JOURNAL OF POLYMEll SCIENCE VOL. XXVIII, PAGES 569-585 (1958)

Diffusion-Controlled Stress Relaxation in Polymers. 111. Stress Relaxation in a Swelling Polymer

AKIRA KISHIMOTO and HIItOSHI E’UJITA, Physical Chemistry Laboratory, Department of Fisheries, University of Kyoto, Maizuru, Japan

INTRODUCTION

Use of chemical relaxation methods for the study of softening or the degradation process of a polymer accompanying sorption of a gas or a vapor usually involves the assumption that the relaxation of stress observed is produced only by a chemical scission of interchain bonds of polymer net- ivork due to a specific action of the penetrant molecule and that any physical process, such as molecular configurational change, is negligible. A theory of stress-relaxation accompanying sorption was presented in Part I of this series,’ based on a simple scission mechanism involving a bimolecular kinetic reaction. For relaxation in non-crosslinked linear amorphous polymers accompanying sorption of a vapor, hon-ever, this scission theory is no longer applicable, because in such systems the action, which has been called the “plasticizi’ng effect” in Part I1 of this series,2 should be a predominant con- trolling factor if the temperature is well above the glass-transition point of the pure polymer. That is, in such systems the enhancement of rate of configurational change of chain molecules in the presence of low molecular weight molecules should be rate-determining. Tho purpose of the present paper is to work out an approximate theory of stress-relaxation in such polymers accompanying sorption on the basis of experimental information about this plasticizing effect obtained in I’art I1 of this series and to check its validity and practical applicability on experimental data for the systems polymethyl acrylate-water and polymethyl acrylate-methanol a t 40°C.

THEORETICAL

Let us consider an experimcnt as follom. A film of a linear amorphous polymer is allowed to relax in uucuo a t constant strain. When the stress reached a certain value corresponding to rubbery consistency, a vapor con- trolled to a given pressure is quickly introduced into the test chamber in which the relaxomcter is fixcd, and the measurement of stress change is continued, keeping the elongatioii of the film unchanged. The vapor reservoir is designed so large that 110 pressure v:iriation owurs i i i the test,

369

570 A. KISHIMOTO AND 11. FUJITA

chamber as the film absorbs the vapor. The following is an attempt to correlate the stress-relaxation process in the new environment with t,he kinetics of the sorption of the vapor into the film.

First, for the sake of simplicity, let us assume that the relaxation merh- nnism involved in the film is represented by a single Mas\\-ell model \I ith :I spring constant E' and a relaxation time 7 0 in its dry state. It is assumed that the film is so thin that non-uniform distribution of penetrant in it is replaced by an over-all concentration C(t) as a function of time t . The origin of t is here taken a t the time of introduction of vapor into the test chamber. From the results reported in Part I1 of this series i t is assumed that E is independ- ent of C t o a first approximation for dilute enough C, and that the relaxa- tion time TO is changed by a factor a, as the overall concentration of pene- trant in the polymer is changed from zero to C . Here a, is a function of C only for a given pair of polymer and penetrant a t a fixed temperature. Since C changes with t , the relaxation time of the model also changes with t in accordance with the relation:

The over-all stress of the film a t time t is denoted byf(t).

7 1 = TOUc (1)

where T~ is the relaxation time a t time t .

yo in such a Maxwell mechanical model is then written in the form: The differential equation for stress-relaxation a t constant elongation

or

1 ntegratiig this with the cwndition l h t T , is here a function of t and given hy equation (1) yields

where f(0) stands for the stress a t the instant when the vapor was intro- duced. By applying a similar argument as above for the relaxation in vacuo, it is shown that

where t* denotes the interval of time from the sudden application of constant elongation yo to the introduction of vapor into the test chamber. Com- bination of equations (3) and (4) gives

DIF’E’USION-CONTROLLED STRESS RELAXATION. I11 571

If, in place of it single Maxwell model, use is made of a generalized Maxwell mechanical model with a continuous distribution of relaxation times, fii(~,,), equation (5) is generalized to give

