Molecular interactions in polymer solutions— I. The influence of volume effects on the second virial coefficient

MOLECULAR INTERACTIONS IN POLYMER SOLUTIONS-I. THE INFLUENCE OF VOLUME EFFECTS ON THE SECOND VIRIAL COEFFICIENT* O. B. PTITSYN and YU. E. EIZNER The I n s t i t u t e of High Molecular Weight Compounds, U.S.S.R. Academy of Scienoes

(Received 17May

1959)

THE expression for the second virial coeffioient of polymer solutions can be given in the form [1,2]: Nav A,= - F(z) (1)

2~

where Na is Avogadro's number M 0 the molecular weight of a chain segment; v, the effective included volume of the chain segment, which becomes zero at Flory's

z= [ ~ 1 "/' I/~v

O-point,

\ 2n ]

~ a'

(2)

N is the number of segments in a chain and a, the bond length between neighbouring segments. Using equation (2), equation (1) can be re-written in the form:

.4,=4~'/,N. (__R,)'/, ~(~)

(3)

.M2

where a~----R2/R~, R~ and ~o are the meanradii of gyration of the chain in the given solvent and in an ideal,solvent, respectively, and: Z

~(~)= -~ F(z).

(4)

The relation between ~ and z is obtained from the theory of intra-molecular interactions of the remote order (volume effects).* The theory of Flory, Kriegbaum and Orofino [2-4], which substitutes a cloud of segments (whose distribution with respect to the centre of mass is Gaussian) for the real chain, gives [4]: - -

Fr, xo (z)--~ 31/~n

z

In

1 + - - .

4

~'~

-- 1.15z+

(5)

* Vysokomol. soedin. 1: No. 8, 1200-1206, 1959. The theories used below establish a relation between z and ~,A=(h~/blo)q,and n o t bet,ween z and ~=(R2/R~)'/z. However, since the difference between a and ~h is small (in particular, when z is small, ~ = 1 + 134z/105, and a~= 1+4z/3) we will assume in further work t h a t ~ ' ~ h .

474

O . B . PTITSYlq and YU. E. EIZl~ER

The substitution of an exact distribution function (which follows from the Gaussian distribution of the inter-segmental spacing) for the Gaussian distribution, carried o u t b y Isihara and K o y a m a [7], did not lead to any essential change in the theory [8]. At the same time, the strict theory, of the second virial coefficient which applies near the 0-point) [1], [9-11] leads to the function: Fexact (Z)= 1 --2.865z+ . . . .

(6)

which is appreciably different from (5). To eradicate this discrepancy Kasassa and lV[arkovic[ 12, 13] substituted a Gaussian distribution with respect to a point of one arbitrary intermolecular contact for the Gauseian distribution of segments with respect of the centre of mass; at the same time calculating approximately the bonding of segments into a chain. Their result, which automatically gave the correct coefficient of z in the expansion of F(z), appears approximately in the form: 1 --e -~.~zl~

FK~(z)--~

5.68 z/~3

(7)

In the derivation of equation (7) the intra-molecular volume effects were not taken directly into consideration. The authors of work [13] calculated approximately the influence of the volume effect on A 2 and assumed that the latter increases ~ times* the length of the link which connects the neighbouring segments of the chain. As a result of this z / ~ occurs in equation (7) instead of z. All the same, in practice the volume effects lead to a non-Gaussian distribution function for the inter-segmental spacing, i.e. their influence cannot be related to the change in length of the link [14]. The present wcrk is dedicated to the calculation of this detail within the bounds of the Kasassa-lKarkovic theory. CALCULATION OF THE VOLUME EFFECT IN THE FUNCTION F ( z )

The non-Gaussian form of the distribution function, for the inter-segmental spaoing in non-ideal solvents, leads to the fact that various characteristics of the maeromoleeule, which have the dimension of length, all depend on the volume effect. In preceeding works we have calculated the influence of this condition on the scattering of light [15,16], hydrodynamic properties [16-18] and dimensions [19] of the macromolecules in solution. ]'he non-Gaussian form of the distribution function appears essentially also in the theory of A2, as can be seen from the examination of the coefficient of z~ in the expalasion of F(z). The coefficient of z2 consists of a quantity which depends on triple intermolecular contacts (equal to 9.726 [9]), and of a quantity which depends on the influence of isolated ifltra-molecular contacts (volume effects) on double inter-molec* Theory of Isihara and Koiama t~grees with the theory of Flory and Krigbaum, if we first make an analogous assumption.

