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P.NEIGI University of Missouri-Rolla Rolla, Missouri

Marcel Dekker, Inc.

New York· Basel- Hong Kong

Library of Congress Cataloging-in-Publication Data Diffusion in polymers / edited by P. Neogi. p. cm. - (Plastics engineering; 32) Includes bibliographical references and index. ISBN 0-8247-9530-X (alk. paper) 1. Polymers-Permeability. I. Neogi, P. (Partho). II . Series: Plastics engineering (Marcel Dekker, Inc.) ; 32. QD381.9.P45D53 1996 668.9---<.lc20

95-51156

eIP

The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the address below. This book is printed on acid-free paper. Copyright © 1996 by Marcel Dekker, Inc. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permissiol) in writing from the publisher. Marc~l Dekker, Inc. 270 Madison Avenue, New York, New York 10016

Current printing (last digit):

10 9 8 7 6 5 4 3 2 1 PRINTED IN THE UNITED STATES OF AMERICA

Preface

To most researchers in the area of diffusion in polymers, the 1968 book Diffusion in Polymers by J. Crank and G. S. Park is a very familiar and most appreciated one. An important reason for its success, and one that will never revisit this area again, is that the book appeared when research activity was about to explode with the advent of membrane separations, barrier membranes, new needs to study polymer devolatilization, and so on. It is now both out of print and out of date, as is one update of Polymer Permeability edited by 1. Comyn. The two books, Membrane Handbook edited by W. S. Ho and K. K. Sirkar and Polymer Gel Separation Membranes edited by D. R. Paul and Y. P. Yampol'skii, stress diffusion only as a precursor to studying separations. Another, Diffusion in and Through Polymers by W. R. Vieth remains in the mainstream of diffusion in polymers. This book began with the realization that fundamental changes have taken place in this area. Diffusivity is no longer a phenomenological coefficient and very fum validation from molecular theories now exists for Fick's law. Highspeed computers have become available that, in principle, can be used to calculate these diffusivities. In practice the results are few, but present a very important view of the shape of things to come. The key results, however, are provided by real-world phenomenology, whether it concerns understanding the matrix of the solid polymers or predicting and correlating the diffusivities of small molecules. These are presented to complement the more abstract concepts. The molecular interpretations are not foregone, but at the same time numerical accuracy is the more important criterion. iii

iv

PREFACE

Another development lies in the area of transport phenomena. It is no longer possible to be content with mechanisms-in-words, because mathematical restrictions now exist to quantify constraints rising out of thermodynamics, mass, momentum, and energy and species balances, and their methods of solutions have become more transparent. In particular, conventional transport phenomena used to address fluids had three important assumptions: homogeneity, isotropy, and local equilibrium. None of these applies to solid polymers uniformly. Some progress has been made in addressing these special effects. A third development lies in advances in understanding the polymer matrix, covering the physical chemistry of solid state and architectures at the molecular level or at the scale of the membrane. Even in " structureless" melts, the study of molecular conformations has proved to be critical. This book examines these aspects and will serve chemical engineers who are involved in separations, controlled release, development of barrier membranes, and transport phenomena in general; chemists, both physical chemists for some of the same reasons and those who synthesize and evaluate new materials; and finally physicists, to whom we owe the development of the molecular theories. P. Neogi

Contents

Preface Contributors

iii ix

Chapter 1. Diffusion in Homogeneous Media 1. M. D. MacElroy L II. III.

IV.

Chapter 2.

L II. III. IV. V.

Introduction Diffusion Fundamentals Simulation and Modeling of Diffusion in Fluid/Solid Systems Concluding Comments References

Molecular Simulations of Sorption and Diffusion in Amorphous Polymers Doros N. Theodorou Introduction Characterization of Structure and Molecular Motion in Amorphous Polymers Prediction of Sorption Thermodynamics Prediction of Diffusivity Conclusions and Future Directions References

1

1 3 13

62 63

67 67 72

91 104

137 139 v

vi

CONTENTS

Chapter 3. I.

II. III. IV. V. Chapter 4.

Free-Volume Theory 1. L. Duda and John M. Zielinski

143

Introduction Free-Volume Concepts Diffusion Above the Glass Transition Temperature The Influence of the Glass Transition More Complex Systems References

143 145 146 156 163 169

Transport Phenomena in Polymer Membranes P Neogi

173

I. Introduction II. Mathematical Methods III. Non-Fickian Diffusion IV. Change of Phase V. Multiphase, Multicomponent, and Inhomogeneous

VI. Chapter 5.

I.

