntially OII thc dcpce with which molecules probe a.ttractive and repulsive porlions of the inlerniolccular potential. From this line of arguments, one then expects an inversion temperature Tin,(, p) ( p is the density) to exist along which b = 0, thus separating regions in thermodynamic state space that are characterized by a positive or negative Joule-Thomson effect? respectively. These notions are readily confirmed by treating a bulk van rlcr Wads fluid [200]. Wc will cxtcnrl these considerations to a coilhied vaii dcr Wads fluid I d o w in Sctrtioii 5.7.5. Unfort#unately, previous work is almost exclusively concerned with the inversion temperature in the limit of vanishing gas density, Z,, (0). The inversion temperature can be linked to the second virial coefficient, which can be measured [210] or computed from rigorous stat,istical physical expressions [211) with moderate effort. Currently, oiily the fairly recent study of Heyes and Llagiino is conccrncd with the densitmydcpcndcncc of t8hcinversion tcmpcraturc from a inolc~ulitr(i.c., statistical physical) p c r s p c c t h [212]. Thcsc authors compute the inversion temperature from isothermal isobaric molecular dynamics simulations of the LJ (12$6)fluid over a wide range of densities and analyze their results through various ctquations of state. All these considerations apply strictly to homogeneous bulk gases, that is, for gases in containers of macroscopic dimensions. Under this proviso only a vanishingly small portion of t,hr gas will he pcrt,iirl)ad by thc intmaction of its rriolcculcs with t,lic coritaiiicr walls rriakiiig tliis interaction incoruequentid for gas properties. However, Rybolt [213] and Pierotti and Rybolt (2141 have studied the Joule-Thomson effect in aerosols consisting of finely dispersed carbon powder in argon gas. In such an aerosol the ratio of solidsurface area to volume becomes large so that gas solid interactions can no longer be ignored (see? for example, Fig. 7.1). Ry applying the concepts of statistical physics, Picrotti and R.yholt (2141 dcrivcd an cxprmsion for the Joule-Thomson coefficient, in terms of gas gas and gas solid virial coefficients. An analysis of the adsorption dat,a of Cole et al. [215] shows that the JouleThomson effect. can be enhanced by up to an order of magnitude over that observed in pure bulk gases depending on powder concentration (i.e., t,he relative contribution of gas solid interactions), and this enhancenient may have practical implications for refrigeration devices. I Jnfortunately, all this

The Joule-Thomson effect

260

work is again restricted to thr liriiit of low gas density. From a more general perspective the interaction of soft condensed matter with solid substrates is of great importance whenever it is desirable to, say, miniaturize mechanical machines. As we demonstrated in Section 5.6, the presence of such substrates has profound consequences for thermophysical properties of soft condcnscd rriattcr arid especially so if tlic coiifiricnicrit is to spaces of nanoscopic dimensions. The availa.bilit,yof a variety of techniques to design aiid to construct devices on a naiio- to rnicrometler lengthscale in a controlled manner has also given birth to a flourishing new field in applied science referred t,o as “microfahrication technology” or “microengineeriiig” [49].A particularly interesting example in the current context are microminiatme rcfrigcrators [216: 21 71. By rncaris of photolithographir tmhniqiics fine riozzlcs and chariiicls can Lc clcsigricd in a coiit,rollcdfashion 011 a microrneter lengthscale through which a gas can flow such t-hat t,he Joule-Thomson effect can be employed for cooling purposes. Through microminiature refrigeration, superconducting electronic devices including fast A/D converters, precision voltage standards, and singlc chip, high-speed logical Josephson devices can be cooled efficiently 12171. Thus, all these examples illustrat,e t,hat t h t corisidcratioii of thc .loul+Thonison effect, undcr nanoconfincmcnt coiidibiorls is of broader than just acaclcrnic intcrcst aiid may very well havvc practical applications in the future.

5.7.2

Model system

The model we shall be eniploying below to investigate various aspects of the Joule-Thornson effect consists of a “simple” fluid confined between the chemically horriogerieous and planar substrates of a slit-pore separated by a fixed distance sz. For t,his system we may cast the configurational energv as

where ufl is given in Eq. (5.38)aud @Ik] in Eq.(5.71) where we are again using the LJ (12,6) potential [see Eq. (5.24)] for the iriteriiiolecular int,eraction potcntinl IL in Eq. (5.39). To investigate the inipact of the chemical nat,ure of the (homogeneous) substrate, two different cases are studied. In the “strong” model A, fluid substrate interactions are described by @ ]! ( z ; s,) as introduced in Eq. (5.71). In addition. the “weak model B is considered in which fluid substrate

261

Confined fluids with short-range potentials

iritcractioris are purely repulsive; that. is [cf. Eq. (5.71))

(5.125) Because of the absence of any fluid substrate attraction, the fluid in rnodcl B cannot wet the confining substrates. In addition to classic fluids with interacting molecules, we shall also consider below the ideal quantum gas of Dosons and Fermions. The ideal quantum gnscs arc confincd by plnnc parallcl, striic tnrrlass, and chamically horiiogcricous substratcs rcprcscntcd by (2:s,)

=

{

I:

--Efs,

4

1.1

24

2-A0 5

1.1 5 4

2-u

5 4 2 -0 2 - x(7

121

(5.126)

This fluid substrate potential is chosen because it accounts for attractive as well as for repulsive interactions, it is short-range, and it permits an analytical t,reatiricnt of corifiiicrricrit effects as wc shall dcrrioiistratc below in Section 5.7.4.3.

5.7.3 Thermodynamic considerations 5.7.3.1

Joule-Thomson coefficient and inversion temperature

Tha central qnantity in thc cont,cxt of this chapter is the Joule-Thomson coefficient. which we dcfirie bv arialogy wit,h its bulk couritcrpart [scc Q. (5.123)] as

(5.127)

It is positive if the confined fluid is cooled (dT < 0) upon transverse compression ( d q > 0) and negative if the fluid is heated instead. According to the assertions a t the beginning of Section 5.7.1, the key thermodynamic potential in the current context is the erithalpy 7-1, which we obtain as a Legendre transform (see Section 1.5) of the internal energy via d3-1

d (U- 7/1As20)= TdS + pdN - ASddTIl

+ 71AOd.3,

(5.128)

where for the current model Q. (1.22) for dU applies. From the exact differential for the enthalpy in Eq. (5.128) we readily conclude that the set { S, A', q,s.} specifies the natural variables of 3-1. To proceed we immediately restrict the discussion t o a situation in which the fluid lamella is

262

The Joule-Thomson effect

corifiricd to a slit-pore of fixcd porc width s, = corist arid coritairis a fixed number of molwiiles N = const. Moreover, froni the definition of the JouleThoinsori coefficient in Eq. (5.127) it, is clear that we need to establish a relation between 'H 011 the one hand and the variables T and 711 on the other hand. 1% accomplish this via

dS=

(g)

N , ~ l,,qz l

dT+(E)

dq, T.N,n,

N , s,

= const

(5.129)

At this point, it is convcnicnt, toodefine a specialized isostrrss (q = const) heat capacity

(5.130) to eliminate the first term on the right side of Eq. (5.129) by some in princzple measurable quantity. The second term in Eq: (5.129) can be replaced through a Maxwell rclation [see Eq. (A.7)]. Thcreforc, wr need to introduce yct ariotlicr Lcgcridrc traiisforrn of thc iiitcriial oricrgy (scc Scction 1.5)

(5.131) which can hc iiiterpretcd a specialized Gihhs potential depending on {TIN,T I [ s.} , as its sct of riatual variablcs. Applying Eq. (A.7) to Eq.(5.131) we realize that

(5.132) where all is the (tramverse) expansivity of the confined lamella. Replacing the partial derivatives on the right side of Eq. (5.129) by Eqs. (5.130) and

263

Confined fluids with short-range Dotentials

(5.132) and rcalixiIig that thc .Joulc!Thornson proccss is carried out under isenthalpic conditions (i.e., d3-I = 0) we can rearrange Eq. (5.128) such that dT =

(g) 1-(

143~0

d q = -(1 - Tall) d q . CII

N , s , = const

(5.133)

which shows that widar thc: currciit coiiditioris T is solcly a function of 711. With the definitions = N / (As&) and cf = q / N we can differentiate the previous expression with respect to 711to obtain [see Eq. (S.l27)]

(5.134) Because all coefficients in Eq. (5.134) are psitivt: dcfiriite wc obtaiii as a thermodynamic expression for the inversion temperature (611= 0)

1

T", = QII

5.7.3.2

(5.135)

Consistency relation

For subsequent checks on the MC simulations, from which we seek to determine T,,,, it will prove coiivenient to derive a consistency relation that must hold rcgardlcss of molcciilar rlatails of tho sprrific. rnodcl iindcr c*onsidc:rat,ion. The derivatiori starts by assuming that an cquatiori of state 71(T,A ) (fixed N , s,) exists such that

(5.136) Focusing on thermodynamic transformations such that q = const (i.e., d q = 0) we can rearrange this expression to give

(5.137) where we employed the definitions of IZII and all in Eqs. (1.81) and (5.132), rcspoctivcly. Inserting the equation of state Eq.(5.136) into &. (5.129) and using also Eqs. (5.130) and (5.132), we obtain

The Joule-Thomson effect

264

wfierc Eqs. (1.81) arid (5.137) liavc also Lccii crnploycd. Dcfiriiiig the isostrairi heat capacity by aiialogy with Eq. (5.130) as

c,

G

T

(g)

(5.139)

N,.4,a,

we obtain from Eq. (5.138) t,he desired consistency relation

(5.140) whcrc c r

= c,/N

5.7.4

The limit of low densities

and ji is defined as above.

5.7.4.1 Virial expansion We now turn t o a microscopic t,rcatmcnt. of the .loiilc-Thomson cffcct and begin with the limit of vanishing density. Tho treatmerit, below is very similar to the one presented in Section 3.2.2 where we derived molecular expressions for the first few virial coefficients of the onedimensional hard-rod fluid. Here it is important to realize that a mechanical expression for the grand potential exists for a fluid confined to a slit,-pore with chemically structured substrate surfaces as we demonstrated in Section 1.6.1 [see Eq. (1.65)]. Combining this expression with the molecnlnr cxprcssion givan in Eq. (2.81) wc may writc

where we have expanded exp (-z) into a power series. Moreover, we write as a power series

-=

0 0 -

(5.142) in terms of the activity z = e x p ( p / k ~ T ) A -[see ~ ~ Eq. (2.120)]. In the previous expression we denote the configuration integral [see Eq. (2.112)] by 2 , to emphasize its implicit dependeiice on the number of molecules. Eqnittiori (5.141 ) siiggvsts .that, it, slioiild 1 ) possihlo ~ to c~xpaid711in a powc:r series in the activity i w well. Tliiis, we employ again Eq. (3.22),which we insert into Eq. (5.141) to obtain

1 - , 4 S d (b,z

+ b?) + V

2

b : z 2

+ a (z3)

(5.143)

Confined fluids with short-range potentials

265

where we rcplaccd iii the origirial Eq. (3.22), Tb by 711 a i d rctain terms oiily up to second order in z. Comparison with Eq. (5.142) irniiiediately gives (5.144a) (5.14411) Unfortunately, the original expansion of 71 in terms of the activity 3 is soinewhat awkward in practice. Instead we would prefer an expansion of 711in tcrms of thc nican density 7 of thc confinad fluid. This can hr ;tccomplishcd by first-notiiig from Eq. (5.142) that we riiay write

where we also used Eqs. (5.141) and (3.22). Expressing now the activity in terms of a power sarics in j j by the ansatz (cf. Eq. (3.26)J

z = alp

+ u g 2 + 0 (p3)

(5.146)

we obtain from Eq. (5.145) the expression [cf. Eq. (3.27)]

where we retain terms only up to second order in p2. Equating in this expression terms of equal power in iij on both sides of the equation, we can express the unknown coefficieuts and a2 in terms of the known constants bl and

bzs

a1 =

1 -bl

(5d48a) (5.148b)

Thus, replacing the expansion coefficients in Eq. (5.146) via Eqs. (5.148) and inserting the resulting expression into Eq. (3.22) we obtain [cf. Eq. (3.29)]

The Joule-Thomson effect

266

(5.150) is thc secoiid virial cocfficicut of the confincd fluid, which appcars to be solcly a function of temperature via Eqs. (5.158) and (5.163). Notice the similarity between &. (5.150) and its counterpart Eq. (3.30a) for the one-dimensional hard-rod fluid.

5.7.4.2

Inversion temperature

W e can now derive an expression for t,he inversion temperature that is valid in the limit of sufficiently low densitks. Therefore we differentiate [see Eq. (S.149)]

whcrc wc may rcplacc q / k ~ T via Eq. (5.149). Aftcr milltiplying both sirlcs of thc resulting cxpressiori by T , dividing t h a n by j5, arid rcmrraiiging terrris we arrive a t

(5.152)

(5.153) where the definition of the exparisivity in Eq. (5.132) has also l m n used. Using this expression we finally obtain

(5.154) Comparing this expression with the thermodynamic one for 611 defined in Eq. (5.134) it is clear t,hat the inversion temperat,ure can be obtained here from the zero of

Confined fluids with shortrange potentials

267

which liw a lucid gconictrical intcrprctation in that it defi~ics7',,, at that point a t which a tangent, to B2 (7') through the origin (term 11) touches that curve (term I). Finally, in closing this section we notice that the inversion temperat#ure defined by Eq. (5.155) is expected to depend on the presence and chemical nature of the solid substrate even in the limit of vanishing density a t least in principle (see Section 5.7.5). This is because Z1 and 22 depend on the fluid siibstmtc potential [scc Eqs. (5.158) and (5.161)],which is, in tiirn, cxpcctcd to affect. B2 (T) through Eq. (5.150). hloreover, we note that, because the above treatment, is valid only in the limit i j -+ 0 [and because B2 (2') # S ( i j ) ] ,the inversion temperature does not depend on thc mean density of the confined fluid.

5.7.4.3

Confined ideal quantum gas

The simplest system one might, cunsider in the context of the Joule-Thomson cffect is the idral gas. A s we showrd in Eq. (5.139) the equation of steakof t.hc idcal gas in tlic classic lzmzt is givcii by

$ = -ijkBT

(5.156)

Using this expression it is easv to verify from Eq. (5.153) that

(5.157) Hence, it follows from Eq. (5.134) that. 611 = 0 regardless of the thcrmu dynamic condit,ions considered, which is in accord with standard textbook knowledge. However, at. the molecular level, symmetry properties of the quantum mechanical wave functioii give rise to deviations from the classic behavior as we showed in Section 2.5. These deviations may be interpreted as a net repulsion (Fernii-Dirac gas) or attraction (Bose-Einstein gas) between the rnolecults. As w(: ciiiphwixtrtl in Swtion 2.5.3,qii;mtiiui cffccts arc iriaxirnixcd in semiclassic ideal gases. From this point, of view, it then seer~issensible to addre.ss the following quest,ions:

1. Does a JouleThonison effect exist in idea.1quantum gases? 2. What is the role of confinement t,o iianoscopic volumes'?

In this section we shall answer both questions by considering an ideal quantum gas (of Fermions or Bosons) confined to a slit-pore with chemically

268

The Joule-Thomson effect (2;sd)

homogcnmiis solid siirfaccs rcprcscntcd hy thc potcntinl in Eq. (5.126). For t,hc ideal quantum gas

dnfinnd

(5.158) where we shall use the subscript, to indicate the number of molecules in the system; that is, is1 is the single-particle configuration integrd. Note that for a biilk systmi Z1 = Asd t)cc.aiisa in this ('as(:CP,,Ikl ( z ; s,~) vanishcs by definition. For thc potential introduced iu Eq. (5.126), Eq. (5.158) can be rewrit,ten more explicitly as

(5.159) The evaluation of the remaining integrals then becomes trivial, a.nd we obtain 21 =

Am

{5

[(A

-

1) exp

(5)+ - A]

l}

(5.160)

as a closed expressiori for the single-particle coilfiguration integral. As cxpected, the bulk expression Z1 = Aszo is recovered from Eq. (5.160) in the limit sd + x. The semiclassic expression for the two-body configuratioual integral follows from h s . (2.110) - (2.112) as

where for a confined den2 (quantum) gas (5.162)

Confined fluids with short-range potentials

269

(zl;s ~ is) again given by Eq. (5.126). In Eq. (5.161) and below, the and upper symbol of the sliortliand notation, ‘3” always refers to a Boson gas, whereas the lower symbol refers to a gas of Fermions instead. Sote also, that in the classic limit, exp(-2nr12/A2) M 0 such that. Z: = 2 2 regardless of whether the ideal gas is confined by solid substrates. To evaluate the double integral in Eq. (5.161) it is advantageous to change variables ,according to T I , ~2 -+ T ~T , = T I - r2 and to employ cylindrical d:c dg dz = dot,J dg dp dz whcre tho datcrminant roordinatcs such that, d r : of the Jacobian for this transformation det J = p. Moreover, we realize from Eq. (5.150) that we need to compute the difference 2 2 - 2: to calculate the second virial coefficient of the confined quantum gas. Hence. by immediately carrying out the trivial integrations over 2 1 , 91, and 4 we obtain

-.9*0/2

x

r

.

L

‘‘D‘

”]-”[-m dtexp

-8.Op-Zl

l

2

1

k=l

J

2

k= I

@g ( z + zi: szo) (5.163)

by noting that the summand Z: can be expressed in terms of the first two integrals times the prefactor if we use the same set of coordinates. If we then pull out the factor exp (-27rz2/A2) from the last integral in Eq. (5.163), the remaining int.egal over p ran inimediately he solved. With thc aid of the trarisforrriatioii p 4 IL z p2 thc rcrriairiirig iritcgral bccornes

Next we consider

(5.165)

270

The Joule-Thomson effect

+

whcrc wc liavc used thc traiisformation .? ---t z‘ G z 21. Focusing on a physical sitiiation in which T and m are not too small we may assume A to be sufficiently small such that cxp [-2a (z’ - z1)2/A2]differs appreciably from 0 only if I t ’ - 21 I N 0, t,hat is, if particles 1 and 2 are very close to each other as far as their ;-coordinates are concerned. Notice that this approximation is consistent with our semiclassic treatment in Section 2.5.3. We may then approxiniat,c t.hc Gaiissian fiinction in t,hc prcvioiis cxprcssion l y t,hc Dirac &function lscc Ey. (B.75)] arid write

(5.16G) where wc replaced t.he intxgration l i m h ksa l y f o o on account. of the sharpness of the Dim: ii-fuiictioii and thc fact tliat, +swPI (2’;.sd) diverges to

infinity as Iz’J -, s,0/2 - c7 [see Eq. (5.126)]. Putting all this together we finally realize that

This turns out, to be very similar to the expression for 21 given in Eq. (5.159). Hence, we can immediately carry out thc remainiiig integration to obtain

z2-z1 =f-

23/2

{

[(A - 1)cxp

(3) - A] + I} k-nT

(5.168)

Inserting this expression together with Eq. (5.160) into Eq. (5.150) we yield6 6The expressioii in Q. (5.169) is correct except for a factor of 2 S + 1 due the total spin S of the ideal quantum gas, which we tiavc ignored from the very beginning for simplicity (see Section 2.5.1 and Ref. 19).

271

Confined fluids with short-range potentials

A3 A3 ( 2 a / a 4 [(A - 1)exp (22) - A] + 1 = 7-f 25/2{(2a/sa) [(A - I>exp (x)- A] + 1)' -

B2 = T-

(z; A, s,,o)

(5.169) for the second virial coefficient of a confined ideal gas of Bosons (-) and Ferinions (+), respectively, where 1/x = kBT/Efs is a dimeiisionless (i.e., "rcdiiccd))) tcmpcratiirc. For Eq. (5.169) to be physicdly mealiingful, the function f ( 5 ;A> s,,~) must not have any poles. Hence, the paramet,er A must be in a range such that the denominator of f (z; A, s,,") has no zeros. These zeros are obtained as a solution of the expression exp (z) =

A - s*/20 A-1

(5.170)

Obviously, for 5 2 0 taliisrelatioil is ineaningful oiily if A 2 1. Moreover, because the left side of the previous expression cannot become negative, the denominator of f (z; A, s,~) cannot have any zeros if X 5 sa/2a. Therefore, the range of physically meaningful values of A is bounded from above and below according to the inequality

l
2a

(5.171)

which is consistent with the definition of &! ( 2 ;s,") in Eq. (5.126). Recalling from Eq. (5.126) that. Efs determines the slrength of fluid s u b strate attraction, we first focus 011 the case sp = O (i.e., z = 0), that is a slit-pore with "hard," repulsive solid surfaces for which

I' (0; A, S&)

=

1 21 1 - 2AO/S,"

(5.172)

on account of the inequality stated in Eq. (5.171). In l3q. (5.172) the equality holds in the bulk, that is, in the limit s d + 00. In this latter case we obtain from Eq. (5.169) the well-known result [see footnote 6 above)

B2

A3

= 7-

25/2

(5.173)

for the second virial coefficient of a bulk gas composed of either Bosom (+) or Ferniions (-). In other words, the function f (z; A, s,,~) in Eq. (5.169) is a quantitative measure of confinement effect,s on the secoiid virial coefficient. Let us now turn to cases where we have attractive fluid substrate interactions in addition to repulsion; that is. wc now focus on cases where c'h > 0

The Joule-Thomson effect

272

(see Eq. (5.126)]. In this (-i~sc' 5 > 0 aid E(4. (5.171) irriplics tlie inequalitics

(A

-

l)cxp(2.r) > 0 2a 1-A> 0

(5.1742~) (5.174b)

Szo

so that

1(z;A,

S,[,)

>0

(5.175)

because the denomiliator uf f (x;A, sZ0) is always positive. To gain some more insight, into the effect of variations of temperature and/or fluid subst,rate attraction it is necessary to investigate the dependence of f (2; A, s,,") on x. This becomes possible by considering small variations of x aroiind somc rcfercncc vahic .xO by rxpanding f (L;A, sd) in a Taylor scries arourid this referencx value xo > 0. Because of the defiiiitioii of the variable z (see above) this may he considered either as an expamion in terms of Efs or, alternat.ively, 1/ T . More specifically, we write

=

f (so;A ?S f i ) + Sxf'

(zo; A, .a)

(5.176)

whore we have truncated the Taylor series after the linear term, which is always possible because the sins11 quantity bx G z-zo << 1 is at our disposal. From E<1. (5.1G9), it is somewhat trdions hut straightforward to vcrify t.hat,

5 0

(5.177)

which follows because exp ( 2 ~ ) exp (z) 2 0 for all x 2 0 and because the denominator is positive for all valiies of A satifying the inequality given in Eq. (5.171). Hcnrc, it. follows that

I ( t i + 1 ; A, Sd) 5 J

(La, A, Sd),

6.1; = .c,+1 - z,> 0

(5.178)

The intcrprctntion of this inequality is rliiitc st,raight,forward. First, bemuse of the definition of the variable T , 6z > 0 corresponds either ta an increase of fluid substrate attractivity (i.e., an increase of or, alternatively, to a decrease in temperature T. Second, because z cannot become negative by definition, f ( O : A . s d ) given in Eq. (5.172) is an upper limit for the confinement-induced shift of the second virial coefficient of confined ideal quantum gases relativr to its lmlk value. The change in f (5;A, s d )

Confined fluids with short-range potentials

273

can, however, be rca1izr.d in tliffcreiit ways. On account of the definition of 2 the inequality in Eq. (5.178)permits us to concliide t,hat ancwasing the attractivity of the substrate at any given fixed temperature T reduces the confinement-induced enhancenient of the (magnitude of) t8hesecond virial coefficient as predicted by Eq. (5.172). Likewise, for a given attractivity of the substrate [i.e., for fixed Efs, see Eq. (5.126)],reducing the temperature (ix., T,+1 < T,) also rcdiicas the shift, of B2 rclativv to its h l k valiic caiiscd by the presence of "hard" repulsive substrates. Bcc.susc of this analysis it is conceivable that the function f (x:A, s,") will generally satisfy

This implies that a temperature T exists at which El2 for the confined ideal quantum gas intersects B2 for its bulk counterpart. However, confinement by attractive (or repulsive) solid surfaces cannot change the sign of Rz. In other words, B2 for a confined gas of Bosorls will always be whereas it will always be positive for a gas of Feniiions. hiloreover, the monotonicity of f (0; A, sd) as wcll as that of A crziisc B2 t,o dccrcasa moiiot,onically toward zcro with incrcasiiig tcrnpcrat,urc. This ohscrvation is importaiit, hccausc it also pcrinits us to concliidc that it will iiot bc possible to const#ructa taligelit through the origiii a t any point of the curve B2 (3") no matter whether we consider a bulk or confined ideal quantum gas and irrespective of whether t,he quantum particles are Fermions or Bosons. In other words, for the ideal (bulk and confined) quantum gases, an inversion temperature Tnvdoes not exist because l?q. (5.155)does not have a solut.ion. However: the re,zder should note that a JouleThornson effect docs cxist, as pointcad out in Sact,ion 5.7.1,namcly a diliit,c gas of Bosons is always coolcd upon ail iscntlialpic cxpansioii (& ( T ) < O ) , wlicrcas a gas of Fermions is always heated during this process (B2( T )> 0). The ext.ent to which this happens is modified in a nontrivial way by confinement according to the above discussion. 5.7.4.4

Nonideal classic fluids

Dcspitr thc insight, gaincd by considcring the confined idcal quantum gas as a model system, the model itself is rather special in that it ignores fluid fluid interactions altogether. Hence, we now turn to nonideal, classic fluids in which the total configurational potential energy is given by Eq. (5.124)with the fluid substrate interaction as represented by models A and B according to the description in Section 5.7.2.In addition, we assume that for nonvanishing fluid fluid interactions the factor r / A >> 1 such that quantum effects can be

274

The Joule-Thomson effect

ignored. Uiidcr thcsc conditioils

and by similar reasoning as before [see Eq. (5.1G3)]

(5.181) similar to the expressions given in Eqs. (5.158) a.nd (5.163) above. However: because of the form of U R and @IL] ( z ; s , o ) for models A and B (see Section 5.7.2)these integals cannot be evaluated analytically, but they are amenable to a nunierical evaluation usiiig standard quadrature techniques. This finally parrnits a nnmclrical dciilation of thc second virial coefficient I?, from Eq. (5.150)on which the subsequent results for the inversion temperature in the limit of vanishingly small gas densities will be based.

5.7.5

Confined fluids at moderate densities

The above considerations are only valid in the limit of very small gas densities. However, in general the inversion temperature can be expected to depend on density as well. To incorporate the density dependence we have to go hcyond the second virial coefficient, in our axpansion of 7 1 1 in Eq. (5.149). Coilvidering larger densities of the confined gas, the virial expansion of T I I would need to involve many more terins if such a power series in i j at all converges. Hence! to calculate the inversion temperature a t higher densities up to the critical density of the confined gas, an alternative approach is required. It becomes possible by employing a mean-field description of the confined fluid, which we discussed in Section 4.2.2. Differentiating q / k g T givexi in Eq. (4.28)we obtniri [cf. Eq. (5.151)1

275

Conflned fluids with short-range potentials

Replacing on the left side of this expression q / k ~ T as before by employing again the equation of state in Eq. (4.28), it is a simple matt,er to show that (5.183) where wc also used Eq. (5.153) for tlic cxpansivity. Usiiig the mean-field expression for a11and the thermodynamic definition for the inversion temperature it, requires nothing but straightforward algebra to demonstrate that a t the mean-field level (5.184) which shows that the inversion temperatmuredepends on the density of the confined gas as anticipated. However. in the limit 7 4 0, the mean-field treatment must be consistent with the one developed in Section 5.7.4. From J3q. (5.184) we see that in this limit, h n kgTnv(7)= kBT,,,, (0)= 2aP (I) (5.185)

b

j3-0

This latter expression can be derived independently by expanding in the mean-field equation of state [see Eq. (4.28)] the term 1/ (1 - hp) (bp >> 1) in a MacLaiirin series according to 1

--

1 - bp

+ C (bij)k 00

-

1

k=l

( 5.186)

Inserting this expansion into the meaii-field equatiou of state mid considering only terms up to second order in density, one can show that (5.187) which may bc compared with Eq. (5.149) to ronclucle that the sccorid virial coefficient at. the mean-field level is given by (5.188) Inserting this expression into F4. (5.155), we find

The Joule-Thomson effect

276

froin which Q. (5.185) follows iinnicdiataly, tlicrcby proviiig the consistelicy bet,ween the current mean-fidd theoretical treatment and the virial expansion in the limit, of vanishing density. As wc alrcady dcinoristrzrtc.d that tlic inca~i-fieldtrcatirierit developed in Section 4.2.2 is capable of describing, for instance. capillary condensation in iianoscopic porous media in a qualitatively correct fashion (see Sectioii 4.2.4), the above discussion permits 11sto draw some important preliminary conclusions concerning the JouleThomson effect in confined fluids. These conclusions. bolstered further by corresponding M C data t o be presented below in Scctions 5.7.8 and 5.7.3, ran lw siiminarizcd its follows: 1. The inversion temperature decreases with increasing density. This follows frorii hs.(5.184) and (5.185) from which t8heinequality

is rcadily dodiiccd. Thc cqud sign holds in thc limit of vanishing dcnsity.

2. The inversion temperature of a coiifiried gas becomes lower the more

severely confined is the gas. This follows froin Eq. (5.184) and the ~ in Eq. (4.24) which turns out to become smaller the definition of L L (0 smallor is s, (ix., th: sriiallvr is <). This iinplics that a gas that niight be cooled during an isenthalpic expansion in a wider porous medium (611 < 0) may get heated in a narrower porous medium instead (611 > 0). At the mean-field level, the magnitude of the associated confinementiiiduced shift of the iiiwrsion temperature is given quantitatively by the terrii in brackets in Eq. (4.24).

3. The inversion temperature does not, depend on the chemical nature of the substrate because neither ap(<)nor b in Eqs. (5.184) or (5.185) depend 011 any parameter describing a specific substrate. Similar conclusions could .not be drawn on the basis of the much more involved expression for l?2 (7') in Fq. (5.150). 4. Existence of an inversion temperature is solely linked to attractive fluid fluid interactions modulated by confinement. In the absence of these attractions, up = 0 and therefore ByF(T) = b. If this expression is inserted into Eq. (5.155), oiie realizes that T,,, = 0 is the only possible solution.

(c)

277

Confined fluids with short-range potentials

5.7.6

Exact treatment of the Joule-Thomson coefficient

Evcii though the rncari-field trcatiricrit iii t,hc prcccdiiig scctioii lcd to sonic d e tailed insight, concerning the impact of coiifiiiemerit,oil the iriversioii temperature! the analysis in both Sections 5.7.4 a,nd 5.7.5 appears to be somewhat hampered in the sense that it was either limited to very low gas densities or that it was based on a meaii-field assumption explicitly stated in Eqs. (4.19) and (4.20). To test the predictions of the two previous approaches, we need to tackle the JouleThomson effect l y an approach that is free of any additional assumptions. In tliis regard tire MC techiiiquc is again ideally suited because MC simulations should be regarded its a firsbprincipleu method according to the opening discussion of t.his chapter in Section 5.1. As we showed in Eq. (5.135), a determination of the inversion temperahre Tn,essent.ially requires computation of the cxpansivity al~. Therefore, we begin by deriving an exact expression suitable for an evaliiat,ion in the siibsequent MC simulations.

5.7.6.1

Partition function

As we showed in Section 2.5.4, thermal averages in isostress isostrain ensembles can be related through a Laplace transformation. Hence, for the ronjiigatc strcxs 711and strain A , wc may cmploy Ecl. (2.121) and rhangr variables according t o rzz --, 7 1 1 aiid sZ --+ A giving

x / d r N O (rN: A ) cxp [-'I

tB;A)]

(5.191)

where we also replaced the grand canonical partition function Z,I by the partition function of the ca.rioiiical ensemble because we are concerned with systems accommodating a fixed number of fluid molecules. As, on the other hand,

( 0 (711))=

c/

d r N O ( r NA;) p ( r N : A )

(5.192)

A

must also hold, where p ( r NA;) is the probability density for a specific configuration r N of fluid molecules in a lamella exposed to a compressional

The Joule-Thomson effect

278

(dilational) st.rain proportioiial to A. a corriparisoil with Eq. (5.191) suggcsts that. p

( T N ; A)

1

= N!A3”T11.~1

[-

U(

In Eq. (5.193) the normalizatiou condition

/

1

- TIAS& kBT

T ~A;)

(5.193)

,-

d r N p ( v N ;A ) = 1

(5.194)

suggesting that. the partition fiinction is given by

where we tacitly assumed that. summation .on and integration over .4 are equivalent operations. In the thermodynamic limit we may apply the maximum term method (see Appendix B.4) to write

=

TIA*Rzo

c+F]

(T.N ,A‘, s,)

(5.196)

Taking the logarithm of this expression we have from Eqs. (2.79),(5.131), and the Legendre transform 3 U - TS that

is the relevant thermodynamic. potential where here and below we shall drop thc siihscript, “cl” of Yll as well as t h c mtmisk as a siiperscript of A to ease thc.iiotaliorial burden.

5.7.6.2

Thermal averages and their fluctuations

Differentiating Eq. (5.197) with respect to T, we obt,ain

279

Confined fluids with short-ranne potentials

whcrc thc I‘ar right side follows from J3q. (5.131). Thc second tcrrn 011 the right side of this expression may be further evaluated by realizing that

-

( H ( r N ; A ) ) + -3=N-k ~ H T 2 T

(5.199)

where we introduce for convenience

H (rN:,4)

G

U

(P; A ) - 7ll.4~~0

(S.200)

and with it. a statistical expression for the enthalpy ‘H that was obtained in &. (5.128) as a Legeridre tramform of the. internal energy

(5.201) From Eq. (5.198) and the therinodynamic definition of it is straightforward to realize that CII =

a

i3 In Tll

aT (knTZF)

= {.}\T

a

AT

CIJ in

3Nk~ (H ( r N ;A ) ) + 2

Eq. (5.130), (5.202)

To evaluate the part,ial derivative on the far right side of this expression we rcwrit,e it morc explicitly as

A

(5.203) Thus, putting together these last two expressions, we obtaixi

(5.204) as an exact statistical physical expression for the isostress heat capacity where H is defined in Eq. (5.200).

280 Ail

The Joule-Thomson effect

arialogous cxprcssioii may bc dcrivcd by wing Yll [see Eq. (5.196)] in

Eq. (5.202), which gives

(5.205) Using the expression for Q we can apply the above considerations to obtain an expression parallel to that given in Eq. (5.204) replacing, however, H ( r NA; ) by U ( r N);. This way (5.206) which we identify as the isostrain heat capacity defined thern~odynamically in Eq. (5.139). This would become immediately apparent if we replace in Eq. (5.197), 911 -, .F and Y;l Q [see Eq. (2.79)] and repeat the above analysis stop by step. Besides t,he heat capacities CJI and c,, the isothermal compressibility lcll --f

From these thermodynamic definitions and the statistical expression for the generalized Gibbs potential in Eq. (5.197). it is straightforward t o show that

-

-'.Lo [ ( A 2 )- ( A ) 2 ] ~ B(T A)

(5.208)

which differs i11 ail idmcstirig way froill thc parallel cxprcssion that would be obtained in the grand canonical ensemble from Eqs. (1.81) and (2.75) where the transverse compressibility is expressed in terms of fluctuations in the number of molecules rather than the shape of the lamella. However, the key quantity in the context of the JoiileThomson effect is the expansivity defined in Eq. (5.132). From the thermodynamic definition and t,hc statkticni physical Eqs. (5.197) and (5.198), it is immcdiatdy clear that

281

Confined fluids with short-range potentials where wc also used Eq. (5.139). It call then be shown that

a In Yll -kB(,,)

I.l\q

1 1 = T N!A3NY~~ szo -s?;(A)

-

JdrGexp[N

H ( r NA; kgT

'1

(5.210)

which follows directly from the definition of H ( r N ;A) in Eq. (5.200) and the definition of the probability density p ( r N.4) ; in Fx. (5.193). For the second term in Eq. (5.209), we obtain by straightforward differentiation

xCJd"qexP [- I] 3H

A

H ( r NA; kBT

Combining these two expressions with the thermodynamic definition of all, we see that (5.212) obtains without further ado. In Table 5.5 we summarize the results for the various response coefficients (ZIJ , K I J cq, and c,.

5.7.7

Isostress isostrain ensemble MC simulations

According to the above discussiori the distribution of microstates in the current, m i x d isastras isostrnin cnscmblc is governcd by the probability density given in Eq. (5.193). The similarity between the present probability densit.y and the one relevant in the closely related ensemble discussed in Section 5.2.4 suggests we should design an adapted Metropolis algorithm closely related to the one described in that section. In fact, from the detailed discussion in Section 5.2.4, it turns out that we just need to replace the substrate separation s, by the area A. More specifically, we need to replace Eqs. (5.46)-(5.48)

The Joule-Thomson effect

282

Table 5.5: Overview of fluctuation-related response coefficients in the mixed isostress iswtrain ensemble at constant T . N , q,and s d (see text for details of the derivations).

Coefficient CII f:,

Kill

QII

Expression See Equation 3 N k ~ / 2 [(H2) - (H)Z]/ k . ~ i ' ' ~ (5.204) 3 N k ~ / 2+ [(V) - (Cr)2] /X:BT~ (5.206) S& [(142) - (.4)2] / ((14) ~ B T ) (5.208) [(HA) - (H) / ( ( A )kBT2) (5.2 12)

+

(5.213a)

(5.2 13b) (5.2 13c) where, by analogy with Eqs. (5.49), (5.2 14a) (5.214b) Again we associate the random process ~ " - 1 --+ gn (see Appendix E.l) with probability (5.215) II3 3 iniri [ 1, cxp (7.A))

<-

where we realize a change in A according to exp ( r A ) > exp ( T A ) 5

E

accept change in area reject change in area

(5.216a) (5.216b)

according to a modified Metropolis criterion if T A < 0. where [ is a pseudorandom numbcr zw hcforc:; if, on thcl othcr hand, T A 0 thc prorcss An-l A,, is accepted without. further ado. As before in Section 5.2.4 generation of a sequence of configurations p r e ceeds in two steps. Step 1 is again identical with step 1 of the corresponding adapted Metropolis algorithm for GCEMC siinulatioils (see Section 5.2.2): a random displacement of a single molecule governed by a transition probability Ill [see Eq. (5.11)]. In step 2 the area A = s,ss is changed according

>

--f

283

Confined fluids with short-range potentials

to

such that [cf. Eq. (5.53)]

) ( ($) ( -:z

F: =

=

?I-

1

1

.

.~x,n/+i,n-l

Sy.n/y.n-l

)

(5.218)

Because in step 2 of the present algorithm the entire set of N 2- and ycoordinates a.re changed at once, steps 1 and 2 are carried out wit.h a frequency N : 1. To demonstrate the validity of t,hese operations in generating properly a numerical representatioii of a Markov chain of configurations, the t8hermodyriamic consistency relation in Eq. (5.140) turns out to be particularly useful. It can be employed to calculate c,, froni the various response coefficients list,ed in Table 5.5 that we cahilatc as ensemble averages in the ciirrcnt isost,ress isostxairi eiiserrible simulations. We also riote from l3q. (5.206) that c, might, as well be calculated in corresponding MC simulations in the canonical ensemble by fixing the dimensions of the simulation cell to (s,) and (sy) dettermined as ensemble averages in the isostress isostrain ensemble simulations. Entries in Table 5.6 show that the agreemcnt between Eqs. (5.140) and (5.206) is always better than 5%, which seeins remarkable in view of the rclativc!ly witlc rang(: of tlic!riiicitlyri~n~ic~ stiLt(!s cwisitlcrcd.

5.7.8

Inversion temperature at low density

Attending now to T,, as the key quantity of this study, it seems sensible to begin discussing the limit of vanishing fluid density. In this limit T,, (0) is obtained as a (numerical) solution of Eq. (5.155), where B2 (T) is obtained from Eqs. (5.158), (5.161), ,wcl (5.150) by numerical integration. Resiilts for 112 ( T )are plotted in Figs. 5.27 for various cascs studied. Generally speaking, over the temperature range plotted, B2 (7') is a monotonically increasing function of T, where &(T) < 0 below the Boyle temperature T ! l eand &(T) > 0 otherwise; a t T = T B ~&(T) ~ ~ =~ 0, and the fluid behaves like an ideal gas [see Eq. (5.149)] (disregarding, of course, higher-order virial coefficients). Confinenient causes B2 (T) to be shifted with respect to the bulk curve. If the substrate potential is wettable [model A, sce Fig. 5.27(a)], B2 ( T ) is

The Joule-Thomson effect

284

I 1

0

m" -1

-2 2

4

6

T 1

0.5

m"

0

-0.5

-1

2

3

4

5

6

T Figure 5.27: Second virial coefficient Bz (T) as a function of temperature T for bulk (-), model A (- . -), and model B (0) ( s d = 10 for confined fluids). Curves are obtained by numerical integration (see text). Intersections between Bz (T) and the solid horizontal line defiac the Boyls temperattmr. (b) As (a) but for bulk (-) and model A for R,O = 10 (- - -), s d = 20 (O),arid sZo = 50 ( 0 ) . Inset shows & ( T ) for bulk ( 0 ) and the tangent through the origin (- - -) defining the inversion temperature demarcated by the vert.ica1 arrow [see Eq. (5.155)).

285

Confined fluids with short-range potentials

Table 5.6: Compctl.is0n of results for isostrain heat capacity from consistency relation Eq. (5.140) with directly computed values from canonical ensemble (CE) [seeeq. (5.206)]for model A (see text).

