On the swelling of polymers in vapours of low molecular liquids as exemplified by block copolymers of polystyrene with polybutadiene

1478

P . I . ZUBOV et al. REFERENCES

1. L. I. KOMAROVA, S. N. SALAZKIN, V. V. KORSHAK, S. Y. VINOGRADOVA, V. I. NIKOLAICHIK, Ye. E. ZABOROVSKAYA and L A. BULGAKOVA, Vysokomol. soyed. BI6: 718, 1974 (Not translated in Polymer Sci. U.S.S.R.) 2. V. V. KORSHAK, S. V. VINOGRADOVA, G. L. SLONIMSKH~ L. N. B ~ A , S.' N. SALAZKIN, A. A. ASKADSKII, Ya. S. VYGODSKII, V. I. BURKOVICH, Z. V. GERASHCHENKO, V. F. BLINOV a n d I. A. BULGAKOVA, U.S.S.R. Pat. 526641, 1974; Byull. izob., No. 32, p. 78, 1976 3. S. N. SALAZKIN, K a n d i d a t s k a y a disscrtatsiya, Moscow, D. I. Mendeleyeva I n s t i t u t e of Chem a n d Technol., 1965 4. V. V. KORSHAK, S. V. VINOGRADOVA and S. N. SALAZKIN, Vysokomol. soyed. 4: 339, 1962 (Not translated in Polymer Sci. U.S.S.R.) 5. A. A. ASKADSKII, F i z i k o k h i m i y a poliarylatov (Physicochemistry of Polyarylates). Izd. " K h i m i y a " , 1968 6. L. N. BELKINA, Voprosy radioelektroniki, seriya obshchetekhnieheskaya, :No. 12, p. 169, 1973 7. G. A. DUBOV a n d V. R. REGEL, Zh. tekhn, fiziki 25: 2543, 1955 8. G. L. SLONIMSKII a n d A. A. ASKADSKII, Mekhanika polimerov, No. 1, 36, 1965 9. A. A. ASKADSKII, Deformatsiya polimerov (Deformation of Polymers). Izd. " K h i m i y a " , 1973

.:Polymer ScienceU.S.S.R. Vol. 22, No. 6, pp. 1478-1492, 1980

Printed In Poland

0032-3950/80/061478-15507.50[0 1981 Pergamon Pre~ :Ltd.

ON THE SWELLING OF POLYMERS IN VAPOURS OF LOW MOLECULAR LIQUIDS AS EXEMPLIFIED BY BLOCK COPOLYMERS OF POLYSTYRENE WITH POLYBUTADIENE* P . I . ZUBOV, Y u . I . ~¢LkTVEYEV, A . A . ASKADSKII a n d T . A . ANDRYUSHCHENKO Physical Chemistry Institute, U.S.S.R. A c a d e m y of Sciences (Received 18 April 1979)

The swelling of polymers in vapours of low molecular liquids has been analyzed, taking into consideration relaxation phenomena accompanying this process. The idea of a time-dependent diffusion coefficient has been introduced, and the kinetics of tho process have been analyzed on the basis of a model proposed for describing the viscoelastic behaviour of polymeric bodies. I t is shown t h a t all the kinetic peculiarities

* Vysokomol. soyed. A22: No. 6, 1347-1358, 1980.

Swelling of polymers in vapours

1479

of the process may be described if relaxation phenomena during sorption are taken into account. It was also found that the penetration of liquid molecules into a polymer gives rise to degeneration of long stress relaxation times.

