Diffusion in polymers

". ".

".

,. .,' 2

,;

10'· ").,,

'0'· 10'"

'0'·

,

mnmBY

!!NEOBI

o

N

s mlTEDBY

P.NEIGI University of Missouri-Rolla Rolla, Missouri

Marcel Dekker, Inc.

New York· Basel- Hong Kong

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

= ri(t)

+

+ !l.t)

= Viet)

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

I1t Viet)

(50a)

and

viet

1 2

(SOb)

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

tciw = -

ViO. riO

- -2-

vm

+

1

2

vm

2

{(ViO ' riO)

2

+ ViO

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

2

(51)

DIFFUSION IN HOMOGENEOUS MEDIA

19

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

V:r = Ivil ~

v: = Iv;j [(1 z

v:e= IVil [(1

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

(53b)

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

(53c)

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

20

MAcELROY

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

z:;:O

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

DIFFUSION IN HOMOGENEOUS MEDIA

4. 5.

21

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

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

1

M

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

(54)

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

22

MACELROY

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

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

[u7)(jBt) u7)(0»)

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

1

=2

N

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

(55a)

i= l

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

(55b)

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

1.0

z o

>-f=

f-U -z u::>

glL. w

>z o

8~ N-.I

-w -'0::

~ ~

N

~

~

0

z

-0.2

0

-0.4

\

,/

"- ' - ----- /

---

/

-06 '0

1.00

(b)

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

1.00

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

24

MAcELROY

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

25

DIFFUSION IN HOMOGENEOUS MEDIA

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

1.2

1.0 ,,4 /J

"

9

(z)

DaK

.-

-...- •

/

0.8

DaM

/'

~ .....

, I

0.6

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

I /

,~

I

-+t-l-l/

0.2

00

0.2

0.6

0.4

A' 0.8

, I

r

I

LO

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

=

=

26

MAcELROY

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

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

3

2

CT,w

(9)

(Rp _ rl)3f CT1w

(rl) -

(3)

(rl)

]

(56)

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

DIFFUSION IN HOMOGENEOUS MEDIA

27

(0)

(b)

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

by the solid in Eq. (49) to give

....s

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

ridiW(R p - ri) ri dri

(57)

j#

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

= R p cos (21Tk) NR (k

= 1,

.. . , N R ; I =

-00

, .. . , + (0)

(58)

28

MAcELROY

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

=-

±

r ij dij _

j=1 r ij drij j#i

:i:

r ij dij _

j=1 r ij drij

~ diW(Rp + (Jiw/2 ri

r i)

(59)

dri

jES

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

DIFFUSION IN HOMOGENEOUS MEDIA

29

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

30

MACELROY

d 2

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

0.10

\

\ \

0.08

,

\

\

0.06

0.04

1 I' \ \

I I

IrI\/1/

0.02

\ I \

0.2

0.8

1.0

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

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

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

DIFFUSION IN HOMOGENEOUS MEDIA

31

and vice versa. Also noting that in general

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

j(Z)(i = 1

2)

"

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

=-

(61a)

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

, kBT dz

(61b) T

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

Dl

= -kBT (Z) nl

LII

n 1K"2 n"2 K l

(Z»)

(62a)

- - L 12

-

and (62b) with

L:;) given by

L:;)

=

_1_1x : (l) DIM

kaTIL (l ) 1 - n L (l)

= n IL(l)22

2

(65a)

12

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

J I'

D (') dn I"

"

(66a)

dz

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

D'z)I' -- D"1 ) -1 D 1·

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

1

1

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

(67a) (67b)

1·1

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

(67c)

nr

and (l)

(z)

(l) 1"1'

-

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

L(l)/N2) f[ ur (l) lY.Lt*l·

(67d)

33

DlFFUSION IN HOMOGENEOUS MEDIA

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

L~Z?). = _1_

VkBT

= _1_

r (v~Z?(t)v~?(O»

Jo

(!) Jo('" \/i

VkoT N

dt

v~Z)(t)V~Z)(O») dt

(68b)

;=)

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

(68a)

--->0

00,

Eq. (67a) simplifies to (69)

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

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

34

MA CELROY

d 10

3.5

5 4

2

3

3.0

2.5

(z)

2.0

D 1*

D1*,B

1.5

, ,

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

\ 0 .5

b

"-"-

\

'-....

'-

----

I

'~ ....

f

- ~

P

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

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

35

DIFFUSION IN HOMOGENEOUS MEDIA

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

, \

D(z)_ I DI-,B

\ \

0 .10 0.08 0.06

I

I,:\11

\

9

\ I, \0

0 .04 0 .03

13 \

I

. I

\1

12

0.02

\

,"/

'.':

10

\

\

\

\

'.

\8 \

\

0.010 0.008 0.006

\

0.004 0 .003 0.002 0 .001

0

1.0

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

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

36

MACELROY

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

: i~ (v~Z)(t)vnO»

(70a) dt

(70b)

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

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

DIFFUSION IN HOMOGENEOUS MEDIA

37

4 .0

3.0

2.0

(z)

\f--1\

1.0

\ \

\

Dl

\

\ \

D 12 ,B

0 0

0.2

0.4 "-

2

\

\f

I

\

-1.0

, ,, I

0.8

0.6

\

I

\

I

I

-2.0

-3.0

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

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

B.