T

1.50 1.50 1.50 1.50

1.SO

1.50 2.00 2.00 3.00 3.00

P

0.577 0.514 0.461 0.378 0.203 0.287 0.532 0.478 0.409 0.267

-11

1.50 1.00 0.75 0.50 0.35

0.25 2.00 1.50 2.00 1.00

all

0.384 0.532 0.676 1.028 1.256 1.305 0.292 0.375 0.233 0.336

KII

0.173 0.327 0.522 1.224 2.581 4.526 0.173 0.282 0.255 0.748

c; 4.243 4.488 4.765 5.345 5.279 4.970 3.784 3.977 3.320 3.369

c," c," (CE) 2.027 2.040 1.962 1.943 1.917 1.933 1.920 1.921 2.085 2.016 2.190 2.111 1.931 1.933 1.886 1.861 1.758 1.763 1.660 1.684

shifted to more positive vdues irrespective of T. If the substrate is nonwettable [model B, scc Fig. 5.27(a)), 112 ( T ) trirns out to bc smaller than for niodel A for T 5 4; it is even lower t,han t.hc second virial coefficient in the bulk for T 5 2. For high temperatures, however, t,he plots in Fig:5.27(a) show that, for niodel B, I32 (T) exceeds all other curves (T 2 6). The different temperature dependence of I32 (7') between various models illustrates the impact. of wettability of the substrate on thermophysical quantities of confined fluids in the h i t , of low densitim. If, on the other hand: the degee of confinement decreases (i.e., with increasing sd), B2 (T) for a confined fluid is expected to approach its bulk counterpart because of the diminishing influence of fluid substrate interactions. This notion is confirmed 13s. the plots in Fig. 5.27(b). However, it swiiis worthwhile t:rnphasizing thiit, CVCII for thc largest substrate separation studied ( s a = 50), I32 (T) for co~ifinetland bulk fluids differ slightly but significantly even though the range of the fluid substrat,e interaction potential does not exceed a distance of a few molecular diameters from either substrate so that, t,he dominant portion of the confined phase is not, subjected to interactions with that substrate. The remarkably large range of distances over which substrateinduccd effects prevail was also rioted with respect to adsorpt#ioriplienoniena in the subcritical regiiric [218]. For the curves plotted in Figs. 5.27(a), T,,, (0) is calculated from differential equation Eq. (5.155). Result,s are compiled in Table 5.7 for models A and B and sz0 = 10. They show that Tinv(0) is higher for a hydrophobic

The Joule-Thomson effect

286

Table 5.7: Inversion t.emperature in the vanishing-density limit [see Eq. (5.155), Fig. 5.27(b)]. Model

B A

A A A A Bulk

szo T n v ( 0 )

10.0 5.0 10.0 20.0 50.0 100.0 00

4.985 4.464 4.841 5.051 5.177 5.217 5.259

TBoyle

2.681 2.300 2.521 2.660 2.747 2.776 2.805

Tbyle/Tnv(O)

1.860 1.941 1.920 1.899 1.885 1.879 1.875

substrate (i.e., if fhdd substrate intmactions are purely repulsive). Table 5.7 also indicates that. in the limit, s , ~ -+ 00 the bulk inversion temperature is approached in accord with the plots in Fig. 5.27(b). From the tnean-field expressions Eqs. (4.24) and (5.185) one expects the difference

(5.219) The plot. in Fig. 5.28 shows that data compiled in Table 5.7 are consistent, with this scaling relation except for .$& = 5.0 where the assumption of homogeneity of the confined fluid, on which the mean-field theory is based (see Section 4.2.2),can hardly be expected to be valid. The results in this section therefore confirm our expectation that the inversion temperature should dcpcnd on tho substrate separation and that, it, lmcomcs higher the more severely confinrd is the fluid (i.c7tlie siiiallcr s , ~becomes). However, it seems worthwhile stressing Lhat the relateion bet,ween Boyle and inversion temperatures is only approximately described by the meanfield t,heory. For example, the mean-field Eqs. (5.185) and (5.188) predict T ~ q y l ~ /(0) T , ,=~2 irrespective of szo, but entries in Table 5.7 show that this ratio is lower and depends on s,O as well as on the chemical nature of the siibst,rat,o. This clcarly indicatm that. tha mcan-fcld trcatmant dcvclopcd in Sections 4.2.2 and 5.7.5 is not fully adequate as one would have expected. However, the deviation from t,he limit,ing value T B ~ ~ I(0)~ =/ T 2 does ~ ~ not exceed 6.5% for s,O = 100, where the rnean-field treatment is expected to work best. This. on t,he other hand, shows that mean-field theory is quite useful to understand the behavior of confined fluids at least froin a qualitative point of view.

287

Confined fluids with short-range Dotentials

0

0.05

0.1

-1

0.15

0.2

0.25

=2

Figure 5.28: ATnv(0;sd) as a function of inverse substrate separation l/sd in the limit of vanishing fluid densit,y [see Eq. (5.219)). According to mean-field theory, data points should fall on a straight line through the origin (seetext). The straight solid line is a fit to the data points using only entries for l/sd = 0,O.Ol.

5.7.9

Density dependence of the inversion temperature

In accord with the niean-field theory developed in Section 5.7.5.the inversion temperature, however, does depend on the density of the fluid. This can be seen from plots of Tall - 1 in Fig. 5.29 based on isostress isostraiii ensemble simulations. Regardless of T, Tcrll/ks - 1 turns out to be a nonmonotonic function of density. I t has a maxinium that increases and shifts to lower dcnsitics with ticcroasiiig tcmpcratim:. In thc limit 7 -+ 0, oiic' cxpccts all curves to approach zero according to [see Eq. (5.155)]

(5.220)

288

The JouleThomson effect

whcrc tlic sccoiid lint! follows with tlic liclp of tlic mcari-fic?ld Eq. (5.188) aft.cr ] ( [@Byf( T ) << 1). The last line of Eq. (5.220) expanding [ 1 + 27B;'' ( T ) -1 is obtained by coilsidering only t,hc leading term of the expansion where Eq. (5.155) has also heen used. Thus, in the limit 7 + 0, the curves in Fig. 5.29 become straight lines wliose slope is determined by the hard core of thc fluid molccnlcs and thcir invcrsion tcmpcrat,nrc in t,hc limit, of vanishing density. For the cast!! piotked T,, (0) / T > 1 (see Table 5.7) so that the slope of Tail - 1 should be positive for low densities, which is confirmed by the plots in Fig. 5.29. One also expect,s from Eq. (5.220) the slope of Tall - 1 to be snialler for higher temperatures, which is also confirmed by Fig. 5.29.

I

1

I

I

I

0.8

0.6 4

0.4

I 0 -0.2 -0.4

0

0.1

0.2

0.3

0.4

0.5

0.6

Figure 5.29: Plots of T n ~ l -1 u functions of avcrage fluid deusity Ti = N/( A )s a for I' = 1.50 ( 0 ) .I' = 2.00 (0),I' = 2.50 (m), T = 3.00 (0),and T = 3.50 ( A ) from MC siniulutions in the isost,ress isostrdn ensemble for rnodel A (s2 = 10). Solid lines are fits of Eq. (5.183) to simulation data. Intersections with dashed horizontal line define inversion temperature z,, [see Eq. (5.135)].

Solid lines in Fig. 5.29 represent fits of the mean-field Eq. (5.183) to the simulation data taking a,,, (<) arid b as fitting parameters. Although these fits represent the simulation data remarkably well? both parameters turn out to

Confined fluids with short-range potentials

289

bc tcriipcraturc dcpcndcnt unlike thc mcan-field trcatmcrit in Section 4.2.2 suggests. Therefore it. is not possible to extract any information about the location of the critical point of the confined (or bulk) fluid from the relations between critical temperature and density on the one hand and q, (t)and 6 on the other hand. However: in view of the rather coinplex variation of a11 with temperatiuc and density, t,hc mean-field approach is still very iiscfiil hcraiisc it perriiits an estirriatc of the invcrsiori tcnipcraturc frorii a11 analytic cxprcssion [see Eq. (5.184)] a t moderate computational expense. The computed inversion temperatures are plotted in Figs. 5.30 for various situat,ions. In accord with the plots in Fig. 5.29, the inversion temperature depends strongly on the density. For example, plots in Fig. 5.30(a) show that, over a density < 0.5, T,,, changes by about a factor of 3. Over wide range of 0.1 < ranges of tcmpcratiirc and dcnsity t,hc data arc: agilili wcll rcprwcntd by thc mcaii-fioltl oxprc:ssioii in &. (5.184). The curves in Fig. 5.30(a) for models A and B appear to be shifted downward compared with the bulk because of confinement. However, a more subtle phenonienon can be seen by comparing the plots referring to wettable and noiiwettable substrates. Here one notices that for high densities the inversion temperature is generally lower for the hydrophobic substrate (model R) coniparcd with the hydrophilic onc (modcl A). Howcvcr, in thc limit, of vanishing density, the invcrsioii tcrnpcraturc is highcr for the iionwcttahle compared with the wettable sukstrate (see Table 5.7). Therefore, the curves q,, (7) for models A and B must, iiit.ersected a t a sufficiently low density. From the fit of Eq. (5.184) to t,he simulation data, the intersection is located at j j N 0.08 and T 21 4.2. This is roughly also the temperature at which BZ(2') for the two models intersect [see Fig. 5.28(a)]. If the siit)st,ratc separation incrcwcs one cxpcct,s the inversion tcmpcraturc of tlic: coiifincd fluid to eventually coincide with the bulk inversion temperature irrespechw of the density. This notion is supported by plots of T q - 1 for sd = 10, 20, and bulk in Fig. 5.31(a). The maximum of this curve and its 1oca.tion are shifted toward the b d k curve with increasing substrate separation and so does Ti,,. Similar plots are obtained for other teniperatures, thus permitting one to construct the plot in Fig. 5.30(b) parallel to the one in Fig. 5.30(;1). As before for (0) onc cxpccts AT,, (F,sd) = Tn,p (PI00) - T,, ( p ,s,o) a s;' from t,he plot in Fig. 5.30(b) and the nieanfield expressions in Eqs. (4.24) and (5.185). For sd = 20, for instance, a depression of q,,,of about 4% compared with the bulk value is deduced from Fig. 5.30(b). Thus, even if fluids are confined to spaces of mesoscopic dimension, the confinement-induced depression of the inversion temperature should in principle be accessible experimentally given the accuracy with which the

z,,

290

The Joule-Thomson effect

0

0.1

0.2

0.3

0.4

0.5

0.6

0

0.1

0.2

0.3

0.4

0.5

0.6

5

4

2

&-

3

2

1

P Figure 5.30: (a) Density dependence of inversion temperature from MIEMC simulations: ( 0 ) bulk, (0) model A; and (0) model B (sd = 10). Solid lines represent fits of Eq. (5.184) to simulation data. Data points for ij = 0 are obtained from Eq. (5.155)and were not, included in the fit. (b) As (a) but for bulk ( 0 )and model A [sa= 10 (M), sd = 20 (O)].

Lattice Monte Carlo simulations

29 1

1.5

1

'

rl

I-

0.5

tl

0

-0.5

0

0.1

0.2

0.3

0.4

0.6

0.5

0.7

P Figure 5.31: As Fig. 5.29 but for model A, T = 1.50, and (O), and bulk (D).

8a) =

10 ( O ) , . 9 d = 20

phase behavior of confined fluids can nowadays be determined [31].

5.8 5.8.1

Lattice Monte Carlo simulations Advantages and disadvantages of lattice models

The discussioii of the Joule-Thomson effect in the previous sect8ionclearly showed that it is advantageous in theoretical treatmeiits of confined fluids to tackle a given physical problem by a combination of different methods. This was illiistratcd in Section 5.7 whcrc we cmqhycd a virial expansion of the equation of state, a van der Waals type of equation of state, and M C simulations in the specialized mixed isostress isostrairi ensemble to investigate various aspects of the impact of confinement on the JoulcThomson effect. The mean-field approach was particularly useful because it could predict certain trends on the basis of analytic equations. However, the mean-field treatment developed in Sections 4.2.2 and 5.7.5 is hampered by the assump

292

Lattice Monte Carlo simulations

tion of Iioinogcncit,y of t,hc coiilincd fluid [scc Eq. (4.20) (which is not really justified as the discussion in Section 5.3.4 shows). The assumption of homogeneity can be abandoned if the continuous rnean-field t.reatment. is replaced by a discrete treatment where the positions of fluid molecules are restricted to nodes on a lattice. The discussion in Section 5.4.2and 5.6.5 showed that the mean-field lattice. density fiinc.t,ional khcory dcvclopcd in Scction 4.3 was criicial in iinravcling t h complex pliasr behavior of fluids confi~icdby chciiiicadly dccoratcd substrat>csurfaces. A similar deep understanding of the phase behavior would not have been possible on the basis of siniulat.ion results alone. Nevertheless? the relation between these MC data and the lattice density funct*ional results remained oidy qualitative on account of the continuous models employed in the computer simulations. Thus, we aim at a more quantitative comparison bct,wccn MC simiilat,ions arid nl(!iLn-ficl(l lat tica rlonsity fiiiictioiial thcorv in the closing section of ttliis cliaptcr. This Iieiiig the primary goal of the subsequent discussion we would also like to emphasize two other! perhaps more practical: aspects. On a,ccount of the rigidity of the underlying lattice it, seems inconceivable to develop mixed isostress isostrain ensembles suitable for lattice MC simula,tions. On the other hand. lattice simulations are computationally much less demanding because molcculcs can occiipv only discrctc posi tioiis in spscc. Hciice, the riumbcr of configuratioiis possiblc oii a lalt ice is grcatly reduced coiriparcd with simulations of continuous niodel systems. Another aspect of lattice iiiodels coiiceriw the determination of phase behavior. As far as continuous models were coiiceriied we emphasized already that an iiivestigation of phase transitioiis in such models usually requires a “mechanical” representation of the relevant thermodynamic potential in tcrms of one or mom clcmcnt,s of t,hc microscopic stress t,cnsor. Thc existciire of such a tiiccliariical rcprcseritatioii was lirikcd inevitably t o syinmctry considerations in Section l . G , whcrc i l was also pointed out that such a mechanical expression may not exist at all. In this case a determination of the thermodynamic potential requires thermodynamic integration along some suitable path in thermodynamic state space, which may turn out to be computationally demanding. For lat,tice models mcchanicnl cxprassions for thcrrnotlynamic potcntials are out of the question regardless of whether the lattice fluid possesses a sufficiently high symmetry in the sense of our discussion in Section 1.6. The reason is again the incompressihi1it.y of the lat,tice so that one always has to resort to thertnodyiiamic integration techniques in MC sirnulations of lattice fluids. A clear advantage of a lattice model, on the other hand. lies in the fact,

Confined fluids with short-range potentials

293

that one may exparid the partition fuiictioii in thc lirnit,T --+ 00 as we s1iowc:d in Section 4.3.2 to obtain a closed analytic expression for thermodynamic state functions, which may in turn be used as a suitable starting point in a thermodynamic integration procedure with moderate coniputational effort. In the limit T + 0, on the other hand, the mean-field treatment becomes exact and we already showed in Section 4.5 that closed analytic expressions for the grand potential may b a dcrivctl [sea Eki. (4.94)] so that a sccond starting poiiit for a thcrmocl-pimiic iiitcgratioii scheinc exists. These cxyrcssioiis have no counterpart- as far as continuous models are concerned.

5.8.2

Grand canonical ensemble Monte Carlo simulations

For pedagogic reasons it. seenis sensible to consider a fluid confined to a slit-pore with chemically het.erogeneous substrates to make contact with the parallel mean-field calculations described in Section 4.3. As in that section we employ a simple cubic lattice of N sites. In accord with our previous notation, L? represents a configuration of fluid molecules where the (doublevalued, discrete) element,s of the N-dimensional vertor & are represented by Eq. (4.51). Moleciiles of thc (pure) lattice fluid intcract with cach other via a squarc-wcll potontial wlicrc thc width of the attractive well is equal to the lattice constant t . The fluid substrate interaction is modelled according to Eqs. (4.48). To minimize finitmesize effects, periodic boundary conditions are applied a t the planes z = 1, n, and y = 1, n,,,such that a molecule located at the plane rr = 1 interacts with its nearest neighbors on lattice site,s characterized by u = n, and vice vcrsa where u = 2.y. We treat, the latticc fluid as an open tharmodynamic system represented microscopically by the grand canonical ensemble. In this ensemble the probability densit,y for the occupation of a given site 7: on the lattice is given by (5.221) where Z is thc partition fimction of the grand canonical ensemble introdiiccd in Eq. (4.54a). It seems worthwhile to emphasize the difference between this expression and the corresponding one for continuous model systems given in Eq. (5.6). A comparison reveals that in Eq. (5.221) the factor 1/N! is absent because fluid molecules on the lattice are distinguishable on account of the specific site to which they are restricted. Second, the factor of l/A3N in Eq. (5.6) is also missing. This is because molecules on the lattice have

294

Lattice Monte Car10 simulations

rio kiiietic aicrby, wliich is cvidcnt from thc! Harriiltoriiari function dcfiricd in Eq. (4.51). Moreover, note that Eq. (5.221) depends on the single-particle Hamilte nian function unlike its counterpart in Eq. (5.6), which is governed by the configurational energy U (FN;N ). Depending on the range of the interniolecular interaction potentials on which I/ (FN;N) depends, many intermolecular intcractioris havc to bc considcrcd. For t,hc ciirrcnt simplc cuhic l a t t i x , on the other haiid, only six nearest neighbors contrihutc to the fluid fluid part of h. because of the short-range squarewell potentials governing the interinolecular interactions. Hence! the coinputational effort in generating new configurations on the lattice is inarginal compared wi t,h that, required by off-lattice simulations. This is true even if similarly short-range potentials would he employed in off-lattice simulations because inolecules can move contiiiiiorisly in space. Thrcforc, niolcculcs arc! rcstrictd to lat,ticc sitcs, a i d consequently, step 1 of the adapted Metropolis algorithm in the grand canonica.1 ensemble for continuous model systems (i.e., random displacement of molecules) can be abandoiied altogether (see Section 5.2.2). 1nstea.d the adapted Metropolis algorithm for lattice fluids proceeds as follows. The n/ sites of the lattice are visited consecutively. If a specific site is occupied (i.e., si = l ) , an attempt is madc! t,o change the valiic! of the occupation nwiibcr to s, = 0 (empty site) . Tlic associatxd clia~igciii thc sin&-particle Hamiltoriiaii function is calculated from (5.222) Ah. G h ( ~ i , -~ h) (s~,,+I) where the single-particle Hamiltonian fiinction is defined in Eqs. (4.51) and (4.54b). Employing the importance sarnpliiig concept (see Section 5.2.l), we accept. a local change in the occupation number with a probability

-

- -

II = rniri

-

[1, exp (-Ai/k.T)]

(5.223)

hased on the principle of clet,ailcd balaiice as we discussed in detail in Section 5.2.2. As before, xi attempt to switch the occupation number between its two values at, lattice site i is not immediately rejected if the associated A'iE > 0. In this case we draw a random number distributed uniformly on the interval [O?I] and apply the adapted Metropolis criterion, exp (-AK/kBT)

>

exp (-A'iE/k,T)

5

<{

<

accept change reject change

Si,n

(5.224a)

-, si,,,

(5.224h)

siqn-1

si,,,-l

The sequence of N attempts t.o chaiige the occupation numbers constitutes a MC cycle.

Confined fluids with short-range potentials

5.8.3

295

Thermodynamic integration for lattice fluids

Based on the above numerical procedure, we shall eventually generate a distribution of microstates (i.e., configuratioils ?t on the lattice) corresponding to a niiniinum of the grand potential R defined in Eq. (4.55) (or its associated density w z R/N). For thc siihscqiient disriission of phase behavior of thc latticx fluid, thc absolutc valuc of w would bc rcquircd according to the discussion in Section 1.7. However, as we pointed out in Section 5.8.1, such a calculation is not straightforward because we cannot directly evaluate the sum over configurations in Eq. (4.55) (i.e.; the grand canonical partition function) nor does a mechanical expression for w exist on account of the rigidity of the underlying lattice. Hence, we must resort to thermodynamic integration following idem originally proposed by nindcr 12191. Wc bcgin by rcaliziiig that Iscc- Eq. (1.59)]

dR

=

-SdT - Ndp

(5.225)

because the lattice is rigid and therefore V0Tr (Tda) = 0. Thus, for p = dS = (as/&''),,d T so that we niay write

CO11St:

-

I(%) 7

T

T

s(T.~)=s(o,~)+/(') =O

0

dT'

d ~ ' = P

(5.226)

P

0

whcrc! S (0, p ) vanishes arcording to the third law of thormodynamics. The far right. side of the previous expression obtains because [219]

(g>,= f (S),

where

(5.227)

UP = u - /LAT

(5.228)

Inscrtirig Eqs. (5.226) and (5.227) int,o Eq. (5.225) permits

I L ~to

calciilate

(5.229) An alternative expression is obtained if we integrate dS from T' = down to the desired temperature T' = T ; that, is T

1/

l

00

296

Lattice Monte Car10 simulations

which follows aftcr partial iiit.egratioii arid l y changitig variables according l/T’. Hence, inserting Eq. (5.230)into Eq. (5.225),we obtain to T’ ---$

(5.231) where for our lattice model S (00, p ) = AfkB ln 2 because at infinite teinperature intermolecular interactions heconie irrelevant and each lattice site can cithcr hc occupicd or ampty [219J. Last but not lwst, wc 1c:alim: that 1)c:caiisc: of [so(: Eq. (5.225)]

N --

-

- -P

(5.232)

a third route for thermodynamic integration exists. Integrating Eq. (5.232) we may write iJ

(T.112) = w (T.p1) -

1

IJ 1

i j (T,p ) d p

(5.233)

and calculate the grand-potential density w (T,p2) provided we know its value for T and p1 and know the equation of state, that is, i j (T,p ) . Equations (5.229)-(5.233)provide the basis for the thermodynamic integration scheme ernployed in Section 5.8.4. However, it is worthwhile noting at this point that key quantities such as i j and U in Eqs. (5.229)-(5.233) can he calciilated readily as thermal averages of s, and 3-1 (&) , respectively. Thtsci grand c:alioiiic.al onsc:ml)lc~avoragc’s will I)(% c*al(*iilatt.ctl via GCEMC simulations described in Section 5.8.2.

5.8.4

Comparison with mean-field density functional theory

To apply the thermodynamic integration procedure introduced above in Section 5.8.3,we need to know roughly where to expect coexistence lines because one miist, not intagrate across t,hcsc liiics wharc qiiant,itirs like 2.4 and j5 change discontinuously. The mean-field results presented in Section 4.5 are taken as a guideline in this sense. In most cases we base our integration on isothermal paths using Eq. (5.233)starting at high (path I) or low values of p1 (path 11). We realize that if p1 is sufficiently large! we can expect, the lattice fluid to be in its liquid stab with all lattice sites completely filled; that is, { p l } = 1 regardless of T. In this case, R (pl, T) is given by R’ ( p ) at T = 0

Confined fluids with short-range potentials

297

(see Table 4.1). If, oii tho otlicr hand, I1.l is sufficiently sriia11: wc will always end up in the thermodynaniically st,able gas phase. Hence, in this case we approximate R ( p l ,T) by RR ( p ) = 0 a.t T = 0 because in the gas phase all lattice sites arc empty. We now havc analytic expressions for R (PI.T), we can calculate i j = (N) /n/from GCEMC: and we can perform the integration in Eq. (5.233) numericallv. We carry out this procedure until we reach a discont.iniioiis change in 7 (ix., (N)), indicating that t,hc latti(:c fluid tmomcu; unstable. Typical data are prcwmled in Fig. 5.32. From Fig. 5.32(b) one notices that, over a cerlain range of temperat'ures, i j turns out to be a double-valued functioii of p; that is, one may observe hysteresis. Hysteresis refers to the fact that somewhere in the t,emperature range where high(er)- or low(er)-density branches of i5 overlap one of the two phases will be metastable, the other one being thermodynamically (i.e.! globally) sttable. On tho basis of t,hc awociatcd w (p.,T) in Fig. 5.32(a), we can rcatlily ciiscriniiriatc: 1)ctwc:cu thc two, thcrcby idcntifying tlio thcrmodynamically stable phase a t this given temperatmure.Moreover, one may locate pap.defined in Eq. (1.76~1)[see Fig. 5.32(a)]. Repeating this procedure for all teinperatures 0 5 T 5 T:fl, we may eventuaJly construct the coexistence line bet.ween phases a and 0. Some attention has t o be paid to the vicinity of the critical points. Not only are density fluct,uations incrcnsing enormously as oiic approaches T,"", but the dcrisitics of the two brandies of thc adsorption isothcrm bcco111c more and more alike, and therefore, it is much more difficult to distinguish between coexisting phases. If one were to locate the critical points with high precision, one would eventually have to employ more sophisticated simulation techniques [220, 2211 as, for example, Wolff's algorithm, which applies in particular to the current (Ising-type of) system [118, 1191. However, as we arc not. intcrcstd in locating critical points wit,h high precision, it, is sufficient to dcfinc the critical poiiit as that tcrripcraturc at,which plots of i j vcrsus p do not exhibit any hysteresis. For the bridge phase, where the one-phase region is triangular in shape bounded by, say, the coexistence lines pgb(T) and p"' (T)(see Fig. 4.10, for example), it turns out to be useful to first. integrate U according to Eq. (5.231) to some temperature T 5 niin (T,"',T;') and then complete the integration cmploying &. (5.233) to higher and lower vdiics of p isothmmnlly until one hits the chemical potentials a t which i j changes to a higher and lower value, respectively. Based on these considerations we arc eventually in a position to construct p x ( T ) for the case previously investigated on the basis of the mean-field approach. Comparing 1 1 ~(T) from the mean-field calculations with GCEMC data in Figs. 4.10: we see that the ovrrall topology of the phase diagrams

298

Lattice Monte Carlo simulations

Figure 5.32: (a) Plots of grand-potential density w as function of 11, for T = 0.73,nx = 20, n, = 10, ns = n, = 10, Efw. = 0, and Efn = 1.5; (*) liquid, ( 0 ) bridge, and ( 0 )gas phase. (b) As (a) but for mean density p; vertical lines represent thermodynamic phase transitions between states represented by (m). Dotted, dashed, and full lines are int.ended to guide the eye.

Confined fluids with short-range potentials

299

is quite well rcprcscrited by the rncari-field trcatriicnt of H (.") . However. there are difference&. In general, one notices that all GCEMMC coexistence lines terminate at lower (critical) temperatures compared with their meanfield analogs, which is a well-known phenomenon in the bulk [15]. Comparison of the mean-field data with their analogs from GCEMC (see Fig. 4.10) also reveals somewhat mote subtle differences. These differences conccrn thr: absrnw of thr: rat8hcrshort, coc!xist,cnc*c.lincs p"' (T) and pgd (T). Thcsc somewhat. rmrc ar'c'aric phasc wcxistciiccs turn out to bc stabilized by the mean-field approximation relative to the GCEhlC data because the former underestimates density fluctuations by disregarding them altogether. However, apart from these differences. it is iioteworthy that the mean-field approxirnat+ionyields px (T) in excellent agreement with the GCEMC data for those ranges of thermodynamic states where both approaches exhibit first--ordrr phase transit,ions.

Reviews in Computational Chemistry Kenny B. Lipkowitz &Thomas R. Cundari Copyright 02007 by John Wiley & Sons, Inc

Chapter 6 Confined fluids with long-range interactions 6.1

Introductory remarks

In all fluid model systems discussed in Chapters 3-5 the molecules interact with each ot-hcr via short-range potentials which decay with t,hc iiitcrinolecular distance T as T - ~cliaracteristic of dispersive interactions bet,ween polarizable molecules or even faster as in the nearest-neighbor lattice models discussed in Sections 4.3 or 5.8. There is, however, an increasing interest. in complex systems governed by long-range interactions’ such as the electrostatic (Coulomb) interactions between charged particles ( T - ’ ) or the dipole dipole interactions ( T - ~ )between particles with perinanent electric or inagnctic dipole moments. Indeed, Coulombic interactions are important or even dominant in almost all biological systcms, such as proteins, DNA, or charged nicmbrancs. Dipolar interactioiq 011 the other hand, play a proriiineiit. role in plimpholipid bilayers [222, 2231, and they are always important because of the omnipresence of dipolar water molecules in biological tissues arid electrolyte solutions. Apart from biological and electrocheniical syst,enis?there is a wealth of technologically important subst,ances where long-range interwt,ions play a central role. An example are polyelectrolybes, where the charged nature is the key ingredient for their functionality [224]. Dipole dipole interactions, on the other hand, are of fundamental importance in colloidal systems I2251 such as ferrofliiids [226-228], which are dispersions of ferromagnetic nanoparticks. Other cxai~iplcsarc inagnct,ic colloids corisistiiig of supcrparmiaguctic ‘Strictly speaking, the tern1 “long range” refers to interaction potentials decaying slower than T - ~ .

30 1

302

Introductory remarks

particles aid clcctrorlicological fluids 12291 rcprcscntcd by colloidal dispcrsions of polarizable particles. In many applications involving such systems, one is faced with some sort of spatial confincmcnt. Examples arc ccll rncmbrancs, soap l-)iibhlcs,alcrtro1yt.e solutions near charged surfaces, proteins near (charged) membranes, polyelectrolyte films [230], and thin films of magnetic colloids or ferrofluids [231]. Although this admittedly incomplete list illustrates the great importance of long-range electrostatic interactions, their correct treatment in computer simulations poses a highly nontrivial problem. This is because in any computer simulation one is inevitably restricted t o microscopically sniall systcms whcxc thc niinilwr of molwiiIc*sis always 11iany orders of magnitiidc smaller than in a macroscopic sample where typically N = 0 Indeed, long-range interactions still represent a. coinputatioiial challenge cvcn for (largc) bulk fliiirlsj which arc?periodic: in all t,hrcc: spatial dimcnsioiis. The reason is that thc coiivcritioiial strategy established for fluids with shortrange interactions: namcly tootruncate the potentials a t some cut-off distance T,, leads to serious inacciiracies and artifacts. when applied to Coiilonibic or dipolar syst,ems. hi view of this dilemma, several techniques to treat the long-range interactions in a more reliable way have bwn proposed. An early example is the reaction-field met.hod [232. 2331 for dipolar systems. Wit,hin this method, the interactions arc trimcatcd at a distance T, from each particlc, but (coiit#raryto a siiiiplc trmicat,ion) thc riicdiurn beyond T , is taken into account as a dielectric continuum characterized by a dielectric constant CRF [140]. Apart froin the approximation m d e via the replacement of microscopic interactions by a macroscopic “reaction field,” a further drawback of this method is that CRF has to be chosen close to the dielectric constant of the fluid itself. The latter, however, is a priori unknown. Other techniques proposed niorc r( ntly arc the fast. miilt.ipolr mcthod, part.icleniesh metohods [234-236], and tlic: Lckiicr irictliocl 12571, which is primarily intcndcd for simulations of thin films2 For two rerzsons we fociis in this chapter on yet a different technique, the so-rallcd Ewald siinirnat,ion mct,hod. First, Ewald slims arc nowadays the most widely used and accepted method t.o handle long-range interactions [140, 2381 at least as far a5 bulk systeins a,re concerned. Second, the formulation of Ewald sums for confined systeiiis is straightforward as we shall demonstrate below and in t,he accompanying Appendix F to which we refer for a detailed discussioii of the derivation of the relevant equations. Moreover, during t,he last few years, there have been substantial improvements tliat. led to an optimization of these methods sucli that nowadays they arc *For an overview of other techniqum to treat long-range interactions, see,e.g., Ref. 238.

303

Three-dimensional Ewald summation

iiot only accurate but also cornputatiorially cfficicnt. Our goal in this chapter is to present, a simple and physically meaningful derivation of various Ewald summation techniques. For a mathematically more rigorous presentation, we refer the reader to the original papers by de Leeuw et al. [239-2411. As an introduction, and for pedagogical reasons, we start in Section 6.2.1 with khc cmcrgy of Conlomhir systems in thrcc spatial dimensions. B a d on thc resultiiig Ewald sum we tlien derive steyby-stcy corresponding forinulas for bulk dipolar systems (see Section 6.2.2) and for systems in slab geometry where we consider point charges and dipoles confined by either insulating (see Sectioii 6.4) or coiiducting walls (see Section 6.5). Mathematical details of the derivations and explicit Ewald expressions for forces, torques, and stress tensors can he found in Appendix F. Illustrating applications are presented in Scctions 6.4 aiid 6.5.

6.2 6.2.1

Three-dimensional Ewald summation Ionic systems

We start by considering an ionic system consisting of N charged particles forming a (rectangular) cell of volume V = s,sysz. We assume t8hat the N system as a whole is electrically neutral, that is, CtZ1 q, = 0: where qI are the individual charges. We are interested in the Coulomb contribution to the potential energy of this N-particle system, which can be written as

where @ ( T , ) is the electrostatic poteiit.ia1at the position of particle i. We inimediately specialize to a bulk-like situation where the central cell is surrounded by periodic replicas in all three spatial dimensions. In this case, tzheelectrostatic potential is given byJ

+ ( r j )=

C'C N

{n} j = 1

qj lrij

+ nl

(6.2)

where ~ i=j ri - rj is the connecting vector between particles i and j and {n}is a set, of latticc vectors of the rectangular supcrlatticc gencratcd by the 3Throughout this chapter we employ Gaussian units to keep notation as compact possible.

BS

304

Three-dimensional Ewald summation

pcriodic rcplicatioii of thc origiiial cell, that is t.o say, n = (nxsx, n.ysy,n,.s,) (%, ny, n, E Z). The prime attached to the first, summation sign in Eq. (6.2) indicates that the terin j = i is omitted for n =O, i.e., within the central cell. For the derivation of the Ewald method it is important to consider the charge density p i ( r ) corresponding to the electrostatic potential @ ( r i )given in Q. (6.2). The link between these quantities is provided by Poisson’s aquation 12421

Coinbixiing Eqs. (6.2) and (6.3) we scc that t,hccharge distributioii generating the potential in Eq. (6.2) may be perceived as a sum of Dirac &functions (see Appendix B.6.2)

c n

pi ( T I ) =

(n} j = I

qj6 ( T I - r j

+ n)

(6.4)

where, as before, the sum over lattice vectors excludes t.he term corresponding to j = i if n = 0 . We note in passing that it is this restriction of the lattice sum, which causes the charge distribution to depend on (particle) index i. For each point charge qi involved in Eq. (6.4): the resiilting elsct!rostat.ic potential decays in proportion to thc invcrsc distance [sce Eq. (6.2)], such tha.t the lattice sum buried in the expression for t,he total energy Uc [see Eq. (S.l)] converges ra.t.her slowly. In view of this dilemma, the central idea of the Ewald summation techniques is to rewrite the &like char e density in Eq. (6.4) as a sum of bhree contributions, pi(1) (r‘),p Y ) ( r ’ ) , and pi (TI ). Each

8,

one of these yields an electrostatic potential whose convergence is controllable by a fcw parainctcrs. In a first, st,ap wc assnciatc with cach &function a diffiisa cloiid of point charges 9j of opposite sigxi located a t T I = r j - n. It is coxiveriient (yct not crucial) to represent these clouds by Gaussiails; that is

where a controls the width of the Gaussian, which is normalized such that

(6.6) We note in passing that this approach follows in spirit the general definition of the Dirac &function via Gaussian dist,ributioris presented in Appendix B.6.2.

Confined fluids with long-range potentials

305

Adding tlic Gaussiaris to thc original chargc distribution in Eq. (6.4) yiclds

pi(1) (r’)=

which may be perceived as a set of screened charges. Their potential d c cays much more rapidly than that of the original point charges. Indeed, as we demonstrate explicitly in Appendix F.1.1.1, t,he electrostatic potential corresponding to pi1) ( T ‘ ) is given by

where erfc(y) is the complementary error function [ll, 371 that, decreases monotonically as y increases. Hence, its decay with increasing interpart-icle separation is controllable by the parameter a. In practice, a is usually chosen such that erfc(g) is esseritially zero for separations larger than half the shortest side length of the central cell. In this case, only neighbors within the central cell, ( n = 0 ) have to be taken into account in the lattice swnmation, that is. in the summation over { n } in Eq. (6.8). As a iiext step we have to take care of the electrostatic potential contribution of the Gaussian charge clouds themselves, N

(n) j = l

wharc thc right sidc follows by siibst,rac:t,ing Eq. (6.7) from Eq. (6.4). Wr recall that thc sum over lattice vectors in Eq. (6.9) excludes the tern1 .j 1: if n = 0 , which is the Gaussian located at r, where we wish to calculate the total electrostatic potential @ ( r l ) . In what follows it turns out to be convenient to first retain the so-called self-term in the sum over lattice vectors, which describes the interaction of a charge cloud with itself. Clearly, this contribution is unphysical and needs to be corrected for at the end of the derivation. We thus split the right side

Three-dimensional Ewald summation

306

of I3q. (6.9) into two contributioiis, riairicly N (n} j = 1

=

N

x q j n} j = 1

(5) 3

exp [-a2(r’- r j

+TX)~]

(6.10a)

(6.10b) such that the lattice sum appearing in Eq. (6.10a) may now be carried out without any restriction. As a consequence, the charge distribution ~ ( ~ ) is (r’) independent of index i . We are thus dealing with a distribution of smooth, Gaussian ciiargc clouds of total chargr~+9j which vary periodically in syacc. This, in turn. suggests carrying out the calculation of the corresponding contribution to the electrostatic potential, d 2 ) ( r i )in , the reciprocal space m, E spanned by lattice vectors k = (2lrm,/s,, 2n7ny/s,, 2lrni,/s,) (mx,my, Z). As shown explicitly in Appendices F.1.1.2 and F.1.1.3, the resulting potential is given by

The first, term on the right side of Q. (6.11) represerit,s contributions from the related to non-ze7w wavevectors (k # 0 ) . Each Fourier coefficients of d2)(ri) suminand is weighted by a Gaussian of widt.h (2a)-’,such that damping is controlled by the wavenumber k = Ikl. With illcreasing k the summands vanish increasingly rapidly, as long as Q is finite and not too large. This is thc grcat iulvantagc of rcprcscnting t.hc pot,cntial from t,hc Gaussian hackground in Fourier rather than real-space where the convergence would be much slower. The last term on the right side of Q. (6.11) is the so-called long-range (2) representing the Fourier coefficients for k = 0 in the expotential @r,R(ri) pansion in Eq. (6.11). As argued ill Appciidix F.1.1.3, @ri(ri)is sensitive to the specific boundary conditions cniployed in an actual computer simulation. Assuming that the total volume (central cell plus periodic replicas) is a large sphere surrounded by a dielectric continuum and characterized by a dielectric constant t‘? the long-range contribution is given by (6.12)

307

C o d n e d fluids with long-range potentials

whcrc

N

M = C9jrj

(6.13)

3=1

is the total dipole nioment of the syst,em. Finally, we need t o consider the potential related to the charge distribution p y ) ( r ' ) [scc Eq. (6.10h)], which corwcts the (miphysical) sclf-interaction included in (ri). As detailed in Appendix F. 1.1./I,

(6.14) We are now in a position to write down an expression for the total potential energy of the three-dimensional Coulomb system [see Eq. (6.l)] within the Ewald formiilation. Adding all contributions t,o the total clertrostatic potential arid insertiIig thc result, iIit,o Ecl. (6.1) wc obtain

[JP = [Jg + [JS+ Ug + [JgR

(6.15)

On thc right side of Eq. (6.15), aid rwult from real- [scc Eq. (6.S)] and Fourier-space (k # 0 ) contaibutions [see Eq. (6.ll)l of the electrostatic potential, respect,ively, that is

(6.16~~)

N

N

= -2nx x C w e x p a=]

j=l k#O

k2

[-21

exp[-ik . r i j ]

(6.16b)

Tho remaining terms on the right sidr of Eq. (6.15). U g R and U g follow from the long-range [see Eq. (6.12)] and self-contributions [see R. (6.14)] a~

308

Three-dimensional Ewald summation

where A I - [MI. At this point some comments on t8heuse of Eq. (6.15) in actual computer simulations seein to be appropriate. For simplicit,y, we coiisider a system in a cubic sirnulation box; that. is, s, = sy = s, = s . As mentioned [see discussion following Eq. (6.8)], the parameter cr is usually chosen large enough such that only neighboring molecules within the central cell (n= 0 ) need to be considered to compute t,hc r(d-spa(:c contribiit,ion (SCC Eq. (6.16a)I of the total Ewald ciicrby. In practice, typical valu~.uof the scrccriiiig parameter are (.LS x 5 - 7, corresponding to a cut-off of the real-space interactions at. T, x s/2. The resulting double sum is then evaluated with periodic bouiidary conditions in all spatial directioiis combined with the minimuin image convention (see Section 5.2.2). A s far as the Fourier-space contribution to the Ewald sum for the configiirat.ional potcntkd energy is conccriictl [scc Eq. (6.16b)], t,hc slim ovcr wavcvcctors is triiiiratctl at, a cut-oH wavciiurritm x:, = (2.rr/.s) witli m: = niax ( 1 . 2 + rn? + m;). Note, however, that the convergence of the

fi

tn.

,my .TI,

Fourier part depends on Q via the Gaussian damping function exp[-k2/4a2]. The larger is a (that is, the short-er is rC); the larger In: needs t o be chosen to rnake sure that ternis with lkl > kc can iiideed be neglected. Thus, the challenge in applying Ewald smiimations is to find the right balance between the number of ternis evaluated in real and Fouriar-space, respectively. In practice, t,ypic:al valiics of rri: x 30 - 60, rorrwponding to a summation ovcr 1000 - 1500 tiiffereiit wave vectors (i.e.: wave vectors riot related by syrnriie try). We also note that the Fourier part can be rewritteri’niore coinpactly (6.18) where we have inCroduced the complex quantity IV

(6.1Y)

and ?(k) is the complex conjugate of iz(k). Eqiiation (6.18) implim that, t.hc Foiiricr-space rontxibiition of thc cncrgy can be evaluated for each wavevector as a product of two single-particle sums. This is of great importance in terns of computational speed, because for each wavevector its evaluation boils down to coniputing 2N terms [see Eqs. (6.18) and (6.19)] instead of N 2 terins a5 in Eq. (6.16b). We finally note that. niost of the recent sirnulations of ionic systems are carried out under so-called “condiictiiig” or “tin-foil” boundary coiiditions

Confined fluids with long-range potentials

309

corresporiding to tlic choicc c.’ -- DO in t.hc loiig-range part of the Ewdd CIIergy [see Eq. (6.17a)l. With this choice, the long-range contribution can be completely neglected. On the ot,her hand, with “vacuum” boundary conditions ( d = l), the long-range term is nonzero and positive, implying that a situation characterized by M # 0 is associated with an energy penalty. This is consistent with our niacroscopic considerations in Appendix F . l .1.3, where wc cmphassizc! that, a polarized sphcro in vaciiiiin cxpcricnrcs a dcpolarizing field acting against tlic polarization within tlic sphcrc [scc Eq. (F.42)]. Up to this point we h w e focused on the (total) energy related to (threedimensional) ionic systems, which is particularly important in MC simulations. However, one may also be interested in perforiiiing MD simulations: where, unlike in MC, the evolution of the syst*ernis not governed by changes in the configurational potential energy (see Section 5.2) but rather by the forces determining the mottion of the particles. Explicit cxprcssions for the Couloinbic coritribiition to tlic forccs, as w d as cxprcssioiis for various coinponents of the stress tensor (see Section 1.2.2) within the Ewald formulation are.givcii in Appendix F.1.2.

6.2.2

From point charges to point dipoles

Based on our derivation of the Ewald energy of a Coulombic system, it is quite straightforward to formulate the analog for a system of point dipoles p,, i = I , . . . , N. Assliming rzci hcforc. a periodic rcpliration of the central cell iii all tlirrc- spatial directions, t h total cricrgy of the bulk dipolar systciii is given by

@=f

”

I

UDD t = l 3=1 { n }

(ry + n v Pt, P J )

(6.20)

where the dipole dipole interaction between two particles i and j is defined bv

(6.21) The easiest derivation of an Ewald expression for the lattice sum in Eq. (6.20) parallel t,o the one in Eq. (6.15) for t,he ionic fliiid is based on the ob.wrvation that ’uDD (:MI IN: rc:writton il,s

(6.22) where Q(rij) is the electrostatic potential of a unit point charge (i.e., a charge 9i = + l ) , namely 1 Q(T,) =(6.23) 1’. .

v

310

Three-dimensional Ewald summation

whcrc rij G Irijl. Thus, from a forrrial point of view, the pair energy ~ L L tween a pair of dipoles can be derived from the Coulomb interaction potential between a pair of charges, ucc ( r i j ) = qiqj/r,j through the transformation q, -+ (pi Vi) and 9j + (pj . Vj). The same applies, of course? t o the total configurational potential energy in h.(6.20), which we can rewrite as

-

We now recall that the key idea in deriving the Ewald expression for Coulombic systems was to divide the charge distribution and the resulting electrostatic potential into several inrlepcndent,ly convcrging contributions. Explicit expressions for the $-functions corresponding to real-space, Fourier, and long-range contribubions to the Ewald' expr&sion for the Coulomb energy can easily be extracted from Eqs. (6.16) and (6.17a), yielding

47T

*p(rij)

1

= -cpexp k#O

[-51

exp[-ik.rij]

(6.25,) (6.25b) (6.254

Because of Eq. (6.22) each of these furictions generates a contribution to the Ewald expression for the total configurational potential energy of a dipolar ~ y s t e m .By ~ complete analogy with the Coulombic case, we can thus write [see Eq. (S.l5)]

'The self-part plays A different role in khis context, and will be discussed later.

