The moving boundary problem in diffusion in polymer-solvent systems

The moving boundary

651

17. J. ~ Y , V y ~ k o u p r u ~ svoMv~ polimerov. (The V i l o o . e l u ~ ProFile8 of Polymers.) Foreign Literature PubliAhln~ House, 1963 (Rusum translation) 18. A. ~ ~md R. B ~ R l t ~ Tr~m. Faraday Soc. 5: 116, 1955 19. R. M. VAI$I~IN, VyBokomol. moyed. |: 1220, 1964 20. A. A. TAG~t, V. Ye. DKEV&L' and R. A. lgH~k~NA, Vysokomol. soyed. 5: 432, 1963 21. A. A. TAGER and A, I. P O D i A ] K , Vysokomol. soyed. 1: 87, 1963

M

MOVING BOUNDARY PROBLEM IN DIFFUSION IN POLYMERSOLVENT SYSTEMS* R. M. VASENIN, A. YE. CHALYKH and V. I. KOROBKO Moscow Technological Institute of L~ght Industry (Re~ved 15 M~y 1964)

IN CONTACT between a polymer of limited swelling capacity solvent the solvent front moves into the depth of the polymer [1-5] and the boundary between the phases moves in the opposite direction. The first boundary is called the optical boundary and the second the phase boundary. Similar moving boundaries arise when the diffusing substance combines irreversibly with the medium [6, 7] or with substances specially incorporated into the polymer [8-10]. From the mathematical point of view these phenomena are equivalent and constitute the general diffusion problem of moving boundaries, treated by a number of authors [8, 11-16]. Movement of the pha~e boun&zry in polymer-solvent systems has long been well known and is usually defined by the term "swelling", however the term "Kirkendoll effect" is named after the discoverer of this effect in alloys [17]. The kinetics of the swelling of polymers have been studied in considerable detail, but movement of the phase boundary in particular has received little study

[1, 19-24]. Flow of the diffusing substance in relation to the walls of the cell (C-section) is connected with flow at the phase boundary (O-section) by the equation [25] (J1)c=(J1)o~-ClUoc

(1)

where c1 is the solvent concentration at the plmse boundary and Uoc is the rate of movement of the phase boundary. The flow of polymer, (Js)o in relation to the O-section is zero, therefore

~oc~-(Jz)c/Cz

(2)

where cmis the concentration of polymer at the phase boundary and (Ja)c is its flow with respect to the C-section. If when the components are mixed there is * Vysokomol. soyed. 7: No. 4, 593-600, 1965.

652

R . M . VAs~I_~ eta/.

no volume change the C-section coincides with the fixed-volume or V-section, for which the following conditions are satisfied

,(J1)v+L(G)v=0

(3)

where vl and v2 are the partial specific volumes o f the components. From (3) it follows that (J~)c-----(12)v---- --~1 (J1)v/~ 2.

(4)

Substituting the value of the flow from (4) in (2) we obtain

(sj The flQw of solvent through the phase boundary is given by (J1)v---- -- Dv ( & d t z ) ~ x

(6)

where (&/$~)xffix is t h e :solvent concentration gradient at the phase boundary and D v is the coefficient of mutual diffusion of the system. Combining (5) and (6) we find

Uoc =dX/dt=' Dv/(1 --elL)

(7)

Equation (7) is easily simplified by substituting the unidirectional coefficient D 1 for the coefficient of mutual diffusion

D~:Dv/(1--cl~).

(8)

From (7) and (8) we obtain

d ldt=-

(9)

The subscripts to ~, D and c are omitted for simplification. In order to find the concentration gradient at z-----X it is necessary to solve the equation

&l,~t =,,~/ar(D(~/&)).

