11, No. 6, November-December 1978
Phase Separation in Macromolecular Multicomponent Systems 1161
P. R. Couchman and F. E. Karasz, Macromolecules, 11, 117 (1978). A. V. Lesikar, to be published. W. J. MacKnight, J. Stoelting, and F. E. Karasz, Adu. Chem. Ser., No. 99, 29 (1971). P. J. Flory, J. Am. Chem. SOC., 87,1833 (1965);B. E. Eichinger and P. J. Flory, Trans. Faraday SOC.,64, 2035 (1968). A. F. Yee, Polym. Eng. Sci., 17, 213 (1977). S. T. Wellinghoff, J. L. Koenig, and E. Baer, J . Polym. Sci., Polym. Phys. Ed., 15, 1913 (1977). G. Gee, Q. Reu., Chem. SOC.,1, 265 (1947). J. R. Fried, Ph.D. Dissertation, Polymer Science and Engineering, University of Massachusetts, 1976; see also J. R. Fried, F. E. Karasz, and W. J. MacKnight, Macromolecules, 11, 150 (1978). L. A. Wood, J . Polym. Sci., 28, 319 (1958). R. F. Boyer, J . Macromol. Sci., Phys., 7, 487 (1973). J. M. Pochan, C. L. Beatty, and D. F. Hinman, to be published. J. J. Hickman and R. M. Ikeda, J . Polym. Sci., Polym. Phys. Ed., 11,1713 (1973);G. A. Zakrzewski, Polymer, 14,347 (1973); B. G. Rinby, J . Polym. Sci., Polym. S y m p . , 51, 89 (1975); Y. J. Shur and B. Rinby, J . Appl. Polym. Sci., 19, 2143 (1975);
T. K. Kwei, T. Nishi, and R. F. Roberts, Macromolecules, 7, 667 (1974). W. J. MacKnight, F. E. Karasz, and J. R. Fried in “Polymer Blends”,S. Newman and D. R. Paul, Eds., Academic, New York, N.Y., 1977. A. R. Shultz and B. M. Gendron, J . Appl. Polym. Sei., 16, 461 (1972). J. Stoelting, F. E. Karasz, and IV. J. MacKnight, P o l ~ mEng. Sci., 10, 133 (1970). W. M. Prest, Jr., and R. S. Porter, J . PolJrn. Sci.. Polym. Phys. Ed., 10, 1639 (1972). A. R. Shultz and B. M. Beach. Macromolecules. 7. 902 (1974). A. R. Shultz and B. M. Gendron, J . Macromol. SCL., Chem., 8, 175 (1974). N. E. Weeks, F. E. Karasz, and W.J. MacKnight, J . Appl. Phjs , 48, 4068 (1977). R. E. Wetton, W. J. MacKnight, J. R. Fried, and F E. Karasz, Macromolecules, 11, 158 (1978). The theoretical predictions for the other compatible PPO/ copolymer blend (67.1 mol 5‘0 pC1S) investigated l)> Fried16are very close to those for the blend of Figure 2, a> are the experimental results.
Thermodynamic Perturbation Theory of Phase Separation in Macromolecular Multicomponent Systems. 2. Concentration Dependenceil Hans Craubner Znstitut fur Physikalische Chemie der Uniuersitat Dusseldorf, Dusseldorf, West Germany. Received October 4 , 1977 ABSTRACT: A perturbation-theoretical approach to the concentration dependence of phase separation in quasiternary macromolecular solutions consisting of polymer, solvent, and nonsolvent is developed. It results in a linear semilogarithmic phase equation yF* = A , - u,* In c,,, where yF* denotes the precipitant fraction (volume fraction of the precipitant at the cloud point) and co the initial polymer concentration in g/cm3. Here, A, and crc* are constants with A, as the precipitant fraction at co = 1 g/cm3 and ut* the inverse value of the change of the relative perturbation density (Ap*?F,,). The theory is verified experimentally at 25 “ C by investigation of the three different systems: (a) polystyrene-benzene-methanol; (b) poly(methy1 methacrylate)-benzene-cyclohexane; ( c ) poly(t-caprolactam)-m-cresol-petroleum ether. Fast turbidimetric titration measurements on the basis of the dynamic volume pulse technique were used. The perturbation-theoretical approach is confirmed furthermore by numerous experimental results taken from the literature. The results are compared critically with the earlier approaches on the basis of statistical thermodynamics, partition equilibria. and solution equilibria. In this context, the effect of the constant ratio of the solvent/nonsolvent in the gel phase is interpreted theoretically.