The influence of volume effects o n the second virial coefficient

475

ular contactS. The trivial effect of this influence, summarized as the change from z to z/~ in the linear term, gives the contribution to the coefficient of zs as equal to 5.73 [9] (since ~2__ 1-}-4z/3 [28]). Moreover, the exact effect of the influence of a small proportion of the intra-molecul~r contacts [10, 11] gave the value of 8.78 as the lower estimate of this contribution. Thus in practice z should be changed n o t i n t o z/a~ but into Z/~e3ff(see [19]), where ~ e f f ~ . This can be related to the sharpening of the maximum of the distribution function for the inter-segmental spacing, which leads to decreased probability of inter-molecular contacts in comparison with the Gaussian distribution function. In order to define ~eff, it is necessary to estinmte the influence o~ the volume effect on double inter-molecular contacts with the contribution of all the intramolecular contacts. The general expression for F(z), in an approximation which considers only double inter-molecular contacts, has the form [13]:

F(z)= l-- -~l ~P(Oj,.h)i.i,~-

....

(8)

_4. h)f(rifj.)drid, .~ .~ f(ri,

(9)

i~,i. $~,Js

where: ~(Ojt,j:)i,,iffi

~

f

rilf~ ffi ri,j:

is the probability of a contact between segments Jl and J2 (belonging to molecules 1 and 2), with the condition that segments ix and i i are found to be in contact:

f(ro)

is the distribution function for the spacing r 0 between the segments i a n d j .

f(ro) is a Ga.ussian function, and P(Oj,, ~,)id, ___(3/2na~)'/,×(Ijl_il ]+ ]j2_i~ I)-'/: [13], which leads to equation (6).

In the absence of volume effects, -4

The distribution function f(r~), including the volume effect, can be. obtained approximately from Peterlin's [21], [22], distribution function for the distanoe between the ends of the chain [19 [23]], by substituting (j--i) for the last N. We have: ar~ij

f(rij)= 2nlj~ ila~]~'1'e '"-ila~[ - - } ( 3

[lJ-iIVh'-/~--z -f-Fen

1 - 4 z ~

ri----L] • (h0S)'/'j

(10)

Substituting equation (10) in equations (8) and (9)"we obtain: F(z) ~ 1 - 2.865 z ( 1 - 4.39 :) + . . . .

(11)

That is, the contribution to the coefficient of z~ depending on the influence of the volume effect on double contacts, is equal to 12.58. Applying F(z) in the form 1-2-865z/~z, we obtain from equation (11): 2

~eff = 1+2.93z-- . . .

,(12)

In order to extend this equation to the region of large z, we substitute z/~l! for z in the right hand side of equation (12), where the magnitude of ~ , (obtained

476

O . B . PTITSYN and Yu. E. EIZN~.R

from work [19]) effectively represents the influence of the degree of the entanglement on intra-molecular volume effects. I t is seen that: 2.93 z ~ ,

(13)

~e6H--~esff~1.04 z.

(14)

~f,=l~

where:

[19]

Finally, instead of equ&tion (7) we obtain: r.

F(z)~

S

I - - e - ~" e S z / % f !

~,

5"68 /~¢eff

,

(15)

where ~ ia rebated to equations (13) and (14). Curve of T(~), constructed from equations (4) and (15) is shown on Fig. 1, where the relation between z and ~ is defined [19] by equation (14) and the equation: ~s_~=

4z ~a ~ __~,

(16)

eteff

which correctly reproduces the linear and quadratic terms in the expansion of in powers of z. On the same Figure are sho~n for comparisort the curves of Flory, Kriegbaum and Orofino, ~Firxo(~), and of Kasassa and Markovic PxM(~), constructed from equations (4), and (5) ar/d (7) respectively. In the Flory, Krigbaum and Orofmo theory, the relation between z and ~, defined by the following equation [2, 14] was used: ~ _ ~ . = 31/-~ z, 2

(17)

and in the theory of Kasass~ and Markovic--by the equation [20, 24]: ~5_~

4z

= ~, 3

(18)

which gives correctly the linear term in the expansion of ~ in terms of z. DISCUSSION OF RESULTS AND COMPARISON WITH EXPERIMENT