Confined fluids with long-range potentials

311

(6.27h)

-

27r M2 2f‘

+1 v

(6.27~)

where the functions R and C are defined as [140] exp(-a2r2) C ( r ,N )

[

1

+ erfc (nr)

(6.28a)

ldl? 1 2ar = - - (3 + 2n2r2)exp (-n2r2)+ 3erfc ( o r ) r dr r5 fi (6.28b)

I n addition, in Eq. (6.2711))

312

Ewald summation for c o d n e d fluids

is again tlic total dipole riioiricrit of thc maiiy-particlc: systcin in tlic central cell and 11.1 is its magnitude. Similar to ionic systems: the long-range term vanishes for the choice c' = 00, reflecting the fact that a conducting environment prevents formation of surface charges and thus suppresses a depolarizing field inside the central cell. Finally, the self-part, of the dipolar Ewald energy, Ui$,cannot be derived by applying the Nabla operator as for the other terms, because the corresponding self-term for charged systems is already a constant. Specifically, oiic rcalizcs f'roiri Eq. (6.17b) that tlic Coulomb sclf-cncrgy caii bc rewritten as (6.31a) (6.31t)) Thus, the lransformation qi -+ p, . V, wodd lead to the erroneous result U s = 0. We defer the correct derivation of the self-term to Appendix F.2.1 where we show that (6.32) Regarding the use of Eq. (6.26) in practice we note that the same comments made earlier apply here as well [see discussion after Eq. (6.17)]. A rlatailad discussion of optimal choiccs for thc Ewald paramatars cy and rri? for dipolar systerns can be found in Rcfs. 243 aiid 244. Filially, roaders who are interested in performing hlD simulations of' dipolar fluids are referred to Appendix F.2.2 where we present. explicit expressions for forces and torques associated with the three-dimensional Ewald sum [see Eq. (6.26)]. Moreover, explicit expressions for various coiiipoiients of the stress tensor can be found in Appendix F.2.3.

6.3

Ewald summation for confined fluids

Having understood the concepts of Ewald summation techniques for threedimensional bulk systenis, we now turn to systems that are Jinite in a t least oiie spatial dimension. We focus on a slab-like geometry where t,he fluid is confined by two plane. parallel and structureless solid surfaces separated by a distance s, along the z-axis of the coordinate system and of infinite extent in the x-y plane (see also Section 1.3.2). Hence. for the time being, we shall be

Confined fluids with long-range potentials

313

dealing with thc situatiori arialyzcd in Section 1.3.2 from a purely therrnodynamic perspective where a rectangular fluid lamella of area A = s,sy and height s, is repeated periodically in the 2- and y-directions but, obviously, not in the direction perpendicular to the walls. Note, that we iniplicitly assume that the fluid lamella considered here can also be deformed in the zarid y-directions by external agents and therefore the thermodynamic analysis of Scction 1.3.2 docs not dircctJy apply hcrc lwcaiisc wc cxplicitly assiimc in that, wction that tlic 1;unclla is oiily subject to a corriprcssiorial strain in the direction normal to the solid surfaces.5 Not surprisingly, tlic rcduction of spatial syriirnctry rcwlers thc! trcatiiicnt of long-range fliiid fluid interactions significantly more involved than in the three-dimensional case. This concerns not only the derivation of appropriate (Ewald or other) expressions for energies, forces, and torques, but also the computational effort needed in actual calculations. An additional issue here concerns thc choice of appropriate boundary conditions at the solid surfaces. Quantitatively speaking, these can be char€", which, on account of the finiteness acterized by another dielectric constmalit of the system in the &-direction, may (and will) in general differ from E' charactmizing the infinite surrouiidiiigs in the X-YJ plarie in which the fluid laiiiella is ei~ibedded. For exai~iple,a t.ypicd question iii elect,rodiemistry is t o what extent "image charges" induced by an ionic fliiid in a. confining conducting (i.e., metallic) substrate (8' = 00) affect the structure of the confined fluid itself. Consequently, computer siniulations of confined fluids with long-range interactions are still quite challenging, and there is an ongoing discussion of how the computational niathods can hc optirnizcd [231, 2381. In light of this background we discuss in this section three different a g proaches to Ewald summation in slab geometries. The first and second of t h s c iiicitliotls arc ;tlq)rol)riitt.cifor fliiids confinad t)c!t,wcicsi insdating s u b strates (i-e., d' = 1). They differ from one aiiotlier in the rriatheiiiatical rigor of their derivation. They are also quite different in the computational effort required to evaluate the final expressions in an actual computer simulation. The third approach presented in this section is capable of dealing with system confined by met,allic solid surfaces (characterized by 'c = ,m).

6.3.1

The rigorous approach

Based on the ideas discussed in Section b.2, various rigorous extensions to slab-like systems have been proposed [245-2511. Because t,hese systems are 5A suitable adjustment. of the therrriociynamic description presented in Section 1.3.2 to the sit.uat,ionconsidercd here is. however, both feasible and straightforward.

314

Ewald summation for confined fluids

infiiiite in two dimensioiis, wc bcgiri by corlsidcriiig N cliargcd particlcs irr a central cell extended periodically along the z- and y- directions. Thus t.he the configurational potential energy is given by

where, in contrast to the corresponding expressions for bulk systems [see Eqs. (6.1) and (6.2)], the lattice veclors {n} are quasi two-dimensional, that is n = (nll,O)with nil = ( ~ S , , ~ ~ ~Starting S ~ ) . from Eq. (6.33), the corresponding Ewald formulation can be worked out in close analogy with the treatment detailed in Section 6.2 for three-dimensioiial (bulk) fluids. As before the main step consists of rewriting the (&like) charge distribution p(r) corrcspoiiding tooEq. (6.83) as a slim of three rontributkms Irf., Eq. (6.7), (6.10)] which are then considered separately. This procedure is sketched in Appendix F.3.1. One finally obtains

(6.34) where the real-space contribution has the same form as before in the threedimensional case and is thus given by the right side of Eq. (6.16a) taking, of course, n = (nl1,O).The same holds for the self-part, which is given by Eq. (6.17b). The Fourier contribution, however, takes a more complicated form, namely

where &j = R, - Rj and R, = (xi,yi)is the projection of the position vector of particle i onto the x-y plane. Furthermore, zjj = z, - zj and kll = (2?rmx/sx,27rn~,/s,) is a two-dimensional reciprocal 1a.tticevector (i.e., exp[ikll rill] = I). Finally, kll = IkllI and

In terms of computer time, evaluation of the Fourier contribution to the tot.al configurational potential energy, iJ$: requires significantly more effort than

Confined fluids with long-range potentials

315

in the thr~~dirnensional case [SLY: Eqs. (6.18) and (6.19) for cornparisori]. The reason is that. the double s u m over all pairs of particles in Eq. (6.35) cannot be rewritten as a product of two single-particle sums. The corresponding expression for a system of point dipoles is even niore involved as one may verify from Eq. (F.115) given in Appendix F.3.1.2. Thus, for both tlie ionic and the dipolar systems, the actual use of the rigoroiisly dcrivrd Ewald siimmation for slab systcms leads to a siibstantial iricrcasc in coriiputer tirnc. Orie way of dealiiig with this problem would bc to employ precalculated tables [252] for potential energies (and forces) on a three-dimensional spatial grid amended by a suitable iiiterpolatioii scheme. Another strategy is to employ approximate methods such as the one presented in t,he subsequent Section 6.3.2.

6.3.2

Slab-adapted Ewald summation

In the following we discuss a modification of the well-established threedimensional Ewald sum siiitahle for simiilat,ions of syst,ems of point charges or dipolcs with a slablike gconictry [252-2571 with irisulatirig substrates (i.e., E” = 1). Within this method the slali geometry is taken into account by incorporat*inga vacuum region along the t-direction in the basic cell. Specifically, we choose a t-etragonal cell with lateral dimensions s, = sy = s (area A = s2), and perpendicular dimension d, = s,+s= ys. where s, is the wall separation and svac is the width of the vacuum region that can be controlled via thr dilatation factor y. The crit,irc ccll with voliiinc V = s2dA = ys3 is replicated in all three spatial directions, which results in a system of parallel h i d slabs alternating with vacuum slabs in the z-direction (see Fig. 6.1). The entire system is then placed into a medium of infinite dielectric constant that prevents formation of surface charges at the outer boundaries. Thus, we can employ the coiiventional three-dimensional Ewald sums [see Eqs. (6.15) and (6.26) for the Coulonibic and dipolar case, respectively] with “tin-foil” boiindarv conditions (i.c., r’ - 00). It is clear, however, that the slabadapted version can mimic the desired single-slab situation only if interactions between periodic images of the slab are neglible, which rigorously requires,s ---t x! (i.c., y -, 00). Because this can hardly be realized. we follow Yeh and Berkowitz [254] and supplement, the Ewald energy by a correction term, implicitly assuming that svac is large enough so that the remaining intcrslab interactions can be handled on the basis of continuum theory. Suppose we have a systcrri of N point cliarges, qt = 0) but carries a net dipole moment which is globally neutral ( i t . . Cfr’=, in the z-direction. hi, = C z ,q,z,. Because of tlie periodic replication, Mz is mirrored by the neighboring slabs, which will therefore act as parallel

Ewald summation for conftned fluids

316

Figure 6.1: Sketch of the simulation cell (front view) employed in the slabadapted three-dimensional Ewald sum.

capacitor plates -with -surface charges P, = h./,/V. These charges generate a capacit,or field, E = E,e^,, within the ct:ntral cell where

-

I*&

(6.37)

= 47l1;

The corresponding contribution to the energy per volume follows as

-

I I<..

1

- = --P,E, V 2

-

N N 271 =- - ~ ~ q l q J z , z j 1'2

r=l

(6.38)

.)=I I

The above considerations suggest considering IJc as the energy contribution due to"inters1ab interactions" resulting from the periodic replication of the

317

Confined fluids with long-range potentials

systcrn in the z-direction. Clcarly, thcsc intcrslab iiitcractioris are an unwanted phenomenon. We therefore introduce an energy correction,

(6.39) which we add to the conventional three-dimensional Ewald sum for point charges with conducting boundaries (i.e., c' = x)to give

(6.40) Equation (6.40) is our final expression for thc energy of a charged system between insiilating walls within t,hc slahax3apted three-dimensional Ewald summation rncthod. OII t8hcright side, the contribution [I$' is defined by Eqs. (6.15), (6.16), and (6.17b).G The reader should also realize that, for the current system, the volume V appearing in the Fourier part of the energy [see Eq. (6.16b)Jincludes the CTLCIIUII~ space; that is, V = s2dz = ys3. Moreover, here let=,

k = 2n

( ) %/S

mY/s

(6.41)

%/Y

Because the correction term in Eq. (6.40) is proportional to M:, the development of a net dipole moment in the z-direction leads to an increase in the total configurational potential energy. This is consistent with oiie's physical intuition if one recalls from elementary electrostatics that an infinitely extended slab polarized along the slab iiormal experiences a depolarizing field equal to -47rPZ = -47~A/,/v act,ing against, the polarization [242]. In fact, [see Eq. (6.3'3))is nothing hut tlic cricrgy pcnalty due to thc dcpolarizirig field. It is then straightforward to formulate a slab-adapted thrte-dimensional Ewald sum for a system of point dipoles. In this case,

(6.42) and the energy correction compc~~satiiig for intcrslab intc!ract.ions follows from Eqs. (6.37)-(6.39) its

(6.43) 'Notice that the long-range part given in Q. (6.178) vanishes because of our current. choice f' = :x.

318

Ewald summation for confined fluids

which has exactly the same foriri as for thc corrcsporiding ionic systcrn [scc Eq. (6.39)). The total energy of the dipolar slab system is then calculated from 2lr + -A{. &? (6.44) "f;lab -- [J,3d 1- [J,, [J;,*kI

I!JE

I+m

V

rign

where is given in Eqs. (6.26)-(6.32) [noting that = 0 for our current choice e' = 00: see h.(6.27c)I. Explicit expressions for the forces, torques. and the normal component of the stress tensor related to Eqs. (6.40) and (6.44),rcspcctivcly, call be fourid in Appcridix F.3.2. In practice, most applications of thc slab-adaptcd Ewald sum in three dimensions employ vacuum spaces that are three to five times thicker than the original substrate separation [252-2571. The energies obtained with the approxirnat,e Ewald sum then coincide almost perfectly with those of the rigorous method discussed in Section 6.3.1. For a systematic discussion of the errors involved in the s l a h a d a p t d version, we refer the interested reader to Rcf. 256.

6.3.3

Reliability of the slab-adapted Ewald method

To deir1onstrat.e the reliability of tlie slabadapted Ewald niethod introduced in the preceding Section 6.3.2. we present in the following results from lattice calculations [257]. Specifically, we consider a slab composed of dipolar spheres of diameter (T located a t the sites of a face-centered cubic (fcc) lattice. The lattice vectors are T = (1/2) ( I x , ,Z Iz). where P is the lattice ronstant, (fixed such that the reduced density pa3 = 4a3/t3 = l.O), and {In} ((r = x,y,x) arc idcgcrs with I, t 1, i lZ OVCII. An infinitdy cxteiitlcd slab is then realized by setting -00 < l , , l , < x and 1, = 0, . , i t , - 1 with nz being the number of fcc layers in ,--direction. We evaluate the system's total dipolar energy for the following situations:

1. all dipoles point along lhe z-axis and 2. all dipoles point along the z-axis. The latter, somewhat unphysical situation lias been included t,o investigate the importance of the correction term given in Eq. (6.43). Clearly, the correction is irrelevant for case 1. For both situations, we calculate the (dimensionless) dipolar energy per 2 via t8heslahadapted three-dimensional Ewald sum7 [see particle, 0 3 U ~ / pN, Eq. (6.44)]and with the rigorous Ewald method for dipolar slab systems 'Calculations were performed with the Ewald parameters 0.9 = 7.0 and rn: = 80.

319

Confined fluids with long-range potentials

s”

I

* *

m L

-I

3O 2 4 6 8 1012

nZ

Figure 6.2: Dimensionless energy per particle for dipolar crystalline (fcc) slabs as a function of the number of lattice layers, assuming perfect order along the z-axis the (a) and dong the z-axis (b). Included are results from direct summation (O), rigorous Ewdd sum for slab syst,rms (A) ( s w Appcirdix F.3.1.21, and the slabadapted three-dimensional Ewald sum ( x ) [see Eq. (6.44)]. Part (b) additionally includes results from the latter method when the correction term [see Eq. (6.43)] is neglmted (*).

as forniulated in Appendix F.3.1.2. The results are compared in Fig. 6.2. Obviously. the two different Ewald methods are fully consistent both for cases 1 and 2. Figure 6.2(b) also includes results obtained with the s l a h adapted tlircediniensiond Ewalcl sum but without thc corrcction tcrin given in Eq. (6.43). Clearly, neglecting this term causes erroiieous results, although the dilatation factor controlling the space bet!ween two neighboring slabs has already been set to y = 10. Thus, we conclude that it is essential to incorporate the correction to achieve consistency between the slabadapted three-dimensional and the rigorous Ewald sum. Also shown in Figs. 6.2 are results obtained via direct summation of the dipolar interactions. Using the fact that particles within a given layer contribute equally, the total configurational potential energy for case 2 can

320

Insulating solid substrates

bc casts as

where N / n , is the number of dipoles per layer and EZ(i,)is the x-component of the field E acting on a dipole in lattice plane 1,. Without loss of generality we may assume that this dipole is located a t a lattice site ri = (1/2) (0.0, l z ) . Then the field due to the other dipoles ( j # i with pj = p2&) located at r j = ( 1 / 2 ) (li.l$, l:) is given by

(6.46)

Similar expressions arc obtained for case 1. Data plotted in Figs. 6.2 have been obtained by truncating the sums over 1, aiid 1, a t f5000, which yields convergent results as long as n, 5 4. Comparing these results with those from the two Ewald methods, we conclude that not, oiily the rigorous Ewald summation, but also the slab-adapted three-dimensional version provide quasicxact, rcsiilt,s for tho dipolar cnrrgy.

6.4

Insulating solid substrates

6.4.1 Dipolar interactions and normal stress Having established the accuracy and reliability of the slabadapted t h r e e dimensional Ewald method, we present in this paragraph numerical results froni GCEMC siimilations (SICC Scction 5.2.2) for a confined Stothnaycr fluid.' The particles then irit,eract with each other via both the long-range, 'The Stockniayer fluid consists of spherical particles with embedded (permanent) point dipoles.

Confined fluids with long-range potentials

321

anisotropic dipole dipole iritcractioiis IhDD givcii in Eq. (6.21) arid the (sphcrical) LJ (12,6) potent,ial given in Eq. (5.24). The Stockmayer fluid is frequently used to model polar molecular fluids with permanent, electric dipole moments such as chloroform, for example [258, 2591. However, more recently, Stockmayer fluids have also been employed to model ferrofluids, which consist of (colloidal) particles with permanent magnetic dipole moments in carrier liquids likc watm or oil. For technical reasons cxylairicd in Scctioii 5.2.2, the GCEMC siitiulatioiis have been performed using a slightly modified Stockmayer potentmiddefined bY t~ ( r i j t pi, ~ j =)'W.R ( 1 . u ) + '~JDD (rij,pi,/+) (6.47) where the short-range contribution is given in Eq. (5.39). In the calculatio~~s below we choose a cut-off radius of r, = 2.5 for the short-range intera~tions.~ Fliiid molccnlcs arc confined l y two planar solid walls scparated by a distance s, along the z-axis of the coordiiiate system arid of illfinite cxteiit, iii the z-y plane. Assuming that fluid particles and atonls composing the wall interact via the LJ (12,6) potentid given in Eq. (5.24) and averaging over the subspaces -00 < z 5 - 4 2 and 4 2 5 z < 00 occupied by substrate particles, the fluid substrate potential follows as [cf. Eq. (5.71)] cp' k1 ( z ; s z )= --E/),(T 45

(6.48)

whcrc thc plus and the minus sign rrfcr to the lower (k = 1) and upper (k = 2) wall, respectively. In Eq. (6.48), tlie reduced density of solid particles is set to psa3 = 1. The GCEMC results presented in this section refer to Stockmayer fluids at a temperature T = 1.60 and dipole moment m = 2.0, which are typical for real polar molecular fluids [259]. The chemical potential is set equal to p = -19.30, so that the bulk fluid has an average mean density of j j N 0.6 Kecping thcsc parameters fixd, we invcstigatcd systcins with substrate soparatiow s, in thc rangc 1.7 5 s, 5 10.0. In the following we focus on the normal stress, T,,, or rather on the disjoining pressure f (s,) defined in Eq. (5.57) for reasons given in Section 5.3.2. Numerical results for f(s,) are presented in Fig. 6.3, where the dipolar contribution to T,, has been calculated according to the microscopic expressions given in Eqs. (F.124)-(F.126) in Appendix F.3.2. As before for confined fluids witohshort-range interaction potentials, the disjoining pressure f (s,) has damped oscillatory character with the oscillations vanishing when the confined systems become bulk-like at large wall 'Note that, no siirh cut,-off is applied t o t,hr long-rang~part of t,he potential.

322

Insulating solid substrates

2 1.5 c .Stockmayer

1 h

aN

iis

-0 0.5

0

-1

2

4

3

5

S

2

Figure 6.3: CCEMC results for the disjoining pressure f ( s , ) [see J3q. (5.57)] for a typical polar (Stockmayer) fluid as a function of the wall separation (data have been obtained at T = 1.60, m.= 2.0, and /L = -19.30 corresponding to an average bulk density 5 i = 0.G). Also shown are corresponding results for an atomic LJ (12,6) fluid at the same temperature and bulk density.

separations s,. This behavior agrees, a t least on a qualitative level, with that observed for simpler systenis interacting via spherically symmetric POtentials only (see Section 5.3.3, Fig. 5.3). To illustrate that agreement, we have included in Fig. 6.3 GCEMC data for a pure LJ (12,6) fluid where the chemical potential has been fixed at a value chosen such that both the LJ (12,6) and t,hc St,ockmaycr bulk fluids haw thc samc avcragc density (ix., p = -13.50 for the Corresponding LJ fluid). Upon decreasing the.subst,rate separation from larger values, the function f ( s , ) first is essentially constant for both systems down to wall separations s, M 50. Only a t smaller separations does f(s,) exhibit pronounced oscillations with a period of roughly one molecular “diameter.” One also sees that maxima and minima of f(s,) are more pronouiiced in the LJ (12,6) fluid where the positions of the extrenia

Confined fluids with long-range potentials

323

are slightly out of phase relative to f (sZ) for the Stockriiaycr fluid. 011thc other hand, for both systems, the oscillations are accompanied by a steplike increase of the average adorption "rate" (N) /A, which is shown in Fig. 6.4.

3

2

k 1

Stockmayer

04

S

2

Figure 6.4: Adsorption rate of a Stockmayer fluid and a comparable LJ fluid as a function of the wall separation (parameters as in Fig. 6.3).

Based on previous studies on the relations between the oscillations in the disjoining pressure and structural changes (see Section 5.3.4), we interpret these. features as fingerprints of stratification, which, to reiterate: is the tendency of fluid particles to arrange themselves in individual layers (strata) parallel to the confining walls. In this sense, maxinia in the plot of f ( s , ) indicatc thc disappcaranro (formation) of complcto laycrs of fluid molwulns upon decreasing (increasing) s,. At separations corresponding to minima of f(s,),on the other hand, particles are accommodated comfortably between the substrates. At other separations (particularly around the maxima), the fluid structure will be more or less frustmted (i.e., disordered). Although this interpretation of the qualitative behavior holds for both the Stoclcniayer and the LJ (12.6) fluid, it is also interesting to compare the actual

324

Insulating solid substrates

values of f( sz),which arc indcvd quitrcdiHcrcrciitafor tlic two rriodcls cousidcrcd. In particular, the oscillations in the LJ (12,6) fluid are characterized by much larger amplitudes. suggesting that the overall repulsion bet$weenthe particles is stronger thaii in the dipolar system. In other words, it seems that, even for strongly confined dipolar fluids. similar to what is found in bulk dipolar systems [260], the anisotropic dipolar interactions manifest themselves as a n d attraction.

Figure 6.5: Cont,ribut,ionsto the normal stress of a Stockmaycr fluid as a function of the wall separation (parameters as in Fig. 6.3). ( 0 ) rzs," (0) rLS,(*) r z , (0) 722.

That this is indeed the case can be seen from Fig. 6.5 where we have plotted separately the three contributions to the normal stress of the Stockrnayer fluid, (6.49) 7 2 2 = 7-rn SR + 72D 2 + 722 FS It is seen that the oscillatory character of the norrnal stress is visible in all contributions to T~~at. sufficiently small substrate separations. One also

Confined fluids with long-range potentials

325

notes a differcncc in sigu ainong the various individual contributions. For example, T: is positive regardless of the substrate separation, indicating that the dipolar interactions tend to decrease t8he(magnitude of the) normal stress. Finally, in closing, it is worth noting that the features described above are not restricted to the specific thermodynamic conditions and not even to the specific model system we have considered here. Rather thev should be pcrccived as generic fcatiircs of confined dipolar licliiids.

6.4.2

Orientational order in confined dipolar fluids

As a second application of the Ewald summation for systems with slab geometry we now consider strongly coupled confined dipolar liquids, that is, liquids where the dipolar iritcractioris dominate thc systcin’s structurc and phase behavior. Examplw for sudi systeins are ferrocolloidal filrris or cow fined molecular liquids with strong dipole moments and relatively weak van der Waals forces or hydrogen bonds [231]. .The model we consider in thc current section is similar to the Stockmayer fluid discussed in Section 6.4.1. However, here we replace the LJ interaction, which includes attractive dispersion interactions [see Eq. (5.24)], by the purely repulsive soft-sphere (SS) potential, (6.50) In a similar spirit we neglect the attractive part of the fluid substrate potential introduced in Eq. (6.48). Choosing this somewhat. mininialist,ic model has thc advantagc of permitting 11s to stiidy more directly t,hc intcrplay between spatial confinemciit arid tlic long-range, anisotropic dipolar intcractions. In fact, recent research on the bulk dipolar soft-sphere (DSS) (and related) model fluids demonstrated already that the specific properties of the dipolar interaction, combined with lack of long-range translational order can lead to new and unexpected physical behavior [260, 2611. In particular, it was shown that dense systems of dipolar spheres may develop spontaneous polarization, yiclding A liquid statc wit.h long-range fcrroelcct-ric order [262-268]. Given the appearance of an orientationally ordered, yet liquid-like phase in bulk dipolar fluids, a main question addressed by us in Refs. 257 and 269, was whether spontaneous order of this type may also exist in nanoscopic slit-pores where it, is not a priori clear whether confinement may support or inhibit ferroelectric order. To this end we performed MC simulations in the nixed isostress isostrain ensemble introduced in Section 5.7 where a thermodynamic state is specified

326

Insulating solid substrates

by tlic set { T,N , 711,sd}: that is, wc arc holding constarit the nmiibcr of molecules N rather than the-chemical potential paof the confined fluid. As an immediate consequence, the confined fluid can no longer be viewed as being in thermodynamic equilibrium with a bulk reservoir because their chemical potentials may (and very likely will) differ if the number of molecules in the confined system remains constant. Although this may seem to be a disadvantage with rc3spcct. to situations cncoimt,crcd frcqiicntJy in laboratory cxperiinerits, wc ncvcrthclcss yrcfer this cnscinblc over the grand canonical one. The reason is that, for the confined DSS fluid, thermodynamic states of interest in the current context are expected to be relatively dense, such that transferring particles between the reservoir and the confined system niight be very difficult. This implies that in parallel laboratory experiments the exchange of matter between the confined fluid and a bulk reservoir is insignificant or that a bulk phase in thc seiisc of tlic SFA cxperinient (sec Section 5.3.1) may not even be present a t all. Such a situation is realized, for example, in the SANS experiments described in Section 4.8.1 where a binary fluid mixture is confined to a nanoporous medium (i.e., pellets) without being in contact with a bulk reservoir. The algorithm by which a (nmnerical representation of a) Markov chain is generated in this ensemble is discussed in detail in Section 5.7.7. However, for thc DSS fluid. wc nccd to alsn include random rotations of the molcculcs as part, of tho canonical substcp of tlic algorithiii in additioii to thoir ranclorn displacement (see Section 5.2.2). All simulations were performed for T = 1.35 and m = 3.0. These parameters have already been employed in earlier siniulations of bulk DSS fluids for which long-range parallel order at siifficientJy high densities or pressures was ohserved [262-2651. Our MC results for the confined dipolar fluids indicate that spontaneous order does iridecd occur owr a certain rauge of wall separations R,. This can be seen from Figs. 6.6-6.8 for a system where s, = 7. Specifically, in Fig. 6.6, we have plotted the global order parameter (6.51) as a function of the applied parallel pressure 41, where d is the global director. The latter is obtained from the simulations; the znstuntuneous value of z i s the eigenvector corresponding to the largest eigenvalue of the order-parameter h

327

Conflned fluids with long-range potentials

matrix [140j

- 1 Q=2

N

( 3 & . j& i=l

x)

(6.52)

In the remainder of this section, we deliberately chose to deviate from our standard treatment of mechanical work in terms of stresses rather than pressure tensor elements. Even though we foster the former treatment throughout this book, the latter seems a bit, inore intiiitivc? in thc: ciirront context, especially with regard to existing literature 011 the bulk DSS fluid. However, we remind the reader that pressure tensor P and stress tensor T are trivially related through the relation P = - T .

Figure 6.6: Order parameter PI and internal energy (inset) versus applied parallel pressure for a DSS fluid confiiied to a slit-pore system with wall separation sl; = 7u. Results are obtained with N = 256 ( 0 , A) and N = 500 (0, A) particles. Temperature and dipole moment are fixed at T = 1.35 and 7n.= 3.0.

Starting from 41= 0.1 (71 = -0.1) and increasing (decreasing) the pressure (stress) up to 41rz 2.0, the order parameter remains small at first,

328

Insulating solid substrates

indicating that tlic systciii is globally isotropic ovcr this prcssurc range. Upon further compression, however, PI increases sharply and for pi1 > 2.6 one o h serves fairly large values of the order parameter PI 2 0.8 that depend only weakly on the number of particles employed in the simulation. As all sirnulations were started from randomly oriented states, we conclude that states characterized by a large transverse pressure are indeed spontaneously polarixcd. They comport?witaha tmc fcrroelactric phasc. The presence of an isotropic ferroelectric transition is also reflected by the total configurational potential energy per particle (U)/ N plott,ed in the inset of Fig. 6.6. Increasing 41from the initial smaller values, (U)/ N first increases, but then begins to decrease at a transverse pressure of about 171 = 2.0 whcrc f’l bcgins to risc rathcr sharply. Clcarly, thc dccreasc of (I/) / N can oiily be caused by the dipolar interactions; because tlie shortrange fluid fluid and t,he fluid subst,rate polentials are purely repulsive. As for ‘(simple”fluids that have only translational degrees of freedom the structure of the confined DSS fluid is inhomogeneous on account of stratification (scc Section 5.3.4). Becausc of the additional rotational rlcgecs of frccdcmi, howc:vcr, tlic structure of the DSS fluid may be riiorc complcx as sna.pshots from the MC simulations in Fig. 6.7 illustrate. In the left part. of that figure, a snapshot is presented for a globally isotropic system, whereas the right part, shows a snapshot for an orientat,ionally ordered phase. For the sake of clarity only molecules in one contact layer (i.e., the layers of niolecules closest to one of the walls) are plotted.

Figure 6.7: Left: Snapshot of thc contact l q w at pi1 = 1.0 (isotropic phase). = 5.0 (spontaneously polarized phase). Right: Snapshot of the contact layer at The thick arrow labeled by “ d denotes the direction of the global director in the z-y plane.

Confined fluids with long-range potentials

329

The globally isotropic state is rharacterhcd by Ihc appearance of clusters and chains of particles with essentially random orientations, as expected for a dilute, strongly coupled dipolar fluid. In this regard, a main effect of spatial confinement is that the dipolar particles tend to form chains with in-plane rather than out-of-plane orientation. This anisotropy is even more pronounced in the orientationally ordered state, which is characterized by long, txscntially straight chains t,hat arc aligncd along a direction within the plane and parallel with the substrates. Closer irispcction reveals that spheres forming neighboring chains tend to arrange themselves in an out of registry conformation. This way they avoid side side configurations where neighboring dipoles would prefer to be antiparallel rather than parallel as indicated by the snapshot plotted in the right part of Fig. 6.7. The structure perpendicular to the chains is generally more open, and an analysis of the rorrcsponding in-plane correlation functions [257] shows that thc ordered system still cxliibits a liquid-likc structure within tho layers. This is similar to the confined fluid’s bulk counterpart , which is also characterized by strongly anisotropic, yet short-range spatial correlations in the ferroelectric nematic state [262, 263, 2651.

A n important difference between the confined system and the bulk, however, coiireriis the tlieriiiodyiiarnic conditions related to the onset of lorigrange parallel order. In fact, based on the data plotted in the two parts of Fig. 6.8, we conclude that in the confined system the onset of order occurs at somewhat lower pressures/densities, indicating that the walls promote rather than inhibit spontaneous orientational order. This result is, at least at first sight, rather surprising, because the substrates in the current system do not couple directly to the fluid particle dipole moments. A rationale for this shift of tlie onset of spontaneous order in the confined relative to the bulk fluid is offered in Ref. 257 where we basically employ entropic arguments.

As we argue in this latter work, the global director ;can point in any direction on the unit sphere in the bulk. In confined systems, on the other

hand, the presence of walls inhibits order in the normal direction, as can bc iindorstood b.v simplc (macroscopir) cncrgy considrrations. Indccd, with insulatiiig walls, order in the normal directioii would lead to surface charges on the iiiner sides of the confining walls. These would in turn generate a demagnetizing/depolarizing field in a dirt-tion opposite to that. of the total dipole moment of the fluid. Therefore, the director d is restricted to point in a direction parallel to the walls; within this (r-yj) plaiie, however, the orientation of i i s then arbitrary. Ths effect is reflected by the snapshot in the bottom part of Fig. 6.7. Because of this restriction, which is absent in the bulk as we already mentioned. orientatiorial fluctuations (and there-

330

L-

Insulating solid substrates

a"

P Figure 6 . 8 Order parameter PI as a function of the external parallel (bulk) pressure (a) and of the average density (b) for the confined system at sZ = 7.0 ( 0 ) and the bulk system (0).

fore the orientational entropy) are supressed to a certain extent even in the isotropic phase and the ferroelecbric transition occurs at significantly lower pressures/densities. This argimcntntion is fnrthcr si1pported by thc predictions of a simplo mean-field theory of the ferroelectric transition, which was originally presented in Ref. 257. Within this theory, we neglect any stratification (i.e., inhoinogeneitiss of the local density) as well as any oscillations in the order paranieter (which are indeed observed in the coinput,er simulations). We also neglect nontrivial interparticle correlations. Our system can then be viewed as a system composed of N uncoml(r.ted dipolar particles individually interact.ing with thc rncan ficld

(6.53)

Confined fluids with long-range potentials

331

whcrc tlie sccorid t.crin in parciithcscs is a. corrcctiori of thc bulk mean field (EMF = 4?rpP1pe^,/3) due tmoconfinement (2571. Without, loss of generality, we have set the director equal to the unit vector e?, along the x-axis. Because particles do riot interact, the Boltzinann factor weighting the individual dipole orientations (wi)is determined only by the field energy, - p ( w , ) -EhgF. Consequently, PI can be obtained from the self-consistency relation (6.54)

In Q. (6.54), J ciw rcprcscnts a11 iiitctgration over all aIigular coordinates of the particles, i.e., w = ( 0 , ~ )The . computer simulation results (see Ref. 257) and the upper part of Fig. 6.7, however, suggest that dipole orientations along the z-axis are already highly disfavored in the isotropic phase. Within the mean-field theory, this phenomenon ran only be formulated in a highly idealized manner. naniely by assuining that the dipoles are completely restricted t+opoint pcrpcndiciilar to thc walls (i.a., CY = 7r/2). Tha sclf-consistcnry equation (6.54) then reduces to

where f(P1) = p p P l p 2 (4x/3 - ?r0/(23,)] arid In(z) is the modified Bessel function of order n.[141]. Expanding the right-hand side of Eq. (6.55) around PI = 0, we find that the system is polarized at densities above a critical density 2k~T (6.56) Pc,p = p2 (4?r/3 - na/2sz) whereas for the corresponding bulk this density is [270, 2711 (6.57) Comparing these two expressions. one sees that, pc,p < pC,b at, all finite pore widths, in qualitative agreement to OUT MC results for s, = 7. Quantitatively speaking, however, the mean-field theory turns out to be poor. as expected from corresponding bulk results for dipolar fluids [225]. This observation is not surprising in view of the strong orientational correlations between dipoles in the ferroelectric phase.

332

6.5

Conducting solid substrates

Conducting solid substrates

6.5.1 The boundary-value problem in electrostatics So far our discussion of coilfined fluids with long-range electrostatic interactions has been restricted to the case of insulating walls characterized by d’ = 1. However, in many cases, one might also want t o consider dielectric interfaces (c” > l), which are important, in biological systems such as ion channels in proteins biit aLso for self-assembled monolayers composed of organic materials (2721. A special case in this context are condiict.ing (i.e., mct,allic) intcrfaws for which c’’ = m. Thcy play a ccntxal rolc in clcctrochemical problems dealing with the structure and transport properties of electrolytes close to metallic electrodes [273]. Another t,ype of system where conducting walls are esc9ential are electrorheological fluids (2731. These are colloidal dispersions of dielectric particles where structural and rheological properties can be controlled by electric fields resulting from two metallic electrodes [229]. From a theoretical point, of view, the crucial difference between a system confined by conducting substrates rather than insulating ones is that the charges and dipoles in the original slablike system create “image charges” and “image dipoles,” respectively, wit.hin the confining metal [242]. These images are, of course, merely theoretical constructs (rather than real charges or dipoles) that allow one to solve the electrostatic boundary value problem. To obtain this solution, one realizes that. tha M~wwcllequations require the electrostatic potential to be coiist,ant oti tlic surfacx of t,hc substrates; that is, the tangential part of t4hefield must vanish. In practice, the existence of images implies that there are additional interactions t80be taken into account such that the treatment of liquids in the vicinity of conducting substrates seerris even more cornplicated than the situations considered before (see Section 6.4). Surprisingly, however, the energy of a system hct.u:ccn condricting silhstrat,cs can he mappcd onto a problcm with three-dimensional periodicity [275, 2761, which can suhsequently be treated by conventional Ewaki summation methods such as the ones presented in Section 6.2. In Section 6.5.2 (and the corresponding Appendix F.3.3), we describe this niappiiig explicitly for systems of point charges. We then gem eralize our treatment to dipolar systems. This section closes with a discussion of representative numerical results.

Confined fluids with long-range potentials

333

Figure 6.9: Sketch of the cffwt of conducting walls on two chargcd particles. (a) Presence of one conducting wall implies creation of one image charge per particle. (b) Two conducting walls yicld a11 infiiiitc numbcr of imagcs per particle, where one group of images has charges of the same sign as the original charge, whereas the other group is characterized by opposite charges. The s%ructurein the zdirection can then be considered as an infinite periodic replication of t.he extended cell (original charges plus one set of images) marked by the thick frame.

6.5.2

Image charges in metals

We start by considering a single particle with charge 9a at some position ri between two conducting (i.e., metallic) solid substrates of infinite thickness. To kccp the notational burden to a minimum, wc dcviatc from thc rcmnindcr of this text in that we are assuming the substrate surface3 to be located at planes z = 0 and z = s, (instead of z = fs,/2). According to t,he rules of electrostatics [242], which state that, the tangential part of the electric field must vanish on conducting surfaces, the effect on one conducting solid surface (e.g., the upper one at z = s,) consists of creating an image charge within the metal. As illustrated in Fig. 6.9(a). the position of the image is

334 -

Conducting solid substrates

ri

+ 2 (sz - zi)Z,, and its cliargc is QI = -qi

(6.58)

The total electrostatic potential in front of the substrat-e is given by the S U Y ~ . of the potentials caused by the original charge ( q i ) phis that due t o its image

(4.

The prcscncc of a second coridiictirig snhst,ratc at. z = 0 changc. the situation drastically because not only the original particle but also its u p per image are mirrored at, the lower substrate such that, they create image charges. These images locat,ed at positions z < 0 in turn induce new images in t.he upper substrate, and so forth ad infinituni [see Fig. 6.9 (b)]. Therefore, each charge generates an infinite number of images. The first group of images (which includes the one ment.ioned in t,he beginning) is characterized by chargcs (1; = - q i located at Ti

= Ti

+ 2 (n..Sz -

Zj)

n+. = 0, f l , f 2 , . . .

i?’,

(6.59)

The second group of images has the same charge as the original one; that is, Q = qi. These images are located at,

TI’ = ri + 2n,s,Z2,

.n,= fl! f 2 . . . .

(6.60)

Consider now IV particles confined to a slit-pore with metallic substrate surfaces. The total configurational potential energy of this system is then obtained from (6.61) is the electrostatic potential arising from the images of particle where aR(ri)

i, whereas cPa(ri) represents the contributions from particles j # .I and from

their images. Using Eqs. (6.58)-(6.60), these potelitids are given by

N

X I *

(13

IT13

nZ=-m IT,]

+2(

e

w z

+ 2%%e^,I + ZJ) ZZl

1

(6.62b)

where the asterisk is attached to the first summation sign in Eq. (6.62a) to indicate that, the term with nz = 0 is omitted. Moreover, in writing

Confined fluids with long-range Dotentials

335

the surnriiands, we used the fact tliat sumrnatioii over n, is cquivalcrit to a summation over -72,. Insert-ing Eq. (6.62a) into Eq. (6.61), we obtain after a straightforward rearrangement of terms

(6.63) wherc the priiiic attached to the first sum ovcr n, signifies that the term n, = 0 is omitted only for i = j. Finally, if the central cell comprising N particles is replicated along the xand y- directions, we obtain a slab-like system confined between conducting walls. Introducing now threedimensional lattice vectors Ti=

( :: )

(6.64)

271,s~

and replacing in Eq. (6.63) the siinis over n, l y thrcodinicnsional sums over the sct of latticc vcctors {E},tlic total coiifigurational potential cncrgy of the system may be cast. as

The lattice sums in Eq. (6.65) reflect the fact that. t,he Coulonibic systeni between conducting walls has, in a wav, three-dimensional periodicity. The basic cell of this three-dimensional array contains the original cell with the N particles pliu the first sat, of imagcs, that, is, the N images resulting from the prcscncc of just, t.hc lowcr wall [scc Fig. 6.Y(b)]. In fmk, as wt: show txplicitly in Appendix F.3.3, the energy of the extended system with a total of 2N charges, is directly linked to Uc by the relation

Up'ex,

Uc: = -up= I

(6.66) 2 Thus, in a computer sirnulation with conducting interfaces, one only needs to calculate the energy (or forces) in the extended system, which turns out to be twice thc original one. However, the current approach has thc great advantage that it can take into account the t,hree-dimensional periodicity of the extended system. Therefore, the conventional three-dimensional Ewald summation technique [see Eq. (6.15)] can be employed. As a consequence, simulations of systems between conducting interfaces are typically much faster than corresponding siinulatioils between insulating substrates 011 account of the simplifying three-dimensional as opposed to the slab geometry.

336

6.5.3

Conducting solid substrates

Dipolar fluids

The above considerations for point charges can readily be generalized to dipolar systems between two conducting surfaces [277]. This follows again from the principles of elementary electrostatics, which tell us that each dipole p, within the original basis cell creates two (infinitely large) groups of image dipoles. The first group is located at positions given in Ekl. (6.59), where'"

(6.67) The second group of images is located a t positions defined by Eq. (6.60) and dipole moments pr = p2. Coiisider now N dipoles in the basic cell and replicate this cell in directions parallel to the walls. Then the total configurational potcntial energy ran bn writtm as (scc Appcndix F.3.3.2)

Finally, using essent,ially the same arguments as for the charged systeni, we can show that (see Appendix F.3.3.2)

(6.69) where U z V e xis the total configurational potcntial energy of an extended system with a basis cell comprising the set of the N original dipoles plus the first set of image dipoles. Based on relation (6.69), we can again employ the conventional three-dimensional Ewald sun1 [see Eq. (6.26)] t o calculate the energy of the slab syst,eni between conrliirting walls. 'To see this, consider the dipole aq an arrangenient of two charges of opposite sign aiid separated by some sinall distance and realize t,hat each charge creates an image charge of opposite sign.