It should be noted that E(r0) represents the distribution of relaxation times of the given polymer a t its dry state. Equation (6) may be written in terms of the relaxation modulus M ( t ) to yield

So far i t has been assumed that penetrant distribution in the polymer is uniform so that C may be treated as a function of t only. Actually, when a vapor diffuses into a polymer film, a nonuniform concentration distribution is set up in the film and its shape changes with time depending upon the nature of the given polymer-vapor pair as well as upon physical conditions of the particular experiment. This implies that C , and hence a,, must actually be treated as a function of both time and position in the film. Let us assume that the diffusion occurs only in the direction of film thickness and take the space coordinate x in this direction with its origin on the cen- ter plane of the film. Application of equation (7) to each infinitesimally thin layer of the film perpendicular to the x axis and summation over the film thickness provides

where 1 is one-half of the thickness of the film.

in the previous study2 to give The form of the concentration shift factor a, as a function of C was found

PC f O ( f 0 + PC)

In a, = - (9)

provided that C is sufficiently small and the temperature of the system is well above the glass transition point of the pure polymer. Here f o is the fractional free volume of the dry polymer at the given temperature and P is a parameter characteristic of the nature of the given polymer-vapor pair. For such low concentrations of vapor as are treated in the present experi- ments described below, the term PC in the denominator of equation (9) may be neglected in comparison withfo with the result

572 A. KISLIIMO'I'O AND 11. FUJl'lA

In a, = - PC:'j"' = - 6C

or

(10) -6C (I, = c

where

6 = B/.fo2 (11)

The value of 6 is actually the slope of the initial tangent to a plot of In a, us. C, which may be determined experimentally using the procedure described previously.2

Substitution of equation (10) into equation (8) gives

To effect the integrations involved in the right side of equation (la), it is first necessary to represent C explicitly as a function of t and 2. This re- quires integration of the ordinary Fick diffusion equation for a plane sheet

0

T

C.

c

Fig. 1. Coiicentration-distanc:c: curve in a plane sheet. Solid lino refers to the actual distribution and dashed line to the approxima- tion.

under relevant initial and boundary conditions. Such integration also requires definite information about the diffusion coefficient D of the given penetrant in the given polymer. Substitution of such mathematical solu- tion for C(x , t ) into equation (la), integration with respect to both t and 2, and further integration for TC with the kn0n.n form of leads finally to the numerical values of relazxatioii modulus AIZ(t) its :t function of t . The results thus obtained niay he c*onip:~red ivith e\;prrimc~iit:il data to check thc wlidity of thc thcoreticxl tmitnir l t t t pwheiitt'd h o r t ~ . I I o ~ v ( ~ r e r , this method of conipnrison )Jt%\\-ec.n thwry :md (.upcrimelit ini*oli cs :L tremen- dous aniount of nuniericd work and nxiy not necessarily be of general inter-

DIFFUSION-CONTROI.LED STRESS RELAXATION. I11 553

est. For this reason, resort is here made to an approximation which niakes an analytic treatment tractable and thus niakes the evaluation of D from M ( t ) data possible. This approximation consists of replacing the actual concentration distribution in the film by a step function as shown by a dashed line in Figure 1. The substituted step function for C(z,t) may be characterized by two parameters Cand x*, where c i s the height of the step and x* is the TT coordinate of the step point. Thus we have for one half sidc of the film

c = c for 7 > x > x*

c = o for .c* > z > 0 (13)

It is obvious from the geometry of the problem that the identical situation holds for the other half side of the film. It is assumed further that the value of c is characteristic of the given pair of polymer and vapor and is kept constant during the process, while z* moves with time from the surface to the center of the film in such a way that the total areas enclosed by the true distribution and the step distribution remain equal with each other a t every instant.

Substitution of equation (13) into equation (12) and coiisiderntioii that c i s independent of t yields

M ( t ) = (1 - .y) *lf,,(f* + r q + 2* JT((f* + t ) (14) 1

with

As readily understood, Mo(l ) represents the relaxation modulus curve of the pure polymer a t the given temperature.