As shown in Fig. 1, the theory of .4 2, proposed in this work, agrees almost entirely with the theory of Kasassa and Markovic [13], when ~ is near to unity, but with an increase in ~ it leads to much greater values of A~ (with given chain sizes). When ~ ~ c~ the proposed theory gives ~ 0.58, from which, when ~ -, ~ , AsMI/(RI)~2=7.7 × 1024. At the same time the theory of Kasassa aud Markovic gives ~----0.18, from which ~--. c~ gives AcM2/(R'~)312=2.4× l02~. We would like to point out that, as Olbrecht [9] has shown the lower, limit of the value ofW ~° is 0.28, whence A~M2/(RI) $z > 3.8 x 1024. We see that our theory,

The influence of volume effects on the second virial coefficient

477

as distinct from the theory of Kasassa and Markovic, ~atisfies this requirement. It is necessary to emphasize, as seen from Fig. 1, that the asymptotic behaviour ~(=) ,~ AzMS/(Rs) "/, is not achieved for normal ~ ~ 1-5. Therefore the approximate constancy of the quantity AsM/[r/], observed in the experiment, which is approximately proportional to AzMS/(Rz) s/2, is explained not by approaching the asymptotic value of ~, but by the weak dependence of ~ on ~, found within the limits of the rather considerable experimental error, in the normal range of ~ (in good solvents). When ~--=-1.3-- 1.5, Y~(~)-----0.24--0.32. Using Flory's equation: (.~.).,, [T/] = ~. 6'/,

--, M

(19)

in which ~ = 2 . 1 x I0 ~, (which applies approximately in good solvents [I'/],[18],) we find that, in this interval of ~, A2M/[#] increases from 1.2 to 1.7. The experimental results collected by Olbrecht [9], from a series of authors, and referring approximately to the same interval of ~, gives AzM/[~]= 1.44-0.2. We note that the maximum value of AsM/[#], reached when ~ ~ 1.3, is equal to 0.9, which agrees with the Kasassa-M~rkovic theory. On Fig. 1, are also added the experimental results re|ating to both good and bad solvents. In works on polystyrene [25-27] and poly~obutylene [25] in good solvents, As, M and [#] were measured and ~ was determined from [#] using the equation =8~-[#]/[#]e which follows from the equation (19). Values of[#]e, obtained previously [28], were used in the above. For the polystyrene fraction, for which there was no data, [~]0 was calculated from the equation [~]e----0.82× 10-s | / ~ , [28]. In practice, as has been shown by us earlier, Flory's coefficient @ depends on =. Because of this, the use of this equation leads to errors. However, in view of the dependence of W on ~, these error~ are not important in the given case. This is confirmed both by our discussion and by the agreement between the experimental data of the authors (see Fig. 1 ) a n d t h e works of Shultz [29] on polyvinyl acetate in methyiethylketone, and Krigbaum and Carpenter [30], on polystyrene in cydohexane (in which not [#], but E ~ was measured from the scattering of light by Zimm's method). The contribotion of the dependence of @ on ~ would have led to a sinai] displacement of the experimental points to the right. The values of R z, necessary, for the calculation of W(~) from experimental data for A~ and M, were calculated by us from equation (19). We see that the experimental results of A 2 for macromolecules in good solvents are described much better by the proposed theory than by the Kasassa-Markovic theory. The dotted curve on Fig. 1, corresponds to the Flory, Krigbaum, Orofino theory. In the region of common good solvents (~ ~ 1.5) this theory leads to a value of ~J(~), which is close to that given by our theory. All the same, when ~ is near to unity and when ~ ~ 2, the theory of Flory, Krigbaum and Orofino leads to essentially low results. ~Vhen ~ ~ 1, this theory gives the initial slope of the curve Y~(~) as approximately half the slope given by the Kasassa-Markovic theory and our theory, which when ~ ~ 1 becomes the same as the exact theory. In other words,

478

O. B. PT~TSY~ and Yu. E. EIZNER

the Flory.Krigbaum-Orofino theory indicates a markedly weaker growth of A.: with an increase in temperature, near the 0-point, than the exact theo~T. Fig. 1, and especially Fig. 2, on which the dependence of A~ on temperature obtained in work [30] is given directly, shows that the experimental data in that region agrees ~..¢

o6~-

!

~ - a

.....,'..~,.,¢..~.

.

0 o I~

.~t

O-C

~*

.o

OI

1

4 .I

I- °-° .

,j/ 0[0~ / Fro.

.

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.

.