Diffusion in Rigid Random Media

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

38

MACELROY 1.0

0.8

,

-.;, \

D(z) a

\

0.6

\ \

11, \

D1a,B

\ \

0.4

,

~, \ \'tJ:\

0.2

\

./-~\\ \ ,.

\\, _;( \ ,, "

\

0._,0'

00

0.2

0.4

0.6

\

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

1.0

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

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

DIFFUSION IN HOMOGENEOUS MEDIA

39

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

40

MAcELROY

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

== DfK

= lim! 1 -'"

3

r

Jo(V(T) . v(O»

dT

(71)

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

DIFFUSION IN HOMOGENEOUS MEDIA

(0)

(e)

41

(b)

(d)

Figure 11

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

ulations, and indeed only a numerical factor is involved in the final evaluation of the diffusion coefficient (Mason and Chapman, 1962). The general form of the VCF and in particular its long-time behavior as determined by storing the particle velocity as a function of time is unaffected by the collision dynamics. Early work by Alley (1979) on a two-dimensional overlapping disk analog of Fig. lla demonstrated this, and this was confirmed recently for the threedimensional systems of interest here (MacElroy, 1992, unpublished). Selected results for the diffusion coefficient DfM of the low-pressure non adsorbing gas diffusing within the overlapping spheres models are shown in Fig. 12 (for the random overlapping spheres medium illustrated in Fig. lla) and in Fig. 13 (the data represented by the full circles in this figure correspond to freeparticle diffusion within the connected overlapping system illustrated in Fig. llb). Results for the nonoverlapping systems may be found in Park and MacElroy (1989). The Boltzmann diffusion coefficient cited in Fig. 12 corresponds to the low-density limit for the solid (I.e., n: = nsjI =0 .500

12

14

16

18 20 2224262830

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

47

DIFFUSION IN HOMOGENEOUS MEDIA

(c)

4

2

*'u- Ie? 8

I

6 4

IjI =0.367

2

10

4 10

10

12

14

16

18

20 22 24 26 28 30

t*

2

8

(d)

6 4

...... *

-

U

2

I

I 1(53

I/!

=0.035

8 6 4

10

14

18

22 26 30

36 42 48 56 6472

t* Figure 14

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

48

MACELROY

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

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

(78)

where 8 = j-l/(2v

+

j-l -

(79)

-y)

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

~ =

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

(80) (81)

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

xr

2

l'+'Y

= constant

+ 00)

(83a)

00)

(83b)

(x

-+

(x

-+ -

(x

-+

0)

(83c)

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

49

DIFFUSION IN HOMOGENEOUS MEDIA

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

J, = -

i Inr J=l

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

~\IJiT) dT

(i = 1, ... , v)

(84a) which simplifies to V

J, = -

L J~) 'Pit -

(i

T)VTl-LlT) dT

= 1, ... , v)

(84b)

J=l

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

at

i' 0

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

az

D

az

(85)

50

MA CELROY

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

(p(s)

= (f)

sw(s) 2 1 - m(s)

(86)

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

(87)

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

(88)

51

DIFFUSION IN HOMOGENEOUS MEDIA I" (1-

0.10

o

0 .1

1'» 0 .5

0.4

0 .3

0 .2

0.6

0 .095

0.09

10-2

•• D 1*(t·)

8

I

6

0.085

4

- C 1• 1·(t*) 2 0.08

10-3 8

6

0.075 0

0.1 I*(HI)

0.2

2

;5

4

5

6

7 8 9 10

,*

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

52

MACELROY

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

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

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

Jo

. Ue(O) dt

(89)

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

Jo \ i

= ~ ('" / 3N

;=. i'l-J

viet) .

i

Vj(O)) dt

j =j j#

e

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

53

DIFFUSION IN HOMOGENEOUS MEDIA

e o

0.1

02

0 .3

0.4

0 .5

0.6

0.7

0 .8

0.9

1.0

0.4

0.2

0.1 0 .08 0 .06

0 .04

0 .02

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

0 .8 0 .01 0.008

F4'IY~---

0 .6

e

~.

-

~

0.006 0 .4 0 .004

e

S

02

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

E>

Figure 17

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

54

MAcELROY

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

c.

Diffusion in Amorphous Polymers

If the microcavities in an amorphous polymeric medium are significantly larger

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

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

DIFFUSION IN HOMOGENEOUS MEDIA

55

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

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

or

(95b)

and 5

Vol>

= kol>

L

a"

cos" (

~

1.4

E

'0

79

1.3

~ ....•

(J

~

1.2

0)

0..

CIj

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

E

~ co M"-

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

1.4

i I.j\···········~

~

E

NPT Monte Carlo

~

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

1.2

'0

>

(J

~

~

1.1

C7s.450K

CIj

1.0 0

30

(b)

Figure 3

Table 1

60

90

120

Pressure (MPa) Continued.

Conformational Characteristics of Tetracosane Chains

Chain property

Bulk NPT Monte Carlo'

Continuous unperturbedh

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

350 43 0.659 0.394 0.532 0.110

354 44 0.666 0.406 0.524 0.104

PI Pit PIli PgIg

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

80

THEODOROU

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

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

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

81

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

82

THEODOROU

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

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

83

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

84

THEODOROU 0 .4 10·\

~.

