Diffusion-controlled stress relaxation in polymers. II. Stress relaxation in swollen polymers

JOT JRNAL OF POLYMER SCIENCE VOI,. XXVITT, PAGES 5 1.7-567 (1958)

Diffusion-Controlled Stress Relaxation in Polymers. 11. Stress Relaxation in Swollen Polymers

HIROSHI FUJITA and AKIRA RISHIMOTO, Physical Chemistry Laboratory, Departirwrit of Fisheries, Univrrsily 05 Kyoto, Jlaizuru, Jnpaii

INTRODUCTION

When a polymeric solid absorbs a vapor of low molecular weight, the solid is gradually softened. This process may quantitatively be followed by keeping the solid at constant extension during the sorption and observ- ing simultaneously the decay of stress produced as a function of time. Since whatever mechanism is involved, the rate of decay of the stress supported by a volume element of the solid may depend upon the amount of penetrant therein present, the over-all rate of stress decay of the polymer accompanying sorption should be diff usion-controlled. If a reasonable assumption can be made of the rate of molecular relaxation as a function of concentration of penetrant, it is in principle possible to predict the over-all softening process of the polymer accompanying sorption from separate information on the diffusion behavior of the given polymer-penetrant system. Two representative mechanisms may be considered concerning the molecular structural relaxation caused by penetration of small molecules. One is the scission of interchain bonds by a chemical action of the penetrant. This mechanism may appear only when the polymer contains intermolecular bonds specific for the penetrant concerned. The other mechanism of molecular relaxation accompanying sorption is the “plasticizing effect.” When small molecules penetrate into interchain spaces of the polymer network (the driving force would be the osmotic pressure resulting froni the concentration gradient between the molecular domain concerned and its surround- ings), the average distance between the chains is increased and as a result segment gains more free volume for thermal fluctuations. The increased interchain spacing causes the intermolecular attraction restricting the thermal motion of segments to decrease, and this also makes the segmental motion more free. All these effects will make configurational change of the polymer chains easier than in the less swollen state. Since the force to deform a polymer by a given amount in a given time depends upon the ease with which the polymer molecules change their configuration to accommo- date themselves to a new strained state, this means that the polymer solid becomes more feasible for deformation as it is swollen by penetrafit. In other words, the polymer becomes softened as it swells. In general, the

547

548 ir . I’UJITA AND A. KISIIIMOTO

two mechanisms mentioned above act concomitantly and cooperatively to make the polymer structure loosen when it is attacked by penetrant molecules. However, the relative importance of the two effects may be different for different polymers and also dependent upon conditions adopted in particular experiments. The simplest case is that in which only one of the two mechanisms is responsible for the observed behavior. Xon-crosslinked linear amorphous polymer solids above their second-order transition temperature probably fit the case where the plasticizing effect is the major factor. On the other hand, polymers crosslinked by penetrant-sensitive bonds may be treated in terms of the scission mechanism, provided the temperature is far helow their second-order transition; a t such temperatures the thermal motion of chain segments is practically “frozen in.”

In a previous paper of this series,’ a theory was presented of the diffusion-controlled stress relaxation in terms of the scission mechanism, and it was found that the behavior of the system Amilan (nylon 6)-water a t temperatures below the second-order transition of the polymer can be described quite satisfactorily in terms of the theoretical model presented. This theory assumes the diffusion coefficient of the penetrant to be dependent on concentration only. The system Amilan-water was confirmed to obey this assumption. Similar experiments carried out on the system polyvinyl alcohol-water revealed a pronounced deviation from the behavior predicted by the theory. It was suggested that this deviation from theory may be ascribed to a remarkable time-dependent nature of the sorption process of the system which could not be pertinently taken into the theoretical treatment.

It is of interest to work out a corresponding theory for the diffusion- controlled stress relaxation in polymers in terms of the plasticizing mechanism. For this purpose, i t is first necessary to establish an appropriate kinetic equation which represents the rate of stress decay due to this mechanism as a function of concentration of penetrant present. By combining this equation with the ordinary diffusion equation it would be possible to obtain an equation which describes the process of relaxation of overall stress in a polymer accompanying slow absorption of a penetrant.

Information useful for establishing the kinetic equation mentioned above may be ohtained by studying how the stress-relaxation process of non-crosslinked linear amorphous polymers is affected by the presence of given amounts of penetrant. This paper presents results of a study undertaken to obtain this sort of information using the systems polyvinyl acetate-water, polyvinyl acetate-methanol, polymethyl acrylate-water, and polymethyl acrylate-methanol. These systems were a11 studied at temperatures above the second-order transitions of the dry polymers to ensure that only the plasticizing effect was controlling. The derivation of the theoretical equation for the softening process accompanying sorption in terms of the experimental information from this study will be presented in the subsequent part of this series.

DII‘I’USION-CONTHOLL~D STRESS RELAXATION. 11 549

EXPERIMENTAL

Materials

The polyvinyl acetate (P\.Ac) aiid polymethyl acrylate (PJlX) used iti the present study were furnished from the Institute for Chemical Re- search, University of Kyoto, through the courtesy of Professor H. Kawai of the Department of Textile Chemistry, University of Kyoto. Detailed stress-relaxation data on this PVAc sample in its dry state hare recently been reported by this laboratory.2

Its viscosity-average molecular weight J?fo was evaluated from the intrinsic viscosity in acetone at 20°C. using Sakurada-Chiba’s equation :3

The PVAc used was not purified further.

1 ~ 3 = 4.52 x 10-4 ~ ~ 0 . 6 2

where [ v ] is expressed in 100 ml./g. The molecular weight obtained is 3.5 X lo5, in good agreement with the value of 3.58 X lo5 reported pre- \iously for this same sample.2 The degree of acetylation of the sample was determined by the ordinary method and was found to be 97 mole-yo.

