Diffusion in polymer diluent systems
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F o r t s c h r . H o c h p o l y m .  F o r s c h . , ]3d. 3, S. 1   4 7 (1961)
Diffusion in PolymerDiluent Systems
By
HIROSHI ~'UJITA1
Physical Chemistry Laboratory, Department of Fisheries,
IZyoto University, Maizuru, Japan
With 18 Figures
Table o5 Contents
Page
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
I. F u n d a m e n t a l s . . . . . . . . . . . . . . . . . . . . . . . . . . 2
II. F i c k i a n Sorption . . . . . . . . . . . . . . . . . . . . . . . . . 4
IIl. N o n  F i c k i a n Sorption . . . . . . . . . . . . . . . . . . . . . . . 13
IV. P e r m e a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
V. Interpretation of Diffusion Coefficient D a t a . . . . . . . . . . . . . . 31
l~eferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Introduction
The diffusion of small molecules in polymeric solids has been a subject
in which relatively little interest has been shown b y the polymer chemist,
in contrast to its counterpart, i.e., the diffusion of macromolecules in
dilute solutions. However, during the past ten years there has been a
great accumulation of important data on this subject, both experimental
and theoretical, and it has become apparent that in many cases diffusion
in polymers exhibits features which cannot be expected from classical
theories and that such departures are related t o the molecular structure
characteristic of polymeric solids and gels. Also there have been a
number of important contributions to the procedures b y which diffusion
coefficients of given systems can be determined accurately from experi
ment. I t is impossible, and apparently beyond the author's ability, to
treat all these recent investigations in the limited space allowed. So,
in this article, the author wishes to discuss some selected topics with
which he has a relatively greater acquaintance but which he feels are
of fundamental importance for understanding the current situation in
this field of polymer research. Thus the present paper is a kind of personal
note, rather than a balanced review of diverse aspects of recent diffusion
studies.
x Present address: Department of Polymer Science, Osak~ University, Nakano
shima, Osaka, Japan.
Fortschr. Hochpolym.Forsch., Bd. 3 1
2 H. ~'uj IT2~:
The systems discussed below are restricted to ones in which tile
penetrant 1 is a solvent or a "plasticizer" for a given polymer and in
which its concentration is sufficiently small in comparison with the con
centration of polymer. Such systems m a y be referred to as extremely
concentrated solutions of polymer. With this restriction we shall not
treat here diffusion processes of gaseous substances, such as oxygen and
carbon dioxide.
I. F u n d a m e n t a l s
1.1 Diffusion coefficients
Diffusion is defined as the process in which components are transported
from one part of a mixture to another as a result of random molecular
motion. Phenomenologically, it can be most rigorously treated b y the
proper application of the thermodynamics of irreversible processes
[DE GROOT (1951); HOOYMAN (1955); GOSTING (1956)].
We designate the penetrant as component 1 and the polymer as
component 2. If the mixture is incompressible and no volume change
occurs on mixing of the two components and if the diffusion takes place
only in one direction x, it can be shown that distribution of component i
(i = 1,2) in the system during isothermal diffusion is governed b y a
differential equation of the form:
a~,lat = (a/ax) [D (ac,/a~)], (1)
where c~ is the concentration of component i expressed in grams per cc.
of polymerpenetrant mixture, t is time, and D is a quantity called the
mutual diffusion coefficient of the system [HARTLEY and CRANK (1949)1.
The point is t h a t in Eqs. (1) for both components 1 and 2 there appears
only a single "diffusion coefficient". Equation (1) is usually called the
Fick diffusion equation or the Fick second law of diffusion. Although this
is generally taken as the basis for analyzing d a t a of diffusion in one
dimension, it m u s t be recognized t h a t it strictly applies only for two
component systems in which the partial specific volumes of the compo
nents are independent of pressure (thus the mixture is incompressible)
and also of composition (thus no volume change occurs on mixing).
To a good approximation these conditions are fulfilled when the amount
of one component relative to the other is very small, as in the case of
dilute solutions or in extremely concentrated solutions which will be
treated below. In passing, we wish to point out that Eq. (1) is no longer
applicable for systems containing more than two components [GosTING
(1956)].
x The term "penetrant" will be used here to designate lowmolecularweight
substances which diffuse in given polymeric solids.
Diffusion in PolymerDiluent Systems 3
According to the thermodynamics of irreversible processes, the
quantity D is equal to (D1) r or (D2)v. Here (D;) r is defined b y the relation
(J~)r =(Di)v(3cdOx), where (Jt)v is the flow (for flux) of component
i relative to a plane (perpendicular to the direction of diffusion) moving
with the local center of volume. The (Di)v is termed the volumefixed
diffusion coefficient of component i. In general, the values of (D~)v and
(D~)v are different from one another, but when the partial specific
volumes of the two components are constant both have the same value.
The D in Eq. (1) denotes this coincident vMue of (D1)v and (D~)r. Several
other kinds of diffusion coefficient m a y be introduced b y defining the
flow of each component in terms of other frames of reference. For
example, the massfixed diffusion coefficient, (Di)~1, of component i is
defined b y the relation (Ji)M =  (D~)M (OcdOx), where (J~)M is the flow
of component i relative to the plane moving with the local center of mass.
One m a y choose as the reference frame a plane which moves with the
local velocity of component 2, the polymer component in the present
case. Then the polymerfixed diffusion coefficients m a y be defined for
penetrant and polymer components. The value of this coefficient for the
polymer component is identically zero, but that for the penetrant
component, denoted here by (D1)P, is nonzero in general. Any two of
these various diffusion coefficients are related from one another by a
simple mathematical equation [WENDTand GOSTING (1960); FUJITA
(1961)]. Thus D and (D~)p are related by the equation:
(D1)F = D/(1   ~1 cl), (2)
where vl is the partial specific volume of component 1, i.e., the penetrant
component. Between (Di)M and D there is a relation of the form:
(Ol) M = O [1   (q/e)]/(1   vx cl), (3)
where ~ is the density of the penetrantpolymer mixture. If D is eliminated
from these equations, an expression which relates (D1)e to (D1)M may be
derived. I t is a simple matter to show from these relations that (D~)ao,
(D1)M, and D all converge to the same value at the limit of zero penetrant
concentration. This limiting value shall be denoted b y a symbol D 0. It is
important to observe that Eqs. (2) and (3) are valid for onedimensional
diffusion. It appears that no corresponding equation is as yet known for
diffusion in higher dimensions.
1.2 S o r p t i o n a n d p e r m e a t i o n m e t h o d s
One of the central problems in the study of diffusion is to evaluate D
for a given system as a function of such parameters as penetrant con
centration and temperature. For polymerpenetrant systems with which
we are concerned in this article two experimental methods are typical
for this purpose. They are the sorption method and the permeation method.
1"
4 H. FUJITA:
In a sorption experiment, a film of a given polymer is exposed to vapor
of a given penetrant substance at a given pressure and the gain or loss in
weight of the film is measured as a function of time. In the present paper,
the term "sorption" will be used loosely for both absorption and desorp
tion, unless otherwise specified. In the usual absorption experiment the
film is initially free of penetrant, b u t in some cases the process of absorp
tion is studied with a film initially equilibrated at a nonzero v a p o r
pressure. I{OKES, LONG and HOARD (1952) have referred to this latter
type of absorption as the "interval" type. This term, however, will not
be used in the present paper. Most desorption experiments measure
processes from nonzero initial concentrations to the zero final concen
tration, and hence are of the "integral" type in the sense of KOKES et al.
(1952).
In a permeation experiment, the amount of penetrant v a p o r flowed
through a film of a given polymer is measured as a function of timeunder
the condition t h a t one surface of the film is allowed to get in contact with
penetrant v a p o r at a constant pressure and the other surface is ex
posed to vacuum.
II. F i c k i a n S o r p t i o n
2.1 Definitions
According to the thermodynamics of irreversible processes, the mutual
diffusion coefficient D m a y be a function of penetrant concentration c1,
position x, and time t. In the present chapter we shall discuss sorption
behavior of systems in which D varies with c1 only, and shall use the
notation D (c,) to indicate this condition. I t is assumed t h a t the sample
film is so thin t h a t diffusion takes place effectively in the direction of its
thickness. At the beginning of an absorption or a desorption experiment
the film is conditioned so that cl is uniform everywhere in it. This initial
concentration is denoted b y cl ~ Then we have
c~ = q o ( _ L/2 < x < L/2, t = o ) , (4)
where L is the thickness of the film. The origin of x has been taken on the
central surface of the film. The boundary condition for c1 in general use
for the mathematical study of sorption processes on polymers is t h a t the
penetrant concentrations at both surfaces of the film attain a certain
value cx~176
instantaneously when the film is exposed to v a p o r and t h a t
this value is maintained during the course of sorption so long as the
pressure, fi, of the ambient vapor remains constant [CRANK (1956)].
This boundary condition is termed the condition of constant surface
concentration, and is represented mathematically b y
(x = L/2
c 1 = cl ~176 a n d   L / 2 , t > 0), (5)
Diffusion in PolymerDiluent Systems 5
provided the film does not swell (or shrink) during the course of absorp
tion (or desorption). In reality, this assumption does not hold, and there
fore Eq. (5) is not applicable. However, inclusion of the dimensional
change of the film during sorption (which means to treat L in Eq. (5)
as a function of time) makes solution of Eq. (1) a formidably difficult
problem. Probably, for systems in which total amounts of absorbed or
desorbed penetrant are small compared with the total mass of the polymer
this effect is not too important, although it is not altogether negligible.
For these reasons most of the current theories on sorption processes in
polymerdiluent systems of the type considered here neglect the
variation of L with time.
Very thorough investigations of Eq. (1) subject to conditions (4) and
(5) have been made b y CRANK and others for various assumed forms
of D (cl), of which an excellent s u m m a r y has been given b y CRANK (1956).
The information we need here is not the detailed mathematical expressions
for such solutions of Eq. (1) but the characteristic features of sorption
processes predicted from this set of equations. Customarily, the sorption
processes in which D is a function of c1 only and the initial and boundary
conditions are given b y Eqs. (4) and (5) are referred to as of the "Fickian"
type. Moreover, it is often said t h a t such processes are controlled b y the
Fickian diffusion mechanism 1.
In section 2.3 we will summarize some representative features of this
type of sorption process.
2.2 Representation of sorption data
D a t a obtained from an absorption (or a desorption) experiment are
the amounts of a given penetrant substance absorbed in (desorbed from)
a given polymer film as a function of time t. Usually, the amounts per
unit volume of dry polymer are computed 2, denoted b y M (t), and plotted
against (t) II,.
The resulting curve is termed the absorption (or desorption) curve,
or more generally the sorption curve. However, in order to interpret
correctly experimental results in terms of Eq. (1) it is necessary to plot
M' (1), the weight of sorbed penetrant per unit Volume of the swelling or
deswelling film, against (t) 1/'. This requirement arises from the fact that
1 Since Fick's first and second laws of diffusion are valid independent of whether
2) is a function of cl only or not and also of the form of initial and boundary con
ditions of a particular experiment, it is quite inadequate to specify this particular
type of sorption as Fickian. The term "'Fickian" should be applied more generally
to all mass transport phenomena which are governed by Eq. (1), i. e., the Fick
diffusion equation.
