Diffusion in Polymers

DIFFUSION POLYMERS NEOGI University of Missouri-Rolla Rolla, Missouri

Marcel Dekker, Inc.

New York.

Basel Hong Kong

Library o f 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). 11. Series: Plastics engineering (Marcel Dekker, Inc.) ;32. QD381.9.P45D53 1996 668.94~20

95-51156 CIP

The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special SalesProfessional 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 permission in writing from the publisher. Marcel Dekker, Inc. 270 Madison Avenue, New York, New York 10016 Current printing (last digit): l0987654321 PRINTED IN THE UNITED STATES OF AMERICA

Preface

To most researchers in the area of diffusionin polymers, the 1968 book Diffusion in Polymers by J. Crank and G. S. Park is a very familiar and most appreciated one. 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 withtheadvent of membrane separations, barriermembranes,newneeds to study polymer devolatilization, and on. It is now both out of print and out of date, as is one updateof Polymer Permeabilityedited by J. Comyn. The books, MembraneHandbook edited by W. S. H0 and K K. Sirkar and Polymer Gel Separation Membranesedited by D.R. Paul andY. P. Yampol’skii, stress diffusion only as a precursor to studying separations. Another, D i m i o n 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 firm validation from moleculartheoriesnow exists Fick’s law. Highspeed computers have become available that, in principle, can be used to calculate these difhsivities. In practice the results are few, but presentavery 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 withmechanisms-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 examinesthese aspects and will serve chemical engineers whoare 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.

l? Neogi

Contents

iii

Preface Contributors Chapter

ix

Diffusion in Homogeneous Media J. M. D. MacElroy

I. Introduction Diffusion Fundamentals III. Simulation and Modeling of Diffusion in Fluid/Solid Systems IV. Concluding Comments References II.

1 3 13 62 63

Chapter 2. Molecular Simulations of Sorption and Diffusion in Amorphous Polymers Doros N. Theodorou

I. Introduction Characterization of Structure and Molecular Motion in Amorphous Polymers 111. Prediction of Sorption Thermodynamics IV. Prediction of Diffusivity V. Conclusions and Future Directions References

67

II.

72 91 104 137 139 V

vi

CONTENTS 3.

J. L. Duda and John M. Zielimki I. Introduction

11. Free-Volume Concepts III. Diffusion Above the Glass Transition Temperature

Iv. The Influence of the Glass Transition V. More Complex Systems References 173

l? Neogi I. 11. 111. Iv. V.

Introduction Mathematical Methods Non-Fickian Diffusion Change Phase Multiphase, Multicomponent, and Inhomogeneous Systems VI. Conclusions References 5. 21 1

Sei-ichi Manabe I. 11. 111. IV.

Introduction Structural Characteristics of Polymer Solids Thermal Motion of Polymer Chains in a Solid Correlation Between Chemical Structure, Composition, and Penetrant Transport V. Effects of Fine Structure (Crystallinity, Orientation, etc.) of a Polymer Solid on Permeation Properties VI. Concluding Remarks References 251

Peter E Green I. 11. 111. Iv.

v.

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

vii

CONTENTS VI.

Index

Concluding Remarks References

295

303

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Contributors

The Pennsylvania State University, University Park, Pennsylvania SandiaNationalLaboratories,Albuquerque,

NewMexico

University College Dublin, Dublin, Ireland Fukuoka Women’s University, Fukuoka, Japan University of Missouri-Rolla, Rolla, Missouri University of Patras,Patras,Greece

M. Pennsylvania

Air Products & Chemicals, Inc., Allentown,

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This Page Intentionally Left Blank

I 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 al., 1979; Chase, 1984a,b; Belter et al., 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 posed 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 al., 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 onthe time evolutionof 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 ofthe 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; Stem 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 I1 the flux equations for homogeneous fluids are initially considered with reference to formulations basedonbothnonequilibriumthermodynamics(phenomenological description) andnonequilibrium statistical mechanics (moleculardescription). 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 systemsare presented and their limitations discussed. In Section I11 the novel methodology of molecular simulation, particularly molecular dynamics, and its application to diffusion in fluidholid 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

MOGENEOUS INDIFFUSION

MEDLA

3

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

for The conservation equationfor 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

aPi "at

v

piui

where and ui 2re the local mass density and velocity of component i at r and at time The groupof terms piui is the flux of i relative a stationarylaboratory frame of reference, i.e.,

JI

= piui

(2)

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 this convective flow. The definition one employs for the velocity of the bulk convective motion is largely a matter of convenience, andone of the most common frames of reference is based on the center-of-mass velocity of the fluid at r and at time t which, for a v-component fluid, is given by

Thus, defining the mass diffusion flux of component i relative to the center of mass of the fluid as

JIb

=

- U)

(4)

where the superscript b refers to the barycentric frame of reference, Eq. (1) may be written as

MAcELROY

4

In this equation component i,

~a "_

Dt

at

is the local fluiddensity xi is the mass fraction of and D/Dt is the substantive derivative

+ U - v

Historically, the mass fluxes J[ and

J,!b

and the corresponding molar fluxes

= niui

(6)

= ni(ui - U)

(7)

or

JP

where ni is the local molar concentration of i, have been investigated either phenomenologically via nonequilibriumthermodynamics [see, e.g., Prigogine (1961),deGrootandMazur(1963),andHanley(1969)l or theoretically via kinetic theory [e.g., Hirschfelder et al. (1954), Hanley (1969), and Chapman and Cowling (1970)l. Although kinetic theory can, in principle, provide exactresults 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 controversy for many years. However, developments in nonequilibrium statistical mechanics stemming from the early work of Green (1952,1954), Kubo(1957), and Kubo etal. (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 Onsager reciprocal relations [for details the reader is referred to Prigogine (1961), deGroot and Mazur (1963), and Hanley (1969)l:

Postulate I: Microscopically large though macroscopically smallvolume 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

where Ji and Xi are the conjugate f l u e s and forces.

MEDIA MOGENEOUS IN DIFFUSION

5

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

J=LX where L is the matrix

phenomenological (kinetic) coefficientsL,.

Onsager's reciprocal relations: with a suitable choice conjugate forces and fluxes, the matrix phenomenological coefficients in Eq. is symmetric, i.e., L, = Lji. For isotropic mediaandunder 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 matrixL is a scalar and theforces andfluxes 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 jlu may be selected arbitrarily without affecting the rate of entropy production. The condition of mechanical equilibrium considerablysimplifies 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:

where V T p j- Fjand T"VT are the thermodynamic driving forces for mass and heat transfer as prescribed by theform of the entropy production equation arising from Postulate I. [ A s 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 notatmechanicalequilibrium.] p j and Fj are the chemical potential of component j and the external force acting on component j @er mole), respectively, and V, is the gradient operator at constant temperature T. Equation is the principal result of nonequilibrium thermodynamics that will be employed in this chapter. The implications this expression from the point of view of nonequilibrium statistical mechanics wereinvestigated in detail

MAcELROY

6

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 formalismin transport processes,andthey also extended Mori’smethod 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 Ji 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

c NI

ji& c) =

vli exp(-ik

r,i)

I= 1

where vli and rli are the center-of-mass velocity and position of particle I of component i at time t, X,& t ) is the wavevector-dependent diffusional thermodynamic force for component j given earlier in Eq. (lo), and L;@, t ) is an “after-effect” function for diffusion that is related to but not equal to the phenomenologicalcoefficient L, definedearlier. The distinction betweenthe Green-Kubo linear response theory (Green 1952, 1954; Kubo, 1957; Kubo et al., 1985) (which leads to a comparatively simple expressionfor L , and which we specialize to below) and Mori’s projection operator formalism lies in the random component j@, t) of the flux that appears explicitly on the right-hand side of (llb) and also implicitly in the coefficients L # , t), i.e., 1

Li(k, t ) = -k (j#, t) jf(-k, 0)) k k2kBT

where the angular brackets represent averaging overthe unperturbed equilibrium canonical ensemble. The term inside the angular bracketsin 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. (llb) characterizes the decay of microscopic fluctuations in the medium. Mori (1965) showed that in thelimit k + 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

MEDIA MOGENEOUS IN DIFFUSION to the corresponding expressionprovidedby 1952, 1954; Kubo, 1957; Kubo et al., 1985),

linear response theory (Green,

i.e., a kinetic coefficient determined by the mechanical properties j , in contrast to the random components of these fluxes that are involved in the generalized expression for the coefficients Lk(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 lengthscales 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& W ) = J: exp(-iwt)L:,(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 f in Eq. (llb) may be neglectedand 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

where qij(t)is the velocity correlation function (VCF)

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 nonequilibriumsystem should not vary significantly over length scales on the order of molecular dimensions and (2) the time scales of interest should belonger 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 L, 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

8

MAcELROY

Although this equation is frequently cited and employed in the literature, in this chapter the Green-Kubo form provided in (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 importantconclusions of their work is that whilethe kinetic coefficients in different flow-frames maybe determined from a knowledge of the coefficients L, [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

where Mkis the molecular weight of component k. The molar diffusion fluxes J: in this frame are again given by (10) with L, replaced by R, and with L , replaced by a similar expression for Riq [see Altenberger et al. (1987) for details]. Further comment onthe coefficientsL, is postponed until later, although at this point it is worthwile considering an additional result [originally derived by Mason and Viehland (1978)l stemming from Eq. (10) and the concomitant condition of mechanical equilibrium. Dividing Eq.(8)by ni and subtracting from this resultthe corresopnding expression for component k, one finds

Now consider the condition of mechanical equilibriumas expressedby the Gibbs-Duhem equation

with

where P is the local pressure in the system. TreatingV+,, - F, as the dependent driving force in Eq. (17), then Eq. (16) may be written as

MOGENEOUS EDIA IN DIFFUSION

9

where

Now multiplying J2q. (19) across by ni?Ik/n2Dik,summing over all species, and defining

where Dikis the mutual dif€usion coefficient for the pair of species i and k, then the flux equations may be written in the Stefan-Maxwell form

(V+, - Fi) - kTiV In T

(i = 1, . . . , v)

where kTiis the thermal diffusion ratio for component i and is given by kTi

=

c k=l

d'iq

- n&kq

n2Dik

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

where 6, is the Kronecker delta. The Stefan-Maxwell equation [M. (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 diffusioncoefficients of amulticomponentfluid in homogeneous fluidholid media.

for 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

MAcELROY

10

medium itself also treated as one ofthe 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 fluidholid systems not far removed from equilibrium, the flux equations are therefore

(i = 1, .. ., V, m) (25) or

(i = 1, . . . , U, m) (26) with L, and D , 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 thematrix of the material making up the solid medium conforms to the condition of isotropicity. If this is nottrue,thenthe 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

(i = 1, . . . , v, m) (27) or

d In T dz

- kn -

(i = 1, . . . , v, m) (28)

DIFFUSION IN HOMOGENEOUS MEDIA

l1

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 haveappearedin the literature over the last few decades,andthe reader is referred to a number of articles dealing withrecent developments in this subject (Reyes andJensen, 1985; Nicholson et al., 1988;Sahimi,1988; Zhangand Seaton, 1992). However, there are a few points worth noting with regard to the application of the above equations to anisotropic pore structures. 1. The influence of the solid arises explicitly through the terms L? and D?. At first glance this type of formulation might appear to becounterintuitive, 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 includethe solid phase as a component in the di€€usion 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 al. (1963)l. 2. A fundamentalproblemdoes 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 and -+ 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 al. (1988). In particular, Schoen et al. demonstrated that although linear response theory does indeed providemeaningfuldiffusion coefficients for diffusion parallel to the surface of thesolid 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 (SuhandMacElroy, 1986) arediscussed to further illustrate this point. The above comments, coupled with the earlier remarks on the theoretical results provided by Mori usingprojection operator theory, lead to the following summary of limitations associated with Eqs. (25)-(28): 1. The fluidholid system must be isotropic ifEqs.(25)and(26)

are to be considered applicable. For solid structures that are strongly anisotropic locally, equations similar to Eqs. (27) and (28) maybe used along axeswithin the medium that are translationally invariant.

MAcELROY

12

2. Density and temperature variations within a microscopic volumeelement of

the medium should be negligible, and the time scales for molecular relaxation processes should bemuch 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 mobilityof the solid phase both at the macroscopic level (i.e., U,) and at the microscopic level [v& in Eq. (14b)l. Furthermore, in view of the inverse dependence of D , on the number density of thesystem, 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):

where = n,,i.e., the numberdensityof 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

and

(i = 1,

..., V)

(32)

The Gibbs-Duhem equation [Eq. (l?] 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 J, = 0 and Lh 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 L, = 0 and in many cases for single-component difision 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), andMasonandMalinauskas (1983)l. In Section 111, 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 computethe VCFs appearing in Eq.

MEDM MOGENEOUS INDIFFUSION

13

(14b)and hence the kinetic coefficients L,. 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 (19831, and the most important of these advantages fromthe point of view of transport in homogeneous media are summarized as follows: 1. Microscopic properties, which cannot bereadilymeasuredexperimentally 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 neverreplace actual experimental measurements, it can, in conjunction with theoretical modeling and limited experimental data, serve as an accurate andpowerful 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, theexpense associated with detailed experimental measurements of transport in such systems. The discussion in Section III is presented in three parts. As an introduction to those whoare not familiar with molecular dynamics simulation, computations based an idealized pore geometry are considered first. The pore geometry employed here is the cylindrical pore model that has been widely used in theequilibrium properties offluids in oretical studies of boththetransportand 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.

SIMULATION AND MODELING OF DIFFUSION IN FLUID/SOLID SYSTEMS

A. Diffusion in an Idealized Pore GeometryThe Cylindrical Pore For a cylindrical pore of length >> R,, where R, is the pore radius, and assuming that the solid phase is immobile, Eqs. (31) and (32) with L, = 0 are

14

MAcELROY

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

or

(i = 1, . . . , v)

(34)

limiting cases of these equations are considered here: a single-component pore fluid and a binary mixture. For simplicity it isalso assumed that noexternal forces act on the fluid components. 1. Single-ComponentPoreFluid The diffusion flux of a pure fluid f is given by

or

Jr)= - --

","IT

(36)

, , in local equilibrium with For a bulk external phase at a chemical potential p the pore fluid at z, = = p,O(T) k,T In tB, where tBis the activity of the bulk fluid that satisfies the limiting condition CB + n, as n, + 0. Substituting this expression into Eq. (36) gives

+

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

Recall that n, 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 (i.e., the pore wallis the bound-

DIFFUSION IN HOMOGENEOUS MEDIA

15

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

If the gas does not adsorb on the pore walls, the diffusion coefficient appearing in this equation is equivalent to D X , 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 Dm is simply (Kennard, 1938)

where is molecular mean speed (8kT/mn)'" andf is 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 K' = 1. When the s u e of the gas moleculesis taken into consideration, these expressions require modification. For example, for spherical nonadsorbing gas moleculesof diameter uf,E q . (40a) is rewritten as (Suh and MacElroy, 1986)

')

Dm=- (2 VRp(l - A) where A = uf/2Rp. The partition coefficient under these conditions also depends on A in accordance with (Ferry, 1936)

Note that the net effect of these modifications is to introduce the factor (1 A)3 into the Knudsen permeability DX& and therefore, even for the smallest gas particles (e.g., helium), steric effects cannot be neglected for diffusion in pores of diameter less than approximately nm. For adsorbing gases and vapors and for liquids, there are, unfortunately, no general tractable expressions for either D Z or Kf (or for the bulk phase activity CB), and approximate theoretical results exist in only a limited number of cases. Theoretical developments in this area may be guided, however, by direct computer simulation of the configurational and dynamical properties of the pore fluid, and it is this method of approach to which we now turn. In general the kinetic coefficient LE) and hence the diffusion coefficient DZ for the pure micropore fluid may be expressed as follows using Eq. (14):

16

MAcELROY

and

where $(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.,

Also recall that the angular brackets in the above expressions represent averaging over an equilibrium ensemble and therefore n, is the local equilibrium number density 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 is very simply related to fluctuations in the velocity field of these particles. Since this diffusion coefficient is also an intensive property, the integral J: (uf)(t)uf)(O)) dt in the aboveequations is an extensive property and disappears in the limit N A variety of molecular simulation techniquesare in current use that are based on sampling fromeither equilibrium ornonequilibrium ensembles, andthe 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)l. 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 involvedin the application of the equilibrium MD method to fluidholid systems, the diffusing fluid molecules are assumed to be simple structureless spherical particles. models for the interparticle interactions in the fluid are considered: (1)the hard-sphere interaction potential

and (2) the Lennard-Jones (12-6) interaction potential +ij(rij) =

k i jZl [);(

-

($1

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

17

MEDM MOGENEOUS INDIFFUSION

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 derivedfromEq.(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 potentialsimilar to the above expressionsfor 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, - = v; dt

For the hard-sphereinteractiongivenbyEq. (45) the force, F,,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 rij = uij,then, it is readily shown that the time to collision, given that the initial positions of the particles are rioand rjo and their precollisional velocities are vio and vjo, is given by

(vijo r,, <

(48a)

-

where vijo = vjo - vio and r,, = rjo - r,. The condition vijo rijo 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 m,mi m,Av, = - 2 -(V;,n mi + mj

k) k

where AV, is the change in the center-of-mass velocity of particle i, mi 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 AV,. [Other hard-core particle

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 andvary 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

jti

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 (4%) in finite-difference form. A number of finite-difference algorithms have been proposed [see Allen andTildesley(1987)for details], one of themost popular of which is known as the “velocity Verlet” algorithm (Swope et al., 1982). This procedure relates the particle positions and velocities at time I + At to the corresponding values and the accelerations at time t according to ri(t + At) = ri(t) + At v,(t)

+ -21 At”,@)

(50a)

and vi(t

+ At) = v,(t) + -21 At [a,(t) + ai(t + At)]

For atomic fluids the time step to be used in the above equations typically lies in the range lo-” At S. This rangeof values usually ensures reliable conservation of energyduring agiven simulation run(AllenandTildesley, 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

DIFFUSION IN HOMOGENEOUS MEDIA

19

where vioand rioare 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,andthe results reportedby Suh andMacElroy (1986) for the limiting cases of specular reflection and cosine law diffuse scattering [f= and 1, respectively, in Eq. are considered here. For specular reflection, only thevelocity components inthe plane the pore cross section are changed during the collision:

Avi = - 2 7(vio riw) with riw= R, For cosine law diffuse scattering, however, all three components of thevelocity are altered.Here only elastic diffuse scattering is 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

where 5, and tz are random numbers that are uniformly distributed on the interval (0,l). The computation of the individual particle trajectories using the above equations is generallysupplemented 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)l. 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 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 computationscan be carried out for only a finite numberof particles, the major difficulty to overcome in simulations of homogeneousmedia is the condition associatedwith 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-homogeneoussystem is to consider the cell to be surrounded on all sides by images of itself. Surprisingly, with fundamental cells containing as few astens or hundreds of fluid particles this approach can provide

MACELROY

20

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 procedureis illustrated schematically in Fig. 1for 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 R, and half-length L of the fundamental unit of the pore in either an orderly [see, e.g.,HeinbuchandFischer(1987)l randommanner(SuhandMacElroy, 1986), the procedure selected generally depending on the required density of the fluid. The initial velocities ofthe particles are thenusuallyassigned by randomly selecting components from the Maxwell-Boltzmann velocity distribution function [seeAllenandTildesley(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 1. 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 timeandthe 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.) The momenta of the colliding particles are changed in accordance with Eqs. (48b), (52), or

z=o

2

L

j

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

MEDIA MOGENEOUS INDIFFUSION

21

4. The collision times for the particles involved in step 3 are reevaluated. 5. Theabove 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 lo4 collisions per particle inlength were found to provide statistics of sufficient accuracy for subsequent evaluation of the diffusion coefficients using Eqs. (42)-(44). The equilibriumtime-correlation function (uf)(t) uf)(O)) appearing in these equations is readily evaluated by sorting the stored trajectory data into equally spaced time intervals 6t, and, using the ergodic hypothesis, the ensemble-averaged VCF is given by

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 approachingthe 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 L$) or the diffusion coefficient DC 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 = tMnx(which, in many cases of interest, is significantly shorter than the total length of the trajectory) with little or no loss in accuracy inthe 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)l. For the moment it will be assumed that the VCF is zero at or beyond t-; we return to the problem of long-time tails in Section IILB. A number of typical center-of-mass VCFs for the pure hard-sphere porefluid 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 II.A.2). The VCFs in the plane of the pore cross section are simply obtained from the

22

MAcELROY

velocity components of the center-of-mass of the fluid using the expression

The results shown in Figs. 2 and are normalized to 1.0 at zero time, and the dimensionless time, T*, in these figures is in units of 2RP(1- A)/ij. 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, i.e., the center-of-mass VCF simplifies to

Due to the absence of interparticle collisions, crosswhere k = xy or correlations for the individual particle velocities are nonexistent as implied by

2 Normalizedvelocityautocorrelationfunctionsforfree-molecule(Knudsen) diffusion versus reduced timeT*. (-) Theoretical VCF in the axial direction for diffuse scattering; moleculardynamics axial VCFfordiffusescattering; (- - -) and (- - -) molecular dynamics VCFs in the plane of the pore cross section for diffuse and specularscattering,respectively.[ReproducedfromSuhandMacElroy with permission.]

DIFFUSION IN HOMOGENEOUSMEDU

1

I

I

(a)

-

-

"

"

_.-

-0.2

-

-

-0.60

\

/

1

0.25

/

1

0.50

I

0.75

T*

-

-

-

-

-0.4

I

0.25

1

0.50

I

0.75

T* 3 Normalizedvelocityautocorrelationfunctions vs. reducedtime for X = 0.21 and n& (= n,&) = 0.6. (a) Specular reflection; @) diffuse reflection. Curve 1, VCF corresponding to L?!,.; curve 2, VCF corresponding to L$); short- and long-dash curves, VCFs in the plane the pore cross section for L!?/. and L$'), respectively. [Reproduced from Suh and MacElroy (1986), with permission.]

MAcELROY

24

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 component of the VCF for particle/pore wall specular reflection (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(*)(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 = 0. 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 difhse scattering, and its integral over time provides the Knudsen diffusion coefficient given in Eq. (40) with = 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 shownin Fig. 2 is not a simple exponential as shownby 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 1II.B. The fluidcenter-of-massVCFs for motion in the plane of the pore cross section shown in Fig. 2 (nf 0 and hence n, 0) and in Fig. at a higher (liquidlike) reduced bulk density demonstrate oneof 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 that the oscillatory behavior of the 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 provide zero valuesfor 0%’. Such values for 02)should be viewed as questionable in light ofthevery strong local inhomogeneities involved in the fluid density, and due consideration should be given to a more in-depth analysis based on the wavenumber dependenceof the kinetic coefficientspredictedbythe projection operator formalism of Mori (1965). Unfortunately, there is at present no simple way of evaluating the pro-

[Es.

-

-

DIFFUSION IN HOMOGENEOUS MEDIA

25

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 the results of linear transport theory to diffusion in translationally 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, are selected results for D$ for a pure fluid f as a function Xf = uf/2R, at a fixed bulk phase reduced density n& = n,u: = 0.4054. (The open circle and open square results are for the individual species in a binary mixture at the same bulk density with h,/A, = These results are in the dimensionless form D$/D,, where DK is given by Eq. (40) (with f = In the limit At + (the hard spherical fluid particles approach the size the pore), diffusion within the pore is described solely by free-particle motion (a result that is independent of density). In the opposite limit, h, + 0, the diffusion mechanism is usually referred to as viscous slip, and the coefficient D$ under these conditions is a function of fluid density, decreasing with increasing density (Suh and MacElroy, MacElroy and Suh, It was also shown by MacElroy and Suh

1

x, Reduced axial di€fusion coefficient relative the membrane as a function of particle reduced radius. Single-component system (a= R.(0)and (0)Results for the solvent (a= and the solute (a= 2), respectively, in the binary system. [Reproduced from MacElroy and Suh with permission.]

MAcELROY

26

that for pore sizes less than approximately one-tenth the diameter of the fluid particles (typically R, nm)thetransportofa dense fluid or gas ina 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, and, as illustrated by MacElroy and Suh 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, Peterson et al., although the transport properties have received attention in afew studies (Heinbuch and Fischer, MacElroy and Suh, [Transport characteristics have also been investigated via MD simulation for London-van der Waals fluids confined within slit-shaped pores (Schoen et al., Magda et al., Usually the particle-pore wall potential function is represented by a two-body interaction in’whichthe solid is treated as a smeared continuum of Lennard-Jones interaction sites. Heinbuch and Fischer 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. 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,

where ri is the radial positionofthefluid particle within the pore, eiWis the potential minimum for interaction between the fluid particle and a single Lennard-Jones site in the solid, urnis the corresponding Lennard-Jones size parameter, and nw is the number of Lennard-Jones sites per unit volume in the solid phase. The two functions f9’(ri)and f”’(ri)are polynomials in ri (Nicholson, and in the limit R, Eq. simplifies to the potential function frequently used in the modeling of sorption on flat surfaces (Steele, It is also important to note that the definition of R, in Eq. differs from that involved in the hard-core interactions discussed above. This difference is readily seen by comparing Figs. and 5a. In molecular dynamics simulations of a Lennard-Jones pore fluid whose interactions with the pore wall are described by Fq. the total force experienced by a given fluid particle i is obtained by including the force field exerted

-

DIFFUSION IN HOMOGENEOUSMEDH

27

5 Model cylindricalpore structures. (a) Particle-pore wall continuum interactions. The hatched region r R, represents the inner repulsioncore of the solid surface atoms.@) Structured pore wall. &a1 positions 1 and 2 are referred as the pore window and polygonal cage, respectively. (Reproduced from MacElroy and Suh (1989), with permission.) by the solid in Eq. (49) to give N

--

= mia,(t) = j=l

rij drij

-

ri&,(R, - r,) ri dri

jfi

where riis the two-dimensional vector coordinateof 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 E is predicted to be infinitely large. Only if one introduces amechanism 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 introducedby 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

S) = R, cos

(k = 1,

.. . , NR; Z

=

. . ., +

(58)

MAcELROY

28

where NRis the number of surface atoms in a polygonal ring and U, is the axial spacing of the rings (NRis 12 for the diagram shown in Fig. 5b). In the work reported by MacElroy and Suh (1989) the interactions between these surface atoms andthefluid particles in the pore were described by a Lennard-Jones potential function similar to Eq. (46) with eij = eiwand uij= ai,. 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, uiw/2 - ri, and the total force on a given fluid particle i for this structured system is therefore

+

j#i

jES

In the molecular dynamics simulation of a Lennard-Jones pore fluid subject or Eq. (59), it is again quite clear to forces of the type described by J3q. that only a relatively small number of particles (= 102-103) may be considered in the finite-difference solution ofNewton’s equations of motion [e.g., using Eqs. (50)].Periodic boundaries at z = +L are therefore employed to minimize the influence of edge effects onthe 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 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 R,, for a spherically symmetric interaction between particles i and j in a condensed phase is taken to lie between 2.50, and 3.50,. The larger the value of Rcjj,the more closely will the simulated fluid approach the physical behavior of the model fluid [e.g., the Lennard-Jones (126) fluid characterized by Eq. (46)]; however, the upper limit in R,, is usually governed by the CPU time required to compute all of the force contributions within the spherical volume rij Rcij.This CPU requirement increases as N:, where N, 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 rij and the computations are not seriously influenced by the truncation at R,, [e.g., at a relative separation of 2.50,~ Eq. (46) provides r$ij(Rcij)= and at 3.50, the potential is -0.002eij]. Additional tricks of the trade such as shifted potentials (to ensure energy conservation in the microcanonical ensembleMD simulations), neighborhood lists, cell linked lists, etc. [details of which may be found inAllenandTildesley

29

DIFFUSION IN HOMOGENEOUS MEDM

(1987)] should also be incorporated in the simulation code to improvethe computational efficiency during program execution. typical simulation run for a Lennard-Jones fluid confined within either of &e model poresillustrated in Fig, 5 would proceed as follows. As described earlier for the hard-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 botential + 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 =lo4, where, asnoted earlier, At usually lies intherange 10”’ c: At < S. After equilibrium has been achieved, rescaIing is terminated, and during the subsequent computations the particle trajectories evolve at fixed energy. the equilibrium system no drift in the average kinetic temperature T = (1/3NkB (Xrnp?) will be observed, although fluctuations in Cuj?lNon the order of l/ N should be present. During the equilibrium trajectory (usually sampled for approximately lo5 time steps), the particle positions and velocities are stored at equispaced intervals st for subsequent evaluation of the VCFs using Eq. (54) and of the diffusion coefficients using

2

Results for the axial diffusion coefficient D% for the structured pore shown in Fig. 5b for a range of values of N R [see (58)] and at a fixed bulk liquid density n, = 0.61~;~ and temperature T = l.l5(~ff/kB) are reported by MacElroy and Suh (1989). These data are reproduced in Fig. 6, where the reduced form DEID, is plotted as a function of X = uf/2RFE.The Knudsen diffusion coefficient involved here corresponds to Eq. (40) with f = 1, and the pore radius is defined as the effective quantity RFR. 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 X obtained here hasa 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% in the limit h for a realistic liquidlsolid 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 h 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 E observed in Fig. 6 under

-

30

MAcELROY

d

0.10 -

0.080.06-

t

III

0.04-

T

1

/

Figure 6 Reduced axial diffusion coefficient relative to the membrane vs. X for the structured pore wall (Fig. 5b). Dn(is the free-molecule (Knudsen) diffusion coefficient, and the upper abscissa, d, is the diametric distance between pore wall surface atoms in units of Ur. [Reproduced from MacElroy 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., VTFJ

-

F, = (VTF, -

FJ)B

(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

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

where D':)and DF) are the Fickian diffusion coefficients for the two species and are related to the microscopic properties of the pore fluid as

and

with L:;) given by

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\$, D:$,and 01'1(MacElroy and Suh, 1987): 1

1 IM

or

with similar expressions for DF). x1 and x2 are the mole fractions of the two species within the pore.

MAcELROY

32 The mutualdiffusioncoefficientsmay kinetic coefficients L$):

also be expressedin

terms of the

and

D:: = k,T~(')l/n,Ly~

(65c)

where IL"I = Ly:Lfi - L??. 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 l * , and in view of the equivalence of molecular properties we have Kl. = Kl and D$M = DYA. For this model binary system, Eq. (61) for the tracer simplifies to

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

and

33

DIFFUSION IN HOMOGENEOUS MEDIA where Llf' is given by

(42) and

"m

Note that in the thermodynamic limit N

D:? = D:) = v ~ , T L ~ ? , .

-

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 L$) and hence the diffusion coefficientD$M= D$. The latter coefficient, which is of particular importance in the engineering design of adsorbers, membrane separations modules, etc., may also be measured via gravimetric or volumetric sorption experiments, which in turn may be used as corroborative evidence for the validity of aproposedmolecularmodel for diffusionin microporous media and membranes. We return to the distinction between DI'! and D$ later in Section III.B, and for the moment we examinethe characteristic behavior of D?? 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 tobe the tracer, a secondary averaging is permitted as shown in Eq. (68b). This secondary averaging can lead to very accurate results for DI".',in contrast to the membrane/fluidmutualdiffusioncoefficient D$, which involvesa single measure the influence of the collective motion of the pore fluid as a whole. Accurate determination of DE 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-Jonesliquid 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 theverysignificanteffect 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 (h increases), the diffusion coefficient for the tracer in the atomically structured pore drops rapidly in agreement with the general trend

34

MAcELROY

3.5

IO I

1

3

2

I

l

I

2.5

7 Reducedaxialdiffusioncoefficientforthetracer as afunction in pores with smooth walls. MD simulation; (0)Davis-Enskog kinetic theory [Davis (19871 andFischer-Methfessel(1980)approximation; (0) Davis-Enskogkinetictheoryand bulk fluid approximation; Lower dashed curve, The empirical correlation of Sattertield etal.(1973). d asinFigure 6. [ReproducedfromMacElroyandSuh(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 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[i.e., Eq. (52)] or cosine law diffuse scattering [Eqs. (53a-c)].