The PMA was purified by precipitation from 2Y0 acetone solution by addition of water as nonsolvent. The viscosity-average molecular weight of the purified PMA was calculated from the intrinsic viscosity in acetone a t 30°C. using Ito-Shimizu-Suzuki’s equation l4

[?] = 1.47 x 10-3 J?fuo.52

The value obtained is 1.1 x lo6. Films of these polymers were made by slow evaporation of the solvent

from 2% acetone solution of the polymer cast on a mercury surface. The films were first dried in air and then vacuum dried for about a day, both a t room temperature. Residual solvent was removed by leaching the dry film in distilled water for a period from 3 to 7 days. The film mas again dried under reduced pressure a t room temperature to constant weight. Since our previous investigation on polyvinyl acetate had indicated that sufficient annealing of the sample was quite important to obtain reproducible viscoelastic data, the polymer films thus obtained mere made sufficiently anneal by keeping them a t an elevated temperature (60°C. for PVAc and 40°C. for PMA) for a long enough interval of time. The annealed samples were stored over P205 iii a desiccator. Film thickness waq varied in a range from 3.5 x lop3 to 17.1 X cm., dependiiig oil the purposes of respective measurements as well as the conditions of a particular experiment.

The methanol used as a penetratit ~ v a s purified by the fractional distilla- tion from a reagent grade product after the removal of acetone by KOH and of n-:itt.r l)y :inhydroiis potns.ium c:irhotiate. In view of the finding of l,oti,o n i i t l ‘I’honipsoiiJ that i h ~ txte of‘ m y t i o i l of acetolie in polyvinyl wt.tat? lincl kwtw protiouiicdly iiifliit~iic~ecl l)g ;i h t i i a l l :tinoiltit of \rater present i t t thtb peii(+raiit, the twno\.:il of natw from the methanol was made as thoroughly and carefully as posiible. The water used a h mother penetrant was a usual distilled water.

550 €1. FUJlTA AND A. KISEIIMOTO

Apparatus and Procedure

An apparatus was specially designed to make possible measurements of stress relaxation in a polymer film under controlled pressures of various low molecular weight vapors. This was constructed by joining a relasometer unit to the main body of a usual sorption apparatus. Figure 1 shows a schematic diagram of the entire apparatus. The sorption units were essentially similar to those designed by Prager and Long.G A Westinghouse tungsten wire kindly furnished by the National Bureau of Standards in Washington was used to make weighing spiral springs for sorption measurements. The relasometer was a modification of an apparatus designed by Berry' in his study on the stress relaxation in vulcanized rubbers accompanying degradation due to

Their force constants were from 3 to 5 mg. mm.-l.

TO VACUUM PUMP

Fig. 1. Schematic diagram of the apparatus including both sorption and rclaxation measuring units. (a) relaxonieter unit; (b) sorption unit; (c, d) water bath; ( e ) vapor generating unit; (f) vapor reservoir; (8) mercury manometer.

oxidation. The details of its mechanism and the procedure for actual use will be described elsewhere.* The relaxometer was tightly fixed in a specially designed glass tube, which, in turn, was placed near the sorption tube and connected through its side arm to the main glass path of the sorption apparatus. Both sorption and relasometer tubes were immersed in a thermostatted water bath controlled to within +0.2"C. The vapor generating unit was also immersed in another thermostatted water bath controlled to the same degree of accuracy. The entire apparatus was mounted in a large air temperature bath controlled to within f 1°C.

Two films of appropriate size and thickness of the given polynier were introduced in the sorption and relaxation tubes, respectively; one was suspended from the tungsten spiral spring and the other was fixed between the upper and lower clamps of the relaxonieter a t strain-free state. After the path from the vapor reservoir to these measuring units had been closed, the two tubes were evacuated to a pressure of mm. Hg. The path to

A schematic diagram of it is shown in Figure 2 .

DIFFUSION-CONTROLLED STRESS RELAXATION. I1 551

the vacuum pump was then closed, and the vapor of the given pressure was allowed to flow into the evacuated tubes from the reservoir. The accompanying weight change of the film was followed as a function of time by measuring the estension of the spring by means of a travelling microscope. One to two hours after the sorption equilihrium had heen reached, the film in the relasometer tube was quickly stretched by a given amount and the

13

- 14

l O C M

13 13

Fig. 2. Details of the relaxonictcr unit. ( I , $ ) Upper and lower clamps; (3) platforill; (4) screw rod; (5) sample film; (6) helical spring; (7) stopper; (8) leaf spring; (9) rubber joint; (10) plate stand; (11) rubber supporter; (12) glass tube; (IS) glass key; (14) glass plug with mercury trough; (15) side arm.

subsequent decay of stress was followed as a function of time according to the manual operation to be described elsewhere.8 In this way, both sorption and stress relaxation were determined under identical conditions of vapor pressure and temperature.

For water used as a penetrant the vapor pressure was kept to within + l n m . Hg during a run for all the cases studied. The corresponding figure for methanol used as another penetrant was f 3 mm. Hg. The travelling microscope permitted reading to the accuracy of 0.01 mm. This cor- responds to the accuracy of 0.034.05 mg. for weighing and to the accuracy

552 11. FUJlTA AND A. KISliIMOTO

of 3.97 mg. for force determination. All glass stopcocks were kept vacuum tight using a Dow silicone grease. The glass key was connected to the main unit of the relaxometer in the tube through a mercury sealing trough to obviate leakage accompanying many turns of i t during a relaxation run.

Measurements on I’VAc were carried out a t 40 and 50°C. for water and a t 40°C. for methanol. Those on PMA were made a t 20 and 40°C. for water and a t 40°C. for methanol. The water studies covered the complete range of relative pressure from 0 to 100% a t all temperatures treated, while the methanol studies treated only limited ranges of relative pressure. This was due to the fact that methanol is a solvent for I’VAc and a swelling agent for PMA. The measurements a t lOOyo R.H. were undertaken by immersing the sample in pure water.