Mathematical solutions to the Fick diffusion equation in one dimension, subject to the conditions that the solid is initially free of penetrant and its surfaces reach instantaneously an equilibrium concentration Co when the solid is exposed to a penetrant vapor, show that in the initial stage of either absorption or desorption in the system with D dependent on C only, the amount of penetrant absorhed or desorl)ctl, Q, changes linearly with thc square root of time. Thus

where Q m is the equilibrium penetrant uptake by the solid under given experimental conditions, and D is an integral diffusion coefficient defined by

- 1 D = -1 D ( C ) dC co (17)

For II independent of C, reduces to the true diffusion coefficient D.

5.74 A. KISHIMOTO AND H. FUJITA

With the substituted step concentration distribution in Figure 1, Q is written as

Q = 2C(I - x*)

Qm = 2Col

(18)

( 1 ! I )

On the other hand, we have, obviously,

Coni1)ining erlii,zti~is (1 G ) , (1 8), ant1 (19) gives

(30) X* 1 - = l - X d t

where X is

Substitution of equation (20) into equation (14) results in

M(t ) = Mdt* + t ) + xdt [M,(t* + e G ) - Mo(t* + 111 (22)

From this equation the relaxation modulus curve accompanying sorption, M ( t ) , may be predicted, provided that Mo(t), the relaxation modulus of the pure polymer a t the same temperature, the values of 6, Co, c, D, and 1 characteristic of the given pair of polymer and vapor and of experimental conditions used, and the value of t* chosen for the particular measurement are known and specified. Conversely, it is possible to evaluate D from an experimental M ( t ) curve with known informations about all quantities involved in equation (22) except for D. To do this we plot M(t) - Mo(t* + t ) against (t)”’[Mo(t* + esct) - Mb(t* + t ) ] . Equation (22) indicates this plot to be a straight line passing through the coordinate origin and a slope A. The value of D may then be obtained from X using equation (21) and known values of Cot C and 1. The value of Co corresponding to the vapor pressure used can simply be found from the sorption isotherm deter- mined separately. It is important to note that equation (22) is applicable only for the initial stage of relaxation accompanying sorption, because it resorts to equation (16) which is usually valid only before the film absorbs a vapor by about one-half of the equilibrium uptake. Mention also should be made that the simple square root time law of sorption given by equation (16) holds only for D dependent only on C. Therefore, the above method of evaluation of D from stress-relaxation is no longer applicable to systems which exhibit a non-Fickian diffusion behavior, that is, such systems in which D depends not only on concentration but also on time and other fac- tors.

The remaining problem to make the above treatment practical is the evaluation of the parameter c. Unfortunately, no definite condition is available on which the value of c can be determined uniquely under given experimental conditions. So far as the theoretical model used is concerned,

DIFFUSION-CONTROT,T,ED STRESS RELAXATION. I11 575

i t is only a parameter which can be chosen entirely arbitrarily with the obvious restriction that its value should physically be equal to or smaller than the equilibrium surface concentration Co. However, to make the theoretical treatment presented above generally usable, some definite way for the evaluation of c must be worked out. The following is an attempt in this direction, though i t is not entirely satisfactory.

Our idea is to choose c as a concentration nverngr with the f x t o r c6r nsn weight. This m a n s that

This eqiint,ion may be written

( 2 3 )

where

u = scc ( 2 5 )

For the early stage of diffusion into a plane sheet, the medium may I)? regarded as semi-infinite, so that equation (24) is approximated hy

(26)

If the diffusion coefficient D is independent of C , the concentration-distance curve in a semi-infinite medium, subject to the initial condition that the medium initially is free of penetrant and the boundary condition that the concentration a t the surface is maintained a t a constant value Co during the process, is given by

(27) C <-- = 1 - q x > C"

where

2 = .1*?2dI)t

:ind @ ( z ) is tlie error frinction of z clefiiicd :is

Introduction of equation (27) into equation (26) yields

(28)

(29)

I t is shown analytically that in the limit when u goes to zero the ratio c/Co approaches 0.586, and in the limit when u increases indefinitely this ratio tends to unity. For intermediate values of u the integral in the right

576 A. KISTTIMOTO AND II. FUJITA

side of equation (30) must he evitIt1:itd nuin~~ric:~lly to obtain c/Co as a function of CJ. The solid line nith 11 = cwnst:int in k'igure 2 shows the results of such calculations. I'rovidcd that L> for the given system is independent of C, the value of c for any given set of Co and 6 may be inter- polated from this curve.