O~ A previous occasion we investigated the mechanical relaxation behaviour of block copolymers of polystyrene (PS) with polybutadiene, and showed that th¢; ~tructure and properties of the copolymers may readily be modified by me~ns of particular precipitants and solvents used in the preparation of films [1, 2f. In addition, in earlier work [1] we carried out a prior.il~vestigatiou of sorption by block copolymer films in relatio~ to the prehistory of their preparation. The present paper relates to a detailed analysis of swelling and sorption processes occurring in the block copolymer specimens. It, is clear from the experimental results that the amount of solvent, adsorbed b y the block copolymers is so large that the sorption process cannot take place without considerable swelling of the specimens. Accordingly the latter term will be employed below. On a previous occasion we suggested [3] that the diffusion of low molecular weight liquids or vapours of the these liquids in polymers is determined not only by the entry of sorbate molecules into pores of polymers, but also by conformational regrouping of the macromolccules, i.e. it is' also related to relaxation processes accompanying diffusion processes. Later on [4, 5] we considered the problem of changes in the relaxation behaviour of polymers whilst swelling, as well as changes in conformational sets of the macromolecules. In addition, a model description of the swelling kinetics was developed. Among experimental work that has been carried out in this field there is that reported in [6] relating to a study of the swelling of rubbers in a number of solvents differing in respect to their polarity. It is noteworthy that conformational regrouping of macromoleculos during swelling may lead to a degeneration of long relaxation times. In the case of a polymer that is in the stressed state, the penetration of solvent vapours leads to local (intermolecular) bond scission, and makes for corfformational regrouping of macromoleculcs in the ordering direction. This may entail the expulsion of solvent from a polymeric specimen. These factors all influence both the magnitude and the kinetics of the swelling process and lead to a distinctive type of kinetic curves. Lot us now a t t e m p t to describe kinetic curves for swelling, taking the above factors into account. To do so we have to solve a differential equation corresponding to the second Fickian law. I t is known [7] that to solve the differential equation OC --D Of

OC2 - Ox2

for a plate (film) bounded b y planes of x-~O and x = l under the boundary conditions: C-~CI if x-~O for all t values, C-~C2 if x ~ l for all t values, C : f ( x ) if

P . I . Zvsov et al.

1480

t----O for O < x < / , we have an expression of the type of

~nx ( C(x,t)=C,-p(C2-- C , ) ~x - - 2 ~ C'c°snn--e'sin--[-exp

Dn~t't~

t

2 oo . nnx

[" D n ~ t 1 f

+i~ ~mTexPL

,nx'

_1 d f ( x ' ) s i n - T - d X '

z,~

(1)

o

I n the case where ift=O,f(x)=O and CI=C~-----Co, solution (1) will be expressed as:

4

~o

sin s __

c(~, t)=c.- ,:-coX. - - ~

2

nnx

[

SinTexp (,

Dn 2 nnt~

i~ /

(~)

The amount of substance absorbed by a plate (film) will be $

0

Expression (3) found with C(x, t), d e t e r m i n e d i n accordance with formula (2), and with a constant diffnsion coefficient, does not allow accurate descriptions of the swelling of polymeric specimens. Let us now demonstrate t h a t the experimental data on the swelling of polymeric films m a y be described by a differential equation with a variable'diffusion coefficient of the type of aC a2C -- =D(t) (4) ~t dx n To solve equation (4) we have to find an explicit type of expression for the diffusiort coefficient. Let us define the diffusion coefficient as was done by Einstein for Brownian movement [8], using the set of equations

f . - ~ =0.

(5)

I vv--D av = 0 , ( ax

(6).

where f is a force acting on a diffusing particle; v is the number of diffusion partides; v is the velocity of a diffusing particle; P is the osmotic pressure. According $o Einstein, in the ease of Br0wnian movement

P = ~v

R

T, y=enrtrv

Now from the set of equations (5), (6) we obtain the known expression for ths-

Swelling of polymers in vapours

148I

diffusion coefficient

RT 1 N 6~r' where ~ is the viscosity of the liquid, r is the particle radius. To take account, of the relaxation behaviour of polymers one has to find a relationship between the velocity of movement of a diffusing particle v in equation (6) and the parameters of a model enabling one to describe thermodynamic properties of polymers and their reaction to an external influence (dynamic and static). As a model in this case let us consider two parallel-connected AleksandrovLazurkin elements, as portrayed in Fig. 1.

If

_J

k,1Eft° k31Ea If Fro. 1. Model for describing relaxation and diffusion effects in polymeric bodies.