RESULTS

Sorption

Figure 3 shows the sorption isotherm data on the system 1’VAc-water at 40°C. and those on the system PMA-water a t 20 and 40°C. The dat,a of Thompson and Longg on the system PVAc-water a t 40°C. are included

Fig. 3. Sorption isotherms of the svstcms f’17Ac-wntrr at 40°C. and PMA-water at Thompson arid 1,ong’s data on the svstem PVAc-uater at 20 and 40°C.

40°C. Other symbols: present data. Solid circles:

for comparison. Our values agree well with theirs, except in t.he region of high relative humidity. The equilibrium regains of water in I’MA are practically the same a t 20 and 40°C. when compared at,equal relativehumid- ities. This smie tarelid ws coufirmed for the d:tt:L on the system I’VAc- water at 40 :mcl 50°C. In view of‘ the gre:it difficdty of miking reIi:ihl(: experiments at saturation pressure, t,he equilihriwii \\nter csontent, :tt t.liis limit was estimated by suitable extrapolation from the values a t lower

DIFFUSION-CONTROLLED STHESS IIELAXATION. 11 553

pressures. This procedure yielded a value of 5.4 X g. (water)/g.- (dry polymer) for PVAc at either 40 or 50"C., and a value of 3.2 X g./g. for PAfA a t either 20 and 40°C. These extrapolated values were adopted as the equilibrium water regains for the respective polymers a t 100% R.H.

Isotherms of the systems I'VAc-methanol and PMA-niethanol were determined only in a range of low relative pressure, and i t was found that the equilibrium regains were far larger, in either of the systems, than those in the corresponding water systems. Where comparison was made, the shapes of the isotherms on both water and methanol systems were similar.

Integral sorption measurements on PVAc were carried out in the range from 25 to 87% R.H. for water, and in the range of pressures from 36 to

1.0

8 \" gr 0.5

0 75

Fig. 4. Integral sorption curves for the system I'VAc-water a t 40°C. aiid 48 mm. Hg vapor pressurr and for the figstem PVAr-methanol at 40°C. and 130 mm. Hg vapor prcwwrc. Oprn cirvles, adsorption. Solid rircles, desorption.

130 nun. Hg for methanol, both a t 40°C. By way of example, the Q/Qm zw. (t)"*/2X plots for absorption and desorption a t a particular external pressure are shown in Figure 4, where the solid line refers to water and the dashed line to methanol. Q is the amount of penetrant sorbed a t a time t (min.), Q m is its equilibrium value, and X is one-half of the thickness of the film (in em.). In both cases bho\vn, the initial parts of' absorption and desorption plots are linear, indicating the diffusion to be IJickian. The coin- cidence of the absorption and desorption curves observed for water a t this particular pressure was found to hoId d 5 0 a t other pressures studied. In addition, it was confirmed that the initial slope of such a coincident curve was prartically independent of equilibrium water regain. These facts suggest the diffusion codFic.ietrt, of witcr in I'V.Ic. at this trtnperature to tw c.onc.entrLttioti-iiidepellclelit 't'his 1 w i l t iz i i i agwenieiit I\ it11 tlir fiiidiiig of Long and Thonipsoiil" for the h m i c system. Contr:iry to the vase of water, the initial slopes of absorption and desorptioii plots for methanol

554 II. FUJITA AND A. KISIIIMOTO

are markedly different from one another. The similar measurements at other pressures showed that these slopes were strongly dependent upon equilibrium methanol regain ; specifically, they both increased sharply ivith increasing external vapor pressure. This behavior suggests that the diffusioii coefficient of niethanol in 1'V.k- a t this temperature depends strongly on concentration. This is consistent with the view that the diffusion of organic vapors in linear polymer above their second-order transition is Fickian and concentration-dependent. l1

The general features of integral sorption behavior of PMA for water and for methanol were essentially similar to those of PVA for the same pene- trants. The process was Fickian in all cases studied. For water the diffusion coefficient was independent of concentration, while for methanol the diffusion was markedly concentration-dependent. The evaluation of these diffusion coefficients as a function of concentration is presented in a subsequent section.

8 0 ,

-iOg t JMINS. )

Fig. 5. Relaxation modulus curves, E(1), of swollen PVAc. Solid lines with open circles, PVAc-water at 4OOC. (solid circles with vertical segment indicate duplicate run). Solid line with solid circles, PVAc-water a t 50°C. Dashed lines with double circles, PVAc-methanol a t 40°C. Numbers attached to lines refer to vapor pressure used in each run; 1 (0% R.H.), 2 (Z% R.H.), 3 (33% R.H.), 4 (49% R.H.), 5 (55% R.H.), 6 (63% R.H.), 7 (69% R.H.), 8 (82% R.H.), 9 (85?6 R.H.), 10 (90% R.H.), 11 (93% R.H.), 12 (100% R.H.); 1' (62 mm. Hg), 2' (77 mm. Hg), 3' (125m1n. Hg); 1" (45% R.H.), 2" (100% R.H.).

DIFFUSION-CONTROLLED STnESS HELAXATION. I1 555

Stress Relaxation

Figure 5 shows complete stress-relaxation data on the system PVAc- water a t 40°C. as a function of relative humidity, along with partial data on the same system a t 50°C. and those on the system PVAc-methanol a t 40°C. In this figure, t,he ordinate is the logarithm of relaxation modulus

8 - log t ( M I N S . )

Fig. 6. Master relaxation modulus curve for pure PVAc a t 40 and 50°C. Solid lines, Chain line, data data from water studies.

from previous experiments on dry sample.2 Dashed line, data from methanol studies.

Fig. i i z . Concentration shift factor a, as a function of equilibrium water rcgltin C' for PVAc at 40 and 5OOC.

Fig. i b . Concentration shift factor a, as a function of equilibrium methanol regain C for PVAc a t 40°C.