Recent work' demonstrates that 1) in many polymer-organic vapor sys- tems exhibits a concentration dcpendcnw fitted :Ldt>quately by an esponen- tial relation such that

(31)

where Do is the value of D a t zero vapor concentration and k is a parameter (positive) characteristic of a polymer-vapor system. Solutions of the Fick diffusion equation with D given by equation (31) for a semi-infinite medium

D = Do esp (kc)

0.9

0.8

t" 0.7 I&

/ D = const. I

1 0.6

0.5 0 5 10

-6

Fig. 2. Relation between F / C , and u. Numbers attached to the lines are valiies of eup (kcb) .

subject to the initial and boundary conditions as mentioned above are avail- able oiily in a graphical or a iiumeric:d form for several particular values of k4 Substitution of such solutions in equation (26) and integration pro- vides c/Co DS. u curves as a function of dimensionless parameter kCo. The curve with D = constant in Figure 2 should be a particular case for k = 0. The two upper lines in Figure 2 show examples of such curves for exp (kCo) = 5 and exp (IcCc) = 95, respectively. For the calculations of these curves the coilcentration-distance curves given in Crank's book (p. 270) have been used.4 When the value of k is known in advance for the given polymer-vapor system, the required value of c for a set of Co and 6 may be determined by suitable interpolation from a family of such c/Co us.

curves. However, in the case when we are interested in evaluating D or from stress-relaxation data in terms of the method described above, k is

not the quantity known in advance but the one that should be evaluated as

DIFFUSION-CONTROLLED S'I'RESS RELAXATION. 111 577

a result of the analysis concerned. This situation suggests application of a trial and error method or of a method of successive approximations to find a correct value of C for a given system, i.e., an appropriate value of k is assumed, Cis interpolated from a family of ~ / C O us. u plots for given values of Cc and 6, D is determined as a function of Ca from stress-relaxation data in terms of the theory presented, the D us. C relation is calculated, and a first approximation to the k value is evaluated (provided the calculated D us. C relation obeys an exponential curve). The new k value is used to repeat the above cycle of calculations, and a second approximation to the k value may be found. This iteration is continued until the successive k values converge to the desired accuracy.

In order to make this successive approximation practical, a family of c/Co us. u curves must be obtained so that satisfactory interpolation of required c values can be made for given Co and 6 when k is given. Unfor- tunately, no sufficient data are yet available which permit extensive evalu- ation of such a family of curves and, in addition, it is apparent that such evaluation requires a considerable amount of numerical work. For this reason, in the present analysis of experimental data we shall tentatively evaluate c for given sets of Co and 6 from the curve with D = constant in Figure 2. It will be shown that, at least as far as the systems studied in this work are concerned, this drastic procedure gives values more than expected.

EXPERIMENTAL

To test the theory developed, data of stress-relaxation accompanying sorption were obtained on the systems polymethyl acrylate-water and poly- methyl acrylate-methanol a t 40°C. (about 35°C. above the glass transition temperature of the pure polymer). The polymethyl acrylate used for this study was the same as that employed in Part I1 of this series. These poly- mer-vapor systems were chosen not only because their shift factor a, data were available, but also because the relaxation modulus curve of the pure PMA, Mo(t), in the rubbery region had been determined completely.2 The viscosity-average molecular weight of the purified PMA as determined from the intrinsic viscosity in acetone a t 30°C. was 1.1 X lo6. Prepara- tion of the sample films used was effected in exactly the same procedure as employed previously.2 Two films, 20 X em. and 8.8 X cm. thick, were studied in order to examine whether the stress-relaxation be- havior in question is truly diffusion-controlled or not.