DIFFUSION IN HOMOGENEOUSMEDU

35

8 Reduced axial diffusion coefficient for the traceras 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 valueof N Rin Eq. (58). [Reproduced from MacElroy and Suh with permission.] For diffusion in binary pore fluid mixtures molecularly disparate species, one the most important questions that frequently arise concerns the relative importance of cross-coupling effects; i.e., can the cross-kinetic coefficients Lyi and L';1 appearing in Eqs. (62a,b) be neglected? supplementary question is then usually posed If the cross-coefficients are neglected, can one assume that the coefficients L?; and 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 ofthese questions are readily answered

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 al., 1954; MasonandMalinauskas, 1983); however, for micropore fluids and particularly dense fluids or liquids, answers to these questions are not easily obtained. 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 0) the solute diffusion coefficient (62b)l simplifies to

-

k*T D , =-L!:

(704

I n2

=

(v!)(t)v!)(O))dt

(70b)

It has been assumed here that the contribution (KJK,) L$ in Eq. (62b) remains finite in this limit. A similar simplification does not result for the solvent diffusion coefficient [Eq.(62a)l unless K2/K, 0 and/or Lfi 0, and neither of these conditions will be generally true. Thesimulation results reported by MacElroyand Suh (1987) for both D? and D:) inhard-sphereporefluid mixtures are shown in Figs. 9 and 10, where &,B in the mutual diffusion coefficient of the pair 1-2 in the bulk homogeneous fluid. W Oimportant trends are observed in Fig. 9: (1)For large pores (A2 0), the rate of diffusion of the solvent within the pore is much larger than correspondingrates in the bulk phase [as discussed by MacElroy and Suh (1987), this phenomenon is generally sociated 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 X , and the magnitude of hl/A2(which equals0.6 in the present case). For therange of pore sizes in which D? is observed to be negative, it was also shown by MacElroy and Suh (1987) that KJK, > 1, and since the mutualdiffusioncoefficients DfA and D?; are always positive it is seen from Eq. (64b) that it is this ratio that governs the sign on D?). Finally, from Fig. it is of interest to note that for large pores (and “nonadsorbing” solutes) the solute diffusion coefficient reduced by the bulk-phase solutelsolvent mutual diffusion coefficient is a universal function of the solute reduced radius h,; i.e., the ratio D$)/D,%B, where is the solute, is independent

-

+.

DIFFUSION IN HOMOGENEOUS MEDM

Reduced axial porediffusioncoefficientforthesolventasafunctionof soluteparticlereducedradius.[ReproducedfromMasonandChapman, permission.]

37

with

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)l. For small pores (h, h2 > l),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 RigidRandom 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 insuch studies is the manner in which interpore connectivity in a macroscopic random medium can influence the overall transport process. Two independent though comple-

38

MAcELROY

0.8

0.6

-

0.2 -

Reducedaxialporediffusioncoefficient for thesoluteasafunction of solute reduced radius. Simulation results for the single-component system (a= l*) with n& = 0.4054; (0)simulation results for the solute (a= 2) in the binary system with n;, = 0.4, n& = 0.01; (--) continuum-mechanical theory [58,591; (- - -) Eq. (40b) (a = l * and f = 1.0) reducedby D,l..8; (- - -) Eq. (40b) (a = 2) reducedby D,ze. [Reproduced from Mason and Chapman, (1962), with permission.]

mentary approaches have been employed in the last 20 years to investigate the effects 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 al. (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 etal.(1983),Chiew et al. (1985), Torquato (1986), Park and MacElroy(1989),MacElroyandRaghavan(1990, 1991), andRaghavanand MacElroy (1991)l. In the lattice models, a “bond” joining any two nodes or “sites” in the lattice usually represents an individual pore channel (frequently assumedto be cylindrical in shape) that is connected to other pores inthe networkatthe respective nodes. These models are very versatile (individual bonds may be assumed to represent pores 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 IILA). 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 correspondingto 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 anno non ball^^ solid, whereas the secondmodel 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: If it is assumedthat the medium is isotropic, then the flux expressions are obtained using Eqs. and rather than Eqs. and 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. and 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 on the dielectric permeability of particle suspensions [for a review of later improvements on Maxwell’s original formulation, see Barrer, Chapter in Crank andPark andmorerecently statistical methods (Prager, Weissberg, have been employed with some success. These models, however, are applicable primarily to systemsin 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 molecularproperties 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 nearapercolationtransition. 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 11 diffusion in polymers. In a number of recent articles, both Monte Carlo (MC) (Nakano and Evans, 1983; Abassi et al., 1983) and molecular dynamics (MD) (Park and MacElroy, 1989; MacElroyandRaghavan, 1990, 1991;RaghavanandMacElroy, 1991; MacElroy and Tomlin, 1992) simulationresults 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-Jones 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 inFig. 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 Extensive computations over a wide range of porosities and for ensembles containingvery long trajectories were conducted in these studies in order to clearly determine the long-time behaviorof the VCF appearing in the free-particle diffusion coefficient

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 MDmethod employed byPark and MacElroy (1989) involved astraightforward 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 (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-

41

DIFFUSION IN HOMOGENEOUSMEDL4

11 Schematicrepresentations of theoverlappingandnonoverlappingspheres models. The solid phase is indicated by shaded regions, and the full circle corresponds to a fluid particle. [From Park and MacElroy 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. l l a demonstrated this, and this was confirmedrecently for thethreedimensional systems of interest here (MacElroy, 1992, unpublished). Selected results for the diffusion coefficient of the low-pressure nonadsorbing gas diffusing within the overlapping spheres models are shown in Fig. 12 (for the random overlapping spheres medium illustrated in Fig. l l a ) and in Fig. 13 (the data represented by the full circles in this figure correspond to freeparticle diffusion within theconnected overlappingsystem illustrated in Fig. llb). Results for the nonoverlappingsystems may befound in Park andMacElroy(1989). The Boltzmanndiffusioncoefficient cited inFig. 12 corresponds to the low-density limit for the solid (i.e., n: = n p ; 0) and is

-

42

MAcELROY

\cI 1.00 0.50 0.25

0.05 0.025

The free-particle diffusion coefficient reduced by its Boltzmann value (left ordinate axis) and the open porosity relative to $ (right ordinate axis) as a function of the reduced density (or porosity) for the random overlapping system (see Fig.lla). 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 the(Gotze et al. (1981a,b)); repeated ring kinetic theory (Masters and Keyes (1982)). [Reproduced from Park and MacElroy (1989), with permission.]

given by

where U 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 overlapping spheres medium are also determined byboth U and n: according to

DIFFUSION IN HOMOGENEOUS MEDIA

43

(RANDOM) 0 I

0.2

0.4

0.6

0.8

1.2

'

I

I

a0.644

1

-

0.200-

-

-

-

-

0.080 0.060,-

0.020.

'

0.008. 0.006

-

2.0

13 The diffusion coefficient reduced by its value for = 0 as a function of fluid particle size (lower abscissa) or particle reduced size = UDR, (upper abscissa) in overlapping systems with = (0)Random overlapping system (Fig. lla); connected overlapping system (Fig. llb). [Reproduced from Park and MacElroy (1989), with permission.]

and

44

MAcELROY

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

The simple relationship between this expression and the point particle Knudsen diffision coefficient given in (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)f(MasonandChapman,1962), where f 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 $T it is possible to immediately read off the ratio D,/D, and hence obtain the “tortuo~ity’~-~~rrected diffusion coefficient Dm. Note that the lower abscissa in Fig. 12 is to be used only when the diffusing particle has zero volume, i.e., U = 0. example of the application of the procedure described aboveis illustrated in Fig. 13 for a medium with a comparatively high total porosity JIT = 0.644. Also shown in this figure are simulation results for the more realistic model depicted in Fig. l l b , which takes into consideration the effect of solid particle connectedness in real media [it should be noted that for JIT < 0.5 the random and connected models in Figs. l l a and l l b are equivalent because the random overlapping spheresmedium satisfies the criterion of complete solid connectivity under these conditions [Shante and Kirkpatrick, 1971; Kirkpatrick, 1973; Haan and Zwanzig, 1977)l. At the porosity JIT = 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 DC differ by only 10%). For small particle sizes the dependence of D , on U 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 U. This transition is generally referred to as the percolation threshold, and for the three-dimensional models shown in Fig. 11its position depends not only on the size of the diffusing particle but also on the porosity (or bulk density as illustratedinFig. 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

45

MEDM MOGENEOUS INDIFFUSION

transition will not be observed abovethe 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

W ) = (v@) v(O))/(v2(0)) One of the most important points to note concerning the long-time behavior 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 theVCF results from particle backscattering after an initial positively correlated forward momentum, and the negative correlation persists to long times dueto 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 al., 1981a,b; Masters and Keyes, 1982; Ernst et al., 1984; Machta et al., 1984) for diffusion in random media, i.e., lim (v(t)

v(0)) =

"I

-

a

-

(76)

re

where a and are density-dependent parameters. Inserting this expression into Eq. (71) gives DfM(t)= DfM

+

-

>'(

1)

tP-1

(77)

in which the time t is significantly larger than the time required for the VCF to pass through its minimum (for theresults 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-

46

MAcELROY

9 = 0.500

o.21'\L 0

0.035

-0.21

1 2

0

IC3 86-

2U

'

10-4-

8-

9=0.500

t" Normalized velocity autocorrelation function for the random overlapping system. (a) Short-time behavior for several porosities. (b) The long-time tail for$ = 0.5. MD simulation; (-) fit a t*-" power law decay [the reduced timet* is in units of the mean free time T~ = 3D&', where D,, is given by Eq. (75)]. (- - -) Modecoupling theory Emst et al. (1984) and Machta et al. (1984).

DIFFUSION IN HOMOGENEOUS MEDU

I

1

I

I

18

47

l

X)

14 Continued (c) As in (b), but for

= 0.367. (d) The long-time tail for MD simulation. (-) leastsquares fit to a powerlawdecay (p = 1.57). [Reproduced from Park and MacElroy (1989), with permission.]

= 0.035.

MAcELROY

48

tion that p # 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 VCF on diffusion in random media is considered by Muralidhar et al. (1990)l. For high-porosity media and small gas particles, the power law exponent f3 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. ll), while for conditions at or near the percolation threshold the value of p 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)

where

6 = cI./(2v

+

cI.

-

y)

(79)

and $c is the critical voidage below which the penetrant is localized ($c = 0.035 for the random overlapping system as shown in Fig. 12). The parameters k, v, and y are the critical exponents appearing in the expressions

and

P($)

($

-

$JY

($

-

$c+)

(82)

where 5 is the correlation length and P($) is the percolation probability, which, in the present case, is equivalent to the relative magnitude of the percolating void fraction, $”, to the void fraction $ (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:

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 DM(t) = l/t, while at the transition [Eq. (83c)], Eq. (78)

DIFFUSION IN HOMOGENEOUS MEDLA

49

simplifies to Eq. (77) with the stationary-state diffusion coefficient DTM= 0 and p-1=s. At high porosities and/or for small diffusing particles, the comparatively large value of p 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. ( l l ) , i.e.,

which simplifies to

for a rigid random medium. The q,(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)l. 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,

MAcELROY

50

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

where ( P ) is mean square jump distance. The coefficient D is equal to Dm,and D4and 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 this system (Fig. lla). The time-dependent behavior suggested by Eq. (77) was also observed in molecular dynamics studies adsorbing Lennard-Jones fluids [Eq. (46)] confined within the micropores of a model silica medium (MacElroy and Raghavan, 1990, 1991),and oneexample of this is providedinFig. 15 for the tracer coefficient D,.(t),

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. l l d . 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 the solid via a Lennard-Jones (126) potential. The relative size, A, 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 Fig. 15. Finally, to return a point madeearlier with regardto the relative importance of the cross-kinetic coefficients L,(i # 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

D,.=

3I

(v&)

- v,.(O))

dt

51

DIFFUSION IN HOMOGENEOUS MEDIA

D**@*)

"

0.2

2

Tiie-dependent tracer diffusion coefficient for a model porous silica medium saturated with a Lennard-Jones liquid.).( 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 = 1.8 2 0.1 t* is in units of &G, and Dl.(t*) is in units of I s i o G f , where is the mass of the Lennard-Jones particles, ernis the potential well depth for interactions between the fluid particles and the oxygen atoms of the silica medium, and lsiois the Si0 bond length insilica (0.162 nm). [Reproduced from MacElroy and Raghavan (1991), with permission.]

52

MAcELROY

16 simulated silica microsphere. Each of the solid spheres shown in Fig. l l d is modeled as one of these. The open circles are nonbridging surface oxygens, and the shaded region illustrates the exposed interior bridging MacElroy and Raghavan with permission.]

oxygens. [Reproduced from

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

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

Note that Dl. and D , are equal only in the limit -D 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%

DIFFUSION IN HOMOGENEOUS MEDIA

53

0.8

?

0.6

t D

0.002

0

0.2

0.2

17 Reduceddiffusioncoefficients of theadsorbingvaporas a function of fractional pore loading. Tracer diffusion coefficient; (0)total adsorbate dif€usion coefficient; (-) EnskogtheoryMacElroyandKelly (1985). The diffusion coefficients are in units of Z s i , , G f . The inset illustrates the relative importance cross-effects inthemicroporefluid.[ReproducedfromMacElroyandRaghavan with permission.]

54

MAcELROY

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

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 L,, and the diffusion velocity of the polymer, J,, is not zero (swelling or shrinkage may be observed) unless external constraints are imposed on the system. For a fluidholid system that is not subject to external forces, conservation of total linear momentum within the medium at any given instant requires

where m, and v,(t) may be considered to be the mass andcenter-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 atomicspecies is implied onthe left-hand side of (90)]. The kinetic coefficients defined in Eq. (14) are then interrelated by

and Eq. (31) may be rewritten as

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 (92) simplifies to

with Dm 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 MEDLA

55

sorption or membrane separation processes. Another diffusion coefficient that may be defined for single penetrant systems is the tracer or se!f-diffusion coefficient, and it is this parameter that is most frequently reported in MD simulation studies (and measured experimentallyvia 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), i.e., J1.

= - D,.Vn,.

(94)

with the tracer diffusion coefficient, Dl., given by (88). One of the major difficulties involved in the simulation of polymers is the availability (or lack a reliable model for the potential interactions arising not only between the penetrant and the polymer macromoleculesbut 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 al., 1989, 1991; Sonnenburg et al., 1990; Takeuchi andOkazaki, 1990; Takeuchi, 1990a,b; Takeuchi et al.,1990),polypropylene (Muller-Plathe, 1992), and polyisobutylene (Muller-Platheet al., 1992)] withthe exception of the recent work of Sok et al. (1992). The simplifying features of the alkane systemlie in the absence of polar or electrostatic interactions and the observation that the interactions between nonbonded atoms (-C- or -H) or sites (“CH, or C H r )on the chains are adequately represented by the (shortrange) Lennard-Jones potential[Eq.(46)]. The potential functions associated withintrachain vibrations and rotations are also frequently described by the comparatively simple forms v b

kb 2

= - (b - bo)*

and <

V, = k,

2

a, COS” (+)

n=O

where b is the bond length between two neighboring atoms or sites (with an equilibrium value of b,), 8 is the angle between two adjacent bonds (with its equilibrium value equal to and is the dihedral angle definedby three successive bond vectors in the polymer chain. [A concise description of these potential functions andthe equations for the forces onagivenatom or site arising from these interactions may be found in Appendix C of the text by Allen and Tildesley (1987).]Othermathematical 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 simulationsof polymer/penetrant systems, in which the penetrants have also usually been modeled as Lennard-Jones fluids [Eq. (46)], havebeenconductedmost frequently in one of possible ways: (1) The trajectories of each atom and/orsite 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 pairwiseLennard-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 ofbonds constrained] by incorporating constraint forces in Newton’s equations of motion that act along the bonds. Conceptually and computationally the unconstrained formulationis the easier ofthe two approaches, although difficulties arise owing tothemuchhigher frequencies usually involved in the bond vibrations in contrast to the comparatively low frequency of the torsional rotations. 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 thekindgiveninEq.(50) with comparatively large timesteps. 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 concerningthe reliability of the activation energies for diffusion determined via unconstrained molecular dynamics (Trohalaki et al., 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 (ina number of versions of this method the bond angles, 8, are also fixed). The method was first proposed in the 1970s (Ryckaert et al., 1977), and improvements and variants on the original technique weresummarized 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)l. 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 phasecoordinates of the chains are obtained to within a given tolerance. This algorithm (the original form of which is called SHAKE) ormodifications of it have been successfullyusedinthe simulation of both small molecular structures and complex macromolecules. Although molecular simulations have been undertaken over the last 15 years or for a variety of polymer systems [see Roe (1991) and references cited therein], it is only very recently that the firstMD simulation results for diffusion

DIFFUSION IN HOMOGENEOUS MEDIA

57

in polymers have been reported (Trohalaki et al., 1989, 1991; Sonnenburg et al., 1990; Takeuchi and Okazaki, 1990;Takeuchi, 1990a,b; Takeuchiet al., 1990; Muller-Plathe, 1992; Muller-Plathe et al., 1992; Sok et al., 1992). In the first few of these studies (Trohalaki et al., 1989,1991;Sonnenburget al., 1990; Takeuchiand Okazaki,1990; Takeuchi,1990a,b;Takeuchi et al., 1990), the unconstrained MD technique described above (method 1) was employed, and the polymer investigated was amorphouspolyethylene. 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, 02, and CO2 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(Cohenand Turnbull, 1959) andmolecular models (Pace and Datyner, 1979a,b,c) to both diffusion and polymer relaxation processes was confirmed. l h o 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 (Trohalaki et al., 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 consistentwith 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 NpEensemble)ratherthanconstant-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 (&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) couldlead to asignificantreductionintheself-diffusioncoefficientof 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 (H2, 02,and C&) diffusing in atactic polypropylene above the glass tran-

58

MAcELROY

sition temperature. This polymer is known to be 100% amorphous, and therefore any questions concerning crystallinity areavoided.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)l. Although no direct comparison could be made with experiment because of the absence of data, the self-diffusion coefficients obtained by Muller-Plathefor O2 and CH, were in good agreement with experimental results for a number of other polymeric materials. This was particularly true for C&, for which the simulated and experimental diffusivities were in very close agreement. However, for the H,/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 polyisobutylene (PIB).In this work it was clearlydemonstratedthat a united-atom representation of the methyl groups in PIB [(-CH2-C(CH,)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 displacementof 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

where rl.(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 heliumparticles are undergoing

MEDIA MOGENEOUS INDIFFUSION

59

s L

.t;

* L

W

0

t

-2.0 -1 .o

0.0

1.o

2.0

3.0

4.0

m,,WPs)

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. with permission.]

anomalous diffusion. It is also of interest to note that the exponent 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 etal. with reference to the stochastic MC simulations carried outby 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 onthe 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 fromits initial position Ir(t) - r(0)l. His published results for the three penetrants H,, O,, and CH., 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

t @S) Relativedisplacementsofrepresentativemolecules of H,, O,, and CH, diffusing in atactic polystyrene. For clarity, the ordinate axes for 0, and H, 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 onemicrocavity to neighboring holes. Similar observations were reported by Sok et al. (1992) for He (fluidlike tracer diffusion) and CH., (latticelike tracer diffusion) in rubbery polydimethylsiloxane(PDMS), and it is also worth noting that in the case 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 T, are not possible at the present time. In only one case has an attemptbeenmade 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 T, it is possiblethat the (slow)thermalmotion 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 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 MEDU

,

I

1=6.0 PS

1=10.1 PS

1=11 .l PS

1112.1 PS

1112.5 PS

143.3

1=16.1 ps

143.7 PS

1~14.1PS

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 (IgSOa), with permission.]

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 throughwhichthe 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 anysignificantenergybarrier.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.

W. 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 liquid and solid behavior over the last four decades, has been described in some detail, and a number of applications for simple fluids confinedwithin 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 dif€usion 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@) thatgive 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 the flux equations in glassy polymers.

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Abassi, M. H., J. W. Evans, and I. S. Abramson (1983). AZChE J., 29, 617. Alder, B. J., and T. E. Wainwright (1970). Phys. Rev., Al, 18. Allen, M.P.,andD. J.Tildesley(1987). ComputerSimulation 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). J. Chem. Phys., 86, 2909. Bioseparations-Downstream ProBelter, P.A., E. L. Cussler,andW.-S.Hu(1988). cessing 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). TheMathematicalTheoryofNon-Uniform Gases, 3rd ed., Cambridge Univ. Press, London. Chase, H. A. (1984a). Chem. Eng. Sci., 39, 1099. Chase, H. A. (1984b). J. Chromatogr, 297, 179. Chiew, Y. C., G. Stell, and E. D. Glandt (1985). J. Chem. Phys., 83, 761. Cohen, M. H., and D. 'hrnbull(l959). J. Chem. Phys., 31, 1164. Crank, J., and G. S. Park, Eds. (1968). Diffusion in Polymers, Academic, New York. Davis, H. T. (1987). J. Chem. Phys., 86, 1474. deGroot, S. R., and P. M m r (1963). Non-Equilibrium Thermodynamics,North-Holland, Amsterdam. Drioli, E., and M. Nakagaki, Eds. (1986).Membranes and Membrane Processes,Plenum, New York. Emst, M.H., J. Machta, J. R. Dorfman, and H. van Beijeren (1984). J. Stat. Phys., 34, 477. Ferry, J. D. (1936). J. 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). J. Chem. Phys., 20, 1281. Green, M. S. (1954). J. Chem. Phys., 22, 398. Gusev, A., and U. W. Suter (1992). Polym. Prepr, ACS Division of Polymer Chemistry, Washington, DC, 33, 631. Haan, S. W., and R. Zwanzig (1977). J. Phys. A: Math. Gen., 1547. Hanley, H. J. M., Ed. (1969). Trunsport 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. O., C. F. Curtiss, and R. B. Bird (1954). The Molecular Theory of Gases and Liquids, Wiley, New York. Ho, F. G.,andW. Strieder (1980). J. Chem. Phys., 73, 6296. Juntgen, H., K. Knoblauch, and K Harder (1981). Fuel, 817. Kennard, E. H. (1938). Kinetic Theory of Gases, McGraw-Hill, New York. Kim, W.-T., A. R. Altenberger, and J. S. Dahler (1992). J. Chem. Phys., 97, 8653. Kirkpatrick, S. (1973). Rev. Mod. Phys., 45, 574.

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Klafter, J., R. J. Rubin, and M. F. Shlesinger, Eds. Transport and Relaxation in Random Materials, World Scientific, Singapore. Kubo, R. J. Phys. Soc. Jpn. 12, Kubo,R.,M.Toda,andN.Hashitsume Statistical Physics 11. Nonequilibrium Statistical Mechanics,Springer-Verlag, Berlin. MacElroy, J. M. D., and J. J. Kelly AIChE J., 31, MacElroy, J. M. D., and K. Raghavan J. Chem. Phys., 93, MacElroy, J. M. D., and K.Raghavan J. Chem. Soc. Faraday Trans., 87, MacElroy, J. M. D., and S.-H. Suh Mol. Phys., 60, MacElroy, J. M. D., and S.-H. Suh Mol. Simul., 2, MacElroy, J. M. D., and B. Tomlin Proc. 3rd ZChemE Res. Symp., Dublin, Machta, J., M. H. Emst, H. van Beijeren, and J. R. Dorfman J. Stat. Phys., 35, Magda, J. J., M. Tirrell, and H. T. Davis J. Chem. Phys., 83, Mason, E. A., and S. Chapman J. Chem. Phys., 36, Mason, E. A., and A. P. Malinauskas Gas Transport in PorousMedia: The Dusty Gas Model, Elsevier, Amsterdam. Mason, E. A., and L. A. Wehland J. Chem. Phys., 68, Mason, E. A., R. B. Evans 111, and G. M. Watson J. Chem. Phys., 38, Masters, A., and T. Keyes Phys. Rev., A26, Maxwell, C. Treatise on Electricity and Magnetism, Vol. I, Oxford Univ. Press, London. Montroll, E. W., and G. H. Weiss J. Math. Phys., 6, Mori, H. Prog. Theor. Phys., 33, Muller-Plathe, F. J. Chem. Phys., 96, Muller-Plathe, F., S. C. Rogers, and W. F.van Gunsteren Chem. Phys.Lett., 199, Muralidhar,R.,D.Ramkrishna,H.Nakanishi,and

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Nakano, Y., and J. W.Evans J. Chem. Phys., 78, Neogi, P. AZChE J., 29, Neogi, P. AZChE J., 29, Nicholson, D. J. Chem. Soc. Faraday Trans. Z, 71, Nicholson, D., and N.G. Parsonage Computer Simulation and the Statistical Mechanics of Adsorption, Academic, New York. Nicholson, D., J. K Petrou, and J. H. Petropoulos Chem. Eng. Sci., 43, Norde, W. Adv. Colloid Znterface Sci., 25, Pace, R. J., and A. Datyner J. Polym. Sci., Polym. Phys. Ed., 17, Pace, R. J., and A. Datyner J. Polym. Sci., Polym. Phys. Ed., 17, Pace, R. J., and A. Datyner J. Polym. Sci., Polym. Phys. Ed., 17, Park, 1.-A., and J. M. D. MacElroy Mol. Simul., 2, Peterson, B. K., and K E. Gubbins Mol. Phys., 62, Peterson, B. K., K. E. Gubbins, G. S. Heffelfinger, U. Marini Bettolo Marconi, and F. van Swol(l988). J. Chem. Phys., 88, Prager, S. Physica, 29,

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Prigogine, I. Introduction to the Thermodynamics of Irreversible Processes,2nd ed., Wiley-Interscience, New York. Mol. Simul., 8, Raghavan, K., and J. M. D. MacElroy Reyes, S., and K F. Jensen Chem. Eng. Sci, 40, Rigby, D., and R. J. Roe J. Chem. Phys., Roe,R.J.,Ed. Computer Simulation of Polymers, Prentice-Hall,Englewood Cliffs, NJ. Principles of Adsorption and Adsorption Processes, WileyRuthven,D.M. Interscience, New York. Ryckaert, J. P., G. Ciccotti, and H. J. C. Berendsen J. Comp. Phys., Sahimi, M. Chem. Eng. Sci., 43, Satterfield, C. N. Heterogeneous Catalysis in Practice, McGraw-Hill, New York. Satterfield, C. N., C. K. Colton, and W. H. Pitcher, Jr. AIChEJ., 19, Schoen, M., J. H. Cushman, D. J. Diestler, and C. L. Rhykerd, Jr. J. Chem. Phys., Shante, V. K. S., and S. Kirkpatrick Adv. Phys., Sirkar, K. K, and D. R. Lloyd, Eds. New Membrane Materials and Processes for Separation, AIChE Symp. Ser., No. Sok,R.M.,H. C. Berendsen, and W.F. van Gunsteren J. Chem. Phys., 96, Sonnenburg, J., J. Gao, and J. H. Weiner Macromolecules, 23, Steele, W. A. The Interaction of Gases with Solid Surfaces, Pergamon, Oxford. Stem, S. A., and H.L. Frisch Annu. Rev. Mater. Sci., 11, Suh, S.-H. and J. M. D. MacElroy Mol. Phys., 58, 445. Swope, W.C., H. C. Andersen, P. H. Berens, and K.R. Wilson J. Chem. Phys., Takeuchi, H. J. Chem. Phys., 93, Takeuchi, H. J. Chem. Phys., 93, Takeuchi, H.,and K Okazaki J. Chem. Phys., 92, Takeuchi, H.,R. Roe, and J. E. Mark J. Chem. Phys., 93, Theodorou, D.N.,and U.W. Suter Macromolecules, Torquato, S. J. Stat. Phys., 45, Trohalaki, S., D. Rigby, A. Kloczkowski, J. E. Mark, and R. J. Roe

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

INTRODUCTION A.

Scope of Molecular Simulations

Many technologically importantprocesses 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 bamers to atmosphericgases. To solve the polymer design problem successfully, one needsto relate the chemical constitution of the polymer (setduring 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 andmorphology)and sorption andpermeation properties. One wayis direct experimental measurement; a second way is the formulation and use of phenomenological correlations based on experimental evidence froma wide variety of systems. third way to structure-property relations, which constitutes the main focus of this chapter, is the development and application of theories and simulation techniquesthat 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

68

THEODOROU

gas solubility inpolymermelts, the dual-modesorptiontheory for polymer glasses, and activation energy and free-volume approaches to the diffusivity of pure or mixed gases in polymers. 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 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 thatcorrectly capture the salient features of molecular geometry and energetics can helpidentify expected changes in performance with changes in chemical constitution. This is valuable for addressing ‘‘what if ” questions of a materials design nature without going throughcostlytrial-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 ofmethoddevelopmentandvalidation 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 SI-TIONS

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69

such systems. We briefly discuss the statistical mechanical principles that can be used to construct simulation techniquesfor 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 II. Section I11 deals with the prediction sorption thermodynamics; theproblem of calculating the sorption isotherminapolymer 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 conclusionsand 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 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 Nppolymer chains occupying a volume V. The potential energy of this system is a highly convoluted function 'Tp(rp)of the coordinates of all atoms constituting the chains, which we symbolizecollectively as a vector r,. For macroscopic specimens, the dimensionality of the configuration space from which r, takes values is on the order of Avogadro's number. In the schematic of Fig. 1 we display the entire configuration space asa single axis, for graphical simplicity; the potential energy hypersurface 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(-pV,), where = l/k,T. 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 'T(rp, is modified. The different conhowever, are again Boltzmann-distributed. From the point figurations (r,, 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, theonly difference being the involatility

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N. V Const.

Schematicrepresentationofthepotentialenergy V, asafunction of the microscopic degrees of freedom r, 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 andlong-time-scalemotion 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, 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 below TBare 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

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by formulating the phaseequilibrium problemseparatelyineach disjoint r, region of configuration space and then averaging the results according to the weight withwhich 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 reduce barrier heights between r, 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 gOvemed by the penetrant sue and interactions with the polymer as well as by the shape, she, connectivity, and time scales of thermal rearrangement of Unoccupied space within the polymer. In a high-temperature melt (T >> Td, 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. in the free-volume picture 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 thepenetrant’s presence, leads to the formation 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 diff3sion. 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 T,, on the otherhand, onewould 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 (i.e., 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 Tpis 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 es-

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Figure 2 Schematic representation of amorphous cell and periodic boundary conditions. Wo-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.

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sentially unperturbed by interactions between topologically distant sites along the chain. a consequence, the mean square radius of gyration (S’) of a long chain in the bulk is related to the number of skeletal bonds n and to the skeletal the characteristic ratio, is a chain bond length 4 as (S’) = (1/6)nCJ2, where 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)’”, that different images of the Same chain do not interact significantly with each other, as such interaction may distort the long-range conformationalcharacteristics 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 chainsdo not have the opportunity to relax their long-range conformation, meaningfulresults 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 noncyclic 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‘ 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’ direction when examined at length scales substantially larger than L .

2. MolecularMechanics The objective of molecular mechanics (MM) is to generate static minimum energy configurations at prescribed density, corresponding to the local minima of V , 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-bondgrowthofthe parent chains in the amorphous Cell, observing periodic boundary conditions. This growing procedurecan be based On FloY’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 VPWith 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 finally introduces attractive interactions aswell (Theodorouand Suter, 1985). 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

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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 aglassysystemthat is locallyarrestedin the neighborhoodof 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, Mansfield and Theodorou, Onthe other hand, static minimumenergy structures generatedby 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 throughnumerical integration of the equations ofmotion for all the degrees of freedom. Originally developed in the microcanonical (constant number of molecules, volume, and total energy, or ensemble, the MD method and isothermalcan now be applied straightforwardly in the canonical isobaric (NPT) 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, The MD method has been applied widely to united-atom polyethylene-like systems by Rigby and Roe 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). asset of molecular dynamics is that it provides directly a wealthof detailed informationon 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 timesof 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-

(m

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ployed in MD vitrification are on the order of 10'l-lO'o K/s. Some problems associated with the generation of amorphous model structures through MD have been addressed by McKechnie et al. (1992).