E(t) in dynes/cm.2 and the abscissa is the logarithm of time t in minutes. The relasation modulus was calculated from measured stress divided by the cross-sectional area of the swollen polymer film a t strain-free state. This area was computed from that of the pure polymer using the corresponding

556 Ir. FUJITA AND A. KISIIIMOTO

equilibrium vapor regain on the assumptions that the swelling had occurred isotropically and that the swollen volume was the sum of the volume of dry polymer and that of vapor ab~orhed .~ Reproducibility of stress measurements mas checked a t 30°C. and 6:370 It.H., with quite satisfactory results. Reliable data were not obtained in the initial stages of re1ax:ttioii ( L e . , in

- log t ( M I N S . )

Fig. 8. Master relaxation modulus curve for pure PhlA a t 40°C. derived from experiments on swollen samples.

- CXIOZ(G./G.)

Fig. 9a. Concentration shift factor a, as a function of equilibrium wat,er regain C for I'MA at 20 and 40°C.

Fig. 96. Concentration shift factor a, as a function of equilibrium methanol regain C for PNIA at 40°C.

the region of time smaller than about 10 min.) a t low vapor pressures. In this region the rate of stress relaxatioii was too rapid to be followed accu- rately by means of our manually operating relauometer.

It is seen from Figure .5 that, the family of re1:txation curves shown is quite similar to that ohtniiied for thc: pure polymer by changing the temperature of mensurement over a range. As a matter of fact, as is the case with data from varying temperature measurements, the individual curves in

DIFFUSION-CON'TROLLl3D STRESS RE1,AYATION. I t .is7

Figure 5 can be shifted horizontally by appropriate amounts to form :I single composite curve. The three curves in Figure G present such composite relaxation curves on PVAc, all reduced to the dry states a t the respective temperatures from the water data and the methanol data. It is seen that a t a fixed temperature (40°C.) both water and methanol studies give an identical composite curve within the limit of experimental accuracy. Furthermore, the curve from the water studies a t 50°C. can be superposed to that a t 40°C. by further horizontal shift along the log time axis. The shift factor a t a fixed temperature depends upon the sort of polymer-vapor pair as well as the equilibrium vapor regain in the given polymer. In 1;ig- ure 7a the logarithm of shift factor a, from the water studies a t 40 and 50°C. is plotted against the equilibrium water regain, and in Fig. 72, the corresponding data from the methanol studies a t 40°C. are shown as a function of equilibrium methanol regain.

The behavior essentially similar to that described above for the systems PVAc-water and 1'VAc-methanol n as observed on the corresponding systems with PMA as the polymer a t the tn-o temperatures studied. The horizontal shifts of the experimental relaxation curves for different vapor concentrations a t a fixed temperature resulted in a single composite curve for all temperatures and vapors treated. The composite curves from different pairs of temperature and vapor were further superposed from one another by horizontal displacemeiits along the log time axis. In Figure 8 the final composite relaxation reduced to pure PMA a t 40°C. is shown on a double logarithm graph. Figures 9a and 96 give the shift factor a, vs. equilibrium regain plots for the system I'MA-water a t 20 and 40°C. and for the system PMA-methanol a t 40"C., respectively.

DISCUSSION

Sorption

has demonstrated the fact that the sorption process of water or organic vapors into linear high polymers above their second-order transition temperature follows the type usually referred to as Fickian. That is, the observed process can be best interpreted in terms of the mathematical solution of the Fick equation with a diffusion coefficient dependent on concentration only, subject to the instantaneous equilibrium surface condition. The sorption data on either PVAc or I'MA presented above conform well to this general rule. This is expected because all the present experiments were conducted a t temperatures above the transition points of the respective polymers a t their pure state; literature reports that the transition temperature of pure PVAc lies between 28 and 30°C." and that of pure PMA is around 3"C.,12 provided the molecular weight is sufficiently high.

Several m e t h o d ~ ~ ~ . ~ * have recently been proposed to evaluate the diffusion coefficient as a function of concentration from a series of Fickian sorption data. Figure 10 shows the diffusion coefficient D vs. concentration C rela-

Recent

558 H. FUJITA AND A. KISHIMOTO

tions obtained for water and methanol in PVAc and I'MA, all at 40°C. by applying some of those methods. The average diffusion coefficient D us. concentration data obtained by Long and associate for the systems PVAc- acetone" and PVAc-methan~l~~ at 40°C. were analyzed to derive the true diffusion coefficient us. concentration relations, and the results obtained are included in Figure 10 for comparison. It is rather striking that for both polymers the diffusion coefficient of water is practically concentration- independent over the entire range. That of methanol in either polymer

0 2 6 8 10 - CXlO'(G./G.)

Fig. 10. Diffusion coefficient us. concentration relations on PVAc and PhTA nt 40°C. Solid lines refer to PVAc and dashed linps to PMA.

depends pronouncedly on concent.ration in an exponential fashion. The D us. C curve for acetone in 1'VAc approximately follows an exponential relation with a steeper slope than for methanol in the same polymer. From this set of D us. C curves on PVAc, one may notice a general trend that for a given polymer at a fixed temperature (above its second-order transition point) the dependence of D on C becomes stronger as the D value at zero concentration becomes smaller. Park showed from experiments on the systems polystyrene-halomethanes that D values at the limit of zero penetrant concentration decrease exponentially with increasing molecular volume of penetrant.I6 It is shown that the present data on PVAc approximately obey this empirical relationship. The values of D for water in PVAc a t 40°C. yield 6.5 X This value may be compared favorably to the mean value of 7.4 x cm.2/min.

cm.2/min. as a mean value.


derived from the data of Long and Thompson1o on the same system at the same temperature. The limiting D values of methanol at zero concentration in PVAc a t 40°C. are 6.4 X cm.2/min. from our data and 8.4 X lo+ cm.2/min. from Kokes and Long's data. l5