(10)

The solutions of this for a constant coefficient of diffusion are known [11], They can also be used when D is dependent on the composition of the system. Using the known solutions of (10) for different boundary it is possible to find the solution of the problem in which we are interested. For a semi-infinite body with a constant surface concentration, ce, the rate of movement of the phase boundary is given by • dX/dt----- --c~ (D/~) 1l~. (11) Integration of (11) gives

From (12) it follows that change in the phase-boundary coordinate is proportional to the square root of the time and is in the direction opposite to the direction of flow of the diffusing solvent.

The moving boundezy

663

Experiment shows [23] that equation (12) is applie~le to erosslinked elMtomers but not to polymerB that are crystalline or in the glassy state. Figure 1 shows results for the polyvinylalcohol (PVA)-water system. The X/l-time curves are S-shaped. Curves of this type are associated with variation in surface conventration with time. This changes for two reasons: it increases as a result of absorption and decreases as a result of diffusion of the solvent into the bulk of the polymer. Penetration of solvent molecules into the surf~co layer of the polymer gives rise to local stresses that are r e l e a ~ l by change in the conformation of the macromolecules and possibly by movement of the structural elements of the polymer.

0

,~

12 t~ m/n

8

I. 0'5

•

..... I

•

e~

••eo e r e •

:. .~.. -• •.."• 5

o'Z

..

•e

°e° ~ •

o

1

•

..

4...

.

.

•••

g2::: •l • •

Z

4 " ee° !

.

• . •

:-.

0

• • ••

• o•

..

,::. ~.~"

/.

Z

• e• •

••

.."

;rF

•

:* i

3 ~rn/n

FIO. 1. Movement of the phase b o u n d a r y in the polyvinylalcohol-water system: 1--60, 2--50, 3--40, 4--30, 5--20°; 4,4', 4 " - - e q u a l time of storage in a disiccator.

The rate of stress relaxation is dependent on the physical and phase state of the system. The problem is complex because increase in the concentration of solvent in the surface layer on the one hand leads to an increase in stress, and on the other hand to an increase in the rate of relaxation as a result of cllange in the mobility of the chain~. I t may therefore be assumed that relaxation et~ects will have the greatest influence in the initial stages of the diffuslon process. Assuming a relaxation mechanism of sorption, for the quantity of substance absorbed per unit surface area we obtain.

M----M,(1--• -t/')

(13)

where z is the average relaxation time and M. is the equilibrium quantity of absorbed solvent.

654

R.M. V A s ~ st a/.

Since stress relaxation affects only the initial stages of the process all but the first two terms of tbe series solution of the exponent in equation (13) may be neglected. The absorbed substance diffuses from the surface layer into the bulk of the polymer. In the initial stages this process may be regarded as diffusion from an infinitesimally thin layer into semi-infinite space. Solution of the diffusion equation with a source of constant potentiaJ, equal to Mj/z, leads to the following expression for the surface concentration:

c,= 2c,ltllS/(~D )ll~

(14)

where Me----eeland 1 is the film thickness. Thus the surface concentration is proportional to the square root of the time, and the constant of proportionality is determined by the value of the coefficient of diffusion and the average relaxation time. The surface concentration increases more slowly the higher the rate of diffusion and the lower the relaxation rate. Using this solution we obtain

dX/dt=--~l/~.

(15)

The rate of movement of the phase boundary is constant. Consequently the change in the boundary coordinate is directly proportional to time (Fig. 2)

(le) From tJais equation it follows that under these conditions movement of the phase boundary is controlled by stress relaxation, not by the diffusion process.

/2 0

l

Z

I/me,min

J

FIe. 2. Movement of the phase boundary during the first moments. Systems: polyvinylalcohol-water: 1,60, 2--50, 3--40, 4--30°; polyvinylaleohol-water/acetonemixture (8 : 2): 5-- 60, 6-- 50, 7-- 40°; polyvinylaleohol-water/aeetonor-i~cture(6 : 4): 8-- 50, 9--40, 10-- $0°.