C

.Q 0. 3 1:5 m

10.2

10.3

E :J

(5

>

0.2

10"

Q)

:0

"", ~t ~

.!:: Q)

I

10.5

·iii en

0

0.5

1.0

1.5

2.0

2.5

Q)

0 0

m 0.1

o

0.5

1.0

1.5

2.0

2.5

penetrant radius (A)

Figure 5

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

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

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

85

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

86

THEODOROU

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

(a)

(b)

(c)

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

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

87

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

C.

Characteristic Times of Molecular Motion

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

88

THEODOROU

for dihedral angles, R,it) as

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

(1)

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

M.(t)

= (u(t)

. u(O»

(2)

and

M 2(t) =

~ (3[u(t)

. u(OW - 1)

(3)

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

89

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

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

90

THEODOROU

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

Pk(t) =

2: exp [-ik . rit)]

(5)

j=l

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

Fs(k, t) = Ns \

~ exp {-i N,

k [rit) - riO)]}

)

(7)

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

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

91

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

III.

PREDICTION OF SORPTION THERMODYNAMICS

A.

Statistical Mechanics of Sorption in a Compressible, Involatile Medium

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

I

Pressure P

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

x,II

r

1 X

1 t il

2 , .. ·,

exp (-

(va) '~lN~nl

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

P T)

c>'

N al 2'

...

N al c ·

;=2

k0 TZ inlr ; •

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

a )]

(16)

where

x

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

X

/.2

k. T Z;

_ _ 1-,---_ CXp

"

(Va) 1~2 N~",

Pya) L exp [-

( - -k. T

(""

1, '" (r) • ] d 3 ..f , N;II , r -

k. T

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

f

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

. . .

d 3 r i 'Ii- l

(18)

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

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

95

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

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

NcrtAJ

,::2

N,.

,.2

p (- I3PV)

0

f

d"" " " . . .

f d3N"" "e -~"(n

. ... "')

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

~ ~!~~A

r

dV exp(-J3PV)

f

d

3N "' Pr p

f

d

3N

N'A rA

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

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

"V (rp , r A )

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

Nt'

NI ,

L L ;=)

1'=;+ I

NA

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

L "V~In(rAk) k=1

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

96

THEODOROU

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

(NA) =

i

-

NA

(~ dVJ d3NpnP rp J

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

1

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

~

where

Z(Np, NAl P, T) =

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

J

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

(24) •• • ,

rANA)]

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

97

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

P k. T

1"

dVexp ( - j3PV)

l~ dVexp(-

X

J J d 'N"" rp

j3PV)

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

J

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

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

J

exp

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

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

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

i'" i'" i'"

V

(V) =

J exp (-I3Pv) J

dV

dV

dV

exp (-I3Pv)

d

3NpnP

rp

d 3NpnP r p

exp (- I3Pv)

i'"

dV

J

d

exp (- I3Pv)

exp [-I3'V p(rp)]

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

3NpnP

J

rp

J

3 d r A l nA

d 3Npnpr p

exp [-I3'V p(rp)]

exp [ - I3'Vp(rp)]

98

THEODOROU

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

f f dVexp(-~PV) f f

l~ dV exp(- ~PV)

f

d'N"" r.

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

d lN"" r,.

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

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

A ,)]

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

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

(27)

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

-

1

H A, p

.

XA

== hm -

P_O f A

•

= hm p_o

(NA ) - Np P

=

(V)

N p k8 T

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

(28)

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

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

99

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

[Z~(T,

P) _ (PVA)] RT

(inaT n P)

(30) XA.cq

where z~(T, P) = Pv~/RT is the compressibility factor of pure A in the fluid phase, VA is the partial molar volume of A in the polymer phase, and the notation computed over all N E walkers through the Einstein equation, Eq. (49). Note that simulation schemes advancing the time in equal intervals are also possible (Termonia and Smith, 1987). If the rate constants k;_i are small, then the time steps I1t taken by the simulation are long, and thus the simulation permits accessing times and displacements that may be several orders of magnitude larger than the ones accessible through atomistic MD. Thus, the long-time problem of MD is solved. For implementing this model of diffusion as a sequence of infrequent events, however, it is necessary that one have a good idea of (1) the macro states and the representative nodal points {r;}; (2) the equilibrium probability distribution {P~q } of the macrostates; and (3) the connectivity and rate constants {k;_J governing transitions between the macrostates. In the next section we discuss how to obtain these quantities from the detailed potential energy hypersurface "V(rp, r A ) based on multidimensional transition-state theory.

2.

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

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

127

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

(66b)

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

xie

xie,

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

(67b)

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

.......

~

Su .... ....

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

....

I

,x.

JI

, J2

\

... .... \

"'5-

----

X. /3

l

-

, ',

s "xi4

X ik

... ....

-.S l~

\

S

.,.

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

....

.... \

I

6

~5

//

,\

}

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

xIi

xIt;

~ ~

g ~

a

c::

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

129

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

It passes through the saddle point xt :

C (xt ) 2.