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 r, occurs with frequency proportional exp["Vp(rp)], 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 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). initial configuration for starting the MCsimulation can be obtained through one of several strategies. Vacatello et al. (1980) started their pioneering simulations of liquid triacontane by placingchains onatetrahedrallattice, whereas Boyd (1989) employed a crystal, which was melted during theMC equilibration stage, as an initial configuration for his simulations of liquid tetracosane. Minimum energy configurations from molecular mechanics have also been used (Dodd et al., 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,becauseofthe 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 al., 1992), an end section of random length is cut from a randomly chosenchain and regrown bondby 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 al., 1993), on the other hand, employs coordinated torsions around seven adjacent skeletal bonds to change the conformation of chains withoutaffecting bond lengths and bond angles. Moves that involve cut-

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ting chains and rejoining the resulting segments inadifferentway,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 circumventthe 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 appliedinavariety 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). A variety of properties can be predicted from an amorphouspolymer model and tested against experimental evidence to confirm that the model provides a reasonable representation of reality. The pair distribution functions g(.) 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 onecan predict small-angle neutron diffraction patterns obtained experimentally by selective labeling of single chains and thus check long-range conformationalcharacteristics 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-difisivity of chains and viscosity in the melt and frequency and activation energy of relaxational motions, can be compared to scat-

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1.t

-

0.7

4 I

2z

CA

v A

-0.2

-1.1

c20

-2.0

(4

4

8

12

1

)

Magnitude of wave vector (i-')

Figure 3 Structural and thermodynamic predictions from isothermal-isobaric MC 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 SORPTlON AND DIFFUSION

Figure 3

79

Continued.

Table 1 Conformational Characteristics of Tetracosane Chains Chain property

Bulk NPT Monte Carlo"

Continuous unperturbedh ~~

350 43 0.659 0.394 0.532 0.110

354 44 0.666 0.406 0.524 0.104

"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.

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3a compares the k-weighted structure factor for simulated C, at 315 K and 1 bar (curve) against accurate neutron diffraction measurements (points) (Habenschuss and Narten, 1990). Table 1 displays a comparison of several C, singlechainconformational characteristics, namely the rootmean square radiusof gyration ( s ’ ) ’ ~ ,the rms end-to-end distance (r’)’’’, the mean fraction p I of skeletal bonds in a trans conformation, the mean fractions p f fand p s of pairs of adjacent bonds in trans-trans and trans-gauche conformations, and the mean fraction pfg,of triplets of successive bonds in a trans-gauche-trans conformationagainstthe corresponding predictions for unperturbedchains. Figure 3b shows simulation predictions for the specific volume of C& and C& at 450 K as a function of pressure (triangles) compared with experimental values(squares and dotted lines) (Dee et al., 1992). The comparisonsindicate 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 1I.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 theinternal 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 polymermatrices through positron annihilationlifetime measurements have appeared recently (Malhotra and Pethrick, 1983; Kluin et al., 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 ofcharacterizing void space in model structures, it is meaningful to considereach 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 ro. 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

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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 @oddand Theodorou, 1991; Greenfieldand Theodorou, 1993). Conversely, the unoccupied volume of a polymer configuration coincides with its accessible volume in the limit 1-2 + 0. 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 becomeavailable. At some critical penetrantradius, r,, an infinitely extended cluster appears that spans the entire periodic array of boxes representing the polymer; that is,percolation accessible volume occurs throughout the model polymer. The percolation threshold value r, 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 r, in primary boxes different L can be used to deduce the percolation threshold in the limit L For glassy atactic polypropylene, the average r, in the infinite box limit is around 0.9 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 [S(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. 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 threefoldor higher overlaps are observed are considered fully occupied. exact analytical solution of

+

-?,

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THEODOROU

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

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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). Unoccupiedand accessible volume calculations have beenconductedon modelglassy atactic polypropylene(&PP),poly(viny1 chloride) (PVC),and bisphenol A polycarbonate (PC), as well as on model melts of united-atom polyethylene(PE)and &-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 2 0.001 for &-PP and 0.39 2 for PC. The accessible volume fraction is a monotonically decreasing convex functionof penetrant radius (see Fig. 5). In glassy &PP, Arizzi et al. report accessible volume fractions of 0.16,0.08, and for He, 02,and NZ,respectively; the corresponding values in PC are 0.19 for He, 0.12 for Oz, and 0.11 for N2. It is interesting to compare theavailabilityof free volume between a glassy polymer and an atomic glass; random close-packed (rcp) configurations of spheres constitute a reasonable modelfor the latter. Figure 5 shows such a comparison between&PP and an rcp structure consisting of 2 diameter spheres. This diameter is commensurate with that of methylene, methine, and methyl segments in the propylene model; the atomic “gran~larity~~ of the rcp and polymer structures is thus comparable. Although the unoccupied volume of the rcp structure is somewhat higher, its accessible volume falls more rapidly with increasing penetrant diameter than thatof the polymer. The macromolecular constitution of the polymer gives rise to a broader distribution of void sues, which can accommodate larger penetrants. This is also reflected in the percolation characteristics of accessible volume in the two types of glassy structures: r, for the rcp structure is around 0.55 which is significantly smaller than the value of 0.9 for the polymer glass. For both the rcp and polymer glass tures, the accessible volume fraction at the percolation threshold is in the range 0.02-0.04, close to the value observed for the “Swisscheese” model(see Chapter 1 of this book). For given penetrant radius, the accessible volume is distributed spatially into clusters (see Fig. 4). Several techniqueshave 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

1u5

I....I...,I....I....I....I Ob

1.0

1.5

2.0

2.5

0 0

0.5

1.o

1.5

2.0

2.5

penetrant radius 5 Accessible volume fraction as a function penetrant size in atactic polyand propylene as obtainedfrom Monte Carlo simulations the polymer in the glassy melt (0) states (Greenfield and Theodorou, 1993). The accessible volume fraction a random close-packed (rcp) structure 2 A diameter spheres, representative an atomic glass, is also shown The diameter the rcp spheres is roughly equal to the van der Waals diameter methyl, methylene, and methine units constituting the polymer.

et al., 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 volumespheres of polymer atoms, then the tetrahedron is inaccessible; otherwise, the tetrahedron has a pocket of accessible volume in its interior. W Oaccessible tetrahedra are said to be connected when they share a face (triangle) that is not completely blocked by the excluded volumespheres polymer atoms. Uninhibited passage of the penetrant from the interior one tetrahedron into the other through the shared face is thus possible. Accessible tetrahedra can be grouped into sets of connectedtetrahedrausinga simple connectivityalgorithm(Greenfieldand Theodorou, 1993). A cluster of accessible volume is simply the union 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

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For given penetrant radius r i , 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 foundin clusters of volume to v dv. py(v;r i ) is generally a decreasing function of v; its reliable determination requires analysis of a large number of configurations. Computed plots of versus v seem to exhibit some fine structure (local extrema) related to specific intermolecular packing patterns that depend on the detailed geometry, conformationalpreferences, and interactionforces among chains (Greenfield and Theodorou, 1993); their general appearance is inreasonableagreementwith positron lifetime spectroscopy results (Deng and Jean, 1993). The average cluster size can be characterized through the average cluster volume (v), (first moment of or through the root mean square cluster radius of gyration (Greenfield and Theodorou, 1993). In the range of penetrant radii r: > r,, the average size of accessible clusters is a smoothly decreasing function of r: (compare Fig. The shape of accessible clusters can be characterized by comparing the principal axes of their radius of gyration tensor; in the case of &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:l:l. picture such as Fig. 4a suggests that a penetrant molecule of sufficiently large r l , 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 dif€usion of the penetrant would proceed by infrequent jumps from cluster to cluster through short-lived passages opening momentarily between the clusters (see also Section IV.D). The likely location of such passages that could open up through thermal fluctuations and thus act as difhsion pathways for the penetrant can be identified by examining the accessible volume distribution at a value of the penetrant radius equal to, e.g., r,, for which long-range connectivity is bound to be established.Suchanexaminationrevealsthe coordinationnumber of a cluster, i.e., the most probable number of clusters with which a given cluster is connected through difhsion 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. moreelaborateandcomputationallymuchmoretime-consumingapproach for identifying clusters and passages between themis to analyze the potential energy field experiencedby 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 & connected by bottleneck regions ca. 5-10 long and 1-2 in diameter, in agreement with the geometric analysis (seeFig. 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 connectivityof the network of clusters is a functionof the time scale over which the network is examined. We return to this point in Section lYD.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 &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, andthe void distribution can be characterized permanent over the effective time scale spanned by the simulation. In the

Figure Evolution of thedistribution of volumeaccessibletoHewithinatactic polypropylene, as obtained from long MC simulations of the polymer. (a) Polymer glass configuration at K. (b) Configuration obtained from (a) after million attempted K. (c) Polymer melt configuration 400 at K (d) Configuration MC moves at1 bar and obtained from (c) after 10 million attempted moves at 1bar 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 geometricconsiderations discussed in this section are useful as a prelude andguide to the more elaborate (and computationally much more demanding) energy-based approaches discussed in Section 1V.D.

C. Characteristic

limes 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”3-10”4 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 p-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 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”2-10”o S. As temperature drops they slow downdramaticallyandbecomeincreasingly cooperative, theassociated 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(PlazekandNgai,1991).Finally,large-scale conformational rearrangement, reorientation, and self-diffusionof 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 chainlength, 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 ILB). Takeuchi and Roe (1991a,b) quantified the rate at which individual torsion angles lose memory of their initial values by defining anautocorrelation function

88

THEODOROU

for dihedral angles, R,(t) as

where the ensemble averagesare taken over all skeletal bonds and Over all time origins along the MD simulation. Rm(t) starts at 1 and decays to zero in a stretched exponential fashion. In the melt, the time 7, at which R, has decayed to l / e is studied as a function of temperature. In an infinite molecular weight PE melt at 300 K, Takeuchi and Roe found T+ = 24 PS; in the high-temperature melt, decreased with increasing T with an activation energy of roughly 3.75 kcal/mol, which is comparable to the trans-gauche torsional barrier. As ternperature was reducedtoward T, (which, as definedthrough the break inthe volume versus temperaturecurve, is 201 K for this polymer at a cooling rate of 1.67 10” Ws), was found to increase dramatically. A KWW fit to Eq. (1) at T, gave a relaxation time = 1.39 ns. Within the glass the rate of well-towell conformational transitions was found to be on the order of 1 ns“ 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 T, (Mansfield and Theodorou, 1991). The latter simulation indicated a large degree of spatial heterogeneity in terms of the ability of bonds to isomerizeconformationally. The glass wasfound 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.Torsionalmobility is enhancednear chain ends, although “soft spots” are likely to be found away from ends as well. One canenvision 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 ii be such a unit vector. As a result of thermal motion, the vector’s orientation at time t, ii(t), will be different from its original orientation ii(0). One can form the two time correlation functions = (ii(t) * ii(0))

(2)

and 1 M$) = - (3[ii(t) ii(O)]’ 2 m

-

1)

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

89

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

Me(t) = exp [-(t/+~e)’]

(4)

For a given choice of Q the ratio T,/T~is always larger than 1, its exact magnitude depending on the mechanism of the orientational relaxation. Smith andYoon (1994) chose Q as unit vectors along the pendant C-H bonds of a tridecane liquid. For the C-H bonds attached to the central carbon chains, their explicit atom MD simulations gave T~ = 10 PS at 312 K with an activation energy of 4 kcal/mol. The value of + T ~was found to besignificantly 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 I3C N M R spin-lattice relaxationexperiments. The behavior T~ and T~ for pendant bonds tracked the behavior of T,, all three times being of the same order magnitude. Takeuchi and Roe (1991a,b)studied three Q vectors with their united-atom modelof 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, T~ being approximately equal to T+ and 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 + T ~ being roughly 3807, at 300 K. This anisotropy of orientational 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 apparentrelaxation times for a and b were on the order of 1 ns (Takeuchi and Roe, 1991b; Mansfield and Theodorou, 1991). Bondreorientation 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 < 1 obtained from fitting (4) to the relaxation functions M&) and M&) of a, b, or pendant bond vectors indicate considerable cooperativity of reorientational motion in the melt = 0.45-0.5) and a very high degree of cooperativity in the glass = 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 T,) obtained from dynamic light scattering (Fytas and Ngai, 1988) and two-dimensional NMR (Schaefer et al., 1990) measurements. The latter times are comparable to the ones obtained from mechanical measurements (Fytas and Ngai,

90

THEODOROU

1988) of the relaxation. The question of how to predict the time scales of relaxation in the rubbery and glassystates 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 r,Q) be the position of segment j at time t. The Fourier component of the instantaneous density corresponding to wavevector k at time t is

The time evolution of F($ t) reveals how density fluctuations occurring over length scale 2dlkl 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 F&, t ) defined as

F&

t) =

L

Ns

(2

exp { -i k [rj(t) - rj(0)]}

,=l

The time evolution of F&, t ) reveals how individual segments lose memory of their original position through motions occurring at a length scale 21~44.Its Fourier transform with respect to time, S& 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 F& t), and the characteristic frequencies associated with the relaxation at a given length scale can be extracted from S&, 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 al., 1990). Similar investigations were conducted by Takeuchi and Okazaki for a model polymer (Takeuchi and Okazaki, 1993b); the decay of F& t) can be observed only at length scales commensurate with the distance between neighboring carbons on different chains (k = 1.5 A”); at this length scale, and at the glass temperature determined from the V(T) behavior of the model polymer at a cooling rate of 1.67 10” K/s, the

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91

relaxation time determined from a KWW fit to F& 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.

PREDICTION OF SORPTIONTHERMODYNAMICS A. Statistical Mechanics of Sorption in a Compressible, Involatile Medium The phase equilibrium between a polymer and a multicomponent fluid mixture can be predictedfrommolecularlevel structure and interactions basedon straightforward principles of statistical mechanics. formulate this phase equilibrium problem in a general way, consider the system shown in Fig. Phase p (the fluid phase) and phase (the polymer phase) together constitute a closed systematconstanttemperature T andpressure P . The system consists of c components. Component 1 is the polymer chains. Components 2, . . . , c are 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 = N4 NP (i = 1, 2, . . . , is fixed. Each of the components can be exchanged freely between the and phases, however. Also, the phase is free to expand (swell) against the phase.

+

Phase Can exchange energy and mass with phase p Can expandlcontract against phasep

7 Thermodynamic system considered in the phase equilibrium formulation of Section IILA.

THEODOROU

92

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 the fluid phase on apolymer-free basis. If c = 2, T and P suffice for specifying the intensive state of the two phases. Let denote collectively the vector of microscopic degrees freedom phase a for given NY, N;, . . . ,N:. f l u = (V", ru)encompasses the volume phase a and the Cartesian coordinatesfor all atoms all molecules constituting that phase. Similarly, will denote the vector of microscopic degrees of freedom of phase for given NB, N!, . . . ,N f . The unnormalized joint Probability density governing the configurations of the two phases is C n U

(x, ...,N:, NB, .. . ,N!;

pN1. . . N c P T

a@)

flu,

C

-- PuN r . . .N:PT (am)

.NFPF i=l

S (Ng

+ NB - Ni)

where

(W)=

1

x! -

n

A:,o,,,

all alom

exp{-p["ir(x,

.. . ,NZ; S ) + P T ] }

stands for the unnormalized configuration-space probability density of phase a inthe isothermal-isobaric ensemble (McQuarrie, 1976) with "ir the potential energy function. The product of thermal wavelengthsA raised to the third power arises from integration over all momentum space; it contains as many terms as there are atoms in phase a intheconfigurationconsidered. The term .NFPT similarly stands for the unnormalized configuration-space probability density phase It is useful to consider the projected probability density for phase a alone, integrated over all possible configurations that phase can assume. This is ,

(x,. . . ,

.

.

N P . . N:PT

= Pu

-

=

N Y . . .NzPT

Pu

am)

I

Qp(N1 - K

.N,-N:PT 7

-

7

(n?

Nc - K,P7 T )

(10)

where Q pis the isothermal-isobaric partition function phase at the indicated numbers molecules, pressure, and temperature. We now specialize to the case where component1 is present in phase only in minute quantities. This would be expected, as the vapor pressure of a pure

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93

long-chain polymer is extremely low:

NY = N I

and

N! =

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

NP c< Ni

and

NY = Ni

(12)

In view of Eqs. (11) and (12), the logarithm of the quantity QP(Nl- NY, N, - N:, P, T ) appearing in (10) can be written as

...,

In QP(O, NZ - N;, . . . ,N, - N:, P, T )

=

-p@(O, NZ,. . . ,N,, P,

+

x

(T, P, )'X

NY

(13)

i=2

where stands for the Gibbs energy of the p phase and p! for the chemical potential of component i in that phase. The composition vectorxP at which the chemical potentials are evaluated has components

Substituting Eq. (13) into Eq. (lo), we can write the normalized configurationspace distribution of phase a as ..

( q , N;,

. ..,NZ, W )

n

(a")

exp

Y!(T, p, x"]

(15)

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

THEODOROU

94

the amount of polymer NI remains fixed atall times while the volume is allowed to fluctuate. In applications it is more convenient to use the fugacities f!(T, P , )'X in place of the chemical potentials FB(T, P, xB) (Prausnitz et al., 1986). Substituting N F m fromEq. we canrewrite (15) as '

,,N~fg..

.lk (NZ7

U

' '

*

7

K7

v", rU)

where

(17)

In Eqs. (16) and (17), ni is the number of atoms of which a molecule of species i is composed. For a given combination of {Nq} values, the vector of atomic positions r" has dimensionality Nqn,. is an intramolecular configurational integral for a molecule of type i, calculated as

Zi"'"

E

I

exp

(al,

. .. , ri ni-l)]d3rild3ri2 -

d3rini-l

(18)

Note that the integral of (18) is taken over only ni - 1 atomic positions; the three degrees of freedom corresponding to overall translation of the molecule are notintegratedover. The term is theintramolecularpotentialenergy of a molecule of type in the ideal gas state the total potential energy of phase is merelyasum of such terms. For a monatomic species, 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 p. In the following we

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

95

drop the symbol (Y with the understanding that we focus on the polymer phase. Some condensation of notation can be achieved by introducing the quantity

has dimensions of (volume)"''. Rewriting the vector of atomic positions r as (r,, r,, . . . , r,), where ri is the 3Nini-dimensional vector of atomic positions of all molecules of type i, we recast Eq. (16) as .

(N2, ...,N,

V, rl,

. .. , d

pNlf2 . has dimensions of (volume)"-'Nci~lN'"', as expected from the fact that it is a probability density in the variables {N2, . . . , N,, V, rl, . , rc}. 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, . . . ,N, at a fixed composition xp, can be extracted from the probability density of (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 in place of 1 (polymer) and 2 (sorbate), respectively. The probability density function for the polymer phase, Eq. (20), reduces to '

..

V, r ~ r,A )

-

exp(-PPv) exp [-Pv(rp,

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

where the 3n,-dimensional vector rpiencompasses the position vectors of all atoms constituting the macromolecule i and the 3n,-dimensional vector r, en-

THEODOROU

96

compasses the position vectors of all atoms in penetrant molecule k. A classical flexible model (G6 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 Npand 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 rpiand carried out as in a pure polymer simulation); translations, rotations, and conformational rearrangements of the sorbate molecules (bringing about changes in rh); insertion of an A molecule (increasing N A by 1); deletion of an A molecule (decreasing by 1);and dilations/ contractions of the simulation box (changing 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 NpfAPT 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 of the polymer phase (providing a direct measure of swelling phenomena) and the average number of sorbate molecules (N,,) present in the polymer at eachpressure, providing the sorption isotherm.Results fromsuch a Monte Carlo simulation approach for the direct prediction of the sorption isotherm have not yet been reported, but the approach hasbeen developed (Boone, 1995). The slopeof the sorption isotherm in thelimit 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 phaseat equilibrium under given Np,P, T is r

N~

( N ~=)

dV

r

d3Npprp d3NAnArA p N ” f l ( ~ , , V, r,, rA)

NA=O

where

d3r,

- - d3r,,

exp[-pT(rp, rA1,. .., rmJ1

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

97

is the configurational integral in the isothermal-isobaric ensemble for Npmolecules of polymer and molecules of penetrant, withdimensions of (VOIUme)NPnP+NWA+l 0. In this limit, -,0, Consider now the polymer phase in the limit P andby Eq. + Equation (23) thenreduces to +

1

dVexp (-pPV)

I I I

- pV:”(ril) - p‘VpA(rp,

d3N*prp d3”ArAl exp

dVexp(-pPV)d3N*prp

I

exp [-pVp(rp)] exp

dxmA-’)r:,

The term V:h in Eq. (25) is the intramolecular potential energy of a single sorbate molecule it is a function of only - 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 ril for the 3(n, dimensional vector (mll, rA,z,. . . , It follows that rAl = @Al, The term ‘ V p encompasses all intramolecular and intermolecular interactions of the macromolecules constituting the polymer matrix, while “ri, is the sum of all intermolecular interactions between the penetrant and the polymer chains. [Compare more explicit notation of Eq. (22).] The average volumeof the pure polymer phase at T and P is obtained the isothermal-isobaric ensemble as

-

1

dV exp (-PPI‘)

I

d3N’”prp exp [-

pvp(rp)]

98

THEODOROU

where we have rewritten the volume V as an integral over the three-dimensional domain spanned by the position vector rAlnA. Combining Eqs. (25) and (26), and recognizing that P/kB T is the molecular density c! in the fluid phase, which behaves as an ideal gas in the considered limit P we obtain

-

As defined in Eq. (27), S,, 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 + 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, weightedaccording 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 1I.B. The low-pressure solubility can be expressed per unit mass rather than per unit volume ofpolymer. Invoking the definition of the Henry’s law constant HhP(Prausnitz et al., 1986),

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 Widom insertions of a “test” penetrant molecule. Heat of mixing effects between the polymer and the penetrant can readilybe analyzed in the NpfAPT 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 mixing:

Qs(T7

p 7

XA)

h!( T7 p ) - z~(T7p7

where h2 and SA are the molar enthalpy 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

where ze(T, P) = PvB/RTis the compressibility factor of pure A in the fluid phase, is the partial molarvolume of A in the polymer phase, andthe notation "eq" indicates that the derivative is taken under conditions of thermodynamic equilibrium. It is possible to calculate Qs from molecular-level information as

+

Qs = (h! - h&um) RT

+ NAvo ('V!p)ig

where and molar the areenthalpy and average potential energy per molecule of pure A in the ideal gas state, NAvo is Avogadro's number, and the brackets denote averagesin the NpfAPT 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 Q*,O

- k J + ('V.Y)ig+ W P+ PV)NpR

"

NAVO

-

+" I r +, " I r Z + PV] exp(-

PsrAp))widom

(exP(-P"IrM))wihm In the above discussion of the statistical mechanics of sorption it was implicitly assumed that the polymer matrix is in thermodynamic equilibrium, that its configurations are distributed with a probability density proportional to exp PP(rp)+ PV)]. If the matrix is glassy, this density distribution holds separately within each a large number of disjoint regions in (rp,V) space, in which the glass islocally"lockedin" (compare Fig. 1). pointedoutin Section I.B, the relative weight with which each such region contributes to the thermodynamics of sorption at low sorbate concentration dependson the history of formation of the glass. In a simulation, for example, regions around local

THEODOROU

l00

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 becomemutually accessible, theconfiguration-space distribution equilibrating betweenthem.Whenthis 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 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 r, are neglected. Penetrant molecules are assumed to reside within sites. A site is defined as a region in the three-dimensional positionspace that encompasses a local minimum of the potential energy olr(rAl;rp)under the considered fixed r,. 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. All interactions between penetrant molecules occupying different sites in the polymer are neglected; i.e., all terms olry of the potential energy function of Eq. are set to zero. With these assumptions, the configurational integrals in Eq. tuted by

can be substi-

where K is the total number of sites in the considered polymer matrix and the configurational integral for site j , calculated as

is

MOLECULAR SIMULATIONS

OF SORPTION AND DIFFUSION

l01

The primes on the summation symbols in Eq. (32) indicate that the sites (jl,j2’ . . . jNJ must all be different from each other. Eq. gives Upon substitution in Eq.

(34) j=

For the structureless spherical sorbates considered here,

Thus, the sorption isotherm emergesfrom the Gusev-Suter theoretical approach as a sum of Langmuir terms, each associated with a particular site in the static polymermatrix.Different sites are characterized by different bp The overall Henry’s constant is inverselyproportionalto Xzlbp As long as bj << 1, the Langmuir term contributed by site j can be approximated by a straight line. The predictedisothermthen is expected to look like a sum of Henry’s lawand 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/bisphenolA polycarbonate. Sites j were identified bycomputing T(rAl; rp) at the nodes of a three-dimensional grid of lo7points running through each static structure. Theborders of sites were determinedthroughthreedimensional steepest descent constructions initiated at each of the grid points. (See also SectionN.D.)The site configurational integrals of Eq. were obtainednumerically from thepotentialenergy values at the grid points.In describing the dispersive part of the interaction between penetrant and polymer atoms, Gusevand 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 amountwas broken up into contributions from families of sites with different l/bj, according to Eq. (34). Sites with llbj in the range 50-500 bar were found to be most important; this range lies significantly higher than the

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value l/bj == 10 bar obtained from the dual-mode sorption model. Clearly, the Gusev-Sutermodelcannotpredict swelling andplasticizationphenomenaat 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 ofsmall 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. Muller-Plathe (1991b) was the first to apply the Widom insertion technique to estimate the solubility So through the canonical ensemble counterpart of (27). He studied H,, NZ, and Oz, 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 NVT MD for 200 PS, and then subjected to a 2 ns longNVT MD production run; configurations fromthis run were used for the Widom insertions. Results were reported in theform of free energies per= -k,T In S,. Predicted values of pa were found to decrease in the correct order He > H, > NZ > 0, > CH, (order of increasing condensability). The absolute values of pa, however, were lower (more favorable for sorption) by 5-10 kJ/molthan values obtainedfromexperimental data fromsorptionin ethylene-propylene copolymer and in the amorphous fraction of isotactic polypropylene. In fact, for NZ and the more condensable gases it is predicted that So > 1, contrary to the experimental finding S,, 1. Differences in pa 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 suchaveraging 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 Muller-Plathe (1991b) for polypropylene is too attractive. Whether the potential parameterization is satisfactory could be judged from the predicted PVT behavior ofthepurepolymer;unfortunately,however,pressureresultsarenotreported

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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 1I.B). Miiller-Plathe finds that decreases linearly with the Lennard-Jones well depth 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 al. (1992) also report computations of the solubility gaseous He and CH, in a polydimethylsiloxane (PDMS) liquid in the Henry’s law region. S, and 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 Widom insertions, a map of the accessible volume was maintained using a 100 100 100cubic grid running throughthe polymer. As in the work of Miiller-Plathe, predicted values were too low (attractive) by 7.8 kJ/mol for CH, and by 3.6 kJ/mol for He. In contrast to the solubilities, diffusivities were predicted in excellent agreement with experiment. (Seealso Section NB.) Qualitatively,the simulation reproduced the experimental finding that the diffusivity He in PDMS exceeds that of CH,, but the overall permeability of CH, is higher owing to its higher solubility. remarkable set Monte Carlo computations of alkane solubilities in a polyethylene melt was conducted by de Pablo et al. (1993). In these studies the linear polyethylene matrix was represented as a liquid of GI chains undergoing NPT MC simulation, and penetrants in the range were examined. The solubilityinthe Henry’s lawregion was computed by theWidominsertion technique, Eq. (28). For the penetrant sizes examined (de Pablo et al., 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 (28); an extremely large number of random insertions would have to be carried out to obtain a good estimate of the solubility. remedy this problem, de Pablo et al. 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 recastinawaythat takes into account thisbiased sampling and thus serve as a basis for an efficient computation of the solubility. Figure 8 displays the weight fraction Henry’s constant H , = HkpMp/MA(where MPand M,, are the molecular weights of the polymer and the penetrant, respectively) estimated by de Pablo et al. (1993) at 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 thePVT properties of long alkanes, are inexcellentagreementwithexperiment. The same authors conducted calculations solubility at higher pressures, outside the Henry’s law regime. In these calculations they used Panagiotopoulos’s Gibbs ensemble, de-

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Data of Maloney Prausnitz(1976) A CCB ghost molccule. NIT ensemble

I

I

I

4

6

8

10

Number of Carbon Atoms Henry’s constants for alkanes in molten polyethylene as predicted through continuum-configurational-bias Widom insertions within a polymer matrix undergoing NPT MC simulation (de Pablo et al., 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 (8) for two phases coexisting at equilibrium at specifiedpressure,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 al. used the CCB scheme [CCBG method (Laso et al., 1992)l. The bias introduced by the use of CCB-based exchanges is removed by appropriatedesign of the Monte Carlo selection criteria. Computations the solubility of pentane in polyethylene by CCBG clearly show deviations from Henry’s law at pressures above 10 bar.

PREDICTION Statistical Mechanics of Diffusion A thorough review of the statistical mechanical formulation diffusion is given by MacElroy (Chapter 1, this volume). Inthis section we summarize somebasic 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 temperatureand pressure and free of any external force fields. The system as a whole does not move; i.e., macroscopically, the center of mass the system remains fixed in space. There is, however, a mac-

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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 (i = P) to denote the macroscopic flux of component i in a coordinate framethat remains fixed with respect to the center of mass of the system; this coordinate frame is usedthroughout our analysis. is measured in molecules per square meter per second. We use the symbol to denote the microscopic (molecular) current of component i, that is, the mechanical quantity

where vei 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 ouranalysis. Linear response theory leads to the following equation for the thermodynamic flux in terms of the thermodynamic forces ( V P A ) ~and ~ ( V F B ) ~ where ~, stands for the chemical potential of species i, in joules per molecule, and V is the volume of the considered macroscopic elementof the system:

denote ensemble averages in an equilibrium The angular brackets in E q . 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 in terms of ( V F A ) ~only. , ~ The Gibbs-Duhem equation, applied to the considered macroscopic element of the system (assumed to be in local thermodynamic equilibrium) leads to (vFA)T,P

where -JA

+ XP (vFP)T.P

=

is the mole fraction of species i. Combining Eqs. =

1

V ksTxp [XP

1

( W ) .i,(t)) dt - xA

1

and (38) yields

(j.40) .jp(i)> dt] (VCLA)T,.P

An entirely analogous equation holds for component P.

