The effects of heterogeneities on memory-dependent diffusion

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The effects of heterogeneities on memory-dependent diffusion

Farhad Adib, P. Neogi

To cite this version:

Farhad Adib, P. Neogi. The effects of heterogeneities on memory-dependent diffusion. Journal de

Physique II, EDP Sciences, 1993, 3 (7), pp.1109-1120. �10.1051/jp2:1993181�. �jpa-00247886�

J. Phys. II Fiance 3 (1993) l109-l120 JULY1993, PAGE l109

Classification Physic-s Abstiacts

61.40K 66.30

The effects of heterogeneities

_on

memory-dependent diffusion

Farhad Adib and P.

Neogi (*)

Chemical

Engineering Department, University

of Missouri-Rolla, Rolla, ^MO 65401, U-S-A-

(Received 2 Novembei 1992, accepted 25 March 1993)

Abstract. Case II diffusion is often _seen in

glassy

polymers, where the _mass

uptake

in

sorption

is

proportional

to time _t instead

of,fi.

_A

memory

dependent

diffusion is needed to

explain

such effects, where the relaxation function used _to describe the memory effect has _a characteristic time.

The ratio of this time _to the overall diffusion times is the diffusional Deborah number.

Simple

models show that _case II results when the Deborah number is around _one, that is, when the _two time scales _are comparable. Under

investigation

_are the

possible

effects of the fact that the

glassy

polymers _are heterogeneous _over molecular scales. The averaging form

given

by Dimarzio and Sanchez has been used to obtain the

averaged

_response. The calculated

dynamics

of

sorption

show that whereas _case II is still observed, the

long

term tails

change dramatically

from the

oscillatory

to

torpid,

to chaotic, which _are all observed in the

experiments.

The Deborah number defined here in

a self-consistent _manner

collapses

in those _cases, but _{causes no} other ill-effects.

Diffusion in

glassy polymers

is often non-Fickian

[1-3].

The class of

problems

addressed here is the non-Fickian

sorption

called _case II. Here the fractional _mass

uptake

^is

proportional

to time t, where in classical systems ^{it should follow t~/~}

[4].

It is well-known that _case II _can be

predicted

if the flux has _a

fading

_memory, where different functional forms for the memory

(within

the bounds of what is

admissible)

all lead to case II

sufficiently

_away from

equilibrium.

However, the

large-time

_response in these

systems

^is

dependent

on the tail of the relaxation function used to model memory. We seek to

identify

_some connection between the tails and the

variety

of

experimentally

observed behaviors at

large

times. The

simplest

model for relaxation is

[4, 5]

J> =

j'~(t t')) _{(x, t')dt' (1)}

where j_, ^{is the}

flux,

_c is the

concentration,

t and t' _are the

present

^{time and} some other time in the past.

Only

a one dimensional _case is

being

considered. Here, _~ is _an overall relaxation

function

given by

~1(t)=D,3(t)+ (Do-D~)xit) (2)

(*) Author _to whom all

correspondence

should be addressed.

where D~ ^and

D~

_are the initial and final diffusivities and _x

(t)

is the relaxation function. The relaxation time is defined _as

T =

j~

^tx

^(t

^dt

⁽³⁾

where

a~

i

= x

(t)

dt

(4)

o

Together

with the conservation

equation

()

=

)j, ₍₅₁

and the initial and the

boundary

conditions _on the membrane of

c ₌ 0 _{at t}

=

0

(6)

c = c~ ^at x =

0

(7)

~~

= 0 at ,< =

~

(symmetry (8)

a-t 2

the

sorption problem

becomes defined. One still needs _to know

x(t),

and the

subject

of

investigation

here is _as to what the realistic forms for _{x can} be. In

addition,

the

boundary

condition

equation (7)

can also be

time/memory dependent,

based on the _same features that

give

rise _{to a} memory

dependent

diffusion. This has _not been considered here both because of

simplicity

and due to the fact that such modifications _can be carried _out much

along

the lines used _to

develop

the relaxation function

x(t).