2 Often the weight of sorbed penetrant per unit weight of dry polymer is
employed. This value is different from M (t) by a constant factor equal to the
density of the dry polymer.
6 H. FUJITA:
the concentration q in Eq. (1) is expressed in terms of the weight per
unit volume of polymerpenetrant mixture. Most of the existing reports,
however, adopt the approximation M ( t ) = M'(t) to represent the ex
perimental data obtained. For systems in which penetrant concentrations
are sufficiently dilute, this approximation probably does not introduce
significant errors into the evaluation and interpretation of the experiment.
Though not verified rigorously, the inaccuracy introduced by this
approximation would be of the same order as that caused by neglecting
the change in L with time during sorption (see Eq. (5)).
Experiments show that when no change in ambient vapor pressure p
occurs during the sorption process, M (t) approaches a limiting value as
time increases. When this limiting value is reached, the film absorbs or
desorbs no more penetrant and is at thermodynamic equilibrium with the
ambient vapor. This is the state called sorption equilibrium. The value
of M (t) for this state is denoted by Moo. Often the ratio M (t)]Moo is
plotted against (t)',l,/L and the resulting curve is called the reduced sorp
tion curve. This form of representation of data is convenient for theo
retical analyses.
2.3 Features of the F i c k i a n sorption
Basic features of sorption processes of the Fickian type have been
clarified by CRANK and coworkers through extensive mathematical
studies of Eq. (1). The following gives a summary, of the features of
particular importance.
(a) Both absorption and desorption curves are linear in the region
of small values of the abscissa. For absorption the linear region is obtained
over 60% or more of Moo. When D increases markedly with c1 the ab
sorption is linear almost up to the equilibrium;
(b) Above the linear portions both absorption and desorption curves
are always concave against the abscissa axis;
(c) The shape of absorption curve is not very sensitive to the depend
ence on concentration of D. It is often well approximated by the ab
sorption curve for a constant D, even when D varies appreciably with c1.
On the other hand, the desorption curve is rather markedly affected by
the D vs. c1 relationship;
(d) In both absorption and desorption the concentration distributions
in the film are greatly influenced by the functional form of D (q) ;
(e) When the initial and final concentrations (cl~ and qoo) are fixed,
the reduced absorption curves for films of different thickness all coincide
with each other, yielding a single curve. This applies for the corresponding
family of reduced desorption curves;
(f) The single absorption curve so obtained is always above the cor
responding single desorption curve when D is an increasing function
Diffusion in PolymerDiluenf:Systems 7
of % Both coincide over the entire range of the abscissa when and only
when D is constant. The difference between the two curves becomes more
appreciable as D increases more sharply with cl in the range from ci ~ to
Cl~176If D (q) passes through a m a x i m u m at a certain value of cI between
the given cl ~ and cl ~176
the two curves m a y intersect at some position;
(g) For absorptions from a fixed initial concentration to various
final concentrations the initial slopes of the reduced absorption curves
are larger as the final concentrations are higher, if D increases mono
tonically with c~;
(h) The same relation as (g) applies for reduced desorption curves
obtained from the experiments in which the final concentration is fixed
and the initial conditions are different.
/.0 f ._..o ~
ebs.~S/.,I
.r
~20
j ub~.O~/VSGni~H~
des./qSG~
0m ~ g
I f
S 10 IS 20
Fig. 1. Rexluced absorption and desorption curves (of the Fickian type) for the system polyisobutylenepropane
at 35 ~ C. Ta ke n from PRAG~R and LONG (1951)
Sorption curves consistent with these criteria, especially with (a),
(b), (e) and (f), have been observed for a number of polymerorganic
diluent systems, when the measurements were made at temperatures
well above the glass transition temperatures of the respective systems.
B y way of example, paired absorption and desorption curves obtained
for the system polyisobutylenepropane at a5 ~ C. are shown in Fig. 1
[PI~AGER and LONG (1951)]. Here b y paired absorption and desorption
curves is m e a n t a couple of absorption and desorption curves which
cover the same range of penetrant concentration. In actual cases,
especially when experimental data are to be determined for a variety of
external conditions, it would be almost impractical to investigate, for
given conditions, whether the individual sorption curves fulfil all of
these criteria for the Fickian sorption. Therefore, it is a usual practice
to regard a given sorption curve as of the Fickian type when it has an
overall shape conforming to features (a) and (b). In the case when paired
absorption and desorption curves are available and, furthermore, there
is reason to believe t h a t D of the system increases with q, reference to
feature (f) m a y be of use to check the conclusion. For a more definite
conclusion i t is recommended to perform experiments with films of
8 H. FUJITA:
different thicknesses and to examine whether the data give a single curve
when plotted in the reduced form.
Probably one of the most significant findings is that with organic
penetrants this type of sorption was observed only when a given system
was initially (in the case of absorption) and finally (in the case of desorp
tion) in the rubbery state. This suggests that the two conditions basic to
the Fickian behavior m a y be related to the molecular features which
distinguish a nonglassy polymer from a glassy polymer. However, it
appears that the above rule does not always apply for polymerwater
systems (perhaps, more generally, polymernonsolvent systems). For
example, KISHIMOTO, MAEKAWA and ~'UJITA (1960) have reported that
both absorption and desorption of water in polyvinyl acetate were
Fickian down to temperatures somewhat below the glass transition point
of the polymer.
Finally, one m a y remark an interesting contribution due to KISHI
MOTO and MATSUMOTO (unpublished), who showed with polyvinyl
acetate and polymethyl acrylate that criteria (a) and (b) are not sufficient
to conclude a given sorption curve as of the Fickian type when the
measurement is made at temperatures slightly above the glass transition
of the polymer: they found that the sorption curves obtained at such a
temperature with films of different thicknesses were not reduced to a
single curve, even though each of them had a shape expected for Fickian
sorption. This experimental finding indicates that more deliberation
than is generally conceived is necessary to conclude a given sorption
curve as Fickian when we are concerned with the region near the glass
transition.
2.4 M e t h o d s for the evaluation of D as a function of c1
Various methods have been proposed for the evaluation of D as a
function of penetrant concentration from sorption measurements. They
all are applicable only for sorption data of the Fickian type, and m a y be
classified into two groups. Methods belonging to one group utilize data
for the initial slope of the reduced sorption curve, while the ones belonging
to the other group resort to rates at which M (t) approaches the equi
librium value Moo. Since little work has yet been done to adapt the
approachtoequilibrium data for concentrationdependent D, the sub
sequent discussion will be confined to methods of the former group only.
The initial slope of the reduced absorption curve is denoted b y I~ and
that of the reduced desorption curve b y Ia. These are generally functions
of the initial concentration cl ~ and the final concentration cl~176 of a parti
cular experiment. In most work undertaken to determine D, measure
ments are done in such a way that cl ~ = 0 for absorption and qoo = 0
for desorption. In these cases, Ia = 1~(cl ~176and Id Ia(Cl~176The methods
Diffusion in PolymerDiluent Systems 9
described below are concerned with initial slope data of these types.
We define two apparent diffusion coefficients, D a and D a, b y
D~ = (~/16) [I~(c1~)] ~, Oa = (~/16) [Ia(cl~ (6)
Theory ECRANK(1956) ] shows that if D is independent of c1both D~ and Dd
are also independent of cl ~176
and Cl~ respectively, and are equal to D.
Thus
D~ = Da = D (case of constant D). (7)
Hence, in this case, the measurement of either absorption or desorption
allows straightforward calculation of D of the given system. For concen
trationdependent D the following approximation was first suggested
[CRANK and PARK (1949)] :
r
Da(cl ~176= .D(cl ~) ~ ( l / q ~) f D ( c l ) d q . (8)
0
Here D is the quantity called the integral diffusion coefficient; this
represents an average of D values over the range of concentration from
zero to cl ~176
According to Eq. (8), the required D (q) is obtained b y graphi
cal differentiation of a plot for Da(q ~176cx~176
vs. Cl~. This method is simple
and quick to use, but it has been shown that it is satisfactory only for
D varying mildly with q. As a better approximation, the arithmetic
mean of D~ and D e for paired absorption and desorption curves was
assumed to equal D [PRAGER and LONG (1951) ; CRANK (1956)]. For such
a pair of sorption curves the initial concentration for desorption is
equal to the final concentration for absorption. Therefore, the new
approximation equation m a y be written
(1/2) [D (Qoo) + Da (q~)] = D (q~). (9)
This indicates that the required D (ca) is determined b y graphical differen
tiation of a plot for (1/2) IDa(ca ~176+ (Da(ca~176 ~176
vs. Cl~ It is not a
serious disadvantage of this method that both Da and D a must be deter
mined experimentally, since, in most work, the measurement of an
absorption curve is followed b y the determination of the corresponding
desorption.
Subsequently, on the basis of very detailed calculations, CRANK
(1955) has demonstrated that D~(cl~176is approximated very accurately
by the equation:
Cl oo
Da(q ~) = (5/3)(cl~162
% f (q)'/' D ( q ) d q , (10)
0
provided that D is an increasing function of % For example, when D
varies with q in a linear or an exponential fashion, Eq. (10) holds to
within an accuracy of one percent over the range of D varying 200fold.
10 H. FUJITA:
He also has shown that for D increasing with c1 the following equation
holds accurately:
Cl~
Da(q ~ = 1.85 (cl~ is5 f (q~  c,)~ (cl) dci. (l 1)
0
Comparison of Eqs. (10) and (11) shows that the initial slopes for ab
sorption and desorption are controlled by different weighted averages
of D. Also, we see that
c these new equations allow
determination of D from
either absorption or desorp
tion experiments only. In
fact, Eq. (11) is the only
1./S/
7 means whereby D can be
evaluated from desorption
experiments alone. The usc
I of Eq. (10) is as simple and
K straightforward as with
2/I
Eqs. (8) and (9). Namely,
,%8 we plot values of (3]5)
(Q~176 Du(cl ~176against Qo~,
differentiate the resulting
/ curve graphically, and then
divide the derivatives b y
(cl~176
'/'. To solve Eq. (11)
~ for D we need a somewhat
F more elaborate technique,
for which details will not be
given here to save space.
These new CRANK equations
may be utilized to investi
y io is 20 gate whether a given system
6 *lOe [g'/cc afpo/ymer§ exhibits Fickian sorptions
Fig. 2. Plots of various diffusion coefficients against penetrant at a given temperature. To
concentration for the system polymcthyl acrylate   benzene
at30~ O DfromDa. 9 D f r o m D a .  thise n d w e m a y determine
from steadystate permeabilities). Taken from unpublished
paperof Kisnl~oxo and E~r paired absorption a n d d e 
sorption curves for a series
of values of cl ~176 in the concentration range concerned and evaluate
D vs. c1 relations from those data b y using Eqs. (10) and (11). If the two
relations derived agree from one another within the required limit of
accuracy, we m a y conclude quite safely that all the measured sorption
processes were Fiekian. This is because Eqs. (10) and (11) should give a
consistent result when and probably only when the sorption is of this type.