Confined fluids with long-range potentials

6.5.4

337

Metallic substrates and ferroelectricity

Given the appearance of spontaneous ferroelectric order in confined DSS fluids bet,ween insulating walls (see Section 6.4.2), it is interesting to consider the dcpcndcncc of this phasc t,ransitioiis on t,hc wall boiindary conditions. To this crid we have rcpcatcd tlic calculations dmcribcd iii Section 6.4.2 for a strongly coupled DSS fluid confined between two conducting walls, using Monte Carlo simulations in t,he (Ar,s,, 41, T ) ensemble with N = 500 particles. The dipolar interactions were treated on t,he basis of E!q. (6.69). To compare with our previous results corresponding to the case of insulating walls (see Section 6.4.2), the reduced temperature! dipole moment, and wall scparaztion have heen set, to thr! sama val~iesas hcfore where T = 1.35, m.= 3.0, arid s, = 7. Nurnerical results for the order parameter PI as function of the applied transverse (ix., parallal) presslire 4, are plott,cd in Fig. 6.10, where we have included corresponding data oblained with insulating walls (see Fig. 6.6) as a reference. Clcarly, the confined fluid between metallic walls does exhibit spontaneous ferroelectric ordering a t sufficiently high pressures, as does its counterpart between insulating walls. Moreover: the pressure range in which the ferroelectric order develops upon compressing the fluid from the dilute limit is essentially the same for the two wall boundary conditions considered. Finally, inspection of the global director d^ (which is a result. of the simulations) indicates that metallic substrates (as do insulating ones) support ordering yamllel to these substrates; that is, d^ has directions within the x-g plane a t all pressures considered. Thus: one would conclude that the dielectric contarit characterizing the confining walls has only marginal influence on the phase transition at least at the wall separation coilsidered here. The same conclusion may be drawn from data for the total configurational potential energy plotted in Fig. 6.11. Upon increasing 41 from zero, the energy first rises in both systems as a consequence of the increasing repulsion between thc particles. In this prmsiira range, thc numerical values of lI for metallic walls on the one liarid, and insulatiiig walls 011 the other hand, are essentially indistinguishable. Further compression then yields a sharp decrease of U that can be a.ttributed to the decrease of dipolar energy due to orientational ordering (see Fig. 6.10). Within the ferroelectric phase the energies then increase again (upon increasing the pressure), with the values of U at high pressures being somewhat larger in the metallic case. On the other hand, local propertics such as t,hc density profilcs and local order parameters turii out to bc ricarly idcrit,ical [277]. Given the strong siniilarity of both the qualitative behavior and the ac-

338

Conducting solid substrates

Oo2L/(

nn

"*"O

,

1

2

3

I

I

I , ,

.4

I

I I

,

5

6

Figure 6.10: Order parameter PI as a function of the applied parallel pressure for a DSS fluid (T = 1.35, m = 3.0) confined between metallic walls with separation

s, = 7 (stars). Also shown are corresponding results for insulating walls (open circles) from Fig. 6.6 (N = 500).

tual therniophysical properties of the confined DSS fluid between metallic and insdating walls, one might wonder whether correct, treatment of wall bouiidary coiiditioris is iiiiport,aiit at all. One qucstion appcaririg in this context concerns the influence of the substrate separation s,. In particular, would we expect the same similarities (observed at. s, = 7) to also occiir in more confined systems characterized by snialler values of s,? To get some insight into these questions we have performed various lattice rnlciilations siniilnr in spirit to t,hosc dcscrihcd in Section 6.3.3. Spccifically, we have considered (infinitely extended) slabs composed of dipolar part.icles located a l the sites of a facecentered cubic (fcc) lattice with (reduced) density prcc = 1.0. We have then employed the Ewald sum for dipolar systems between metallic walls [see Eq. (6.69)] to calculate t,he total dipolar energy 0, for various configurations characterized by perfect orientational order. Numerical results for r l D as a function of the number of lattice layers n, are

Confined fluids with long-range potentials

339

5

Figure 6.11: Tot,al configurational cnmw U for DSS fluids between metallir (stars) and insulating (open circles) walls, respectively. Parameters are the same as in Fig. 6.10. given in Fig. 6.12, where Part (a) compares the energies of a system oriented along the x-axis (i.e., d 11 k) with those of a system oriented along the z-axis (i.e., d 11 i ) . It, is observed that, regardless of the actual value of %, the energy WD related to an ordering parallel to the walls is smaller than that related to perpendicular ordering. This explains why the MC simulations a t s, = 7 described above predict spontaneous ordering parallel to the walls. However, we also observe from Fig. 6.12(a) that, the actual diffcrcncas betwccn thc two ordering directions are large only for very thin films and decrease with increasing film thickness. This is in marked contrast to t,he corresponding energies for systems with insulating walls plotted in Fig. 6.2 revealing that perpendicular ordering (between insulatiiig walls) is energetically unfavorable even for macroscopically thick slabs. We can understand these differences as a conscqueiicc of depolarizing fields that, arise for perpendicular ordering

340

Conducting solid substrates

Figure 6.12: Dimensionless energy per particle for dipolar crystalline (fcc) slabs with perfect orientational order. Part (a) contains data for systerns between metallic walls with order in the z-direction (solid circles) and the z-direction (open triangles), respectively. Part (b) compares data corresponding to metallic (solid circles) and insulating (open squares) walls for systems ordered in the z-direction.

between insulating walls but not for metallic walls. Finally, wc comparc in Fig. 6.12 (b) thc eucrgics related to parallcl ordcriug (d 11 5)for thc two wall bomiclary c.ontlit,ious c:onsitlcrc:d. It is obscrvctl that the boundary condit4ionshave a large effect only at, very small values of 71, and become increasingly unimportant upon increasing n, toward the bulk limit n, + 00. This may explain why the energy values obtained in our MC simulations a t sz (which roughly corresponds to n, M 6) within the ferroelectric phase are quite similar. Finally, the lattice energies plotted in Fig. 6.12(b) also reflrct, that thc pcrfertly ordered system (d 11 2) between irisulating walls is gcricrally c1iaractcrkl;cd by mallcr cricrgy valucs (as co~npared with the nietallic case), whidi is again consistent, with our computer simulation results obtained at the largest, pressure considered (see Fig. 6.11).

Reviews in Computational Chemistry Kenny B. Lipkowitz &Thomas R. Cundari Copyright 02007 by John Wiley & Sons, Inc

Chapter 7 Statistical mechanics of disordered confined fluids 7.1

Introductory remarks

So far we have considered only situations where the fluid is confined to a single pore. However, real porous solids often consist of an interconnected network of pores of various sizes and shapes [4]. Prominent examples of such disordered porous materials are inesoporous glasses such as Vycor and CPG (controlled pore glass), which are formed by spinodal decomposition of a riiixture and subsequent removal of one coniponent. Contrary to mesoporous glasses, which are characterized by relatively low porosity’ of 30 to 60 percent, aerogels are formed by extremely dilute disordered net-worksof microscopic particles that occupy only a very srnall portion of the total volume (porosity 95 to 98 percent). An example is presented in Fig. 7.1. Mesoporous materials are of importance for a wide range of technical applications such as gas storage, separation processes, and heterogeneous catalysis and much progress lias been ~nadcin tho dtsigri, syithcsis, aiid charac:torizrttiori of materials with novel properties [278, 2791. In this chapter, we are interested in the influence of such a disordered material on the striicture a i d phase behavior of an adsorbed fluid. Interest in this topic was stimulated by the intense experimental research on phase transitions in disordered media in the 1990s. The experimental or N2 [131, 132, 2801, fluid studies have involved “simple” fluids such as mixtures (e.g., isobutyrir acid and water [lls]),and anisotropic fluids such as namatic liqnid crystals (281-2841 One main roncliision from these stiidie^s was that fluids in dilutc aerogels can iiidccd undcrgo true pliasc transitions, ‘The porosity is the volume fraction of the space available for the adsorbed fluid.

34 1

342

Statistical mechanics of disordered conflned fluids

Figure 7.1: Inner structure of a carbon aerogel. From R. Emmerich, http://idw-

online.de/public/pmid-42335/zeige-pm. html

whereas the existence of phase transitions in low-porosity mesoporous glasses still scrms qnita controversial [28S, 2861. Thercforc, niost thmrctical shidic?; focus on the highly dilute case. Indeed, corresponding experiments have indicated that even extremely dilute media with porosities as high as 99.9 percent. can alter the phase behavior of the adsorbed fluid drastically compa.red wit8h the bulk. Typical effects observed in systeins with condensation and/or deniixing phase transitions [115, 131, 132, 2801 are shifts of the critical temperature toward significantly smaller values, an accompanying shift of the critical density (or composition), and a substantial narrowing of the coexistence curves. For nematic liquid krystals in silica aerogels, exparimentzi have indicated t.hat tlie isotropic neinatic transition survives, but the long-range orieiit,ational (nematic) order occurring in the bulk is replaced by short-range or “quasi”long range order [281-2841. One may therefore expect similar effect,sin ot,her fluids displaying orientat,ional order. Compared with the large amount of experimental information, the theoretical understanding of fluids in highly dilute porous media like aerogels is fa.r less developed. The challenge in this context is to understand the influence of the quenched (frozen) disorder realized by the nearly random aerogel network on the fluids properties. One of the earliest attempts to niodel this

Quenched-annealed models

343

situation was a study by Brochard arid de Gcrixics [287, 2881, who suggested considering the adsorbed fluid as an experimental realization of the so-called random field Ising model (RFIM) [289]. The main idea here is that the local preferential attraction of the fluid by t,he solid surface within a pore, combined with the disordered charact.er of the pore structure, induces a spatially random perturbation of the chemical potential. The latter can be represented aii a local random rnagnctic field in the Ising pictwo. A major drawback of the RFIM, however, is that it focuses entirely on the aspect of disorder, whereas confinement plays no role. To account for this problem, more recent, theoretical studies, and computer simulations! of fluids in disordered media employ the concept of a quenched-annealed (QA) mixt-ure [290! 2911. Here, the fluid molecules (the annealed species) equilibrabe in a “mat,rix” consisting of partrticlcsqiionchcd in a disordcrcd configuration. Thus, QA inodels coiribiiie both disorder aiid confinenient, thc latter being guaranteed by the finite size of the matrix particles. In additmion,preferential adsorption can be realized by assuming attractive (or other, more complex) interactions between fluid and matrix particles. In what follows we first introduce in some more detail the concept of QA models and the rcsiilting appcarancc of doiiblc avcrgm. Wc thcn prcscnt. the foundations of the so-called replica integ+al equation theory, a theoretical formalism appropriate for calculating two-particle correlation functions and thermodynamics quantities of QA systems, which are homogeneous on average. The last part of the chapter is devoted to applications of the replica integral equations, with an emphasis on fluids with long-range dipolar interact ions.

7.2 -Quenched-annealedmodels In the framework of QA models, the disordered medium, such as the one depicted in Fig. 7.1, is modeled as a matrix consisting of N,,, particles. The latter are frozen in place (quenched) according to a distribution P (QNm), where QNm= (01,. . . ,QN,}denotes the set of matrix particle coordinates. In the simplest case (e.g., hard-sphere matrices), these quenched variables are just, the particle positions R,. Howcvcr, one may also consider thc case of Iriatrix particles with interrial dcgrccs of freedom, such as a charge or an orientation. In the latter case, the coordinates are &, = (R+,R,), with R, being the set of Euler angles defining the particle orientation. For the theoretical formalism t o be described it is convenient to choose P(QNm)as an equilibrium canonical distrihution established at some

344

Statistical mechanics of disordered confined fluids

tcriipcrature To,

where

u m m (QNm)

CC

Nm-1

=

Nm

“mm ( Q t , Q j )

(7.2)

&=I )=“I

is thc configurational potential tmcrgy govmiing tlic distribution of thc niatrix particles (assuming pairwise additive interactions), and 2 , =

[

J dQNmexp - uinin

(QNnl)

kBTo

1

(7.3)

is the corresponding configuration integral. The physical significance of To is that of a quenching temperature, which is a temperature at which matrix particles in a given equilibriuni configuration’are suddenly “frozen” into their nctmd positions in that configurat,ion. The nobtion dQNm. . . indicatm an iiitcgratioii over the set of matrix particle coordinates. In writing Eq. (7.3) we have neglected combina torial prefactors because matrix partsicksare not permitted to move, thus making them distinguishable through their spatial arrangement. We now imagine that the free space left by the quenched matrix particles is occupied by a fluid of Nf mobile particles. The fluid particle coordinates, qi, arc thus anncalcd variables that, can cquilihratc for a given configuration ( “realization" ) of the inatrix. Again, OIIC way consider sirriplc fluids with only translational degrees of freedom, where qi = T * . However, one may also consider anisotropic fluids for which qi = ( T ~ , w * )where , the set { w i } are Eider angles specifying the orientmation.To complete the description of the QA model, one needs to specify the interactions between fluid particles and those between the fluid and the matrix. We assuine again pair-wise additive intmactions t=l

j=i+l

for the fluid fluid interactions where qNf is thc set of fluid variables and

(7.5) accounts for the fluid matrix interactions.

Quenched-annealed models

345

Bwausc of the quciiclicd nature of thc matrix, the evaluation of mi equilib rium property A of the adsorbed fluid representring, for example, its internal energy or pair correlation function is not at all straightforward. To see this, consider first the thermal average involving the fluid rriirroscopic variables, q,. For a given realization (i.e., configuration) QNmof the matrix, and a given temperature T , this thermal average is defined as

where

is the configurational integral of the fluid. Equations (7.6) and (7.7) have been formulated for an adsorbed fluid with a fixed number of particles (canonical ensemble), but, they can be easily generalized to a grand canonical treatment where the fluid is coupled to a reservoir such that the particle number fluctuates around some average value (see Chapter 2). The latter situation is. in fact, ralcvant in inany cxpcrimcnts of fluids in disorrlcrcd wrogrls [290;. In this chaptcr we conccntratc 011 the caiioriical cnscniblc to kccp tlic notation as compact yrs possible. Formulas relevant, to a grand canonical description are given in Appendix G. 1. From a practical point of view, the thermal averages defined by Eq. (7.6) arc iiot very rncauingful as they dcpcnd 0x1 th: specific rcalizatioii of the matrix. Therefore one needs to supplement, the thermal average by a "disorder average" over matrix configurations, yielding the double average

[(

. .).I

=/dQN1n(...)OP(QNm)

(74

The problem in evaluating this double average is that the two sets of variables involved, qNfand QNm,are not treated on equal footing as in conventional st,at,ist,icalphysics. Instcad, as indicatcd by Eq. (7.8), the thcrmal avcragc has to be performed befon the disorder average is carried out. One way is to employ computer simulations where both t.ypes of averages can be directly evaluated. However, for complex interactions this procedure will be extremely time-consuming, especially at low temperatures where the number of matrix realizations required for the disorder average increases strongly (see, for example, R.ef. 292).

346

Statistical mechanics of disordered conflned fluids

7.3 The introduction of replicas The appearance of double averages of the type just discussed is characteristic not oiily for the QA mixtures considcred in this chapter, but also it is a generic feature of systems with quenched disorder. Prominent examples, which were extciisivcly studied in the 1970s a i d 198Os, arc spin glasses (293, 2941 and randorri-field systems "4. Work 011 these system has established thr SCF called replica method, which allows one to circumvent the double averages by relating the original disordered system to an artificial, yet, fully annealed "replicated" system [294]. Essentially the same methods can also be applied to QA mixtures, as first realized by Given arid Stell [295-2971. To introduce the replica, concept. we consider an arbitrary physical quantity expressible iii thr. forin of Ey. (7.8), such the internal energy of the

(7.9) Using Eqs. (7.G) and (7.8), the double average over one pair term appearing on the right, side of Eq. (7.9), that is. for example, the term corresponding to i = 1. j = 2, can be written as

We now multiply both the numeratm and the denoniiriator of the last term in t8hcintegrand by ZG-': whcrc 71,is an arbitrary integer. Becausc ZQ involves an integral over the N f fluid particles coordinates [see EQ. (7.7)], this multiplication implies that we are iiitroducing n - 1 copies (i.e., replicas) of the fluid particles. Assigning an arbitrary index a' to the variables ( q l , q z ) a p pearing in the numerator of Eq. (7.10), and iilserting Eq. (7.1) for P (QNm), Fq. (7.10) may be recast as

where we have used the iiotatiori J d q 2 = dql, d g , . . . dqNra for the integrations related to the copy of the fluid with index Q (a = 1 , . . . , n).

The introduction of replicas

347

Furtlierrnore, (Irep appearing iii J3q. (7.11) is givcri as

Inspection of Eq. (7.12) indicat.es that the replicated system introduced by the multiplication trick is a (n 1)-component mixture composed of the matrix particles and dtogcthcr 71, idcntical copim (i.c., rcplicas) of thc fluid particles. Each of the copies interacts with the matxix particles, but, there are no interactions among different copies. Returning to Eq. (7.11), one now uses the fact. that the denominator a p pearing 011 the right-hand side, ZmZ&which still depends on the realization QNm, beconies independent of QNnl in the limit n ---t 0. More specifically, one has (7.13) liin ZlnZG = Z,,, lirri ZG = Zm

+

n-0

n-4

The same limiting value is obt,aiIicd when taking thc limit n. 0 of thc configuration integral of the replicated that is if we consider --$

(7.14) because n+0 lim

Zrep=

1

dQNm exp

(-

z,,,

u m ~ ~ ~ N m=) )

(7.15)

where we have used the definition of [Ire, given in Eq. (7.12)r, The equivalence of thc two limits allows onc to rawritc Eq. (7.11) as

(7.16)

)LP

where (. . .)rep is a conventional canonical ensemble average in the replicated system. Combining Eqs. (7.16) and (7.9) one immediately obtains

(5 & Nr-1

'ff

= n+O lim

Nf

'ff

(q*Qt7436'

= lim U,.,, rep

(7.17)

n-ro

*See Appendix G.l for the corrcsponding formula in the grand canonical ensemble.

348

Statistical mechanics of disordered confined fluids

whcrc U;:, is the iiitcnial energy of a subsystem formed by the fluid particles of copy a’. Equation (7.17) is a representative example showing how one can calculate, in principle, a physical quantity of the QA mixture ulzthout evaluating the cumbersome double average. One first calculates the corresponding quantity in the replicated system. The latter contains more (that is, n, 1) components, but it is conceptionally easier to handle because all particles are mobile. This can be realized from Eq. (7.14), which shows that the coordinates of the ~natrixparticlcs aiid tlic fluid particle replicas are treated on equal footing in the replicated system. By letking n then go to zero, one eventually arrives a t the quantity of interest. Froin a practical point of view, however, it is clear that the procedure described above is highly nontrivial: Apart. from the necessity to deal with mixtures of n+ 1 components, the way to carry out the limit n + 0 in practice is by no means straightforward. One method t o deal with these difficulties is the replica integral equation formalism, which we will introduce Section 7.5. Howevcr, before doiiig this we first introduce the key coiicepts of thc replica integral equations, which are the two-part icle correlation functions of the QA system.

+

7.4

Correlation functions and fluctuations in the disordered fluid

From now on wc focus 011 situatioiis whcrc tlic! fluid adsorLcd by a disordcrcd matrix is both honiogeneous and isotropic after averaging over different matrix configurati~ns.~ In such a situation, the fluid’s singlet density is just a constant; that is,

(7.18) The expression after the first equal sign in Eq. (7.18) provides the statistical definition of the singlet deiisitv in the disordered system where d ( q - qt) = 6 ( r - ri) for a simple fluid without internal degrees of freedom, whereas 6 (q - q,) = b (T - T , ) b (w- u ~ for ) anisotropic fluid particles. The second ’Note that the fluid structure for a specrfic realization will usually be highly irihornogeneous and/or anisotropic.

Correlation functions and fluctuations in the disordered fluid

349

equal sign in Eq. (7.18) rcpreserits thc link to the singlet density in the replicated system, which can found by using the same strategies introduced in Section 7.3. Finally, jj = Nf/V for a simple fluid, whereas 7 = (4n)-’Nr/V for anisotropic particles. Given that. the singlet density is just a constant, the most important quantities characterizing the local structure witchinthe adsorbed fluid are the two-particle correlation functions. We start by considering the pair correlation function gm(q, q‘) between two fluid particles or, equivalently, the corrmponding total correlation function he ( q ,q’) = !iff( q ,q’) - 1. Thc statistical definition of the latt,er is given l y the generalization of the corresponding definition for equilibrated systems [30],

’bating thc doiihle average on the right-hand side as described in Section 7.3, one finds

-2

= p

. lim h::, (q,q’)

n41

(7.20)

where hz:, is the total correlation function between two fluid particles of the same copy in the replicated system. A practical way to actually calculate the total correlation function as well as various other functions to be introduced below will be presented in Section 7.5. Here we note that the function hr (q,q’) alone is already sufficient to calculate the internal (fluid-fluid) energy of the disordered fluid. In fact, combining Eqs. (7.9) and (7.19), one ha (7.21) by complete analogy with the corresponding bulk fluid relation [30]. The next correlation function we consider is characteristic for a quenchedannealed system in the sense t,hat it vanishes for conventional, fully annealed fluids. This is the so-called blocked correlation function hb(q,q‘) defined by

350

Statistical mechanics of disordered confined fluids

For coriveritioiial fluids the outcr (disorder) avcragc of thc first term on thc right side is absent and each thermal average equals the singlet density. Thus: hb = 0 for systems without quenched disorder. In the presence of disorder, on the other hand, the blocked correlation function is usually nonzero, because the singlet density for a particular realization, ( C z ,b (q - pi)) can be Q’

znhomogeneousand thus very different from its disorder average, p. Thus, ht, can be interpreted as a measure of matrix-induced fluctuations of the local density. The relation of the blocked correlation function t o the replicated system is somewhat different from the cases discussed before because of the appearance of t,wo thermal averages siiperordinatd by the disorder average in Eq. (7.22). Thc final rcsrilt, (S(V Appcndix G.2 for a derivation) is givtm bY (7.23) 0Z P h h (q,q’) = liln I($ ( q , q’) , 71-0

where h? (q,q’) is the total correlation fuiiction between two fluid particles of different copies in the replicated system. The total and the blocked correlation function introduced above are alr c d y sufficicnt to dcscrilic thc structnrc within thc adsorbcd fluid. Howcvcr, to descritx tlicrrrial fluctuatioiis wc nccd to introcluc-c two additional corrcL lntion functions. The first one is the response function G‘K ( q , q’) defined as

=

pb ( q - q‘)

+

$12hc

(q.? q’)

(7.24)

where the second member of the equation defines the so-called ”connected” Correlation function, h,(q, q’). Combining Eq. (7.24) with the definitions in Eqs. (7.19) and (7.22)>one sees that. t,he connected function is related to the total and blocked correlation function via

where the far right side introduces the connection to the replicated system [see Eqs. (7.20) and (7.23)]. To see the iinportance of the connected and response function for thermal fluctuations, we present two examples. The first one concerns fluctuations

Correlation functions and fluctuations in the disordered fluid

351

of tlic riurribcr of fluid partsicks Nf in an adsorbed fluid coupled to a particle reservoir (grand canonical ensemble). These fluctuations are commonly measured by the isothermal compressibility, KT. A definition of this quantity within the framework of statistical therrnodynarnics of disordered systems is given by [298] KT =

[(Nf?Q - (Nf)%] PkBT [(Nf)Q] 1

-

(7.26)

where we remind the reader that [. . .] signifies the average Over matrix r e p rcscntations" [SCC &a. (7.8)]. Coinbiriirig the abovc equation with Eq. (7.24). one obtains

~CTP~CBT = -J d q J d q ' c K VP

(q.q') = 1 +

f JdqJdq'h.

(q,q'> (7.27)

where the second line shows that it is the spatial (and angular) integral over the. connected correlation function that determines the conipressibility. As a second example we coiisider the dielectric constant CD of a dipolar fluid, where each particle carries a permanent dipole moment, p , and therefore q = ( r , w ) . The dielectric constant measures fluctuations of the total clipolc riioiiicrit M = C,pz arid is defined as 12'3'31

assuming the canonical ensemble. Writing the total moment as

6 ( r - r i ) b ( w- w i ) p ( w )

(7.29)

and employing Eq. (7.24), Eq. (7.28) can be cast as

(7.30) 4See Eqs. (3.37) and (5.78)for corresponding expressionsin the bulk and ordered porous niedia, respectively.

352

Statistical mechanics of disordered conflned fluids

111writing tlic first liiic of Eq. (7.30) wc uscd thc fact that, in a systcin that is on average homogeneous and isotropic, correlation functions depend on the positions of tvhefluid molecules only via the separation vector T = T - T’. This allows us to replace the double integral / d r .f dr’ . . . by V .f dF . . . after carrying out the trivial integration over T ‘ . The integral over orientations on the second line of Eq. (7.30) is a projection of the connected correlation fiiiwtioii onto t,hcscalar prodiict, p ( w) p (w’). This quitiitity can tw (!vid11;tt*(~tl bv using a rotationally invariarit expansion of hC [258]. Finally, we note that the (angle-averaged)structure factor of the adsorbed fluid S (k),which is accessible in scattering experiments, contains both a blocked and a connected part. To see this we start from the expression +

= 1

‘J J

+V

dq

dq’exp [ik. (T - T ’ ) )hm ( q ,q’)

.

(7.31)

where on the second line we have used the statistical definition of the total fluid-fluid correlatioii function given in Eq. (7.19). The structure factor can be rewritten in terrris of the Fourier transform of hff ( q ,q’) defined by

-

Itrr(k7 w , w‘) =

J

d;r;h,, (7,w , w’) exp (ik. T )

(7.32)

where we assume a fluid with both translational and orientational degrees of freedom as the most general case. For a simple fluid the above relation simplifies to

-hff (k)=

/

dfhm (7;)exp (ik. ?;) = 47r

/

sin (kr) dF2hm ( T ) -

kT

(7.33)

as we argue in Appendix G.4 where k and T; are the magnitude of the wavevector and the separation vector, respectively, Inserting Eq. (7.32) into Eq. (7.31) we obtain

where the second line results from the definition (7.25) of the connected correlation function.. Equation (7.34) shows that iiideed both the connected

Integral equations

353

arid thc blocked correlatiorls arc involved ill thc structurc factor of thc 4sorbed fluid. As a consequence, the long-wavelength (small angle) limit of the structure factor, S ( k + 0), does not coincide with the (reduced) isothermal compressibility given in Eq. (7.27), which may be expressed as

(7.35) This discrepancy is in cont,rast to conventional fluids where S (k --t 0) = KTpkBT (30).

7.5

Integral equations

We now turn t o the actual calculation of the correlation functions introduced in the preceding section. Our strategy is based on the fact that all particles in the niulticomponent replicated system are mobile. This allows the application of standard liquid st,atcl approaches such as integral cqriation theories [30] as has first bccn rcalizcd by Givcii arid Stcll [295-297]. Thc only s c ~ rious complication is the limit n + 0 relating the replicated to the original disordered system [see, for example, Eq. (7.20)]. In this chaptcr wc will dcal with this problem bv starting from integral equations for the (n, l)-component, mixture and assuming then permutation symmetry bet,ween the replicas. Thereby the n-dependence in the equations becomes isolated, which finally allows us to take the limit n,-+ 0 relatively easily. Of course, an implicit assumption of this procedure is that the permutation symmetry between the replica indices is ynsenred even for non-integer values in the range n, < 1. Breaking the replica symmetry does indeed occur in several disordered systems with low-temperature glassy states [293, 294. However, in this context, we are only interested in the description of the (high-t,cinperatiirc) fluid phase! whcrr! t.hc assiiinption of prtr;clrvat,ion of rcplica syrnmt:try for all TI. is rcaoriat)lo.

+

7.5.1

Replica Ornstein-Zernike equations

At the core of any integral equation approach we have the (exact) OrnsteinZernike (OZ) equation (3001 relating the total correlation function(s) of a given fluid t o the so-called direct correlation function(s). For the replicated system at hand, the OZ equation is that of a multicomponent mixture [30],

354

Statistical mechanics of disordered confined fluids

namcly

where the component indices i, j , and k can assume 0,1,. . . ,n. Here 0 r e p rcscnts the mobile matrix particlcs, 1, . . . ,n the copies of fluid part,iclcs, and the c,, are the corresporidiiig direct, corrcla tion functions. The convolution integrals in Eq. (7.36) can be circumvented by introducing Fourier transforms [see J3q. (7.32)]of all correlation fiinctions. The real-spa.ce 0 2 equation (7.36) then transforms into

where the symbol “@I” denotes both multiplication and an integral over the orientation of the third particle if that particle possesses orientational degrees of freedom. The important point about the Fourier-transformed OZ equation (7.37) is that it, dccouplm with rcspcct to k. In a next step we iriakc use of tlic fact that the ri. copies of the fluid particles are zdenticul (this is obvious from the introduct,ion of the replicated system described in Section 7.3). As a consequence, there is permutation symmetry between the replica indices. This implies for the fluid fluid and fluid niat,rix/matrix fluid correlations

where f = h or c and a’ is an arbitrary replica index. Furthermore, the correlat&ioiisbetween dzflelant fliiid copies have the symmetry

(7.39) Using relations (7.38) and (7.39) along with the symmetries of the singlet density, pi = Fa,,1: = 1. . . . ,71, and writing the OZ equations [see Eq. (7.37)]

Integral equations

355

where we have dropped the arguments of the correlation functions to emphasize the structure of the equations. Indeed, inspectmionof Eqs. (7.40) shows that the n-dependence of the correlation functions has now become isolated. This process allows us to perform the last step of our derivation, that is, the limit n, + 0 relating the replicated to the original disordcrcd system. Using the dcfinitinns (7.20), (7.23), and (7.25) of t,hc t,otal, blocked, and connected correlation function, respectively, and introducing -rep rep the notations lamm = lim,,o h, , pm = limn,,oPo, hf, = lini,,o hato, and hmf = lim,,o h,:; one obtains5 I

-

h,, = hmf = hf1, =

-

ha =

-

hb =

- -Cmm + Pm/lmm - 8 cmm -Cmf + ijmhnlm 8 C , f + F hnir @ Zc

- + Fmhfni - @ Crm

Zmm

cff + jTrllhrm(r3 C,nf

-

(7.4 lb)

1

(7.41d)

+ j j h*c 8 Efm

[

i 7 L,@+L,@Eb

(7.41~~)

pm&m 8 Emf + PEA.@ 6 +is [hb8Eff- 231,@4

(7.4 1c)

Eb

(7.41e)

Finally, suhtrwting thc last two of thcsc! cqiiations from each ot,hcr, one finds for the connected correlation function

h, =

-

+Fhc QZc

5The same notation is used for the various direct correlation functions.

(7.42)

356

Statistical mechanics of disordered confined fluids

Together Eqs. (7.41) a~id(7.42) form the rcplica-symmetric Oriistciri-Zcrriikc (RSOZ) equations first derived by Given and Stell in 1992 [295-2971. They are mact relationships! as are the 0 2 equations for conventional fluids. One specific feature of the RSOZ equations is the decoupling of the matrix structure from the fluid correlations [see Eq. (7.41a)l. This reflects that the matrix is quenched aiid thus not influenced by the structure of the adsorbed fluid. Conscqiicntlg, t,hc matrix ciorrclatioms servo as input to thc theory. In fact, oiie riiay CVCII cmploy cxpcrimerital data (c.g., froiri iicutrori scat,tcring) for the matrix structure factor, which is rclated to the matrix correla.tions (aswming a mat,rix without rotational degrees of freedom) by Smm(k) E 1 Pm &nIn(k). The direct correlation fiinction between matrix particles then 1 follows from Eq. (7.41a), which implies 1 pInhInln(k)= (1 - Pni &In(k))- . The above example already indicates that the practical solution of the whole set of RSOZ equations becomes particularly easy for simple QA s p tenis where the correlations appearing in (7.41a)-(7.42) depend onlv on the wavenumber. For molecular fliiids and matrices, on the other hand, the angledependence of the correlations can be handled by usiiig rotationally invariant expansions. This procedure, which is outlined in Appendix G.3, results in a system of linear RSOZ equations for t-hc corrclatiori fuiictiori cocfficicnts.

+

7.5.2

+

-

Closure relationships

The RSOZ equations Eqs. (7.41) and (7.42) still involve both the total and the direct correlation functions. Therefore, appropriate closure expressions relating the correlation functions to the pair potentials are needed t o calculate the correlation functions at given densities aiid temperatures. Typically, onc uscs standard closure cxprcssions familiar from bulk liquid statc theory [SO]. One sliould notc, 1iowc:vc.r, that, the pc:rforrnancc of tlic-sc closures for disordered fluids can clearly not he taken for granted. Instead, they need to be tested for each new model system under consideration. 111 the following discussion wc present as an example closure expressions appropriate for systems where both fluid and matrix particles are spherical and have fixed diameters (“hard cores”) of (fluid) and T(, (matrix). The corresponding fluid fluid, fluid matrix, and matrix matrix interactions then contain (apart from other contributions) the hard-sphere(HS) potential

(7.43)

+

where cqn, = (q am)/2 and

T

is the separation of the particles.

357

Integral equations

Following thc stratcgy dcscribtxl at thc bcgiiiniiig of Scctiori 7.5, we start by considering closure relations for the replicated system, employing the notation introduced in &. (7.36). The (exact) hard-core conditions can be written as CZP

(q,q’) = -1 -

vgp( q ,q’)

r < om

7

< Of, r < mfm T < Of

c;p(q,q’) = -1 - q : ; p ( q , q ’ ) , (..rep l)i ( q , q ‘ ) = -1 - r/Zp ( q , q ’ )7

7’

c:i“p(q,q‘) = -1 - q ; * e p ( q , q ’ ) ,

(7.44a) (7.44b) (7.444 (7.44d)

where i = 1 , . . . ,n,.Furt,hermore, qre*’= hre*- y e p with h8$io,%,ii)= -1 for thr sapnration rango considcrcd, rcflrcting t,hc fact, that the corresponding pair correlation functioiis are zero. Notc that there is no hard-core condition for the correla.tionsbetween particles of different fluid copies, (i # j ) because these particles do not int,eract, as may be realized from the configurational potential energy of the replicated system in Eq. (7.12). We also note that each of the relations in Eq. (7.44) involves only one species of particles. Therefore, there is no explicit n-dependence and we can directly perform the limit ?L -.+ 0, yiclding ‘:H(frri,mtn)

( q ,q’) = -1 - r/~(ftii,inin)(ql q‘) 7

7’

< of(fm,m)

(7.45)

and Cmf (qt q’) = ct,(q’,q ) For separations out,side t,he hard core, the direct correlation funct.ions have to be approximated. “Classic” closure approximations recently applied to QA models are t,he Percus-Yevick (PY) closure [301],the mean spherical approximation (MSA) [302],and the hypernetted chain (HNC) closure (301. None of these relations, when formulated for the replicated system, contaiils any coupling bct,wccn diffcrcnt species, arid wc can rlircctly procwrl to the limit n, 0. Thc PY closurc thm implics --f

Cmm

( q ,q’) = [ I + h m m (q1 q’)l

x

cb

(q,q’)

{

1 - exp

0? v r

[-

ILinm

( q ,q’)

kB T O

, r > IT,,,

(7.46a)

(7.46~)

Note the appearance of the quenching temperature To (instead of 7’) for the matrix correlations [first member of Eq. (7.46)]. This is a consequence of

358

Statistical mechanics of disordered confined fluids

thc prcfactor T/To in front of the matrix coutribution to tlic configurat8ional potent,ial energy of the replicated system [see Eq. (7.12)]. The PY closure has frequently been applied to various hard-sphere QA models [303], for which the first relation in Eq. (7.46) reduces to CR(fm,mm)

(7.47)

( q , q‘) = 0

that is, outside the hard core. A drawback of the PY closure is that the hlockd direct correlation funct,ion Cb is zero for d l separations and that the blocked total correlation function hb resulting from combining Eq. (7.46) with the RSOZ equat,ions turns out to be very small. This finding is in contrast to results from coniput,er simulat,ions [304], where h b has significant values especially at. srriall separations [304]. Next, consider the MSA, defined by (7.48a) C:R(rrn)

( q ,q’) =

1

- p ~ ( r m )(q7

ke T

d)

1

T

> nf(fm)

(7.48b)

(7.48c) Vr The MSA is linear in the pair potentials and has been applied to a variety of QA models with electrostatic interactions [305-3071. However, concerning thc hlockad correlations, thc MSA has tha samc drawbacks as thc PY closiirc (to which the MSA, ill fact, rerliicrs for pure hard-core ~iiodels).As a final example, we preseiit, the HNC closure defined by q,(q,q’) = O,

+hmm

1

cqrlII) ( q ,q’) = --liir(irl,)

kRT

+ h ( m l )

( q ,d )

r > om

9

(q.q’) - 111 [ 1

(qld )

.

r

+

(7.49a)

+ hr(rtli) ( q ,Q’)]

> Cf(fm)

(7.4913)

ch(q~q’) = -In [ I -!-hb (q*q’)] hl, ( 4 , q ’ )

vr (7.49c) Clearly, the HNC yields nonzero blocked correlations. Moreover, it is a particiilarly successful approximation for long-range, elect.rostatic interactions appearing in ionic and dipolar QA models (see Section 7.7 for a discussion of specific applications). 7

7.6 Thermodynamics of the replicated fluid So far we have focused on the calculatioii of correlation functions in the disordered system, from which one may extract structural features as well

359

Thermodynamics of the redicated fluid

susceptibilities (see Section 7.4). Howwer, in the context of phase b e havior, one may be also interested in thermodynamic quantities such as the free energy, pressure, or chemical potential. In equilibrium fluids, all of these quantities can be related (assuming pair-wise additive potentials) to the usual pair correlation function(s) accessible by integral equation theory [30]. Not surprisingly, these relations become much more involved for Q A systcms (3081 anrl wc not,c in rulvancc. that explicit. cxprcssions for thermodynamic quantities can only be obtained for specific iiiodcl systeins and closure approximations. In the following discussion, we therefore restrict ourselves to the relations between the thermodynamics of the original and the replicated syst*ern,and refer to the literature for specific systems. We start by considering the free energy, which is defined by ils

where ZQ is given in Eq. (7.7) and we have used the definition of the disorder average (7.8) in writing the second line of Eq. (7.50). To introduce the replicated system, we employ the mathematical ident.ity 111~.=

d

(7.51)

liiri -xn n-+O dn

which can be proved by using the identity z" G exp[nlns] and a Taylor expansion of the exponential, which yields xn = 1 n l n x 0 (n2). A derivative with respect to n: followed bv taking the limit n -+ 0, then leads to Eq. (7.51). Identifying with .T in Eq. (7.51) the configuration integral ZQ anrl inserting the latter relation into Eq. (7.50), one obtains

+

-

1

d

- lini -Zrep Z, n+o dn

+

(7.52)

where we have used t,he definitions given in Eqs. (7.12) and (7.14) of the configurational potential energy and the configurabion integral of the replicated system. Finally, introducing the free energy of this replicated system

Frep = -kBT

111

Zrep

(753)

360

Statistical mechanics of disordered confined fluids

and wing Eq. (7.15), o w firids

F

=

d lim -Frep n+O dn

(7.54)

Equation (7.54) represents a practical guide to calculate the free energy providcd that 1. the free energy of the (annealed) replicated system can be expressed by the correlation functions of the replicated system in a closed manner and that

2. the dependence of Frep on n,can be isolated such that the niatheinatical operation limn-o d/dn . . . can indeed be performed. Whether tlhese reqiiircment,s can be metzdepends on the model considered and on tlie closurc rclation involved for the calculatioii of the correlation functions. Examples for which Eq. (7.54) has actually been used pertain to the class of simple QA systems, that is, QA systems with no rotational degree of freedom where the interaction potent.ials contain a spherical hard-core contribution plus (at most) an attractive perturbation. For such systems, the free energy has been calculated on the basis of correlatioii functions in the mean spherical approxiniation (or an optimized random-phase approximation) [114, 2‘981. Wc now t.urn t,o the prc?jsiirc P and the chemical pdentinl pf of thc adsorbed fluid. CorJdcriiig first thc (n + l)-c*orriponcnt rcplicatcd syst,crri one has

= -Pre,V

+ pEpN,,, + .pi“,

Nf

(7.55)

where Prepis the pressure of the replicated system, and pr: and ,uEp are the chemical potentials of one fluid copy and that of the matrix particles, respectively. Also, we have used the s p m c t r y between the fluid copies in writing thc third twin on thc second linc nf Eq. (7.55). Combining F ~ s(7.55) . a i d (7.54): wc obtain (7.56) Based on Eq. (7.56),one finds for the pressure P and the (fluid) chemical

Applications

361

potciitid Irf of the origirial systciii

(7.57b) Quations (7.57a) and (7.57b) provide two ways to calculate the pressure and chemical potential. The first one is to perform the appropriate dcrivativc of the free energy, assuming that the latter can be evaluated for all states of interest. The second, more direct way is to employ the relations to the pressure and chemical potentials in the replicated syst,em. This second strategy is particularly useful for the calculation of pf because limn-+op c , can be cast in closed form for a variety of model systems and closure relations, including the HNC approximation for molecular fluids [309, 3101. The pressure is more difficult because of the prcsciiw of the scc'orid, rnatrix-rclatcd teriii oil thc riglit, side of Eq. (7.56). Finally, wc riotc without proof that both prcssurc arid chciiiical potmeritid can also be obtained by integrating the compressibility given in Eq. (7.27). Explicitly, one has [308]

For a derivation of these formulas we refer to Refs. 308 wid 311.

7.7

Applications

7.7.1 Model systems The earliest applications of the replica integral equation approach date back to the beginning of the 1990s. They focused on quite simple QA systems such as hard-sphere (HS) and LJ (12,G) fluids in HS matrices (see, for example: Refs. 4, 286, 200, 298, 303, 312, and 313 for rcvicws). From a technical point of view: these studies have shown that the replica integral equat,ions yield accurat,e correlation functions compared with parallel computer simulation results 1292, 303, 314, 3151. Moreover, concerning phase behavior, it turned out that the simple LJ (12,G) fluid in HS matrices already displays features also observed in experiments of fluids confined to aerogels [131, 1321. These features concern shifts of the vapor liquid critical temperature toward values

362

Statistical mechanics of disordered conflned fluids

significantly smaller thaii in the bulk, an ucconiyaniying iricrcasc of thc crit*ical density, and a narrowing of the coexistence curve. Motivated by this success, a series of inore recent replica int,egral equation studies has focused on the effects of more realistic features of both the adsorbed fluid a i d its interactions with matrix. Examples are studies of the influence of teinplated matrix materials [316], associating fluids [317], LJ mixtiires [114, 3111. and QA systems with ionic int,cractions [905, 306, 3183201. Howcver, until rcccutly, only one study [300] has bccn available on QA systems with angledepondent (specifically anisotropic: steric) interactions. In t8hischapter we discuss recent replica integral equation results for an adsorbed dipolar model fluid [307, 310, 3211. Specifically, we consider the socalled Stockmayer fluid consisting of spherical particles interacting with each other via both the (spherically symmetric) 1,J (22.6) potential [see Eq. (5.24)] and long-range, anisotropic dipolc dipolc intcractions gcnoratd by point dipolc‘ /A, cinbcddd in tlic c:cntcr of t,hc particlrs. The dipolar iritcraction is given by Eq. (6.21). and the potential energy between a pair of Stockmayer molecules is given by

The hard core iu Eq. (7.59) lias bccn irriposcd for iiurricrical corivcuicricc. As a consequence, it is mainly the va,n der Wads-like attractive (rather than the repulsive) part. of the LJ (12,6) potential (a T - - ~ ) that contributes to the fluid fluid pot.entia1. The strength of the dipolar relative to the attractive LJ interactions is conveniently measured by the “reduced” (i.e. , dimensionless) dipole moment m = p / m . Depending on this parameter, the Stockmayer fluid may serve as a siniplc modcl for polar molccnlar fliiids (258, 2591 (small m.)or for fcrrofluids [227, 2283 (largc m).Hcrc we consider a system with dipole moment rn = 2, which is a value typical for moderately polar molecular fluids [259]such as chloroform. For this value of rn, GCEMC simulations have been presented in Section 6.4.1. In what follows we discilss the phase behavior of the Stockmayer fluid in the presence of disordered matrices of increasing complexity. All results are h a s d on a variant, of the HNC qiint,ion [see Eq. (7.40)1, which yiclds very good results for bulk dipolar fluids [268, 3221. Moreover, subsequent studies of dipolar hard-sphere (DHS) fluids [defined by Eq. (7.59) with ‘ULJ = 01 in disordered matrices by Fernaud et al. [323, 3241 have revealed a very good performance of the HNC closure compared wit,h parallel computer simulation results. The integral equations are solved numerically with ail iteration procedure. To handle the multiple angle-dependence of the correlations

Applications

363

functions, one crnploys cxpansioiis in strcallcd rotational iiivariants, using the same steps that are well established for bulk dipolar fluids [268, 3221. In particular, the RSOZ equations can be formulated as a system of linear equations for the expansion coefficients. This is outlined in Appendix G.3. For further details concerning the numerical procedure, we refer to Refs. [310] and [307].