Stress Relaxation

It has been shown above that for all polymer-diluent pairs studied in this work, stress-relaxation curves :it constant temperature for different diluent concentrations can be reduced to a single master relaxation curve tiy proper shift along the log time axis. This time-concentration reduction property can be accounted for if i t is assumed that all relaxation mechanisms-at least those contributing to the mechanical behavior in the rubbery consist- ency investigated in this work-have the same dependence on diluent concentration in the polymer. This means that all relaxation times are changed by a common factor a, when the diluent concentration of the system is changed from zero to C. If this assumption is combined with the postulate that each type of relaxation mechanism contributes in propor- tion to the mass of polymer contained per unit volume of the system, i t is shown, according to the procedure used by Ferry,l7 that at fixed temperature the relaxation modulus curve E(t) of a swollen polymer with diluent concentration C is represented by

and that the steady-flow viscosity q by

Here Eo(t) and 70 are the relaxation modulus curve and the steady-flow viscosity of the pure polymer at the same temperature, respectively. The diluent concentration C is expressed in grams of diluent per unit gram of dry polymer. ps and pp are the densities of the diluent and the dry polymer, respectively. It is shown from equation (1) that relaxation modulus curves (plotted on a log time axis) for different diluent concentrations, after hav- ing been corrected for the concentration factor 1 + (pp /p . ) C, can be superposed with each other by shifting them horizontally by amounts of log a,. The resulting curve should give the relaxation modulus curve of the pure polymer, Eo(t), a t the given temperature. Equation (2) indicates that the value of a, for this horizontal shift can be obtained simply from the ratio of the steady-flow viscosities of the pure polymer and the swollen polymer of concentration C. Conversely, if the values of al, needed to make E(t) curves a t different C superpose to obtain a single composite curve for the pure polymer are determined experimentally, the concentration dependence of the steady-flow viscosity of' the given polymer-diluent system a t the given temperature can be evaluated.

All the relaxation modulus data obtained in this investigation were made to superpose to form smooth relaxation curves of pure polymers a t given temperatures without applying vertical shift due to the concentration fac-

560 11. NJJITA AND A. KISHIMOTO

tor. This came out of the fact that in the ranges of diluent concentration treated in this work the concentration factor was so close to unity that the correction of this sort was practically within the accuracy of stress deter- minations by the experimental technique used. Application of this correction was actually made to all data, but no essential alteration was found on the final composite relaxation modulus curves. It is apparent that this correction becomes important in treating data on systems containing much more diluent than studied here. However, it is rather doubtful if the postulate made to derive this concentration factor is applicable over an extended range from pure polymer to dilute solutions.'*

The superposed curves in Figures 6 and 8 demonstrate that the familiar time-temperature superposition principle of linear viscoelastic behavior, as evidenced recently on many polymers, holds well for the Eo(t) curves derived above from the time-concentration reduction scheme. That is, the Eo(t) curves derived a t different temperatures can be superposed with each other by further horizontal shift to a particular temperature chosen as the reference state. Figure 6 includes for comparison the relaxation modulus curve a t 40°C. derived from the previously reported relaxation modulus data2 at 75°C. on the same dry PVAc sample in terms of the Wil- liams-Ferry equationlg for the temperature shift factor aT. It is seen that the Eo(t) curves derived from temperature and concentration reduction pro- cedures agree quite well with each other, except in the regions of sufficiently small and large time-scale. The slight deviation of the previous data from the present in the initial zone of the rubbery region is admittedly due to the insufficient moisture control in the previous experiment on the pure polymer.

Williams, Landel, and Ferrylg showed that a remarkably accurate empirical equation for the temperature shift factor aT can be derived from Doo- little's free volume equation for the viscosity of simple liquids.20 We now try to obtain an equation for the concentration shift factor a, from Doo- little's equation using a similar procedure.

The Doolittle viscosity equation as modified by Williams, Landel, and Ferry to fit. supercooled systems may he written

lnq = InA + l/f (3)

where 11 is the steady-flow viscosity, f is the fractional free volume of the system, and A is a parameter dependent on the nature of the given sub- stance. When this equation is applied to swollen polymers, f may depend primarily on both temperature T and diluent concentration C. So it is convenient for this case to write equation (3) in the form:

lnq = 1nA + l/f(T,C) (4)

For the pure polymer this becomes

In qo = In A + l/f(T,O) (5)


Combination of equation (2 ) with equations (4) and (5) leads to

1% ac = 1% + (P , /PS)Cl + (1/2.303) [l/f(T,C) - MT,O)I (6)

For such sufficiently low diluent concentrations as treated in the present experiments, equation (6) may be simplified to

log U, = (1/2.303) [l/f(T,C) - l / f (T ,O)] (7) The dependence of free volume on temperature in pure polymers is usu-

ally assumed to he due to the diff erelice between the thermal expansion coefficients above and below the second-order transition temperature, aZ. l9

This leads to

f(T,o) = f(T,O, 0) + m z ( T - Tgo) (8)

where T,O stands for the second-order transition temperature of the pure polymer.

No corresponding information is yet available about the concentration dependence of free volume a t fixed temperature. It would be reasonable to assume that the free volume of the system increases linearly with diluent concentration in the range of sufficiently lorn concentrations. This results in

f(T,C) = f(T,O) + PC (9) Here p is a parameter representing the contribution of the given diluent to the increase of free volume. While 0 should be independent of C to a first approximation, it may depend on the sort of polymer-solvent pair.

Substitution of equation (9) into equation (7) and rearrangement yields

2.303f(T,0)2 log a, 1 + 2.303f(T,O) log a, = pc -

Thus, a plot of the quantity in the left side of equation (10) against C should give a straight line passing through the coordinate origin with a slope of p. The value of f(T,O) can be calculated from equation (8), provided f(T,O,O), 012, and Too are known for the given polymer. Here we may use forf(T,O,O) and a2 the “universal” values as suggested by Williams, Landel, and Ferry,’9 i.e.,f(Too, 0) = 0.025 and a2 = 4.8 x lop4 (deg.-I). For the two polymers treated in this study we take Tg0 = 28°C. for PVAc and Tgo = 3°C. for PMA from literature data.