Diffusion in PolymerDiluent Systems 11
Several other methods are available for the determination of D from
sorption data. Some of them involve no restriction for the form of D (cl),
as is the case with the methods described above, but the others are
applicable only for D changing with c1 in a linear or an exponential
fashion. For these the reader should consult CP.ANK'S monograph (1956).
Which of these existing methods should be chosen is mainly dependent
on the desired precision of the results to be obtained and partly on the
simplicity of its use. In the author's experience, Eq. (10) appears to be
eminently accurate and
simple, and therefore its 5 I I
general use is strongly rec ~s j55oc.r dsoc
ommended.
2.5 D vs. c 1 relations for cj f ~/. / /~, //soc
polymer diluent systems
/
The application of Eqs.
(10) and (11) is illustrated
in Fig. 2, where the data for
/ H///
D~ and Da taken on the sys
tem polymethyl acrylate
benzene at 30 ~C. and the D
values calculated from them
)
b y means of these equa
tions are plotted loga
rithmically against % It is  2
seen that the change in D
with q amounts to more
than 900fold between the
extremes of the concentra "/00 s 10 /S
tion range indicated. Such a
pronounced dependence on Fig. 3. Plots of D against penetrattt concentration for the
system polymethyl acrylate   ethyl acetate at different
concentration of D is not temperatures [FuJITA, KISHIMOTO and MATSUMOTO {1960)]
exceptional of this particu
lar polymersolvent combination but rather common in many polymer
solvent systems. B y way of example, the data on the system polymethyl
acrylateethyl acetate at various temperatures are given in Fig. 3 [FUJITA,
KISHIMOTOand MATSUMOTO(1960) ]. This family of curves displays several
important features which appear to be characteristic of amorphous poly
mersolvent systems at temperatures above Tu~ where Tu ~ denotes the
glass transition temperature of a given polymer in the dry state. (1) As the
temperature is raised, the value of D at a given c1 increases and its depend
ence on concentration becomes less appreciable. (2) At relatively high
temperatures log D varies approximately linearly with % but the log D
12 H. FOJITA:
vs. cl plots a t lower temperatures show downward curvature in the region
of small values of c1. (3) This curvature becomes more noticeable and the
minimum concentration for the linear log D vs. cl relation shifts toward
the high concentration region, as the temperature approaches T , ~
Owing to this strong curvature there is increasing difficulty at tempera
tures not greatly removed from Tg ~ to extrapolate log D to zero penetrant
concentration. Previously, when measurements were not extended down
to sufficiently low concen
trations, it was often assumcd
t h a t the linear log D vs. c1
  o .  ~. :0 ~ plots obtained at relatively
~% r176
high concentrations could be
extrapolated to zero concen
oq.O 3o."oC

tration for evaluating D o .
Since this assumption now
appears not to hold generally,
2o,oc
except at temperatures far
~s.s
o20oc above Tw~, some formerly
reported values of D 0, espe
o "s ~ cially those of systems near
Tw~, have to be accepted
:/0oC
with reservation.
The general features of
the D vs. Cl relations for
I amorphous polymersolvent
0.s LO i.s
c/x/Oe [ff/cc oP po/y/'#e/'+wu/ee]_
systems slightly above or
below Tw~ are not fully elu
Fig. 4, Plots of D against penetrant concentration for the
system polymethyl acrylate water at different temperatures. cidated. This is mainly due to
Taken from KtsmMoTo, MAEKAWAand FOIITA (1960)
the fact that, as will be ex
plained in the next section,
the sorption processes in glassy polymers are not Fickian and hence the
methods for the determination of D (cl) presented in the previous section
can no longer be applied. In principle, D values of such systems could
be determined from steadystate permeation measurements, but
I4ISltn~tOTO (unpublished) has shown t h a t the analysis of permeation
data on glassy systems is complicated b y factors which are not yet
fully resolved.
For crystalline polymers the available data are yet so limited that we
are not in a position to be able to discuss the general character of their D
vs. c1 relations. However, it is of interest to note that for diffusion of
various organic vapors in polyethylene films ROGERS et al. (1~~ have
recently deduced D (actually, D) vs. c1 plots which were quite similar in
character to those shown in Fig. 2 and 3.
Diffusion in PolymerDiluent Systems 13
Work on polymerwater systems indicates that when the affinity of
water to the polymer is very low, i.e., when water is essentially a non
solvent for the polymer, the mutual diffusion coefficients of the systems
show slight or no concentration dependence and are less temperature
dependent than in polymerorganic solvent systems. As an example we
give data on the system polymethyl acrylatewater in Fig. 4 [KISHIMOTO,
MAEKAWA and FUJITA (1960)]. On comparison of Fig. 3 and 4 one finds
how markedly not only the form of D as a function of ca but also the
magnitude of D itself are affected b y the kind of penetrant substance.
The behavior of D (ca) illustrated in Fig. 4 is probably characteristic of
polymernonsolvent systems. This implies that if water were a solvent
or a good "swelling agent" for a given polymer, such as in the case of
polyvinyl alcohol or of cellophane, the D of the system should have been
appreciably concentrationdependent, as generally found with polymer
organic solvent systems.
III. N o n  F i c k i a n S o r p t i o n
3.1 N o n  F i c k i a n a n o m a l i e s
Sorption processes which did not conform to the Fickian type have
been frequently observed for a variety of polymersolvent systems [for
example, MANDELKERN and LONG (1951); KOKES, LONG and HOARD
(1952); DRECHSEL, HOARD and LONG (1953); LONG and KOKES (1953);
PARK (1952; 1953)]. The most fundamental fact deduced from these
observations is that such processes were encountered only when a given
system was studied at temperatures below T~~ or, more precisely, the
glass transition point, Tg, of a given polymerpenetrant mixture. The
types of reported deviations from Fickian features are so numerous that
here we must be content with summarizing some representatives of them
[see also CRANK and PARK (1951)] :
(a') In the region of small values of (t) V• both absorption and de
sorption curves are not linear;
(b') The absorption curve has an inflection point;
(c') Despite the expectation that D should increase with c1 in the
region of ca concerned, the paired absorption and desorption plots inter
sect; the relatively high initial rate of desorption followed b y a very
slow rate leads to intersection of the two curves;
(d') Paired absorption and desorption curves which have the coinci
dent initial slopes do not coincide over the entire region of (t)'/,;
(e') The initial slopes of absorption curves from different initial
concentrations to a fixed concentration pass through a maximum as the
initial concentration increases;
(f') Absorptionordesorptioncurvesobtained fromvaryingthickness ex
periments cannot be reduced to a single curve when plotted against (t)'h/L.
14 H. FUJITA :
Some of these teatures are illustrated in the actual data shown in
Fig. 5 [quoted in CRANK (1953)], It should be remarked that for a given
system not always do all of these features appear simultaneously.
Sorption curves which exhibit
"* /d
f any of these and other types of devi
ations from Fickian features are usu
f o r ~ ally called "nonFickian" processes
or sometimes' 'anomalous" processes.
In recent years, a number of inter
pretations have been presented for
the origins of nonFickian features,
JO 100 ISO but none of them are yet wholly
(f/me V~ s~d)ll~ satisfactory. In what follows, we
Fig. 5. Typical nonFickian absorption and de describe basic ideas of some represent
sorption of methylene chloride in polystyrene at
2 5 ~ C. Taken from CRANK (1953) ative theories and typical sorption
features predicted from them. In so
doing, we shall not necessarily follow the historical sequence in which
they appeared but rather attempt to group similar ideas in one scheme.
3.2 The timedependent diffusion coefficient
The conditions basic to the Fickian sorption were that (1)D is a
function of ct only and (2) a constant surface concentration is main
tained during sorption. So long as we wishes to retain the Fick diffusion
equation as the basis of the discussion, any attempt for the theoretical
interpretation of nonFickian characteristics must abandon either or
both of these conditions. In this section we give a brief account of a
theory which involves an alternation of condition (1). It is due originally
to CP~ANKand PARK (1951).
T h e y considered that condition (1) implies that any rearrangement of
polymer molecules accompanying diffusion takes place very rapidly com
pared with the rate of diffusion. If this is the case, it follows that mutual
diffusion coefficients of rubbery polymeric systems would be purely
concentrationdependent, since, as is wellknown from the study of
viscoelastic behavior, the segmental motion of polymer molecules in
such a system is very rapid. On the other hand, this motion in glassy
polymer systems should be a relatively slow process, leading to the
expectation that the D of such a system must depend not only on q but
also on other factors. These considerations are consistent with the obser
vation that above Tg sorption processes are generally Fickian and below
T~ the processes are always nonFickian. CRANK and PARK supposed
that under the condition that polymer segments rearrange themselves
slowly, the value of D in a volume element of the system does not attain
Diffusion in PolymerDiluent Systems 15
its equilibrium value D,(q) instantaneously when the penetrant concen
tration in the volume element is brought to a value ca but will get nearer
to D,(q) as the concentration remains longer at this particular value.
As the simplest expression describing this situation they assumed a
kinetic equation of the form:
aD/Ot = a (q) [De(ca)  D]. (12)
Here at (ca) is a ratedetermining factor for the approach of D to De(q)
and m a y be a function of ca, as indicated. One m a y expect that x becomes
larger as the polymer segments more rapidly change configurations. Inte
gration of Eq. (12), with ca fixed, gives
D = D , ( q )  [D,(q)  D,(q)] exp [ ~ (q)t], (la)
where Di(q) is the value of D obtained at the instant when the concen
tration in the volume element has become c1, and m a y be a function of c1.
Equation (13) indicates that under the conditions considered D is an
explicit function of both ca and t, and therefore m a y be expressed b y the
symbol D(q,t). Then it follows from Eq. (13) that D,(q)= D(q, oo).
Equation (12) is concerned with the volume element in which ca remains
constant during the approach of D to De. For a more general case in
which ca is changing with time we have
OD]Ot = [dD~(q)/dcx] (Oq/Ot) + re(c1)[De(ca)  D ] . (14)
This kinetic equation was originally proposed b y CRANK (1953), who
referred to the first term as the instantaneous part and the second term
as the slow part of the time dependence of D. He considered that these
parts are associated with the instantaneous and retarded deformations
of a polymer molecule occurring when it is subject to an external force.
In accordance with CRANK and PARK (1951), the diffusion process govern
ed b y a diffusion coefficient depending explicitly on time is generally
termed the timedeibendent diffusion or the historydependent diffusion.