I. Introduction This p a p e r is concerned with a perturbation-theoretical approach to the initial concentration dependence of phase separation in macromolecular m u l t i c o m p o n e n t s y s t e m s after the a d d i t i o n of nonsolvent (precipitant), and w i t h a p p l i c a t i o n s of the t h e o r y . The a p p r o a c h is b a s e d o n concepts that I have developed in previous The a s s u m p t i o n s and basic equations have been s e t out in the first p a p e r of t h i s ~ e r i e s . ~ Firstly, it is s h o w n how t h e p e r t u r b a t i o n relations c a n b e applied t o calculating t h e functional dependence of t h e p r e c i p i t a n t fraction (volume fraction of the nonsolvent at the precipitation threshold, yp*) from the original polymer c o n c e n t r a t i o n , t a k i n g i n t o a c c o u n t t h e macromolecular p h a s e concentrations. T h e n , t h e t h e o r y is a p p l i e d to ‘This paper is dedicated to Professor Maurice L. Huggins on the occasion of his 80th birthday, but publication unfortunately had to be delayed. *Presented at the 76th Annual General Meeting of the Deutsche Bunsen-Gesellschaftfur Physikalische Chemie in Braunschweig, May 19-21, 1977.
0024-9297/78/2211-1l61$01.00/0
various m u l t i c o m p o n e n t macromolecular systems. T h e results obtained a r e scrutinized and c o m p a r e d critically w i t h earlier a t t e m p t s at p r o b l e m solution s u c h as, e.g., p a r t i t i o n equilibria4,j and solution equilibria.61i
11. Fundamentals: The Perturbation-Theoretical Approach of Phase Separation in Fluid Macromolecular M u l t i c o m p o n e n t S y s t e m s A d d i t i o n of nonsolvent to a homogeneous macromolecular solution m a y cause its disintegration b y polymer precipitation. T h e s i m p l e s t s y s t e m h e r e is now a fluid heterogeneous t e r n a r y one. Generally, however. solutions of macromolecules i n solvent/nonsolvent m i x t u r e s a r e q u a s i t e r n a r y m u l t i c o m p o n e n t systems. I n t h e following, the a s s u m p t i o n s and t h e n o t a t i o n of ref 3 a r e u s e d , L e t the finite set of a multicomponent macromolecular solution of nonelectrolytes IK0 = (1,2 , ..., LA,...,P,,..., F,,, F,+l,..., r) be p e r t u r b e d b y t h e a d m i x t u r e of a nonsolvent F, @ IKo. T h e n , t h e system passes from t h e reference state IKo IK = (1,2 , ..., LA,..., P,, ..., F d ,..., r ] = IKO { F J ,F, E IK 3 IK,J(i,X,x) E N,b,A,,l).L e t t h e statistical polymer c o m p o n e n t in t h e p r e c i p i t a t i o n e q u i l i b r i u m w h i c h is
-
C 1978 American Chemical Society
+
1162 Craubner
Macromolecules
perturbed be the main component P, E IP C !KO,the perturbing precipitant be the secondary component of the first kind F, E IF C IK, and the remaining members of set IK be the additional components of the second kind IK" = IK - (P,,F,); (P,,FJ IK'q3. The corresponding r-tuples of volume fractions are (see ref. 3, eq 5): yo = (yl, YZ,..., Y L ..., ~ ~YP,,...,Y F ~ - ~ Y ,F ~ + ..., ~ , Y r ) and = (71, ~ 2 :..., yLI, ..., yp,, ..., y F x , ..., y r ) . In this context, the subscripts P,,LA,and F, designate the respective polymer, solvent, and nonsolvent components. In the following, since there is no risk of confusion, the simplified perturbation notation will be used (see ref 3, section IV.l). As pointed out in ref 3, the investigation of the most important quasiternary systems can be restricted ( E E r > 3, > 1, = F= 1) - - -to the analysis of ternary systems ( K = r = 3, P = 1, = F = 1). Hence, for (a-p) phase separation in (quasi-) ternary macromolecular solutions the difference of the perturbed chemical phase potentials of the polymer App(*-fi' and the related change of the chemical polymer potentials of the unperturbed reference states App(0)are linearly combined with the precipitant fraction yF RT &,(a-8) = 4 ~ p ( O+ ) A F * ~ F Y F= & p ( O ) + FYF (1) Equation 1can be assumed to be a first-order perturbation series expansion in powers of (small) YF. The product terms on the right-hand sides mean the perturbation. Hereby, &*pF denotes the (a+) change of the perturbation density, u* the dimensionless inverse relative phase change of the perturbation density, and yF the relative strength of perturbation by the addition of nonsolvent. Its minimum is Y ~ = 0 ,and~its maximum ~ yF,max = 1. In this context, regarding the reference states of the sol ( y p " 0) and gel phases ( y p @ 11, it was borne in mind that the original (quasi-) binary macromolecular solutions are highly diluted ( y p ( 0 ) 0). Therefore, one can assume ~~"(0) = ~ ~ (and 0 )p p @ ( 0 )= pop where pp(0) = y p ( ~ odenotes ) the chemical polymer potential of the original (quasi-) binary solution and pop is the chemical standard polymer potential (see ref 3). For practical applications these facts are in accordance with the two limit cases discussed in ref 3, Section 111.2. Hence, according to ref 3, the application of eq 9, 11, 12 to eq 21b results approximately in (pop(0) = pop + R T In bop(0)) A p L , ( o ) = ~ ~ (- 0~~p) = bop + R T In rdo) - FOP = RT(ln y p ( 0 ) + In b ~ p ( O ) )
-
Introducing these expressions on the left-hand sides of eq 1 to 3, after transformation one obtains for the precipitant fraction (yp(0)bop(O) = ~~(0)): nm
Here, a p ( 0 )is the polymer activity of the original solution and bopDa the polymer limit activity coefficient of the sol phase. In the case of the most important quasiternary macromolecular solutions (E r > 3, > 1, E = F = I), subscripts P, in lieu of P should formally be used.
F
111. Perturbation-Theoretical Approach to the Concentration Dependence of Phase Separation in Fluid Macromolecular Multicomponent Systems T o determine the concentration dependence of phase separation in (quasi-) ternary macromolecular solutions, eq 3 or 6 can be used according to choice. At phase equilibrium and constant molecular weight (M) of the statistical polymer component involved, according to eq 1 and 15 of ref 3, it is ((T,p,M,w) = constant; w = injection rate)
PI?" = Ps B
0
-
-
Combining eq 15 of ref 3 and eq 1, one obtains App(m-fl)= RT[ln y p ( 0 ) + In bop(0)I + SP*PFYF = RT RT[ln y p ( 0 ) + In bop(0)I + FYF
aI?a= a$
(7b)
U
Ys"fI?" = YsPfsP
(7c)
Hence, from eq 21a of ref 3 and eq 5c it follows that App("-8)= 0. Combining this result with eq 3, after rearrangement one obtains yF* = -
RT ~
[In bOp(0) + In yp(0)I =
Sp*PF.~
where yF* denotes the precipitant fraction at the cloud point. Going over from the volume fraction to the initial weight concentration of the polymer (co = yp(0)ppg(cm3; pp = partial density of the polymer), we arrive at the linear semilogarithmic function
according to ref 3, Section 111.2. Here,
bop(0)= f p / f p = exp[(bop - POP)/RTI= exp[-(AEfip - AEpp)/RT1 designates the limit Dolvmer activitv coefficient of initglly highly dilutid [quasi-) binary macromolecular solution. Applying the analogous considerations, including the two thermodynamic limit cases cited in ref 3, to the sol and gel phases, it results (see ref 3, eq 1,9, 11, 12; ~ o =ppopp ~ = pop; yp" 0;ypb 1): p p o a= p o p
-+ -
R T In upa = pop"
+ R T In y p a = p o p + RT(1n y p a + In bop") (5a)
This is the phase equation when (T,p,h4,w) = constant and
A, =
r,*
In b p / b O p ( O ) )
(10)
In eq 8-10 the perturbation parameters Ap*pF,, and uc* were labeled by the subscript c. A , designates the precipitant fraction a t co = 1 g/cm3. The corresponding volume fraction is equal to the dimensionless quantity of Yp(0) = l l / P P l = IOPI. Plotting yF* vs. In co = 2.303 log co, one should get a linear decreasing function of the slope dyF*/d In co = -u,*. Accordingly, the decrease of the precipitant fraction a t incipient phase separation should be proportional to the