The reason w h y a model of this t y p e was selected is that it enables us to describe two transitions (a- and ?-transitions) that take place in all polymers subjected to dynamic tests, as well as the stress relaxation and creep curves, and the thermodynamic properties. The validity of this model for describing the mechanical relaxation properties of polymers will be discussed below. Let us now analyze the movement of a diffusing particle in a polymer, which we will treat as a high-viscosity liquid described b y the proposed model (Fig. 1). An analysis of the model can be seen in the Supplement below. The velocity of a particle v determined on the basis of expression (34) (see Supplement) will be:

or

v(t)=

_fa~e-*le °*_{_otaae-tle ,*

(k,q-k,)0*

(7)'

Substituting expression (7) into the set of equations (5), (6), we obtain the diffusion

1482

P . i . ZuBov

et a/.

coefficient for vapours in polymer

D(t):--Do(a2e -t]o'°+aaae -tIe'°),

(8)

where

RT Do ~

1

--

~v (kl+k~)O*

The diffusion coefficient depends on geometrical dimensions. The character of geometrical dimensions is found by an extreme transition to Einstein's formula. I t should be noted t h a t the set of equations (5), (6) for determining D m a y be used in amorphous polymers in cases where 0<<01 and 0<<02, where 0 is the collision time for atoms of a macromolecule with a diffusing substance, 01 and 03 are characteristic time lags (in relaxation) for macromolecules (quasi static conditions). The quasi static condition is practically invariably satisfied in polymers, since 0 is considerably less t h a n the time required for structure rearrangement of an entire macromolecule, or of its individual parts, to take place. Let us now refine the solution of diffusion equation (2) so as to allow for the diffusion coefficient being time-dependent, in line with expression (8). To find the time component of the concentration, we have the equation d In C(t) - -------tD(t), dt

(9)

where 2-----(zn/l) ~, n = I, 2 .... while D(t) is determined in accordance with (8). The dependence of C (t) found using equation (9) will be expressed as sxn~2 R T

(10)

C(t)= Ae-(-£) --~-/~t), where A is a constant of integration, al -~-a~e-t ~ ' *-4-ase -tIe'• f(t)= , kl + ka

(11)

and a general solution of diffusion equation (4) under the initial and boundary conditions considered above will be written as sin ~ C(t) 4: 2 unx [" [ •n \ ' R T Co ----1-- ,-, L - -n sin --/- exp k -- (\ T / . JT $ ( I )

-I Jz

(12)

The a m o u n t of substance absorbed by a plate in time t is given by formula (3) with C(x, t) and D (t) determined from expressions (12) and (8) respectively GO

'

lf(t,=~-~2G'o/~ ( 2 n + l ) , { e x p [ - - ( ~

--exp

T )

T-)

2n+I\'RT-

--ff f(/).~

f{o)j"] (13)

Swelling of polymers ia vapours

1483

or in the dimensionless form

1

o

{exp[

f 2n+ RT -~,~--z 1)2 ~-f(°)] -

M(t)

~' (2n-t- 1)?

~]'
/ 2n+l\eI~T_ ZO (2n-F')2 ( 1 _ fexpl-__ ) L

_

2n-F1

-]

2RT (14)

where 1

G1

f(O)= k ~ k--3'

f(oo ) = k,-~-Ic~

With t = 0

(15)

D (0)------Do(a,+o~aa)

RT

Sinee-~-f(t)<< 1, it, follows that

M(t)

f(O)--l(t)

a2(l -e-t/e ,') 3Fa3(1_e-t/s. °)

M(oo)

f(0)--f(m)

a2-Faa

(Io)

or M(t)

M(oo)

,~__A 1 ( 1 _ e_tle ,) _~_A 2(1 _

_t/o,,) '

(17)

where AI--

a2

a2--~-a 3

;

A2--

a3

a 2-J-a 3

Thus if the model proposed b y us correctly reflects the basic regularities of relaxation and diffusion processes in polymers, the kinetic curves of sorption (swelling) ought to be described by the sum of the exponents with two characteristic time lags 0* and 0". It will be demonstrated below that equation (17) does correctly convey the shape of the kinetic curves. Let us consider the experimental data according to which maxima appear oa kinetic curves for sorption. It is seen from the formulae presented above that relaxation processes in polymers cannot be responsible for the appearance of maxima on M(t) type kinetic curves for sorption. Maxima appearing on the M(t) curves could be due to pha~e transitions or other structural transitions occurring under the influence of molecules absorbed during experiments. These transitions are due to crystallization, or to some other form of ordering. The phenomenon in question is described within the framework of the proposed model. Certainly, in the case of a phase transi-