The apparatus and procedure used for the measurements were similar to those described previously.2 A strip of the sample film fixed between the upper and lower clamps of the relaxometer a t strain-free state was dried in the relaxometer tube under a reduced pressure of mm. Hg for 5 hours. The film was then quickly stretched by a given amount and the subsequent decay of stress was followed as a function of time in vucuo, keeping the elongation constant. After 20 minutes, when the stress had

578 A. KISIIIMOTO AND 11. FUJITA

reached a value characteristic of rubbery consistency of linear amorphous polymers, the given vapor controlled to a given pressure was quickly intro- duced into the relaxation tube from the vapor reservoir and the change in stress was followed, as before, as a function of time. All measurements were conducted in an air temperature bath a t 40°C. for both water and methanol systems.

RESULTS AND DISCUSSION

Figure 3 shows a family of stress-relaxation curves accompanying sorp- tion on the system PMA-water a t 40°C. as a function of equilibrium water regain Co (g. water absorbed/g. dry PMA). In this figure, the ordinate is

3

Fig. 3. Diffusion-controlled stress relaxation curves on the system PMA-water at 40°C.

the relaxation modulus M(t ) in dynes/cm.2 and the abscissa is the loga- rithm of time t in minutes. The origin of time t is taken a t the time of intro- duction of vapor into the relaxation tube, in conformity with the theoretical argument presented. The relaxation behavior in vacuo before the intro- duction of vapor checked very satisfactorily with that reported in the pre- vious paper,2 and is not shown here. Stress values after the introduction of vapor have to be corrected for the progressive expansion of the cross-sec- tional area of the film as diffusion proceeds. However, since equilibrium vapor uptakes were quite small in the range of pressures studied for either water or methanol system (about 0.04 g./g. a t largest), such correction of the cross-sectional area accompanying sorption was ignored in the present treatment of experimental data. Thus, the relaxation moduli shown here

DIFFUSION-CONTROL1,ED STRESS RELAXATION. 111 579

were calculated by simply dividing measured stresses by the cross-sectional area of the dry polymer film a t strain-free state. The relaxation modulus curve corresponding to C, = 0 (dry polymer) in Figure 3 has been replotted from the data of Part I1 by transforming the origin of time by 20 minutes to the new one. Figure 3 demonstrates how the rate of stress decay be- comes fast as the film absorbs water vapor; the higher the external pres- sure of water vapor the faster the initial rate of relaxation results, as should be expected.

1

-+ log t (MINS.)

Fig. 4. Diffusion-controlled stress relaxation curves on the system PMA-methanol :tt 40°C. Open circles refer to thin film and solid circles to thick film.

The stress-relaxation behavior in the PMA accompanying sorption of methanol was essentially similar to that for the system PMA-water. The data obtained on the system PMA-methanol at 40°C. are plotted in Figure 4. For this system the effect of film thickness was investigated by making measurements on samples of different thicknesses (20 X cm. and 8.8 X cm., respectively) under an identical pressure of methanol vapor. Two relaxation curves with the same Co = 3.9 X g./g., shown in Fig- ure 4, are the results of such measurements; the curve with open circles refers to the thin film and that with solid circles to the thick film. It is

580 A. KISIIIMOTO AND H. FIJJITA

seen that rate of relaxation in the thick film is slon-er than that in the thin film over the entire range of time scale shown, but the curves appear to coincide after sufficiently long runs. These facts suggest that the relaxa- tion of stress observed is doubtlessly diffusion-controlled; as well known, the rate of vapor uptake by a solid sheet is pronouncedly influenced by sam- ple thickness, provided that the diffusion is rate-determining.

It is of interest to notice that the curves for sm4ing qpstems in Figures 3 and 4 tend to become parallel to the c'urve for the pure polymer after a sufficiently long interval of time from the iiitrodiiction of vapor. This asymptotic behavior of relaxation curves of swelling polymers confirms the time-concentration superposition as demonstrated in the previous study2 of this series on polymers a t swelling equilibrium. In fact, it is found that the values of log a,, estimated from the limiting horizontal distances be- tween the curves of smelling polymers and that of pure polymer in Figures 3 and 4, fall exactly on the corresponding log a, us. C curves as obtained in Part I1 directly from this same PhlA sample swollen with either water or methanol.