To obtain quantitative information about sorption processes con
trolled b y timedependent diffusion CRANK (1953) solved numerically
Eq. (1), coupled with Eqs. (4), (5) and (14), for some assumed forms of
Di(q), D,(q) and ~ (q). His results agreed reasonably with many typical
nonFiekian features known at that time [PARK (1953)]. However, when
a new type of nonFickian behavior, now generally called the "twostage"
type, was discovered in 1953 b y LONG and his coworkers, it soon became
evident that the concept of timedependent diffusion was too simple to
explain every nonFickian behavior. This situation remains unaltered at
present, and so we shall not go further into this subject.
16 H. FUJITA :
3.3 Variable surface concentration
The absorption of vapor b y the surface layer of a polymer film will
necessitate some rearrangement of the polymer molecules, and it is
reasonable to consider that the more active the segmental motion of
polymer chains becomes, the more rapidly the surface layer takes up
penetrant to the equilibrium concentration. This implies that the surface
concentration gradually approaches an equilibrium value at a finite rate
which m a y depend upon the rate of relaxation motions of the polymer
molecules. CRANK and PARK (1951) expressed this situation b y the equa
tion:
clS = qo + (clOO_ clO)(1  eat), (15)
where qs is the surface concentration and fl is a rate parameter. It should
be noted that when fl is infinitely large, this equation reduces to the
condition of constant surface concentration, Eq. (5).
With Eq. (15) as the boundary condition, CRANK and PARK calculated
absorption curves from Eq. (1) for a system in which D is independent of
concentration. The curves obtained represented well some typical non
Fickian features, such as the sigmoid behavior, known at that time.
However, it is shown that this simple equation for the timedependent
surface concentration fails to give the "twostage" behavior which will
be discussed in the next section.
Recently, LONG and RICHMAN (1960) have shown that both sigmoid
and twostage sorption curves can be derived from Eq. (1) (with D de
pendent on c1 only) if, in place of Eq. (15), one assumes for cx*a somewhat
more general equation of the form:
c 1 ' = c 1 ~ + (cl ~  q 0 (1  eS 0 , (16) 1
where cl* is a new parameter and is assumed for absorption to be greater
than the initial concentration cx~ The time dependence of cxs represented
b y this equation consists of an initial sudden increase from q~ to cl*
followed b y a gradual approach to cl ~. For sorption of methyl iodide
into cellulose acetate films at 40 ~ C. LONG and RICHMAN (1960) actually
observed a surface concentration which essentially followed this type of
behavior. In a companion article [RIcH~mN and LONG (1960)], they also
have given definitive evidence for the validity of the assumption that in
rubbery polymers the equilibrium concentration is attained instanta
neously at the film surface. These experimental findings lead to the
important conclusion that many of the nonFickian anomalies charac
teristic of glassy polymer systems m a y be ascribed, if not all, to the slow
establishment of the equilibrium concentration at the surface of the sample.
1 T h e y also discussed t h e b o u n d a r y e q u a t i o n of t h e form, Cl~ = Cl~  ~ /~ t,
Diffusion in PolymerDiluent Systems 17
It is quite plausible that this phenomenon is a reflection of the slow re
laxation of polymer molecules in glassy systems. Since in a polymer system
there are many relaxation mechanisms having different relaxation times,
the general expression for el* corrcspondingly should be a sum of time
dependent terms having as many rate parameters as there are different
relaxation times. One m a y consider that Eq. (16) is a special case of such
a general expression when the entire spectrum of relaxation mechanisms
is a p p r o ~ m a t e d b y two mechanisms, one having zero relaxation time
(which allows of a sudden increase of c1' from Cl~ to ci') and the other
having a finite relaxation time (which causes a gradual approach of c1'
to Qoo). However, it is hazardous to interpret the parameter fl as being a
direct measure of the relaxation time of the corresponding mechanism.
3.4 Twostage behavior and successive differential sorptions
The discovery of the twostage behavior is ascribed to LOl~G and his
associates [LONG, B AGLEY and ~VILKENS (1953); BAGLEY and LONG
(1955)]. When they performed successive absorption experiments with
the systems cellulose acetate
acetone and cellulose acetate
methanol at temperatures be
low the glass transition point "Cans/i
b*" .. . ' ~
of this polymer, the absorption
curves starting with initial
concentrations above a certain
~1 ' ~ f ',
value followed processes of a
type which had not been no % i
ticed previously. Figure 6
shows this new type of absorp
Fig. 6. Schematic diagram of an absorption curve of the
tion curve schematically. B y twostage type
the successive absorption ex
periment is meant a serial experiment in which a dry polymer film
is first exposed to vapor of pressure Pl until sorption equilibrium is
reached and then exposed to the vapor of a higher pressure Ps until
the new equilibrium is attained, etc. This process m a y be continued
for as m a n y steps as are of interest. Thus in this type of experiment
the initial concentration of the nth absorption step is equal to the final
concentration of the n1th step. In most experiments of this type, the
concentration increments of successive steps are chosen sufficiently small
so that the mean penetrant concentration in the film remains practically
constant during each absorption step. In the present paper, we refer to an
experiment of this type as the "differential" type. Contrary to this, an
experiment in which the difference between the initial and final concen
trations is relatively large will be referred to as the "integral" type.
Fortschr. Hochpolym.Forsch., Bd. 3 2
18 H. FUJITA:
This definition of the integral sorption differs from the one proposed by
KoxEs et al. (1952), who termed a sorption experiment with zero initial
or final concentration the integral type. It should be noted that, in reality,
no definite borderline can be drawn between the integral and differential
experiments defined as above.
The curve shown in Fig. 6 consists of two portions of distinctly diffe
rent nature, i.e., a portion of the Fiekian type from 0 to B via A and a
/a'
PPessum l'nterml[T~T~ HT]
131/ ~" lqO
f 130
IZV
J" 13~t
~" 130
f__. /IS m, IZ~/
I10 i 112
100 ~ 110
.90 ~ /00
S" f ~ f 
i f
f f
j f
"T
gO
dO '~
2S
,,
"
#0
gg
SO
.....
~
f 0 ~ 25
J
/0 20 30 ~0
f~e [,>7r~i~])~2
Fig. 7. Successive differential absorptions of mcthyI acetate in polyn~ethyI rnethaerylate at 30~ C. Taken
from KISHIMOTO,FLrJITA,ODANI, KURATAand TAMURA(1960)
portion of sigmoid shape from B to a final equilibrium D via C. This
twostep structure gives the reason why this type of sorption curve is
termed the "twostage" type. The initial Fickian portion is referred to as
the first (or initial) stage and the subsequent sigmoid portion as the second
stage. Often we observe that the first stage is terminated with a very flat
curve before the second stage appears. In such a case we m a y stop con
tinuing the measurement at this flat end of the first stage, misunderstand
ing that the sorption has already reached equilibrium. However, this flat
portion never represents the true equilibrium to be reached under given
external conditions, and thus sometimes it is referred to as the "quasi
equilibrium" state.
The twostage sorption behavior is not a unique characteristic of the
cellulosic systems investigated b y LONG et al. Subsequent studies, b y
D i f f u s i o n lit P o l y m e r  D i l u e n t Systems 19
NEWNS (1956) and many others, have observed it for a number of polymer
penetrant systems, including both amorphous and crystalline materials.
At present, it is taken as one of the most general sorption features of
glassy polymer systems. By way of example, we reproduce here successive
differential absorption data obtained by us [KISHIMOTO,:FUJITA, ODANI,
KURATA and TAMURA (1960)] in Fig. 7 and 8. These data manifest
$0 P:essure /'n/e/'val~t~H~
f I~9.9 ~ ISG.3
f IqLO ~ Itr163
./
e ~
130.0 ~
ll~.g ~
I~1.0
130.0
JO
t~ 79.0 ~ 9s
/ 18.: . :~.1
8S "~ 18.1
0 ~ 8.s
1 1
0 10 20 30 ~ 50
Fig. 8. Successive differential absorptions of acetone in cellulose nitrate at 25~ C. Taken from ICasmMoIo,
Ft:jIT^, OvAnl, KURArAand TAMVRA{1960)
important differences of successive absorption curves between amorphous
and crystalline polymers.
One can see that each differential absorption curve changes its shape
systematically as the initial concentration of each step becomes higher.
The general trend of this change is similar in the two systems. For the
first several steps covering low penetrant concentrations the curves all
have sigmoid shape and the point of inflection on each curve shifts to the
region of small values of time as the initial concentration increases. The
sigmoid character disappears at a certain initial concentration, for which
the differential sorption shows a behavior resembling Fickian sorption.
2*
20 H. FUJITA:
However, the limited linear initial region and the very slow approach to
equilibrium indicate that this curve is not truly Fickian. KISHIMOTO
et al. have referred to this type of curve as "pseudoFickian". Next to
this curve there appears a twostage curve, and as the initial concentration
is increased, the second stage portion shifts to the short time region,
gains a higher rate, and gives a greater contribution to the total concen
tration increment of each step. When a certain concentration is reached,
the second stage portion predominates the entire process and the two=
stage character disappears. In the polymethyl methacrylate system the
differential curve at this concentration looks again pseudoFickian,
whereas in the cellulose nitrate system it is of the sigmoid type. In the
former system, further rise of the initial concentration changes this
pseudoFickian curve to a curve having a shape of the Fickian type, and
above a concentration of 0.12 g/g all curves look Fickian. It is of interest
to note that this critical concentration for the transition from non
Fickian to Fickian behavior corresponds to the methyl acetate content
at which this polymersolvent system undergoes glass transition at the
temperature of the experiment. Similar relations had been early recogn
ized by LONG et al. for other amorphous polymersolvent systems [KOKES,
LONG and HOARD (1952) ; LONG and KOKES (1953)]. However, it should
be remarked that, according to recent work, absorption is not always
Fickian in the region not greatly removed from the critical concentration
even if it has a shape conforming to the Fickian type. This is equivalent
to saying that at temperatures not far above T a the sorption is still
nonFickian. The cellulose nitrate data do not show such a critical
concentration, but the increase in initial concentration merely gives rise
to a vertical shift of a sigmoid sorption curve with its shape practically
unchanged.
The character of the polymethyl methacrylate data is essentially
similar to that found for systems atactie polystyrenebenzene at 25 ~
35 ~ and 50 ~ C. [KISHIMOTO,FUJITA,ODANI,KURATAand TAMURA (1960);
ODANI, I~IDA, KURATA and TAMURA (1961)] and also atactic polysty
renemethyl ethyl ketone at 25 ~ C. [ODANI, HAYASHI and TAMURA
(1961)1, and appears to be fairly general for amorphous polymersolvent
systems in the glassy state. On the other hand, the cellulose nitrate data
shown in Fig. 8 appear to manifest features characteristic of crystalline
polymersolvent systems. For example, the earlier data of NEWl~S (1956)
on the system regenerated cellulosewater (in this case, water is not the
solvent but merely a swellingagent) and recent studies for several
crystalline polymers all show essentially similar characters Esee KISHI
MOTO, I~U]'ITA, ODANI, KURATA and TAMURA (1960)]. To arrive at a more
definite conclusion, however, more extensive experimental data are
needed.