1484

P . I . Z u s o v eta/.

tion the time lag 8-* oo and the amount of substance absorbed are, according t o equation (17), greatly reduced. In the extreme case, where crystallization in solvent vapours results in an ideal crystal, the entire amount of solvent absorbed m a y be released from polymer, which is accompanied b y the appearance of a deep maximum on M (0 curves. In practice one finds intermittent shallow maxima and minima appearing (Fig. 2) along with a marked reduction in M ( 0 . To investigate the cause of M(t) variations let us determine the frequency of these variations, using the experimental data. Since the variations are occurring in a closed volume, they could be due to standing waves appearing in the cyclindrical cavity of a MacBain balance during regular charging of the latter with vapours. Using appropriate formulae we have now to determine whether the waves in question are expansible waves or diffusion waves. The condition for emergence of a standing wave of the former t y p e may be expressed as

~L=~,

(18)

where K is the wave number for the type of waves being excited, L is the eyclindrical cavity length for the MacBain balance. Taking the frequency of fluctuations in M from the experimental data, let us determine the phase velocity for sound in the vapours. The variation period

x/rn 4t

0"I~ 042

0.08 ~

3

0.0# f

2 I Io

I zo

if"

FiG. 2. Kinetic curves for isooctane vapour sorption for specimens of the PS-polybutadiene block copolymer containing 30% polystyrene blocks. PIP=-----0-2 (1, 1'); (~4 (2, 2'); (F6 (3, 3'); 0"8 (4, 4'); specimens 1-4 prepared from solutions; 1'-4'--from dispersions.

S w e l l i n g of p o l y m e r s in v a p o u I ~

1485

is in our ease T0=320 rain (Fig. 2). From the condition for the appearance of standing waves (18) we obtain the wave number x=n/L (if L----0.5 m, ic==6.28 m-l). I f the standing wave bears an expansible character, then ~:y==c)/c, where c~300 m/see, oJ:2n/To~3×lO -4 sec -1, ~-y--10 --e m 1. Thus the wave Ira, let consideration cannot be an expansible one. According to [.9] we have in the case of a diffusion wave :x ~:=x/i~J/D*, , where ,,J is the cyclic frequency for natural oscillations, and D* is the diffusion coefficient. Now D*=2z~/To×~ 2. In our case (T0=320 min), D*~0.08 em 2 scc-'. which is in k ~ p i n g with the diffusion coefficient for isooctane vapours. The value of D* may also be estimated in accordance with the kinetic theory of gases. I t is known t h a t D*--

1 3

~.12no,~

(19)

where ~ is the average rate o f t h e r m a l m o t i o n f o r v a p o u r molecules, n o is the nu n:-

ber of molecules in 1 cm a, s is the effective cross section of a molecule. This being sO

~= ,/SkT,

(~,)i

7~m

where/~ is Boltzmann's oonstant, T is the absolute temperature and m is the molecular mass. Let us calculate the effective area s on the basis of the van der Waals volume for the isooetane molecule ~ A V~, in accordance with the data in [10] i

~AV~----'lS0.7 AS;

s=34.2 A 2

i

Now at P/Ps~0.6 the value of D given by formula (I9) is 0.09 cm" sec -l, i.e. it agrees with the value of D* determined from kinetic curves for swelling based on the mass variations period. We therefore have good agreement between D* values obtained by an independent method. This leads us to a definite conclusion t h a t diffusion standing waves appear in the MaeBain cylindrical ca~ity, a n d t h a t it,is these waves t h a t give rise to variations in the mass of a specimen during swelling (given t h a t P/P,=eonst.). Now t h a t the wave has been identified, an expression for determining the concentration on filling the cavity will be written as M(t, x) -~ (Mooq-Ap' sin kxe~ ) exp .... where x is the coordinate for a specimen in the ~ a c B a i n balance, P ' is the pressure amplitude. Evidence confirming t h a t the mass variations are due to diffusion type standing waves could be seen in the results of a special experiment, in addition to the foregoing calculations. Having found out how the pressure amplitude P ' may be kept constant from one experiment to another, the mass amplitude m a y be varied for a specimen by moving it in the cylindrical cavity of the balance. The

