Macromolecules 1989,22, 879-890
gram through the Stanford Center for Materials Research. We would like to thank Dr. W. T. T a n g and Dr. Georges Hadziioannou (IBM Almaden Research Center) for providing the end-tagged polystyrene. Registry No. PS, 9003-53-6;1-pentylpyrene, 80655-41-0.
References and Notes J. Chem. Phys. 1969,51, 5720. (2) Chuang, T. J.; Eisenthal, K. B. Chem. Phys. Lett. 1971,11, 368. (3) Fleming, G. R.;Morris, J. M.; Robinson, G. W. Chem. Phys. 1976,17,91. (4) von Jena. A.: Lessinn. H. E. Chem. Phvs. 1979.40. 245. ($ Moog, R. S . ; Ediger, M. D.; Boxer, S. G.;-Fayer,M.D. J. Chem. Phys. 1982,86,4694. (6) Gochanour, C. R.; Fayer, M. D. J. Chem. Phys. 1981,85,1989. (7) Miller, R. J. D.; Pierre, M.; Fayer, M. D. J.Chem. Phys. 1983, 78,5138. (8) Peterson, K. A.; Fayer, M. D. J. Chem. Phys. 1986,85,4702. (9) Peterson, K. A,; Zimmt, M. B.; Linse, S.; Domingue, R. P.; Fayer, M. D. Macromolecules 1987,20, 168. (10) Fredrickson, G. H.; Andersen, H. C.; Frank, C. W. Macromolecules 1984,17,54. (11) Forster, Th. Ann. Phys. 1948,2,55. (12) Forster, Th. 2. Naturforsch., A: Astrophys., Phys. Phys. Chem. 1949,4,321. (13) Gochanour, C. R.; Andersen, H. C.; Fayer, M. D. J. Chem. Phys. 1979,70,4254. (1) Eisenthal, K. B.; Drexhage, K. H.
879
(14) Knoester, J.; van Himbergen, J. E. J. Chem. Phys. 1984,81, 4380. (15) Carter, T.P.; Fearey, B. L.; Hayes, J. M.; Small, G. J. Chem. Phys. Lett. 1983,102,272. (16) Thijssen, H. P. H.; Van Den Berg, R.; Volker, S. Chem. Phys. Lett. 1983,97,295. (17) Jankowiak, R.;Biissler, H. Chem. Phys. 1983,79,57. (18) Peter, G.; Bawler, H.; Schrof, W.; Port, H. Chem. Phys. 1985, 94,445. (19) Peterson, K.A.; Stein, A. D.; Fayer, M. D., to be published. (20) Ediger, M. D.; Domingue, R. P.; Peterson, K. A,; Fayer, M. D. Macromolecules 1985,18,1182. (21) Tang, W. T.;Hadziioannou, G.; Smith, B. A.; Frank, C. W. Polymer 1988,29,1718. (22) Oliver, N.H.; Pecora, R.; Ouano, A. C. Macromolecules 1985, 18,2208. (23) Kawski, A.; Weyna, I.; Kojro, Z.; Kubicki, A. 2. Naturforch, A: Phys., Phys. Chem., Kosmophys. 1983,38A,1103. (24) Fredrickson, G. H.J. Chem. Phys., in press. (25) Galinin, M. D. Tr. Fiz.Inst. im. P. N. Lebedeva, Akad. Nauk. SSSR 1950,5,341. (26) Craver, F. W.; Knox, R. S. Mol. Phys. 1971,22,385. (27) Brito Cruz, C. H.; Fork, R. L.; Knox, W. H.; Shank, C. V. Chem. Phys. Lett. 1986,132,341. (28) Huber, D. L.Phys. Rev. B: Condens. Matter 1979,20,2307. (29) Huber, D. L. Phys. Rev. B: Condens. Matter 1979,20,5333. (30) Fredrickson, G.H.; Frank, C. W. Macromolecules 1983, 16, 1198. (31) Avis, P.; Porter, G. J. Chem. soc., Faraday Trans. 2 1974,70, 1057. (32) Johnson, G. E. Macromolecules 1980,13,839.