Phase Separation in Macromolecular Multicomponent Systems 1163
Vol. 11, No, 6, November-December 1978
Table I Solution and Precipitation Conditions to Polystyrene, Poly(methy1 methacrylate), and PolvIe-camolactam) a t 25 " C "
reciprocal relative perturbation d e n s i t y uc*, as well as to the relative increase of t h e initial polymer concentration, -dYF* = u,* dco/co. According to e q 4 and in addition to pp and uc* in eq 9 and 10, the polymer limit activity coefficient of t h e original (quasi-) b i n a r y macromolecular solution bop(0)is a f u r t h e r c o n s t a n t . In connection w i t h A, which m a y b e obtained experimentally as t h e intercept at co = 1 g/cm3, bop(0)c a n b e d e t e r m i n e d at known polymer d e n s i t y acc o r d i n g to
In bop(0)= In pp
-
A,/u,*
(IW
poly(methy1 poly(epolystyrene methacrylate) caprolactam) type U
%
solvent precipitant c,
x 103 ( gl cm.3)
Therefore, all t h e constants of eq 9--11 can be d e t e r m i n e d experimentally. S t a r t i n g the p e r t u r b a t i o n calculations from eq 7b,c we arrive at t h e same expressions for t h e c o n c e n t r a t i o n dep e n d e n c e of yF* as given b y e q 9. B e y o n d that, however, one o b t a i n s a d d i t i o n a l information w i t h r e g a r d t o the "interior" state of p h a s e equilibria in (quasi-) t e r n a r y macromolecular multicomponent systems, because it follows
H e r e , the right-hand side is obtained b y applying t h e t w o thermodynamic limit cases cited in ref 3, Section 111.2 (rp" 0, f p a boppa; yp0 1,f p d 1). E q u a t i o n 1 2 means a c o n s t a n t r a t i o of the e q u i l i b r i u m p o l y m e r v o l u m e fractions in t h e sol and gel phase irrespective of t h e initial p o l y m e r c o n c e n t r a t i o n co. In the case of a q u a s i t e r n a r y macromolecular solution w i t h a finite polymer s u b s e t IP of cardinal n u m b e r ? > 1,e q 12 is valid obviously for each statistical polymer c o m p o n e n t , and subscripts P, instead
- -
-
+
of P must be used. Finally, o n account of t h e concentration e q 6 of ref 3 one can calculate the course of t h e b o u n d a r y curves of p h a s e s e p a r a t i o n i n macromolecular m u l t i c o m p o n e n t s y s t e m s according t o
1= c(Yp,* Ji'h
1=
+ yFx*
t'):,I?
(x> 3; cI,x,X) E I N r b x A ) ) ( 1 3 4 yp*
+ yF* + TI,*,
( R = 3;
=
F=
= 1) (13b)
H e r e , t h e asterisks designate t h e cloud-point concentration of the various c o m p o n e n t s of a m i x t u r e , calculated acc o r d i n g to eq 1 4 of t h e following section IV.
IV. Experimental Section The precipitant fractions were determined by fast turbidimetric titration, according to the dynamic volume pulse technique (see ref 9 and 10). For that purpose, a thermostated semiautomatic recording turbidity measuring device for the fast optical precipitation analysis of polymer solutions was applied as previously d e ~ c r i b e d .The ~ precipitations were carried out at injection rates of ca. w = 20 cm3/min. For the computation of yF*, the linear portion of a recorder curve was extrapolated to the point of intersection with the time axis (abscissa) which is correlated, via the injection rate zi,,with the injected precipitant volume (VF). The measuring periods were less than a minute. The measurements were carried out with white light at 25 "C, using solutions of polystyrene, poly(methy1 methacrylate), and poly(6-caprolactam). They were adapted to optimum precipitation conditions by the preaddition of certain amounts of precipitant. Hereby, "tilted states" (in German "Kippzustande") were approached, just within the cloud-point t h r e s h ~ l d s . ~ J ~ The turbidimetric precipitation conditions are summarized in Table I. Details of the evaluations are given in ref 9. The accuracy equals 0.1% (cf. ref 9). Mixing additivity of the volumes was
adaptation ratio
fractionated