1486

P.I. ZuBov et al.

same result is obtainable by varying conditions under which the cavity is filled with sorbate vapours. Let us now turn to the problem of analyzing experimental data on the sorption of vapours of solvents in block copolymers. Experiments were carried out using two series of specimens. Films for the first series were prepared from solution in benzene, and those for the second series were prepared from a mixture of solvent and selective precipitant (isooctane) for the polystyrene blocks, the precipitant content being such as to ensure that a block copoIymer specimen in the mixture in question will form a dispersion. Tho block copolymer specimens used b y us differed also with respect to their composition, and contained from 20 to 62 % polystyrene phase. Sorption tests were carried out whilst gradually chan~ing the relative pressure P/Ps, and so at every successive stage (apart from the first) a specimen already contained a definite amount of solvent.* Figure 2 shows the •/m plots, where x is the amount of solvent absorbed at each stage, and m is the specimen mass, including dry polymer mass and the mass of solvent absorbed in all subsequent stages. Figure 2 shows the kinetic curves of sorption of isooctane vapours for films of the block eopolymer containing 30% polystyrene, prepared from solutions and from dispersions. Isooctane is a selective solvent for polybut.adiene blocks, and is a precipitant for polystyrene blocks; consequently, sorbate molecules penetrate into regions of the polybutadiene matrix. On a previous occasion we showed that block copolymer specimens prepared from solution are more rigid than those prepared from a dispersion, since the presence of a selective' precipitant in the system results in a contortion of the polystyrene blocks, which play a lesser role in developing a set of mechanical relaxation properties [1, 2]. The prehistory in the preparation of specimens also has a very important bearing on-the sorption of selective solvent vapours. During the sorption of isooctane vapours, the specimens prepared from dispersions are characterized b y high current and equilibrium values of K/m. In addition, specimens prepared from dispersion have maxima appearing on the kinetic curves in the ease of high PIPs ratios. Let us now analyze the sorption kinetics in the light of these considerations. Our calculations show that a maximum for two time lags is sufficient to allow an accurate description of the kinetic curves'of sorption prior to the advent of the equilibrium in line with equation (17). The following are the time lags 0" and 8", and the relaxation times ~1 and ~2 for block copolymers of PS with P B with a 6 2 ~ PS content. Determinations of T1 and v~ were based on stress relaxation curves with e--~30°/o, while O* and ~4" * The authors thank V. I. Sidorenko for kindly measuring the sorption kinetics.

Swelling of polymers in vapouI~

1487

determinations were based on the kinetic curves of sorl)tion in benzene

(PIPs

=0.4). Time, min Specimen prepared from a solution Specimen prepared from a dispersion

TI 76 31

8" 79 41

r2 5.5 3.4

0,* 14 6

The estimated M(t) values fit the experimental kinetic curvcs satisfactorily (Fig. 2). This means t h a t the model proposed by us for (lescriptiolm of the relaxation behaviour of polymers, in the form of two parallel-connected AlcksandrovLazurkin elements, is quite acceptable in the case under consideration. I f relaxation processes in polymers (lo in fact determine the rate of diffusion of low-molecular liquids in the latter, the time lags, determined from the sorptiou data, ought to agree with relaxation times determined experimentally fi'om stress relaxation measurements using the same model. To verii)" this assumptioll one has first of all to be sure t h a t stress relaxation curvcs are described satisfactorily by the model in question.

o', ~,.q/cmz 90-

60 ~ _ _ ~ 30

231 I

qO

I'

'