=0

(68a)

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

I

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

(68b)

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

xg,

V.C(x) . g(x)

= 0,

(68c)

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

•

(x - xt )

=0

(69)

xg

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

130

THEODOROU

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

dVx (

J

.E{i}

J

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

p NVT

(x, p)

(70)

o ·p>O

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

(

TST

ki-j

1

= (2~hy/2

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

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

(

(71)

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

xE{i}

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

MOLECULAR SIMULATIONS OF SORPTION AND DIFFUSION

131

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

xt

L

d .-\x exp [- f3"V(x»)

(2f3 1T)"2

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

r

J'l

d 'x exp [- f3"V(x»)

L

r - Ix exp [- f3"V(x»)

5.

r

L

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

Sl

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

J (I )

L

d ,- Ix exp [- f3"V(x»)

5,

L

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

x ... x

s.

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

L..

d ,- Ix exp [- f3"V(x»)

1 t1 X il
V;1> and V; are clearly independent of X. Furthermore, Vrentas and Vrentas (1991b) suggest that the distribution of free volume in the polymer and the size of the jumping unit of the polymer chain do not depend on X, in which case neither does -y, V); , nor V;. The above approximations lead to the result that ~ is also independent

164

DUDA AND ZIELINSKI

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

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

(25)

~(T,O)

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

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

D j = Do exp RT

A

W jVFHj

A

W

(26)

FH

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

FREE-VOLUME THEORY

165

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

1.

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

B.

Multicomponent Diffusion

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

J

·2

J

= -D

api _ D 11

= -D

ax

12

api _ D 21

ax

22

ap2

ax

(27)

ap2

(28)

ax

VJI + Vzi2 + V..J3 = 0

(29)

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

166

DUDA AND ZIELINSKI

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

DI

(30)

D I2 -

0

(31)

D22 - D2

(32)

D 21 -

(33)

0

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

WI

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

W3

V;~13)

(34)

YI'H

wl~d~13 +

V; + W3 V;~23)

W2

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

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

(35)

- Tg2 )

(36)

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

FREE- VOLUME THEORY

167

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

C.

Block Copolymers

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

168

DUDA AND ZIELINSKI

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

-V

FH

V

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

(37)

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

(-E)

DI = Do exp exp ( RT

- )'[Wl

V; +

W2(W2a£12.

V;.

+

W2b£12bV ; b)])

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

VFH

(40)

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

FREE-VOLUME THEORY

169

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

REFERENCES Arnould, D., and R. L. Laurence (1992). Ind. Eng. Chem. Res., 31, 218. Arnould, 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, and R. G. Wissinger (1989). J. Polym. Sci., B: Polym. Phys., 27,2105. Barbari, T. A, W. J. Koros, and D. R. Paul (1988). J. Polym. Sci., B: Polym. Phys., 26, 729. Barrer, R. M., J. A Barrie, 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. A and J. O. Simpson (1989). Proc. 18th N. Am. Thermal Anal. Soc., 1, 366. Blum, F. D., B. Durairaj , and A 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. Thmbull (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, A , D. E. Axelson, J. J. Dechter, B. Borah, and L. Mandelkern (1985). J. Polym. Sci. B: Polym. Phys., 23, 367-385. DiBenedetto, A 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). AlChE J. 28, 297. Ferdinand, A, 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). Forstschr. 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. 1., D. DeKee, and B. Harrison (1992). Chem. Eng. Sci., 47, 1525. Hadj-Romdhane, I. (1994). Polymer-solvent diffusion and equilibrium parameters by inverse gas-liquid chromatography, Ph.D. Thesis, Pennsylvania State Univ. Hadj Romdhane, 1. and R. P. Danner (1993). AIChE J ., 39, 625. Haward, R. N. (1970). J. Macromol. Sci. Rev. Macromol. Chem. , C4, 191. Huggins, M. L. (1942a). J . Am. Chem. Soc ., 64, 1712. Huggins, M. L. (1942b). J. Phys. Chem., 46, 15l. Iwai, Y., S. Maruyama, M. Fujimoto, S. Miyamoto, and Y. Arai (1989). Polym. Eng. Sci., 29(12), 773-776. Kaelble, D. H. (1969). In Rheology, Vol. 5, F. R. Eirich, Ed., Academic, New York, p. 223. Kinning, D. 1., E. L. Thomas, and J. M. Ottino (1987). Macromolecules, 20, 1129. Lipscomb, G. G. (1990). AIChE J. , 36(10), 1505. Macedo, P. B. and T. A. Litovitz (1965). J. Chem. Phys. , 42, 245. Michaels, A. S., W. R. Vieth, and H. Bixler (1963). J. Polym. Lett., 1, 19. Pace, R. J. and A. Datyner (1979). J. Polym. Sci., B., Polym. Phys., 17, 437-451. Paw lisch, C. A. (1985). Measurement of the diffusive and thermodynamic interaction parameters of a solute in a polymer melt using capillary column inverse gas chromatography, Ph.D. Thesis, Univ. Massachusetts, Amherst, MA. Raucher, D., and M. D. Sefcik (1983). ACS Symp. Ser., 223, 11I. Reid, R. c., J. M. Prausnitz, and T. K Sherwood (1977). The Properties of Gases and Liquids, 3rd ed., McGraw-Hill, New York. Rein, D. H., R. F. Baddour, and R. E. Cohen (1992). J. Appl. Polym. Sci., 45, 1223. Sax, J. E., and J. M. Ottino (1985). Polymer, 26, 1073. Spiess, H. W. (1990). Polym. Prepr., 31, 103. Vieth, W. R., and K J. Sladek (1965). J. Colloid Sci., 20, 1014. Vrentas, 1. S. and J. L. Duda (1977a). J. Polym. Sci. , 15, 403. Vrentas, J. S. and J. L. Duda (1977b). J. Polym. Sci., 15,417. Vrentas, J. S. and J. L. Duda (1978). J. Appl. Polym. Sci., 22, 2325. Vrentas, J. S., and J. L. Duda (1986). Diffusion, in Encyclopedia of Polymer Science and Engineering, Vol. 5, H. F. Mark, N. M. Bikales, C. G. Overberger, and G. Menges, Eds., Wiley, New York. Vrentas, J. S. and C. M. Vrentas (1989). Macromolecules, 22, 2264. Vrentas, 1. S. and C. M. Vrentas (1991a). Macromolecules, 24, 2404. Vrentas, J. S., and C. M. Vrentas (1991b). J. Appl. Polym. Sci., 42, 1931. Vrentas, J. S. and C. M. Vrentas (1992). J. Polym. Sci. : B Polym. Phys., 30, 1005. Vrentas, J. S., and C. M. Vrentas (1993). Macromolecules, 26, 1277. Vrentas, J. S., H. T. Liu, and 1. L. Duda (1980). J . Appl. Polym. Sci. , 25, 1297. Vrentas, J. S., J. L. Duda, and H. C. Ling (1984). J. Polym. Sci.: B Polym. Phys., 22, 459. Vrentas, J. S., J. L. Duda, and A. C. Hou (1985a). J . Appl. Polym. Sci., 31, 739. Vrentas, J. S., J. L. Duda, and H. C. Ling (1985b). J. Appl. Polym. Sci., 30, 4499. Vrentas, J. S., J. L. Duda, and H. C. Ling (1988). Macromolecules, 21, 1470. Vrentas, J. S., C. W. Chu, M. C. Drake, and E. von Meerwall (1989). J. Polym. Sci., B: Polym. Phys., 27, 1179.