The moving boundary

655

From experimental measurements the time interval during which change in the surface concentration and in the average relaxation time was determined b y means of equation (16). The results are presented in Table 1. TABLE 1. T m ~ FOR ESTABLISHMENT OF THE EQUILIBRIUM SURFACE CONCENTRATION, to, AND THE AVERAGE RELAXATION TIME, v, IN THE DIFFUSION OF SOLVENT INTO A POLYMER

(Polymer-- polyvinylalcohol) l

Ce,g/era31 Solvent

T,

°C

I~0 ~,

T, s e e

sec

sol;fent

Dr

cm2/sec

Water

20 30 40 50 60

240 240 180 150 120

0"595 0.62 0,645 0"665 0"685

1190 620 387 260 160

0.9 1.2 1.8 2.4 3.4

Water-acetone 8 : 2

20 40 50 60

240 180 120 120

0"48 0"56 0"588 0"615

2280 840 500 320

0.16 0-4 0.6 0.9

Water-acetone 6 : 4

30 40 50

300 240 180

0"435 0-472 0"525

1300 710 570

0.15 0.23 0.32

Both magnitudes are dependent on temperature and on the chemical potential of the solvent. The log ~--I/T curve is linear. The energy of activation for relaxation is independent of the composition of the solvent and is 9 kcal/mole. The energy of activation for diffusion is a function of composition and increases with decrease in the chemical potential of the solvent. For the PVA-water system it is 5 kcal/mole and for the PVA-water/acetone (8 : 2} system it is 7 kcal/mole. Movement of the optical boundary has received almost no s t u d y [1-5], hence the nature, causes and site of origin of the optical boundary are not known. I n order to throw light on these problems a study was made of the nature, rate of movement and site of origin of the boundary in polymers in different phase and physical states (in soluble polymers and in polymers of limited swelling capacity, in amorphous a n d crystalline polymers, in elastomers and organic glasses and in some systems above and below the glass temperature). The microscopic and interferometric [26] methods were used. The first method was used to study movement and the second for localization of the optical boundary on the diffusion coordinate. The results are presented in Table 2. I n the majority of cases studied by us a Well defined optical boundary is formed. I t consists of a dark band of width varying from a few to tens of microns

Benzene Ethylbenzene Toluene m -Xylene CC14

Benzene

Polystyrene M---- 1-2 x 106

Polystyrene

Ethylcellulose

M - ~ 2 . 5 × 104

Polyvinylacetate M=6×IO t

!

Benzene i E t h a n o l (96~/o)j

Diehloroethane Acetone Benzene Butylacetate CC14

Polyvinylchloride

SKN-40

Benzene Ethylbenzene m-Xylene Benzene

Solvent

Polyisobutylene M = 8.65 x 106

Polymer

13

8 8 10 8 5

6 18 12 7 3

v

Dv

v

30°

Dv

v

0.25 0.45

4"6

1-8 1-8 2-8 1"8 0.7

1"3 0"25

4

1 9

1"8

1-8 1.8

16

10 11 16 10 6

9 20 16 10 4

10 11 8 9

1.3 0"7

7-1

2.8 3"4 7-1 2"8 1

2.3 11 7-1 2-8 0.45

2"8 3"4 1 "8 2-3

10 6

18

11 14 18 ll 10

12 23 18 15 5

13 14 12 12

Systems with soluble polymers

20°

40°

2.8 1

9

3"4 5-5 9 3-4 2.8

4 15 9 6 0"7

4-6 5"5 4 4

Dv

I

I 13 7

23

13 17 23 13 18

28 22 20 6

14

16

v

50°

4.6 1.3

15

4.6 8 15 4.6 9

22 13-5 11 1

5-5

7"1

DV

log D o,

--2-4 0.7 1.1 --2"4 7.8

2-1 -- 1.9 --0-8 2.33 --0-8

--1.8

--0.23 0.7

cm=/sce

5.6

]

is c o n v e x

--3-1

6.1 --1.8 The curve o f log D - - 1 / T

5.8 10 10 5.8 20

12 7 7.5 14 9

6.7

8.8 10

cal/mole

E X 10 a,

TABLE 2. RATE OF MOVEMENT OF THE OPTICAL BOUNDARY v ( X 1 0 8 e r a / r a i n 1/2) AND THE COEFFICIENTS OF MUTUAL DIFFUSION D v ( × l 0 T cmS/see IN POLYMER-SOLVENT SYSTEMS