We now are in a position to compare the theory presented above with the experimental data. In what follon-s, values of the integral diffusion co- efficient b are derived from M ( t ) data in terms of the theory and are com- pared with those obtained from sorption experiments. To do this me define functions P(t ) and Q ( t ) such that

P(t) = AT" (f* + t ) - flf(t)

Q(t) = cdt / I ) [Jf" (1* + t ) - M" (t* + P S C t ) ]

The evaluation of P(t) as a function of t can readily be made by direct sub- stitution of the M ( t ) data given above and of the M,(t) curve reported previously.2 Here the value of t* is taken as 20 minutes in accordance with the condition adopted in the present experiments. To calculate Q ( t ) , values of 6 pertinent to the polymer-vapor systems studied must be assigned. These are obtained from the shift factor data presented in Part 11, and we have

6 = 150 for PMA-water system

6 = 110 for PMA-methanol system

The values of c for given sets of Co and 6 are tentatively interpolated from the curve with D = constant in Figure 2

Typical examples of P(t ) us. Q ( t ) plots obtained in this way are given in Figure 5. In this figure, the curves with solid circles refer to data from the thick film (20 X cm. thick) and the one with open circles to the thin film (8.8 X The two curves from methanol studies cor- respond to an identical vapor pressure (hence, to the same Co = 3.9 x 10-2 g./g.) and indicate how the thickness of the film influences the process con- cerned. It is seen from the figure that, in conformity with the theoretical

em. thick).

prediction, all the curves shown are linear in the range of small &(t ) , i.e., of small t . It is shown that the length of this linear region is closely related with the interval of time in which sorption process obeys the di law represented by equation (16). This explains why the linear region is so limited in the plot for water; previous data on sorption of water in PMA indicate that the rate of diffusion of water in this polymer a t 40°C. is so rapid that the & sorption law holds only for several minutes from the start of an experiment, and i t is confirmed that this interval of time just cor- responds to the limited range of Q ( t ) in n-hich the plot for water in Figure 5 follows a linear relation. The d:Lshed straight lines in Figure 5 have been drawn to help understand the ranges in whirh the predicted linear behavior

0 0.25 0.50 0.75 1.00 1.25 - Q( f ) x 10-9 (DYNESMIN.4. CM13)

Fig. 5 . Test of the linearity betneerl P( t ) and Q ( t ) on the systems PMA-water and PMA-methanol at 40°C. Sgmhols a r ( a thr same as those in Fig. 4.

holds. The theory indicates that the slope of this straight line should be ( 2 / d i ) f i (Co le ) , so that it must be independent of film thickness 1. This requires, in turn, that the initial portions of the two curves for meth- anol in Figure 5 superpose. The fact that this is not the case with the present data may partly be attributed to experimental errors. However, i t is quite probable that this rcflects the failure of the present approxima- tion to take the effect of film thickness into mathematical considerations. Further improvement of the theory is thus evidently indicated. However, a slight improvement in the agreement of the two slopes in question can be attained if we take into account the fact that the width to thickness ratio in the sample strip of thick film was as small as about 10, so that penetra- tion of vapor from its lateral surfaces was not completely negligible. The corresponding ratio in the sample strip of thin filni was so large, about 70, that no concern was necessary about thi- effect, a t least, within the accu- racy of the present treatment. Xssuniing for simplicity that penetrations of vapor from the main and lateral surfaces of a given strip occur independ- cntly from one another in thc carly stage of diffubion, it is shown that I

582 A. KIS€iIMOTO AND €1. FUJITA

in equation (21) may approximately be replaced by an effective thickness 1’ defined by

where S , and S , are the areas of the main and lateral surfaces of the strip, respectively. Introduction of actual values of S , and SL and recalculation of Q(t) in terms of the effective thickness results in a slope of the initial straight line for the thick film that is about 10% less than thak calculatcd

-5

‘ 0 2 4 6

cxlUz (G./C.)