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

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

I.

INTRODUCTION

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

ac

at

a.

= -

ax Jx

(1)

173

174

NEOGI

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

~~ = :x (D ::)

(3)

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

M, = 1 _ M=

~ 1T

± m=O

1

(2m + 1)

2 exp [ - 4D(2m

~

L

1)21T2t

J

(4)

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

2:1 = 1

-

8

1T2

[4D1T2 ---u- t in ]

exp -

TRANSPORT PHENOMENA IN POLYMER MEMBRANES

175

or (5) It has been assumed that

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

M,

-

M~

=

8 Dt2"L (

112 )

[

'TT-

I12

+

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

--112

]

(6)

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

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

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

= 0.01224L 21D

(9a)

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

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

J1. _ Dt _ ! LeI - L2 6

_~ ~ 'TT2

ft

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

(10)

176

NEOGI

1.0

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

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

TRANSPORT PHENOMENA IN POLYMER MEMBRANES

177

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

Q, Dt 1 --=-- -

(13a)

or L d(Q,) = DH PI dt

=P

(13b)

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

LQ, (L2) -DH t - PI

6D

(13c)

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

178

NEOGI

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

II.

MATHEMATICAL METHODS

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

ae at

a2e dX

- = D -2

(14)

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

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

(15)

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

A.

Concentration-Dependent Diffusivities

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

TRANSPORT PHENOMENA IN POLYMER MEMBRANES

179

o • L = 3.53 X 10-3cm L = 7.0 X 10-'cm 9 , L = 1.2 xIO-'-'?~. ~61

6rf

8

/o~ 0.5

rr6~~~./ ,._e-

~ ::::e

___ 0

/,,0

6-~

oes .

o-.~ .... ~ ,.

/0

#. • .,

o

~"'I

L..

---.:...--'

0 .6

u l J'

04 0 .2

0 0

2

3

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

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

TRANSPORT PHENOMENA IN POLYMER MEMBRANES

203

Figure 10

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

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

NEOGI

204

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

C.

NMR Self-Diffusion Coefficients

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

TRANSPORT PHENOMENA IN POLYMER 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 NMR self-diffusion coefficient was the same as the mutual diffusion coefficient measured with an ultracentrifuge. The last observation is exciting but needs more investigation.

VI.

CONCLUSIONS

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

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

I.

A.

INTRODUCTION

Scope and Context of the Presentation

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

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

B.

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

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

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

~-

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

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213

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

C.

General Description of Supermolecular Structure of Polymer Solids

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

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~~'"

~~

Dissolution/diffusion flow -Free volume

model

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

}

capillaf}'

model

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

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

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

Target size (nm)

0.1-1.0

Second (conformation)

1-10

Third (fine structure)

10-102

Fourth (aggregation structure)

102 _ 104

Fifth (morphological structure)

104 _106

Structure description

Evaluation methods'

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

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

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

Optical microscopy, light scattering

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

215

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

II.

STRUCTURAL CHARACTERISTICS OF POLYMER SOLIDS

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

Free volume model

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

SUPERMOLECULAR STRUCTURE OF POLYMER SOLIDS

A.