7.7.2 Dipolar fluids in simple matrices The simplest disordered mediuin is realized by a niatrix consisting of positionally quenched hard spheres. In this caw, both the matrix matrix and the resulting fluid matrix interaction are given by Eq. (7.43). For simplicity we assume that matrix and fluid particles have the same diameter. Tha inflitcncc of the IIS matrix on thc phase bchavior of an adsorbed Stockmayer fluid is checked most easily by invcstigat,ion of the stability limits (spinodals) of the isotropic high-temperature phase. Indeed, localization of true phase cocxistence lilies is significantly more difficult because of the lack of a closed expression for the pressure within the replica HNC approach. Typical results for HS matrices of different porosity are displayed in Fig. 7.2 (3101. The S-like shape of the corresponding bulk curve reflects the presence of two typcs of phase transit,ions within the density range considered: a gas liquid transition appearing a1 low and intermediate fluid densities (corresponding to the hill in the spinodal) and an isotropic-teferroelectric (IF) transition occurring at higher densities. These RHNC predictions are consistent with computer simulation results on Stockmayer fluids. This concerns in particular the vapor liquid critical teniperature, the RHNC estimate for which is very close to the corresponding GCEMC estimate [268]. Condensation/cvaporation in the biilk is signalad l y a divcrgcncc of thc isothermal compressibility K T . For adsorbed fluids, we find clear evidence for a gas liquid transition only at very small matrix densities, e.g., at pm = 0.1, which corresponds t o a porosity typical for a silica aerogel (where gas liquid transitions have indeed been observed experimentally (131, 1321). In more dense matrices, the cornpressilJilit,yremains small for all temperatures considered, which is consident with recent. lattice fluid studies of Kierlik et al. (285, 286) qucstionirig the cxistcncc of gat-liquid trarsitioiis in dciisc disordcred porous materials regardless of the nature of the fluid. These considerations motivated us to focus mainly on the dilute matrix case in what follows. In particular, one observes from Fig. 7.2 that the (gas liquid) critical point of this system is displaced toward lower temperature and lower deiisity where the latter shift essentially disappears if additional attractions between fluid particles and t.he matrix are included in the model [310].

Statistical mechanics of disordered conflned fluids

364

2 1.8

1.6

h 1.4 1.2 1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 7.2: Replica HNC results for the temperatures corresponding to the stability limits of the homogeneous isotropic phase of confined Stockmayer fluids (and for the bulk) as a function of the renormdid fluid density (9 = 7rpu3/6). Curva are labeled wording to values of the reduced matrix densit.y pm.

These trends are similar to what is observed in simpler model fluids with purcly sphcrirally symmctrir iiit,critctions[298, 3131, which is to somc cxtcnt expected because the gay liqiud transition in Stockmayer fluids is mainly driven by the isotropic LJ (12,G) interactions underlying this model. W e show in Ref. 307 that the main effects of HS matrices on the condensation can be reproduced when the dipolar model fluid is Bppraximated by a fluid with angleatreraged dipolar interactions that are not only spherically symmetric but also short-ranged (they decay in proportion to T - ~for T -, 00). This notion is particularly importarit for future siniulation studies on adsorbed dipolar fluids.

At high fluid densities, bulk Stockmayer fluids exhibit an IF transition, which is signaled by a divergence of the dielectric constant CD [seeEq.(7.30)]. Results for CD are displayed in Fig. 7.3,which suggests that the IF transition

365

Applications

240

r l

1

I

1

1

I

0.1

I

I

I

200

1

I

160 CI

w

120

80 40 0

0.3

0.4

0.5

0.6

0.7

0.8

1IT Figure 7.3: Dielectric constant ED versus (inverse) temperature for Stockniayer fluids at fixed fluid density p = 0.7. Curves are labeled according to values of the matrix density.

occurring in bulk Stockxnayer fluids survives regardless of the matrix porosity [310]. Moreover: the transition temperatures indicated by the divergences of CD increase with increasing matrix density. To shed some light on this puzzling rcsiilt, wc!pcrforinad a dc!tdlcd st,iidy comparing partly qiianchcd and fully equilibrated mixtmes [310]. From this study it turns out that the shift of transition temperatures is mainly caused by the restricted volume nccessible, which leads to an increased tendency of the fluid to form ferroelectric clusters in matrix-free regions of space. Similar conclusions have also been drawn later for a related model system, namely a DHS fluid in an HS matrix [323].

366

Statistical mechanics of disordered confined fluids

7.7.3 Dipolar fluids in complex matrices More dramatic effects arise when the perturbation induced by the disordered matrix couples directly to thc dipole rnonients of the fluid particles. Charged matrix particles provide an example. Their inipact on a DHS fluid has been studied by Fernaud et al. [323]. They report a significant decrease of the dielectric const,ant arid an enhanced teiidency of dipoles to form aggregates at low densities. Anot,har int,crcsiting case are dipolar fluid matrix intmwtions whcrc each fluid particle i‘fc(:ls’’ h t , h tlic dipole ficlds of its fluid neighbors and the additional dipole fields arising from the adsorbing medium. Understanding t,he resulting interplay (or competition) between these interactions is relevant not. only from an academic point of view but also from the perspective of adsorption processes in experimental sysbems. Indeed, dipolar fluid matrix interactions play a central role in purification processes such as liquid chromatography where polar liquids are adsorbed by disordered rnatmials ciompthsscd of molcciilos wit,h polar hondgroups [325]. The siriiplcst matrix gcricrat irig disordered dipolar ficlds corisists of a syst,em of DHS, which are quenched from an equilibrium fluid configuration at, quenching temperature To. At, t,his temperature, the coupling between two matrix particles is given by

whcrc I L D D ( r ,pL,,,l, p,,2) is thc dipolar intcractioii bctwecn two rriatrix particles. Throughout this work we ilSSiime the interaction strength & , / k ~ T a ~ t,o have values such that the matrix is homogeneous and isotropic on average, that is. when averaged over different matrix configurations. Finally, the resulting fluid matrix coupling at. temperature T is given by

on the stability limits of a polar StockThe effect of the variatioii of maycr fluid, which irnplics a variation of t,ha dipolar fluid matrix conpling, is illustrated in the upper part of Fig 7.4. All results correspond to dilute matrices that do not suppress the gas liquid transition but lead to a significant. shift of that tramition. In particular, the critical temperature decreases with increasing pm, whereas the critical de1isit.y increases. This characteristic effect is referred to as “preferential adsorption” in other contexts. The replica integral equation results t h i s demonstrate, at a microscopic level. that polar

367

Applications

h

27

I

I

I

26

I .5

I

I

I

I

25

n w

24

0.5

23 22 21

-

Prn I

I

I

I

I

;

-

2

I

2

Prn Figure 7.4: Top: Replica HNC predictions for the stability limits of the homogeneous isotropic phase of Stockmayer fluids adsorbed to disordered DHS matrices of density p,,, = 0.1. Curves are labeled according to the reduced matrix dipole moment p L / k ~ T o (the o ~ pure HS matrix corresponds to pm = 0). Bottom: Dielectric constant of a dense adsorbed fluid as a function of t,he matrix dipole moment (T = 0.5,p = 0.7, pm = 0.1). The inset shows the integrated blocking part of the dipole dipole correlation function.

368

Statistical mechanics of disordered confined fluids

iritcrwtionu (arising, for cxamplc, fro111polar headgroups on the rrioleculw) between a fluid and the disordered adsorbing material act essentially as a net attraction, as long as dilute or moderately dense fluids are considered. This notion is further supported by results for fluids with angle-averaged dipolar fluid matrix interactions as discussed in detail in Ref. [307]. At high fluid densities, the directional dependence of the dipolar fluidmatrix intrractions doniinatcs thc proprrtics of thc ndsorhcd flilid. Specifically, oric obwrvw that cvcri smdl values of 11: ykld a pronounced decrease of the dielectric constant ED (see lower plot in Fig. 7.4). This r m l t reflects the decreasing ability of the fluid to respond to an external field. Additional signatures of the disturbance of dielectric properties are the appearance of a blocked part of the dipole dipole correlation function (see inset in lower part of Fig. 7.4) and, more directly, the growth of magnitude and range of the dipolar corralation fiiiiction bctwccn thr fluid and thr matrix (scc Rcf. [307] for dctails). Not surprisingly, thim effects beconic particularly pronounced at low temperatures. where bulk Stockmayer fluids as well m Stockniayer fluids in neutral matrices exhibit an IF phase transition. A clear sign for such an instability is a divergence of the dielectric constant, which is observed only for relatively weak dipolar fluid matrix interactions, i.e., fluid matrix interactions that are significantly weaker than those between fluid particles. This divergence suggmts that larger fidds dcstroy the ferroelectric ordcriIig as one miglit, iridccrl expect 0x1 physical grounds. However, cvcn for thc weakly disturbed systems, it seems likely that. the frozen dipolar matrix fields influence the structure inside the low-temperature ferroelectric state. Based on work on related systems such as nematic liquid crystals in disordered silica matrices [281-2841 (where the silica particles induce local ordering fields), one could imagine that the long-range ferroelectric ordering typical for bulk fluids is rcplaccd bv some typr of short-range or “qiiasi”-long-ranp order charactcrizcd by a power-law dccay of tlir corrclution fuuctions. In our opiiiion, a closer investigation of the nature of these low-temperature systems is presently beyond the replica-RHNC approach. This opinion is based on technical reasons because the 0 2 equations become too complex. In addition, more studies, and in particular simulation data, are required to test and improve the closure approximations under strongly coupled conditions (including a disciwsioii of rcplica-sytnmctry hrcnking [294]).

Reviews in Computational Chemistry Kenny B. Lipkowitz &Thomas R. Cundari Copyright 02007 by John Wiley & Sons, Inc

Appendix A Mathematical aspects of equilibrium thermodynamics A.l

The trace of a matrix product

Consider two quadratic n x n matrices A and B with elements a,j and bij, respectively. The trace of their product is defined as the sun1 of the diagonal componcnts of matrix C = AB; that is, n

n

n

n

n

i=l

C (BA)jj n

=

j=l

Tr (BA) = 'IYG

(A4

where = BA. Equation ( A . l ) also proves that, under the trace operation, matrices commute; that is, their order does not matter unlike for the matrix product itself where, of course, holds in general. Exceptioiis are special cases where A and/or B represent the unit or zero matrices. Conimutation of matrices under the trace operation are important in Section 1.5 in the context of Legendre transforms of thermodynamic potentials of confined phase.

A.2

Legendre transformation

A convenient way of switching between various sets of natural variables (see Section 1.5) is provided by the concept of Legendre transformation. We

369

370

Legendre transformation

follow licrc thc iriorc cor~iprclicrisivcdiscussion prcscutcd in Chapter 5.2 of the excellent text by Callen 1121 to which the interested reader is referred for more detail. Siipposc~a function J (21,-cz, . . . ,.ck) = J' (x)exists siich that the domain D ( f ) C Rk. Hence, the point (f,x) lies on a (k 1)-dimensional hypersurface (i.e., a surface in a (k + 2)-dimensional space). The slope of this hypersurface in the it11 direction is given by

+

whcrc we introduce the notation "{.} \xi'' to indicnt,c that cxccpt for :ci all oblier (k - 1) variables arc being hcld constant upon tliff'crerkiating f . The goal then is to use some f; t,o specify f itself without sacrificing any information contained in the original representation of the hypersurface through 2.

In this venture the only problem is t.hat an infinite number of functions fk. In other words. the simple representation of f in terms of (some of) the { f:} is not unique but ambiguous. A unique representation is, however, feasible by realizing that the h-ypersurface may equally well be represented by either the vector x sat,isfving the rdation J = J (x)or thr cnvclopc of tangant hypcrplancs. Tlic fanlily of tmigcnt liypcrplaiics may bc c:liaractcrizcrl by the intercept of a new hyperplane g ,

f (x)+ c with c E R exist,, all of which are giving rise to the same

where fT = (fi, 1;. . . . , f;) is the transpose of the vector f. Equation (A.4) is the basic identity of the Legendre transformation. Just like the original function selects a subset of points from Rk7 the intercept selects a subset of tangent planes. This equivalence is essentially ail expression of duality between the more conventional point and the so-called Pliicker' line geometry in mi~ltidimciisionnlspace. The relation between J and y is bijective except for a sign difference. To realize the bijectivity. consider the differential

lJiilius Pliicker (1801-1868), Professor of Matheiiiatirs arid Physics at the Rheinisclie

Friedrich-Wilhehris-Univeritat Bonn. He developed line geometry which substitutes the

straight line for the point as the basic geometrical unit in space.

371

Euler's theorem

because d f = f'da:. The sign diff(:rcncc follows iImncdiatdy from Q. (A.5) because

gives back the negative of the original set of variables. One should also realize that it is iiot necessary to replace the entire set of original variables by the new ones. Rather it is possible to choose an arbitrary subset of variables { x k ' } E {xk} and perform the Legendre transformation on it. Last but not least, suppose f (21,. . . , Z k ) exists such that f and its first and sccond partial derivatives arr coatiniioiis in thr ncighborhood of a point P = (z1, . . . , . z k ) E W k . Uridcr this prcsuyposition, the thcorcrri of Schwarz holds, which states that for t$hemixed second-order partial derivatives the order of differentiation is irrelevant,: t,hat is,

Applying Eq. (A.7) to thermodynamic state functions (instead of a general function f) gives rise to the celebrated blaxwell relations. They can be used to express certain quantities that arc hard to measure or control in a laboratory experiment in terms of "mechanical" variables such as a set of strmscs arid strains arid their t*cmpcraturcarid dciisity depcndcncc.

A.3 Euler's theorem When obtaining closed cxprrssions for t4hcrmodvnamicpot,cnt,ials, the concept, of hoiiiogciieity of fuiictioiis is of kcy iniportaricc. Suppose a function f ( t l , 2 2 : . . . 2,) exists such that, its domain D (f) C W". This function is called homogeneous of degree k if it satisfies the equation

f (Aq,

AI2!.

. . , Ax,,)

= Akj

(51,22,.

. . ,z,) ,

AEW>O

(A.8)

Differentiating both sides of Eq. (A.8) with respect t,o A, we obtain Euler's theorem ,

-=

Ex,(-)af ?I

2=1

a(AxJ

= kAk--'f

{ }\z*

(A.9)

which holds for all k 2 1. In particiilar, for t.hc sprrial caw k = 1. wr obtain from the previous expression

(A.lO)

372

Euler’s theorem

as a spccializcd Eulcr’s tlicorciii for l~o~nogcncous fuiictions of dcgrcc 1.

Reviews in Computational Chemistry Kenny B. Lipkowitz &Thomas R. Cundari Copyright 02007 by John Wiley & Sons, Inc

Appendix B Mathematical aspects of statist ical thermodynamics B.l

Stirling’s approximation

Consider the function

(x)defined through the expression

r

E

/

cxp (-1) LZ-’dl

(B.1)

0

where it, is easy to verify that by partrial integration that

r (1) = 1. It is also straightforward to show exp (-t) tTdt =

(x)

0

Thus, for z = N E N,we may write by recursively applying Eq. (B.2). Considcr now thc intcgral

A’! =

I 0

exp(-l+ 1InN)dt =

7 0

exp[-Ng(t)]dt

(B.4)

where it is clear from Eq. (B.l) the function g (t) must have a niaximum for some t = t,, because the integrand consists of a product of a monotonically decreasing [i.e., exp (-t)] and increasing function (i.e., t N ) .respectively.

373

Elements of function theory

374

Bccausc of t,hc definition of (1) givcii in Eq. (B.4)’a straightforward calcuilation gives t,,, = h’. Expanding g ( t ) around the maximum at g ( N ) in a Taylor series and retaining terms only up to second order, we find

where the far right side follows from the definition of 9 ( 1 ) in Eq. (B.4). From Eqs. (B.4) and (B.5)’ w(?find

M!

N

exp ( NIn N - N )

Irn (exp

= exp (A’ In N - N )

exp

)

(t - N)2

2N

(-&)

dt

dt

where we changed variables according to f -+ t = f - N between the first and tlic sccontl liiics arid took thc lower liiriit ol iritcgratioii to -m because the integrand in the second line is extremely peaked around 0 for large N. Thus, in this limit, we have Stirling‘s approximation from the previous expression; In N != N In N - N 0 (In A‘) (13.7)

F=

+

where the last term may he neglected in the limit N -, 00. The reader may verify that, even for relatively small N = 100 [compared with the therinodynaniic limit in statistical physical applications where N = 0 ( or even niore], the value of In N! calculated froin Eq. (B.7) deviates by less than 1% from thc cxart rcsiilt,, which ilhistrat,csthr poww of Stirling’s approximation.

B.2 Elements of function theory B.2.1

The Cauchy-Riemann differential equations

Consider a complex function

f(z) 2

= PI+)

+w(2)

= x+?y,

zEC:s,yER

(B.8s) (B.8b)

where the real arid iriiagiriary parts arc’ giwri by IL ( z ) = Ref ( z ) )arid v ( z ) = Iinf ( z ) , respectively. We define the first derivative of f ( z ) throiigh

f’( z ) =

h i

A2-0

J’(z

+ Az) - J’ AZ

(2)

Mathematical aspects of statistical thermodynamics

375

which is f o m a l l y equivalent to the dcfiiiitioii of tlic: first. derivative of a real function. Functions f ( z ) for which the above limit exists at some point z are called dzerentiable; functions for which this limit exists in some area B of the complex plane are called unalgtic or mgular. However: because of J3q. (B.8b) there is essentially an infinite nuniber of ways to approach the point z bv making Az smaller. This nonuniqueness of thc limit, in Eq. (B.9) is a conscqiiciicc of thc fact, t,hat,onc is piirsuing a, pat,h in t4hecorriplcx plane wlicrc an iIifiiiitc dcgencracy of such paths exists. For the limit in Eq. (B.9) to exist, it. is necessary that the specific path Az -+ 0 in the complex plane be irreleva.nt. Let us consider two distinguished paths, one along the real and the other one along the imagiriarv axis of the coinplex plane. The first of these is characterized by Az = Ax and Ay = 0 so that we have from Eqs. (B.8b) and (B.9)

f’(z)

+

+

f ( X A X iy) - f (z + 29) = lini AZ Ar-4 Ax u (X + Ax + il~)+ 1 : (X ~ + AX + Zy) - u (J + iy) - iv ( X + iy) = lim Ax-0 AX u ( X + AX + ZIJ) - IL (.E + i y ) = lim Ax+O An: V ( S + AX +iy) - V ( X + i y ) +i lim Ax-4 Ax 5

liin

f ( Z + Az) - f (2)

Az+O

(B.lO) Along the second path in the complex plane we have Ax = 0 so that AZ = iAy. By exactly the same reasoniiig one can then show that in this case 1 all j ’ ( z ) - -1

+a!/ a?/ &I

(B.ll)

Accortliiig to our suppositioii tlic two cxprcssions in Qs. (B.lO) aiid ( B . l l ) must be equal. For this equality to be reached: the real and imaginary parts of both expressions to be equal which gives rise to the so-called CauchyRiemann differential equations (B.12a) (B.12b) Theorem B.l then follows.

376

Elements of function theory

Theorem B.l A functaon .f ( z ) dcJined u w r u rlu71ium R ( f ) C C is analytic i f its partial derivatives with rasped to z and y exist and the Cauchy -Riemann dafemntial equations are satisfied.

Proof B.l Consider. AS ( z ) = Au (2,y) + i A v (x.y )

(B.13)

where

Au

E

u (x+ AT.y + A y ) - u (x,y) = -Ax i3U

8z

so that

-

+ -dAU y dY

(R.14a)

(B.15)

A2

from which

follows, which obviously is independent of the specific path A z + 0 in the complex plane, which completes the proof of Theorem B. 1. q.e.d.

B.2.2 The method of steepest descent Lct us itssu~~ic' a complcx function .f ( z ) exists [scc Eqs. (B.8)] such thata 1. f ( 2 ) is analytic (set. Appendix B.2.1) in a domain B of the complex plane. 2.

-m at thc cud of a path (J, whcrc C pcrtaiiis to thc domain B but is arbitrary because f ( z ) is supposed to be analytic (see

/ ( z ) --+

Theorem B.l).

With this function we now seek t o evaluate t8heiiit,egral [cf. Eq. (2.22)]

lirri I (N)

N-CC

G

lirri N-CG

J

C

dr: cxp [Nf

(2)]

(B.17)

377

Mathematical aspects of statistical thermodynamics

~ l iexMorcovw, supposc a poirit z = zo exists such that. R c j ( z ) a ~ ~ u i r iiw tremum and Jinf ( z ) = u (zolyo) = vo ‘Y const. Because of the definition of f ( z ) [see Eqs. (B.8)], this also implies that the integrand in Eq. (B.17) assumes a maximum at the point I = zo. The necessary condition for an extremum of Ref ( z ) to exist, may be stat’ed more explicitly as

(B.18) To find out what is the nature of the extremum, we consider (B.19a) (B.19b) where we obtain the right. side by using the Cauchy-Rieniann differential equa.tions [see Eqs. (B.12)). Because the order of different.iationis irrelevant: the previous expressions can be coinbined to give (B.20) which tells us that the curvature of the twedimensional surface u (ZO,yo) along t,he real axis is equal to t,hc riirvat,iuc of u (.cOl yo) along the imaginary axis. In other words, tlic poiid z = zo is a saddlc point,. To evaluate the integral in Eq.(13.17) we now specify the path C according to two criteria, riariiely 1. C should pass through t8hesaddle point such that u (TO,yo) becomes ma.ximum. 2. Along C , 71 (50,yo) = 2’0 N const. Thcse conditions cause C to be the path of “steepest descent” from the saddle point. To achieve this result we need to establish a relation between the rcal and imaginary parts of f ( z ) .

Theorem B.2 Ref ( z ) = u (r,g) and I n l f ( z ) = through v u (z.g) * vv (z,y) = 0

71

(x,y) are related (B.21)

378

Elements of function theory

Proof B.2 Using thc Cuuchy-Rtcrriuiin cliflcreiLtiu1 cquations, one wulizes t h d ihr - - -au auGu -- --ax ay ay; aya y

au

ax

completing the proof. q.e.d

Theorem B.3 Consider (I fmction, g (x,y) specifying surface; then V y Ig ( 5 0 . yo) = con,st.

0,tioo-dirn.e7~sionel

Proof B.3

where rT = (,;I. y ) is a two-dim.ensiona1 vector. For special cuts through the surfuce y sutisfyiny g ( 5 0 , yo) = const, d g = 0 , and therefore Vg . d r = 0 so that the vectora Vg and d r must be orthogonal. q.e.d. Because of Theorem B.3, V v is perpendicular to the line v0. Theorem B.2, on the other hand, tells IIS that V u and Vu are orthogonal, so that any line 21 = const niust also be tangential to Vu. Thus, lilies along which t~ = const corrcspond to thc stccpcst, drscont from thc sadrllo. Let us now expand f ( 2 ) in a Taylor series around 2 = 20 according to

w1ic:rv wc’ rc!tain tcrnw only 111) 4 o sc~*ontlorclcr in z - z0 and thc: lincm tc:rni is missing because of Eq. (B 18) aid our supposition u = Z~O(i.e., ai*/r3z = 0) from which f’(z0) = 0 inimediat,ely follows with the help of Theorem B.l. Inserting Eq. (B.22) into Eq. (B.17), we obtain

= exp[NJ(~o)]/dzexp

i;.

‘1

( 2 - 20) [-1Nlf”(zn)l 1

(B.23)

where the second line follows because J (z) has it,s rnaxirnuin at z = zo, c = 210 = const. Therefore f ” ( 2 0 ) < 0 and is a real quantity. Let us also assunie that in the immediate vicinity of z0 the path C can be chosen SO that it is parallel t,o the real &xis and then z - :0 z z - 20. Under this assumption

379

Mathematical aspects of statistical thermodynamics

arid bccilusc we arc int,crcstcd in tlic iutcgral in thc: limit N replace &.. . dz 4 J-”, . . . dz, which then gives

we nlay

--t r ~ ) ,

(R.24)

where l3q. (B.102) has also been used.

B.2.3 Gaufl’s integral theorem in two dimensions To prove Theorem B.5 in the subsequent Appendix B.2.4, we first need to prove another t,heorem from GauR. Suppose we are given two functions f (x.y) and g ( 5 ,y). We assume both f (2,y) and g (x, y) as well as their first derivatives to be continuous in a simply connected domain D where D is bounded by a piece-wise continuous curve C. Specifically we assume D to bo rcproscnted by thc sct

D

= { (x,y)[ a

5 z 5 b v !Dl (z) I y I !D2 (x)} c R2

such that. the contour C of D is described by the functions Alternatively we niay specify

D

= { ( 3 ,I/)[ c5

\kl

5 d V @i (y) 5 1 I@z ( 9 ) ) C It2

(x) and

(B.25) \k2

(x).

(B.26)

as indicated in Fig. B . l . Theorem B.4 by GauB then asserts the following.

Theorem B.4

where it is important that, the contour C is traversed in a counterclockwise fashion as indicat,ed in Fig. B.1. Gad’s integral theorem can be proved as follows.

Proof B.4 Consider first the integral [see Eq. (B.25)] (B.28)

380

Elements of function theory

Y4

I : a

b

X

X

Figure B.l: Sketch of a siniply connected domain D bounded by a piecewise continuous curve C. The rontour C is traversed in a cminterclockwise fashion as indicated by the arrows. In (a) the contour is described by the functions Q1 ( r )and 9 2 (z) in the interval { z)a 5 z 5 b}, whereas in (b) the contour is specified alternat,ivclv by the fiinctions @, (y) and (y) dafinrd on thc int,arval { yI c < y 5 d } .

Mathematical asDects of statistical thermodynamics

38 1

Noticing that the function .Q (z, 3) iy the antiderivative of the integrand on the right side of the previous exprtwion, we have

n

h

n

a

b

after interchanging the limits of integration on the second integral. Because the functions 9 1 (2;) an,d \kz (:c) describing the contour C appear as arguments of g , it is clear that the two conventional integrnls appea7ing on the right side of the previous eqression are equivalent to the line integral along C except for a sign difference because C is traversed in a clockwise rather than a counterclockwisefashion (see Fig. B.l). Hence, we have (B.30)

By the same token we m.ay write

because J ( x ,y) is the antiderivative of i )j/i):c.As before the two conventaonal integrals appeariny on the right sade of the pvttrious expression represent the line integral along C. However. in this case C is traversed in a coun,terclockwise fashion (see Fi.9. B . l ) i71 accord with our original supposition, so that we m.ay write (B.32)

Putting together Eqs. (B.30) and (B.32) yields Eq. (B.27) which completes the proof of Theomm B.4. q.e.d. We note in passing that GauB’s theorem applies to domains in arbitrary dimensions. For example, in electrostatic or hydrodynamic problems, one is frequently confronted with the change of charge or mass density inside a three-dimensional volume. Using GauB’s theorem this change is equivalent to the net flux through the surface of this three-dimensional volume.

382

Elements of function theory

B.2.4 Cauchy integrals and the Laurent series Coilsidcr a cornplcx furict,iorif (z) as dcfiiied iii Eqs. (B.8).

Theorem B.5 Iff ( z ) is analytic in some simply comected domain 13, the integral over J (z) along some closed path C pertaining to D vanishes. Proof B.5

f

dzf (z) = =

+ f kdril(r.s) dr

[?I.

(z, y)

(x,y)] (d.7: + id!/)

- / d y u ( x , y ) + i ~ d y u ( x . y ) + i fdxz.i(z,y)

Using GauJl’s theorem [.see Eq. as

(B.Y?)]we may rewrite the original integral

A s f (2) is analytic, the Cauchy-Riemann diflerential equations are satisfied so that each of the two integrals above vanishes identically rqgardless of the specijic choicc of the path C.q.c.d. Consider now a domain D ,which is no longer simply connected. Such a situation arises if f ( z ) is analytic everywhere in D except at some point 2 = z1 where J ( z ) is supposed to have a singularity (see Fig. B.2). Then, in fact, thc intcgal around any closccl path in I) siirrounding the singularity does iiot vanish but, one iriay iristeacl define the so-called residue (B.33) which vanishes in the absence of such a singularity according to Theoreni B.5. Howcvcr, similar to tlic proof of Thcwrcni B.5it can I c shown that thc prccise path along which lhe residue is calculated is irrelevanl for its value. Suppose now f ( z ) is analytic across a simply connected domain; then it is immediately clear that, if we pick a point z = 20 in that domain, the quanthy f ( z )/ ( z - 20) will have a singularity a t that point. Because of the above, the integral over f ( z ) / ( z - zo) along aiiy closed path surrounding z = 20 will have some nonzero value that we seek to calculate. Because the closed path C surrounding z = zo is arbitrary, we take it to be a circle of

383

Mathematical aspects of statistical thermodynamics

Figure B.2: Sketch of closed contours Cl,C2,and C, in the complex plane surrounding a singularity at z1. Arrows indicate the direction in which the paths along the contours C1,Czl and Cgare traversed. radius R around that point,; that. is: z = a Thus, we obt.ain

2n

= i / d l j [to 0

+ R exp (it) or dz = iR exp (it)dl,.

+ Rexp (il)] "go27rif (a)

(B.34)

where we set R = 0 in the lwt step because the path C surrounding z = ZIJ is arbitrary and may therefore be taken to be a circle of radius R = 0 without loss of generality. With the transformation a + z and zo C we recover Cauchy's integral forniula from the previous expression,

-

(B.35)

384

Elements of function theory

Figure B.3: Two circles K I and K2 enclosing a point z = 3. Also shown is a closed contour C, in the complex plane where the arrows iiidicate the direction in which the path dong the contour is traversed; z is a point enclosed by C,.

Consider now the Cauchy integral (see Fig. B.3)

wherr I(.) is analytic cvrrywhcrc in the domain surroiindcd by thc ciirvc C, and K1 and K z are two two circles w indicakd in Fig. B.3. Outside the domain surrounded by t,hecurve CRand inside the circle K z , .f ( z ) is supposed to have a singularity at a. point z = t o . The negative sign in front of the second integral is a consequence of the fact that along the second circle the integration is performed in a clockwise fashion. Became the direction along the paths along Cgconnecting the circles Kl and Kz are traversed in opposite

Mathematical aspects of statistical thermodynamics

385

directions, one can XbWC that, coritributiorls to the integral aloiig <Jg froin the connection between K1 and I(z vanish. The first integral in Eq. (B.36) can be rewritten as

where wc used thc fact that

2 n=O

1

.z? I = -, 1 - .c

1.1 < 1

(B.38)

as one can verify by expanding the right side of Eq. (B.38) in a Maclaurin series. In a similar fashion we can rewrite the second integral in Eq. (B.36) as

whiew we used ttc fact that thc curvc surrouiiclitig thc singularity is arbitrary and may therefore coincide with the curve K 1 in l3q. (B.37) wilhout loss of generality. The two expressions in Eqs. (B.37) and (B.39) may thus be combined to the so-called Laurent. series expansion m

(B.40a)

(B.40b) The Laurent expansion is very useful in analyzing the nature of singularities. However, a discussion of this aspect would go way beyond the scope of this Appendix. Therefore, we emphasize only the relation of the Laurent series

386

Lagrangian multipliers

to t,hc rcsiduc introduced in Eq. (I3.33). Frorii thc coclficicnts of thc Laurcnt expansion, it is immediately obvious that

(B.41) is identical with the expression in J3q. (B.33).

B.3

Lagrangian multipliers

The problem of finding extrema of a function f ( T I , 2 2 , . . . , x , ~ z ) f (2)subject to 71 constraints may he solved by using the iiicthod of Lagrangian multipliers. In the absence of such coiist8raintms the necessary condition for the existence of extrema may be stated as

(B.42) from which

f‘ = 0

(B.43)

follows bccaiisc d z is arbitrary. If a sct of n coiist.raiiittjspocifictl by

p,(z)=O,

j = 1 - ..., n

(R.44)

has to be satisfied simultaneously, the variables {xi} are riot independent anymore and Eq. (B.43) is no longer the correct necessary condition for the cxistciicc of tliesc cxtrcma. However, consider a new function

I: (4 = P (4+

c n

A,%

($1

(B.45)

,=1

=O.

see Eq (B.44)

where {A,} is a, set of n undetermined Lagrangian multipliers. The necessary condition for the existence of an extremum of F may be stated as

Notice that zeros of d F in Eq. (B.46) immediately satisfy the constraints specified in Eq. (B.44) unlike its counterpart df in Eq. (B.42). If we now

387

Mathematical aspects of statistical thermodynamics

spwify tlic Lagrangian rriultiplicrs such that, thc! tcrin iii brackets 1. . .I v m ishes identically, da: may assume arbitrary values and the necessary condition for the existence of an extremum can be restated as

where the matrix

cpl

is given by

and AT ( A * , A2. . . . , An) is (the t,ranspose of) a vector of n elements. Whet,her the set, of linear equations given in Eq. (B.47) has a solution (vector) A depends on the ranks r and R of the functional matrices c p l and cp', mspcctivdy, which must, satisfy

for such

it

solution to exist.

which we obtain by adding the vector -f as the rightmost column to the original matrix c p l . Hence, as cp' is an m x n,matrix. @' is an rn x (n+ 1) matrix. The latter can therefore either have rank R = 7- 1 or R = T . In the former case it would be possible to form at least one nonvanishing subdctcrminant of order (n+ 1) x (71 + I ) of thc inatrix 9 (nssnining 7th > 7 1 ) . HCIIC'C, oric iiccds Lo verify that this is impossible. That R = T for matrix @' can be dcrnorwtratcd if oiic cinploys the following properties of determinants, which we sumniarize without proof as follows:

+

2. Multiplying a determinant by some number A is equivalent to multiplying all elements in one of its rows (columns) by that number.

388

Lagrangian multipliers

3. A dctcrminant rcrriains uiicliaiigctl if OIIC Iiiultiplics one of its rows (columns) by some number A and adds it to another one. 4. A determinant vanishes if any two rows (columns) are ident.ical.

Because of statement 1, all others hold for rows as well as for columns of a detcrminant . Consider now the (n, 1) x ( n 1) subdeterniinant of matrix @' and multiply it by a product. of numbers according to

+

+

(B.51) which follows with the aid of statement, 2 above. From skatement 3, it, is then clear t.hat if we add columns 1 to n - 1 to the n.th column t8hedeterminant remains unchanged. The nth column then contains elements

1 = k, ...,k + n + I j=l

Howcvcr, bccniisc. of Eq. (B.47) I,hr rlcmcntfsof thc ntJi rollinin arc idcritical to the ones forming column n 1 regardless of k such that all ( n 1) x ( n+ 1) subdeterminarits of @' vanish on account of statement 4 of our list of elementary properties of deterniinants. This, in tiirIi, proves that the rank of matrix is equal to the rank of matrix 9'. If, in addition, one can show Lhat R = 7' = n, a unique solution of Eq. (B.47) exists for the set of Lagrangian multipliers { A J } . However, whether the rank of the functional matrix cp' equals the number of constraints depends on the specific problem and needs to be investigated separately for each case under consideration.

+

+'

+

389

Maximum term method

B.4 Maximum term method Consider the sum

M i= I

(B.52)

where we assume that all summands s, > 0. Suppose a term smox exists. Then the inequality Smm

must hold. Let now R",

< s < Ms,,

= 0 (P)

so that the inequality in Eq. (B.53) may be rephrased as

O(M) < l n S < O ( M ) + l n M

(B.53)

(B.54) (B.55)

by taking the (natural) logarithm of all teriiis appcariiig in &I. (B.53).Hence: in the limit M 4 00, In M becomes negligible cornpared with 0 (M). Therefore, in this limit, In S is bounded froin above and below by the same number 0 (A!). Hence, we find In S = Ins,,, (B.56)

meaning that the sum in Eq. (B.52) can be replaced by its maximum term.

B.5

Basis sets and the canonical ensemble part it ion function

B .5.1 Parseval's equation

Suppose we are given a complex function f (x)defined on an interval [a,b]
n=l

(B.57)

where the expansion coefficients {c,~} E C in general. From Eq. (B.57)we immediately have

390

Basis sets and the canonical ensemble partition function

whcrc we cicfiiic the scalar product through h

a

and the asterisk is used to iiidicatc the coinplcx coiljugate of thc funct.ion y (x).Because the fiinctioIis { w i ( x ) } are orthonormal, that is, (iom

Iwn ) = &nn

(B.60)

where

0, in f n 1, in. = n is the Kroiiecker syinbol, we have from Eq. (B.58)

(B.61)

Consider next the function (B.63) mid compute

n=l m=l

n=l

n=l

n= 1

n=1

N

(B.64) In t,he derivation of Eq.(B.64) we used Eqs. (B.62), (B.60), and the fact that [see Eq. (B.62)] cz = (flp,). I n t,he limit N -, 00, Eq. (B.64) constitutes Bessel’s inequality. If the equality holds in Eq. (B.64)’ we obtain Parseval’s equat.ion (B.65) 71

A special case of Eq. (B.65) is obtained if the function f itself is normalized to 1; that is,

cc;cm =1 11

(B.66)

Mathematical aspects of statistical thermodynamics

B.5.2

39 1

Proof of Eq. (2.41)

To prove the validity of Eq. (2.41) wc wsiimc? that the expansion of cigenfuiirtions iii tcxmis of a coniplctc orthonorinal basis exists (SCC Eq. (2.40)] so that, we have (B.67) ajt = (d% I4j )

from h.(B.62). Let us furthermore assume that in addition to Eq. (2.40) the expansion

is also possible where { b k l } is a set of constants similar t,o the expansion coefficients {uj,} in Eq. (2.40). From the discussion in Appendix B.5.1, it is immediately clear that (B.69) bkl = ( h I d j k ) = a;k whcre thc far right, side follows with thc aid of Eq. (13.67) and from the fact, that tlic functions { # l } form a coiriplctc ortliorioririal basis. Equatioii (B.69) shows that the coefficients { b k l } are related to the set of expansion coefficients { a k l } by taking the complex conjugate and reversing the order of t.he indices. Hence, it, follows froin Eq. (B.66) that

(B.70) 1

1

We now employ the expansion in Eq. (2.40) to replace the set of basis functions {&} on the far right side of Eq. (2.41) so t,hat.

Because the functions

A

($1)

are eigenfunctions of H , we have [see Eq. (2.39)]

(B.72) and therefore

=

C 1

exp ( - ~ 2 E i )

(B.73)

T h e classic limit of quantum statistics

392

Using this rcsiilt arid suinrning both sidcs of Eq. (B.71) over i, we find

=I

= C e x p (-A&)

=

Q

(B.74)

I

where we changed the summation index on the second line, wed F3q. (B.70), and the definition of Q in Eq. (2.38). Comparing Eq. (B.74) with Eq. (2.39) completes the proof of Eq. (2.41).

B.6

The classic limit of quantum statistics

B.6.1 The Dirac b-function To prove Eq. (2.97) in the subsequent Appendix B.6.2 we first need to introduce a “fiinctliontt

b (x - b) = lim

7

exp [-a ( x - b)2]

11’00

(B.75)

known as the Dirac &function’ which we introduce here through the limiting process applied to Gaussian distributions where we realize from Eqs. (B.75) and (B.102) that 00

(B.76) Note t,hat 6 ( x - b) is not really a function in the ordinary sense because

x#b z = h

0.

d ( x - b) =

00

(B.77)

tliat, is, the value of (r (x - h) is infinity and cliangcs discontiiiuously at .r = h. However, because of Eq. (B.76) the integral

7

-m

dn:f(n:)b(r-b)=

i‘

h-

dxf(r)b(r-b)

(B.78)

1

‘The “fuuction” d (x - b) was named after the English physicist Paul Adrien Maurice Dirac (1901-1984) in honor of his contributions to the development of quantum mechanics.

Mathematical aspects of statistical thermodynamics

393

is a well-defined mathematical object whore c. << 1 is a vanishiugly sirid1 quantity. We may change the limits of int#egration on the right side of Eq. (B.78) because b (x - 6) is nonzero only over a vanishingly small interval around b. We may now argue on the basis of Fig. B.4 that the inequality

.m (6 - a,) 5

1

dz f (x)5 A 1 (6 - u )

(B.79)

will always be satisfied if A4 = max f(x) and m = min f(x) are the valasxsb

a<x
ues of f (x) a t the absolute maximum and minimum in the i n t , e r d [u,b], respectively. Hcrice, some E [a, b] will exist such that

<

b

b

(B.80) n

U

represented by the hatched area in Fig. B.4. Applying this logic to the integral on the right side of Eq. (B.78) we may write

/

b+e

b-

b+c

dx f ( r )6 (z - 6) = f (t) dz 6 (x - 6) = f (t)

(B.81)

I

b-t

t

=I

because of Eq. (B.76). Hence? taking the limit e previous expression and Eq. (B.78) that

-,

0 it follows from the

T d x f ( x ) b ( x - b )= f ( b )

(B.82)

-bo

after replacing the integrabion limits by fx, which we may do on account of the “sharpness” of b (x - 6) [i.e., because of Eq. (B.77)]. By analogous reasoning it follows that in three diniensions

(B.83) holds where in Cartesian coordinates

6 ( r - T ’ ) = b .( - z‘)6 (y - y’) 6 ( 2 - 2’)

(B.84)

The classic limit of quantum statistics

394

5

a

b

X

Figure B.4: Sketch of a nonmonotonic function f (z)having an absolute maximum and minimum of height, M and m, respectively. The hatched region corresponds t.0 the area f (0(6 - a).

An interesting relation is obtained by considering the Fourier transform of f ( T ) . By analogy with Eqs. (2.84) and (2.87) we may write f(k)

=

S (T.’)

=

1

-/dT’f

(W3

/

(TI)

cxp ( - i k . T I )

d k f ( k )exp (ik . r.’)

(B.85a)

(B.85b)

where the first equation constitutes the Fourier transform of f ( T ) and the second one the inverse transformation. Tiiserting Eq. (B.85h) into Q. (B.85a)

395

Mathematical amects of statistical thermodynamics

wc obtain

f (r’) =

[/

1 dkdrf ( r )exp (-ik . r )exp (ik. (q3.

- &/drf(r)

T’)

{/dkcxp[-ik.(r-r’)]

Comparing the second line of the previous expression with Eq. (B.83) it is clear that (B.87) p * (T - r’)] b ( r - r‘)= - dk e : ~ [-ik

w3 ’ J

is the Fourier representation of the Dirac &function.

B.6.2 Let set

Proof of the completeness relation

11s now expand a coinplex fiinct,ion f ( r )in tcrnis of sonic ort,honormal of furictioiis { cpi ( r ) }[cf. Appcridix B.5.1) according to

(B.88) t= 1

where {G} E C as in Eq. (B.57). A s the basis {cp, (r)}is supposed t,o consist of orthonormal functions

/

00

dr’cp; (r’)f ( T ’ ) =

c, r=l

/

C c& 00

dr’cpi (r‘)cp, (r‘)=

= c3

(B.89)

1=l

where hzJ dcnotes the Kronerker synibol. Upon substituting Eq. (B.89) back into the original expansion in Eq. (B.88), we obtain

Comparing the right side of this expression with Eq. (B.82), we coiiclude that 00

b (T - T ’ ) =

c r=l

p: ( r )pi (r’)

(B.91)

thus proving J3y. (2.97). Finally, take as a specific example the eigenfunctions of the momentum operator (see Section 2.5) as a basis; that is cp,

(TI)

= exp (ik,. r’)

(B.92)

T h e classic limit of quantum statistics

396

whcrc k, is again taken to be the wave vector associated with the rIioiricrituiri of part,icle i . It then follows from Eq. (B.91) that

b (T - T ‘ ) =

30

exp !-ik*. (T - T ’ ) ]

(B.93)

z= I

Replacing in this expres4on the suiiimittioii by

xi

integration ac:c-ordingto

we recover the Fourier representation of the Dirac &function given above in Eq. (B.87). However, the reader should appreciate the fact that Eq. (B.91) is far more general than Eq. (B.93) because it holds for an arbitmry complete set of orthonormal functions. From that perspective it may thus be concluded that, Eq. (B.93) is nothing hiit. R st,at,cincnt ahoiit t,hc complctcncss of thc special basis represented by Eq. (B.92).