80 Time rain

I

r

120

/60

Fin. 3. Stress relaxation curves for the block copolymer containing 62% polystyrene blocks with deformation values of l0 (1), 15 (2) and 30% (3). On checking the validity of formula (36) in the light of experimental findings it is seen t h a t the formula does satisfactorily describe stress relaxation curves for the block copolymers under study. Figure 3 shows the stress relaxation curves for the block copolymer containing 620/o PS. The calculated points fit the experimental curves reasonably well. Relaxation times determined for various specimens on the basis of expression (36) are presented above. I t is apparent t h a t the relaxation times agree satisfactorily with time lags 0* and 8* obtained on the basis of the sorption data. This means t h a t the swelling mechanism is in fact related to relaxation processes taking place under conditions of conformational regrouping of macromolecules. On a previous occasion [1] we said t h a t in systems where relaxation processes do not go to completion, swelling m a y take place more rapidly, and m a y attain higher equilibrium values. This was verified experimentally in the absorption

1488

P.I. ZuBov et ~.

, f benzene vapours by the polystyrene-polybutadiene block copolymer. Moreover, it has once again to be noted that benzene is a good solvent for both polybutadiene and polystyrene blocks, and so far-reaching conformational transitions m a y take place during the sorption process. 075 0", kg/cm 2

ll'

0.12

5O

O.Og

40

0 06

30

g.

0.03 ,

I,

: , ,

,

L,,

I0

,

"

,

2o

15 "/'[me, m/n

,

20 V'-{"

Fro. 4

30

Fio. 5

~ o . 4. Kinetic curves of sorption for PS-polybutadiono block copolymer specimens ab P/P,=0"4: 1, 1'--62% PS, swelling in benzene; 2, 2'--30% PS, swelling in isooctano; 3, 3'--62% PS, swelling in i=ooctane; specimens/-J--prepared from solutions; 1"-3"--from dispersions. F1o. 5. Stress relaxation curves for block copolymer (62% PS) specimens prepared from solutions (1) and dispersions (1') measured in isooctane. I n the present instance we took as the sorbate isooctane, which is a selective solvent only for polybutadiene fragments, and m a y penetrate only the polybutadiene matrix. For this reason the sorption kinetics and the equilibrium degree ~)f sorption are largely dependent on the rigidity of the system. A less rigid specimen prepared from a dispersion exhibits both a higher rate of swelling and a higher equilibrium level for the amount of solvent absorbed. The pattern changes if isooctane is replaced b y benzene (Fig. 4). The more strained (rigid) specimen has a higher equilibrium value for the amount of solvent absorbed. The entry of sorbate molecules into polymer ought to result in a degeneration , f long relaxation times, as was pointed out in [3, 6]. This idea called for experi" R E L A ~ r & T I O N TIMES T~ (MIN) D~XJ~RMINED B Y T H E T O B O L ' S K I [ - M U R A K A M I

FS-FB Block copolymor specimens From From From From

solution a dispersion solution a dispersion

METHOD :FOR

B L O C K COPOLYM'ER S P E C I M E N S

Tesb conditions In air ,, In isooctane

i

r, 934.8 1535.7

14.2

7.42

0.75

161.2

9.05

0.73 1.79

0.062 0.067 4.150 0.135

Swelling of poly~ncrs in vapours

1489

mental verification. As we showed in a previous paper [2], it call be said t h a t if a discrete relaxation time spectrum determined in accordance with the Tobol'skii-Murakami method, is used to describe relaxation curves, there will need to be five discrete relaxation times within the limits of duration (180 min) of the relaxation process. In view of this it seemed appropriate to carry out stress relaxation experiments ia the medium i~l which sorption takes place, i.e. under conditions where P I P s - 1. Figure 5 shows the relaxation curves t h a t were plotted under these conditions. 5tress relaxatioll takes place rapidly in the initial stages of the process, after which stress relaxation ceases and the onset of equilibrium conditions appears. An tmalysis of the discrete relaxatiotl time spectrum, using the Tobol'skii-Murakami method results in only two values for short relaxation times. A degeneration of the long relaxation times is implied (see tabulated values). Thus it appears in the light of these findings t h a t the swelling of polymers in vapours of low molecular liquids is undoubtedly related to relaxation phenomena (conformational rearrangement) leading to swelling characterized b y kinetic curves of a particular type. The swelling kinetics are described satisfactorily in terms of the proposed model containing two characteristic time lags. At the ~amo time the sorption process is accompanied by a degermration of long relaxation times in the system. All these considerations point to a predominating relaxation mechanism in the swelling of polymers. SUPPLEMENT