II. III. IV. V. VI. Chapter 6.

I. II. III. IV. V.

173 178 184 195

~~~

1~

Conclusions References

205 205

Supermolecular Structure of Polymer Solids and Its Effects on Penetrant Transport Sei-ichi Manabe

211

Introduction Structural Characteristics of Polymer Solids Thermal Motion of Polymer Chains in a Solid Correlation Between Chemical Structure, Composition, and Penetrant Transport Effects of Fine Structure (Crystallinity, Orientation, etc.) of a Polymer Solid on Permeation Properties Concluding Remarks References

211 216 233 241 244 249 249

Translational Dynamics of Macromolecules in Melts Peter F. Green

251

Introduction Translational Dynamics in Homopolymer Melts Diffusion of Chains of Differing Architectures Interdiffusion Diffusion in Block Copolymers

251 255 276 281 288

vii

CONTENTS

VI.

Index

Concluding Remarks References

295 298 303

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Contributors

J. L. Duda Pennsylvania

The Pennsylvania State University, University Park,

Peter F. Green

Sandia National Laboratories, Albuquerque, New Mexico

J. M. D. MacElroy Sei-ichi Manabe P. Neogi

University College Dublin, Dublin, Ireland Fukuoka Women 's University, Fukuoka, Japan

University of Missouri-Rolla, Rolla, Missouri

Doros N. Theodorou John M. Zielinski Pennsylvania

University of Patras, Patras, Greece Air Products & Chemicals, Inc., Allentown,

ix

This Page Intentionally Left Blank

DIFFUSION IN POLYMERS

This Page Intentionally Left Blank

1 Diffusion in Homogeneous Media J. M. D. MacElroy University College Dublin Dublin, Ireland

I.

INTRODUCTION

Recent advances in separation science and technology and in reaction engineering owe their origin, in part, to the development of specialized solid materials that interact kinetically as well as thermodynamically in a unique and controlled manner with multicomponent fluid mixtures. This ongoing technological growth has taken place in parallel with an improvement in our understanding of the fundamental properties of fluids in contact with solids. Noteworthy examples in the chemical and biochemical process industries include energy-efficient and nondestructive separation of molecular and macromolecular solutions by sorption onto solid substrates (Ruthven, 1984; Chase, 1984a,b; Norde, 1986; Yang, 1987), membrane separation of gases and liquids (Turbak, 1981; Drioli and Nakagaki, 1986; White and Pintauro, 1986; Sirkar and Lloyd, 1988), and chromatographic separation of multicomponent mixtures (Yau et aI., 1979; Chase, 1984a,b; Belter et aI., 1988; Brown and Hartwick, 1989). The fundamental mechanisms that govern the behavior of fluid/solid systems are also central to research and development in such diverse areas as enhanced oil recovery, toxic waste treatment, textile manufacturing, food technology, and biomedical engineering, and although significant progress has been achieved much still remains to be done. The characterization of fluid/solid systems is particularly difficult when the dispersion of the components of the fluid within the solid medium is determined 1

2

MACELROY

solely by intimate details of the molecular structure of both the fluid and the solid. For example, the very high sorptive specificity of some rigid microporous materials is directly related to the geometrical and topological constraints imposed by the pore structure on the components of the adsorbing fluid. Solids that fall into this class include the zeolites (Weisz, 1973; Satterfield, 1980; Ruthven, 1984), which are cyrstalline media possessing pore apertures in the range of 0.3-1.0 nm, with the actual aperture size depending on the origin and/or method of manufacture of a given zeolite. Another example of a rigid medium that exhibits a high degree of selectivity is molecular sieving carbon, which contains local pore bottlenecks smaller than 0.5 nm (Juntgen et ai., 1981). The specificity of this material is most clearly demonstrated by its ability to separate nitrogen from air. The mechanism for the separation process is kinetic in origin in that the diffusion rates of oxygen and nitrogen within the pores of molecular sieving carbon usually differ by a factor of 10 or more even though the sizes of the molecules of these two species differ by only a few percent. When the "solid" material is also nonrigid, the analysis of diffusion is much more complicated. At a given temperature one is confronted with the need for detailed information on the time evolution of the size, shape, and number of the microvoids or cavities locally within the medium as well as the required characterization of the fluid-solid intermolecular interactions. The temperature dependence of the translational, rotational, and intramolecular motion of the membrane atoms and particles and the concomitant existence of phase transitions (glassy amorphous states to rubbery or liquid crystalline states and vice versa) further complicates the description. In view of the rapidly growing technological importance of materials of this type, particularly polymers, much effort has been expended in elucidating the numerous subtle effects associated with these intramembrane characteristics (Crank and Park, 1968; Stern and Frisch, 1981; Vieth, 1991; Roe, 1991). The material presented and discussed in this chapter is primarily of an introductory nature, and later chapters in the book should be consulted for details of more specific methods of analysis and applications. The general framework of the presentation provided here takes the following form. In Section II the flux equations for homogeneous fluids are initially considered with reference to formulations based on both nonequilibrium thermodynamics (phenomenological description) and nonequilibrium statistical mechanics (molecular description). These equations generally form the basis for the development of the flux relations for porous media and membranes, and in closing Section II the diffusion equations for such systems are presented and their limitations discussed. In Section III the novel methodology of molecular simulation, particularly molecular dynamics, and its application to diffusion in fluid/solid systems are of primary concern. Examples and applications are described for three different methods of modeling the internal structure of permeable media: (1) idealized pore shapes in

DIFFUSION IN HOMOGENEOUS MEDIA

3

rigid media, (2) random bicontinuous media with a stationary solid phase, and (3) random media with a mobile solid phase (polymers). Finally, in Section IV concluding comments are provided.

II.

DIFFUSION FUNDAMENTALS

A.

Flux Equations for Homogeneous Fluids

The conservation equation for component i within an infinitesimally small volume element of a nonuniform, homogeneous system centered at r at time t in the absence of chemical reactions is given by

at = - V· ap ·

Pilli

(1)

where P i and lli 2re the local mass density and velocity of component i at rand at time t. The group of terms Pilli is the flux of i relative to a stationary laboratory frame of reference, i.e.,

J: =

(2)

Pilli

and is generally considered to be composed of two terms: (1) a convective contribution arising from the local bulk motion of the fluid and (2) the residual microscopic thermal motion of the molecules of component i relative to this convective flow. The definition one employs for the velocity of the bulk convective motion is largely a matter of convenience, and one of the most common frames of reference is based on the center-of-mass velocity of the fluid at rand at time t which, for a v-component fluid, is given by v

2:

II

Pilli

= -'--=--;=1

(3)

Thus, defining the mass diffusion flux of component i relative to the center of mass of the fluid as (4) where the superscript b refers to the barycentric frame of reference, Eq. (1) may be written as DXi

P Dt

=-

V

. Ji,b

(5)

MAcELROY

4

In this equation P is the local fluid density L 7=1 Pi, Xi is the mass fraction of component i, pJp, and DIDt is the substantive derivative D Dt

a

-=-+u·V

at

Historically, the mass fluxes J; and J;b and the corresponding molar fluxes (6) or J~ = ni(ui - u)

(7)

where n i is the local molar concentration of i, have been investigated either phenomenologically via nonequilibrium thermodynamics [see, e.g., Prigogine (1961), deGroot and Mazur (1963), and Hanley (1969)] or theoretically via kinetic theory [e.g., Hirschfelder et al. (1954), Hanley (1969), and Chapman and Cowling (1970)]. Although kinetic theory can, in principle, provide exact results for the fluxes in terms of the transport parameters and driving forces for mass transfer, tractable expressions may be obtained in only a limited number of cases (e.g., low-density gases). Nonequilibrium thermodynamics, on the other hand, provides a general framework within which a consistent set of forces may be prescribed for fluxes defined for various frames of reference. This approach, however, is not without its own limitations, and the phenomenology of the original theory of irreversible processes was a matter of controversy for many years. However, developments in nonequilibrium statistical mechanics stemming from the early work of Green (1952, 1954), Kubo (1957), and Kubo et al. (1985) using linear response theory and, later, the projection operator formalism of Mori (1965) have to some extent alleviated suspicions regarding the applicability of irreversible thermodynamics. In the following the general tenets of nonequilibrium thermodynamics are briefly summarized and a number of exact results provided by nonequilibrium statistical mechanics are cited. Nonequilibrium thermodynamics is founded on two postulates in conjunction with the On sager reciprocal relations [for details the reader is referred to Prigogine (1961), deGroot and Mazur (1963), and Hanley (1969)]: Postulate I: Microscopically large though macroscopically small volume elements of a nonequilibrium system are themselves in local equilibrium, and therefore the fundamental relations of equilibrium thermodynamics are locally applicable. The rate of entropy production under these conditions is given by

(8) where J i and Xi are the conjugate fluxes and forces.

DIFFUSION IN HOMOGENEOUS MEDIA

5

Postulate II: For systems not too far removed from equilibrium, the fluxes of heat, mass, and momentum are linear homogeneous functions of the thermodynamic driving forces arising from the gradients of temperature, chemical potential, and the components of velocity J=LX

(9)

where L is the matrix of phenomenological (kinetic) coefficients L jj •

Onsager's reciprocal relations: With a suitable choice of conjugate forces and fluxes , the matrix of phenomenological coefficients in Eq. (9) is symmetric, i.e., L jj = L j i • For isotropic media and under conditions of mechanical equilibrium, two additional theorems may also be invoked in the general definition of the fluxes given in Eq. (9). The first is Curie's theorem, which states that in an isotropic medium the matrix L is a scalar and the forces and fluxes in Eq. (9) are therefore of the same tensorial rank. This theorem is assumed to apply for the homogeneous systems under consideration in this chapter. The second theorem is that proposed by Prigogine, which states that at mechanical equilibrium, the flowframe reference velocity in the definition of the diffusion flux may be selected arbitrarily without affecting the rate of entropy production. The condition of mechanical equilibrium considerably simplifies the analysis of diffusion, and since this condition is usually involved in experimental diffusion measurements it is appropriate to employ it here. Furthermore, in light of Prigogine's theorem it is also particularly convenient to consider the diffusion fluxes in a laboratory fixed frame of reference, in which case Eq. (9) provides the following expression for the molar diffusion flux of component i: (10)

where. VT f.1j - Fj and T - 1VT are the thermodynamic driving forces for mass and heat transfer as prescribed by the form of the entropy production equation arising from Postulate I. [As an aside it is noted that the momentum driving force does not appear directly in Eq. (10) because of its tensorial rank. This, however, does not preclude an indirect influence of inertial and viscous effects on the flux of material in systems that are not at mechanical equilibrium.] f.1j and Fj are the chemical potential of component j and the external force acting on component j (per mole), respectively, and VT is the gradient operator at constant temperature T. Equation (10) is the principal result of nonequilibrium thermodynamics that will be employed in this chapter. The implications of this expression from the point of view of nonequilibrium statistical mechanics were investigated in detail

6

MA CELROY

by Altenberger et al. (1987) and Kim et al. (1992), and it is worthwhile at this point to outline a number of the salient features of these studies and prior theoretical developments in statistical mechanics. Altenberger and coworkers summarized much of the earlier work of Mori (1965) on the projection operator formalism in transport processes, and they also extended Mori's method to frames of reference other than the laboratory frame and the fluid center-of-mass frame. The essence of Mori's theory is that the molecular fluxes of the species in a multicomponent system (and therefore the macroscopic fluxes J i defined above) are determined by a random component that is orthogonal to the space spanned by the density and temperature fluctuations within the medium and an induced or systematic contribution arising from the decay of these fluctuations . In the interests of brevity, only the results for isothermal conditions are considered here, in which case the molecular diffusion flux of component i is given by N/

ji(k, t) =

L

Vii

exp( - ik . r li)

(11 a)

/=1

(l1b) where Vii and r li are the center-of-mass velocity and position of particle I of component i at time t, Xj(k, t) is the wavevector-dependent diffusional thermodynamic force for component j given earlier in Eq. (10), and L;l k, t) is an " after-effect" function for diffusion that is related to but not equal to the phenomenological coefficient L ij defined earlier. The distinction between the Green-Kubo linear response theory (Green 1952, 1954; Kubo, 1957; Kubo et aI. , 1985) (which leads to a comparatively simple expression for Lij and which we specialize to below) and Mori's projection operator formalism lies in the random component j ;(k, t) of the flux that appears explicitly on the right-hand side of Eq. (l1b) and also implicitly in the coefficients L;l k, t), i.e.,

L;lk, t) = e~BT k

. (j;(k, t) jj( - k, 0) . k

(12)

where the angular brackets represent averaging over the unperturbed eqUilibrium canonical ensemble. The term inside the angular brackets in this equation is the time-correlation function of the random components of the microscopic fluxes of components i and j, and the second term in Eq. (l1b) characterizes the decay of microscopic fluctuations in the medium. Mori (1965) showed that in the limit k -0- 0 (the subsystem volume, V, is much larger than the scale of local molecular density inhomogeneities) the frequency-dependent form of Eq. (12) simplifies

7

DIFFUSION IN HOMOGENEOUS MEDIA

to the corresponding expression provided by linear response theory (Green, 1952, 1954; Kubo, 1957; Kubo et aI., 1985),

Lij(w) = _1_ 3VkB T

Jor exp( - iwt) (j;(t) . jj(O)

dt

(13)

i.e., a kinetic coefficient determined by the mechanical properties jk in contrast to the random components of these fluxes that are involved in the generalized expression for the coefficients L;j(k, t) in Eq. (12). Equation (13) is much easier to evaluate and is of primary concern in this chapter. Fortunately this expression is essentially exact except when variations in composition over length scales on the order of the molecular dimensions within the fluid are of interest, in which case nonlinear contributions in the thermodynamic forces that appear explicitly in the general frequency-dependent form of L;j(k, w) = f ~ exp( - iwt)L;j(k, t)dt (Mori, 1965) need to be taken into consideration. Mori also showed that in the low-frequency limit w - 0 the random contribution j ; in Eq. (Ub) may be neglected and under these conditions the diffusion flux within a microscopically large but macroscopically small volume element of the nonuniform system may be expressed in the form given earlier in Eq. (10) with the phenomenological coefficients given by L ..

= _1_

'I

3 VkBT

r 'P{t) dt Jo

(14a)

'I

where 'Pit) is the velocity correlation function (VCF) (14b) The zero frequency limit corresponds to time scales that are significantly longer than the decay times of the VCFs appearing in Eq. (14a), and therefore the principal restrictions involved in the application of the linear flux relations given in Eq. (10) with Lij given by Eq. (14a) are that (1) the local thermodynamic properties in the nonequilibrium system should not vary significantly over length scales on the order of molecular dimensions and (2) the time scales of interest should be longer than the characteristic relaxation times for molecular processes. In many situations of interest these restrictions are not crucial and the linear relations coupled with the Green-Kubo integrals for Lij provide an accurate description of diffusion in nonuniform homogeneous fluids . Another mathematical form for the kinetic coefficients originally proposed by Einstein is obtained by carrying out the integration indicated in Eq. (14a). One finds

I N.

Lij =

6~BT ~ :t \~

Nj

[rki(t) - rki(O)] .

~

) [r,it) - r'j(O)]

(14c)

8

MACELROY

Although this equation is frequently cited and employed in the literature, in this chapter the Green-Kubo form provided in Eq. (14a) is favored in view of its significant theoretical and experimental interest. As noted above, Altenberger et al. (1987) also provide expressions for the multicomponent diffusion fluxes in a variety of frames of reference, and one of the most important conclusions of their work is that while the kinetic coefficients in different flow-frames may be determined from a knowledge of the coefficients Lij [the laboratory fixed frame parameters provided in Eq. (14)], the reverse is not true. Of particular interest are the kinetic coefficients for the center-of-mass frame of reference, which may be determined from the fixed frame coefficients using the relationship (15) where Mk is the molecular weight of component k. The molar diffusion fluxes J~ in this frame are again given by Eq. (10) with L ij replaced by D.ij and with L iq replaced by a similar expression for D.iq [see Altenberger et al. (1987) for details]. Further comment on the coefficients Lij is postponed until later, although at this point it is worthwile considering an additional result [originally derived by Mason and Vie hi and (1978)] stemming from Eq. (10) and the concomitant condition of mechanical equilibrium. Dividing Eq. (8) by ni and subtracting from this result the corresopnding expression for component k, one finds (16) Now consider the condition of mechanical equilibrium as expressed by the Gibbs- Duhem equation v

2: nk(VTf.1k -

Fk ) = 0

(17)

k= 1

with (18) where P is the local pressure in the system. Treating VTf.1v - F v as the dependent driving force in Eq. (17), then Eq. (16) may be written as (19)

DIFFUSION IN HOMOGENEOUS MEDIA

9

where (20) Now multiplying Eq. (19) across by nink/n2Dib summing over all species, and defining (21) where Dik is the mutual diffusion coefficient for the pair of species i and k, then the flux equations may be written in the Stefan- Maxwell form

(i

= 1, ... , v) (22)

where kTi is the thermal diffusion ratio for component i and is given by ~ nJ-iq - nLkq

kTi = L.J A~ l

(23)

-.....:....,2,...-~~

n D ik

From the definition given in Eq. (21) and the condition that the dependent driving force is VTil-v - F v, it is easily shown that the mutual diffusion coefficients and the kinetic coefficients are interrelated by the expression

i: ~l

nv(nALij - nLkj) - nj(nJ-iv - nLkv) = ~ n nin"Dik kBT

(i,j =1= v)

(24)

where 8ij is the Kronecker delta. The Stefan-Maxwell equation [Eq. (22)] is equivalent to Eq. (10) and is a very useful way of expressing the diffusion fluxes in multicomponent systems. In particular, this equation has been widely used in the development of models for diffusion in porous media, most notably the Dusty Gas model [Mason and Malinauskas (1983)], and in the next section the underlying principles of the latter approach are employed in the formulation of general expressions for the diffusion coefficients of a multicomponent fluid in homogeneous fluid/solid media.

B.

Flux Equations for Fluid/Solid Systems

If the volume element of the fluid/solid system in which the concept of local equilibrium can be considered applicable includes both the fluid components and particles of the solid material, then all of the equations discussed in the previous section can be used to describe diffusion within the solid with the solid

10

MACELROY

medium itself also treated as one of the components in the multicomponent system. For the solids of interest here, i.e., microporous media and polymeric materials, local inhomogeneities usuaLLy exist only over length scales on the order of atomic or molecular dimensions, and therefore it is assumed here as a working premise that the concept of local equilibrium applies. For fluid/solid systems not far removed from equilibrium, the flux equations are therefore

= 1, . . . , v, m)

(25)

(i = 1, .. . , v, m)

(26)

(i or

with Lij and Dij given by Eqs. (14) and (24), respectively. For convenience the contribution arising from the solid (component m) has been separated out. In treating the solid as a component of the mixture it is assumed implicitly in Eqs. (25) and (26) that the matrix of the material making up the solid medium conforms to the condition of isotropicity. If this is not true, then the above equations may still be considered applicable locally within the medium as long as the anisotropic character of the local solid structure is taken into account. For example, a very simple model that is frequently employed in the analysis of diffusion in porous membranes is the cylindrical pore model; i.e., the solid structure of the medium is assumed to form long cylindrical cavities along the axial (z) direction of which the components of the fluid are allowed to diffuse. Equations (25) and (26) may be employed under these conditions to predict the axial diffusion fluxes in a given pore using

J~z) = _ ~ L~;) (dlJ-j I j=l

dz

- FY» )

T

_ L(z) U"

(dlJ-m dz

I _F(Z»m) _ L~~)T dT dz l'

(i = 1, ... , v, m)

(27)

or

ninm ( (z) _ (z» 2D(z) U, Um n im

+ ~ nink ( (z) _ (Z» L.J 2D(z) U, hi n ik

Uk

= _

_"dlnT "T, dz

~ k T n

lJ

(dlJ-i d Z

I _FZ'») T

(i = 1, ... , v, m)

(28)

DIFFUSION IN HOMOGENEOUS MEDIA

11

These results can then be incorporated into a network model for the pores of the medium to estimate the macroscopic mass transfer rates through the membrane. It is not the intention here to review the various network models that have appeared in the literature over the last few decades, and the reader is referred to a number of articles dealing with recent developments in this subject (Reyes and Jensen, 1985; Nicholson et aI., 1988; Sahimi, 1988; Zhang and Seaton, 1992). However, there are a few points worth noting with regard to the application of the above equations to anisotropic pore structures. The influence of the solid arises explicitly through the terms L~~ and D~~. At first glance this type of formulation might appear to be counterintuitive, i.e., the pore fluid should be considered separately and a boundary value problem should be solved. For microporous systems, however, it is much more convenient to include the solid phase as a component in the diffusion equations, as permitted by Postulate I, even if the atoms of the solid assume a geometrically ordered configuration [for further comments on this aspect of transport in model pores, see Mason et aI. (1963)]. 2. A fundamental problem does arise, however, in applying linear response theory to porous media that are locally anisotropic due to the limiting condition of zero wavenumber that is implicit in Eqs. (27) and (28) [see the discussion following Eq. (12)]. Consider the following question: For very fine pores, how can one obtain a sensible measure of the macroscopic diffusion parameters if, owing to the very dimensions of the micropores, the linear response kinetic coefficients in the limit k ->0 0 and w ->0 0 are, under certain conditions, nonexistent? The difficulties posed by this question were clearly illustrated in the work of Vertenstein and Ronis (1986, 1987) and by Schoen et aI. (1988). In particular, Schoen et aI. demonstrated that although linear response theory does indeed provide meaningful diffusion coefficients for diffusion parallel to the surface of the solid material on either side of a slit-shaped pore regardless of the width of the slit, this is not generally true for diffusion normal to the pore walls. Later in this chapter results obtained from an application of linear response theory to diffusion in cylindrical pores (Suh and MacElroy, 1986) are discussed to further illustrate this point.

1.

The above comments, coupled with the earlier remarks on the theoretical results provided by Mori using projection operator theory, lead to the following summary of limitations associated with Eqs. (25)-(28): 1.

The fluid/solid system must be isotropic if Eqs. (25) and (26) are to be considered applicable. For solid structures that are strongly anisotropic locally, equations similar to Eqs. (27) and (28) may be used along axes within the medium that are translationally invariant.

12 2.

MA cELROY

Density and temperature variations within a microscopic volume element of the medium should be negligible, and the time scales for molecular relaxation processes should be much shorter than the macroscopic time scales of interest in the diffusion measurements.

In many situations of interest these limitations do not significantly influence the prediction of transport rates in homogeneous media. The application of Eqs. (25)-(28) does, however, require consideration of the mobility of the solid phase both at the macroscopic level (i.e., u m ) and at the microscopic level [vt",(t) in Eq. (14b)]. Furthermore, in view of the inverse dependence of Dij on the number density of the system, n, as predicted generally by kinetic theory, it is convenient to redefine the pair diffusion coefficients for the system as follows (Mason and Viehland, 1978; Mason and Malinauskas, 1983): (i=l, . . . , v)

(29)

(i,j = 1, .. . , v)

(30)

where nf = L ~= l ni, i.e., the number density of the fluid within the medium (fluid particles per unit volume of the fluid/solid system as a whole). With these definitions and specializing to isothermal conditions, then Eqs. (25) and (26) may be rewritten for the fluid species as

(i = 1, ... , v)

(31)

and

(i = 1, ..., v)

(32)

The Gibbs-Duhem equation [Eq. (17)] has been used to simplify Eq. (31). If the particles or atoms of the solid phase are assumed to be stationary ("rigid" media), then Eq. (31) simplifies further because Jm = 0 and Lim is also zero as indicated by Eq. (14). Equations (31) and (32) have been widely used in the literature in the development of correlative models for diffusion in porous media and polymeric materials (frequently subject to the assumption Lim = 0 and in many cases for single-component diffusion only), and the reader is referred to these sources for full details of the modeling techniques in current use [see, e.g., Crank and Park (1968), Stem and Frisch (1981), Vieth (1991), and Mason and Malinauskas (1983)]. In Section III, a methodology that has been developed over the last decade is reviewed. This approach involves direct molecular dynamics simulation of confined fluids in model systems to compute the VCFs appearing in Eq.

DIFFUSION IN HOMOGENEOUS MEDIA

13

(14b) and hence the kinetic coefficients Lij. The rapid advances in computer technology over the last 15-20 years, and particularly the advent of supercomputers and more recently dedicated desktop workstations, now permit "exact" determination of the transport (as well as equilibrium) properties for a wide variety of systems. The advantages of computer simulation as a means for investigating the behavior of fluids and solids are clearly demonstrated in each of the works that have appeared in the last decade [see, e.g., Roe (1991), Nicholson and Parsonage (1982), and Allen and Tildesley (1987)], and the most important of these advantages from the point of view of transport in homogeneous media are summarized as follows: Microscopic properties, which cannot be readily measured experimentally but which are nonetheless central to a physical understanding of the underlying mechanisms for fluid transport within solid media, are accessible via molecular simulation. 2. Although simulation can never replace actual experimental measurements, it can, in conjunction with theoretical modeling and limited experimental data, serve as an accurate and powerful predictive tool for extrapolating beyond the range of possible laboratory measurements. This aspect of molecular simulation is particularly important for microporous media and membranes in view of the complexity and, in many cases, the expense associated with detailed experimental measurements of transport in such systems. 1.

The discussion in Section m is presented in three parts. As an introduction to those who are not familiar with molecular dynamics simulation, computations based on an idealized pore geometry are considered first. The pore geometry employed here is the cylindrical pore model that has been widely used in theoretical studies of both the transport and equilibrium properties of fluids in membranes. This is followed by a summary of work on random media in which the particles and atoms of the solid phase are held stationary, and finally recent developments in the simulation and prediction of diffusion in amorphous polymers are discussed.

III.

SIMULATION AND MODELING OF DIFFUSION IN FLUID/SOLID SYSTEMS

A.

Diffusion in an Idealized Pore GeometryThe Cylindrical Pore

For a cylindrical pore of length 2L » Rp, where Rp is the pore radius, and assuming that the solid phase is immobile, Eqs. (31) and (32) with Lim = 0 are

applicable for diffusion in the axial (z) direction along the pore,

j~z) = _ i L~;) (djJ.j I - F}'» ) dz j=l

(i

= 1, . . . , v)

(33)

T

or

j~Z) + ~

~

D (z) iM

k= J

_1_ ( 1)(Z)

n~ ik

1 (z) _

n/(J ,

, 1 (Z»

niJ k

= _

~

k T B

(djJ.i l _ d Z

T

F(Z» ) '

(i = 1, . . . , v)

(34)

Two limiting cases of these equations are considered here: a single-component pore fluid and a binary mixture. For simplicity it is also assumed that no external forces act on the fluid components.

1.

Single-Component Pore Fluid The diffusion flux of a pure fluid f is given by _ _ L (z) j (z) r rr

djJ.r dz

I

(35)

1"

or

j~Z) = _ nrD~ djJ.f I kBT dz

(36)

T

For a bulk external phase at a chemical potential jJ.rn in local equilibrium with the pore fluid at z, jJ.r = jJ.rn = jJ.0(1) + kBT In ~B' where ~B is the activity of the bulk fluid that satisfies the limiting condition ~ B -+ nrn as nrn -+ O. Substituting this expression into Eq. (36) gives _ D j (z) r - nf

(z) d fM

In ~B dz

I

(37a)

T

=-

(Z)

D fM

d In ~B dnr dlnnr dz

---

__ d z) K -

fM

r

d In ~B dnfB d In nfB dz

(37b) (37c)

where Kr is the partition coefficient for the fluid f defined by

Kr = nr/nrn

(38)

Recall that nr is the local number density of the pore fluid, and for single-pore analyses of the type under consideration here this number density traditionally has units of fluid particles per unit pore volume (Le., the pore wall is the bound-

DIFFUSION IN HOMOGENEOUS MEDIA

15

ary of the control volume under investigation). Equation (37b) corresponds to the Darken equation for the diffusion flux , while Eq. (37c) takes on a particularly simple form in the ideal gas limit nm - 0,

f

z) _ _ D (z) K

r -

1M

dnm r dz

(39)

If the gas does not adsorb on the pore walls, the diffusion coefficient appearing in this equation is equivalent to D fK , the Knudsen or free-particle diffusion coefficient for the gas f. If the diffusing gas particles are treated as points, then

the functional form of DfK is simply (Kennard, 1938) D

fK

=

(2 -f f)

~3

vR

(40a)

P

where v is molecular mean speed (8kT/7rm) 1{J. andfis the fraction of gas particles that are reflected from the pore walls according to the cosine law for diffuse scattering. Also note that for point gas particles the partition coefficient Kr = 1. When the size of the gas molecules is taken into consideration, these expressions require modification. For example, for spherical nonadsorbing gas molecules of diameter
2(2 - f)

= 3 -f-

vRp(l - A)

(40b)

where A =
L~~) =:r~ B

[ (u~Z)(t)u~Z)(O) 0

dt

(42)

16

MAcELROY

and

D~ =N i~ (u~Z)(t)u~Z)(O»

dt

(43)

where u~z)(t) is the instantaneous center-of-mass velocity of the pore fluid as a whole in the axial direction of the pore at time t, i.e., 1

L v~Z)(t) N

u~Z)(t) = -

N

(44)

i= 1

Also recall that the angular brackets in the above expressions represent averaging over an equilibrium ensemble and therefore nr is the local equilibrium number density of the pore fluid . Furthermore, N is the number of fluid particles in the locally equilibrated pore fluid, and it is of interest to observe that the diffusion coefficient D~ is very simply related to fluctuations in the velocity field of these particles. Since this diffusion coefficient is also an intensive property, the integral f~ (u~Z)(t)u~Z)(O» dt in the above equations is an extensive property and disappears in the limit N -+ 00 . A variety of molecular simulation techniques are in current use that are based on sampling from either equilibrium or nonequilibrium ensembles, and the computational procedures involved in these methods have been described in detail in a number of very readable texts [see, e.g., Nicholson and Parsonage (1982), Allen and Tildesley (1987), and Roe (1991)]. In this chapter, only the method most frequently employed in the computation of the VCFs appearing in Eqs. (14), (42), and (43)-namely, equilibrium molecular dynamics (MD) in the microcanonical ensemble (fixed particle number N , fixed volume V, and fixed energy E)-is considered. Furthermore, since the primary objective here is to illustrate the principles involved in the application of the equilibrium MD method to fluid/solid systems, the diffusing fluid molecules are assumed to be simple structureless spherical particles. TWo models for the interparticle interactions in the fluid are considered: (1) the hard-sphere interaction potential

'" () = {oo, 0,

'l' ij r ij

r ij

<

(Jij

r ij ~ (Jij

(45)

and (2) the Lennard-lones (12-6) interaction potential
=

4Ei{ (::yz- (:i~YJ

(46)

where, in both cases, r ij is the relative separation of particles i and j , (Jij is the relative separation of the particles when the potential energy becomes positive (repulsion), and Eij in Eq. (46) is the potential energy minimum for attraction (pairwise additivity of the interactions is also assumed here).

17

DIFFUSION IN HOMOGENEOUS MEDIA

The principal objective in the (classical) MD method is to solve Newton 's equations of motion for the center-of-mass of each of the fluid particles in the system subject to the interparticle forces derived from Eq. (45) or Eq. (46) and the forces associated with interactions between the fluid and the pore walls. The latter interactions will also involve a hard-core potential and a Londonvan der Waals potential similar to the above expressions for the respective fluids; however, before discussing the details of these interactions it is worthwhile to briefly outline the method of solution of Newton 's equations of motion for each particle i:

dr j

- = v·

(47a)

dv 1 - ' = aj = - Fj dt mj

(47b)

dt

'

For the hard-sphere interaction given by Eq. (45) the force, F j , acting on particle i is zero between collisions and is impulsive during any given collision. Noting that a collision takes place when the relative separation of the colliding pair of particles is rjj = (Jjj' then, it is readily shown that the time to collision, given that the initial positions of the particles are r jO and r jO and their precollisional velocities are V jO and VjO , is given by

(VjjO • r jjO

< 0)

(48a)

where V jjO = vjO - V iO and r jjO = r jO - r jO . The condition V jjO • r ijO < 0 simply refers to the fact that the particles must be approaching one another for the collision to occur. Also note that a collision is not predicted if the group of terms under the square root is less than zero (the particles bypass each other). When the particles collide, their momenta and energies are changed in accordance with the momentum and energy conservation laws of physics subject to possible constraints associated with the external forces acting on the system. For a system of particles that are not subject to external forces and that obey the normal rules of specular scattering (smooth hard spheres), conservation of total linear momentum and energy for the colliding pair provides (48b) where AV j is the change in the center-of-mass velocity of particle i, m is the particle mass, and k is the unit vector along the line of centers at collision. A similar equation of opposite sign is obtained for AVj. [Other hard-core particle j

MAcELROY

18

models are discussed by Allen and Tildesley (1987), and the reader is referred to this source for details.] For Lennard-Jones fluids, the forces Fi in Eq. (47b) are generally nonzero at all times and vary continuously with time as the particles move within the potential field exerted by their respective neighbors. For pairwise additive interactions in homogeneous Lennard-Jones fluids, the force on a given particle i is determined by (49)

with ij given by Eq. (46). At a given time t the position of each of the particles in the system is known, and therefore the forces (or particle accelerations) can be determined using the above expression. The positions and velocities of the particles a short time later may then be calculated by expressing Eqs. (47a) and (47b) in finite-difference form. A number of finite-difference algorithms have been proposed [see Allen and Tildesley (1987) for details] , one of the most popular of which is known as the " velocity Verlet" algorithm (Swope et aI., 1982). This procedure relates the particle positions and velocities at time t + !l.t to the corresponding values and the accelerations at time t according to

ri(t

1

+ "2 !l.t 2ai(t)

+ !l.t)

= ri(t)

+

+ !l.t)

= Viet)

+ - !l.t [ai(t) + ai(t + !l.t)]

I1t Viet)

(50a)

and

viet

1 2

(SOb)

For atomic fluids the time step to be used in the above equations typically lies in the range 10- 15 < !l.t < 10- 14 s. This range of values ~sually ensures reliable conservation of energy during a given simulation run (Allen and Tildesley, 1987). For hard-sphere or Lennard-Jones fluids confined within cylindrical pores, the only additional contribution that needs to be included in the equations of motion for the particles is the fluid particle-pore wall interaction. In the following the conditions appropriate for the hard-core system are considered in detail first and later we return to the simulation conditions for the Lennard-Jones pore fluid. Hard-Core Interactions. For particle-pore wall hard-core collisions, the time to collision is given by

tciw = -

ViO. riO

- -2-

vm

+

1

2

vm

2

{(ViO ' riO)

2

+ ViO

2 ]}I12 [(Rp - -2 ) - riO (J'f

2

(51)

DIFFUSION IN HOMOGENEOUS MEDIA

19

where V iQ and riO are the two-dimensional vector contributions to the initial velocity and the initial position of particle i in the xy plane of the pore cross section and the origin of the coordinate frame lies on the pore axis. The evaluation of the velocity change on collision depends on the mode of scattering assumed, and the results reported by Sub and MacElroy (1986) for the two limiting cases of specular reflection and cosine law diffuse scattering [f = 0 and 1, respectively, in Eq. (40)] are considered here. For specular reflection, only the velocity components in the plane of the pore cross section are changed during the collision: (52) with r iw = R p - (Tr/2. For cosine law diffuse scattering, however, all three components of the velocity are altered. Here only elastic diffuse scattering is of concern, in which case the kinetic energy of the colliding particle is conserved during the collision and, as shown by Suh and MacElroy (1986), the postcollisional components of the particle velocity in cylindrical coordinates are

V:r = Ivil ~

v: = Iv;j [(1 z

v:e= IVil [(1

(53a) ~1) COS(21T~2)]1 !2

(53b)

- ~L) sin(21T~2)r!2

(53c)

where ~1 and ~2 are random numbers that are uniformly distributed on the interval (0,1). The computation of the individual particle trajectories using the above equations is generally supplemented with one or more simplifying computational devices to alleviate the burden of the calculations [a number of " tricks of the trade" are described in detail by Allen and Tildesley (1987)]. The most important of these devices is common to nearly all molecular simulatins and involves periodic imaging of a fundamental cell containing a finite number of particles. It was clearly recognized in the 1950s and 1960s by the pioneers of molecular simulation methods that no computer in existence at that time or indeed at any time in the foreseeable future could determine the trajectories for a system of macroscopic volume containing billions of particles. Since computations can be carried out for only a finite number of particles, the major difficulty to overcome in simulations of homogeneous media is the condition associated with the boundary of the simulation cell. A simple impenetrable wall is clearly out of the question due to the severe distortion such a boundary would induce on the particle phase coordinates, and a straightforward method for mimicking the behavior of the pseudo-homogeneous system is to consider the cell to be surrounded on all sides by images of itself. Surprisingly, with fundamental cells containing as few as tens or hundreds of fluid particles this approach can provide

20

MAcELROY

particle trajectory data that are sufficiently accurate for evaluation of the thermodynamic and transport properties of bulk macroscopic fluids. (For a number of interfacial or near-critical states, this, unfortunately, is not the case due to the long-range correlations involved in such systems, and care must be exercised in the selection of the size of the simulation cell.) The imaging procedure is illustrated schematically in Fig. 1 for the cylindrical pore under discussion here, and for the hard-core system a given simulation run would typically proceed as follows. The number of particles, N, to be employed in the simulation are placed within the fixed volume defined by the radius Rp and half-length L of the fundamental unit of the pore in either an orderly [see, e.g., Heinbuch and Fischer (1987)] or random manner (Suh and MacElroy, 1986), the procedure selected generally depending on the required density of the fluid . The initial velocities of the particles are then usually assigned by randomly selecting components from the Maxwell-Boltzmann velocity distribution function [see Allen and Tildesley (1987) for details] subject to fixed energy and zero net fluid momentum. The particle trajectories are then traced in a sequence of steps in which The minimum collision time predicted by either Eq. (48a) or Eq. (51) determines the next collision. 2. All of the particles in the fundamental cell are moved through this minimum collision time and the collision takes place. (If, during this process, any particle moves out of the fundamental cell across the boundaries at ±L, then it reappears in the cell at the opposite boundary.) 3. The momenta of the colliding particles are changed in accordance with Eqs. (48b) , (52), or (53). 1.

z:;:O

Figure 1 Fundamental unit of the cylindrical pore. The filled spheres on either side of the fundamental cell (below z = -Land above z = +L) are images of the shaded particles shown inside the cell.

DIFFUSION IN HOMOGENEOUS MEDIA

4. 5.

21

The collision times for the particles involved in step 3 are reevaluated. The above steps are repeated.

After an initial equilibration period lasting approximately 500- 1000 collisions per particle, this sequence of computations continues until a sufficiently large statistical sample of the (equilibrium) particle positions and velocities has been recorded. Suh and MacElroy (1986) found that fundamental cells containing 200 particles were representative of the macroscopic thermodynamic system, and equilibrium trajectories in the microcanonical ensemble approximately 104 collisions per particle in length were found to provide statistics of sufficient accuracy for subsequent evaluation of the diffusion coefficients using Eqs. (42)-(44). The equilibrium time-correlation function M Z)(t) u~z)(O» appearing in these equations is readily evaluated by sorting the stored trajectory data into equally spaced time intervals at, and, using the ergodic hypothesis, the ensemble-averaged VCF is given by

1

M

(u~z)(t)u~z)(O» = (u~Z)(jat) u~z)(O» = M ~ u~z)[(j + k)St]u~z)(kat)

(54)

where M is the number of independent time origins employed in the averaging process for a given value of j. For small j (short times), M will generally be very large, and very accurate evaluation of the time-correlation function can be achieved. For large j , i.e., times approaching the length of the trajectory, M will necessarily be a small number, and for this reason the statistical error in the computation of the time-correlation function at long times will be large. Frequently, this limitation does not play a significant role in the evaluation of the kinetic coefficient LWor the diffusion coefficient D~ because the VCF usually decays rapidly to zero. Under these conditions the upper limit of infinity in Eqs. (42) and (43) may be replaced by a time t = tMAX (which, in many cases of interest, is significantly shorter than the total length of the trajectory) with little or no loss in accuracy in the numerical integration involved in these expressions. If the VCF does not decay rapidly to zero, it is still possible to obtain reliable estimates for the transport coefficients by suitable extrapolation of the long-time tail of the VCF, although this does require some prior knowledge or information concerning the expected time-dependent behavior of the long-time tail [e.g. , via scaling theories (Havlin and Ben-Avraham, 1987)]. For the moment it will be assumed that the VCF is zero at or beyond tMAX ; we return to the problem of long-time tails in Section III.B. A number of typical center-of-mass VCFs for the pure hard-sphere pore fluid both in the axial direction and in the plane of the pore cross section are illustrated in Figs. 2 and 3 (Suh and MacElroy, 1986). (Also shown in these figures are the VCFs for tracer particles, which are discussed in Section 1I.A.2). The VCFs in the plane of the pore cross section are simply obtained from the x, y

22

MACELROY

velocity components of the center-of-mass of the fluid using the expression (u~XY)(t) u~XY)(O)

1 =2 ([ u~X)(jBt) uy)(O)] +

[u7)(jBt) u7)(0»)

The results shown in Figs. 2 and 3 are normalized to 1.0 at zero time, and the dimensionless time, T*, in these figures is in units of 2Rp(l - A)/V. The VCFs illustrated in Fig. 2 correspond to the dilute gas (i.e., Knudsen) limit, in which case, by definition, the diffusing particles never collide with one another, Le., the center-of-mass VCF simplifies to lim (U~k)(t) U~k)(O) tfr_O

1

=2

N

L (V~k)(t) V~k)(O) N

(55a)

i= l

=(V(k)(t) V(k)(O)

(55b)

where k = xy or z. Due to the absence of interparticle collisions, crosscorrelations for the individual particle velocities are nonexistent as implied by

1.0

z o

>-f=

f-U -z u::>

glL. w

>z o

8~ N-.I

-w -'0::
00 Zf-

~

Figure 2 Normalized velocity autocorrelation functions for free-molecule (Knudsen) diffusion versus reduced time T*. (--) Theoretical VCF in the axial direction for diffuse scattering; (e) molecular dynamics axial VCF for diffuse scattering; (- - -) and (- - -) molecular dynamics VCFs in the plane of the pore cross section for diffuse and specular scattering, respectively. [Reproduced from Suh and MacElroy (1986), with permission.]

DIFFUSION IN HOMOGENEOUS MEDIA

LO

23

(0 )

""

U

;>

~ ~

N

~

~

0

z

-0.2

0

-0.4

\

,/

"- ' - ----- /

---

/

-06 '0

1.00

(b)

o -0..2 -0..4 -0..6 0

1.00

T* Figure 3 Normalized velocity autocorrelation functions vs. reduced time T * for A = 0.21 and II~I (= limO}) = 0.6. (a) Specular reflection; (b) diffuse reflection. Curve 1, VCF corresponding to L\z),.; curve 2, VCF corresponding to L~~); short- and long-dash curves, VCFs in the plane of the pore cross section for L\?/. and L\7), respectively. [Reproduced from Suh and MacElroy (1986), with permission.]

24

MAcELROY

Eq. (55a), and, as indicated by Eq. (55b), the VCF under these conditions is equivalent to the VCF for a single, isolated particle (N = 1) diffusing within the pore. Note that the axial (z) component of the VCF for particle/pore wall specular reflection [Eq. (52)] is not explicitly shown in Fig. 2. The normalized axial VCF in this case must be equal to 1.0 at all times because v(Z)(t) is unchanged during a collision with the pore wall. Also note that for these specular reflection conditions the diffusion coefficient predicted by Eq. (43) is infinite, in accord with the result predicted by Eq. (40) when f = O. For cosine law diffuse scattering, the axial component of the momentum of the particle is not conserved during pore wall collisions (one can hypothesize the existence of an external clamping force on the solid that holds the pore wall stationary during any given collision, and it is this force that would be required in the balance equations to reinstate conservation of momentum). In this case the axial momentum of any given particle is subject to " memory" loss during collision with the pore wall, and for this reason the axial VCF shown in Fig. 2 for diffuse scattering decays to zero with increasing time. The solid line shown in this figure is the theoretical (as opposed to simulation) prediction of the VCF for diffuse scattering, and its integral over time provides the Knudsen diffusion coefficient given in Eq. (40) with f = 1. It is of interest to note that although an exponentially decaying VCF is frequently assumed in approximate theories of diffusion, the decay in the axial VCF shown in Fig. 2 is not a simple exponential as shown by Suh and MacElroy (1986). Indeed, a simple exponential decay rarely describes the true temporal behavior of the VCF for a wide variety of systems [even for homogeneous dilute gases (Alder and Wainwright, 1970)], and care must be exercised when interpreting relaxation time constants obtained assuming pseudo-exponential decay. A particularly important example of nonexponential behavior that is believed to be of direct relevance to rigid glassy polymers is considered in Section III.B. The fluid center-of-mass VCFs for motion in the plane of the pore cross section shown in Fig. 2 (Ilr - 0 and hence nCB - 0) and in Fig. 3 at a higher (liquidlike) reduced bulk density demonstrate one of the shortcomings discussed earlier with regard to the prediction by linear response theory of local diffusion coefficients in anisotropic systems. It is clear from Figs. 2 and 3 that the oscillatory behavior of the xy VCF will lead to diffusion coefficients that are completely different from the axial results, and in fact it is readily shown that integration of the dashed curves shown in Fig. 2 and the long dashed curves in Fig. 3 over the time range from 0 to 00 provide zero values for D~-Z) . Such values for D ~) should be viewed as questionable in light of the very strong local inhomogeneities involved in the fluid density, and due consideration should be given to a more in-depth analysis based on the wavenumber dependence of the kinetic coefficients predicted by the projection operator formalism of Mori (1965). Unfortunately, there is at present no simple way of evaluating the pro-

25

DIFFUSION IN HOMOGENEOUS MEDIA

jected random fluxes appearing in this theory, and until further work in this area is undertaken it will be necessary, as noted earlier, to restrict the application of the results of linear transport theory to diffusion in translation ally invariant systems (in the present case, in the axial direction of the cylindrical pore). The data represented by the filled circles illustrated in Fig. 4 (MacElroy and Suh, 1987) are selected results for D~ for a pure fluid f as a function of Ac = CTr/2Rp at a fixed bulk phase reduced density n;t = nrnCT; = 0.4054. (The open circle and open square results are for the individual species in a binary mixture at the same bulk density with A/A2 = 0.6.) These results are in the dimensionless form D~lDfK' where DCK is given by Eq. (40) (with f = 1). In the limit Ac ~ 1 (the hard spherical fluid particles approach the size of the pore), diffusion within the pore is described solely by free-particle motion (a result that is independent of density). In the opposite limjt, Ac ~ 0, the diffusion mechanism is usually referred to as viscous slip, and the coefficient D~ under these conditions is a function of fluid density, decreasing with increasing density (Suh and MacElroy, 1986; MacElroy and Suh, 1987). It was also shown by MacElroy and Suh (1987)

1.2

1.0 ,,4 /J

"

9

(z)

DaK

.-

-...- •

/

0.8

DaM

/'

~ .....

, I

0.6

a... 1>--0-.0 0.4

I /

,~

I

-+t-l-l/

0.2

00

0.2

0.6

0.4

A' 0.8

, I

r

I

LO

Aa Figure 4 Reduced axial diffusion coefficient relative to the membrane as a function of particle reduced radius. (e) Single-component system (ex = f) . (0) and (0) Results for the solvent (ex 1) and the solute (ex 2), respectively, in the binary system. [Reproduced from MacElroy and Suh (1987), with permission.]

=

=

26

MAcELROY

that for pore sizes less than approximately one-tenth the diameter of the fluid particles (typically Rp $ 2.5 nm) the transport of a dense fluid or gas in a pressure gradient is primarily determined by slip flow and not by shear flow; i.e., the Hagen-Poiseuille equation or Darcy's equation is not applicable in very fine pores. This has long been known for dilute gases (Kennard, 1938), and, as illustrated by MacElroy and Suh (1987), it is now possible to quantify the range of validity of continuum formulations such as the Hagen-Poiseuille equation for dense fluids and liquids using molecular simulation techniques. London-van der Waals Interactions. For the cylindrical pore model a number of different particle-pore wall interaction potentials have been investigated, primarily with the equilibrium properties of the pore fluid in mind (peterson and Gubbins, 1987; Peterson et aI., 1988), although the transport properties have received attention in a few studies (Heinbuch and Fischer, 1987; MacElroy and Suh, 1989). [Transport characteristics have also been investigated via MD simulation for London-van der Waals fluids confined within slit-shaped pores (Schoen et aI., 1988; Magda et aI., 1985).] Usually the particle- pore wall potential function is represented by a two-body interaction in 'Which the solid is treated as a smeared continuum of Lennard-Jones interaction sites. Heinbuch and Fischer (1987) employed a layered structure of concentric cylindrical shells of smeared solid atoms in MD simulations of an adsorbing Lennard-Jones vapor. However, the most common representation is that of a continuum solid that is devoid of any internal or surface structure, and for a pore fluid characterized by the potential given in Eq. (46) it has been shown that the potential energy for interaction between a fluid particle i and the pore wall in this case is given by (Nicholson, 1975) ,w(Rp - rl) =

""3 Elw n w CTIW [ 15 (Rp _ rl)9f 21T

3

2

CT,w

(9)

(Rp _ rl)3f CT1w

(rl) -

(3)

(rl)

]

(56)

where r i is the radial position of the fluid particle within the pore, E iW is the potential minimum for interaction between the fluid particle and a single Lennard-Jones site in the solid, CIiW is the corresponding Lennard-Jones size parameter, and Ilw is the number of Lennard-Jones sites per unit volume in the solid phase. The two functions j9)(ri) and p )(ri) are polynomials in ri (Nicholson, 1975), and in the limit Rp - 00, Eq. (56) simplifies to the 9-3 potential function frequently used in the modeling of sorption on flat surfaces (Steele, 1974). It is also important to note that the definition of Rp in Eq. (56) differs from that involved in the hard-core interactions discussed above. This difference is readily seen by comparing Figs. 1 and 5a. In molecular dynamics simulations of a Lennard-Jones pore fluid whose interactions with the pore wall are described by Eq. (56), the total force experienced by a given fluid particle i is obtained by including the force field exerted

DIFFUSION IN HOMOGENEOUS MEDIA

27

(0)

(b)

Figure 5 Model cylindrical pore structures. (a) Particle-pore wall continuum interactions. The hatched region r < Rp represents the inner repulsion core of the solid surface atoms. (b) Structured pore wall. Axial positions 1 and 2 are referred to as the pore window and polygonal cage, respectively. (Reproduced from MacElroy and Suh (1989), with permission.)

by the solid in Eq. (49) to give

....s

r ij dij Fi = miai(t) = - L.J - j=l rij drij

ridiW(R p - ri) ri dri

(57)

j#

where r i is the two-dimensional vector coordinate of the particle in the plane of the pore cross section; i.e., for the smooth pore wall there is no axial force component on the fluid arising from interactions with the solid phase. Therefore, as in the case of the hard, specularly reflecting pore wall discussed earlier, the axial momentum of the fluid is also conserved here and D ~ is predicted to be infinitely large. Only if one introduces a mechanism for axial backscattering during interaction with the solid phase will a finite diffusion coefficient be observed, and the simplest way to achieve this is to incorporate a discrete atomic or molecular structure in the pore wall. Such a structure was introduced by MacElroy and Suh (1989) [and in the slit-pore studies reported by Schoen et al. (1988)] that is represented by a single periodic layer of surface atoms {S} whose coordinates are given by rj(j E S)

= R p cos (21Tk) NR (k

= 1,

.. . , N R ; I =

-00

, .. . , + (0)

(58)

28

MAcELROY

where NR is the number of surface atoms in a polygonal ring and (Jw is the axial spacing of the rings (NR is 12 for the diagram shown in Fig. 5b). In the work reported by MacEIroy and Suh (1989) the interactions between these surface atoms and the fluid particles in the pore were described by a Lennard-lones potential function similar to Eq. (46) with Eij = EiW and (Jij = (JiW Furthermore, as implied by the diagram in Fig. 5b, Eq. (56) was employed to characterize the interaction of the pore fluid with the solid beyond the radial position r = R p + (Jiw/2 - r i, and the total force on a given fluid particle i for this structured system is therefore Fi = miai(t) .

=-

±

r ij dij _

j=1 r ij drij j#i

:i:

r ij dij _

j=1 r ij drij

~ diW(Rp + (Jiw/2 ri

r i)

(59)

dri

jES

In the molecular dynamics simulation of a Lennard-lones pore fluid subject to forces of the type described by Eq. (57) or Eq. (59), it is again quite clear that only a relatively small number of particles (= 102 _10 3 ) may be considered in the finite-difference solution of Newton's equations of motion [e.g., using Eqs. (50)]. Periodic boundaries at z = ±L are therefore employed to minimize the influence of edge effects on the properties determined from the particle trajectories. For van der Waals interactions of this kind, an additional problem arises that is not encountered in purely repulsive hard-core systems, namely, the long-ranged nature of the interaction itself. In principal, this would imply that a very large simulation cell should be employed, and as this is generally not feasible it is necessary to truncate the range of any given interaction at a point that is at least as small as half a characteristic dimension of the fundamental simulation cell [this limit arises from the minimum image convention; see Nicholson and Parsonage (1982) and Allen and Tildesley (1987) for details] . Traditionally the cutoff point or radius Rcij for a spherically symmetric interaction between particles i and j in a condensed phase is taken to lie between 2.5(Jij and 3.5(Jij' The larger the value of R cij , the more closely will the simulated fluid approach the physical behavior of the model fluid [e.g., the Lennard-lones (126) fluid characterized by Eq. (46)]; however, the upper limit in RCij is usually governed by the CPU time required to compute all of the force contributions within the spherical volume r ij :5 R cij . This CPU requirement increases as N~, where Nc is the number of particles within the cutoff volume. Fortunately, for London-van der Waals interactions of the type given by Eq. (46), the interaction approaches zero rapidly with increasing r ij and the computations are not seriously influenced by the truncation at Rcij [e.g., at a relative separation of 2.5(Jij' Eq. (46) provides ij(Rcij) = - 0.016E ij , and at 3.5(Jij the potential is - 0.002Eij]' Additional tricks of the trade such as shifted potentials (to ensure energy conservation in the microcanonical ensemble MD simulations), neighborhood lists, cell linked lists, etc. [details of which may be found in Allen and Tildesley

DIFFUSION IN HOMOGENEOUS MEDIA

29

(1987)] should also be incorporated in the simulation code to improve the computational efficiency during program execution. A typical simulation run for a Lennard-Jones fluid confined within either of the two model pores illustrated in Fig. 5 would proceed as follows. As described earlier for the bard-core system, the N fluid particles are placed in the pore either randomly or in an ordered manner, and their initial velocities are assigned from the Maxwell-Boltzmann velocity distribution function at the desired temperature of the simulation. The total energy (potential + kinetic) is again a fixed quantity, and therefore during the initial stages of the simulation the temperature (which is determined by the kinetic energy of the particles) will vary as the fluid relaxes toward equilibrium. This necessitates rescaling the individual particle velocities during the equilibration period to return the system to the desired temperature. The number of time steps involved in the finite-difference calculations during this equilibration period is typically ""104 , where, as noted earlier, fl.! usually lies in the range 10- 15 < fl.! < 10- 14 s. After equilibrium has been achieved, rescaling is terminated, and during the subsequent computations the particle trajectories evolve at fixed energy. In the equilibrium system no drift in the average kinetic temperature T = (1/3N~ (~miv7> will be observed, although fluctuations in L,V~/N on the order of llyN should be present. During the equilibrium trajectory (usually sampled for approximately 105 time steps), the particle positions and velocities are stored at equispaced intervals 31 for subsequent evaluation of the VCFs using Eq. (54) and of the diffusion coefficients using Eq. (43). Results for the axial diffusion coefficient D~ for the structured pore shown in Fig. 5b for a range of values of NR [see Eq. (58)] and at a fixed bulk liquid density nfB = 0.60'(3 and temperature T = 1.15(Ecr/kB) are reported by MacElroy and Suh (1989). These data are reproduced in Fig. 6, where the reduced form D~/DfK is plotted as a function of A = a f/2Rpeff' The Knudsen diffusion coefficient involved here corresponds to Eq. (40) with f = 1, and tbe pore radius is defined as the effective quantity R pef(' This effective pore radius is itself determined by using the definition of a dividing surface at the pore wall, which is consistent with the definition of the dividing surface for a smooth, hard wall, and therefore the magnitude of A obtained here has a one-to-one correspondence with the definition given earlier for hard-core interactions [see MacElroy and Suh (1989) for details]. One of the most important aspects of the results shown in Fig. 6 is that they confirm the existence of viscous slip (nonzero D\2 in the limit A -+ 0) for a realistic liquid/solid interface. It may therefore be concluded that, in general, in addition to shear flow, slip flow should not be neglected as a viable mechanism for transport in micropores. In the range of reduced radii A 2: 0.5, the fluid particles within the pore diffuse in single file and the continuum concepts of shear and slip are no longer tenable. The trends in D~ observed in Fig. 6 under

30

MACELROY

d 2

O.l2....---....:.r---T--T---r-------y------,

0.10

\

\ \

0.08

,

\

\

0.06

0.04

1 I' \ \

I I

IrI\/1/

0.02

\ I \

0.2

0.8

1.0

Figure 6 Reduced axial diffusion coefficient relative to the membrane vs. A for the structured pore wall (Fig. 5b). DfK is the free-molecule (Knudsen) diffusion coefficient, and the upper abscissa, d, is the diametric distance between pore wall surface atoms in units of (Jr. [Reproduced from Mac Elroy and Suh (1989), with permission.]

these conditions are notably similar to those for the hard-sphere pore fluid illustrated in Fig. 4.

2. Binary Mixtures For a micropore fluid in local equilibrium with a bulk fluid mixture, the thermodynamic forces appearing in Eqs. (33) and (34) may be replaced by equivalent bulk-phase thermodynamic forces, i.e., (60) Furthermore, using the Gibbs-Duhem equation [Eq. (17)] for a binary bulkphase mixture of components 1 and 2, it is readily shown that the micropore driving forces are interrelated by

DIFFUSION IN HOMOGENEOUS MEDIA

31

and vice versa. Also noting that in general

n, = K,n,B where K, is the partition (or distribution) coefficient for component i, the flux equations for the two species in the axial direction of the cylindrical pore are given by Eq. (33) as

j(Z)(i = 1

2)

"

D(Z) ~ dfJ-, I , kBT dz T

=-

(61a)

= _ D(Z)K ~ dfJ-,BI '

, kBT dz

(61b) T

where D\Z) and D~Z) are the Fickian diffusion coefficients for the two species and are related to the microscopic properties of the pore fluid as (z)

Dl

= -kBT (Z) nl

LII

n 1K"2 n"2 K l

(Z»)

(62a)

- - L 12

-

and (62b) with

L:;) given by

L:;)

=

_1_1x
0

A=I

dt

(63)

/=1

The Stefan-Maxwell form of the flux equations may also be employed to express the Fickian diffusion coefficients defined above in terms of the mutual diffusion coefficients D~z~, D~~, and DW (MacElroy and Suh, 1987):

(j\Z»)] 1 1 +1 [ D(z)=D(Z) D(Z) I - x 1 1 + j~Z) I

1M

1"2

(64a)

-'-

or (Z)

DI

1 + xl(D~~/DiZi)(l - Kz/KI )

= l/D\=;'" +

l/Dizi [1 - x l (1 -

with similar expressions for Di). species within the pore.

XI

D~z~/D\Z~)]

and

X2

(64b)

are the mole fractions of the two

32

MAcELROY

The mutual diffusion coefficients may also be expressed in terms of the kinetic coefficients L ~;>: (l) DIM

kaTIL (l ) 1 - n L (l)

= n IL(l)22

2

(65a)

12

(65b) and (65c) where IL(')I = L~';L ;~ - L~~2. The above results simplify when the special case of self-diffusion is under consideration. In this case component 1 may be taken as the "solvent" and component 2 is defined as the tracer, which, in principle, is at infinite dilution and has the same molecular properties as the solvent particles. For clarity the tracer is defined here as component 1 *, and in view of the equivalence of molecular properties we have K = KI and D \'}M = D \'~. For this model binary system, Eq. (61) for the tracer" simplifies to (l) _ _ -

J I'

D (') dn I"

"

(66a)

dz

(66b) and the expressions for the Fickian and mutual diffusion coefficients reduce to the following, in which it is assumed that N,• = 1 and thus NI = N - 1 (Suh and MacElroy, 1986):

D'z)I' -- D"1 ) -1 D 1·

2) Vle a T~ - L(l)/N N _ 1 (L (Z) 1'1' rr

1

1

-(l)- +D(l) - D(l) l *M

(67a) (67b)

1·1

(') _ kaT L(') D I' M ff

(67c)

nr

and (l)

(z)

(l) 1"1'

-

DI'I = VleaT L[r (LL(z) _ rr

L(l)/N2) f[ ur (l) lY.Lt*l·

(67d)

33

DlFFUSION IN HOMOGENEOUS MEDIA

where L ~~ is given by Eq. (42) and

L~Z?). = _1_

VkBT

= _1_

r (v~Z?(t)v~?(O»

Jo

(!) Jo('" \/i

VkoT N

dt

v~Z)(t)V~Z)(O») dt

(68b)

;=)

Note that in the thermodynamic limit N D~=? = D~z) = VkoTL~Z?l '

(68a)

--->0

00,

Eq. (67a) simplifies to (69)

It is this parameter (or more specifically its directionally averaged value for a

randomly oriented pore network) that is measured via nuclear magnetic resonance spectroscopy of radioactive tracer studies, and when these are complemented with molecular simulation results it should be possible to accurately predict M~) and hence the diffusion coefficient D\z?M = D~ . The latter coefficient, which is of particular importance in the engineering design of adsorbers, membrane separations modules, etc., may also be measured via gravimetric or volumetric sorption experiments, which in turn may be used as corroborative evidence for the validity of a proposed molecular model for diffusion in microporous media and membranes. We return to the distinction between D\=2 and D~ later in Section III.B, and for the moment we examine the characteristic behavior of Dl! alone for tracer diffusion subject to specular or diffuse scattering interactions with the cylindrical pore wall. The determination of the tracer diffusion coefficient from the MD simulation data for a pure micropore fluid involves a straightforward application of Eq. (54) to a single particle in the system, and as each individual particle may be independently considered to be the tracer, a secondary averaging is permitted as shown in Eq. (68b). This secondary averaging can lead to very accurate results for D~z2 , in contrast to the membrane/fluid mutual diffusion coefficient D~, which involves a single measure of the influence of the collective motion of the pore fluid as a whole. Accurate determination of D~ usually requires MD trajectories that are at least an order of magnitUde longer than those needed to obtain tracer diffusivities of similar accuracy. Sample results for tracer diffusion in a Lennard-lones liquid at a fixed chemical potential confined within the smooth-walled and structured cylindrical pores illustrated in Fig. 5 are provided in Figs. 7 and 8, respectively (MacElroy and Suh, 1989). These results demonstrate the very significant effect axial backscattering has on the diffusion mechanism and the need for a reliable atomistic model of the solid phase (or, at the very least, some provision for axial backscattering) when conducting computer simulations of micropore fluids. As the pore size decreases (A increases), the diffusion coefficient for the tracer in the atomically structured pore drops rapidly in agreement with the general trend Z

34

MA CELROY

d 10

3.5

5 4

2

3

3.0

2.5

(z)

2.0

D 1*

D1*,B

1.5

, ,

~--------------~\-------- ?-\

\ 0 .5

b

"-"-

\

'-....

'-

----

I

'~ ....

f

- ~

P

Figure 7 Reduced axial diffusion coefficient for the tracer as a function A in pores with smooth walls. (e) MD simulation; (0) Davis - Enskog kinetic theory [Davis (1987)] and Fischer- Methfessel (1980) approximation; (0) Davis - Enskog kinetic theory and bulk fluid approximation; Lower dashed curve, The empirical correlation of Satterfield ct al. (1973). d as in Figure 6. [Reproduced from MacElroy and Suh (1989), with permission. ]

expected in physically realistic situations. The straight solid line shown in Fig. 8 is, in fact, an empirical correlation obtained by Satterfield et al. (1973) from a regression analysis on the diffusion coefficients for a variety of dilute aqueous and nonaqueous solutions in microporous alumina [it is of interest to note that a similar correlation has also been suggested to describe steric effects in polymers; see, e.g., Pace and Datyner (1979a,b,c)]. Somewhat similar results were reported by Suh and MacElroy (1986) for tracer diffusion in a hard-sphere pore fluid subject to either specular reflection at the pore wall [Le., Eq. (52)] or cosine law diffuse scattering [Eqs. (53a-c)].

35

DIFFUSION IN HOMOGENEOUS MEDIA

d 2 2 .0 1.0 0 .8 0.6 0.4 0 .3 0 .2

, \

D(z)_ I DI-,B

\ \

0 .10 0.08 0.06

I

I,:\11

\

9

\ I, \0

0 .04 0 .03

13 \

I

. I

\1

12

0.02

\

,"/

'.':

10

\

\

\

\

'.

\8 \

\

0.010 0.008 0.006

\

0.004 0 .003 0.002 0 .001

0

1.0

Figure 8 Reduced axial diffusion coefficient for the tracer as a function of A in pores with structured walls. All symbols are as in Fig. 7. The numbers next to the simulation points refer to the value of NR in Eq. (58). [Reproduced from MacElroy and' Suh (1989), with permission.]

For diffusion in binary pore fluid mixtures of molecularly disparate species, one of the most important questions that frequently arise concerns the relative importance of cross-coupling effects; i.e., can the cross-kinetic coefficients Lizi and L~zl appearing in Eqs. (62a,b) be neglected? A supplementary question is then usually posed: If the cross-coefficients are neglected, can one assume that the coefficients L\1 and L~l are simply related to their pure component values? (The simplest approximation here is to assume that these coefficients are equal to the pure component parameters.) Both of these questions are readily answered

36

MACELROY

for low-density gas mixtures in macroporous media because reliable molecular predictions can be made in such cases (Chapman and Cowling, 1970; Hirschfelder et aI., 1954; Mason and Malinauskas, 1983); however, for micropore fluids and particularly dense fluids or liquids, answers to these questions are not easily obtained. As a rule (particularly in view of the negative answers usually implied for dilute gases) one should not neglect cross-effects unless independent evidence exists to support the assumption that these terms are negligible. Even for dilute solutions care must be exercised as illustrated by the MD simulation results for a dilute binary hard-sphere dense fluid mixture reported by MacElroy and Suh (1987). Taking components 1 and 2 as the solvent and solute, respectively, for dilute solutions (n2 -+ 0) the solute diffusion coefficient [Eq. (62b)] simplifies to (z) kBT (z) D2 :: - L 22 n2

: i~ (v~Z)(t)vnO»

(70a) dt

(70b)

It has been assumed here that the contribution (K j /K2 ) L~z? in Eq. (62b) remains

finite in this limit. A similar simplification does not result for the solvent diffusion coefficient [Eq. (62a)] unless K2/Kl -+ 0 and/or L\1 -.. 0, and neither of these conditions will be generally true. The simulation results reported by MacElroy and Suh (1987) for both D\z) and D~) in hard-sphere pore fluid mixtures are shown in Figs. 9 and 10, where D 12.B in the mutual diffusion coefficient of the pair 1-2 in the bulk homogeneous fluid. 1Wo important trends are observed in Fig. 9: (1) For large pores 0\2-.. 0), the rate of diffusion of the solvent within the pore is much larger than corresponding rates in the bulk phase [as discussed by MacElroy and Suh (1987), this phenomenon is generally associated with diffusive slip, which in turn is intimately related to osmotic transport pphenomena] ; and (2) for very small pores the solvent Fickian diffusion coefficent is negative, and this is due solely to the influence of the cross-coupling term in Eq. (62a). The latter effect arises from configurational constraints within the pore fluid associated with both the magnitude of A2 and the magnitude of A,/A2 (which equals 0.6 in the present case). For the range of pore sizes in which D\z) is observed to be negative, it was also shown by MacElroy and Suh (1987) that KJKJ > 1, and since the mutual diffusion coefficients D~~ and D\Zi are always positive it is seen from Eq. (64b) that it is this ratio that governs the sign on D\z). Finally, from Fig. 10 it is of interest to note that for large pores (and " nonadsorbing" solutes) the solute diffusion coefficient reduced by the bulk-phase solute/solvent mutual diffusion coefficient is a universal function of the solute reduced radius Aa; i.e., the ratio D~)/Dla,B ' where (l is the solute, is independent

DIFFUSION IN HOMOGENEOUS MEDIA

37

4 .0

3.0

2.0

(z)

\f--1\

1.0

\ \

\

Dl

\

\ \

D 12 ,B

0 0

0.2

0.4 "-

2

\

\f

I

\

-1.0

, ,, I

0.8

0.6

\

I

\

I

I

-2.0

-3.0

Figure 9 Reduced axial pore diffusion coefficient for the solvent as a function of solute particle reduced radius. [Reproduced from Mason and Chapman, (1962), with permission.]

of the size of the solvent particles, an observation that is in agreement with experimental measurements (Satterfield et al., 1973). This is further confirmed by the solid line shown in Fig. 10, which corresponds to the theoretical predictions for diffusion of a solute of finite size in a continuum solvent [see, e.g., Anderson and Quinn (1974) and Brenner and Gaydos (1977)]. For small pores (A, + 11.2 > 1), however, the pseudocontinuum assumption for the solvent breaks down, and the intrinsic particulate or molecular structure of each component in the pore fluid needs to be taken into consideration.

B.

Diffusion in Rigid Random Media

Single-pore analyses of the type discussed above can provide valuable insight into a variety of properties of fluids confined within cavities of molecular dimensions. However, one very important aspect that is not covered in such studies is the manner in which interpore connectivity in a macroscopic random medium can influence the overall transport process. 1\vo independent though comple-

38

MACELROY 1.0

0.8

,

-.;, \

D(z) a

\

0.6

\ \

11, \

D1a,B

\ \

0.4

,

~, \ \'tJ:\

0.2

\

./-~\\ \ ,.

\\, _;( \ ,, "

\

0._,0'

00

0.2

0.4

0.6

\

'~\ ~,o---, \ 0.8

1.0

Aa Figure 10 Reduced axial pore diffusion coefficient for the solute as a function of solute reduced radius. (. ) Simulation results for the single-component system (ex = 1*) with nft = 0.4054; (D) simulation results for the solute (ex = 2) in the binary system with ntu = 0.4, lit II = 0.01; ( - - ) continuum-mechanical theory [58, 59]; ( - - - ) Eq. (40b) (ex = 1* and f = 1.0) reduced by D ....Il ; ( - - - ) Eq. (40b) (ex = 2) reduced by D l 2,s. [Reproduced from Mason and Chapman, (1962), with permission.]

mentary approaches have been employed in the last 20 years to investigate the effects of pore space topology on both the equilibrium and transport properties of fluids in random media, the first approach being based on lattice models of the pore network [e.g., Shante and Kirkpatrick (1971), Kirkpatrick (1973), Reyes and Jensen (1985), Nicholson et a1. (1988), Sahimi (1988), and Zhang and Seaton (1992)], while the second views the random medium (pore space and solid phase) as a continuum [see, e.g., Haan and Zwanzig (1977), Nakano and Evans (1983), Abassi et a1. (1983), Chiew et a1. (1985), Torquato (1986), Park and MacElroy (1989), MacElroy and Raghavan (1990, 1991), and Raghavan and MacElroy (1991)]. In the lattice models, a "bond" joining any two nodes or " sites" in the lattice usuaIJy represents an individual pore channel (frequently assumed to be cylindrical in shape) that is connected to other pores in the network at the respective nodes. These models are very versatile (individual bonds may be assumed to represent pores of different sizes or shapes, the nodes may be assumed to have zero or nonzero volume, the site or bond coordination number may be varied locally or globally, etc.) and are particularly popular in

DIFFUSION IN HOMOGENEOUS MEDIA

39

studies relating to percolation phenomena in disordered systems. A primary distinction between lattice models and continuum models is the assumption in the former that the topology and structure of the pore network can be treated independently of the underlying physics of the diffusion mechanism (the mechanistic problem itself is usually considered for each bond individually using equations similar to those provided in Section UI.A). In a continuum description of a random medium the structure and topology of the system and the diffusion (or percolation) process are usually modeled simultaneously. The solid phase may, for example, be represented by inclusions randomly distributed in space, with the interinclusion void space corresponding to the volume accessible to the diffusing fluid components, or the inverse description may be employed in which the void region corresponds to the space occupied by the (interconnected) inclusions. If the inclusions are spherical, the first model is sometimes called a "cannonball " solid, whereas the second model is referred to as a " Swiss cheese" medium. The flux equations for diffusion in this case are essentially the same as those described in Section III.A, with the following exceptions: (1) If it is assumed that the medium is isotropic, then the flux expressions are obtained using Eqs. (31) and (32) rather than Eqs. (33) and (34), and each of the kinetic coefficients (and hence the diffusion coefficients) is obtained from averages over all three Cartesian coordinates [as, for example, in Eq. (14)]; and (2) the influence of the structure, shape, and topology of the pore space is implicitly taken into consideration in the determination of the transport parameters themselves. For a wide variety of permeable media, and particularly for gels and polymer films, it is believed that continuum rather than lattice models better represent the structure of the material, and for this reason only continuum models are considered in this section. Furthermore, since the microscopic structures of gels and polymers frequently have the appearance of a network of beaded particles (e.g., interconnected colloidal particles or monomer units), it is reasonable to model the backbone of the medium as solid inclusions randomly distributed in space. Early theoretical studies in this area were primarily based on the work of Maxwell (1873) on the dielectric permeability of particle suspensions [for a review of later improvements on Maxwell's original formulation, see Barrer, Chapter 6 in Crank and Park (1968)], and more recently statistical methods (prager, 1963; Weissberg, 1963) have been employed with some success. These models, however, are applicable primarily to systems in which the diffusing fluid within the void space can be treated as a continuum (i.e., the size of the fluid particles relative to the size of a typical cavity in the void region is inconsequential) and are therefore restricted to macroporous media or to heterogeneous composites. As may be clearly inferred from the results discussed earlier for idealized microcapillaries, in microporous media or in the amorphous regions of dense polymer films the molecular properties of the components in the system

40

MAcELROY

must be taken into consideration. Another limitation of these models and also of recent variational formulations of gas diffusion in random media (Ho and Strieder, 1980) is their inability to predict the properties of the permeating fluid at or near a percolation transition. As we will see below, there is reason to believe that the temporal evolution of the microscopic properties of severely hindered diffusing species has a direct bearing on case II diffusion in polymers. In a number of recent articles, both Monte Carlo (MC) (Nakano and Evans, 1983; Abassi et aI., 1983) and molecular dynamics (MD) (Park and MacElroy, 1989; MacElroy and Raghavan, 1990, 1991; Raghavan and MacElroy, 1991; MacElroy and Tomlin, 1992) simulation results have been reported for diffusion of gases in random media. The random media modeled in these studies were composed of assemblies of solid spheres randomly distributed in space, and the gas phase was usually considered to be nonadsorbing, although MD studies of adsorbing Lennard-lones vapors were also reported by MacElroy and Raghavan (1990, 1991). In the following only the results obtained via molecular dynamics are discussed, both for reasons of brevity and in view of the real-time analysis involved in such computations, and the reader is referred to the work of Nakano and Evans (1983) and Abassi et al. (1983) for details on the MC technique. In the studies reported by Park and MacElroy (1989), the four model solidsphere assemblies illustrated in Fig. 11 were investigated, and the pore fluid simulated was a single nonadsorbing diffusing hard-sphere particle. These simulations therefore correspond to Knudsen diffusion, and to further simplify the computations the solid spheres in each of the models shown in Fig. 11 were chosen to be of uniform radius cr•. Extensive computations over a wide range of porosities and for ensembles containing very long trajectories were conducted in these studies in order to clearly determine the long-time behavior of the VCF appearing in the free-particle diffusion coefficient DfM

== DfK

= lim! 1 -'"

3

r

Jo(V(T) . v(O»

dT

(71)

where v(t) is the velocity of the diffusing particle at time t. By definition the (stationary-state) diffusion coefficient is independent of time; however, it is convenient in a number of specific cases to consider a time-dependent diffusion coefficient defined by the integral to the right of the limit in Eq. (71), and we return to this below. The MD method employed by Park and MacElroy (1989) involved a straightforward application of Eq. (48a) to predict the time to collision for the diffusing particle and the immobile solid spheres illustrated in Fig. 11. For simplicity the postcollisional velocities were evaluated using Eq. (48b) (with the mass of the solid spheres, m., set equal to infinity) instead of the more realistic cosine law reflection condition represented by Eq. (53). It should be emphasized here, however, that this does not have a significant influence on the outcome of the sim-

DIFFUSION IN HOMOGENEOUS MEDIA

(0)

(e)

41

(b)

(d)

Figure 11

Schematic representations of the overlapping and nonoverlapping spheres models. The solid phase is indicated by shaded regions, and the full circle corresponds to a fluid particle. [From Park and MacElroy (1989), with permission.]

ulations, and indeed only a numerical factor is involved in the final evaluation of the diffusion coefficient (Mason and Chapman, 1962). The general form of the VCF and in particular its long-time behavior as determined by storing the particle velocity as a function of time is unaffected by the collision dynamics. Early work by Alley (1979) on a two-dimensional overlapping disk analog of Fig. lla demonstrated this, and this was confirmed recently for the threedimensional systems of interest here (MacElroy, 1992, unpublished). Selected results for the diffusion coefficient DfM of the low-pressure non adsorbing gas diffusing within the overlapping spheres models are shown in Fig. 12 (for the random overlapping spheres medium illustrated in Fig. lla) and in Fig. 13 (the data represented by the full circles in this figure correspond to freeparticle diffusion within the connected overlapping system illustrated in Fig. llb). Results for the nonoverlapping systems may be found in Park and MacElroy (1989). The Boltzmann diffusion coefficient cited in Fig. 12 corresponds to the low-density limit for the solid (I.e., n: = ns
42

MACELROY

1.00

0.50

'"

0 .25

0 .025

0 .05

0.10

1.0

~,

" , \ ' ,,

0 .8

'\''\

'\ '\

,

0 .6 DIM

I

....

,

'\2

DB

\

0.4

\

\

0 .2

0

0

0.2

T"'0

•

\

\

\

\

\

0.6

0.4

\ 0 .8

1.0

n.* Figure 12 The free-particle diffusion coefficient reduced by its Boltzmann value (left ordinate axis) and the open porosity relative to IjJ (right ordinate axis) as a function of the reduced density (or porosity) for the random overlapping system (see Fig. 11a). (e) MD simulation; 1, kinetic theory for moderately dense gases (van Leeuwen and Weijland (1967) and Weijland and van Leeuwen (1968)); 2, memory function/mode coupling theory (Gotze et a1. (1981a,b)); 3, repeated ring kinetic theory (Masters and Keyes (1982)). [Reproduced from Park and MacElroy (1989), with permission.]

given by (72) where (J" is the diameter of the diffusing hard-sphere particle. Furthermore, since the mean pore radius and the void space accessible to this particle in the random accordoverlapping spheres medium are also determined by both (J" and ing to

n:

(73)

43

DIFFUSION IN HOMOGENEOUS MEDIA

A (RANDOM) 0.2

0.4

I

I

I

0

0.2

0 1.000 0 .800 0.600 0.400

~

,0

1.0

A (CONNECTED) 0 .8

0.6

1.2

I

I

1

1.0

'It - 0 .644 \

\

\

, \

\

0

\ \

\

0

DIM

I

,,

0 .200

(0)

0 .8

i

0.4

'0

DIM

0 .6

0 .100 0 .080

\

\

\ \ \

0 .060

\

0

0 .040

\

\

\

\

0.020

0

\

\

\ (, I

0 .010 0 .008

\ I

0.006 0 .004 0

I 0 I

0.4

1.2

0 .8

1.6

2.0

012 The diffusion coefficient reduced by its value for rr = 0 as a function of fluid particle size (lower abscissa) or particle reduced size A = rr/2Rp (upper abscissa) in overlapping systems with I\IT = 0.644. (0) Random overlapping system (Fig. 11a); (e ) connected overlapping system (Fig. 11b). [Reproduced from Park and MacElroy (1989), with permission.]