B.6.3 Quantum corrections due to wave-function symmetry As we argued in Section 2.5.3, expressioris of the form

arise in the second term of Eq. (2.101) on account, of permutations. To evaluate these integrals, we employ spherical polar coordinat.es and rewrite the previous expression niore explicitly as

30

=

27r /p2exp n

(-L) 2711k~T d p / e x p (T) dx 1

-1

fiJpexp (-L) sin (X) dp 00

-

r

0

2mkBT

(B.94)

where the factor 2 x arises because we immediately carried out the integration over t,he angle 6 in the first line of Eq. (B.94). 011the second line we

397

Mathematical aspects of statistical thermodynamics

iiitroducd tlic new variablc .r cosw. which pcrrriits us to carry out the second integrat,ion over the angle cp. On the t$hirdline we finally use Euler’s representation of complex numbers: tha
(B.95)

The final expression in Eq. (B.94) can be simplified by partial integration where we note that

= -mkBTexp

(--

p2

2711,kHT

)+C

(B.96)

where we used the transformation p -+ j 7 = p2 arid C is an integration constant. Hence, partial integration of Eq. (B.94) leads to m

J 0

m ..

Y p exp (-ni’) sin (py) dp = 2a

J exp (--u.p’) cos ( p y )dp

(B.97)

0

where we simplified the notation by introducing a G 1/2mk~Tand y = r / h . Using again Eq. (B.95) we realize that the integral on the right side of Er4. (B.97) nay bv rcwrittcii as

where we used the substitutions

With the transformation z‘

iy

z

G

1’--

-,’

= -

iY p+G

+

2a

(B.99,) (B.99b)

Z = -2, the second integral in Eq. (B.98)

The classic limit of auantum statistics

398

lm z

Rez=-oo -1---------

Figure B.5: Plot of the integration path in the complex plane used to evaluate the integral in &. (B.98). Individual portions of the integration pat,h labeled I-IV refer to contributions from integrals given in &. (B.101).

may hr rcwrittan so that its right, sidc bccomrs

-exp 2 -

(--):

[

- 00- iy/2u

-ry/za mre

xp(-azz)di-

J

-*Y/%

exp ( -n.ip) dz

2 30

(B.100)

Ly12a

after interchanging the integration limits on the second integral. This last, integral may he (valuated using function theoretical argiments. Consider closcd intqyatioii path in thv cornplcx plam as iIidicatad in Fig. B.5. Integrating exp ( - a z 2 ) along this path we may write

J

r-ry/2o

-x - t y / z a 4

exp ( - a 2 dz) " Path I

I

+

7 7 , exp (-aZ") dz

+

exp (-m2) dz

CQ-lY/lll

+ I

Path I1

Path Ill

-m-ly/2u

-m

rxp (--a:') dz = 0 Path IV

(B.lO1)

399

Mathematical amects of statistical thermodynamics

This irittcgral must vanish bccausc the iiitqptioii path criclosrrs an area in the complex plane in which the integrand does not have any poles according to Theorem B.5, which we proved in Appendix B.2.4 above. Moreover, we notice that integrals I1 and IV cancel each other because the integration is carried out parallel to the complex axis but the direction of the integration path is reversed between both integrals. This observation leaves us with integrals I and IV, which may t)c rcarrangcd to give oo-iy/2a

J

exp (-nz2) dz =

-oo-iy/2c

exp (-nz2) dz = -m

8

(B.102)

Thus, from Eqs. (B.94), (B.97)' (B.98): and (B.102) we conclude that

whire we used the definitions of n and y given above as well as the expression for the de Broglie wavelength given in l3q. (2.103).

B.6.4

Quantum corrections to the Hamiltonian function

In Section 2.5.3 we derived the semiclassic expression for the canonical partition function [see Eq. (2.110)] based on the assumption that at, sufficiently high temperatures we may replace the Hamiltonian operator by its classic analog, the Hamiltoniari fiinction [see Eq. (2.100)]. In this section we will sketch a more refined treatment of the serniclassic theory developed in Section 2.5 originally due t.o Hill and presented in detail in his classical work on s t a t i s t i d mechanics 13261. Bccaiisr: of Hill's rlaar and dctailcrl cxposition and twcause we need the firial result rnaiiily as a justification to treat confined fluids by means of classic st,a.t,isticalthermodyna.mics, we will just briefly outline the key ideas of Hill's treatment for reasons of coniplet,eness of the current. work. Our starting point is the expression

-

exp ( - , ~ f i )exp ( i k N r ~= )ex;

( - 3 ~exp ) ( i k .~r

~w () r ~ p N, ;p)

(B.104) where we use conventional notation p l/lc~Tbecause in the end we wish to focus on the high-temperature limit 7'-' + 0. The key idea then is to determine the correction U J( r N , p N ; such that in that limit the quantum

=

a)

T h e classic limit of quantum statistics

400

mechanical arialysis riiay Ijc atbandoncd iii favor of a classic one [scc also

Eq. (2.100)]. Defining in Eq. (B.104): exp (-!3k)through its power-series expansion [see Eq. (2.39)],we niav derive Bloch’s equation,

a

-exp (-0fi) a;?

exp (ik” . T ” ) = -H exp ( - P f i ) exp (ik” . T ” ) A

Replacing in Bloch’s equation the term exp

(-pfi)

exp (zk”

- T“)

(B.105) according

to ECq. (B.104). we eventually arrive at. n partial differential equation for the unknown fiinrtion 71) (r”? p”; i‘j). This calciilation requires a considerable airiouut of tedious yet straightforward algebraic riiariipulations dct8ailcdin Hill‘s book (see Eqs. (16.16)-(16.23) in Ref. 326). We shall therefore skip those steps of the derivation and just jump to t,he final result.,

ilW 8P

=

h2 [v”.(v%) 2m +j3’

pvN.( V ~ I -J 2p ) ( v ~ w ). (vNu) ( v ~ u(v”u)] ) -

*

+-ih [(V”W). p N - PW (VNU). p”] m.

A4

(a)

(B.106)

T

where we introduce shorthand iiotation (V”) = (VI,V2,.. . VN) and Vi = a/&,. In deriving Eq.(B.106) we employed the space representation of the Harniltonian operator [see Eq. (2.95)],and the fact that the classic Hamiltonian function can be split, into kinetic- and potential-energy contributions according to Eq. (2.100). Tcrriis proportioiial to V Nin Eq. (D.1OG) arisc from the kinetic part of applied to the product of terms on the right side of Eq. (B.104) (using, of course, the product rule of conventional calculus). Equation (B.lOG) can be formally integrated to give

fi

satisfying t-he boundary condition U P(rN1 k”;0) = 1 such t,hat in the limit of infinite temperature we recover Eq. (2.100) directly from Eq. (B.104). TO solve the integral equation in Eq. (B.107), we cxpand w ( r N , p N ; pin) the spirit of the Wenzel-KramersBrillouiri (WKB) formalism and write (B.108)

Mathematical aspects of statistical thermodynamics

401

whcrc thc set of unkriowri cxpansioxi fuxiclions {q ( r Np,N ;;?)} cilii be d e terrnined by inserting the ansntt int,o Eq. (B.107), which gives

-2P' (VNal). (VNU)+ 0 '' (VNU)2] dp' .

w

(B.109) Equating in this expression terms of equal power in h, one immediately finds that N N (B.llO) O Q ( r . p ;$) = 1 Inserting this result back into Eq. (B.109), the first-order function is obtained as

which is 110 lorigcr constaxit. but dcpciids on t,fic configuration of rnoleculcs and their momenta. In a similar fashion the second-order function is obtained as

and so forth where the higher-order functions quickly become intractable on account of their rapidly increasing complexity.

T h e classic limit of quantum statistics

402

0 1 1 thc basis of Ihcsc Iirst Ecw terms ill tfic cxpansion in Ey. (B.108), or10 eventually arrives at, an improved approximation of Q in Eq. (2.110) if one replaces Eq. (2.100) by

[l

+

CL2

( r N , p N ; P )ti2

+ . . .]

(B.113) In writing Eq. (B.113) we have already neglected the term proportional to a1 ( r N , p N p). ; Because this coefficient is linear in p N , it will vanish upon integration over nioinentuni subspace [see, for exaniple, Eq. (2.1ll)]. From Eq. (B.113) one realizes that the correction 0.2

( r N , p N~ ;3 h2 ) = 0 (A2)

(B.114)

which depends also on derivatives of the total configurational potential energy [see Eq. (B.112)]. Hence, the correction vanishes for an ideal gas where by definitioii I/ ( r N )= 0. In this latter case only the semiclassic correction due to synimctry propcrt.ics of the wavc fiinction survives [see Eq. (2. I 1 O)].

Reviews in Computational Chemistry Kenny B. Lipkowitz &Thomas R. Cundari Copyright 02007 by John Wiley & Sons, Inc

Appendix C Mathematical aspects of one-dimensional hard-rod fluids C .1 Distance between mirror images -

In this Appendix we will demonstrate that the distance PIP, bet,ween a pair of points PI = rT = ( z I , 2 2 ) and f 2 = r: (12,y2) remains unaltered upon reflection at some straight line in the z-y plane. Consider the situation depicted in Fig. C.l where the two points are separated by distances

-

We wish to demonstrate that the lcngth of the line fl P2 remains unaltered if the points PI and f 2 are reflected at the line y = --n: as indicated in Fig. C.l. arv r c l a t d through In gcncral tlic poiiits pi and

whew the mapping ri -+ rl is cffcctcd l y the orthogonal transformation matrix c m a sina A ( a )= sincr -coscr such that a/2 is t,he angle bet-ween the straight line of reflection and the positive z-axis as shown in Fig. C.1 for the special case ~ 1 = : Hence? from

..4

403

404

Distance between mirror images

IY

p*

\ \

,/\

L a12 \

X

\

\

\

\

\ \ \ \

Figure C.l: Two points PI = ( 2 1 , y1) and 9 ( z g , y2) are transformed into cxxresponding points Pi (z{,yi) and Pi (rh,&hrough a reflection at the straight line y = --z (- - -). The lines 99 arid Pj'Pi have lengths 7-12 and 7-i2, respectively (---).

Eqs. (C.2) and (C.3), it is immediately evident, that -sins cosa

T,cos a + ,vI sin Q --xZsin 0 + y, cos cr

,

I =

1,2

and therefore

Inserting this into Eq. (C.lb), we obtain

4 2 = J(.; -

1 2

21)

+ (xi- Z1) 1 2 -

J(X2

2

- q)

+ (y2 - 91)2 =

TI2

(C.7)

405

Integral transformations

by comparison with Eq. ( C l a ) , which coiriplctcs tlic proof. HC'IICC, we conclude that any function f(r12) depending only on the distance between a pair of points will equal 1(ri2)if the two points are reflected at, any line in the z-y plane.

C.2 Integral transformations 4 z2

JI,

-sz'2

I

Figure C.2: Sketch of the domain J (see text) over which a function f ( 2 1 2 ) is to be integrated. The direction of integration is indicated by the arrows where subdomains JI and JII are defined in Eqs. (C.10).

Consider now the integral

where 1112 = 12. - z1I such that f (q2) is invariant under reflecction along the line z2 = -21 indicat,d in Fig. C.2. Wc split t,hc domain

J

E{

( z ~ , z ~ ) I - L/2 5 21 5 L/2 v -L/2 5 ~2 5 L/2}

(C.9)

406

Integral transformations

into two subdomains, nilr~dz;

a i d write

12

as a sum of two integrals according to L/2

-L/L

L/2

L/2

-22

-L/2

-L/2

-22

P' " 11

/

111

where for region of integration is .TI and ,JII, respectively. Changing now in thc: firsl irit4cgraltlic order of' integralion, we inay write 1-12

I,=

J -L/2

LI'L

LIZ

L/2

J dzl J d z 2 f ( z l 2 ) = ~ ;

dz2 J d z l j ( z 1 2 ) =

-L/2

-22

(C.12)

-21

where the integration is still performed over the triangular area JI. Because f ( z 1 2 ) does not change if we reflect each pair of points at the line 22 = -21 according to the discussion in Appendix C.l, we may define the integral Ip alternatively over the subdomain JII and wri te

/ / LI'2

If =

L/2

d z ~ dz2f

-il2

21

(212)

=

dzl -Ll2

7

/

LI2 d 2 2 f (212) =

-LIZ

-L/2

-22

d s

d z l f (212) -L/2

(C.13)

where we used the fact that z1 arid z 2 may treated as dummy iiidices and may therefore he interchanged. From Eq. (C.13) it is then clear that, 1 x 1

If =

111

(C.14)

(C.15) follows without further ado. Hence, because our function f (212) depends only on the clzstcince z12 between a pair of point,s rather than on z1 and 22. we see from Eqs. (C.ll)-(C.13) that 22

(C.16) -t/2

4

2

Mathematical aspects of one-dimensional hard-rod fluids

407

Wc now cniploy tllis result, to prow by corriplctc iriductioii that in general (i.e., for arbitrary n E N) L/2

I,

= n.!

J

dzn,

-ti2

Therefore, we

J

-22

-;n

dz,,-l..

.

-L/2

J

dtlf

(C.17)

-L/2

U S S ~ L T I thst I,~

I,-,

=

J

dzn-,

-L / 2

J

dzn-2

...

J

dzlf

-L / 2

-L / 2

where the function f is supposed to depend only on distances zij = Izj - zil between any pair of points i and j. From Eq. (C.18) we see that L/2

I, =

J

-L / 2

L/2

dz,,In-l

= (n- I)!

J

-L/2

L/2

dztl

J

J

-2,

dzn-1

-L/2

-1

-L / 2

-22

d ~ n - 2 . ..

J

-I 4 2

dzlf

(C.19)

which we niay split into two terms according to

Comider now tohesecond siimrnand in Eq. (C.20) and interchange the order of integration betwecii the first two of its integrals similar to the t r a m formation between Eqs. (C.11) and (C.12). One obtains

(C.21)

409

Gaussian density distribution

at lhc next. stagc. Thus, one may contiriuc until uo pair of intcgals 11the same upper limit of integration. At this point, t,he original I, has been decomposed into n identical terms and may therefore be rewritten as

We therefore conclude that the form of Z,,-l hypothesizedin Eq. (C.18) implies Eq. ((2.24) for Morcovcr, the ca.sc TI, = 2; provcd explicitly abovc, is nothing but, a special case of I,, as one realizes by comparing Eqs. (C.16) and (C.24)so that E!q. (C.24)completes our proof. Finally, wc stress that the decomposition of I, according to Eq. (C.24) is based on the explicit assumption that the integrand f depends only on distances z,j and not on t,he individual coordinates z, and zj as independent, variables.

r,.

C.3

Gaussian density distribution

C.3.1

Limiting behavior of the probability density

In the grand canonical ensemble, the probability of finding N molecules in a (constant) volunie V at some fixed temperature T may be cast as

(C.25) where C is a normalization constant. Assuming that P ( N ) becomes maximum for the most probable number of molecules N = TIwe may expand In P ( N ) about this value in a Taylor series according to l n P ( N ) = hlP(hT)+

In I'

(N-N)

410

Gaussian density distribution

where the first-order derivative vaxiishcs 1,ccause we are expaiidiIig P ( N ) about, its maximum. From &. (C.25)it is also clear that $111

P

8In

Q

(C.27)

However, the fact that, the first-order term in the Taylor exp,msion vanishes equips us with an atlditioiial rclatioii, riamely

(C.28) N=E

so that we can rewrite Eq. (C.26) as

/'(N)

=

/'(N)cxp

[-%( N -",'I

(C.2 9 4 (C.29b)

In addition, we require P (N) to be properly normalized; t.hat is,

1-m bo

m

/P(AN)dN 0

=

P ( A N ) d ( A N )=

x.

7

P(AN)d(AN)

1

(C.30)

-00

where A N = N As P ( A N ) can be expected to be peaked fairly + 00 in the thermodynaniic limit, we sharply at A N = 0, and because may change the Iowcr integration limit to --oo without affecting the integral. The normalizat.ion condition leads to [see Eq. (B. 102)]

w

((2.31) With the aid of P ( N ) we can also calculate the average number of molecules via

1

(N)

=

--P

=

1 00

0.3

iVIj(N)dAN =

-m

JANP(N)dAN+E

(AN

i

+N) / ' ( N ) d A N

P(N)dAN

=w

(C.32)

41 1

Mathematical aspects of one-dimensional hard-rod fluids

whcrc thc first irittcgal vaIiishcs bccausc P (N) is ail CVCU function of A N and the second integral is nothing but the normalizat,ion condition. To determine the constant a in Eq. (C.29b),we consider the so-called second central moment of P ( N ) ;that. is,

J

00

((AN)')

=

(AN)' P ( N )dAN

-6/ [-%

-a3

=

.m

2a d T da

exp

(N - N)']d A N

(C.33)

-00

which is an identity as one easily verify by performing the differentiation of the integral and using the definition of P ( N ) in Eq. (C.29a) together with Eq. (C.31). However, because of Eq. (B.102) the previous expresion may be ram&
((AN)') =

-/--\i" 2n d K da

a.

=1

(C.34)

n

from wliicli

1 ((AN)') = ( ( N - ( N ) ) ' ) = ( N ' ) - (N)' = - = 0; (1

follows with the aid of Eq. (2.75). Hence, we may rewrite

&.

(C.35) (C.29a) as

(C.36)

To analyze the sirnulabion data in Section (5.4.2) for the isothermal compressibility, it. turns out to be more convenient to transform variables in this last expression according to N , (N) -+ p, 7 to obtain an analogous expression for the density rlistribiit,ion P ( p ) in a slit-pora.

C.3.2 Moivre-Laplace approximation

An altcrnativc roiitc to Q. (C.36) procacds as follows. Siipposc WP take a system of constant, volume at, fixed T and p. Let us perform an experiment on this system that allows us to determine the particle number N. As under conditions of fixed T and p, but where A' may vary, there is a certain probability p that the result of our measurement will be positive, that is, it will give preciselg N. Clearly, the outcome of the experiment may also be negative; that is, there is a probability 1 - p that we will measure some

412

Gaussian density distribution

other particle iiurribcr but N . We now crivision to repeat, our cxperimciit. n times. In this sequence of n measurements m are assumed to be positive (i.e., they have as a result N particles in the system) and therefore n - m measurements will yield a negative result. The probability to rrieasurc A' in the entzre sequence of n experiments is then given by

p, (n1)=

( ) , ;

= p"l(1 - p ) -

=

n! p'n ( 1 - p)"-" m!(n,- m)!

(C.37)

where the combinatorial factor arises because the precise sequence of experimcnts with positivc: and ncgat,ivc oiitcom! docs not. matt,cr in the scqucricc. Equa,tion (C.37) represents the so-called Bernoulli distribution. Wit,h the aid of Eq. (B.6)we may rewrite the cornbinatorial factor as rn! (71n! - nz)! - J 27r711 (n, " -7n) ( m ~ )

¶ t

(

n.n rri )n-m

(C.38)

such that Eq. (C.37) may he recast, as

(C.39) We now seek an approximate expression for Eq. (C.38) valid in the limit of a very large number of experiments; that is, we focus on the limit n -, 00. To derive this approximate exprcssiori we introduce

(C.40) as ail auxiliary quantity for t he subscqucmt dcxivation. Using the previous expression for C, it is st,raightforward to realize that

x In (1 -

<J"-) n.(l p) -

(C.41)

Mathematical asDects of one-dimensional hard-rod fluids

413

hi tlic limit n,+ m wc realize from thc definition of in Eq. (C.40) that we may approximate the logarithmic functions in the previous expression by expanding them in a Taylor series According to l l l ( l + q / ~ )

N

l-p c{;-Ty

('1-p

n(1 -p)

+...

(C.42a)

2 n(1 - p )

- . . . (C.42h)

which we truncat,e after the quadratic term. Inserting these expressions on the far right side of El. (C.41), we yield

(C.43)

(C.44) Neglecting in these expressions terms proportional to l / f i (because we focus on the limit n m), we obtain from Eqs. (C.40), (C.41), and (C.43) the approximate relation --f

R.eirserting this approximation into the right side of Eq. (C.39), we finally arrive at the R/loivre-Ilaplace approximation, nainely

P,

21

Jvcxp 2mn (n - m)

[

'1

--1 (rn - 7 ~ p ) 2 np(1 - p)

(C.46)

To cst,ahlish a conncct,ion bat,wcan Fqs. (C.4G) and (C.aG), lat us introduce nP = ( N ) n p ( 1 - p ) = ( N * ) - ( N ) 2 = f7;

(C.474 (C.47b)

414

Gaussian density distribution

whcrc wc used the dcfiiiitioii of t hc variancc of the parliclenunibcr distribution introduced in E.1. (2.75). hioreover, as we defined a t the outset' p = m/n as the probability of successful measurements (i.e.. measurements that, have as their result the desired number of molecules N),we may rewrite the prcfactor of the exponential function in Eq. (C.46) as [see Eq. (C.47b)l

((2.48) Hence, we recover Eq. (C.36) using in Eq. (C.46) the expressions given in Eq. (C.47) and (C.48). We emphasize t,hat no specific form of a partition fii~c*t,ion was itivok(v1 in t h c*iirrc:iit clc~iv;lt,ionof El. (C.36).

Reviews in Computational Chemistry Kenny B. Lipkowitz &Thomas R. Cundari Copyright 02007 by John Wiley & Sons, Inc

Appendix D Mathematical aspects of mean-field theories D.l

Van der Waals model for confined fluids

D.l.l

Evaluation of the double integral in Eq. (4.21)

Transforinirig variables in the double integral in Eq. (4.24) from { r l , r 1 2 }we . have froin the definition of up ( 6 )

{ T I , 73) to

where 2; (z1) denotes the 21-dependent volume restricted by the hard cores of fluid molecules and by the hard walls. In the last line of Eq. (D.l), cylindrical coordinates T, p , and fl are introduced for convenience and we immediately carried out the one trivial integration over the angle 0 5 0 5 2x. We assume s,o > 2 (ufwt OH). Thew thc: int,c:grttt.iorion z1 1)rwks down into tliroc ri~ng(-s:

415

Van der Waals model for conflned fluids

416

3.

S&

- (Ofw

+ gff)< 21 < s,o

- Ofw.

In turn, the integrations on z and p can be broken into either two or three regions a1 =

szo

- 2Ufw

, “fw

-

1

12 (S,” - 21 - O h )

1

+-24

[

1

}I‘-

2

(S,” - 20fw - uff)

0;

Mathematical aspects of mean-field theories

41’7

Y

Figure D.l: Newton’s iterative method to locate the zero of a function y = f (20)= 0 through a tangent construction (see text). From Eqs. (D.2) and (D.3), we a t last obtain

(D.4

Equation (4.24) follows imrncdiately from Eqs. (D.4) and (4.25)and the definit,ion of <.

D.1.2

Newton’s method

To find the zero(s) of an analytic function f (x)defined on a donlain D (f)G

W, one may

employ an interative technique due to Newton. Suppose an interval [a,b] exists such that {).( is monotonic, continuous. and differentiable

418

Van der Waals model for confined fluids

cvcrywhcrc iii [a,b1 (scc Fig. D.l). Morcovcr assumt. a valuc x o E In,bl such that, f ( x 0 ) = 0. Consider now ail arbztmry poiIi1 .J*)E [n,!4 f :rO. At this poiiit. t,hc tangent to f ( ~is)given by

where f’ ( ~ ( ~ 15) df (s)/dxlr=T(,). Froni this expression we may cdrulate thc zero of thc tangcnl (see Fig. D.1)

Eqiiatioii (D.6) is t,hc basis of Newton‘s it,crat,ivcschcmc. In the next, iterat’,ion we replace d*) on tlic riglit, sidt. of Eq. (D.6) by .r(’+’) arid obtain a iicw estimate J ( ’ + ~ )until 1T(71+1)

-

5

(D.7)

where 6 << 1 is some predefined small number. Because of the properties of f (x)specified at the outset, one obtains

if t,he inequality stated in Eq. (D.7) is satisfied. The accuracy of this algorithm dcpcnds on whctlicr J’ ( . r ) is known ana1yt2ically. An cxtcnsion to functions depending on more than just a single variable is straightforward. In applying Newton’s method it is crucial tliat J (x)is rrionotoiiic iii the int,erval [a,b ] . If this is not, the casel an iterative solution of Eq. (D.6) may iiot converge to the correct value of xo. hi this case, Newton’s method may, however, still work if the intial value di) is selected by an “educat.ed guess”; that, is, .di) needs to be chosen sufficiently close thoxo where f (x)is still monotonic.

D.1.3

Maxwell’s constraint

The densitiw of the coexisting gas

(D.9) and liqnid

(D.lO)

Mathematical asDects of mean-field theories

419

inust satisfy Maxwell's constraint 13271

we assume the gas density

p:

= Pb to he fixed; the density of coexisting

liquid arid thc cocxistcricc tcmpcraturc arc unknowii. Thcrcfore, wc must solve Eqs. (4.43) and ( D . l l ) .szrndtcmeoo.usZy for TOb and A. We solve the equations by the following procedure: 1. Compiitc P(T,) from El. (4.29) for fixcd perature T,/Tcb = 1.

2. Compute pE(T) 5 calriilate

pb

/)b/P&

5 1 and initial tcm-

and pk(T,) from Eq. (4.44) and use these to

3. Obtain a new estimate for the coexistence temperature via Newton's = T, - f (T,)/ f' (T,)[see Eq. (D.6)], where method as

x+,

is the (partial) derivative o f f with respect to T . 4. Replace

Ti by Tir,,, and return to step 1.

The procedure is halted when If (7;t) I 5 At the end it is sensible to check whcthcr p: (T,) = P b , as spccificd at the outset, and that I' (T,,,p") = P (Tn,A) for the coexisting phases, so that TOb ( p b ) 2 T,. Convergence is achieved after five or six iterations. The resulting bulk coexistence curve is plotted in Fig. 4.5. To illustrate the effect of confinement, we also plot in that same figure the pore coexistence curve for sz = 50 determined by the procedure detailed above using, however, the correspondiiig equation of state for the pore fluid given in Eq. (4.28).

420

Lattice models

D.2

Lattice models

D.2.1

Numerical solution of Eq. (4.86)

To solve Eq. (4.86) we employ the Jacobi-Newton iteration technique, which proceeds iteratively in an alternating sequence of "local" and "global" minimization steps. Let be the local density at lattice site i in the i t h local and thc It.h global minimization step. A local estimate for the corresponding niinimiim valiic of $Ik*' is obtained via Newton's mcthod [see Eq. (D.ci)];that is

&'

~

(D.14) From Eqs. (4.83) we have for the functional derivative

(D.15) and t,hcreforc

f'

(&")= -

kBT p y (1 -

&.')

(D.16)

It is important. to realize that throughout each local minimization cycle the densities at, nearat-neighbor sites of site i represented by the set {p;''} reinah fixed at, tlic initial valucs assigiied to tlicrri a t the bcgiriiiirig of the local cycle. The iterative solution of Eq. (D.14) is halted if [see &. (D.7)]

(D.17) is satisfied, which is typically achieved in approximately two to three iterations. Local minimization is performed by visiting each lattice site 2' consecutively, and the local cycle ends orice all sites have been considered. "Global" minimization then involves updating the local density of the en.tire latticc according to pa0,1+1 -. p;k + l , l

~

,i - 1, . . . , A f

(D.18)

thus providing iiew initial valucs for the next. lmul mininiiziltiori cycle [by setting 1 + 1 --+ 1 and returning to Eq. (D.l4)]. Global minimization is carried out, until [see Eq. (D.7)]

(D.19)

Mathematical aspects of mean-fleld theories

42 1

which is achicved ill roughly 100 stcps. Oiire local and global niinimzation have converged for the ciirrent, T and

p according to Eqs. (D.17) and (D.19), we may calculate the set of grand

potentials {!-la} for all morphologies from Eq. (4.83) and repeat the iterative procedure for a slightly different chemical potantial. Once the curves R" versus p have been obtained for all morphologies {M"} and the given temperature T , we determine pa" by numerically solving Eq. (1.7Ga). We then increase the temperature by a small amount bT and repeat the above proccdiirc where some care has to hc taken in thc vicinity of rritiral points because the grand potentials for coexisting phases a and /3 become increasingly similar. Hence: we chose 6T = 2.5 x (in dimensionless units) as soon as the density difference Ap"p 5 0.05. To initiate the iterative scheme, suitable starting solutions for Eq. (D.14) are obviously required. These soliltions are provided by the morphologies Ma at T = 0 for which pap can be calciilnted analytically from the expressions for W compiled in Table 4.1.

D.2.2

Binary mixtures

D.2.2.1

The mean-field Hamiltonian function

Hcrc we work out expressions for thc niimbcr NAA ( N B B )of A-A (B-B) pairs. These pairs are directly conriected sitcs: both of which are occupied by a molecule of species A (B). Likewise, expressions for NAB(s) and the total number of molecules of species A and B at the solid substrate, NAW (s) and NBW (s), may be derived easily. The resulting expressions presented in Eqs. (4.118)-(4.121) contain terins that can be cast as

(D.20b) (D.20~)

Lattice models

422 at thc incan-ficlrl lcvcl. In addition,

arise where the summation over the four nearest neighbors n.n. of lattice site k can be carried out explicit-ly if Eqs. (4.126) aiid (4.127) arc invoked. Finally,

c

Sk.1

=

np1nu

(D.22a)

np1

(D.22b)

k

Cs:,1= k

may bc cmploycd in Eqs. (2.4) and (2.5) of Rcf. 84 to rcplacc thc sum over sitcs in lattiw plarics 1 = 1,z. Hence, Eqs. (D.20) as well as (D.22) permit us to write down the meanfield expressions

+plpi-l

(m,r + 1) (m1-1

+ 1) + 4pf (ml + l)'] (D.23a)

423

Mathematical aspects of mean-field theories

which follow after somewhat t,cdious but straightforward algebraic manipulations. Replacing now in Eq. (4.123) No$ (s) and N,w (s) ( o , P = A, B) by their mean-field counterparts given in Eqs. (D.23) eventually yields

as the mean-field analog of H given in Eq. (4.123). In Q. (D.24) the first. two sums account for the interaction between a pair of molecules of species A and B, respectively. Likewise, the third sun1 represents t,he contribution of A-B attractions to the mean-field Hamiltonian. The next two terms represent the interaction between a molecule of species A and B wit.h the solid substrate and the lrrst, t,erni coiiples the system to an (infinit,elylarge) external reservoir of matter. Morcovw, it. is casy to verify that Eq. (D.24) tlcgcnoratcs to thc expression

+

ns

Ewn[P1(1 - ml) +

for a pure fluid (rn = 0, XB = XAW = 1. cally honiogeneous substrates.

P Z O

-7741

- nCPlCl

025)

1=1

E

= CAB) confined between chemi-

Lattice models

424 D.2.2.2

Derivation of Eqs. (4.132)

To dcrivo Eqs. (4.132) wo clopart from Eq. (4.130), which may bc writtm tnore explicitly as

(D.26) in the lirriit n,{q}-+ oc using Stirling's approxirnatiorl (see Eq. (B.7)]. Using [SCF: Eqs. (4.126) and (4.127)]

(D.27a) (D.27b) so that Eq. (D.26) niay he recast as

Hence, together with Eqs. (4.131) and (D.24):'Q.(D.28) yields

-

l

-z

c L

PlP

1=1

where the intrinsic freeenergy density is defined as

(D.29)

425

Mathematical aspects of mean-ffeld theories

arid the ciitropy dciisity R ( p ,m) as wcll as t.hc various iritcriial ciiergies UA A (p,m ) ,UBB ( p ,m ) ,and 'UAB ( p , m) are given by t,he expressions

+

z

pi { (1 1=1

I

+ mi)In (1 + ml)+ (1 - mr)In (1 - m ~ ) }(D.31a)

(D.31b)

(D.3lc)

(D.31d)

In Eq. (D.31a) the factor proportional to ln 2 arroiints for the distinguishability of molecules of species A aiid B (i.e., their different "color"). Following Pini et al. the color of molecules may be given a more physical interpretation by interpreting it as a "spinlike variable in addition to translational degrees of freedom so that their mutual interaction depends both on their relative position and on their 'internal' state, iiainely whether the interacting particles belong t o the same species or not" [328]. Differentiating Eqs. (D.29) according to Eqs. (4.132) we find after somewhat lengthy but straightforward algebraic manipulations our desired result,

426

Lattice models

&h1 , k = -/~‘+k

+;

c

[Pk+l

(1

(w+, + 1) + pk-1

+ m k ) In (1+ m k )

(mk-1

+ 1) + 4Pk (mk+ I)]

(D.32h)

=

where we also used Eqs. (D.31) and introduced p* p - p - l In 2, which takes into account the trivial contribution from the different “c01or’~of molecules pertaining to one or the other species. In Eqs. (D.32), 6ij is the Kronecker symbol. For the special case of a symmetric binary mixture ( , y ~= 1) confined between rionselect,ive solid surfaces ( x , v = 1)1 Eqs. (D.32) simplify

427

Mathematical amects of mean-field theories

corisiderably aid wc! obtain

[

Pk l-pk

-ji -I-

f3-’

+-2

(1+ mk)ln(l + r n k )

’I

111 -

+ (1 - mk)ln(l - 7 n k )

11 (D.33b)

where EW

= EAW

and

(D.34a) (D.34b) D.2.2.3

Numerical solution of Eq. (4.143)

As we restrict ourselves to nearest-neighbor interactions, depend only the set of variables {pj.tnj (k- 1 5 j 5 k + 1 ) as one can verify from h s . (D.32). Hence, D in Eq. (4.142) has a band structure where all elements

011

(D.35a) (D.35b) Moreover, as w (P)is continuous arid differentiable, we have

(D.36) that is, D is symmetric with respect to its main diagonal. As in this work we focus on planar, cliemically homogeneous substrates, an additional symmetry exists for the local order parameters with respect to a (virtual) midplane

Lattice models

428

thc lattice thal rriiglit coincidc with Therefore, if z is odcl we conclude thak oil

a11 actual

latticc plaiic if z is odd.

where 4) = (z 1)/2. If, 011 t,he other hand, t is even t

where now. of course, @ = z / 2 . To simplify the subsequent. cfscussion we restrict ourselves to the case of even z where we note in passing t,hat, for sufficiently large z the distinction between odd and even numbers of lattice planes becomes irrelevant. Then we may reorganize the 2z elenients of vect.ors z,f , arid V such that the resilting niatrix D has point synimetry with respect to an inversion center. More specifically, we rnav express D formally as

iifi

D=(B A )

(D.39)

where elements of the submatrices are related t,hrough

It is then easy

t,o verify

that

which coiitairis only four iioiizcro chiicnts. Siniilarly, thc last two rows of submatrix can he cast as (D.42)

Mathematical aspects of mean-field theories

429

-

O m then rcalizcs that, in A thc clciricwt (D.43)

-

appears, whereas in B the conjugate element

ae+l

-=

aP,

3'lW

a2W

-

8Peanz*+l

ame+laP@

(D.44)

ariscs. Similar considcrations apply to thc pair of rlcmrnts i?hf/ap+ and iIhf+'/iIp@ as well as to tlie correspoiidiiig two pairs of elements on the last, rows of and B. Moreover, it is apparent from symmetry properties stated in Eqs. (D.38) ( z even) that both dh:/dp* and t3h:+'/ap@ are acting on the same element of the vector x - xo in Eq. (4.141) such that l3q. (D.39) car1 bc rccast, as

D'=

(2

x'

0

0 A')

(D.45)

x

where 0 is the z x t zero ma.trix, is identical with except for the last two elements in the two bottoni rows, that is [see Eq. (D.42)],

... 0 +

... 0

x) aPa + 1

(D.46) and t.he relation het,ween the new siibmatrices A' and A' is the same as that between arid A [see Eq. (D.40a)l. Because of these symmetry considerations we may replace Eq. (4.143) by I

z

-

x1+1 = -2- * f

(%a)

+ si = 6Zi + Zi

(D.47)

where the [transpose of the) z-dimensional vect,orx

FT

(Pl,~nl,-..,P.$,m,) fT (Z) = (hi.h i : . . . ,hf: h;) =

(D.48a) (D.48b)

and the z x t matrix 2 replaces the 22 x 22 matrix D, which considerably reduces the numerical efforts necessary in solving the original Eq. (4.141) iteratively. In practice, starting from a suitable solution 35," for a given phase Pa,we solve Eq. (D.47) iteratively until (Zi+l)l 5 10-l': which requires typically

If

Lattice models

430

102-103itcrations. Uiidcr this conditioii, &+I is a(n approximate, nmiicrical) solution of the equation f ( 2 )= 0 (D.49) In general, we are not only interested in solutions of Eq. (D.49) but also, more specifically, in those solutions satisfying Eq. (1.76a), which defines the chemical potential a t coexistence pap between phases (i.e., morphologies) M” and ML’ for a given temperature T . To determine p”’j at a slightly different temperatiirc T’ = T + dT, wr cxpand w in R Taylor series aroiind some cheniical potential / i t , say, so t,hat

-

- wa*p( p i ) - p:’@ (pt - & + I )

,

T’ = const

(D.50)

where we dropped all other arguments l o ease the notational burden. The second line of Eq. (D.50) follows from

dw

z

k p l = -p

(D.51)

1=1

where Eqs. (4.132)and (1.78) have also been employed. Assuming Eq. (1.76a) to hold for pi+l, wc can solvc Eq. (D.50) for li.i+l to obtain

(D.52) thus providing an iterative scheme to calculate the chemical potential a t coexistence. This scheme may be initiated by setting initially pi = p:’ at the prcvioiis beniperakirc To and calciilating p;” from 3?’*0 at that temperature. However, ZOyBwill no longer be solutions of Eq. (D.49) at T’ and p i + l . Therefore, we solve Eqs. (D.49) and (D.52) until Jbp(= - pp” 5 lo-”. Hence, for a given temperature, Ma and Ma coexist at a chemical potential pRp G p,+l. However, the associated woo does not necessarily correspond to the absolute but inay instead represent only a relative minimum of the grandpotential density. If, on the other hand, for any pair a,p, the grand-potential dcrisity assuiiics its ylobal IIiiriiiiiuIIi, M” and Ml’ arc thewnodynamically stable phases a t coexistence. The range of temperatures and chemical potentials over which this condition is satisfied defines the coexistence line p.:’ ( T ) between M” arid M’ (see Section 1.7).

Reviews in Computational Chemistry Kenny B. Lipkowitz &Thomas R. Cundari Copyright 02007 by John Wiley & Sons, Inc

Appendix E Mathematical aspects of Monte Carlo simulations E.l

Stochastic processes

E. 1.1 The Chapman-Kolmogoroff equation Let y ( t ) be a general random process, that is, a process incompletely determined a t any given time t. Specific examples are discussed in the context of MC simulations in Section 5.2. The rmdoin process can be described by a set of probability distributions { P,} where, for example, P2 ( y l t ~y2t2) , dyldyz is tho probability of finding :ti1 in thc intcrval [!ill?jl t djyl] at t,imc I = bl and in the interval [yz, 92 dy2] at another time t = t2. Thus, the set { Pn}forms a hierarchy of probability distributions describing y (t) in increasingly greater detail the larger is n. The simplest, random process is completely stochastic so that one may write, for example,

+

p 2 (?/Ifl>?/2f2) = Pl

(!/lh) 6(YZt2)

F.1) However, here we are concerned with a slight.ly more complex random process known as a Markov process, characterized by the equation p 2 (Pltl. Y2t2) =

PI ( y l t l ) h'l

I Y2t2)

(E.2) where K1 (yltl I y2t2) is the conditzonal probability of finding y in the interval [yz, y~+ dyz] at time t = t 2 provided y = y1 at an earlier time t = tl < tz. Some properties relevant to the current discussion are listed below: (?/dl

431

Stochastic processes

432

2. The conditional probability serves as some sort of propagator in t#hatit controls the temporal evoliitioii of y ( t ) in the sense of (E.4)

3. The probabilily dist.ribut,ions satisfy a stalionari ty condit.ion. In particular, 1’1

p2

(ah)

(Yltl.

Y2t2)

= PI =

P2

(Yl)

(E.5a)

(31,y 2 ; t 2 - t ] )

(E.5b)

4. Most importantly, Markov processes have. a “onestep memory”. That is, to find y in the interval (Y,~, yftr d y J at t = tn depends only on the realization y = y n - l at the immediately preceding time t = tn-l but is independent of all earlier realizations y = ym at times 1 = t,, where 1 5 m 5 n - 2. Mathematically speaking this can be cast as

+

Kn-1

(YI~I.. . . * ~ i 1 - 1 t n - 1I

vntn) =

1(1 (

I

~ n - ~ t nY - n~ l n )

(E.6)

433

Mathematical aspects of Monte Car10 simulations

Because of property 3 an altcrnativc forrriiilation of thc prcvious expression is given by

hi (yn-2 I Yn; t + 7 ) =

J 1(1

(yn-2

I Yn-1: t) KI ( ~ n - 1I g n ; 7 )dyn-1

(E.10)

where t = t,-l - tn-2 arid r = tn - tTt-l. Equation (E.lO) is the celelmted Chapman-Kolmogoroff equation in the theory of stochastic processes.

E.1.2

The Principle of Detailed Balance

Suppose a m a l l time interval rc exists such that, during r,, yn-1 changes without strongly affecting K1 (yn-2 I yn;t 7,) in Eq. (E.lO). In the liinit rc 4 0 wf may then expand the left, side of Eq. (E.10) in a Taylor series according to

+

Inserting this expression into Eq. (E.10), we find after rearranging terms

1

(E.12)

At, this point. it is convenient to introduce the transition probability per time interval rc via

4 (yn - 1

I yn.) = r-rc lim

1 r

-KI (yn

1

1~

n T ;)

(E.13)

satisfying

(E.14) where we also employed the fact that, the c:onditional probability is normalized (SCC pr0pcrt.y 1. introdiiccd in Section E. 1.I]. Miilt,iplying the left sidc of Eq. (E.12) by ?I' (!jTL-2t,-2) (sc:c Scctiori E.l.l) arid intcgratiug over gTt-2, we may write

(E.15)

434

Stochastic processes

whcrc wc introduced a iicw variable t’ E t t tn-2 arid irivokcd propertics 2 and 3 from Section E.l.l as indicated. By a similar token we may recast, the first term on the right side of Eq. (E.12) as

where we also used the definition of d, in J3q. (E.13). Treating the second term on the right site of Eq. (E.12) on equal footing, we obtain

where in t,he last. step we employed the definition of 1/rCgiven by Eq. (E.14). Putting together Eqs. (E.15)-(E.17), we finally arrive a.t,

(replacing the tirne variable trivially according to t’ 4 t ) . hi the context of the discussion of MC simulations in Section (5.2), the stationary solution of Eq. (E.18),

a. = o

ap1 ( Y l l t )

(E.19)

is of particular interest. because it leads to a special formulation of the Principle of Detailed Balance, namely

where ll is the transition probability associat,edwith t,he change yn-1 ++ yn. Equation (E.20) reflects the condition of riiicroscopic reversibility, which we already introduced in Eq. (E.14).