OI

v

w

2

v

30 ° Dv

(continu~) w

v

4 5 4

0.254 0.21 0.23

Water

Methanol Ethanol

Polyvinylaleohol M = 7 × 104 P o l y a m i d e A K 60/40

* w is the degree of swe,l.ng (weight fraction of solvent).

2

1.1

4-5

0.29 0.24 0.256

0.7 1-4 0.9

2.2

0.2'

7

2

0-26 0.33

0.6 0.02

6

0.286

0.166

0.324

0-41

0.355

5

5

9

1.4

1.4

4"5

I Dv

Systems with polymers of limited swellability

w

20 °

Benzene

Acetone

Solvent

Polyvinylchloridc

M ~ 1-2 × 105

Polystyrene

Polymer

TABLE

0-362

0"5

0"36

w

3"5 2 6

6.7 8

I1

~v

log D O

curve

0.93

em2/see

o f log D-- lIT is c o n v e x 6 --2-6

The

10.7

kcal/mole

o~ o

6t~8

R . M . V A s ~ N ~ et a/.

depending on the duration of the experiment and on the nature of the components. Between crossed Nicols it is seen as a light band against a dark background. In polymer films containing fissures or that crack during swelling (polystyrene, polyvinylchloride), the boundary gradually becomes uneven. The surface irregularity favours diffusion of the solvent from the fissures into the depth of the polymer. In crystalline polymers (PVA) birefringence arises between the phase and optical boundaries, and this disappears after removal of the solvent. The optical boundary is clearly seen as a contour enveloping the polymer specimen in the interference pattern (Fig. 3).

FIQ. 3. Interference p a t t e r n for the polyvinylacetate-butyl acetate system. Polymer on the left, solvent on the right, and swollen, dissolving polymer between the two.

Concentration and concentration gradient distribution curves are shown in Figure 4. Comparison of Figures 3 and 4 shows that the position of the optical boundary corresponds to the point of inflexion in the distribution curve and to the break concentration gradient curve. Similar interference patterns were obtained at different temperatures and in various systems. The position of the point of inflexion on the ordinate is affected neither by the duration of the experiment nor by temperature. Its position is dependent only on the nature of the components. From the point of view of geometrical optics the appearance of an optical boundary is explained by total internal reflection of light from the surface dividing slightly swollen polymer from a solution of the polymer, where there is a sharp change in optical properties. The distribution curves for the polymer-solvent systems are not Gaussian [11], they are convex in the intermediate concentration region. This is explained by

The moving boundary

-05

I

0

;"-.

0~5

659

I-0

x, mm

F i e . 4. Curve of distribution of concentration (1) and of concentration gradient (2)

for the same system. the dependence of the coefficient of diffusion on concentration. The coefficient of mutual diffusion is lowest in a polymer in the condensed state and reaches its maximal value when the volume fraction of solvent is in the region of 0.4-0.6. Therefore in this region an accumulation of solvent arises, from which it slowly diffuses into the polymer. As s result, at a certain concentration c=, a point of inflexion arises in the distribution curve. The convexity of the c-x curves is greater the greater the difference between the coefficients of dit~usionin the polymer and in the polymer solution. In the rubber-benzene system [27] these coefficients differ by a factor of approximately 10, therefore in rubber (NR, SKS-30)*benzene systems a faint boundary is observed only during the first minute after contact between the solvent and polymer. I f the polymer is previously subjected to swelling in the solvent vapour (to about 10%) the distribution curves are concave toward the abscissa and an optical boundary does not arise. The concentration in the plane corresponfling to the surface of total internal reflection can be determined approximately from the c-z curve. For the polyvinylacetste-buty] acetate system it corresponds to a volume fraction of 0.1, and for the other systems it is a little lower. It is not possible to determine this concentration accurately because of the large error in calculation at the extremities of the distribution curves. The rate of movement of the optical boundary is determined by diffusion of the solvent into the polymer. For quantitative description of this phenomenon it is necessary to find the relationship between thecoordinate z = X , at which the concentration is %, and the duration of the experiment. Diffusion into a pure polymer is a slow process and it may be considered that the position of~the optical boundary corresponds to a section on both sides of which the solvent content is constant. By analogy with the previous argument we find from (1)