On

simplification,

the dimensionless

equations

become

~~ ^T ~2

=

~z~(T T')

0

(f, T')

dT'

(9)

~~

o

if~

1«

^~

^u(T)

^{dT =}

⁽¹⁰⁾

o

a~ x~(T)dT

=

(11)

o

a~

De

=

TXD(T)

dT

(12)

o

~1~ t

w&

(T)

₊

(i

_w

x~(T) (13)

where

~ ~ ~ D~ _~

~2

T

=

,

= -, f

= -, ~~ = ~, ^{w =}

~

^{and xD #}

j

(L~/4 Do

co L/2

(2 Do/L )

_o o

The diffusional Deborah number is

De

= ~

~

(14)

(L

/4

Do)

N° 7 DIFFUSION IN POLYMERS

and the fractional _mass

uptake

measured in

sorption experiments,

is

jf

~= ~d~. (15)

~w

0

The

simple

relaxation function

given by Neogi [5] predicts

in addition _{to case}

II, decaying

overshoots and oscillations in the fractional _mass

uptake [6],

which is also _seen

experimentally.

Much wider class of behaviors _are encountered in the

sorption experiments.

In _{one case} Vrentas, Duda and Hou

[7]

observed a maximum, followed

by

a minimum and then _an increase. The

experiments

stop at this stage.

Consequently,

_one is not certain if the systems represent oscillations with

decay predicted by

Adib and

Neogi [6].

Nevertheless the latter authors show that the data _can be fitted _to their

predicted

_curves and the parameters lie in the

right

_range. The data of Franson and

Peppas [8]

show _a maximum followed

by

_a

decay

with

no

signs

of

reaching

_a

minimum,

etc. In contrast the

system

^studied

by Lyubimova

and Frenkel

[9]

is

wildly oscillating.

Identifying

the range of relaxation functions that _can be used is _a difficult _one. A _more realistic

goal

would be _to characterize the moments, such _as the Deborah number in

equation (12).

In that

Vrentas,

Jarzebski and Duda

[10]

_were first _to define _a Deborah number suitable for

describing

diffusion. Their

premise

was that the diffusion and

rheology

_were

coupled,

and thus the relaxation time _was

T',

the first moment of the bulk shear modulus.

Against

the

diffusional time

L~/4 Do,

their Deborah number is De'

=

T'/(L~/4 _Do

_).

_Obviously

in viscoelas- tic solids their Deborah number is

unbounded,

which does not mean that _case II _{cannot occur} in such systems. ^{The definition of the relaxation} time/Deborah number

given

here is self-

contained, but does _not exclude the

possibility

that

they

cannot be unbounded _{or even}

negative

in very realistic relaxation functions. In fact _even in viscoelastic fluids _can the relaxation time

T' _can be unbounded

[I Il.

Thus the

generalized

Deborah number here has _no

physical

meaning

but

provides

_{a means} for classification.

Glassy

state and a model.

Glassy polymers

_are not at

equilibrium

but relax

slowly

towards it. If the

nonequilibrium regions

are

mapped

with _a property ^{called internal order then} conventional treatments in

classical irreversible

thermodynamics

leads _to

equations (I)

and

(2).

This form has been very successful in

explaining

the data _on non-Fickian diffusion.

The other

important

feature in

glassy polymers

is that it is

heterogeneous

at

microscopic length

scales. The

only proof

of this _comes from the

solubility

studies

[3],

where the

solubilities _are shown _to have _two

contributions,

_one from the solid matrix and the other from defect-like _« microvoids _». Since the microvoids cannot be _seen,

they

must not be much different from the interstitial voids and in that the model for

glassy polymers

_{as a}

single phase heterogeneous

continuum is _{a very} reasonable _one. To model diffusion in such systems a

random walk model is invoked where there is _a

prescribed waiting

time distribution at every step

[12].

Case II diffusion is

predicted [13],

and _at very

large

times Fickian diffusion is

recovered. The

particular

class of

problems

is called the continuous time random walk

(CTRW).

In _a different

light

it appears ^to be less

appropriate

_a model for

heterogeneities.

If the

waiting

time distribution is _a Poisson process then the diffusion _can be shown to be Fickian _at all times.

The property ^{of the Poisson} process of _course is that the

probability

that _a random walker will take _a step ^{between times} t and _{t +} At is A At, where A is a constant. That

is,

this

probability

is

independent

of time and hence memory.

Using

_any other function for the

waiting

time

effectively

amounts to

using

_a

time/memory dependent

A.

Why

time should be the tag

indicating heterogeneity

is _not clear.

The _more attractive method is that

given by

^Dimarzio and Sanchez

[14],

who suggest ^{that in}

a

heterogeneous

^{medium the} fundamental time constant should have _a

distribution,

and the response observed is that

averaged

_over such _a distribution.