217

Supermolecular Structure of Crystalline Polymer Solids

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

218

MANABE

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

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

,

,

I

I

, , I

I

I

I

I

I

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

I I I

,, I

c

I

I I 1 I I I

I I I

I

I

I,_______1 I

(a)

(b)

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

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

SUPERMOLECULAR STRUCTURE OF POLYMER SOLIDS

219

Figure 5

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

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

B.

Supermolecular Structure of Noncrystalline Polymer Solids

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

220

MANABE

Table 2

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

0.60 0.26 0.05

IRe

Densityd

0.25 0.28

0.39 0.27 0.25

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

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

c o

:;:;

1

:J .0

·c ..... •«l! "0

>. u

C
g

Amorphous leathery

I

rtilt~ I I

I

I

~

1\

I \

I \ \ \ \ \

Tg

Tm

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

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

1. 2.

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

SUPERMOLECULAR STRUCTURE OF POLYMER SOLIDS 3. 4.

A.

235

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

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

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

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

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

(9)

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

B.

Glass Transition and Segmental Micro-Brownian Motion

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

236

MANABE

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

TT)

Iog ( TT.

=

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

(10)

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

SUPERMOLECULAR STRUCTURE OF POLYMER SOLIDS Table 5

Glass Transition Temperature Tg for Various Polymer Solids

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

C.

237

Evaluation method

Tg (K)

Dilatometry Refractive index Dilatometry Dilatometry Dilatometry

249 318 148, 243 302 233

Dilatometry Dilatometry Dilatometry Dilatometry Dilatometry

253 263 160, 400 279 249

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

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

Local Twisting Motion of Polymer Chains in an Amorphous Region

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

238

MANABE

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

D.

Other Molecular Motions of a Polymer Chain

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

VJ

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

Table 6

'fmc

Polymer

(K)

!:lH. (kl/mol)

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

70

10

60-160 Pc), where P is the degree of polymerization of the host chains. Consequently, there should be a correction of DRop , which varies in a manner that may be described by the Stokes-Einstein relation. This correction is predicted to vary as D CR - N 1!2p - 3 • There have been a variety of other theories developed to understand the effect of the matrix on diffusion (Klein, 1986; Wantanabe and Tirrell, 1989; Viovy, 1985; Graessley, 1984; des Cloizeaux, 1988a,b, 1990, 1992; Rubenstein et aI. , 1987; Rubenstein

Figure 9 The configuration of the chain changes as the topology of its environment changes when a constraint is released.

TRANSLATIONAL DYNAMICS IN MELTS

271

and Colby, 1988; Doi et aI. , 1987). With the exception of Klein 's work, these theories address the question of viscoelasticity primarily. Most of the constraint release theories suggest that the correction to D Rcp varies as DCR - P- 3 . Klein, on the contrary, argued that the dependence is somewhat stronger, DCR _ p - 512 • Klein's argument is based on the fact that a single chain may provide more than one constraint on the N-mer chain. Graessley's theory, for example, as discussed later, assumes that each P-mer chain accounts for one constraint. The effect of the interdependence of the constraints is to enhance the constraint release (tube renewal) contribution; hence DCR - p - 512 • Hess (1988b) also addressed the question of constraint release in melts using a many-body approach and suggested that DCR - P- J • It turns out that there are no experimental data on the effect of constraint release of the diffusion of linear chains that support Hess 's finding. Below we describe Graessley 's contribution. The basic idea is that each chain in the system is undergoing a reptative motion with a characteristic reptation time given by 'Td(M) for the probe chain and 'TiP) for the matrix chains. Both the reptation and the constraint release processes are assumed to be independent; therefore, the total diffusion coefficient of the N-mer chain is now

D* = D Rcp

+ DCR

(24)

where D CR , as described above, is the contribution of the host environment. It has been argued (Wantanabe and Tirrell, 1991) that the assumption that the two processes are independent is valid only if the conformation of the tube and that of the chain trapped in it remain Gaussian during each successive step. The constraints in this model are considered in an idealized manner where the diffusion of the N-mer chain occurs on a cubic lattice with z effective constraints (the removal of an arbitrary constraint will not necessarily contribute to alteration of the primitive path) per step along the primitive path. Constraints should relax at a rate proportional to 'Td • The mean waiting time for the release of the first of the z constraints is defined as

'Tw = [

[yet)]' dt

(25)

According to Graessley, the contribution of constraint release to the diffusion coefficient of the chain is (26)

In the case of a single chain diffusing into a single-component monodisperse host, the mean waiting time takes on a value of'Tw = (1T2/12)''Td.

272

GREEN

Note that an approximation is made whereby ~(t) is represented by the first, and by far the most dominant, term in the series. It follows that the complete diffusion coefficient for a linear flexible chain of molecular weight M diffusing into a monodisperse host environment of molecular weight Mp is given by

D,

=Do(M -2 + nCRM~M/M!)