Diffusion in PolymerDiluent Systems
By
HIROSHI ~'UJITA1
Physical Chemistry Laboratory, Department of Fisheries,
IZyoto University, Maizuru, Japan
With 18 Figures
Table o5 Contents
Page
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
I. F u n d a m e n t a l s . . . . . . . . . . . . . . . . . . . . . . . . . . 2
II. F i c k i a n Sorption . . . . . . . . . . . . . . . . . . . . . . . . . 4
IIl. N o n  F i c k i a n Sorption . . . . . . . . . . . . . . . . . . . . . . . 13
IV. P e r m e a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
V. Interpretation of Diffusion Coefficient D a t a . . . . . . . . . . . . . . 31
l~eferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Introduction
The diffusion of small molecules in polymeric solids has been a subject
in which relatively little interest has been shown b y the polymer chemist,
in contrast to its counterpart, i.e., the diffusion of macromolecules in
dilute solutions. However, during the past ten years there has been a
great accumulation of important data on this subject, both experimental
and theoretical, and it has become apparent that in many cases diffusion
in polymers exhibits features which cannot be expected from classical
theories and that such departures are related t o the molecular structure
characteristic of polymeric solids and gels. Also there have been a
number of important contributions to the procedures b y which diffusion
coefficients of given systems can be determined accurately from experi
ment. I t is impossible, and apparently beyond the author's ability, to
treat all these recent investigations in the limited space allowed. So,
in this article, the author wishes to discuss some selected topics with
which he has a relatively greater acquaintance but which he feels are
of fundamental importance for understanding the current situation in
this field of polymer research. Thus the present paper is a kind of personal
note, rather than a balanced review of diverse aspects of recent diffusion
studies.
x Present address: Department of Polymer Science, Osak~ University, Nakano
shima, Osaka, Japan.
Fortschr. Hochpolym.Forsch., Bd. 3 1
2 H. ~'uj IT2~:
The systems discussed below are restricted to ones in which tile
penetrant 1 is a solvent or a "plasticizer" for a given polymer and in
which its concentration is sufficiently small in comparison with the con
centration of polymer. Such systems m a y be referred to as extremely
concentrated solutions of polymer. With this restriction we shall not
treat here diffusion processes of gaseous substances, such as oxygen and
carbon dioxide.
I. F u n d a m e n t a l s
1.1 Diffusion coefficients
Diffusion is defined as the process in which components are transported
from one part of a mixture to another as a result of random molecular
motion. Phenomenologically, it can be most rigorously treated b y the
proper application of the thermodynamics of irreversible processes
[DE GROOT (1951); HOOYMAN (1955); GOSTING (1956)].
We designate the penetrant as component 1 and the polymer as
component 2. If the mixture is incompressible and no volume change
occurs on mixing of the two components and if the diffusion takes place
only in one direction x, it can be shown that distribution of component i
(i = 1,2) in the system during isothermal diffusion is governed b y a
differential equation of the form:
a~,lat = (a/ax) [D (ac,/a~)], (1)
where c~ is the concentration of component i expressed in grams per cc.
of polymerpenetrant mixture, t is time, and D is a quantity called the
mutual diffusion coefficient of the system [HARTLEY and CRANK (1949)1.
The point is t h a t in Eqs. (1) for both components 1 and 2 there appears
only a single "diffusion coefficient". Equation (1) is usually called the
Fick diffusion equation or the Fick second law of diffusion. Although this
is generally taken as the basis for analyzing d a t a of diffusion in one
dimension, it m u s t be recognized t h a t it strictly applies only for two
component systems in which the partial specific volumes of the compo
nents are independent of pressure (thus the mixture is incompressible)
and also of composition (thus no volume change occurs on mixing).
To a good approximation these conditions are fulfilled when the amount
of one component relative to the other is very small, as in the case of
dilute solutions or in extremely concentrated solutions which will be
treated below. In passing, we wish to point out that Eq. (1) is no longer
applicable for systems containing more than two components [GosTING
(1956)].
x The term "penetrant" will be used here to designate lowmolecularweight
substances which diffuse in given polymeric solids.
Diffusion in PolymerDiluent Systems 3
According to the thermodynamics of irreversible processes, the
quantity D is equal to (D1) r or (D2)v. Here (D;) r is defined b y the relation
(J~)r =(Di)v(3cdOx), where (Jt)v is the flow (for flux) of component
i relative to a plane (perpendicular to the direction of diffusion) moving
with the local center of volume. The (Di)v is termed the volumefixed
diffusion coefficient of component i. In general, the values of (D~)v and
(D~)v are different from one another, but when the partial specific
volumes of the two components are constant both have the same value.
The D in Eq. (1) denotes this coincident vMue of (D1)v and (D~)r. Several
other kinds of diffusion coefficient m a y be introduced b y defining the
flow of each component in terms of other frames of reference. For
example, the massfixed diffusion coefficient, (Di)~1, of component i is
defined b y the relation (Ji)M =  (D~)M (OcdOx), where (J~)M is the flow
of component i relative to the plane moving with the local center of mass.
One m a y choose as the reference frame a plane which moves with the
local velocity of component 2, the polymer component in the present
case. Then the polymerfixed diffusion coefficients m a y be defined for
penetrant and polymer components. The value of this coefficient for the
polymer component is identically zero, but that for the penetrant
component, denoted here by (D1)P, is nonzero in general. Any two of
these various diffusion coefficients are related from one another by a
simple mathematical equation [WENDTand GOSTING (1960); FUJITA
(1961)]. Thus D and (D~)p are related by the equation:
(D1)F = D/(1   ~1 cl), (2)
where vl is the partial specific volume of component 1, i.e., the penetrant
component. Between (Di)M and D there is a relation of the form:
(Ol) M = O [1   (q/e)]/(1   vx cl), (3)
where ~ is the density of the penetrantpolymer mixture. If D is eliminated
from these equations, an expression which relates (D1)e to (D1)M may be
derived. I t is a simple matter to show from these relations that (D~)ao,
(D1)M, and D all converge to the same value at the limit of zero penetrant
concentration. This limiting value shall be denoted b y a symbol D 0. It is
important to observe that Eqs. (2) and (3) are valid for onedimensional
diffusion. It appears that no corresponding equation is as yet known for
diffusion in higher dimensions.
1.2 S o r p t i o n a n d p e r m e a t i o n m e t h o d s
One of the central problems in the study of diffusion is to evaluate D
for a given system as a function of such parameters as penetrant con
centration and temperature. For polymerpenetrant systems with which
we are concerned in this article two experimental methods are typical
for this purpose. They are the sorption method and the permeation method.
1"
4 H. FUJITA:
In a sorption experiment, a film of a given polymer is exposed to vapor
of a given penetrant substance at a given pressure and the gain or loss in
weight of the film is measured as a function of time. In the present paper,
the term "sorption" will be used loosely for both absorption and desorp
tion, unless otherwise specified. In the usual absorption experiment the
film is initially free of penetrant, b u t in some cases the process of absorp
tion is studied with a film initially equilibrated at a nonzero v a p o r
pressure. I{OKES, LONG and HOARD (1952) have referred to this latter
type of absorption as the "interval" type. This term, however, will not
be used in the present paper. Most desorption experiments measure
processes from nonzero initial concentrations to the zero final concen
tration, and hence are of the "integral" type in the sense of KOKES et al.
(1952).
In a permeation experiment, the amount of penetrant v a p o r flowed
through a film of a given polymer is measured as a function of timeunder
the condition t h a t one surface of the film is allowed to get in contact with
penetrant v a p o r at a constant pressure and the other surface is ex
posed to vacuum.
II. F i c k i a n S o r p t i o n
2.1 Definitions
According to the thermodynamics of irreversible processes, the mutual
diffusion coefficient D m a y be a function of penetrant concentration c1,
position x, and time t. In the present chapter we shall discuss sorption
behavior of systems in which D varies with c1 only, and shall use the
notation D (c,) to indicate this condition. I t is assumed t h a t the sample
film is so thin t h a t diffusion takes place effectively in the direction of its
thickness. At the beginning of an absorption or a desorption experiment
the film is conditioned so that cl is uniform everywhere in it. This initial
concentration is denoted b y cl ~ Then we have
c~ = q o ( _ L/2 < x < L/2, t = o ) , (4)
where L is the thickness of the film. The origin of x has been taken on the
central surface of the film. The boundary condition for c1 in general use
for the mathematical study of sorption processes on polymers is t h a t the
penetrant concentrations at both surfaces of the film attain a certain
value cx~176
instantaneously when the film is exposed to v a p o r and t h a t
this value is maintained during the course of sorption so long as the
pressure, fi, of the ambient vapor remains constant [CRANK (1956)].
This boundary condition is termed the condition of constant surface
concentration, and is represented mathematically b y
(x = L/2
c 1 = cl ~176 a n d   L / 2 , t > 0), (5)
Diffusion in PolymerDiluent Systems 5
provided the film does not swell (or shrink) during the course of absorp
tion (or desorption). In reality, this assumption does not hold, and there
fore Eq. (5) is not applicable. However, inclusion of the dimensional
change of the film during sorption (which means to treat L in Eq. (5)
as a function of time) makes solution of Eq. (1) a formidably difficult
problem. Probably, for systems in which total amounts of absorbed or
desorbed penetrant are small compared with the total mass of the polymer
this effect is not too important, although it is not altogether negligible.
For these reasons most of the current theories on sorption processes in
polymerdiluent systems of the type considered here neglect the
variation of L with time.
Very thorough investigations of Eq. (1) subject to conditions (4) and
(5) have been made b y CRANK and others for various assumed forms
of D (cl), of which an excellent s u m m a r y has been given b y CRANK (1956).
The information we need here is not the detailed mathematical expressions
for such solutions of Eq. (1) but the characteristic features of sorption
processes predicted from this set of equations. Customarily, the sorption
processes in which D is a function of c1 only and the initial and boundary
conditions are given b y Eqs. (4) and (5) are referred to as of the "Fickian"
type. Moreover, it is often said t h a t such processes are controlled b y the
Fickian diffusion mechanism 1.
In section 2.3 we will summarize some representative features of this
type of sorption process.
2.2 Representation of sorption data
D a t a obtained from an absorption (or a desorption) experiment are
the amounts of a given penetrant substance absorbed in (desorbed from)
a given polymer film as a function of time t. Usually, the amounts per
unit volume of dry polymer are computed 2, denoted b y M (t), and plotted
against (t) II,.
The resulting curve is termed the absorption (or desorption) curve,
or more generally the sorption curve. However, in order to interpret
correctly experimental results in terms of Eq. (1) it is necessary to plot
M' (1), the weight of sorbed penetrant per unit Volume of the swelling or
deswelling film, against (t) 1/'. This requirement arises from the fact that
1 Since Fick's first and second laws of diffusion are valid independent of whether
2) is a function of cl only or not and also of the form of initial and boundary con
ditions of a particular experiment, it is quite inadequate to specify this particular
type of sorption as Fickian. The term "'Fickian" should be applied more generally
to all mass transport phenomena which are governed by Eq. (1), i. e., the Fick
diffusion equation.