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Since there is no macroscopic displacement of mass (flow) in the considered volume element, the fluxes J A and JPmust satisfy the condition

with m, and mp 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

The binary d i h i v i t y or transport diffusivity D in our system is defined through Fick's first law (de Groot and Mazur, 1984; Bird et al., 1960):

where is the mass density and wi is the mass fraction of species i inthe considered macroscopic volume element. Comparing Eqs. (39) and (43) and substituting the chemical potential p,A in terms of the fugacity f A (Prausnitz et al., 1986), we obtain a microscopic expression for the binary diffusivity:

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 interdifjizsion current

This expression is

+

where N = NA Np. A form of Eq. (46) was presented by Hansen and McDonald (1986). (Note that a factor of 1/N is missing from the equation given in that reference.) The equivalence of Eqs. (44) and (46) can be readily shown on the

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AND DIFFUSION

-

basis of Eq. (42), taking into account that the equilibrium average(i,(O) 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

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

with rm,&) 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-dimivity 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 meansquare displacement of a molecule linearly with time at long times. The self-diffusivity is extracted from the proportionality constant in that relationship:

In extracting from an equilibrium MD simulation, one usually averages Eq. (49) over all molecules 4 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-difhsivities cantherefore be obtainedwith 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)]:

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 polymermatrix and (at high

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concentrations) with other penetrant molecules.A plot of the normalized velocity autocorrelation function (veA(0) VeA(t))/(lVeJ), 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 D,, and D,,p.Substituting the microscopic currents into Eq. (44) from their definition, (36), we obtain

Consider the system in the limit where A is infinitely dilute in P * 0). In this limit, the cross-correlation terms between the two species in Eq. (51) drop. Also, different molecules 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 111), and thus the thermodynamic derivative in (51) reduces to unity. Invoking Eq. (50), we conclude that lim D = lim D,, wn-0

Equation (52) is a rigorous statistical mechanical result. 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 different species) are negligible. This assumption leads to

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

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109

-

300K

-0.5

c

0

0.4

0.8

1.2

1.6

2

t (PS)

Normalizedvelocityautocorrelationfunctions for amethanepenetrantin amorphous polyethylene, as accumulated from PS long trajectories at 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 cageis less mobile. The undulations at long times make clear the computational difficulties in calculating the self-diffusivity as a time integral of the velocity autocorrelationfunctionthroughtheGreen-Kuhorelation.[ReproducedfromPantandBoyd with permission.]

that the thermodynamic correction term (a In $/a In xi)T,pis the same for both species (Gibbs-Duhem equation), one readily arrives at

Equation (54) is used frequently to express the composition dependenceof the binary diffusivity in liquid mixtures. In a polymer/penetrant system, the self-diffusivity Ds,pof polymer chains is typically much smaller than that of penetrant molecules; its contribution to (54) can thus be neglected. Equation (53) can be rewritten in a more practically useful form as

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110

The self-diffusivity of the penetrant, D,, binary diffusivity.

plays a dominant role in shaping the

EquilibriumMolecularDynamicsSimulations SimplifiedPolyethylene-LikeModels 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 Ds,* 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 withinthe 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 carried out microcanonical MD simulations of 20 small spherical moleculesresembling oxygen in 30 united-atom 20-mer molecules of “polymethylene.” The model representation of RigbyandRoe was used for the chain molecules. The predicted self-difisivity 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 energyunderconstantpressure, indicating that the thermal expansion of the polymer (increase inaccessible volume with temperature) drastically affects diffusion. The activation energy underconstant 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, v , 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 foundto correlate well with v, according to the simple Fujita equation

D,, = Ad exp (- &/vf)

(56)

(see Fig. The relaxation frequency of internal rotations, which is on the order of S, also exhibits a dependence on v, of the form of Eq. (56); this dependence,however, is considerablyweakerthanthat of the selfdiffusivity. Under the high-temperature melt conditions corresponding tothe simulations of Takeuchi and Okazaki, T+ seems dominated by the height of intramolecular

MOLECULAR SIMULATIONS OF SORPTION

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111

t 2.4

2.8

3.2

l / Vf

Dependence on theself-diffusioncoefficientofsmall,oxygen-likemolecules 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 andare given inreducedunits.[ReproducedfromTakeuchiandOkazaki 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 the same density, where the torsional barriers were artificially removed. T,,,inthismodified chain liquid was found to be shorter by more than three orders 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 v, in the original system and in the barrier-less system were almost identical. 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. interpretation of these results in the light of the Brandt theory of diffusion was proposed by Petropoulos (1994). The effects of matrix chain lengthwere briefly examinedby Takeuchi (1990b) by comparing the results the above simulations with MD trajectories in a PE-

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0.25

0.19 1/T

Temperaturedependence 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 torsional barriers, filled symbols). All data are atthesamepressureandare giveninreducedunits.[ReproducedfromTakeuchiand 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,, and T+ were found to be lower by a factor of 2 in the long-chain system. The changein 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-difhsivity, Takeuchi et al. (1990) undertook MD simulations in a series of infinite-chain matrices in which the equilibrium value of the bond angle 8 was varied systematically between 100' and 150". All these "mutated" PE-like systems and the original system (equilibrium 0 = 109.5') were studied under the same overall fractional unoccupied volume v, and temperature. Changes in 8 induced significant changes in the packing of chains. Increasing brought about a decrease in the most probable intermolecular distance r,,, between mers andsharpened the structure of the polymer liquid, as seen from the intermolecular pair distribution functions. The tendency

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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 nottruly equilibrated; it was strongly affected by the artificial boundary condition used A local unoccupiedvolume to create the infinite-chain models (see Fig. 2b). fraction &(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 v: fv(rm) was used as a basis for comparing the unoccupied volume distributions of the different polymer models. v: was found to be a decreasing function of 8, reflecting the tight, parallel alignment of chains at large 8 values. The penetrant self-diffusion coefficient was found to increase roughly linearly with the chain spacing r,,, and to obey a Fujita-type equation of the form of (56) as a function of v:. Given that segmental mobility, as quantified through 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. attempt to correlate 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 1I.B.) MD simulations of diffusion of two different spherical penetrants (roughly corresponding to 0, and He) were performed in three different infinite-chain polymer matrices (characterized by different 8 values) over a variety of densities (therefore, of free-volume fractions). In parallel, the accessible volume distributions at a penetrant radius r, 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 n, present at percolation as D,, = A, exp (-B,, n,), where the constants A,, and B, 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 asThis result is suspect, as it indicates that the volumeweighted density distribution of clusters pv(v;r,)should be anincreasing function of v, contradicting recent extensive studies in a similar polymer (Greenfield and Theodorou, 1993). Subject to this caveat, the quantity log (Ds, min) was found vary linearly with the fraction a 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,, is proportional to both the kinetic velocity of

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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,-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 formulationfor 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 temperaturedependence in the network than in thesimple 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 crossoveroccurred. 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 1I.B.)

2. More Realistic Melt Models The latest equilibrium MD work on gas diffusion in polymer melts and rubbery polymers has been characterized bya trend to use more elaborate representations the polymer matrix and penetrant species, in an effort to predict self-diffusion coefficients quantitatively. In parallel, the mechanismof motion of the penetrant has been explored through detailed analysis of MD trajectories. Muller-Plathe (1991a) simulated 20 methane molecules in a system of 20 united-atom pentacontane (Cso)chains at 300 K using the parameterization providedby 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 mainlythroughinfrequent jumps oflength less thanthepenetrant 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, Muller-Platheconsidered ascaling factor A multiplying the penetrant/polymer potential The logarithm of D,, was found to decrease linearly with A in the range 0.1 A 1.0, and the associated correlation is

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proposed as a means for estimating diffusional characteristics efficiently through simulation. A more recent investigation (Muller-Plathe, 1992) focused on H,, O,, and CH., in atactic polypropylene at 300 K, generated by an MM method ('I%eodorouand Suter, 1985) and subsequently relaxed by molecular dynamics. In comparison with PE, this polymer has the advantage being noncrystallizable, which makes the validation 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 (CH,) exhibited the most expressed hopping motion; the motion of H, 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 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 H, and 0, were overpredictedby factors roughly 8 and 3, respectively, whereas that CH, was very close to experimental. This trend improved agreement with experiment with increasing penetrant radius led Muller-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 0, were compared in amorphouspolyisobutyleneusing a fully explicit representation and a representation wherein methylsare lumped into united atoms. Itwas concluded that the use of the united-atom approximation leads to an overestimationof the self-diffusivity and that a fully explicit polymer model is necessary for estimating thediffusivitycorrectly. The interpretationproposed is thatunited-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 away for larger penetrants; this seems supported by the fact that the diffusivity of CH,was correctly captured in a polypropylene model with united-atom methyls. No information is given, however, on whetherthe models invoked were capable of reproducing the experimentalPVT thermodynamicsof 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 CH., 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 Muller-Plathe. Through analysis and

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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-diffbsivities were in excellent agreement with experiment (especially for CH,), 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 0, 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 (AUA) 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 AUA 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 (r') 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 ( ( r 2 )a t'), which causes the mean square displacement to rise very quickly to a finite offset. This is followed by a rela-

L

n

0 Q)

-r2

-9.0

7

-9.5

-

m

E

W

0 0

x -10.0 :

+ 0

-10.5

BPE BPE BPE LPE LPE

(MB) (LWH) (KH) (MB)

(L)

simulation WLF f i t

- 1 1.0

-11.5 1.5

2

2.5

3

3.5

4

1000/T 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 necessa-ry. The temperature dependence of both simulation and experimental points is of a WLF type. [Reproduced from Pant and Boyd (1993), with permission.]

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tively extended regime, wherein the mean square displacement growsless than linearly with time (i.e., (r’) f with v < l),but where the rate of growth d(r2)/ dt exceeds that observed in the long-time diffusive regime (6D,,,). The origin of this subdiffusive region, which Pant and Boyd refer to as the “cage effect” region, is discussed in more detail in Section W.D. At low temperatures (e.g., 280 K inPE), this subdiffusiveregion extends out to times on the order of 200-300 ps ((?)ln = 3-4 A), beyond which a clearly diffusive region (d(?)/ dt = const. = 6D,$ 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,, over successivetime intervals of duration They show that if is chosen long enough, successive changesin the averaged position of the penetrant are uncorrelated, i.e., the averaged trajectory behaves like a random walk; 7 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 displacementof 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 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. Incontrast to PE, PIB displayeda 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. NonequilibriumMolecularDynamics The simulation method of nonequilibrium molecular dynamics (NEMD) was applied for the first time only very recently to polymer/penetrant systems by Muller-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

G

119

20.0

15.0

0.0

100

200

400

700

800

70.0

60.0 50.0

0.0

Displacement history for a single methane penetrant molecule in polyethylene. (a) T = 300 K. The squared displacement is computed between successive 16 PS positionalaverages of thepenetrant. @) T = 400 K. Thesquareddisplacementis computed between successive 7.5 PS positional averages of the penetrant. [Reproduced from Pant and Boyd with permission.]

force. In the case of diffusion, the most commondriving 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

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of the transport diffusivity D . In liquids (Evans and Morris, and microporous sorbents (Maginn et al., 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. 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 Fick's law, as a relationship between mass flux and chemical potential:

The externally imposed force F in NEMD plays the role of the negative gradient in chemical potential. The flux J, is calculated as the steady-state nonequilibrium ensemble-average molecular flux (jA(t Equation then leads to

-

In Eq. 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 anisotropic diffusion, all elements DUBof the diffusion tensor can be computed by applying the driving force in the direction and measuring the steady-state flux in the a direction. simpliIn thelimit of infinitely dilute penetrant in thepolymer, fies to

Miiller-Plathe et al. employed Eq. to estimate the diffusivity of H,, He, and 0, in PIB 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,. Results from the NEMD simulation were not very encouraging. The value of (jA(t * m))neq, as computed by averaging over PS long pieces of the simulation, was found to fluctuatewidely, 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 foundto rise artificially as a result of theimposition 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 transportdifhsivity at high occupancies via Eq.

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AND DIFFUSION

121

D. Transition State Theory-Based Prediction of Low Concentration Diffusivity As temperature is reduced toward and below Tg,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 of this. Consider the diffusion of COz gas in glassy (unplasticized) PVC at low concentration and a temperatureof 25°C. To obtain a reliable diffusivity from MD, one should let the penetrant molecules move for sufficient time to sample the frozenin inhomogeneitiesof 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 consideredsystem has beenmeasured as D , = 2.4 cm% (Berensand Huvard, 1987). This means that in order to safely estimate the diffusivity, one would need an equilibrium MD simulation of duration at least t=-= 6 Dr,A

(5

(2.4

A), cm2/s)

= 1.7

S

= 170 ns

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, 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 (60) unless all these different runs were conducted on a massively parallel machine. A molecule larger than CO, would require even more time for the first-principles estimation of its self-diffusivity through equilibrium MD. As already pointed out inSection 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 exhaustsitself 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 toexecute jumps between two sites in the polymer. The unoccupied volume in the immediate vicinity of this molecule did

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not show any significant deviation from the unoccupied volume around molecules thatremained “trappedy7 during the entire duration of thesimulation. Furthermore, the potential energy felt by the jumping molecule [i.e., its contribution to v A p ( r A , 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 emphasizeherethat this jump process is notconfined to diffusion in polymers below T,. As shown by the subsequent simulations of Muller-Plathe (1992), Muller-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 (rp, 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 (GG and Scheraga, 1976). Let v = NPnp + 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 T(rp,rA)of the polymer/penetrant system. The projections of the state minima onto the subspace rpof 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

Macrostate State Transition

123

j

(a)

(a) Schematic of a penetrant jump in a low-temperature amorphous model polymer matrix. two-dimensional representation is given, for clarity. Shaded regions are occupied by chain segments, and periodic boundary conditions are indicated. The system polymerlpenetrant, initially in macrostate i, passes into macrostate j by crossing adividinghypersurfacethatseparatesthetwomacrostatesandcontainsafirst-order saddle point (transition state) of the potential energy. Both penetrant (translational, orientational) and polymer degrees freedom change along the reaction coordinate. Note the changes in the clusters unoccupiedvolumebeforeandafterthejump. (b) Simple electrical analog the Poisson process jumping between macrostates. If the reduced probability pi of beinginamacrostate is associatedwithanelectricalpotential,the is network of macrostatesbehaves as anelectricalnetworkwhereineachmacrostate associatedwithacapacitance p:' andeachconnectionbetweenmacrostateswitha resistance llk, .

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We will use the term macrostate to refer to a collection of neighboring states separated by energy barriers that are low relative to k,T. The system spends most of its time confined within macrostates. The infrequent diffusive jumps of the penetrant constitute transitions from one macrostateto another across a bottleneck in "V(rp,r,) separating the macrostates. The projection of a macrostate onto the subspace r, 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 II.B on the pure polymer provide areasonable approximation of these three-dimensional projections of macrostates. With each macrostate i we associate a position (site) vector ri in three-dimensional space that is representative of the position of the penetrant in macrostate i. A convenient choice for ri is the Boltzmann-weighted average of r, over all configurations belonging to the macrostate. The evolution of the polymer/penetrant systemin time is viewed as a Poisson process(Feller, 1957) consisting of successive uncorrelated jumps between neighboring macrostates. With each jump i -P j is associated a first-order rate constant ki+ Consider an (in general nonequilibrium) ensemble of penetrant/ polymer systems that at time t = 0 conform to a specified butotherwise arbitrary distribution among macrostates. As time elapses, this distribution evolves through transitions between macrostates occurring in the individual systems of the ensemble. Let p i ( t ) be the probability finding a system the ensemble in macrostate i at time t. The quantities pi(t) evolve according to the master equations

At very long times, the ensemble reaches its equilibrium distribution, wherein the probability of each macrostate is pTq. The equilibrium probabilities of each macrostate obey the condition of microscopic reversibility (detailed balance)

In view of Eq. (62) and the normalization condition X#:q = 1, it is clear that in a system with a total of m macrostates only (m + 2)(m - 1)/2 of the quantities {kidj},@?} are independent. (Inpractice most of the independent rate constants ki,j are zero, as they correspond to pairs of macrostates that are not connected.) The average residence timein macrostate i at equilibrium is

MOLECULAR SIMULATIONS

O F SORPTIONAND DIFFUSION

125

Combining Eqs. (61) and (62), we find that the reduced probabilities p$) = pi(t)/ppqobey the evolution equations

Equation (64) suggests an interesting electrical analog (see Fig. 14b). The network of macrostates can bemapped onto athree-dimensionalnetwork with nodes at the representative points {ri}.With each node i is associated a capacitance p?, and with each pair of connected nodes a resistance l/kjj, with k, defined in (62). The reduced probability distribution {pi(t)}evolves exactly as the electrostatic potential in the electrical network. The transient solution to the master equation, (61), can be used to extract thepenetrantself-diffusivity.Forexample, the evolutionof (pj(t)} could be tracked under the initial condition pj(0) = Sij by integrating the master equation forward in time. If the behavior of the system is diffusive, then the profile ( p j (t)}, after appropriate smoothing to eliminate the consequences of differences in p;' among different macrostates, should evolve asexp[-(rj - ri)'/(6DSAt)], from which D, 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 wasused 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: Consider a three-dimensional network of a large number m of sites placed at positions ri, i = 1, . . . ,m,with connectivity defined by the rate constants {kidj}.If the macrostates have been determined throughanalysis of a model polymer/penetrant system with periodic boundary conditions (see Section IV.D.2), then the network can be formed by periodic continuation 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 (ppq}. 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 Ni(t)be the number of random walkers that find themselves in site i at time t.

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126

4.

5. 6.

7.

For each site i that is occupied at the current time t, calculate the expected fluxes Ri,(t) = Ni(t)ki, to all sites with which it is connected. Also, compute the overall flux R(t) = XiXj Ridj(t)and the probabilities qi-j(t) = Ri+(t)/R(t). Select a random number 5 [0, 1).Choose the time for occurrence of the next elementary hop event in the network as At = ln(1 - Q/R(t). Choose the type of the next elementary hop event by picking one of the possible transitions i j according to the probabilities qi-j(f). Of the Ni(t) walkers present in site i, pick one with probability l/N,(t) and move it to site j . Advance the simulation time by At. Update the array, keeping track of the current positions of all walkers to reflect the implemented hop. Update the occupancy numbers Ni(t + At) = Ni(t) - 1 and Nj(t + At) = Nj(t) + 1. Return to step 3 to implement the next hop event.

The outcome fromperforming this stochastic simulation overa large number of steps is a set of trajectories r&) for all NErandom wakers. The self-diffusivity Ds,A = D is estimatedfromthemean square displacement ([rk(t)- r,(O)]') computed over all NEwalkers through the Einstein equation, Eq. (49). Note that simulation schemes advancingthe time in equal intervals are also possible (Termonia and Smith, 1987). If the rate constants ki,j are small, then the time steps At taken by the simulation are long, and thus the simulation permits accessing timesand 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 macrostates and the representative nodal points {ri};(2) the equilibrium probability distribution Mq} of themacrostates;and (3) theconnectivityand rate constants {ki,} governing transitions between the macrostates. In the next section we discuss how to obtain these quantities from the detailed potential energy hypersurface "cr(rp,rA)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 v microscopic degrees of freedom specifying the position vectors (rp,rA)of all polymer atoms and of the penetrant molecule. For convenience in the subsequent analysis, we will use the mass-weighted coordinates (Vineyard 1957) In rpij, xpij = mpi,

X,

= min

(65)

MOLECULAR SIMULATIONS

OF SORPTION AND DIFFUSION

127

where the subscript i j denotes the jth atom of the ith polymer molecule.We use the notation x = (xp, x*) 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 - 1)-dimensional hypersurfaces of high potential energy relative to k,T. A macrostate i contains one or more states, each of which is constructed around a local minimum of "Ir(x). The states within a macrostate are mutually accessible over barriers that are low relative to k,T. We denote the minima in macrostate i by xh, xf2, . . . (see Fig. 15). At each such minimum,

and positive definite 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 and $e be two nearest-neighbor minima in the two macrostates between which such atransitioncanoccur.By the nearest-neighborproperty of $k and $e7 there will be at least one first-order saddle point x: between them, at which the gradient of "Ir vanishes and the Hessian matrix H of second derivatives has one negative and v - 1 positive eigenvalues: g(x;) = 0 H(xi) has one negative eigenvalue eigenvector nu

(67a) with associated (6%)

The saddle point or transition state x: is the highest energy point on the lowest energy passage betweenxfk and xis,. construct this passage, or transition path, which is a line in v-dimensional space, one can initiate two steepest descent constructions at x,: one with direction +nu and the other with direction -nu. Each such construction can be carried out in small steps 6x. For example, starting at x: one can displace the configuration toward x: by a small vector 6x = nu From point x: + 6x one can trace the steepest descent path leading to xfk as a series of successive steps ax = - (g/lg))6s. A similar construction can be carried out toward x;~. The resulting transition path is represented in Fig. 15 as a dot-dashed line.

128

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MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

129

The dividing surface separating states x i and xis,, and therefore macrostates i and j , is a (U - 1)-dimensional hypersurface with equation C(x) = 0, that has the following properties (Sevick et al., 1993): 1. It passes through the saddle point x;:

c (x;) = 0

(684

2. At x;, it is normal to the eigenvector correspondingto the negative eigenvalue of the Hessian:

As a consequence, the dividing surface and the transition path are normal to each other. 3. At all points other than x;, vector: V,C(X)

-

=

the dividing surface is tangent to the gradient

x # x;

(684

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

-

C(x) = nu (x - x;) = 0

(69)

In locating the saddle points x: and the associated reaction paths and dividing surfaces, the geometric analysis of accessible volume clusters described in Section I1.B serves as a useful starting point (Greenfield, 1995). Envision 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 "V with respect to the penetrant degrees of freedom, x, keeping all polymer degrees of freedom x, fixed and equal to those of the bulk polymer. Well-established algorithms for the numerical determinationof 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, that lie in concentric spheres of progressively increasing radius around the penetrant),

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130

until further expansion of the set of degrees of freedom leaves the estimates of x: and 'T(xi) unchanged. Once x i has been identified in this way, the transition path and the states x; and x; 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 S, we can proceed to estimate the rate constants kF7 and k , y through v-dimensional transition-state theory 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(VoterandDoll, 1985; June et al., 1991), whenever the polymer/penetrant system finds itself on the dividing hypersurface between i and j with net momentum directedfrom i to j , a successful transition between i and j will occur. Under this assumption (Sevicket al., 1993; Voter and Doll, 1985) the rate constant can be expressed as

np>o

d"p [n(x)

P1 8 [C(X)l

PXC(X)l

(x7

P-

P)

(70)

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, (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)

I

Equation (71) expresses k z as aratio 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 al., 1991). For the highly dimensional problem of the diffusive jump in all relevant degrees freedom of the penetrant and the polymer, the direct sampling of configuration space points uniformly distributed in S, 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 - 1)-dimensional hyperplanes S1, S,, . .., S, . ,S" normal to the transition path starting at xfk and ending at x: (see Fig. 16). We also consider a thin v-dimensional slice of macrostate {i} around the hyperplane S1, of thickness Ax&; we use the symbolism A{i}, to denote this thin domain. Equation (71) can be written as

..

J1,

d"'x exp [-PV(x)l

(2PT)'R

kZ =

l,

d"'x exp

d"'x exp[-PV(x)]

The ratio of v-dimensional configurational integrals in Eq. (72) equals the probabilitythat a system that samples macrostate i according to theequilibrium probability density of the canonical ensemble visits the narrow strip A{i}k. It can be determined through anNVT MD or a Metropolis NVT MC simulationof the polymer/penetrant system confinedwithin macrostate i. The fraction of configurations sampled by such a simulation that lie in A{i}k provides an estimate of this ratio. compute each one of the ratios of (v - 1)-dimensional integrals over successive hyperplanes S,-l and S, appearing in Eq. (72), we consider the vector Sx, along the transition path between these hyperplanes. If S,.-l and S, are spaced closely enough, they are practically parallel to each other and normal to Sx,. Foreachconfiguration x lying onthere is a configuration x + Sx, lying on S, and vice versa. The ratio of configurational integrals can then be

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C(x) =

. ..

16 Sequence of (v - 1)-dimensionalhyperplanes S,, S,, and v-dimensional domain A{i}k used for the free-energy perturbation calculation of the jump rate constant k?; based on multidimensional transition state theory.

written as

d""Ix exp [-

L

exp {

,

+

-

exp = +=P

+

-

The ratio of (U - 1)-dimensional integrals over hyperplanes S,,,-, and S,,, be computed as an ensemble average over a MD Metropolis MC simulation

MOLECULAR SIMULATIONS OF SORPTION

AND DIFFUSION

133

of the system confined to hyperplane S,,,-.l.The calculation of kiAj through (72) is thus possible through a v-dimensional and aseries of (v - 1)-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 fromi to j but subsequently crosses back and ultimately thermalizes in i, are well possible. The actual transition rate constant ki-j can thus be expressed as ki+ = k z 6-j

(74)

where k z is the transition-state theory estimate, obtained as described above, while Jdj is a dynamical correction factor accounting for recrossing events. $-j can be estimated by undertaking short MD runs of the system initiated on the bottleneck surface Su. 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, June et al., NotethattheMD simulations needed to estimate 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

I],

d’x exP t-B~(x)l

p;q

=

z

li.1

li} -1

d’x exP

I-PWol

d’x exp [-pT(x)]

(75)

d’x exp [-p”lr(x)]

It can be accumulated in thecourse of a Widom testparticle insertion calculation [compare Section IKA, 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, aMetropolis MC simulation of the penetranVpolymer 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.

pioneering TST-based study of the diffusion of argon in amorphous polyethylene was presented by Jagodic et al. Computational limitations prevented the generation of realistic model polymer structures, the results from this study were not very conclusive.Recently,a set of thoroughTST-based calculations of high predictive value was carried out by Gusev et al.

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These investigators calculated the infinite-dilution diffusivity of He, HZ, and NZ in rubbery polyisobutylene and glassy polycarbonate at 300 K, representing the polymer matrices by rigid minimum energy configurations obtained through molecular mechanics. With thespherical representation used for all penetrants, the X space is three-dimensional, spanned by the three translational degrees of freedom r, of the penetrant, all of which are associated with the same mass. This greatly simplifies the TST calculation of the rate constants k z . No distinction was made between states and macrostates; i.e., all local minima of V(rP) were identified exhaustively andused to definemacrostates,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 space into states and the identification of dividing surfaces were carried out by constructing regular grids of small cubic elementsor 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 al., 1991). The three-dimensional and two-dimensional configurational integrals needed for the evaluation of p:q and k z via Eq. (71) were obtained through direct numerical integration over the values of T(r$ 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,, D through the generation of kinetic Monte Carlo trajectories for the jump process. The computational requirementsof 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 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. 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 IV.D.2,Gusev and Suter assumed a quadratic dependence of the polymerpolymer contribution to the potential energy Vp(rp)of the form

-

where rPkis 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 (Ai)

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shoulddependontemperatureandatomtype. (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 “elastic7’ fluctuations the polymer matrix are fast in comparison to the time between diffusive jumps, whereas more drastic motional processesassociated 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 aregoverned by athree-dimensional potential mean force A(rA), obtained by integrating out all polymer degrees of freedom in the configuration-spaceprobability density pNw (rp, rJ. By virtue of Eq. (76), this potential of mean force is pairwise additive, consisting penetrant-polymer atom contributions the form - rpks0() 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 al., 1993), usingA(rA) in place of Critical to this fluctuating matrix approach are the values (Ai). These are estimated from short (5 PS duration) atomistic MD simulations of the pure polymer matrix. The (A:) values obtained from these simulations are found to increase with time, following the approximate scaling log ((Ai)”) log In view this, it is decided to use in the TST calculations the value of (Ai) 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 (Ai) is initially postulated and used to compute the rate constants k z ; the residence time distribution is extractedfrom the rate constants, and the time at which it goes through a maximum is identified as andthe whole procedure is anew value of (Ai) is determinedfrom this repeated. The self-consistent calculation is found to convergein practically one iteration. In the case of He and H,, was approximately 0.15 PS and the corresponding (Ai)’” was taken as 0.22 A. In the case of 0, and NZ, was approximately 1 PS and (A:)’” was 0.46 The diffusivities extracted from this fluctuating matrix approach are within a factor 2 from experimental values for all penetrant molecules examined, indicating that the “trapping” experienced by the larger penetrants in static model matrices was eliminated. 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 tenths nanoseconds to tenths of microseconds. “anomalous diffusion” regime, wherein the mean square displacement of the penetrant scales roughly as (r’) m to.’,is observed at short times, and it is only in the longest time portion of the stochastic trajectories that the diffusive scaling (r’) t sets in. A strikingly similar conclusion was reached through atomistic MD the same model systems (He and in polyisobutylene) at room temperature carried out to very long times (Muller-Plathe et al., 1992b) and also in the MD simulations of Pant and Boyd (1993). The crossoverbetween anomalous and diffusive regimes

THEODOROU

136

-1

-

-2 -14

I

I

I

I

I

I

l

-12

-11

-10

-9

-8

-7

-6

l -5

-4

Meansquaredisplacementversustimecurves for heliumandoxygenin polyisobutylene at 300K, 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 diffusionregime in low-temperature polymers is many orders of magnitude longerthanthe “ballistic” regime seen in small molecularweight liquid simulations. A qualitative interpretation of the anomalous diffusion regime based on the jump picture of diffusion can be provided bythe following argument: As pointed out in Section II.B., the connectivity of the network of accessible volume clusters or macrostates seenby the penetrantmolecule is a function of the time scale of observation, owing to the wide distribution of kidj 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 withinisolated groups of a few clusters that are connected via diffusion pathways of high ki+ It is an interesting question whether these clusters be-

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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 acritical time scale, percolation of the network of clusters occurs. A random walk on isolated groups sites or on a percolating collection 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 belonger the slower the motionof the penetrant (i.e., the lower the values of ki+j); this is confirmed by the simulaton findings (see Fig. 17).

V. CONCLUSIONS AND FUTUREDIRECTIONS In recent years there has beenintense computer modeling andsimulation 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 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 equilibriummolecular 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 approachesbased on transition-state theory. Most of this work has focused on chemically simple penetrants (He, H2, 02,N,, CH,) 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 wherediffusion is not too slow (D 1 cm%), MD simulation provided good quantitative estimates of the selfdiffusivity whenever sufficientcare 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 dealingwith diffusion at low temperatures, where atomistic MD becomes impractical.

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There is stilt 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 interms of their ability to predict scattering, spectroscopic, equation-of-state, sorption, and diffusion measurementsis becoming increasinglyobvious. ability to predictallthese 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,andplasticization effects in polymer glasses, has hardly been addressed from a fist-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 hasnot been investigated sufficientlywith 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 couldbetreatedwithmultidimensional TST; explicit incorporation of the polymer degrees of freedom in calculating diffusion pathways would be imperative in this case (see Section IVD), 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 motionin a mediumcapable of exerting Langevin andfrictional 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 conductedto 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 I1 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 withsolvent, become available. Dealing with the long-time relaxationbehavior of polymers in a way thatcombines computational

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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 macrostates, 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 agood example of suchahierarchy. 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.

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. 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. 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. also appreciate the patience of our Editor, Professor Partho Neogi, with my repeated delays in finishing the manuscript.

Allen, M. P., and D. J. Tildesley Computer Simulation of Liquids, Clarendon Press, Oxford. Arizzi, S., P. H. Mott, and U.W. Suter J. Polym. Sci., Polym. Phys. Ed., 30,415. Baker, J. J. Compuf. Chem., 7, Barrat, J. L., J.-N., Roux, and J.-P. Hansen Chem. Phys., 149, Berens, R., and G.S. Huvard Interaction of polymers with near-critical carbon dioxide, Presented at the Annual Meeting of the AIChE, New York, Nov. Bird,R.B., W. E. Stewart, and E. N. Lightfoot Transport Phenomena, Wiley, New York.

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Boone, T. D. Ph.D. Thesis, Univ. California, Berkeley. Boyd, R. H. Macromolecules, 22, Boyd, R. H., and K V. Pant In Computer Simulation of Polymers, R. J. Roe, Ed., Prentice-Hall, Englewood Cliffs, NJ, p. Boyd, R. H., and P. V. K Pant Macromolecules, Czerminsky, R., and Elber, R. J. Chem. Phys., 92,5580. Dee, G. T., T. Ougizawa, and D. J. Walsh Polymer, 33, de Groot, S. R.,andP.Mazur Non-Equilibrium Thermodynamics, Dover, New York. Deng, Q., and Y. Jean Macromolecules, 26, de Pablo, J. J.,M. Laso, andU.W. Suter J. Chem. Phys., 96, de Pablo, J. J., M. Laso, U.W. Suter, and H. D. Cochran Fluid Phase Equilibria, 83,

Dodd, L. R. and D. N. Theodorou Mol. Phys., 72, Dodd, L. R., and D. N. Theodorou Adv. Polym. Sci., 116, Dodd, L. R., T. D. Boone, and D. N. Theodorou Mol. Phys., 78, Ediger, M.D. and D. B. Adolf Adv. Polym. Sci., 116, Eisenberg, A. In Physical Properties of Polymers, J. E.Mark,A.Eisenberg, W.W. Graessley, L. Mandelkern, and J. L. Koenig, Eds., American Chemical ciety, Washington, DC, Chap. p. Elber, R. J. Chem. Phys., 93, Evans, D. J., and G. P. Morriss Statistical Mechanics ofNonequilibrium Liquids, Academic, London. Feller, W. An Introduction to Probability Theory and Its Applications, Vol. II, Wiley, New York. Flory, P. J. Statistical Mechanics of Chain Molecules, Wiley, New York. Fytas, G., and K. L. Ngai Macromolecules, 21, Go, N.,and H. Scheraga (1976). Macromolecules, 9,534. Greenfield, M. L. Ph.D. Thesis, in progress, Univ. California, Berkeley. Macromolecules, 26, Greenfield, M. L., and D. N. Theodorou Gusev, A. A., and U. W. Suter Phys. Rev. A , 43, Gusev, A. A., and U. W. Suter J. Chem. Phys., 99, Gusev, A. A., S. Arizzi, and U. W. Suter J. Chem. Phys., 99, Habenschuss, A., and A. H. Narten J. Chem. Phys., 92, Hansen, J. P.,and I. R. McDonald Theory of Simple Liquids, 2nd ed., Academic, New York, p. Jagodic, F., B. BorStnik, and A. Aiman Macrornol. Chem., 173, June, R. J., A. T. Bell, and D. N. Theodorou J. Phys. Chem., 95, Kluin, J.-E., Yu, S. Vleeshouwers, J. D. McGervey, A. M. Jamieson, R. Simha, and K. Sommer Macromolecules, 26, Laso, M., J. J. de Pablo, and U. W. Suter J. Chem. Phys., 97, Lijwen, H., J.-P. Hansen, and J.-N. Roux Phys. Rev. A, 44, Ludovice, P.J.,andW. W. Suter In Encyclopedia of Polymer Science and Engineering, 2nd ed., J. I. Kroschwitz, Ed., Wiley, New York, Suppl. Vol. p. McKechnie, J. I., D. Brown, and J. H. R. Clarke Macromolecules, 25,

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3 Free-Volume Theory J.