435

Chemically striped substrates

E.2

Chemically striped substrates

Because we are concerned in this tutorial with the effects of chemical heterogeneity at the nanoscale on the behavior of the confined film, we expect the details of the atomic structure not to matter greatly for our purpose. Thercforc, we adopt a coritiriuuiri rcprcsciitatiori of tlic iuteractiou of a film riiolecule with the substrate, which we obtain by averaging the f i l ~substrak ~i interaction potential over positions of substrate atoms in the x-y plane. The resulting continuurn potential can be expressed as

d’](.r,z)

= nA

2 2 jdy‘{

7n=-m

+

m’=O-

J

OD

-- d s / % ? M x

J

dduf,,,

1.(

- r’l)

-ux/2+7na,

da/Z+msx

dx‘u,, ( ~-rr’l)

-&/Z+mSx

+ J

Rx/Z+Wx

A/Z+msx

dz’uf,,

- T‘I)

(IT

1

(E.21)

In Eq. (E.21), nA = 2/@ is the areal density of the (100) plane of the fcc lattice. The position of a filni molecule is denoted by r , and T’ = (x‘:y’, z’ = f s 4 2 f ni‘be) represents the position of a substrate atom. where “-,I rcfcrs to thc lowcr (A: = l ) , I‘+” to the iippcr (k = 2) siibstrate, and be is the spacing bctwccri succcssivc crystallographic plaiics in thc f r dirwtion. We note that, because all features of the substrate at the atomic scale have been washed out in dkl,our continuum model cannot, account properly for solid formation, which, AS mentioned briefly at the beginning of Section 5.4, is strongly influenced by the atomic structure of the substrate. By interchanging the order of integration and introducing the traiisforination

z’+ .2.” = x - 2’ y’

4

y”

2’ + 2’’

= y - y’

z - ( f . 4 2f m’6e)

=

(E.22)

we can rewrite the integrals on the right side of Eq. (E.21) as

1

dz’

0

J”

-00

r-b

dy’u (Ir - r’l) = -4e

J

2-a

00

dx1’

J dy’’

--oo

[( 2 ) TI’

l2 -

‘1

(2) T I‘

(E.23)

436

Chemically striped substrates

whcrc a arid b rcfcr to integration limits arid 11 and c correspond to TLY, arid cfs or bo ufwand qW, depending on a and b. The definite integral over yrr can he found in standard tabulations (see, for example, 210. 60 in Ref. 141). Thus, Eq. (E.23) simplifies to

1= 2

T d d r [Il (d’,zrr;Ct,? s x 9s,)

- /2 (zrr: zrr;d,, sx,s z ) ]

(E.24)

z-a

where

(E.25a) (E.25h)

R G arr2 + z 1r2

(E.254

The remaining integration over zlr can also be carried out analytically (see, for example, No. 244 in Ref. 141). A tiresome computation yields

z-a

.T-a

I .r”=.r- a

Molecular expressions for stresses

437

where the diniensionless quantity S is given by

R

S = -Z l t 2

(E.28)

To simplify the exqxessions, we define the auxiliary function

E.3

Molecular expressions for stresses

E.3.1

Normal component of stress tensor

E.3.1.1

Virial expression

To evaluate the partial derivative of the configuration integral in Eq. (5.62): we employ an approach suggestredby Hill [21) and transform coordinates according to [see also Eq. (5.7)] zi

+

z = za/s,

This permits us to recast the previous expresion as [see Eq. (5.69)]

(E.30)

Molecular expressions for stresses

438

whcrc fr = fJFF t f/FS. Thc partial derivative ill Eq. (E.31) riiay be cvduatcd according t,o the product rule, that is, y' = dv 2471' (y = 7121): where in the current case

+

11

=

s,N

nJ

(E.32a) Srl2

8x12

'11

i=l

dxi

- 4 2

J

-.?$.I2

112

dyi

J

-112

dsexp

(-"> kBT

(E.32b)

It then follows that

where V E Ass, and Eq. (5.6) have also been used. Equation (E.33) represciits the idcal-gas c.ontribution to the iioriiial cornponcnt of the strcss tcnsor. To compute the derivative 11' we realize that U = UFF Urn depends on s, because UFS x dk]depends on t, = ?is, and ~ J F Fdepends on [see Eq. (5.69)]

+

Ht:nc:c:, wc firitl froni Eq. (E.31)

Mathematical aspects of Monte Car10 simulations after rcvcrtirig the trarisforrnatiori Zi

-+ .=i

439

[sec Eq. (E.30)Jwhcrc

where (E.36) is the force exerted by particle i on substrate k and W,, is Clausius' virial (211. Equation (E.35b) follows by noting from Eq. (5.68) that.

(E.37) so that

(E.38) and therefore

Molecular expressions for stresses

440 Hciicc we obtain

(E.40a)

(E.40h)

E.3.1.2

Force expression

A different expression for Tzz can he obtained directly from Eq. (5.62) without the transformatioii of coordinates. Therefore, it is convenient to recast the configmition intcgral as 13291

J

9,/2

=

dwi

(21)

(E.41)

-.V,/2

where

By applying Leibniz's rule for the differentiation of an iiikgral [ll]

(E.43)

Mathematical aspects of Monte Car10 simulations

441

(E.44) which is a dircct conscqiiancc of t,hc divcrgcncc of thc function A (z, z ) in that limit [sec Eqs. (5.68), (5.69), (E.25r), (E.26), (E.27), (E.28), and (E.29)]. Defining therefore

(E.45b) the above analysis may be repeated N - 1 times. We finally obtain

-

(Fill)

-

(Fi2])

2Ad

(E.46)

where (E.47) and wc rioticc h a t .

(E.48)

which follows directly from Eq. (5.68). Equation (E.46) constitutes the socalled force expression for the normal component of the stress tensor.

Molecular expressions for stresses

442

E.3.2

Shear stress

E.3.2.1 Virial expression To dcrivc a molcc.ii1ar cxprcwion for t.hc shear stress, wc hcgin by realizing that [see J3q. (1.66)]

where we also employed E$. (2.112) arid (2.120). To proceed let. us write the configuration integral inore explicitly as

which incorporates the effect, of deforming the fliiid lamella in t,he x-direction. It is coriveriicnt t,o elirniriatc the shc.ar-straiii dcpcridciicc of thc iritcgratiori limits through the transformation (E.51) so that we can cast the partial derivative of the configuration integral in Eq. (E.49) as

x oxp

(-">

kBT

(E.52)

The shear strain then appears only in the argument of

tig ( T i j )

via

It is t,hen an easy rnatter to apply the derivation detailed in Appendix E.3.1.1 to show that, id

Txz

-

0

(E.54a) (E.54b)

Mathematical aspects of Monte Carlo simulations

443

wherc IVzx is defiiied analogously to Pi’= in Eq.(E.40a). For tlic corrcspoiiding fluid suhst,rat,e contribution to bhe shear stress, we find 2

N

(E.55) k = l *=I

where F$ is the z-component of the force exerted by fluid molecule i on substrate k. Introducing [see Eq. (5.103)]

5:“’ =

xi

zy21 =

5 2

+(2Zi + s,) 2% QSxn

QSXO +(2Zi - s z ) 2s,

(E.56a) (E.56b)

it is straightforward to vcrify froin Eqs. (E.49): (E.52), (E.55), aid (E.56) that

(E.57) where the probability density in the grand canoncical ensemble is defined in Eq. (E.40n).

E.3.2.2

Force expression

Here we derive a molecular expression for the shear stress parallel to the “force” expression for the compressional stress derived in Appendix E.3.1.2. Howww, horc wv (mploy it sliglitly tlifkrcwt dcfinition of thr auxiliary fiiiict,ioris { g z } . which we introduce via

nJ

sy/2

z=

I.=

where

1

-s y / 2

SJ2

dy,

J dt;

-8.12

S,/2+~2S,0(2Z.+S.)/28,

-s,/2+rm,o

S

( 22,+sz)/2n.

dzigi

(21, y i , .

. . , Y N , a,. . . ,. Z N ) (E.58)

(E.59)

Molecular expressions for stresses

444

HCIICC,we rriay again apply Lci\mix's rule for thc differentiation of a para111eter integral to obtain

+

[

91 (z,

= -s,/2

+ QS,o

( 2 q + s,) /2s,)

- 91(21= sX/2 + C V S , ~(221 + s,) /2~,)

] -:

(E.60)

whcrc we clroppctl thc: argimiciits ?/I, . . . ,?jN aiid z1, . . . , ZN to sirriplify the notation. The reader should note that, t,hese last two terms do not vanish separately as we argued in Eq. (E.43)because U does not necessarily diverge as,^ (2z1 s,) /2s, but at, X I = fs,/2

91 ( X I= -~,/2

+ +

QS,~

+ (221 + s,) /2s,)

= 91 (21 =sX/2

+

C Y S , ~(

+

2 ~ 1 s,) /2s,)

(E.61)

because U is periodic in x on account of periodic boundary conditions. Thus, as boforc in Scction (E.3.1.2) wc may rcpeat t,hc ahovc argurnont N - 1 times to finally arrivc ats

az - a (QSXO)

-

n-'

sy/2

J ~ B T .

x exp

1=1

--.py/2

8.12

dpi

(-&)

J

-.9./2

J

a,/2+nR.u(2zi+a.)/2n.

dzi

dxi

-~,/2+aa,o(2z,+s,)/2s.

aU,

a (asxo) (E.62)

Because of Eq. (5.103) we may rewritc the partial derivative in the previous cxprmion as

-

whcrc + H k -- 1 and - H k: -- 2, rmpcct,ivcly. In Eq. (E.63), F, is the z-component. of the instantaneous net, force exerted by the confined fluid on the substrates.

E.3.3

Virial expression for the bulk pressure

By an approach similar to the one outlined in Section E.3.1.1, we niay derive a molecular expression for the bulk pressure Pb following again the original

445

Mathematical aspects of Monte Car10 simulations

dcrivation of Hill [21]. As wc pointed out in Scctioii 1.3.1,thc bulk fluid is homogeneous and isotropic on account, of the absence of any external fields. Hence, we rnay write Eq. (1.29) alternatively as

(E.64)

V = A,os, = Aasy = Alasz

which rcflccts t,his symmctry of t h biilk fliiid. Moreover, wc rcalizc from Eqs. (1.31) and (1.60) t.hat.

(E.65) where we also ernployed Eq. (2.81). From Eq. (E.64) we realize that,

a

l

l a

a

l a

(E.66)

Hence, by analogy with Eq. (E.31),we may write

(E.68) is introduced as convenient shorthand notation where we transformed to unit-cube coordinates via Ni

a,

(E.69)

I

-4

- Ni/.Sa

By the same logic applied before to Eq. (E.31): it is a simple matter to show that (E.70) where the ideal-gas contribution PLd follows if Q. (E.64) is also employed. involves terms of The configurational cont,ribution l o the bulk pressure, the form [see Q. (E.35a)l XJFF- Waa --(E.71) as,, 2s,,

el

Molecular expressions for stresses

446 whcrc i-1

Hence, when summed over combined leading to

Q

' '3

j#i=1

(E.72)

and because of Eq. (E.64) these terms may be

(E.73) where

N

N

(E.74) i = l J#z=I

Together Eqs. (E.70). (E.73), arid (E.74) constitute the virial expression For the bulk pressure, which parallels Eq. (5.63) for the stress exerted by a confin,ed fluid on the planar substrates of a slit-pore.

Reviews in Computational Chemistry Kenny B. Lipkowitz &Thomas R. Cundari Copyright 02007 by John Wiley & Sons, Inc

Appendix F Mathematical aspects of Ewald sumrnation F. 1 Three-dimensional Coulombic systems F.l.l Energy contributions in Ewald formulation F. 1.1.1 Real-space contribution

To derivc Eq. (6.8) for tho rod-spaw part, of tho Ewald potcnt#ial, wc start from Eq. (6.7) for the set of scrcciied charges axid apply Poissori’s formula [see Eq. (6.3)]. This gives

where the first term in parentlieses is the usual Coulomb potent>ialand

(F.2)

is the potential due to a Gaussian charge cloud (total charge - q j ) located at T~ - n [see Eq. (6.5)].To evaluat.e the integral on the far right side of Eq. (F.2), we transform variables according to T’ -+R = T’ - rj n, which givw

+

In Eq. (F.3) the integration is carried out over the entire three-dimensional space. 447

448

Three-dimensional Coulombic systems

Thc inost coiivwicnt way of doing this is to trarisforrn to spherical coordinates R = lRl, 8, and (9, where 0 and q are the polar and azimuthal angles, respectively, associated with the orientation of R in a spacefixed coordinate system. We may split the integral over R into two contributions from regions characterized by the inequalities (F.4a) (F.4b) This separation of the integral can be effected by using an expansion in terms of spherical harmonics {y,tn}[258)valid for arbitrary vectors TI and r2'; that is,

and cp, are polar and azimuthal angles associated with vectors rl 2 respcctivcly. , Notkc that thc complex conjugate q:n= &,-m. In Eq. (F.5), T< ( r , ) is the magnilade of the smaller (larger) vector of the pair r1 and r2. Setting T I = R and 7 3 = rU + n,and inserting Eq. (F.5) into Eq. (F.3), one realizes that oiily terms characterized by 1 = m = 0 (with Yo0 = l/&) survive because [258]

where and ~

r9i

2n

1

(F.6) Equation (F.3) can therefore be rewritten as

where the first integral appearing in brackets can he recast by wing 2

dRR exp ( - a 2 R 2 ) =

S 2 n

-

'See Eq. (3.70)in Ref. 242.

"I z

2a aa

---

dRexp (-cr2R2)

[

0

a 1[duexp

2aaa

a

(-u2)]

449

Mathematical aspects of Ewald summation

whcrc we have employed tlic defiiiitioii of ~ l i ccrror functiori 111, 37, 3301,

(F.9) to arrive at the third line of be carried out by using

E.1. (F.8). The remaining part,ial derivative can (F.lO)

which follows immediately from the Eq. (F.9) and Leibniz’s rule for t,he differentiation of a parameter integral [ 11, 330). One finally obtains

The second integral in Eq. (F.7) gives dRRexp (-a 2 R2 ) = --

2tr2

2

T d R g e x p (-a2R2) = 1 exp ( - a 2 x 2 ) 2u2

X

(F.12) Inserting Eqs. (F.ll) and (F.12) into Eq. (F.7). and replacing z by l r i j n1 the elech-ostatic potential at ri due to one Gaussian located atbrj - n # ~i reduces to

+

(F.13) Finally, inserting Eq. (F.13) into our initial Eq. (F.l) a.nd using the identity 1 - erf (9) = erfc (y)

(F.14)

wc eventually arrive ataEq. (6.8).

F.1.1.2

Fourier-space contribution for nonzero wavevectors

To evaluate the electrostatic potential ck(2)(r) [see Eq. (6.11)] from the peri-

odic Gaussian charge distribution P ( ~ ) ( T[see ) Eq. (6.10a)], it is most convenient, to start from Laplace’s equation [242], which says that

ACp(2)(~) =-4~p(~)(~)

(F.15)

450

Three-dimensional Coulombic systems

where A = V . V is t h Laplacc operator. The Laplacc cquation is cqiiivalent to Poisson's equation [see Eq. (6.3)) and follows directly from the first Maxwell equation of electrostatics,

v . E(T)= 47rp(r)

(F.16)

E ( r )= -V@(T)

(F.17)

using the definition for the electric field E. Roin Eq. (F.15)it is evident that the Laplace equation is a second-order differential equation that can be solved convetiiently in Fourier spacc. To this (xidl w c cxparitl tlic cliarge distribution and the corresponding potential according to the (discrete) Fourier series

where k is a vector of thca reciprocal lattice related to the set of real-spacc lattice vectors {n}[see text. above J3q. (6.11)], and the quantities g2)(k)and 6(2)(k)are Fourier coefficients of the charge distribution and the potential. rmpcctivcly. Thcisc Foiiricr rocfficicnt,s can bc obtaincd from the corrcspondirig rcal-space quantities via

g2)(k) =

Ky,

/

drp(2)( T ) exp ( - i k

.T )

(F.19a)

(F.19b) where KF is the volunie of thc m t z w system consisting of the basic cell plus its periodic replicas. Thus, &, = Vnce,l,where n,l1 is the total number of cells. Inserting thc cxpnnsions (F.18) into Eq. (F.15),wc havc

(F.20)

Mathematical aspects of Ewald summation

451

We now rccall that. Fouricr expalisions (F.18) are orthogorial cxparisioiis (sce, e.g., Ref. 242). It follows that, each summand on the second line of Eq. (F.20) has to be equal to its counterpart, on the third line so that (F.21) which is Laplace’s equation in Fourier space. Thus, given t8heFourier coefficients of the charge distribution (see below): we can easily calculate from Eq. (F.21) all Fourier coefficients of the corresponding potential, except its contrihiition at, k = 0, which will he disciissed in the siibsequent Appciiclix F. 1.1.3. Replacing in Eq.(F.lab), 6(2) (k)bv tfic expression givcri in E ~ J (F.21) . permits 11s to calculate t,he desired potential a(*)( r ) . Having in mind this strategy we start by evaluating the Fourier CCF efficients of p(2)(k). Inserting the explicit expression for p @ ) ( r ) given in Eq. (6.10~~) into Eq. (F.l9a), we have

$”(k)>= 1

(5) 3

C C qN j S d c e x p [ - i k . r - a 2 ( r - r j + n ) 2 ] { n } j=l

(F.22) The spatial integral on the right side is a standard (three-dimensional) Gaussian intcgal and can be carried oiit. analytically [330]. Using, in addition, the relatioii exp ( - i k . n)= 1 (wliicli defines k as a reciprocal lattice vector), one finds

1

- - exp

v

(-5)c N

qj exp (-ik rj)

(F.23)

j=l

where the second line has been obtained by employing the relation 1 -. 1 1

(F.24)

Inserting Eq. (F.23) iiit.0 Eq. (F.21) t,heii yields

Finally, inserting the coefficients (F.25) into Eq. (F.18b) together with T = ~i gives the first term on the right side of Eq. (6.11), which is the contribution ( r i ) .The missing “long-range” term related to the electrostatic potential to the special case k = 0 is discussed in the subsequent Appendix F.1.1.3.

452

Three-dimensional Coulombic systems

F.1.1.3 Long-range contribution

@fi(ri),

Evaluation of the long-range part of the electrostatic potential, which results from the long-wavelength limit (k = 0) of the corresponding Fourier expansion! is the “trickiest.” part) in the derivation of the Ewald expression for tohealsctrostatic potential of a Coulombir system. Thc p r o k lem is iriirncdiatcly apparciit frorii Laplacc’s cquation iii E’ouricr space [see Eq. (F.21)] which, when solved for $)’ (k) directly at k = 0, yields a divergent result because of the factor l/k2. Fortunately, we are not really interested in the value of %c2) (k)for k = 0. To realize the irrelevance of the (k)at, k = 0, consider thc corresponding cncrgy corit2ribut8ion value of

-

(F.26) where [see Eq. (F.19b)l

(F.27) is the spatial integral over the potential which must be independent of (particle) index j . Now recall that we are dealing with a globally neutral system, N rrieauirig that XI=, 9j = 0. Consequently, ( 0 ) vmislics regardless of the actual value of $2) (0). Thus, in the following discussion we focus on the limit k + 0 of the full product %(2) (k)axp (ik . r ) appearing in Eq. (F.18b). More cxpliritly, given that wc arc dealing with mi isotropic systcin whcrc thc direction of the wavevector k should not mat,ter, we consider the angleaveraged quantity 1 aLn( r ) = lini 4lr k-0 (2)

where

8k

and

qk

1

2n

J d cos 8, J dVk6(’) ( I C )exp (ik . -1

T)

0

(F.28) duk&(’) (k)exp (ik. r) 4lr k-0 are the angles specifying the orientation of k and wk =

( o k ~(Pk).

To cvaluatc t.hc riglit, sidc of J3q. (F.28) wc coiisidcbr first thc chargcL density coefficients fi(2)(k)that. give rise t.0 the potential @(’) for small, but nonvanishing k. Expanding these coefficients in a Taylor series around k = 0 and using Laplare’s equation [see Eq.(F.21)], we obtain I

(k)= - [g2)lo -I- k . V k i;’2)(k)10 + j k k . V k V k g2)(k)I0+ 0 (k3)] 41r

k2

1

(F.29)

Mathematical aspects of Ewald summation

453

we riow considcr the lowest-order expansion coc:fficicrits of F2)( I C ) appcaring on the right side of Eq. (F.29). Using the general definition (F.19a) for the Fourier coefficients and performing the required derivatives, we obtain (F.30a)

The quantities on the right, side of Eqs. (F.30) have a simple and lucid physical interpretation in terms of the midtipole moments of the charge distribution p(2)( T ) [242]. Indeed, Q@)is nothing but the monopole moment, P(') is the dipole moment, and the second-rank teilsor A(2)is related closely to the quadrupole moment. Explicit expressions for these quantities can be easily obtained by inscrtirig Eq. (6.lOa) into Eqs. (F.30) and carrying out the (Gaussian) spatial integrals. For the monopole, this procedure gives

(F.31) where we used Eq. (F.24). Thus, the riionopole ~iiome~it~ vanishes due to the global charge neutrality of the system. The dipole moment of t.he charge distribution ~ ( ~ )coincides ( r ) with that, of the original delta-like distribution in Eq. (6.4); that is,

1

N {n) j = 1

M

v

(F.32)

In writing the last member of Eq. (F.32) we have used the definition M = Cy!,q,rj for the total dipole monient of the central cell aiid the fact that each replicated cell has exactly the same total dipole moment. Using similar arguments we obtain for the cartesiari components (A(2)),l(k?b = x, y, or z)

454

Three-dimensional Coulombic systems

of thc sccond-rank tcrisor A(2),

1 = (D(2))k, V

(F.33)

We proceed by inserting the nonvanishing multipole moments defined in Eqs. (F.30)-(F.33) into the expansion in Eq. (F.29),which gives $(2)

4n k2V

(k)= -2-k.

M

-

27r -kD(')k k2V

+ 0 (k)

(F.34)

As we emphasized before we are interested in the long-wavelength limit of g(2)(k)times the phase factm cxp (ik . T ) . Expanding the l a t h in a Taylor

series around k = 0, that is

exp (ik. r ) = 1

+ ik

T

-

-21 (k. T

+ o (k3)

) ~

(F.35)

and combining this expansioii with Eq. (F.34), we obtain

6(2)(k)exp (ik . T )

=

- i - k4n. k2V 4a +-(k k2V

*

2T M - -kD(2)k k2V T ) (k . M ) - i (k

1

T)

2a

2n -kD(2)k k2V (F.3G)

We now consider separately the terms on the right side of l3q. (F.36), focusing on the question whether they contribute to the desired (angle-averaged) ( T ) [defined in Eq. (F.28)]. The first. t,erm depends on l / k and potential 0. However, as this may therefore seem to diverge as we take the limit k first term also contains k . M . it vanishes already €or nonvanishing k because of the angle average in Eq. (F.28). To see this result, we note that the scalar product of two arbitrary unit vectors iz and b can be expressed in terms of spherical harmonics as [258)

-

(F.37)

455

Mathematical aspects of Ewald summation Thcreforc, Jdwkk. M = k [MI

3

6

Jdw*Y;,

(Ldk) Ylm( w M ) =

0,

(F.38)

m=-1

where we have also used Eq. (F.6). The next term on the right. side of Eq. (F.36) is constant in I; and involves the product kD(2)k, which does not immediately vanish if averaged over orientations. Nevertheless, we can safely neglect this term. The reason is that it is independent of the position of particle i, with the immediate consequence that the corrwponding energy contrihiition vanishes dire to thc global charge neutrality of the systeni [see text below Eq. (F.26)]. The third term on the right side of Eq. (F.36) contains the product, (k . r) (k . M). It has an explicit, positional dependence even after performing the orientational average. Indeed, expanding both scalar products according to Eq. (F.37) and using the orthogonality of spherical harmonics given by [258]

(F.39) we find

4lr 3

= -r.M

(F.40)

where the last line has been ohtairied by using Eq. (F.37) in reverse direction. The subsequent terms on the right side of Eq. (F.36) can be ignored because they are at least proportional to k arid therefore vanish in the limit k + 0. Thus, the potential ( r )reduces to [see Eq. (F.28)]

@i:i

(F.41) The above expression for the long-range part of the electrostatic potential is consistent with a well-known result from macroscopic electrostatics regarding

456

Three-dimensional Coulombic systems

the average electric field irisidc a largc sphcrc coiitaining an (arbitrary) charge distribution. This field is given hy [242]

-

E = - -4lrp

(F.42)

3

where P is the polarization of the sphere. Clearly, F is independent of the radius of the sphere. Moreover, it is constant within the sphere, implying that the corresponding electrostatic potential is given by

* ( r )-

4x 3

-r.F- -r.P

(F.33)

Wc: iiow recall that our systeiii is rcprmmtctl by one unit cell that, is replicatcd in all three spatial directions. Thus, we can indeed take our system to be a (macroscopically) large sphere. As a consequence, the quantity P ca.n be identified with the quantity P(*)/C',,,= M / V appearing in Eqs. (F.30b) arid (F.32). We therefore conclude that t,he poteiitial m(r)is identical with long-range potential QLR (2) ( r )given in Eq. (F:41). The above considerations are useful because they permit one Lo understand from a macroscopic perspective why a long-range contribution to the electrostatic potential should arise. Moreover, they are particularly helpful because they indicate a strategy to introduce different boundary conditions i1it.o the Ewald summation technique. Indeed, the physical picture t o which Eqs. (F.41)-(F.43) corrcspond is that thc (macroscopically) largc! spharc! is surroundcd by a V~CUUIII. In this cwc, any polarizatiori in tlic splicrc: will generate surface charges a t t,he interface between the sphere and the vacuum, and these charges in turn generat,e the average (or depolarization) field given in Eq. (F.42). If: on the other hand, the sphere is surrounded by a dielectricum with dielectric constant 8 , the average field inside the sphere has to be corrected by tlie so-called reaction field [331],

2 (d - 1) 47r ERF= 2c'+ 1 -3P

(F.44)

which, as expected, vanishes for the special case 6' = 1 (i.e., in the vacuuiii). Combining Eqs. (F.42) and (F.44), the total average field inside the sphere thcn hrcomcs

(F.45) Inserting Eq. (F.45) into Eq. (F.43) and taking r = ri, one obtains the final expression for the long-range contribution of the electrostatic potential given in Eq. (6.12).

457

Mathematical amects of Ewald summation

F. 1.1.4

Self-contribution

The self-part of the Ewald electrostatic potential given in Eq. (6.14) can be derived in a fashion similar to our derivation of the real-space contribution in Appendix F . l . l . l . Starting from Poisson's forniula [see Eq. (6.3)) and inserting Eq. (&lob) for thc charge dansitv p(3) ( T I ) , wa have

The threedimensional integral on the far right side of Eq. (F.46) can be evaluated bv transforming variables according to T' 4 R = T' - r, followed by a transformation t o polar coordinates. This gives 13301

(F.47) which can be easily be evaluated in closed form to give Eq. (6.14). Finally, it seems worth noting that the self-part can also be derived directly from Eq. (F.13) rcprcscritiiig thc potmtial (Pj,,, (ri)caused by a Gaussian located at, rj - n # ri. Indeed, considering @j,na t n = 0, one obtains

erf (arij)

- lim qi Tlj-+0

2

2a

Tij

- - -0

Tij'0

= -91-

2a

fi

J;r

3 f i

3 2

rij + 0 (r:)

- cp(3) ( T i )

(6.14)

ill thc limit rij + 0 whcrc we havc used the first few terms iri expansion of the error function erf (x)around z = 0 given by 2 erf (z) = -z

J;;

-

2 -x3 3J;;

+~ ( 2 )

(F.48) i\

Taylor

(F.49)

F.1.2 Force and stress tensor components

F.1.2.1

Force components

Based on Eqs. (6.15)-(6.17b) we can also dcrivc: thc: corrcspotiding cxprcs sions for the force acting 011 particle i ,

F$ = -q*Vt@( ~ z )

(F.50)

Because of Eqs. (6.16a) and (6.17a) we can split the total force into a sum of three individual contributions, namely

Fg

= FZ.i

+ F%,* + Fc"dLR,,

(F.51)

Three-dimensional Coulombic systems

458

The reader should rcalixc that thc sclf-part rriakcs 110 contribution bccausc the summand in Eq. (6.17h) is independent of the coordinates of particle i. Considering the individual contributions to thc total force separately, we

obtain after straightforward differentiation

(F.52a)

FEni =

-q,V,

" I= rj.C v 4n9jT-j (2d + 1) j=l

4nM -qi IT (2€' 1)

+

(F.52~)

In writing h s . (F.52a) and (F.52b) we have taken into account that the operator Vi appearing in the original force expression [see &. (F.50)] can be replaced by its counterpart Vij with respect to the distance vector r i j = ri - rj whcrc, of coiirsc, (F.53) aid ~ i = j Irijl also hold. In deriving Eq. (F.52a) we also used Eqs. (F.lO) and (F.14). Moreover, the last line of Eq. (F.52b) has been obtained using ikcxp(-ik.r,,) = i k c o s ( k . r i j )

+ ksin(k.rij)

(F.54)

where the cosine term (coiitrary to sine term) changes sign upon inversion, that is, k -, -k, and therefore vanishes in the sum over all wavevectors.

.

Mathematical aspects of Ewald summation

459

F.1.2.2 Stress tensor components By analogy wit8hAppendix E.3 we derive molecular expressions for various (diagonal) components of the stress tensor rTr(y = x, y, or z) by realizing that we may write (F.55) in the grand canonical ensemble where r$ is given in Eq. (E.33). From the definition of the Clausius virial [see Eq. (E.35)] and Eq. (6.15) for the total configurational potential energy of the three-dimensional Coulomb system in Ewald formulation, we have

because U g is a constant that does not depend on the actual configuration [see Eq. (6.17b)l. To evaluate the partial derivatives on the right side of Eq.. (F.5G), it. tiirns out to bc convmiant to t.ransform to imit,-ciibc coordinates via (F.57) Consider thc first tcrni on the right side of Eq. (F.50;). From Eq. (6.16a) we obtain

cc N

- i)S,

-

12

N

a

erfc (aJrij+ nl)

i=l j=l C{n} ’ q i q j K

N

N

Irij

-tnl

erfc (alr,j Irij

+ nl)

+ n1

2

which follows with tho aid of Eqs. (F.10) aid (F.14). Wo now noticc that, because of Eq. (F.57) lrij

+ nl

= 4s;

(%j

+

nx)2

+

S; ( & j

+7

~

+ ~ (5j ) +~n2)2 S:

(F.59)

In thc: previous cxprmsion we uscd the fact that the lattice vectors n = (nXs,,n,,sy, n,s,). Therefore,

460 whcrc

Three-dimensional Coulombic systems

Z, is a unit, vcctor

iii

thc y-dircctioii a i d yij = Fij . Z-, so that

follows without further ado. Turning t o the second term on the right side of Eqs. (F.56), we realize that a ( k ) is indcpcndcnt of {s,} t)ccaiisc cach tmm in tlic slim [SCC Q. (6.19)] can bc written as

-

exp ( - i k . ri) = exp [-27ri (,mxEi+ myci + m,B)]

(F.62)

where we used the defiriitioii of the wavevectors k [see text before Eq. (6.1l)]. This leaves us with

1

x ( k . Z,)2 17i.(k)I2

(F.63)

from Eq. (6.18) where we have used the fact that

V

= A,~s,

(F.64)

and

(F.65) from which

8k-

as,

_-

J(mx/s.)2

follows directly where

+

27r -nz2 _y (my/sy)2 (nz,/sz)2

+

k - e ^ , =27rm, S?

-

( k .i57)21

k

s?

(F.66)

(F.67)

Three-dimensional dipolar system

461

is thc projcctioii of thc wavcvcctor k onto thc y-axis (i.c.. thc ~-componerit of k). To evaluate the third contribution to W7,in Eq. (F.56),we realize from Eq. (6.17a) that.

(F.68) In writing the first term on the right side of Eq. (F.68) we introduced the projection of the total dipole moment. M [see Eq. (6.13)] onto the y-axis, namely N

N

(F.69) Equation (F.68) may he rewritten to givc

-= ---1 8%

27r

I's, 2€' + 1

["2

- 2 ( M * Z,)*]

(F.70)

for the long-range contribution to \Vz,c. Finally, putsting all this together we have from Eqs. (F.55), (F.56), (F.61), (F.63), arid (F.70) the somewhat lengthy expression

1 v 2

2n 2t'

+ 1 ([

M-~2 ( M .g 7 ) * ] ) ,

y = x, y, or z

(F.71)

for the diagonal components of the stress tensor in a Coulombic bulk system.

F.2 Three-dimensional dipolar system F.2.1

Self-energy

We now derive an expression for the self-contribution to the dipolar energy in Ewald formulation given in Q. (6.32) by recalling that the corresponding

462

Three-dimensional dipolar system

Coulorrilic coritributiou [we Eq. (6.17b)l rcsults from the interaction of the charges qi at r, with the cor~espondiiigGaussian charge clouds centered at T, and representing a total charge of -q,. Moreover, we have seen at the end of Appendix F.1.1.4 that for a given particle i the self-part of the electrostatic potential can be calculated from the potential generated by a Gaussian at rJ by taking the limit T , -+ ~ 0. Keeping this observation in mind and replacing thc chargc-s 4% by oparators pa. V, as siiggcst,cd I)y Eq. (6.22). wc find thc following prcscriptioii to calculate tlic dipolar self-contribution

~ ) hv its Taylor expansion for small distances Approximating erf ( ~ 7 . i /riJ given in Eq. (F.48),we obtain

rij

from which

(F.74) follows immediately by inserting Eq. (F.73) into Eq. (F.72) and taking the double limit. Equation (F.74) is identical to Eq. (6.32). In Eq. (F.73) we used the fact that

and Eq. (F.53). Therefore,

F.2.2

Force and torque

According to Eqs. (6.26)-(6.32) the total force on particle i can be expressed as a suin of two cont,ributions, namely

F$

= FG,i

+ FE,i

(F.77)

Mathematical asDects of Ewald summation

463

bccause both long-rangc contributions arid sclf-coritributioris in Eqs. ( 6 . 2 7 ~ ) and (6.32) turn out to be independent of the position of particle i and therefore do not contribute to the force. From Eq. ( 6 . 2 7 ~it~ )follows that. N Fs,i =

C' {(pi p j ) R

-vij

*

j = l {n}

- [pi *

(rij

+ n ) ][pj

*

(rij

+ nlia)

(~rij

+ n ) ]C (Irij+ nly 0))

(F-78)

where the functions B and C are defined in Eqs. (6.28a). Transforniing variables according to T i j -+ r = rij nlnoting that V, = Vij, and that

+

r d v, = --

(F.79)

r dr direct diffcrcntiat,ion on the right side of Fd. (F.78) gives

(F.80) where the function

-

U(r1LY) =

D (rla) is defined as [see Eqs. (6.28a)I

1dC - I d

rdr

-

1dB

y%(Fz)

(15 + 100%'

+ 40"')

+

exp (-a2r2) 15erfc (ar)

(F.81)

Thc Foiiricr-sparc contxibution follows from Eq. (6.27b) as

(F.82) where we employed Eq. (6.29). Differentiating in Eq. (F.82) with respect to r,j gives (see Appendix F.1.2 for the parallel derivation in the Coulombic case) 47r

FD3dFi. = -

C p1 exp (-2) c a k ( p i .k)(pj . k)exp (-ik. k.2

k#O

=

c

47r 7 21 cxp k#O

(-5) c

ri,)

j=l

N

j=l

k (pi . k)(pj . k)sin (k. r i j )(F.83)

464

Three-dimensional dipolar system

-

whcrc the last line is obtained via Eq. (F.54). Finally, using sin (x - ?I) = sin ir cos y-cos z sin y and the definitions of real and imaginary parts of M (k) given in Eq. (6.29) w e obtain

(k)](pi . k)k

- cos (k . ri)I&

(F.84)

The t,orquc acting on particlc .i is clcfirictl l y [I401 T D3 ,d~-=

@gi

-Pi

x

(F.85)

(vp,@!i)

where is the energy of particle i. Froin Eqs. (6.27) we realize that can he written as a siini of three t.erms, namely @gi=

@?Rj

@gi

(F.86)

-k @ g F , i -k @ k 3 , , i

and the differentiation is perforined with respect to Eqs. (6.26)-(6.32) we realize that

pi.

Referring back to (F.87)

where

[

+

x cos (k . r,)R e E (k) sin (k

ri)I m E ( k ) ]

(F.88b) (F.88~)

In Eq. (F.88a) we iise again the shorthand notattion r as bcforc [sco below Eq. (F.78)].

F.2.3

G

rij + n and

T

=

(TI

Stress tensor

As before in Section F.1.2.2 diagonal coniponents of the stress tensor of a dipolar fluid can be obtained from the relation (F.89)

Mathematical asDects of Ewald summation

465

ri!,

wherc the idcal-gas conitributiori is giveri in Eq. (E.33), invoking again the grand canonical ensemble for convenience. By analogy with Eq. (F.56) we have W,,,D =

a(JF as ,

=

aU,3d,

aU&

+as, + asy as, ,

y = x, y, or z

(F.90)

Based on the same transformat,ion to iinit-cube coordinate9 employed before [see Eq. (F.57)]. we realize. that the dependence on s, is buried in the argument of the functions B and C [see Eqs. (6.28a)I and in the factors P,,~. (riJ n) as far as UZR is concerned. Differentiating t,hese terms with respect to sy,it is easy to verify that terrns of the form

+

arisc whcrc pi^) stands for either pi or p j . Employing also the relation ani&ig thc functions R , C,arid D (sce Eqs. (0;.28a), (F.81)] as wcll as &. (F.60), it is a simple matter to show that

where we again transformed variables according to 1'

=

.1.1

~ i--fj

P =

+ n and

~ i j

Turning to the Fourier-space contribution next, we immediately see that

Ug contains a factor

1 1 --t?xp

V k2

(-5)

that has already been considered in the derivation of rz,CF in &. (F.63). We are then left with a derivative of the function z ( k ) [see Eq. (6.29)] which depends on s, because of Eq. (F.66). Introducing

Slab geometry

466 oiic obtains

Finally, the long-range contribution to the energy [see Eq. (6.27c)I gives rise to a stress contribution

(F.94) The reader should appreciate the difference between the previous expression and the last term on the right side of‘ Eq. (F.71). This difference arises N bccausc, for a dipolar systcm, Ad, = pi . i?, is independent of s7. whereas for a Coulonh systcin, hJ7 dcpcn& on s7 as one can verify from Eq. (F.69). The diagonal component of the total stress tensor is then o h tained by adding the t,hree contributions given in Eqs. (F.91), (F.93), and (F.94) [see Eqs. (F.89) and (F.90)].

F.3

Slab geometry

F.3.1 Rigorous expressions F.3.1.1 Point charges To derive Eq. (6.34) for a system of point charges in slab geometry, we proceed in a fashion analogous to the one employed for bulk systems in Section 6.2.1 and Appendix F . l . In other words, we divide the original charge density related to J3q. (6.33)

into three contributions correspoiidiiig to a set of screened charges pi(1) ( r ’ ) , a periodic set of charge clouds screening those original ones p(2) ( T ‘ ) , and a self-contribution p,!“’ ( r ’ )describing the interaction of each charge cloud with itself. We cliomc thc charge clouds to Lc spherical Gaussians’ with that the 2See Ref. 248 for other choices.

467

Mathematical aspects of Ewald summation

t h e contributiorls of tlic cliargc distribution arc

[

x exp -a2

(R' - Rj + rill)

1'

(F.96a)

(F.96b)

(F.96c) which is completely analogous to the bulk expressions given in Eqs. (6.7), (6.iOa), and (6.10b). respectively. Thus, we can immediately write down expressions for bhe potentials related to pi1) ( T I ) and pi3' ( T I ) [see Eqs. (6.8) and (6.14)]; that is,

(F.97a) d 3 ) (Ti)

=

2a

(F.97b)

-9iJ;;,

where in Eq. (F.97a), n = (nll,O). However, the potential @(') ( T I ) related to p(2)( T I ) differs from its bulk counterpart [see Eq. (6.14)] because the basic simulation cell of the current slab system is repeated in only two (of the three) spatial dimensions. Nevcrthclcss, wc can still apply our lmsic strat,cgv datailcd in Appcndix F. 1.1.2 to find the cxplicit exprcssioii for a(') (r'). We start by expanding the potential in Fourier space according to

1exp [ik~l- R] J d$6(') 00

(T)

=

5 2n

kil

(k)exp [ik,z]

(F.98)

-m

where kll = (2nmx/sx, 2nnt,/s,) are two-dimensional reciprocal lattice vectors (i.e., exp [ikll . n11] = l),whereas the vectors k appearing as arguments of the coefficients 6(2)(k)are still three-dimensional. Equation (F.98) follows from its three-dimensional counterpart (F.19b) if we replace in the full

Slab geometry

468

Ck. . .

-

Ck,'&Ck, . ..

t-he partial summation over tlie discrete variable k, by an integration, that is Ck.. . . (Arcz)-1 m dk, . . . with Ak, = 27r/s,. This is consistent with viewing the system in slab geometry as thrcc-dimcnsiond wit,h a basic. ccll lxconiing anfin,atch/ largc in tho Z-dirLxAion (i.c-.,s, + 00) such that Akz -+ 0. The Fourier coefficients &(')(k) appearing in Eq. (F.98) are linked to the corresponding coefficients of the charge density via the Fourier-transformed Laplace equation [see Eq. (F.21)] SUIII

=

q2)(k) = -

'J + (--.;.-)

v,,

1

- exp V

d r exp (-zk - T ) p(')

k;

k:

s-,

(f)

qj exp ( -zk,zj) exp

(-ikll . Rj)

j=l

(F.99) where we have inserted Eq. (F.96b) to obtain the second line of Eq. (F.99). Coinbirling Eqs. (F.99) and (F.21) and inserting the resulting Fourier coefficients &*)(k) into the expansion in Eq. (F.98), we find

~t)+~

whcrc thc SUIII in llic first liric is rcstrictcd to no~ixcrowavevectors kll aid contains coiitrii)utions froxu the long-wavclcngtli liiiiit (see ~xlow).111 Eq. (F.lOO).the integral over the continuous variable kz gives [248]

(F.lO1)

where the function f (kll, z - z j , a) lias been defined in Eq.(6.36). ( z - z j ) is defined From EQs. (F.lOO) and (F.lO1) it. follows that as N

Thus, we consider tlie behavior of the function f(kll,z - zj, a) for small wavenumbers kll. To this end we perform a Taylor expansion of both the

Mathematical amects of Ewald summation

469

cxporiciitids arid the coriiplcineritary crror fuiictions appearing OII the right side of Eq. (6.36). In this expmsion, the lowest-order Taylor coefficients of erfc (y) = 1 - erf ( 9 ) follow immediately from Eq. (F.49). We also note the relation erf (-y) = -erf (y), which follows from the definition of the error function in Eq. (F.9). Expanding f (kll, z - z,, a ) [see Eq. (6.36)] around kll = 0, we obtain

where we retain only linear terms in kll. Conibining the previous expression with Eq. (F.102) yields

Inspecting the right side of Eq. (F.104) we see that the first term in parentheses is constant,. This term is irrelevant because of global charge neutraliy (i.e., Cc,qi = 0). We therefore obtain from Eqs. (F.lOO) and (F.104) as a final expression for the potential from the set of Gaussians

2fi

--

N (12

exp[-a 2 ( z - zj)']

(F.105) The corresponding contribution to the energy is

(F.106) which coincides with Eq. (6.35).

470

Slab geometry

F.3.1.2 Point dipoles The rigorous Ewald sum for a slab-like system of point dipoles follows from the corresponding expression for Conlonibic systems [see Eqs. (6.34) and (6.35)). The derivation proceeds in a fashion similar t o the one already rlisciissed for hiilk syst,rms in Section 6.2.2. That is, we replace thci charges 4i arid qj in trhc cncrgy cxprcssiotis by the operators (pt . Vi) and (pJ. Vj). As a result,

where both real-space and self-part. have the same form as in the threedimensional case and are thus given by Eqs. (6.27a) and (6.32), respectively.