2 Often the weight of sorbed penetrant per unit weight of dry polymer is
employed. This value is different from M (t) by a constant factor equal to the
density of the dry polymer.
6 H. FUJITA:
the concentration q in Eq. (1) is expressed in terms of the weight per
unit volume of polymerpenetrant mixture. Most of the existing reports,
however, adopt the approximation M ( t ) = M'(t) to represent the ex
perimental data obtained. For systems in which penetrant concentrations
are sufficiently dilute, this approximation probably does not introduce
significant errors into the evaluation and interpretation of the experiment.
Though not verified rigorously, the inaccuracy introduced by this
approximation would be of the same order as that caused by neglecting
the change in L with time during sorption (see Eq. (5)).
Experiments show that when no change in ambient vapor pressure p
occurs during the sorption process, M (t) approaches a limiting value as
time increases. When this limiting value is reached, the film absorbs or
desorbs no more penetrant and is at thermodynamic equilibrium with the
ambient vapor. This is the state called sorption equilibrium. The value
of M (t) for this state is denoted by Moo. Often the ratio M (t)]Moo is
plotted against (t)',l,/L and the resulting curve is called the reduced sorp
tion curve. This form of representation of data is convenient for theo
retical analyses.
2.3 Features of the F i c k i a n sorption
Basic features of sorption processes of the Fickian type have been
clarified by CRANK and coworkers through extensive mathematical
studies of Eq. (1). The following gives a summary, of the features of
particular importance.
(a) Both absorption and desorption curves are linear in the region
of small values of the abscissa. For absorption the linear region is obtained
over 60% or more of Moo. When D increases markedly with c1 the ab
sorption is linear almost up to the equilibrium;
(b) Above the linear portions both absorption and desorption curves
are always concave against the abscissa axis;
(c) The shape of absorption curve is not very sensitive to the depend
ence on concentration of D. It is often well approximated by the ab
sorption curve for a constant D, even when D varies appreciably with c1.
On the other hand, the desorption curve is rather markedly affected by
the D vs. c1 relationship;
(d) In both absorption and desorption the concentration distributions
in the film are greatly influenced by the functional form of D (q) ;
(e) When the initial and final concentrations (cl~ and qoo) are fixed,
the reduced absorption curves for films of different thickness all coincide
with each other, yielding a single curve. This applies for the corresponding
family of reduced desorption curves;
(f) The single absorption curve so obtained is always above the cor
responding single desorption curve when D is an increasing function
Diffusion in PolymerDiluenf:Systems 7
of % Both coincide over the entire range of the abscissa when and only
when D is constant. The difference between the two curves becomes more
appreciable as D increases more sharply with cl in the range from ci ~ to
Cl~176If D (q) passes through a m a x i m u m at a certain value of cI between
the given cl ~ and cl ~176
the two curves m a y intersect at some position;
(g) For absorptions from a fixed initial concentration to various
final concentrations the initial slopes of the reduced absorption curves
are larger as the final concentrations are higher, if D increases mono
tonically with c~;
(h) The same relation as (g) applies for reduced desorption curves
obtained from the experiments in which the final concentration is fixed
and the initial conditions are different.
/.0 f ._..o ~
ebs.~S/.,I
.r
~20
j ub~.O~/VSGni~H~
des./qSG~
0m ~ g
I f
S 10 IS 20
Fig. 1. Rexluced absorption and desorption curves (of the Fickian type) for the system polyisobutylenepropane
at 35 ~ C. Ta ke n from PRAG~R and LONG (1951)
Sorption curves consistent with these criteria, especially with (a),
(b), (e) and (f), have been observed for a number of polymerorganic
diluent systems, when the measurements were made at temperatures
well above the glass transition temperatures of the respective systems.
B y way of example, paired absorption and desorption curves obtained
for the system polyisobutylenepropane at a5 ~ C. are shown in Fig. 1
[PI~AGER and LONG (1951)]. Here b y paired absorption and desorption
curves is m e a n t a couple of absorption and desorption curves which
cover the same range of penetrant concentration. In actual cases,
especially when experimental data are to be determined for a variety of
external conditions, it would be almost impractical to investigate, for
given conditions, whether the individual sorption curves fulfil all of
these criteria for the Fickian sorption. Therefore, it is a usual practice
to regard a given sorption curve as of the Fickian type when it has an
overall shape conforming to features (a) and (b). In the case when paired
absorption and desorption curves are available and, furthermore, there
is reason to believe t h a t D of the system increases with q, reference to
feature (f) m a y be of use to check the conclusion. For a more definite
conclusion i t is recommended to perform experiments with films of
8 H. FUJITA:
different thicknesses and to examine whether the data give a single curve
when plotted in the reduced form.
Probably one of the most significant findings is that with organic
penetrants this type of sorption was observed only when a given system
was initially (in the case of absorption) and finally (in the case of desorp
tion) in the rubbery state. This suggests that the two conditions basic to
the Fickian behavior m a y be related to the molecular features which
distinguish a nonglassy polymer from a glassy polymer. However, it
appears that the above rule does not always apply for polymerwater
systems (perhaps, more generally, polymernonsolvent systems). For
example, KISHIMOTO, MAEKAWA and ~'UJITA (1960) have reported that
both absorption and desorption of water in polyvinyl acetate were
Fickian down to temperatures somewhat below the glass transition point
of the polymer.
Finally, one m a y remark an interesting contribution due to KISHI
MOTO and MATSUMOTO (unpublished), who showed with polyvinyl
acetate and polymethyl acrylate that criteria (a) and (b) are not sufficient
to conclude a given sorption curve as of the Fickian type when the
measurement is made at temperatures slightly above the glass transition
of the polymer: they found that the sorption curves obtained at such a
temperature with films of different thicknesses were not reduced to a
single curve, even though each of them had a shape expected for Fickian
sorption. This experimental finding indicates that more deliberation
than is generally conceived is necessary to conclude a given sorption
curve as Fickian when we are concerned with the region near the glass
transition.
2.4 M e t h o d s for the evaluation of D as a function of c1
Various methods have been proposed for the evaluation of D as a
function of penetrant concentration from sorption measurements. They
all are applicable only for sorption data of the Fickian type, and m a y be
classified into two groups. Methods belonging to one group utilize data
for the initial slope of the reduced sorption curve, while the ones belonging
to the other group resort to rates at which M (t) approaches the equi
librium value Moo. Since little work has yet been done to adapt the
approachtoequilibrium data for concentrationdependent D, the sub
sequent discussion will be confined to methods of the former group only.
The initial slope of the reduced absorption curve is denoted b y I~ and
that of the reduced desorption curve b y Ia. These are generally functions
of the initial concentration cl ~ and the final concentration cl~176 of a parti
cular experiment. In most work undertaken to determine D, measure
ments are done in such a way that cl ~ = 0 for absorption and qoo = 0
for desorption. In these cases, Ia = 1~(cl ~176and Id Ia(Cl~176The methods
Diffusion in PolymerDiluent Systems 9
described below are concerned with initial slope data of these types.
We define two apparent diffusion coefficients, D a and D a, b y
D~ = (~/16) [I~(c1~)] ~, Oa = (~/16) [Ia(cl~ (6)
Theory ECRANK(1956) ] shows that if D is independent of c1both D~ and Dd
are also independent of cl ~176
and Cl~ respectively, and are equal to D.
Thus
D~ = Da = D (case of constant D). (7)
Hence, in this case, the measurement of either absorption or desorption
allows straightforward calculation of D of the given system. For concen
trationdependent D the following approximation was first suggested
[CRANK and PARK (1949)] :
r
Da(cl ~176= .D(cl ~) ~ ( l / q ~) f D ( c l ) d q . (8)
0
Here D is the quantity called the integral diffusion coefficient; this
represents an average of D values over the range of concentration from
zero to cl ~176
According to Eq. (8), the required D (q) is obtained b y graphi
cal differentiation of a plot for Da(q ~176cx~176
vs. Cl~. This method is simple
and quick to use, but it has been shown that it is satisfactory only for
D varying mildly with q. As a better approximation, the arithmetic
mean of D~ and D e for paired absorption and desorption curves was
assumed to equal D [PRAGER and LONG (1951) ; CRANK (1956)]. For such
a pair of sorption curves the initial concentration for desorption is
equal to the final concentration for absorption. Therefore, the new
approximation equation m a y be written
(1/2) [D (Qoo) + Da (q~)] = D (q~). (9)
This indicates that the required D (ca) is determined b y graphical differen
tiation of a plot for (1/2) IDa(ca ~176+ (Da(ca~176 ~176
vs. Cl~ It is not a
serious disadvantage of this method that both Da and D a must be deter
mined experimentally, since, in most work, the measurement of an
absorption curve is followed b y the determination of the corresponding
desorption.
Subsequently, on the basis of very detailed calculations, CRANK
(1955) has demonstrated that D~(cl~176is approximated very accurately
by the equation:
Cl oo
Da(q ~) = (5/3)(cl~162
% f (q)'/' D ( q ) d q , (10)
0
provided that D is an increasing function of % For example, when D
varies with q in a linear or an exponential fashion, Eq. (10) holds to
within an accuracy of one percent over the range of D varying 200fold.
10 H. FUJITA:
He also has shown that for D increasing with c1 the following equation
holds accurately:
Cl~
Da(q ~ = 1.85 (cl~ is5 f (q~  c,)~ (cl) dci. (l 1)
0
Comparison of Eqs. (10) and (11) shows that the initial slopes for ab
sorption and desorption are controlled by different weighted averages
of D. Also, we see that
c these new equations allow
determination of D from
either absorption or desorp
tion experiments only. In
fact, Eq. (11) is the only
1./S/
7 means whereby D can be
evaluated from desorption
experiments alone. The usc
I of Eq. (10) is as simple and
K straightforward as with
2/I
Eqs. (8) and (9). Namely,
,%8 we plot values of (3]5)
(Q~176 Du(cl ~176against Qo~,
differentiate the resulting
/ curve graphically, and then
divide the derivatives b y
(cl~176
'/'. To solve Eq. (11)
~ for D we need a somewhat
F more elaborate technique,
for which details will not be
given here to save space.
These new CRANK equations
may be utilized to investi
y io is 20 gate whether a given system
6 *lOe [g'/cc afpo/ymer§ exhibits Fickian sorptions
Fig. 2. Plots of various diffusion coefficients against penetrant at a given temperature. To
concentration for the system polymcthyl acrylate   benzene
at30~ O DfromDa. 9 D f r o m D a .  thise n d w e m a y determine
from steadystate permeabilities). Taken from unpublished
paperof Kisnl~oxo and E~r paired absorption a n d d e 
sorption curves for a series
of values of cl ~176 in the concentration range concerned and evaluate
D vs. c1 relations from those data b y using Eqs. (10) and (11). If the two
relations derived agree from one another within the required limit of
accuracy, we m a y conclude quite safely that all the measured sorption
processes were Fiekian. This is because Eqs. (10) and (11) should give a
consistent result when and probably only when the sorption is of this type.