The concentration shift factor data shown in Figures 7a, 76, 9a, and 9b are substituted in the left side of equation (10) together with the corresponding values of f(T,O), and the values obtained are plotted against equilibrium regain C in Figures l la (1’VAc-water), l l b (PVAc-methanol), 12a (PMA-water), and 12b (PMA-methanol). It is seen from these figures that our a, data on the four pairs of polymer and diluent fit well to equation (10) at all temperatures studied, except in the region of C smaller than about 0.01 g./g. on the system PVAc-water a t 40°C. At present, it is not clear to us why this particular system deviates from the predicted behavior a t these


0.015

w b 0.010 _ I

low concentrations of water. Values of 0 were determined from average straight lines drawn in these graphs, and are recorded in Table I. It appears from Figures l l a and 12a that the values of 0 for a given pair of polymer and diluent are practically independent of temperature. Of course, the ranges of temperature treated in this study are so limited that this fact must be accepted with some reservation. For both PVAc and PMA the B values of

I I I I I I

Fig. lla. Test of equation (10) on the system PVAc-water a t 40" (open circles) and 50°C. (solid circles).

Fig. llb. Test of equation (10) on the system PVAc-methanol a t 40°C.

Fig. 12a. Test of equation (10) on the system PMA-water at 20" (open circlrs) anti 40°C. (solid circles).

Fig. 12b. Test of equation (10) on the system !?VIA-methanol a t 40°C.

water are larger than those of methanol. This fact implies that water is more effective than methanol to get these amorphous linear polymers softened. I n connection with this, it is of interest to note that water dif- fuses more rapidly in these polymers than does methanol when compared at the same temperature. It is not clear whether this correlation between diffusion and plasticizing effect is significant or not.


TABLE I Values of f l for the Four Pairs of Polymer and Diluent Studied

Polymer

Polyvinyl acetate

Polymethyl acrylate

From equations (8) and (9) we have

with

Diluent Water Methanol

40 and 50°C. 40°C.

20 and 40°C. 4OOC. 0.37 0.28

0.30 0 .23

Fox and FloryZ1 considered that the second-order transition of a given polymer system occurs at a temperature at which the fractional free volume of the system reaches a certain critical value. Equation (8) indicates that this critical fractional free volume is given byf(T,O, 0). This value should also be applied to systems containing diluents. Thus, we find from equation (11) that the second-order transition temperature of a system with diluent concentration C should be To given by equation (la), provided both a2 and p are independent of temperature. Equation (12) indicates that this To should be a linearly decreasing function of C with a slope equal to @/az. It is important to note that this relation may hold only for sufficiently small concentrations of diluent. If the To us. C data on a given system of polymer and diluent obeys this linear relation and, more- over, if the value of a2 for the system is known, we may evaluate @ from the slope of such linear relation. It would be of considerable interest to compare the /3 value obtained in this way from T, us. C data with that derived separately from stress-relaxation data in terms of the time-concentration reduction scheme. Unfortunately, no To data to make this comparison possible are available for the polymer-diluent pairs investigated in this study. However, extensive data for To as a function of C are available from the literature on a variety of polymer-diluent pairs other than those herein studied. Careful survey of those data indicates that T, decreases linearly with C at sufficiently low concentrations, in agreement with the prediction from equation (12). By assuming for a2 the “universal” value as quoted above, the ,8 values of some amorphous linear polymer-diluent systems have been calculated from those data, and are recorded in Table 11. The very detailed data on polymethyl methacrylate and on polystyrene by Jenckel and HeuschZ2 were particularly useful to complete this table. I t is seen that the p values of these systems are in the range from about 0.3 down to 0.05. A general trend is observed that P for a given polymer decreases with increasing molecular weight of diluent, M . However, no

564 1%. FUJITA AND A. KISHIMOTO

Polymer

Polyvinyl acetate Polystyrene

TABLE I1 Values of Calculated from Ye us. C Data

Diluent

Acetonyl acetone Tricresyl-phosphate C ~ H S N O ~ CsH5CH3 C6H6

CHnClz CHCli ccl4 CS2 Methyl acetate Ethyl acetate n-Butyl acetate Amy1 butyrate

C6H6 Polymethyl methacrylate CHCI,

C~HSCH, C6H5X02 CHjCOCHI C4HgOH Methyl acetate Ethyl acetate Propyl acetate Butyl acetate Diethylphthalate

P

0.14 0.13 0 . 1 6 0.20 0.20 0.22 0.18 0.16 0.13 0.27 0.28 0.28 0 .22 0 .22 0.08 0.15 0.15 0 .14 0.25 0.19 0.25 0 .24 0.24 0.24 0.07 (120°C.) 0.05 (140°C.)

Ref.

11 22 22 22 22 24 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 23”

a Values determined from reported viscosity us. concentration data by application of equation (10).

unique relation can be established between P and M , but P’s for diluents of different chemical constituents but with the same molecular weight are generally different from one another. This suggests that the plasticizing efficiency of a diluent for a given polymer, which is characterized by p, depends not only on the size of diluent molecules but also on its specific interaction (either physical or chemical) with the polymer chain element.

Grateful acknowledgment is made for the kindness of Professor H. Kawai of the De- partment of Textile Chemistry, University of Kyoto, who prepared the PMA sample specifically for this investigation under carefully controlled polymerization conditions. Thanks are also due Dr. H. Leaderman and Dr. S. L. Madorsky of the National Bureau of Standards, who supplied US with a fine tungsten wire from which the sorption spring balance used in this work was made according to their relevant indications. This investigation was supported in part by a grant from the cooperative research program on the physical properties of high polymers under the supervision of Professor J. Furuichi of the Department of Physics, Hokkaido University.