They

derived _one such distribution

by assuming

that the relaxation time had _an activation energy, and the distribution of the activation energy was

given by

the Boltzmann distribution. The

averaging

turned _a

simple

relaxation function of

exponential decay

to « stretched

exponential

_».

Consequently,

in the

first set of calculations we use stretched

exponentials

as a means for

modelling

the

heterogeneous

system. ^{These lead} us to _some

pathologies

which

together

with the

experimental

observations _on real systems, ^lead us to very

specific properties

that the

averaged

relaxation functions should have.

(In

that _{we are}

guided by

considerable information from CTRW _on the

classification of

waiting

time distributions

by

their

large-time tails,

that

is,

the mathematics of

how to make such choices _are known

[15-17].) Unfortunately

the

glassy polymer

is _{not an}

equilibrated

_system, and thus _we have _no constraints _on the distribution functions for the relaxation times, and

eventually

_on the final relaxation function.

Dimarzio and Sanchez

[14]

have warned that _one should know where to make the

averaging.

This is very relevant when the basic process has _{no «} time _{constant »,} that is, the effective parameter ^{is also time}

dependent.

Such processes are not covered

by equation (2),

in which

case we need _to go a little

deeper.

If _a

particle

is released _at the

origin

at zero time in _a

symmetric

_system then its

probability density

function under

equation (1)

would be

given by

~~

=

l'

^{~ ~~}

) ^~'~

^~

^r~

^~~

(r, t')j

^dt'

⁽¹⁶⁾

at

~ ^r Jr Jr

where the _van Hove function

G,

which is

normalized,

vanishes far away. It is

possible

to show

[18]

that

equation (16)

leads to _a variance

(r~(t))

=

~

) ₍₁₇₎

s

where overbars indicate

Laplace

transforms and _s is its variable.

The variance is related _to the

velocity

autocorrelation function

(r2ji))

=

juji). ujo)j j18)

s

Hence _one obtains

A

=

(U(t), u(o)) (19)

That is, in _an

isotropic

_system the relaxation function is the

velocity

autocorrelation function.

This opens up very

interesting possibilities,

for

Zwanzig

and Bixon

[19]

which have shown that in molecular fluids the

velocity

autocorrelation function

decays

to zero

only

after it passes

through negative regions.

In that the _area under the _curve, where it is

negative,

can be small, but its contribution to the _moment in

equation (3)

_can be

large

and

negative

as the

negative

parts occur at

large

times. To

proceed

_{a step}

further, negative

moments can be

envisaged

in

more

general velocity

autocorrelations functions of this kind. The main conclusion however is that the evolution of the

velocity

autocorrelation function is

govemed by

_an

equation containing

the memory function

[15].

^As a

preview

we note here that

equation (2)

results

only

when there is _{no memory.} Some results of

averaging

^{this different} class of

problems

are also considered.

N° 7 DIFFUSION IN POLYMERS 13

The

key

_aspect uncovered here is that the diffusional Deborah numbers need not

always

be

positive,

_nonzero and

finite,

in that it has

only

_a mathematical definition. Some basis for

why

^it may

happen

to be _so, and _some basis for

choosing

relaxation functions _are

given

here.

Finally

a note of caution is added that _not all

apparently

well behaved kernels have _a solution in such

problems [20].

The calculations _are

given

next.

CTRW,based models.

Following

earlier discussion

equations (9-15)

when

averaged by

the distribution

given by

Dimarzio and Sanchez

[14]

would lead _to relaxation functions which _are stretched exponen- tials. The fraction _mass

uptake

characteristics in these _{cases are} discussed first.

Following

the

decomposition

of ~~ into _a Fickian part ^and a non-Fickian part ⁱⁿ

equation (13)

it is

appropriate

to relate x~ ^to a dimensionless T. Thus

xD #

~

j

exp(- )

~j ⁽²⁰⁾

~~ ~ ^e

fl

where

r(x)

is the gamma function. For

p

= I, Adib and

Neogi [6]

have

given

the exact solution. Two other values of

p

have been selected

here, p

₌ 2 and

p

₌ 1/2.

Equations (9),

11

3)

and

(20)

_are solved

numerically subject

_to the initial and

boundary

conditions of

equations (6-8).

The

resulting profiles

_are

integrated following equation (15)

_to get ^the fractional _mass

uptakes.