Diffusion in PolymerDiluent Systems 11
Several other methods are available for the determination of D from
sorption data. Some of them involve no restriction for the form of D (cl),
as is the case with the methods described above, but the others are
applicable only for D changing with c1 in a linear or an exponential
fashion. For these the reader should consult CP.ANK'S monograph (1956).
Which of these existing methods should be chosen is mainly dependent
on the desired precision of the results to be obtained and partly on the
simplicity of its use. In the author's experience, Eq. (10) appears to be
eminently accurate and
simple, and therefore its 5 I I
general use is strongly rec ~s j55oc.r dsoc
ommended.
2.5 D vs. c 1 relations for cj f ~/. / /~, //soc
polymer diluent systems
/
The application of Eqs.
(10) and (11) is illustrated
in Fig. 2, where the data for
/ H///
D~ and Da taken on the sys
tem polymethyl acrylate
benzene at 30 ~C. and the D
values calculated from them
)
b y means of these equa
tions are plotted loga
rithmically against % It is  2
seen that the change in D
with q amounts to more
than 900fold between the
extremes of the concentra "/00 s 10 /S
tion range indicated. Such a
pronounced dependence on Fig. 3. Plots of D against penetrattt concentration for the
system polymethyl acrylate   ethyl acetate at different
concentration of D is not temperatures [FuJITA, KISHIMOTO and MATSUMOTO {1960)]
exceptional of this particu
lar polymersolvent combination but rather common in many polymer
solvent systems. B y way of example, the data on the system polymethyl
acrylateethyl acetate at various temperatures are given in Fig. 3 [FUJITA,
KISHIMOTOand MATSUMOTO(1960) ]. This family of curves displays several
important features which appear to be characteristic of amorphous poly
mersolvent systems at temperatures above Tu~ where Tu ~ denotes the
glass transition temperature of a given polymer in the dry state. (1) As the
temperature is raised, the value of D at a given c1 increases and its depend
ence on concentration becomes less appreciable. (2) At relatively high
temperatures log D varies approximately linearly with % but the log D
12 H. FOJITA:
vs. cl plots a t lower temperatures show downward curvature in the region
of small values of c1. (3) This curvature becomes more noticeable and the
minimum concentration for the linear log D vs. cl relation shifts toward
the high concentration region, as the temperature approaches T , ~
Owing to this strong curvature there is increasing difficulty at tempera
tures not greatly removed from Tg ~ to extrapolate log D to zero penetrant
concentration. Previously, when measurements were not extended down
to sufficiently low concen
trations, it was often assumcd
t h a t the linear log D vs. c1
  o .  ~. :0 ~ plots obtained at relatively
~% r176
high concentrations could be
extrapolated to zero concen
oq.O 3o."oC

tration for evaluating D o .
Since this assumption now
appears not to hold generally,
2o,oc
except at temperatures far
~s.s
o20oc above Tw~, some formerly
reported values of D 0, espe
o "s ~ cially those of systems near
Tw~, have to be accepted
:/0oC
with reservation.
The general features of
the D vs. Cl relations for
I amorphous polymersolvent
0.s LO i.s
c/x/Oe [ff/cc oP po/y/'#e/'+wu/ee]_
systems slightly above or
below Tw~ are not fully elu
Fig. 4, Plots of D against penetrant concentration for the
system polymethyl acrylate water at different temperatures. cidated. This is mainly due to
Taken from KtsmMoTo, MAEKAWAand FOIITA (1960)
the fact that, as will be ex
plained in the next section,
the sorption processes in glassy polymers are not Fickian and hence the
methods for the determination of D (cl) presented in the previous section
can no longer be applied. In principle, D values of such systems could
be determined from steadystate permeation measurements, but
I4ISltn~tOTO (unpublished) has shown t h a t the analysis of permeation
data on glassy systems is complicated b y factors which are not yet
fully resolved.
For crystalline polymers the available data are yet so limited that we
are not in a position to be able to discuss the general character of their D
vs. c1 relations. However, it is of interest to note that for diffusion of
various organic vapors in polyethylene films ROGERS et al. (1~~ have
recently deduced D (actually, D) vs. c1 plots which were quite similar in
character to those shown in Fig. 2 and 3.
Diffusion in PolymerDiluent Systems 13
Work on polymerwater systems indicates that when the affinity of
water to the polymer is very low, i.e., when water is essentially a non
solvent for the polymer, the mutual diffusion coefficients of the systems
show slight or no concentration dependence and are less temperature
dependent than in polymerorganic solvent systems. As an example we
give data on the system polymethyl acrylatewater in Fig. 4 [KISHIMOTO,
MAEKAWA and FUJITA (1960)]. On comparison of Fig. 3 and 4 one finds
how markedly not only the form of D as a function of ca but also the
magnitude of D itself are affected b y the kind of penetrant substance.
The behavior of D (ca) illustrated in Fig. 4 is probably characteristic of
polymernonsolvent systems. This implies that if water were a solvent
or a good "swelling agent" for a given polymer, such as in the case of
polyvinyl alcohol or of cellophane, the D of the system should have been
appreciably concentrationdependent, as generally found with polymer
organic solvent systems.
III. N o n  F i c k i a n S o r p t i o n
3.1 N o n  F i c k i a n a n o m a l i e s
Sorption processes which did not conform to the Fickian type have
been frequently observed for a variety of polymersolvent systems [for
example, MANDELKERN and LONG (1951); KOKES, LONG and HOARD
(1952); DRECHSEL, HOARD and LONG (1953); LONG and KOKES (1953);
PARK (1952; 1953)]. The most fundamental fact deduced from these
observations is that such processes were encountered only when a given
system was studied at temperatures below T~~ or, more precisely, the
glass transition point, Tg, of a given polymerpenetrant mixture. The
types of reported deviations from Fickian features are so numerous that
here we must be content with summarizing some representatives of them
[see also CRANK and PARK (1951)] :
(a') In the region of small values of (t) V• both absorption and de
sorption curves are not linear;
(b') The absorption curve has an inflection point;
(c') Despite the expectation that D should increase with c1 in the
region of ca concerned, the paired absorption and desorption plots inter
sect; the relatively high initial rate of desorption followed b y a very
slow rate leads to intersection of the two curves;
(d') Paired absorption and desorption curves which have the coinci
dent initial slopes do not coincide over the entire region of (t)'/,;
(e') The initial slopes of absorption curves from different initial
concentrations to a fixed concentration pass through a maximum as the
initial concentration increases;
(f') Absorptionordesorptioncurvesobtained fromvaryingthickness ex
periments cannot be reduced to a single curve when plotted against (t)'h/L.
14 H. FUJITA :
Some of these teatures are illustrated in the actual data shown in
Fig. 5 [quoted in CRANK (1953)], It should be remarked that for a given
system not always do all of these features appear simultaneously.
Sorption curves which exhibit
"* /d
f any of these and other types of devi
ations from Fickian features are usu
f o r ~ ally called "nonFickian" processes
or sometimes' 'anomalous" processes.
In recent years, a number of inter
pretations have been presented for
the origins of nonFickian features,
JO 100 ISO but none of them are yet wholly
(f/me V~ s~d)ll~ satisfactory. In what follows, we
Fig. 5. Typical nonFickian absorption and de describe basic ideas of some represent
sorption of methylene chloride in polystyrene at
2 5 ~ C. Taken from CRANK (1953) ative theories and typical sorption
features predicted from them. In so
doing, we shall not necessarily follow the historical sequence in which
they appeared but rather attempt to group similar ideas in one scheme.
3.2 The timedependent diffusion coefficient
The conditions basic to the Fickian sorption were that (1)D is a
function of ct only and (2) a constant surface concentration is main
tained during sorption. So long as we wishes to retain the Fick diffusion
equation as the basis of the discussion, any attempt for the theoretical
interpretation of nonFickian characteristics must abandon either or
both of these conditions. In this section we give a brief account of a
theory which involves an alternation of condition (1). It is due originally
to CP~ANKand PARK (1951).
T h e y considered that condition (1) implies that any rearrangement of
polymer molecules accompanying diffusion takes place very rapidly com
pared with the rate of diffusion. If this is the case, it follows that mutual
diffusion coefficients of rubbery polymeric systems would be purely
concentrationdependent, since, as is wellknown from the study of
viscoelastic behavior, the segmental motion of polymer molecules in
such a system is very rapid. On the other hand, this motion in glassy
polymer systems should be a relatively slow process, leading to the
expectation that the D of such a system must depend not only on q but
also on other factors. These considerations are consistent with the obser
vation that above Tg sorption processes are generally Fickian and below
T~ the processes are always nonFickian. CRANK and PARK supposed
that under the condition that polymer segments rearrange themselves
slowly, the value of D in a volume element of the system does not attain
Diffusion in PolymerDiluent Systems 15
its equilibrium value D,(q) instantaneously when the penetrant concen
tration in the volume element is brought to a value ca but will get nearer
to D,(q) as the concentration remains longer at this particular value.
As the simplest expression describing this situation they assumed a
kinetic equation of the form:
aD/Ot = a (q) [De(ca)  D]. (12)
Here at (ca) is a ratedetermining factor for the approach of D to De(q)
and m a y be a function of ca, as indicated. One m a y expect that x becomes
larger as the polymer segments more rapidly change configurations. Inte
gration of Eq. (12), with ca fixed, gives
D = D , ( q )  [D,(q)  D,(q)] exp [ ~ (q)t], (la)
where Di(q) is the value of D obtained at the instant when the concen
tration in the volume element has become c1, and m a y be a function of c1.
Equation (13) indicates that under the conditions considered D is an
explicit function of both ca and t, and therefore m a y be expressed b y the
symbol D(q,t). Then it follows from Eq. (13) that D,(q)= D(q, oo).
Equation (12) is concerned with the volume element in which ca remains
constant during the approach of D to De. For a more general case in
which ca is changing with time we have
OD]Ot = [dD~(q)/dcx] (Oq/Ot) + re(c1)[De(ca)  D ] . (14)
This kinetic equation was originally proposed b y CRANK (1953), who
referred to the first term as the instantaneous part and the second term
as the slow part of the time dependence of D. He considered that these
parts are associated with the instantaneous and retarded deformations
of a polymer molecule occurring when it is subject to an external force.
In accordance with CRANK and PARK (1951), the diffusion process govern
ed b y a diffusion coefficient depending explicitly on time is generally
termed the timedeibendent diffusion or the historydependent diffusion.