Figure 13

and (74a) (74b)

44

MAcELROY

where S is the surface area per unit volume of the medium and IjIT is the total porosity available to a point particle, then Eq. (72) may also be written as

2

D B

=-\iR 3

p

(1

+

1 a/2as )2

(75)

The simple relationship between this expression and the point particle Knudsen diffusion coefficient given in Eq. (40) is quite apparent. Recall, however, that Eq. (75) corresponds to specular collision dynamics, and to obtain the expression for cosine law elastic scattering the right-hand side of Eq. (75) is simply divided by 1 + (4/9)[ (Mason and Chapman, 1962), where [ is the fraction of gas particles undergoing diffuse scattering during collision with the solid spheres. This modification may be used in conjunction with Fig. 12 to obtain estimates of the free-molecule diffusion coefficient for nonadsorbing gases in amorphous porous materials including rigid glassy polymers. The upper abscissa in Fig. 12 corresponds to Eq. (74), and therefore for a gas particle of a specific molecular size and for a membrane of a known total porosity 1jI1' it is possible to immediately read off the ratio D IM/DB and hence obtain the "tortuosity " -corrected diffusion coefficient D IM' Note that the lower abscissa in Fig. 12 is to be used only when the diffusing particle has zero volume, i.e., a = O. An example of the application of the procedure described above is illustrated in Fig. 13 for a medium with a comparatively high total porosity 1jI1' = 0.644. Also shown in this figure are simulation results for the more realistic model depicted in Fig. llb, which takes into consideration the effect of solid particle connectedness in real media [it should be noted that for IjIT < 0.5 the random and connected models in Figs. lla and llb are equivalent because the random overlapping spheres medium satisfies the criterion of complete solid connectivity under these conditions [Shante and Kirkpatrick, 1971; Kirkpatrick, 1973; Haan and Zwanzig, 1977)]. At the porosity IjIT = 0.644 the differences between the two systems are significant only when the size of the diffusing gas particle is greater than one-third the average pore size (the point particle coefficients D~ differ by only 10%). For small particle sizes the dependence of DIM on a is approximately exponential, in agreement with experimental observations for gas diffusion in a variety of polymeric materials [see Pace and Datyner (1979a,b,c)]. However, for large gas particles (gas particles that are commensurate in size with typical monomer units in polymer macromolecules), the diffusion coefficient is extremely small and is predicted to go to zero at a specific value of a. This transition is generally referred to as the percolation threshold, and for the three-dimensional models shown in Fig. 11 its position depends not only on the size of the diffusing particle but also on the porosity (or bulk density as illustrated in Fig. 12) of the rigid medium. The word rigid is italicized here to emphasize the fact that in dense polymeric media the effects of a percolation

DIFFUSION IN HOMOGENEOUS MEDIA

45

transition will not be observed above the glass transition. Under these conditions the intrinsic mobility of the polymer chains will preclude the existence of percolation phenomena and the diffusion coefficients for comparatively large nonadsorbing penetrant molecules will be significantly greater than the predicted values for the rigid systems shown in Figs. 12 and 13. However, at or below the glass transition the characterization of the diffusive properties of polymer/ penetrant systems will, at least in part, be assisted by studies of diffusion in rigid media. Of particular interest in this respect is the temporal behavior of the VCF, and it is this to which we now turn. The VCFs for a number of the diffusion coefficients reported in Fig. 12 are provided in Fig. 14. These correlation functions are plotted here in the normalized form defined by

C(t) = (v(t) . v(O)/(V2(O) One of the most important points to note concerning the long-time behavior of the VCFs is the increasing depth of the negative tail and its persistence to very long times as the porosity of the medium decreases and/or as the size of the diffusing particle increases. The minimum in the VCF results from particle backscattering after an initial positively correlated forward momentum, and the negative correlation persists to long times due to a greater than average probability for the particle to return to the origin of its trajectory. This is particularly true at low porosities and/or for large diffusing particle sizes where partial or even complete caging of the particle amplifies this effect. When it is further noted that the diffusion coefficient is obtained from the integral of the VCF over time, the relative importance of the long-time tails shown in Fig. 14 becomes abundantly clear. The simulation results for these tails also confirm the power law time dependence predicted by approximate kinetic or mode-coupling theories (Gotze et aI., 1981a,b; Masters and Keyes, 1982; Ernst et aI., 1984; Machta et aI., 1984) for diffusion in random media, i.e., . a hm (v(t) . v(O) = - 13 I -~ t

(76)

where a and 13 are density-dependent parameters. Inserting this expression into Eq. (71) gives D

t -D fM() - fM

+ 3(13 a-

1)

(1) -t~- l

(77)

in which the time t is significantly larger than the time required for the VCF to pass through its minimum (for the results shown in Fig. 14, the power law decay of the VCF is seen to characterize the long-time behavior for times greater than approximately 15 average collision times). It has been assumed in this integra-

MAcELROY

46 (0)

.--

-

*

>jI-0.500

u

0 .135 0 .035 -0.2 -0.4 0

2

8

6

4

10

t*

10- 3 8

(b)

6

4

2

,

;.!; U I

10- 4

,,

8 6

"

4

2

>jI =0 .500

12

14

16

18 20 2224262830

t* Figure 14 Normalized velocity autocorrelation function for the random overlapping system. (a) Short-time behavior for several porosities. (b) The long-time tail for 1\1 = 0.5. (e) MD simulation; (--) fit to a t* - 2.S power law decay [the reduced time t* is in units of the mean free time Tc = 3Duly2, where Do is given by Eq. (75)]. ( - - - ) Modecoupling theory Ernst et a!. (1984) and Machta et a!. (1984).

47

DIFFUSION IN HOMOGENEOUS MEDIA

(c)

4

2

*'u- Ie? 8

I

6 4

IjI =0.367

2

10

4 10

10

12

14

16

18

20 22 24 26 28 30

t*

2

8

(d)

6 4

...... *

-

U

2

I

I 1(53

I/!

=0.035

8 6 4

10

14

18

22 26 30

36 42 48 56 6472

t* Figure 14

Continued (c) As in (b), but for IjJ = 0.367. (d) The long-time tail for IjJ = 0.035. (e) MD simulation. ( - ) least squares fit to a (t* r ~ power law decay (13 = 1.57). [Reproduced from Park and MacElroy (1989), with permission.]

48

MACELROY

tion that ~ i= 1, and this is indeed the case for the three-dimensional media under consideration here. [A novel perspective of the influence of the power law exponent on the long-time tail of the VeF on diffusion in random media is considered by Muralidhar et al. (1990)]. For high-porosity media and small gas particles, the power law exponent ~ is 2.5 (this result is generally true at low densities for a class of random media known as Lorentz gases [see Ernst et al. (1984) and Machta et al. (1984)], which includes each of the models shown in Fig. 11), while for conditions at or near the percolation threshold the value of ~ is 1.57 according to the simulation results given in Fig. 14d. Although the latter result does not have the same generality as the high-porosity value of 2.5, it can be related to a set of "universal" constants (critical exponents) via scaling theory. In the vicinity of the percolation threshold the time-dependent diffusion coefficient predicted by scaling theory at long times takes the general form (Havlin and Ben-Avraham, 1987) D",(I) =

~ 4>[ (tc - 1) I""]

(78)

where 8 = j-l/(2v

+

j-l -

(79)

-y)

and l\Jc is the critical voidage below which the penetrant is localized (l\Jc = 0.035 for the random overlapping system as shown in Fig. 12). The parameters j-l, v, and -yare the critical exponents appearing in the expressions DfM =

~ =

(l\J - l\J')1'1l\J -l\JJ-"

(80) (81)

and (82) where ~ is the correlation length and P(l\J) is the percolation probability, which, in the present case, is equivalent to the relative magnitude of the percolating void fraction, l\Jo, to the void fraction l\J (results for this quantity are provided in Fig. 12 for the random overlapping spheres model). The function (x) in Eq. (78) has the following properties: (x) = xl'= (-

xr

2

l'+'Y

= constant

+ 00)

(83a)

00)

(83b)

(x

-+

(x

-+ -

(x

-+

0)

(83c)

The limiting condition in Eq. (83a) applies above the percolation threshold and leads to Eq. (80) on substitution into Eq. (78). Below the percolation threshold, Eq. (83b) provides DfM(t) = 1/t, while at the transition [Eq. (83c)], Eq. (78)

49

DIFFUSION IN HOMOGENEOUS MEDIA

simplifies to Eq. (77) with the stationary-state diffusion coefficient DfM = 0 and ~ - 1 = 8. At high porosities and/or for small diffusing particles, the comparatively large value of ~ results in a rapid disappearance of the long-time tail on the VCF, and hence this tail does not seriously interfere with the assumed validity of the stationary form of the Fickian diffusion flux. This is not true, however, when the diffusing particle is severely hindered in its motion and its time-dependent diffusion coefficient is described by the scaling form in Eq. (78). Under these conditions the power law exponent on the VCF is generally found to be much smaller than 2.5, and the very slow decay of the VCF raises serious questions concerning the applicability of simple constitutive forms of the type provided in Eq. (38) [or indeed the more general case of Eqs. (31) and (32)]. The conclusion to be drawn from these observations is that for conditions that give rise to anomalous diffusion in homogeneous media (as this non-Fickian behavior is generally called) it is necessary to generalize Eq. (31) to a time-dependent convolution form as implied by the zero wavevector limit of Eq. (11), i.e.,

J, = -

i Inr J=l

J ['Pit - T) - n 'P,m(t - T)] nm

~\IJiT) dT

(i = 1, ... , v)

(84a) which simplifies to V

J, = -

L J~) 'Pit -

(i

T)VTl-LlT) dT

= 1, ... , v)

(84b)

J=l

for a rigid random medium. The 'PIJ(t) are the VCFs defined in Eq. (14b). It is noteworthy that modified flux equations similar to Eq. (84) have been proposed for rigid media (Lorentz gases) of the type shown in Fig. 11 (Alley, 1979) and for glassy polymers [see Neogi (1983a,b) and Chapter 5 of this text]. The analysis for Lorentz gases described by Alley (1979) is of interest from the molecular point of view in that it employs the concept of a waiting time distribution (Montroll and Weiss, 1965) in a continuous-time random walk theory for the free-particle diffusion process [see also Klafter et al. (1986)]. At each step in the random walk the diffusing particle is assumed to be subject to a waiting time that results from partial entrapment in holes and dead-end pores within the medium, and this mechanism is not unrelated to the view that penetrant diffusion in glassy polymers is governed primarily by a sequence of activated jumps in which slow segmental motion and relaxation of the polymer chains can retard the jump frequency. The modified form of Fick's second law derived by Alley (1979) is, in one dimension, -anf =

at

i' 0

a4nf(T) 'P(t - T) [a:!nf(T) -+ D4 - + . .. ] dT 4 2

az

D

az

(85)

50

MA CELROY

where 'f'(t) is the single-particle VCF (v(z)(t) v(z)(O» , which is related to the waiting time distribution function m(t) via the Laplace transform

(p(s)

= (f)

sw(s) 2 1 - m(s)

(86)

where (f) is mean square jump distance. The coefficient D is equal to D fM , and D4 and higher order terms are known as the Burnett coefficients. In the zero wavenumber limit, the Burnett terms may be neglected. Equation (85) was verified by Alley (1979) via molecular dynamics simulation for a two-dimensional random overlapping hard disk Lorentz gas, and recently (MacElroy and Tomlin, 1992; MacElroy, 1992, unpublished), its validity was further confirmed for the three-dimensional analog of this system (Fig. 11a). The time-dependent behavior suggested by Eq. (77) was also observed in molecular dynamics studies of adsorbing Lennard-Jones fluids [Eq. (46)] confined within the micropores of a model silica medium (MacElroy and Raghavan, 1990, 1991), and one example of this is provided in Fig. 15 for the tracer coefficient D I.(t),

(87)

The results shown in Fig. 15 are for a liquid-filled void space in a model nonoverlapping connected spheres medium similar to that shown in Fig. 11d. In this case each of the solid (silica) spheres is atomistically modeled as illustrated in Fig. 16 [for details see MacElroy and Raghavan (1990)], with the fluid particles interacting with the individual atoms of the solid via a Lennard-Jones (126) potential. The relative size, 1.., of the fluid particles and the pores within the medium in these studies is approximately 0.2, and therefore anomalous effects associated with the percolation threshold are not evident here. There is, however, a significant long-time tail on the VCF as illustrated in the inset of Fig. 15. Finally, to return to a point made earlier with regard to the relative importance of the cross-kinetic coefficients L;j(i :F j), results also provided by MacElroy and Raghavan (1991) for sorbed vapor diffusion over the entire range of pore-filling conditions are reproduced in Fig. 17. The filled circles in this figure correspond to the tracer diffusion coefficient

(88)

51

DIFFUSION IN HOMOGENEOUS MEDIA I" (1-

0.10

o

0 .1

1'» 0 .5

0.4

0 .3

0 .2

0.6

0 .095

0.09

10-2

•• D 1*(t·)

8

I

6

0.085

4

- C 1• 1·(t*) 2 0.08

10-3 8

6

0.075 0

0.1 I*(HI)

0.2

2

;5

4

5

6

7 8 9 10

,*

Figure 15 Time-dependent tracer diffusion coefficient for a model porous silica medium saturated with a Lennard-Jones liquid. (e) Results obtained from integration of the VCF; (0) results obtained from the long-time slope of the mean-square displacement [Eq. (14c) at finite times]. The inset shows a power law fit of the normalized VCF for the tracer, and the solid lines in both the main figure and the inset are for 13 = 1.8 :t 0.1 t· is in units of l SiO V m,/Ero, and D,.(t*) is in units of l SiO V Ero/mr, where mr is the mass of the Lennard-Jones particles, Ero is the potential well depth for interactions between the fluid particles and the oxygen atoms of the silica medium, and l SiO is the SiO bond length in silica (0.162 nm). [Reproduced from MacElroy and Raghavan (1991), with permission.]

52

MACELROY

Figure 16 A simulated silica microsphere. Each of the solid spheres shown in Fig. lld is modeled as one of these. The open circles are nonbridging surface oxygens, and the shaded region illustrates the exposed interior bridging oxygens. [Reproduced from MacElroy and Raghavan (1991), with permission.]

and the open circles correspond to the results for the total sorbate diffusion coefficient DfM =

l!.- (oo (ur(t) 3

Jo

. Ue(O) dt

(89)

The inset in Fig. 17 also provides estimates of the relative magnitude of the cross-coupling effects in terms of the ratio (DfM - D j. )/DfM . The numerator in this ratio is simply DfM - D ••

Jo \ i

= ~ ('" / 3N

;=. i'l-J

viet) .

i

Vj(O)) dt

j =j j#

e

Note that D •• and DfM are equal only in the limit -+ 0 and that at any nonzero concentration cross-effects are always present. The results plotted in the inset of Fig. 17 demonstrate that these cross-effects can contribute as much as 70%

53

DIFFUSION IN HOMOGENEOUS MEDIA

e o

0.1

02

0 .3

0.4

0 .5

0.6

0.7

0 .8

0.9

1.0

0.4

0.2

0.1 0 .08 0 .06

0 .04

0 .02

.--_--.-_ _. - _ - ,_ _--,-_-; 1.0

0 .8 0 .01 0.008

F4'IY~---

0 .6

e

~.

-

~

0.006 0 .4 0 .004

e

S

02

0.002 L_l-_l-_...I-_JJL-_l.-_....I-_---''--_....I-_--I 0 o 0.1 0.2 0.3 0 0.2 0.4 0 .6 0 .8 1.0

E>

Figure 17

Reduced diffusion coefficients of the adsorbing vapor as a function of fractional pore loading. (e) Tracer diffusion coefficient; (0) total adsorbate diffusion coefficient; (--) Enskog theory MacEtroy and Kelly (1985). The diffusion coefficients are in units of 15 ;0 V Ero/mr. The inset illustrates the relative importance of cross-effects in the micropore fluid. [Reproduced from MacElroy and Raghavan (1991), with permission .]

54

MAcELROY

to the overall diffusion coefficient at saturation for the system considered by MacElroy and Raghavan (1991).

c.

Diffusion in Amorphous Polymers

If the microcavities in an amorphous polymeric medium are significantly larger

than the penetrant molecules, then the assumption of solid-phase rigidity should not be an unreasonable approximation. However, for large solute species and/ or high polymer densities, the diffusion mechanism is widely considered to be governed by the formation of holes via local motion of the macromolecular chains, and it is clear that under these conditions the dynamics of the chain segments should be included in the overall model. The flux equations now include the kinetic coefficients Lim, and the diffusion velocity of the polymer, Jm, is not zero (swelling or shrinkage may be observed) unless external constraints are imposed on the system. For a fluid/solid system that is not subject to external forces, conservation of total linear momentum within the medium at any given instant requires (90) where mm and vkm(t) may be considered to be the mass and center-of-mass velocity, respectively, of a polymer chain, a monomer unit, or the individual atoms of the macromolecule [in which case an additional sum over atomic species is implied on the left-hand side of Eq. (90)]. The kinetic coefficients defined in Eq. (14) are then interrelated by (91) and Eq. (31) may be rewritten as (92) Modeling of single-component diffusion has been of primary concern in the literature, and for this reason the discussion below is restricted to such systems. In this case Eq. (92) simplifies to (93)

with DfM again given by Eq. (89). This coefficient quantifies the collective diffusion of the penetrant within the medium, and as noted earlier it is collective properties of this type that need to be addressed in the engineering design of

DIFFUSION IN HOMOGENEOUS MEDIA

55

sorption or membrane separation processes. Another diffusion coefficient that may be defined for single penetrant systems is the tracer or self-diffusion coefficient, and it is this parameter that is most frequently reported in MD simulation studies (and measured experimentally via NMR) of diffusion in polymers. Using Eq. (92), it is readily shown that the tracer diffusion flux is given by the vector form of Eq. (66), Le., (94) with the tracer diffusion coefficient, D 1. , given by Eq. (88). One of the major difficulties involved in the simulation of polymers is the availability (or lack of) a reliable model for the potential interactions arising not only between the penetrant and the polymer macromolecules but also, and most important, the intramolecular interactions within the polymer chains themselves. To date essentially all of the molecular dynamics simulation studies conducted on diffusion in polymers have modeled the polymer chains as alkane structures [polyethylene (Trohalaki et aI. , 1989, 1991; Sonnenburg et aI. , 1990; Takeuchi and Okazaki, 1990; Takeuchi, 1990a,b; Takeuchi et aI., 1990), polypropylene (Muller-Plathe, 1992), and polyisobutylene (Muller-Plathe et aI. , 1992)] with the exception of the recent work of Sok et aL (1992). The simplifying features of the alkane system lie in the absence of polar or electrostatic interactions and the observation that the interactions between nonbonded atoms (-C- or -H) or sites (-CH3 or -CH2-) on the chains are adequately represented by the (shortrange) Lennard-Jones potential [Eq. (46)]. The potential functions associated with intrachain vibrations and rotations are also frequently described by the comparatively simple forms (95a)

ke 2 Vo = - (cos e - cos eo) 2

or

(95b)

and 5

Vol>

= kol>

L

a"

cos" (
(95 c)

n=O

where b is the bond length between two neighboring atoms or sites (with an equilibrium value of bo), e is the angle between two adjacent bonds (with its equilibrium value equal to eo), and is the dihedral angle defined by three successive bond vectors in the polymer chain. [A concise description of these potential functions and the equations for the forces on a given atom or site arising from these interactions may be found in Appendix C of the text by Allen and Tildesley (1987).] Other mathematical expressions reproducing the same

56

MAcELROY

general functional behavior have been employed for the torsional potential V", primarily as a matter of computational convenience. Molecular dynamics simulations of polymer/penetrant systems, in which the penetrants have also usually been modeled as Lennard-Jones fluids [Eq. (46)] , have been conducted most frequently in one of two possible ways: (1) The trajectories of each atom and/or site in the system are computed using Newton's equations as written in Eqs. (47a) and (47b) for unconstrained motion with the forces acting on particle or site i determined by the pairwise Lennard-Jones interaction potentials and the potential functions (or equivalent forms) provided in Eqs. (95), or (2) the lengths of the bonds within the polymer chains are fixed [thus eliminating Eq. (95a) and reducing the number of degrees of freedom by the number of bonds constrained] by incorporating constraint forces in Newton's equations of motion that act along the bonds. Conceptually and computationally the unconstrained formulation is the easier of the two approaches, although difficulties arise owing to the much higher frequencies usually involved in the bond vibrations in contrast to the comparatively low frequency of the torsional rotations. To overcome this, realistic estimates of the spring constant kb are reduced by a factor of 7 (Rigby and Roe, 1987) and the equations of motion are solved using finite-difference methods of the kind given in Eq. (50) with comparatively large time steps. The relative importance of the assumption involved in the lower kb value for diffusion of penetrants in polymers has not been quantitatively assessed as yet, and this has raised questions concerning the reliability of the activation energies for diffusion determined via unconstrained molecular dynamics (Trohalaki et aI. , 1991). The second method, involving constraint dynamics, makes the not unreasonable assumption that, in view of the very high frequency of the bond vibrations, the bond lengths over moderate time scales are effectively constant (in a number of versions of this method the bond angles, 6, are also fixed). The method was first proposed in the 1970s (Ryckaert et aI., 1977), and improvements and variants on the original technique were summarized by Allen and Tildesley (1987). In effect, at the beginning of a given time step the atoms or sites of the polymer chain are assumed to move along free, unconstrained trajectories subject only to intermolecular forces [derived from Eq. (46)] and intramolecular forces [derived from Eqs. (95b) and (95c)]. At the end of the time step, the system is "shaken" by imposing the constraint forces in an iterative manner, and the instantaneous configurational and dynamical phase coordinates of the chains are obtained to within a given tolerance. This algorithm (the original form of which is called SHAKE) or modifications of it have been successfully used in the simulation of both small molecular structures and complex macromolecules. Although molecular simulations have been undertaken over the last 15 years or so for a variety of polymer systems [see Roe (1991) and references cited therein], it is only very recently that the first MD simulation results for diffusion

DIFFUSION IN HOMOGENEOUS MEDIA

57

in polymers have been reported (Trohalaki et aI., 1989, 1991; Sonnenburg et aI., 1990; Takeuchi and Okazaki, 1990; Takeuchi, 1990a,b; Takeuchi et aI., 1990; Muller-Plathe, 1992; Muller-Plathe et aI., 1992; Sok et aI. , 1992). In the first few of these studies (Trohalaki et aI., 1989, 1991; Sonnenburg et aI., 1990; Takeuchi and Okazaki, 1990; Takeuchi, 1990a,b; Takeuchi et aI. , 1990), the unconstrained MD technique described above (method 1) was employed, and the polymer investigated was amorphous polyethylene. The simulations reported by Trohalaki et al. (1989, 1991), Sonnenburg et al. (1990), Takeuchi and Okazaki (1990), Takeuchi (1990b), and Takeuchi et al. (1990) were carried out at conditions well above the glass transition temperature, Tg , and it was found that even for a system of " polyethylene" chains containing as few as 20 methylene units, the computed self-diffusion coefficients for a number of rare gases, Oz, and CO z were in fair agreement with experiment. Furthermore, it was shown that the degree of chain flexibility and, in particular, intramolecular torsional rotations was a controlling factor in the rate of penetrant diffusion and the applicability of free-volume theory (Cohen and Turnbull, 1959) and molecular models (Pace and Datyner, 1979a,b,c) to both diffusion and polymer relaxation processes was confirmed. Two systematic effects were observed, however: (1) The diffusion coefficients determined via simulation were approximately three to four times larger than those observed experimentally and (2) in a number of cases the activation energies were a factor of 2 lower than the experimental values. In one study (Troha\aki et aI., 1991) the latter effect was associated with density differences between the simulated chain system and the experimental density of amorphous polyethylene; however, in the work reported by Takeuchi and Okazaki (1990), densities consistent with experimental values were employed, and it was shown that the source of error in the estimation of the activation energies lay with the simulation method itself in that constant-pressure simulations (the NpE ensemble) rather than constant-volume conditions (the usual NVE ensemble) are required. In an effort to rationalize the systematically higher diffusivities obtained irrespective of the ensemble considered, Takeuchi (1990b) investigated the effects of the chain length of the " polyethylene" molecules and found that for chains of infinite length (in contrast to the Czo chains considered in prior studies) only a marginal improvement in the simulation results could be obtained. He suggested that the experimental diffusivities may be influenced by the degree of crystallinity of polyethylene, i.e., crystalline inclusions embedded in the amorphous region in a manner analogous to the models illustrated in Fig. 11 (with the solid spheres corresponding to the crystallites) could lead to a significant reduction in the self-diffusion coefficient of the penetrant. To clarify the possible effects of crystallinity as well as a number of other issues, Muller-Plathe (1992) conducted MD simulations for a number of penetrants (Hz, Oz, and CIL) diffusing in atactic polypropylene above the glass tran-

58

MAcELROY

sit ion temperature. This polymer is known to be 100% amorphous, and therefore any questions concerning crystallinity are avoided. Furthermore, the polymer model investigated by Muller-Plathe (1992) was semiatomistic (improving on the structureless site model for the methyl and methylene groups employed in the earlier polyethylene simulations), and the fixed bond length constraint dynamics method described above was employed (thus eliminating the problem associated with the weak spring constant kb employed in the unconstrained MD method). As additional safeguards against the introduction of extraneous errors, Muller-Plathe simulated a very long polymer chain at the appropriate bulk density of amorphous polypropylene and employed carefully equilibrated polymer configurations to initiate the trajectories [see also Theodorou and Suter (1985)]. Although no direct comparison could be made with experiment because of the absence of data, the self-diffusion coefficients obtained by Muller-Plathe for O2 and CIL were in good agreement with experimental results for a number of other polymeric materials. This was particularly true for CH4 , for which the simulated and experimental diffusivities were in very close agreement. However, for the H2/polypropylene system the simulation diffusion coefficient was an order of magnitude higher than all available experimental data, and although statistical sampling in the MD simulation work may be a contributing factor the reason for this disparity remains an open question. One possible reason for the higher than expected diffusion coefficients sometimes observed in MD simulations is inferred from the very recent results reported by Muller-Plathe et al. (1992) for oxygen diffusion in rubbery poly isobutylene (PIB). In this work it was clearly demonstrated that a united-atom representation of the methyl groups in PIB [(-CHz-C(CH3)2-)"] gave rise to diffusion rates that were almost two orders of magnitude larger than the corresponding rates in an all-atom model of the polymer. Although differences of this order are generally larger than those usually encountered in comparative studies of united-atom (smeared) and all-atom simulations, they are consistent with documented results for other systems [e.g., compare Figs. 7 and 8; see also MacElroy and Raghavan (1990) for comparative work on microporous silica]. Another significant outcome of additional simulations conducted by MullerPlathe et al. (1992) for helium tracer diffusion in PIB is illustrated in Fig. 18. In this figure the ensemble-averaged mean square displacement of the He atoms diffusing in an all-atom model of PIB is plotted as a function of time, and when it is noted that the time-dependent diffusion coefficient defined in Eq. (87) may also be expressed, using Eq. (14c), as DJ.(t)

= .! dd

6 t


(96)

where r l.(t) is the position of the tracer particle at time t, then it is clear from the short-time behavior shown in Fig. 18 that the helium particles are undergoing

59

DIFFUSION IN HOMOGENEOUS MEDIA 3.0

2.0 N

J\

1.0

-.:-

-

'-'

*

0.0

I ..

v

'-'

o

00

.3

-1 .0

-2.0 -1 .0

L-_~_---'-

__

0.0

~_--'-_~

__

1.0

.L.-_~_--'

2.0

__

3.0

~_

4.0

log,o(Vps)

Figure 18 Mean-square displacement of helium atoms in amorphous polyisobutylene obtained via MD with an all-atom force field for the polymer. [Reproduced from MullerPlathe et al. (1992), with permission.]

anomalous diffusion. It is also of interest to note that the exponent 13 in the anomalous regime [see Eqs. (76) and (77)] is found to be 1.50, which compares very favorably with the exponents reported by Park and MacElroy (1989) for diffusion in rigid random media [a similar observation was made by MullerPlathe et al. with reference to the stochastic MC simulations carried out by Gusev and Suter (1992) for helium diffusion in static polycarbonate structures]. At long times the results for the mean square displacement shown in Fig. 18 assume a linear (Fickian) relationship with time. In addition to the diffusion coefficients themselves, the MD simulation data provide a wealth of information on the microscopic properties of polymer/penetrant systems, and an interesting perspective on the diffusion mechanism is provided by the results reported by Muller-Plathe (1992) for the time-dependent behavior of the magnitude of the relative displacement of a given penetrant molecule from its initial position Ir(t) - r(O)I . His published results for the three penetrants H2 , O2 , and CH4 are reproduced in Fig. 19, and the jumps observed for methane (and to a lesser extent oxygen) confirm the hopping mechanism postulated for diffusion in rubbery polymers. The small size of the hydrogen

60

MACELROY 8 .0

E C

Hl 6.0

'-'

~

'-'

s..

4.0

'Z' '-' :..

2.0

0.0 -==_ _ 0.0

~_---.J'--

500.0

_ _ _ _--'_ _ _ _ _--'_ _ _ _ _--'

1000.0

1500.0

2000.0

t (PS) Figure 19 Relative displacements of representative molecules of H2 , O2 , and CH. diffusing in atactic polystyrene. For clarity, the ordinate axes for O2 and H 2 are offset by 1 nm and 2 nm, respectively. [Reproduced from Muller-Plathe (1992), with permission.]

molecule results in a fluidlike behavior with a comparatively short time involved between jumps from one microcavity to neighboring holes. Similar observations were reported by Sok et al. (1992) for He (fluidlike tracer diffusion) and CH. (lattice like tracer diffusion) in rubbery polydimethylsiloxane (PDMS), and it is also worth noting that in the case of both penetrants very good agreement was found between the simulation and experimental results for the tracer diffusion coefficients. Each of the above-cited studies has addressed self-diffusion in polymer systems above the glass transition temperature, and it is believed that comparable MD studies below Tg are not possible at the present time. In only one case has an attempt been made to elucidate the transport mechanism below the glass transition temperature (Takeuchi, 1990a) where it was shown that although rigid cage structures are a predominant feature of polymeric materials below Tg it is possible that the (slow) thermal motion of local chain segments can induce changes in the shape of the microcavities in the system without changing the free volume. This is illustrated in Fig. 20, where the sequence of events taking place during the jump of an oxygen molecule in a model polyethlene medium is clearly demonstrated (note that the time origin indicated in this figure is not

61

DIFFUSION IN HOMOGENEOUS MEDIA

~

(a)

(d)

1=12.1 ps

1=6.0 ps

sA

1=13.3 ps

1=16.1 ps

(e)

(b)

J----------i

(g)~

sA

J----------i

1=12.5 ps

1=10.1 ps q

(Q

(c)

1=12.9·ps

1=14.1 ps

Figure 20 The temporal evolution of thc contours of the potential energy surface for an oxygen molecule as it jumps from one microcavity to a neighboring one in glassy polyethylene. [Reproduced from Takeuchi (1990a), with permission.]

62

MACELROY

the origin of the trajectory itself; the true origin was 70 ps earlier). Under favorable conditions the shapes of neighboring cavities are distorted sufficiently to form an interconnecting channel through which the oxygen molecule can easily diffuse. The jump actually occurs between steps (d) and (g), and it is important to observe that the potential energy of the oxygen molecule is unchanged as it moves from one cavity to the next, i.e., the oxygen molecule itself is not subject to any significant energy barrier. Unfortunately, a preliminary analysis of the conformational dynamics of the local chain motion during the jump failed to reveal the underlying cause for the formation of the channel, and further work is needed to clarify the mechanism involved.

IV.

CONCLUDING COMMENTS

It is now possible to formally express the diffusion equations for fluids in homogeneous media in terms of the microscopic properties of the fluid/solid system. Difficulties arise, however, when one attempts to extract mathematically tractable results for use in engineering design. Approximate theories based on free-volume and molecular concepts are currently employed for this purpose although it is recognized that future developments in this area will rely heavily on an improved understanding of the molecular phenomena involved. In this chapter one technique, molecular dynamics simulation, which has contributed significantly to our understanding of liquid and solid behavior over the last four decades, has been described in some detail, and a number of applications for simple fluids confined within idealized micropores, random media, and polymers have been discussed. The MD method can serve as an alternative and complementary means for investigating diffusional behavior in fluid/solid systems, and in view of its intrinsic microscopic character it can provide valuable information concerning molecular properties that cannot be readily measured experimentally. Although current computational facilities would appear to limit the application of the MD technique to diffusion in rubbery polymers, in the next chapter of this book complementary methods employing both MD and stochastic simulation techniques that may extend the scope of computational nonequilibrium statistical mechanics to the glassy state are discussed. One of the most challenging aspects of transport in polymers that still needs to be addressed is the molecular mechanism(s) that give rise to anomalous and case II diffusion below the glass transition temperature. MD simulation of diffusion in homogeneous random media at or near a percolation threshold demonstrates the importance of low-frequency molecular relaxation phenomena that can influence the constitutive form of the flux equations over macroscopic time scales. This may have a direct bearing on the non-Fickian character of the flux equations in glassy polymers.

DIFFUSION IN HOMOGENEOUS MEDIA

63

REFERENCES Abassi, M. H., J. W. Evans, and I. S. Abramson (1983). AIChE 1.,29,617. Alder, B. J., and T. E. Wainwright (1970). Phys. Rev., AI, 18. Allen, M. P., and D. J. Tildesley (1987). Computer Simulation of Liquids, Clarendon Press, Oxford. Alley, W. E. (1979). Studies in molecular dynamics of the friction coefficient and the Lorentz gas, Ph.D. Thesis, Univ. California, Davis. Altenberger, A. R., J. S. Dahler, and M. Tirrell (1987) . 1. Chem. Phys., 86, 2909. Belter, P. A., E. L. Cussler, and W.-S. Hu (1988). Bioseparations-Dowllstream Processing for Biotechnology, Wiley, New York. Brown, P. R., and R. A. Hartwick, Eds. (1989). High Performance Liquid Chromatography, Wiley, New York. Chapman, S., and T. G. Cowling (1970). The Mathematical Theory of Non-Uniform Gases, 3rd ed., Cambridge Univ. Press, London. Chase, H. A. (1984a). Chem. Eng. Sci., 39, 1099. Chase, H. A. (1984b). 1. Chromatogr., 297,179. Chiew, Y. C., G. Stell, and E. D. Glandt (1985).1. Chem. Phys., 83, 76l. Cohen, M. H., and D. Turnbull (1959). 1. Chem. Phys., 31, 1164. Crank, 1., and G. S. Park, Eds. (1968). Diffusion ill Polymers, Academic, New York. Davis, H. T. (1987).1. Chem. Phys., 86, 1474. deGroot, S. R., and P. Mazur (1963). NOli-Equilibrium Thermodynamics, North-Holland, Amsterdam. Drioli, E., and M. Nakagaki, Eds. (1986). Membranes and Membrane Processes, Plenum, New York. Ernst, M. H., J. Machta, J. R. Dorfman, and H. van Beijeren (1984).1. Stat. Phys., 34, 477. Ferry, J. D. (1936).1. Gen. Physiol., 20, 95. Fischer, J., and M. Methfessel (1980). Phys. Rev., A22, 2836. Gotze, w., E. Leutheusser, and S. Yip (1981a). Phys. Rev. , A23, 2634. Gotze, w., E. Leutheusser, and S. Yip (1981b). Phys. Rev., A24, 1008. Green, M. S. (1952).1. Chem. Phys., 20, 128l. Green, M. S. (1954).1. Chem. Phys., 22, 398. Gusev, A., and U. W. Suter (1992). Polym. Prepr., ACS Division of Polymer Chemistry, Washington, DC, 33, 63l. Haan, S. w., and R. Zwanzig (1977). 1. Phys. A : Math. Gen., 10, 1547. Hanley, H. J. M., Ed. (1969). Transport Phenomena in Fluids, Marcel Dekker, New York. Havlin, S., and D. Ben-Avraham (1987). Adv. Phys., 36, 695. Heinbuch, U., and J. Fischer (1987). Mol. Simul., 1, 109. Hirschfelder, J. 0. , C. F. Curtiss, and R. B. Bird (1954). The Molecular Theory of Gases and Liquids, Wiley, New York. Ho, F. G., and W. Strieder (1980).1. Chem. Phys., 73, 6296. Juntgen, H., K. Knoblauch, and K. Harder (1981). Fuel, 60, 817. Kennard, E. H. (1938). Kinetic Theory of Gases, McGraw-Hill, New York. Kim, w.-T., A. R. Altenberger, and J. S. Dahler (1992) . 1. Chem. Phys. , 97, 8653. Kirkpatrick, S. (1973). Rev. Mod. Phys. , 45, 574.

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Prigogine, I. (1961). Introduction to the Thermodynamics of Irreversible Processes, 2nd ed., Wiley-Interscience, New York. Raghavan, K , and J. M. D. MacElroy (1991). Mol. Simul. , 8, 93. Reyes, S., and K F. Jensen (1985). Chem. Eng. Sci. , 40, 1723. Rigby, D., and R. J. Roe (1987). J. Chem. Phys., 87, 7285. Roe, R. J., Ed. (1991). Computer Simulation of Polymers, Prentice-Hall, Englewood Cliffs, NJ. Ruthven, D. M. (1984). Principles of Adsorption and Adsorption Processes, WileyInterscience, New York. Ryckaert, J. P. , G. Ciccotti, and H. J. C. Berendsen (1977). J. Compo Phys., 23, 327. Sahimi, M. (1988). Chem. Eng. Sci., 43, 298l. Satterfield, C. N. (1980). Heterogeneous Catalysis in Practice, McGraw-Hill, New York. Satterfield, C. N., C. K Colton, and W. H. Pitcher, Jr. (1973). AlChE J., 19, 628. Schoen, M. , J. H. Cushman, D . J. Diestler, and C. L. Rhykerd, Jr. (1988). J. Chem. Phys., 88, 1394. Shante, V. K S., and S. Kirkpatrick (1971). Adv. Phys., 20, 325. Sirkar, K K, and D. R. Lloyd, Eds. (1988). New Membrane Materials and Processes for Separation, AIChE Symp. Ser., No. 261, 84. Sok, R. M., H. J. C. Berendsen, and W. F. van Gunsteren (1992). J. Chem. Phys., 96, 4699. Sonnenburg, J., J. Gao, and J. H. Weiner (1990). Macromolecules, 23, 4653. Steele, W. A (1974). The Interaction of Gases with Solid Surfaces, Pergamon, Oxford. Stem, S. A, and H. L. Frisch (1981). Annu. Rev. Mater. Sci., 11, 523. Suh, S.-H. and J. M. D. MacElroy (1986). Mol. Phys., 58, 445 . Swope, W. C., H. C. Andersen, P. H. Berens, and K R. Wilson (1982). J. Chem. Phys., 76,637. Takeuchi, H. (1990a). J. Chem. Phys., 93, 2062. Takeuchi, H. (1990b). J. Chem. Phys., 93, 4490. Takeuchi, H., and K Okazaki (1990). J. Chem. Phys., 92, 5643. Takeuchi, H., R. J. Roe, and J. E. Mark (1990). J. Chem. Phys. , 93, 9042. Theodorou, D. N., and U. W. Suter (1985). Macromolecules, 18, 1467. Torquato, S. (1986). J. Stat. Phys., 45, 843. Trohalaki, S., D. Rigby, A Kloczkowski, J. E. Mark, and R. J. Roe (1989). Polym. Prepr., 30(2),23. Trohalaki, S., A Kloczkowski, J. E. Mark, D. Rigby, and R. J. Roe (1991). In Computer Simulation of Polymers, R. J. Roe, Ed., Prentice-Hall, Englewood Cliffs, NJ, Chap. 17. Thrbak, A F., Ed. (1981). Synthetic Membranes, Vols. I and II, American Chemical Society, Washington, DC. van Leeuwen, J. M. J., and A Weijland (1967). Physica, 36, 457. Vertenstein, M., and D. Ronis (1986). J. Chem. Phys., 85, 1628. Vertenstein, M., and D. Ronis (1987). J. Chem. Phys., 87, 5457. Vieth, W. R. (1991). Diffusion In and Through Polymers, Hanser, Munich. Weijland, A, and J. M. J. van Leeuwen (1968). Physica, 38, 35. Weissberg, H. L. (1963). J. Appl. Phys., 34, 2636.

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2 Molecular Simulations of Sorption and Diffusion in Amorphous Polymers Doros N. Theodorou University of Palras Palras, Greece

I. A.

INTRODUCTION Scope of Molecular Simulations

Many technologically important processes rely upon the design of polymers with tailored characteristics of permeability and selectivity toward fluid molecules. Examples include gas separation with polymeric membranes, food packaging using plastic films, and encapsulation of electronic components in polymers that act as barriers to atmospheric gases. To solve the polymer design problem successfully, one needs to relate the chemical constitution of the polymer (set during synthesis) and its morphology (set during processing) to the sorption isotherms and diffusivities of fluid molecules within it. There are several ways of establishing quantitative relations between polymer structure (chemical constitution and morphology) and sorption and permeation properties. One way is direct experimental measurement; a second way is the formulation and use of phenomenological correlations based on experimental evidence from a wide variety of systems. A third way to structure-property relations, which constitutes the main focus of this chapter, is the development and application of theories and simulation techniques that rely directly upon fundamental molecular science. Theories start from a more or less detailed model of the polymer/penetrant system and proceed to derive functional expressions for sorption isotherms or diffusivities through statistical mechanical analysis aided by the introduction of judicious approximations. Examples of theories are the lattice fluid theory of

67

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THEODOROU

gas solubility in polymer melts, the dual-mode sorption theory for polymer glasses, and activation energy and free-volume approaches to the diffusivity of pure or mixed gases in polymers. An excellent review of theoretical work on sorption and diffusion in polymers has been given by Petropoulos (1994). A theory is typically cast in closed form as a set of algebraic, integral, or differential equations into which the chemical ' ' personality" of the polymer and penetrant enter through a handful of parameters. To afford such a closed formulation, one typically has to introduce simplifying approximations in the mathematical treatment or invoke relatively crude models for the microscopic representation of the system at hand. Computer simulations, on the other hand, can be thought of as numerical solutions of the full statistical mechanics given a model for the molecular geometry and interaction energetics for all molecular species present. Simulations, or "computer experiments, " involve the generation of configurations representing a particular material system, from which structural, thermodynamic, and transport properties are estimated. In principle, simulations can provide " exact" results for a given model representation of the polymer/penetrant system. In practice, compute time considerations necessitate the introduction of approximations in simulation work as well, but typically these approximations are less severe than the ones invoked in theories. Molecular simulations can improve our ability to describe and predict sorption and diffusion phenomena in polymers in several ways. First, simulations can elucidate the molecular mechanisms underlying the macroscopic behavior of polymer/penetrant systems and thereby serve as a guide for the development of better theories. Direct comparisons between the results of simulations and theories based on exactly the same molecular representation are an excellent way to assess the legitimacy and implications of the approximations invoked in the theories. Second, simulations based on models that correctly capture the salient features of molecular geometry and energetics can help identify expected changes in performance with changes in chemical constitution. This is valuable for addressing " what if" questions of a materials design nature without going through costly trial-and-error laboratory synthesis and testing of properties. Third, simulations that rely on sufficiently detailed molecular models can actually be used for the quantitative prediction of solubility and diffusivity values directly from chemical constitution. Although the field of polymer molecular simulations is still at a stage of method development and validation against experimental evidence from existing systems, there is every expectation that it will develop into a powerful framework for the design of new materials. In this chapter we review recent molecular simulation work relating to the sorption and diffusion of gases in polymer melts and amorphous glassy polymers. As practically all published simulation work to date has been performed on relatively simple amorphous systems in which permeation is found macroscopically to proceed by a solution-Fickian diffusion mechanism, we focus on

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

69

such systems. We briefly discuss the statistical mechanical principles that can be used to construct simulation techniques for sorption and diffusion and summarize what has been learned from the application of such techniques to date. In the remainder of this introduction we discuss qualitatively some molecular aspects of sorption and diffusion in polymers. Simulation techniques for characterizing the internal structure and mobility of amorphous polymers are presented in Section ll. Section III deals with the prediction of sorption thermodynamics; the problem of calculating the sorption isotherm in a polymer is formulated in a general way, and existing applications are reviewed. The problem of predicting diffusivities is discussed in Section IV; both molecular dynamics simulations and transition state theory-based techniques for species that move slowly through the polymer matrix are presented. Finally, Section V summarizes some conclusions and future directions.

B.

Some Molecular Aspects of Sorption and Diffusion in Amorphous Polymers

The solubility of a penetrant in a polymer is expected to depend on the nature and magnitude of polymer-penetrant interactions in relation to polymer-polymer and penetrant-penetrant interactions, as well as on the distribution of shapes and sizes of open spaces formed among chains within the polymer, where penetrant moelcules can reside. Thus, a prerequisite for predicting the solubility is that one be able to describe the distribution of molecular configurations taken on by the penetrant/polymer system. For melt systems in thermodynamic equilibrium, this distribution of configurations is well established by equilibrium statistical mechanics. Consider, for example, a pure melt of N r polymer chains occupying a volume V. The potential energy of this system is a highly convoluted function "V p(rp) of the coordinates of all atoms constituting the chains, which we symbolize collectively as a vector rp . For macroscopic specimens, the dimensionality of the configuration space from which rl' takes values is on the order of Avogadro 's number. In the schematic of Fig. 1 we display the entire configuration space as a single axis, for graphical simplicity; the potential energy hypersurface "V p(rp) is thus plotted as a curve. When the temperature T is sufficiently high for the polymer to be an equilibrium melt, every point in configuration space is visited with a frequency proportional to the Boltzmann factor exp(-I3"V p), where 13 = l/kBT. When NA molecules of a penetrant are present in the polymer melt, the dimensionality of the configuration space is augmented and the potential energy hypersurface "V(rp, r A ) is modified. The different configurations (rr, r A ) , however, are again Boltzmann-distributed. From the point of view of simulation, the solubility prediction problem in a gas/melt system is thus conceptually analogous to the problem of predicting vapor-liquid equilibrium of low molecular weight mixtures, the only difference being the involatility

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THEODOROU

N. V Const.

rp

Figure 1 Schematic representation of the potential energy 'VI' as a function of the microscopic degrees of freedom rp for an amorphous polymer region of volume V containing N p macromolecules. In the glassy state, the polymer configuration is confined to fluctuate in the vicinity of local minima of the potential energy, as shown by the bold arrows.

of the polymer and the formidable computational challenges associated with the large-length-scale structure and long-time-scale motion of macromolecular systems. Glassy polymers, on the other hand, are not in thermodynamic equilibrium. It is conceptually useful to think of the configuration in a pure polymer glass as arrested, so that it can fluctuate within a small region of configuration space in the vicinity of a local minimum or of a relatively small set of local minima of the potential energy. Such regions are displayed with bold arrows in Fig. 1. Transitions of the configuration from one such region into another (constituting the phenomenon of structural relaxation) are severely inhibited by high energy barriers, which cannot be overcome over ordinary time scales at the prevailing temperature. [Volume relaxation times for physical aging 20°C below Tg are on the order of years (Eisenberg, 1984).] These regions of configuration space in which a glassy system may be trapped are thus effectively disjoint over ordinary experimental time scales. How likely it is to find the local configuration of a macroscopic polymer sample residing in each of these regions is no longer dictated by the Boltzmann distribution but rather is strongly influenced by the formation history of the glass (e.g., the cooling rate). The assumption of Boltzmann-distributed configurations locally within each region is legitimate (in fact, it defines the regions) but breaks down if extended across barriers. When a gaseous penetrant is sorbed in the glass at low concentration, this picture remains valid; an estimate of the solubility can be obtained

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

71

by formulating the phase equilibrium problem separately in each disjoint rp region of configuration space and then averaging the results according to the weight with which each region contributes to the overall glass thermodynamics. At high concentrations, however, it is expected that a strongly interacting penetrant may modify the potential energy hypersurface sufficiently to reduce barrier heights between rp regions that were disjoint in the pure polymer glass and start allowing transitions between these regions over ordinary time scales. The probability distribution in configuration space will start shifting toward the global equilibrium distribution characteristic of a melt and eventually lead to devitrification of the polymer. This is a qualitative molecular interpretation of plasticization effects often observed at high penetrant activities. From the point of view of molecular simulation, the problem of sampling the relevant fluctuations that shape sorption equilibria in an intrinsically nonequilibrium glassy system is a serious conceptual and computational challenge. The diffusivity of a dilute penetrant in an amorphous polymer matrix is governed by the penetrant size and interactions with the polymer as well as by the shape, size, connectivity, and time scales of thermal rearrangement of unoccupied space within the polymer. In a high-temperature melt (T » TJ, openings among chains that are capable of accommodating the penetrant undergo rapid redistribution in space. One can envision that the penetrant is "carried along" by density fluctuations caused by the thermal motion of surrounding chains. As in the free-volume picture of liquid-state diffusion, one could envision that a penetrant molecule resides in a certain position of the polymer matrix until the motion of surrounding chains, modified by the penetrant's presence, leads to the formation of a cavity, at a distance commensurate with the penetrant's diameter, into which the penetrant can move. After the move, the cavity in which the penetrant was originally accommodated is closed. A succession of such small random moves of the penetrant constitutes diffusion. In this high-temperature melt limit, formation of a cavity of sufficient size to accommodate the penetrant can be viewed as the rate-controlling step for a move. The time scale between moves is thus set by the relaxation time of density fluctuations on the length scale of the penetrant diameter within the polymer matrix. At temperatures below Tg , on the other hand, one would expect the molecular picture of diffusive jumps to be substantially different. The distribution of open spaces within the configurationally arrested glassy matrix is more or less permanent (long-lived). One can envision a network of preexisting cavities, the magnitude and shape of which fluctuates somewhat with thermal motion and is modulated by the possible presence of a penetrant molecule. A dilute penetrant spends most of its time rattling within a cavity and occasionally jumps from cavity to cavity through a "window" that opens instantaneously via fluctuations in "soft" regions (Le., regions of lower density or enhanced molecular mobility) between the cavities. The overall diffusivity in the glass would thus depend on

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the distribution of distances and connectivities among the cavities as well as on the magnitude and distribution of rate constants governing the infrequent jumps of the penetrant between adjacent cavities. In the following sections we shall see how the simple conceptual picture outlined above is substantiated by molecular simulations.

II.

CHARACTERIZATION OF STRUCTURE AND MOLECULAR MOTION IN AMORPHOUS POLYMERS

A.

Methods of Generating Amorphous Polymer Configurations

As pointed out in the qualitative discussion of Section I, the ability to represent the molecular level structure and mobility of amorphous polymers is a prerequisite for simulating sorption and diffusion in them. In this section we discuss existing computational methodologies for generating model amorphous polymers and analyzing the distribution of unoccupied space and the short-time molecular motion of chains. We confine our discussion to relatively detailed models in continuous space, in which the bonded geometry (bond lengths, bond angles) and interaction energetics of chains are represented realistically; only such models are conducive to a quantitative study of diffusion. In these models chains are represented as groups of interaction sites, i.e., atomic nuclei and centers of charge associated with polar groups. The potential energy "If' p is expressed as a sum of bond and bond angle distortion terms, torsional potentials, as well as intermolecular and intramolecular exclusion, dispersion, electrostatic, and charge transfer interactions that depend on the coordinates rp of the interaction sites. Groups of atoms may be lumped into single interaction sites, as in the "united-atom" model of linear polyethylene, consisting of a sequence of methylenes capped by methyls at the two ends. The parameterization of the potential energy is based on fitting experimental structural and thermodynamic properties of low molecular weight analogs of the polymer or on ab initio calculations of partial charges and specific interactions between polar groups (Allen and Tildesley, 1987; Ludovice and Suter, 1989).

1. The Amorphous Cell" As in simulations of small molecular weight liquids (Allen and Tildesley, 1987), the model system considered in amorphous polymer simulations is a box, or cell, filled with chains and characterized by three-dimensional periodic boundary conditions (see Fig. 2a). Currently, amorphous cells of edge length 20-50 A containing 500-5000 interaction sites are studied with conventional computational resources. According to Flory's "random coil hypothesis," for which ample experimental evidence has been collected through neutron scattering, the conformation of chains in an equilibrium melt or amorphous glass remains esII

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

73

(a)

(b)

Figure 2

Schematic representation of amorphous cell and periodic boundary conditions. 1Wo-dimensional projections of two three-dimensional periodic model configurations are shown. The primary simulation cell lies at the center and is surrounded by eight images of itself. (a) Finite molecular weight system; the contents of the amorphous cell are formed from four "parent" chains. (b) Infinite molecular weight system; the contents of the amorphous cell are formed from a single chain with no ends; three images of this chain go through the primary box.

74

THEODOROU

sentially unperturbed by interactions between topologically distant sites along the chain. As a consequence, the mean square radius of gyration (S 2) of a long chain in the bulk is related to the number of skeletal bonds n and to the skeletal bond length e as (S 2) = (1/6)nC~e 2 , where C~, the characteristic ratio, is a chain length-independent constant that can be predicted by conformational analysis of short sections of the chain (Flory, 1969). Strictly, the edge length L of the periodic box must fulfill the condition L > 2(S2)112, so that different images of the same chain do not interact significantly with each other, as such interaction may distort the long-range conformational characteristics of chains; this imposes an upper limit on the molecular weight that can be simulated with a cell of given dimensions. In glassy polymer simulations, however, where chains do not have the opportunity to relax their long-range conformation, meaningful results have been obtained with as few as one " parent" chain per box, provided care is taken in the initial preparation of the cell so that the conformation of the chain is close to unperturbed (Theodorou and Suter, 1985). Infinite chain length models have also been constructed to examine properties in the absence of chain end effects (Weber and Helfand, 1979). The model system of Fig. 2b, for example, is entirely constructed from images of a single noncyc\ic chain with no ends. In the preparation of such models, care is taken so that the basic periodic elements of which the system is formed (e.g. , the chain section between A and A I in Fig. 2b) are close to unperturbed; clearly, however, this cannot be said about the entire infinite chain, which is strongly oriented in the AA I direction when examined at length scales substantially larger than L.

2. Molecular Mechanics The objective of molecular mechanics (MM) is to generate static minimum energy configurations at prescribed density, corresponding to the local minima of OV p shown in Fig. 1. The technique was developed to model glassy amorphous polymers (Theodorou and Suter, 1985). One method for creating a minimum energy configuration starts by generating an initial guess configuration through bond-by-bond growth of the parent chains in the amorphous cell, observing periodic boundary conditions. This growing procedure can be based on Flory 's rotational isomeric state model for unperturbed chains, modified so as to avoid intra- and intermolecular excluded volume interactions. The initial guess configuration is then subjected to minimization of the total potential energy "V' p with respect to all microscopic degrees of freedom. The minimization is most efficiently accomplished in a stagewise fashion, wherein one starts with purely repulsive interatomic potentials of reduced range to relax the most severe excluded volume overlaps, gradually enlarges the atomic radii to their actual size, and finalJy introduces attractive interactions as well (Theodorou and Suter, 1985). An advantage of the MM technique is its speed; a model glassy configuration can be arrived at within a few minutes to an hour of CPU time on a

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

75

Cray-Y/MP vector machine, i.e., with two orders of magnitude less computer time than is required by methods that incorporate thermal motion. A disadvantage of molecular mechanics is the fact that thermal fluctuations are not explicitly accounted for and that the system density has to be set a priori; thus, only properties of a glassy system that is locally arrested in the neighborhood of potential energy minima can be examined. Also, the procedure followed in generating minimum energy configurations does not correspond to a well-defined history of vitrification. Nevertheless, arithmetic averaging of the properties of minimum energy configurations (which is based on the assumption that such configurations are generated by molecular mechanics with a probability distribution comparable to that encountered in a real-life glassy polymer) has led to encouraging predictions of structure, elastic constants, and surface thermodynamic properties (Theodorou and Suter, 1986; Mansfield and Theodorou, 1990). On the other hand, static minimum energy structures generated by MM are satisfactory as starting configurations for molecular dynamics and Monte Carlo simulations that incorporate thermal motion.

3. Molecular Dynamics Molecular dynamics (MD) tracks the temporal evolution of a microscopic model system through numerical integration of the equations of motion for all the degrees of freedom. Originally developed in the microcanonical (constant number of molecules, volume, and total energy, or NVE) ensemble, the MD method can now be applied straightforwardly in the canonical (NVT) and isothermalisobaric (NPl) ensembles as well. Constraints on the microscopic degrees of freedom, such as constancy of bond lengths and bond angles, can also be handled (Allen and Tildesley, 1987). The MD method has been applied widely to united-atom polyethylene-like systems by Rigby and Roe (1987, 1988, 1989). To create a model melt system, an MD simulation of monomeric segments was conducted, in the course of which bonded forces were gradually "turned on" to form chains. Glassy configurations were obtained from the melt through a series of MD runs at progressively lower temperatures (the cooling rates were too fast to allow crystallization). An asset of molecular dynamics is that it provides directly a wealth of detailed information on short-time dynamical processes in the polymer. Its major limitation is that, since it faithfully mimics molecular motion in actual amorphous polymer systems, it is fully subject to the bottlenecks that limit this motion. Hundreds of CPU hours on a vector supercomputer are required to simulate a nanosecond of actual motion with atomistic MD. In view of the much longer relaxation times of actual long-chain polymer melts, the mere equilibration of a model melt by MD becomes problematic, and results are not trustworthy if one starts from an improbable initial configuration. Model glassy configurations formed by MD cooling of the melt have a well-defined history, which, however, is very far from that used in glass formation experiments; the cooling rates em-

76

THEODOROU

ployed in MD vitrification are on the order of 1011_1010 Kls. Some problems associated with the generation of amorphous model structures through MD have been addressed by McKechnie et al. (1992). 4. Monte Carlo The objective of a Monte Carlo (MC) simulation is to generate a large number of configurations of the microscopic model system under study that conform to the probability distribution dictated by the macroscopic constraints imposed on the system. For example, a Monte Carlo simulation of a melt of N chains in volume V at temperature T generates a large sequence of configurations, in which each configuration rp occurs with frequency proportional to exp[ - 'VI'(r•.)], as dictated by the canonical ensemble. At each step of the MC simulation one attempts to generate a new configuration from the current configuration through an elementary move. The attempted move is either accepted or rejected according to selection criteria designed so that the resulting sequence (or Markov chain) of configurations asymptotically samples the probability distribution of the ensemble of interest. Thermodynamic properties are calculated as averages over the sampled configurations (Allen and Tildesley, 1987). An initial configuration for starting the MC simulation can be obtained through one of several strategies. Vacatello et al. (1980) started their pioneering simulations of liquid triacontane by placing chains on a tetrahedral lattice, whereas Boyd (1989) employed a crystal, which was melted during the MC equilibration stage, as an initial configuration for his simulations of liquid tetracosane. Minimum energy configurations from molecular mechanics have also been used (Dodd et aI., 1993). To accelerate convergence of an MC simulation with elaborate atomistic models, Vacatello recently proposed using equilibrated configurations of a simple model of tangent sphere chains with the same endto-end distance as the actual chains as a starting point (Vacatello, 1992). Designing elementary moves for a polymer MC simulation is more of a problem than for a small-molecule simulation, because of the constraints imposed by bonds and bond angles. Rotations around bonds near the chain ends have been used along with " reptation " moves, wherein a terminal segment of a chain is deleted and a new terminal segment is appended on the other end at a random torsion angle. In the recently developed continuum configurational bias (CCB) algorithm (de Pablo et aI., 1992), an end section of random length is cut off from a randomly chosen chain and regrown bond by bond by selectively placing segments in regions where they are not likely to experience excluded volume interactions with other chains; the bias associated with this selective growth procedure is removed with appropriate selection criteria. MC simulation with concerted rotations (Dodd et at , 1993), on the other hand, employs coordinated torsions around seven adjacent skeletal bonds to change the conformation of chains without affecting bond lengths and bond angles. Moves that involve cut-

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

77

ting chains and rejoining the resulting segments in a different way, thereby drastically altering the system configuration, are currently being explored as an extension of the concerted rotation move. An advantage of the MC method is that, by judicious choice of the elementary moves, one can circumvent the bottlenecks in configuration space that inhibit molecular relaxation in real polymer systems and in MD simulations and thus effect rapid equilibration of a multichain model system. There are strong indications that MC schemes under development today afford order-of-magnitude savings in CPU time relative to molecular dynamics as means of preparing equilibrated melt systems. In addition, the MC method can readily be applied in a variety of ensembles, including ensembles in which the total number of particles of some species is allowed to fluctuate; this is of strategic significance in simulations of sorption (see below). A disadvantage of MC simulation is that it does not provide true dynamical information. When the moves employed mimic dynamical processes that can actually take place in the polymer, one can establish a loose correspondence between attempted moves per interaction site and elapsed time that allows conclusions about long-time dynamical processes to be extracted (" dynamic Monte Carlo" approaches). Furthermore, in using MC simulations to predict the thermodynamics of configurationally arrested glassy systems, one has to rely upon appropriately chosen " quasi-dynamic" moves to stay confined within the relevant regions of configuration space and avoid tunneling through physically insurmountable energy barriers (compare Fig. 1).

5. Validation of Amorphous Polymer Models A variety of properties can be predicted from an amorphous polymer model and tested against experimental evidence to confirm that the model provides a reasonable representation of reality. The pair distribution functions g(r) for all site pairs in the system can be obtained readily from the configurations generated in the course of a simulation (Allen and Tildesley, 1987). Note that the location of the first peak in the intermolecular part of the pair distribution function between skeletal atoms gives a direct measure of the average distance between chain backbones in the amorphous bulk. Through Fourier transformation of the pair distribution functions one can predict X-ray, neutron, or electron diffraction patterns that can be compared to experimental wide-angle patterns from the bulk polymer. Similarly, from the intramolecular pair density function one can predict small-angle neutron diffraction patterns obtained experimentally by selective labeling of single chains and thus check long-range conformational characteristics of the chains in the model sample. The ability to predict PVT behavior in the melt is a rather stringent test of the model representation used in a simulation and is a prerequisite for reliable studies of sorption and diffusion. Dynamical properties, such as self-diffusivity of chains and viscosity in the melt and frequency and activation energy of relaxational motions, can be compared to scat-

78

THEODOROU

-g

"""' 0.7

~

~ -0.2

-1.1

-2.00~-----':-4--'--~-:!:-8--'--~:-'::12~-.L.------:-l16 (a)

Magnitude of wave vector (A.-I)

Figure 3 Structural and thermodynamic predictions from isothermal-isobaric Me simulations of linear alkane liquids. (a) Diffraction pattern compared against experiment; (b) equation-of-state properties compared against experiment. See text for details. tering, rheological, and spectroscopic measurements. Such comparisons are currently practiced only to a limited extent, owing to the difficulties in accessing long-time dynamics with atomistic simulations. Smith and Boyd (1992) compared activation energies for side-group rotations in their model glassy polymers against experimental evidence, with favorable results. More recent work by Smith and Yoon (1994) demonstrated that a well-calibrated explicit atom potential, when used in equilibrated MD simulations of a high-temperature tridecane melt, can reproduce chain self-diffusivities and correlation times for C-H bond reorientational motion in excellent agreement with experiment. A united-atom potential is less successful in capturing the local segmental dynamics, although it can reproduce equation-of-state behavior, scattering patterns, and chain selfdiffusivities quite well. As an indication of the agreement that can be achieved between simulation estimates on the one hand and structural and thermodynamic measurements on the other, we display some results from MC simulations of long-chain linear alkane liquids in Fig. 3 and Table 1. A consistent united-atom representation was used for the simulations (Dodd and Theodorou, 1994). Figure

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

E

1.6r---------~--------------------~--------~ ·0··

~

co 1.5

'S~ 0)

~t

Experiment NPT Monte Carlo

~

· •..... 4···· .•.....A .... ......•.....1'.......•......•.....K ....

;:J

>

~

1.4

E

'0

79

1.3

~ ....•

(J

~

1.2

0)

0..

CIj

1.1 ';;-0-----::3~0:------67::0:------9:t-:0:---------.J120 Pressure (MPa)

E

~ co M"-

1.5.--------,---~---.,.-----.,....-------, ··0·· Experiment

1.4

i I.j\···········~

~

E

NPT Monte Carlo

~

......•.... .....•.......•....rl! ........···•···.. 4 .....•....•

1.2

'0

>

(J

~

~

1.1

C7s.450K

CIj

1.0 0

30

(b)

Figure 3

Table 1

60

90

120

Pressure (MPa) Continued.

Conformational Characteristics of Tetracosane Chains

Chain property

Bulk NPT Monte Carlo'

Continuous unperturbedh

(r2). 1\2 (S2),1\2

350 43 0.659 0.394 0.532 0.110

354 44 0.666 0.406 0.524 0.104

PI Pit PIli PgIg

'Sampled in the course of an isothermal-isobaric Monte Carlo simulation of the bulk liquid. hUnperturbed chains governed by the same intermolecular potentials but not experiencing nonlocal interactions.

80

THEODOROU

3a compares the k-weighted structure factor for simulated Czo at 315 K and 1 bar (curve) against accurate neutron diffraction measurements (points) (Habenschuss and Narten, 1990). Table 1 displays a comparison of several Cz4 singlechain conformational characteristics, namely the root mean square radius of gyration (S 2)112, the rms end-to-end distance (r 2)112, the mean fraction P, of skeletal bonds in a trailS conformation, the mean fractions p" and P,g of pairs of adjacent bonds in trans-trans and trans- gauche conformations, and the mean fraction P,g, of triplets of successive bonds in a trans-gauche- trans conformation against the corresponding predictions for unperturbed chains. Figure 3b shows simulation predictions for the specific volume of Cz4 and ~8 at 450 K as a function of pressure (triangles) compared with experimental values (squares and dotted lines) (Dee et aI. , 1992). The comparisons indicate that the simulation is well-equilibrated and free of model system size effects and that the model representation employed is reasonable.

B. Accessible Volume and Its Distribution Atomistic model configurations obtained through the simulation techniques described in Section II.A can serve as a starting point for characterizing the free spaces within the polymer where a penetrant molecule can reside. Such geometry-based (as opposed to energy-based) analyses of the internal structure of amorphous polymers can be conducted with little computational expense once the model configurations are available. A main objective of these analyses is to relate the magnitude and distribution of "free volume, " which has played a central role in theories of sorption and diffusion, to chain chemical constitution and architecture. Experimental efforts to determine the distribution of unoccupied volume in polymer matrices through positron annihilation lifetime measurements have appeared recently (Malhotra and Pethrick, 1983; Kluin et aI. , 1993; Deng and Jean, 1993); the geometrical analysis of model structures is helpful in the interpretation of such measurements. Several definitions have been used for " free volume" in theoretical work. For the purpose of characterizing void space in model structures, it is meaningful to consider each interaction site (atom) on the polymer chains or on the penetrant molecule as a hard sphere of diameter equal to its van der Waals radius r Oo We use the term unoccupied volume to refer to the volume of the three-dimensional domain composed of points within a configuration that lie outside the van der Waals spheres of all polymer atoms. The term accessible volume refers to the volume of the domain composed of points that can be occupied by the center of mass of the penetrant molecule without any overlap between the van der Waals spheres of the penetrant and those of the polymer atoms. Our discussion here is confined to spherical penetrant molecules represented as single interaction sites. It is emphasized that the analytical techniques we discuss are applied

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

81

to the pure polymer matrix. The penetrant molecule is used as a geometrical probe of the internal structure of the polymer, which does not respond to the penetrant in any way. The determination of the volume, within a given configuration, that is accessible to a given penetrant can be conveniently conducted as follows. The radii of all polymer atoms are augmented by a length equal to the penetrant radius r~, and the unoccupied volume of the resulting model system, in which each atom i is represented through its "excluded volume sphere" of radius r ~ + r~, is calculated (Dodd and Theodorou, 1991; Greenfield and Theodorou, 1993). Conversely, the unoccupied volume of a polymer configuration coincides with its accessible volume in the limit r~ -+ O. Figure 4 depicts the threedimensional domains within a glassy atactic polypropylene configuration that are accessible to helium (r~ = 1.28 A) and to argon (r~ = 1.91 A) (Greenfield and Theodorou, 1993). In both cases the accessible volume consists of disjoint clusters. As the penetrant radius is reduced, the accessible clusters grow in size; tentacle-like protrusions on the periphery of different clusters come together, causing pairs of clusters to merge into a larger cluster. At the same time, new clusters become available. At some critical penetrant radius, re , an infinitely extended cluster appears that spans the entire periodic array of boxes representing the polymer; that is, percolation of accessible volume occurs throughout the model polymer. The percolation threshold value re varies somewhat from configuration to configuration; it also depends on the edge length L of the primary box, smaller boxes being easier to percolate. A systematic study of re in primary boxes of different L can be used to deduce the percolation threshold in the limit L -+ 00 . For glassy atactic polypropylene, the average re in the infinite box limit is around 0.9 A, i.e., smaller than the radius of any gaseous penetrant that might permeate the polymer (Greenfield and Theodorou, 1993). The calculation of unoccupied volume is complicated by the fact that bond lengths are typically small relative to van der Waals radii, and thus the van der Waals spheres of atoms along a chain interpenetrate profusely. This interpenetration is even more pronounced in the case of excluded volume spheres used for the calculation of accessible volume. Shah et al. (1989) introduced a Monte Carlo integration technique for the determination of accessible volume. The technique consists in choosing a large number [0(106)] of points randomly in the simulation box and determining what fraction of these points lie outside the excluded volume spheres of all polymer atoms. A computationally more efficient approximate technique was introduced in the pioneering work of Arizzi et al. (1992). In this technique, the model configuration is partitioned into tetrahedra of nearest-neighbor atoms, and the accessible volume is computed separately in each tetrahedron by an analytical procedure that accounts for twofold overlaps between excluded volume spheres; tetrahedra in which threefold or higher overlaps are observed are considered fully occupied. An exact analytical solution of

82

THEODOROU

Figure 4 Three-dimensional depiction of the volume within a model configuration of glassy atactic polypropylene that is accessible to helium (top) and argon (bottom). The edge length of the model box is approximately 23 A. Different clusters of free volume are displayed in different colors (shown here in shades of gniy). Periodic boundary conditions are evident.

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

83

this problem, based on an efficient algorithm that accounts for sphere overlaps of any order (Dodd and Theodorou, 1991), was presented recently (Greenfield and Theodorou, 1993). Unoccupied and accessible volume calculations have been conducted on model glassy atactic polypropylene (at-PP), poly(vinyl chloride) (PVC), and bisphenol A polycarbonate (PC), as well as on model melts of united-atom polyethylene (PE) and at-PP. The unoccupied volume fraction in the glassy polymers is found to be around 0.35, the exact value depending on the nature of the polymer. For example, Arizzi et al. report 0.354 ± 0.001 for at-PP and 0.39 ± 0.038 for Pc. The accessible volume fraction is a monotonically decreasing convex function of penetrant radius (see Fig. 5). In glassy at-PP, Arizzi et al. report accessible volume fractions of 0.16, 0.08, and 0.07 for He, O2 , and N2, respectively; the corresponding values in PC are 0.19 for He, 0.12 for O2 , and 0.11 for N2 • It is interesting to compare the availability of free volume between a glassy polymer and an atomic glass; random close-packed (rcp) configurations of spheres constitute a reasonable model for the latter. Figure 5 shows such a comparison between at-PP and an rcp structure consisting of 2 A diameter spheres. This diameter is commensurate with that of methylene, methine, and methyl segments in the propylene model; the atomic "granularity" of the rcp and polymer structures is thus comparable. Although the unoccupied volume of the rcp structure is somewhat higher, its accessible volume falls off more rapidly with increasing penetrant diameter than that of the polymer. The macromolecular constitution of the polymer gives rise to a broader distribution of void sizes, which can accommodate larger penetrants. This is also reflected in the percolation characteristics of accessible volume in the two types of glassy structures: rc for the rcp structure is around 0.55 A, which is significantly smaller than the value of 0.9 Afor the polymer glass. For both the rcp and polymer glass structures, the accessible volume fraction at the percolation threshold is in the range 0.02-0.04, close to the value observed for the "Swiss cheese" model (see Chapter 1 of this book). For given penetrant radius, the accessible volume is distributed spatially into clusters (see Fig. 4). Several techniques have been used to quantify the size and shape distribution of these clusters. Boyd and Pant (1991a) chose to examine the distribution of the radii of the largest spheres that can be inscribed within tetrahedral interstitial sites formed by polymer atoms. Arizzi et al. (1992) introduced a rigorous procedure for defining clusters of accessible volume that relies on Delaunay tessellation of the model polymer configurations. In a three-dimensional Delaunay tessellation, an arbitrary collection of points (atomic centers) is partitioned completely into tetrahedra, each tetrahedron having four nearest-neighbor points as its apices; the circumsphere of a Delaunay tetrahedron does not contain any other points inside it. Fast algorithms for performing the Delaunay tessellation and its dual Voronoi tessellation are available (Tanemura

84

THEODOROU 0 .4 10·\

~.

C

.Q 0. 3 1:5 m

10.2

10.3

E :J

(5

>

0.2

10"

Q)

:0

"", ~t ~

.!:: Q)

I

10.5

·iii en

0

0.5

1.0

1.5

2.0

2.5

Q)

0 0

m 0.1

o

0.5

1.0

1.5

2.0

2.5

penetrant radius (A)

Figure 5

Accessible volume fraction as a function of penetrant size in atactic polypropylene as obtained from Monte Carlo simulations of the polymer in the glassy (X) and melt (0 ) states (Greenfield and Theodorou, 1993). The accessible volume fraction of a random close-packed (rcp) structure of 2 A diameter spheres, representative of an atomic glass, is also shown (0). The diameter of the rep spheres is roughly equal to the van der Waals diameter of methyl, methylene, and methine units constituting the polymer.

et aI., 1983). Delaunay tetrahedra are an excellent means for identifying interstices of accessible volume within a polymer configuration. For a given r~, if the interior of a tetrahedron is completely filled by the excluded volume spheres of polymer atoms, then the tetrahedron is inaccessible; otherwise, the tetrahedron has a pocket of accessible volume in its interior. 1\vo accessible tetrahedra are said to be connected when they share a face (triangle) that is not completely blocked by the excluded volume spheres of polymer atoms. Uninhibited passage of the penetrant from the interior of one tetrahedron into the other through the shared face is thus possible. Accessible tetrahedra can be grouped into sets of connected tetrahedra using a simple connectivity algorithm (Greenfield and Theodorou, 1993). A cluster of accessible volume is simply the union of the accessible volumes of such a set of connected tetrahedra. Although fast and accurate, Delaunay tessellation is not the only tessellation whereby clusters of unoccupied volume can be analyzed. Takeuchi and Okazaki (1993a), for example, partitioned their MD configurations into cubic elements for the same purpose.

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

85

For given penetrant radius r~ , one can define a volume-weighted probability density distribution of accessible cluster volumes, Pv(v; r~ ), such that Pv(v; r~ ) dv equals the fraction of the total accessible volume found in clusters of volume v to v + dv. Pv(v; r~ ) is generally a decreasing function of v; its reliable determination requires analysis of a large number of configurations. Computed plots of Pv versus v seem to exhibit some fine structure (local extrema) related to specific intermolecular packing patterns that depend on the detailed geometry, conformational preferences, and interaction forces among chains (Greenfield and Theodorou, 1993); their general appearance is in reasonable agreement with positron lifetime spectroscopy results (Deng and Jean, 1993). The average cluster size can be characterized through the average cluster volume (v)v (first moment of P,.) or through the root mean square cluster radius of gyration (Greenfield and Theodorou, 1993). In the range of penetrant radii r~ > r e , the average size of accessible clusters is a smoothly decreasing function of r~ (compare Fig. 4). The shape of accessible clusters can be characterized by comparing the principal axes of their radius of gyration tensor; in the case of at-PP, the asphericity of the clusters was found to be comparable to that of an ellipsoid of revolution with principal axis lengths in the ratio 2:1:1. A picture such as Fig. 4a suggests that a penetrant molecule of sufficiently large r ~, sorbed at low concentration within a glassy polymer, would spend most of its time confined in the interior of small disjoint clusters of accessible volume. The diffusion of the penetrant would proceed by infrequent jumps from cluster to cluster through short-lived passages opening momentarily between the clusters (see also Section IVD). The likely location of such passages that could open up through thermal fluctuations and thus act as diffusion pathways for the penetrant can be identified by examining the accessible volume distribution at a value of the penetrant radius equal to, e.g., re , for which long-range connectivity is bound to be established. Such an examination reveals the coordination number of a cluster, i.e., the most probable number of clusters with which a given cluster is connected through diffusion pathways. For Ar and He in glassy at-PP (Greenfield and Theodorou, 1993), the coordination number of a cluster is found to follow rather broad distributions with most probable values of 4 and 2, respectively. A more elaborate and computationally much more time-consuming approach for identifying clusters and passages between them is to analyze the potential energy field experienced by a penetrant at every point in a glassy polymer configuration. Such an energetic analysis of He in PC, conducted by Gusev et al. (1993), revealed accessible clusters of diameter 5-10 A, connected by bottleneck regions ca. 5-10 Along and 1-2 Ain diameter, in agreement with the geometric analysis (see Fig. 4). It should be emphasized that the rate constants for passage from cluster to cluster through a diffusion pathway follow a broad distribution, and therefore the connectivity of the network of clusters is a function of the time scale over which the network is examined. We return to this point in Section IV.D.3.