Lot us now analyze deformation for the model depicted in Fig. 1. Since the medium under study is a one-dimensional one, the displacement of x (t) under the action of a constant force f, duo to osmotic pressure, will, according to [10], be expressed as 1,0(0 = k, [d-- ],.0i0]

d

H,.(,)]k7

(21)

whore ~ l = x s - - x t . Transposing system (21) we obtain fl.o(t)=k2d

kafl,o(t) ~ dc kl ~7~1dr

~, dj',o(t) k~ dt

dc ~, dfl.o(t) k,[.f-.h.o(t)] "~-~l,l -dr z- ks dt k3 Differentiating the first, equation in set (22) f--fl.o(t) = k , d - -

df,.o(t) - - - eft

~]¢$

de dt

k, df,,o(t) kl dt

d2c dt 2

th d~fl.o(t) kl d~2

(22)

(23)

From the first equation in set (22) we also have f a.o(t)=a~ + b

d~

--c

dfl.o(t) dt

(24)

1490

P.I.

et al.

Zu~ov

where

a=--

ksl

;

tlsl

b=

tI, c = - -

;

th

(25)

S u b s t i t u t i n g e q u a t i o n (24) i n t o t h e s e c o n d e q u a t i o n in set (22), we o b t a i n

f--ae--b~t ÷ c

dl,j~(t) : a ' e T b ' - ~d8 +c' ~f,.o(t__ 2) dt

(26)

where

k ,l

th l k, ;

k4 ;

l+k--.

th c'-- k~-f-kt

~+k-~

(27)

L e t u s n o w r e w r i t e (26) as

de

.f=e(a+a')+ -dr (b+b')+

~

(c'--c)

(2s)

F r o m (28) w.e h a v e

de

f--e(a-~-a')-- -~ (b+b')

d],.o(t) dt

(29)

CrmC

and de • d~l.o(t)

d'e (a-}-a')- ~-~ (b+b')

dt s

(30)

c"-- c

S u b s t i t u t i n g e q u a t i o n s (29) a n d (30) i n t o (23), w e o b t a i n de

( k"'f+)- e~( a + a ' ) - - d ~ (c'--c b+b)l

-----k.l~ - ~ . l dt'--d'e

de dZe th ~ (a-}-a')-- ~-~ (b+b') R

kI

(31)

C'--C

Let us now introduce the notation

,

+ a*=

-

-

;

b*=--

( 1 :'--

Ct--C

"

;

C'--C

(32) d*= --~h/--

t12 (b + b') kl (c'-c)

U s i n g t h i s n o t a t i o n we r e w r i t e e x p r e s s i o n (31) as de

d2e

a* +b*e-',-c* ~ -~d* ~-~ = 0

(33)

Swelling of polymers in vapours

1491

A solution of differential equation (33) under b o u n d a r y conditions of

f

/

8 (0) ~ (k, -~-/c~) ,-0

and ~-~ = dt

/

i~-~-i-tl..) l,,.o

yields the following relation of deformation to time

(I:~ -~ k~) l~ (t) : f (a~ + a~e-~/°~ * + a~e-tlO. *) ,

(34)

where

k, ~- k,

(~h q- ~h)

- - 1+ l/k,-F 1/k,

~'2 ~

l 1 / k ~ - Ilk,

)-2--)q

kl -r'ka __ __(kl-~-ks))'1 .!_ (~h +~h) (k~+k2) aa =:

;.I 1

1 l/k, ' l/lc3~- l/lc,

).s--2l I 1)-x 1/kx~-l/k2 -~- 1/ka+ l/k~

al~--

0:= ~ ; +

( ~ - ~ . - e*/ ;

/+.,.. 0:= ~-; - ~(2d*),

h~

E q u a t i o n (34) describes creep in polymers, given t h a t f = c o n s t . I n the case of stress relaxation, in view of the parallel connection of the A l e k s a n d r o v - L a z u r k i n elements, one has to ~ i t o [10] a(t)=~x(t)-Fa2(t) =