DIFFUSION-CONTHOLLED STRESS RELAXATION. I1 565

References I . A. Kishimoto and H. Pujita, Kolloid-Z., 150, 24 (1957). 2. :<. I. Sakurada and T. Chiba, J . Chem. SOC. Japan (Znd. Chenz. Sect.), 47, 135 (1944). 4. H. Ito, S. Shimizu, and S. Suzuki, ibid. , 59,930 (1956). 5. F. A. 1,ong and 1,. J . Thompson, J . Polymer Sci., 14, 321 (1954). 6. S. Prager and F. A. Long, J . Am. Chem. SOC., 73,4072 (1951). 7. J. P. Berry and W. F. Watson, J . Polymer Sci., 18,201 (1955). 8. K. Ninomiya, A. Kishimoto, and H. Fujita, Chem. High Polymers (Japan), 14,

9. L. J. Thompson and F. A. Long, J . Am. Chcna. SOC., 76, 5886 (1954).

K. Xinomiya and H. Fujita, J . Colloid Sci., 12,204 (1957).

504 (1957).

10. F. A. Long and L. J. Thompson, J . Polymer Sci., 15,413 (1955). 11. R. J. Kokes, F. A. Long, and J. L. Hoard, J . Chem. Phys., 20, 1711 (1952). 12. E. Jenckel, Z. physik. Chem. (Leipzig), A190, 24 (1941) (cited from Ref. 19, p.

13. H. Fujita and A. Kishimoto, Textile Research J., 22, 94 (1952); 23, 59 (1953). 14. J. Crank, Trans. Faraday SOC., 51, 1632 (1955). 15. R. J. Kokes and F. A. Long, J . Am. Chem. Soc., 75,6142 (1953). 16. G. S. Park, Trans. Faraday SOC., 46, 684 (1950); 47, 1007 (1951). 17. J . D. Ferry, J . Am. Chem. Soc., 72, 3746 (1950). 18. J. D. Ferry, E. R. Fitzgerald, RI . F. Johnson, and L. D. Grandine, Jr., J . Appl.

Phys., 22, 717 (1951); F. E. Helders, J. D. Ferry, H. Markovitz, and L. J. Zapas, J . Phys. Chem., 60, 1575 (1956).

19. M. L. Williams, R. F. Landel, and J. D. Ferry, J . Am. Chem. Soc., 77,3701 (1955). 20. A. K. Doolittle, J. Appi. Phys., 22, 1471 (1951); 23,236 (1952). 21. T. G Fox and P. J. Flory, J . Appl. Phys., 21, 581 (1950); J . Phys. & Colloid

22. E. Jenckel and R. Heusch, Kolloid-Z., 130,89 (1953). 23. F. Bueche, J. Appl. Phys., 26,738 (1955). 24. F. A. Long and R. J. Kokes, J . Am. Chem. SOC., 75,2232 (1953).

3703).

Chem., 55, 221 (1951).

Synopsis

Stress relaxation in the systems PVAc (polyvinyl acetate)-water, PVAc-methanol, PhIA (polymethyi acry1ate)-water, and PRIA-methanol was measured a t several temperatures above the glass-transition points of the pure polymers in ranges of sufficiently low concentrations of diluents. I t was found that the time-concentration superposition principle as proposed by Ferry is applied successfully to thcsc data to obtain master relaxation curves of the pure polymcrs a t fixed temperatures. The master relaxation curves so obtained a t different tcmperatures u-ere further superposed with cach other by horizontal shift along the log time axis, in agreement with the time-temperature superposition principle. The concentration shift factor a, as a function of diluent concentration C a t fixed temperature was well interpreted in terms of the modified Doolittle’s free volume equation for the viscosity of supercooled systems on the assumption that the fractional free volume of the system increases linearly with C. The proportionality coefficient, p, of this linear relation provides a measure of the plasticizing effect of a given diluent on a given polymer. It was found that p is practically independent of temperature a t sufficient.ly small C and is larger for xater than for methanol in both polymers studied. On thc basis of Doolittle’s viscosity equation modified to account for the concentration effect, it is shon-n that the glass-transition temperature Yg of a given polymer-diluent pair decreases with C i n a linear f:tshion in the range of small C. The value of p cwi also be calculated from the slope of this linear relation. Values of p determined in terms of this procedure from literature data on a variety of polymer-diluent pairs are given in a tabular form. For a given polymer the p values generally become


smaller as the size of diluent molecule is increased, but no unique relation can be obtained between the two quantities. Equilibrium sorption and integral sorption processes were studied on the four pairs of polymer and diluent mentioned above, not only to obtain data needed for analyzing the present stress-relaxation data but also to find information which will serve a study described in Part I11 of this series which is concerned with diffusion-controlled stress-relaxation in amorphous linear polymers. Some correlation (though not definite) appears to exist between the diffusion coefficient D (at zero concentration) and the plasticizing parameter p for a given polymer. This may be expected because both D and p are quantities depending on the frequency a t which polymer segments jump cooperatively in the presence of a low molecular weight diluent.

Rbum6 On a mesure les phenombnes de tension-relaxation dans les systbmes acetate de poly-