86

THEODOROU

It is informative to track thermal fluctuations in the distribution of accessible volume. Results from such a study, conducted in the course of long MC simulations of an at-PP glass and melt, are shown in Fig. 6. In the glass (top), one clearly sees that the distribution of accessible volume changes very little. The clusters present at the beginning of the MC run are surviving at the end of the run; the configuration is locked in, and the void distribution can be characterized as permanent over the effective time scale spanned by the simulation. In the

(a)

(b)

(c)

(d) Figure 6 Evolution of the distribution of volume accessible to He within atactic polypropylene, as obtained from long MC simulations of the polymer. (a) Polymer glass configuration at 233 K. (b) Configuration obtained from (a) after 10 million attempted MC moves at 1 bar and 233 K. (c) Polymer melt configuration at 400 K. (d) Configuration obtained from (c) after 10 million attempted moves at 1 bar and 400 K.

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

87

melt (bottom), and over the same number of MC moves, one sees a dramatic change in the accessible volume distribution. The void space has reorganized to such an extent that it is impossible to recognize clusters in the bottom right snapshot as having evolved from clusters in the bottom left snapshot. Melt clusters are transitory. This difference in the rate of accessible volume redistribution in the high-temperature versus low-temperature amorphous polymers has important implications for the mechanism of diffusion. Although not capable of providing predictions for sorption isotherms and diffusion coefficients, the simple geometric considerations discussed in this section are useful as a prelude and guide to the more elaborate (and computationally much more demanding) energy-based approaches discussed in Section IV.D.

C.

Characteristic Times of Molecular Motion

Molecular motion in amorphous polymers is governed by a wide spectrum of characteristic times. Bond and bond angle vibrations occur on time scales on the order of 10- 13 _10- 14 s, which are relatively insensitive to molecular packing. The rates of rapid rotations of pendant groups exhibit an Arrhenius temperature dependence; they are associated with [3-relaxation phenomena observed in lowtemperature glasses. Librations of skeletal bonds in their energy wells and conformational transitions across torsional energy barriers are accomplished through localized distortions of the chain backbones involving on the order of 10 bonds along a chain. The rate of such "segmental " motions is very sensitive to density. In a high-temperature melt they are are relatively uninhibited by surrounding chains, occurring over time scales of 10- 12 _10- 10 s. As temperature drops they slow down dramatically and become increasingly cooperative, the associated a-relaxation functions usually being fit to a stretched exponential KohlrauschWilliams-Watts (KWW) form and the temperature dependence of the observed relaxation time following a manifestly non-Arrhenius Williams-Landel-Ferry (WLF) equation (Plazek and Ngai, 1991). Finally, large-scale conformational rearrangement, reorientation, and self-diffusion of chains are frozen-in in a glass but present in a melt. The characteristic times for such large-scale conformational rearrangements lie in the " terminal" region of the relaxation spectrum; they are very sensitive to chain length, their chain length dependence being described rather satisfactorily by the Rouse and reptation models in unentangled and entangled polymer melts, respectively. (See Chapter 6 by P. F. Green.) Here we discuss briefly some findings about local segmental motions in an amorphous polymer matrix, as obtained from MD simulations. These motions are particularly relevant to diffusion of small penetrants through the matrix, as they dictate the thermal fluctuation of accessible volume clusters and diffusion pathways (compare Section II.B). Takeuchi and Roe (1991a,b) quantified the rate at which individual torsion angles lose memory of their initial values by defining an autocorrelation function

88

THEODOROU

for dihedral angles, R,it) as

R (t) = (cos <\>(t) cos <\>(0» - (cos <\>(O)? (cos2 <\>(0» - (cos <\>(O)?

(1)

where the ensemble averages are taken over all skeletal bonds and over all time origins along the MD simulation. R(t) starts at 1 and decays to zero in a stretched exponential fashion. In the melt, the time 1"<1> at which R has decayed to lie is studied as a function of temperature. In an infinite molecular weight PE melt at 300 K, Takeuchi and Roe found 1"<1> = 24 ps; in the high-temperature melt, 1"<1> decreased with increasing T with an activation energy of roughly 3.75 kcal/mol, which is comparable to the trans-gauche torsional barrier. As temperature was reduced toward Tg (which, as defined through the break in the volume versus temperature curve, is 201 K for this polymer at a cooling rate of 1.67 X 1011 K!s), 1"<1> was found to increase dramatically. A KWW fit to Eq. (1) at Tg gave a relaxation time 1"<1> = 1.39 ns. Within the glass the rate of well-towell conformational transitions was found to be on the order of 1 ns- 1 down to a temperature of 148 K. Comparable time scales for conformational transitions have been found in an MD simulation of a short-chain atactic polypropylene glass near its Tg (Mansfield and Theodorou, 1991). The latter simulation indicated a large degree of spatial heterogeneity in terms of the ability of bonds to isomerize conformationally. The glass was found to contain isolated " soft spots," where conformational transitions occur at rates comparable to those seen in polymer melts, surrounded by a "stiff" continuum, where no transitions are observed over hundreds of picoseconds. Torsional mobility is enhanced near chain ends, although "soft spots" are likely to be found away from ends as well. One can envision that with increasing temperature the size and connectivity of "soft spots" increases at the expense of the stiff region until the spots percolate through the glassy bulk, signaling the devitrification of the polymer. Another way of studying segmental mobility in MD simulations is to track the orientational decorrelation of characteristic unit vectors rigidly embedded in the chains. The correlation times for such motion are measurable with NMR, dielectric relaxation, electron spin resonance, photon correlation, and fluorescence spectroscopy. Let u be such a unit vector. As a result of thermal motion, the vector's orientation at time t, u(t), will be different from its original orientation u(O). One can form the two time correlation functions

M.(t)

= (u(t)

. u(O»

(2)

and

M 2(t) =

~ (3[u(t)

. u(OW - 1)

(3)

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

89

The time decay of these is not exponential; it is well described by a KWW expression (4)

For a given choice of t1 the ratio Tir2 is always larger than 1, its exact magnitude depending on the mechanism of the orientational relaxation. Smith and Yoon (1994) chose t1 as unit vectors along the pendant C-H bonds of a tridecane liquid. For the C-H bonds attached to the central carbon of chains, their explicit atom MD simulations gave T2 = 10 ps at 312 K with an activation energy of 4 kcal/moi. The value of T2 was found to be significantly shorter for the five carbons near each chain end, dropping to 2.5 ps at the terminal methyls. All these observations are in excellent agreement with 13C NMR spin-lattice relaxation experiments. The behavior of TJ and T2 for pendant bonds tracked the behavior of T, all three times being of the same order of magnitude. Takeuchi and Roe (1991a,b) studied three t1 vectors with their united-atom model of polyethylene. Vector a is directed along the bisector of a skeletal bond angle in the plane of two adjacent skeletal bonds; vector c is normal to a in the plane of the bond angle and thus points along the direction of the chain backbone; finally, the " out-of-plane" vector b is the cross product of c and a. In the melt, the vectors a and b that are directed normal to the backbone were found to relax with comparable rates, b being somewhat faster. Their behavior was strongly correlated with bond angle relaxation, TJ being approximately equal to T and T2 being 0.3-0.5 T at all temperatures studied. In contrast to a and b, vector c (oriented along the backbone) was found to relax dramatically slower, its TJ being roughly 38O'T at 300 K. This anisotropy of orienta tiona I relaxation is comparable to but stronger than that observed in Brownian dynamics simulations of isolated polymer chains in solution (Ediger and Adolf, 1994). In the glass, the KWW apparent relaxation times for a and b were on the order of 1 ns (Takeuchi and Roe, 1991b; Mansfield and Theodorou, 1991). Bond reorientation angle distributions in the glassy polymer revealed two mechanisms as responsible for the decorrelation of a and b. One is rotational diffusion, the other a jumplike process wherein the bond direction changes abruptly by an angle comparable to the distance between conformational energy wells. The low values of the stretching exponent (3 < 1 obtained from fitting Eq. (4) to the relaxation functions MJ(t) and M2(t) of a, b, or pendant bond vectors indicate considerable cooperativity of reorientational motion in the melt ((3 = 0.45 - 0.5) and a very high degree of cooperativity in the glass ((3 = 0.2). The limited duration of the MD simulations (= 1 ns), however, does not permit the reliable prediction of correlation times (seconds to hundreds of seconds near Tg) obtained from dynamic light scattering (Fytas and Ngai, 1988) and two-dimensional NMR (Schaefer et aI., 1990) measurements. The latter times are comparable to the ones obtained from mechanical measurements (Fytas and Ngai,

90

THEODOROU

1988) of the a relaxation. The question of how to predict the time scales of a relaxation in the rubbery and glassy states reliably through molecular simulation is extremely important but still unresolved. Perhaps a more complete way to characterize density fluctuations in the amorphous polymer bulk is to accumulate the intermediate scattering function F(k, t), i.e., the Fourier transform of the density-density correlation function of the amorphous polymer. Consider a polymer consisting of segments of one type (e.g., polyethylene in a united-atom representation). Let rit) be the position of segment j at time t. The Fourier component of the instantaneous density corresponding to wavevector k at time t is N,

Pk(t) =

2: exp [-ik . rit)]

(5)

j=l

where Ns the total number of segments in the system. The intermediate scattering function is (6) The time evolution of F(k, t) reveals how density fluctuations occurring over length scale 2'lT/lkl disappear through thermal motion. The Fourier transform of F(k, t) is the dynamic structure factor S(k, w); it is measurable through coherent inelastic neutron scattering. Perhaps more interesting than F(k, t) itself is its self-part Fs(k, t) defined as 1 /

Fs(k, t) = Ns \

~ exp {-i N,

k [rit) - riO)]}

)

(7)

The time evolution of Fs(k, t) reveals how individual segments lose memory of their original position through motions occurring at a length scale 2'lT/lkl. Its Fourier transform with respect to time, Ss(k, w), is measurable through incoherent neutron scattering. The structural "slowing down " occurring as a liquid is cooled toward the glass temperature can be detected in the time decay of Fs(k, t), and the characteristic frequencies associated with the a relaxation at a given length scale can be extracted from Ss(k, t). This approach has been useful in studying the kinetic glass transition in colloidal suspensions of spherical particles, where MD simulation results have been compared with the predictions of mode-coupling theory (Barrat et aI., 1990). Similar investigations were conducted by Takeuchi and Okazaki for a model polymer (Takeuchi and Okazaki, 1993b); the decay of Fs(k, t) can be observed only at length scales commensurate with the distance between neighboring carbons on different chains (k = 1.5 A- I); at this length scale, and at the glass temperature determined from the V(l) behavior of the model polymer at a cooling rate of 1.67 X 1011 K/s, the

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

91

a relaxation time determined from a KWW fit to Fs(k, t) was around 0.75 ns. The relaxation time was strongly dependent on temperature. The investigation of structural relaxation over larger length scales is limited by the heavy computational requirements of MD.

III.

PREDICTION OF SORPTION THERMODYNAMICS

A.

Statistical Mechanics of Sorption in a Compressible, Involatile Medium

The phase equilibrium between a polymer and a mUlticomponent fluid mixture can be predicted from molecular level structure and interactions based on straightforward principles of statistical mechanics. To formulate this phase equilibrium problem in a general way, consider the system shown in Fig. 7. Phase 13 (the fluid phase) and phase a (the polymer phase) together constitute a closed system at constant temperature T and pressure P. The system consists of c components. Component 1 is the polymer chains. Components 2, 3, ... , care the fluid molecules whose sorption in the polymer we wish to describe. The total number of molecules of each component in the two-phase system, Ni = N~ + N r (i = 1, 2, . .. , c) is fixed . Each of the components can be exchanged freely between the a and 13 phases, however. Also, the a phase is free to expand (swell) against the 13 phase.

I

Pressure P

- Ie,================:==JI Phase ~

x,II <:: O

xl. X3P,..., xPc

--

-=- _-~~ = iempe~ature T /

Phase a Can exchange energy and mass with phase ~ Can expand/contract against phase ~

Figure 7 Thermodynamic system considered in the phase equilibrium formulation of Section IIl.A.

92

THEODOROU

By the Gibbs rule, the intensive properties of the two phases are fully specified if one fixes c independently variable intensive properties in the system. As such it is convenient to choose P, T, and c - 2 mole fractions describing the composition of the fluid phase on a polymer-free basis. If c = 2, T and P suffice for specifying the intensive state of the two phases. Let 0 " denote collectively the vector of microscopic degrees of freedom of phase ex for given N~, N ; , . .. , N~. 0 " :; (V" , r") encompasses the volume of phase ex and the Cartesian coordinates for all atoms of all molecules constituting that phase. Similarly, O~ will denote the vector of microscopic degrees of freedom of phase 13 for given N ~, Nt ... , N~. The unnormalized joint probability density governing the configurations of the two phases is N1 ... N, I'T

P

(N"1 , ·

··,

N "c ,

N~

b'

O

N ~' '

,

no "

no ~)

c, ~£,~£.

c

= p ~f ... N~PT (0") p ~r .. N~PT (O~)

TI 8 (N~ + N~ -

N j)

(8)

;=1

where

p ~f

.

N~rr (0") =

1

~! . . ·Fc!

TI A!,om all

aloms

stands for the unnormalized configuration-space probability density of phase ex in the isothermal-isobaric ensemble (McQuarrie, 1976) with "If the potential energy function. The product of thermal wavelengths A raised to the third power arises from integration over all momentum space; it contains as many terms as there are atoms in phase ex in the configuration considered. The term p ~r ... N~1'1' similarly stands for the unnormalized configuration-space probability density of phase 13. It is useful to consider the projected probability density for phase ex alone, integrated over all possible configurations that phase 13 can assume. This is PN,,'

. . . N, PT

=

(N"h

. . • ,

N"C) ~" no")

p~I' . N~I'1' (0")

f dn~ p~'-NI' . N,-N~PT (O~)

:; p~f .. N ~P'J' (0") Q~(N, - N~ , ... , Nc - N~, P, T)

(10)

where Q~ is the isothermal-isobaric partition function of phase 13 at the indicated numbers of molecules, pressure, and temperature. We now specialize to the case where component 1 is present in phase 13 only in minute quantities. This would be expected, as the vapor pressure of a pure

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

93

long-chain polymer is extremely low:

N~ =

and

°

(11)

Furthermore, we will assume that phase 13 is very much larger in extent than a ; therefore, for all components other than 1, N~

(12)

and

«Ni

In view of Eqs. (11) and (12), the logarithm of the quantity Ne - N ~ , P, T) appearing in Eq. (10) can be written as

=

(a In Q~(O, N aN.

In Q (0 N N P T) _ ~ ~ , 2, ·· ·, e, , ?

2 , ••• ,

.=2

• • • ,

N e , P, 1)

N e , P,

+ 13 L

Nf

-

N~,

T))

I

e

= -I3G~(O, N 2 ,

Q~(NI

fLr (T, P, x~)

. .. ,

N ?"

•

Hj .P,T

(13)

i=2

where G ~ stands for the Gibbs energy of the 13 phase and J..Lr for the chemical potential of component i in that phase. The composition vector x~ at which the chemical potentials are evaluated has components

x~ == (0,

N2 e

N3 ,

' . ..

e

LNi LNi ;=2

;=2

'~)

(14)

LNi ;=2

Substituting Eq. (13) into Eq. (10), we can write the normalized configurationspace distribution of phase a as a PNalNZ .. N.PT (Nah N 2 ,

.• . ,

N a11 , 01£ £loa) e

p ~I~ .. . N~PT (oa)

IT exp [I3N~ J..L r(T, P, X~)] ;=2

(15) Letting the extent of phase 13 become indefinitely large [Eq. (12)], we can substitute the upper limits of the summations over numbers of molecules in Eq. (15) by infinity. In this limit, the composition of phase 13 becomes constant, given by Eq. (14). This composition enters Eq. (15) exclusively through the chemical potentials J..L r. The probability density for phase a defined by Eg. (15) is reminiscent of the grand canonical ensemble, the only difference being that

94

THEODOROU

the amount of polymer NJ remains fixed at all times while the volume is allowed to fluctuate. In applications it is more convenient to use the fugacities H(T, P, x~) in place of the chemical potentials J.L f (T, P, xII) (Prausnitz et aI., 1986). Substituting p ~IN~ ... N~P,/, from Eq. (9), we can rewrite Eq. (15) as ' . . /g,,'I' (N a . ra va a) a: 2 ,···,lV c " r P NI/~

=~(Na .....

I>

r

1 X

1 t il

2 , .. ·,

exp (-

(va) '~lN~nl

(1) Il [(va)";H]Nt c

P T)

c>'

N al 2'

...

N al c ·

;=2

k0 TZ inlr ; •

pya) exp [ _ _1 'V(r koT koT

a )]

(16)

where

x

[dVaIT [(va)", L~JfIt o

X

/.2

k. T Z;

_ _ 1-,---_ CXp

"

(Va) 1~2 N~",

Pya) L exp [-

( - -k. T

(""

1, '" (r) • ] d 3 ..f , N;II , r -

k. T

(17) In Eqs. (16) and (17), n; is the number of atoms of which a molecule of species i is composed. For a given combination of {N~ } values, the vector of atomic positions r a has dimensionality 3 2:~=1 N~ n;. z ;nl,. is an intramolecular configurational integral for a molecule of type i, calculated as Z iinl,. -=

f

inlra ( 3 3 exp [(.lOjr -..., v i r ih ... , r; n j- J)] d 'il d' r i2

. . .

d 3 r i 'Ii- l

(18)

Note that the integral of Eq. (18) is taken over only n; - 1 atomic positions; the three degrees of freedom corresponding to overall translation of the molecule are not integrated over. The term 'V ;nlra is the intramolecular potential energy of a molecule of type i; in the ideal gas state the total potential energy of phase ~ is merely a sum of such terms. For a monatomic species, z ;nlr. is simply 1. The symbol beneath the integral over all atomic positions in Eq. (17) indicates that the limits of integration for each atom coincide with the boundaries of phase a. One should note that the c - 1 fugacities appearing in Eq. (17) are all functions of the c - 2 independent mole fractions specifying the composition of phase ~ and of temperature and pressure. By the requirements of phase equilibrium, the fugacities of all species are the same in phases a and ~. In the following we

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

95

drop the symbol a with the understanding that we focus on the polymer phase. Some condensation of notation can be achieved by introducing the quantity (19) ~ has dimensions of (volumer"'. Rewriting the vector of atomic positions r as (rl> r 2, . . . , r e), where r j is the 3Njn j-dimensional vector of atomic positions of all molecules of type i, we recast Eq. (16) as

! ...! (n ~) (n ~~;) [dvex Nl'JJ

NcrtAJ

,::2

N,.

,.2

p (- I3PV)

0

f

d"" " " . . .

f d3N"" "e -~"(n

. ... "')

(20) p N,I> .. · f.,pT has dimensions of (volumerl - l:N'i.'N~'i, as expected from the fact that it is a probability density in the variables {N2' .. . , N e , V, r 1, • •• , re}. All thermodynamic properties pertaining to the equilibrium at T and P between the polymer phase, originally consisting of component 1, and the fluid phase, consisting of components 2, 3, ... , Ne at a fixed composition x~, can be extracted from the probability density of Eq. (20). In the following discussion we specialize to the case of a pure sorbate (c = 2). For greater clarity we use the subscripts P and A in place of 1 (polymer) and 2 (sorbate), respectively. The probability density function for the polymer phase, Eq. (20), reduces to

~ ~!~~A

r

dV exp(-J3PV)

f

d

3N "' Pr p

f

d

3N

N'A rA

exp [-J3"V(rp, r A)]

(21) The potential energy function of the polymer/sorbate system can be written in general as Np

"V (rp , r A )

= L "V~"m (rpj) + i_ I

Nt'

NI ,

L L ;=)

1'=;+ I

NA

"V~''' (rpi' r pi .) +

L "V~In(rAk) k=1

(22) where the 3np-dimensional vector rpj encompasses the position vectors of all atoms constituting the macromolecule i and the 3nA -dimensional vector r Ak en-

96

THEODOROU

compasses the position vectors of all atoms in penetrant molecule k. A classical flexible model (Go and Scheraga, 1976) is assumed for the description of the configuration of all molecules. Of particular interest is the prediction of the sorption isotherm of A in the polymer phase. This could be accomplished through a series of Monte Carlo simulations in the NpfAPT ensemble, all carried out at the same amount of polymer N p and temperature T. Each simulation would be at a different P value. Given P and T, the fugacity f A is known through the equation of state of the pure fluid sorbate A. The MC simulation would employ the following elementary moves: translation, rotation, and conformational rearrangement of a polymer chain (bringing about changes in r pi and carried out as in a pure polymer simulation); translations, rotations, and conformational rearrangements of the sorbate molecules (bringing about changes in rAj); insertion of an A molecule (increasing N A by 1); deletion of an A molecule (decreasing N A by 1); and dilations/ contractions of the simulation box (changing V) . The acceptance criteria to be used with each of these moves can be extracted directly (Allen and Tildesley, 1987) from the probability density, Eq. (21). In fact, such an NdAPT simulation can be viewed as a hybrid between isothermal-isobaric and grand canonical MC methods (Allen and Tildesley, 1987). Observables would include the average volume (V) of the polymer phase (providing a direct measure of swelling phenomena) and the average number of sorbate molecules (NA ) present in the polymer at each pressure, providing the sorption isotherm. Results from such a Monte Carlo simulation approach for the direct prediction of the sorption isotherm have not yet been reported, but the approach has been developed (Boone, 1995). The slope of the sorption isotherm in the limit of very low pressures (Henry's law region) can be obtained through a simpler calculation. To derive an expression for the Henry's constant, one can think as follows. From Eq. (21) the average number of A particles present in the polymer phase at equilibrium under given N p , P , T is

(NA) =

i

-

NA

(~ dVJ d3NpnP rp J

d3NAnArA pNI'fAIYT(Nru V, rio, r A )

1

Z(Nr, 1, P, 1) + (2~~/2!) Z(Np, 2, P, 1) + ... =Z(Np, 0, P, 1) + ~ Z(Np, 1, P, 1) + (~~/2!) Z(NI" 2, P, 1) + .. . ~A

~

where

Z(Np, NAl P, T) =

L~ dVexp(-j3PV) Jd 3Npnp rl' Jd 3rA1 X

J

d 3 r A2'" d 3r ANA exp[ -j3'V(rp, r A lI

(24) •• • ,

rANA)]

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

97

is the configurational integral in the isothermal-isobaric ensemble for N p molecules of polymer and NA molecules of penetrant, with dimensions of (volume) Npnp+NApIIA+ I. Consider now the polymer phase in the limit P -+ O. In this limit, fA -+ 0, and by Eq. (19), ~ -+ O. Equation (23) then reduces to \ _Z('-c N:..:.. ,., ~l,_Pc.... ' 1),.,'. ( NAJ = ~ Z(Np, 0, P, 1)

P k. T

1"

dVexp ( - j3PV)

l~ dVexp(-

X

J J d 'N"" rp

j3PV)

d"'ArA, exp [ - j3'VrCr p)

J

d'N"" rp exp [- j3'Vp(r,.)]

- j3'V~' ''(r~,) -

J

exp

j3'V"A(r p, r A,)1

l-j3'V~' '' (r~,)] d3("A-')r~, (25)

The term 'V~'ra in Eq. (25) is the intramolecular potential energy of a single sorbate molecule 1; it is a function of only nA - 1 atomic coordinates, since it is invariant to rigid translations of the molecule [compare Eq. (18) and following discussion]. We have substituted the shorthand notation r~1 for the 3(nA - 1)dimensional vector (rAil' r A1 2, .. . , rAInA- I)' It follows that rAI == (r~1> rAIn.)' The term 'V p encompasses all intramolecular and intermolecular interactions of the macromolecules constituting the polymer matrix, while 'VI'A is the sum of all intermolecular interactions between the penetrant and the polymer chains. [Compare more explicit notation of Eq. (22).] The average volume of the pure polymer phase at T and P is obtained from the isothermal-isobaric ensemble as

i'" i'" i'"

V

(V) =

J exp (-I3Pv) J

dV

dV

dV

exp (-I3Pv)

d

3NpnP

rp

d 3NpnP r p

exp (- I3Pv)

i'"

dV

J

d

exp (- I3Pv)

exp [-I3'V p(rp)]

exp [ - I3'V p(rp)] (26)

3NpnP

J

rp

J

3 d r A l nA

d 3Npnpr p

exp [-I3'V p(rp)]

exp [ - I3'Vp(rp)]

98

THEODOROU

where we have rewritten the volume Vas an integral over the three-dimensional domain spanned by the position vector r Al nA • Combining Eqs. (25) and (26), and recognizing that P/kn T is the molecular density c ~ in the fluid phase, which behaves as an ideal gas in the considered limit P - 0, we obtain . (NJ So= h m ( ) -p J'- .{} V c"

f f dVexp(-~PV) f f

l~ dV exp(- ~PV)

f

d'N"" r.

d'"" rA , exp[ - W'V.(r. ) -

d lN"" r,.

~'V~'~(r~,)] exp[ - ~'V AP(r,., r

d'"" r A , exp[ - I3'V p(rp) -

A ,)]

~'V~' ''(r~,)l

= (exp[ - ~'VAj.(r,., r''')])Wldom

(27)

As defined in Eq. (27), So is a dimensionless partition coefficient equal to the ratio of molecular concentrations of penetrant in the polymer phase and in the pure sorbate phase in the limit P - 0; it is a direct measure of the low-pressure solubility of the penetrant in the polymer. Equation (27) expresses this partition coefficient as a Widom " test particle insertion" average (Allen and Tildesley, 1987). The averaged quantity is the Boltzmann factor of the potential energy of interaction between a single penetrant molecule and the polymer matrix. The average is taken over all polymer configurations, weighted according to the NPT ensemble of the pure polymer; over all internal configurations of the penetrant, weighted by the Boltzmann factor of the corresponding intramolecular energy; and over all translational degrees of freedom (positions of insertion of the penetrant in the polymer). The latter average is purely spatial; i.e., positions of insertion are chosen randomly from within the pure polymer phase without the polymer' 'feeling" the presence of the penetrant. Note that for a model system in which the penetrant is spherical and interacts with all polymer atoms through hard-sphere repulsive forces only, the partition coefficient of Eq. (27) would reduce to the accessible volume fraction discussed in Section II.B. The low-pressure solubility can be expressed per unit mass rather than per unit volume of polymer. Invoking the definition of the Henry's law constant H A •I, (Prausnitz et aI., 1986),

-

1

H A, p

.

XA

== hm -

P_O f A

•

= hm p_o

(NA ) - Np P

=

(V)

N p k8 T

(exp[ -Inr AP(rl" r AI )])W,dOO1

(28)

Equations (27) and (28) are useful for calculating the low-pressure solubility of the penetrant through Monte Carlo or molecular dynamics simulations of the pure polymer with Wid om insertions of a "test" penetrant molecule. Heat of mixing effects between the polymer and the penetrant can readily be analyzed in the NdAPT ensemble. We define the differential heat of sorption of

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

99

the penetrant at given composition of the polymer phase as the negative of the partial molar heat of mixing: (29) where h ~ and hA are the molar enthalpy of A in the (pure) fluid phase and the partial molar enthalpy of A in the polymer phase, respectively. Straightforward thermodynamic analysis leads to the Clausius-Clapeyron equation QiT, P; xA ) = RT

[Z~(T,

P) _ (PVA)] RT

(inaT n P)

(30) XA.cq

where z~(T, P) = Pv~/RT is the compressibility factor of pure A in the fluid phase, VA is the partial molar volume of A in the polymer phase, and the notation
where h1..p ure and ('V~~rn);g are the molar enthalpy and average potential energy per molecule of pure A in the ideal gas state, N Avo is Avogadro's number, and the brackets denote averages in the N pfAPT ensemble. It is thus possible to calculate the differential heat of sorption by monitoring correlations between the number of penetrant molecules and the microscopic enthalpy in the course of a simulation of the polymer phase carried out in this ensemble. In the Henry 's law region Eq. (31) reduces to

<[Vp + 'V AP + 'V~\ra + PV] exp( - /3'VAI')w;dom (exp(- /3'VAP)W;dom In the above discussion of the statistical mechanics of sorption it was implicitly assumed that the polymer matrix is in thermodynamic equilibrium, so that its configurations are distributed with a probability density proportional to exp [-/3 (Vp(rp) + PV)]. If the matrix is glassy, this density distribution holds separately within each of a large number of disjoint regions in (rp , V) space, in which the glass is locally " locked in" (compare Fig. 1). As pointed out in Section I.B, the relative weight with which each such region contributes to the thermodynamics of sorption at low sorbate concentration depends on the history of formation of the glass. In a simulation, for example, regions around local

100

THEODOROU

minima generated via molecular mechanics through quenching melt configurations are typically assigned equal weights. As the sorbate concentration rises, more and more of these originally disjoint regions become mutually accessible, the configuration-space distribution equilibrating between them. When this equilibration percolates through the configuration space of the polymer/penetrant system, plasticization of the glass occurs. A well-designed Monte Carlo simulation should capture this gradual unlocking of the glassy structure with the moves introduced to sample configuration space.

B.

A Theoretical Treatment of Sorption Thermodynamics Based on Static Model Configurations

In this and the following subsection we review existing theoretical and simulation work on the prediction of sorption thermodynamics in polymer glasses and melts, using the formalism of the previous subsection as a guide. Gusev and Suter (1991) presented an early theoretical treatment of the sorption thermodynamics of small spherical gas molecules in glassy polymers that employs atomistically detailed configurations generated by molecular mechanics. The following assumptions are introduced: The glassy polymer configurations are static; any effects of thermal motion and of the presence of the penetrant on rp are neglected. 2. Penetrant molecules are assumed to reside within sites. A site is defined as a region in the three-dimensional position space r AI that encompasses a local minimum of the potential energy V(rAl; rp) under the considered fixed rp' A given site can be occupied by at most one penetrant molecule; this exclusion principle is the origin for the term " spatial Fermi gas" attributed to the penetrant phase. Exchange of a given molecule between different sites is possible. 3. All interactions between penetrant molecules occupying different sites in the polymer are neglected; i.e., all terms v ~t.r of the potential energy function of Eq. (22) are set to zero.

1.

With these assumptions, the configurational integrals in Eq. (23) can be substituted by (32) where K is the total number of sites in the considered polymer matrix and the configurational integral for site j, calculated as

Z; =

L

3

exp [-V PA(rA1; r p)] d r AI

I

Z; is (33)

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

101

The primes on the summation symbols in Eq. (32) indicate that the sites (jll h, . . . ,jN;) must all be different from each other. Upon substitution in Eg. (23), Eg. (32) gives

i NA~A (i' i' ... i' NA •

N,,=l

NA!

h=l

h= 1

Z}, Z}2 . . . ZJN')

jN,o'

(NA )=---------------------------------

(34) For the structure less spherical sorbates considered here,

£AZ; = .b.-

kBT

f

exp [-j3'V PA (rAJ; rp)] d 3rA1

== bdA

(35)

V

J

Thus, the sorption isotherm emerges from the Gusev-Suter theoretical approach as a sum of Langmuir terms, each associated with a particular site in the static polymer matrix. Different sites are characterized by different bj • The overall Henry's constant is inversely proportional to 'i:,f= lbj • As long as bj « 1, the Langmuir term contributed by site j can be approximated by a straight line. The predicted isotherm then is expected to look like a sum of Henry's law and Langmuir terms, as proposed by the "dual-mode sorption" model (petropoulos, 1994) at not too high pressures. Thus, the energetic and entropic heterogeneity of the sites presented by the polymer to the penetrant is proposed as the origin of dual-mode sorption behavior. Gusev and Suter applied this formalism to the system methane/bisphenol A polycarbonate. Sites j were identified by computing 'V(rA1 ; rp) at the nodes of a three-dimensional grid of 107 points running through each static structure. The borders of sites were determined through threedimensional steepest descent constructions initiated at each of the grid points. (See also Section IV.D.) The site configurational integrals of Eg. (33) were obtained numerically from the potential energy values at the grid points. In describing the dispersive part of the interaction between penetrant and polymer atoms, Gusev and Suter introduced an adjustable parameter kM that characterizes deviations from the Lorentz- Berthelot rule. Reasonable agreement with experiment at 300 K and pressures up to 60 bar was obtained with kM = 0.5. The total sorbed amount was broken up into contributions from families of sites with different l!bj , according to Eq. (34). Sites with l!bj in the range 50- 500 bar were found to be most important; this range lies significantly higher than the

102

THEODOROU

value l/bj = 10 bar obtained from the dual-mode sorption model. Clearly, the Gusev- Suter model cannot predict swelling and plasticization phenomena at high penetrant activities, because it postulates a static polymer matrix. The quantity kM is little more than a fitting parameter that absorbs inadequacies of the approximations invoked. Nevertheless, the model constitutes an interesting and computationally inexpensive first attempt to capture the physics of high-occupancy sorption in glassy polymers through molecular modeling.

C.

Molecular Dynamics and Monte Carlo Simulations of Sorption Equilibria

We now discuss attempts to predict the solubility of small penetrants in polymer glasses and melts through atomistic simulations that provide a full numerical solution to the statistical mechanical formalism of phase equilibrium discussed above. Simulation work on solubility has been far less extensive than on diffusivity; this is somewhat surprising, as a good understanding of thermodynamics is usually a prerequisite for dynamical studies. Miiller-Plathe (1991b) was the first to apply the Widom insertion technique to estimate the solubility So through the canonical ensemble counterpart of Eq. (27). He studied Hz, N z, and O2 , represented explicitly as diatomic dumbbells, within atactic polypropylene at 300 K, which is roughly 45 K above the experimental glass temperature. Samples of the amorphous polymer were generated by molecular mechanics, relaxed with NVF MD for 200 ps, and then subjected to a 2 ns long NVT MD production run; configurations from this run were used for the Widom insertions. Results were reported in the form of free energies j.Lex = -kBT In So. Predicted values of j.Lcx were found to decrease in the correct order He > H2 > N2 > O2 > CH4 (order of increasing condensability). The absolute values of j.L"", however, were lower (more favorable for sorption) by 5-10 kllmol than values obtained from experimental data from sorption in ethylene-propylene copolymer and in the amorphous fraction of isotactic polypropylene. In fact, for N2 and the more condensable gases it is predicted that So > 1, contrary to the experimental finding So < 1. Differences in j.L"" between different penetrants were in more reasonable agreement with experiment. All results were based on a single starting configuration for the polymer. A more reliable approach would be to average over many different starting configurations; it seems unlikely, however, that such averaging would bring the computed estimates much closer to experimental values. The disparity between prediction and experiment could result from the use of inaccurate potential parameters. In fact, more recent work (Boone, 1995) indicates that the parameter set used by Miiller-Plathe (1991b) for polypropylene is too attractive. Whether the potential parameterization is satisfactory could be judged from the predicted PIT behavior of the pure polymer; unfortunately, however, pressure results are not reported

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

103

along with the sorption study. The computational effort needed for estimating the solubility increases significantly with increasing penetrant size, as expected from the much smaller volumes accessible to larger penetrants (compare Section II.B). Miiller-Plathe finds that j.L'" decreases linearly with the Lennard-Jones well depth E of penetrant-penetrant interactions. This correlation is reminiscent of the linear relation between the logarithm of low-pressure solubility and the penetrant's heat of vaporization suggested by regular solution theory (Petropoulos, 1994). Sok et ai. (1992) also report computations of the solubility of gaseous He and CH4 in a polydimethylsiloxane (PDMS) liquid in the Henry 's law region. So and j.Lex were computed through Widom insertions [see Eq. (27)] in a model PDMS liquid of degree of polymerization 30 undergoing NPT MD simulation. To enhance the efficiency of Widom insertions, a map of the accessible volume was maintained using a 100 X 100 X 100 cubic grid running through the polymer. As in the work of Miiller-Plathe, predicted j.Lex values were too low (attractive) by 7.8 kJ/mol for CH4 and by 3.6 kJ/mol for He. In contrast to the solubilities, diffusivities were predicted in excellent agreement with experiment. (See also Section IV.B.) Qualitatively, the simulation reproduced the experimental finding that the diffusivity of He in PDMS exceeds that of CH4 , but the overall permeability of CH4 is higher owing to its higher solubility. A remarkable set of Monte Carlo computations of alkane solubilities in a polyethylene melt was conducted by de Pablo et ai. (1993). In these studies the linear polyethylene matrix was represented as a liquid of c,. chains undergoing NPT MC simulation, and penetrants in the range Cz-Cs were examined. The solubility in the Henry's law region was computed by the Widom insertion technique, Eq. (28). For the penetrant sizes examined (de Pablo et aI., 1993), a random insertion will almost certainly lead to severely repulsive overlaps with the polymer chains and thus contribute negligibly to the ensemble averages of Eq. (28); an extremely large number of random insertions would have to be carried out to obtain a good estimate of the solubility. To remedy this problem, de Pablo et ai. employed biased insertions based on the CCB scheme. Rather than being thrown randomly into the matrix, the articulated penetrant molecule is "threaded," bond by bond, through accessible regions of the matrix; Eq. (28) can be straightforwardly recast in a way that takes into account this biased sampling and thus serve as a basis for an efficient computation of the solubility. Figure 8 displays the weight fraction Henry's constant Hw = HA.rMp/MA (where Mp and MA are the molecular weights of the polymer and the penetrant, respectively) estimated by de Pablo et ai. (1993) at 1 bar and temperatures of 423 and 513 K as a function of the penetrant chain length. The simulation estimates, based on a potential parameterization that reproduces the PVI properties of long alkanes, are in excellent agreement with experiment. The same authors conducted calculations of solubility at higher pressures, outside the Henry's law regime. In these calculations they used Panagiotopoulos's Gibbs ensemble, de-

104

THEODOROU

8 o

~6

.D ,-...

'-"

,

/~

~

:r::

'-"

.sI

+

4

Data of Maloney & Pmusnitz (1976) & CCB ghost molecule. NPT ensemble

T=423 K

+

. - - - T=513K

+

0

+

6 8 Number of Carbon Atoms

4

10

Figure 8 Henry's constants for alkanes in molten polyethylene as predicted through continuum-configurational-bias Widom insertions within a polymer matrix undergoing NPT Me simulation (de Pablo et aI., 1993) and as measured experimentally by Maloney and Prausnitz. The abscissa is the number of carbon atoms in the penetrant molecule.

signed to sample the configuration-space probability density of Eq. (8) for two phases coexisting at equilibrium at specified pressure, temperature, and total amounts of all species present. In order to make the particle exchange moves between the two phases that are required by this method computationally feasible, de Pablo et a1. used the CCB scheme [CCBG method (Laso et aI., 1992)]. The bias introduced by the use of CCB-based exchanges is removed by appropriate design of the Monte Carlo selection criteria. Computations of the solubility of pentane in polyethylene by CCBG clearly show deviations from Henry ' s law at pressures above 10 bar.

IV. PREDICTION OF DIFFUSIVITY A. Statistical Mechanics of Diffusion A thorough review of the statistical mechanical formulation of diffusion is given by MacElroy (Chapter 1, this VOlume). In this section we summarize some basic relations that are useful in extracting the binary diffusivity and the self-diffusivity from molecular simulations. Consider a binary system consisting of a polymer P and a penetrant A. The entire system is at a uniform temperature and pressure and free of any external force fields. The system as a whole does not move; i.e., macroscopically, the center of mass of the system remains fixed in space. There is, however, a mac-

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

105

roscopic spatial variation in composition that keeps the system away from thermodynamic equilibrium and sets up macroscopically observable fluxes of the two components. We use the symbol Ji (i = A, P) to denote the macroscopic flux of component i in a coordinate frame that remains fixed with respect to the center of mass of the system; this coordinate frame is used throughout our analysis. Ji is measured in molecules per square meter per second. We use the symbol j; to denote the microscopic (molecular) current of component i, that is, the mechanical quantity (36)

e

where V (i is the center-of-mass velocity vector of molecule of species i and Ni is the total number of molecules of species i at a given time within a macroscopic volume element of the system on which we focus our analysis. Linear response theory leads to the following equation for the thermodynamic flux J i in terms of the thermodynamic forces (V IA-A)1W and (V IA-B)r,p, where lA-i stands for the chemical potential of species i, in joules per molecule, and V is the volume of the considered macroscopic element of the system:

-J" = {3/kBT

i~ 0 ,,(0) . j A(t»

+ {_1_

3V kBT

r

dt} (VIA-"h-/'

Jo 0,,(0) . j,,(t»

dt} (VIA-Ph,I'

(37)

The angular brackets in Eq. (37) denote ensemble averages in an equilibrium system that finds itself at the same temperature, pressure, and average composition as the considered volume element of the nonequilibrium system at the considered time. Equation (37) follows directly from combining Eqs. (10) and (14a) of MacElroy (Chapter 1) for the case considered here. It is useful to cast J" in terms of (VIA-A)r,1' only, The Gibbs-Duhem equation, applied to the considered macroscopic element of the system (assumed to be in local thermodynamic equilibrium) leads to (38) where

Xi

is the mole fraction of species i. Combining Eqs. (37) and (38) yields

- J" =3 V :oTXp [xp

f f

(j,,(0) . j A(t) dt - x"

f f

(UO) . jp(t) dt] (\1 fLAh,P

(39)

An entirely analogous equation holds for component P:

- J" = 3V :nTxA [X"

(MO) . j p(t) dt - XI'

(j1'(0) . Mt) dt] (\1fLl'h/'

(40)

106

THEODOROU

Since there is no macroscopic displacement of mass (flow) in the considered volume element, the fluxes J A and J P must satisfy the condition for all t

(41)

with rnA and rnp the masses of a penetrant molecule and a polymer molecule, respectively. Using Eqs. (39) and (40), one can translate the macroscopic condition, Eq. (41), into the microscopic form

i~

([XpjA(O) - x",MO)] . [rn,JA(t) + ml'jp(t)]) dt

=0

(42)

The binary diffusivity or transport diffusivity D in our system is defined through Fick's first law (de Groot and Mazur, 1984; Bird et aI., 1960): (43) where p is the mass density and W i is the mass fraction of species i in the considered macroscopic volume element. Comparing Eqs. (39) and (43) and substituting the chemical potential /-LA in terms of the fugacity fA (Prausnitz et aI., 1986), we obtain a microscopic expression for the binary diffusivity:

(aa InIn fA)

D = -1

XA

W I'

-

XA

T.P

i~ (jA(O)

- 1 [ XI' 3NA

. jp(t)

i~ (jA ' (O)

.

. JA(t) dt

0

dtJ

(44)

The binary diffusivity emerges as a product of a term that depends on the thermodynamics of sorption and the system composition and a second term that incorporates the time integrals of time correlation functions of the microscopic currents of the two components. A more symmetric expression for D can be obtained in terms of the microscopic interdiffusion current (45)

This expression is D = _1_ XAX p

(aa InIn fA) XA

~ T.P

r

3N Jo

(jC(O) . nt) dt

(46)

where N = NA + N p • A form of Eq. (46) was presented by Hansen and McDonald (1986). (Note that a factor of liN is missing from the equation given in that reference.) The equivalence of Eqs. (44) and (46) can be readily shown on the

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

107

basis of Eq. (42), taking into account that the equilibrium average (jAO) . j/,(t» is an even function of time. Equation (46) for the binary diffusivity can be converted from the GreenKubo form to the corresponding Einstein form (Hansen and McDonald, 1986) by invoking the general equilibrium equality

r/

Jo \

St(t)St(O) \ dt = lim

/

1_ 00

.!.2t \/[sfJ.(t) -

sfJ.(O)f \ /

(47)

where sfJ. is any scalar dynamical quantity and the dots denote time derivatives. The Einstein form of Eq. (46) is D = NxAxp

fA) hm. -1 \ {[remAt) -

a In(a In x A

1'./'

1_ 00

6t

rem.p(t)]

(48) with rem..{t) the center of mass of all molecules of species i at time t. The Einstein forms are generally preferable to the Green-Kubo forms for use in equilibrium MD simulations, as they circumvent the need to integrate the time correlation functions, whose long-time tails suffer from noise due to limited sample size (Allen and Tidesley, 1987). The self-diffusivity of species i provides a measure for the displacement of individual molecules of i as a result of random thermal motion. In a system exhibiting diffusive behavior, the mean square displacement of a molecule grows linearly with time at long times. The self-diffusivity is extracted from the proportionality constant in that relationship:

Ds.; = lim 1_'"

{.!.6t \/[re;(t) - re;(OW)}

(49)

In extracting Ds.; from an equilibrium MD simulation, one usually averages Eq. (49) over all molecules of species i. Thus, one has a much larger sample size than is available for the calculation of the collective property D through Eq. (48). Self-diffusivities can therefore be obtained with much better accuracy than binary diffusivities from equilibrium dynamic simulations. The Einstein form, Eq. (49), is equivalent to the Green-Kubo form [compare Eq. (47)] :

e

Ds,; =

~ i'"(v e;(t) . ve;{O»

dt

(50)

The velocity autocorrelation function appearing within the time integral of Eq. (50) expresses how the penetrant molecule "loses memory " of its original motional state through interactions (collisions) with the polymer matrix and (at high

108

THEODOROU

concentrations) with other penetrant molecules. A plot of the normalized velocity autocorrelation function (VeA(O) . VeA(t»/(lv eAI 2 ), as accumulated from equilibrium MD simulations of methane in polyethylene (PE) (Pant and Boyd, 1993), is shown in Figure 9. Since self-diffusivities can be obtained much more readily from simulation, there is incentive for expressing D in terms of DS,A and Ds,p. Substituting the microscopic currents into Eq. (44) from their definition, Eq. (36), we obtain

(a In fA)

1

D =

a In X

Wp

T.P

A

x {Xp [~ +

1 3NA

L~ (VeA(O) . VeA(t»

dt

~ e'~H L~ (VeA(O) . Ve'A(t»

- XA

[~ ~ L~ (VeA(O) . vkP(t»

dtJ dtJ}

(51)

Consider the system in the limit where A is infinitely dilute in P (XA> W A -+ 0). In this limit, the cross-correlation terms between the two species in Eq. (51) drop. Also, different molecules of species A are very far apart; thus, one would expect the velocity correlation terms between different molecules of species A in Eq. (51) to reduce to zero. Furthermore, the thermodynamics of the polymer/ penetrant system in this limit conforms to Henry ' s law (see Section III), and thus the thermodynamic derivative in Eq. (51) reduces to unity. Invoking Eq. (50), we conclude that (52)

lim D = lim D s•A wA-o

Wl\-O

Equation (52) is a rigorous statistical mechanical result. An approximate relation between the binary and the self-diffusivities at any composition may be arrived at by assuming that all velocity correlations between different molecules in Eq. (51) (be they of the same or different species) are negligible. This assumption leads to

D = -XI' Wp

(a- In- fA -) a In X

A

TP

DSA

(53)

,

Under the same approximation, Eq. (53) holds with the subscripts P and A reversed. Combining this counterpart of Eq. (53) with Eq. (53), and recognizing

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

109

-3OOK ---e-- 400 K

.gc u

.E 0.5

.g

1 1il 0

o

~

-0.5

o

1.2

0.8

0.4

t

1.6

2

Cps)

Figure 9 Normalized velocity autocorrelation functions for a methane penetrant in amorphous polyethylene, as accumulated from 100 ps long trajectories at two temperatures. The sign reversal at short times results from the first collision of the penetrant with the cage of polymer atoms surrounding it; it is much sharper at the lower temperature, where the cage is less mobile. The undulations at long times make clear the computational difficulties in calculating the self-diffusivity as a time integral of the velocity autocorrelation function through the Green-Kubo relation. [Reproduced from Pant and Boyd (1993), with permission.)

that the thermodynamic correction term (a In f/a In Xi)1;t. is the same for both species (Gibbs-Duhem equation), one readily arrives at (54) Equation (54) is used frequently to express the composition dependence of the binary diffusivity in liquid mixtures. In a polymer/penetrant system, the self-diffusivity D •.•, of polymer chains is typically much smaller than that of penetrant molecules; its contribution to Eq. (54) can thus be neglected. Equation (53) can be rewritten in a more practically useful form as D

=

(aa InIntA) W

A

D TP

s,A

(55)

110

THEODOROU

The self-diffusivity of the penetrant, D s•A , plays a dominant role in shaping the binary diffusivity.

B.

Equilibrium Molecular Dynamics Simulations

1. Simplified Polyethylene-Like Models Over the past few years there has been much interest in conducting equilibrium MD simulations of amorphous polymer/penetrant systems for the purpose of predicting the self-diffusivity D S•A and elucidating microscopic mechanistic aspects of diffusion. For small molecular weight penetrants in melts or rubbery polymers, diffusivities are often high enough to be captured within the maximum time spans that can be simulated with conventional molecular dynamics (tens of nanoseconds) with present-day computational resources. Early MD studies of gas diffusion employed united-atom alkane- or PE-like models for the polymer. They did not lay much emphasis on the calibration of potential parameters; in fact, the PVT behavior that these simulations yielded was in poor agreement with experiment (density too low for given pressure and temperature), and predicted values of diffusion coefficients were quite far from experimental values. Nevertheless, they revealed interesting relations between self-diffusivity and polymer structure. Takeuchi and Okazaki (1990) carried out microcanonical MD simulations of 20 small spherical molecules resembling oxygen in 30 united-atom 20-mer molecules of "polymethy lene." The model representation of Rigby and Roe (1988, 1989, 1991) was used for the chain molecules. The predicted self-diffusivity was studied as a function of temperature, unoccupied volume, and chain stiffness. The activation energy for selfdiffusion under constant density was found to be approximately twice as high as the activation energy under constant pressure, indicating that the thermal expansion of the polymer (increase in accessible volume with temperature) drastically affects diffusion. The activation energy under constant pressure was found to correlate linearly with the preexponential factor in the Arrhenius expression for the self-diffusivity, as experimentally observed for many polymers. Fractional unoccupied volumes, Vr, were accumulated by subtracting an estimate for the hard-core volume of chains from the total volume of the model system. The self-diffusivity was found to correlate well with Vr according to the simple Fujita equation (56) (see Fig. 10). The relaxation frequency 1h4> of internal rotations, which is on the order of 1011 s, also exhibits a dependence on Vr of the form of Eq. (56); this dependence, however, is considerably weaker than that of the selfdiffusivity. Under the high-temperature melt conditions corresponding to the simulations of Takeuchi and Okazaki, T4> seems dominated by the height of intramolecular

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

\

-3

111

n

\

Q

\?,

o c -4

\

-5

2.4

3.2

2 .8

\ 3 .6

1/ V r

Figure 10

Dependence on the self-diffusion coefficient of small, oxygen-like molecules in a polyethylene-like liquid on the fractional free volume of the liquid, as obtained from equilibrium molecular dynamics simulations. All data are at the same temperature and are given in reduced units. [Reproduced from Takeuchi and Okazaki (1990), with permission.]

torsional barriers. To study the influence of the dynamic flexibility of chains on self-diffusion, Takeuchi and Okazaki performed simulations in a model chain liquid of the same density, where the torsional barriers were artificially removed. 1"", in this modified chain liquid was found to be shorter by more than three orders of magnitude and to exhibit an activation energy one fifth of that exhibited in the presence of the barriers. Upon removal of the torsional barriers, the activation energy for self-diffusion under constant density dropped by a factor of 2, although the overall fractional unoccupied volumes Vr in the original system and in the barrier-less system were almost identical. A similar decrease in the activation energy for diffusion was observed under conditions of constant pressure (see Fig. 11). The diffusivity in the high-temperature melt was thus found to depend on the dynamics of redistribution of unoccupied volume and not only on the amount of unoccupied volume. An interpretation of these results in the light of the Brandt theory of diffusion was proposed by Petropoulos (1994). The effects of matrix chain length were briefly examined by Takeuchi (1990b) by comparing the results of the above simulations with MD trajectories in a PE-

112

THEODOROU

-2 . 5

-3.0

0

-c:

-3.5

-4 . 0

-4.5 0 .19

0.21

0 .23

0.25

1/ T

Figure 11 Temperature dependence of the self-diffusivity of oxygen-like molecules in a polyethylene-like liquid subject to full torsional barriers (open symbols), and in a similar liquid of freely rotating chains (no torsional barriers, filled symbols). All data are at the same pressure and are given in reduced units. [Reproduced from Takeuchi and Okazaki (1990), with permission.]

Like matrix of infinite chain length at the same composition, number density, and temperature. After correcting for the difference in unoccupied volume between the two systems, D S•A and T <j> were found to be lower by a factor of 2 in the long-chain system. The change in D A was thought to stem in part from the difference in chain packing between the two systems. To elucidate the effects of chain packing and the concomitant distribution of unoccupied volume on penetrant self-diffusivity, Takeuchi et al. (1990) undertook MD simulations in a series of infinite-chain matrices in which the equilibrium value of the bond angle e was varied systematically between 100° and 150°. All these "mutated" PE-like systems and the original system (equilibrium e = 109.5°) were studied under the same overall fractional unoccupied volume Vr and temperature. Changes in e induced significant changes in the packing of chains. Increasing e brought about a decrease in the most probable intermolecular distance rmax between mers and sharpened the structure of the polymer liquid, as seen from the intermolecular pair distribution functions. The tendency S•

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

113

for parallel alignment of neighboring chains also increased, as seen from the intermolecular orientation correlation function for skeletal chords. In other words, opening up the bond angles resulted in a tendency for chains to bundle together more tightly. As pointed out by Takeuchi et al. (1990), however, the conformational distribution of the chains in these simulations was not truly equilibrated; it was strongly affected by the artificial boundary condition used to create the infinite-chain models (see Fig. 2b). A local unoccupied volume fraction fv(r) was accumulated as a function of distance r from the center of a reference mer; to define fv(r) , the average hard-core volume of mers within a shell centered at the reference mer of inner radius equal to the mer radius and outer radius r, was subtracted from the volume of the shell. The value vi == h(rmax) was used as a basis for comparing the unoccupied volume distributions of the different polymer models. vi was found to be a decreasing function of e, reflecting the tight, parallel alignment of chains at large e values. The penetrant self-diffusion coefficient was found to increase roughly linearly with the chain spacing rmax and to obey a Fujita-type equation of the form of Eq. (56) as a function of vi. Given that segmental mobility, as quantified through 'T<\> ' was comparable among all model polymers, this effect was viewed as a consequence of differences in the spatial distribution of unoccupied volume, as opposed to differences in its dynamics. An attempt to correlate D S•A with the accessible volume distribution of the polymer matrix at a probe radius corresponding to percolation was presented by Takeuchi and Okazaki (1993a). (See Section II.B.) MD simulations of diffusion of two different spherical penetrants (roughly corresponding to O2 and He) were performed in three different infinite-chain polymer matrices (characterized by different e values) over a variety of densities (therefore, of free-volume fractions). In parallel, the accessible volume distributions at a penetrant radius rc corresponding to percolation in each polymer matrix were analyzed based on MD simulations of the pure matrix. For each penetrant, the self-diffusivity was found to correlate with the total number of clusters nc present at percolation as D A = An exp (-Ell nc), where the constants An and En depend on the penetrant but not on the matrix (i.e., are universal for all matrices examined). The numberweighted density distribution of cluster volumes v at percolation was found to depend on v as V - M I. This result is suspect, as it indicates that the volumeweighted density distribution of clusters piv;rc) should be an increasing function of v, contradicting recent extensive studies in a similar polymer (Greenfield and Theodorou, 1993). Subject to this caveat, the quantity log (D A m~) was found to vary linearly with the fraction <X of accessible volume at percolation that is contained in clusters larger than the hard-core volume of the penetrant. Data from both penetrants in all polymers collapsed onto a single line when plotted in this way. This result was interpreted as lending support to the free-volume theory, according to which D A is proportional to both the kinetic velocity of S•

S•

S•

114

THEODOROU

the penetrant and the probability of finding an accessible hole of sufficient volume for the penetrant to hop into. Molecular dynamics simulations of diffusion of spherical CO 2-like molecules in a PE-like matrix corresponding to the model of Rigby and Roe were also carried out by Trohalaki et al. (1989, 1991, 1992). The formulation for obtaining the binary diffusivity from the self-diffusivities of penetrant and polymer was applied in this work. Sonnenburg et al. (1990) simulated the diffusion of spherical penetrants in an idealized polymer network in the rubbery regime. The temperature dependence of D exhibited Arrhenius behavior, while its dependence on the penetrant diameter agreed qualitatively with experimental observations. The diffusion exhibited stronger temperature dependence in the network than in the simple liquid formed by eliminating all covalent bonds. At low density values, diffusion was faster in the simple liquid than in the network, as one would expect from the reduction in atomic mobility caused by the covalent bonds. At a certain value of the density, however, a crossover occurred. Beyond that value, diffusion proceeds at a measurable rate in the network, while it is practically absent in the simple liquid. This somewhat unintuitive result seems to originate in the intrinsically more inhomogeneous distribution of accessible volume in a polymer as compared to a monomeric substance (compare Fig. 5 and associated discussion in Section II.B.)

2. More Realistic Melt Models The latest equilibrium MD work on gas diffusion in polymer melts and rubbery polymers has been characterized by a trend to use more elaborate representations of the polymer matrix and penetrant species, in an effort to predict self-diffusion coefficients quantitatively. In parallel, the mechanism of motion of the penetrant has been explored through detailed analysis of MD trajectories. Miiller-Plathe (1991a) simulated 20 methane molecules in a system of 20 united-atom pentacontane (Cso) chains at 300 K using the parameterization provided by a commercial simulation package. The configurations of the chain liquid were relaxed through energy minimization and MD. Diffusion was found to proceed through a " hopping" mechanism, the displacement of the penetrant occurring mainly through infrequent jumps of length less than the penetrant diameter. The duration of a jump event (a few picoseconds) was very short relative to the time spent between jumps. Fast recrossings of a penetrant back into its original position following a jump were also observed. The infrequent nature of jump events makes the estimation of the self-diffusivity problematic, as very long simulations would be required to accumulate sufficient statistics. To address this problem, Miiller-Plathe considered a scaling factor A. multiplying the penetrant/polymer potential "If AP. The logarithm of D A was found to decrease linearly with A. in the range 0.1 :5 A. :5 1.0, and the associated correlation is S•

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

115

proposed as a means for estimating diffusional characteristics efficiently through simulation. A more recent investigation (Miiller-Plathe, 1992) focused on H 2, O2 , and C~ in atactic polypropylene at 300 K, generated by an MM method (Theodorou and Suter, 1985) and subsequently relaxed by molecular dynamics. In comparison with PE, this polymer has the advantage of being noncrystallizable, which makes the validation of the simulation results straightforward, subject to the availability of experimental data. A rather detailed model was used, with united-atom methyl groups but with all other atoms treated explicitly, and production runs as long as 2 ns were conducted. Of the three penetrants studied, the largest (C~) exhibited the most expressed hopping motion; the motion of H2 was more akin to a continuous sequence of small random displacements, as one finds in a liquid. (See Fig. 16 of MacElroy, Chapter 1, this volume.) The logarithm of the predicted self-diffusion coefficient was found to drop almost linearly with increasing penetrant diameter, in agreement with some empirical correlations. Comparison with available experimental data from atactic polypropylene and similar polymers led to the conclusion that the diffusivities of H2 and O2 were overpredicted by factors of roughJy 8 and 3, respectively, whereas that of CH4 was very close to experimental. This trend of improved agreement with experiment with increasing penetrant radius led Miiller-Plathe et al. (1992a) to investigate the influence of the potential representation of the polymer on the estimated diffusion coefficients. Predictions for the self-diffusivity of O2 were compared in amorphous polyisobutylene using a fully explicit representation and a representation wherein methyls are lumped into united atoms. It was concluded that the use of the united-atom approximation leads to an overestimation of the self-diffusivity and that a fully explicit polymer model is necessary for estimating the diffusivity correctly. The interpretation proposed is that united-atom groups give rise to artificially large interstitial sites within the polymer, thereby accelerating diffusion. This sensitivity to the polymer representation should be greatest for small penetrants that can explore small interstitial cavities and should go away for larger penetrants; this seems supported by the fact that the diffusivity of CH4 was correctly captured in a polypropylene model with united-atom methyls. No information is given, however, on whether the models invoked were capable of reproducing the experimental PIT thermodynamics of the pure polymer, which should be a prerequisite for any simulation that aims at capturing sorption and diffusion phenomena quantitatively. Sok et al. (1992) simulated the transport of He and C~ in polydimethylsiloxane (PDMS) at 300 K. A rather detailed model with united-atom methyls but with all other atoms explicit, incorporating both Lennard-Jones and Coulombic interactions among partial charges, was employed for the chains. In the flexible, mobile PDMS, significant diffusion can be observed over relatively short MD trajectories (= 250 ps). The penetrants (especially methane) exhibited a jumplike process analogous to the one observed by Miiller-Plathe. Through analysis and

116

THEODOROU

visualization of trajectories, it was confirmed that the penetrant spends most of its time within fluctuating cavities in the polymer liquid. A jump occurs when fluctuations in the chains momentarily open a channel between two cavities. Eliminating the motion of the chains (i.e., freezing the polymer configuration) led to trapping of both penetrants, which were only able to "rattle" within their cavities. The importance of fluctuations of the polymer matrix for diffusion was thus confirmed. The predicted self-diffusivities were in excellent agreement with experiment (especially for CH4 ), although, as mentioned in Section III.B, the predicted solubilities were much less satisfactory. The motion of penetrant molecules within cavities and the jumps between cavities have been probed in detail for a two-site O2 model in polyisobutylene (PIB) by Miiller-Plathe and van Gunsteren (1992). Cavities were found to be wide enough to allow relatively uninhibited motion of the O2 molecule while at the same time firmly opposing its long-range translational motion. Penetrant molecules participating in a "jump" (defined as a translation by more than 5 A over 1 ps) were found to be translationally "hotter" than average by 350 ± 100 K, which is taken as an indication that the molecule has to overcome an energy barrier to perform the jump. This interpretation is not conclusive, however, as jumping molecules fulfil a minimum velocity requirement and are therefore hotter by definition. Both end-on hops (with the molecular axis parallel to the direction of translational motion) and edge-on hops were observed. Boyd and Pant (1991) studied the diffusion of methane in PIB and PE using a united-atom representation. Model matrices consisting of 24 skeletal unit long PIB and PE chains were created starting from a crystalline arrangement by reptation-based NPT MC (Boyd, 1989) followed by NVT MD. The simulations allowed for interesting structural comparisons between the two polymers. Whereas the intermolecular pair distribution functions in PE reveal a distance of - 5.5 A between the backbones of neighboring chains, the corresponding distance in PIB exceeds 7 A. Chains are compared to cylinders with a dense core and less dense surface region; the larger diameter of the core in PIB explains the better packing and higher mass density in that molecule. The predicted self-diffusion coefficients were found to be very sensitive to the matrix density. Diffusion was predicted to be slower in PIB than in PE, as seen experimentally. The absolute values of D, however, were overpredicted and the corresponding activation energies underpredicted by roughly a factor of 4. To investigate this disparity between predicted and observed values of the diffusivity in recent MD work, Pant and Boyd (1992) conducted a study of both the diffusivity and the polymer melt's equation-of-state behavior as a function of the potential parameterization used for the chains. Chains of infinite molecular weight were used in this investigation. The conclusion was that past united-atom simulations of PE had been conducted with potentials leading to incorrect equation-of-state behavior (e.g., highly negative pressures at experimental densities); the use of

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

117

an anisotropic united-atom (AVA) potential that can capture the PVT behavior of the matrix was found to also give very reasonable values for the self-diffusivity. Thus, the ability of a potential to describe the thermodynamic properties of the pure polymer was established as a prerequisite for the quantitative prediction of penetrant diffusivity. A thorough investigation of methane diffusion in PE and PIB at a variety of temperatures, using an AVA potential representation that can correctly capture PVT properties, was presented by Pant and Boyd (1993). The predicted diffusivities and activation energies in both polymers are in excellent agreement with experiment. (In the case of low-temperature PE, an analysis based on effective medium theory was conducted to extract the diffusivity in amorphous regions from experimental data using the crystalline volume fraction of the polymer.) Diffusivities in amorphous PE were found to follow a distinctly non-Arrhenius temperature dependence that can be described by a WLF-type expression (see Fig. 12). Diffusivities in PIB, on the other hand, exhibited an Arrhenius temperature dependence. The mean square displacement (r2) as a function of time was found to exhibit three regimes. At very short times (tenths of a picosecond), fast ballistic motion of the penetrant is observed between collisions with polymer atoms «r2) ex: (2), which causes the mean square displacement to rise very quickly to a finite offset. This is followed by a rela-

-7.5 -8.0

..-. u Q)

......!!!..

5

-8.5 -9.0 0 0

-9.5

x

C bi)

.3

-10.0

m

-10.5

•

+

BPE (MB) BPE (LWH) BPE (KH) LPE (MB) LPE (L) simulation WLFfit

-11.0 -11.5 1.5

2

2.5

3

3.5

4

1000 IT

Figure 12

Diffusion constants for methane in amorphous PE from both experiment and MD simulation. Filled circles: simulation results. Other symbols: experimental values from various sources, corrected for crystallinity wherever necessaTy. The temperature dependence of both simulation and experimental points is of a WLF type. [Reproduced from Pant and Boyd (1993), with permission.]

118

THEODOROU

tively extended regime, wherein the mean square displacement grows less than linearly with time (Le., (r2) oc t with v < 1), but where the rate of growth d(r2)/ dt exceeds that observed in the long-time diffusive regime (6D s•A ). The origin of this sub diffusive region, which Pant and Boyd refer to as the " cage effect" region, is discussed in more detail in Section IIV.D. At low temperatures (e.g., 280 K in PE), this subdiffusive region extends out to times on the order of 200-300 ps «r2 )112 = 3-4 A), beyond which a clearly diffusive region (d(r2)/ dt = const. = 6D s•A ) sets in. Pant and Boyd (1993) undertake a very informative analysis of the penetrant's motion as a function of temperature. To free the data of the noise associated with " rattling" within clusters of accessible volume, they average the penetrant position r A over successive time intervals of duration T. They show that if T is chosen long enough, successive changes in the averaged position of the penetrant are un correlated, i.e., the averaged trajectory behaves like a random walk; T values of 16 ps at 300 K and of 7.5 ps at 400 K are sufficient for this to happen. Plots of the squared displacement of the penetrant between successive steps along such an averaged trajectory appear strikingly different at different temperatures. In the low-temperature melt (Fig. 13a), one sees infrequent large jumps (as long as 6 A in the figure) separated by long periods of quiescence. Jumps longer than the radius of the penetrant account for 76% of the total diffusive progress of methane in PE at 280 K. In the hightemperature melt, on the other hand (Fig. 13b), jumplike displacements of the penetrant become very frequent. The penetrant is no longer trapped in clusters of accessible volume for long periods of time but rather seems to be carried along by rapid fluctuations of the accessible volume. (Compare also Fig. 6 and associated discussion.) This change in mechanism of the diffusion process with changing temperature in an amorphous matrix, identified by Pant and Boyd, is one of the most important results obtained through atomistic MD simulations of amorphous polymers. In contrast to PE, PIB displayed a behavior comparable to that illustrated in Fig. 13a throughout the temperature range studied (350600 K). The rate of redistribution of accessible volume in that polymer was very slow, and the onset of the diffusive regime took longer to be established than in PE at comparable temperatures.

C.

Nonequilibrium Molecular Dynamics

The simulation method of non equilibrium molecular dynamics (NEMD) was applied for the first time only very recently to polymer/penetrant systems by Miiller-Plathe et al. (1993). In contrast to equilibrium MD, NEMD imposes an external driving force that keeps the model system away from equilibrium and measures directly the flux elicited by this force . If the imposed force is small enough for the system to remain in the linear response regime, the associated transport coefficient can be obtained directly from the ratio of flux to driving

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

119

35.0 30.0 25.0

N

20.0

c;::!

15.0

-

-S 10.0 5.0 0.0 100

0

200

300

400

500

600

700

500

600

700

800

t (ps)

(a)

70.0 60.0 50.0

N

40.0

...

30.0

-S N

-

20.0 10.0 0.0 0 (b)

100

200

3<Xl

400 I (pS)

800

Figure 13 Displacement history for a single methane penetrant molecule in polyethylene. (a) T = 300 K. The squared displacement is computed between successive 16 ps positional averages of the penetrant. (b) T = 400 K. The squared displacement is computed between successive 7.5 ps positional averages of the penetrant. [Reproduced from Pant and Boyd (1993), with permission.]

r

force . In the case of diffusion, the most common driving force used is a spatially uniform force field F acting directly on each molecule. In an equimolar binary mixture, for example, equal and opposite forces may be applied on molecules of the two species, as if the molecules were bearing equal and opposite charges or " colors" in a uniform electric or "color" field; the system center of mass remains fixed, and the steady-state fluxes of the two species provide an estimate

120

THEODOROU

of the transport diffusivity D . In liquids (Evans and Morris, 1990) and microporous sorbents (Maginn et aI. , 1993), NEMD routes to D have proved more effective than equilibrium routes to the same property based on accumulating equilibrium autocorrelation functions of molecular currents or species center-ofmass displacements [compare Eqs. (46)-(48)] . Owing to their collective nature, these equilibrium quantities can be obtained with only limited accuracy. To understand the basis of NEMD methods, it is useful to rewrite Pick's law, Eq. (43), as a relationship between mass flux and chemical potential: A

p

W W mAJ A = - pD -

1

(

) ( -1-) (Vj.LA).rp ks T .

[(a In f A)/(a In xA)h-.p

Xp

(57)

The externally imposed force F in NEMD plays the role of the negative gradient in chemical potential. The flux J A is calculated as the steady-state nonequilibrium ensemble-average molecular flux (jA(t - oo)neq. Equation (57) then leads to D--X-p Wp

(a-a IIn- f-A) n

XA

1;1'

-VksT -F- (J.A ,a ( t_oo)ncq N A

(58)

a

In Eq. (58) it is assumed that the driving force is applied in the a direction and that the resulting molecular flux is in the same direction. In the case of aniso-

tropic diffusion, all elements Da~ of the diffusion tensor can be computed by applying the driving force in the [3 direction and measuring the steady-state flux in the a direction. In the limit of infinitely dilute penetrant in the polymer, Eq. (58) simplifies to (59) Miiller-Plathe et al. (1993) employed Eq. (59) to estimate the diffusivity of H2 , He, and O2 in pm through NEMD simulations. They applied no force on the polymer molecules but rather chose to force half the penetrant molecules with F and the other half with - F; the velocities of the latter half were inverted in computing j A' Results from the NEMD simulation were not very encouraging. The value of (j A(t - oo)ncq, as computed by averaging over 30 ps long pieces of the simulation, was found to fluctuate widely, especially at large field strengths. The linear response regime broke down at relatively low values of F, and the kinetic energy of penetrant molecules was found to rise artificially as a result of the imposition of F, It is likely that future applications of NEMD employing many configurations of the polymer and averaging over long times will lead to better results. Of particular interest for future NEMD investigations would be the problem of predicting the transport diffusivity at high occupancies via Eq. (58).

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

D.

121

Transition State Theory-Based Prediction of Low Concentration Diffusivity

1. Jumplike Nature of Low-Temperature Diffusion As temperature is reduced toward and below T g , penetrant diffusion through an amorphous polymer matrix becomes too slow to be predictable by MD simulations. A simple " back of the envelope" calculation can convince us of this. Consider the diffusion of CO 2 gas in glassy (un plasticized) PVC at low concentration and a temperature of 25°C. To obtain a reliable diffusivity from MD, one should let the penetrant molecules move for sufficient time to sample the frozenin inhomogeneities of the glassy polymer structure. For this to happen, the translational displacement of molecules must be at least equal to the mean distance between neighboring clusters of accessible volume, which we can most conservatively estimate as 5 A (compare Fig. 6). Experimentally, the diffusivity in the considered system has been measured as D s,A = 2.4 X 10- 9 cm 2js (Berens and Huvard, 1987). This means that in order to safely estimate the diffusivity, one would need an equilibrium MD simulation of duration at least (r2?

t

=- - = 6 D S•A

(5

;.y

6 (2.4 X 10

9

cm2 js)

= 1.7

X

10- 7 s

= 170 ns

(60)

which exceeds by an order of magnitude the duration of the longest atomistic polymer MD simulations that have been conducted. Actually, the MD simulation should be conducted over a variety of starting positions of the penetrant and over a variety of configurations of the matrix, so as to sample the wide distribution of local environments experienced by the penetrant. This would multiply the time required to estimate D by a couple of orders of magnitude relative to Eq. (60) unless all these different runs were conducted on a massively parallel machine. A molecule larger than CO2 would require even more time for the first-principles estimation of its self-diffusivity through equilibrium MD. As already pointed out in Section IV.B, the reason that diffusion through such a low-temperature amorphous polymer matrix is so slow is that the penetrant spends most of its time "trapped" within clusters of accessible volume in the matrix and only infrequently jumps from cluster to cluster. A brute-force MD simulation exhausts itself in tracking the relatively uninteresting "rattling" motions of the penetrant within a cluster but is much too short to accumulate sufficient statistics on the jumps that govern diffusion. This was shown characteristically in the work of Takeuchi (1990a), which was the first to explore the mechanism of the jump process in a glassy polymer. Oxygen-like penetrant molecules in a polyethylene-like, infinite molecular weight matrix were simulated through atomistic MD 30°C below the simulated glass temperature. Only a single penetrant molecule was seen to execute jumps between two sites in the polymer. The unoccupied volume in the immediate vicinity of this molecule did

122

THEODOROU

not show any significant deviation from the unoccupied volume around molecules that remained " trapped" during the entire duration of the simulation. Furthermore, the potential energy felt by the jumping molecule (i.e., its contribution to "If" AP(rA' rp)] did not show signs of overcoming a barrier. Motions of the polymer matrix were very important in effecting a jump. Takeuchi has given a vivid graphical representation of a jump process in terms of the evolution of the contours of potential energy experienced by the penetrant. [See Fig. 17 of Chapter 1, this volume).] The process starts with the generation of a channel connecting two clusters of accessible volume through thermal fluctuations. The channel is quickly traversed by the molecule and subsequently disappears. The jump length is on the order of 5 A, and the overall duration of the process is around 10 ps. We emphasize here that this jump process is not confined to diffusion in polymers below Tg • As shown by the subsequent simulations of Miiller-Plathe (1992), Miiller-Plathe and van Gunsteren (1992), Pant and Boyd (1993), and Sok et al. (1992), it also occurs in low-temperature melts and rubbery polymers. When the temperature is sufficiently low for the distribution of accessible volume clusters to remain relatively unchanged over the time scale required for the penetrant to move by the mean distance between clusters, a jumplike situation should ensue (compare Fig. 6). The above observations on the mechanism of low-temperature diffusion of small molecules in glassy polymers suggest a transition state theory description of diffusion as a sequence of infrequent jump events. Each jump event involves a relatively small number of degrees of freedom in the configuration space (r p , r A) of the polymer/penetrant system, as shown schematically in Fig. 14a. To facilitate our discussion, we consider a structureless spherical penetrant in a model glassy polymer matrix with three-dimensional periodic boundary conditions, represented by the flexible chain model (Go and Scheraga, 1976). Let v = 3 N p np + 3 - 3 = 3 Npnp , the total number of degrees of freedom of this system. (The 3 is subtracted because the total potential energy is invariant to rigid translations of the whole system. The three degrees of freedom of the penetrant are defined relative to a reference atom in the polymer matrix.) We will use the term state to refer to the region around a local minimum of the potential energy "If"(rp, r A) of the polymer/penetrant system. The projections of the state minima onto the subspace rp of degrees of freedom of the configurationally arrested polymer will be close to, but not identical with, the minima about which the pure polymer configuration fluctuates in the absence of penetrant (compare Fig. 1). In other words, the likely polymer/penetrant configurations will differ somewhat from the pure polymer configurations because polymer degrees of freedom around the penetrant are locally modified to accommodate the penetrant; the larger the penetrant, the more extensive this modification (see Fig. 14a).

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

--

-Macrostate i

123

Transition State

Macrostate j

(a)

(b)

Figure 14 (a) Schematic of a penetrant jump in a low-temperature amorphous model polymer matrix. A two-dimensional representation is given, for clarity. Shaded regions are occupied by chain segments, and periodic boundary conditions are indicated. The system polymer/penetrant, initiaJly in macrostate i, passes into macrostate j by crossing a dividing hypersurface that separates the two macrostates and contains a first-order saddle point (transition state) of the potential energy. Both penetrant (translational, orientational) and polymer degrees of freedom change along the reaction coordinate. Note the changes in the clusters of unoccupied volume before and after the jump. (b) Simple electrical analog of the Poisson process of jumping between macrostates. If the reduced probability Pi of being in a macrostate is associated with lip electrical potential, the network of macrostates behaves as an electrical network wherein each macros tate is associated with a capacitance P~q and each connection between macrostates with a resistance l{kij •

124

THEODOROU

We will use the term macrostate to refer to a collection of neighboring states separated by energy barriers that are low relative to kBT. The system spends most of its time confined within macrostates. The infrequent diffusive jumps of the penetrant constitute transitions from one macrostate to another across a bottleneck in OV(rp , r A ) separating the macrostates. The projection of a macrostate onto the subspace r A of the penetrant degrees of freedom will be a confined three-dimensional domain. The clusters of accessible volume determined by carrying out the geometrical construction of Section n.B on the pure polymer provide a reasonable approximation of these three-dimensional projections of macrostates. With each macrostate i we associate a position (site) vector r j in three-dimensional space that is representative of the position of the penetrant in macrostate i. A convenient choice for rj is the Boltzmann-weighted average of r A over all configurations belonging to the macrostate. The evolution of the polymer/penetrant system in time is viewed as a Poisson process (Feller, 1957) consisting of successive uncorrelated jumps between neighboring macrostates. With each jump i -+ j is associated a first-order rate constant k j • Consider an (in general non equilibrium) ensemble of penetrant/ polymer systems that at time t = 0 conform to a specified but otherwise arbitrary distribution among macrostates. As time elapses, this distribution evolves through transitions between macrostates occurring in the individual systems of the ensemble. Let p j(t) be the probability of finding a system of the ensemble in macrostate i at time t. The quantities p;(t) evolve according to the master equations j_

(61)