Duda

The Pennsylvania State University, University Park, Pennsylvania

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

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 distributionandaveragemolecular 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 offilms, coatings, andfoams.Inaddition, distinctive molecular diffusion behavior is essential for miscellaneous polymer products such as bamer 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 diffusioncoefficients for systems composed of two or more low molecular weight materials (Reid et al., 1977). These techniquesare 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

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In mixtures of low molecular weight molecules, for example, molecular migration is typically a weak function of temperature and concentration, whereas in polymericsystems diffusional transport canbe significantlyenhanced(or diminished) by varying either condition. The most pronounced effects are observed near the glass transition temperature (Ta, where a 1% increase of solvent weight fraction in a polymer solution can increase D by three orders of magnitude. Diffusion coefficients ranging between and cm’ls have been reported, and apparent activation energies (ED)greater than 50 kcal/mol have been determined (Vrentasand Duda, 1986).Consequently,significantexperimentation 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 aseither 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. 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 polymersand small-molecule solvents. In Sections II and 111, diffusion within concentratedpolymer solutions above TBof the mixture is emphasized, whereas SectionIV 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 occurat conditions that are described by classical (Fickian) diffusion.

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145

FREE-VOLUMECONCEPTS The presumption that molecular transport is regulated by free volume was first introduced by Cohen andTurnbull (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. FromtheCohenand Turnbull perspective,thehard-sphere molecules that constitute an idealizedliquid exist in cavities (or cages) formedby nearest neighbors. Thus, the total volume the liquid could bedivided 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 the diffusional transport mechanism was successfully completed when the cavity a molecule left behindwas occupied by a neighboring molecule. Translational motion, according toCohen and Turnbull, did not require a molecule toattain 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-volumeelementswithin the liquid. If the volume of each hard sphere is denoted V*, then the occupied volume in a liquid containing N spheres is Additionally, if the average free volume per sphere is given by V', then the total volume of the liquid, V,, can be expressed as V, = Nv'. 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

+

D =

exp(-yV*/v')

(1)

In this expression, D is the self-diffusion coefficient 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 ToluenePoly(viny1 acetate) System

Parameter D,,

lo4 4.82' (cm2/s)

X

K,,/Y lo3 K) K& lo4 (cm3/g K) KZ - T ~ (K) I - T82 (K) y; (cm3/g) V ; (cm3/g)

5

E (cal/mol)

0.393 1.45

4.33 -86.32 -258.2 0.728 0.82 0.0

'Do = 4.82 X cm'/s is estimatedusingphysical properties pure toluene(Zielinski and Duda, 1992a). Analysis of toluenePVAc mutual diffusion data at 40°C yields a Do value 10.36 lo" cm*/ S. The effect of Do on diffusion coefficient predictions is illustrated in Fig. 1.

between free-volume elements (i.e., 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 tothe free volume per molecule in the liquid. While relatively simplistic, this framework constitutes the forerunner of the free-volume models presented in the following 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 (Vrentasand Duda, 1977a,b; Vrentas et al., 1993; Duda et al.,1982; Zielinski andDuda, 1992a,b). The Vrentas-Duda approach, generally accepted as one of the most successful theories for describing molecular diffusion in concentrated polymer solutions, is the main focus of this chapter.

DIFFUSION ABOVETHE GLASS TRANSITION TEMPERATURE The Cohen and Turnbull development continues to serve as a valuable foundation for modem theoretical frameworks that successfully describe difisional behavior in concentrated polymer solutions. The formalism, as originally de-

147

FmE-VOLUME THEORY

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

-.

D l= D,, exp(-rVl/Vm)

v:

(2)

where is the critical molar free volume required for a jumping unit of species is the free volume per mole of all individual jumping units of 1 to migrate 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 accommodateentire 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 (Amould and Laurence, 1992; Vrentas et al., 1985a). To generalize the Cohen and Turnbull theory to describe motion in binary liquids, Vrentas and Duda used the relationship

-

v,

=

v,

%H

(moles of jumping units)/g

-

ol/Mlj + wz/Mzi

where , p is the specific hole free volume of a liquid with a weight fraction mi of species i, and with jumping unit molecular weights of M,(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,, is equal to the total molecular weight of the solvent. In contrast, if species 2 is a polymer chain, then Ma is a relatively small fraction of the total molecular weight. Combining Eqs. (2) and (3) results in an expression for solvent self-diffusioninapolymer solution, namely,

c;

-.

where is the specific hole free volume of component i requiredfor a diffusive -* step and = Vli/V2+ A critical step in’ implementing free-volume concepts to describe transport in polymer solutions involves quantifyingthe specific free volume,. ,p As a first approximation, onemight 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 ZIELINSslyl

and Turnbull). The specific occupied volume of a liquid is generally defined as the specific volume of the equilibrium liquid at 0 K [Q"(O)]. Hence, the specific free volume of a species as a function of temperature is given by

cm= Q(T) - P(0) where is the specific volume of an equilibrium liquid at any temperature T. Although not directly measurable, 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 overcominga precise activation energy barrier. incorporate this condition, Vrentas and Duda followed the concepts of Kaelble (1969) and divided the free volume into two types. One portion, denoted the interstitialfree 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 as the specific hole free volume and precludesdetermination of frommeasurements 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 arelationship between the hole free volume and well-defined volumetric characteristics of the pure components in solution:

vFH

For polymer solutions, K,, and K2,denote free-volume parameters for the solvent,while K,,and are free-volumeparameters for thepolymer. The glass transition temperature of species i is given by Tgi.Throughout the ensuing development, subscript l 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 parametersare directly related to the parameters in the Williams-Landel-Ferry (WLF) equation, which has been used extensively for correlating polymer viscosity data (Williams et al., 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.Macedoand Litovitz (1965) andChung (1966) challenged this conceptual notion and suggested that a jumping unit must

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overcome the attractive forces with adjoining moleculesprior to a diffusive step. According totheir viewpoint, therefore, theappropriate expression for the preexponential term in Eq. (4) is

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

ED = R T a2 In ( D y ) OlP

In concentrated polymer solutions where the availability of hole free volume limits translational mobility, the preexponential factor Dol is often less dependent on temperature than the exponential term related to free volume. In many cases Dol can be taken as a constant. When all of these proposed modifications to the original Cohen and lhmbull theory are incorporated, the following expression emergesfor the self-diffusion coefficient a solvent in a polymer solution:

analogous expressionfor the polymer self-diffusion coefficient, D2,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 quantityrequiredindefiningmosttechnological applications. Typically, molecular theories present expressions for the self-diffusion coefficients (Dl and D2) rather than D . Thus, it is highly desirable to express D in terms of D, and D2 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

AND

DUDA

ZIELINKI

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

Here, p,, is the chemical potential of the solvent. Although many thermodynamic theories are available to determine the concentration dependence of the solvent chemical potential, the Rory-Huggins theory has provided an adequate representation of polymer solution thermodynamicsin many cases 1942; Huggins, 1942a,b). Consequently, Eq. (10) is often rewritten as

D = Dl(1 - W2(1- 2x44

(11)

where is the solvent volume fraction in the solution and x is the polymersolvent interaction parameter. Combining Eqs. (g), and (11) results inanexpression for the mutual binary diffusion coefficient in a polymer/solvent system. While this relationship contains numerous parameters, each one possessesphysical significance. In other words, these parameters are not added to simply facilitate empirical fits to data. The total number of parameters needed to define a polymer/solvent system can be reduced to 10 by grouping some together. The key resulting variables are K&, K,, - T,,, Kl2Iy,K2, - Tgz, x, Do,E, and 5, each ofwhich can be estimated a priori without the use of any diffusivity data. Information that is requiredhere to determine all of the parameters to estimate D includes the following:

e;,

1. Chemical structure of both the solvent and the polymer 2. Viscosity versus temperaturedata for both pure components, or comparable variable-temperaturerelaxation data (Zielinski et al., 1992; Spiess, 1990; Bidstrup and Simpson, 1989; Blum et al., 1986; Dekmezian et al., 1985) 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,. Comparison of theoretical predictions with experimental data suggests that some parameters, particularly D, and 5, 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 al., 1989). The solid lines in Fig. 1 correspond to predictions the mutual diffusion coefficient for the toluene/poly(vinylacetate) system based on techniques of

FREE-VOLUMETHEORY

151

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 were used to obtain a better estimate of Do. The resulting semipredictive dashed lines shown in Fig. 1indicate that the theory does anexcellent 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 coefficientswithout the use of any experimental diffusivity data is, at best, precarious.

"""----""_

-"

0

A

0.1

0.2

0.3

0.5

Comparison of mutualdiffusioncoefficientspredictedfromfree-volume theory with experimental data for toluene/poly(vinyl acetate) system40,80, at 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.

DUDA AND ZIELINSH

152

The primary hurdle in applying free-volume models has been the accurate estimation of the jumping unit molar volumes, and required to calculate 6. 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 canassume 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. a first approximation, can be estimated as the molar volume of the solvent at 0 K [denoted here as R(O)] if the entire molecule jumps as a unit. Experimental evidence reveals that molecular shape also influences the effective size of the jumping unit. 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, have been much more elusive. However, 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 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). correlation of this type is anticipated to yield poor values of 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 concentrationdependence of diffusioncoefficients is significantlyenhanced 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 generallyacceptedthat the theory providesan excellent framework for the correlation ofdiffusivity data andthatwithlimited 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 followingrules of thumb are a direct result of the free-volume formalism:

vlj vzj,

vlj

vzj,

vzj

vZj

vzj

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

FREE-VOLUME

153

state of the system that remove it from T, (e.g., increases in solvent concentration or system temperature) result in a decrease in the apparent activation energy. Similarly, the concentration dependenceof the mutual binary diffusion coefficient is greatest near T, 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 smallmolecules, which possess larger free volumes than their neighboring polymers, will increase the overall free volume available for molecular migration. 2. In general, the temperature and concentration dependence 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 adiffusive 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 fked gases in polymer systems are very weak functions concentration and can often be viewed as independentof 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 T, 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. 4. In contrast to observed behaviorin dilute polymer solutions, themutual 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 freethe volume associated with segments in the interior sections of the chain, the increase in the number 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 most commercial polymers do not significantly influence diffusion in concentrated polymer systems in the regime processing interest. 5. The influence additives on diffusion characteristics is directly related to the contribution the additives to the free volume of the system. For example, plasticizers significantly enhance the available free volume, which leads to a decrease in TB and anenhancement inthe rate molecular migration. The addition of impermeable or solid additives (such as fillers) maynotsignificantlymodifythe 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.

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154

6. In most cases, the free volume associated with the solvent is significantly

greater than the free volumeof 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 coefficientwith increasing solvent levels. This is particularlytruewhen 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 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-volumetheory is that molecular interactions between the polymer chain and the solvent molecules are not explicitly considered. Specific polymer-solvent interactions are, however, intro(1) Equation (10) accounts for molecular interactions ducedintwoways: through the chemical potential of the solvent in the solution, and (2) the activation energy, E , in the most general formulationreflects 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 hasbeen discussed by several investigators (Macedo and Litovitz, 1965; Vrentas et al., 1980; VrentasandVrentas, 1993; Zielinski andDuda,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 < T,, 100°C), the diffusion process is dominated by the scarcity of free volume, and specific molecular interactions are of secondary importance. This conclusionmay 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

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 FujitaandVrentas-Duda models havetheir origin in the older Cohen-Turnbull formalism. However, several significant differences exist between the two models. From a practical point 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 groundsthat this difference renders the VrentasDuda approach more consistent with the original Cohen-nmbull 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 comparison, the Vrentas-Duda model can be rearranged to the form reminiscent of the Fujita model (Vrentas et al., 1993):

where &(O) denotes the solvent self-difhsion coefficient at zero solvent concentration, R is the fractional hole free volume of pure component i, and 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 that no volume changes arise upon mixing. Similarly, the following equation can be derived from the Fujita theory:

e

where the free-volume parameters of the theory are &,f;, and In using the Fujita expression, it is common practice to correlate experimental data to determine f& and f a d . A physical interpretation 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: fl = fi,f2 = f2, and Bd = -&@. When related in this fashion, Eqs. (12) and

DUDA AND ZIELINSKI

156 (14) reveal that the two theories are identical when

"This analysis demonstrates that the Vrentas-Duda expression reduces to the Fujita model when certain restrictions are imposed. For example, therelationship 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 areapproximately 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 utilityof the Vrentas-Duda theory arises from its ability to serve as a predictive model without the use of diffusivity data from the polymer/solvent systemof 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. Thefundamental 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.

W.

THE INFLUENCE OF THE GLASS TRANSITION

As amorphousrubbers are cooled, the motionof individual polymer chains becomes 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 frequencyof 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 equilibriumconformationswithin commonlyreferencedtime scales. Consequently, extra hole free volume becomes trapped withinthe polymer as it is cooled through the glass transition. Although the rate of molecular motion within glassy polymers prevents volumerelaxation from reaching equilibrium, molecular motionsare not completely eliminated within a glass. Density fluctuations persist and consequently necessitate redistribution of the hole free volume.

FREE-VOLUME

THEORY

157

LIQUID VOLU

J

>

z

3 a

Tg

TEMPERATURE

2 Characteristics transition temperature (TA.

thevolume

of apolymeraboveandbelowtheglass

The formalisms presented in the preceding section would be appropriate for elucidating and modeling moleculartransport in a glassy polymerif 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 nonequilibrium 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 TB is basedon the concepts previously discussed (VrentasandDuda, Vrentas and Vrentas, Vrentas et al., Accurate prediction of diffusion in the glassy state requires quantification of the extra holefree volume trapped within the glassy state. With this accomplished, the free-volume formalism developed for difhsion in the rubbery state can be

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DUDA AND ZIELINKI

modified to encompass mutualdiffusion 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 = Do e x p ( g ) exp(

pm + o,~Q;) )Q

The symbols in this equation retain their meanings from the last section, although @m is used to denote the hole free volume of the mixture, rather than to emphasize the glassy state. In addition, Q accounts for the concentration dependence of the solvent chemical potential [see Eqs. (10) and (ll)]. Only the magnitudes ofand Q areexpected to be affected by the glass transition. The polymer/solvent mixture below T, 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

vm7

i",(T .c Tgm)= ",Pm1(T) + Wz*m(T)

(17)

Here pm constitutes the hole free volume of the equilibrium polymer plus the excess hole free volume trapped due to the glass transition:

where the excess hole free volume of the polymer, be estimated in terms of the difference between the thermal expansion coefficients of the equilibrium rubber and the glassy polymer (a,and aZg, respectively). Thus,

where @ @ ), is the specific volume of the equilibrium polymer at TBm' The corresponding expression for the hole free volume in the glassy state, therefore, becomes

%(T < T,) = d t m ( T ) +

(20)

+ wz%(Tgm)(azg - a)(T - T,)

vm,

where and are the hole free volumes of the pure species in the equilibrium state and are related to the free-volume parametersKl,, Kzl, K12,and KZ [Eq (6)], which can be estimated from pure-component viscosity data. In summary, the hole free volume, for a glassy polymer/solvent system can beestimated from the equilibrium hole free volumes of the pure components in the system, and information concerning - aZ,T,, and @(TV). In addition, the overlap factor, y, has to be determinedindependently. Techniques are

FREE-VOLUME THEORY

159

available for estimating the TV dependence on solvent concentration (Chow, 1980). Similarly, e ( T ) is available for most commercial polymers from either direct measurements or correlations. Consequently, 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 secondparameter required to estimate dBusion 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 withinpolymer solutions above the glass transition, the equilibrium theory does not adequately describe thermodynamicsin 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. The dual-mode theory of gas sorption (Barrer et al., 1958; Michaels et al., 1963; Vieth and Sladek, 1965) has been used extensively to describe the

solubility of gases in glassy polymers. The fundamental assumptionof the theory is the existence of two distinct solute populations withinthe 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 TA. Local equilibrium between these two solute populations is presumed to be maintained throughout the polymer matrix. 2. 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 moleculehas onpolymer structural rearrangement and subsequent volumetric change. Starting with the free-volume concepts introducedhere,Ganeshand coworkers (1992) proposedafree-volume-based lattice modelthat predicts both gas sorption and transport in glassy polymers. 4. 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 lowsorptionbigh-bulk modulus limit. 5. 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 asthe dual-mode sorption model with one fewer adjustable parameter.

160

DUDA AND ZIELINSKl

6. The gas-polymer-matrix model (Raucher and Sefcik, 1983) assumesa 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 morephysical insight into the sorption process (Barbari et al., 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(methy1 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. apparent paradox arises herein: Polymer lutions are inevitably not at equilibrium below T,,, and yet equilibrium thermodynamicconcepts 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 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 topenetrant 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 modelsto address both rubbery and glassy polymers, inthis section wecompare the fundamental differences between thetwo approaches. In dual-mode sorption @MS), certain regions of the polymer matrix do not relax, owing to extensive chain entanglements that prohibit molecular reorientation. Some segmentsof the free volume are subsequently trapped in the matrix and are not continuously redistributed by random thermal fluctuations. These locally trapped free-volume pockets donot 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

FREE-VOLUMETHEORY

161

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 - ag.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 semipredictive for glassy polymers, because parameters such as azgdepend 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 + 0), Eqs. (16) and (20) reduce to

and

Y

=

e)

[Ku+ X(T - T@)]

where

h = 1 - (012 - a z g ) / K l z

(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, Dl. These relationships suggest that diffusion of a trace amount of solvent relies upon X , which reflects the extra hole free volume trapped in the glassy state. Aside from X , the remaining parametersare identical to those used in modeling diffusion in polymer melts. Numerous experimental investigations using various analytical techniques reveal that the temperature behaviorof the diffusion coefficient is consistent with predictions of Eqs. (21)-(23) when the glass transition temperature is traversed (Coulandin et al., 1985; Amould, 1989; Hadj Romdhane, 1994). These studies confirm that a single value of X 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 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 Tgz. Although free-volume transport theory has not been extensively evaluatedfor diffusion at finite solvent concentrations at temperatures belowT,, it continues to lend itself to qualitative interpretations consistent with experimental observations. for the case of diffusion above the glass transition, we delineate the qualitative behavior of diffusion in glassy systems suggestedby the free-volume formalisms:

1. Contrary to intuitive beliefs, the rate of diffusion (or the diffusion coefficient) does notabruptly decrease as a system cools to the glassystate. Rather, free-volume theory predicts that a fixed solvent concentration the diffusioncoefficient is a continuousfunction of temperatureas T, is traversed. 2. The apparent activation energy for diffusion, ED [see Eq. undergoes a step change at T,. For a given polymer, the change in ED will be greater for larger penetrant molecules. Consequently, it is possible that measure-

1Oa/(K,,-T,,+T)

(K")

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).

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163

ments of the diffusion of small molecules, such as permanent gases, will not register any E,, change at Tp, whereas larger molecules, such as solvent vapors, will exhibit significant increases in E,, as the temperature is lowered through Tm In addition, the change in ED at T p will decrease as solvent concentration increases, i.e., the influence of theglass transition on diffusion is most apparent for trace amounts of large solvent molecules in the polymer matrix. 3. 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 > then the addition of solvent increases the hole free volume andcorrespondingly increases thediffusivity.However,the addition of solvent also lowers Tp 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 loweringTp. This, however, is not always the case, andthus the mutual binary coefficient candecrease withincreasing solvent concentration(Vrentas et al., 1988; VrentasandVrentas,19921. Solvents or penetrants that significantly relax the glassy matrix structure withoutenhancing the free volume of the mixture are referredto 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.

(vml vm),

V. MORECOMPLEXSYSTEMS 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 Vrentas and Vrentas (1991b) proposed modifying this theory to address solvent diffusion in cross-linked polymers.In their development, systemsmust possess a relatively low degree of cross-linking, X , such that 50 or more monomers separate neighboring cross-link points.In this limit, as afirst approximation, manyof the parameters in (16) be considered to be independent of the degree of cross-linking. For example, the solvent properties and p: 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 notdependon X, inwhich case neither does - polymer chaindo V,;, nor p;. The above approximations leadto the result that 5 is also independent

vm.

vml,v,*l,

164

AND

DUDA

ZIELINKI

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 D,, 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 polymerinfluencesonly the diffusioncoefficient through the thermodynamics of the polymer-solvent interactions and the specific hole free volume of the polymer, pm2.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 anticipatedbehavior 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 pm2for cross-linked polymers. One way to do presumes that the hole free volume of a cross-linked material at a given temperature, pm2(T,X),is a fractional portion of the hole free volume of the polymer in the absence of cross-links, i.e.,

Vrentas and Vrentas (1991b) presented both theoretical and experimental evidence to indicate that is virtually independent of temperature but is related to the specific volumes of the pure cross-linkedanduncross-linkedpolymer, @(T,X) and @(T,O), respectively:

Consequently, the influence of chemical cross-linking on the polymer free volume can be characterized by a single parameter, 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 (col+ 0), Eq. (9) takes the form

When = 1, this relationship correctly reduces to the expression for solvent self-diffusioninanuncross-linkedpolymer, whichhas a hole free volume pm2(T,0). Experimental measurementsprobing the influenceof temperature, concentration, degree of cross-linking, and solvent size on Dl (in the limit of

FREE-VOLUME

165

-

0) are all relatively consistent with the predictions of Eq. (26). Consequently, the incorporation of only one new parameter, is required to modify the conventional free-volumetheory 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:

o1

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, ED,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. 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 (ol 0) the dependence of the diffusivity on penetrant concentration increases as the cross-link density increases.

-

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: aP1

j l = -Dll - - D,2 -

ax

+ Pj2+

ax

=0

(29)

where j i is the mass diffusive fluxof species i relative to volume average velocity, is the mass density of species i, and plis partial specific volume of species i.

AND

166

DUDA

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, Dij, that describe mutual diffusioninamulticomponent 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 in some concentration interval near the pure polymer limit (w3 l), the following expressions can be obtained: -+

Dl1 Dl* D22

-+

D1

(30)

0

1)

D2

(3 (32)

D21 0 (33) where D l 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 (Dll and DZ2)are significantly larger than the cross-diffusion coefficients (Dl2and DZl)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 forthe solvent self-diffusion coefficient [Eq. can be readily modified for a ternary system. First, the distribution of the available hole free volume amongall the jumping units of solvent 1, solvent 2, and the polymermust 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 +

where D , is the preexponential factor for component i, KIi and KZ are freevolume parameters for component i, and is the average hole free volume per gram of mixture.

vm

FREE-VOLUME

167

As for binary polymer/solvent systems, the parameters in the free-volume expressions for D , and D2 can be experimentally determined from volumetric, viscosity, and diffusivity data for single-component or binary systems. Recall that for each of the three components can be estimated fromgroup contribution techniques for estimating the equilibrium liquid volume of a component at 0 K The parameters K l i / y and KZi- Tgi for each component can also be discerned from viscosity data. Finally, Do,, &,Dm,and can be attained from diffusivity data for the binary polymer/solvent systems. Ferguson and von Meerwall (1980) were the first investigators to modifythefree-volumetheory 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 diffusionunder 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 al. (1985b). C. Block 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 oneparticular 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 copolymersare chains composed of two (or more) monomer species inwhich 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 measuring (and modeling) the rate of diffusion in microphase-separated block copolymers (Rein et al., 1992; Ferdinand andSpringer, 1989; Csernica et al.,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 (OD”).This

168

DUDA AND ZIELINSKl

section, therefore, focuses on solvent self-diffusion within atwo-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 pm denotes the specific hole free volume in a block copolymerlsolvent mixture, then the molar free volume available for molecular transport can be written as

Here, oiis the weight fraction of component i (i = 1 or 2), and and 0 2 b are the weight fractions of blocks A and B within the copolymer, respectively. The molecular weightsof the jumping units for the solvent, polymer A, and polymer B are given by M,, MZja,and M2,%,respectively. o1and o2sum to unity, as do 02a and 0 2 b . Substitution of Eq. (37) into (2) and introduction of the overlap factor (7) and the energy to break free from neighbors ( E ) yields the transport expression for solvent diffusioninahomogeneous block copolymer melt. The resultant expression can becast into a form comparable to that provided earlier if

and

Then

where (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 (i.e., 02a= 0 2 b = 1, or = B), then Eq. (40) correctly collapses to the original expression for solvent self-diffusion in a homopolymer [see Eq. This relationship atthe very least suggests self-consistency with the methodology adopted. Equation (40) is expected to describe solvent self-diffusion inany monomer block copolymer irrespective of molecular architecture [e.g., AB diblocks, ABA triblocks, or (AB),, multiblocks], provided the copolymers are homogeneous melts. The difference in transport characteristics for the various architectures is expected to arise in the vmly term.

FREE-VOLUME THEORY

169

Previous researchers have investigated the influence of microstructural morphologyand chemical composition on transport within microphase-separated block copolymersand blends (e.g., Kinning et al., 1987; Sax and Ottino, 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 rubberyphasethrough which solvent molecules can diffuse with relative ease. Therefore, perhapsa 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.

Amould, D., and R. L. Laurence (1992). Znd. Eng. Chem. Res., 31, 218. Amould, D. D. (1989). Capillary column inverse gas chromatography for the study of diffusion in polymer-solvent systems, Ph.D. Thesis, Univ. Massachusetts, Amherst,

MA. Astarita, G., M. E. Paulaitis, andR. G. Wissinger (1989). J. Polym. Sci., B: Polym Phys., 27, 2105.

Barbari, T.

W. J. Koros, and D. R. Paul (1988). J. Polym. Sci., B: Polym. Phys., 26,

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Barrer, R. M.,J. Bame, and J. Slater (1958). J. Polym. Sci., 27, 177. Bearman, R. J. (1961). J. Phys. Chem., 65, 1961. Berry, G. C., and T. G. Fox (1968). Adv. Polym. Sci., 5, 261. Bidstrup, S. and J. Simpson (1989). Proc. 18th N.Am. ThermalAnal. Soc., 1,366. Blum, F. D., B. Durairaj, and S. Padmanabhan (1986). J. Polym. Sci. B: Polym. Phys., 24,493.

Brandt, W. (1955). Phys. Rev., 98, 243. Bueche, F. (1962). Physical Properties of Polymers, Interscience, New York. Chow, T. S. (1980). Macromolecules, 24, 2404. Chung, H. S. (1966). J. Chem. Phys., 44, 1362. Cohen, M. H., and D. 'hrnbull (1959). J. Chem. Phys., 31, 1164. Coulandin, J., D. Ehlich, H. Sillescu, and C. H. Wang (1985). Macromolecules, 18, 587. Csernica, J., R. F. Baddour, and R. E. Cohen (1987). Macromolecules, 20, 2468. Dekmezian, D. E. Axelson, J. J. Dechter, B. Borah, and L. Mandelkern (1985). J. Polym. Sci. B: Polym. Phys., 23, 367-385. DiBenedetto, T. (1963). J. Polym. Sci., A , 1; 3477. Duda, J. L.,Y. C. Ni, and J. S. Vrentas (1979). Macromolecules, 12, 459. Duda, J. L., J. S. Vrentas, S. T. Ju, and H. T.Liu (1982). AIChE J. 28, 297. Ferdinand, and J. Springer (1989). Colloid Polym. Sci., 267, 1057. Ferguson, R. D., and E. von Meerwall (1980). J. Polym. Sci., B., Polym. Phys., 18. Flory, P. J. (1942). J. Chem Phys., 10, 51. Fujita, H. (1961). Forsfschr.Hochpolym.-Forsch., 3, 1.

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Ganesh, K., R. Nagarajan, and J. L. Duda (1992). Ind. Eng. Chem Res., 31, 746. GUO, C.J., 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, I. and R. P. Danner (1993). AIChE J., 39, 625. Haward, R. N. (1970). J. Macromol. Sci. Rev. Mucromol. Chem., C4, 191. Huggins, M. L. (1942a). J. Am. Chem. Soc., 1712. Huggins, M. L. (1942b). J. Phys. Chem., 46, 151. Iwai, Y., S. Maruyama,M.Fujimoto, 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. J., E. L. Thomas, and J. M. Ottino (1987). Macromolecules, 20, 1129. Lipscomb, G. G. (1990). AIChEJ., 36(10), 1505. Macedo, P. B. and T. A. Litovitz (1965). J. Chem Phys., 42, 245. Michaels, A. 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. Pawlisch, 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, 111. 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. J. E., and J. M. Ottino (1985). Polymer, 26, 1073. Spiess, K.W. (1990). Polym. Prepr., 31, 103. Vieth, W. R., and K. J. Sladek (1965). J. Colloid Sci., 20, 1014. Vrentas, J. 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, J. and C. M. Vrentas (1991a). Macromolecules, 24, 2404. Vrentas, J. 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. and C. M. Vrentas (1993). Macromolecules, 26, 1277. Vrentas, J. H. T. Liu, and J. L. Duda (1980). J. Appl. Polym. Sci., 25, 1297. Vrentas, J. J. L. Duda, and H. C. Ling (1984). J. Polym. Sci: B Polym. Phys., 22, 459. Vrentas, J. J. L. Duda, and A. C. Hou (1985a). J. Appl. Polym. Sci., 31, 739. Vrentas, J. 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. C. W. Chu, M. C. Drake, and E. von Meerwall (1989). J. Polym. Sci., B: Polym. Phys., 27, 1179.

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Vrentas, J. S., C. M. Vrentas, and J. L. Duda (1993). Polym. J., 25(1), 99-101. 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 J. Am. Chem. Soc., 77, 3701. Zielinski, J. M.and J. L. Duda (1992a). AIChE J., 38, 405. Zielinski, J. M., and J. L. Duda (1992b). J. Polym. Sci., B: Polym. Phys., 30, 1081. Zielinski, J. M., J. Benesi, and J. L. Duda (1992). Znd. Eng. Chem. Res., 31, 2146.