Numerical

integration

is difficult. As

w - 0 and T

~

0,

the

equation

becomes

hyperbolic

and

explicit

finite difference

(forward

time central space,

FTCS)

scheme used here

collapses.

It is

important

to

recognize

that the dimensionless

diffusivity

increases from

w to I. In

principle sufficiently

small AT for _a stable scheme is arrived _at

by taking

the dimensionless

diffusivity

to be _w and the

stability

ratio

=

1/2. This AT is _so small that the total time taken is

prohibitively large.

If the

stability

ratio is taken _to be 1/2

by using

a dimensionless

diffusivity

of I, one needs to

integrate

_{over a} lesser number of

time-steps

but the system ^is

expected

to be unstable

initially.

It turns out not to be the _case because of the fact that

diffusivity,

and

damping,

increases with every time step.

However,

the scheme still

requires

that the initial

diffusivity

be

sufficiently large

and _w

=

0.I has been used

throughout.

The results for the three values of p have been shown in

figure

I,

together

with the exact solution for p ₌ I. A cumulative _error of 0.02 is observed and the _error is

generally

found _to

be of this order. In

figure

I, De =1.0 has been used in all _cases in

keeping

with the

observation that it is around De

= 1.0 that _case II is _seen

[6].

The above results have been

plotted

ⁱⁿ

log-log

in

figure

2 and show that the

slope

is

practically

close _to

1.0,

that

is,

show _a

case II type ^behavior. Thus, the

stretching only changes

the _nature of the overshoots. For p ₌ 1/2,

although

the overshoot

persists,

the oscillations _can be _{seen to} have

disappeared

^for all

practical

_purposes. From

equations (12)

and

(20)

it is _seen that _as p _-

0,

De becomes

unbounded. Since oscillations in _case II _are

rarely

seen in the

experiments,

the observation that oscillations die _{out as} the first moment

(that

is,

De)

becomes unbounded is of considerable

importance.

This leads to another well known form for the relaxation function studied in CTRW which is also _an average

~ j~AJ

xD =

~j ^lap

exp

(21)

"~~~j

"

where _a,

p,

and A _are

positive

constants and all less than I-o- At

large

times

[17]

1 3

'. 2 J=2 _,, ',

1 i

,

o o

o. o

M~ ^o.7 fl='/2

~

o.«

o. s

o. 4

o. >

o. 2 a>ialytical sol>ition furl =

o i

o. o

o 1 2 3 4 5

T

Fig.

I. Fractional _mass

uptake

is shown _{as a} function of dimensionless time T

using

stretched

exponentials,

equation (20), _as the relaxation function. The dashed line is the exact solution from Adib and

Neogi

[6] for p I. Values of

w =

0. and De

=

1.0 have been used.

o

J=2

M I

loo fi

i

" Maa

= lf2

2

'~_~

_~ i i

logT

Fig.

^2. The data from

figure

are plotted in

Iog-log.

The

slopes

are nearly 1.0 in all _cases.

N° 7 DIFFUSION IN POLYMERS II15

xD

decays

as

T~~~

^' where H

= In

p/In

A. For A

~ p one has

De

=

"

j'

^~

₁₂₂₎

but for A

~

p,

De is undefined. For A

=

I, equation (22) collapses

to the

simple exponential

studied

by

Adib and

Neogi [6],

for which exact solution exists.

Many

cases have been studied

by

^Adib

[21],

all of which show _case II, overshoots and

decaying

oscillations. It _can be

supposed

that

equation (22)

tums to a power law at such

large

times that the system ^{is almost}

equilibrated

and shows _no

distinguishing

features.

Consequently

_we look for relaxation functions where the first moment

diverges

at a faster rate

(17]

:

xD =

) _eT/«12

_erfc

J[ ₍₂₃₎

«

where

i~

_{erfc is the}

complementary

_error function of second order

[22]

and

i~

erfc

,&

=

II

+ 2 _x

)

erfc

,&

^~

,&

_{e~ '}

₍₂₄₎

4

,$

Equation (23)

can be normalized

following equation (11),

but all

higher

moments are

unbounded

[17].

In

figure

3 this response has been shown for _a

= I, for other values of _a the variable T should be

replaced

with Tla in the

figure.

The result shows _an overshoot followed

by

_a

decay exactly

^{like in the} data

by

Franson and

Peppas [8].

However the result in

figure

3

was found to be far _more

sharply peaked

than the data in _{most cases.} In _{one case} the

comparison

_was

quantitatively

reasonable.