~ I. Relation: ~(~) = (~/~/:.~)AIM:/(R~)

0 ~I,

from ~=(R:IRi)~h (I) proposed theory; (2) asymptote to this curve; (3) theory Kasassa-Markovie; (4) theory Flory-Krigbaum-Orofino. Experimental results: (a) polystyrene in toluene [2527]; (b) polystyrene in methylethyl ketone [27]; (c) poly~obutylene in cyclohexane [25]: (d) polyvinylacetate in methylethyl ketone [29]; (c) polystyrene in cyc/ohexane [30].

JlO

320

3JO r. "~

Fie,. 2. Dependence of A: cn temperature near the 0point: (1) proposed theory; (2) theory Kasaesa-Markovic; (3) theory Flory-Krigbaum-Orofhm; circles, experimental results (polystyrene in ~yclohexane [30].)

with our theory much better than with the Flory-Krigbaum-Orofino theory. Values of a and R ~, necessary for the construction of the theoretical curves on Fig. 2, were taken from the smoothed--otit curve of R~(T) given in the work [30]. * When a is large, the Flory-Krigbaum Orofino theory leads to very large values of ~(a), which also does not agree with experiment. Thus from the data of Eskin [31], it can be seen that when a:= 2.4, ~(a) 0.3~),while the Flory-Krigbaum-Orofino t h e o r y ~(a)----0.72 (our t h e o r y and the Kmsassa-Markovic t h e o r y give 0.49 a n d 0.18, respectively). I t is necessary to m e n t i o n t h a t a n y increase in the e x a c t ness of the F l o r y - K r i g b a u m - O r o f i n o t h e o r y (exchange of e q u a t i o n (17) b y t h e more e x a c t e q u a t i o n (18), and also the c o n t r i b u t i o n of the n o n - h o m o g e n i t y of the e n t a n g l e m e n t ) leads to an i m p r o v e d a g r e e m e n t between t h e o r y and e x p e r i m e n t in the region a w a y from unity. * In the series of works (see e.g. [13]), which compared the theory wish experiment near the 0-point, the values of ~0 calculated from the equation ~a=[q]/[r/]0,.were used. Thus, when ~ depends on a particularly strongly namely near the 0-point [16-18], [301, such comparison becomes untrue.

The influence of volume effecte on the second viri~l coefficient

479

It is known t h a t A 2 decreases with an increase in the molecalar weight, moreover, the experimental data usually appears in the form A 2 ~ M - L From equation (3) we have: 1

d In ~(~) l ,

where e is the degree index in the equation R Z ~ M I + ' , which is related to ~ by the equatmn 8= ( ~ - - 1}/(5~~ - 3) * [ 18]. On Fig. 3 is shown the dependence, of ? on ~, /"

~-a

~2

o-

.~--

b

,,

"\.

/

i

I

"

I

I

0 Q03 0.I0 ~15 0.20e FIG. 3. Dependence of y on e. (1) proposed theory; (2) theory Flory-KrigbaumOrofino; (3) theory Kasassa-Markovie. Experimental results: (a.) polystyrene in toluene [25]; (b) the same from [32]; (c) polyi~obutylene in eydohexane [25].

which follows from equation (20) and from the expression for ~u(~) in the theories of Flory-Krigbaum-Orofino, Kasas~-Markowic and ours. On tim same sketch are shown experimental results from works [25] and [32], where e was defined by the dependence of[g] on M, in the hypothesis that [q] ~ (R~)8/2.~" We see the experimental results lie between the theoretical curves obtained from the theories of Kasassa-Markovic and our theory; while'theory Flory-Krigbaum-Orofino clearly underestimates the dependence of A 2 on M. CONCLUSIONS

(1) More exact calculation (when coml)ared with preceeding theories) of the influence of intra-nmlecular volume effects on the second virial coefficient of polymer solutions leads to an essential change in theory. (2) Agreement of the values of A~, obtained in good solvents with given chain dimension, appears to be appreciably better with the proposed theory t h a n with the theory of Kasassa and MarkovJc, while the dependence of A2 on molecular weight is noticeably smaller than in the above-mentioned theory. * This equation follows from the expr~sion ~s--~a=const. z. increasing the exact hess of this relation by considering the contribution of non-homogeneous entanglement, r19]. has little effect on the relation between e and ~. In w o r k [321 the data on [q] is absent, ther(;fore the appropriate values of e were taken from w o r k [25], in which polystyrene in toluene, in a similar molecular weight range, was investigated.