Theory of Semidilute Solutions of Polymer Mixtures in a Common Solvent: Blob Picture and Pseudobinary Approximation Akira Onuki Research Institute for Fundamental Physics, Kyoto University, Kyoto 606, Japan
Takeji Hashimoto* Department of Polymer Chemistry, Kyoto University, Kyoto 606, Japan. Received April 11, 1988;Revised Manuscript Received June 28, 1988
ABSTRACT A renormalization group theory is used to devise a formula for the effective interaction parameter Xeff between different species of polymers, 1 and 2, in a common solvent. It describes the overall behavior as a function of the polymer volume fraction 4, the solvent quality, and the microscopic (bare) interaction parameter x12 The polymer-polymer phase transition is then examined by using a generalized pseudobinary approximation. Its critical line is shown to extend into a semidilute region, 4 > d-. Here 4- is a minimum volume fraction of the order of the overlapping volume fraction $* in a 9 solvent and considerably larger than q5* in a good solvent. As 4 @-, deviations from the Flory-Huggins predictions on the phase transition become enhanced. Particularly Xeff saturates into a value proportional to a negative fractional power of the molecular weight and the critical value of xlz diverges as 4 &in. These are consistent with previous experiments and Monte Carlo simulations. However, as the solvent quality is decreased, the fluctuation effects of 4 become important and, as a result, the critical line will reach a tricritical point or a critical end point Our theory is also used to explain spinodal decomposition processes in ternary systems. a t 4 larger than 4h,.
-
1. Introduction A great number of phase-separation experiments have been performed on ternary systems consisting of polymer 1, polymer 2, and a In such ternary systems
there are four independent thermodynamic variables. Here, for simplicity, let the pressure be fixed. Then we have three variables and can envisage the phase diagram i n a three-dimensional space.*v5 We have critical lines, two-phase coexistence surfaces, and lines of triple points (three-phase coexistence lines) formed by the intersection of two coexistence surfaces. Generally, lines of triple points end at a critical end point at which two of the three phases become identical. A tricritical point is an intersection point of two critical lines and is generally a special point on a line of critical end points i n a four-dimensional space (the
-
pressure being included).4q5 Such a phase diagram has been rather extensively studied for a mean-field model which can mimic a ternary system of low molecular weight fluids (the model is given by (1.1)below if N1 = N 2 = l).%'In our polymer systems, however, the phase diagram should strongly depend on new parameters, the molecular weights, and its understanding remains very insufficient. To get some idea let us consider a binary mixture of two polymers. There, we have a polymer-polymer phase separation which can be easily triggered even with very small repulsion -between two polymer^.^,^ On the other hand, i n a binary system of polymer + solvent, a phase separation occurs T h, ~u s , we find the with decreasing the solvent q ~ a l i t y . ~ corresponding three critical points i n the dilute limits of one component. T h e y extend, forming critical lines into
002~-9297/89/2222-0879~01.50/0 0 1989 American Chemical Society
Macromolecules, Vol. 22, No. 2, 1989
880 Onuki and Hashimoto
the three-dimensional thermodynamic variable space. The global geometry of the phase diagram is then of great interest.1° Previous data have been analyzed on the basis of the Flory-Huggins mixing free energy1-3r9J1J2 1 1 F O / b T = -41 In $1 + -42 In $2 + In + Nl N2 XlS41$8 + X 2 S 4 2 A + x124142 (1.1) This is the free energy per lattice site with a microscopic linear size a. Nl and N2 are the polymerization indices, $l and Cb2 are the volume fractions of polymers 1 and 2, respectively, and $* = 1 - $1 - $z is the solvent volume fraction. When data were analyzed on the basis of (l.l), the polymer-polymer interaction parameter x12turned out to be very small and dependent on the molecular weights and the composition^.^ In original theories the parameters xla,xZs,and x12were considered to represent microscopic two-body interactions between monomers and hence should be independent of the molecular weight and the compositions. Further, to see the implication of (1.1)on phase separation, let us suppose xlS = xZsand N1 = N2 = N and consider a polymer-polymer phase transition in which the order parameter is the relative composition X $J$, $ = $1 #2 being the total polymer volume fraction. The solvent quality is assumed to not be poor. Then (1.1)can be written as 1 FO/kBT In 4 + 4 s $8 + xls$$s] +
+
[E$
In X
1 + -(1 - X) In (1 - X) + x12$X(1- X) N
1
(1.2) The second term of (1.2) means that the critical point is given by X = X, = ' I zand