1 2

1 i

i o

o e

o o

o 7

Mt

I

°'~

o s

o 4

o 3

o a

o i

o. o

o i a 3 s s 7 o e lo

T

Fig. 3. Fractional _mass uptake is known _{as a} function of dimensionless time T

using

equation (23) _as the relaxation function.

Memory

^{functions and} molecular

phenomena.

Evolution of

velocity

autocorrelation function is traced

through

_{a memory} function

[18]

_as

I

-

II ^{~~~~ t') /l(t)dt}

^~~~~

Under

Laplace

transform and with _~

(0)

= 1, _one has

iz

=

126)

s +

K~

where the memory function

K~

^{can be found}

through

simulation

studies,

neutron

scattering

_or

assessed

through

models. In the present case

K~

^{is written} as a function of T and with the

given decomposition

of _{~, one} takes

kD

_~

(27)

s +

K~

Two models for

K~

^{have been}

given by

Beme

[18]

which _can be

generalized

to stretched

exponentials

T

)~

K~

= a e ^"

(28)

where _a and _{a are} constants. Berne refers to p ₌ I _as Lorentzian memory and p

=

2 _as Gaussian. An additional case of

p

=

1/2 is included

here,

that

is,

_{one may} take the Lorentzian

as the base _case, and the stretched

exponential

form _as that is obtained

by averaging.

It is

important

to note that _an

exponentially decaying

relaxation function of

equation (2),

_can be obtained from

equation (25)

if

K~

^is

given by

the Dirac delta

function,

that

is,

if there is _no memory.

Further, K~

^of

equations (25-27) plays

the role of _an inverse time constant, except that it is

time/history dependent.

It _seems

appropriate

to average ^at

equation (25)

as it

provides

the

key dynamical

_step in this process. ^The

product

an is

proportional

to

temperature

with _a

positive proportionality

constant. Thus the individual terms will also be

positive.

The value of

p

not

only

_governs the

length

of

tails, they

also affect the short-time memory in that the

slopes

at T

=

0 _{are zero} for p

= 2, finite and _nonzero for p

= I and infinite for p ₌ ^{1/2. These} are

respectively

the accelerations

(or decelerations)

_{at zero} time, that

is,

_as the molecule enters the

polymer. Using Laplace

transforms

[23]

and

equations (27)

and

(28)

one has

~q

~

~/~

~

3/2 _~~

~~

~~~~

~

,~

'

~

~~~ ~~~~

~

~

j i P

"

j30)

s + a

~

^s~a~

= a

' "

a e ^~ erfc ^"~ p ₌ 2

(31)

2 2

and

kD

_# ^~

~"j

fl

=

1/2

(32)

~ ~ ~

~~

2

~/~

~^{~ ~~} ~~~~

2,~

N° 7 DIFFUSION IN POLYMERS II17

s+a~~ (33)

"~"

p=1

~2

~~-lS+a~

j3~)

~p

au

;

p=2.

gr

~ ~

"f

~ ~

~aS)

s+~ ""e

_~~~

2

The first _moments of

K~

are

A ₌ 12 _au

~;

_p

=

1/2

(35)

=aa~; _{p=1 (36)}

~~2

=

f

ⁱ

^fl

^{= 2}

⁽³⁷⁾

and the first _moments of xD ^are

jj

12 _au ~)

(38)

p

₌ 1/2 D~

"

2 _au ^'

~39)

(1-aa~),

p=1

~

au ^'

au ~

~40)

=

~ aa~

'

~

~

One

peculiarity

of these

equations

need _to be made obvious first. If p

= I, and _one wishes _to

study

the effect of memory then A ₌ in

equation (36)

would be the

right

value to choose.

However, one finds in

equation (36)

that it would lead _to De

=

0. In

general

the forms for De show

clearly

their

capacity

to take

negative

values _as well. In _contrast A is

always positive.

Of the four constants _{a, a,} A and

De, only

two are

independent.

Before

giving

numbers _to these it is necessary to

provide

_{some reasons} behind

choosing

a

procedure

for

doing

so. One method would be to assume that all

temporal phenomena

have

representative

time-scales. Such time-scales would be the first moments, in which _case A and De _are

expected

to be

positive,

finite and _nonzero. Values of A and De _were made _to vary over three orders of

magnitude.

The

resulting plots

for the fractional _mass

uptakes

showed _no

peculiarities

over the _ones

already

encountered

[21].