(27)

where Do was defined earlier as Do = (4/15)MoM.kBT/~. It is clear from this result that the correction to the simple reptation prediction becomes less significant as Mp increases. It is interesting to note that in the case of self-diffusion (M = P), D. has a slightly higher magnitude that D* , which becomes significant, particularly at lower M. At sufficiently high M, D * = Ds. Green and coworkers have shown that Eq. (27) provides a very good description of the diffusion in d-PS chains into PS hosts of varying molecular weights. Shown in Fig. lOa are data that have been fit with Eq. (27). Only one adjustable parameter has been used, n CR , which was found to have a constant value of 11. Forward recoil spectrometry measurements by Green et al. (1984) on PMMA melts are also well described by Eq. (27) using a value of n CR =11 (Fig. lOb). IRD measurements in polyethylene (Von Seggren, 1991) also indicate that Eq. (27) provides a good description of the data. It was, however, found

10-11

10-12

"*~

10-13

10-14

~

0,

~ 'al ~~ \~\

oo -

0_

o

.-

~:;-.-.

~'-o-o

0-

--

\!'-----

T~ \ t::.-t::._t::.

t::.-

.....

10-15 104

(a)

'-."'-~ 108

105

p

Figure 10 (a) Data showing the effects of constraint release of d-PS in PS at 170°C. The lines drawn through the data were computed using Eq. (28). (0) M = 55,000; (e ) M = 110,000; (0 ) M =255,000; (_) M =520,000; (D.) M =915,000; (£.) M = 1,800,000. [Data of Green et al. (1984).1

TRANSLATIONAL DYNAMICS IN MELTS

273

that although O!CR was the only adjustable parameter, its value varied with the molecular weight of the d-PS diffusant. Studies of the polypropylene system did not provide strong evidence for the reliability of Eq. (27) (Smith, 1982; Smith et aI., 1984), but other studies supported this prediction (Tead and Kramer, 1988; Antonietti and Sillescu, 1986). The effects of constraint release on the tracer diffusion of a homopolymer into miscible blends have been investigated in PS/pVME (Green, 1991) and in PS/pXE (Composto et aI., 1992) systems. The results were found to be well described by a generalized form of the equation

D* DR(q»

-- = 1

kMM;(2) + ------,......:..-'-'------:[~q>

+ (1 - q>)t['vq> + (1 - q»]P3

(28)

that accounts for a host of two components, 1 and 2. In this equation, 'Y = T(1)/ T(2) and ~ = [Mc(1)/M.(2)] l!2. In the absence of the second component (i.e,. q> = 0), Eq. (28) reverts to Eq. (27). The data in Fig. 11 represents the diffusion of d-PS of M = 200,000 (circles) and of M = 520,000 (squares) into a blends of PS of molecular weight P = 1.8 X 106 with 40% PVME, where the PVME

..

10 1

c.

~

*~

10 0

10.ll~0n.4,---L-~--L-~~~lLOI5----~--~~-L~~1~0' ~

p

Figure 10 Continued (b) Constraint release data of d-PMMA (M PMMA of molecular weight P at 180a C. [Data of Green et al. (1984).]

= 519,000)

into

274

GREEN 10-12

10-13

C

10-14

1 0-16 L--J.---&--'-J....&...u..aJ'---""--'--L-L.u..u.L_--'--'-..J....J~LI.I

104

105

p

106

107

Figure 11 Data showing the constraint release of d-PS [Ce) M = 200,000; C-) M = 520,000] into miscible blends of PS of molecular weight P with 40% PVME of fixed molecular weight P = 145,000_ [Data of Green et al _ (1991).]

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

2. Tube Length Fluctuations In addition to the constraint release process, the N-mer chain is capable of undergoing other relaxation processes not described by the original reptation model such as fluctuations in the length, L, of the primitive path. It has been suggested that these fluctuations in L might account for the discrepancy between the experimental and predicted power law dependence of 1]0 (Doi, 1983; Doi and Edwards, 1986). Doi has shown that the average fluctuations

(M} )/(L)

= Z - l12

(29)

Recall that Z is the number of steps on the primitive path (Z = L/a). Considering that 1]0 is proportional to the longest relaxation time, one might incorporate the fluctuations in length in calculating the new relaxation time. According to Doi, (30)

TRANSLATIONAL DYNAMICS IN MELTS

275

where k is a numerical constant that is close to unity. This result is obtained by noting that 'Td = L 2/DRo and 'Td(F) = (L - !:::.L)2/DRO. Based on the result in Eq. (30), the new 110, which incorporates the chain length fluctuations, is (31)