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

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 mathematicalmodeling ends a little too quickly. In this chapter the origins of the key equations, their representations, and methods solution are analyzed first. The unusualvariants of the equations importance to this area are discussed next. 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 phenomenaare 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

NEOGI

174

where c is the concentration of the diffusing species, t is the time, and j x is the flux in the direction, which is along the thickness of a membrane and the only direction in which mass transfer is taking place. If is the area of the face of the membrane, then Ala >> L, the membrane width, which makesthe 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 volumeof mixing is nonzero,and the excess volume of mixing is almost always neglectedin polymers, with somejustification. 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,

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

!?=a (.E) at ax

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 maintainedat 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. leads to

4 D ( h L ; l)’nit] (4) M, 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 decreasedrastically with m. If a halftime is defined as t = t,, where MIIM, = 112, then Eq. (4) can be approximated as ”

1 2

8 ,IT

4Dr2

TRANSPORT PHENOMENA

IN POLPNER MEMBRANES

175

or

It has been assumed that tIn 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

+ 2%

3

M, = 8 @ ) I R

(-l)m ierfc

m=O

"I

where ierfc is the integral of the error function (Crank, 1975, p. 375). At short times, Eq. (6) approximates to 1R

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

Equations (5) and (8) are respectively

tln = 0.01224L2/D

(94

and

tln= 0.01227L2/D

J

Obviously,Eq. ( gives a good description of the solution, which is that M,/ M, is linear in t/L past the half-time but away from equilibrium as shownin Fig. la. The experimental data are plotted against g t / L , 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 from one another. Then the gas is introduced on the upstream side and kept at a constant pressure p l . On the downstream side the pressure p 2 slowly rises as the permeant is being stored. However, the magnitudes of the pressures are such that p , >> p2(t) = Under those conditions the total amount that has permeated through the membranes Qt is given by

Ql Dt """_ LC, - L'

1 6

1

,=,

NEOGI

1 (a)Thefractionalmassuptakeversus t”/L inasorptionexperiment.The diffusivityiscalculatedfromthehalf-timeortheinitialslope. (b) Theresultsfrom permeationexperimentsare 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, Cl

= HP,

(11)

where H is the Henry’s law coefficient. If V, is the volume of the container downstream, then under the ideal gas law

~ 2 V 2= (&APT

(12)

TRANSPORT PHENOMENA IN

POLKVER MEMBRANES

177

where A is the area of the membrane. That is, p 2 can be monitored to get Q, as a function of time. At large times Eq. (10) becomes

et_"_ Dt 1 LHp, L2

6

or

This is exactly 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

Equations (10) and (13c) are shown in Fig. lb. The asymptote given by Eq. (13c) makes an intercept L216D 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 sufficientlyhigh solubilities are suitable for the gravimetric measurementsusedinthe sorption experiments. Consequently there are only a very few systems for which both sorption and permeation results have been reported. few examples have been given by Stem et al. (1983), Kulkami and Stem (1983), and Subramamian et al. (1989). 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 (CrankandPark, 1968; Rogers, 1985; Vieth, 1991). Corrections that need to be consideredwhen the reservoir pressure changes due to dissolution have been quantified. In improving the scope of the experiments, the emphasis appears to have beenlaid on measuring mass or pressure more accurately, on measuring changesin the dimensions of the membrane, and on the ability of the system to go tohigher 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).] very dif-

NEOGI

178

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-changehorption experiments, whichare discussed later.

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

Equation (14) is similar to the heat conduction problem where the temperature replaces concentration and the thermal diffusivity replaces D. The book onheat conduction by Carslaw and Jaeger (1959) and one on difhsion by Crank (1975) cover a great range of solutions to equations such as (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. becomes

m.

-=-(-).+Dac aD ac at ac ax

a2c

ax2

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

D = Doemc

(16)

where Doand are constants. The impact of concentration dependence ondiffusivities is discussed next.

Concentration-DependentDiffusivities A critical theorem in this area is one due to Boltzmann (1894), who showed that even for this case (7) holds away from equilibrium. example is shown in Fig. 2. The apparent diffusivity, obtained using an equation such as Eq. (9, 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 POLYMER IN MEMBRANES

179

.T~//L XI m in 1Wcm 2 The data of Kishimoto for methyl acetate in poly(methy1 acrylate) at 35"C, as reproduced by Fujita (1968).

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NEOGI

Diffusivities of solutes in solid polymers are almost always givenbyan 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 lutions. These solutions tend to become more approximate asthe concentration dependence becomessteeper and to become more exact as dfisivities become less concentration-dependent. Neogi (1988) obtained an approximate solution for the case when in Eq. (16) is very large. Many of the characteristics obtainedthere are well known frombefore.Boththeoldandnew results are summarized below. 1. Boltzmann's scaling is upheld.It is seen that the fractional massuptake when plotted against t is almost linear. However, the telltale sign that it is actually Fickian is observed in thefact that thefractional mass uptake curves make a slope of 90" 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 happensbecause 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 oflowest concentration. Calculated concentrationprofiles showing selfsharpening are illustrated in Fig. 3. It should be noted that (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 fractiondesorbedshouldbeequaltothefractional 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. 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 c,. This is the integral sorption. If the step change is made from ci to c,, where ci is very close to c, then it could be assumed that the diffusivity can be approximated to be a constant and evaluated at a mean concentration, say (1/2) (c, + ci).This is the differential sorption and provides informationon diffusivity as a function of concentration.

TRANSPORT PHENOMENA IN POLWER MEMBRANES

181

~/(4w,l)~

3 Numericalsolutions of concentrationprofiles(dimensionless) in asemiinfinite system where the diffusivityis concentration-dependent,are shown. The numbers on plot are the different values dimensionless a defined in Eq. (16). [Reproduced from Crank (1975), with permission.]

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 continvariable one has a discrete set of points {xj}, and instead of time t one has the set {ti}. Likewise, the concentrations become c, and the derivatives become

and

Inserting @S. (17) and (18) into

(14) and rearranging, one has

where the increments At = ti+l - ti and = xicl - xi are constants. If the initial concentration profiles ckj are known, then Eq. (19) provides the means

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for moving up in time to ci+lj. Such iterations can be continued until the numerical approximation to equilibrium or steadystate is achieved. It is obvious from Eqs. (17) and (18) that the basic representation (forward time central space, FTCS) in this finite-difference scheme is an approximate one that improves asthe size of the increments is decreased. However, the price that is paid in this process is that the computationsbecomeimpossibly 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 lJ),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 At/(Ax)’ 1/2 (Carnahan et al., 1969), the system is stable. This causes considerable difficulties. If Eq. (19) is cast into dimensionless form where goes from 0 to 1, and there are 20 subdivisions, then = 0.05. The dimensionless diffusivity is 1, and the stability criterion allows At = 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 onthe 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 At 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 At are required there. However, another problem arises in that often in semi-implicit schemes an algebraic equation has to be solvedor 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 systemthe errors in the steps shown in (17)and (18) mayaddup. There is no wayof predicting the cumulative error. Finally, stability can break down dueto boundary conditions, a feature that has not been explored in the literature in detail (Roache, 1982). In recent yearsanother method hasbecome 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 ofthismethod lies inthe fact that its programming is generalizedandnot

+

-

TRANSPORT PHENOMENA POLYMER IN MEMBRANES

183

tailored to a specific problem, and it is about the only method that can handle complex geometries:three-dimensionalandasymmetric.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).

Boundary One importantclass of problems involve systems where the position of a boundis not known beforehand and has to be obtained as part of the solution. In particular, the boundarycan move as a function of time that also has be determined. For simplicity it is assumed below that the boundary lies on the plane, that is, perpendicular to the 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,

pyv: - v ) = pyv: - v)

(20)

species balance,

and energy balance,

~1 c1 (I v , - v ) - aT1 ~ " = n~ nc (n v-,v ) - P -aT'1 +HR

ax

ax

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 v, is the velocity inside a phase. As constructed here, all velocities have only 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,providedthat 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 v, 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. integral heat balance is used [which incorporates the jump heat balance of Eq. (22)], an integral balance of polymer

NEOGI

184

is used[which incorporates (21) for the polymerandthe fact that it is nonvolatile], and the jump balancefor 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 al., 1990). The basic scheme for solving these problems requires a well-defined initial state. Assumingthis initial position, the firstupdating of the concentrationprofile 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.

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 al., 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 ofnonFickian 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 ofFickian origin, showninFig. 3, move at a speed proportional to One can generalize these types of behavior as

4 = k"t" M, where

n>l n=l 112 n 1 n = 112 n < 112

supercase I1 (Jacques et al., 1974) case II (Alfrey, 1965) anomalous (Haga, 1982) classicalJFickian (Boltzmann, 1894) pseudo-Fickian (Liu and Neogi, 1991)

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 onecalled a two-step response. The physical mechanisms behind the non-Fickian behavior are only now beginning to emerge and are notcomplete. One mechanism is thatofcoupledmassand 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 al., 1959), which has a low glass transition temperature of

t

case I1

4 Thedifferentclasses of non-Fickian sorption.

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A. Memory-Dependent Sorption To explain case 11, it becomes apparent that the parabolic form of Eq. 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 equationis 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 =-

p(t

ac

- t') -

ax

t') dt'

where a simple form of relaxation function would be p(t) = DiS(t)

+

~

where Di and D, are the initial and final diffusivities, S(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 should decay with t - t'; this is called the fading memory. It is possible to see that the relaxation time is =

tp(t)dt

(26)

and that

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

which predicts adiscontinuity where the front travels with aconstantspeed

W).

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 POLYUER 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 howto apply laws whenthe 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 I1 is observed to blendinFickiandiffusionwithchanges in theconditions of the experiments (Baird et al., 1971; Hopfenberg et al., 1969,1970).Vrentasetal. (1975) suggest that if diffusion is coupled with elastic effects, then a Deborah

= =

too Time [. hrsl

5 ThedataofHopfenbergetal. (1969) forn-pentaneinbiaxiallydrawn polystyrene. Some crazing also takes place, but Hopfenberg et al. showed later (1970) that crazing and case I1 were not directly related. (Reproduced with permission.)

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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 L’ID, 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 11 is seen. In Neogi’s (1983b) model as well, it is possible to define a Deborah number, which is d(L’/D). Case 11 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, (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 anelastic 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 differentmembranethicknesses L do not collapse into a single curve when plotted against V t I L . These could be quantified, as well as case I1 itself. However, the index n in Eq. (23) inthe linear model can be shown to varybetween 112 and 1 only,andnot 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 p(c, t). For, when we insert into Eq. (24) the function c(x, t), which is clearly understandable asthe concentration atpoint at time t, becomes c(& t - t ’ ) . We know that c(x, t ’ ) is the concentration at point at a previous time t‘, but we do not know what c(x, t - t ’ ) is. Therefore, no number can be

TRANSPORT PHENOMENA POLYMER IN MEMBRANES

189

supplied for that quantity. The remedy is to use an expansion ac a2c -+q.-+...=-

at

at2

where 7 and D can be functions of concentration. Substituting Eq. (24)into Eq. (1) and inverting by using Laplace transform leads to Eq. (29). Camera-Roda and Sarti (1990) solved such an equation using only the first two terms on the left in Eq. (29). This truncation limits the solution large times. The sorption experiments of Vrentas et al. (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 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). Thismoves the Deborah numbers away from 1, and obviously one needs to ascertain what these effective Deborah numbers become during the computations.

Sorption The first model in this class that could be comparedwith experiments wasgiven by Kim and Neogi (1984). They followed an earlier formalism put forward by Larch6 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 is affected. This in turn affects the flux, which can be written as DC a p

="- RT ax

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 but those of the out, their flux exwhole system. As a result, Larch6 and Cahn (1982) point pression is not local anymore, which goes againstthe usual assumptionthat

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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 Duming 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 ofhow to define, quantify, or measure “internal stresses” that occur inglassypolymersremainsunresolved. 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

=

-

qv1

tr(4

(31)

where U is the stress, tr denotes trace, v1 is the specific volume of the solute, and q is the change in the volume due to concentration changes. The chemical potential in the absence of stresses is 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 continueto 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 showninFig. 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 responseoccurred 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 couldnot 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

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191

0.75

0.50

0.25

0.2

0.5

0.6

0.7

0.8 0.9

Thetheoreticalresults of Neogi et al. onswelling-inducedeffects fitted to the data of Long and Richman 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 is dimensionless time. effective diffusivity has been used insteadof a concentration-dependent one. [Reproduced from Neogi with permission of the American Institute of Chemical Engineers. AIChE. All rights reserved.]

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ductile failure in these materials. Rogers reported such effects in polyethylene where the amorphous regions are rubbery, and Durning and Rebenfeld reported similar effects in poly(ethy1ene terephthalate) where the amorphous regions are glassy. It becomes evidentthat more detailed stress-strain constitutive equations can be worked into the theory, and the theory itself can be given a broader basis thanthat given by Larch6andCahn Lustig et al. 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. 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 Fickian diffusion was observed at lower concentrations, and sigmoidal at high concentrations. However, more detailed differential sorption data given by Odani et al. andKishimoto et al. show thatroughly 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 far that can explain the sigmoidal behavior at the very lowest concentrations. At suchlow concentrations one peculiarity of glassy polymersis summarized by its dual-mode solubility isotherm

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, 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., Patton et al., The question now is, do these holes have an effect on transport? Michaels et al. 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 thanthatin the polymer. Petropolous suggested that holes would contribute to the flux whatever

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

where D , is the diffusivity in the polymer andDHis that of the adsorbedspecies, and c, and C , are concentrations there. D,c,,/RT and DHcHIRT are the two mobilities, and the last term is the driving force, the same for both. Paul and Koros showed how to combine Eqs. and (32) to get a complete expression for flux. Subramanian et al. used that expression to comparethe experimental data withtheory to get a value for DH, which was found tobe less than D,. In another study,TshudyandvonFrankenberg used the holes and adsorption as a sourcelsink term in the conservation equation. The adsorptiondesorption kinetics are also included. The model was notlikedbecausethe contribution of the holes to the flux was made even more obscure. Vieth 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 diffusantin the microvoids. Barrer and Fredrickson andHelfand put together molecular models to justify and generalize a flux expression such as that given in (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. compareone set of experiments with this generalized model, they find thatthe 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: DHin 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 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 randomlyplaced-areaccounted for in transport phenomenathroughaveraging methods. Obviously, in a randomly heterogeneous systemthe 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 to the equations for the average responses. Sangani attempted to average the dual sorption model together with Petropoulos’smodel 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.

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Neogi attemptedtoformulate theproblem using a more detailed model. There are four resistances to mass transfer to a hole: diffusion in the polymer, adsorption-desorption resistance the inpolymer, 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, which was originallyconstructed by Maxwelland applies to systemswith a small volume fraction of holes. l k o 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 properdescriptionfor the sorption experiments. The results show sigmoidal sorption. The main parameter there is theratio 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, thenthis non-Fickian effect will disappear atsufficientlyhigh concentrations whensaturation is reached. The anomalous effect is due to hole filling and disappears when the holes are filled. All of this occurs atlowconcentrations.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. predicts pseudo-Fickian sorption (see Equation A very different kind ofaveraging wasintroduced by Di Marzio and Sanchez 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 were able to show that the simple exponentialrelaxation becomes “stretched exponential”exp[ - ( t / ~ ) ’ ] on averaging. These ideas were used by Adib and Neogi to average the relaxation discussed earlier. Whereas case I1 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. cannot exist, that is, will give negative or infinite values!

TRANSPORT PHENOMENA IN

POLWER MEMBRANES

195

7 The theoretical results of Neogi on hole-filling effects at very small concentrations as fitted to the experimental data of Odani et al. and Kishimoto et and isotactic (2) polystyrene at al. The systems are benzene in atactic Here 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.

W. 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 al., 1990). Diffusion of gases through a lamellar composite, where one material has a high

NEOGZ 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, diffusionprocess, etc. Only these two are discussed below.

A.

ReverseOsmosis 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. inorganic salt such asmagnesium perchlorate was also added. The procedure used was as follows. A thin film of about 250 pm thickness was layered onto a surface kept at -10 to 0°Cand after a few minutes was plunged into hot waterandkept there for over an hour. The surface that was exposedtothe atmosphere developed a dense skin -0.1 pm 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 chargedevelopson 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 NaC1) from another with no salt. Now if these pores are negatively charged, thenthe “solubility” (andhence the permeability) of the Na.’ioninsucha pore will be veryhighandthatof the Cl- ion will be verylow.But in the absence of an externally applied voltage difference there will be no current, and the fluxes ofthe two ions will be equal. The overall effect is that there is virtually no net flow of NaCl across the membrane (Cl- 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 R 0 membranes that can be consultedfor additional details (Sourirajan, 1970, 1977;Merten, 1966; TurbackJ981; 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 R 0 membranes. Kesting (1971) discussed qualita-

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tively the nature of precipitation/aggregation in the evaporating solution. The most important partof his discussion concerns the relationship between swelling and the structure of the precipitate. Acetone, being a goodsolvent for cellulose acetate, swells it; water, being a nonsolvent, 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 regardingthe origin of the asymmetry. In fact, this question was not studied until Sirkar et al. (1978) showed experimentally thatit 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 changein the densities,which in turn 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-InducedCrystallization 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(ethy1ene 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 the edges of the lamellae,

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and hence if r is the radius of the spherulite, then

dr -=G dt

(34)

The term G can be expressed as

G = vzGoexp(- AEDIRT) exp(-AF*/RT)

(35) where v2 is the volume fraction of the polymer, Go is a rate constant, AEDis the transport activation energy (that needed to bring the polymer to the growing interface), and AF* 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

ED = C,T/(C2 -I- T

-

Tg)

(36) where C, and C, are universal constants. The other activation energy, AF*,is a more complicated function of different constants of the system plus the melting point T,. 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(-kt")

(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
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 hE, 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 N2, 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 + (v) = - j*

ax

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

-=

at

j*

(39)

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

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c/?. Ap x 10- (s''Z/m) 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

one

The solution, when the input is a pulse, is

TRANSPORT PHENOMENA IN POLYMER MEMBRANES

201

where 6 is the Dirac delta function and M is the total mass in the pulse. The elution time at = L is L(l 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-liquidchromatography (GLC), a viscous liquid is used to coatthe 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 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 overlapin 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 D, without knowledge of Taylor's (1953) work inthis 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.Golayarguedthat one should perform chromatographyin 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 peakshape is shown in Fig. 9.ArnouldandLawrence(1992)reported the vast amounts of data on such diffusivities thathavebeen thus obtained anddiscussed their impact on the current theories of diffusivities. We are entering a yet newer era in the use 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. 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.

+

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

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I

0

l

1

I

2

9 Comparison between the theoretical (a) and experimental (b) elution curves. The polymer is polystyrene, and the vapor is benzene; the system at is 130°C. [Reprinted with permission from Pawlisch and Lawrence (1988). Copyright 1988 American Oemical 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 the phases is dilute, were discussed briefly in Section 1II.C These originate from the work of Maxwell and have been used by Robeson et al. (1973) to compare permeation data with theoretical predictions. The problem 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

POLYUER MEMBRANES

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

203

phases.

passes through another many times in an intimate way. Thus, this brings up the second point, thatif one hastwo different samples, bothhaving 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 discussed some attempts at quantifying diffusion in such systems. The new methods of analyzing such systems were usedby Ottino and coworkers (Sax and Ottino, Ottino and Shah, Shah et al., 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 highfractions it is the continuous phase. In between, both phases are continuous, that is, the system becomesbicontinuous. 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 If we v01 % [according to one model (Park and MacElroy, 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 of the silver has been removed. After that thesilver stops being a continuous phase and the conductivity falls dramatically. (In this connection it should be noted

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that in two dimensions one could go down to l%.) 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 flakestouch one another at theedges,thena continuous impermeable surface is formed that prevents any diffusant from passing through, that is, a perfect insulation to permeation is formed. Actuallythere 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 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-DiffusionCoefficients 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 followingis a brief review of how these measurements are made and an attempt to explain them based on the d f i s i o n coefficients encountered here. Some nuclei have magnetic dipoles: 'H, 'H, I3C, etc. They orient in a magnetic field and absorb energy from an rf (radio-frequency) souce provided that the frequency correspondsto 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 sed 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 periodoverwhichafirst 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 (due to diffusion) in between the application of the radio frequencies, then the echo strength will be attenuated. It has been possible to relate the attenuation diffusion. The reviews by von Meerwall (1985), Stilbs (1986), and Urger et al. (1988) cover experimentaland 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 al. (1990) also

TRANSPORT PHENOMENA POLYMER IN MEMBRANES

205

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 N M R 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 atthephenomenon 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 in the future.

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Stilbs, P. (1986). Prog. NMR Spectrosc., 19, Subramanian, S., J. C.Heydweiller,and S. A. Stem(1989). J. Polym. Phys. Ed., 27, 1209. Tamura, M., K Yamada, and H. Odani (1963). Rep. ProF. Polym. Phys. Jpn., 6, 163. Taylor, G. I. (1953). Proc. Royal (London),A219, 186. Truesdell, C., andW.No11(1965). In Handbook of Physics, Vol. IW3, S. Fliigge, Ed., Springer-Verlag, Berlin. Tsay, C. S., and A. J. McHugh (1991). J. Polym. Sci. Polym. Phys. Ed., 29, 1261. Tshudy, J. A., and C. von Frankenberg (1973). J. Polym. Sci. A-2, 11, 2027. Turback, A. F., Ed. (1981). Synthetic Membranes, Vols. I and 11, ACS Symp. Set, ACS, Washington, DC, pp. 153-154. Vieth, W. R. (1991). Diffiion In and Through Polymers, Hanser, New York. von Meerwall, E. D. (1985). Rubber Chem. Tech., 58, 527. Vrentas, J. S., and J. L. Duda (1979). AIChE J., 25, 1. Vrentas, J. S., C. M. Jarzebski, and J. L. Duda (1975). AZChEJ., 24, 894. Vrentas, J. S., J. L. Duda, and A.-C. Hou (1984a). J. Appl. Polymer Sci., 29, 399. 18, 161. Vrentas, J. S., J. L. Duda, S. T. Ju, and L.-W. Ni (1984b). J. Membrane Vrentas, J. S., J. L. Duda, and W. J. Huang (1986). Macromolecules, 19, 1718. Windle, A. H. (1985). In Polymer Permeability, J. Comyn,Ed.,Elsevier,NewYork, 1985, p. 75. Yasuda,H.,andC.E.Lamaze(1970). In Membranes from Cellulose and Cellulose Derivatives, A. F. Turback, Ed., Interscience, New York, p. 157. Yoon, D. Y., and P. J. (1979). Disc. Faraday 68, 288. Zachmann, H. G., and G. Konrad (1968). Macromol. Chem., 118, 189.

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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. INTRODUCTION A. Scope and Context

the Presentation

The penetration of small molecules into a polymer solid or liquid frequently plays an important role inindustrial processes such asthe spinning and finishing processes in fiber manufacturing (T.akeda and Nukushina, 1963) and the casting and finishing processes in membrane manufacturing (Manabe et al., 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 al., 1976) and cuprammonium-regenerated cellulose hollow fiber (Fujioka and Manabe, 1995), works as to orient the hydroxy group in the direction parallel to the surface plane of a film or a fiber, whereas waterwould 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 suchsmall-molecule effects contribute to the completion of the fine structure of the polymer solid andthe specific permeation of the molecules 21l

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212

through the solid, this chapter deals only with the supermolecular structure of the polymer solid (see Section 11) 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.

6. Methodology for Development of Novel Materials in the Field of Penetrant Transport Studies of the correlation of penetrant transport andthe conditions of preparation of the solid polymer have been carried out rigorously, andsome 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 al., 1967). These studies can be classified intoa blackbox that connects phenomena without causal sequence. When the demandfor penetrant transport becomes complicated as seen in the case of virus removal filters (Manabe, 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 supermolecular structure and physical properties. When we suceed in determining the quantitative relationship between structure and properties-the causal sequence, 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.

) ~ e m b n eManufac (Forming Mechanism) Correlation betweenpreparation condition and pore chsract. FineStructure)

I(Transport M e c k i )

!;rope ;tis Version ISocial Needs

technical t e n i n d o g y

I

Figure Flowchart for relating social needs and membrane manufacturing: The numbers 1, 2, and indicate the stream development a novelmembranethat 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 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 ondependingonthe size ofthedomainnecessary for characterizing experimentallythe structure inquestion. 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 alonga 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 lo3 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 witha single flowmechanism as denoted,and the broken line indicates the actual general penetrant transport. Because the transport the penetrants occurs mainly from the environmental liquid or gas to the inside the polymer solid, a higher order structural factor always contributes to the transport. When the penetrant molecule is small, the rate-determined region transport is related to the lower order structural factor. For example, dissolution/diffusional flow is affected much more by the first-order structural factor (i.e., chemical structure) than by a higher order factor. In the case 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 the transport may change from case to case. Table 1 summarizes the classification supermolecular structure and lists the characterization methods for the various structural orders. A first-order struc-

214

“E Generalpenetranttransport

DissolutionMiffusion flow -Free Surfacediffusional

volume model

flw

2 Schematicrepresentationofrelationshipbetweensupermolecularstructure 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 difhaction 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

Definition of Supermolecular Structure and Methods

of Evaluating It

arget Structural Structure description Evaluation methods. (nm) sizeorder First (chemical structure of monomer) Second (conformation) Third (fine structure)

Fourth (aggregation structure)

Fifth (morphological structure)

Chemical structure, moleculal Spectrometry (NMR, weight and its distribution, IR, vis, W, MS), chromatography configuration and tacticity, composition of copolymer W , GPC,"LC), light scattering, and blend and its viscometry distribution. Conformation, local chain NMR, IR, LS, 1-10 diffractometry (Xorientation, crystal ray, electron), CD, structure, amorphous ORD, EM, AFM structure Orientation (chain axis, NMR (broad-line), 10-10' IR, LS (smallcrystal plane), interfacial angle), X-ray structure, crystal size, diffraction, EM crystallinity, crystal perfection, molecular @EM, optical packing (density, regularity), lateral order microscopy, thermal analysis @TA, DSC), viscoelastometry (dynamic, static), refractometry Electron microscopy, lo2-lo4 Aggregation structure (particle size, porosity, optical microscopy, degree of amalgamation), fibril and microfibril (size, viscoelastometry, permeability, length, orientation), porosimetry, microvoid, pore structure (pore size and its adsorption isothermometry distribution, pore shape, porosity) Optical microscopy, 104-106 Shape of membrane (plane, hollow fiber), symmetrical light scattering and asymmetrical membrane, complex membrane, dynamic membrane, liquid membrane

0.1-1.0

'NMR = Nuclear magnetic resonance, IR = infrared, vis = visible, UV = ultraviolet, MS = mass spectrometry,LC = liquidchromatography,GPC = gel permeationchromatography, TLC = thin layerchromatography, LS = light scattering, CD = circular dichroism, ORD = optical rotary dispersion, EM = electronmicroscopy, AFM = atomicforcemicroscopy,SEM = scanning EM, TEM = transmissionEM,DTA = differentialthermalanalysis, DSC = differentialscanning calorimetry.

215

216

W X B E

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

II. STRUCTURALCHARACTERISTICS OF POLYMER SOLIDS 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 ILC. 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.

Capillary model

Free volume model

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

A.

OF POLXMER SOLIDS

21 7

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 cannotcrystallize 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 becauseof 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 at A noncrystalline polymer solid has crystallinity ofnot more than This definition is based on actual crystallinity andnotonthe polymer’s chemical structure. Since crystallinity varies depending onthe methods of evaluationsuch as X-raydiffraction,density,andinfrared absorption, the X-raydiffraction (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 on, except polymers with side chains whose number of carbon atoms ranges between four (polyhexene-l) and eight (polydecene-l), isotactic poly-4methylpentene-l, polytetrafluoroethylene, Nylon 6 and Nylon 66, polyethylene terephthalate (PET), polyacetal, cellulose, and poly-pura-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 polymerssuch as atactic polystyrene, atactic polymethyl methacrylate, polyacrylonitrile, polyvinyl acetate, and polyvinyl chloride; polymers whosechemical compositionis complicated with bulky side chains such as polycarbonateand many kinds of copolymers; and polymers with low meltingpoints such as cis-174-polybutadieneand cis-1,4-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 thenoncrystallineregion. For example, thenoncrystalline 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, 1970~).

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218

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 micelle 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. molecular chain is nm thickness in the folded-chain crystal, and this type of folded within ca. 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., 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). Underhigh 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 extendedchain 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 andnoncrystalline 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

--

...

"

. A . ? . .

n

C

..-

.b..

" "

4 mica1 molecular arrangement in a crystal. (a) Fringed micelle; chain crystal; (c) extended chain crystal.

folded-

SUPERMOLECULAR STRUCTURE OF POLYMER SOLIDS

5 Takayanagi’smodel(Takayanagi, 1967) forrepresentingfinestructure crystallinepolymersolid.Blackarea:noncrystallineregion.Whitearea:crystalline region.

219

of

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 onthe assumption that thecontent 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 microBrownian motion) and are represented by the packing density of the molecular chain with the same regularity (Manabe and Kamide, 1984).

0

MANME

220

Table 2 Content of Noncrystalline Region' with Three Methods of Evaluating Crystallinity Polymer terephthalate Polyethylene 0.27 Polyethylene 0.28 Regenerated cellulose

IF

Densityd

0.25

0.26

-

0.25

'Given by 1 - crystallinity. bWide-angle X-ray diffraction method. lnfrared 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.

Schematic representation of frequency distribution of molecularchainsin the region characterized by a given regularity and a given packing density. Curves1, 2, and show the values of a perfect amorphous polymer, a noncrystalline polymer, and a crystallinepolymer,respectively. N, S, andContheregularityordinateindicate 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. Nodules 2-4 nm in size and grain boundaries of 1-2 nm were observed andfound to possess a little regularity,andthe 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 foundto coincide with that of achain 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 II.C and 1II.A will deal with the structures achieved under different preparation conditions.

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 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.

Modelrepresentingthestructure a noncrystallineregion(Yeh, OD, GB, and IG denoteordereddomain,grainboundary,andintergrainregions, respectively.

1972).

MANABE

222

of 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 al., 1990). Figure 8 shows two typical structural models representing the noncrystalline region in a fiber. The fringed micelle model (Fig. Sa) 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, polyacrylonitrile, and aramide 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

l

T

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

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 chainkeeps 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 thefibrilandmicrofibrilmodel (Peterlin, 1969). The pm. The filament is considered diameter of a filament ranges from 10 to to be composed of fibrils having sizes between 100 and 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.

'\

\ \

\ \

l

I

I

I

/

/ /

Schematicrepresentation of the fibril and microfibril model proposed for melt-spun crystalline polymer (Peterlin, 1969). A filament pm in diameter is composed of many fibrils having diameters between 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.

UANABE

224

The change in supermolecular structure along the radial direction is also a very important factor for the permeation molecules (i.e., 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 alongthe radial direction (Manabe et al., 1980). The microfibril is subdivided into fine tips having widths of 5-10 nm and lengths 0.3-0.5 pm. The chain orientation (c axis orientation) denoted by the length the arrows c and the plane orientation denoted by the length the arrows b decrease toward the center of the fiber.Fibers prepared by thewet-spinning process show changes similar to that of Kevlar 49. For example, the cuprammonium-regenerated cellulose fiber Bemberg showsmaximum crystallinity at ca 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. rlR (Manabe et al., 1980). The change in porosity for various kinds hollow fibers has been clarified (Kamideet al., 1989). Meltspun fibers such as polyethylene terephthalate also show a similar change, especially in the case the fiber manufactured through high-speed take-up(Kuriki et al., 1985).

5-10 nm

a block

\

‘a fiber a microfibril

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 stateof the microfibrils between5 and nm 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 pm. 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.

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 distinguishbetweenundrawnandmultiaxiallydrawnfilms from the chain axis orientation because the latter is generally drawn as to be isotropic. The mechanical properties of drawn films are superior to those of an undrawn film. film frequently shows the orientation of a specific plane near the film surface. Thistype 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 environmental conditions (Haraguchi et al., 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 thefilm surface (Kajiyama et al., 1995). The molecular conformation and molecular aggregationin a thin membrane such as a biomembrane play the main role in molecular transport through the membrane.

In artificial membranes, although the structure of homogeneous membranes that have no observable pores under an electron microscopecan 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 al., 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 r and 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,, and pore density N are given by the

+

226

. . ,

.