u,o=ax *

J1)v/C

NR = Natural rubber SKS-30 -- styrene butadiene polymer, 3 0 ~ styrene

0 7)

660

R . M . VASE~IN e$ a/.

where u~ is the rate of movement of the optical boundary with respect to the walls of the diffusion cell. Combining (6) and (17) we obtain d X /dt~- -- Dv/c ~ (8Cl/5X)x= x

(18)

Making use of the solution of (10) for a constant coefficient of diffusion we find [8] gl/2ae~'erf o~--(ce - c x ) / c x

(19)

where err a is the Gaussian integral error, and the constant a is given by = X/(4Dvt)

= v/(4Dv)

(20)

The term v~-X/tl/2in equation (20) can give rise to a nonlinear rate of movement of the optical boundary. With exclusion of the initial period the experimentally determined, non-linear rate is constant. From (20) it follows that Dv=v~/(2~) 2

(21)

Analysis of the experimental results showed that for systems with soluble polymers the denominator in equation (21) is close to six and for polymers with limited swelling capacity it is close to three. These values of the denominator were used to calculate the coefficient of mutual diffusion. A "compensation" effect was found between the energy of activation and the pre-exponential factor. Study of the rate of movement of the optical boundary is a simple and convenient method of determination of the rates of swelling and solution of polymers and of the integral coefficient of mutual diffusion. By its aid it is possible to determlue, easily and rapidly, the average depth of penetration of a solvent into a polymer after a given time or after an interval of time in which the degree of swelling has reached a given value. CONCLUSIONS

(1) A study has been made of the movement of the phase boundary in the systems PVA-water and PVA-water-acetone. The slow movement of the phase boundary at the commencement of the process is controlled by the clmnge in the surface concentration with time. (2) There are two causes of change in the surface corlcentration, namely sorption and diffusion. The first controls the rate of change of the conformation of the macromolecules and possibly of movement of the structural elements of the polymer. I t is shown t h a t to the first approximation the surface concentration is proportional to the square r o o t of the time. (3) The average relaxation time has been determined. The lower the temperature and the lower the chemical potential of the solvent the greater is the average relaxation time.

The moving boundary

661

(4) T h e presence of a n optical b o u n d a r y has been d e t e c t e d in p o l y m e r s in different p h ~ e a n d physical states (in soluble p o l y m e r s a n d in p o l y m e r s of limited swelling capacity, in a m o r p h o u s a n d crystalline p o l y m e r s a n d in elastomers a n d organic glasses, a b o v e a n d below t h e glass t e m p e r a t u r e ) . (5) T h e optical b o u n d a r y has b e e n localized precisely o n t h e diffusion coordin a t e b y a n i n t e r f e r o m e t r i c micro-method. (6) As a result o f t h e strong d e p e n d e n c e of t h e coefficient of diffusion on conc e n t r a t i o n t w o regions differing sharply in optical properties arise in t h e system. T h e optical b o u n d a r y is t h e result o f t o t a l i n t e r n a l reflection f r o m t h e surface s e p a r a t i n g these regions. T r a ~ / a ~ by E. O. P m J ~ P s

REFERENCES 1. 2. 3. 4. 5, 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

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