At very long times, the ensemble reaches its equilibrium distribution, wherein the probability of each macrostate is P ~q. The equilibrium probabilities of each macrostate obey the condition of microscopic reversibility (detailed balance) (62) In view of Eq. (62) and the normalization condition ~;JJ~q = 1, it is clear that in a system with a total of m macro states only (m + 2)(m - 1)/2 of the quantities {k j_J, {P~q} are independent. (In practice most of the independent rate constants k j _ j are zero, as they correspond to pairs of macrostates that are not connected.) The average residence time in macrostate i at equilibrium is 1

'T . = - - I

Lki-j j

(63)

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

125

Combining Eqs. (61) and (62), we find that the reduced probabilities P;(t) obey the evolution equations

==

p;(t)/p~q

cq dp; _ ~k (-) p; dt - - L.J ;j P; - Pj

(64)

Equation (64) suggests an interesting electrical analog (see Fig. 14b). The network of macrostates can be mapped onto a three-dimensional network with nodes at the representative points {r;}. With each node i is associated a capacitance P~q, and with each pair of connected nodes a resistance l/k;j' with k;j defined in Eq. (62). The reduced probability distribution {P;(t)} evolves exactly as the electrostatic potential in the electrical network. The transient solution to the master equation, Eq. (61), can be used to extract the penetrant self-diffusivity. For example, the evolution of {Pit)} could be tracked under the initial condition pj(O) = 8;j by integrating the master equation forward in time. If the behavior of the system is diffusive, then the profile {pj (t)} , after appropriate smoothing to eliminate the consequences of differences in p? among different macrostates, should evolve as exp[ - (rj - r;?/(6D S•A t)], from which D A can be extracted. A kinetic Monte Carlo method for extracting the self-diffusivity from the mean square displacement of the penetrant under conditions of equilibrium was used by June et al. (1991) in the context of diffusion in zeolites. This method directly simulates the continuous time- discrete space Markov process described by Eq. (61) on an ensemble of model systems. For the problem discussed here, it would proceed as follows: S•

1.

2.

Consider a three-dimensional network of a large number m of sites placed at positions r;, i = 1, ... , m, with connectivity defined by the rate constants {k;_j}. If the macrostates have been determined through analysis of a model polymer/penetrant system with periodic boundary conditions (see Section IY.D.2), then the network can be formed by periodic continuation of the macrostates within the primary box. (As an example, consider the network formed by periodic continuation of the box in Fig. 14b.) Distribute a large number NE » m of random walkers on the sites of the network according to the equilibrium probability distribution {P~q } . Multiple occupancy of sites is allowed in this deployment of the random walkers. The walkers will be allowed to hop between sites without interacting with each other (i.e., they will behave as ghost particles toward each other). Each random walker summarily represents a system in the ensemble of polymer/ penetrant systems whose temporal evolution we want to track with kinetic MC simulation. Let N;(t) be the number of random walkers that find themselves in site i at time t.

126

THEODOROU

3.

For each site i that is occupied at the current time t, calculate the expected fluxes Ri-j(t) = N;(t)ki-j to aU sites j with which it is connected. Also, compute the overall flux R(t) = L;Lj R;_lt) and the probabilities qi-j(t) = Ri-j(t)/R(t) . 4. Select a random number ~ E [0, 1). Choose the time for occurrence of the next elementary hop event in the network as I1t = In(l - ~)/R(t). Choose the type of the next elementary hop event by picking one of the possible transitions i -+ j according to the probabilities q;_j(t). 5. Of the N;(t) walkers present in site i, pick one with probability l/N,{t) and move it to site j. 6. Advance the simulation time by I1t. Update the array, keeping track of the current positions of all walkers to reflect the implemented hop. Update the occupancy numbers N,{t + I1t) = N;(t) - 1 and Nlt + I1t) = Nj(t) + 1. 7. Return to step 3 to implement the next hop event. The outcome from performing this stochastic simulation over a large number of steps is a set of trajectories rk(t) for all NErandom walkers. The self-diffusivity D S•A == D is estimated from the mean square displacement ([rlt) - riOW> computed over all N E walkers through the Einstein equation, Eq. (49). Note that simulation schemes advancing the time in equal intervals are also possible (Termonia and Smith, 1987). If the rate constants k;_i are small, then the time steps I1t taken by the simulation are long, and thus the simulation permits accessing times and displacements that may be several orders of magnitude larger than the ones accessible through atomistic MD. Thus, the long-time problem of MD is solved. For implementing this model of diffusion as a sequence of infrequent events, however, it is necessary that one have a good idea of (1) the macro states and the representative nodal points {r;}; (2) the equilibrium probability distribution {P~q } of the macrostates; and (3) the connectivity and rate constants {k;_J governing transitions between the macrostates. In the next section we discuss how to obtain these quantities from the detailed potential energy hypersurface "V(rp, r A ) based on multidimensional transition-state theory.

2.

Multidimensional Transition-State Theory Formulation of a Diffusive Jump Consider a model polymer matrix containing one penetrant molecule. As discussed in Section D.1, the system can be described in terms of 11 microscopic degrees of freedom specifying the position vectors (rp, r A ) of all polymer atoms and of the penetrant molecule. For convenience in the subsequent analysis, we will use the mass-weighted coordinates (Vineyard 1957) (65)

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

127

where the subscript ij denotes the jth atom of the ith polymer molecule. We use the notation x == (x p , xA ) for the v-dimensional vector of mass-weighted coordinates needed to describe the microscopic configuration of the system. By definition, a macrostate is a region in x-space surrounded by (v - I)-dimensional hypersurfaces of high potential energy relative to kaT. A macrostate i contains one or more states, each of which is constructed around a local minimum of "V(x). The states within a macrostate are mutually accessible over barriers that are low relative to kaT. We denote the minima in macrostate i by X71' X72, .. . (see Fig. 15). At each such minimum, (66a) and positive definite

(66b)

The system can move readily among the states within a given macrostate. Transitions between different macrostates (e.g., macrostates i and j), however, can occur only infrequently along relatively few high-energy paths, each such path connecting a state in macrostate i to a nearest neighbor state in macrostate j. Let x7k and be two nearest-neighbor minima in the two macrostates between which such a transition can occur. By the nearest-neighbor property of x7k and there will be at least one first-order saddle point xt between them, at which the gradient of "V vanishes and the Hessian matrix H of second derivatives has one negative and v - I positive eigenvalues:

xie

xie,

(67a) H(xg) has one negative eigenvalue A:;'g with associated eigenvector n U

(67b)

The saddle point or transition state xt is the highest energy point on the lowest energy passage between X7k and xle . To construct this passage, or transition path, which is a line in v-dimensional space, one can initiate two steepest descent constructions at xt , one with direction +nu and the other with direction -nu. Each such construction can be carried out in small steps 8x. For example, starting at xt one can displace the configuration toward xt. by a small vector &x = n U 8s. From point xt + 8x one can trace the steepest descent path leading to X7k as a series of successive steps &x = - (gJlgl)8s. A similar construction can be carried out toward xle. The resulting transition path is represented in Fig. 15 as a dot-dashed line.

.......

~

Su .... ....

....." X------. S ' S... ......

....

I

,x.

JI

, J2

\

... .... \

"'5-

----

X. /3

l

-

, ',

s "xi4

X ik

... ....

-.S l~

\

S

.,.

........ .... S .... I

....

.... \

I

6

~5

//

,\

}

Figure 15 Schematic representation of two adjacent macrostates i and j . Although the picture is two-dimensional, each macrostate should be envisioned as a domain in the v-dimensional configuration space of the polymer/penetrant system. Each macrostate contains a number of local minima (x7h xb" ... ; xlt, xk, ...) of the potential energy function 'V(x). The solid line surrounding the macrostates is a (v - I)-dimensional contour of constant 'V(x). Broken lines indicate low-energy pathways between states within a macrostate. The dot-dashed line traces the transition path connecting state x7k of macrostate i to state of macrostate j . This path controls the passage between the two macrostates. The transition path passes through a saddle point of 'V(x), labeled x~. Su is the (v-I)-dimensional dividing hypersurface separating x7k from here it is approximated as a hyperplane.

xIi

xIt;

~ ~

g ~

a

c::

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

129

The dividing surface separating states Xfk and x;e, and therefore macrostates i and j , is a (v - I)-dimensional hypersurface with equation C(x) = 0, that has the following properties (Sevick et aI., 1993): 1.

It passes through the saddle point xt :

C (xt ) 2.

=0

(68a)

At xt , it is normal to the eigenvector corresponding to the negative eigenvalue of the Hessian:

I

V.C(x) -n U \V.C(x)\ x=x~ -

(68b)

As a consequence, the dividing surface and the transition path are normal to each other. the dividing surface is tangent to the gradient 3. At all points other than vector:

xg,

V.C(x) . g(x)

= 0,

(68c)

The lowest energy region of the dividing surface in the vicinity of xt contributes mostly to transitions between the two macrostates (see below). We expect that this region will be well approximated by a hyperplane Su through xt drawn normal to the direction nU. In the following we use this approximation; i.e., we represent the dividing surface by C(x) = n U

•

(x - xt )

=0

(69)

xg

In locating the saddle points and the associated reaction paths and dividing surfaces, the geometric analysis of accessible volume clusters described in Section II.B serves as a useful starting point (Greenfield, 1995). Envision two clusters, i and j , that have been identified by the geometrical analysis of the pure polymer matrix using as a probe the penetrant molecule of interest. By carrying out the geometrical analysis for progressively smaller probe radii, it is possible to identify a "neck" connecting clusters i and j (compare Fig. 4). Using this neck point as an initial guess, one can locate the closest saddle point of "If with respect to the penetrant degrees of freedom, XA> keeping all polymer degrees of freedom X p fixed and equal to those of the bulk polymer. Well-established algorithms for the numerical determination of the closest saddle point are available (Baker, 1986). Using this three-dimensional saddle point as an initial guess, one can progressively augment the set of degrees of freedom with respect to which the saddle point is calculated (e.g., including more and more of the X p that lie in concentric spheres of progressively increasing radius around the penetrant),

130

THEODOROU

until further expansion of the set of degrees of freedom leaves the estimates of xt and 'V(xt ) unchanged. Once xt has been identified in this way, the transition path and the states xf and xJ can be located through a steepest descent construction in the entire x space, as described above. Note that this steepest descent construction fully accounts for changes in the polymer matrix induced by the presence of the penetrant. Having identified the transition path between macrostates i and j and the associated dividing surface Su, we can proceed to estimate the rate constants k;:T and k]:1 through v-dimensional transition-state theory (TST). In the following, it is assumed that only one transition path contributes significantly to the flux between macrostates i and j. The analysis can be readily generalized for multiple diffusion paths. According to the TST approximation (Voter and Doll, 1985; June et aI., 1991), whenever the polymer/penetrant system finds itself on the dividing hypersurface between i and j with net momentum directed from ito j, a successful transition between i and j will occur. Under this assumption (Sevick et aI., 1993; Voter and Doll, 1985) the rate constant can be expressed as ki _ i = (

dVx (

J

.E{i}

J

dVp [n(x) . p] 8 [C(x)] IV.C(x)1

p NVT

(x, p)

(70)

o ·p>O

where p is the vector of mass-weighted momenta conjugate to x, n(x) the unit vector normal to the dividing surface at position x, p NVT (x, p) the canonical ensemble probability density in phase space, and the Dirac delta function selects configurations on the dividing surface. Upon performing all momentum space integrations, one obtains from Eq. (70)

(

TST

ki-j

1

= (2~hy/2

. d Vx 8[C(x)]IV.C(x)1 exp [-!3'V(X)])

---1-------J. E{ I}

(

(71)

d Vx exp [-[3'V(x)]

xE{i}

Equation (71) expresses k~J as a ratio of two configurational integrals: one taken over the dividing surface, the other over the entire macrostate {i} . For systems with small dimensionality v, the configurational integrals of Eq. (71) can be computed directly by MC integration (June et aI., 1991). For the highly dimensional problem of the diffusive jump in all relevant degrees of freedom of the penetrant and the polymer, the direct sampling of configuration space points uniformly distributed in Su or in {i} is not straightforward, and a free-energy perturbation technique along the transition path (Elber, 1990; Czerminsky and Elber, 1990) is more appropriate. To implement this technique, we consider a

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

131

set of closely spaced (v - I)-dimensional hyperplanes SI> S2, . . . , Sm- l, Sm, . .. , Su normal to the transition path starting at X7k and ending at (see Fig. 16). We also consider a thin v-dimensional slice of macrostate {i} around the hyperplane SI, of thickness Llxik ; we use the symbolism ~{ih to denote this thin domain. Equation (71) can be written as

xt

L

d .-\x exp [- f3"V(x»)

(2f3 1T)"2

kJ:I = --:-------

r

J'l

d 'x exp [- f3"V(x»)

L

r - Ix exp [- f3"V(x»)

5.

r

L

d ,- Ix exp [- f3"V(x)]

Sl

------d "x exp [- f3"V(x)]

J (I )

L

d ,- Ix exp [- f3"V(x»)

5,

L

d "-I x exp [- f3"V(x)]

x ... x

s.

x ... . - - : - - - - - - - -

L..

d ,- Ix exp [- f3"V(x»)

1 t1 X il<

1

d "x exp [- f3"V(x»)

d ,-Ix exp [- f3"V(x»)

L

d , - IX exp [- f3"V(x»)

s,

d{II'

f

L.

------d 'x exp [- f3"V(x)]

L

d ' - Ix exp [- f3"V(x)]

s,

(i)

L x

L

d ,- Ix exp [-f3"V(x)]

X ...

s""

L..

d,-I x exp l - f3"V(x)]

d , - Ix exp [- (W(x»)

x .. . x

Su

~------

L-.

(72)

d '-I x exp [- f3"V(x»)

The ratio of v-dimensional configurational integrals in Eq. (72) equals the probability that a system that samples macros tate i according to the equilibrium probability density of the canonical ensemble visits the narrow strip ~{ih . It can be determined through an NVI' MD or a Metropolis NVI' Me simulation of the polymer/penetrant system confined within macrostate i. The fraction of configurations sampled by such a simulation that lie in ~{ih provides an estimate of this ratio. To compute each one of the ratios of (v - I)-dimensional integrals over successive hyperplanes Sm- 1 and Sm appearing in Eq. (72), we consider the vector 8xm along the transition path between these hyperplanes. If Sm- 1 and Sm are spaced closely enough, they are practically parallel to each other and normal to 8xm • For each configuration x lying on Sm- h there is a configuration x + 8xm lying on Sm, and vice versa. The ratio of configurational integrals can then be

132

THEODOROU

... s

u

-- -~ C(X)=o Figure 16 Sequence of (v - I)-dimensional hyperplanes S" S2, . . . and v-dimensional domain ~{ih used for the free-energy perturbation calculation of the jump rate constant k~J based on multidimensional transition state theory.

written as

L

d "- Ix exp [-13'V(x)]

Lm -I d"- Ix exp [- 13'V(x)] Lm -l d"- Ix exp [- 13'V(x + 8x

m )]

=

L-l

d"- Ix exp [-13'V(x)]

Lm -l d"- Ix exp [-13'V(x)] exp {-13['V(x + 8x

m) -

'V(x)]}

=

Lm -l d"- Ix exp [-13'V(x)] = (exp {-13['V(x + 8x 'V(X)]})Sm_l m)

-

(73)

The ratio of (v - 1)-dimensional integrals over hyperplanes Sm- I and Sm can be computed as an ensemble average over a MD or Metropolis Me simulation

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

133

of the system confined to hyperplane Sm- l ' The calculation of ki-j through Eq. (72) is thus possible through a v-dimensional and a series of (v - I)-dimensional MD or Metropolis MC simulations. Transition-state theory is not exact. In reality, not all dynamical trajectories penetrating Su in the direction from i to j effect a successful transition from i to j . Recrossing events, wherein the system passes from ito j but subsequently crosses back and ultimately thermalizes in i, are well possible. The actual transition rate constant k i _ j can thus be expressed as (74)

where k~T is the transition-state theory estimate, obtained as described above, while h-j < 1 is a dynamical correction factor accounting for recrossing events. h-j can be estimated by undertaking short MD runs of the system initiated on the bottleneck surface SUo The fraction of such trajectories that ultimately thermalize in the macrostate toward which they were moving when initiated provides the dynamical correction factor (Voter and Doll, 1985; June et al., 1991). Note that the MD simulations needed to estimate h-j are much less compute intensive than "brute force " MD; once placed on the bottleneck hypersurface Su, the system will quickly move toward a macrostate and thermalize in it. The equilibrium probability of each macrostate i is obtained as

f.

f

d"x exp [-WV(x)]

cq

Pi =

{I}

d"x exp [-\3"V(x)]

__ {i}_ _ _ _ _ __

.f,; L}

d''x exp [-\3"V(x)]

=

J

(75)

d "x exp [- \3"V(x)]

It can be accumulated in the course of a Widom test particle insertion calculation [compare Section 1II.A, Eq. (27)] in which contributions to the solubility from configurations in which the penetrant resides in different clusters of accessible volume (corresponding to different macrostates) are accumulated separately and finally divided by the total solubility. Alternatively, a Metropolis MC simulation of the penetrant/polymer system can be employed that provides for exchange of the penetrant among different clusters; the fraction of configurations sampled by such a Metropolis calculation that belong to macrostate i is P~q.

3.

Applications of the TST Approach for the Prediction of Diffusivity A pioneering TST-based study of the diffusion of argon in amorphous polyethylene was presented by Jagodic et a1. (1973). Computational limitations prevented the generation of realistic model polymer structures, so the results from this study were not very conclusive. Recently, a set of thorough TST-based calculations of high predictive value was carried out by Gusev et a1. (1993).

134

THEODOROU

These investigators calculated the infinite-dilution diffusivity of He, H 2, Ar, O2, and N2 in rubbery polyisobutylene and glassy poly carbon ate at 300 K, representing the polymer matrices by rigid minimum energy configurations obtained through molecular mechanics. With the spherical representation used for all penetrants, the x space is three-dimensional, spanned by the three translational degrees of freedom r A of the penetrant, all of which are associated with the same mass. This greatly simplifies the TST calculation of the rate constants k;:}. No distinction was made between states and macrostates; i.e., all local minima of "V(rp) were identified exhaustively and used to define macrostates, between which the rate constants were computed. The number of local minima identified was quite large (approximately 50 for the largest penetrants in a model structure of - 30 A edge length). The tessellation of r A space into states and the identification of dividing surfaces were carried out by constructing regular grids of small cubic elements or voxels through the polymer configurations. Each dividing surface was thus represented as a collection of contiguous square facets oriented normal to one of the three coordinate axes (see also June et ai. , 1991). The three-dimensional and two-dimensional configurational integrals needed for the evaluation of P~q and kJ:1; via Eq. (71) were obtained through direct numerical integration over the values of "V(r A) at the node points of the grid, taking into account the increase in area of the dividing surfaces due to the finite size of the voxels. The results of the TST analysis were used to accumulate residence time distributions of the penetrants in the states and to estimate D s,A == D through the generation of kinetic Monte Carlo trajectories for the jump process. The computational requirements of the approach were very modest relative to " brute force " MD, permitting the generation of millisecond-long stochastic trajectories. For He, the diffusivities predicted were within a factor of 3 of the experimental values for both polymers. For the larger penetrants, however, the diffusivities were underestimated by three to five orders of magnitude as a result of the rigid representation of the matrix. An improved approximate TST formulation that takes into account thermal motion in the polymer matrix was presented by Gusev and Suter (1993). To circumvent the complexity of the full-blown TST formulation outlined in Section IY.D.2, Gusev and Suter assumed a quadratic dependence of the polymerpolymer contribution to the potential energy "V p(rp) of the form (76) where rpk is the position vector of the kth atom of the polymer and rpk,O the equilibrium position of the same atom in the static minimum energy configuration. Note that in this Debye-type approximation, atoms are assumed to move isotropically and independently of each other. The mean square deviation (a;>

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

135

should depend on temperature and atom type. (The latter dependence is neglected in this implementation; i.e., a mean square positional deviation over all atom types is used.) The implicit assumption is introduced that " elastic" fluctuations of the polymer matrix are fast in comparison to the time between diffusive jumps, whereas more drastic motional processes associated with structural relaxation in the polymer occur over times much longer than the time between jumps. On the assumption of this double time scale separation, the transitions between states are governed by a three-dimensional potential of mean force A(r A ) , obtained by integrating out all polymer degrees of freedom in the configuration-space probability density p NVT (r p , r A). By virtue of Eq. (76), this potential of mean force is pairwise additive, consisting of penetrant-polymer atom contributions of the form A k(lr A - r Pk.ol) that can be computed directly from the penetrant-polymer atom Lennard-Jones potentials once (ai) is known. The TST calculation then proceeds as in the static polymer case (Gusev et aI. , 1993), usingA(rA) in place of 'V(rA). Critical to this fluctuating matrix approach are the values of (a~) . These are estimated from short (5 ps duration) atomistic MD simulations of the pure polymer matrix. The (ai) values obtained from these simulations are found to increase with time, following the approximate scaling log «a~) l!2) ex: log t. In view of this, it is decided to use in the TST calculations the value of (M> that corresponds to the most probable residence time T of the penetrant in a state. This introduces a self-consistent character in the calculation: A value for (a ~> is initially postulated and used to compute the rate constants kJ:';; the residence time distribution is extracted from the rate constants, and the time at which it goes through a maximum is identified as T; a new value of (ai) is determined from this T, and the whole procedure is repeated. The self-consistent calculation is found to converge in practically one iteration. In the case of He and H2, T was approximately 0.15 ps and the corresponding (M>I!2 was taken as 0.22 A. In the case of O2 and N2 , T was approximately 1 ps and (a~>1!2 was 0.46 A. The diffusivities extracted from this fluctuating matrix approach are within a factor of 2 from experimental values for all penetrant molecules examined, indicating that the "trapping" experienced by the larger penetrants in static model matrices was eliminated. An interesting observation from the Gusev and Suter simulations is that the motion of the penetrant molecule is not diffusive up to times on the order of tenths of nanoseconds to tenths of microseconds. An " anomalous diffusion " regime, wherein the mean square displacement of the penetrant scales roughly as (r2) ex to.s, is observed at short times, and it is only in the longest time portion of the stochastic trajectories that the diffusive scaling (r2) ex t sets in. A strikingly similar conclusion was reached through atomistic MD of the same model systems (He and O2 in polyisobutylene) at room temperature carried out to very long times (MU11er-Plathe et aI., 1992b) and also in the MD simulations of Pant and Boyd (1993). The crossover between anomalous and diffusive regimes

136

THEODOROU 7 6

5 4 ~

N

N

'
3

~ ...........

2

----00 0 .......

0

o

o

-1

-2

.

• ••

~--~--~

-14

-13

____

-12

~

-11

__-L____L -__ - 10

- 9

~

-8

__~____~__-L__~

-7

-6

-5

-4

log(t/s) Figure 17 Mean square displacement versus time curves for helium and oxygen in polyisobutylene at 300 K, as computed from the transition-state theory approach of Gusev and Suter (1993) using a Gaussian model for the polymer fluctuations. The data, displayed in logarithmic coordinates, are averages over 1500 independent kinetic Monte Carlo simulation paths. " Anomalous diffusion" behavior is seen out to time scales of tenths of nanoseconds to tenths of microseconds. [Reproduced from Gusev and Suter (1993), with permission.]

is seen characteristically in Fig. 17. Clearly, the time span of the anomalous diffusion regime in low-temperature polymers is many orders of magnitude longer than the "ballistic" regime seen in small molecular weight liquid simulations. A qualitative interpretation of the anomalous diffusion regime based on the jump picture of diffusion can be provided by the following argument: As pointed out in Section II.B., the connectivity of the network of accessible volume clusters or macrostates seen by the penetrant molecule is a function of the time scale of observation, owing to the wide distribution of k i _ i values. At very short times, the network consists of isolated clusters in which the penetrant rattles. At somewhat longer times, the penetrant can execute short length scale (5- 10 A.) jump motions within isolated groups of a few clusters that are connected via diffusion pathways of high k i _ i . It is an interesting question whether these clusters be-

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

137

tween which transitions are most facile tend to align along the backbones of chains, as suggested by the Pace and Datyner theory (Dr. J. Petropoulos, personal communication). As the time scale of observation increases, the clusters that are mutually accessible become more and more connected until, at a critical time scale, percolation of the network of clusters occurs. A random walk on isolated groups of sites or on a percolating collection of sites is known to be subdiffusive (Stauffer, 1985). It is only for times that are long compared to the time scale for percolation that the network is well connected and that diffusive behavior is observed. By this argument, the time span of the anomalous regime should be longer the slower the motion of the penetrant (i.e., the lower the values of ki-j); this is confirmed by the simulaton findings (see Fig. 17).

v.

CONCLUSIONS AND FUTURE DIRECTIONS

In recent years there has been intense computer modeling and simulation activity in the areas of sorption and diffusion of small molecular weight gases in amorphous polymers. Calculations carried out include the geometrical characterization of accessible volume in a pure polymer matrix and of its distribution and rearrangement with thermal motion; the calculation of Henry's constants characterizing the sorption thermodynamics in melts at low pressures through the Widom test particle insertion method; an approximate theoretical treatment of sorption in glassy polymers based on static, energy-minimized model-polymer structures; numerous equilibrium molecular dynamics and a nonequilibrium MD investigation of the self-diffusion of gases in melts and rubbery polymers; and the estimation of the low-concentration diffusivity of gases in glassy and lowtemperature rubbery polymers through approaches based on transition-state theory. Most of this work has focused on chemically simple penetrants (He, H2 , O2 , N2, CIL) and polymers (polyethylene, polypropylene, polyisobutylene), although some more complicated matrices (polycarbonate, polydimethylsiloxane) have also been examined. The atomistic MD simulation work has revealed a wealth of mechanistic information on how diffusion takes place at various temperatures; this information has been useful in testing theoretical ideas and phenomenological correlations proposed in the past. In systems where diffusion is not too slow (D ;::: 10- 7 cm2/s), MD simulation provided good quantitative estimates of the selfdiffusivity whenever sufficient care was taken in the model representation of the polymer and its interactions with the penetrant. The estimation of Henry ' s constants has been generally less successful, with some notable exceptions (alkanes in polyethylene). Approximate TST-based formulations based on the picture of diffusion as a succession of jumps in the matrix have proved very promising in dealing with diffusion at low temperatures, where atomistic MD becomes impractical.

138

THEODOROU

There is still' much to be done before a general framework of theoretical and simulation methodologies becomes available for use in the efficient firstprinciples estimation of sorption and diffusion phenomena and therefore in the " molecular engineering design" of materials with tailored separation and barrier properties. The very generation of well-equilibrated model polymer melts or of glassy configurations that correspond to realistic formation histories is still a challenge, owing to the very long relaxation times present in these systems. New Monte Carlo algorithms for the bold exploration of configuration space, fast multipole algorithms for the rapid summation of interactions, and the use of parallel machines are promising for alleviating this problem. The need to validate simulation approaches in terms of their ability to predict scattering, spectroscopic, equation-of-state, sorption, and diffusion measurements is becoming increasingly obvious. An ability to predict all these properties in known systems using a consistent potential representation is a prerequisite for using molecular modeling as a design tool. The source of the difficulty in predicting low-pressure sorption thermodynamics accurately has to be clarified. Furthermore, the problem of sorption equilibria at high penetrant activities, including dual-mode sorption behavior, swelling, and plasticization effects in polymer glasses, has hardly been addressed from a first-principles prediction point of view. Real-life applications often have to do with large, complex, or strongly interacting solvent or plasticizer molecules whose thermodynamic and transport behavior has not been investigated sufficiently with molecular modeling. In view of the very long relaxation times characterizing such systems, it is unlikely that " brute force" MD will be very useful, and coarse-grained approaches become imperative. The low-concentration diffusion of large rigid penetrants (e.g., benzene) in an amorphous matrix could be treated with multidimensional TST; explicit incorporation of the polymer degrees of freedom in calculating diffusion pathways would be imperative in this case (see Section IY.D), as the time scale separation postulated by simpler treatments does not hold. For large, flexible penetrants with many torsional degrees of freedom, a picture of diffusion as Brownian motion in a medium capable of exerting Langevin and frictional forces on the penetrant would seem more appropriate. A quantitative connection between the " friction factor " invoked in such a coarse-grained description and the detailed potential energy hypersurface of the system must be established. All atomistic modeling work conducted to date on diffusion in polymers has to do with systems that, macroscopically, exhibit Fickian behavior. In the long run, the quantitative prediction of anomalous and case II diffusion from molecular level information should be addressed. This can be accomplished only if methods for predicting the time spectrum for structural relaxation in glassy polymer matrices, pure or swollen with solvent, become available. Dealing with the long-time relaxation behavior of polymers in a way that combines computational

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

139

tractability with quantitative accuracy is today a grand challenge for polymer molecular modeling. The wide spectra of length and time scales present in polymer systems call for hierarchical modeling approaches consisting of several interfacing " modules," each module being a theoretical formulation or simulation algorithm that can capture phenomena over a limited range of scales. Modules should receive input from lower level (smaller scale) modules and provide input to higher level ones. The transition-state theory-based approach to low-temperature diffusion, combining geometrical and energetic analysis of configurations of the polymer/ penetrant systems to identify macros tates, calculation of equilibrium occupancy probabilities for and transition rate constants between the macrostates through MC or MD techniques, and extraction of the diffusivity from long stochastic trajectories, provides a good example of such a hierarchy. The conventional atomistic MD and MC simulation techniques will always be useful for establishing the ultimate connection to chemical constitution at the base of this hierarchy but are unlikely to solve the property prediction problem alone. The need for coupling these techniques to more coarse-grained theoretical and simulation approaches cannot be overemphasized.

ACKNOWLEDGMENTS I wish to thank my collaborators, Dr. Krishna Pant, Mike Greenfield, and Travis Boone, for stimulating discussions and for providing some of the research results discussed in this chapter. I am grateful to Dr. John H. Petropoulos for cultivating my interest in diffusion in polymers and sharing his deep insights on the subject with me. Randy Snurr is thanked for his thorough reading of the manuscript and his suggestions. I appreciate the permission given to me by Professors Ulrich w. Suter and Juan J. de Pablo and by Drs. Hisao Takeuchi and Krishna Pant to include copies of their figures in the chapter. I also appreciate the patience of our Editor, Professor Partho Neogi, with my repeated delays in finishing the manuscript.

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Takeuchi, H., and R.-J. Roe (1991b). J. Chem. Phys. , 94, 7458. Takeuchi, H., R.-J. Roe, and 1. E. Mark (1990). J. Chem. Phys., 93, 9042. Tanemura, M., T. Ogawa, and N. Ogita (1983). J. Comput. Phys., 51, 191. Termonia, Y , and P. Smith (1987). Macromolecules, 20, 835 . Theodorou, D. N., and U. W. Suter (1985). Macromolecules, 18, 1467. Theodorou, D. N., and U. W. Suter (1986). Macromolecules, 19, 139. Trohalaki, S., D. Rigby, A KJoczowski, J. E. Mark, and R. J. Roe (1989). Polym. Prepr. (Am. Chem. Soc., Div. Polym. Chem.), 30(2), 23. Trohalaki, S., A KJoczowski, and J. E. Mark (1991). In Computer Simulation of Polymers, R.-J. Roe, Ed., Prentice-Hall, Englewood Cliffs, NJ, Chap. 17, p. 220. Trohalaki, S., D. Rigby, A KJoczowski, J. E. Mark, and R. J. Roe (1992). PolYIn. Prepr. (Am. Chem. Soc., Div. Polym. Chem.), 33(1), 629. Vacatello, M. (1992). Polym. Prepr. (Am. Chem. Soc., Div. Polym. Chem.), 33(1), 529. Vacatello, M., G. Avitabile, P. Corradini, and A J. Tuzi (1980). J. Chem. Phys., 73, 543 (1980). Vineyard, G. H. (1957). J. Phys. Chem. Solids, 3, 121. Voter, A , and J. Doll (1985). J . Chem. Phys., 82, 80. Weber, T. A , and E. Helfand (1979). J. Chem. Phys., 71, 4760.

3 Free-Volume Theory J. L. Duda The Pennsylvania State University, University Park, Pennsylvania

John M. Zielinski Air Products & Chemicals, Inc., Allentown, Pennsylvania

I.

INTRODUCTION

The phenomenon of small-molecule mobility in macromolecular materials dictates the effectiveness of polymerization reactors as well as the physical and chemical characteristics of the polymer produced. The molecular weight distribution and average molecular weights, for example, are among the physical properties influenced by the diffusion-controlled termination step of free-radical polymerization reactions. Other polymer processing operations affected by molecular transport include devolatilization, mixing of plasticizers (or other additives), and formation of films, coatings, and foams . In addition, distinctive molecular diffusion behavior is essential for miscellaneous polymer products such as barrier materials, controlled drug delivery systems, and membranes for separation processes. The fundamental physical property required to design and optimize the processing operations is the mutual diffusion coefficient, D . Numerous techniques have been developed to correlate and predict mutual diffusion coefficients for systems composed of two or more low molecular weight materials (Reid et al., 1977). These techniques are not, however, suitable for systems in which one of the species possesses chainlike characteristics such as a synthetic or natural polymer. The failure to extend theoretical developments that accurately describe mass transfer in systems of low molecular weight species to macromolecular systems stems from characteristics unique to polymeric materials. 143

144

DUDA AND ZIELINSKI

In mixtures of low molecular weight molecules, for example, molecular migration is typically a weak function of temperature and concentration, whereas in polymeric systems diffusional transport can be significantly enhanced (or diminished) by varying either condition. The most pronounced effects are observed near the glass transition temperature (TJ, where a 1 % increase of solvent weight fraction in a polymer solution can increase D by three orders of magnitude. Diffusion coefficients ranging between 10- 16 and 10- 5 cm2 js have been reported, and apparent activation energies (ED) greater than 50 kcal!mol have been determined (Vrentas and Duda, 1986). Consequently, significant experimentation is required to quantify and optimize a processing operation governed by molecular transport. In addition to temperature and composition, diffusion in polymers is controlled by morphological features such as crystallinity and cross-linking, both of which tend to reduce molecular mobility. Although these complexities inhibit theoretical analysis of mass transfer, experimental measurements of D are, in general, not any more laborious than those of amorphous materials. Under certain conditions, however, transport in even thermoplastic amorphous polymers does not follow the laws of classical molecular diffusion. This class of transport is often designated as anomalous (or non-Fickian) and is not discussed further in the context of this chapter. This topic, however, is addressed thoroughly in Chapter 5. Depending on experimental conditions, polymeric materials may exhibit a variety of mechanical properties ranging anywhere from brittle (glassy) to deformable (rubbery). Thus, they may be viewed as either highly viscous, molten liquids or relatively low viscosity solutions. Several theoretical formalisms have been developed that attempt to describe and predict mutual diffusion in both the rubbery and glassy states. An example of a successful transport theory in this vein is given by Vrentas and Duda (1977a,b). The model accurately describes diffusion both above and below Tg , through the presumption that transport is controlled by the availability of free volume within a system. In this chapter, the fundamental concepts of free-volume theory are introduced in the context of the Vrentas-Duda formalisms . These models are subsequently employed to correlate (and predict from material characteristics) mutual binary diffusion coefficients for solutions of amorphous polymers and small-molecule solvents. In Sections II and III, diffusion within concentrated polymer solutions above Tg of the mixture is emphasized, whereas Section IV focuses on transport in the glassy state. Section V addresses the ways in which diffusional behavior is affected by additional considerations such as chemical cross-linking, block copolymers, antiplasticization, and multicomponents. Throughout the chapter, we implicitly assume that transport processes occur at conditions that are described by classical (Fickian) diffusion.

FREE-VOLUME THEORY

II.

145

FREE-VOLUME CONCEPTS

The presumption that molecular transport is regulated by free volume was first introduced by Cohen and Turnbull (1959). Although their conceptualization was initially believed to be suitable only for liquids that could be envisioned as an ensemble of uniform hard spheres, the greatest impact of their development, ironically, has been on describing mass transfer in solutions consisting of long polymer chains mixed with small solvent molecules. From the Cohen and Turnbull perspective, the hard-sphere molecules that constitute an idealized liquid exist in cavities (or cages) formed by nearest neighbors. Thus, the total volume of the liquid could be divided into two components: occupied volume and free volume. Despite possessing the innate ability to vibrate within its cage, each sphere was presumed incapable of migration until natural thermal fluctuations caused a hole (or vacancy) to form adjacent to its cage. The hole had to be sufficiently large to permit a significant displacement of a spherical molecule. A single step of the diffusional transport mechanism was successfully completed when the cavity a molecule left behind was occupied by a neighboring molecule. Translational motion, according to Cohen and Turnbull, did not require a molecule to attain a prerequisite energy level to overcome an activation energy barrier. Rather than creating holes by physically displacing nearest neighbors, as suggested by the activation energy approach (DiBenedetto, 1963; Brandt, 1955; Pace and Datyner, 1979), molecular transport was presumed to rely on the continuous redistribution of free-volume elements within the liquid. IT the volume of each hard sphere is denoted V *, then the occupied volume in a liquid containing N spheres is NV* . Additionally, if the average free volume per sphere is given by V f, then the total volume of the liquid, VL , can be expressed as VL = NV* + NV f • The migration (or self-diffusion) rate of a hard sphere is therefore proportional to the probability of finding a hole of volume V * or larger adjacent to the sphere. When free-volume voids form adjacent to spherical molecules by means of natural thermal fluctuations within a liquid, the molecules undergo a single step in the diffusion process. By describing the dispersion of free-volume elements within a liquid in mathematical terms, Cohen and Turnbull developed a distribution function that provides the probability of finding a free-volume hole of a specific size. The diffusion coefficient, considered proportional to the probability of finding a hole of volume V * or larger, may be written as (1)

In this expression, D is the self-diffusion coefficient of the molecules, V* is the minimum volume hole size into which a molecule can jump, and 'Y is a numerical factor between 0.5 and 1.0 that is introduced to account for the overlap

146

DUDA AND ZIELINSKI

Table 1 Free-Volume Theory Parameters for Toluene/Poly(vinyl acetate) System Parameter

Tol/PVAc

Do X 104 (cm2/s)

4.82'

X K llh X 101 (cm1/g K) K 12f-., X 104 (cm3/g K) K 21 - T gl (K) K 22 - Tg2 (K) V; (cm1/g) V; (cm1/g)

0.393 1.45

~

E (cal/mol)

4.33 -86.32 -258.2 0.917 0.728 0.82 0.0

'Do = 4.82 x 10- 4 cm2,s is estimated using physical properties of pure toluene (Zielinski and Duda, 1992a). Analysis of toluene/PVAc mutual diffusion data at 40°C yields a Do value of 10.36 X 10- ' cm', s. The effect of Do on diffusion coefficient pred ictions is illustrated in Fig. 1.

between free-volume elements (Le., free volume shared by neighboring molecules). The proportionality constant A was considered by Cohen and Turnbull to be related to the gas kinetic velocity. This formalism indicates that the self-diffusion coefficient is an exponential function of the ratio of the size of the diffusion molecule to the free volume per molecule in the liquid. While relatively simplistic, this framework constitutes the forerunner of the free-volume models presented in the foHowing sections for polymer/solvent solutions. Numerous investigators have extended the freevolume concepts outlined here, and several of these developments have been incorporated into the theory of Vrentas, Duda, and coworkers (Vrentas and Duda, 1977a,b; Vrentas et aI., 1993; Duda et aI., 1982; Zielinski and Duda, 1992a,b). The Vrentas- Duda approach, generaHy accepted as one of the most successful theories for describing molecular diffusion in concentrated polymer solutions, is the main focus of this chapter.

III.

DIFFUSION ABOVE THE GLASS TRANSITION TEMPERATURE

The Cohen and TurnbuH development continues to serve as a valuable foundation for modern theoretical frameworks that successfully describe diffusional behavior in concentrated polymer solutions. The formalism, as originaHy de-

147

FREE-VOLUME THEORY

rived, provides a relationship between the system free volume and the selfdiffusion coefficient D J for a one-component liquid. This relation can be readily extended to describe self-diffusion of a single species in a binary mixture:

(2) where V; is the critical molar free volume required for a jumping unit of species of 1 to migrate VFH is the free volume per mole of all individual jumping units in the solution, and DOl is a temperature-independent constant. In the original Cohen and Turnbull representation, a jumping unit was designated as a single hard-sphere molecule undergoing diffusion. In solutions of real molecules, particularly in mixtures of macromolecules, however, an individual molecule can be composed of multiple jumping units covalently bonded together. Free-volume holes that readily accommodate entire polymer molecules simply do not form. Instead, polymer chain migration is envisioned to result from numerous jumps of small segments along the polymer chain. To complicate matters further, low molecular weight molecules of sufficient size and flexibility are also capable of migrating by a mode, reminiscent of polymers, that involves coordinated motion between several parts of the molecule (Arnould and Laurence, 1992; Vrentas et aI., 1985a). To generalize the Cohen and Turnbull theory to describe motion in binary liquids, Vrentas and Duda used the relationship

-

VFH FIl

V

= (moles of jumping units)/g = wl/M1j

VFH + WJM2j

(3)

where VFH is the specific hole free volume of a liquid with a weight fraction W i of species i, and with jumping unit molecular weights of Mij(i = 1 or 2). In a binary solution, the hole free volume is distributed equally among all the jumping units. In cases for which species 1 is a simple molecule (such as benzene), the jumping unit molecular weight, M Jj , is equal to the total molecular weight of the solvent. In contrast, if species 2 is a polymer chain, then M2j is a relatively small fraction of the total molecular weight. Combining Eqs. (2) and (3) results in an expression for solvent self-diffusion in a polymer solution, namely, (4) where V; is the specific hole free volume of component i required for a diffusive step and ~ = V;j;V;j' A critical step in implementing free-volume concepts to describe transport in polymer solutions involves quantifying the specific free volume, VFlI' As a first approximation, one might assume that the total volume of a liquid is composed of two parts, the occupied volume and the free volume (as suggested by Cohen

148

DUDA AND ZIELINSKI

and Turnbull). The specific occupied volume of a liquid is generally defined as the specific volume of the equilibrium liquid at 0 K [fO(O)]. Hence, the specific free volume of a species as a function of temperature is given by (5) where V(T) is the specific volume of an equilibrium liquid at any temperature T. Although not directly measurable, V(O) can be estimated by the group contribution methods discussed by Haward (1970), provided the chemical composition of the species of interest is known. Even though Eq. (5) can be used to approximate the total free volume in a liquid, some question still exists as to whether this free volume is the same as the one considered in the Cohen and Turnbull framework. It is possible that the total free volume cannot be continuously redistributed without overcoming a precise activation energy barrier. To incorporate this condition, Vrentas and Duda followed the concepts of Kaelble (1969) and divided the free volume into two types. One portion, denoted the interstitial free volume, requires a large redistribution energy and is thus not implicated in facilitating transport through the mixture. The remaining free-volume allotment, which is presumed to dictate molecular transport, is termed to the hole free volume and is redistributed effortlessly. The Vrentas-Duda theory, therefore, regards VFH as the specific hole free volume and precludes determination of VFH from measurements of the specific volume and estimates of the occupied volume, as suggested by Eq. (5). To circumvent this problem, Vrentas and Duda adopted the ideology espoused by Berry and Fox (1968) and developed a relationship between the hole free volume and well-defined volumetric characteristics of the pure components in solution: (6) For polymer solutions, K I I and K ZI denote free-volume parameters for the solvent, while KJ2 and Kzz are free-volume parameters for the polymer. The glass transition temperature of species i is given by Tgi • Throughout the ensuing development, subscript 1 refers to the solvent and 2 to the polymer. These freevolume parameters can be determined from pure-component viscosity data, as a function of temperature, for the individual components in the solution. Furthermore, these free-volume parameters are directly related to the parameters in the Williams-Landel-Ferry (WLF) equation, which has been used extensively for correlating polymer viscosity data (Williams et aI., 1955). In the original Cohen - Turnbull framework, a molecular jump does not involve any activation energy but is solely related to the probability of locating a sufficiently large free-volume void. Macedo and Litovitz (1965) and Chung (1966) challenged this conceptual notion and suggested that a jumping unit must

FREE-VOLUME THEORY

149

overcome the attractive forces with adjoining molecules prior to a diffusive step. According to their viewpoint, therefore, the appropriate expression for the preexponential term in Eq. (4) is

DOl = Do exp( - E/RT)

(7)

In this expression, E is the activation energy required for a jumping unit to break free from its neighbors before it can move into a contiguous free-volume void. This quantity should not be confused with the activation energy for diffusion (ED)' which is defined as

2(a D)

ED = RT -In-

aT

(8) WI ,/'

In concentrated polymer solutions where the availability of hole free volume limits translational mobility, the pre exponential factor DOl is often less dependent on temperature than the exponential term related to free volume. In many cases D Ol can be taken as a constant. When all of these proposed modifications to the original Cohen and Turnbull theory are incorporated, the following expression emerges for the self-diffusion coefficient of a solvent in a polymer solution: (9) An analogous expression for the polymer self-diffusion coefficient, D 2 , was developed by Vrentas and Duda (1977b) based on the entanglement theory for polymer solutions proposed by Bueche (1962). Although self-diffusion coefficients are intrinsic properties of a given solution and can be measured directly by techniques such as nuclear magnetic resonance (NMR) and radioactive labeling, the mutual binary diffusion coefficient, D, reflects the mass transfer rate required for the design and development of polymer processes and products. Although self-diffusion coefficients constitute a measure of the mobility of the various species in a homogeneous solution, the mutual binary diffusion coefficient provides the rate at which concentration gradients within a mixture dissipate and is the critical quantity required in defining most technological applications. Typically, molecular theories present expressions for the self-diffusion coefficients (Dl and D 2 ) rather than D . Thus, it is highly desirable to express D in terms of Dl and D2 An indisputable relationship between the friction coefficients that relate these diffusion coefficients does not, however, presently exist. Incorporating the work of Bearman (1961), Duda et al. (1979) proposed an approx-

150

DUDA AND ZIELINSKI

imation for low solvent concentrations that couples D to the self-diffusion coefficients for polymer/solvent systems:

D

= Djwlwz (all-I) RT

aW l

(10) T,P

Here, Il-l is the chemical potential of the solvent. Although many thermodynamic theories are available to determine the concentration dependence of the solvent chemical potential, the Flory-Huggins theory has provided an adequate representation of polymer solution thermodynamics in many cases (Flory, 1942; Huggins, 1942a,b). Consequently, Eq. (10) is often rewritten as (11) where
1. Chemical structure of both the solvent and the polymer 2. Viscosity versus temperature data for both pure components, or comparable variable-temperature relaxation data (Zielinski et aI., 1992; Spiess, 1990; Bidstrup and Simpson, 1989; Blum et aI., 1986; Dekmezian et aI., 1985) 3. Density data for the pure solvent 4. Critical volume of the solvent 5. The Flory-Huggins X parameter or some other information characterizing thermodynamic compatibility between the polymer and solvent such as solubility parameters 6, Glass transition temperature of the polymer, T g2 Comparison of theoretical predictions with experimental data suggests that some parameters, particularly Do and ~, are critical for accurate predictions whereas other parameters such as E and X are of secondary importance. For example, reasonably good predictions have been attained by assuming that E equals zero as a first approximation (Vrentas et aI., 1989). The solid lines in Fig. 1 correspond to predictions of the mutual diffusion coefficient for the toluene/poly(vinyl acetate) system based on techniques of

151

FREE- VOLUME THEORY

Zielinski and Duda (1992a). This system is representative of many in that although the predictions are reasonable they may be nonetheless inadequate for the design of a polymer processing operation. As one might expect, the availability of even a limited number of diffusion coefficient data points significantly enhances the predictive capabilities of the theory. To illustrate this point, the data at 40°C were used to obtain a better estimate of Do. The resulting semipredictive dashed lines shown in Fig. 1 indicate that the theory does an excellent job of correlating diffusivity data and extrapolating to other experimental conditions (e.g., 80°C and 110°C in Fig. 1). In contrast, the prediction of mutual binary diffusion coefficients without the use of any experimental diffusivity data is, at best, precarious.

10-5 ~---------------------------------,

---- - - - - ------~~-9-- --- ----~-~~

---

-....... -

10- 7

t il

E ()

10-8

Q

10-9

10-11

L-~

0.0

o

T=40°C

II

T=80°C

o

T=110ac

___ L_ _ _ _ _ _L __ _ _ __ L_ _ _ _ _ _L __ _ _ _

0 .1

0.3

0.2

0.4

~

0 .5

W, Figure 1 Comparison of mutual diffusion coefficients predicted from free-volume theory with experimental data for toluene/poly(vinyl acetate) system at 40,80, and 110°C. Solid lines are based on the purely predictive techniques of Zielinski and Duda (1992a). Dashed lines use data at 40°C to determine Do.

152

DUDA AND ZIELINSKI

The primary hurdle in applying free-volume models has been the accurate estimation of the jumping unit molar volumes, V ii and V 2i , required to calculate ~. A survey of numerous studies indicates that solvents can be divided in two categories. Spherical, rigid molecules belong to the first (and most elemental) class and diffuse as single units. Larger solvent molecules can assume the shapes of rods or chains and constitute the second category. In this case, Mij is less than the total molecular weight of the solvent. As a first approximation, Vii can be estimated as the molar volume of the solvent at 0 K [denoted here as Vi(O)] if the entire molecule jumps as a unit. Experimental evidence reveals that molecular shape also influences the effective size of the jumping unit. Guo and coworkers (1992) developed an empirical relation between the effective molar volume of solvent jumping units and weighted averages of the principal dimensions of the solvent molecule. Methodologies sought for estimating the size of the jumping unit of the polymer chain, V 2i , have been much more elusive. However, V 2i can be estimated from diffusivity data for any solvent in the polymer of interest because the size of the polymer jumping unit is a characteristic feature of the polymer and is therefore independent of the solvent. The only known protocol available to estimate V2i without the use of any diffusivity data is based on the correlation of the jumping unit molar volume with polymer stiffness as a given by the glass transition temperature (Zielinski and Duda, 1992a). A correlation of this type is anticipated to yield poor values of V2i for polymers of high stiffness that nonetheless exhibit low glass transition temperatures, such as polybutadiene. Thus, we emphasize once again that the theory's ability to predict temperature and concentration dependence of diffusion coefficients is significantly enhanced through the use of a few data points. The free-volume theory for diffusion in polymer systems can be applied on several levels. It is generally accepted that the theory provides an excellent framework for the correlation of diffusivity data and that with limited data, diffusion coefficients can be predicted at temperatures and solution concentrations where data are not available. At the present state of development, however, the predictive capabilities of the theory are inadequate for some applications. Nonetheless, this is the only theory currently available that can predict mutual binary diffusion coefficients in concentrated polymer solutions to within an order of magnitude. The theory can also provide a basis for a qualitative understanding of the influence of such variables as temperature, concentration, and solvent molecular size on diffusional behavior. The following rules of thumb are a direct result of the free-volume formalism: 1.

The apparent activation energy of the mutual binary diffusion process is a strong function of temperature and concentration and increases as the solution approaches Tg of the system [see Eq. (8)]. In contrast, changes in the

FREE-VOLUME THEORY

2.

3.

4.

5.

153

state of the system that remove it from Tg (e.g., increases in solvent concentration or system temperature) result in a decrease in the apparent activation energy. Similarly, the concentration dependence of the mutual binary diffusion coefficient is greatest near Tg and decreases with increasing temperature and concentration. All of these trends are related to the fact that increasing the temperature of the system or adding small molecules, which possess larger free volumes than their neighboring polymers, will increase the overall free volume available for molecular migration. In general, the temperature and concentration dependence of diffusion coefficients for polymer/solvent systems increases with solvent size. This is particularly true for large, spherical solid molecules, which are expected to move as single jumping units. The probability of a large molecule locating sufficient free volume to take a diffusive step is relatively low; consequently, small changes in the available free volume caused by increases in temperature or solvent concentration can have a substantial impact on the diffusion coefficient. In contrast, the diffusivities for systems of small molecules such as fixed gases in polymer systems are very weak functions of concentration and can often be viewed as independent of concentration. Similar arguments suggest that the apparent activation energy for such systems is relatively low and independent of temperature and concentration. Elastomers or rubbers at conditions far above their Tg possess a large fractional hole free volume. Thus, diffusion coefficients under these circumstances are not only large in magnitude but are also relatively weak functions of concentration and temperature. In contrast to observed behavior in dilute polymer solutions, the mutual binary diffusion in concentrated polymer solutions is a very weak function of the molecular weight of the polymer. Although the free volume associated with the chain segments at the end of each chain is greater than the free volume associated with segments in the interior sections of the chain, the increase in the number of end units does not significantly impact the overall free volume until relatively low molecular weight polymers are involved. In general, variations in molecular weight distribution and in the average molecular weight of most commercial polymers do not significantly influence diffusion in concentrated polymer systems in the regime of processing interest. The influence of additives on diffusion characteristics is directly related to the contribution of the additives to the free volume of the system. For example, plasticizers significantly enhance the available free volume, which leads to a decrease in Tg and an enhancement in the rate of molecular migration. The addition of impermeable or solid additives (such as fillers) may not significantly modify the available free volume but will cause a significant decrease in the diffusivity due to tortuosity effects, i.e., an increase in the path length along which a molecular must travel.

154 6.

DUDA AND ZIELINSKI In most cases, the free volume associated with the solvent is significantly greater than the free volume of the pure polymer, and thus the mutual binary diffusion coefficient increases with the addition of solvent. As mentioned above, this rate of change of the diffusivity with solvent concentration will be greatest when the system is close to its glass transition and when large solvents diffuse as single units. Thermodynamic forces, however, serve to decrease the mutual diffusion coefficient with increasing solvent levels. This is particularly true when the solvent is a poor solvent for the polymer. Consequently, we have two forces acting in opposition. The net result is a maximum in the mutual binary diffusion coefficient as a function of solvent concentration that shifts to lower solvent concentrations as temperature is increased. In rare cases when a polymer/solvent system occurs at temperatures far above the system glass transition, the thermodynamic forces dominate and the mutual binary diffusion coefficient will actually decrease with increasing solvent concentration over the entire concentration range (Iwai et al., 1989). In contrast, the solvent self-diffusion coefficient, which does not contain the thermodynamic term, will generally increase with solvent concentration.

One concern often expressed regarding free-volume theory is that molecular interactions between the polymer chain and the solvent molecules are not explicitly considered. Specific polymer-solvent interactions are, however, introduced in two ways: (1) Equation (10) accounts for molecular interactions through the chemical potential of the solvent in the solution, and (2) the activation energy, E, in the most general formulation reflects the interaction between the solvent jumping unit and its neighbor. Clearly, the energy required for a solvent molecule to break the interactions with its neighbors may depend upon whether the neighbors are other solvent molecules or segments of the polymer chain, and the possibility that E is a function of solvent concentration has been discussed by several investigators (Macedo and Litovitz, 1965; Vrentas et aI., 1980; Vrentas and Vrentas, 1993; Zielinski and Duda, 1992b). Although the incorporation of a concentration-dependent E will improve the correlation of diffusivity data, it can be argued that this is strictly an empirical procedure that merely introduces more adjustable parameters. At the present time, the activation energy and its concentration dependence have not been related to fundamental molecular interactions. From all available experimental data, for concentrated polymer solutions in the vicinity of the glass transition (T < Tg2 + lOO°C), the diffusion process is dominated by the scarcity of free volume, and specific molecular interactions are of secondary importance. This conclusion may be biased, though, due to the fact that the great majority of accessible diffusivity data correspond to systems that do not exhibit strong interactions such as hydrogen bonding.

FREE-VOLUME THEORY

155

Although many recent studies are based on the Vrentas-Duda model, this discipline was dominated for several decades by the Fujita theory (Fujita, 1961). Both the Fujita and Vrentas-Duda models have their origin in the older Cohen-Turnbull formalism. However, several significant differences exist between the two models. From a practical point of view, the principal distinguishing feature is that the Fujita theory is purely correlative and is incapable of extrapolating results beyond the range of available data. Consequently, it has no predictive capabilities. Perhaps the greatest difference conceptually is that the Fujita theory is based on the free volume per unit volume of solution, whereas the Vrentas-Duda theory is based on the average free volume per jumping unit (Vrentas et al., 1993). It could be argued on conceptual grounds that this difference renders the VrentasDuda approach more consistent with the original Cohen- Turnbull concept. A more important question to be answered, though, is whether the Fujita theory can be related directly to the Vrentas-Duda theory. For the sake of comparison, the Vrentas-Duda model can be rearranged to the form reminiscent of the Fujita model (Vrentas et aI., 1993):

(12) (13) where D,(O) denotes the solvent self-diffusion coefficient at zero solvent concentration, /; is the fractional hole free volume of pure component i, and V? is the specific volume of pure component i at the temperature of interest. This relationship is based on the assumption that the partial specific volumes of the solvent and polymer are independent of composition so that no volume changes arise upon mixing. Similarly, the following equation can be derived from the Fujita theory:

(14) where the free-volume parameters of the theory are Ed, h, and k In using the Fujita expression, it is common practice to correlate experimental data to determine NEd and /JBd. A physical interpretation of these parameters is not necessary in employing Eq. (14) as a correlative model. However, the similarity of the two theories becomes evident if the following relationships are assumed: h =tv /2=h, and Ed = 'YV;/~. When related in this fashion, Eqs. (12) and

156

DUDA AND ZIELINSKI

(14) reveal that the two theories are identical when ~V;

V;

~ =~

(15)

This analysis demonstrates that the Vrentas-Duda expression reduces to the Fujita model when certain restrictions are imposed. For example, the relationship given by Eq. (15) is satisfied when the jumping units of the polymer and solvent are identical and the specific volumes of the two species are approximately equal. Under these restricted conditions, the two theories are identical in form and have the same number of parameters. Although both theories are excellent correlative tools, the utility of the Vrentas-Duda theory arises from its ability to serve as a predictive model without the use of diffusivity data from the polymer/solvent system of interest. Unlike the Fujita formalism, the Vrentas- Duda theory is also capable of describing diffusional behavior over broad ranges of temperature and concentration from limited diffusivity measurements. The fundamental strength of the VrentasDuda formalism is based on the determination of the free volume associated with each jumping unit as an integral part of the generalization of the original Cohen-Turnbull theory to describe solvent self diffusion.

IV.

THE INFLUENCE OF THE GLASS TRANSITION

As amorphous rubbers are cooled, the motion of individual polymer chains becomes so constrained that the cooling rate becomes faster than the rate at which the polymer sample can volumetrically relax. The resulting nonequilibrium condition is referred to as the glassy state. Thus, the passage from the rubbery to glassy states is denoted the glass transition. The glass transition is a dynamic phenomenon and occurs over a temperature increment whose range is influenced by the thermal and mechanical history of the polymer as well as the rate of cooling and/or frequency of the experiment. A schematic representation of the idealized volumetric behavior of a polymer above and below its Tg is illustrated in Fig. 2. At T > Tg, polymer chains are capable of achieving equilibrium configurations, whereas polymer segments in the glassy state do not have sufficient mobility to attain equilibrium conformations within commonly referenced time scales. Consequently, extra hole free volume becomes trapped within the polymer as it is cooled through the glass transition. Although the rate of molecular motion within glassy polymers prevents volume relaxation from reaching equilibrium, molecular motions are not completely eliminated within a glass. Density fluctuations persist and consequently necessitate redistribution of the hole free volume.

157

FREE-VOLUME THEORY

EQUILIBRIUM LIQUID VQLUME

w ~

3o >

0: W

::

~ oQ.

INTERSTITIAL FREE VOLUME

9 TEMPERATURE

Figure 2 Characteristics of the volume of a polymer above and below the glass transition temperature (TJ.

The formalisms presented in the preceding section would be appropriate for elucidating and modeling molecular transport in a glassy polymer if the polymer were given sufficient time to relax to its equilibrium state. Due to the long time scale required for glassy polymers to relax fully, however, diffusion in glassy polymers typically occurs under non equilibrium conditions wherein a polymer sample possesses more hole free volume than it would at equilibrium. Consequently, predictions of the availability of hole free volume in glassy polymers based on transport measurements made in an equilibrium rubbery state are always appreciably low and consequently lead to predictions of diffusion coefficients in the glassy state that are lower than those measured experimentally. We emphasize that although additional hole free volume is effectively trapped in a glassy polymer, this volume continues to be redistributed throughout the sample. Within the time frame of interest, the rate of volumetric collapse due to relaxation is negligible. The development presented below for diffusion below Tg is based on the concepts previously discussed (Vrentas and Duda, 1978; Vrentas and Vrentas, 1992; Vrentas et ai., 1980, 1988). Accurate prediction of diffusion in the glassy state requires quantification of the extra hole free volume trapped within the glassy state. With this accomplished, the free-volume formalism developed for diffusion in the rubbery state can be

158

DUDA AND ZIELINSKI

modified to encompass mutual diffusion below the glass transition of the solvent/ polymer mixture, Tgm' A general relationship between the mutual diffusion coefficient, hole free volume, and the polymer-solvent thermodynamic interaction can be written as

D-Dexp (-E) (-'Y(W1V; + W2~V;») Q RT exp -

V~

0

(16)

The symbols in this equation retain their meanings from the last section, although V~ is used to denote the hole free volume of the mixture, rather than Vfll, to emphasize the glassy state. In addition, Q accounts for the concentration dependence of the solvent chemical potential [see Eqs. (10) and (11)]. Only the magnitudes of V~ and Q are expected to be affected by the glass transition. The polymer/solvent mixture below Tgm is considered as a combination of the glassy polymer at the temperature of interest with a liquid solvent at equilibrium. Additivity of hole free volume yields (17) Here V~2 constitutes the hole free volume of the equilibrium polymer plus the excess hole free volume trapped due to the glass transition: (18)

where the excess hole free volume of the polymer, y.~, can be estimated in terms of the difference between the thermal expansion coefficients of the equilibrium rubber and the glassy polymer (et2 and et2g, respectively). Thus, (19) where ~(Tgm ) is the specific volume of the equilibrium polymer at Tgm. The corresponding expression for the hole free volume in the glassy state, therefore, becomes

V&fll(T < Tgm ) = WIVflll(T) + W2Vfll2 (T)

+

(20)

w2~ (Tgm )(et2g - et)(T - Tgm)

where Vfll1 and Vfll2 are the hole free volumes of the pure species in the equilibrium state and are related to the free-volume parameters Kl h K2h K 12 , and K22 [Eq. (6)], which can be estimated from pure-component viscosity data. In summary, the hole free volume, V~, for a glassy polymer/solvent system can be estimated from the equilibrium hole free volumes of the pure components in the system, and information concerning et2g - et2, Tgm' and ~(Tgm ). In addition, the overlap factor, 'Y, has to be determined independently. Techniques are

FREE-VOLUME THEORY

159

available for estimating the TgJD dependence on solvent concentration (Chow, 1980). Similarly, ~(T) is available for most commercial polymers from either direct measurements or correlations. Consequently, C(2g - C(2 becomes the critical parameter for quantifying the extra hole free volume in a glassy state, which can depend strongly on the processing history of the system. The second parameter required to estimate diffusion coefficients in the glassy state from free-volume concepts is Q. Despite the usefulness of the FloryHuggins theory in reflecting thermodynamic interactions between polymer and solvent molecules within polymer solutions above the glass transition, the equilibrium theory does not adequately describe thermodynamics in nonequilibrium glassy systems. If thermodynamic data such as vapor-liquid equilibrium measurements are available, Q can be calculated explicitly. Alternatively, correlative and predictive formalisms that accurately portray polymer-solvent thermodynamic interactions can be used to estimate Q. Numerous sorption models have been proposed in the literature, and a few representative ones are listed below: 1.

2.

3.

4.

5.

The dual-mode theory of gas sorption (Barrer et aI., 1958; Michaels et al., 1963; Vieth and Sladek, 1965) has been used extensively to describe the solubility of gases in glassy polymers. The fundamental assumption of the theory is the existence of two distinct solute populations within the polymer matrix (i.e., gas molecules that are either sorbed by an ordinary dissolution mechanism or reside in preexisting voids trapped within the glassy polymer upon cooling below Tg). Local equilibrium between these two solute populations is presumed to be maintained throughout the polymer matrix. The Vrentas sorption model (Vrentas and Vrentas, 1989, 1991a) differs from the dual-mode sorption model in that it considers the existence of a single sorption environment. In addition, the model accounts for the effect a penetrant molecule has on polymer structural rearrangement and subsequent volumetric change. Starting with the free-volume concepts introduced here, Ganesh and coworkers (1992) proposed a free-volume-based lattice model that predicts both gas sorption and transport in glassy polymers. Lipscomb (1990) described penetrant sorption as the combination of solid deformation followed by mixing with the penetrant. The sorption isotherms predicted from this theory are consistent with experimental data, and the model parameters can be related to those in the dual model at the lowsorption/high-bulk modulus limit. Weiss et al. (1992) modified the dual-mode sorption model to be a multisite model, based on a single continuous distribution of Langmuir isotherms. Their model correlates data nearly as well as the dual-mode sorption model with one fewer adjustable parameter.

160

6.

7.

DUDA AND ZIELINSKI

The gas-polymer-matrix model (Raucher and Sefcik, 1983) assumes a single sorption environment and correlates data equally as well as the dualmode sorption model. Correlations of dual-mode parameters with material properties, however, seem to provide more physical insight into the sorption process (Barbari et aI. , 1988). Astarita et al. (1989) incorporate an internal state variable, the free volume, into a lattice model of polymer solutions. Their theory represents the sorption of carbon dioxide in poly(methyl methacrylate) well.

Since many alternatives exist for estimating the polymer- solvent thermodynamic interactions, Q is often readily available when mutual binary diffusion coefficients in glassy polymer systems are to be estimated from the free-volume theory for molecular diffusion. An apparent paradox arises herein: Polymer solutions are inevitably not at equilibrium below Tgm , and yet equilibrium thermodynamic concepts are suggested to estimate Q in the glassy state. The assumption often introduced to address this seeming contradiction is that once the nonequilibrium glassy structure forms (regardless of the temperature and solvent concentration), the structure remains unchanged during the time frame of interest. Relaxation of the polymer structure toward the equilibrium liquid structure will unquestionably influence not only the thermodynamic relationship of the system but also the availability of hole free volume. Consequently, the developments presented here are limited to systems that exhibit classical Fickian diffusion. The applicability of the free-volume approach to penetrant diffusion in glassy polymers has often been questioned. Primarily, the debate addresses the consistency of the theory with the physical concept associated with dual-mode sorption. Of the sorption models alluded to earlier, the dual sorption model is the most widely implemented for gas sorption in glassy polymers. For molecular transport in rubbers, however, the free-volume approach described here is more applicable. Since the Vrentas sorption model uses the same principles exercised in developing the diffusion models to address both rubbery and glassy polymers, in this section we compare the fundamental differences between the two approaches. In dual-mode sorption (DMS), certain regions of the polymer matrix do not relax, owing to extensive chain entanglements that prohibit molecular reorientation. Some segments of the free volume are subsequently trapped in the matrix and are not continuously redistributed by random thermal fluctuations. These locally trapped free-volume pockets do not contribute appreciably to the vacant space available for transport, in which case not all of the excess free volume associated with the glassy state is available for molecular transport. From this point of view, the excess free volume associated with the glassy state could be divided into two parts consisting of (1) excess hole free volume

161

FREE-VOLUME THEORY

and (2) excess interstitial free volume. The principal difference in physical states conceptualized by the DMS and free-volume approaches is reflected in the parameter a 2g - a g o In free-volume theory, all the excess free volume in the glassy state contributes to the hole free volume. This viewpoint is contrary to some variations of the DMS approach that suggest that all of the excess free volume is immobile and is trapped as interstitial free volume. At the present time, sufficient experimental data are lacking to ascertain with any certainty if some of the excess free volume in the glassy state should be identified as interstitial free volume. Vrentas and Vrentas (1992) demonstrated that the predictions of Eqs. (16) and (20) are qualitatively consistent with observations of diffusion in glassy polymer/solvent systems. Unlike diffusion above Tg , the free-volume theory has not been extensively evaluated for glassy systems. At its present state of development, the theory is sernipredictive for glassy polymers, because parameters such as a 2g depend on the specific system history and cannot be related to purecomponent properties alone. The most extensive data available for the evaluation of this theory at T < Tg exist for diffusion of trace amounts of solvent in a glassy polymer. In this limit (as Wj - 0), Eqs. (16) and (20) reduce to D

= D = Do exp j

(-E) RT

exp (-'Y~V;) -'gV FH2

(21)

and (22) where (23) In this limit, by definition, the mutual binary diffusion coefficient, D, is equal to the self-diffusion coefficient of the solvent in the polymer, D j • These rela-

tionships suggest that diffusion of a trace amount of solvent relies upon A, which reflects the extra hole free volume trapped in the glassy state. Aside from A, the remaining parameters are identical to those used in modeling diffusion in polymer melts. Numerous experimental investigations using various analytical techniques reveal that the temperature behavior of the diffusion coefficient is consistent with predictions of Eqs. (21)-(23) when the glass transition temperature is traversed (Coulandin et ai., 1985; Amould, 1989; Hadj Romdhane, 1994). These studies confirm that a single value of A can be used with success to characterize the diffusion coefficients of different solvents in a single glassy polymer. This be-

162

DUDA AND ZIELINSKI

havior is expected from Eq. (23), which specifies that A is a characteristic quantity of the pure polymer and therefore must be independent of the penetrant. Representative data correlations are provided in Fig. 3 for the diffusion of toluene in polystyrene above and below Tg2 • Although free-volume transport theory has not been extensively evaluated for diffusion at finite solvent concentrations at temperatures below Tgm' it continues to lend itself to qualitative interpretations consistent with experimental observations. As for the case of diffusion above the glass transition, we delineate the qualitative behavior of diffusion in glassy systems suggested by the free-volume formalisms: Contrary to intuitive beliefs, the rate of diffusion (or the diffusion coefficient) does not abruptly decrease as a system cools to the glassy state. Rather, free-volume theory predicts that a fixed solvent concentration the diffusion coefficient is a continuous function of temperature as Tgm is traversed. 2. The apparent activation energy for diffusion, ED [see Eq. (8)], undergoes a step change at Tgm. For a given polymer, the change in ED will be greater for larger penetrant molecules. Consequently, it is possible that measure-

1.

10 -6

,

10 -7

.,

T =100·C 10 -" :

-...

10-9

()

10 - 10

I I)

"'E ~

0

-

1\=0

10-"

1\=0.30

10- 12 10- 13 10- 1• 0

20

40

60

80

Figure 3 Free-volume theory correlation of data for the diffusion of toluene in polystyrene above and below the glass transition temperature. Data are from Pawlisch (1985) and Hadj Romdhane and Danner (1993).

FREE-VOLUME THEORY

3.

163

ments of the diffusion of small molecules, such as permanent gases, will not register any ED change at Tgm' whereas larger molecules, such as solvent vapors, will exhibit significant increases in ED as the temperature is lowered through Tgm. In addition, the change in ED at Tgm will decrease as solvent concentration increases, i.e., the influence of the glass transition on diffusion is most apparent for trace amounts of large solvent molecules in the polymer matrix. Equations (16) and (20) relate solvent concentration to diffusivity. If, as is usually the case, the hole free volume of the solvent is greater than that of the polymer (VFHl > VFH2 ), then the addition of solvent increases the hole free volume and correspondingly increases the diffusivity. However, the addition of solvent also lowers Tgm and consequently reduces the excess hole free volume associated with the glassy state [see Eq. (19)]. The addition of hole free volume from the solvent often outweighs the loss of excess hole free volume associated with lowering Tgm' This, however, is not always the case, and thus the mutual binary coefficient can decrease with increasing solvent concentration (Vrentas et aI., 1988; Vrentas and Vrentas, 1992]. Solvents or penetrants that significantly relax the glassy matrix structure without enhancing the free volume of the mixture are referred to as antiplasticizers.

4.

The rate of diffusion in a glassy polymer system is strongly influenced by the processing history of the polymer. Aging or annealing cause a densification of the polymer and a reduction in excess hole free volume. The freevolume framework suggests that as a glassy polymer ages and relaxes toward its equilibrium state, diffusional rates become slower.

V.

MORE COMPLEX SYSTEMS A. Lightly Cross-linked Systems

Equation (9) can be considered a general form of the free-volume theory. This relationship can be modified to describe diffusion for specific classes of systems through development of appropriate relationships for the parameter VFH • Vrentas and Vrentas (1991b) proposed modifying this theory to address solvent diffusion in cross-linked polymers. In their development, systems must possess a relatively low degree of cross-linking, X, such that 50 or more monomers separate neighboring cross-link points. In this limit, as a first approximation, many of the parameters in Eq. (16) can be considered to be independent of the degree of cross-linking. For example, the solvent properties VFH 1> V;1> and V; are clearly independent of X. Furthermore, Vrentas and Vrentas (1991b) suggest that the distribution of free volume in the polymer and the size of the jumping unit of the polymer chain do not depend on X, in which case neither does -y, V); , nor V;. The above approximations lead to the result that ~ is also independent

164

DUDA AND ZIELINSKI

of X. When the cross-link points are sufficiently spaced, the jumping rates of the solvent molecules and the interactions between these molecules and their neighbors (which are represented by Do and E) will not be influenced by the presence of the cross-links. Following this line of reasoning, one is led to conclude that cross-linking a polymer influences only the diffusion coefficient through the thermodynamics of the polymer-solvent interactions and the specific hole free volume of the polymer, VFH Z• The addition of chemical cross-links to a polymer will obviously inhibit the segmental and molecular motion of the chains that arise due to thermal fluctuations; consequently, the free volume of a cross-linked polymer is expected to be less than that of a non-cross-linked polymer. This anticipated behavior is substantiated by the fact that cross-linking increases the polymer density. Furthermore, it seems reasonable to assume that the loss of free volume associated with the formation of cross-links reduces the hole free volume, which dictates solvent transport, and that the occupied and interstitial free volumes of the polymer are both independent of X. Thus, the theory must be appropriately modified to incorporate the influence of X as well as temperature on VFH2 for cross-linked polymers. One way to do so presumes that the hole free volume of a cross-linked material at a given temperature, Vmz(T,x), is a fractional portion of the hole free volume of the polymer in the absence of cross-links, i.e. , (24) Vrentas and Vrentas (1991b) presented both theoretical and experimental evidence to indicate that 8 is virtually independent of temperature but is related to the specific volumes of the pure cross-linked and uncross-linked polymer, ~(T,x) and ~(T,O), respectively:

8 = ~FHZ(T,X) = ~(T,x) VFHz(T,O)

(25)

~(T,O)