(1.3) 4 being held fixed. This relation will be shown to hold only under special conditions. Afterward, de Gennes13 showed that values of xlz obtained by fitting (1.1)to phase-separation data of ref 3 are proportional to N-1/2. He pointed out, using scaling arguments, that contact between two polymers should be much weaker than expected in the simple Flory-Huggins ~cheme.~J~ Recently, considerable progress has been made in renormalization group analysis of ternary sy~tems.'~-'' It has shown that interactions between unlike polymer species must be greatly modified in the dilute and semidilute regions, $ > 1. The derivation of (3.18) will be deferred to Appendix B. In comparing our results with experiments, x12is a very important parameter. As a first approximation we regard it to coincide with Flory's parameter in the melt case. Note that this cannot be justified here because our scheme is valid only in the case 4 1 from (3.20) and c
V 0.01
to
XSD
is defined by XSD = x / ( ~ -v 1) z 0.27
-
(3.23)
While the previous theories14J6J7show Xblob z x l 2 4as~~ suming uo u*, our formula 3.22 is applicable even in the case uo 0 and the condition of strong segregation is expressed as9 6th 5 1 (3.43) while the condition of weak segregation is 0 6th 5 1
= (a/83[fblob + finhl +
(3.46)
**e
with finh
1 = 36X(1 - X)t21VXI2
(3.47)
This term has the same form as the gradient term for the melt case derived from the RPA a p p r o x i m a t i ~ n .Then ~~~~ the correlation function of X, = Jdr e"*'X(r) is given by
The thermal correlation length [th
&h
(3.50)
In particular, in the weak segregation case (3.44) near the critical point, &h grows in the mean field theory as
-
RGl'thl-1'2
-
(3.51)
where Nl = N2 = N has been assumed and RG ,$(N/g)'l2 is the gyration radius of a chain. On the basis of (3.47) the surface tension u can be easily estimated. In the strong segregation case 6th 5 1, (3.43), we have31
-
kBT(Xblob)"2/62
(3.52)
(3.53)
-
kBTF2(N/g)2'26th2e
(3.54)
where 3 z 0.625 is the Ising critical exponent for the correlation length. The coefficient in front of 6th25 has been determined such that (3.53) and (3.54) coincide at eth = g/N. Broseta et a1.18 studied u by assuming (i) the strong segregation condition, (ii) the good-solvent limit fo 1, and (iii) the symmetric condition N1 = N2. Our formulas 3.52-3.54 can predict the overall behavior of u in a much wider parameter region. To illustrate relationships to the previous results, we rewrite formula 3.52 for the strong segregation case
-
UU2/kBT
-
-
[x12/(@+ x12/u*12)11/2d'z
in 0 solvent (3.55a) [XlZ/(fO
+ x12/u*12)11/2f02-x4x in good solvent (3.55b)
where x = ('l2x + 2v)/(3v - 1) = 1.65 and use has been made of (3.4b) and (3.22) for good solvent and (3.7) and 3.24) for 8 solvent. Notice the following: (i) Equation 3.55a reduces to the mean-field result by Helfand et al., u a 41,5,if 4 2 xlz/u*lz, while it yields u a 42for relatively strong repulsion x12 2 (ii) The exponent x in the good-solvent case was already calculated in ref 18. However, our formula can predict the behavior of the coefficient in front of 4x for small f,,. 4. Spinodal Decomposition in Ternary Systems A. Theory. We consider the spinodal decomposition in the case N1 = N2 = N. The system is assumed to be at the critical composition X = 'I2 before quenching. It becomes unstable after Xblob is increased above XblobC = %IN. For simplicity we first assume 0 < 6th 0 and A < 0, T, decreases drastically with decreasing 4 W (particularly as 4 4min).The factor W in (4.1) accounts for the renormalization effect important for 4 4min.If W = 1, (4.1) is the usual Flory-Huggins result. Next we consider q., in the 0-solvent case. Assuming T - T, to be small, we have from (3.42) and (3.25b) 6th
= '/4NW(aXiz/aT)c(T - Tc) = W(8 In xlz/a5"l,(T - T,)
-
(4.3)
Macromolecules, Vol. 22, No. 2, 1989
886 Onuki and Hashimoto
where the derivative should be taken at T = T, and use has been made of l/24Nx12= 1 +. xlz/iiu*lz = W-l at the critical point. Because the coefficient in front of T - T, in t t h is a very important parameter, representing the distance from the criticality for a given T - T,, we here express it as a function of 4, again assuming the form x12 =A+BIT tth
= (2/BN4)[1 - f/zAN4W2(Tc- 2')
(4.4)
This form is not much different from the Flory-Huggings result (which is obtained by setting W = 1) and the coefficient tends to a constant, 2/BN4,h, as 4 whereas T, 0: W as 4 $,in. On the other hand, in the good-solvent case we have (4.5) t t h = W@ In XlZ/aT),(T - T,)
-
-
where the temperature dependence of fo has been neglected and W is redefined as
W = 1 - Ao(fo/4)5/4+~s~
(4.6)
qm(~)/qm(o)
-
7-a
(4.14)
The exponent a is about 1/4 if the Binder-Stauffer cluster dynamics is applicable to this case.39 However, for 7 > 7, >> 1,the growth of the domains is affected by the velocity field induced by surface tensi0n~O9~~ and then qm(7)
- (?/W
(4.15)
Here (T is estimated as (3.53) in the mean-field approximation and q is the macroscopic viscosity. The reptation theory shows
-
-
(kBT/