where k' is a new constant. It is important to point out that Eq. (30) approaches the reptation prediction at extremely large values of M. At smaller values of M, these corrections are important. Experimentally, Colby et al. (1987) have shown that 110 - N 3 .4 for MIMe as great as 150; beyond MIMe - 200, departures from the N 3 .4 dependence that are consistent with N 3 are observed. While the data of Colby et al. suggest that in the limit of very large MIMe the viscosity should approach M 3 , there is a suggestion that the tube length fluctuations may not fully account for the discrepancy (O'Connor and Ball, 1992). This is due, in part, to the large error associated with measurement of the viscosity of the very high M polymers. O' Connor and Ball (1992) later revisited this problem. As mentioned earlier, the Doi-Edwards model considers the chain diffusing along the tube in accordance with one-dimensional Rouse behavior. Only a single diffusion coefficient, D RQ , and only one Rouse mode, the longest relaxation time, are considered to be relevant. Furthermore, fluctuations in chain length are also ignored. O'Connor and Ball recognized that any description of the dynamics of the chain should incorporate the full Rouse relaxation behavior, particularly of the chain ends. This is important because the release of stresses in the tube occurs at the current tail (defined by the direction of motion of the chain) of the chain whereas that due to the head of the chain is not important, provided there are no major fluctuations in contour length. By carefully rescaling the relaxation times and all the length scales in the problem, they were able to express the positions of the chain ends in terms of independent coordinates. Consequently, the behavior of the chain ends could be described in terms of independent Rouse modes. With the use of only two material-dependent parameters, which are easily measured, G ~) and the monomeric friction factor ~, O' Connor and Ball (1992) showed, through a computer simulation, that the outstanding discrepancies (110M, J~O) G ~) prdiction, etc.) in dependence could be accounted for. One of the important results of this work is that the chain contour fluctuations described by Doi and Edwards (1986) could not account for the viscosity power law discrepancy. In fact, O'Connor and Ball demonstrated that by combining the constraint release effects with their corrections they could account for the dependence of the viscosity on complete magnitude and molecular weight, entangled and unentangled. The M 3.4 power law dependence was accounted for in a number of polymer systems; at very large M the M 3 power law dependence is recovered. Furthermore, the discrepancy in the value of J ~O) G ~) was also resolved. It is my opinion that this is not the end of the story! There are details of the

276

GREEN

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

III.

DIFFUSION OF CHAINS OF DIFFERING ARCHITECTURES

A.

Branched Molecules

Branched molecules, in the presence of fixed objects, are unable to undergo a strict reptation process. Their motion is facilitated primarily through fluctuations in contour length. In a host of linear reptating chains, the constraint release process is expected to play an important role, more important than in the diffusion of linear chains. The question of the diffusion of star-shaped molecules was first addressed by deGennes (1979) and subsequently by a number of other authors (Graessley, 1982; Helfand and Pearson, 1983; Klein, 1986; Pearson and Helfand, 1984; Doi and Kunuzu, 1980). For the sake of clarity we begin with the motion of a star molecule of f = 3 arms, the schematic of which is shown in Fig. 12. The star undergoes translation in the presence of fixed obstacles. It follows that translational motion of the star can occur only by fluctuations in arm length Ls. For the star in Fig. 12a to move a distance a, a step on the primitive path, arm 1 must retract to the node without crossing any obstacles. We can calculate the relaxation time, T., for such a process. Doi argued that if the probability distribution of a chain of N. segments, Ls - (L s), is Gaussian, then the motion of L. can be considered to be Brownian and occurring within a harmonic potential

U(L s) =

Nsb2 (Ls (23)(kBT)

(32)

(Ls» 2

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

(a)

-

(b)

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

TRANSLATIONAL DYNAMICS IN MELTS

277

is given by (33)

Ts "" To exp[(3/2)Ns(b/an

An alternative manner in which one might arrive at the same result, as shown by deGennes, is to consider the probability of an arm of Ns segments retracting along its own contour without enclosing any obstacles. Such a probability is P(Ns) ex exp( - "{Ns)

(34)

where "{ is a constant that depends on M •. The rate at which the arm retracts, deGennes argues, should be given by (35) Therefore, (36)

Ts ex Td(N.) exp("{Ns) There have been a number of predictions for the form of Eq. (36) where Ts - N ! exp( -"{Ns)

(37)

The exponent k has been assigned values of 3 (deGennes, 1979; Doi and Kunuzu, 1980); 0 (Graessley, 1982); 3/2 (pearson and Helfand, 1983); and 1.9 ± 0.1 (Needs and Edwards, 1983). When the arm completely retracts (Fig. 12c), the center of mass of the chain diffuses a distance equivalent to the primitive path. During this process, the chain is forced to drag the other two arms that same distance. Since the diffusion coefficient is defined as (38) Eq. (38) predicts that 2

D"or "" a To

exp[-3(~)2] 2N. a

(39)

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

U' =

~ (f -

2)kB TNs

GY

(L s - (L s)?

(40)

Therefore the general diffusion coefficient is D sta ,

""

a

2

To exp

[-3"2"

(b)2]

(f- 2)Ns ~

(41)

GREEN

278

In general, one might write the diffusion coefficient for an f-arm star as (42) where C h C, and C3 are constants. We may now consider the situation in which the star chains diffuse into a matrix of linear flexible chains of molecular weight Mp. The constraint release process will play a significant role in the translational dynamics of this system. Under these conditions the total diffusion coefficient is (43) where D CR is given by Eq. (25). Few measurements have been done in this area (Kline et ai., 1983; Bartels et ai., 1986; Antonietti and Sillescu, 1986; Shull et ai., 1988; Crist et aI., 1989; Fleischer, 1985). Experimentally it is well established that star molecules diffuse much more slowly than linear chains of comparable dimensions and that the diffusion rate of the stars depends exponentially on the length of the arm. The actual molecular weight dependence of the pre exponential is less certain at this point. Many of the data appear to be well described by assuming that the pre exponential factor is independent of molecular weight. Shull showed that D star varies exponentially with the molecular weight per arm for stars of different lengths diffusing into rnicrogel matrices (Fig. 13). These micro gels relax suffi-

10. 10

10. 11

4