252

Three typical porous polymeric membranes. (a) Amembrane with straightloRpores/cmz surface through cylindrical pores. In general, the pore density is about area,andthethickness d islessthan 30 pm. asymmetricalmembrane.This membrane is composed 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 -70 nm including the top surface is considered to be the active layer for the particle separation. (c)The multilayer structural membrane. The wholeof the membrane is composedof particles with a size of ranging between 50 and 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:

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 al., 1990a), and we can easily obtain information of pore structure as a function of radial coordinates (Kamide et al., 1989). Many evaluation methods (Kamide and Manabe, 1980) for N(r) have been proposed far: (1)mercury intrusion (MI) method, (2) bubble pressure and pressure dependence of the filtration rate offluid(liquid and/or gas) (BP method), (3) permselectivity 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 immersingliquid, 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 al., 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 formsof 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 gives the mean pore radius r, using the following equation derived through the Hagen-Poiseuille equation,

where P, is the porosityof the membrane, d is its thickness, and-q is the viscosity of the liquid. The value of rf corresponds to (r3r.,)lRfor the membrane with straight-through cylindrical pores. When the surface area of the membrane including pore and membrane surface S, is observed, we can calculate the mean

228

1

.-

0.5

0 Pore

radius

/pm

Pore radius /pm

Comparison of poresizedistributionfunctions 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 caseof spherical pores. Curves1 and 2 are the valuesof the front and back surfaces of the membrane. @)N(r) values obtained through EM, MI, and BP methods. Pores are assumed to be straight-through cylindrical pores in all cases.

pore radius r, (corresponding to r2)as

The permeability coefficient a gas gives the specific mean pore radii corresponding to r,, r,, and (r3r4)1R (Nohmi et al., 1978). The porosity P, can also be evaluated directly through the apparent density using the equation

p, = 1 - PalPp

0

where the subscript attached to P, indicates that the P, value is evaluated through the experimental value and pp is the density the polymer solid

SUPERMOLECULAR STRUCTURE

OF POLYMER SOLIDS

employed in the membrane. The porosityobtained method, P,, is calculated using the equation

229

through the immersion

where W, and W,are the weights of themembraneand the liquid used for immersion, and ps is the density of the liquid. Table shows a comparison of the values obtained by various methods mentioned above for the two types of porous membranes shown in Fig. 5.11a and llc. When the pore structure becomes complicated as in the case of aregenerated cellulose membrane, the difference in the characteristic values obtained through several methods becomes more obvious. The maximum pore radius r,, is evaluated through the bubbling method. The bubbling pressure D b 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,, is calculated as

r,, = 2 u J h p b

(9)

where u2 is the interfacial tension between the two fluids at the interface. The value of r,, is often employed as the quality control parameter for the membrane and as the integritytest value for the sterilization filterandthe 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.Fromthe 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 employedin the molding.Here,wet,dry,andmelt moldings aredefined 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.

'

I

I

**m

y-9

000

+ m +

99q

000

O + d 0 0 0

230

SUPERMOLECULAR STRUCTURE

231

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, lidification 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 ina solution. There are four important solidification mechanisms: crystallization, fine particle generation,vitrification,andpolymerpolymer two-phase separation. As for crystallization, there are two mainsteps, nucleation and crystal growth. Nucleation occurs in a sporadic and/or predetermined manner. The number nuclei generated by nucleation decreases withan elevation 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

Molding

hod

solid Polymer

ary

Wet Crystalline

Porous

Primary and secondary particles

particleparticle"

Crystal-Crystal-

Dry Melt

?

?

Phase separationb, crystallization and drawing

lization lization

mary Primary Wet Noncrystalline

Amalgamation Primary

Dry

Melt

Primary particles

particles particles

Primary and secondary particles

gelation particles or

Primary and secondary particles

Vltlifl-

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

Vitrification

Phase separation, dissolution

232

M A B E C i rcular pore

1

I

Noncircularpore

l nside

..

13 Schematicrepresentation of thepore-formingprocessstartingfromcelluloseacetatesolution(Kamideetal., 1977). Thefilledcirclesanddomainsarethe 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."Interfacialsurf"and"Inside"indicatethecross-sectionalview 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 andoligomers. Crystal growth proceeds along onedimension (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 inpolypropylene.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 generationof fine particles has been observed in cases of both crystalline and noncrystalline polymer solids. In the wet-molding process in particular, the supermolecularstructure of the polymer solid be interpreted as the manifold kinds of aggregations of the particles. The polymer solid is constructed 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 POLXMER SOLIDS

233

membranes (Kamide et al., 1977). This processis an exampleof the microphase separation method (Manabe et al., 1973) for preparing porous polymericmaterials. 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 nuclei with a high concentrationof polymer, and these nuclei grow into primary particles having sizes ranging from 10 to 50 nm. The primary particles collide with each other due totheir 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 highercontent of the swelling agents and poor solvents than that 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 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 al. (1990b) used this model to study regenerated cellulose hollow fibers.

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 permolecular 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 orvacant 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 shearmodulus G’ and the dynamic loss tangent, tan S, are typical representatives of the dynamic viscoelasticity. The

W A B E

234

Tg

Temperature

Comparison of thethermalmotionofpolymersolids of crystalline and amorphous polymer solids. Full line: a crystalline polymer solid. Broken line: an amorphous polymer solid. a,,local twisting motion of a main chain in a crystalline region; cxa, segmental micro-Brownian motion in an amorphous region; pa,local twisting motion of a main chain in an amorphous region; %, free rotation or local twisting motion of a side chain in an amorphous region; ymc,free rotational motion of an end methyl group. T, and T,,, aretheglasstransitiontemperatureandthemeltingtemperature, respectively.

existence of the melting point and the dynamic absorption (peak of tan a, originated by the thermal motion of the molecular chains in a crystal distinguishes a crystalline polymersolid from an amorphous one.Many dynamic absorptions are common in crystalline and amorphous polymers.These are, starting from the higher temperature side, the absorption a, that originates in the micro-Brownian motion of polymer segments located in the neighborhood of the glass transition temperature Tg, the absorption Pa due to the local twisting motion of a main chain, the absorption or due to the free rotational and/ or local twisting motion of a side chain,and the absorption ymcdue to the rotational and/or other types of motion of a methyl group. The generalempirical rules obtained far are as follows:

1. The larger the moving unit is, the higher the temperature of the corresponding absorption. 2. 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

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.

A.

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

From the viewpoint of thermodynamics, many kinds 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

T: = AH/M where AH and M are the differences in enthalpy and entropy between the melt and the crystalline state and the contribution 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 T,(Z) of a crystal thickness l is given by

where and 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 andliquid crystal phase transition. The transition temperatures are given by equations similar to In this case the differences in enthalpy and entropy arethe 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(Washizuetal., The first-order transition characteristics of a polymer liquid crystal have been discussed in detail for various polyphosphazens (Schneideret al.,

B. Glass Transition and Segmental Micro-Brownian Motion Part of a polymer melt of a crystalline polymer or most of the melt 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

" B E

apparentlysimilar to the second-order phase transition, the glass transition should be regarded as arelaxationphenomenoninwhich segmental microBrownian motion plays the principal part. This transition is the most important one for a noncrystalline polymer solid. The glass transition temperature T, 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 a,absorption. The temperature dependence of the relaxation time T is represented by theWLF equation given by Eq. (10) as a function of the temperature difference T - TB.

):(

log

-17.44(T - T,) + T - T,

= 51.6

where T~ and,.T are the relaxation times at temperatures T and T,, respectively. Equation is known to fit the data down to about TB.Equation indicates that when the temperature approachesTB,the relaxation time increases abruptly. Under certain limiting conditions T, can beregarded as the constant representing the material. Table 5 lists the values of TBfor representative polymer solids. The size of the chain segments that initiate their micro-Brownian motion at T, is estimated to be less than carbon atoms (Nakayama et al., 1977) and more than 20 (Manabe et al., 1969), evaluated from theapparent activation energyof a, absorptionand from 2 to nm inlengthevaluatedfrom 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 T, penetrant transport becomes very difficult. The diffusion of a dye molecule correspondsto 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 T,. Many physical properties other than transport properties change drastically at T,. The dynamic properties including dynamic modulus G' and tan 6, the dielectric properties, and gas adsorption isothermsare examples. The apparent activation energy of a,absorption ranges between 150 and 850 kJ/mol, 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 D,, extrapolated to infinite temperature. The value of Do is known to closely depend on T,, decreasing with increases in T,.

aluation

237

SUPERMOLECULAR STRUCTURE OF POLYMER SOLIDS Table 5 Glass Transition Temperature TEfor Various Polymer Solids Polymer Polybutene-l Polychlorotrifluoroethylene

Polyethylene Poly-4-methylpentene-1 Polypentene-l Polypropylene Atactic Isotactic Polytetrafluorethylene Polymethyl acrylate Polyethyl acrylate Polymethyl methacrylate Syndiotactic Isotactic Polystyrene Polyacrylonitrile Polyvinyl acetate Polyvinyl alcohol Polyvinyl chloride Nylon 6 Nylon 6, 6 Polyethylene terephthalate Cellulose triacetate

TE

(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

C. Local Twisting Motion of Polymer Chains in an Amorphous Region

(3.

The dynamic absorption caused by local twisting of polymer chains in an amorphous region has been named absorption. The peak temperature of Pa absorption is usually located between and below TB'The apparent activation energy generally ranges between 40 and kJ/mol. 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 thediffusion coefficient due to the addition of small molecules may be related to the disappearance of the absorption caused by this addition (Robeson, Whena 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

WAL3E

is poor and the van der Waals interaction is dominant (Fujioka and Manabe, 1995). Some of these moleculesprevent the absorptionlocatedatabout -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 amorphousregionthat is related to thedevelopmentofthe absorption. Table 6 summarizes the peak temperature T,,, of a, and Pa absorptions and the apparent activation energy A H , for various polymer solids including other dynamic absorptions described below. Since the values of T,, and A H , 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. Figure15 shows the temperature dependenceof 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 al., 1970b). The polymer employed is isotactic poly-a-olefin with unbranched side chains. If we define the number of carbon atomsin the side chain asN, then if N 6 (see Fig. Ea), the dynamic Pa(mc),and a,(,)can be observed in the temperature range absorptions a,, between -180 and +lOO"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:(,cl,a:(+ and Pa(%)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 1/(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 kJ/mol depending onthe value of N. The absorption Pa(sc), whose mechanism is a local twisting motion in a side chain, can be observed when N > and becomes more pronounced with an increase in N. These features of Pa(,) are quite similar to those of 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 endof the main chain or side chain shows mechanical absorption between -220 and -120"C, and the activation energy

+

Table 6 Peak Temperatures of Various Viscoelastic Absorptions Under an Isochronal Measurement at 110 Hz and Apparent Activation Energy AHa for Some 'Qpical Polymer Solids ~~

Yrne

Tmax

aa(sc)y

ma

Polymer

6

(k.J/mol)

Polybutene-1 Polyethylene Polypropylene Polymethylacrylate Polyethylacrylate Polymethyl methacrylate Polystyrene Poly acrylonitrile 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

Trnm

6

260

210

32

Pa~sc,

AH* (kJ/mol)

a.

Pa

Tmm

m

a

00

(kJ/mol)

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 92

40

37

Tm,

m

a

(m

orJ/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

E

B h

"E

240

. . I

-100 0 Temperature I'C

'

I

.

-100 0 Temperature I"C

Temperaturedependence 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 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); IP'I'D 1 = isotactic polytetradecene-l (N= 12); IPHD 1 = isotactic polyhexadecene-l (N= 14); P O D 1 = isotactic polyoctadecene-l (N= 16); PE = polyethylene. (b) The value of N is less than 6. PP 1 = isotactic polypentene-l (N = PB 1 = isotactic polybutene-l (N= 2); PP = isotactic polypropylene (N = 1).

SUPERMOLECULAR STRUCTURE OF POLYMER SOLIDS

241

Plots of melting temperature and peak temperature of various viscoelastic absorptions versus reciprocal of number of carbon atoms of a monomer, 1/(N + 2), for isotactic poly-a-olefins with unbranched side chains. mp, melting point; 4,absorption caused by segmental microBrownian motion; 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.

and 20 kJ/mol. As the moving unit is very small, this abranges between sorption 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 future research.

IV. CORRELATIONBETWEENCHEMICALSTRUCTURE, COMPOSITION, AND PENETRANT TRANSPORT A.

MolecularInteraction

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 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

" B E

the folding back of the chain. This case ofmolecular interaction shouldbe regarded as intermolecular interaction. Interactions betweennonbonded 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 (instantaneousdipole-instantaneous dipole interaction). The potentialenergy U caused by the van der Waals interaction is expressed in Eq. (11) as a function of the distance r between atoms:

U = - A/r6 -k B/r",

n=

., . ,

where A and B are constants determined bythe 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 regularityof the molecular packing in a crystal. The second stmctural 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 A H a of the rate process. Both T, and A H a 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 - A b . Here, A b 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 -Ape is given by -

A b = RT In([P],/[P],)

where R is the gas constant, T the temperature in kelvins, and [P], and [P], are the concentrations of the penetrant in the polymer solid andin 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 [P],/[P], = K = exp(-ApdRT)

(13)

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

(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 coefficientD , is theoretically given by

D, = ( 4 K ) D o + D,

(15)

where D,, is the diffusion coefficient of the penetrant in the medium without the polymer solid, is the weight of the medium in the capillaries per unit weight of the polymer solid, and D,, is the diffusion coefficient of the penetrant in the solid. The value of D , decreases with anincrease 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 moleculealso moves through a polymersolid 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 andwhenthe pore size is less than 20 nmand larger than4 nm, surface diffusional transport dominates (Kamide et al., 1983). In this type of transport, the adsorptionandtwo-dimensionaldiffusioncoefficientshould be taken into account. Affinity works mainly on the adsorption. When thepore size

244

" B E

is less than 2 MI, 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 al., 1993), polyester (Sheu and Chern, 1989), polycarbonate (Hellums et al., 1992), polysulfone (Aitkekn et al., 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 thediffusion 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, thep-structure in polypeptide, andplane zigzag, etc. In general, the configuration governs the conformation. The regular form of chainconformationandconfiguration results inacrystal or aliquidcrystal. Penetrant transport is influenced by the existence of these types of crystals as mentioned in Section 11, and an increase in the regularity of the chain conformation and the chain configuration gives rise to a decrease in penetrant transport. When the conformationofthechain molecule constituting amembrane 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 al.,1993). The conformational changeof a protein molecule may play a very important role in ion transport across a biomembrane.

V. EFFECTS OFFINESTRUCTURE(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-

OF POLYMER SOLIDS

SUPERMOLECULAR STRUCTURE

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

-

)OK,

(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 g. Both the crystallinity 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 valuesof 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)

x

D = D,/Py

(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 acrystalline 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 the extended chain molecules through which most molecules do not diffuse. (19). These changes may reflect the change in the value in 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:ina 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, an increase in the perfection of the crystal, and 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 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 relaxation of the change in the moremaining strain, (2) a decrease in the size of the region, lecular chain aggregation into a closer packing, and a decrease inorientation. 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 In general, heat treatment changes the supermolecular structure 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 T,, the permeability decreases and the permselectivity increases in polyimide thinfilms (Pfromm and Kovos, Figure 17 shows the dependence of the dye exhaustion by polyethylene terephthalate fibers spun at 5, 6, 7, and km/min on the temperature of the heat treatment T, for 1 S at constant length (Kuriki et al., The dye molecule employed as a disperse dye of 1,5-diamino-2-bromo-4,8-dihydroxyanthraquinone with a molecular weight of The dye exhaustion shows a slight minimum at ca. 150°C and then increases with the elevation of T,. 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 viscoelasticities of the solid under the coexisting solvent (Fujioka and Manabe, 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

Or

150 200 Annealing temperaturelOC

247

250

17 Change in exhaustion of adispersedyewithannealingtemperature for polyethylene terephthalate fibers spun at various speeds ranging between 4 and 9 km/ min (Kuriki et al. 1985). 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 amountsof 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 polymerblends (Manabe et al., 1969). Figure 18 shows the temperature dependence of the dynamic viscoelasticity of a polymer blend of nitrile butadiene rubberandpolyvinyl chloride after treatmentwith 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

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-21 ' -50

-50

0

Temperature I0C

0

50

50

Tempetat we l OC

Temperature dependence of dynamic modulus E' and loss modulus E" for polymer blend of polyvinyl chloride/nitrile butadiene rubber(PVCN3R) after treatment withsolvent(Manabeetal.,1969).The PVC/NBR ratio = 77/23byweight. (a)The solventisdimethylformamide,anditscontentis 6 and 14 wt %. @)The solventis chloroform, and its content is 7 and 20 %. 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 al., 1969). The permeability coefficients of toluene, methylcyclohexane, and isooctane before treatment in one case were 15.3 lo-',14.3 and 1.4 lo-' g cm/@ cm*),respectively. After the treatment at 100°C, and coefficients were 59.4 lo-', 46.2 lo-', and 21.2 g cm/(h cm2).

VI. CONCLUDINGREMARKS 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 probemicroscopy (SPM), atomic force microscopy, friction force microscopy, and magnetic force microscopy are very powerful tools for investigating supermolecular structure. correlate supermolecular structure and penetrant transport, we must employan adequate transportmodel. Although the free-volume modelandthe capillary model and their combinations have been proposed 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 apolymersolidguidedbytheoryshouldbedevelopedinthe 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,andW. J. Kovos (1990). ICOM '90, Preprint, p. 821. Aptel, P., and I. Cabasso (1980). J. Appl. Polymer Sci., 25, 1969. Fujioka, R., and Manabe (1995). Polym. Prep. Jpn., 44(5), 759. Haraguchi, H., T. Kajiyama, and M. Takayanagi (1976). Rep. Prog. Phys. Jpn., 16,303. Hellums, K. W.,W. J. Kovos, and J. C. Schmidhauser (1992). J. Membrane Sci., 67, 75. Iijima, H.,and Manabe (1983). Japanese Patent, Tokkai Sho 58-183907. Kajiyama, T., K Tanaka, I.Ohki, S. R. Ge, S. Yoon, and A. Takahara (1995), Macromolecules, 27, 7932. Kamide, K., and Manabe (1978). Sen-i Gakkaishi, 34, T53. Kamide, K., and S. Manabe (1980). In Ultrafiltration Membranesand Applications, A. R. Cooper, Ed., Plenum, New York, p. 173. Kamide, K.,and Manabe (1985). In Materials Science ofsynthetic Membranes, D.R. Lloyd, Ed., American Chemical Society, Washington,DC, p. 197. Kamide, K.,and S. Manabe (1986). Polymer J., 167. Kamide, K., S. Manabe, T. Matsui,T.Sakamoto,and Kajita (1977). Kobunshi Ronbunshu, 34, 205. Kamide, K., Manabe, T. Nohmi, H. Makino, H. Marita, and T. Kawai (1983). Polymer J., 15, 179. Kamide, K, Nakamura, T. Akedo, and Manabe (1989). Polymer J., 21, 241.

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Kawaguchi, T. (1962). InKoubunshi no Bussei, T. Kawai, T. Kawaguchi, and A. Miyake, Eds., Kagakudoujin, Tokyo, p. 141. Kesting, R. E. (1985). Synthetic Polymeric Membranes, Wiley, New York. Kesting, R. E., and A. Manefee (1969). Kolloid 230, 241. Kuriki, T., K. Kamide, S. Manabe, and M. Iwata (1985). Sen-i Gakkaishi, 38, T85. Manabe, S. (1992). InAnimal Cell Technology: BasicandApplied 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, 1581. Manabe, S., and M. Takayanagi (1970~).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, 391. 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. Alcalay (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, Okamoto, K., N. Umeo, S. Okamyo, K. Tanaka, and H. Kita (1993). Chem. Lett., 225. Peterlin, A. (1969). J. Polym. Sci., A-2, 7, 1151. Pfromm, P. H., and W. J. Kovos (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. ScL 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, 1241. T S U N ~ ~ 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., B6, 465.

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

INTRODUCTION A flexible polymer chain is capable of assuming an enormous number of pos-

sible configurations by rotations of chemical bonds. Such local motions, which occur on time scales on the order of lo-" 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 its translational diffusivity is determined primarily by N,the number of monomer segments that compose a single chain, and 5. The intent 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 ratherremarkable behavior that distinguishes themfrom other materials (Ferry, 1980). One important feature can be seen inthetime dependence of the stress relaxation of a melt that has been subjected to a strain. A sudden imposition of a shear strain, y, 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 toequilibrium through some Brownian process. It is convenient to describe the time dependence of such a 251

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GREEN

process by defining a shear-stress relaxation modulus, G(t) = u(t)/y, where u(t) is the time-dependent shear stress thatreflects the structure and interactions between the chains in the system (Ferry, Graessley, Figure 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, Ne, where Ne is the number of monomers betweenentanglements, G(t) becomesindependent of t, the plateauregion. 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 tozero. Here the chains assume equilibriumconformations through a translational (Brownian) process. Properties such as the viscosity are also affectedby entanglements. For chains shorter than a critical number of segments, N,, where N, is a material-dependent property(empirically N, W,),the viscosity in thelimit of zero shear, qo Figure 2 shows a plot N , while for N > N,, qo ~ 3 . 4 ’ 0 . 2 (Berry and Fox, of the molecular weight dependence of the viscosity of polyethylene. It is important to note that for N N,, qois observed to vary as N after corrections are made to achieve a constant free volume state. The triangles represent the un-

-

-

I Schematic

/

Log t thetimedependence

a melt. C: is the plateau modulus, and

melt.

of thestressrelaxationmodulus,

G(t),

M and M* are the molecular weights of the

253

TRANSLATIONALD W M I C S IN MELTS 14

12

10

6

4

2

0

-2 2

3

4

6

8

log 2 Molecularweightdependenceof the zeroshearrateviscosityofpolyethFor N < N,, it varies as N (0) after the ylene. For N > N,,the viscosity varies as N3.4. raw data (A)are adjusted for chain end effects. This behavior is typical. [Data of Colby et al. (1987).]

corrected data. The N3.4dependence remains unchanged for N > N,. This behavior is typical. The foregoingdiscussion provides the framework withinwhich we can begin to understand translational dynamics in polymer melts or concentrated systems. Because the large number of configurations that a polymer chain canassume, 1969). the shape of a single polymer chain can be described statistically Consequently, a description the dynamical process necessarily exploits this simplification. In an attempt to understand the dynamics of chains in solution,

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Rouse(1953)arguedthat 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, 5. As shown later, what ultimately emerges from such a picture is that the translational diffusion coefficient DRo 5 8 ” and that the longest relaxation time of the chain T~~ 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 thechain. The chain translates in a manner that simulates the motion of a snake. Reptation was later introduced by Doi and Edwards (1978a,b, 1986) 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, andexperimental data on diffusion and viscoelasticity support the new predictions. One might emphasize at this point that calculations the viscoelastic properties of polymers based on this picture, while qualitatively correct, do have shortcomings. In this chapter we begin with a detaileddescriptionof 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). discussion of experimental tests and the extent to which the results support the predictions follow. Finally, a discussion of interdifhsion in two-component miscible blends is presented. Here the concept of the Flory interaction parameter

-

-

TRANSLATIONAL DWAMICS MELTS IN

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 determinednot only by N and &,, but also by x. When x < 0, the interdiffusion coefficient, D , is enhanced overthe 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 HOMOPOLYMERMELTS

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 p o i and Edwards, 1986). This is primarily because itneglects excluded volumeand hydrodynamic interactions, both of which are assumed to be screened in melts. The central assumption in the Rouse model is thatthe dynamics of a chain are governed by localized interactions along the chain. Each chain is imagined to be divided into a series of submolecules, each composed of a number of monomers. Each submolecule is represented by a bead, and thebeads are connected by Hookeansprings, where each spring constant is 3kBT/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 5. The motion of the beads can be described by a linearized Langevin equation. In essence, therefore, we now have an equation thatdescribes the Brownian motion ofa series of coupled oscillators. A transformation is subsequently made 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, T,.,corresponding to one normal mode. The essential predictions follow. The diffusion coefficient of the center of mass is

DRo= k,T(N{)”

(1)

One might recognize that q.(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

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where P(t) = R,.,(f)-RN(0) and r1is 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, r l , and is given by rRo =

= (b2/3~2k,T) yV2

(3)

The viscosity in the limit of zero shear is predicted to be rlo

- N5

(4)

As mentioned earlier, this model is expected to provide a reasonable description of the dynamics of unentangled chains in the regime N < N,.

Shown in Fig. is a schematic of a labeled chain composed of N segments in a meltsurroundedby other chains, where each ofthe 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 theN-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 p o i 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

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.

TRANSLATIONALDW'ICS

ii)

--- """""--

257

IN MELTS

---*

b)

+ \

PRIMITIVE PATH

(a) A schematicdemonstratingthemechanism of diffusionfor 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 the chain, which results in changes in the primitive path. 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 q, 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 a constant contour length L. One can define the number steps on the primitive path as Z; hence L = Zu, where a is the length of a step along the path. The diffusive motions along

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258

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, it follows that

L = Nb2/a

(5)

One can define the end-to-end vector of the chain as P(t) = R(L, t) - R(0, t). The time-dependent correlation function of the end-to-end vector is given by (P(t) P(0)). It is proportional to +(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),

(W)- W)) =

(6)

and

The longest relaxation time, ?d

rd,

is calculated as

= (b4/~2kBTa2)

In contrast to the longest Rouse relaxation time, the reptation relaxation time, ?d, varies as N3 as opposed to NZ. 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 32. It is interesting to note that the tube disengagement time, ?d, could be calculated from a simple scaling relation by noting that

L2/&

(9)

which, using Eq. (5), yields 7d

(b4/kBTa2)

(10)

The diffusion coefficient of a chain moving by a reptation process could be derived by noting that during the time interval ?d the center of mass of the chain should have moved a distance equivalent to the radius of gyration of the chain, N1”u,

DRcp= NU2/7d

(11)

Therefore,

D , = (kBTu2/3b2() NZ

(12)

LTS

INTRANSLATIONAL DI%M."IC

259

equivalent form of this equation may be realized by recognizing that the tube diameter U can be written as (Graessley, 1980; Graessley and Edwards, 1981) = (4/5)Nbz(M/Me)

(13)

where M = MOA' (Mo is the molecular weight of a monomer on the chain and M, is the molecular weight between entanglements). It follows that

D , = Do"'

(14)

where D, = (4/15)M0M,kBT/5.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 in a highly entangled melt varies as M-', independent of the length of the P-mer chains in the host environment(recall the initial assumption that P >> We now use the above results to calculate qo that we can show howDRep can beexpressed in terms of viscoelastic parameters (Graessley, 1980). The viscosity can be obtained from the relationship = G:

W) dt

Upon integrating this expression one arrives at

This result shows that the viscosity is determined by the longest relaxation time [To GN(O)'T~]. It fOllOWS that

We show later that upon substitution of the appropriate viscoelastic parameters, this result yields very reasonable predictions for the magnitude of DRepfor different polymers.

C. Measurement of Diffusion in Polymer Melts 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

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square displacementof the center of mass 1 6t

D = lim - ([R(r) - R(O)]*)

(18)

where R is thepositionof the center of mass ofthe diffusant and r 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. Spica1 diffusivities in polymer melts range from to cm2/s,which means that one needs experimentaltechniqueswith sufficientlygood depth resolution to measuresmall diffusion coefficientsin reasonable time scales. As an illustration of this point, one might consider attempting to measureadiffusioncoefficient of a molecule thatdiffused for one day. If the diffusion coefficient is lo-’ cm2/s, then the mean displacement is 2.3 cm.On the other hand, if D* = 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 al., 1978; GreenandDoyle, 1990; Greenetal.,1985),forward recoil spectrometry (FRES), and elastic recoil detection (ERD)] (Mills et al., 1984; Green et al., 1986; Green and Doyle; 1990), nuclear reaction analysis (NRA) (Chaturvedi, 1990),fluorescencerecovery after patternphotobleaching (FRAPP) (Smith, 1982; Smith et al., 1984), forced Rayleigh scattering (FRS) (Antonietti et al., 1984), small-angle neutron scattering ( S A N S ) (Bartels et al., 1984, 1986), and nuclear magnetic resonance(NMR) (Fleisher, 1983, 1984, 1992;von Meerwall, 1991; McCall et al.,1986). The range of techniques is notlimited 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 D W M I C S 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 dmsivities. The ion beam analysis techniques, andIRM, 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 extract D* or D,. 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. analysis of the yield and energy of the appropriate recoils gives the desired concentration versus depth profile. In NRA, the natureandenergyof the incident particles is carefully chosen 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 & and are ideally suited for measuring midrange to very slow diffusion coefficients (10-10-10”6 Cm”S).

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 > lo-’ cm’/s.) 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. is capable of measuring D$ in the range of 10-8-10”6 cm2/s. While FRAPP and FRS require labeling of the appropriate species, they do not yield directly the concentration profiles of the d&sant. 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

GREEN dependence of this process yields D. The concept of FRS is similar to FR4PP except that a photochromatic dye is used in place of the fluorophore. In this technique two intersecting laser beams producespatially 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 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 pulses results in a spin echo a given amplitude. If the nuclei in the region of the sample into which the magnetic field is applied are undergoing adiffusion process, then the amplitudeis 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 cmz/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 conventionaltransmission Fourier transform infrared spectroscopy (Highet al., 1992), scanning electron microscopy X-ray fluorescence (Gilmore et al., 1980), infrared attenuated total reflectance (IRATR) spectroscopy (Van Alsten and Lustig, 1992), dynamic light scattering (Murschall et al., 1986), and a modified optical schlieren technique (MOST) (Composto et al., 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 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 al., 1956) usinga 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 D MELTS W M I C S IN

263

using infrared microdensitometry. Figure 5a showsa double logarithmicplot of the self-diffusion of polyethylene from different authors. While virtually all the published data for polyetheylene follow the M-' dependence predicted by reptation (Bartels et al., 1984; Kimmich and Bachus, 1982; Bachus and Kimmich, 1983; Fleisher, 1984, 1985, 1987; McCall et al., 1959; Klein and Briscoe, 1979; Klein et al., 1983; Peterlin, 1983; Zupancic et al., 1985), the actual magnitudes di€fer in some cases. The discrepancy is associated with the NMR data. Some of the NMR data, like those of Pearson et al. (1987), appear to be higher than the other sets of data determined using SANS (Bartels et al., 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 S A N S and IRM after they were corrected for polydispersity. The data of Pearson et al. 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 174-polybutadiene, which have some minor microstructural differences that differentiate them from PE (inclusion of some 172-polybutadiene). The different sets of data on polystyrene also exhibit an M-' dependence (Bueche, 1957; Kumugai et al., 1979; Kimmich and Bachus, 1982; Bachus and Kimmich, 1983; Green et al., 1985, 1986; Green and Doyle, 1990; Antonietti et al., 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 thanthat for polyethylene. There are additional results for other pure homopolymer systems, such as poly(methy1 methacrylate) (PMMA) (Green et al., 1988, 1989), that also show M-' dependencies for other pure homopolymer systems (Fig. 5c). There should be a word of caution about comparingD* and D,. The M-' 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 measuresthe diffusion ofchains of molecularweight M into a hostof 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 whereM M,. Data on PE exhibit an M-' dependence (Pearson et al., 1986), as expected from the Rouse model. This result is realized after free-volume corrections have beenmade. RecentNMR experiments ofselfdiffusion in polydimethylsiloxane (PDMS) show that for M < M,, D* M", which is also consistent with the Rouse prediction (Crosgrove et al., 1992). There remain some noteworthy observations that are disturbing. In PDMS (Crosgrove et al., 1992), the molecular weight regime where M > M,, it was determined that D M-", where a is appreciably less than 2. Another note-

-

-

264

GREEN

n

1o - ' O

I

10-l2

1

0

0

'

1 o2

(a)

O A\0

1

1

o3

I

I

1 o4

1 o5

1 o6

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 prefactor is Do = 0.006 whereas the measured prefactor is 0.0075 cm'/s [ G i = 2 X lo6 dyn/cm' (Ferry, 1980); (R') = 1 X cm', p = 0.96 g/cm3, M, = 31,000, and rlO(Mc)= 1200 (Fox and Berry, 1968)l. In the case of polyethylene, at 176"C, the experimental value of the prefactor ranges from 0.32 cm2/s(SANS) to 0.82 cm'/s (IRM), while the calculated value is 0.34 cm2/s [ G i = 2 X lo7 dyn/cm2, (R') = 1 X cm2, p = 0.767 g/cm3, M , = 3800, and qo(Mc)= 0.3 (Bartels et al., 1984)l. At 175"C, Pearson et al. 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. 'lbo systems have been investigated.