Consequently, the influence of chemical cross-linking on the polymer free volume can be characterized by a single parameter, 8, which is determined directly from volumetric data on both the cross-linked and uncross-linked polymer. Following the Vrentas and Vrentas (1991b) development in the limit of a trace amount of solvent in a polymer (WI -+ 0), Eq. (9) takes the form

] [-EJ exp [-"'{(WI+V; +Z8Vwz~V;) z(T,0)

D j = Do exp RT

A

W jVFHj

A

W

(26)

FH

When 8 = 1, this relationship correctly reduces to the expression for solvent self-diffusion in an uncross-linked polymer, which has a hole free volume VFII2(T,O). Experimental measurements probing the influence of temperature, concentration, degree of cross-linking, and solvent size on DI (in the limit of

FREE-VOLUME THEORY

165

WI 0) are all relatively consistent with the predictions of Eq. (26). Consequently, the incorporation of only one new parameter, 8, is required to modify the conventional free-volume theory to address diffusive transport in amorphous, lightly cross-linked systems. Although the theory for cross-linked systems has not been extensively evaluated, the following qualitative behavior is suggested by this free-volume formalism:

1.

The diffusion coefficient decreases with increasing cross-link density, and this increase can be quite significant. Furthermore, the decrease in the diffusivity due to the cross-linking is more pronounced for larger penetrants. 2. The activation energy for diffusion, E D, increases with increasing degree of cross-linking. Furthermore, the influence of cross-linking on ED is enhanced for larger penetrants. Consequently, the activation energy for diffusion in a polymer with a particular cross-link density increases with increasing penetrant size. 3. The diffusion coefficient in a cross-linked polymer increases as the solvent concentration is increased, since low molecular weight solvents or polymers bring more hole free volume to the system. The influence of concentration on the diffusivity is more pronounced in cross-linked systems, and in the pure polymer limit (WI - 0) the dependence of the diffusivity on penetrant concentration increases as the cross-link density increases.

B.

Multicomponent Diffusion

Numerous polymer processes and applications involve multicomponent diffusion such as in the formation of many coatings, devolatilization of solvent mixtures, and membrane separation of two or more species. Vrentas et al. (1984, 1985b) considered the free-volume framework for the case of two solvents diffusing through a polymer during devolatilization. From basic diffusion theories, four independent diffusion coefficients are required to describe fully the molecular fluxes of all the species in a ternary system. For the case of mutual diffusion in a ternary system, the mass diffusive fluxes relative to a volume-average velocity can be related to the concentration gradients in the solution by four diffusion coefficients: ·I

J

·2

J

= -D

api _ D 11

= -D

ax

12

api _ D 21

ax

22

ap2

ax

(27)

ap2

(28)

ax

VJI + Vzi2 + V..J3 = 0

(29)

where ji is the mass diffusive flux of species i relative to volume average velocity, Pi is the mass density of species i, and VI is partial specific volume of species i.

166

DUDA AND ZIELINSKI

The free-volume theory, like most fundamental theories of diffusion, results in expressions that describe the self-diffusion coefficient of a species in solution. Relating self-diffusion coefficients to the diffusivities, D;j' that describe mutual diffusion in a multicomponent system is nontrivial. In fact, Bearman (1961) showed that no unique relationship exists between self-diffusion coefficients and mutual diffusion coefficients. Vrentas et al. (1985b) showed that for the case of two solvents (species 1 and 2) in a polymer (species 3) in some concentration interval near the pure polymer limit (W3 - 1), the following expressions can be obtained: Du -

DI

(30)

D I2 -

0

(31)

D22 - D2

(32)

D 21 -

(33)

0

where D I and D2 are the self-diffusion coefficients for solvents 1 and 2, respectively. In close proximity to the pure polymer limit, the principal diffusion coefficients (Du and D 22 ) are significantly larger than the cross-diffusion coefficients (D12 and D 21 ) and are approximately equal to the self-diffusion coefficients of the two solvents. Consequently, multicomponent diffusion taking place in a process such as devolatilization that involves low concentrations of the constituent solvents can be conveniently analyzed with only the self-diffusion coefficients of the two solvents. The basic free-volume expression for the solvent self-diffusion coefficient [Eq. (9)] can be readily modified for a ternary system. First, the distribution of the available hole free volume among all the jumping units of solvent 1, solvent 2, and the polymer must be considered. In addition, the available hole free volume must include contributions from the two solvents as well as from the polymer. The resulting free-volume expressions for the solvent self-diffusion coefficients in a ternary system of two solvents and a polymer are DI = D Ol exp ( - "'I

WI

V; + WS';~13 /~23 + f,

W3

V;~13)

(34)

YI'H

wl~d~13 +

V; + W3 V;~23)

W2

D2 = D02 exp ( -"'I ......:.=...::.:..:;---"i'-'=----=---=-== VPH

VFH = wJKu(K21 + T - Tg I ) + W2K I2(K22 + T + W3K I3(K23 + T - Tg3 )

(35)

- Tg2 )

(36)

where Do; is the preexponential factor for component i, K I ; and Ku are freevolume parameters for component i, and VI'H is the average hole free volume per gram of mixture.

FREE- VOLUME THEORY

167

As for binary polymer/solvent systems, the parameters in the free-volume expressions for Dl and D2 can be experimentally determined from volumetric, viscosity, and diffusivity data for single-component or binary systems. Recall that V; for each of the three components can be estimated from group contribution techniques for estimating the equilibrium liquid volume of a component at 0 K. The parameters Klih and K2i - Tgi for each component can also be discerned from viscosity data. Finally, DOl> ~13' D o2 , and ~23 can be attained from diffusivity data for the binary polymer/solvent systems. Ferguson and von Meerwall (1980) were the first investigators to modify the free-volume theory to describe a ternary system and correlate the self diffusion of two solvents in a ternary polymer solution. In conclusion of this section, a fairly straightforward modification of the freevolume approach leads to formalisms for the self-diffusion coefficients of two solvents in a two-solvent/polymer ternary system. These self-diffusion coefficients can be used to describe mutual ternary diffusion under conditions of relatively low solvent concentrations. Extension of the present theory to describe mutual diffusion for the entire concentration range in a ternary system would require a suitable approximation regarding the relationship between friction coefficients as discussed by Bearman (1961) and Vrentas et ai. (1985b).

C.

Block Copolymers

Chemical modification of polymer matrices provides enhanced opportunities to regulate and tailor materials for diverse technologies such as packaging, coatings, and separations. Within the last 20 years one particular class of chemically altered polymers has received significant experimental and theoretical attention due to the unique characteristics of the chemical species; they are block copolymers. As the name implies, block copolymers are chains composed of two (or more) monomer species in which long linear sequences of like monomer units are covalently bonded together. This physical attribute leads to interesting morphological repercussions because phase separation is restricted by covalent bonds. Microphase separation produces a variety of ordered morphologies and occurs in block copolymers when (1) sufficient thermodynamic incompatibility exists between the blocks and (2) the blocks are long enough to self-assemble into microdomain structures. The macroscopic physical attributes of block copolymers can be finely tuned through specific tailoring of the microstructure. Several studies have focused on measuring (and modeling) the rate of diffusion in microphase-separated block copolymers (Rein et aI., 1992; Ferdinand and Springer, 1989; Csernica et aI., 1987). As far as we know, however, no attempt has addressed transport in homogeneous, or disordered, block copolymers at conditions well removed from the order-disorder transition (ODT). This

168

DUDA AND ZIELINSKI

section, therefore, focuses on solvent self-diffusion within a two-component, homogeneous block copolymer melt. In the following analysis, 1 and 2 refer to the solvent and polymer, respectively, while 2a and 2b correspond to blocks A and B of the copolymer. If VFH denotes the specific hole free volume in a block copolymer/solvent mixture, then the molar free volume available for molecular transport can be written as

-V

FH

V

= - - - - - - : . :FH .:..----wl/Mlj + W2(W2jM2j. + w2JM2}b)

(37)

Here, Wi is the weight fraction of component i (i = 1 or 2), and W2a and W2b are the weight fractions of blocks A and B within the copolymer, respectively. The molecular weights of the jumping units for the solvent, polymer A, and polymer B are given by M lj, M 2j.. and M 2}b' respectively. WI and W2 sum to unity, as do W2. and W2b' Substitution of Eq. (37) into Eq. (2) and introduction of the overlap factor ()') and the energy to break free from neighbors (E) yields the transport expression for solvent diffusion in a homogeneous block copolymer melt. The resultant expression can be cast into a form comparable to that provided earlier if (38) and (39) Then

(-E)

DI = Do exp exp ( RT

- )'[Wl

V; +

W2(W2a£12.

V;.

+

W2b£12bV ; b)])

-':';"'~-=----=-="" A =:""-=----'==-"":":;'::":'

VFH

(40)

where V~ (k = a or b) is the specific volume of block k in the copolymer at 0 K. In the limit that the copolymer becomes a single-component homopolymer (Le., W2a = 0, W2b = 1, or A = B), then Eq. (40) correctly collapses to the original expression for solvent self-diffusion in a homopolymer [see Eq. (9)]. This relationship at the very least suggests self-consistency with the methodology adopted. Equation (40) is expected to describe solvent self-diffusion in any twomonomer AlB block copolymer irrespective of molecular architecture [e.g., AB diblocks, ABA triblocks, or (AB)n multiblocks], provided the copolymers are homogeneous melts. The difference in transport characteristics for the various architectures is expected to arise in the VHlh term.

FREE-VOLUME THEORY

169

Previous researchers have investigated the influence of microstructural morphology and chemical composition on transport within microphase-separated block copolymers and blends (e.g., Kinning et aI., 1987; Sax and Gttino, 1985). Although not elaborated upon here, we speculate that the mechanism of diffusion in microphase-separated block copolymers may be reminiscent of that in semicrystalline polymers. This analogy is particularly appropriate when one of the phases is a glassy polymer with low diffusion characteristics and the other is a continuous rubbery phase through which solvent molecules can diffuse with relative ease. Therefore, perhaps a simple modification of Eq. (40) to reflect the reduction in transport rate as a result of tortuosity may be adequate to describe the effect of morphology on solvent transport in ordered systems.

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Ganesh, K, R. Nagarajan, and J. L. Duda (1992). Ind. Eng. Chem. Res. , 31, 746. Guo, C. 1., D. DeKee, and B. Harrison (1992). Chem. Eng. Sci., 47, 1525. Hadj-Romdhane, I. (1994). Polymer-solvent diffusion and equilibrium parameters by inverse gas-liquid chromatography, Ph.D. Thesis, Pennsylvania State Univ. Hadj Romdhane, 1. and R. P. Danner (1993). AIChE J ., 39, 625. Haward, R. N. (1970). J. Macromol. Sci. Rev. Macromol. Chem. , C4, 191. Huggins, M. L. (1942a). J . Am. Chem. Soc ., 64, 1712. Huggins, M. L. (1942b). J. Phys. Chem., 46, 15l. Iwai, Y., S. Maruyama, M. Fujimoto, S. Miyamoto, and Y. Arai (1989). Polym. Eng. Sci., 29(12), 773-776. Kaelble, D. H. (1969). In Rheology, Vol. 5, F. R. Eirich, Ed., Academic, New York, p. 223. Kinning, D. 1., E. L. Thomas, and J. M. Ottino (1987). Macromolecules, 20, 1129. Lipscomb, G. G. (1990). AIChE J. , 36(10), 1505. Macedo, P. B. and T. A. Litovitz (1965). J. Chem. Phys. , 42, 245. Michaels, A. S., W. R. Vieth, and H. Bixler (1963). J. Polym. Lett., 1, 19. Pace, R. J. and A. Datyner (1979). J. Polym. Sci., B., Polym. Phys., 17, 437-451. Paw lisch, C. A. (1985). Measurement of the diffusive and thermodynamic interaction parameters of a solute in a polymer melt using capillary column inverse gas chromatography, Ph.D. Thesis, Univ. Massachusetts, Amherst, MA. Raucher, D., and M. D. Sefcik (1983). ACS Symp. Ser., 223, 11I. Reid, R. c., J. M. Prausnitz, and T. K Sherwood (1977). The Properties of Gases and Liquids, 3rd ed., McGraw-Hill, New York. Rein, D. H., R. F. Baddour, and R. E. Cohen (1992). J. Appl. Polym. Sci., 45, 1223. Sax, J. E., and J. M. Ottino (1985). Polymer, 26, 1073. Spiess, H. W. (1990). Polym. Prepr., 31, 103. Vieth, W. R., and K J. Sladek (1965). J. Colloid Sci., 20, 1014. Vrentas, 1. S. and J. L. Duda (1977a). J. Polym. Sci. , 15, 403. Vrentas, J. S. and J. L. Duda (1977b). J. Polym. Sci., 15,417. Vrentas, J. S. and J. L. Duda (1978). J. Appl. Polym. Sci., 22, 2325. Vrentas, J. S., and J. L. Duda (1986). Diffusion, in Encyclopedia of Polymer Science and Engineering, Vol. 5, H. F. Mark, N. M. Bikales, C. G. Overberger, and G. Menges, Eds., Wiley, New York. Vrentas, J. S. and C. M. Vrentas (1989). Macromolecules, 22, 2264. Vrentas, 1. S. and C. M. Vrentas (1991a). Macromolecules, 24, 2404. Vrentas, J. S., and C. M. Vrentas (1991b). J. Appl. Polym. Sci., 42, 1931. Vrentas, J. S. and C. M. Vrentas (1992). J. Polym. Sci. : B Polym. Phys., 30, 1005. Vrentas, J. S., and C. M. Vrentas (1993). Macromolecules, 26, 1277. Vrentas, J. S., H. T. Liu, and 1. L. Duda (1980). J . Appl. Polym. Sci. , 25, 1297. Vrentas, J. S., J. L. Duda, and H. C. Ling (1984). J. Polym. Sci.: B Polym. Phys., 22, 459. Vrentas, J. S., J. L. Duda, and A. C. Hou (1985a). J . Appl. Polym. Sci., 31, 739. Vrentas, J. S., J. L. Duda, and H. C. Ling (1985b). J. Appl. Polym. Sci., 30, 4499. Vrentas, J. S., J. L. Duda, and H. C. Ling (1988). Macromolecules, 21, 1470. Vrentas, J. S., C. W. Chu, M. C. Drake, and E. von Meerwall (1989). J. Polym. Sci., B: Polym. Phys., 27, 1179.

FREE-VOLUME THEORY

171

Vrentas, J. S., C. M. Vrentas, and J. L. Duda (1993). Polym. J. , 25(1), 99-10l. Weiss, G. H., J. T. Bendler, and M. F. Shlesinger (1992). Macromolecules, 25(2), 990. Williams, M. L., R. F. Landel, and L. D. Ferry (1955). J. Am. Chem. Soc., 77, 370l. Zielinski, J. M. and J. L. Duda (1992a). AlChE J ., 38, 405. Zielinski, J. M., and J. L. Duda (1992b). J. Polym. Sci., B: Polym. Phys., 30, 108l. Zielinski, J. M., A. J . Benesi, and J. L. Duda (1992). Ind. Eng. Chem. Res., 31, 2146.

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4 Transport Phenomena in Polymer Membranes P. Neogi University of Missouri-Rolla Rolla, Missouri

I.

INTRODUCTION

The literature on diffusion is vast and is mainly mathematical. The two books by Crank (1975, 1984), for instance, have become a part of the standard reading material on the subject. Other authors, like Cussler (1976), feel that much on diffusion can be learned without resorting to such endless mathematics. In a very specialized area such as polymers, the conventional mathematical modeling ends a little too quickly. In this chapter the origins of the key equations, their representations, and methods of solution are analyzed first. The unusual variants of the equations of importance to this area are discussed next. An analysis of the role of mass transfer in bringing about morphological changes follows. One of the well-known examples where such changes are engineered for applications is the reverse osmosis membrane, discussed in Section IV. Finally, in Section V, systems that the investigators have put together with applications in mind are reviewed. The special features of these systems are that they contain more than one component or phase and their transport phenomena are quite differently organized. There are pressing reasons to understand them quantitatively. The very new area of measurement of diffusivity with NMR is one example. The conservation of species equation in one dimension (as appropriate in membranes) is given by

ac

at

a.

= -

ax Jx

(1)

173

174

NEOGI

where c is the concentration of the diffusing species, t is the time, and jx is the flux in the x direction, which is along the thickness of a membrane and the only direction in which mass transfer is taking place. If A is the area of the face of the membrane, then A l{2 » L , the membrane width, which makes the dynamics one-dimensional. Equation (1) assumes that there is no convection. Actually, a convective term arises due to a change in the volume as the polymer swells in the presence of the solute. However, this term is not considered as Duda and Vrentas (1965) showed that it has no effect unless the excess volume of mixing is nonzero, and the excess volume of mixing is almost always neglected in polymers, with some justification. The entire problem needs to be converted to a boundary value problem in concentration c, which requires that the flux be related to the concentration field . The simplest way to do this is to employ Fick's law, (2) This form is valid when the solute is dilute (Bird et al., 1960) or when the volume-averaged reference velocity is being used or assumed (Cussler, 1976). Combining Eqs. (1) and (2), one has

~~ = :x (D ::)

(3)

which is essentially the equation that has to be solved subject to appropriate initial and boundary conditions. The object then becomes to describe the measured quantities in terms of diffusivity D, following which the theory and experiments can be compared to back out a number for D . In the simple form of sorption experiments, a membrane is suspended in vacuum. A vapor or gas is then introduced and maintained at a constant pressure. The solute dissolves and diffuses into the membrane, and the weight gain is measured gravimetrically. The data are reported as the fractional mass uptake (with respect to the eqUilibrium value) as a function of time. For constant D, the solution to Eq. (3) leads to

M, = 1 _ M=

~ 1T

± m=O

1

(2m + 1)

2 exp [ - 4D(2m

~

L

1)21T2t

J

(4)

where M, and M= are the mass uptakes at time t and at infinite time and L is the membrane width. Obviously, the solubility can be calculated from M=. It is seen in Eq. (4) that the exponential terms decrease drastically with m. If a halftime is defined as t = t l{2, where M,/M= = 1/2, then Eq. (4) can be approximated as

2:1 = 1

-

8

1T2

[4D1T2 ---u- t in ]

exp -

TRANSPORT PHENOMENA IN POLYMER MEMBRANES

175

or (5) It has been assumed that

tl l2 is sufficiently large that all terms other than the first in the series can be neglected. Equation (4) does not quite show what the solution is like. This is given by another solution in the form

M,

-

M~

=

8 Dt2"L (

112 )

[

'TT-

I12

+

. mL 22:~ (-It terfc 4(Dt) m~O

--112

]

(6)

where ierfc is the integral of the error function (Crank, 1975, p. 375). At short times, Eq. (6) approximates to (7)

that is, only the first term is important. If one insists on calculating the halftime from the approximate equation, Eq. (7), one has (8)

Equations (5) and (8) are respectively tl l2

= 0.01224L 21D

(9a)

and (9b) Obviously, Eq. (7) gives a good description of the solution, which is that M,/ M~ is linear in v'tiL past the half-time but away from equilibrium as shown in

Fig. 1a. The experimental data are ploUed against VlIL, and Eq. (9) is used to calculate D. In the permeation experiments the two sides of the membrane, which are initially under vacuum, are sealed off from one another. Then the gas is introduced on the upstream side and kept at a constant pressure PI' On the downstream side the pressure P2 slowly rises as the permeant is being stored. However, the magnitudes of the pressures are such that P I » P2(t) = O. Under those conditions the total amount that has permeated through the membranes Q, is given by

J1. _ Dt _ ! LeI - L2 6

_~ ~ 'TT2

ft

(-ll ex p [- Dn2'TT2t] n2 L2

(10)

176

NEOGI

1.0

Figure 1 (a) The fractional mass uptake versus t,n.IL in a sorption experiment. The diffusivity is calculated from the half-time or the initial slope. (b) The results from permeation experiments are shown in a form such that the slope at steady state gives permeability and the intercept on the t axis leads to diffusivity.

where c, is the upstream concentration, and, if Henry's law holds, (11) where H is the Henry's law coefficient. If V2 is the volume of the container downstream, then under the ideal gas law (12)

TRANSPORT PHENOMENA IN POLYMER MEMBRANES

177

where A is the area of the membrane. That is, P2 can be monitored to get Q, as a function of time. At large times Eq. (10) becomes

Q, Dt 1 --=-- -

(13a)

or L d(Q,) = DH PI dt

=P

(13b)

This is exactly so at steady state, and P is called the permeability. The units of permeability are cubic centimeters of gas that passes through the membrane at STP per second per atmosphere pressure drop per square centimeter membrane area, times the total membrane thickness in centimeters (Pauly, 1989). The permeability is seen to be a property of the solute-polymer interaction only, and its importance lies in the fact that when two solutes have permeabilities sufficiently removed from one another they can be separated by using a membrane. Equation (13a) can be rewritten as

LQ, (L2) -DH t - PI

6D

(13c)

Equations (10) and (13c) are shown in Fig. lb. The asymptote given by Eq. (13c) makes an intercept of L 2/6D on the t axis and has a slope of DH, which allows one to find both the solubility and the diffusivity. Permeabilities are so low that large pressure differences and membranes of small thicknesses have to be employed. These restraints tend to exclude condensable vapors. In contrast, only vapors of sufficiently high solubilities are suitable for the gravimetric measurements used in the sorption experiments. Consequently there are only a very few systems for which both sorption and permeation results have been reported. A few examples have been given by Stem et al. (1983), Kulkarni and Stem (1983), and Subramamian et al. (1989). A compilation of conventionally measured values was made available by Pauly (1989). The experimental apparatus has changed very little in its basic outline, a feature that is made clear by reviews (Crank and Park, 1968; Rogers, 1985; Vieth, 1991). Corrections that need to be considered when the reservoir pressure changes due to dissolution have been quantified. In improving the scope of the experiments, the emphasis appears to have been laid on measuring mass or pressure more accurately, on measuring changes in the dimensions of the membrane, and on the ability of the system to go to higher pressures or temperatures. [See in particular the systems developed by Stem and coworkers and Koros and coworkers as cited by Vieth. See also Costello and Koros (1992).] A very dif-

178

NEOGI

ferent approach was reported by Vrentas et al. (1984b, 1986) where the input was oscillatory. The results show that the method enjoys additional advantages over the step-change/sorption experiments, which are discussed later.

II.

MATHEMATICAL METHODS

When the diffusivity D is a constant, Eq. (2) becomes

ae at

a2e dX

- = D -2

(14)

Equation (14) is similar to the heat conduction problem where the temperature replaces concentration and the thermal diffusivity replaces D. The book on heat conduction by Carslaw and Jaeger (1959) and one on diffusion by Crank (1975) cover a great range of solutions to equations such as Eq. (14). These include various geometries, initial and boundary conditions, concentrated sources and sinks, simple composites, etc. The solutions are all analytical, or exact, as they are generally called. It becomes a little difficult to solve for the case where the diffusivity is a function of concentration. In that case Eq. (3) becomes 2

de = aD (ae) 2+ D a e at ae ax ax 2

(15)

Crank (1975) provides a general discussion on the subject. Some useful simplifications can be made. One of them is that the diffusivity is an increasing function of concentration, and a simplistic representation is (16) where Do and a are constants. The impact of concentration dependence on diffusivities is discussed next.

A.

Concentration-Dependent Diffusivities

A critical theorem in this area is one due to Boltzmann (1894), who showed that even for this case Eq. (7) holds away from equilibrium. An example is shown in Fig. 2. The apparent diffusivity, obtained using an equation such as Eq. (8), is obviously an average value over the concentration range employed. It is also suggested that in this range there exists one concentration value that corresponds to this diffusivity. However, it has been difficult to find that concentration. The important observation to make here is that as long as the diffusion is Fickian, an effective constant diffusivity can be used to mimic the sorption results. It is shown later that this holds even under more severe conditions (see Figs. 6 and 8).

TRANSPORT PHENOMENA IN POLYMER MEMBRANES

179

o • L = 3.53 X 10-3cm L = 7.0 X 10-'cm 9 , L = 1.2 xIO-'<:m

o•

O~~------~--------~~--------~--------~ o s 10 15 20

JT (min

1/2)

J

1.0

0

t>-'?~. ~61

6rf

8

/o~ 0.5

rr6~~~./ ,._e-

~ ::::e

___ 0

/,,0

6-~

oes .

o-.~ .... ~ ,.

/0

#. • .,

o

~"'I

L..
S

10

{f1L

xl 0-'

1S

20

min I/Ycm)

Figure 2 The data of Kishimoto for methyl acetate in poly(methyl acrylate) at 35°C, as reproduced by Fujita (1968).

180

NEOGI

Diffusivities of solutes in solid polymers are almost always given by an increasing function of concentration as shown in Eq. (16). Specifically, for diffusivities given by Eq. (16) and for concentration-dependent diffusivities in general, Crank (1975) put together a detailed compilation of the approximate solutions. These solutions tend to become more approximate as the concentration dependence becomes steeper and to become more exact as diffusivities become less concentration-dependent. Neogi (1988) obtained an approximate solution for the case when <X in Eq. (16) is very large. Many of the characteristics obtained there are well known from before. Both the old and new results are summarized below. Boltzmann's scaling is upheld . It is seen that the fractional mass uptake when plotted against t is almost linear. However, the telltale sign that it is actually Fickian is observed in the fact that the fractional mass uptake curves make a slope of 900 at the origin. Grayson et al. (1987) and Korsmeyer et al. (1986) appear to have observed such responses experimentally. 2. When diffusivity is a constant, the concentration profiles are seen to change smoothly and gently within the polymer. If the diffusivity is an increasing function of concentration, then a concentration sharpening of the profile takes place and a sharp front is obtained. This happens because for the same concentration gradient, the flux is greater at the greater concentration. As a result, the movement of the concentration front is controlled by the region of lowest concentration. Calculated concentration profiles showing selfsharpening are illustrated in Fig. 3. It should be noted that Eq. (15) is classified as a parabolic equation (Ames, 1965) and can in no case lead to a sharpening to the extent of showing a discontinuous shock, that is, where the profile drops suddenly and discontinuously. 3. In desorption, the fraction desorbed should be equal to the fractional mass uptake if the diffusivity is a constant. If the diffusivity increases with concentration, then the desorption is slower than sorption, as seen in Fig. 2.

1.

Eventually the problem reduces to the fact that it is difficult to obtain the concentration dependence of diffusivity without using a model for diffusivity such as Eq. (16). At least experimentally this problem can be handled partially. In the sorption experiments a step change is given from the initial concentration of zero to Co. This is the integral sorption. If the step change is made from Ci to Co, where Ci is very close to Co, then it could be assumed that the diffusivity can be approximated to be a constant and evaluated at a mean concentration, say (1/2) (co + co). This is the differential sorption and provides information on diffusivity as a function of concentration.

TRANSPORT PHENOMENA IN POLYMER MEMBRANES

181

(,0

0·8 z

0 H 0·6 .... ;:! .... z

'"uz

0·4

0

u

0·2

0·0

-3

-2

Figure 3 Numerical solutions of concentration profiles (dimensionless) in a semiinfinite system where the diffusivity is concentration-dependent, are shown. The numbers on the plot are the different values of dimensionless ex defined in Eq. (16). [Reproduced from Crank (1975), with permission.]

B.

Numerical Solutions

The ability to solve these equations numerically has greatly improved our understanding of the diffusion process, The solutions obtained are approximate. In finite-difference schemes the partial differential equations are converted to algebraic equations (more accurately, difference equations). Instead of the continuous variable x one has a discrete set of points {Xj}, and instead of time tone has the set {til . Likewise, the concentrations become cij and the derivatives become aC

=

C;+ I ,j -

C;,j

+ OeM)

(17)

!1t

at and

_::_~

=

C;,j+ J -

(:)2+

C;,j_ 1

+ 0(&)2

(18)

Inserting Eqs. (17) and (18) into Eq. (14) and rearranging, one has C'·+ J J·

= c·· ',j

D!1t

+- (c ',.j'+ 1 (&)2

-

2c·· ',j

+

c·',j'+ J)

(19)

where the increments !1t = ti+ l - t; and & = xj+ 1 - Xj are constants. If the initial concentration profiles C;,j are known, then Eq. (19) provides the means

182

NEOGI

for moving up in time to Ci+ 1J . Such iterations can be continued until the numerical approximation to equilibrium or steady state is achieved. It is obvious from Eqs. (17) and (18) that the basic representation (forward time central space, FfCS) in this finite-difference scheme is an approximate one that improves as the size of the increments is decreased. However, the price that is paid in this process is that the computations become impossibly lengthy. Besides accuracy, stability provides an important criterion for constructing an effective algorithm. For a number of reasons, errors accumulate at nodes, such as at the (i,j) node. The question that arises now is that of whether this error would propagate. If in the next iteration, at (i + 1,j), this error increases, then the algorithm is unstable. Linear stability analysis for the system given by Eq. (19) shows that when the stability ratio R = D !1t/(tu)2 =:; 1/2 (Carnahan et aI., 1969), the system is stable. This causes considerable difficulties. If Eq. (19) is cast into dimensionless form where x goes from 0 to 1, and there are 20 subdivisions, then tu = 0.05. The dimensionless diffusivity is 1, and the stability criterion allows !1t = 0.00125 as the largest step size in dimensionless time. The diffusion process is best observed around t - 1, which would take 800 iterations in time! To circumvent this problem, various techniques are used, of which the semi-implicit form of algorithm is a common one. Here the procedure requires that one or more terms on the right-hand side in Eq. (19) have a term in i + 1. Among these semi-implicit schemes, the one known as the Crank-Nicholson technique is frequently used. There is a more direct reason for requiring semi-implicit or stable schemes in the case of diffusion in a solid polymer membrane, which is that the diffusivity D changes by a few orders of magnitude in the range of concentration changes. Thus, the value of !1t that keeps the algorithm stable in the explicit scheme when D is moderate no longer works when D increases by a few orders of magnitude; much lower values of !1t are required there. However, another problem arises in that often in semi-implicit schemes an algebraic equation has to be solved or a matrix has to be inverted. If one of the terms in these becomes very small, which will happen in the present problem, the roundoff error can become exceedingly large. Consequently, in such problems, which are called stiff, some independent means of checking the accuracy is desirable. Such mischief does not happen in the explicit schemes, although in any system the errors in the steps shown in Eqs. (17) and (18) may add up. There is no way of predicting the cumulative error. Finally, stability can break down due to boundary conditions, a feature that has not been explored in the literature in detail (Roache, 1982). In recent years another method has become popular. The finite-element method seeks an approximate solution that is exact at selected points. When these points are brought close together, the accuracy increases. The importance of this method lies in the fact that its programming is generalized and not

TRANSPORT PHENOMENA IN POLYMER MEMBRANES

183

tailored to a specific problem, and it is about the only method that can handle complex geometries: three-dimensional and asymmetric. However, it has not seen significant applications in mass transfer. On the other hand, orthogonal collocation has been used extensively to seek the descriptions of many mass transfer operations (Holland and Liapis, 1983).

c.

Moving Boundary

One important class of problems involve systems where the position of a boundary is not known beforehand and has to be obtained as part of the solution. In particular, the boundary can move as a function of time that also has to be determined. For simplicity it is assumed below that the boundary lies on the yz plane, that is, perpendicular to the x direction being studied here. The key feature is the jump boundary conditions at the interface, where many quantities become discontinuous (Slattery, 1981; Miller and Neogi, 1985). These conditions are on the total material balance, p'(v~ -

v) = pn(v~ - v)

(20)

species balance,

, ,

c (vx

, ac'

v) - D ax =

-

II

C

II

(vx

-

v) - D

"

ac" + ax

R

(21)

and energy balance, "

,

pc (vx

-

v) - "d

aT' = ax

"""

p c (vx

-

v) -"dI aT" ax

+ HR

(22)

The superscripts I and II denote the two phases; the above conditions apply at the interface between the two, which moves with a velocity v; and Vx is the velocity inside a phase. As constructed here, all velocities have only x components. Further, c is the concentration, T is the temperature, and c', k', and D' are the specific heat, thermal conductivity, and diffusivity in phase I, etc. R is the rate of generation of the species at the interface, and H is the heat liberated per unit generation. The symbol p denotes density. What these equations represent is the fact that total mass, mass of a species, or energy transfers fully across an interface, provided that the observer also moves with the interface and provided that there is no accumulation (reactions, adsorption, etc.) at the interface. In solids there is no convection, and the Vx terms can be ignored. Moving boundary problems are encountered in the drying of polymer films and have been analyzed by Shah and Porter (1973), who provide a complete but nevertheless simplified analysis. An integral heat balance is used [which incorporates the jump heat balance of Eq. (22)], an integral balance of polymer

184

NEOGI

is used [which incorporates Eq. (21) for the polymer and the fact that it is nonvolatile], and the jump balance for the solute is avoided using the results for the polymer. Actual problems in devolatilization (Biesenberger, 1983) can involve vacuum conditions and lead to nucleation (Albalak et aI., 1990). The basic scheme for solving these problems requires a well-defined initial state. Assuming this initial position, the first updating of the concentration profile in time is done. One boundary condition remains, which is used together with the new concentration profile to update the position of the interface. The issue of numerical stability of the algorithm at the moving interface is a complex one, and Crank (1984) provides a few details.

III.

NON·FICKIAN DIFFUSION

Some of the basic assumptions made about a medium (particularly a fluid medium) break down in solid polymers. These are conditions of isotropy, homogeneity, and local equilibrium. Glassy polymers are not at equilibrium, but they relax slowly toward it (Rehage and Borchard, 1973). Even crystallites in semicrystalline polymers are not at equilibrium (Hoffman et aI., 1975). Stretching beyond the elastic limit gives rise to anisotropy in polymers, which is not restored on heating then cooling, for instance, indicating nonequilibrium configuration. Besides semicrystalline polymers, glassy polymers are also inhomogeneous although over much smaller length scales as determined from solubility studies (Hopfenberg and Stannett, 1973). The net result is that the Boltzmann scalling in sorption is no longer satisfied. There are two basic types of nonFickian sorption. The first of these was identified by Alfrey (1965); the mass uptake in the glassy polymer studied was found to be proportional to time instead of to the expected square foot of time. The concentration front was seen to be very sharp, appearing to be discontinuous, and to move with a constant velocity. Concentration-sharpened fronts of Fickian origin, shown in Fig. 3, move at a speed proportional to C lf2 • One can generalize these types of behavior as M, =kt"

M~

(23)

n

where n> 1 n=1 1/2
supercase II (Jacques et aI., 1974) case II (Alfrey, 1965) anomalous (Haga, 1982) classical/Fickian (Boltzmann, 1894) pseudo-Fickian (Liu and Neogi, 1991)

TRANSPORT PHENOMENA IN POLYMER MEMBRANES

185

Only a few references have been cited here. Windle (1985) has reviewed such data. The second kind of non-Fickian behavior was reported by Bagley and Long (1955) and by Long and Richman (1960). The fractional mass uptake curve was seen to be sigmoidal but with somewhat classical scaling. Figure 4 explains the different types of responses including one called a two-step response. The physical mechanisms behind the non-Fickian behavior are only now beginning to emerge and are not complete. One mechanism is that of coupled mass and momentum transport, coupled through swelling. The other is based on the fact that glassy polymers are inhomogeneous. This, as a mechanism for non-Fickian effects, has seen much less success. Finally, in all the references cited here (and more), the polymer is glassy. Although exceptions are very few, they do exist (Kishimoto and Matsumoto, 1964; Odani, 1968), including even polyethylene (Rogers et aI., 1959), which has a low glass transition temperature of -70°C.

1. 0

o

1.0

~

________________________

o~

_______________________ if

fl/L

s i gmo iri 3 i

c l a ssi cal

1.0

1.0 I I

o~

________________________

two-step

Figure 4

The different classes of non-Fickian sorption.

case II

186

NEOGI

A.

Memory-Dependent Sorption

To explain case II, it becomes apparent that the parabolic form of Eq. (3) will not be sufficient. From the characteristics reported by Alfrey (1965), which are discontinuity in the profile and front moving at constant speed, one concludes that the governing equation would have to be hyperbolic (Ames, 1965). The simplest model would then be a memory-dependent diffusion, where this form of memory is encountered in viscoelasticity (Frederickson, 1964). At short times the governing equation is hyperbolic (corresponding to elastic response), and at large times it is parabolic (corresponding to viscous flow). Neogi (1983b) proposed that the diffusion be expressed as

jx = -

r

Jo

j.L(t - t')

ae (X, t') dt' ax

(24)

where a simple form of relaxation function would be

j.L(t)

= D 8(t) + Do -T j

tJ

D j exp [ - ~

(25)

where D j and Do are the initial and final diffusivities, 8(t) is the Dirac delta function, and T is the relaxation time. In Eq. (24) the concentration gradient at a previous time t' also drives the flux at the present time t. Thus it is memorydependent. The reach of the recall is t - t', and realistically the proportionality factor j.L should decay with t - t'; this is called the fading memory. It is possible to see that the relaxation time is T

=

i~ tj.L(t)dt

(26)

and that (27) This last result can be used to show that Eq. (24) tends to the classical form of Eq. (2) at large times. At short times, and for Do much larger than D j , one has the hyperbolic wave equation

a2e at = 2

Do a2e -;;:-

ax

2

(28)

which predicts a discontinuity where the front travels with a constant speed Y(DJT). Neogi (1983b) justified the model by asserting that the glassy polymer is not at equilibrium but relaxes slowly toward it. The nonequilibrium parts and their

TRANSPORT PHENOMENA IN POLYMER MEMBRANES

187

relaxation are modeled using classical irreversible thermodynamics, and therefore the thermodynamic constraints behind Eq. (24) are satisfied. One important feature of transport phenomena is that although the entire system may not be at equilibrium (global equilibrium), every point is at equilibrium and satisfies the equation of state there. It is this lack of local equilibrium that is modeled using irreversible thermodynamics. We do not know how to apply laws when the local equilibrium does not apply. Consequently, it is assumed that the departure from local equilibrium is sufficiently small that the system of equations can be linearized about the local equilibrium, and there the detailed constraints are known. The model still needs an additional equation describing relaxation, which is generally done phenomenologically. The alternative to memory is to model changes in the matrix as an aging process, which makes the diffusivity an explicit function of time, D(t) (in contrast, memory dependence is often called heredity). This has also shown good results (Petropoulos, 1984; Crank, 1953). However, we have no knowledge of any constraints on the model for the aging phenomenon, and hence it has not been considered further. One example of case II is shown in Fig. 5. Generally speaking, case II is observed to blend in Fickian diffusion with changes in the conditions of the experiments (Baird et ai., 1971; Hopfenberg et ai., 1969, 1970). Vrentas et ai. (1975) suggest that if diffusion is coupled with elastic effects, then a Deborah

P/~ A

= .63 = 40°C

c = 35°C o = 30°C

o = 25°C

OL---------~----------~------

o

50

100

__ 150 _______200 ~----

Time (hrs)

Figure 5 The data of Hopfenberg et al. (1969) for n-pentane in biaxially drawn polystyrene. Some crazing also takes place, but Hopfenberg et aI. showed later (1970) that crazing and case II were not directly related. (Reproduced with permission.)

188

NEOGI

number, which is the ratio between the relaxation time and an overall time, can be defined. They choose the relaxation time from the elastic modulus (actually, their formula applies only to viscoelastic liquids), and the overall time scale as L2/D , where L is the membrane thickness. At small Deborah numbers, the relaxation times are small, the material becomes viscous quickly, and the diffusion is Fickian. At large Deborah numbers, the material is a frozen solid, and diffusion is again Fickian. It is in between that case II is seen. In Neogi's (1983b) model as well, it is possible to define a Deborah number, which is T/(L 2/D). Case II is observed in the vicinity of a Deborah number of 1 (Adib and Neogi, 1987), and diffusion becomes Fickian as we move away from there in either direction. In the Fickian limit, the governing equation, Eq. (14), is analogous to the equation for viscous flow, and in the limit in which case II is predicted, one has Eq. (28), which governs wave propagation in an elastic medium, and in that Neogi's model is also "viscoelastic." Using the linear form, many successful comparisons between theory and experiments have been made (Neogi, 1983b; Adib and Neogi, 1987). These are the sorption results of Vrentas et al. (1984a), which, though Fickian, contain small but decaying oscillations. The data of Odani (1968) and Kishimoto and Matsumoto (1964) on fractional mass uptakes for different membrane thicknesses L do not collapse into a single curve when plotted against Vi/L. These could be quantified, as well as case II itself. However, the index n in Eq. (23) in the linear model can be shown to vary between 1/2 and 1 only, and not beyond this range. This form also does not predict sigmoidal uptake. One important feature of this model is that the solubility (or skin concentration) changes with time. Mehdizadeh and Durning (1990) have shown, using a model derived earlier (Durning and Tabor, 1986) that memory-dependent diffusion can also predict the two-stage sorption. The linear version of their conservation equation is mathematically the same as that presented here, but the expression for the changing solubilities is quite different. Their calculations are done at very low Deborah numbers, and thus the diffusion response is Fickian, except at times too short to have a direct impact. However, it has a significant effect on the solubilities. There is a quick change in solubilities over a short period followed by a Fickian uptake; hence a two-stage response results. It was stressed earlier that diffusion in polymers is strongly concentrationdependent. However, Eqs. (24) and (25) are not derivable unless most of the coefficients are assumed to be independent of concentration. A more complete model can be made based on intuition (Jackle and Frisch, 1985) or through a detailed analysis (Durning and Tabor, 1986). However, problems arise if such a model is cast as J..L(c, t). For, when we insert into Eq. (24) the function c(x, t), which is clearly understandable as the concentration at point x at time t, becomes c(x, t - t'). We know that c(x, t') is the concentration at point x at a previous time t', but we do not know what c(x, t - t') is. Therefore, no number can be

TRANSPORT PHENOMENA IN POLYMER MEMBRANES

189

supplied for that quantity. The remedy is to use an expansion

ac + 1" ac+ ... = ~ (D ac) at at ax ax 2

2

(29)

where 1" and D can be functions of concentration. Substituting Eq. (24) into Eq. (1) and inverting by using Laplace transform leads to Eq. (29). Camera-Rod a and Sarti (1990) solved such an equation using only the first two terms on the left in Eq. (29). This truncation Limits the solution to large times. The sorption experiments of Vrentas et ai. (1984b, 1986) based on oscillatory response become quite important in memory-dependent diffusion. Imagine that the relaxation function in Eq. (24) has no concentration effect, an assumption that will hold in differential sorption in all cases. Then a sinusoidal response leads one to a Fourier transform of the relaxation function as a function of the frequency. Thus a frequency response with small amplitude provides us with a direct means for measuring the relaxation function, where the small-amplitude part plays the same role as in differential sorption, that is, inactivating the concentration dependence. Finally, the difficulties of numerically solving equations that are nearly hyperbolic are enormous. The worst occurs at a Deborah number of 1. The present methods introduce what is called artificial viscosity (Roache, 1982). This moves the Deborah numbers away from 1, and obviously one needs to ascertain what these effective Deborah numbers become during the computations.

B.

Sigmoidal Sorption

The first model in this class that could be compared with experiments was given by Kim and Neogi (1984). They followed an earlier formalism put forward by Larche and Cahn (1982), in which swelling is seen as a strain, and as strain causes stress and stress is the generalization of pressure, the chemical potential fJ, is affected. This in turn affects the flux, which can be written as

. Dc Jx = - RT

afJ,

ax

(30)

Because the swelling at equilibrium causes no stress, the stresses generated are transient. They occur primarily due to the condition of impenetrability; that is, when swelling occurs the region in the membrane tries to take up space that it is not allowed, and this act of exclusion is effected by a force or stress. Now the stresses have to satisfy force balances, which brings into play the shape and the boundary conditions of the overall system. This brings into the expression for the flux not just the properties of the point x, but those of the whole system. As a result, Larche and Cahn (1982) point out, their flux expression is not local anymore, which goes against the usual assumption that

190

NEOGI

fluxes can be described by the properties at a point (Truesdell and Noll, 1965). Carbonell and Sarti (1990) pointed out that some models, such as that of Durning and Tabor (1986), do not satisfy a force balance. Whereas this is desirable, it should also be noted that if the local equilibrium is not satisfied, forces do not always have to balance; thus the practical problem of how to define, quantify, or measure " internal stresses" that occur in glassy polymers remains unresolved. However, if the model is being defined to the point that even the constitutive equations are being specified, then forces ought to be balanced to be at least consistent. The approximate solution of Kim and Neogi (1984) to the sorption problem shows that the diffusion remains Fickian but the solubility changes with time. The chemical potential is made up of the concentration effect and the strain energies and is expressed as (31) where (J" is the stress, tr denotes trace, VI is the specific volume of the solute, and TJ is the change in the volume due to concentration changes. The chemical potential in the absence of stresses is f.I.oo. As the strain energy changes with time, the skin concentration adjusts accordingly such that the overall chemical potential at the surface remains constant and equal to the value of that in the reservoir during sorption. This changing skin concentration has been measured experimentally (Bagley and Long, 1955; Long and Richman, 1960). At large elastic modulus of the material, it appears that when the diffusion process is long over, the solubility can still continue to increase and the concentration profile becomes spatially constant. Such profiles have been measured for water vapor diffusing in epoxy (Wolf, personal communication, 1987). In general, all non-Fickian effects vanish at low concentrations in the swelling model. Neogi et al. (1986) later solved numerically a more detailed model, which showed that the uptake was sigmoidal. This allowed them to quantify the data of Bagley and Long (1955) and Long and Richman (1960), as shown in Fig. 6. They also calculated that large compressive stresses would arise, sometimes large enough to give rise to plastic deformation. Downes and MacKay (1958) observed that in their experiments a Fickian response occurr.ed when the step changes in sorption were small. These were also reproducible. When large step changes were given, sigmoidal uptakes were observed, but once they occurred the older Fickian responses could not be reproduced. Presumably, plastic deformation had changed the matrix. In a similar vein, Tamura et al. (1963) correlated sigmoidal uptakes to yield stresses. In semicrystalline polymers, a polymer molecule can run through more than one crystallite, and the effect is that the crystals are chained to one another. This prevents the amorphous regions from swelling, which gives rise to forces that pull at the anchors. In extreme cases the crystallites unravel, which constitutes

TRANSPORT PHENOMENA IN POLYMER MEMBRANES

191

1.00.

0..75

MC

,/./ ,/

0.25

// /

~

. 0.0.

0.1

'"

./

./'"

./

/'

./

//

./

.!-~

CCOi

~

...

/'

-~ I

I

I

I

I

I

I

I

I

0.2

0..3

CA

0..5

0..6

0..7

0..8

0 .9

1.0

F

T

Figure 6 The theoretical results of Neogi et a1. (1986) on swelling-induced effects fitted to the data of Long and Richman (1960) showing how the skin concentration and mass uptake change with time. The system is methyl iodide in cellulose acetate. Here s is a dimensionless elastic modulus and T is dimensionless time. An effective diffusivity has been used instead of a concentration-dependent one. [Reproduced from Neogi (1992) with permission of the American Institute of Chemical Engineers. © 1986 AlChE. All rights reserved.]

192

NEOGI

ductile failure in these materials. Rogers (1962) reported such effects in polyethylene where the amorphous regions are rubbery, and Durning and Rebenfeld (1984) reported similar effects in poly(ethylene terephthalate) where the amorphous regions are glassy. It becomes evident that more detailed stress-strain constitutive equations can be worked into the theory, and the theory itself can be given a broader basis than that given by Larche and Cahn (1982). Lustig et al. (1992) provided a generalized framework for diffusion including relaxation and stress effects to which all constitutive equations must conform.

C.

Heterogeneities

One key feature observed by Neogi et al. (1986), and one that can be generalized to other systems, is that the effects of swelling die out as the concentrations are lowered. That is probably the reason that in the work by Downes and MacKay (1958) Fickian diffusion was observed at lower concentrations, and sigmoidal at high concentrations. However, more detailed differential sorption data given by Odani et al. (1966) and Kishimoto et al. (1960) show that roughly with decreasing concentrations, sigmoidal (swelling plus elastic effects), then Fickian (disappearance of swelling effects?), two-step (swelling plus viscoelastic effects?), Fickian (disappearance of swelling at low concentrations), and finally a sigmoidal response at very low concentrations are observed. Although there is some confusion as to what happens in the intermediate ranges, there is nevertheless some information. In contrast, no mechanism has been described so far that can explain the sigmoidal behavior at the very lowest concentrations. At such low concentrations one peculiarity of glassy polymers is summarized by its dual-mode solubility isotherm

c = Kop + CH

bp -

1

--

+

bp

(32)

where the first term on the right represents dissolution in the solid polymer and the second term is due to the adsorption on the walls of the microvoids (Hopfenberg and Stannet, 1973). That is, the solid matrix has small holes, called microvoids, and the adsorption is described by a Langmuir adsorption isotherm. Very detailed experimental measurements are now available in support of this solubility model (Chan et al., 1978; Patton et al., 1988). The question now is, do these holes have an effect on transport? Michaels et al. (1963) found that if adsorption equilibrium was assumed to exist at the interfaces, then the holes had no effect on fluxes. Of course, it was also assumed that the diffusivity in the holes was much greater than that in the polymer. Petropolous (1970) suggested that holes would contribute to the flux whatever

TRANSPORT PHENOMENA IN POLYMER MEMBRANES

193

the detailed mechanisms were, and therefore the flux would reduce to

. Jx = -

Dpcp + DHcH af-L RT ax

(33)

where Dp is the diffusivity in the polymer and DII is that of the adsorbed species, and cp and CII are concentrations there. DpcsJRT and DHcH/RT are the two mobilities, and the last term is the driving force, the same for both. Paul and Koros (1976) showed how to combine Eqs. (31) and (32) to get a complete expression for flux . Subramanian et al. (1989) used that expression to compare the experimental data with theory to get a value for D H , which was found to be less than Dp. In another study, Tshudy and von Frankenberg (1973) used the holes and adsorption as a source/sink term in the conservation equation. The adsorptiondesorption kinetics are also included. The model was not liked because the contribution of the holes to the flux was made even more obscure. Vieth (1991) gave a very detailed account of the use made of the above models to interpret experimental data on glassy polymers. No absurdities are encountered, but neither are any insights gained into the nature of mobility of the diffusant in the microvoids. Barrer (1984) and Fredrickson and Helfand (1985) put together molecular models to justify and generalize a flux expression such as that given in Eq. (32). Their results, however, leave one with no clear connection with the more easily understood features of a continuum approach. Further, when Sada et al. (1988) compare one set of experiments with this generalized model, they find that the direct effect of the microvoids on transport is negligible and that an indirect effect in terms of a coupling between holes and the surroundings dominates: DH in Eq. (33) is negligible, and a cross term, not given there, is important! The observation appears at present to be general, and the progress at this point (in view of Petropoulos's original premise) begins to resemble a snake swallowing its tail. Some aspects of the presence of holes on transport phenomena were ignored until the work of Sangani (1986) namely, the matters of space invoked in laying out the holes. Both key observations that result when this is recognized-that near a hole the concentration profiles change drastically and that holes are randomly placed-are accounted for in transport phenomena through averaging methods. Obviously, in a randomly heterogeneous system the detailed profile is complicated and can never be measured. Instead, one measures some kind of average response. We know only the detailed equations, and the averaging processes lead us to the equations for the average responses. Sangani attempted to average the dual sorption model together with Petropoulos's model for the flux. The method used is valid over a large range of volume fractions of the holes, which is a very difficult problem to solve. The final expression, though incomplete, does show that quite a few changes occur on averaging.

194

NEOGI

Neogi (1993) attempted to formulate the problem using a more detailed model. There are four resistances to mass transfer to a hole: diffusion in the polymer, adsorption-desorption resistance in the polymer, adsorptiondesorption resistance at the interface on the side of the hole, and diffusion in the hole. The adsorbed layer can move by surface diffusion as well. (Thus, this approach is a more detailed version of the model of Tshudy and von Frankenberg.) The averaging technique is local volume averaging (Hinch, 1970), which was originally constructed by Maxwell and applies to systems with a small volume fraction of holes. 1\vo kinds of results can be obtained. In the first, the constitutive equation is averaged to yield an effective diffusivity. It shows that to a good approximation the diffusion is Fickian even at unsteady state. This formulation is what one needs to describe steady-state permeability exactly. In the second, the conservation equation is averaged, which is needed to describe the total solute in a polymer, that is, the proper description for the sorption experiments. The results show sigmoidal sorption. The main parameter there is the ratio of the rate of adsorption to the rate of diffusion. The comparison between the theory and the experiments is shown in Fig. 7. This model also uses a linear adsorption relation. If a nonlinear relation is used that has a saturation (the present state of averaging principles does not allow this), such as a Langmuir adsorption isotherm, then this non-Fickian effect will disappear at sufficiently high concentrations when saturation is reached. The anomalous effect is due to hole filling and disappears when the holes are filled . All of this occurs at low concentrations. In contrast, the swelling phenomenon takes place at high concentrations and disappears at low concentrations. When the holes are very small, it becomes difficult to define adsorption. A model for this case, by Kasargod et al. (1995), predicts pseudo-Fickian sorption (see Equation 23). A very different kind of averaging was introduced by Di Marzio and Sanchez (1986). In many mechanisms one can have a forward step and a backward step, each with an activation energy. In a heterogeneous system, there will be a distribution of such energies, which can be used to obtain an average rate process. In dielectric relaxation, Di Marzio and Sanchez (1986) were able to show that the simple exponential relaxation becomes " stretched exponential" exp[ -(ti'r)~ ] on averaging. These ideas were used by Adib and Neogi (1993) to average the relaxation discussed earlier. Whereas case II itself was always exhibited, a great many features were seen in the calculated fractional mass uptake curves at large times. Such features (some of which can get to be a little exotic) have been documented in experiments. Comparison with experiments indicates that more often than not the relaxation time as defined in Eq. (26) cannot exist, that is, will give negative or infinite values!

195

TRANSPORT PHENOMENA IN POLYMER MEMBRANES 1.0

0 .9

0.8

~2

0.7

0 .6

::::8

8

/0.5

::::8~ 0 .4

0 .3

0.2

0 .1

0.3

0 .4

T

o.s

0 .6

1/2

Figure 7 The theoretical results of Neogi (1993) on hole-filling effects at very small concentrations as fitted to the experimental data of Odani et al. (1966) and Kishimoto et al. (1960). The systems are benzene in atactic (1) and isotactic (2) polystyrene at 35°C. Here T is dimensionless time. (Reprinted with the permission of John Wiley and Sons, Inc.)

Whereas there has been a concerted effort to bring together memorydependent diffusion with the effects of swelling on diffusion, there has been relatively little work on incorporating the effects of the heterogeneities that undoubtedly exist in these systems. Some beginnings have been made.

IV.

CHANGE OF PHASE

Phase changes take place during mass transfer operations in a variety of situations. The production or collapse of polymer foams, devolatilization, etc., offer instances in polymer processing where vaporization and nucleation of the vapor phase are of considerable importance (Han and Han, 1990; Albalak et aI., 1990). Diffusion of gases through a lamellar composite, where one material has a high

0.7

196

NEOGI

permeability and the other has a low permeability, can give rise to local supersaturation and nucleation (Graves et al., 1973). However, the most often studied phenomenon is precipitation as it applies to membrane formation. A quite different class of phase transition is polymer crystallization, which can also be promoted by a solute, diffusion process, etc. Only these two are discussed below.

A.

Reverse Osmosis Membranes

Loeb and Sourirajan (1963) made the first reverse osmosis (RO) membrane capable of withstanding the rigors of industrial use. The membrane was made of cellulose acetate (about 22% polymer) in a solution of acetone (68%) and water (10%). Water is a nonsolvent in this case, and acetone is a good solvent. An inorganic salt such as magnesium perchlorate was also added. The procedure used was as follows. A thin film of about 250 f..Lm thickness was layered onto a surface kept at -10 to O°C and after a few minutes was plunged into hot water and kept there for over an hour. The surface that was exposed to the atmosphere developed a dense skin -0.1 f..Lm thick (Riley et al., 1964, 1966) and contained pores as small as 0.3-5 nm (Kotoh and Suzuki, 1981). The rest of the membrane was a spongy mass and formed the backing, whereas the dense skin was needed for desalination (Yasuda and Lamaze, 1970). In water, a charge develops on the surface of the solid polymer, and an electrostatic field results that extends into the aqueous phase. However, such a field is confined to a very thin layer next to the wall, and only if the pores are very fine, such as in the dense film , will the electrostatic field cover the entire pore space. The membrane itself can separate a reservoir with some salt (say NaCl) from another with no salt. Now if these pores are negatively charged, then the "solubility " (and hence the permeability) of the Na+ ion in such a pore will be very high and that of the Cl- ion will be very low. But in the absence of an externally applied voltage difference there will be no current, and the fluxes of the two ions will be equal. The overall effect is that there is virtually no net flow of NaCl across the membrane (CI- ions holding back the Na+ ions) even though there can be a pressure-induced flow of water from the salt side to the pure water side. This is the phenomenon of reverse osmosis (Jacazio et al., 1972), which is also known as hyperfiltration. It is also possible to do regular filtration, but with particles that are very fine, even molecular in dimension, such as proteins (Michaels, 1968). This is called ultrafiltration. There are many books and reviews on the subject of RO membranes that can be consulted for additional details (Sourirajan, 1970, 1977; Merten, 1966; Turback,1981 ; Lonsdale and Podall, 1972; Bungay et al., 1986; Probstein, 1989). The key feature of interest here is the way in which the mass transfer gives rise to the structure of the RO membranes. Kesting (1971) discussed qualita-

TRANSPORT PHENOMENA IN POLYMER MEMBRANES

197

tively the nature of precipitation/aggregation in the evaporating solution. The most important part of his discussion concerns the relationship between swelling and the structure of the precipitate. Acetone, being a good solvent for cellulose acetate, swells it; water, being a non solvent, decreases the swelling. Inorganic salts also increase the swelling. Now, the precipitation from a good solvent takes place rather late, that is, at a higher degree of supersaturation, and the precipitate has large irregular pores. The precipitate from a poor solvent has very small, well-marked holes. Thus, it can be concluded that the acetone-water-salt mix was arrived at as one where the resulting solvent gave the right pore structure. Nothing was suggested regarding the origin of the asymmetry. In fact, this question was not studied until Sirkar et al. (1978) showed experimentally that it took less than 0.01 s of evaporation time for the skin to form. Neogi (1983a) gave different mechanisms for the formation of the skin and the backing. At very short times during the evaporation step the fluxes are very high, which gives rise to a sharp change in the densities, which in tum causes interfacial instability. The shape of the interface becomes bumpy, and some of the pits develop into pores. This chain of events was shown to take place virtually instantaneously. The slower mechanism of backing formation was ascribed to the growth of nuclei. Some effort was made to connect this step to swelling. Ray et al. (1985) also gave a mechanism based on interfacial instability and density effects. This is confined to the effect of density on colloidal forces and that of the latter on flow. The effects of density variation on other phenomena are ignored. Tsay and McHugh (1991) provided detailed calculations that show that the skin formation is over in at most 20 s.

B.

Solvent-Induced Crystallization

Polymers with simple molecular structure, such as polyethylene, never fail to crystallize when cooled from melt. In contrast, atactic polystyrene is ungainly enough that packing problems are enormous, and it never crystallizes. This is, of course, an observation within the cooling rates that are currently available. Poly(ethylene terephthalate) (PET), on the other hand, can be semicrystalline or fully amorphous when it is quenched from melt. One key observation is that when a crystallizable, but presently amorphous, polymer is exposed to solute that forms a solvent for the polymer, crystallization sets in. This phenomenon is called solvent-induced crystallization (SINC). The common polymeric crystal is a lamella. On the edge of the lamella, a polymer molecule attaches itself and weaves up and down onto the face of the lamella. The rate of growth of the sheet is usually denoted by the symbol G. In turn, lamellae are gathered into sheaves, which are pinched in the middle to form spindles. Finally, the empty parts of the spindle are filled to form a spherulite. The outer surface of the spherulite is made up 0'£ the edges of the lamellae,

198

NEOGI

and hence if r is the radius of the spherulite, then dr -=G dt

(34)

The term G can be expressed as G = v2 GO exp(- flED/RT) exp(-M*/RT)

(35)

where 1)2 is the volume fraction of the polymer, Go is a rate constant, flED is the transport activation energy (that needed to bring the polymer to the growing interface), and M* is the free energy for the attachment of the next polymer chain to the growing edge. The first activation energy is identified as the WLF energy (36) where C 1 and C 2 are universal constants. The other activation energy, M*, is a more complicated function of different constants of the system plus the melting point Tm. The nature of crystals and that of their growth rates are covered in more detail by Hoffman et al. (1975). Avrami (1939, 1940) showed how the rate of growth G can be related to the conversion into crystallites with time:

x=

1 -exp( -ktn)

(37)

where n = 3 is a typical value. In SINC it is seen that as the concentration of the solute advances, crystals are formed. If the front advances by a diffusion-controlled mechanism, its rate is seen to be proportional to Vi. The first modeling effort by Zachmann and Konrad (1968) identified that it was necessary to combine crystallization growth with diffusion. Their methods of calculation were revisited by Makarewicz and Wilkes (1978), who among other things came to an important conclusion regarding the mechanism behind SINC: The presence of the solute depresses the glass transition temperature Tg , which increases the value of G in Eq. (35) by orders of magnitUde. The plasticizing effect can be quantified using the KellyBueche equation. The proper transport phenomena problem was solved by Durning and Russell (1985), who solved separate equations for the conservation of solute and the amorphous polymer. Generally there are two limiting cases. In one the diffusion is rate-controlling and crystallization is very rapid; in this case the formation of crystals will increase as Vi. At the other extreme, when the diffusion is rapid and crystallization is slow, the crystallization rate would be given by an Avrami-type equation. The solution by Durning and Russell shows that a great many things can happen in between. One interesting case is when diffusion is rapid and the membrane in a sorption experiment equilibrates first. Then crystallization starts, but as the crystallites are impermeable to the solute, some solute is driven off the membrane. That is, the mass uptake passes through

TRANSPORT PHENOMENA IN POLYMER MEMBRANES

199

a maximum. The comparison between Durning-Russell theory and experiment is shown in Fig. 8. A note of discord has been struck by Mandelkern (1955) and Yoon and Flory (1979). They contend that MD should be the activation energy of the selfdiffusion coefficient of the polymer, and not the WLF energy. It is likely that the main issues raised here will remain intact although the quantification may see some changes in the future.

v.

MULTIPHASE, MULTICOMPONENT, AND INHOMOGENEOUS SYSTEMS

In this section some systems with more than one component or phase are reviewed. A notable exception is the case of coupled fluxes. None have been detected, nor has there been any attempt to search seriously for such effects. The antisymmetric permeability in a membrane with a gradient in the polymer composition has been discussed briefly by Cussler (1976) and will not be considered here. A few of the more active areas are described below.

A.

Chromatography

There are endless types of chromatographies, and the only ones of concern at present are those that lead to the diffusion in polymers of the type considered here. The simplest is gas-solid chromatography, which employs a column containing a solid packing material. An inert gas, such as N z, is the carrier and flows through the packed bed at a steady rate. The sample/species under consideration is introduced as a pulse at the inlet and would normally exit the column at elution time L/(v), where L is the length of the column and (v) is the average velocity of the carrier gas.

ac at

ac = ax

- + (v) -

- j*

(38)

The species under consideration is at very dilute concentrations. However, the material adsorbs on the surface of the packing. In Eq. (38), the mass transferred from the gas phase to the adsorbed phase is given by j*. The axial dispersion has been neglected. In the adsorbed phase the conservation equation is

ar =j*

-

at

(39)

where r is the surface concentration. If local equilibrium holds, then r = Kc, where K is a Henry's law constant. Substituting into Eq. (39) and summing Eqs.

NEOGI

200

40

30

10

a

2.0

4 .0

6.0

1F/2Apx 10" (sl/t/m)

Figure 8

The theoretical observations of Durning and Russel (1985) compared to the data of Overbergh et al. (1975) for methylene chloride diffusing in isotactic polystyrene films at 30°C and different film thicknesses. Effective diffusivities were used. (Reproduced with permission.)

(38) and (39), one has

(1

ac

ac

+ K) - + (v) - = 0 at ax

(40)

The solution, when the input is a pulse, is c = M '0

[t _x(1 (v)+ K)]

(41)

TRANSPORT PHENOMENA IN POLYMER MEMBRANES

201

where 3 is the Dirac delta function and M is the total mass in the pulse. The elution time at x = L is L(1 + K)/(v) (Littlewood, 1970). As K differs from species to species, the separation of species is made possible. More detailed calculations are discussed by Holland and Liapis (1983) . In gas-liquid chromatography (GLC), a viscous liquid is used to coat the solid packing. If the layer is very thin, the above concepts apply. For separation and identification, a mixture is injected at the inlet. Inside the chromatograph the species are separated and elute at the outlet at different times depending on their K values. The form they take is that of a sharp pulse. However, in real systems, axial dispersion plays the role of smearing out the pulse into a bell-shaped curve. If these curves overlap in the eluant, then identification (by their retention times) becomes difficult, and quantitative analysis (area under the curve) becomes impossible. Golay (1958a,b) developed intuitively many features of the dispersion coefficient DL without knowledge of Taylor 's (1953) work in this area: that it caused widening of the peak, that it was inversely proportional to the molecular diffusivity, and, most important, that it decreased strongly with decreasing tube radius. Golay argued that one should perform chromatography in a capillary column with the walls coated with the absorbent. The methods for fabrication, experimental verification of sharpening of the peaks, and a simple model were given. Capillary chromatographs are now available commercially and give the best resolutions. The investigations in the period following this were reviewed by Vieth (1991). Pawlisch and Lawrence (1987, 1988) took this up and provided a more thorough analysis. Much improved equations were used, and capillary chromatography was shown to be an excellent way of obtaining the diffusivities at infinite dilution in the absorbent, a polymer in their case. A comparison between their theoretical calculations and experimentally obtained peak shape is shown in Fig. 9. Arnould and Lawrence (1992) reported the vast amounts of data on such diffusivities that have been thus obtained and discussed their impact on the current theories of diffusivities. We are entering a yet newer era in the use of capillary chromatography. As the layer of polymer is thin, the orientation of the top layer affects the response of the polymer, and the orientation of this layer itself is affected by the fluid on top of it. Two important features emerge, according to Andrade (1988). These are that the surface effects (interfacial phenomena) can be made to govern the overall dynamics and that these vary with the environment.

B.

Polyblends

The problem in transport phenomena that polyblends give rise to is that the medium becomes inhomogeneous in a discontinuous manner. Well-defined do-

202

NEOGI 1.4 1.2

to O.B

.....--... ...JI>

---.:...--'

0 .6

u l J'

04 0 .2

0 0

2

3

t l td Figure 9 Comparison between the theoretical (a) and experimental (b) elution curves. The polymer is polystyrene, and the vapor is benzene; the system is at 130a C. [Reprinted with permission from Pawlisch and Lawrence (1988). Copyright 1988 American Chemical Society.]

mains of the two phases are evident when they are examined locally, but they are distributed in a random fashion . Some methods of analyzing such media, particularly when one of the phases is dilute, were discussed briefly in Section m.e These originate from the work of Maxwell and have been used by Robeson et a1. (1973) to compare permeation data with theoretical predictions. The pro?lem where the two phases are of comparable amounts but one phase is clearly dispersed in another was reviewed by Barrer (1968). The dispersed phase is given a geometric shape such as a cube. The entire system is divided into cells, and each cell has a cube located right in the center. As far as polyblends are concerned, all the above models ignore two key features. The first is that they all require some degree of geometric niceties. However, the shapes and distribution of the phases are actually random, and it is even difficult to say which phase is continuous. In fact, in polyblends, a good blend is described as having an "interprenetrating network." That is, one phase

TRANSPORT PHENOMENA IN POLYMER MEMBRANES

203

Figure 10

Randomly cut polygons (Voronoi tessellations) distributed in two phases. [Reproduced with permission from Kuan et al. (1983).J

passes through another many times in an intimate way. Thus, this brings up the second point, that if one has two different samples, both having the same volume fractions of the two phases, their performances will be different because the detailed layout of the two phases will be different. Consequently, only a discussion of the average response is meaningful. Rogers (1985) discussed some attempts at quantifying diffusion in such systems. The new methods of analyzing such systems were used by OUino and coworkers (Sax and OUino, 1983; Ottino and Shah, 1984; Shah et aI., 1985). Without going into detail, some basic ideas of random systems are analyzed below. In Fig. 10 is shown a two-dimensional system made out of two phases marked with white and black in the figure. Their domains are random: the system was cut up into random polygons, each of which was assigned randomly to be black or white. When the fraction of the black phase is low it becomes the dispersed phase, and at high fractions it is the continuous phase. In between, both phases are continuous, that is, the system becomes bicontinuous. The point of inversion is the percolation threshold. The question now is, what will the overall diffusivity be if the two phases have two different diffusivities? This is the quantity that is calculated in a variety of ways. The above references carry the mathematical details. The phenomenon itself has many applications. Spherical polymer particles are coated with silver and compressed into a film. Since the silver on the surface forms a continuous phase (percolates), the film has a very high conductivity comparable to that of pure silver, even though the amount of silver can be as low as 3.5 vol % [according to one model (Park and MacElroy, 1989)). If we have a pure silver film and we carve out spheres and replace them with polymer randomly, and the spheres can overlap, we can keep doing this until 96.5% of the silver has been removed. After that the silver stops being a continuous phase and the conductivity falls dramatically. (In this connection it should be noted

NEOGI

204

that in two dimensions one could go down to 1%.) The reverse case has been achieved by putting mica flakes in a film. The flakes tend to orient parallel to the film , and no diffusant passes through the flakes. If the flakes touch one another at the edges, then a continuous impermeable surface is formed that prevents any diffusant from passing through, that is, a perfect insulation to permeation is formed. Actually there is no guarantee that the flakes will touch, and hence this system is not perfect. Finally, we note that these methods of calculating averages are confined to linear systems and constant conductivities. They also ignore the possibility of the morphology itself affecting the diffusivity. We know that it could do so even in a single-component system that is semicrystalline through the free-volume effect; this has been shown to be the case in polyethylene (Liu and Neogi, 1988).

C.

NMR Self-Diffusion Coefficients

The measurement of NMR self-diffusion coefficients in polymer systems now is fairly routine [the pulsed gradient spin echo (PSGE) technique in particular], and yet there are still some uncertainties regarding the nature and utility of the quantities that are being measured. The following is a brief review of how these measurements are made and an attempt to explain them based on the diffusion coefficients encountered here. Some nuclei have magnetic dipoles: 'H, 2H, 13C, etc. They orient in a magnetic field and absorb energy from an rf (radio-frequency) souce provided that the frequency corresponds to the characteristic Larmor frequency of the nucleus. That is, with the rf burst it is possible to affect a particular species, whereas in a steady field there is no selectivity of that kind. Very briefly, a steady field is applied, after which an rf burst is 'Jsed to flip the magnetization by 90°. A brief gap follows, after which it is flipped by an additional 180° by a second rf burst. This last maneuver is equivalent to flipping the steady magnetic field by 180°. The period over which a first rf burst is applied is called the dephasing, and the second rf burst constitutes refocusing. Essentially, dephasing induces a phase lag in the spinning magnetic dipole and refocusing restores it, hence the "echo." However, these shifts are also dependent on the steady field strengths. If the steady field strength varies spatially and the nucleus wanders off (due to diffusion) in between the application of the two radio frequencies, then the echo strength will be attenuated. It has been possible to relate the attenuation to diffusion. The reviews by von Meerwall (1985), Stilbs (1986), and Kiirger et a1. (1988) cover experimental and theoretical aspects quite extensively. The one by von Meerwall is exclusively on polymers. In polymer solutions it has been shown that the NMR self-diffusion coefficient of the solute measures the self-diffusion coefficient, as given by Vrentas and Duda (1979), excellently (Pickup and Blum, 1989). Blum et a1. (1990) also

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showed that in their system the NMR self-diffusion coefficient was the same as the mutual diffusion coefficient measured by Duda et al. (1979), only at infinite dilution of the solute. For the other case of diffusion of polymer molecules, Gibbs et al. (1991) showed that for ovalbumin, a globular protein, the NMR self-diffusion coefficient was the same as the mutual diffusion coefficient measured with an ultracentrifuge. The last observation is exciting but needs more investigation.

VI.

CONCLUSIONS

Without doubt this area of physicochemical effects involving diffusion in polymers and their quantification has been a rich one. Particular emphasis should be placed on quantification. In the area of non-Fickian diffusion, one would have no direction or even a coherent way of looking at the phenomenon without models. Other than rationalizing existing data, one could also find new directions to take for making meaningful inventions, as seen with polyblends. In general, there can be very little meaning to investigating the dynamics of diffusion or its applications without including transport models and their solutions. This has proven to be the key feature in the past and will continue to be so in the future .

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5 Supermolecular Structure of Polymer Solids and Its Effects on Penetrant Transport Sei-ichi Manabe Fukuoka Women 's University Fukuoka, Japan

I.

A.

INTRODUCTION

Scope and Context of the Presentation

The penetration of small molecules into a polymer solid or liquid frequently plays an important role in industrial processes such as the spinning and finishing processes in fiber manufacturing (Takeda and Nukushina, 1963) and the casting and finishing processes in membrane manufacturing (Manabe et aI., 1987). The kinds of solvent molecules used in solidifying a polymer through coagulation influences the supermolecular structure of the finished polymer solids. For example, acetone, which is employed for the coagulation of a solution of polyparaphenylene terephthalamide film (Haraguchi et aI., 1976) and cuprammonium-regenerated cellulose hollow fiber (Fujioka and Manabe, 1995), works so as to orient the hydroxy group in the direction parallel to the surface plane of a film or a fiber, whereas water would orient it in the direction perpendicular to the plane. When one solvent component of a polymer solution remains in the polymeric material during the solidification stage, it is found that even though it is finally removed from the polymer solid, this component can penetrate easily through the solid in question as if the solid has retained the memory of that solvent (Iijima and Manabe, 1983). Although such small-molecule effects contribute to the completion of the fine structure of the polymer solid and the specific permeation of the molecules 211

212

MANABE

through the solid, this chapter deals only with the supermolecular structure of the polymer solid (see Section II) relating to the penetrant transport (Sections IV and V), in addition to the thermal motion of the polymer chain in the solid (Section III) and the molecular interaction (Section IV) between the polymer solid and the small molecule.

B.

Methodology for Development of Novel Materials in the Field of Penetrant Transport

Studies of the correlation of penetrant transport and the conditions of preparation of the solid polymer have been carried out rigorously, and some empirical equations for the ideal manufacturing procedure have been proposed in the field of membrane technology (Kesting, 1985) and in that of fibers (Mark et aI., 1967). These studies can be classified into a blackbox that connects phenomena without causal sequence. When the demand for penetrant transport becomes complicated as seen in the case of virus removal filters (Manabe, 1992), this methodology gives only a low level of efficiency in attaining the final goal. Figure 1 summarizes the flowchart that relates social needs and demands and manufacturing conditions through the knowledge of supermolecular structure and physical properties. When we suceed in determining the quantitative relationship between structure and properties-the causal sequence, so to speakthen we can design the fine structure of the polymer solid that will satisfy the demand. The key research area for this methodology is the investigation of supermolecular structure. The research and development for novel materials oriented to the market is carried out along the lines of the flowchart shown in Fig.

Mecranism) ......----' (Transport technical terminology

~-

Figure 1 Flowchart for relating social needs and membrane manufacturing: The numbers 1, 2, and 3 indicate the stream of development of a novel membrane that fits a social need.

SUPERMOLECULAR STRUCTURE OF POLYMER SOLIDS

213

1. Consumer demand may be transformed into physical properties, and then the supermolecular structure that is ideal for getting these properties is designed using the structure-property correlation. This supermolecular structure can be realized by the leading principle governing the mechanism of solidification of polymer liquids or melts. One example of research and development following the chart is demonstrated in the virus removal filter (Manabe, 1992).

C.

General Description of Supermolecular Structure of Polymer Solids