To obtain quantitative information about sorption processes con
trolled b y timedependent diffusion CRANK (1953) solved numerically
Eq. (1), coupled with Eqs. (4), (5) and (14), for some assumed forms of
Di(q), D,(q) and ~ (q). His results agreed reasonably with many typical
nonFiekian features known at that time [PARK (1953)]. However, when
a new type of nonFickian behavior, now generally called the "twostage"
type, was discovered in 1953 b y LONG and his coworkers, it soon became
evident that the concept of timedependent diffusion was too simple to
explain every nonFickian behavior. This situation remains unaltered at
present, and so we shall not go further into this subject.
16 H. FUJITA :
3.3 Variable surface concentration
The absorption of vapor b y the surface layer of a polymer film will
necessitate some rearrangement of the polymer molecules, and it is
reasonable to consider that the more active the segmental motion of
polymer chains becomes, the more rapidly the surface layer takes up
penetrant to the equilibrium concentration. This implies that the surface
concentration gradually approaches an equilibrium value at a finite rate
which m a y depend upon the rate of relaxation motions of the polymer
molecules. CRANK and PARK (1951) expressed this situation b y the equa
tion:
clS = qo + (clOO_ clO)(1  eat), (15)
where qs is the surface concentration and fl is a rate parameter. It should
be noted that when fl is infinitely large, this equation reduces to the
condition of constant surface concentration, Eq. (5).
With Eq. (15) as the boundary condition, CRANK and PARK calculated
absorption curves from Eq. (1) for a system in which D is independent of
concentration. The curves obtained represented well some typical non
Fickian features, such as the sigmoid behavior, known at that time.
However, it is shown that this simple equation for the timedependent
surface concentration fails to give the "twostage" behavior which will
be discussed in the next section.
Recently, LONG and RICHMAN (1960) have shown that both sigmoid
and twostage sorption curves can be derived from Eq. (1) (with D de
pendent on c1 only) if, in place of Eq. (15), one assumes for cx*a somewhat
more general equation of the form:
c 1 ' = c 1 ~ + (cl ~  q 0 (1  eS 0 , (16) 1
where cl* is a new parameter and is assumed for absorption to be greater
than the initial concentration cx~ The time dependence of cxs represented
b y this equation consists of an initial sudden increase from q~ to cl*
followed b y a gradual approach to cl ~. For sorption of methyl iodide
into cellulose acetate films at 40 ~ C. LONG and RICHMAN (1960) actually
observed a surface concentration which essentially followed this type of
behavior. In a companion article [RIcH~mN and LONG (1960)], they also
have given definitive evidence for the validity of the assumption that in
rubbery polymers the equilibrium concentration is attained instanta
neously at the film surface. These experimental findings lead to the
important conclusion that many of the nonFickian anomalies charac
teristic of glassy polymer systems m a y be ascribed, if not all, to the slow
establishment of the equilibrium concentration at the surface of the sample.
1 T h e y also discussed t h e b o u n d a r y e q u a t i o n of t h e form, Cl~ = Cl~  ~ /~ t,
Diffusion in PolymerDiluent Systems 17
It is quite plausible that this phenomenon is a reflection of the slow re
laxation of polymer molecules in glassy systems. Since in a polymer system
there are many relaxation mechanisms having different relaxation times,
the general expression for el* corrcspondingly should be a sum of time
dependent terms having as many rate parameters as there are different
relaxation times. One m a y consider that Eq. (16) is a special case of such
a general expression when the entire spectrum of relaxation mechanisms
is a p p r o ~ m a t e d b y two mechanisms, one having zero relaxation time
(which allows of a sudden increase of c1' from Cl~ to ci') and the other
having a finite relaxation time (which causes a gradual approach of c1'
to Qoo). However, it is hazardous to interpret the parameter fl as being a
direct measure of the relaxation time of the corresponding mechanism.
3.4 Twostage behavior and successive differential sorptions
The discovery of the twostage behavior is ascribed to LOl~G and his
associates [LONG, B AGLEY and ~VILKENS (1953); BAGLEY and LONG
(1955)]. When they performed successive absorption experiments with
the systems cellulose acetate
acetone and cellulose acetate
methanol at temperatures be
low the glass transition point "Cans/i
b*" .. . ' ~
of this polymer, the absorption
curves starting with initial
concentrations above a certain
~1 ' ~ f ',
value followed processes of a
type which had not been no % i
ticed previously. Figure 6
shows this new type of absorp
Fig. 6. Schematic diagram of an absorption curve of the
tion curve schematically. B y twostage type
the successive absorption ex
periment is meant a serial experiment in which a dry polymer film
is first exposed to vapor of pressure Pl until sorption equilibrium is
reached and then exposed to the vapor of a higher pressure Ps until
the new equilibrium is attained, etc. This process m a y be continued
for as m a n y steps as are of interest. Thus in this type of experiment
the initial concentration of the nth absorption step is equal to the final
concentration of the n1th step. In most experiments of this type, the
concentration increments of successive steps are chosen sufficiently small
so that the mean penetrant concentration in the film remains practically
constant during each absorption step. In the present paper, we refer to an
experiment of this type as the "differential" type. Contrary to this, an
experiment in which the difference between the initial and final concen
trations is relatively large will be referred to as the "integral" type.
Fortschr. Hochpolym.Forsch., Bd. 3 2
18 H. FUJITA:
This definition of the integral sorption differs from the one proposed by
KoxEs et al. (1952), who termed a sorption experiment with zero initial
or final concentration the integral type. It should be noted that, in reality,
no definite borderline can be drawn between the integral and differential
experiments defined as above.
The curve shown in Fig. 6 consists of two portions of distinctly diffe
rent nature, i.e., a portion of the Fiekian type from 0 to B via A and a
/a'
PPessum l'nterml[T~T~ HT]
131/ ~" lqO
f 130
IZV
J" 13~t
~" 130
f__. /IS m, IZ~/
I10 i 112
100 ~ 110
.90 ~ /00
S" f ~ f 
i f
f f
j f
"T
gO
dO '~
2S
,,
"
#0
gg
SO
.....
~
f 0 ~ 25
J
/0 20 30 ~0
f~e [,>7r~i~])~2
Fig. 7. Successive differential absorptions of mcthyI acetate in polyn~ethyI rnethaerylate at 30~ C. Taken
from KISHIMOTO,FLrJITA,ODANI, KURATAand TAMURA(1960)
portion of sigmoid shape from B to a final equilibrium D via C. This
twostep structure gives the reason why this type of sorption curve is
termed the "twostage" type. The initial Fickian portion is referred to as
the first (or initial) stage and the subsequent sigmoid portion as the second
stage. Often we observe that the first stage is terminated with a very flat
curve before the second stage appears. In such a case we m a y stop con
tinuing the measurement at this flat end of the first stage, misunderstand
ing that the sorption has already reached equilibrium. However, this flat
portion never represents the true equilibrium to be reached under given
external conditions, and thus sometimes it is referred to as the "quasi
equilibrium" state.
The twostage sorption behavior is not a unique characteristic of the
cellulosic systems investigated b y LONG et al. Subsequent studies, b y
D i f f u s i o n lit P o l y m e r  D i l u e n t Systems 19
NEWNS (1956) and many others, have observed it for a number of polymer
penetrant systems, including both amorphous and crystalline materials.
At present, it is taken as one of the most general sorption features of
glassy polymer systems. By way of example, we reproduce here successive
differential absorption data obtained by us [KISHIMOTO,:FUJITA, ODANI,
KURATA and TAMURA (1960)] in Fig. 7 and 8. These data manifest
$0 P:essure /'n/e/'val~t~H~
f I~9.9 ~ ISG.3
f IqLO ~ Itr163
./
e ~
130.0 ~
ll~.g ~
I~1.0
130.0
JO
t~ 79.0 ~ 9s
/ 18.: . :~.1
8S "~ 18.1
0 ~ 8.s
1 1
0 10 20 30 ~ 50
Fig. 8. Successive differential absorptions of acetone in cellulose nitrate at 25~ C. Taken from ICasmMoIo,
Ft:jIT^, OvAnl, KURArAand TAMVRA{1960)
important differences of successive absorption curves between amorphous
and crystalline polymers.
One can see that each differential absorption curve changes its shape
systematically as the initial concentration of each step becomes higher.
The general trend of this change is similar in the two systems. For the
first several steps covering low penetrant concentrations the curves all
have sigmoid shape and the point of inflection on each curve shifts to the
region of small values of time as the initial concentration increases. The
sigmoid character disappears at a certain initial concentration, for which
the differential sorption shows a behavior resembling Fickian sorption.
2*
20 H. FUJITA:
However, the limited linear initial region and the very slow approach to
equilibrium indicate that this curve is not truly Fickian. KISHIMOTO
et al. have referred to this type of curve as "pseudoFickian". Next to
this curve there appears a twostage curve, and as the initial concentration
is increased, the second stage portion shifts to the short time region,
gains a higher rate, and gives a greater contribution to the total concen
tration increment of each step. When a certain concentration is reached,
the second stage portion predominates the entire process and the two=
stage character disappears. In the polymethyl methacrylate system the
differential curve at this concentration looks again pseudoFickian,
whereas in the cellulose nitrate system it is of the sigmoid type. In the
former system, further rise of the initial concentration changes this
pseudoFickian curve to a curve having a shape of the Fickian type, and
above a concentration of 0.12 g/g all curves look Fickian. It is of interest
to note that this critical concentration for the transition from non
Fickian to Fickian behavior corresponds to the methyl acetate content
at which this polymersolvent system undergoes glass transition at the
temperature of the experiment. Similar relations had been early recogn
ized by LONG et al. for other amorphous polymersolvent systems [KOKES,
LONG and HOARD (1952) ; LONG and KOKES (1953)]. However, it should
be remarked that, according to recent work, absorption is not always
Fickian in the region not greatly removed from the critical concentration
even if it has a shape conforming to the Fickian type. This is equivalent
to saying that at temperatures not far above T a the sorption is still
nonFickian. The cellulose nitrate data do not show such a critical
concentration, but the increase in initial concentration merely gives rise
to a vertical shift of a sigmoid sorption curve with its shape practically
unchanged.
The character of the polymethyl methacrylate data is essentially
similar to that found for systems atactie polystyrenebenzene at 25 ~
35 ~ and 50 ~ C. [KISHIMOTO,FUJITA,ODANI,KURATAand TAMURA (1960);
ODANI, I~IDA, KURATA and TAMURA (1961)] and also atactic polysty
renemethyl ethyl ketone at 25 ~ C. [ODANI, HAYASHI and TAMURA
(1961)1, and appears to be fairly general for amorphous polymersolvent
systems in the glassy state. On the other hand, the cellulose nitrate data
shown in Fig. 8 appear to manifest features characteristic of crystalline
polymersolvent systems. For example, the earlier data of NEWl~S (1956)
on the system regenerated cellulosewater (in this case, water is not the
solvent but merely a swellingagent) and recent studies for several
crystalline polymers all show essentially similar characters Esee KISHI
MOTO, I~U]'ITA, ODANI, KURATA and TAMURA (1960)]. To arrive at a more
definite conclusion, however, more extensive experimental data are
needed.