TRANSLATIONAL DYNAMICS IN MELTS

265

1 0-l2

1 0-13

*

p

10-l~

1 0-15

I

1 0-l6 1 o4

M

I

D'(RBS)

\

1 o4

(c)

1 o6

I o5

(b)

0

D*(ERD)

A

D'(FRS)

X

D' (NRA)

1 o6

1 o5

1 o7

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).]

GREEN Green et al. studied the diffusion of d-PS into compatible mixtures of PS and poly(viny1 methyl ether) (PVME) of varying composition and determined that the D* for the d-PS chains varied as M-' (Fig. 6) (Green, 1991; Green et al., 1991). However, the actual magnitude of D* varied with composition and exhibitedaminimum at approximately the 50% composition regime. Figure 7 shows the composition dependenceof D* for d-PS diffusing into the PSPVME system at a constant temperature abovethe glass transition temperature of each blend (T - TB= The variation is considerable and is not fully understood at this point. Compost0 et al. (1990) studied the tracer diffusion of d-PS and of deuterated poly(xyleny1 ether) (d-PXE) chains into mixtures of PS and PXE of varying composition. Like the PSPVME system, D* of the d-PS and of the dPXE chains diffusing into the mixtures varied as M-*. These values exhibit a strong compositional dependence. What is interesting abuot the composition dependence of D* in bothsystems is that it is highlynonlinear. 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 the components in PSPVME and PSPXE blends show that the seg-

1 0-l2

m D(40%0,1070C)

lo"8

3

TRANSLATIONAL DYNAMICS IN MELTS

10"~

267

~ ~ ~ " " " " " ' " ' ' " '

-0.2

o

0.2

0.4

0.6

0.8

1

cp Figure 7 Compositiondependence thetracerdiffusioncoefficient, D*, d-PS chains diffusing into miscible blends if PSandPVME are at temperatures T - T, = 100°C. [Data Green et al.

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 PMMAIpolyethylene oxide (PEO)mixture using an approach that simultaneously determinesthe infrared dichroism and the birefringence also support these findings (Zawada et al., 1992). Dichroism measurements by Monerrie and coworkers also support these findings (Lefebvre et al., 1984; Faivre et al., 1985). Therefore it is clear thatina 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. TemperatureDependence It is very clear from Eq. (17) that knowledge of the temperature dependenceof qoshouldbe sufficient todetermine the temperaturedependence of The temperature dependenceof DRepcan be determinedin a straightforward manner. Equation (17) can be rearranged yield log(DR,/T)

= c(M)- 1% l o

(19)

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268

The relation that G$) = pRT/M, was substituted into (17) (Graessley, 1980). Thetemperaturedependence of qo is well describedby the Vogel-Fulcher equation log ll0 = A

+ B(T - To)“

(20)

where To and B are the Vogel-Fulcher constants, unique to eachpolymer (Berry and Fox, 1968). It follows that the temperature dependenceof D,,/T should be given by log

(%)l

[(=)

DIl,(T,S

= A’ - B(T - To)-’

where A‘ = B(T,, - To)”. In this equation, T,, is the reference temperature at which DRep(Tnf) is determined. This result suggests that log T/D* should have the same temperature dependence as log qo.It turns out that the experimental data in polystyrene (Green and Kramer, 1986a,b) and in polyisoprene (Nemoto et al., 1984) are consistent with this. Figure 8 shows the temperature dependenceof the diffusion d-PS into PS. It also exhibits the same temperature dependence asthat the inverse

o”*

0-15 5

10

6

B/(T-T ) Temperature dependence the D* d-PS of M = 110,000 into PS.The These are empirical parameters used for the fit are B = 710”C, To = 49”C, T,, = thesameconstantsusedto 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 qoin polybutadiene, as shownby Bartels et al. (1984). In polybutadiene, the temperature dependence of the melt viscosity is where the activation energy for flow is 30 kJ/mol. While independent of M , as expected, it is 8 kJ/mol higher than that for diffusion. As argued by Bartels et al., this can be rationalized in terms of (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 5 varies by orders of magnitude over a temperature range of 100°C. The other temperature-dependent factors in Eq. (17), (R') 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 (Rz) become important. These factors account for the difference in temperaturecoefficient. alternative explanationhasbeensuggested (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. l), the stress relaxation modulus, is not a simple exponential in time; instead it is described by a stretched exponential [G@) exp (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 M -'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 D,, and of G$)are in reasonableagreement withpredictions 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 qo M' while experiments showthat qoactually scales as (Ferry, 1980; Berry and Fox, 1968). Furthermore, while the plateau of G(t) [G(t) = G,(O)+(t)] is correctly predicted, the prediction that the product of the steady-state compliance J.(O) and is

-

is not correct. Experiments show that the product is between 2 and 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 T~ of the N-mer chain. It is clear from this figure that upon removal of constraints on the N-mer chain, the chain canundergo 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. 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 qo P 3 (P > Pc), where P is the degree of polymerization of the host chains. Consequently, there should be a correction of DRcp, which varies in a manner that maybe described by the Stokes-Einstein relation. This correction is predicted to vary as DCR N'nP". There have been a variety of other theories developed 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 al., 1987; Rubenstein

-

-

The configuration of the chain changes as the topology of its environment changes when a constraint is released.

271

TRANSLATIONAL D W M I C S IN MELTS

and Colby, 1988; Doi et al., 1987). With the exception of Klein’s work, these theories address the question of viscoelasticity primarily. Most the constraint release theories suggest that the correction to DRepvaries as DCR P-3. Klein, on the contrary, argued that the dependence is somewhat stronger, DCR P-SR. Klein’s argument is based on the fact that a single chain may provide more than one constraint on the N-merchain. Graessley’s theory, for example, asdiscussed later, assumes that each P-mer chain accountsfor one constraint. The effect of the interdependence the constraints is to enhance the constraint release (tube renewal) contribution; hence D l C R F S R . Hess (1988b) also addressed the question of constraint release in melts using a many-body approach and suggested that D , P”. It turns out that there are no experimental data on the effect 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 byTd(lM) for the probe chain and Td(P) for the matrix chains. Both the reptation and the constraint release processes are assumedto be independent;therefore, the total diffusion coefficient of the N-mer chain is now

-

-

-

-

D* = DRep+ DCR

(24)

where DCR, as described above, is the contibution 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 the chain trapped in it remain Gaussian during each successivestep. The constraints in this model are considered in an idealized manner where the diffusion the N-mer chain occurs on a cubic lattice with effective constraints (the removal of an arbitrary constraint will not necessarily contribute to alteration 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 the first the constraints is defined as

According to Graessley, the contribution of constraint release to the diffusion coefficient of the chain is

In the case of a single chain diffusing into a single-component monodisperse host, the mean waiting time takes on a value of T,,,= (m2/12)2?d.

272

GREEN

Note that an approximationis made whereby +(f) 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 MPis given by

D, = Do(”’

+ acRM2M/Mi)

where Dowas defined earlier as Do= (4/15)MdM,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 thatD*, which becomessignificant, particularly at lower M. At sufficiently high M, D* = D,. 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. 10a are data that have been fit with Eq. (27). Only one adjustable parameter has been used, acR,which was found to have a constant value of 11. Forward recoil spectrometry measurements by Green et al. (1984) on P “ A melts are also well described by Eq. (27) using a value of = 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

I

I

lo7 (a)

P

(a) Data showing the effects of constraint release of d-PS in PS at The lines drawn through the data were computed using Eq. 0) M = M= M= M= (A) M = (A)M = 1,800,000. [Data of Green et al. (1984).]

TRANSLATIONALDYNAMICS MELTS IN

273

thatalthough was the only adjustable parameter, its value varied with the molecular weight of the d-PS diffusant. Studies of the polypropylene systemdid not provide strong evidence for the reliability of Eq. (27) (Smith, 1982; Smith et al., 1984), butother 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 (Compost0 et al., 1992) systems. The results were found to be well described by a generalized form of the equation

that accounts for a host of components, 1and 2. In this equation, = ~ ( 1 ) / ~ ( 2and ) = [Me(1)/Me(2)lm.In the absence of the second component(i.e,. cp = 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 lo6 with PVME, where the PVME

P Continued (b) Constraintrelease data of d-PMMA (M = 519,000) into PMMA of molecular weight P at [Data of Green et al. (1984).]

GREEN

274 10-12

10-13 -

10.14

10-15

-

10-16 104

I

....

I

107105

I ...a

l

106

..

I

I

**ad

P

Figure

Data showing the constraint release of d-PS M= (M) M = 520,0001 into miscible blends of PS of molecular weight P with 40% P V M E of fixed molecular weight P = 145,000. [Data of Green et al.

molecular weight was fixed at M = 145,000. The broken line was calculated [Eq. (28)], with the constants cp = 0.4 and the molecular weight between entanglements of PS taken to be Me(l) = 18,000 and that of PVME, Me(2) = 12,000. The constant k was taken to be equal to aCR = 11. While the values of k were found to be consistent in the PSPVME system, they were found by Compost0 et al. (1992) to vary considerably with composition in the PSPXE! system. A resolution of this situation will await further experiments and theory.

2. TubeLengthFluctuations In addition to the constraint release process,theN-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 qop o i , 1983; Doi and Edwards, 1986). Doi has shown that the average fluctuations

(M2)/@) = Z-In

(29)

Recall that 2 is the number of steps on the primitive path (2= L/u). Considering that qois proportional to the longest relaxation time, one might incorporate the fluctuations in length in calculating the new relaxation time. According to Doi, 7d(F)

m

7d(1 - k/Z1,)’

275

TRANSLATIONALDYNAMICS MELTS IN

where k is a numerical constant that is close to unity. This result is obtained by noting that 7 d L*/&, and 7 d ( F ) (L - AL)’/DR,,.Based on the result in Eq. (30), the new qo,which incorporates the chain length fluctuations, is

-

qom

qo[l - k v 4 e ~ ~ i n 1 3

(31)

where k’ is a new constant. It is important to point out that Eq. approaches the reptation prediction at extremely large values of M . At smaller valuesof M, these corrections are important. Experimentally, Colby et al. (1987) have shown that qo N3.4for M/Me as great as 150; beyond M/Me 200, departures from the N3.4dependence that are consistent with N 3 are observed. While the data of Colby et al. suggest that in the limit of very large M/Me 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 the very high M polymers. O’ConnorandBall(1992) later revisitedthisproblem. 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 R O , 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 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 thechain 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 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$) and the monomeric friction factor O’Connor and Ball (1992) showed, through a computer simulation, that the outstanding discrepancies (qoM, J:)G$) prdiction, etc.) in dependence could be accounted for. One the important results of this work is that the chain contourfluctuations described by DoiandEdwards(1986)couldnot 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 M3.4power law dependence wasaccounted for in a number polymer systems; at very large M the M 3 power law dependence is recovered. Furthermore, the discrepancy in the value of J:)G$) was also resolved. It is my opinion that this is not the end of the story! There are details of the

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276

simulation that are unknown because they were not published. The diffusion of ring and star molecules is discussed below.

BranchedMolecules 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-shapedmolecules wasfirst addressed by deGennes (1979) and subsequently by a number of other authors (Graessley, 1982; Helfand and Pearson, 1983; Klein, 1986; Pearsonand Helfand, 1984; Doi and Kunuzu, 1980). For the sake of clarity we begin with the motion of a star molecule off = 3 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 L,. 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, L, - (Ls), is Gaussian, then the motion L, can be considered to be Brownian and occurring within a harmonic potential

=

k,T (S) (L, -

If one considers this an activated process, then the disengagement time associated with the chain end going from a point (L,) to L, = 0, as shown in Fig. 12b,

12 Schematic depicting the mechanism of diffusion of a star molecule.

TRANSLATIONAL DYNAMICS IN MELTS

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277

is given by T~ exp[(3/2)~,(b/a)~]

(33)

alternative manner in which one might arrive at the same result, as shown by deGennes, is to consider the probability of an arm of N, segments retracting along its contour without enclosingany obstacles. Such a probability is

34)

P(NJ

exp(-yNs)

where y is a constant that depends on M,. The rate at which the arm retracts, deGennes argues, should be given by 1hs= P(Ns)k@J

(35)

Therefore, (36)

7 s

cc

Td(NS) exp(yNs)

There have been a number of predictions for the form of Eq. (36) where 7 s

- N: e x p ( - W

(37)

The exponentk has beenassigned values of 3 (deGennes, 1979;Doi and Kunuzu, 1980); 0 (Graessley, 1982); 312 (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

D

a2/7,

(38)

(38) predicts that

The foregoing result is for a three-arm star. Doi pointed out that as the star diffuses, it needs to withdraw - 2 arms to the branching point. The activation energy for such a process is

Therefore the general diffusion coefficient is

70

278

GREEN

In general, one might write the diffusion coefficient for an farm star as

D,,, = c”Dre,

- 2WeMsI

exp[-s(f

(42)

where c,, c, and c, 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

D* = D,,,, + D m

(43)

where DCRis given by Eq. (25). Few measurements have been done in this area (Kline et al., 1983; Bartels et al., 1986; Antonietti and Sillescu, 1986; Shull et al., 1988; Crist et al., 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 preexponential is less certain at this point. Many of the data appear to be well describedby assuming that the preexponential factor is independent of molecularweight. Shull showedthat D,,, varies exponentially with the molecular weight per arm for stars of different lengths diffusing into microgel matrices (Fig. 13). These microgels relax suffi-

1 0-l1

U

10”‘

l0”

*

0-l 0

10000 20000

30000 40000 50000

M

a

13 length dependence of the D* of afour-armstarmolecule.Thedata are exponential as expected. [Data of Shull et al.

TRANSLATIONAL DWAMICS MELTS IN

279

ciently slowly that they simulate the behavior of linear chains of infinite length. Consequently the arm retractionmechanism is significant. Bartells et al.and Klein et al. also demonstrated this exponential dependenceof the diffusivity of PE star molecules diffusing into high molecular weight matrices. The constraint release mechanism is also shown to be veryimportantin 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. where the constraint release contribution is described by (Klein, 1985)

(44)

DCR= D ~ M - 1 P - 5 R

This is a stronger dependece onP than the expected F 3 . This discrepancy might be related to the interdependence of the constraints discussed by Klein earlier.

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 caseit will be trapped and be able to undergo translational motion,unless, 10

I

Y %c

1

-

0.1 1 0'

1Ob

os

14 Datashowingthe effect of constraintrelease thediffusion of a star molecule of M. = 60,000. D m vanes as P-'.', as [Replotted data Shull et al.

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280

of course, by some constraint release process. There are, however, 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-consc "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. 1 5 ~Therefore . the diffusion coefficient of a ring molecule is DR(NR)

DRep(NR/2)

(45)

Rubenstein (1987) made a similar prediction. If the chain diffuses into a host environment of linear chains of molecular weight M,,, then the total difision coefficient is

D* = ~ ( N R+ )DcR(M~)

(46)

Studies of the d f i s i o n of ring polymers into different host environments have been performed in PS (Mills et al., 1987; Tead et al,. 1992) and in PDMS (Crosgrove et al., 1992). The experimental situation is far less certain for rings than for stars. The NMR measurements (Crosgrove et al., 1992) the selfdiffusion of ring molecules show a molecular weight dependence that is appreciably less than F*. 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 consideringthat the authors foundDs 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 diflFuse 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 al. (1992) show clearly that thereis a lower limit of the ring diffusion coefficient in linear matrices that was not observed by Mills et al. in anearlier study. They showed that for large enough host chains D* became independent of host molecular weight, consistent with the predictions of (25) (Fig. 16). This is reasonably strong evidence in support of a

-

15 Possible configurations of ring molecules in a host of linear chains.

TRANSLATIONALDYNAMICS IN MELTS

U

10”2

n

l

281

O Om

0

m. 1 0-13

0 m

0 0

10”‘

00

0

4

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 al. (1992) replotted.]

reptation type of mechanism. 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 al. are replotted in Fig. 17 as D*M& versus M to illustrate the large deviation. This result is somewhatof 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 addressthe 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.

W.

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, T,, (recall T~ 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

-

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1 o4

1 o5

1 os

17 The D* of d-PS rings into PS microgels of P = 490,000. The data of Teal et al. (1992) are replotted D*Mzgversus 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 T~ 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 thermodynamicsof the system on D can, within the Flory-Huggins model (Flory, be characterized by x. In a two-component system the excess free energy of mixing is proportional to x+A+B, where +A is the volume fraction of component A and +B 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

TRANSLATIONALDW’ICS MELTS

IN

283

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 entropyof mixing for polymers is extremely low, varying as UN, 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 itis expected that the mutual diffusion coefficient will be highly influenced by the value of When < 0, mixing is favored and, as we will see later, the magnitude of the mutualdiffusioncoefficient(the interdiffusion coefficient) is greatly enhanced over the case where x = 0. This behavior can be described as a thermodynamic “acceleration” of D. When > 0, mixing is influencedby the combinatorial entropy such that a system canremain in a region of single-phase stability provided that 0 < < S,where

x.

x

x

As shown later, D(+) will undergo a “thermodynamic slowing down” under

x

these conditions. Of course, when > xS,mixing is not favored. Our discussion on interdiffusion will concentrate on two-component mixtures for simplicity. Wemay begin by considering twopolymer layers, A and B, separated by an interface. Further, we will imagine that the chains are arranged on a quasi-lattice, where eachcell occupies a volume 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, JA, due to the A chains and another, JB, 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 JA # JBand 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-

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portionality are the Onsager (or mobility) coefficients JA

= -MAVpA

J B

= -MBVpB

Jv

= MAVFA

+ MBVCLB

Here we formallyrefer to the compensating bulkflow as a "vacancy" flow and designate the flux as J, Vpi are the gradients in chemical potential. We have formally introduced the assumption that Vc~y= 0, which guarantees that "vacancies" are at equilibrium everywhere. We have also assumedthatthe diagonal terms M m = MBA= 0. The following expressionfor the interdiffusion coefficient was derived by Kramer et al. (1984):

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 segments of JA(tot), across a fixed interface must be conserved, i.e., JA(tot) = "AVPA

+ +(MAVPA + MBVpB)

(51)

and that

Eq. (50) follows. By writing the Onsager coefficients in terms of the Rouse segmental mobilities, Kramer et al. (1984) showed how M. (50) could be expressed in terms of the tracer diffusion coefficients of species A and B in the mixture. The Onsagercoefficients can be written in terms of the segmental mobilities, BA and BB, as

MB = BBcB

(54b)

where C, = +/aand cB = (1 - +)/aare the concentrations of species A and B. These relations follow from the fact that the flux is proportional to the dif-

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IN TRANSLATIONAL DYNAMICS

285

fusional velocities, where the constant of proportionality is the concentration. Alternatively,the flux is also proportional to the chemical potential gradient (generalizedforce), where the Onsagercoefficients are the proportionality constants. Equation (50) &n be written in terms of the tracer diffusion coefficients by noting that the polymer segment mobilities, B, and BB, can be described by the curvilinear Rouse segment mobilities. Therefore,

D(+) = where DT =

- +PT(X.

-

- +)DXNA + +DzNB

(55) (56)

This result was derived independently by Sillescu using a more general approach, where he took advantage of the Hartley-Crank equation. The form of Eq. (56) indicates, for example, that if DX >> D:, then D(+) is controlled by D:, the faster moving species. Therefore, if a Kirkendall-type marker experiment wasperformed, 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 “fast7’-mode theory of diffusion and has been criticized for reasons discussedbelow. 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 difEusive process should be controlled by the more slowly moving species. This led to anotherproposal, the “slow7’-mode theory (Brochard-Wuart et al., Brochard et al., Brochard-Wuart and deGennes, Binder, One can, in fact, arrive at the prediction for the slow-mode theory by invoking the condition that J, = 0. Brochard and coworkers and Binder arrived at the result that the expression for DT is

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 al., subsequent proposal by these authors suggested that the problem is

286

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actually a length scale-dependent one whereby, over a length scale

L = [D? Td(B)]'R 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 tothe slowmode theory (Brochard-Wyart and deGennes, 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. I R M 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 interdfise at distances of macroscopic lengthscales (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 concentrationprofiles 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 al. also addressed this problem but used a different approach. They studied the molecular weight dependenceof the mutual diffusion coefficient at a given concentration in the PSPXE system and determined that D N&?, 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 in PMWpoly(viny1ydine fluoride) (PVDF)and that of Murschall et al. in polyphenylmethylsiloxane (PPMS)PS. As discussed by Composto et al. 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 who used ERD to study interdiffusion in d-PSPS 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 al., The isotopic substitution results in slight differences in segment volumeV and atomic polarizability a between the one polymer and its isotopic counterpart. It has been shown that knowledge of a and V will enable oneto calculate x. Because of this unfavorable segmental interaction, the d-PSPS system exhibits an upper critical solution

-

TRANSLATIONAL DE?VAMICS MELTS IN

287

temperature (UCST). The stability limit for the system that Green and Doyle studied was xs(aC) = 2.1 the degree of polymerizationof the PS chains was 8.7 lo3 and that of d-PS, 9.8 lo3. Green and Doyle showed that the mutual diffusion coefficient in the d-PSPS system experienced a minimum, or critical slowing down, in the middle of the concentration regime at the critical composition. the temperature of the experiment approached the UCST, 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)T1 - 3.2(+0.4)

(59)

which is in excellent agreement with independent measurements using The UCST was determined for this system to be approximately 140°C. The PSPXE system investigated by Composto and coworkers is characterizedby x < 0. In addition to determining the compositionaldependence of interdifhsion, where they showed a thermodynamic “acceleration” of the process (Fig. 19), they also examinedthe temperaturedependence of D. They showed how the value D(+ = 0.55) changed in relation to D* as a function of


18 “Thermodynamic slowing down” Green et al. (1987).]

D in d-PSPS mixtures.[Data

GREEN

288

x

temperature (Fig. 20). For -c 0, D(+) > D*, and as the temperature increased and the system approached the lower critical solution temperature (LCST), x > 0 and D(+) became smaller than D*.

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. equally interesting problem is associated with the diffusion of a homopolymer chain or a chain composed of blocks segments distinct chemical structure, a block copolymer, into a block copolymer host. Below we address these issues. Block copolymers comein a variety ofarchitectures, 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 fractionofpolymer andthe segmental interaction parameter At sufficiently high temperatures and chain lengths, the copolymer

x.

16”’ 0.0

0.2 VOLUME FRACTION PS

Thermodynamic “acceleration” of D in the PSPXE system is represented by diamonds. The circles represent the tracer diffusionof d-PS chains in the mixture and al. the squares that of d-PXE. [Figure reproduced with permission from Composto et (1988).]

TRANSLATIONALDYNAMICS IN MELTS T 600

'

+-

"

289

(K)

560

520 I

-

IO IOOO/(T-T,),

(K-')

20 Temperaturedependence of diffusion. (0)D; D* of d-PS; (0) D* of d-PXE. [Figure reproduced with permission from Composto et al. (1988).]

21 Thedifferentarchitecturesthatblock @) triblock, (c)

blocks.

copolymers possess (a) diblocks;

290

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is completely disordered; hence the microstructure can be characterized as homogeneous. In this regime the radius of gyration of the copolymers R, NIn. As the temperature is lowered (i.e., 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 called strong segregation limit. Here the chains are stretched and the interdomain spacing varies as NZ3.Depending on the value off, the volume fraction of the A segment, different ordered phases may form. With increasing values off, 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 NIn. 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 N, 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 R, = Ninb16, 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 # 112. As the tracer difises into the copolymer host, the A segment located at aposition r experiences a chemical potential p(r) = kBTx+B(r)while that felt by a B monomer is -p(r). The net force that the copolymer experiences froma periodic potential is

F = 2uk&&N(l

-

2f) sin(&)

(60)

where a is the amplitude of the potential (it is assumed to be much less thanunity),and is awavenumberthat characterizes the scale of the compositional variation in the melt, which is formally defined as k,, = ,&ln. The parameter (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

TMSUTZONAL DYNAMICS IN MELTS

291

22 Schematic the diffusing copolymer in different lamellae hosts. (a) The diffusant is small in comparison to the domainsize. (b) The diffusant size is comparable tothe size 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 = D,ZZ+ D,,,@ -

(61)

where is a unit vector in the direction. D,, is the d f i s i o n coefficient in the direction parallel to the lamellae; henceforth it will be denoted by Dl. The prediction for D, is

D, = Dl/[lg(4IZ

(62)

where Io(a)is a modified Bessel function and a = &@(l - 2f). In the limit of small (a l),a single d f i s i o n coefficient is predicted,

DID1 = 1 - (2/3)uz(xN)'(l - 2f)' and when a >> 1 the prediction is

+ O(a4)

(63)

292

GREEN

DID, = (2/3)[1

+T

exp(-2la()]

+---

(64)

These predictions are meant to apply even in the limit where the tracer is a homopolymer = 1 or 0). 2. Thedomain spacing isvery large (x,, << l ) , butthe 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 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

DID, = 1 - 4.6

10-3a4(xNc)4x~ + O(P4)

(65)

-

where P (R, N&x h)’. 3. The domain spacing is comparable to or smaller than that of the tracer (x,, 2 (Figs. 22b and 22c). In this situation, in particular when > 1, more than one period of the potentialis 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 IaxNcl << 1, and for arbitrary values of and f, the diffusion coefficient is predicted to be

D/Dl = 1 - 0.67a2(~Nc)X(x,,,

+ 0[(a~N~)~]

(66)

where K is a dimensionless function of and off. If the tracer is a homopolymer = 0 or l),then at small K FJ 1 - xd3, and at large it crosses over to K FJ 15.8.q-’. 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. I t 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 segment is in the phase and its B segment is in the B phase. The segment then has to difhse across the B domain to get to the adjacent domain (Figs.23band23c). During this process a portion of the segment is stretched considerably beyond normal conditions as the A segment to get completely into the adjacent A phase. This is shown in Figs. 23d and 23e. Figure 23f shows the final configuration.

TRANSLATIONAL DI?VAMICS IN MELTS

23 A schematic of a diblock copolymer chain diffusing across a domain unlike chemical structure.

293

of

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294

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 systemwas performed. Their results showedan anisotropy of diffusion. There were not sufficient results to make a true comparison with theory. Shul et al. (1991) used FRES to study self-dif€usion in poly(ethy1ene propylene)/polyethylethylene (PEP-PEE) copolymer in a temperature range above and below the order-disorder transition They observed no marked change in the temperature dependence diffusion at theODT. They 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 complete understanding of the domain structure of the copolymer host. Green et al. (1989) studied the diffusion 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 P " A (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

I

I

I

2

3

4

N 24 Molecular weight dependence of the diffusion of d-PS into PS (A),and a copolymer of N =

a symmetric copolymer of N = 824

into

295

TRANSLATIONAL DYNAMICS IN MELTS

FREE SURFACE NEAR SURFACE REGION

BULK

25 Domainstructure of an unannealedcopolymer. temperature dependenceof 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 that the chains were forced todiffuse perpendicular to the orientation the lamellae (Fig. 27). They observedthe M-' dependence butalso 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 weightdependence 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 While it appears that the first notable attempts to understand translational dynamics of melts began with Bueche in the mid-1950~~ the truly significant advances in the area were made by deGennes withhis 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 -1Q

-11

-

-12

-13

9,

-

-14

-1 5

-1 6

-17

26 Temperaturedependence into a copolymer (A).

d-PS diffusing

PS (opensymbols)and

path, the primitive path. The tubeis defined to accountfor 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

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;KremerandBinder,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 amatrix 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 evidencethat suggested that the reptation picture may be valid. The work of O’Connor and Ball suggests a similar conclusion. In addition to reptation, a numberof other theories have been proposed. Many of these invoke tube-like concepts and give predictions similar to those of reptation. recenttheorybasedon the introduction of ageneralizedLangevin 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 essental reptation predictions in the long-chain limit. It is clear that reptation continues to bethe 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 hasprovided 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 bedone. There are unresolved questions regarding why anomalous power law exponents are observed in PI and PDMS. Critical tests theories for diffusion chains 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 tracer diffusion. Our understandingof diffusion in block copolymers is clearly very poor in comparison to that 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 theory but also the design of new experimentsthat will give us further insight into the details the dynamical process.

This work wasperformed at Sandia National Laboratories and supported by the U.S. Department Energy under contract number DE-AC04-76DP00789.

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Index

Activation energy, Activity, Adsorption, Affinity, After-effect function, Aging, Anti-plasticizer, Anisotropy, Averaging methods, Axial dispersion,

Binary mixtures, Block copolymers, Bond angle, 55 Bond length, 5 5 , 5 8 Branched polymers, Burnett coefficients,

Cannonball solid, Capillary model,

Case II, Cellulose, cuprammonium regenerated, Cellulose regenerated, Chemical potential, Chromatography, capillary column, gas-liquid, gas-solid, Cluster connectivity, shape, 85 size, percolation, Cohen and Turnbull, Configuration, accessible volume, 80 bias, Monte Carlo, random coil, unoccupied volume, Conjugate force-flux,

303

304

Connected overlapping spheres, Conservation of species equation, Constraint forces, Constraint release, Correlation function end-to-end vector, length, Critical lower critical solution temperature upper critical solution temperature (UCST), Cross-coupling coefficients, Cross-kinetic coefficients, Cross-linked polymers, Crystallinity, Crystallization, polymers,

INDEX [Diffusion] multicomponent, non-Fickian, pure fluids, statistical mechanics, surface, Diffusion coefficient, binary, concentration dependent, Einstein, Fickian, Green-Kubo, interdiffusion coefficient, measurement techniques, mutual, polystyrene/polyvinylmethyl ether, reptation, ring-shaped polymers, self,

Crystallization, solvent induced,

NMR, Darken equation, Deborah number, Desorption, Devolatilization, Diffraction pattern, prediction, Diffuse reflection, Diffusion amorphous polymer, anomalous, collective, flux, hopping mechanism, Knudsen, low temperature, mechanism, memory-dependent, molecular simulation, molecular view,

polyethylene, predicted, star-shaped polymers, tracer diffusion coefficient, polymethyl methacrylate, polystyrene, transition state theory, transport, unentangled chain, Dihedral angle, Dispersion coefficient, Dissolution, Dividing surface, Dry molding, Dual-mode sorption, Ductile failure, Dusty gas model,

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 particle-porewall, 18 Interchain rotation, 55 vibration, 55 Interfacial instability, 197 Interpenetrating network, 202 Interpore connectivity, 37 Isotropy, 10, 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 ultrafiltration, 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

307

INDEX Phenomenological coefficients, Plastic deformation, Plasticization molecular view, glass transition, Percolation, Periodic imaging, Permeability, Permeation, coefficient, Poisson process, Polyblends, Polymers aging, amorphous, annealing, branched, foams, materials, molecular engineering, semiatomistic, semi-crystalline, star-shaped, molecular models, validation of, Poly-a-olefine, Polyethylene,

Polyethyleneterephthalate, Polymethyl methacrylate, Poly-paraphenyleneterephthalamidel Kevlar, Polystyrene, Polyvinyl acetate, Polyvinyl methyl ether, Pore, fluid binary, continuum solvent, mixtures, single component,

Pore density, Pore radius distribution, Porous media, asymmetric membranes, cannonball solid, dusty gas model, interpore connectivity, percolation, 44, Swiss cheese solid, Positron annihilation spectroscopy, Potential energy, Power law, Primary particle, Primitive path,

Radio labeling, Radius, mean, cutoff, maximum pore, pore distribution, Random media, Random overlapping spheres, Random nonoverlapping connected spheres, 50 Random nonoverlapping spheres, Random walk, Relaxation diffusion time, function, diffusional, molecular, segmental, shear stress modulus, stretched exponential, time, longest, Reptation, Monte Carlo, Ring-shaped molecules, Rouse model,

308

Rutherford backscattering spectrometry, 260 relaxation modulus, 252

Scaling theory, 21 Scattering functionintermediate, 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, isotherm, 96 molecular dynamics, 102 molecular simulation, 67 Monte Carlo, 102 sigmoidal, 185, 190 statistical mechanics, 91 thermodynamics, 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 Thermodynamic slowing down, 283, 287 Tie molecule, 223 Times characteristic 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

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

309 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