We define supermolecular structure and fine structure in the following section although these definitions are stricter than those that have been accepted so far. We classify supermolecular structure into many structural factors according to a numbered scheme as the second structural factor, the third structural factor, and so on depending on the size of the domain necessary for characterizing experimentally the structure in question. The supermolecular structure represented by the lower structural order factor is influenced by the chemical structure (this structure is defined as the first structural factor), and the supermolecular with the higher order factor can be controlled by appropriate preparation conditions independent of the chemical structure. Fine structure constitutes the third structural factor in a narrow sense. The structural characteristics belonging to fine structure are, for example, the orientation of chain molecules along a crystal plane, the crystallinity, the crystal size including lamellar thickness, the dispersion state of the crystal and amorphous regions, and the intersurface characteristics. All these are concerned with sizes between 5 and 103 nm. Figure 2 shows a schematic representation of the relationship between supermolecular structure and the mechanisms of penetrant transport through the polymer solid. The arrow shown by a full line indicates the flow of a penetrant with a single flow mechanism as denoted, and the broken line indicates the actual general penetrant transport. Because the transport of the penetrants occurs mainly from the environmental liquid or gas to the inside of the polymer solid, a higher order structural factor always contributes to the transport. When the penetrant molecule is small, the rate-determined region of transport is related to the lower order structural factor. For example, dissolution/diffusional flow is affected much more by the first-order structural factor (Le., chemical structure) than by a higher order factor. In the case of viscous flow, the coagulation structure (the fourth structural factor) dominates penetrant transport and is almost independent of other factors. In the actual case of penetrant transport, the complex situation shown by the broken line in Fig. 2 may occur, and the contribution of each factor to the transport may change from case to case. Table 1 summarizes the classification of supermolecular structure and lists the characterization methods for the various structural orders. A first-order StrUC-

214

MANABE General penetrant transport ~~

~~'"

~~

Dissolution/diffusion flow -Free volume

model

Surface diffusional fleM' Free molecular fl ow (gas) Viscous flow

}

capillaf}'

model

Figure 2 Schematic representation of relationship between supermolecular structure and penetrant transport through a polymer solid. The numbers indicate the order of the structural factor. The area crossed by the full-line arrow denoted by the flow mechanism indicates the contribution of that structural factor to the flow mechanism.

tural factor can be mainly evaluated through resonance-type spectrometries such as infrared or nuclear magnetic resonance spectrometry. The second can be evaluated through diffraction methods such as X-ray and/or electron diffraction and through resonance spectroscopy. The third structural factor can be characterized mainly through electron microscopy and relaxation absorption, and the fourth through optical microscopy and/or the methods of interfacial phenomena such as mercury intrusion porosimetry. The probe-type microscopy developed recently gives information on the second and third structural factors, especially on interfacial structure. When we use the additivity principle we can estimate the properties of the whole system from those of the basic units. For example, the refractive index, optical absorbance, degree of swelling and dissolution, specific heat capacity, latent heat of the first-order transition, thermal expansion coefficient, etc., can be estimated from the values of the components through the additivity law based on weight or volume fractions. On the other hand, the dynamical properties can be calculated through a series or parallel type of combination of components or, more generally, by applying Takayanagi's (Takayanagi, 1967; Manabe and Takayanagi, 1970b) model. In the case of penetrant transport, three types of combination methods have been employed: the additivity law of the component units based on weight fraction, the series connection, and Takayamagi's model type of connection. The type of combination that should be employed is determined

Table 1 Definition of Supermolecular Structure and Methods of Evaluating It Structural order First (chemical structure of monomer)

Target size (nm)

0.1-1.0

Second (conformation)

1-10

Third (fine structure)

10-102

Fourth (aggregation structure)

102 _ 104

Fifth (morphological structure)

104 _106

Structure description

Evaluation methods'

Chemical structure, molecular weight and its distribution, configuration and tacticity, composition of copolymer and blend and its distribution. Conformation, local chain orientation, crystal structure, amorphous structure Orientation (chain axis, crystal plane), interfacial structure, crystal size, crystallinity, crystal perfection, molecular packing (density, regularity), lateral order

Spectrometry (NMR, IR, vis, UV, MS), chromatography (LC, GPC, TLC), light scattering, viscometry NMR, IR, LS, diffractometry (Xray, electron), CD, ORD, EM, AFM NMR (broad-line), IR, LS (smallangle), X-ray diffraction, EM (SEM, TEM), optical microscopy, thermal analysis (DTA, DSC), viscoelastometry (dynamic, static), refractometry Electron microscopy, optical microscopy, viscoelastometry, permeability, porosimetry, adsorption isothermometry

Aggregation structure (particle size, porosity, degree of amalgamation), fibril and microfibril (size, length, orientation), microvoid, pore structure (pore size and its distribution, pore shape, porosity) Shape of membrane (plane, hollow fiber), symmetrical and asymmetrical membrane, complex membrane, dynamic membrane, liquid membrane

Optical microscopy, light scattering

' NMR = Nuclear magnetic resonance, fR = infrared, vis = visible, UV = ultrav iolet, MS = mass spectrometry, LC = liquid chromatography, GPC = gel permeation chromatography, TLC = thin layer chromatography, LS = light scattering, CO = circular dichroism, ORO = optical rotary dispersion, EM = electron microscopy, AFM = atomic force microscopy, SEM = scanning EM, TEM = transmission EM, DTA = differential thermal analysis, DSC = differe ntial scanning calorimetry.

215

216

MANABE

through information on the supermolecular structure of the higher order structural factor.

II.

STRUCTURAL CHARACTERISTICS OF POLYMER SOLIDS

As mentioned in other chapters (see Chapter 3), the penetrants permeate into a polymer solid through the pores (capillary model) and/or through the free volume (free-volume model). We must note that the difference between models exists because of the assumption that pores may exist in the solid. The pores play the main role in transport in the capillary model and are neglected in the case of the free-volume model. In the latter model the fine structure of the amorphous region, where the polymer segments frequently jump from their equilibrium positions to adjacent holes that generate by the random thermal motion of the segments at temperatures above the glass transition temperature Tg , determines the transport. Figure 3 illustrates the structural features of both models and their peculiarities. In the capillary model, the contribution of the chemical structure is taken into account by the adsorption and/or solubility coefficient of the penetrant in the polymer solid. How to characterize the pores is discussed in Section 1I.c. On the other hand, information about fine structure of the noncrystalline region is necessary for the description of the free-volume model because the penetrant molecule can diffuse into the domain with a large free-volume fraction, and this domain is limited only by the amorphous region where the molecular chains are activated by the micro-Brownian motion or the local twisting motion. A more detailed interpretation is given in Section II.B.

Free volume model

Figure 3 Comparison between capillary model and free-volume model from the viewpont of dye uptake. The small dots stand for water molecules, the open circles are dye molecules, and the full lines stand for polymer chains. In the capillary model, the water molecules in a pore make a water channel for a dye molecule to diffuse through. In the free-volume model, the diffusion region is the amorphous region with water molecules where the polymer segments move actively under segmental micro-Brownian motion.

SUPERMOLECULAR STRUCTURE OF POLYMER SOLIDS

A.

217

Supermolecular Structure of Crystalline Polymer Solids

When the chemical structure of a polymer is given, we can distinguish whether or not the polymer is inherently crystallizable. For example, a polymer that has atactic side chains cannot crystallize under normal conditions. On the other hand, even if a polymer is potentially crystallizable judging from its chemical structure, it may have very low crystallinity at room temperature because of its low melting temperature and/or the preparation conditions, prohibiting its crystallization when quenched from the melt, for instance. Then we define a crystalline polymer solid as a polymer solid whose crystallinity is more than 30% at 20o e. A noncrystalline polymer solid has crystal1inity of not more than 30%. This definition is based on actual crysta1linity and not on the polymer' s chemical structure. Since crystallinity varies depending on the methods of evaluation such as X-ray diffraction, density, and infrared absorption, the X-ray diffraction (wide-angle) method is employed here as a standard. In a general way, polyethylene (low density, high density, linear low density), isotactic poly-a-olefins such as isotactic polypropylene and isotactic polybutene1 and so on, except polymers with side chains whose number of carbon atoms ranges between four (polyhexene-l) and eight (polydecene-l), isotactic poly-4methylpentene-l, polytetrafiuoroethylene, Nylon 6 and Nylon 66, polyethylene terephthalate (PET), poly acetal, cellulose, and poly-para-phenylene terephathalamide are crystalline solids. We can also easily prepare noncrystalline solids from PET and from cellulose. The polymers that tend to solidify into noncrystalline solids are, for example, atactic polymers such as atactic polystyrene, atactic polymethyl methacrylate, polyacrylonitrile, polyvinyl acetate, and polyvinyl chloride; polymers whose chemical composition is complicated with bulky side chains such as polycarbonate and many kinds of copolymers; and polymers with low melting points such as cis-l,4-polybutadiene and cis-l,4-polyisoprene. The crystalline polymer solids have very complex supermolecular structures because of the coexistence of crystalline and noncrystalline regions. Although the noncrystalline regin is the principal area for diffusion of penetrants, both the dispersion state of the crystals and its content (crystallinity) directly influence penetrant transport, and the domain boundary of a crystal may affect the transport indirectly. There is a tendency in a crystal to retain its molecular conformation even in the noncrystalline region. For example, the noncrystalline region of isotactic poly-a-olefin polymers shows two peaks in the curve of X-ray intensity of wideangle diffraction versus diffraction angle. The first indicates a distance of ca. 0.4 nm, which corresponds to the distance between the nearest-neighbor side chains, and a longer one that varies depending on the length of the side chains and corresponds to the distance between adjacent main chains. These two are also observed in the crystalline region (Manabe and Takayanagi, 1970c).

218

MANABE

The size of a crystal along the direction of the molecular chain is several tens of manometers, and this value is far less than that of the length of a molecular chain. This fact supports the fringed miceLJe structure shown in Fig. 4a. Many polymer single crystals have been observed. This observation supports the folded-chain crystal model (see Fig. 4b) and has been regarded as a model more widely applicable than the fringed micelle model. A molecular chain is folded within ca. 10 nm thickness in the folded-chain crystal, and this type of crystal is referred to as a lamellar crystal in a polymer solid. The lamellae are observed in a spherulite generated from a polymer melt. When polyethylene melt is crystallized under a high pressure (e.g., >3000 atm), extended chain crystals are generated. In such crystals, chain molecules are disposed parallel to each other with extended form and the crystal thickness corresponds to the length of the chain (see Fig. 4c). Under high shear rates in dilute polyethylene solutions (ca. 0.1 wt % in xylene), folded-chain crystals are first generated and then they are stretched and deformed by shear stress. Finally, the center of the aggregates formed is constructed of an extended chain crystal, and folded crystals that grow perpendicular to the direction of the extended chain crystal are formed at the periphery of the extended chain crystal. This structure is called a shish kebab structure because of its similarity in appearance to the Turkish food of that name. The dispersion state of crystalline and noncrystalline regions is given by Takayanagi's model shown in Fig. 5 (Takayanagi, 1967). The black area indicates the noncrystalline region and the white area the crystalline region, and the

1=1:::-::-::-::-::-=1

,

,

I

I

, , I

I

I

I

I

I

L. .. .... . ... .I I I I I I I

I I I

,, I

c

I

I I 1 I I I

I I I

I

I

I,_______1 I

(a)

(b)

1:.:.:.:..:::.:..: :::.:.1 (c)

Figure 4 Typical molecular arrangement in a crystal. (a) Fringed micelle; (b) foldedchain crystal; (c) extended chain crystal.

SUPERMOLECULAR STRUCTURE OF POLYMER SOLIDS

219

Figure 5

Takayanagi 's model (Takayanagi, 1967) for representing fine structure of crystalline polymer solid. Black area: noncrystalline region. White area: crystalline region.

two regions are connected in parallel and/or in series. Although this model has been proposed to interpret the viscoelastic response, we must take into account this type of connection even in the case of penetrant transport because the penetrant may pass through both regions in the case of a polymer blend.

B.

Supermolecular Structure of Noncrystalline Polymer Solids

When we define the noncrystalline region as the part not belonging to the crystalline region, then its content will vary depending on the evaluation method employed for the crystalline region. Table 2 shows some examples of the extent of the noncrystalline region evaluated on the assumption that the content is equal to 1 - crystallinity. The content cannot be determined definitely. The noncrystalline region is classified into three states according to the packing regularity of the molecular chains: the smectic state of two-dimensional regularity, the nematic state of one-dimensional regularity, and the amorphous state with no dimensional regularity. Additionally, the molecular chains in the noncrystalline region can be characterized by thermal motion (e.g., segmental micro-Brownian motion) and are represented by the packing density of the molecular chain with the same regularity (Manabe and Kamide, 1984).

220

MANABE

Table 2

Content of Noncrystalline Region' with Three Methods of Evaluating Crystallinity Polymer Polyethylene terephthalate Polyethylene Regenerated cellulose

0.60 0.26 0.05

IRe

Densityd

0.25 0.28

0.39 0.27 0.25

' Given by 1 - crystallinity. bWide-angle X-ray diffraction method . eInfrared ray method. dApparent density method.

In Fig. 6 is shown a schematic representation of the noncrystalline region for three types of polymer solids-a typical amorphous polymer solid, a typical noncrystalline polymer solid, and a typical crystalline polymer solid-observed from two different standpoints of regularity and packing density, the latter being the representation of the segmental thermal motion.

c o

:;:;

1

:J .0

·c ..... •«l! "0

>. u

C
J:

Figure 6 Schematic representation of frequency distribution of molecular chains in the region characterized by a given regularity and a given packing density. Curves 1, 2, and 3 show the values of a perfect amorphous polymer, a noncrystalline polymer, and a crystalline polymer, respectively. A, N, S, and C on the regularity ordinate indicate amorphous, nematic, smectic, and crystalline regions, respectively.

SUPERMOLECULAR STRUCTURE OF POLYMER SOLIDS

221

The fine structure of the noncrystalline region has been investigated through electron microscopy for direct observation, electron diffraction, and dark-field imaging. Nqdules 2-4 nm in size and grain boundaries of 1-2 nm were observed and found to possess a little regularity, and the remaining intergrain region was composed of random coil molecules. On the basis of these results, the model shown in Fig. 7 was proposed (Yeh, 1972). The radius of gyration of a molecular chain in a melt was found to coincide with that of a chain in a noncrystalline solid and to be proportional to the square root of its molecular weight. This result supports the conclusion that the molecular chains in a noncrystalline region have the random coil conformation. The supermolecular structure of the noncrystalline region of actual polymer solids such as a fiber and/or a membrane depends on their preparation conditions. Sections 1I.C and lILA will deal with the structures achieved under different preparation conditions.

c.

Supermolecular Structure of Polymer Fibers and Films

Fibers have a fiber axis as a symmetric axis of the supermolecular structure, and the direction of penetrant transport is perpendicular to the fiber axis. On the other hand, the symmetric axis of the supermolecular structure of a film is, in principle, perpendicular to the film surface, and the transport direction coincides with that of the symmetric axis. Consequently, the contribution of supermolecular structure to penetrant transport is different between a fiber and a film, a fact that should be taken into consideration in the series mode and parallel mode, respectively, in the additivity models.

Figure 7 Model representing tbe structure of a noncrystalline region (Yeb, 1972). OD, GB, and IG denote ordered domain, grain boundary, and intergrain regions, respectively.

222

1.

MANABE

Supermolecular Structure of a Fiber

For a fiber, the most important characteristics is the molecular orientation along the fiber axis. The orientation is caused by the spinning process and influences the transport of penetrants. For the removal of viruses using hollow fibers, this orientation should be minimized as it prohibits the generation of void-type pores penetrating the hollow fiber walls (Tsurumi et aI., 1990). Figure 8 shows two typical structural models representing the noncrystalline region in a fiber. The fringed micelle model (Fig. 8a) requires that each molecular chain penetrate both crystalline and noncrystalline regions, resulting in a part of the noncrystalline region having the lateral order in molecular chain packing. Here, the lateral order indicates that the molecular chains pack regularly side by side in some order that makes the distance between homogeneous chains and may even have semihexagonal packing. This model is believed to be applicable to the fibers manufactured by wet and/or dry spinning such as regenerated cellulose, poly acrylonitrile, and ararnide fibers. Another model, shown in Fig. 8b, originates in the folded-chain crystal model. The feature of molecular chain packing is divided into six types: tight loop, loose loop, tie molecule, cilia, floating chain, and extended chain. Tight loops are the chains located on a lamellar crystal that are stretched back to the same lamella. The loose loop also

c ~

l.o.

(a)

(b)

Figure 8 Two typical structural structural models representing the noncrystalline region in a fiber. (a) Fringed micelle model. C and NC stand for the crystalline and noncrystalline regions, respectively. The area denoted by 1.0. indicates the NC region with lateral order. (b) Noncrystalline model based on the folded-chain crystal. 1, tight loop chain; 2, loose loop chain; 3, tie molecule; 4, cilia; 5, floating chain; 6, extended chain.

SUPERMOLECULAR STRUCTURE OF POLYMER SOLIDS

223

starts on a lamella and returns to the same lamella, making a long loose loop. The tie molecule connects two lamellae, and the cilia are molecular chains that arise from one lamella and have the other end free. The floating chain is an isolated amorphous random coil. The extended chain keeps the shape under the aggregation state stretched through multiple lamellae. Fibers prepared by melt spinning have the noncrystalline region shown in Fig. 8b. Figure 9 represents the fibril and microfibril model (peter lin, 1969). The diameter of a filament ranges from 10 to 100 j.Lm. The filament is considered to be composed of fibrils having sizes between 100 and 1000 nm, and these fibrils are composed of microfibrils about 10 nm in size. The supermolecular structure of the microfibril is given by the model shown in Fig. 8.

Figure 9 Schematic representation of the fibril and microfibril model proposed for melt-spun crystalline polymer (peterlin, 1969). A filament 10-100 fLm in diameter is composed of many fibrils having diameters between 100 and 1000 nm. A fibril is composed of many microfibrils with diameters of = 10 nm. The supermolecular structure of a microfibril is shown at the right in this figure under an appropriate enlargement and is similar to that shown in Fig. 8b.

224

MANABE

The change in supermolecular structure along the radial direction is also a very important factor for the permeation of molecules (Le., dyeing) from the surface to the inside of a fiber. Figure 10 shows an example of the change in the supermolecular structure of Kevlar 49 along the radial direction (Manabe et aI., 1980). The microfibril is subdivided into fine tips having widths of 5-10 om and lengths of 0.3-0.5 11m. The chain orientation (c axis orientation) denoted by the length of the arrows c and the plane orientation denoted by the length of the arrows b decrease toward the center of the fiber. Fibers prepared by the wet-spinning process show changes similar to that of Kevlar 49. For example, the cuprammonium-regenerated cellulose fiber Bemberg shows maximum crystallinity at ca 0.7 rlR (where R is the radius of the fiber and r is the radial coordinate measured from the center of the fiber), and the maximum of the molecular chain orientation (b axis orientation) is located at ca. 0.6 rlR (Manabe et aI., 1980). The change in porosity for various kinds of hollow fibers has been clarified (Kamide et aI., 1989). Meltspun fibers such as polyethylene terephthalate also show a similar change, especially in the case of the fiber manufactured through high-speed take-up (Kuriki et aI., 1985).

111\UIII\lII,,~Ur~-05~ a block

Figure 10 Fibril-like block structural model for Kevlar 49. The original model prepared by Manabe et al. (1980) is modified a little in this figure. The figure on the left shows the aggregation state of the microfibrils between 5 and 10 run in width. The degree of orientation of the b axis along the radial direction of a fiber is shown by the length of the arrows labeled b. The figure on the right shows the supermolecular structure in a microfibril cut to a length of less than 0.3-0.5 f.Lm. Here, the length corresponds to the pleated sheets of the c axis. The molecular chain exists in a microfibril in the extended state. The chains in a noncrystalline region shown by wavy lines are also extended along the fiber axis.

SUPERMOLECULAR STRUCTURE OF POLYMER SOLIDS

2.

225

Supermolecular Structure of a Film

For a film, the orientation of the molecular chains can be classified into three types: undrawn, uniaxially drawn, and multiaxially drawn. It is difficult to distinguish between undrawn and multiaxially drawn films from the chain axis orientation because the latter is generally drawn so as to be isotropic. The mechanical properties of drawn films are superior to those of an undrawn film . A film frequently shows the orientation of a specific plane near the film surface. This type of orientation may influence the transport of penetrants because the adsorption/dissolution properties may be governed by the orientation of the active groups, and this orientation near the film surface accompanies the plane orientation. Ultrathin films with a thickness of less than 20 nm show characteristic peculiarities. For example, the direction of the intermolecular hydrogen bonds at the film surface varies depending on environmental conditions (Haraguchi et aI., 1976; Fujioka and Manabe, 1995), and the distribution of the component in the polymer blend is also influenced by the film thickness and the materials in contact with the film surface (Kajiyama et aI., 1995). The molecular conformation and molecular aggregation in a thin membrane such as a biomembrane play the main role in molecular transport through the membrane.

3.

Characterization of Porous Polymeric Membranes

In artificial membranes, although the structure of homogeneous membranes that have no observable pores under an electron microscope can be regarded as having the same type of supermolecular structure as described above for films, the aggregation of polymer chains in the porous membrane (the fourth structural factor in Table 1) should be characterized in advance, as the contribution of a pore to the transport as a whole increases strongly with pore size. Figure 11 illustrates three typical commercially available porous polymeric membranes: a membrane with straight-through cylindrical pores, an asymmetrical membrane, and a membrane with a multilayer structure (Manabe et aI., 1987). All of these have observable pores, and when the membrane is sliced parallel to the surface in ultrathin slices, the section shows the straight-through pore structure. Consequently, in order to characterize a porous membrane, we must first take into account the pore structure in a plane. Here "porous membrane" includes both porous plane-type membranes and porous hollow fibertype membranes. We define the pore radius distribution function N(r) (Kamide and Manabe, 1980) as the total amount of the pore, whose radius ranges between rand r + dr, is given by N(r)dr in a unit surface area. The ith-order mean pore radius ri, the ith order moment Xi' the porosity P r., and pore density N are given by the

226

MANABE

0 0 0

(a)

a

I I

I I

t::~

pTimary particle

(b)

d

(c)

Figure 11 Three typical porous polymeric membranes. (a) A membrane with straightthrough cylindrical pores. In general, the pore density is about 108 pores/cm2 surface area, and the thickness d is less than 30 !-Lm. (b) An asymmetrical membrane. This membrane is composed of two types of pores: very fine pores on the top surface and very large pores (or finger-shaped pores) inside the membrane. The top surface is composed of very fine particles = 10 nm in size (these particles are called primary particles). This thin layer with a thickness of = 70 nm including the top surface is considered to be the active layer for the particle separation. (c) The multilayer structural membrane. The whole of the membrane is composed of particles with a size of 252 ranging between 50 and 3000 nm (these are termed secondary particles). Because of the anisotropic feature of the membrane-manufacturing process, the particles are deformed and amalgamated, resulting in the layer structure shown in cross-sectional view.

SUPERMOLECULAR STRUCTURE OF POLYMER SOLIDS

227

following equations:

ri = Xi = N =

Pre =

f f f f

rW(r)dr /

f

ri- 1N(r)dr

(1)

riN(r)dr

(2)

N(r)dr

(3)

2

(4)

'ITr N(r)dr = 'IT r2 r l N

The scanning electron microscope (e.g., field emission type, FE-SEM) shows the detailed pore structure and at high-resolution power pores more than 5 nm in size can be observed directly. A novel technique for observing the connection between pores has been proposed (Tsurumi et aI., 1990a), and we can easily obtain information of pore structure as a function of radial coordinates (Kamide et aI., 1989). Many evaluation methods (Kamide and Manabe, 1980) for N(r) have been proposed so far: (1) mercury intrusion (MI) method, (2) bubble pressure and pressure dependence of the filtration rate of fluid (liquid and/or gas) (BP method), (3) perm selectivity of various particles (UP), (4) gas permeation (GP) method, (5) electron microscopy (SEM and TEM methods), and others (thermodynamical methods such as melting point of an immersing liquid, evaporation pressure of adsorbed gas using Kelvin 's equation). Figure 12 shows a comparison for N(r) distributions obtained using various methods of measurement (Manabe et aI., 1985). The values of N(r) depend on the method used. Consequently, the features of the evaluation method should be known in advance, and the marginal limit of the method must be taken into account. Some forms of the mean radius ri can be evaluated directly without knowing N(r). The filtration rate per unit filtration area for a liquid, J, under a transmembrane pressure of 6.P gives the mean pore radius rr using the following equation derived through the Hagen-Poiseuille equation,

(5) where P r is the porosity of the membrane, d is its thickness, and 1'\ is the viscosity of the liquid. The value of r r corresponds to (r3r4)112 for the membrane with straight-through cylindrical pores. When the surface area of the membrane including pore and membrane surface Sr is observed, we can calculate the mean

228

MANABE

I

(b)

(al

t,2 I.

, I I. I I

spherical model ------- ellipsoidal model I, front 2,back

•

19 2\

,

:

---- EM method

14

\

u

2,back

·,·: ·\

.... 'E 18

l,front

------ M I method ...........- B P method

·. " .. '\.' .'.",, ,

1 -,

~ CI

o

\

,,

z C1l

o

\

\

....: 17

\

;~

\

\

' ,, \ ,,

\ , '\,

CI

o

,

\

'

\

""\"

,

16L-----~~~--~

a

__~

0 .5

Pore

1.0

11 '--....:.-.--"'---'---'--~--'

a

radius I pm

~5

1·0

Pore radius If m

Figure 12 Comparison of pore size distribution functions N(r) for the porous cuprammonium-regenerated cellulose membrane evaluated through various methods. (a) N(r) obtained through electron microscopy (EM). The broken line was obtained under the assumption that all pores have ellipsoidal cross sections, and the full line represents values obtained in the case of spherical pores. Curves 1 and 2 are the values of the front and back surfaces of the membrane. (b) N(r) values obtained through EM, MI, and BP methods. Pores are assumed to be straight-through cylindrical pores in all cases.

pore radius rs (corresponding to r2) as

rs = 2P,/S,

(6)

The permeability coefficient of a gas gives the specific mean pore radii corresponding to r3, r4, and (r3r4y !2 (Nohmi et aI., 1978). The porosity P, can also be evaluated directly through the apparent density P. using the equation

(7) where the subscript p attached to P, indicates that the P, value is evaluated through the experimental value of P. and Pp is the density of the polymer solid

SUPERMOLECULAR STRUCTURE OF POLYMER SOLIDS

229

employed in the membrane. The porosity obtained through the immersion method, P'" is calculated using the equation (8) where Wp and W, are the weights of the membrane and the liquid used for immersion, and p, is the density of the liquid. Table 3 shows a comparison of the values obtained by various methods mentioned above for the two types of porous membranes shown in Fig. S.lla and llc. When the pore structure becomes complicated as in the case of a regenerated cellulose membrane, the difference in the characteristic values obtained through several methods becomes more obvious. The maximum pore radius rmax is evaluated through the bubbling method. The bubbling pressure d1\ is the pressure at which a small amount of a fluid (liquid and/or gas) passes through the membrane initially in the course of increasing the transmembrane pressure. The value of r max is calculated as (9) where 0"2 is the interfacial tension between the two fluids at the interface. The value of rmax is often employed as the quality control parameter for the membrane and as the integrity test value for the sterilization filter and the virus removal filter.

D.

How to Actualize the Intended Supermolecular Structure

Although the supermolecular structure of a polymer solid is determined in advance as discussed in the previous section, it is only when we know the coagulation mechanism of a polymer chain that a more critical understanding of the structure becomes possible. From the manufacturer ' s aspect, the preparation method and preparation conditions for the solidification of a polymer fluid are essential for effective development of novel polymer solids. Table 4 summarizes the molding method and the solidification mechanism employed in the molding. Here, wet, dry, and melt moldings are defined as follows: Wet molding. A polymer solution contacts another solution, and this contact gives rise to the transport of good solvents from the polymer solution to the other solution and/or of poor solvents in the opposite direction, and finally the polymer solidifies. Dry molding. A polymer solution contacts gases, and evaporation of the good solvents changes the solution into a solid.

tv v" ~

Table 3

Comparison of Pore Characteristics Obtained Through Various Methods'

Membrane Regenerated cellulose membrane

Evaluation methods EM Front Back

MI BP FP App. density Swelling

1'1

1'2

1'3

1',

(r3 . 1', )112

j.Lm

j.Lm

j.Lm

j.Lm

j.Lm

0.30 0.07 0.14

0.50 0.09 0.22

0.67 0.12 0.41

0.81 0.16 0.84

0.74 0.14 0.59

0.21

0.21

0.21

0.21

0.21 0.20

EM MI FP Appl. density Swelling

N

%

Number/ cm2

1.21 1.25 2.04

27.0 4.3 56.0

5.9 X 107 2.1 X 108 5.7 X 108

1.00

9.5

6.0 X 108

71.9 71.0 73 .7 90.3

0.37

SA Polycarbonate membrane Nuclepore® Nu 0.8

Pr

1'.11'3

0.36 0.31

0.46 0.32

0.50 0.33

0.57 0.35

0.53 0.34 0.53

1.13 1.02

21.0 18.0

5.4 X 107 5.5 X 107

22.0 16.0

' Regenerated cellulose membrane prepared through the microphase separation method and polycarbonate membrane prepared through the track-etched method. EM = electron microscopy; MI = mercury intrusion method; BP = bubble pressure and pressure dependence of the filtration rate of fluid s, FP = fi ltration rate method given by Eq. (5); Appl. density = apparent density method given by Eq. (7), Swelling = liquid immersion method given by Eq. (8), SA = membrane surface method given by Eq. (6). Source: Manabe et a!. 1985.

231

SUPERMOLECULAR STRUCTURE OF POLYMER SOLIDS

Melt molding. A polymer melt is cooled to a temperature below the melting temperature and/or the glass transition temperature. During the cooling, solidification by crystallization and/or vitrification occurs.

In the actual molding process, a complex molding method is often employed. For example, the polymer solution is first cast or spun into a gas environment and is then immersed in a solution. There are four important solidification mechanisms: crystallization, fine particle generation, vitrification, and polymerpolymer two-phase separation. As for crystallization, there are two main steps, nucleation and crystal growth. Nucleation occurs in a sporadic and/or predetermined manner. The number of nuclei generated by nucleation decreases with an elevation of the crystallization temperature, resulting in an increase in crystal size. In contrast, the number of nuclei generated through predetermined nucleation remains constant, independent of the crystallization temperature and time ~ but varies with the coexistence

Table 4

Preparation Method and Solidification Mechanism for Crystalline and Noncrystalline Polymer Solids

Polymer solid

Molding method

Crystalline

Wet

Primary particle'

Dry

?

Melt

Crystallization

Crystallization

Phase separationb , crystallization and drawing

Wet

Primary particles

Primary particles

Primary and secondary particles

Dry

Primary particles

Amalgamation or gelation

Primary and secondary particles

Melt

Vitrification

Vitrification

Phase separation, dissolution

Noncrystalline

Fiber

' Fine particle generation through microphase separation. bPolymer solid/polymer solid two-phase separation.

Film Primary particle

?

Porous membrane Primary and secondary particles Primary particles

232

MANABE

Ci rcular pore

LD~ Interfacial surf.

prim.~ secon.• m.p. -aggre. --<.f~ifcJrf. mv.

.::.. ... .. •• .:.

...

II C_~

--m .... -~\'t- ~.- --- - --W'T-

......

Inside

'"

=l~

M'

-~"'.,...( t*+~

Noncircular pore Interfacial surf.

-.~~.:.-

..

Inside

-::;--...

-

-\~: - l>l - d ~

:~

••• •••

....

.:. ,.." ••

••

---

~

Figure 13 Schematic representation of the pore-forming process starting from cellulose acetate solution (Kamide et at, 1977). The filled circles and domains are the polymer-rich phase and/or polymer-concentrated gelatinous phase. The circular and noncircular pores denote that the shape of the pore is a smooth circular arc or an irregular shape, respectively. "Interfacial surf" and "Inside" indicate the cross-sectional view near the interfacial surface of the feed solution and the view at the inside of the membrane, respectively. Prim = primary particle, secon. = secondary particle, m.p. = microphase separation, aggre. = aggregation of the particles, phase inv. = phase inversion, and solidif. = solidification.

of extraneous substances such as remaining catalysts and oligomers. Crystal growth proceeds along one dimension (fiberlike growth) or two dimensions (lamellar growth) or, more likely, in three dimensions (spherulitic growth). When a constant temperature gradient is imposed, growth tends to occur in one dimension along the direction of the temperature gradient as in the case of b-axisoriented crystals in polypropylene. When we carry out the crystallization of polymer melts on a foreign material, many nuclei are generated on the surface and crystals are forced to grow in one dimension directed as a whole perpendicular to the surface (this type of crystals is called a trans crystal). The generation of fine particles has been observed in cases of both crystalline and noncrystalline polymer solids. In the wet-molding process in particular, the supermolecular structure of the polymer solid can be interpreted as the manifold aggregations of the particles. The polymer solid is constructed of two kinds of fine particles, the primary and secondary particles described below. The typical example applied for this particle generation mechanism in the solidification of polymer solids is the case of a porous polymeric membrane. Figure 13 shows a schematic representation of the process of pore formation in cellulose acetate

SUPERMOLECULAR STRUCTURE OF POLYMER SOLIDS

233

membranes (Kamide et aI., 1977). This process is an example of the microphase separation method (Manabe et aI., 1973) for preparing porous polymeric materials. Here, the microphase separation method consists of the following steps: In the wet and/or dry molding process, the homogeneous one-phase solution separates into two phases, a polymer-rich phase and a polymer-lean phase. This separation begins with the formation of nuclei with a high concentration of polymer, and these nuclei grow into primary particles having sizes ranging from 10 to 50 nm. The primary particles collide with each other due to their Brownian motion, and after the collision the particles amalgamate, resulting in the formation of secondary particles. Secondary particles have a size between 50 and several hundred nanometers. Since the particles are rather stable and the change in size appears small, this phase separation is referred to as microphase separation in the membrane-forming process. Similar fine particles have been observed in many kinds of membranes and hollow fibers and in wet-spun textile fibers (Manabe, 1993). The secondary particles do not correspond to the droplet model proposed by Kesting and Manefee (1969). The concept of phase separation with droplets indicates that the interfacial domain of the fine particles (droplets) generated from the polymer solution is composed of the polymer-rich layer and the inside the particle is a solution with a higher content of the swelling agents and poor solvents than that of the outer solution. According to the theoretical approaches of Kamide and Manabe (1985) for the microphase separation method, five fctors govern the final pore structure of a porous polymeric membrane: (1) the number of nuclei, (2) the polymer concentration, (3) the interfacial tension between two phases after phase separation, (4) the rate of diffusion of particles (viscosity, temperature, etc.), and (5) the holding time after reaching the composition that causes phase separation (i.e., the phase separation time). Tsurumi et a1. (1990b) used this model to study regenerated cellulose hollow fibers.

III.

THERMAL MOTION OF POLYMER CHAINS IN A SOLID

Although two distinct models for interpreting molecular diffusion in a polymer solid, the free-volume model and the capillary model, are based solidly on supermolecular structure, we note that the significant feature of this structure is that it is not static but dynamic. The polymer molecules do not keep stationary but exhibit micro-Brownian motion or other thermal motion. The movement of polymer segments gives rise to increases in the free volume or vacant space into which a penetrant can diffuse. Conversely, the penetrant may affect the segmental thermal motion by giving its own free volume to the segment. Figure 14 shows a comparison of the thermal motion of a crystalline and an amorphous polymer solid. The dynamic shear modulus G' and the dynamic loss tangent, tan 8, are typical representatives of the dynamic viscoelasticity. The

234

MANABE

/Crystalline polyrrer '-------'~



g

Amorphous leathery

I

rtilt~ I I

I

I

~

1\

I \

I \ \ \ \ \

Tg

Tm

Temperature Figure 14 Comparison of the thermal motion of polymer solids of crystalline and amorphous polymer solids. Full line: a crystalline polymer solid. Broken line: an amorphous polymer solid. !xc, local twisting motion of a main chain in a crystalline region; !x" segmental micro-Brownian motion in an amorphous region; 13., local twisting motion of a main chain in an amorphous region; cx.c, 1380, free rotation or local twisting motion of a side chain in an amorphous region; 'Ymc, free rotational motion of an end methyl group. Tg and Tm are the glass transition temperature and the melting temperature, respectively.

existence of the melting point and the dynamic absorption (peak of tan 3) IXc originated by the thermal motion of the molecular chains in a crystal distinguishes a crystalline polymer solid from an amorphous one. Many dynamic absorptions are common in crystalline and amorphous polymers. These are, starting from the higher temperature side, the absorption IX. that originates in the micro-Brownian motion of polymer segments located in the neighborhood of the glass transition temperature Tg , the absorption 13. due to the local twisting motion of a main chain, the absorption IXsc or I3sc due to the free rotational and/ or local twisting motion of a side chain, and the absorption 'Y mc due to the rotational and/or other types of motion of a methyl group. The general empirical rules obtained so far are as follows:

1. 2.

The larger the moving unit is, the higher the temperature of the corresponding absorption. When the apparent activation energy of the movement increases, the temperature shifts to a higher value even if the moving unit is the same in size.

SUPERMOLECULAR STRUCTURE OF POLYMER SOLIDS 3. 4.

A.

235

As the intermolecular interaction increases, the absorption from the interacting molecules moves to the higher temperature side. The peak value of the absorption increases with the size of the moving unit and the number of moving units.

First-Order Thermodynamic Transitions (Melting, Crystal Transformation, Liquid Crystal Phase Transition)

From the viewpoint of thermodynamics, many kinds of transitions can be defined. The first-order transition is defined as a transition during which latent heat is generated or absorbed. In the case of melting, the equilibrium melting point T::' is given by

(8) where t:J{ and tlS are the differences in enthalpy and entropy between the melt and the crystalline state and the contribution of the surface energy to the Gibbs free energy is neglected. When the crystal size is not as large, the contribution of the surface energy is not negligible, and the melting point Tm(l) of a crystal of thickness I is given by 2ao )

Tm(l) = T::' ( 1 - !::.h . I

(9)

where ao and !::.h are the surface energy and the heat of fusion per cubic centimeter of crystal. A polymer solid with a high melting temperature has strong intermolecular interaction and high molecular rigidity. Above the melting point, a crystalline polymer behaves very like a liquid, that is, a polymer melt. Other examples of transitions belonging to the first order are crystal transformation and liquid crystal phase transition. The transition temperatures are given by equations similar to Eq. (8). In this case the differences in enthalpy and entropy are the differences between the values before and after the transition. The membrane formed by a mixture of a polymer with a liquid crystal shows changes in the permeation characteristics of the membrane when the permeation temperature crosses the liquid crystal transition temperature (Washizu et al., 1984). The first-order transition characteristics of a polymer liquid crystal have been discussed in detail for various polyphosphazens (Schneider et aI., 1978).

B.

Glass Transition and Segmental Micro-Brownian Motion

Part of a polymer melt of a crystalline polymer or most of the melt of a noncrystalline polymer can be frozen into the glassy state without crystallization. This transition is the glassy transition. At this point, the second derivatives of thermodynamic properties such as the specific heat capacity and the coefficient of thermal expansion show a stepwise change. Although this step change is

236

MANABE

apparently similar to the second-order phase transition, the glass transition should be regarded as a relaxation phenomenon in which segmental microBrownian motion plays the principal part. This transition is the most important one for a noncrystalline polymer solid. The glass transition temperature Tg changes depending on the time scale of measurement and the relaxation time of the segmental motion. Here, the mechanical absorption originated by this motion is called IX. absorption. The temperature dependence of the relaxation time Tis represented by the WLF equation given by Eq. (10) as a function of the temperature difference T - Tg •

TT)

Iog ( TT.

=

-17.44(T - Tg) 51.6 + T - Tg

(10)

where T1' and T1'g are the relaxation times at temperatures T and Tg , respectively. Equation (10) is known to fit the data down to about Tg • Equation (10) indicates that when the temperature approaches Tg , the relaxation time increases abruptly. Under certain limiting conditions Tg can be regarded as the constant representing the material. Table 5 lists the values of Tg for representative polymer solids. The size of the chain segments that initiate their micro-Brownian motion at Tg is estimated to be less than 100 carbon atoms (Nakayama et al., 1977) and more than 20 (Manabe et al., 1969), evaluated from the apparent activation energy of IX. absorption and from 2 to 10 nm in length evaluated from the relationship between the viscoelasticity of a polymer blend and its dispersed state observed through electron microscopy (Manabe et al., 1969). When the size of the penetrant is similar to that of a segment, the transport may be dominated by the segmental motion, and below Tg penetrant transport becomes very difficult. The diffusion of a dye molecule corresponds to this case. Even in the case of a penetrant smaller than a segment, the diffusional flow is influenced by the segmental movement. The free-volume model for diffusion is based on this segmental micro-Brownian motion. For a gas, since the size of the penetrant is far smaller than that of the segment, the temperature dependence of the diffusion coefficient shows the linear relation in the Arrhenius plot indicating no abrupt change at Tg • Many physical properties other than transport properties change drastically at Tg • The dynamic properties including dynamic modulus G' and tan 8, the dielectric properties, and gas adsorption isotherms are examples. The apparent activation energy of IX. absorption ranges between 150 and 850 kllmol, and the activation energy of the dye diffusion into a polymer solid falls in this range. When we plot the diffusion coefficient against the reciprocal of the absolute temperature (Arrhenius plot), we obtain the hypothetical diffusion coefficient Do extrapolated to infinite temperature. The value of Do is known to closely depend on Tg , decreasing with increases in Tg •

SUPERMOLECULAR STRUCTURE OF POLYMER SOLIDS Table 5

Glass Transition Temperature Tg for Various Polymer Solids

Polymer Polybutene-l Polychlorotrifluoroethylene Polyethylene Poly-4-methylpentene-l Polypentene-l Polypropylene Atactic Isotactic Polytetrafluorethylene Poly methyl acrylate Polyethyl acrylate Poly methyl methacrylate Syndiotactic Isotactic Polystyrene Polyacrylonitrile Polyvinyl acetate Polyvinyl alcohol Polyvinyl chloride Nylon 6 Nylon 6, 6 Polyethylene terephthalate Cellulose triacetate

C.

237

Evaluation method

Tg (K)

Dilatometry Refractive index Dilatometry Dilatometry Dilatometry

249 318 148, 243 302 233

Dilatometry Dilatometry Dilatometry Dilatometry Dilatometry

253 263 160, 400 279 249

Dilatometry Dilatometry Dilatometry Dilatometry Dilatometry Dilatometry Dilatometry Dilatometry Differential thermal analysis Dilatometry Dilatometry

388 318 373 378,433 301 358 354 348 330 342 378, 430

Local Twisting Motion of Polymer Chains in an Amorphous Region

The dynamic absorption caused by local twisting of polymer chains in an amorphous region has been named i3. absorption. The peak temperature of i3a absorption is usually located between 50° and 100°C below Tg • The apparent activation energy generally ranges between 40 and 100 kl/mot. Although the contribution of this type of polymer chain motion to penetrant transport has not yet been clarified, the antiplasticizer effect that causes a decrease in the diffusion coefficient due to the addition of small molecules may be related to the disappearance of the i3. absorption caused by this addition (Robeson, 1969). When a liquid with small molecules penetrates into a polymer solid, the noncrystalline region where the molecule can diffuse is dependent on the intermolecular interaction between the small molecules and a polymer chain. For example, benzene and other hydrophobic small molecules can diffuse into the limited domain in an amorphous region where the intermolecular hydrogen bond

238

MANABE

is poor and the van der Waals interaction is dominant (Fujioka and Manabe, 1995). Some of these molecules prevent the 13. absorption located at about -60°C as in the case of a regenerated cellulose solid (Fujioka and Manabe, 1995). This indicates that smaller molecular penetrants can diffuse only into that restricted amorphous region that is related to the development of the 13. absorption. Table 6 summarizes the peak temperature Tmax of a. and f3a absorptions and the apparent activation energy tlH. for various polymer solids including other dynamic absorptions described below. Since the values of T max and tlH. also depend on the supermolecular structure, we need to recognize that the values in this table represent only typical cases.

D.

Other Molecular Motions of a Polymer Chain

When the chemical structure becomes complex as when there are long side chains, then one or more additional dynamic absorptions are generated. Figure 15 shows the temperature dependence of the dynamic tensile modulus E' and loss modulus E" at a measuring frequency of 110 Hz for various polymer solids (Manabe and Takayanagi, 1970c; Manabe et aI., 1970b). The polymer employed is isotactic poly-a-olefin with unbranched side chains. If we define the number of carbon atoms in the side chain as N, then if N < 6 (see Fig. 15a), the dynamic absorptions a c , a .(mc), f3.(mc), and a .(sc) can be observed in the temperature range between -180 and + 100°C. Here the subscripts c and a indicate that the absorptions are caused by the motion of the molecules in the crystalline or amorphous region, respectively. The mc and sc in parentheses indicate main chain and side chain, respectively. When N > 6 (Fig. 15a), the absorptions a~ , a~, a~(mc), a~(",), and f3.(sc) are observed. Figure 16 shows a plot of the peak temperature against 1/(N + 2), where N + 2 corresponds to the number of carbon atoms in a monomer unit. Most of the characteristic temperatures show a linear dependence on lI(N + 2). When we extrapolate these values into zero or 1/3 at the values of 1/(N + 2), we obtain those of polyethylene or polypropylene, respectively. The dynamic absorptions caused by the motion of the side chains have activation energies ranging between 40 and 100 kllmol depending on the value of N. The absorption f3,(sc), whose mechanism is a local twisting motion in a side chain, can be observed when N > 3 and becomes more pronounced with an increase in N. These features of f3a(sc) are quite similar to those of 13. in the case of polyamide (Kawaguchi, 1962). The polymer chain end group is easy to move compared to groups inside the chain. The methyl group at the end of the main chain or side chain shows mechanical absorption between - 220 and -120°C, and the activation energy

VJ

Peak Temperatures of Various Viscoelastic Absorptions Under an Isochronal Measurement at 110 Hz and Apparent Activation Energy t:.H. for Some Typical Polymer Solids

Table 6

'fmc

Polymer

(K)

!:lH. (kl/mol)

Polybutene-l Polyethylene Pol ypropy lene Polymethylacrylate Polyethylacrylate Polymethyl methacrylate Polystyrene Polyacrylonitrile Polyvinyl acetate Polyvinyl alcohol Polyvinyl chloride Nylon 6 Nylon 6, 6 Polyethylene terephthlate Cellulose triacetate Cellulose

70

10

60-160 <60 100 70- 100

10 2 7 1-20

Tmax

13.

et,(se), p ,(se)

Tmax (K)

260

210

32

!:lH. (kllmol)

(K)

230 150 220

62 54 58

190 370

79 87

320 340 230

125 62 46

270 150 160 220

46 37 42 75

320 200 240

208

40

37

92

~

~ tl]

et,

!lH, (kl/mol)

Tm..

~

T max

t:.H,

(K)

(kl/mol)

280 260 290 300 270 410

146 154 191 179 162 516

400 430 310 320 370 350-370 350-390 340-380

537 395 171 229 283 266-374 229-279 341-279

420-450 570 525 385

320-333 1206 287 208

CJ

~

VJ

;5

R ....,

~

~ "tl

~

~

::ti

VJ

~

t3 VJ

tv w '0

MANABE

240

(a)

..

E'

... -~

~~""'-"

.

0

~

-(!)

-w -w

01 0

-2

3

,poo,

4

"10,

"HO', ,pO~

-3

-200

-100

100

0

Temperature I °C

(b) ••• ,......

0

'

f1. (!) o-

w

1. PO-I 2.PP-' l.PS-' , . PP

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

-1

""-.,

...... , ..

'.

'-.

-W

01 0

-2 -3 -100

0

100

Temperature I °C

Figure 15 Temperature dependence of dynamic tensile modulus E' and loss tensile modulus E" for poly-a-olefins with unbranched side chains. (a) The number of carbon atoms in a side chain (N) is more than 6. IPO 1 = isotactic polyoctene-l (N = 6), IPD 1 = isotactic polydecene-l (N = 8); IPDD 1 = isotactic polydodecene-l (N = 10); IPTD 1 = isotactic polytetradecene-l (N = 12); IPHD 1 = isotactic polyhexadecene-l (N = 14); lPOD 1 = isotactic polyoctadecene-l (N = 16); PE = polyethylene. (b) The value of N is less than 6. PP 1 = isotactic polypentene-l (N = 3); PB 1 = isotactic polybutene-l (N = 2); PP = isotactic polypropylene (N = 1).

SUPERMOLECULAR STRUCTURE OF POLYMER SOLIDS

241

200r---------------~

-

uo

-2000L-----'---.L----0......3~

Figure 16 Plots of melting temperature and peak temperature of various viscoelastic absorptions versus reciprocal of number of carbon atoms of a monomer, l/(N + 2), for isotactic poly-a-o\efins with unbranched side chains. mp, melting point; a " absorption caused by segmental micro-Brownian motion; ~a, local twisting motion. The subscripts a and c indicate that the absorption is from the molecules in an amorphous (a) region or crystalline (c) region. The symbols mc and sc in parentheses stand for main chain and side chain, respectively.

ranges between 10 and 20 kllmol. As the moving unit is very small, this absorption may influence a penetrant gas molecule. We have few reports of studies on the relationship between penetrant transport and the molecular motion emphasizing the side chains and the methyl end groups. This topic is open to future research.

IV.

CORRELATION BETWEEN CHEMICAL STRUCTURE, COMPOSITION, AND PENETRANT TRANSPORT

A.

Molecular Interaction

Molecular conformation may be decided by inter- and intramolecular interactions. Intramolecular interaction determines the angle of rotation around a single bond. For example, in the case of n-butane, the trans form is most stable and two gauche forms come in second. Even in the same chain, two atoms in question could be far apart along the chain but close to each other in space due to

242

MANABE

the folding back of the chain. This case of molecular interaction should be regarded as intermolecular interaction. Interactions between nonbonded atoms can be classified into three kinds: hydrogen bond interaction, electrostatic interaction, and van der Waals interaction. The van der Waals interaction is caused by three mechanisms: dipoledipole interaction, dipole-induced dipole interaction, and dispersion forces (instantaneous dipole-instantaneous dipole interaction). The potential energy U caused by the van der Waals interaction is expressed in Eq. (11) as a function of the distance r between atoms: U = - Alr6

+ Blr",

n = 9, ... , 12

(11)

where A and B are constants determined by the combination of atoms interacting. The first term in Eq. (11) represents the effect of the attractive force, and the second that of the repulsive force. Molecular interaction contributes to penetrant transport through both equilibrium and the dynamic rate process. Examples of the former are the molecular chain conformation and the regularity of the molecular packing in a crystal. The second str.uctural factor in Table 1 is a reflection of intermolecular interaction. The molecular interaction between a penetrant molecule and a polymer chain molecule (or segment) governs the solubility of the penetrant molecule in the polymer solid and is given a more detailed description in the next section using the theory of affinity. In a rate process, such as diffusion, the rate of sorption and desorption, the molecular interaction demonstrates its effect through control of the glass transition temperature Tg and the activation energy tlHa of the rate process. Both Tg and tlH. increase with an increase in the interactions.

B.

Affinity and Its Role in Transport Phenomena

In this section the term affinity is defined as the chemical affinity that corresponds to - Llfl-o' Here, LlIJ..o is the difference in the chemical potential between the product (final goal) system and the initial (starting) system. Consequently, affinity is a thermodynamic quantity and is independent of the path from the start to the product. Suppose that a penetrant in a solution is diffusing into a polymer solid. The affinity - Llf.Lo is given by (12) where R is the gas constant, T the temperature in kelvins, and [P]p and [P]I are the concentrations of the penetrant in the polymer solid and in the solution, respectively. Equation (12) is derived under the assumption that the penetrant is soluble in both phases and the solutions are ideal solutions. This equation is

SUPERMOLECULAR STRUCTURE OF POLYMER SOLIDS

243

transformed into

[PV[P]I = K = exp(-AfJ.o/R1)

(13)

where the constant K is the partition coefficient and is taken to be independent of the concentration of the penetrant. In the special case when the concentration of the penetrants is located on the surface of the solid, the relationship between [P]p and [P]I becomes the adsorption isotherm. On the other hand, for the case when the penetrant molecule is in the gas phase, [P]I should be changed to the concentration of the penetrant in the gas phase (Le., partial pressure, denoted as [P]g). Then, in the case of penetrant transport from the gas phase to a polymer solid, Eq. (13) holds if K is replaced by k defined by

(14) The constant k is the solubility coefficient. The value of k increases with the boiling point of the penetrant. Suppose a penetrant in a solution diffuses into a polymer solid through a capillary (the capillary model) and dissolves into the polymer solid with the partition coefficient K. Then the apparent diffusion coefficient Dc is theoretically given by

(15) where Do is the diffusion coefficient of the penetrant in the medium without the polymer solid, a is the weight of the medium in the capillaries per unit weight of the polymer solid, and DA is the diffusion coefficient of the penetrant in the solid. The value of Dc decreases with an increase in K. If a penetrant in a solution diffuses into a polymer solid through the freevolume model, the value of K contributes to the permeation coefficient P as follows :

P=DK

(16)

In this case, the affinity works only on K , resulting in a strong contribution to the penetrant transport. Here, the driving force of the transport is regarded to be the transmembrane pressure originating from the difference in concentration of the penetrant between the two sides of a membrane (i.e., the osmotic pressure). This type of permeation is called the dissolution/diffusion mechanism. A gas molecule also moves through a polymer solid by mechanisms of either the capillary model or the free-volume model depending on the size of the molecule and that of the pore in the solid. When the boiling point of a molecule increases and when the pore size is less than 20 nm and larger than 4 nm, surface diffusional transport dominates (Kamide et aI., 1983). In this type of transport, the adsorption and two-dimensional diffusion coefficient should be taken into account. Affinity works mainly on the adsorption. When the pore size

244

MANABE

is less than 2 run, the dissolution/diffusion mechanism becomes the dominant factor for transport and the permeability coefficient is expressed by P =Dk

(17)

To increase the permeability coefficient of a gas molecule, a bulky side chain is induced into aromatic polymer solids such as polyimide (Okamoto et aI. , 1993), polyester (Sheu and Chern, 1989), polycarbonate (Hellums et aI., 1992), polysulfone (Aitkekn et aI., 1990), and polyphenylene ether (Aguilar-Vega and Paul, 1993), whose glass transition temperatures are high. The side chain increases the free volume, providing additional space for the diffusion of gas molecules.

C.

Effect of Chain Conformation and Chain Configuration on Penetrant Transport

The chain configuration indicates the steric regularity of the chemical bond, i.e., tacticity here. This structural characteristic belongs to the first-order structural factor of Table 1. Chain conformation is the shape of the chain molecule, for example, a random coil, a helix, the l3-structure in polypeptide, and plane zigzag, etc. In general, the configuration governs the conformation. The regular form of chain conformation and configuration results in a crystal or a liquid crystal. Penetrant transport is influenced by the existence of these types of crystals as mentioned in Section II, and an increase in the regularity of the chain conformation and the chain configuration gives rise to a decrease in penetrant transport. When the conformation of the chain molecule constituting a membrane changes, then the transport properties of the membrane change. For example, a polypeptide membrane shows electrical pulses under the concentration gradient of a salt. This is due to a conformational change between a helix and a coil, which depends on the concentration of the salt (Minoura et aI., 1993). The conformational change of a protein molecule may play a very important role in ion transport across a biomembrane.

V.

EFFECTS OF FINE STRUCTURE (CRYSTALLINITY, ORIENTATION, ETC.) OF A POLYMER SOLID ON PERMEATION PROPERTIES

The fine structure treated in this section belongs to the third-order structural factor in Table 1. The structural factor includes the characteristics of a crystal, such a crystal size, crystallinity, crystal orientation, and crystal shape, the dispersion state of crystalline and noncrystalline regions, and the characteristics representing the interfacial aggregation structure. The adsorption inside a crystal region and diffusion through a crystal are considered to be negligible. The effect of fine structure on the permeation prop-

SUPERMOLECULAR STRUCTURE OF POLYMER SOLIDS

245

erties can be described through the noncrystalline region including crystal surfaces. Since the contribution of the monolayer on the surface to the total volume of the noncrystalline region is usually small, the solubility coefficients k and K are expressed as k = (1 - X)k.,

K = (1 - X)K.

(18)

where X is crystallinity and k. and K. are the values of k and K, respectively, in the noncrystalline regions. We note that the solubility coefficient is defined at thermodynamic equilibrium. In contrast, we apply this value to rate processes such as permeation under the assumption that Eq. (18) may hold approximately only on the surface of the polymer solid where the diffusion and permeability coefficients are small. The value of X should be the value at the surface. In the case of the capillary model, the equilibrium is assumed to hold over the entire surface of the capillary. As for the diffusion coefficient, the existence of crystals affects the transport in a complicated manner. For example, crystals in a noncrystalline region give the molecular chains in the region some kind of strain that results in immobilizing the chains. This effect is called the immobilization factor 13. Both the crystallinity X and the size of a crystal hinder the transport of a penetrant, and the path of the penetrant becomes tortuous. The degree of tortuosity varies with the change in the dispersion state of a crystalline region even in the same values of X and the crystal size. This tortuosity factor 'Y should be used to correct the diffusion coefficient. The diffusion coefficient D for a crystalline polymer solid is represented by (Michaels and Parker, 1959) D = D./I3'Y

(19)

where D . is the diffusion coefficient of a penetrant in a noncrystalline region.

A.

Effect of Draw Ratio on Permeability

There are two types of molecular chain orientations, those in a crystalline region and those in a noncrystalline region. The orientation of the crystalline part may contribute to penetrant transport through the variation in the value of 'Y and can give rise to an anisotropic diffusion coefficient. On the other hand, the molecular orientation in a noncrystalline region causes not only the anisotropic feature in the penetrant transport but also the change in the region of diffusion for the penetrant. The increase in the degree of orientation gives rise to an increase in the regularity in a noncrystalline region and also to an increase in the number of the extended chain molecules through which most molecules do not diffuse. These changes may reflect the change in the 13 value in Eq. (19). The drawing of a polymer solid produces both increases in the crystallinity and increases in the degree of the orientation of a molecular chain. According to the results reported by Michaels et al. (1964), the permeability coefficient P

246

MANABE

of a gas molecule decreases with an increase in the draw ratio, but when the drawing approaches the breaking point, P increases abruptly.

B.

Effects of Heat Treatment on Permeability

The effect of heat treatment results in a change in the supermolecular structure and a change in the permeability of a penetrant. Heat treatment is usually performed under three typical conditions: in a system free of tension, constant length, and constant tension. In the crystalline region, the treatment results in (1) an increase in crystallinity, (2) an increase in crystal size, (3) an increase in the perfection of the crystal, and (4) a decrease in crystal orientation. The treatment condition of the free of tension preferentially brings about change 4. The condition of constant tension promotes changes 1-3 and a minimum change in crystal orientation. Constant length in one and/or two dimensions is often employed, and heat treatment then produces changes all four types of changes. In the noncrystalline region, the treatment gives rise to (1) relaxation of the remaining strain, (2) a decrease in the size of the region, (3) change in the molecular chain aggregation into a closer packing, and (4) a decrease in orientation. The treatment free of tension may show all four types of changes most effectively, and treatment under constant tension may work on changes 2 and 3. In general, heat treatment changes the supermolecular structure so as to decrease the transport rate of a penetrant. For example, even in the case when the temperature of the heat treatment is far below Tg , the permeability decreases and the perm selectivity increases in polyimide thin films (Pfromm and Kovos, 1993). Figure 17 shows the dependence of the dye exhaustion by polyethylene terephthalate fibers spun at 4, 5, 6, 7, 8, and 9 km/min on the temperature of the heat treatment Ta for 1 s at constant length (Kuriki et aI., 1985). The dye molecule employed as a disperse dye of 1,5-diamino-2-bromo-4,8-dihydroxyanthraquinone with a molecular weight of 349. The dye exhaustion shows a slight minimum at ca. 150°C and then increases with the elevation of Ta. The heat treatment results in the relaxation of the remaining strain and a decrease in the orientation of the noncrystalline region, and these changes produce loosely and relaxed chain molecules in the diffusible amorphous region of the dye molecule, resulting in a change opposite to the closer packing mentioned for the noncrystalline region.

C.

Effects of Solvent Treatment on Permeability

The prerequisite of the treatment is a solvent molecule that can diffuse readily into a polymer solid. The diffusional region can be evaluated through measurement of the dynamic visco elasticities of the solid under the coexisting solvent (Fujioka and Manabe, 1995). The solvent molecule accelerates the thermal motion of the polymer segment, resulting in effects similar to those of heat treat-

SUPERMOLECULAR STRUCTURE OF POLYMER SOLIDS

247

100

~ 80 c:

0


:l

160

~

~

~

5

<1>

>-

0

40

Or

150 200 Annealing temperature/·C

250

Figure 17 Change in exhaustion of a disperse dye with annealing temperature for polyethylene terephthalate fibers spun at various speeds ranging between 4 and 9 km/ min (Kuriki et al. 1.985). The numbers in this figure are the take-up speed in kilometers per minute. The dye employed was Resolin Blue FBL, and the annealing time was 1 s.

ment. We can expect the following effects from solvent treatment: (1) an increase in crystallinity, (2) an increase in crystal size, (3) an increase in crystal perfection, (4) a decrease in the orientation of the crystalline and noncrystalline regions, (5) the relaxation of the remaining strain in a polymer chain, (6) closer packing of molecular chain aggregation, (7) the dissolving out of a part of the component, (8) swelling by the remaining solvent, and (9) the generation of interfacial molecular orientation. Although these effects influence the penetrant transport, very few workers have succeeded in making a quantitative evaluation of these effects separately. Treatment of an acetate film with boiling water causes a loss in luster. This phenomenon is the result of effect 7, and the dissolving component is a low molecular weight component with small amounts of acetyl groups (Kamide and Manabe, 1978). The water molecule develops the local heterogeneity of the packing density of chain molecules. This type of change accelerates the rate of transport of the dispersed dye molecule. The increase in heterogeneity in the molecular chain aggregation brought about by solvent treatment has been observed in polymer blends (Manabe et ai., 1969). Figure 18 shows the temperature dependence of the dynamic viscoelasticity of a polymer blend of nitrile butadiene rubber and polyvinyl chloride after treatment with a solvent that acts selectively for one component in the polymer blend. The increase in the halfwidth value of the loss modulus E" versus temperature curve indicates an increase in the heterogeneity in the aggregation of component polymers. Solvent

248

MANABE

-2L......J..._ _ _L.....:..._ _...L.-_ -50 0 50 Temperature I·e

-50

0

50

Temperature I·e Figure 18 Temperature dependence of dynamic modulus E' and loss modulus E" for polymer blend of polyvinyl chloride/nitrile butadiene rubber (PVC/NBR) after treatment with solvent (Manabe et aI., 1969). The PVC/NBR ratio = 77/23 by weight. (a) The solvent is dimethylformamide, and its content is 6 and 14 wt %. (b) The solvent is chloroform, and its content is 7 and 20 wt %.

treatment against a porous membrane can impart a composition gradient to the membrane (Aptel and Cabasso, 1980). When polypropylene film is treated with p-xylene at various temperatures, the permeability coefficients for toluene, methylcyclohexane, and isooctane through the film increase with the treatment temperature (Michaels et at., 1969). The permeability coefficients of toluene, methylcyclohexane, and isooctane before treatment in one case were 15.3 X 10- 5 , 14.3 X 10- 5 , and 1.4 X 10- 5 g cm/(h cm2), respectively. After the treatment at 100 C, and coefficients were 59.4 X 10- 5 , 46.2 X 10- 5 , and 21.2 X 10- 5 g cm/(h cm2) . D

SUPERMOLECULAR STRUCTURE OF POLYMER SOLIDS

VI.

249

CONCLUDING REMARKS

The supermolecular structure of a polymer solid is determined in two ways by the chemical structure (the first-order structural factor) and the preparation conditions. Penetrant transport is governed by three functional factors of the supermolecular structure, the thermal motion of polymer chains, and the intermolecular interaction between the penetrant and the polymer chain. Because of the structural complexity and heterogeneity of the solid, it is very difficult to obtain a complete view of the structure. As for the relation to penetrant transport, the evaluation of the second- and third-order structural factors, that is, the characterization of the supermolecular structure on a nanometer scale, is important. The methods of novel microscopies such as scanning probe microscopy (SPM), atomic force microscopy, friction force microscopy, and magnetic force microscopy are very powerful tools for investigating supermolecular structure. To correlate supermolecular structure and penetrant transport, we must employ an adequate transport model. Although the free-volume model and the capillary model and their combinations have been proposed so far, some modifications or the use of new models that take supermolecular structure into account are needed. Combined with advanced theory, a novel preparation method for a polymer solid guided by theory should be developed in the future for preparing solids with the desired penetrant transport properties.

REFERENCES Aguilar-Vega, w., and D. R. Paul (1993). J. Polymer Sci., Polym. Phys. Ed., 31, 1577. Aitkekn, C. L., D. R. Paul, and W. J. Kovos (1990). ICOM '90, Preprint, p. 821. Aptel, P., and I. Cabasso (1980). J. Appl. Polymer Sci. , 25, 1969. Fujioka, R., and S. Manabe (1995). Polym. Prepr. JplI. , 44(5), 759. Haraguchi, H. , T. Kajiyama, and M. Takayanagi (1976). Rep. Progr. Phys. Jpn., 16, 303. Hellums, K. w., W. J. Kovos, and J. C. Schmidhauser (1992). J. Membrane Sci. , 67, 75. Iijima, H., and S. Manabe (1983). Japanese Patent, Tokkai Sho 58-183907. Kajiyama, T., K Tanaka, I. Ohki, S. R. Ge, J. S. Yoon, and A. Takahara (1995), Macromolecules, 27, 7932. Kamide, K , and S. Manabe (1978). Sen-i Gakkaishi, 34, T53. Kamide, K , and S. Manabe (1980). In Ultrafiltration Membranes and Applications, A. R. Cooper, Ed., Plenum, New York, p. 173. Kamide, K , and S. Manabe (1985). In Materials Science of Synthetic Membranes, D. R. Lloyd, Ed., American Chemical Society, Washington, DC, p. 197. Kamide, K , and S. Manabe (1986). Polymer J., 18, 167. Kamide, K , S. Manabe, T. Matsui, T. Sakamoto, and S. Kajita (1977). Kobunshi Ronbunshu, 34, 205. Kamide, K , S. Manabe, T. Nohmi, H. Makino, H. Marita, and T. Kawai (1983). Polymer J., 15, 179. Kamide, K , S. Nakamura, T. Akedo, and S. Manabe (1989). Polymer J., 21, 241 .

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MANABE

Kawaguchi, T. (1962). In Koubunshi no Bussei, T. Kawai, T. Kawaguchi, and A. Miyake, Eds., Kagakudoujin, Tokyo, p. 14l. Kesting, R. E. (1985). Synthetic Polymeric Membranes, Wiley, New York. Kesting, R. E., and A. Manefee (1969). Kolloid Z. , 230, 24l. Kuriki, T., K. Kamide, S. Manabe, and M. Iwata (1985). Sen-i Gakkaishi, 38, T85. Manabe, S. (1992). In Animal Cell Technology: Basic and Applied Aspects, H. Murakami, S. Shirahata, and H. Tachibana, Eds., Kluwer, Boston, p. 87. Manabe, S. (1993). Sen-i Gakkaishi, 49, 195. Manabe, S., and K. Kamide. (1984). Polymer J., 16, 375. Manabe, S., and M. Takayanagi (1970a). Kogyo Kagaku Zasshi, 73, 1572. Manabe, S., and M. Takayanagi (1970b). Kogyo Kagaku Zasshi, 73, 158l. Manabe, S., and M. Takayanagi (1970c). Kogyo Kagaku Zasshi, 73, 1595. Manabe, S., R. Murakami, and M . Takayanagi (1969). Mem. Fac. Eng. Kyushu Univ. , 28,295 . Manabe, S., S. Minami, and M. Takayanagi (1970a). Kogyo Kagaku Zasshi, 73, 1576. Manabe, S., H. Nakamura, S. Uemura, and M. Takayanagi (1970b). Kogyo Kagaku Zasshi, 73, 1587. Manabe, S., E. Osafune, and K. Kamide (1973). Japanese Patent Sho 48-25055 . Manabe, S., S. Kajita, and K. Kamide (1980). Sen-i Kikai Gakkai, 33, T93. Manabe, S., Y. Kamata, and K. Kamide (1985). Polymer J., 17, 775 . Manabe, S., Y. Kamata, H. Iijima, and K. Kamide (1987). Polymer J., 19, 39l. Mark, H. F., S. M. Atlas, and E. Cernia, Eds. (1967). Man Made Fibers, Wiley, New York. Michaels, A. S., and R. B. Parker, Jr. (1959). J. Polymer Sci. , 41, 53. Michaels, A. S., W. R. Vieth, and H. J. Bixter (1964). J. Appl. Polym. Sci., 8, 2735 . Michaels, A. S. , W. R. Vieth, A. S. Hoffman, and H. H. AlcaJay (1969). J. Appl. Polym. Sci., 13, 577. Minoura, N., S. Aiba, and Y. Fujiwara (1993). J. Am. Chem. Soc., 115, 5902. Nakayama, C., K. Kamide, S. Manabe, and T. Sakamoto (1977). Sen-i Gakkaishi, 33, T139. Nohmi, T., S. Manabe, K. Kamide, and T. Kawai (1978). Kobunshi Ronbunshu, 35, 509. Okamoto, K., N. Umeo, S. Okamyo, K. Tanaka, and H. Kita (1993). Chem. Lett. , 225 . Peterlin, A. (1969). J. Polym. Sci., A-2, 7, 115l. Pfromm, P. H ., and W. J. Koyos (1993). ICOM '93, p. 213. Robeson, L. M. (1969). Polym. Eng. Sci., 9, 277. Schneider, N. S., C. R. Desper, and J. J. Beres (1978). In Liquid Crystalline Order Polymers, A. Blumstein, Ed., Academic, New York, p. 299. Sheu, F. R., and R. T. Chern (1989). J. Polym. Sci. Polym. Phys. Ed., 27, 1121. Takayanagi, M. (1967). Pure Appl. Chem., 65, 555 . Takeda, H., and Y. Nukushina (1963). J. Chem. Soc. Jpn. Ind. Chem. Sect., 68, 124l. Tsurumi, T., N. Osawa, H. Hitaka, T. Hirasaki, K. Yamaguchi, S. Manabe, and T. Yamashiki (1990a). Polym. J., 22, 751. Tsurumi, T., T. Sato, N. Osawa, H. Hiraka, T. Hirasaki, K. Yamaguchi, Y. Hamamoto, S. Manabe, T. Yamashiki, and N. Yamamoto (1990b). Polym. J., 22, 1085. Washizu, S., I. Terada, T. Kajiyama, and M. Takayanagi (1984). Polymer J., 16, 307. Yeh, G. S. Y. (1972). J. Macromol. Sci., 86, 465 .

6 Translational Dynamics of Macromolecules in Melts Peter F. Green Sandia National Laboratories Albuquerque, New Mexico

l.

INTRODUCTION

A flexible polymer chain is capable of assuming an enormous number of possible configurations by rotations of chemical bonds. Such local motions, which occur on time scales on the order of 10- 11 s, facilitate the translational motions of single chains in melts. Their effect on the overall dynamics of a macromolecule can be accounted for by a mean monomeric frictional coefficient, ~. The time scale of its translational diffusivity is determined primarily by N, the number of monomer segments that compose a single chain, and ~. The intent of this chapter is to discuss the translational center-of-mass motions that chains in polymer melts undergo in response to thermal agitation. A remarkable feature of the dynamics of long polymer chains is an important interaction that arises from the fact that these one-dimensional objects cannot cross each other. The topological constraints, known as entanglements, have a profound impact on chain dynamics. Because of the entanglement effect, polymer liquids exhibit rather remarkable behavior that distinguishes them from other materials (Ferry, 1980). One important feature can be seen in the time dependence of the stress relaxation of a melt that has been subjected to a strain. A sudden imposition of a shear strain, "I, that displaces the chains in a melt of linear chains from their equilibrium configurations results in a time-dependent relaxation of the system as the chains return to equilibrium through some Brownian process. It is convenient to describe the time dependence of such a

251

252

GREEN

process by defining a shear-stress relaxation modulus, G(t) = cr(t)ly, where cr(t) is the time-dependent shear stress that reflects the structure and interactions between the chains in the system (Ferry, 1980; Graessley, 1974, 1982). Figure 1 depicts the time dependence of the shear modulus. At short times, the reduction of G(t) results from the relaxation of stresses along portions of the individual chains (note that such a process is characterized by a distribution of relaxation times). At intermediate times, for chains that are longer than an effective number of monomers, No> where Ne is the number of monomers between entanglements, G(t) becomes independent of t, the plateau region. This so-called plateau region, characterized by a modulus G~, can be rationalized only by the presence of "entanglements." For the most part, G~) is relatively insensitive to temperature. It is a property of the melt and is determined by structure. While G~) is independent of N, the onset of the terminal region is a strong function of N. This will be addressed later. At long times, in the terminal region, G(t) rapidly relaxes to zero. Here the chains assume equilibrium conformations through a translational (Brownian) process. Properties such as the viscosity are also affected by entanglements. For chains shorter than a critical number of segments, N e, where Nc is a material-dependent property (empirically Ne "" 2Ne), the viscosity in the limit of zero shear, 1)0 N , while for N > N e , 1)0 - N3.4 ~0.2 (Berry and Fox, 1968). Figure 2 shows a plot of the molecular weight dependence of the viscosity of polyethylene. It is important to note that for N < Ne , 1)0 is observed to vary as N after corrections are made to achieve a constant free volume state. The triangles represent the un-

log G(t) Tenninal _ _ _p_la_te_au_ _ _:::::---_.,.,.••• ,.,. •••• G O N

M

>

~

Logt Figure 1 Schematic of the time dependence of the stress relaxation modulus, G(t), of a melt. G ~ is the plateau modulus, and M and M* are the molecular weights of the melt.

253

TRANSLATIONAL DYNAMICS IN MELTS 14

•

12

••

10

....>-

:::III

• 8

0

()

.1

III

->

6

CI

0

4

, IIPDlf .. ~ i

•

••

I

II

2

0

M

c

-2 2

3

4

5

6

7

8

log M

Figure 2

Molecular weight dependence of the zero shear rate viscosity of polyethylene. For N > N c , the viscosity varies as N 3 .4 • For N < N c , it varies as N (0) after the raw data (A.) are adjusted for chain end effects. This behavior is typical. [Data of Colby et al. (1987).]

corrected data. The N 3 .4 dependence remains unchanged for N > N c • This behavior is typical. The foregoing discussion provides the framework within which we can begin to understand translational dynamics in polymer melts or concentrated systems. Because of the large number of configurations that a polymer chain can assume, the shape of a single polymer chain can be described statistically (Flory, 1969). Consequently, a description of the dynamical process necessarily exploits this simplification. In an attempt to understand the dynamics of chains in solution,

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Rouse (1953) argued that a chain could be considered as a series of beads connected by springs. The dynamics of such a chain could be modeled by the Brownian motion of beads whose positions are described by a position vector RN • Each bead is assumed to experience an average frictional force, ~. As shown later, what ultimately emerges from such a picture is that the translational diffusion coefficient D Ro - ~JV- l and that the longest relaxation time of the chain 'fRo - ~JV2. The Rouse model failed miserably at describing translational dynamics in dilute solution primarily because it neglected the effects of intramolecular hydrodynamic and excluded volume interactions. Interestingly, as shown later, it provides the basis for understanding translational dynamics in melts where long-range hydrodynamic and excluded volume interactions are assumed to be screened and where each chain can be described by ideal Gaussian statistics. Before deGennes introduced reptation (deGennes, 1971, 1979) just over 20 years ago, a number of unsuccessful attempts (Bueche, 1968) had been made to understand the dynamics of entangled melts. In reptation, a polymer chain is imagined to undergo one-dimensional translational motion in the presence of a fixed network of obstacles. The motion is facilitated by the flow of a series of "defects" (kinks) along the contour of the chain. The chain translates in a manner that simulates the motion of a snake. Reptation was later introduced by Doi and Edwards (1978a,b, 1986) so that a constitutive equation could be developed to understand the behavior of true melts. They introduced the concept of a tube in order to account for the effect of entanglements. It was noted that although permanent cross-links exist in elastomers, they are not present in linear polymer systems. Therefore, although one might introduce the concept of the tube in linear polymer melts, it has to be acknowledged that the topology of such a tube is time-dependent. We show later that there is a regime of molecular weights over which predictions based on the chain undergoing Brownian motion in a fixed " tube" are in agreement with experiment. Complications that result from the time-dependent topology of the tube can be dealt with mathematically, and experimental data on diffusion and viscoelasticity support the new predictions. One might emphasize at this point that calculations of the viscoelastic properties of polymers based on this picture, while qualitatively correct, do have shortcomings. In this chapter we begin with a detailed description of the Rouse model because it provides the basis for subsequent discussions. This is followed with a discussion of reptation of linear chains in a fixed tube and a subsequent discussion of the effects of the time dependence of the topology of a single chain as it diffuses. This provides the context within which we can begin to describe the Brownian motion of chains of different architectures (rings, stars). A discussion of experimental tests and the extent to which the results support the predictions follow. Finally, a discussion of interdiffusion in two-component miscible blends is presented. Here the concept of the Flory interaction parameter

TRANSLATIONAL DYNAMICS IN MELTS

255

X, a measure of the thermodynamic interactions between the components in the blend, is introduced. It is shown that the time scales of interdiffusion are determined not only by N and ~o, but also by X. When X < 0, the interdiffusion coefficient, D , is enhanced over the value of the tracer diffusion coefficient, D * , of a single chain. This is known as a thermodynamic "acceleration" of D. Similarly, when X > 0, a thermodynamic " slowing down " of D is realized. We conclude with a brief discussion of new developments on diffusion in block copolymer systems and some recent theoretical advances.

II.

TRANSLATIONAL DYNAMICS IN HOMOPOLYMER MELTS

A.

Dynamics of Unentangled Chains: Rouse Model

The Rouse model is expected to provide a reasonable description of the translational dynamics of unentangled chains in melts (Doi and Edwards, 1986). This is primarily because it neglects excluded volume and hydrodynamic interactions, both of which are assumed to be screened in melts. The central assumption in the Rouse model is that the dynamics of a chain are governed by localized interactions along the chain. Each chain is imagined to be divided into a series of sub molecules, each composed of a number of monomers. Each submolecule is represented by a bead, and the beads are connected by Hookean springs, where each spring constant is 3kB T/b2 ; b is the separation between beads. The position of each bead is determined by coordinates {Rl> R2 , R3, • • • , RN } , and each bead experiences an average frictional drag ~. The motion of the beads can be described by a linearized Langevin equation. In essence, therefore, we now have an equation that describes the Brownian motion of a series of coupled oscillators. A transformation is subsequently made so that the system can now be described in terms of normal mode coordinates where each mode behaves independently of the other. The equation is solved to yield a spectrum of N relaxation times, each relaxation time, 'ri, corresponding to one normal mode. The essential predictions follow. The diffusion coefficient of the center of mass is (1)