480

O. B. PTITSYN and YU. E. EIZNER

(3) T h e t h e o r y of F l o r y , K r i g b a u m a n d Orofino in t h e region of n o r m a l values o f t h e e n t a n g l e m e n t coefficient ~, leads to t h e values o f ,4 w h i c h are n e a r to t h o s e predicb~cl b y t h e p r o p o s e d t h e o r y . At t h e s a m e time, it l e a d s to s h a r p l y different r e l u l t s b o t h in b a d s o l v e n t s a n d in regions o f large ~ a n d it also p r e d i c t s m u c h b e t t e r t h e w e a k d e p e n d e n c e o f As on m o l e c u a l r weight, t h a n does o u r t h e o r y . (4) C o m p a r i s o n o f s e v e r a l theories o f A 2 w i t h e x p e r i m e n t a l results r e l a t i n g to b o t h g o o d a n d b a d solvents, shows t h a t t h e p r o p o s e d t h e o r y agrees w i t h e x p e r i m e n t m u c h b e t t e r t h a n t h e t h e o r y o f F l o r y - K r i g b a u m - O r o f i n o , a n d Ca~assaMarkowitz. Trans/at~ /~j Z. KOWSZU~ REFERENCES 1. B. ZIMM, J. Chem. Phys. 14: 164, 1946 2. P. FLORY, Principles of polymer chemistry, Cornell University Pre~, sect. X I I , 1953 3. P. FLORY and W. KRIGBAUM, J. Chem. Phys. 18: 1086, 1950 4. T. O]gOFINO and P. FLORY, J. Chem. Phys. 26: 1067, 1957 5. A. T~IWARA, J. Phys. Soc. J a p a n 5: 201, 1960 6. P. DRRY]g and F. BEULah[E, J. Chem. Phys. 20: 1337, 1952 7. A. I~wTARA and R. KOIAMA, J. Chem. Phys. 25: 712, 1956 8. D. CARFENTF.~g and W. KRIGBAUM, J. Chem. Phys. 28: 513, 1958 9. A. ALBRE~[~JP, J. Chem. Phys. 27: 1002, 1957 10. H. YAMAKAWA, J. Phys. Soc. Japan 13: 87, 1958 11. M. KURATA and H. YAMAKAWA, J. Chem. Phys. 29: 311, 1958 12. B. CASASSA, J. Chem. Phys. 27: 970, 1957 13. E. CASABSA and H. MARKOVITZ, J. Chem Phys. 29: 493, 1958 14. M. V. VOL']g~NSHTEIN and O. B. PTISYN, Uspekh i fiz, nauk 49: 50L, 1953 15. O. B. PTIBYN, Zh. fiz. khimii 81: 1091, 1957 16. O. B. PTIBYN and Iu. E. EIZNER, Vysokomol. soedin. 1: 966, 1959 17. O. B. PTISYN and Iu. E. EIZNER, Zh. fiz. khirnii 32: 2464, 1958 18. O. B. PTISYN and Iu. !¢.. EIZNER, Zh. tekhn, fiziki 29: 1117, 1959 19. O. B. PTISYN, Vysokomol. soedin. 1: 715, 1959 20. B. ZIMM, W. STO(rKMAYER and M. ~ , J. Chem. Phys. 21: 1716, 1953 21. A. PIgTERI2[N, J. Colloid Sci. 10: 587, 1955 22. A. PETERLIN, Bull. Sci. Conseil Acad. R P F Y 2: 98, 1956 23. A. PETEItLIN, Bull. Sci. Conseil Acad. R P F Y 2: 97, 1956 24. W. 8TOCKMAYER, J. Polymer Sci. 15: 595, 1955 25. W. KRIGBAUM and P. FLORY, J. Amer. Chem. Soc. 75: 1775, 1953 26. W. KRIGBAUM, J. Polymer Sci 28: 213, 1958 27. D. CARPENTER and W. KRIGBAUM, J. Chem. Phys. 24: 1041, 1956 28. W. KRIGBAUM and P. FLORY, J. Polymer Sci. 11: 37, 1953 29. A. SCHULTZ, Amer. Chem. Soc. 76: 3422, 1954 30. W. KRIGBAUM and D. CARPENTER, J. Phys. Chem. 59: 1166, 1955 31. V. E. ESILIN, Vysokomol. soedin. I: 138, 1959 32. C. BAWN and M. WAJID, J. Polymer Sci. 12: 1~9, 1953