E~ 8o8_~/~, ExE z E~ ,.;. EaE, E-~-FE---~, E,+E------~,80" E ~E,-- 8o8-' -{- Ea+E,~ 80,

(35)

whoro a(t) is the stress; t - - t i m e , E l , Ez, Ez and E~ are moduli in the model in Fig. 1; 8, is t h e constant deformation

v,---- El + E t ;

~4~ Es-F E-'----,

Transposing equation (35), we now write

a(t)

_*=-t/r~-- * - - , / , , .

*

(36)

whore

E1E,

E~+E.

~°

;

a,*=

-

-

8

E,*E , E a-+-

o ;

~*.=8° ~

E,E4~

+ E,+Ed

Translated by R. J. A. ~ a r REFERENCES 1. P. I. ZUBOV, T. A. ANDRYUSHCHENKO a n d A. A. A S K A D S g H , Vysokomol. soyed. AI9: 2738, 1977 (Translated in P o l y m e r Sci. U.S.S.R. 19: 12, 3163, 1977) 2. T. A. ANDRYUSHCHENKO, A. A. A S K A D S K I I a n d P. I. ZUBOV, Vysokomol. soyed. A21: 2366, (1979 (Translated in P o l y m e r Soi. U.S.S.R. 21: 10, 2616, 1979)

1492

Y~. S. LIPATOV et al.

3. P. I. ZUBOV, D o k t ~ r s k a y a dissertatsiya (Doctorate Dissertation). N I P h C h I (L. Ya. K a r p o v Institute), Moscow, 1949 4. B. A. DOGADKIN and V. Ire. GUL', Dokl. A N SSSR 70: 1017, 1950 5. V. Ye. GUL', Kolloidn. zh. 15: 170, 1953 6. P. I. ZUB0V and M. Ts. ZBEREV, Kolloidn. zh. 18: 679, 1956 7. A. N. TIKHONO¥ and A. A. SAMARSKII, U r a v n e n i y a matematicheskoi fiziki (Mathematical Physics Equations). Izd. " N a u k a " , 1966 8. A. EINSHTEIN, Sbornik nauchnykh t r u d o v (Collection of Research Transactions). Izd. " N a u k a " , 1966 9. L. D. LANDAU and Ye. M. LIFSHITS, Mekhanika sploshnykh sred (Mechanics of Continua) State Press for Techn. a n d Theoret. Lit., 1954 10. A . A . ASKADS]KII, Deformatsiya polimerov (Deformation of Polymers). Izd. " K k i m i y a " , 1973

l~olymer Science U.S.S.R. Vol. 22, No. 6, pp. 1492-1500, 1980

0032-8950/80/061492-09507.50[0

Pflnted In Poland

© 1981 Pergamon Press Ltd.

INVESTIGATION OF MICROPHASE SEPARATION OCCURRING IN INTERPENETRATING POLYMERIC NETWORKS PREPARED FROM POLYURETHANE AND STYRENE-DIVINYLBENZENE COPOLYMER* YU. S. LIPATOV, V. V. Smmov, V. A.

BOGDANOVICH, L. V. KARABANOVA

and L. M. SERG~.Y~.VA High P o l y m e r Institute, U.S.S.R. Academy of Sciences

(Received 23 M a y 1979)

I n t e r p e n e t r a t i n g polymeric networks based on polyurethane and s t y r e n e divinylbenzene copolymer have been prepared. The heterogeneous structure of the networks has been investigated b y the methods of small- a n d wide-angle X - r a y scattering. The e x t e n t of heterogeneity domains has been calculated and the transitional layer thickness obtained, using the experimental data, and in addition the diffusivity of phase boundaries and the degree of segregation of the components have been determined as well as the molecular level of mi~ing of the components. I t is shown t h a t the degree of microphase separation of components in the interp e n e t r a t i n g polymeric networks depends on the amount of the second component in the system.

* V y s o k o m o l . s o y e d . A$2: N o . 6, 1359-1365, 1980.