Fig. 6. Integral diffusion coefficient b us. concentration Co relations on PMA. Solid Circles, data from the present relaxation study. lines, data from the previous study.2

Symbols are the same as those in Fig. 4.

in terms of the actually measured thickness 1. This correction brings tht. slopes of the two curves for methanol in Figure 5 to a slightly better agree- ment. The remaining discrepancy between the two slopes must await an improvement of the theory itself.

Values of integral diffusion coefficient D were computed for all Co studied from the initial slopes of the respective P(t) us. &(t) plots, based on the theo- retical requirement that this slope be equal to (a/&) 45 ( C , / C ) . The value of C for a given set of CC and 6 was obtained, as before, from the curve with D = constant in Figure 2. I n evaluating the slopes of P(t) us. Q ( t ) plots for the thick film, the correction was applied for the film thickness in accordance with the procedure described above. The I) valucs thus 01)- tttined are plottcd against equilibrium regain CO in Figure 6, where the solid

DIFFUSION-CONTROLLED STRESS RELAXAHON. 111 583

circles give the values from the thick film and the open circles from the thin film. The b values for methanol from the two films do not agree with each other, reflecting the disagreement of the corresponding P(t) us. &(t) curves as discussed above. It should be noted that the value of b is proportional to the square of the slope of the initial linear portion of a P(t) us. &(t) plot. This means that a slight change in the latter is greatly magnified in the former. I n Figure 6 the b vs. CO data determined previously from sorption measurements are shown by solid lines for both systems investigated. It is rather striking that the values derived from mechanical data fall so closely to the lines from usual sorption experiments. For the system PMA- water the agreement between the two sets of D values appears particularly surprising. The degree of concentration dependence of obtained for the system PMA-methanol is also quite similar to that found from direct sorption measurements. These agreements may be more or less fortuitous and even more so than expected, considering the crude nature of the present theory, in particular, the drastic approximation involved in the evaluation of c. Nevertheless, i t is considered that the results obtained doubtlessly support the validity of the basic ideas for the present theoretical treat- ment of the stress-relaxation process in amorphous linear polymers accom- panying sorption of a vapor.

The previous finding2 that the diffusion c:oefficieiit Ll in the system PMA- water is practically independent of C justifies the use of the ~ / C O curve for D = constant in Figure 2 for the analysis of stress-relaxation data on this system. However, this is apparently not the case with the system PMA-methanol, and it may be suspected that the fairly good agreement of the two sets of D values obtained above for this system is entirely due to fortuitous selection of c values. This is partially true in so far as no theo- retical justification can be given for the procedure used. However, the agreement obtained may be regarded as an evidence of the fact that values to be determined from mechanical data in terms of the present method are quite insensitive to the values assigned to c. The best way to check this point would be to fix the value of c at the surface concentration CO for any type of concentration dependence of D and to compute the value in exactly the same procedure as described above. The results obtained in this way for the system PMA-methanol are compared below with the values plotted in Figure 6.

co x 102g./g.

1.75 (thin film) 2.40 (thin film) 3.90 (thin film) 3.90 (thick film)

* Expressed in cm.2 min.-'.

b (with C # Co)*

0.81 x 10-6 1.1 1 . 5 1.6 2.4 2.1 2.9

b (with C = CO)*

1.0 x 10-6

Itr is ~ C P I I that, both w f s of all C'o ~ a l u e s investigated.

v:rI~ic?s gi\-p pr:~(di(~ally idriitical results for This implies that for most practical purposes

584 A. KISHIMOTO AND 11. FUJITA

D values may be determined with c set equal to CO, independent of the form of concentration dependence of D to be established. It is not clear, how- ever, why b is so insensitive to the value of c.

Thanks are due Mr. T. Yamada of the Department of Forestry, Faculty of Agricul- ture, University of Kyoto, for his help in constructing the apparatus used. This inves- tigation was partly supported by a grant from the cooperative research project on the physical properties of high polymers under the supervision of Professor J. Furuichi, Department of Physics, Faculty of Science, Hokkaido University.