In _a second method _one may assume that the molecular

phenomenon

comes

first and the rest are derived results. For

p

=

1/2,

_a table of values _are

given

below _:

Table I. Chosen values

of

_a and _a, and calculated values

of

A and De.

~ _l/8

8

1/8 De 125/4 De

= 13/4 De I/4

A 3/128 A 3/16 A 3/2

De

=

2 De I1/2 De

= 95/16

A=3/2 A=12 A=96

8 De

=

95/2 De 767/16 De

= 6 143/128

A 96 A 768 A

= 6 144

It is very

interesting

to note that it takes very little _to make De

negative

and further that it does _not prevent one from

calculating

the fractional _mass

uptakes.

The method used _was

Laplace

transform under which

fif~ ^~ ~sj^T

~C° ~0

f/ f/

_d ^~~~~

i

_{2 ds} ^~~~

where

(~

=

(2

k ₊ I gr/2 and s~ are the _roots of

sj ~

= ⁼

(j (42)

lLDi

where

#~

and

dp~/ds

at _s

= s~

give

_us

_#~~

and

dp~~/ds respectively. Obviously

where

#~~

^is a transcendental

function,

there _are infinite _roots in

equation (42)

for every value of k.

These _{roots were} established

independently by

ZANLYT subroutine from IMSL and

by

the

global homotopy

continuation method

developed by

Choi

[24].

The latter _{assures one} that all

roots can be found. It _was found _more

practical

to

plot

the roots on

Argand diagrams.

This not

only

shows the

completeness

of the roots but also _as the which the

important

_{ones are.} The

significant simplification

comes from the observation that

only

one root, where the

magnitude

of the real part is the

smallest, plays

the _most

important

role. In

fact,

it is the

only

_one that _one needs to be concerned with. As _a result the lower

right

hand _comer of the

Argand diagrams

were

thoroughly

combed for _roots. The accuracy of the method _was tested

by comparing

the solutions _to the stretched

exponential

_cases obtained under this method

against

their numerical solutions in

figure

I. The results show

good

agreement ^{and in the} case where _exact solution is

available the present ^method was more accurate than the numerical solution. One of the

Argand diagrams

is shown in

figure

4.

Only

five of the entries in table I have been used in

figure

5. All

plots

_are

deeply oscillatory,

some fail to show clear indications of

moving

to

equilibrium,

and in _{one case}

la =1/8,

a = I, De

= 2 and A

=

3/2)

the response is almost

aperiodic

with

misshapen peaks

and

crests these _are the characteristics of the responses obtained

by Lyubimova

and Frenkel

[9].

The _reason for the almost chaotic pattern lies in the fact that whereas all the other responses are

characterized

by

a

single

_root

lone

where the real part ^{has the smallest}

magnitude),

this

im(s~)

-8 -7 -6 -S -4 -3 -2 -t o

llc(sLJ

Fig.

4. The

complex

roots of

equations

(42) and (20) for p 1/2 _are shown _on Argand

diagram.

All

roots have negative real pans, ^and as the conjugate is also _a root, only _one quadrant has been used.

N° 7 DIFFUSION IN POLYMERS I 19

1 o

'.? I,a= =l

8 1-s

i. s

~" ~*

''*

~~i'~~~

1 3

1 2

1 i

Mi i.

~°'~

°°

o. o

o 7

O «

o s

o 4

o 3

o a

o t

o.

o i a 3 4 5 e 7 o o lo

T

Fig.

5.- Fractional _mass

uptake

is shown for the _case of p ^{=1/2 in the} memory function

K~ ⁱⁿ

equation

(28). The effect of parameters ^chosen on the Deborah numbers is shown in table1.

particular

_response is

govemed by

two roots of about

equal importance.

The interference between _{two wave} trains _cause near-chaos.

Other _cases,

p

=

I and

p

=

2,

_are available in Adib's

[21]

thesis. The responses are similar but the almost chaotic response discussed

above,

_{was not seen.}

Discussion.

The

simple exponential decay

of the relaxation

[5, 6]

or memory function

[18]

when

averaged

for

heterogeneities

in the _sense of Dimarzio and Sanchez

[14] yield

results which compare well with

experiments.

We _are able _to show for the first time that the

heterogeneities

do have _an effect _on diffusion. In

bringing

these effects

in,

_{we see} that the

ability

of the

simple

models _to

predict

_case II is

unchanged,

but the

predictions

for the

large-time

behavior

improve significantly.