One might recognize that Eq. (1) is essentially the Einstein relation if the parameter N~ is identified as the effective friction coefficient of the chain in its environment. The time correlation function of the end-to-end vector is (P(t) . P(o) = Nb2

2:

+

p ,odd 'IT

e - lp2/Td

P

(2)

256

GREEN

where P(t) = RN(t)-RN(O) and 'T J is the relaxation time of the first mode (note that there are p modes). The relaxation time of the first mode is the longest relaxation time, 'TJ> and is given by 'TRo

=

'T t

= (b 2/3'TT2kBT)

(3)

"{jV2

The viscosity in the limit of zero shear is predicted to be (4) As mentioned earlier, this model is expected to provide a reasonable description of the dynamics of unentangled chains in the regime N < N c •

B.

Dynamics of Entangled Chains: Reptation

Shown in Fig. 3 is a schematic of a labeled chain composed of N segments in a melt surrounded by other chains, where each of the surrounding chains is composed of P segments. For the initial purposes of discussion, the length of the P-mer chains is considered to be infinite. Therefore the N-mer chain is assumed to undergo Brownian motion effectively in a fixed "tube." The motion of the chain occurs by the propagation of stored length, or kinks, as depicted in Fig. 4a (de Gennes, 1979). Curvilinear motion of the chain occurs along an average trajectory that Edwards called the primitive path (Fig. 4b). Edwards defined the primitive path as the shortest distance connecting the ends of the chain that has the same topology as the chain (Doi and Edwards, 1980). There are effectively two time scales in this problem. At short time scales, the dynamics are controlled by rapid motions (propagation of kinks) around the

a)

N-mer P-mer

N-mer b)

-

"tube"

Figure 3 (a) Schematic of a labeled chain of degree of polymerization N in the presence of a host of chains of degree of polymerization P . (b) Schematic of the labeled chain restricted to a tube-like region.

257

TRANSLATIONAL DYNAMICS IN MELTS

a)

'"

_------K---~ -----

~-....... -~/ ,.. /' _-----

i)

ii) ._--------------

---

I·~----

.......

b)

PRIMITIVE PATH

Figure 4 (a) A schematic demonstrating the mechanism of diffusion for a linear flexible chain. (b) A depiction of the chain moving along its primitive path.

primitive path, and at longer time scales they are controlled by the translational motion of the chain, which results in changes in the primitive path. As the chain diffuses in random directions it creates and destroys the ends of the primitive path. The translational motion of the chain is characterized by a time scale 'Td over which the chain loses complete memory of its original tube, or, equivalently, the original primitive path. In the pure reptation calculation where the chain is imagined to diffuse in a fixed environment (tube), fluctuations in the primitive path are ignored; i.e., it is considered to be of a constant contour length L. One can define the number of steps on the primitive path as Z; hence L = Za, where a is the length of a step along the path. The diffusive motions along

258

GREEN

the path are assumed to be well described by the Rouse model, and therefore Eq. (1) provides an adequate description of the motion of the chain in the tube. It is further assumed that the conformation of the primitive path remains Gaussian, at least at large length scales. The mean square end-to-end distance of the primitive chain (the dynamical equivalent of the primitive path) should be the same as that of the Rouse chain, so it follows that (5)

One can define the end-to-end vector of the chain as P(t) = R(L, t) - R(O, t). The time-dependent correlation function of the end-to-end vector is given by (P(t) . P(O». It is proportional to l\J(t), the probability that a segment of the tube that was originally occupied at time t = 0 remains occupied at time t. Here (Doi and Edwards, 1986),

(P(t) . P(O» = Lal\J(t) and

l\J(t)

=L p ,odd

+

e - rp2/T.

'TT

(7)

P

The longest relaxation time, Td

(6)

= (b /TzkBTa 4

Z ) '{)V3

Td,

is calculated as (8)

In contrast to the longest Rouse relaxation time, the reptation relaxation time, 3 Td, varies as N as opposed to N Z. It is interesting to note that because of the effect of entanglements, the actual relaxation time for a very long chain is reduced by a factor of 3Z. It is interesting to note that the tube disengagement time, Td, could be calculated from a simple scaling relation by noting that (9)

which, using Eq. (5), yields Td

=

4

(b /kBTa

Z )

rJ/3

(10)

The diffusion coefficient of a chain moving by a reptation process could be derived by noting that during the time interval Td the center of mass of the chain should have moved a distance equivalent to the radius of gyration of the chain, N i !2 a , so

D Rep = Na 2/Td

(11)

Therefore, (12)

TRANSLATIONAL DYNAMICS IN MELTS

259

An equivalent form of this equation may be realized by recognizing that the tube diameter a can be written as (Graessley, 1980; Graessley and Edwards, 1981)

(13) where M = MoN (Mo is the molecular weight of a monomer on the chain and Me is the molecular weight between entanglements). It follows that (14) where Do = (4/15)MoMc kBT/~. A rigorous derivation of this result may be found elsewhere (Doi and Edwards, 1986). Equation (14) is a very important result that predicts that the center-of-mass diffusion coefficient of a flexible polymer chain diffusing i.n a highly entangled melt varies as M - 2 , independent of the length of the P-mer chains in the host environment (recall the initial assumption that P »N). We now use the above results to calculate Tlo so that we can show how D Rcp can be expressed in terms of viscoelastic parameters (Graessley, 1980). The viscosity can be obtained from the relationship Tlo

= G~ i~ tjJ(t) dt

(15)

Upon integrating this expression one arrives at Tlo =

(~y C~BT) GN(O) (fi3

(16)

This result shows that the viscosity is determined by the longest relaxation time [Tlo = GN(O)'rd]' It follows that

D

rep

= G~) M 2 (R ~») 135

e

M

MJTlo(Mc) M2

(17)

We show later that upon substitution of the appropriate viscoelastic parameters, this result yields very reasonable predictions for the magnitude of D Rep for different polymers.

C.

Measurement of Diffusion in Polymer Melts

1. Techniques At this point it is important to differentiate between tracer diffusion, D*, and self-diffusion, D.; in a later section we address mutual diffusion. The tracer or self-diffusion coefficient of a species A can be defined in terms of the mean

260

GREEN

square displacement of the center of mass D = lim

!

,-..eo 6t

([R(t) - R(O)f)

(18)

where R is the position of the center of mass of the diffusant and t is the diffusion time. In reality, the tracer diffusion coefficient is realized when the molecule has translated a distance that is on the order of a few Rg; at smaller length scales D may have contributions from segmental displacements. In the case of tracer diffusion, one monitors the displacement of a single labeled molecule executing Brownian motion in a given environment. Self-diffusion corresponds to the situation in which the motion of a molecule is monitored in an environment of identical molecules. The driving force for tracer diffusion and self-diffusion is entropic in origin, unlike mutual diffusion, which is driven by gradients in chemical potential. 'TYpical diffusivities in polymer melts range from 10- 16 to 10- 6 cm%, which means that one needs experimental techniques with sufficiently good depth resolution to measure small diffusion coefficients in reasonable time scales. As an illustration of this point, one might consider attempting to measure a diffusion coefficient of a molecule that diffused for one day. If the diffusion coefficient is 10- 5 cm 2/s, then the mean displacement is 2.3 cm. On the other hand, if D* = 10- 16 cm2/s, then the mean displacement is only 207 A. It is reasonable to state that any technique that can be used to determine the concentration profile of a diffusant after some specified time interval can be used to determine a diffusion coefficient. This requires labeling the diffusant in such a manner that the label does not affect the mobility of the diffusant. An alternative approach might involve monitoring a time-dependent process that reflects the motion of the diffusant. Techniques that have been used to measure diffusion in polymer melts include radiolabeling techniques (Crank, 1975), infrared microdensitometry (IRM) (Klein and Briscoe, 1979), ion beam analysis techniques [Rutherford backscattering spectrometry (RBS) (Chu et aI., 1978; Green and Doyle, 1990; Green et aI., 1985), forward recoil spectrometry (FRES), and elastic recoil detection (ERD)] (Mills et aI., 1984; Green et aI., 1986; Green and Doyle; 1990), nuclear reaction analysis (NRA) (Chaturvedi, 1990), fluorescence recovery after pattern photobleaching (FRAPP) (Smith, 1982; Smith et aI., 1984), forced Rayleigh scattering (FRS) (Antonietti et aI., 1984), small-angle neutron scattering (SANS) (Bartels et aI., 1984, 1986), and nuclear magnetic resonance (NMR) (Fleisher, 1983, 1984, 1992; von Meerwall, 1991; McCall et aI., 1986). The range of techniques is not limited to those mentioned, but those are among the most commonly used. We now briefly discuss the essential features of these techniques and point out some of their more important limitations. The interested reader is referred to the original references for further details.

TRANSLATIONAL DYNAMICS IN MELTS

261

The review that follows is necessarily brief. Of these techniques, the radiotracer labeling technique, which involves determining the concentration profile of a radioactively labeled species, is the oldest. At best its depth resolution is on the order of micrometers, and it would not be well suited for measuring the slower diffusivities. The ion beam analysis techniques, SANS and IRM, are techniques that rely on labeling the diffusant in order to determine D. Further, they rely on the time evolution of concentration gradients. These RBS, ERD, NRA, and IRM techniques yield concentration profiles that can then be compared with the theoretically predicted profiles to extract D* or Ds. In RBS, the energy of an incident monoenergetic beam of ions (most diffusion experiments in polymers have been performed using helium ions) is measured after the ions are backscattered from a sample. The energy and yield of the backscattered particles can be analyzed to obtain the concentration versus depth profile of the appropriate species in the target. In ERD, an incident beam of particles causes nuclei to recoil from the target. An analysis of the yield and energy of the appropriate recoils gives the desired concentration versus depth profile. In NRA' the nature and energy of the incident particles is carefully chosen so that when these particles impinge on the appropriate nucleus a nuclear reaction will occur. The yield and energy of the nuclear products can be converted to the concentration versus depth profiles. The ion beam analysis techniques, in general, have excellent depth resolution, 200 - 800 A, and are ideally suited for measuring midrange to very slow diffusion coefficients (10- JO_1O- J6 cm% ). In IRM, a sample that is initially composed of a bilayer of two chemically distinct species (one could be deuterated) that are allowed to interdiffuse at a chosen temperature is microtomed into layers perpendicular to the diffusion direction and analyzed using infrared spectroscopy. The concentration profile from which the diffusion coefficient is extracted is then constructed with the information. The depth resolution of this technique is on the order of micrometers. Hence the IRM is not well suited to study slowly diffusing species. (D *s > 10- 9 cm2/s.) SANS exploits the strong coherent scattering contrast between deuterated and hydrogenated polymers. The samples are multilayered, and the coherent scattering intensity is measured only when the layers have interdiffused. The time dependence of the coherent scattering intensity yields information about the diffusion process. SANS is capable of measuring D,s in the range of 10- 8 _1O- J6 cm2/s. While FRAPP and FRS require labeling of the appropriate species, they do not yield directly the concentration profiles of the diffusant. In FRAPP, fluorophores in a sample are bleached in a spatially periodic pattern by an intense laser beam. The recovery of the fluorescence in the bleached region is monitored by a less intense beam as unbleached molecules diffuse in that region. The time

262

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dependence of this process yields D . The concept of FRS is similar to FRAPP except that a photochromatic dye is used in place of the fluorophore . In this technique two intersecting laser beams produce spatially periodic regions of high and low intensity to create a grid in the sample, a result of the photoisomerization (a reversible process). The time dependence of the decay of intensity of an incident laser beam diffracted from the grating yields D. NMR, unlike the other techniques, does not depend on labeling or on concentration gradients. It depends on the local environment of a nucleus in the presence of an inhomogeneous magnetic field. Pulsed field gradient NMR has been used for the past 30 years to measure diffusion coefficients. Here a sequence of radio-frequency (rf) pulses results in a spin echo of a given amplitude. If the nuclei in the region of the sample into which the magnetic field is applied are undergoing a diffusion process, then the amplitude is attenuated. The approach for extracting the diffusivities exploits this process. Traditionally, this technique has been capable of measuring very fast diffusivities, 10- 5 _ 10- 9 cm2/s in polymers. Recent refinements of the technique may enable one to determine diffusivities as slow as 10- 15 cm2/s (Fleisher, 1992). There are a number of other techniques that have been used to study diffusion in polymers that we will not discuss here. These include conventional transmission Fourier transform infrared spectroscopy (High et aI., 1992), scanning electron microscopy X-ray fluorescence (Gilmore et aI., 1980), infrared attenuated total reflectance (IRATR) spectroscopy (Van Alsten and Lustig, 1992), dynamic light scattering (Murschall et aI. , 1986), and a modified optical schlieren technique (MOST) (Composto et aI., 1990). It is very clear from the foregoing discussion that the choice of technique will depend on the range of diffusion coefficients to be determined, the availability of suitable labels, the depth resolution of the experiment, and the availability of equipment. Other considerations include the ease with which samples can be made. One very important concern is the effect of polydispersity on the result of the actual measurement. The importance of this effect varies among these techniques. For example, the effect of polydispersity on the diffusivity in the NMR experiments may result in an increase in the measured diffusivity by as much as a factor of 5 (Fleischer, 1985) over that obtained using ERD or SANS.

2.

Molecular Weight Dependence of Diffusion

While the earliest known measurements of diffusion of flexible linear chains in melts were performed by Bueche and coworkers (Bueche, 1968; Bueche et aI., 1956) using a radioisotope labeling technique, the first systematic series of measurements designed to test the predictions of reptation were performed by Jacob Klein at the Cavendish Laboratories during the late 1970s (Klein, 1978; Klein and Briscoe, 1979). These measurements were performed on polyethylene (PE)

TRANSLATIONAL DYNAMICS IN MELTS

263

using infrared microdensitometry. Figure 5a shows a double logarithmic plot of the self-diffusion of polyethylene from different authors. While virtually all the published data for polyetheylene follow the M - 2 dependence predicted by reptation (Bartels et aJ., 1984; Kimmich and Bachus, 1982; Bachus and Kimmich, 1983; Fleisher, 1984, 1985, 1987; McCall et aJ., 1959; Klein and Briscoe, 1979; Klein et aI., 1983; Peterlin, 1983; Zupancic et aI., 1985), the actual magnitudes differ in some cases. The discrepancy is associated with the NMR data. Some of the NMR data, like those of Pearson et aI. (1987), appear to be higher than the other sets of data determined using SANS (Bartels et aI., 1984) and IRM (Klein, 1978). It is evidently not related to the technique, as the other sets of NMR data are in agreement with those data obtained using SANS and IRM after they were corrected for polydispersity. The data of Pearson et aJ. were also corrected, but the discrepancy still remains. The difference is therefore not related to polydispersity or to the techniques. One strong possibility if that some of the measurements were done on hydrogenated or deuterated 1,4-polybutadiene, which have some minor microstructural differences that differentiate them from PE (inclusion of some 1,2-polybutadiene). The different sets of data on polystyrene also exhibit an M - 2 dependence (Bueche, 1957; Kumugai et aI., 1979; Kimmich and Bachus, 1982; Bachus and Kimmich, 1983; Green et aI., 1985, 1986; Green and Doyle, 1990; Antonietti et ai., 1984, 1987). The molecular weight dependence of the D* values of PS is shown in Fig. 5b. The agreement between the different sets of data is much better than that for polyethylene. There are additional results for other pure homopolymer systems, such as poly(methyl methacrylate) (PMMA) (Green et aI. , 1988, 1989), that also show M - 2 dependencies for other pure homopolymer systems (Fig. 5c). There should be a word of caution about comparing D* and Ds. The M - 2 dependence is strictly valid in the case of a tracer chain diffusing into a host of sufficiently high molecular weight. In the case of self-diffusion one measures the diffusion of chains of molecular weight M into a host of chains of identical molecular weight. There is a regime of M where D . necessarily becomes larger than the D* of a chain in a matrix of high molecular weight. This is attributed to the constraint release or tube renewal effects (Graessley, 1982; Klein, 1986), which are discussed in a later section. We can now comment of the molecular weight dependence of D* in the molecular weight regime where M < Me. Data on PE exhibit an M - 1 dependence (Pearson et al., 1986), as expected from the Rouse model. This result is realized after free-volume corrections have been made. Recent NMR experiments of selfdiffusion in polydimethylsiloxane (PDMS) show that for M < Me, D * - M - l, which is also consistent with the Rouse prediction (Crosgrove et aI., 1992). There remain some noteworthy observations that are disturbing. In PDMS (Crosgrove et aI., 1992), the molecular weight regime where M > Me, it was determined that D - M - a , where a is appreciably less than 2. Another note-

264

-

.t il

(\IE

GREEN

10- 8

•

u

•

C

~

O.

10- 10

0 0

° 10- 12 2

10

(a)

3

10

4

10

105

6

10

M

Figure 5 (a) A plot showing the molecular weight dependence of tracer diffusion and self-diffusion in polyethylene melts. The circles represent the SANS data (self-diffusion); the triangles represent the IRM (tracer diffusion) results, and the solid straight line is a representation of the data of Pearson et al. using NMR (self-diffusion).

worthy discrepancy is seen in polyisoprene (PI), where the apparent exponent is 3 and not the expected value of 2. These discrepancies remain unresolved (Yu, private communication). Predictions of the magnitude of D based on Eq. (17) are in reasonable agreement with the published diffusion data. In the case of PS, at 174°C, the predicted 2 pre factor is Do = 0.006 whereas the measured prefactor is 0.0075 cm /s [G~ = 16 2 6 2 X 10 dyn/cm2 (Ferry, 1980); (R2) = 1 X 10- cm , p = 0.96 g/cm3 , Me = 31,000, and TJo(MJ = 1200 (Fox and Berry, 1968)]. In the case of polyethylene, at 176°C, the experimental value of the prefactor ranges from 0.32 cm2/s (SANS) to 0.82 cm2/s (IRM), while the calculated value is 0.34 cm 2/s [G~ = 2 X 107 dyn/cmZ, (R2) = 1 X 10- 16 cm2 , p = 0.767 g/cm3 , Me = 3800, and TJo(Me) = 0.3 (Bartels et aI., 1984)]. At 175°C, Pearson et ai. found from the NMR experiments that Do = 1.25 cm2/s. Having discussed tracer diffusion and self-diffusion in pure homopolymer systems, we can now address tracer diffusion of chains into a host composed of two dissimilar yet compatible polymers. lWo systems have been investigated.

TRANSLATIONAL DYNAMICS IN MELTS

265

•

10 (b)

5

10

M

•

O*(RBS)

0

O*(ERO)

&

O*(FRS)

X

0* (NRA)

6

c

(c)

M

Figure 5 Continued (b) FRES measurement of d-PMMA of molecular weight M into high molecular weight h-PMMA (data of Green). (c) Dependence of D* on M for polystyrene using different techniques. [FRS data of Antonietti et al. (1987); RBS data of Green et al. (1984); ERD data of Mills et al. (1984).]

266

GREEN

Green et al. studied the diffusion of d-PS into compatible mixtures of PS and poly(vinyl methyl ether) (PVME) of varying composition and determined that the D* for the d-PS chains varied as M - 2 (Fig. 6) (Green, 1991; Green et al., 1991). However, the actual magnitude of D* varied with composition and exhibited a minimum at approximately the 50% composition regime. Figure 7 shows the composition dependence of D * for d-PS diffusing into the PS/PVME system at a constant temperature above the glass transition temperature of each blend (T - Tg = 100°C). The variation is considerable and is not fully understood at this point. Composto et al. (1990) studied the tracer diffusion of d-PS and of deuterated poJy(xyJenyJ ether) (d-PXE) chains into mixtures of PS and PXE of varying composition. Like the PS/PVME system, D * of the d-PS and of the dPXE chains diffusing into the mixtures varied as M - 2 • These values also exhibit a strong compositional dependence. What is interesting abuot the composition dependence of D * in both systems is that it is highly nonlinear. It is clear nevertheless that the relaxation times of the chains in the blend are considerably slower than in the pure components. Recent NMR measurements of the relaxation of the components in PS/pVME and PSIPXE blends show that the seg-

10. 12

10. 13

•

O*(12% .128°C)

o

O(40%.107°C)

"

O(80%.80°C}

it

C 10. 14

M Figure 6 M dependence for D * of d-PS diffusing into PS/pVME blends of 40% PVME at 107°C (0), 12% PVME at 128°C (_) and 80% PVME at 80°C (A). [Data of Green et al. (1991).]

267

TRANSLATIONAL DYNAMICS IN MELTS

10. 11

-

•

10. 12

t/)

ce---

(.)

•

-II

0

10. ,3

• • •

10. 14 ·0.2

0

0.2

0.4

0 .6

0 .8

q> Figure 7 Composition dependence of the tracer diffusion coefficient, D*, of d-PS chains diffusing into miscible blends if PS and PVME are at temperatures T - T. = 100°C. [Data of Green et aL (1991).]

mental mobilities of the components in the blends are very slow in comparison to mobilities in the pure components (Jones et aL, 1993). In addition, recent measurements of the relaxation of the components of a PMMNpolyethylene oxide (PEO) mixture using an approach that simultaneously determines the infrared dichroism and the birefringence also support these findings (Zawada et aI., 1992). Dichroism measurements by Monerrie and coworkers also support these findings (Lefebvre et aL, 1984; Faivre et aI., 1985). Therefore it is clear that in a compatible mixture the components relax, or undergo translational diffusion, at a rate that is considerably slower than in the mixture. Further, the relaxation varies nonlinearly with the average composition of the blend. There is yet to be a clear explanation of this finding. It could be due, in part, to the composition-dependent interactions between the unlike segments in the mixture.

3. Temperature Dependence It is very clear from Eq. (17) that knowledge of the temperature dependence of 1'\0 should be sufficient to determine the temperature dependence of DRop' The temperature dependence of DROp can be determined in a straightforward manner. Equation (17) can be rearranged to yield 10g(DRep/T)

= C(M)

- log

1]0

(19)

268

GREEN

The relation that G ~) = pRTIMe was substituted into Eq. (17) (Graessley, 1980). The temperature dependence of Tlo is well described by the Vogel-Fulcher equation log Tlo =A

+ B(T

- Tofl

(20)

where To and B are the Vogel-Fulcher constants, unique to each polymer (Berry and Fox, 1968). It follows that the temperature dependence of DReplT should be given by log [( DRep(T) ) (TIc!) ] DRciTref) T

=A'

- B(T - Tof

1

(21)

where A' = B(Tref - Tor J. In this equation, Trer is the reference temperature at which DRep(Trcr) is determined. This result suggests that log TID * should have the same temperature dependence as log Tlo. It turns out that the experimental data in polystyrene (Green and Kramer, 1986a,b) and in poly isoprene (Nemoto et aI., 1984) are consistent with this. Figure 8 shows the temperature dependence of the diffusion of d-PS into PS. It also exhibits the same temperature dependence as that of the inverse

--

I-

•

•

c

• 1 0. 1 5

L...L.........................................-'-'...................1-.L.........................................._

5

6

7

8

9

10

B/{T .. TJ Figure 8 Temperature dependence of the D* of d-PS of M = 110,000 into PS. The empirical parameters used for the fit are B = 710°C, To = 49°C, Trer = 170°C. These are the same constants used to fit the PS viscosity data. [Data replotted from Green and Kramer (1986).]

TRANSLATIONAL DYNAMICS IN MELTS

269

of the viscosity of PS (Green and Kramer, 1986a,b). Another system where a similar observation is made is in PMMA. It should be pointed out that the temperature dependence of log (T/D*) is different from that of log 'T]o in polybutadiene, as shown by Bartels et al. (1984). In polybutadiene, the temperature dependence of the melt viscosity is 'T]o -

exp(EvlRT)

(22)

where the activation energy for flow is 30 kllmol. While independent of M, as expected, it is 8 kllmol higher than that for diffusion. As argued by Bartels et al., this can be rationalized in terms of Eqs. (17) and (12). The temperature dependence of DIT is controlled primarily by the temperature coefficient of the friction factor ~. It turns out that in PS, PI, and PMMA, the temperature dependence of ~ varies by orders of magnitude over a temperature range of lOO°e. The other temperature-dependent factors in Eq. (17), (R2) and the density, vary by only a few percent over the same range. In the case of PBD, however, the temperature dependence of ~ is not very strong; therefore, that of the density and that of (R2) become important. These factors account for the difference in temperature coefficient. An alternative explanation has been suggested (McKenna et al., 1985) for the difference in temperature coefficient. This argument is based on the fact that the entanglement interactions between diffusion and viscosity are different. This is related to the observation that the time dependence of G(t) (Fig. 1), the stress relaxation modulus, is not a simple exponential in time; instead it is described by a stretched exponential [G(t) - exp -(t/1Y -"] (Ngai et al., 1988). The value of the coupling constant n for the viscosity is different from that for diffusion. The data for PS, PI, and PMMA, however, appear to refute this alternative explanation.

D.

Tube Length Fluctuations and Constraint Release

There is considerable experimental evidence in support of the l f2 power law scaling behavior for the diffusion of a long flexible polymer chain in a host environment composed of sufficiently long chains. In addition, measurements of the magnitude of DROp and of G ~) are in reasonable agreement with predictions of reptation. However, there exist a number of inconsistencies between predictions of the pure reptation model and experiments. One discrepancy of great concern is the prediction that 'T]o - M3 while experiments show that 'T]o actually scales as M 3 .4 ;;;O.2 (Ferry, 1980; Berry and Fox, 1968). Furthermore, while the plateau of G(t) [G(t) = GN(O)I\J(t)] is correctly predicted, the prediction that the product of the steady-state compliance 1.(0) and G~O) is (23) is not correct. Experiments show that the product is between 2 and 3. As we mentioned earlier, the idea of treating the chain as diffusing into a fixed tube is

270

GREEN

not strictly valid in many situations. One can imagine a situation in which a long flexible chain diffuses in an environment where the host chains relax at a faster rate than the probe chain. In that case, Eq. (14) would no longer be valid. Figure 9 shows a schematic of the topology of the tube being altered on time scales smaller than the 'Td of the N-mer chain. It is clear from this figure that upon removal of constraints on the N-mer chain, the chain can undergo a lateral displacement, which is otherwise prohibited in the simple reptation picture. One might view this, at one level, as the diffusion of the tube, as the primitive path is altered in the process. Clearly, however, the effect of the mobility of the host environment is to enhance the mobility of the N-mer chain. 1. Constraint Release One of the first attempts to rectify the situation was made by Daoud and deGennes (1979). They noted that in the limit where the molecular weight of the diffusant is very high, the diffusant should diffuse as a coil in an environment whose viscosity should vary as 110 - p 3 (P > Pc), where P is the degree of polymerization of the host chains. Consequently, there should be a correction of DRop , which varies in a manner that may be described by the Stokes-Einstein relation. This correction is predicted to vary as D CR - N 1!2p - 3 • There have been a variety of other theories developed to understand the effect of the matrix on diffusion (Klein, 1986; Wantanabe and Tirrell, 1989; Viovy, 1985; Graessley, 1984; des Cloizeaux, 1988a,b, 1990, 1992; Rubenstein et aI. , 1987; Rubenstein

Figure 9 The configuration of the chain changes as the topology of its environment changes when a constraint is released.

TRANSLATIONAL DYNAMICS IN MELTS

271

and Colby, 1988; Doi et aI. , 1987). With the exception of Klein 's work, these theories address the question of viscoelasticity primarily. Most of the constraint release theories suggest that the correction to D Rcp varies as DCR - P- 3 . Klein, on the contrary, argued that the dependence is somewhat stronger, DCR _ p - 512 • Klein's argument is based on the fact that a single chain may provide more than one constraint on the N-mer chain. Graessley's theory, for example, as discussed later, assumes that each P-mer chain accounts for one constraint. The effect of the interdependence of the constraints is to enhance the constraint release (tube renewal) contribution; hence DCR - p - 512 • Hess (1988b) also addressed the question of constraint release in melts using a many-body approach and suggested that DCR - P- J • It turns out that there are no experimental data on the effect of constraint release of the diffusion of linear chains that support Hess 's finding. Below we describe Graessley 's contribution. The basic idea is that each chain in the system is undergoing a reptative motion with a characteristic reptation time given by 'Td(M) for the probe chain and 'TiP) for the matrix chains. Both the reptation and the constraint release processes are assumed to be independent; therefore, the total diffusion coefficient of the N-mer chain is now

D* = D Rcp

+ DCR

(24)

where D CR , as described above, is the contribution of the host environment. It has been argued (Wantanabe and Tirrell, 1991) that the assumption that the two processes are independent is valid only if the conformation of the tube and that of the chain trapped in it remain Gaussian during each successive step. The constraints in this model are considered in an idealized manner where the diffusion of the N-mer chain occurs on a cubic lattice with z effective constraints (the removal of an arbitrary constraint will not necessarily contribute to alteration of the primitive path) per step along the primitive path. Constraints should relax at a rate proportional to 'Td • The mean waiting time for the release of the first of the z constraints is defined as

'Tw = [

[yet)]' dt

(25)

According to Graessley, the contribution of constraint release to the diffusion coefficient of the chain is (26)

In the case of a single chain diffusing into a single-component monodisperse host, the mean waiting time takes on a value of'Tw = (1T2/12)''Td.

272

GREEN

Note that an approximation is made whereby ~(t) is represented by the first, and by far the most dominant, term in the series. It follows that the complete diffusion coefficient for a linear flexible chain of molecular weight M diffusing into a monodisperse host environment of molecular weight Mp is given by

D,

=Do(M -2 + nCRM~M/M!)

(27)

where Do was defined earlier as Do = (4/15)MoM.kBT/~. It is clear from this result that the correction to the simple reptation prediction becomes less significant as Mp increases. It is interesting to note that in the case of self-diffusion (M = P), D. has a slightly higher magnitude that D* , which becomes significant, particularly at lower M. At sufficiently high M, D * = Ds. Green and coworkers have shown that Eq. (27) provides a very good description of the diffusion in d-PS chains into PS hosts of varying molecular weights. Shown in Fig. lOa are data that have been fit with Eq. (27). Only one adjustable parameter has been used, n CR , which was found to have a constant value of 11. Forward recoil spectrometry measurements by Green et al. (1984) on PMMA melts are also well described by Eq. (27) using a value of n CR =11 (Fig. lOb). IRD measurements in polyethylene (Von Seggren, 1991) also indicate that Eq. (27) provides a good description of the data. It was, however, found

10-11

10-12

"*~

10-13

10-14

~

0,

~ 'al ~~ \~\

oo -

0_

o

.-

~:;-.-.

~'-o-o

0-

--

\!'-----

T~ \ t::.-t::._t::.

t::.-

.....

10-15 104

(a)

'-."'-~ 108

105

p

Figure 10 (a) Data showing the effects of constraint release of d-PS in PS at 170°C. The lines drawn through the data were computed using Eq. (28). (0) M = 55,000; (e ) M = 110,000; (0 ) M =255,000; (_) M =520,000; (D.) M =915,000; (£.) M = 1,800,000. [Data of Green et al. (1984).1

TRANSLATIONAL DYNAMICS IN MELTS

273

that although O!CR was the only adjustable parameter, its value varied with the molecular weight of the d-PS diffusant. Studies of the polypropylene system did not provide strong evidence for the reliability of Eq. (27) (Smith, 1982; Smith et aI., 1984), but other studies supported this prediction (Tead and Kramer, 1988; Antonietti and Sillescu, 1986). The effects of constraint release on the tracer diffusion of a homopolymer into miscible blends have been investigated in PS/pVME (Green, 1991) and in PS/pXE (Composto et aI., 1992) systems. The results were found to be well described by a generalized form of the equation

D* DR(q»

-- = 1

kMM;(2) + ------,......:..-'-'------:[~q>

+ (1 - q>)t['vq> + (1 - q»]P3

(28)

that accounts for a host of two components, 1 and 2. In this equation, 'Y = T(1)/ T(2) and ~ = [Mc(1)/M.(2)] l!2. In the absence of the second component (i.e,. q> = 0), Eq. (28) reverts to Eq. (27). The data in Fig. 11 represents the diffusion of d-PS of M = 200,000 (circles) and of M = 520,000 (squares) into a blends of PS of molecular weight P = 1.8 X 106 with 40% PVME, where the PVME

..

10 1

c.

~

*~

10 0

10.ll~0n.4,---L-~--L-~~~lLOI5----~--~~-L~~1~0' ~

p

Figure 10 Continued (b) Constraint release data of d-PMMA (M PMMA of molecular weight P at 180a C. [Data of Green et al. (1984).]

= 519,000)

into

274

GREEN 10-12

10-13

C

10-14

1 0-16 L--J.---&--'-J....&...u..aJ'---""--'--L-L.u..u.L_--'--'-..J....J~LI.I

104

105

p

106

107

Figure 11 Data showing the constraint release of d-PS [Ce) M = 200,000; C-) M = 520,000] into miscible blends of PS of molecular weight P with 40% PVME of fixed molecular weight P = 145,000_ [Data of Green et al _ (1991).]

molecular weight was fixed at M = 145,000. The broken line was calculated [Eq. (28)] , with the constants 'P = 0.4 and the molecular weight between entanglements of PS taken to be Mc(l) = 18,000 and that of PVME, Mc(2) = 12,000_ The constant k was taken to be equal to ClCR = 11. While the values of k were found to be consistent in the PS/PVME system, they were found by Composto et al. (1992) to vary considerably with composition in the PS/PXE system. A resolution of this situation will await further experiments and theory.

2. Tube Length Fluctuations In addition to the constraint release process, the N-mer chain is capable of undergoing other relaxation processes not described by the original reptation model such as fluctuations in the length, L, of the primitive path. It has been suggested that these fluctuations in L might account for the discrepancy between the experimental and predicted power law dependence of 1]0 (Doi, 1983; Doi and Edwards, 1986). Doi has shown that the average fluctuations

(M} )/(L)

= Z - l12

(29)

Recall that Z is the number of steps on the primitive path (Z = L/a). Considering that 1]0 is proportional to the longest relaxation time, one might incorporate the fluctuations in length in calculating the new relaxation time. According to Doi, (30)

TRANSLATIONAL DYNAMICS IN MELTS

275

where k is a numerical constant that is close to unity. This result is obtained by noting that 'Td = L 2/DRo and 'Td(F) = (L - !:::.L)2/DRO. Based on the result in Eq. (30), the new 110, which incorporates the chain length fluctuations, is (31)

where k' is a new constant. It is important to point out that Eq. (30) approaches the reptation prediction at extremely large values of M. At smaller values of M, these corrections are important. Experimentally, Colby et al. (1987) have shown that 110 - N 3 .4 for MIMe as great as 150; beyond MIMe - 200, departures from the N 3 .4 dependence that are consistent with N 3 are observed. While the data of Colby et al. suggest that in the limit of very large MIMe the viscosity should approach M 3 , there is a suggestion that the tube length fluctuations may not fully account for the discrepancy (O'Connor and Ball, 1992). This is due, in part, to the large error associated with measurement of the viscosity of the very high M polymers. O' Connor and Ball (1992) later revisited this problem. As mentioned earlier, the Doi-Edwards model considers the chain diffusing along the tube in accordance with one-dimensional Rouse behavior. Only a single diffusion coefficient, D RQ , and only one Rouse mode, the longest relaxation time, are considered to be relevant. Furthermore, fluctuations in chain length are also ignored. O'Connor and Ball recognized that any description of the dynamics of the chain should incorporate the full Rouse relaxation behavior, particularly of the chain ends. This is important because the release of stresses in the tube occurs at the current tail (defined by the direction of motion of the chain) of the chain whereas that due to the head of the chain is not important, provided there are no major fluctuations in contour length. By carefully rescaling the relaxation times and all the length scales in the problem, they were able to express the positions of the chain ends in terms of independent coordinates. Consequently, the behavior of the chain ends could be described in terms of independent Rouse modes. With the use of only two material-dependent parameters, which are easily measured, G ~) and the monomeric friction factor ~, O' Connor and Ball (1992) showed, through a computer simulation, that the outstanding discrepancies (110M, J~O) G ~) prdiction, etc.) in dependence could be accounted for. One of the important results of this work is that the chain contour fluctuations described by Doi and Edwards (1986) could not account for the viscosity power law discrepancy. In fact, O'Connor and Ball demonstrated that by combining the constraint release effects with their corrections they could account for the dependence of the viscosity on complete magnitude and molecular weight, entangled and unentangled. The M 3.4 power law dependence was accounted for in a number of polymer systems; at very large M the M 3 power law dependence is recovered. Furthermore, the discrepancy in the value of J ~O) G ~) was also resolved. It is my opinion that this is not the end of the story! There are details of the

276

GREEN

simulation that are unknown because they were not published. The diffusion of ring and star molecules is discussed below.

III.

DIFFUSION OF CHAINS OF DIFFERING ARCHITECTURES

A.

Branched Molecules

Branched molecules, in the presence of fixed objects, are unable to undergo a strict reptation process. Their motion is facilitated primarily through fluctuations in contour length. In a host of linear reptating chains, the constraint release process is expected to play an important role, more important than in the diffusion of linear chains. The question of the diffusion of star-shaped molecules was first addressed by deGennes (1979) and subsequently by a number of other authors (Graessley, 1982; Helfand and Pearson, 1983; Klein, 1986; Pearson and Helfand, 1984; Doi and Kunuzu, 1980). For the sake of clarity we begin with the motion of a star molecule of f = 3 arms, the schematic of which is shown in Fig. 12. The star undergoes translation in the presence of fixed obstacles. It follows that translational motion of the star can occur only by fluctuations in arm length Ls. For the star in Fig. 12a to move a distance a, a step on the primitive path, arm 1 must retract to the node without crossing any obstacles. We can calculate the relaxation time, T., for such a process. Doi argued that if the probability distribution of a chain of N. segments, Ls - (L s), is Gaussian, then the motion of L. can be considered to be Brownian and occurring within a harmonic potential

U(L s) =

Nsb2 (Ls (23)(kBT)

(32)

(Ls» 2

If one considers this an activated process, then the disengagement time associated with the chain end going from a point (Ls) to Ls = 0, as shown in Fig. 12b,

(a)

-

(b)

Figure 12 Schematic depicting the mechanism of diffusion of a star molecule.

TRANSLATIONAL DYNAMICS IN MELTS

277

is given by (33)

Ts "" To exp[(3/2)Ns(b/an

An alternative manner in which one might arrive at the same result, as shown by deGennes, is to consider the probability of an arm of Ns segments retracting along its own contour without enclosing any obstacles. Such a probability is P(Ns) ex exp( - "{Ns)

(34)

where "{ is a constant that depends on M •. The rate at which the arm retracts, deGennes argues, should be given by (35) Therefore, (36)

Ts ex Td(N.) exp("{Ns) There have been a number of predictions for the form of Eq. (36) where Ts - N ! exp( -"{Ns)

(37)

The exponent k has been assigned values of 3 (deGennes, 1979; Doi and Kunuzu, 1980); 0 (Graessley, 1982); 3/2 (pearson and Helfand, 1983); and 1.9 ± 0.1 (Needs and Edwards, 1983). When the arm completely retracts (Fig. 12c), the center of mass of the chain diffuses a distance equivalent to the primitive path. During this process, the chain is forced to drag the other two arms that same distance. Since the diffusion coefficient is defined as (38) Eq. (38) predicts that 2

D"or "" a To

exp[-3(~)2] 2N. a

(39)

The foregoing result is for a three-arm star. Doi pointed out that as the star diffuses, it needs to withdraw f - 2 arms to the branching point. The activation energy for such a process is

U' =

~ (f -

2)kB TNs

GY

(L s - (L s)?

(40)

Therefore the general diffusion coefficient is D sta ,

""

a

2

To exp

[-3"2"

(b)2]

(f- 2)Ns ~

(41)

GREEN

278

In general, one might write the diffusion coefficient for an f-arm star as (42) where C h C, and C3 are constants. We may now consider the situation in which the star chains diffuse into a matrix of linear flexible chains of molecular weight Mp. The constraint release process will play a significant role in the translational dynamics of this system. Under these conditions the total diffusion coefficient is (43) where D CR is given by Eq. (25). Few measurements have been done in this area (Kline et ai., 1983; Bartels et ai., 1986; Antonietti and Sillescu, 1986; Shull et ai., 1988; Crist et aI., 1989; Fleischer, 1985). Experimentally it is well established that star molecules diffuse much more slowly than linear chains of comparable dimensions and that the diffusion rate of the stars depends exponentially on the length of the arm. The actual molecular weight dependence of the pre exponential is less certain at this point. Many of the data appear to be well described by assuming that the pre exponential factor is independent of molecular weight. Shull showed that D star varies exponentially with the molecular weight per arm for stars of different lengths diffusing into rnicrogel matrices (Fig. 13). These micro gels relax suffi-

10. 10

10. 11

4<

C

10. 12

1 0. 13

10.14 '--_--'_ _---1._ _...J..._ _- ' -_ _

o

10000 20000 30000 40000 50000

M

a

Figure 13 Arm length dependence of the D* of a four-arm star molecule. The data are exponential as expected. [Data of Shull et al. (1988).]

TRANSLATIONAL DYNAMICS IN MELTS

279

ciently slowly that they simulate the behavior of linear chains of infinite length. Consequently the arm retraction mechanism is significant. Bartells et al. and Klein et al. also demonstrated this exponential dependence of the diffusivity of PE star molecules diffusing into high molecular weight matrices. The constraint release mechanism is also shown to be very important in facilitating motion in linear matrices of sufficiently short length. As one might anticipate, this release process plays a considerably more significant role in the translational diffusivity of star-shaped molecules. Shown in Fig. 14 is the dependence of the diffusivity of star-branched molecules on the molecular weight of the chains in the matrix. The data are well described by Eq. (43), where the constraint release contribution is described by (Klein, 1985) (44) 3

This is a stronger dependece on P than the expected P - • This discrepancy might be related to the interdependence of the constraints discussed by Klein earlier.

B.

Ring Molecules

As pointed out by Kline (1985), ring molecules can assume a number of conformations as shown in Fig. 15. In Fig. 15a the ring can enclose obstacles, in which case it will be trapped and be able to undergo translational motion, unless, 10 ,--------------,--------------,

II!

N

Q.

C

III

a:

III

0

III

III

III III

•

~

0.1

Figure 14 Data showing the effect of constraint release on the diffusion of a star molecule of M . = 60,000. DCR varies as p -2.S, as shown. [Replotted data of Shull et aJ. (1988).]

280

GREEN

of course, by some constraint release process. There are, however, two untrapped conformations shown in Figs. 15b (ramified configuration) and 15c (linear configuration). The probability that the ring molecule will not be trapped by the chains in the matrix depends on the size of the molecule and varies as e - conS! n R, where NR is the number of segments that compose the molecule. It is therefore expected that a fraction of the chains will be trapped. If the ring is to undergo translational motion, it has to be of the linear configuration as shown in Fig. 15c. Therefore the diffusion coefficient of a ring molecule is (45) Rubenstein (1987) made a similar prediction. If the chain diffuses into a host environment of linear chains of molecular weight M p , then the total diffusion coefficient is (46) Studies of the diffusion of ring polymers into different host environments have been performed in PS (Mills et aI., 1987; Tead et al, . 1992) and in PDMS (Crosgrove et aI., 1992). The experimental situation is far less certain for rings than for stars. The NMR measurements (Crosgrove et aI., 1992) of the selfdiffusion of ring molecules show a molecular weight dependence that is appreciably less than ]V2. The exponent shows no tendency to approach - 2 with increasing N . This is a surprising result. It is possible that the PDMS system is unusual considering that the authors found Ds - N -·, where a < 1, in the regime where an exponent of a = 2 is anticipated. It is also noteworthy that rings of N less than about 20 units diffuse at a rate that is faster than that of linear chains of a comparable number of units. At larger values of N, the rates are comparable. These data are shown in Fig. 16. The PS measurements by Tead et ai. (1992) show clearly that there is a lower limit of the ring diffusion coefficient in linear matrices that was not observed by Mills et ai. in an earlier study. They showed that for large enough host chains D * became independent of host molecular weight, consistent with the predictions of Eq. (25) (Fig. 16). This is reasonably strong evidence in support of a

'[1'

. . . , . .... •

..

' .

"

. '

.. ..

"

..

....

.

..",

'oO

0°

•

.

..

. ....

(a) ' :: . Figure 15

...

. .

.

~ "

.

.

.. .

. . . .

'........ . . '

.. . .. .. .... .. "

.

.....

.

(b)

Possible configurations of ring molecules in a host of linear chains.

TRANSLATIONAL DYNAMICS IN MELTS

10. 12

281

0

O•

...

0

Q 10. 13

• •

0

•

0 0

00

0

Figure 16 Constraint release of ring molecules diffusing into ring matrices (0) and into linear matrices (~ . N dependence of the self-diffusion of PS chains. [Data of Tead et a1. (1992) replotted.]

reptation type of mechanism. An additional point that is worth mentioning is that the diffusivity of linear chains into linear matrices is identical to that into rings of the same molecular weight. The D* of rings into PS microgels of P = 490,000 were found to vary with M with an exponent much larger in magnitude than -2. The data of Teal et at are replotted in Fig. 17 as D *Mri;g versus M to illustrate the large deviation. This result is somewhat of a mystery. It is clear that the diffusivity of ring molecules is an area that awaits further exploration given the limited, and to some degree controversial, data that are currently available. In the section that follows we address the question of interdiffusion in polymers where the diffusivities of the chains become highly dependent of concentration and of the strength of the thermodynamic interactions between them.

IV. INTERDIFFUSION In the foregoing section we showed that the tracer diffusion coefficient of a

single chain into a high molecular weight matrix environment was determined primarily by two parameters, the longest relaxation time, Td (recall Td - '), and the molecular weight. It was also noted that when the chains that compose the matrix are able to relax sufficiently fast, the topology of the tube is altered on time scales faster than the longest relaxation time of the diffusant. Under these

282

GREEN

•

•

•

• •

10. 6

L -____________- L_______________

Figure 17 The D* of d-PS rings into PS microgels of P = 490,000. The data of Teal et al. (1992) are replotted as D*M ri;;g versus M.

circumstances, a correction has to be made to the tracer diffusion coefficient to account for the effect of the host chains. In polymer mixtures the interdiffusion coefficient D is of great practical importance. It influences adhesion and bonding between dissimilar polymers, flow and viscoelastic properties, and phase separation in polymer blends. The interdiffusion coefficient not only depends on Td and N , it is also highly concentration-dependent. Tracer diffusion and self-diffusion coefficients are entropically driven, whereas D is influenced by gradients in chemical potential. The effect of the thermodynamics of the system on D can, within the Flory - Huggins model (Flory, 1953), be characterized by X. In a two-component system the excess free energy of mixing is proportional to XAn, where A is the volume fraction of component A and n is that of component B. It was first pointed out by Flory (1952) that the free energy of mixing per segment can, within a mean field approximation, be described by (47)

283

TRANSLATIONAL DYNAMICS IN MELTS

The first two terms on the right represent the combinational entropy of mixing, and the third represents the enthalpic and noncombinatorial entropy of mixing. It is clear from this equation that the combinatorial entropy of mixing for polymers is extremely low, varying as liN, where N is the degree of polymerization of the chain. This is in contrast to small-molecule systems (recall that N for polymers is typically a few hundred to a few thousand). Therefore it is expected that the mutual diffusion coefficient will be highly influenced by the value of X. When X < 0, mixing is favored and, as we will see later, the magnitude of the mutual diffusion coefficient (the interdiffusion coefficient) is greatly enhanced over the case where X = O. This behavior can be described as a thermodynamic " acceleration" of D . When X > 0, mixing is influenced by the combinatorial entropy such that a system can remain in a region of single-phase stability provided that 0 < X < Xs> where (48) As shown later, D(<\» will undergo a "thermodynamic slowing down" under these conditions. Of course, when X > X.. mixing is not favored. Our discussion on interdiffusion will concentrate on two-component mixtures for simplicity. We may begin by considering two polymer layers, A and B, separated by an interface. Further, we will imagine that the chains are arranged on a quasi-lattice, where each cell occupies a volume n. When the chains diffuse across the interface, an important requirement is that the total flux across the interface must be zero. There is a flux, J A, due to the A chains and another, J B, due to the B chains. It is clear from the early experiments, beginning with the Kirkendall (1948) marker experiments in metallic alloys, that in diffusion experiments of this nature there are other fluxes in addition to the diffusive fluxes, J(species A) and J(species B), present that necessarily compensate in the event that J(species A) J(species B). In the case of metallic alloys, it is a vacancy flow. In these landmark metallic alloy experiments it was demonstrated that the motion of the marker at the original interface of the dissimilar metals provides direct evidence of this vacancy flow. In the case of polymers, one should expect similar behavior. If J A J B and there are no other fluxes present, then a pressure gradient proportional to a gradient in chemical potential will develop at the interface. It is expected that these gradients must be relaxed in the system. For polymers, however, where a lattice does not exist in reality, a vacancy mechanism of the sort that exists in metals would not be appropriate. However, any mechanism that accommodates a bulk flow to release the buildup in pressure should suffice. In this regard we can, as Onsager showed earlier, write down the fluxes as linear combinations of generalized forces (gradients in chemical potential), where the constants of pro-

'*

'*

284

GREEN

portionality are the Onsager (or mobility) coefficients J A = - MA'Vf..LA

(49a)

J B = - MB'Vf..LB

(49b)

J v = MA'Vf..LA + MB'Vf..LB

(49c)

Here we formally refer to the compensating bulk flow as a " vacancy " flow and designate the flux as J v • 'V f..L i are the gradients in chemical potential. We have formally introduced the assumption that 'V f..L v = 0, which guarantees that " vacancies" are at equilibrium everywhere. We have also assumed that the offdiagonal terms MAB = MBA = O. The following expression for the interdiffusion coefficient was derived by Kramer et al. (1984): 1 - D( ( -- MB

+

MB

(Xs - X)

(50)

where XS is defined by Eq. (48). This equation was derived by first obtaining expressions for the chemical potential gradients in terms of gradients in concentration. That was followed by noting that the total flux of segments of A, J A(tot), across a fixed interface must be conserved, i.e.,

JA(tot)

= -MA'Vf..LA + (MA'Vf..LA + MB'Vf..LB)

(51)

and that

(52) Since

a

at

='V[D(]

(53)

Eq. (50) follows. By writing the On sager coefficients in terms of the Rouse segmental mobilities, Kramer et al. (1984) showed how Eq. (50) could be expressed in terms of the tracer diffusion coefficients of species A and B in the mixture. The On sager coefficients can be written in terms of the segmental mobilities, BA and BB' as

(54a) and

(54b) where CA = /0 and CB = (1 - )/0 are the concentrations of species A and B. These relations follow from the fact that the flux is proportional to the dif-

TRANSLATIONAL DYNAMICS IN MELTS

285

fusional velocities, where the constant of proportionality is the concentration. Alternatively, the flux is also proportional to the chemical potential gradient (generalized force), where the Onsager coefficients are the proportionality constants. Equation (50) can be written in terms of the tracer diffusion coefficients by noting that the polymer segment mobilities, BA and BB, can be described by the curvilinear Rouse segment mobilities. Therefore,

D( <\»

= 2<\>(1

- <\> )DT(Xs - X)

(55)

where (56) This result was derived independently by Sillescu (1984, 1987) using a more general approach, where he took advantage of the Hartley-Crank equation. The form of Eq. (56) indicates, for example, that if D! » D~, then D(<\» is controlled by D!, the faster moving species. Therefore, if a Kirkendall-type marker experiment was performed, one would expect that there would be a net flow of species A in the direction of the more slowly diffusing species B and the marker would move in the direction opposite to that which species A (faster) moves. This result has been identified as the "fast" -mode theory of diffusion and has been criticized for reasons discussed below. Long polymer chains are highly entangled and diffuse by reptation, and for this reason it is not natural to define a lattice, as one does in metallic systems. Therefore it is difficult to envision a vacancy mechanism that would alleviate any pressure gradients that might develop. This problem is compounded by the fact that polymers are highly incompressible. Therefore one might imagine a case where for the faster A chain to reptate there must be "free" space ahead of it, and this space can be created only if the slower B chain reptates. Consequently, the diffusive process should be controlled by the more slowly moving species. This led to another proposal, the "slow" -mode theory (Brochard-Wuart et aI., 1983; Brochard et aI., 1984; Brochard-Wuart and deGennes, 1986; Binder, 1983, 1987). One can, in fact, arrive at the prediction for the slow-mode theory by invoking the condition that J v = O. Brochard and coworkers and Binder arrived at the result that the expression for DT is

1 I-<\> <\> -=--+--

DT

D!NA

D~NB

(57)

This equation shows that when D! » D~, the process is dominated by D~, the more slowly diffusing species. As we show later, marker experiments conducted in polystyrene systems are strongly in favor of the fast theory (Green et aI., 1984). A subsequent proposal by these authors suggested that the problem is

286

GREEN

actually a length scale-dependent one whereby, over a length scale

L = [D X 'Td(BW 12

(58)

the more slowly moving species is swelled by the faster diffusing species. This process occurs by the fast-mode process. However, over much longer length scales, or longer times, the diffusion process commences according to the slowmode theory (Brochard-Wyart and deGennes, 1986). We will show that subsequent experiments do not support this view. Further support for the predictions of the fast-mode theory was provided by two additional studies. One study by Jordan et al. addressed the concern of the length scale dependence. IRM was used to determine the concentration profile of polyethylene chains of very different molecular weight, one of 32,000 and the other of 520,000. They were allowed to interdiffuse at distances of macroscopic length scales (on the order of many micrometers). It was determined that the diffusion process was determined by the faster diffusing species, in support of the fast-mode predictions. The concentration profiles were asymmetric, which is consistent with the fact that the faster diffusing species penetrate more deeply into the slower moving species. Concentration profiles were calculated using both predictions, and the results could be rationalized only in terms of the fastmode theory. Composto et ai. (1986, 1987) also addressed this problem but used a different approach. They studied the molecular weight dependence of the mutual diffusion coefficient at a given concentration in the PS/pXE system and determined that D - N;i, a result that can be rationalized only in terms of the fast-mode theory. There have been two noteworthy studies that support the slow theory, the work of Garbella and Wendorf (1986) in PMMNpoly(vinylydine fluoride) (PVDF) and that of Murschall et ai. (1986) in polyphenylmethylsiloxane (PPMS)/PS. As discussed by Composto et ai. (1988), the analysis of the results that enabled them to arrive at their conclusions has been shown to be in error. Therefore, the overwhelming experimental evidence is in support of the original conclusions of the marker experiments of Green and coworkers. The question of thermodynamic slowing down in polymers was first demonstrated by Green and Doyle (1986, 1987), who used ERD to study interdiffusion in d-PS/PS mixtures of sufficiently high molecular weight. It turns out that the replacement of one component in a mixture with its deuterated counterpart can affect the phase equilibrium properties of the mixture. This effect could become important for polymers of sufficiently high molecular weight at finite compositions. The origins of this isotope effect are reasonably well understood (Bates and Wignall, 1986a,b; Bates et aI., 1985). The isotopic substitution results in slight differences in segment volume Vand atomic polarizability IX between the one polymer and its isotopic counterpart. It has been shown that knowledge of IX and V will enable one to calculate x. Because of this unfavorable segmental interaction, the d-PS/PS system exhibits an upper critical solution

TRANSLATIONAL DYNAMICS IN MELTS

287

temperature (VCST) . The stability limit for the system that Green and Doyle studied was Xs(c) = 2.1 X 10- 4 ; the degree of polymerization of the PS chains was 8.7 X 103 and that of d-PS, 9.8 X 103 • Green and Doyle showed that the mutual diffusion coefficient in the d-PS/PS system experienced a minimum, or critical slowing down, in the middle of the concentration regime at the critical composition. As the temperature of the experiment approached the VCST, the degree of slowing down increased. This is shown in Fig. 18. The lines drawn through the data were calculated using the fast-mode prediction [cf. Eq. (50)]. The only fitting parameter was X, whose temperature dependence is well described by

x = 0.22(±0.01)r 1

-

3.2(±0.4)

X

10- 4

(59)

which is in excellent agreement with independent measurements using SANS. The VCST was determined for this system to be approximately 140°C. The PS/pXE system investigated by Composto and coworkers is characterized by X < O. In addition to determining the compositional dependence of interdiffusion, where they showed a thermodynamic " acceleration" of the process (Fig. 19), they also examined the temperature dependence of D. They showed how the value D( = 0.55) changed in relation to D* as a function of

10. 12 ..--_..,--_-._--,,--_-.-_.....,

10 -16 ':-_-'-_-'-_-''-:---::-'--=-_...J o 0.2 0.4 0.6 0.8 LO

Figure 18

" Thermodynamic slowing down" of D in d-PS/PS mixtures. [Data of Green et al. (1987).]

288

GREEN

temperature (Fig. 20). For X < 0, D( <1» > D *, and as the temperature increased and the system approached the lower critical solution temperature (LCST), X > o and D(
v.

DIFFUSION IN BLOCK COPOLYMERS

Thus far the content of this review has been devoted to diffusion of homopolymers of different architectures into homogeneous pure homopolymer or miscible blend host environments. An equally interesting problem is associated with the diffusion of a homopolymer chain or a chain composed of blocks of segments of distinct chemical structure, a block copolymer, into a block copolymer host. Below we address these issues. Block copolymers come in a variety of architectures, which include diblocks, triblocks, and star blocks, shown in Fig. 21. We focus our attention on diblocks, which are composed of two chemically distinct monomers, A and B. The phase behavior of diblocks is determined primarily by the degree of polymerization N , the overall volume fraction of polymer A, and the segmental interaction parameter x. At sufficiently high temperatures and chain lengths, the copolymer

Figure 19 Thermodynamic " acceleration" of D in the PS/PXE system is represented by diamonds. The circles represent the tracer diffusion of d-PS chains in the mixture and the squares that of d-PXE. [Figure reproduced with permission from Composto et al. (1988).]

289

TRANSLATIONAL DYNAMICS IN MELTS

T (K)

600

560

520

480

-13

10

'0 Q)

~ J()14 N

E

u

-I~

-

....... 10

......

~

-16

I-' 10

a -17

10

-18

10

161t'---'----l..---~---'---

4

6

8

10

10001 (T-TCI)), (K- I )

Figure 20 Temperature dependence of diffusion. (0) D; (0) D* of d-PS; (0) D* of d-PXE. [Figure reproduced with permission from Composto et al. (1988).]

/

. ,....

,-

"y~ ~

.

....

(8)

(b)

(c)

Figure 21 The different architectures that block copolymers possess (a) diblocks; (b) triblocks; (c) star blocks.

290

GREEN

is completely disordered; hence the microstructure can be characterized as homogeneous. In this regime the radius of gyration of the copolymers Rg - N 1(2. As the temperature is lowered (Le., X increases) the number of A-B contacts is reduced. Of course, this is accompanied by a loss of combinatorial and conformational entropy. In the limit XN » 1, well-developed ordered phases are formed, and the interfacial region between the phases is narrow. This is the socalled strong segregation limit. Here the chains are stretched and the interdomain spacing varies as N 2I3 • Depending on the value of f, the volume fraction of the A segment, different ordered phases may form . With increasing values of f, the following phases are observed: body-centered cubic arrays of spheres of the minor component in a host of the major component, an ordered array of cylinders of the minor component, an ordered bicontinuous double diamond lattice of the minor component, and finally, when f = 1/2, a lamellar phase is formed. Close to the transition where the system goes from disordered to ordered, the phases are weakly segregated; this is the so-called weak segregation regime. Here the radius of gyration varies as N I12 . At this point the different microstructures still exist, but the interfacial region is comparatively broad. A complete thermodynamic theory was developed by Leibler (1980) that addresses the weak segregation limit of diblock copolymers. Below we discuss the results of a theory developed by Fredrickson and Milner (1990) to address the diffusion of an A-B diblock copolymer of degree of polymerization Nc and volume fraction of A phase, f, into a host composed of an A-B diblock copolymer that forms a lamellar phase. The radius of gyration of the diffusing copolymer is Rg = N~12bI6, where b is the statistical segment length. The A and B blocks are assumed, for simplicity, to have the same segment length, and each monomer occupies a volume b3 • In theory, the lamellae are oriented perpendicular to the z direction, the direction of mobility of the tracer. A number of situations are addressed. 1.

The lamellar spacing of the host is very large in comparison to that of the tracer of f =1= 112. As the tracer diffuses into the copolymer host, the A segment located at a position r experiences a chemical potential j..L(r) = kBTXe(r) while that felt by a B monomer is -j..L(r). The net force that the copolymer experiences from a periodic potential is

F

= 2akJcBTxN(1

- 2f) sin(koz)

(60)

where a is the amplitude of the potential (it is assumed to be much less than unity), and leo is a wavenumber that characterizes the scale of the compositional variation in the melt, which is formally defined as ko = (xol Rl)I12. The parameter Xo (see Fig. 22) is dimensionless and can be varied by changing the molecular weight of the copolymer host or by the addition of homopolymer molecules. As expected, the diffusivity is anisotropic, and

TRANSLATIONAL DYNAMICS IN MELTS

291

a)

b)

c)

Figure 22 Schematic of the diffusing copolymer in different lamellae hosts. (a) The diffusant is small in comparison to the domain size. (b) The diffusant size is comparable to the size of the domain structure. (c) The interdomain spacing is small compared to the dimensions of the diffusant.

the diffusion coefficient is described using a tensor D = Dzzz

+ Dperp('6 - zz)

(61)

where z is a unit vector in the z direction. Dperp is the diffusion coefficient in the direction parallel to the lamellae; henceforth it will be denoted by D 1• The prediction for D z is (62) where 10(0.) is a modified Bessel function and 0. = 2aXN(l - 2f). In the limit of small 0. (0. « 1), a single diffusion coefficient is predicted, (63) and when

0.

» 1 the prediction is

292

GREEN DIDJ = (2/3)[1 +

2.

"IT

lal exp( -2Ial)] + ...

(64)

These predictions are meant to apply even in the limit where the tracer is a homopolymer (f = 1 or 0). The domain spacing is very large (xo « 1), but the tracer is symmetric (f = 1/2). Here the segments of the copolymer experience potentials that are exactly equal and opposite. Therefore the net force on the chain vanishes. The calculation proceeds by using an electrical analogy where the A segment is considered to be negatively charged and the B segments positively charged. The chain is then considered to be in the presence of an electric field. The effective diffusion coefficient is calculated to be (65)

3.

where ~ - (Rg NcaX krY The domain spacing is comparable to or smaller than that of the tracer (xo ~ 1) (Figs. 22b and 22c). In this situation, in particular when Xo > 1, more than one period of the potential is experienced on length scales on the order of the radius of gyration of the tracer as it diffuses. In the calculation the tracer is assumed to behave as a Rouse chain (unentangled). For the case where laxNcl « 1, and for arbitrary values of Xo and f, the diffusion coefficient is predicted to be (66) where K is a dimensionless function of Xo and of f. If the tracer is a homopolymer (f = 0 or 1), then at small xo, K = 1 - xoI3, and at large Xo it crosses over to K = 15.8x~2.

More recently Helfand (1992) considered the case of diffusion in strongly segregated block copolymer hosts. The study, however, concentrated on calculating the activation energy that a copolymer chain must experience as it tries to overcome the force exerted on it as it diffuses through the domains. There were no predictions of diffusion with which one might compare results. Figure 23 shows the schematic of a diblock copolymer attempting to diffuse across a copolymer domain whose dimensions are comparable to its own. It is clear that the segment of the chain that diffuses into the phase with which it is incompatible undergoes considerable stretching during this process. Figure 23a shows the chain initially in a configuration where its A segment is in the A phase and its B segment is in the B phase. The A segment then has to diffuse across the B domain to get to the adjacent A domain (Figs. 23b and 23c). During this process a portion of the A segment is stretched considerably beyond normal conditions as the A segment tries to get completely into the adjacent A phase. This is shown in Figs. 23d and 23e. Figure 23f shows the final configuration.

TRANSLATIONAL DYNAMICS IN MELTS

Figure 23

293

A schematic of a diblock copolymer chain diffusing across a domain of unlike chemical structure.

294

GREEN

Few experiments have been performed that address the diffusion of homopolymer chains or block copolymer chains into copolymer hosts. The most recent is that of Ehlich et al. (1992), where an FRS study of self-diffusion in the PS-PI copolymer system was performed. Their results showed an anisotropy of diffusion. There were not sufficient results to make a true comparison with theory. Shul et al. (1991) used FRES to study self-diffusion in poly(ethylene propylene)/polyethylethylene (pEP-PEE) copolymer in a temperature range above and below the order-disorder transition (OD1). They observed no marked change in the temperature dependence of diffusion at the ODT. They also noted considerable anisotropy in diffusivity. They showed that diffusion perpendicular to the lamellae was orders of magnitude lower than diffusion parallel to the lamellae. It was not possible to make a careful comparison with theory because of a lack of complete understanding of the domain structure of the copolymer host. Green et al. (1989) studied the diffusion of PS and PMMA homopolymer chains into symmetric PS-PMMA diblock copolymer hosts. They showed that for chains whose molecular weight M was equal to or less than half of that of the copolymer host their tracer diffusion coefficient varied as D* = C/M2. C is at least an order of magnitude lower than that measuerd for PS diffusing into PS or of PMMA into PMMA (Fig. 24). This is due, in part, to the fact that the domains of the host are randomly aligned (tortuous), as shown in Fig. 25. The

10·1\

10 -11

10- 13

*~ 10 -14

\ ~.

\'.\-'\\. \0-\~\. -\

10- 15

10- 16

2

3

-\

"-

4

N

Figure 24 Molecular weight dependence of the diffusion of d-PS ioto PS (e ), into a symmetric copolymer of N = 824 (£.), and into a copolymer of N = 3220 (_ ).

295

TRANSLATIONAL DYNAMICS IN MELTS

FREE SURFACE NEAR SURFACE

REGION

BULK

Figure 25

Domain structure of an unannealed copolymer.

temperature dependence of D* for PS diffusing into PS was identical to that of PS diffusing into the copolymer. These data are shown in Fig. 26. The same observation was made for PMMA. From these observations it was concluded that diffusion occurred primarily into the phases where the interactions were favorable. As with the other studies, it was not possible to make quantitative comparisons with theory because of a lack of understanding of the domain structure. Green et al. (1989) also studied the case where the copolymers were annealed for a long period of time to allow the domains to align parallel to the surface so that the chains were forced to diffuse perpendicular to the orientation of the lamellae (Fig. 27). They observed the M - 2 dependence but also observed that the coefficient C ' was appreciably smaller (C ' « C) in this situation than in any of the cases mentioned earlier. This result is not unexpected. Like the other studies, it is not possible to make comparisons with theory because the case addressed by Fredrickson and Milner applies only to short-chain tracers and Green et al. studied the molecular weight dependence of long chains, which are expected to diffuse by reptation. It is clear that considerable more experimental work and theory are needed in this area.

VI.

CONCLUDING REMARKS

While it appears that the first notable attempts to understand translational dynamics of melts began with Bueche in the mid-1950s, the truly significant advances in the area were made by deGennes with his introduction of reptation in 1971 and later by Doi and Edwards in the late 1970s. It is important to reemphasize that reptation is purely phenomenological. It imagines the existence of a chain undergoing curvilinear motion in an imaginary "tube" along a fictitious

296

GREEN - 10

-11

-12

----• t-~

-.

-13

t-

~

•o

-14

-15

-16

-17 4

5

6

7

8

9

10

11

B/(T-T ) o

Figure 26 Temperature dependence of d-PS diffusing into PS (open symbols) and into a copolymer (A).

path, the primitive path. The tube is defined to account for entanglement effects, which in turn were introduced to account for the existence of a rubbery plateau in melts. Despite its phenomenological nature, reptation is of great appeal, primarily because of its simplicity and because of its success at predicting qualitative features about the diffusion and viscoelasticity of melts. Despite the success of reptation, there are some aspects that one might find disturbing. This is, in part, because of its phenomenological nature and because of modifications that involve additional assumptions about the nature of the fictitious tube (constraint release, tube length fluctuations, etc_), which have been made to account for many experimental observations. Considerable computer simulations have been performed to understand the nature of the dynamics of melts and to provide validity for the central assump-

TRANSLATIONAL DYNAMICS IN MELTS

297

FREE SURFACE NEAR SURFACE REGION

BULK

Figure 27 Schematic of the near-surface structure of a copolymer that has been annealed for a longer period of time.

tions of reptation (Kolinski et al., 1986, 1987a,b; Kremer and Binder, 1984; Kremer and Grest, 1990; Muthukumar and Baumgartner, 1989). Although the early Monte Carlo studies of Kolinski et al. found evidence for reptation in the limit where a single chain was allowed to diffuse into a matrix of fixed obstacles, they raised serious doubt about the validity of reptation as a viable mechanism for the dynamics of melts. On the other hand, subsequent molecular dynamics simulations by Kremer and Grest, the first of this type in dense polymer melts, did provide evidence that suggested that the reptation picture may be valid. The work of O'Con~or and Ball suggests a similar conclusion. In addition to reptation, a number of other theories have been proposed. Many of these invoke tube-like concepts and give predictions similar to those of reptation. A recent theory based on the introduction of a generalized Langevin equation to describe the dynamics was developed by Schweizer (1989). The concept of the tube is not introduced, and the effect of the host chains in a melt on the motion of a central chain is accounted for by a generalized dynamic friction function. This function is evaluated using mode-mode coupling ideas that allow one to evaluate correlated collision and feedback processes in dense molecular systems that give rise to confinement-type effects that one might expect in polymer melts. This theory recovers the essen tal reptation predictions in the long-chain limit. It is clear that reptation continues to be the subject of debate. What is indisputable about it is that it is by far the most successful of the theories that have been proposed. The reader is referred to a review by Lodge et al. (1990) that presents a critical evaluation of its consequences. This review has provided an overview of the progress that has been made in diffusion in melts. Earlier reviews of this topic are those of Pearson (1987), Tirrell (1984), Kausch and Tirrell (1989), and Klein (1990). Considerable the-

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oretical and experimental work still needs to be done. There are unresolved questions regarding why anomalous power law exponents are observed in PI and PDMS. Critical tests of theories for diffusion of chains of architectures other than linear chains also need to be pursued. There are also open questions concerning diffusion in miscible blends, particularly the anomalous concentration dependence of tracer diffusion. Our understanding of diffusion in block copolymers is clearly very poor in comparison to that of diffusion in pure homopolymers and blends. In the final analysis, our overall progress in this field will require not only continued activity in the area of theory but also the design of new experiments that will give us further insight into the details of the dynamical process.

ACKNOWLEDGMENT This work was performed at Sandia National Laboratories and supported by the U.S. Department of Energy under contract number DE-AC04-76DP00789.

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Index

Activation energy, 57, 236-238, 240 Activity, 14 Adsorption, 41, 50, 192, 194, 224 Affinity, 241 After-effect function, 6 Aging, 187 Anti-plasticizer, 163 Anisotropy, 10, 11, 24 Averaging methods, 193, 194 Axial dispersion, 200

Binary mixtures, 30 Block copolymers, 167,289-297 Bond angle, 55 Bond length, 55, 58 Branched polymers, 276 Burnett coefficients, 50

Cannonball solid, 39 Capillary model, 216

Case II, 186-189 Cellulose, cuprammonium regenerated, 211, 223 Cellulose regenerated, 211, 220, 223, 233, 237 Chemical potential, 14 Chromatography, 200, 201 capillary column, 201 gas-liquid, 200 gas-solid, 200 Cluster connectivity, 85 shape, 85 size, 85 percolation, 81 Cohen and Turnbull, 146 Configuration, 243 accessible volume, 80 bias, Monte Carlo, 76 random coil, 72 unoccupied volume, 80 Conjugate force-flux, 4

303

304 Connected overlapping spheres, 41 Conservation of species equation, 173 Constraint forces, 56 Constraint release, 58, 270, 278, 280 Correlation function end-to-end vector, 258 length, 48 Critical lower critical solution temperature (LCST), 288 upper critical solution temperature (UCST), 287 Cross-coupling coefficients, 35, 52 Cross-kinetic coefficients, 35, 50 Cross-linked polymers, 163 Crystallinity, 57, 217, 219, 220, 223, 244 Crystallization, polymers, 195, 198, 217, 233, 245 Crystallization, solvent induced, 197

Darken equation, 15 Deborah number, 187-189 Desorption, 180 Devolatilization, 184, 195 Diffraction pattern, prediction, 77 Diffuse reflection, 24 Diffusion amorphous polymer, 67 anomalous, 49, 59 collective, 54 flux, 3, 14, 49, 54 hopping mechanism, 114 Knudsen, 15, 40 low temperature, 122 mechanism, 33, 59 memory-dependent, 186, 194 molecular simulation, 67 molecular view, 71

INDEX [Diffusion] multicomponent, 165 non-Fickian, 49, 184 pure fluids, 14 statistical mechanics, 104 surface, 194 Diffusion coefficient, 242, 244 binary, 106 concentration dependent, 178 Einstein, 107 Fickian, 31, 59 Green-Kubo, 107 interdiffusion coefficient, 255, 282, 285 measurement techniques, 259 mutual, 9, 31 polystyrene/polyvinylmethyl ether, 266, 267, 273 reptation, 258, 259, 267 ring-shaped polymers, 279 self, 32, 55, 57, 107, 111, 149, 259 NMR, 204 polyethylene, 263, 265 predicted, 116 star-shaped polymers, 277 tracer diffusion coefficient, 32, 33, 55, 255, 285 polymethyl methacrylate, 263-265 polystyrene, 265-267 transition state theory, 121 transport, 106 unentangled chain, 254, 255 Dihedral angle, 55 Dispersion coefficient, 201 Dissolution, 213, 214, 224 Dividing surface, 129 Dry molding, 230 Dual-mode sorption, 159, 192 Ductile failure, 192 Dusty gas model, 9

INDEX

Dynamic loss, tangent, 233 Dynamic modulus, 233

Elastic diffuse scattering, 19 Elastic recoil detection, 260 Electrostatic field, 196 Elution times, 200 Ensemble, microcanonical, 16 Equilibrium, local, 4, 9, 14 Equilibrium, mechanical, 8 Exponents critical, 48 scaling, 48 Extended chain, 218, 223

Fiber, 221, 222 Fibril, 223, 224 Fick's law, 174 Finite difference, 18, 181 Crank-Nicholson, 182 forward time central space, 182 semi-implicit, 182 Finite element, 182 Flexibility chain, 57 Flory-Huggins, 150, 255, 285 Fluxes, molecular, 6 Fluctuations, temperature, 6 Foams, polymer, 195 Folded chain, 218, 222, 223 Force balances, 189 Frames reference, 6 Frames, flow, 8 Free volume, 57, 155, 216 Friction factor, monomeric, 251, 254, 258, 269 Fringed micelle, 218, 222 Fujita model, 155

Gibbs-Duhem, 8, 12

305

Glassy polymer, 184 configuration, 70 molecular mechanics, 74 transition, 57, 60, 156, 162, 198, 234-236

Half-time, 174 Heat of sorption, 98 Heat treatment, 245 Henry's law, 98, 176 Hierarchical modeling, 139 Heredity, 187 Hole-filling, 194

Interaction potential hard core, 18 hard sphere, 16 Lennard-Jones, 16, 28, 50 London-van der waal, 26 partic1e-porewall, 18 Interchain rotation, 55 vibration, 55 Interfacial instability, 197 Interpenetrating network, 202 Interpore connectivity, 37 Isotropy, to, 11

Jump balances, 183

Kevlar, 223, 225 Kinetic theory, 4 Kinetic temperature, 29

Lamella, 197, 218, 223 Langmuir adsorption isotherm, 192, 194

306 Laplace transform, 189 Lateral order, 222, 223 Linear response theory, 6, 24 Local twisting motion, 237 Long-time tail, 21, 45, 48, 49 Loop, 223 Loretz gas, 48, 49

Master equation, 124 Maxwell-Boltzmann, 20 Melting, 234 Melt molding, 231 Membranes, 174, 225 desalination, 196 hyperfiltration, 196 reverse osmosis/asymmetric, 196, 197, 225, 226 skin, 196 ul trafiltration, 196 Micro Brownian motion, 233, 235 Microfibril, 223, 224 Micro phase separation, 232 Microscopic reversibility, 124 Microvoids, 192 Mobility, 12 Mode theory fast, 285 slow, 285 Modulus plateau, 252, 268, 269 shear stress relaxation, 252 Molecular dynamics simulations, 12, 28, 40, 55, 62, 75 nonequilibrium, 118 sorption equilibria, 102 unconstrained technique, 57 Molecular model, 72 Molecular simulation amorphous cell, 72 diffusion, 67 mechanics, 74

INDEX [Molecular simulation] scope, 68 sorption, 67 isotherm, 96 Molecular weight critical, 252 Monte Carlo simulation, 40, 76 concerted rotation, 76 configuration bias, 76 Gibbs ensemble, 103 kinetic Monte Carlo, 125 reptation, 76 sorption, 102 Moving boundary problems, 183 Multicomponent diffusion, 165

Nematic, 220 Newton 's laws of motion, 17,56 Nodule, 221 Noncrystalline, 217, 219, 220, 221, 233,245 Nonequilibrium statistical mechanics, 4 Nonequilibrium thermodynamics, 4 Nuclear magnetic resonance, 260, 262 Nuclear reaction analysis, 260 Numerical solution, 181

Onsager coefficients, 283 Onsager reciprocal relationship, 4, 5 Orientation, 223-224, 244, 245 Orientational decorrelation, 88 Orthogonal collocation, 183 Oscillatory, 178, 189

Packing density, 220 Pair distribution function, 77 Partition coefficient, 14, 15, 31, 242

INDEX

Phenomenological coefficients, 5 Plastic deformation, 190 Plasticization molecular view, 71 glass transition, 198 Percolation, 40, 44, 48 Periodic imaging, 19 Permeability, 15, 177, 245, 246 Permeation, 175 coefficient, 242, 243 Poisson process, 124 Polyblends, 201, 247, 248 Polymers aging, 163 amorphous, 54, 197, 220 annealing, 163 branched, 276 foams, 195 materials, 12, 39 molecular engineering, 138 semiatomistic, 58 semi-crystalline, 184, 197 star-shaped, 276 molecular models, 55, 58, 72 validation of, 77 Poly-a-olefine, 217, 237, 239, 240 Polyethylene, 57, 220, 263, 265 Polyethyleneterephthalate, 220, 223, 245, 246 Polymethyl methacrylate, 263-265 Poly-paraphenyleneterephthalamide/ Kevlar,211 Polystyrene, 265-267, 273 Polyvinyl acetate, 151 Polyvinyl methyl ether, 266, 267, 273 Pore, 13, 26, 38, 39 fluid binary, 30 continuum solvent, 37 mixtures, 36 single component, 14

307

Pore density, 227 Pore radius distribution, 227, 228 Porous media, 12 asymmetric membranes, 196, 197, 225, 226 cannonball solid, 39 dusty gas model, 9 interpore connectivity, 37 percolation, 40, 44, 48 Swiss cheese solid, 39 Positron annihilation spectroscopy, 80 Potential energy, 72 Power law, 45, 48 Primary particle, 226, 230 Primitive path, 256

Radio labeling, 260 Radius, mean, 227, 228 cutoff,28 maximum pore, 229 pore distribution, 227, 228 Random media, 37, 38 Random overlapping spheres, 41 Random nonoverlapping connected spheres, 50 Random nonoverlapping spheres, 41 Random walk, 49 Relaxation diffusion time, 257, 271 function, diffusional, 186 molecular, 12 segmental, 89 shear stress modulus, 252 stretched exponential, 89, 194 time, longest, 254 Reptation, 254 Monte Carlo, 76 Ring-shaped molecules, 279 Rouse model, 254, 255

308 Rutherford backscattering spectrometry, 260 relaxation modulus, 252

Scaling theory, 21 Scattering function intermediate, 90 Secondary particle, 226 Segregation limit strong, 290 weak, 290 Self-sharpening, 180 Shear stress, 252 Silica, 50 Simulations, 13, 62 Skin concentration, 190, 191 Slip, flow, 26 diffusive, 36 viscous, 29 Small angle neutron scattering, 260 Smectic, 220 Solubility, 190, 242 Widom test particle, 98 Sorption, 174 differential, 180 heat, 98 dual-mode, 159 Fermi gas, 100 integral, 180 isotherm, 96 molecular dynamics, 102 molecular simulation, 67 Monte Carlo, 102 sigmoidal, 185, 190 statistical mechanics, 91 thermodynarrrics, 91 Specular reflection, 19, 24 Spherulites, 197 Stefan-Maxwell, 9, 31 Strain, 189 Stress, 189

INDEX Stress-strain constitutive equation,

192 Structure multilayer, 225, 226 regularity, 220 Structure factor dynamic, 90 fine, 211, 213, 221 first, 213 fourth, 213, 225 second, 213, 230 third, 214 Swelling, 189, 1944

Takayanagi's model, 214, 218, 219 Tessellation Delaunay, 83 Voronoi, 83, 203 Thermal diffusion ratio, 9 Thermal motion, 233 Thermodynamic acceleration, 283 Thermodynarrric slowing down, 283, 287 Tie molecule, 223 Times c~aracteristic molecular, 87 correlation function, 6, 21 Transition free energy perturbation, 130 Transition state theory of diffusion, 121 macrostate, 124 prediction, 133 transition path, 127 Transition rate constant for jumps, 130 Transport phenomena, 173 Tube, 254 length fluctuation, 274

309

INDEX

Ultracentrifuge, 204

van der Waals, 242 Vector end-to-end, 258 Velocity autocorrelation function , 7, 107 Velocity Verlet, 18 Virus removal filter, 213 Viscosity zero shear, 252, 253, 256, 259, 267, 269 Vogel-Fulcher, 268

Volume accessible, 80 clusters, 81 free, 111, 112 percolation, 81 redistribution, 87, 111 unoccupied, 80

Waiting time distribution, 49, 50 Wave-vector dependence, 6 Wet molding, 230 Williams, Landel and Ferry, 148, 198,235