References 1. A. Kishimoto and H. Fujita, Kolloid-Z., 150, 21 (1957). 2. H. Fujita and A. Kishimoto, J . Polymer Sci., 28, 517 (1!)58). 3. G. Park, Trans. Faraday SOC., 46, 684 (1950); ibid., 47, 1007 (1951); R. J. Kokes

and F. A. Long, J . Ant. Chem. SOC., 76,6142 (1954); S. Prltger and F. A. Long, ibid., 73, 4072 (1951).

4. J. Crank, The Mathematics of Diflusion, Oxford Univ. Press (London), 1956.

Synopsis An approximate theory of the relaxat,ion of stress in amorphous linear polymers ac-

companying sorption of a low molecular weight penetrant is worked out on the assump- tion that the relaxation time of each Maxn-ellian relaxation mechanism involved is changed in the presence of penetrant by a factor dependent on penetrant concentration. A method is derived from the theory w-hich permits approximate evaluation of the in- tegral diffusion coefficient b of the penetrant in the polymer from stress-relaxation data on swelling systems. The theory is checked on experimental data for the systems poly- methyl acrylate-water and polymethyl acrylate-methanol a t 40”C., and it is found that, for both systems studied, the values of b computed from mechanical data agree reason- ably with those evaluated directly from usual sorption experiments. The theory de- scribed presents only a first attempt to the quantitative interpretation of the phenom- enon discussed. Further improvement and modification of it is apparently desirable, in view of some drastic approximations incorporated in its mathematical development.

R6sum6 Une theorie approchee de la relaxation B 1’Btirement au sein de polymkres lineaires

amorphes, due B la sorption d’un agent penetrant de faible poids molCculaire, a Bt6 basee sur l’hypothkse que le temps de relaxation de chaque mecanisme de relaxation de Maxwell est modifie par la presence d’un agent pbnetrant, par un facteur fonction de la concentration de ce dernier. Une methode est deduite de la theorie; e lk permet 1’6valuation approchee du coefficient integral de diffusion b de l’agent phnetrant ail depart de mesures de tension-relaxation sur des systkmes gonfl6s. La theorie est con- trolee par les resultats experimentaux sur des systkmes polyacrylate de m6thyle-eau et polyacrylate de m6thyle-m6thanol B 40°C.; les valeurs de b obtenues aux dbpens de mesures mecaniques s’accordent raisonablement avec celles obtenues directement par des experiences de sorption pour les deux systkmes envisagis. La th6orie ne presente qu’un premier essai B fournir une interpretation quantitative des phenomknes discut6s. Une amelioration subsequente e t sa modification semblent desirables par suite de cer- taines approximations drastiques effectuCes au cours du d6veloppement mathhmatique.

Zusammenfassung Eine annahwnde Theorie der Spanniingsrc4axntion in amorphen linearen I’olymeren,

die die Sorption cines I’enetranten von niedrigem AIolekulargewicht begleitet, n-ird mit

DIE’E’USION-CONTHOLLJ~D STRESS RELAXATION. 111 585

der Voraussetzung ausgearbeitet, dass die Relaxationszeit jedes eingeschlossenen Max- well’schen Relaxationsvorganges in Gegenwart von Penetranten durch einen Faktor geandert wird, der von der Penetrantenkonzentration abhangig ist. Aus der Theorie wird eine Methode abgeleitet, die eine annahernde Abschatzung des integralen Diff u- sionskoeffizienten B des Penetranten in das Polymer aus Epannungs-Relaxations-Daten fur Quellungssysteme erlaubt. Die Methode wird mit experimentellen Daten fur die Systeme Polymethylacrylat-Wasser und Polymethylacrylat-hfethanol bei 40°C. ge- pruft, und es wird gefunden, dass die Werte fur 6, die aus mechanischen Daten berech- net werden, fur die beiden untersuchten Systeme zufriedenstellend mit den direkt aus ublichen Sorptionsexperimenten evaluierten iibereinstimmen. I>ie heschriebene Theorie ist nur ein erster Versuch, zur quantitativen Auslegung der diskutierten Vor- gange. Weitere Verbesserung und Ahanderung ist im Hinblick auf einige drastische Annaherungen, die in ihrer mathematischen Entmicklung eingeschlossen sind, offenbar w[inschenswert.

Received June 19, 1957