vinyle (PVAc)-eau, PVAc-methanol, le polyacrylate de mkthyle (PMA)-eau, e t (PMA)- methanol A plusieurs temperatures au dessus des points de transition vitreuse des polymbres purs dans des domaines de concentrations en diluants suffisament basses. On a trouve que le principe de superposition temps-concentration, propose par Ferry, s’ap- plique avec succhs A ces resultats e t fournit des courbes de relaxation maftresses pour les polymbres purs A des temperatures d6termindes. Ces courbes de relaxation maftresses ainsi obtenues A differentes temperatures ont Bt6 en outra superposees l’une sur l’autre par glissement horizontal le long de l’axe des logarithmes du temps en aCcord avec le principe de superposition temps-temperature. La facteur de glissement de concentration a, est fonction de la concentration en diluant C A une temperature determinee; il a 6tB interpret6 Blegamment au moyen de 1’6quation modifiee du volume libre de Doolittle pour la viscosite des systbmes refroidis, en admettant que la fraction de volume libre du systbme croflineairement avec C. Le coefficient de proportionnalit6,p, de cette relation lineaire fournit une mesure de l’effet plastifiant d’un diluant determine pour un polymere donne. On a trouv6 que p est pratiquement independant de la temperature A faible valeurs de C et est plus grand pour l’eau que pour le methanol pour les deux poly- meres Btudies. Sur la base de 1’Bquation viscosimetrique de Doolittle, modifiee pour rendre compte de l’effet de la concentration, on montre que la temperature de transition vitreuse T , d’un systbme polymbre-diluant d&ro€t avec C de fapon lineaire aux faibles valeurs de C. La valeur de p peut 6tre calculee de la tangente de cette relation linbire. On a rassemble dans un tableau les valeurs de p d6terminCes par ce procede aux d6pens des donnees de la litteratwe sur de nombreuses paires polymhre-diluant. Pour un polymkre donne les valeurs de p decroissent g6n6ralement avec une augmentation de la grandeur de la molecule du diluant; aucune relation n’a pu toutefois 6tre 6tablie entm ces deux quantites. Les processus de sorption ii 1’Bquilibre e t de sorption integrale ont Btk 6tudiEs pour les quatre paires de polymbre-diluant cites ci-dessus, ceci non seulement en vue d’obtenir les resultats necessaires A l’analyse des resultats tension-r6laxation, mais Bgalement afin d’obtenir des renseignements relatifs aux ph6nombnes de tension relaxation limit& par diffusion au sein de polymbres lineaires amorphes; I’btude relative A cet aspect fait l’objet de la partie 111 de ce travail. I1 semble exister une correlation (bien que non d6finie) entre le coefficient de diffusion D ( A concentration nulle) e t la para- metre de plastification B pour un polymbre determine. Ceci peut 6tre attendu puisque aussi bien D que p sont fonctions de la frequence A laquelle les segments polymeriques peuvent se mouvoir conjointement en presence d’un diluant de faible poids mol6culaire.

Zusammenfassung Die Spannungsrelaxation in den Systemen PVAc (Polyviny1acetat)-Wasser, PVAc-

Methanol, PMA(Polymethy1acrylat)-Wasser und PMA-Methanol wurde bei verschiedenen Temperaturen oberhalb der Glas-Umwandlungspunkte reiner Polymere in genii- gend niedrigen Koneentrationsbereichen des Verdiinnungsmittels gemessen. Es wurde


gefunden, dass das Zeit-Konzentration-Uberlagerungsprinzip, welches von Ferry vor- geschlagen wurde, erfolgreich auf diese Daten angewendet werden kann, um allgemeine Relaxationskurven reiner Polymere hei festgesetzten Temperaturen zu erhalten. Die so bei verschiedenen Temperaturen erhaltenen allgemeinen Relaxationskurven wurden n-eiterhin durch horizontale Verschiehung an der log-ZeitcAchse entlang miteinander iiberlagert, in Ubereinstimmung mit, dem Z~itcTemperatur-Uberlagerungsprinzip. Der lionzentrations-Verschiebungsfaktor a, als Funktion der Verdunnungsmittel-Konzen- t,ration C bei festgesetzten Temperaturcn wurde mittels der abgeanderten Freivolumen- Gleichung von Doolittle fur die Viskositat unterkuhlter Systeme gut erklart unter der Voraussetzung, dass das fraktionale freie Volumen des Systems linear mit C zunimmt. Der Proportionalitatskoeffizient p dieser linearen Beziehung gibt eine Messung der Weichmacherwirkung ekes gegebenen Verdunnungsmittels auf ein gegebenes Polymer. Es wurde gefunden, dass p bei genugend kleinem C praktiseh unabhangig von der Tem- peratur ist und in den beiden untersuchten Polymeren fur Wasser grosser als fur Meth- anol ist. Auf der Grundlage der Doolittle’schen Viskositatsgleichung, die abgeandert wurde, urn der Konzentrationswirkung Rechnung zu tragen, wird gezeigt, dass die Glas- Umwandlungstemperatur T, eines gegebenen Polymer-Verdunnungsmittelpaares mit C in h e a r e r Weise im Bereich vom kleinen C abnimmt. Der Wert von p kann auch aus der Neigung dieser linearen Beziehung berechnet werden. Werte von p die mittels dieses Vorganges aus Literaturdaten fur eine Anzahl von Polymer-Verdunnungsmittel- Paaren bestimmt wurden, werden in Tabellenform gegeben. Fur ein gegebenes Polymer werden die p-Werte in allgemeinen kleiner, wenn die Grosse der Verdunnungsmittel- nlolekiile erhoht wird, aber es kann zwischen den beiden Mengen keine einformige Beziehung erhalten werden. Gleichgewichts-Sorptions- und integrale Sorptionsvorgange werden an den vier oben envahnten Polymer-Verdunnungsmittel-Paaren untersucht, nicht nur, um Daten zu erhalten, die zur Analyse der gegenwartigen Spannungs-Relaxa- tions-Daten notig sind, sondern auch, um Auskunft zu erhalten, die fur eine in Teil I11 dieser Arbeit beschriebene Untersuchung nutzlich sein wird, die sich mit durch Diffusion kontrollierte Spannungsrelaxation in amorphen linearen Polymeren befasst. Einige Beziehung (aber nicht definitiv) scheint zwischen dem Diffusionskoeffizienten D (bei Konzentration Null) und dem Weichmacher-Parameter p fur ein gegebenes Polymer zu bestehen. Dies kann erwartet werden, da sowohl D als auch p Mengen sind, die von der Haufigkeit abhangen, mit der Polymersegmente kooperativ in der Gegenwart eines Verdunnungsmittels von niedrigem Molekulargeaicht springen.

R.eceived June 19, 1957

Documents

Diffusion-controlled stress relaxation in polymers. II. Stress relaxation in swollen polymers