It is also _seen that the Deborah numbers _cannot

always

^be

given

_a

physical significance,

but could be used _to

classify large-time

behaviors in

sorption. Possibly

chaotic behavior in

sorption

could

happen only

^{if the Deborah number is}

negative

^{and the lack of overshoots could}

only

^be

due _to unbounded values of the Deborah numbers. It should be noted that

negative

Deborah

numbers _were

only

^{obtained in} the _case where the basic time _{constant was not} defined

unequivocally.

Finally,

_we note that for _some other models that _we have

analyzed

_no solutions _were

obtained.

(That

is, under the

Laplace

^transform method _some roots were found which had

positive

real

parts.)

In

problems

in viscoelastic

flows,

such features _are associated with

catastrophic

events such _as

cracking, separation

from the walls, etc. We _note that such _events

as solute induced

crazing

is sometimes _seen in _case II

[25].

References

Ii PARK G. S., Diffusion in Polymers, J. Crank and G. S. Park Eds. (Academic Press, New York, 1968) _p. 141.

[2] FRISCH H. L., Polymer

Eng.

St-I. 20 (1980) 2.

[3] HOPFENBERG H. B. and STANNETT V. T., The

Physics

of

Glassy Polymers,

R. N. Haward Ed.

(Appl.

Sci., London, 1973) _p. 504.

[4] BOLTzMANN L., Ann. Phys. Chem. 53 (1894) 959.

[5] NEOGI P., A/ChE J. 24 (1983) 829, 833.

[6] ADIB F, and NEOGI P., AIChE J. 33 (1987) 164.

[7] VRENTAS J. S., DUDA J. L. and Hou A.-C., J. Appt. Po/ym. Sci. 29 (1984) 399.

[8] FRANSON N. M. and PEPPAS N. A., J. Appl. Polym. Sci. 28 (1983) 1299.

[9] LYUBIMOVA V. A. and FRENKEL S., Polym. Bull. 21(1984) 229.

[10] VRENTAS J. S., JARzEBSKI C. M. and DUDA J. L., AIChE J. 21(1975) 894.

[I I] GODDARD J. D., Adv. Co/%id Interface Sci. 17 (1982) 241.

[12] SCHER H. and LAX M., Phys. Rev. B 7 (1973) 4491.

[13] STASTNA J., DE KEE D. and HARRISON B., Recent Developments in Structured Continua II, Longman Sci. & Tech. (John Wiley & Sons, Inc., New York, 1990) _p. 56.

[14] Di MARzIO E. A. and SANCHEz I. C.~

Transport

and Relaxation in Random Materials, J. Klafter, R. J. Rubin and M. F.

Shlesinger

Eds. (World Sci.,

Philadelphia,

1986) _p. 253.

jis] J. KLAFTER, R. J. RUBIN and M. F. SHELSINGER. Eds.,

Transport

and Relaxation in Random Materials (World Scientific,

Singapore,

1986).

[16] LINDENBERG K., and WEST B. J., J. Statistic-al

Phys.

42 (1986) 201.

[17] MONTROLL E. W. and SHELSINGER M. F.,

Nonequilibrium

Phenomena II. From Stochastics _to

Hydrodynamics,

E. W. Montroll and J. L. Lebowitz Eds., vol. XI (North-Holland

Physics

Pub., 1984) _p. 46.

[18] BERNE B. J.~

Physical

Chemistry, An Advanced Treatise, D. Henserson Ed. (Academic Press, NY, 1971) _p. 539.

[19] ZWANzIG R. and BIXON M.~ Phys. Rev. A 2 (1970) 2005.

[20] RENARDY M., Ann. Rev. Fluid Mech. 21(1989) 21.

[21] ADIB F., Mathematical Models for Diffusion in

Glassy

Polymers, Ph. D. Thesis, Chemical

Engineering Department, University

of Missouri-Rolla (1991).

[22] CRANK J., The Mathematics of Diffusion~ 2nd Ed. (Oxford

University

Press, London, 1975) p. 376.

[23] BATEMAN H.~ Tables of Integral Transforms _v. I (McGraw-Hill, NY~ 1954) _p. 144-147.

[24] CHoi S. H., The

Applications

of Global Homotopy Continuation Methods to Chemical Process

Flowsheeting

Problems, Ph. D. Thesis, Chemical Engineering

Department,

University ^of Missouri-Rolla (1990).

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