To evaluate the Fourier part, we start by considering the first sum on the right side of Eq. (6.35)involving nonzero wavevectors Icll # 0 . For each pair .i and j and each wavevector kll, thc replmcmcnt of the cliargc?qj hy t8he operator (pj Vj)yields

(F.108)

( F.109) Now we need to differentiate Eq. (F.108) one more time because of the second operator (pi Vi)replacing qi in the original energy expression in Eq. (6.35), +

471

Mathematical aspects of Ewald summation

which yields

where we have also used [see Q. (F.109)]

and the funct,ion

(i:

+ exp (-kllzij) erfc - - a z i j ) ]

(F.112)

Keeping in milid that the total Fourier energy involves a sum over all nonzero wavevectors lcll # 0 [see Q. (6.35)], we may employ symmetry arguments to siinplify the above expressions. For example, both (pi kil) (p, kll) and the function j' (kill z,,, (Y) appaaring on tho first. linc of the right side of Eq. (F.llO) are invariant, against inversion of the wavevectors, that is to say, a replacement of kll + by -kll. Therefore, only the r e d part of exp (ilcll . & j ) [i.e., cos (kll . &j)] will contribute t o the sum over all wavevect.ors. The same is true for the fourth term involving the product (pi Zz) (pj e^J and the function e( k!l,zij a) which is again invariant against inversion of kll. However, the second and third terms on the right. side of Eq. (F.llO) are +

~

-

472

Slab geometry

(F.113) We tiow consider the remaining (second) siiin in the Coiilomb Fourier energy (SLW the right side of Eq. (6.35))involving a double ~ L U I Iover pairs of dipoles z and j . Replacing the products qt and qI by the operators (pi . Vi) and ( p j Vj), respectively, as before yields

Finally, putting all this together we arrive at, the rigorous expression for the Fourier-space contribution to the total configiirational energy of the dipolar system in slab geometry. namely

(F.115)

Mathematical aspects of Ewald summation

473

F.3.2 Force, torque, and stress in systems with slab geometry F.3.2.1

Point charges

For ionic systems, the total Coulomb force acting on particle i within the slabadapted three-dimensional Ewald sum [see Eq. (6.40)] can be cast, as

FEY

=

q z : + F;!: + q : !

(F.116)

whcrc t,hc first, two cont,ribiit,ion.s arc identical to thc: rorrwponding ones in a truly threedimensional systeni and are tlius given by Eqs. (F.52a) and (F.52b), respectively. The last term in Eq. (F.116) arises from the correction ~ Eq. (6.39)]. One obtains term in the Ewald energy, U C , [see

(F.117) Regarding the stress tensor of the systeni, the (Coulomb) components corresponding to the two orthogonal directions parallel to the walls (i.e., y = z,y)can bc calciilatctl exactly as in the thrccdiincnsional caw (sec Appendix F.1.2.2). On thc other hand, tlic noriiial component (y = z ) is given by slab - Tslnb + Tslilb + Tslab (F.118) 7C,zz - C R m CF;n C.c,zz whcrc only t,hc rcal-space part T ~ ~ ' = & T$,= [see Eq. (F.58)]. To cvaluatc thc Fourier-spaw contribution, T&!'&. wc iiotc that, bccause of the artificialelongation of the basis cell in z-direction, neither the wavevectors involved in the Fourier contribution to the Ewald energy [see Eq.(6.18) with the wavevectors given in Eq. (6.41)] nor the volume V depend on sz. The vectors ri = (zz,yi, szZ), on the other hand, depend on s, if we employ scaled z-coordinates as indicated. Differentiation thus yields

+k ( k ) Z ( k ) )

x (g(k)Z(k)

(F.119)

where the quantity Z(k)is defined ill Eq. (6.19) aid

-

b (k)=

N i=l

ikzi exp (-A . ri)

(F.120)

Slab geometry

474

Filially, as Llic cncrgy correctioii tcrrn givcii in J3q. (6.39) dcpends on s, only through the t-components of the position vectors ri, the expression for the corresponding stress in the direction normal to the confining substrates follows as

(F.121) F.3.2.2 Point dipoles For dipolar particlcs, all force contdmtions within the slahczrlaptcd thrwdiinensiorial Ewald sum coiricide with those for truly three-dimensional systems discussed in Appendix F.2.2. This result arises because the correction term to the total dipolar energy [see Eqs. (6.44) and (6.43)] is independent of particle positions. There is, however, a coritribut,ion to the total torque that we need to consider separately. The total torque can be cast as

2g;h = T$$ where %',$ and whereas

+ qg't qgi

(F.122)

are given by the bulk expressions [see Eqs. (F.88)],

(F.123) Turning next to the stress tensor we realize that its normal component mit>hinthe slab-adapted three-dimensional Ewald formalism can be written as a sum of two contributions, namely pD.zz lah - Tdsb DR..zz + 7s1ab DF,zz

(F.124)

because t,he correction term to t,he total configurational potential energy in Eq. (6.43) does not depend on s, (note that t,he V is the volume of the artificial cell includzng the vacuum space in the z-direction). The real-space part. on t.hc right, sidc of El. (F. 124) coincidcs with its t,hrac-dimcnsional aiialog givcn in Ey. (F.91). Tlic Fourier part, T$$'', can be derived along the same lines already discussed below Eq. (F.118). We finally obtain with little ado the expression

x

(v (k)P (k)+ 17" (k)G(k))

(F.125)

Mathematical aspects of Ewald summation

475

N

(k)=

i=l

( p i .k)ilC,z, exp ( - i k f T

~ )

(F.126)

F .3.3 Metallic substrates F.3.3.1 Point charges

Here we derive Eq. (6.66)linking the energy of a slahlike system of point charges between metallic walls to that of an extended system with threedimensional periodicity. The basic cell of the exteiided systeru contains N cliarges ill the original cell plus the first set of images; that is, the N images resulting from the presence of just the lower wall. Positions arid charges of these image particles we-then given by the relations [see Eq. (6.59)with nz = 0)

i = l , . .. , N i = l , . . .! N

ri+N

= ri - 2%&,

qi+N

= -qi.

(F.127a) (F.127b)

Replicating thc cxtcnded hasic cell periodically in all three spatial directions, t,he total energy of the resultiiig system is given by

(F.128) where the lattice vectors ?z are specified in Eq. (6.64) arid the prime at the surii indicates that the term related to 1: = j is oriiitted for Ti = 0. We now split the double slim in Ecl. (F.128) into four terms containing N 1. Particle particle cont,ributions EL1Cj=l,

3. Particle image contributions 4. Image particle contributions

ELl xi!&+,, and 2N Ci=N+l xLl.

476

Slab geometry

Tcrins 1 arid 2 givc thc same rcsult

on(: iriay varify from thc rclatioiis

(F.12%)

and the fact that we slim in Eq. (F.128) over an infinite set of lattice vectors {K}such that the term 4s,n.,E2 on the right side of Qs. (F.129) is irrelevant. By similar reasoning, terms 3 and 4 in the above decomposition give eqnivalant results because of

(F.1304

Putting all this together,

&. (F.128) can be rewritten as

We therefore see that UFYexin Eq. (F.131) for a three-dimensional system with the extended basis cell is indeed exactly twice the energy Uc given in Eq. (6.65). F.3.3.2 Point dipoles

To derive Eq. (6.68) for the total energy of an infiniteslab of dipolar particles between nietallic substrates, we go one step back and consider a situation where thc central cell comprising N particles has not yet been replicated in the .I:- arid ?/-directioiis. Tlic corresponding energy can lie written as N

(F.132)

Mathematical amects of Ewald summation

477

where EpeIf arid qiS arc thc clcctrostatic fields arisiiig from the images of particle i, on the one hand, and from the other particles j and their images, on the other hand. Using short-hand notation (F.133) tbe fields EYlf and E,disfollow as 54

11.=--00

00

Tk=-Oo

(F.134a)

where the asterisk attached to the slims indicates t,hat terms corresponding to n, = 0 haw bcrn omitted. Replicating the original cell now in the z- and 9- directions essentially implies that the sums over the integer variable n, in Eq. (F.134a) have to be replaced by three-dimensional lattice sums over the vectors E introduccd in J3q. (6.64) [scc the arialogous proctdurc for charges described below Eq. (6.62a)l. Inserting the resulting field expressions into Eq. (F.132) and summarizing, one obtains Eq. (6.68) after some tedious but straightforward algebraic manipulations. As a next step we now have to prove Eq. (6.69) where we again proceed as before in the Coulombic case. The basic cell of the extended dipolar system contains N dipoles in the original cell plus the first set of images, which are the N images resulting from the presence of just the lower wall. Positions and orientations of these N image particles are then given by

R.eplicating the basic cell periodically in all three spatial directions, we obtain

478

Slab geometry

for the total configurational potential energy thc cxpressiori

1

cc F' { lTV 2N 2N

=2 r=l j = I

pt ' p J - 3 1111 . ( r l J +~ 1 3

b'J lTtJ + ~1~

+

* (TtJ

+ (F.136)

where the lattice vectors n are specified in Eq. (6.64) arid the prime at the lattice slim indicatcs that i = j is omitted for f i = 0. Separating now the (F.136) i1it.o doublr sum in

a.

1. Particle particle contrilmtions

2. Image iniage contributions

c~N_,

zj=, , N

CtzN+, z5EN+1,

3. Particle image contributioris

N

4. Image particle contributions

2N

2N Cj=N+,, and N Cj=,

one finds that terms 1 and 2 give tht: same result after performing the lattice sum. Indeed, onc can easily show t,hat

whcrc Eq. (F.137b) is idciit,ical with Eq. (F.129b) a11d

where we recall that the terms with 4 ~ ~ s , &are irrelevant because we sum over an infinite set of la,ttice vectors in Eq. (F.136). Moreover, terms 3 and 4 in the above decomposition are also equivalent because of the relations [see also Eq. (F.130b)l

and

We therefore see t,hat the energy Uf7" [see Eq. (F.136)] is indeed exactly twice the energy UD given in E.1. (6.68).

Reviews in Computational Chemistry Kenny B. Lipkowitz &Thomas R. Cundari Copyright 02007 by John Wiley & Sons, Inc

Appendix G Mathematical aspects of the replica formalism G.l

Replica expressions in the grand canonical ensemble

For a fluid coupled to a reservoir with chemical potential p, the thermal average at a given matrix realization (and fixed number of matrix particles) is given [instead of the canonical expression Eq. (7.6)] by

where zf= exp ( - p r / k ~ T is ) the fugacity and

is the matrix-dependent partition sum. Equation (7.8), on the other hand, for the disorder average remains unchanged. Moreover, the replicas can be introduced exactly as in the canonical ensemble discussed in Section 7.3. In particular, the configurational potential energy of the semi-grand canonical replicated syst,em is the same as for the canonical system [see Eq. (7.12)]. The partition function of tho replicated syst.cm in thc grand canonical ensemblc is defined by

479

Derivation of Ea. (7.23)

480

All rclatioiis givcri in Scctioii 7.4 bctwccri correlations in the disordcrcd aiid replicated system, respectively, remain unchanged. Finally, we consider the grand canonica1 potential, 52, of the adsorbed fluid. Proceeding exactly as for the free energy in the canonical ensemble (see Section 7.6), one finds the analogous expression

where R,,

G.2

= -kBT In ZreP.

Derivation of Eq. (7.23)

To derive ELq. (7.23). which relates the blocked correlation function,

hb, to the replicated system, we start from the statistical physical definition of the blocked function given in Eq. (7.22). On t'he right side of this equation, the double average over one of the pair terms, that is, for example, the term with i, = 1, .j = 2. ran hc writtrn as

where we employ the definition.. of the thermal averages and the disorder average given in Eqs. (7.6) and (7.8), respectively, and ,4 (q1) 6 ( q - q1) and B ( 4 2 ) = 6 (q' - q2). We may now introduce the replicated system by multiplying both the numerator and the denominator of the integrand in Eg. ( G . 5 )by Z&'. The numerator then involves alt,ogcthcr 71. niiiltiplc integrals ovcr tha fluid particlc coordinates, qNf. In other words, because of the multiplication by Z2;*, we introduce n - 2 copies of the fluid particle, in addation to the two already present. The already existing copies are represented by the two integrals dq" . . . in Ey.( G . 5 ) .To these two copies we assign arbitrary. yet different indices a' and /3'. The functions A (q1) and B (qz) then become functions of particle 1 of copy a' and particle 2 of copy fl. Inserting then Eq. (G.l) for

481

Molecular fluids

P ( Q N m ) , Eq. (G.5) can bc rcwrittcn as

where we employ the definition of the configurational potential energy of the replicated system in Eq. (7.12). Finally, we use the same "trick" as described in Eq. (7.14) and identify the denominator in Eq. (G.6) with the partition function of the replicated system in the limit n --j 0. This identification gives

= n-0 lim ( A (Qla')B (QzP')),p

I

# /3/

(G.7)

Applying the relation in Eq. (G.7) to cadi pair term in the definition of the blocked correlation function [see Eq. (7.22)] and using Eq. (7.18) for the densit,y, one finally arrives at Eq. (7.23).

G.3

Molecular fluids

The purpose of this section is to reformulate the RSOZ equations in k-space given in Eqs. (7.41a)-(7.42) in an appropriate form for a molecular fluid in a molecular matrix such that. the resulting expressions are still numerically tractable. We specialize to the case of linear molecules, such as spheres with dipolc momcnts or cllipsoidal particks whosc oricnt,at,ioncan bc spccificd by two angles w = (19,p). The correlation functions f7(k,( ~ 1w2) , (where f = h or c, y = mm, mf, fm, ff, b or c) then depend on seven variables altogether. They describe the orientations of the pair of particles, that. of the wavevector, and the wavenumber k = lkl. To handle these variables, we expand each correlation function in an angledependent basis set of rotational invariants [258]. Taking advantage of thc fact that, in an globally isotropic system, t,hc direction of thc wavcvmtor k does not matter, we choose k t,o be parallel to the z-axis of the space-fixed coordinate system. The resuhing '.k-franie" expansion is then defined by ~581 I

Molecular fluids

482 whcrc thc i1ivariaiit.s

contain spherical harmonics K,Awith 1x1 5 min (11,12) and wi and ui are the dipolc oricnt,ations tncitsiirrd with rchpcct, to the wavrvcc-tor k 11 Sz. Invaria c e of the fluid against inversion of k implies that (k) = (k) for all correlatioms y invnlved [258). Moreover. exchange symmetry of the fluidfluid and matrix-matrix potentials imposes the restriction (k) =

%$

p27

c,2(c,b,n,m)

(k). Note, however, that this symmetry does not hold for the correlations involvirig fluid and matrix particles where ek (k)= ?Af (k). fx,fl(c.b.mm) 7Zll

Insc.rtningthc k-frariic expansions (G.8) of t tic corralation turictioiis into the RSOZ Eqs. (7.41a)-(7.42), the angular integral buried in the products @ can be easily perforiiied by using the orthogonality of the spherical harmonics [258] [cf. Eq. (F.39)]. Introducing, for compactness, matrices F, with elements

fb

-

(G.lO) we finally obt,ain

where p = 47~7= Nf/V and pm = 47rijIn = N,n/VWe note that the RSOZ equations in Eqs.(G.1la)-(G.1ld) are decoupled with respect to the wavenumber k and the angular index x. In practice, the matrices H, and 5, have finite dimensions due t o a truncation of the rotationally invariant expansion in Eq. (G.8) at appropriate values of 11 and 12. The RSOZ Eqs. (G.lla)--(G.lld) can thus besolved by standard matrix inversion t.echniques.

483

Proof of Eq. (7.33)

Proof of Eq. (7.33)

G.4

Our starting point in this section is

-

hH (Ic) =

1

[SCC

Eq. (7.33)]

dfh, (P)exp (ikT cos 6)

(G.12)

where we iised the definition of the scalar product of two vectors, namely k . ;F = kPcos 29, where YI is thc angle between k and f . The reader should nole that this form of Eq. (7.33) is a consequence of the fact that, hr depends only on the magnitude (but not on the direction) of 'ii on account of the isotropy and homogeneity of the system under consideration (see discussion in Section 7.4). For a fluid composed of molecules with rotational (F,ul, u2) and the integrals in Eq. (7.32) degrees of freedom, hff (7) -, ran only be rarricd oiit by employing the Raylcigh expansion of the phase factor exp (ik . P) (2581. However, in the current case of a "simple" fluid, h e mogeneity and isotropy may be invoked conveniently to rewrite the previous expression more explicitly as

-

2r r m

J J hK (P)exp (ikFccos6) d9sin6dtF2dP

hff (k)= 0

0

(G.13)

0

using spherical polar coordinates, where the integration over p can be carried out trivially yielding a factor of 27r. The integral over 6 can also be carried out in closed form because h~ (F) does not depend on this angle. Using the transformation z = cos6, we can rewrite the integral over IY in Eq. (G.13) as

I

-1

7I

siri dd29 exp (ik7cos 19) = -

0

dz exp (iIcFx) I

-

1 [exp( - i k ~ )- exp (ik~)] ikF

--

(G.14)

where we uscd Euler's identity cxp (fix) = cos z f i siri z for corriplex numbers. Combining Eq. (G.14) with the integration over cp in Eq. (G.13), we arrive a t the far right side of Eq. (7.33). It is also instructive to consider the limit of vanishing wavenumber k -, 0. In this case sin (ka) (G.15) lim -= lim c'os (kP)= 1 k-0

kP

k-0

Numerical solution of internal equations

484

which follows from dc 1’Hospitd‘srulc for taking th: liriiit of an uridctcrrriiried expression of the form In this caw Eq. (7.33) simplifies to

5.

I

m

hR(o) =~ T J ~ F F ~ ~ ~ ( F )

(G.16)

0

G.5

Numerical solution of integral equations

In this appendix we present solution strategies for integral equations such as the ones discussed in Chapter 7. However, rather than considering the replica-vcrsion of intcgral cqiiations introdiiced in Chapter 7 for disordcrcd fluids wc focus here on the equations related to a fully annealed binary niixture of particles without internal degrees of freedom. This simpler system can be considered as a generic case appropriate to introduce the basic steps of the numerical solution of integral equations. Indeed, based on the numerical solution of an annealed binary mixtnre, it is straightforward to generalize the algorithm for more coniplicated cases such as partly quenched (sphcrical) mixtiircs and c m i inolcciilar (anncalcd or partly qiicnchd) fliiids, where rot ationally invariant expansions of correlation functions lead to integral equations formally equivalent to those of niulticoinponent mixtures [258]. Specifically, we consider a model consisting of two species A and B of spherical particles with diameters QA and CB and isotropic interaction potentials Z L , , ~ ( T ) .As a consequence, the twepoint correlation functions depend only 0x1 the separation T = Irl - r21 between the particles. We are particularly iiitcrtstcd in thc total corrdation function haO(r) or. cquivalciitly, the pair correlation function gaO(r) = h , ~ ( r ) 1 and the direct correlation function C ~ S ( T ) ,which provide together a complete description of the structure and thermodynamics of the mixture [30]. These correlation functions are linked by the exact OZ equations [see also Eq. (7.36) for the general case of a rnulticornporieiit mixture]

+

As mentioned in Eq. (7.36), after introducing Fourier transforms of the correlation functions, the OZ equations simplifies to [see Eqs. (7.32) and (7.33)]

Mathematical aspects of the replica formalism

485

In addition, assurriirig particles with hard-core repulsive interactions, we have trhe exact, core conditions [see Eqs. (7.44)] cap ( r ) = -1 -

van ( r ),

r

< fla4

(G.19)

+

where q12a(r) = hnB( r ) - clxB(r) and oClLj= (go o,j) /2. For separations beyorid the hard core we n e ~ dail approximation, and we coiisider here as an examplary case the (nonlinear) hypernetted cliai~i(HNC) closure introduced in Eqs. (7.49) for the case of disordered fluids. Specializing our treatment to the current binary mixture: the HNC closure reads as follows:

where we stress that these equations are decoupled with respect to the species indices. The same is true for the simpler Percus-Yevick (PY) and mean spherical approximation (MSA) closure approximations introduced in Eqs. (7.46) and (7.48). The god is now to solve &is. (G.18)-(G.20) together for given partial densities pa and temperature T . Apart from SOIIIC iiotablc exceptions such as the PY equations for hard spheres [332-3341 and hard sphere mixtures [335, 3361 and the MSA for charged [337, 3381 or dipolar hard spheres [339], the actual solution has to be done numerically. To this end, all correlation functions are discretized on a lattice with a typical grid width of Ar = 0 . 0 2 ~and typical lattice sizes of A’, = 1000 - 2000 points (corresponding to separat,ions of 10 - 200). Within this prcdcscribcd range the correlation functions can be calculated via an iteration procedure. The flow scheme representing one iterative cycle is depicted schematically in Fig. G.l. 1. Initiation of it-crativc proccdiirc As a starting point of the iteration OIIC defines xi iriitial guess C ~ ~ ( Tfor - ) the direct correlation functions. Indeed, a good initial guess is particularly important for strongly interacting systems, that is, those systems at high densities and/or low temperatures. Most conveniently one may use the final result @ ( T ) of a previous solution at a “nearby” thermodynamic state with similar densities and/or temperature as an initial gicss for thr dcnsitics and tcmpcraturcs of interest,. As an altcrnative, which is particularly useful for hard-core systems, one may take the (PY) direct correlation functions [332-3361 of the underlying hardsphere system as an input for the current iteration. Finally, it seem worth mentioning that for weakly interacting systems (especially for soft-core systems at high temperatures) convergence can often also be achieved with the trivial choice c h ; j ( ~=) 0.

486

Numerical solution of integral equations

new input

Figure G.l: Flow scheme of the numerical solution of the integral equations Eqs. (G.18) and (G.22)] together with the HNC consisting of the 02 equations [&see closure [see Eqs. ((2.19) and (G.20)]. The acronyms FT and FT-' denote Fourier transformation and Fourier inversion, respectively; Ac is defined in Eq. (G.27); and d is a threshold value set according to the desired accuracy.

2. Fourier transformation The next step consists of a Fourier transformation of the direct correlation fiinctions according to the prescription [we also Eq. (7.33)1

(G.21) whcrr tlic Fourier traiisforriis {Fag (k)} arc again tliscrctizal on a latticc with Nk = Nl points and grid width Ak = n / ( N l A r ) . The actual Fourier traiisformation can be performed via fast Fourier transform techniques [175]. Some care licls to be taken due to the discontinuity of the fuiictioris { c , a ( r ) } at separations T = oao, which arise from the corresponding discontinuity of the pair potentials for hard-core systems. To circnmvcnt this problrm onc iwially conipiitcs the Foiuicr transform sepilratcly for thc rcgioiis T < ma,j arid t' > ua0.

3. Solution of the OZ eqtiatioii Having obt.airied the quantities Z,,o (k)!we are now in the position to

487

Mathematical aspects of the replica formalism I

calculate the ~ouricr-transforrrie~ tot,al corrclatioxi functions hap (k)by solving Eq. (G.18). However, for hard-core systems in particular, it is more convenient, to consider instead the function Gap (k) hap (k)cmg(k). These quantities are more suitable for the Fourier inversion (required to solvr thc closiirc relations) brcaiisc thc: rral-space functions qap(r) are continuous at the hard core (i.e., at r = 0 ~ 0 ) . The total correlation functions hap ( r ) ,on the other hand, exhibit a discontinuity (as do the direct correlation functions). To calculate the quantities qUp (k),we first rewrite F,q.(G.18) as I

-

This lattcr equation may hr rcwrittcii more compactly by inbroducing the 2 x 2 matrices

E (k)and e (k) witth elements

Rewriting Eq. (G.22) in terms of

E (k)and (k)yields

E ( k ) = E ( k ) C ( k )+ e ( k ) E ( k ) which may be solved for

E(k)

(G.24)

(k)to give

= e ( k ) e ( k )[l -e(k)]-l

(G.25)

where 1 is the unit matrix. Equation (G.25) involves only 2 x 2 ma-1

trices. Thus, the inverse matrix 1 - 5 (k)] can be calculated anal-ytically so that, Gap (k) can easily be obtained from the elements of matrix E (k) according to it.s definition given in Eq. (G.23a).

[

4. Fourier inversion

Next, wc wish to calculatc thc (continuoiLs) real-space functions 7 1 4 ( r ) . These functions simply follow from the Fourier inversioii

(G.26)

Numerical solution of integral equations

488

5. Solutioii of the closuw exprcssions Using the closure relationships Eq. (G.19) and (G.20) for separations below and above the hard core, we can now calculate the new direct correlation functions cz: ( r ) . We stress that the closure can be solved independently for each index coinbinat,ion ap. Furthermore, note that the HNC closure ( r > orrp)iiivolves the total correlation functions hup ( r ) ,whirh can ho approxiinatod t y tho ciirrcnt, cstiniatcs h,,;,( r ) = CEO ( r )+ qad ( T ) .

6. Accuracy check

As a final step of an it.eration cycle we have t o test the quality of the soliit,ion using an appropriatc sclf-consistcncy critorion. Tho choicc of this criterioii is not uniquc. However, a suitablc quantity to be calculated in this context. is

(G.27) If this quantity is smaller than some predescribed small number 6 (e.g., W 6 ) , one considers tlie solution cE$ ( r ) as being self-consistent and the iteration is stopped. If, on the contrary, Ac > b the iteration is continued using appropriate input as dewribed below. c5 =

7. Constructing new input We have now reached a stage of our solution procedure where updated input is required for the next it+erationcycle. Looking at this problem n%vely, it may seem that the most natural way of starting a new iteration consists of using the output functions c$(T-) of the previous it,ciration IL as ail initial gums for tho ciirrrnt it,rration T L + 1. However, in practice. this procedure often leads to an iiristablc iteration where Ac defined in Eq. (G.27) oscillates between subsequent iteration cycles or even increases. Several strategies to stabilize the iteratioil procedure have been proposed. Here “stable” means that Ac decreases monotonically as the number of iterations increases. The two most important strategies leading to a stable iteration procedure can be sketched as follows: 0

Within the first method the new input of iteration n+ 1 is given as a linear inkrpolation between the input and the outpllt functions of the previous iteration cycle; that is, in,n+l CaO

out,n

( r )= acuO ( r )

+ (1 - a)c$”

(1.)

,

0 < cr

< 1 (G.28)

489

Mathematical asDects of the redica formalism

The optixnal choice of the parmncter n has to be determined ernpirically. For t,ypical calculations, n! = 0.1 - 0.2 and it takes of the order of lo3iterations to obtain satisfactory convergence [340]. 0

The sccorid iiicthod is lmsccl 0x1 xi idca of Ng arid uscs thc "history" of the iteration procedure to construct the new input functions [341]. Specifically, one constructs new input by extrapolating the previous output from at least two or, in niost cases, several previous iterations according to the prescription %:ain'''+' 0 (T)

= [1 -

(72)

-

+a1 (n)c2Jl-l

(TI.)] C$'"

(T)

(T)

+ a2 (n)put,n-2 C@

(1')

(G.29)

where the prefactors a1 (n)and a 2 (n)are determined in each cycle such that Ac given in Eq. (G.27)is minimized. The advantage of this method compared with the one described before is much faster convergence (typicallv only of the order of 10 iterations are required). A t the same time, however, the procedure is more susceptible to iiurnerical instabilitics, iinplyirig that, the quality of the input function has to be better. A more detailed discussion of this method can be found in the originnl pa.per of Ng [341].

Reviews in Computational Chemistry Kenny B. Lipkowitz &Thomas R. Cundari Copyright 02007 by John Wiley & Sons, Inc

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[5] M. Schoen, Computational Methods in Surface an,d Colloid Science, Marcel Dekker, New York, 2000. [6] E. II. Lieb and J. Yngvason, Phys. Rep., 310, 1 (1999). [7] E. H. Lieb and J. Yngvason, Phys. Rep., 310, 669 (1999) [8] S . G. Briwh, The Tempemtun: of History. Phases of Science and Culture in the Nineteenth Century, Burt Franklin, New York, 1977. 191 Lord Kelvin, Nature, 1,551 (1870).

(lo] J. Tyndall, Heat Considered as a Mode of Motion, Longmans and Green, London, 1863. [Ill G. Arfkcn, Mathematical Methods for Physicasts, Acadcmir Prcss, London, 1985. [12] H. B. Callen, Thermodynamics, Wiley, New York, 1960. [13] D. J. Evans and G. P. Morris, Statistical Mwhanics of Nonequilibrium Liquids, Academic Press, London, 1990. [14) H. Goldstein, Klawische Mechanik, Aula-Verlag, Wiesbaden. 1991. 491

492

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509

Reviews in Computational Chemistry Kenny B. Lipkowitz &Thomas R. Cundari Copyright 02007 by John Wiley & Sons, Inc

Index r-fimct,ion, 373 k-framr cxpmsion. 481 X-expansion, 227 activity, 88 adsorption rate, 323 aerogels, 341, 345 annealed variables, 344 atomic force microscopy colloidal probe, 205

elactxic field, 333 insulator, 313, 315, 332, 339 periodic, 186 vacuum? 309

Cardanic formilla, I12 Caucliy intcgral, 383 Cauchy-Riemann equations, 375, 382 Chapman-Kolrnogoroff equation, 183, 184, 433 charge cloud Bernoulli distribution, 83, 412 Gaussian, 305, 447, 466 Bessel inequality, 390 charge distribution binary mixture &like, 314 asymmetric Fouricr sarirs, 460, 451 critical behavior, 167 riiultipolc IIIOIIICII~, 453 experimental data, 165 charge neutrality, 455 grand-potential, 164 condition of, 452 phase behavior in confinement, Clausius virial, 439, 459, 465 167 closure approximation symmetric analytic, 485 decomposition. 157 hypernetted chain, 357, 358, 362, topology of yhasc diagram, 153 485 Bloch equation, 400 mcan Bogoliubov theorem, 121 spherical, 357, 358, 360, 485 Boltzmann factor, 96 optimized random phase, 360 field energy: 331 Percus-Yevick, 357, 358, 485 BoseEitstein statistic, 35 strong coupling condition, 368 symmetry of wave function, 60 coexistence line, 31, 363 boundary conditions colloids, 332 conducting, 300, 313, 315: 332, coinplete induction. 407 475, 476 completeness relation, 64

510

INDEX

coiiiplcx function differentiation, 374 orthonormality, 390 path of steepest descent, 378 residue, 382 saddle point, 377 scalar product, 390 complex riuriibcr Euler representation, 397, 483 configuration, 59 discretization of space, 181 configuration integral, 68 confined hard-sphere gas, 104 f;lctorization. 177 Incan-field, 104, 122, 150 onedimensional, 76, 86 confined fluid correlations, 97 critical-point shift, 100, 107, 342, 361. 363, 366 dipolar fluid orientational order, 325 equation of state onedimensional fluid, 87 homogeneity and isotropy, 14 shear stress, 238 contact layer, 231, 328 contrast, factors and dcnsity profilcs dcpciidcrice on teniperaturc, 171, 174 controlled pore glass, 98, 160, 341 correlation function n-dependence, 355 blocked, 349. 350, 353. 358. 368. 480

connected, 350-352, 355 projection of, 352 direct, 353, 484 in-plane, 329 pair, 349, 484 singlet density, 348, 354

511 total, 349, 350, 353, 484 crit,ical exponent definition, 31 mean-field value, 109 water-iBA, 167 de Broglie wavelength: 66, 188, 399 de I’Hospital’s rule, 484 density distribution finitc-sizc! scaling, 219 Gaussi,an limit, 83, 218 rnonot~ony,83 depolarizing field, 309, 317. 329, 339 det#erminant properties of, 387 dielectric continuum, 302 dielectricconstant, 315,351,364,368, 456 dipolar soft-sphere flnid boundary condition at substrate, 338 irlsulatirig substrate, 337 intermolecular potential, 325 dipole moment, 301, 312, 317, 329, 351, 453, 461, 481 dipole potential, 309, 321 Dirac &function, 64, 87 and Fourier integral, 395, 396 charge distribution, 304 Gauss distribution, 270, 304, 392 in three dimensions, 393 direction cosinr, 9 dirwtor, 326, 329, 337 disjoiiiing pressure, 91, 201, 321 disordered porous materials, 341 displacement t-ensor, 4, 9 distribution of quantum states, 39 mean occupation, 45 variance, 45 electric field, 450, 456

512 clcctrorhcological fluids, 332 electrostatic poteiitial, 303, 305 charge Fourier part, 306 Gaussian distribution, 449 long-range, 306, 451, 452, 456 rcal-spacc, 305, 457 self, 307, 457 dipole Fourier part, 310 long-range, 310 real-space, 310 image charges, 334 macroscopic sphcrr, 456 poiiit clwgc, 309 substrate, 334 ensemble canonical Dirac representation, 63 partition function, 48, 54 qiiantiim corrections, 67, 69 qucllcllcd-tlrlrlc~l~~d, 343 replicated system. 347, 479 concept of, 36 equivalence of, 58 grand canonical acceptance ratio, 230 latticc modrl, 293 partition function, 54, 75, 86, 117, 149 replica formulation, 479 grand mixed isostress isostrain classic limit, 70 constraints, 39 partition fimction, 48, 54 potential, 248 probability distribution, 47, 70 thermodynamic potential, 55 microcanonical, 54 mixed isostress isostrain Laplace transformation, 277

INDEX partition function, 195, 278 potential, 194 probability density, 195, 278 entropy statistical expression for, 51 equipartition theorem, 77 ergodicity, 179 error function, 305, 449, 469 Euler-Lagrange equation, 124, 127 Eiilcr-Mlu*Laiuinformula, 104 Ewald summation genera.1 aspects, 302 parameters: 312, 318 slab geometry, 312, 313 coniputer time, 315 rigorous, 314,318,320,470,472 .slabadapted, 315, 318, 320 three-dimensional Coulomb intcraction, 307, 458 dipolc intcrartion, 309 extended system, 335, 475, 477 expansivit#y,262 fluctuations, 281 mean-field, 275 fast multipole method, 302 Fermi funct,ion, 224 Fermi-Dirac statistic, 35 svminetry of wave function, 60 ferroelectric order, 325, 368 ferrofluid, 301, 362 fluctuations disordcrcd systcm, 351 matrix-induced, 350 thermal, 56, 329, 350 fluid bridge structure, 215, 231, 242 fluid fluid interaction dipole dipole, 309 shifted-force potential, 193 fluid lamella

INDEX tquilibriuin, 17 therrnodyriamic potential free energy, 19 generalized Gibbs, 20 grand potential, 19 fluid solid interaction heterogeneous substrate interaction potential, 211 homogeneous substrate interaction potential, 213 force charge correction, 473 dipole Fourier, 463 red-space, 463 Fourier series, 450 transformation, 61, 352, 354, 394, 487 functional derivative, 124 GauS’s theorem, 379, 382 Gibbs fundamental equation bulk fluids, 13 general expression, 10, 11 Legendre transformation, 22 slit-pore, 15, 16 virtual transformations, 18 Gibbs-Duhern equation symmetry properties, 26 Gibbsian st,atistical mechanics, 95 grand-potential fimctional, 123

513 Hookem regime, 244 hydraulic radius, 165

image charge, 313, 332, 334, 475 image dipole, 332, 336 importance sampling, 183 information reduction, 33, 39, 95 Ising model. 35, 74, 118 random field, 343 isostrairi h a t capcity, 264, 280 isostress heat capacity, 262, 279 isothermal compressibility correlations, 135 density fluctuations confined fluid, 280 disordered system, 351, 361 onedimensional, 81 system-size effect: 219 divergence of, 363 thermodynamic definition, 28 vanishing temperature, 132 Jacobi-Newton iteration, 420 Jacobian matrix, 200 .Joiile-Thomson coefficient dcfinition, 261 response functions, 263 Joule-Thomson effect idea1 quantum gas, 258, 273 Kirkwood-Wigner theory, 63 Kronecker symbol, 390. 395, 426

Lagrange multiplier, 47, 49, 387 Laplare equation. 449-452, 468 Hamiltonian function, 65, 399, 400 Laplace operator, 64, 450 Harniltonian operator, 37, 399 Laplace transformation. 70 space representation, 64, 400 lattice fluid, 35 hard-sphere potential, 356, 366 binary mixture, 145 Heaviside function, 87, 224 bulk critical point, 124 high-temperature expansion, 102,118 grand potential, 295 honiogeneous function, 371 Hamiltonian function, 117

INDEX

514

morphologies a i d riiodulcs bulk: 128 hard subslrate, 128 heterogeneous substra.te, 130 homogeneous substrate, 129 phase diagram bulk, 124 confined fluid, 135 temperature expansion, 120 lat,t,iccniodcl dipolar soft spheres, 338 mean-field, 115 water-iBA, 163 Laurent, series, 43, 385 residue, 44, 386 Legendre transformation biject,ivity, 370 Maxwell relation, 371 point and linc! gcomct.ry, 370 Sdiwarz’s thcororn, 371 thermodynamic potential, 22 Leibniz’s rule, 440, 444, 449 Lekncr method, 302 Lennard-Jones potential, 189, 321, 361: 362, 364 linear equations existence of solutioii, 387 long-range correction wsiirnption of Imnogcncity~ 191, 192 mean-field approximation, 191 long-range interactions computational aspects, 302 Coulomb, 301 dipoledipole, 301 potential cut-off, 302 long-wavelength limit, 468 MacLaurin series, 48, 275 macroscopic electrostatics , 455 Markov Drocess. 184. 431. 432 I

--:

hlarkovchain, 184, 188, 195,212,218, 283 maximum term method, 58. 194, 278 proof of, 389 Maxwell construction, 112 algorithm, 419 hlmcwc:ll equation, 450 mean-ficld approximation binary mixture, 148 ferroelectricity, 330 general concept , 97 pure lattice fluid, 122 reference fluid, 104 reliability, 178 van dts Waals fluid, 105 vanishing temperature, 124 various, 177 mechanical expression bulk fluids, 24 Euler theorein, 23 noncxisteiicc of, 25 slit-port., 24, 25 mechanical stability, 203, 239 mesoyorous glasses, 341 bfetropolis algorithm , 183 classic version, 186 grand canonical, 184. 294 isostrcss ischstrain, 195. 281 rcliability of, 18G mininium image convention, 187 Moivre-Laplace theorem, 83, 413 molecular dynamics (MD), 178 momentum operator, 61 eigenfuiictioii aiid -value, 61, 395 spacc rc:prcscntation, 61 monopole Irlolllerlt, 453 most probable distribution concept, of, 40 Darwin-Fowler theory, 42 plausibility argument for, 41 multinomial expansion! 43

INDEX iriultipolc monicnt, 453. 454

515

discontiiiuous condit.ions of, 28 Newton’s method, 417, 420 ferroelectric, 328, 330, 363, 365 fluid morphology, 216 one-dirnensional hard-rod fluid liquid crystals, 342 equation of state, 77 low-porosity materials, 341 excluded volume and pore denmetastability, 29, 235 sity, 92 spinodal, 363 fluid fluid potential, 75 point cliaxge, 304, 309, 315, 466 isothermal compressibility, 81 point dipole, 315, 317, 320 partic*lcxmiribcrdistribution, 82 Poisson equation, 304. 447, 450, 457 phase transitions, 74 polarization, 456 virial equation, 77 pore occupancy, 87 order parameter Principle Bessel function, 331 Detailed Balance, 183, 434 consistency relation, 331 Equal A Priori Probability, 40 donsity, 149 Probability Conservation, 50 ferroelectricity, 326, 330 Uncertainhy, 34 miscibility, 149 probability transverse pressure, 337 classic versns quantum mechanorder-parameter matrix, 326 ics. 34 ordered mesoporous silica, 100 Ornstein-Zernike equation, 353, 354, quenched-annealed mixture, 343 368, 484 quenched-annealed models numcrical solution, 484 disorder average, 345, 359 replica-symmetric, 356, 481, 482 double average, 345, 346 orthogonal expansion, 451 electrostatic interactions, 358 ort,hogonal transformation, 403 hard spheres, 358 Parstwl’s cquation, 390 thermal average, 345 Parsewl’s theorem, 48 quenching temperature, 344,357.366 particle in a box, 38 randoin-field systcms: 346 particle-mesh method. 302 permutation operator, 60, 62 Rayleigh expansion, 483 perturbation theory, 102, 121, 226 reaction field, 302, 456 phnw diagram, 31 replica phase space, 59 integral equation, 343, 348, 361 phase transition dipolar fluid, 362 boundary condition at substrate, trick, 359 replica method, 346 337 chemical corrugation, 217 replicated system continuous, 31 n-dependence, 357, 360

516 closure relatioils, 357 copies, 346, 349, 354, 357, 480 internal energy, 348, 349 isothermal compressibility, 351 permutation symmetry, 353 semi-grand canonical, 479 symmetry, 354 thermodynamics, 358 response function, 350 rhcology, 332 dynamical, quasistatic approach, 238 fluid bridge, 141, 242 Hooke’s law, 242 master curve, 244 niean-ficld treatment, 138 shear modulus molecular expression, 240 thormodynamics, 239 shear strcss, 23‘3 surface forces apparatus. 238 yield point, 242 rotational invariants, 352, 356, 363, 481, 482 scattering experiment., 352 Schrodinger equation, 37 Schrodinger-Hill t+heory,46 screened charges, 305, 447, 466 shear moduliis microscopic expression, 240 thermodynamic definition, 240 Voigt’s notation, 239 yield poiiit, 243 shear strain morphologies lattice fluid, 138 phase behavior lattice fluid, 138 substrate al igii ment , 239 shear stress

INDEX cquatioii of state, 246 Hooke’s law, 242 iiihomogeneity of fluid, 209 yield point, 242 sinall angle neutron scattering contrast factors, 170 rontrihiitions to scat t,cring int-cnsity, 161 correlation length, 160 small strain approximation, 244, 251 small-strain assumption, 4 soft-sphere potential, 325 sorption experiment, 98 capillary condensation, 113 siirfacc cxc(:sscoverage, 112 lhermodynarnic analysis, 110 spherical harmonics, 448, 482 and scalar product, 454 orthogonality, 455 spin glass. 346 spinodal decomposition, 34 1 sport aiicous polarization, 325 state function, 10 Stirling approxiniation, 41, 88, 424 Stockmaycr potential, 320,362-364, 366 strain, 3 strain tciisor bulk fluids, 13 compression and dilalation, 5 slit-pore, 14, 16, 202 surface forces apparatus, 20 syinmetrg decomposition of, 4 stratification, 74,91, 237, 328, 330 disjoining pressiirc, 323 local density, 206 solvation force, 204, 208 stress and force, 5 stress tensor bulk fluids, 13 Coulombic system

INDEX slab gconictry, 473 threedimensional, 461 dipolar system slab-geometry, 474 three-dimensional, 465 matrix representation, 6 normal component force expression, 203 Storkmayer fluid, 321, 324 virial cxprcssion, 202 sliear component force expression, 239 virial expression, 239 slit-pore, 15, 16 transformation properties, 7 structure factor, 356 Debye term, 162 disordered system, 352 long-wavelength limit, 353 Ornstein-Zernikc term, 162 Porod scatteriiig, 162 scattering intensity, 161 surface forces apparatus, 91 conipressioiial mode, 197 experimental setup, 197 shear mode, 198 dynamical, quasistatic, 238

51’7 rnctastabilit,y, 235 t,hermodynamiw consistency, 264 extxemurn principle, 21 general principles, 1 historical aspects, 2 microscopic: foiindation, 2, 33 natmal variables, 21 skate function, 10 torque, 464, 474 trace, 11, 369 matrix product, 369 UIic:crt;zint,yPrinciplc, 34

van der Waals equation, 107 van der Waals fluid critical exponent, 109 critical point,, 107 rcduccd cquatiori of statc, 108 virial coefficient classic fluid, 273 configuration integral, 266 ideal quantum gas, 271 mean-field, 275 onedimonsional fluid, 78 virial equation, 77, 265, 275 Vycor? 341

Taylor series, 38, 41, 44, 85, 102, 108, Wenzel-Kramers-Brillouin formalism, 121, 152, 272, 378, 409, 413, 400 430, 433, 452, 454, 457, 462. work 468 clicrriical, 10 tcinpcratnrc mechmiira1 absolute, 10 bulk fluids, 13 Boyle, 283 general expression, 11 inversion, 259, 263 slit-pore: 15 low-density limit, 266 surface forces apparatus, 20 mean-field, 275 thermodynamic integration routes, 227. 295 thermodynamic limit, 80

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