Correlation between the combinatory entropy of polymer and ideal liquid solutions S. Saeki Department of Materials Science and Engineering, Fukui University. Fukui 910, Japan (Received 28 April 1992; revised 20 July 1992) A semiempirical equation of combinatory entropy in binary solutions of semiflexible or rod-like polymers in solvent has been derived, based on a correlation between the combinatory entropy in polymer solution derived by Flory, Huggins, Miller and Guggenheim and an ideal simple liquid solution. The expression for the partial entropy of mixing of solvent ASM,~ obtained in this work is given by: ASu, l/k= -In ~b 1 - ~b2(1 - r- 1) d- ~(]) 1 (]12 --]-ln{ [~b 1 + ((a2/r)]/[qb 1 -I- (~b2/r) H- ~b 1 t~2] } + ctO2 ln{ (r- 1+ ~bl)/[~bl + (q~2/r)+ ~ 1(~2] } where ~ is the volume fraction of polymer (i = 2) and solvent (1), r is the number of segments per polymer, k is the Boltzmann constant and ~ is a parameter characterizing the flexibility of polymer chain such that ct=0 for flexible polymers and ct=(1-r-~)/q~ for rigid-rod polymers. The chemical potential of solvent in the polymer solution is given by: #, -- I2° = R T[-- ASu,,/k + zoo + a)~b22] where ;(0 is the polymer-solvent interaction parameter in the Flory-Huggins theory. (Keywords: combinatory entropy; polymer solution; ideal simple liquid solution; Flory-Huggins theory; semiflexible polymer; rod polymer) INTRODUCTION are discussed by DiMarzio 7, while the combinatorial entropy correction for imperfect randomness is calcu- The Flory-Huggins theory is the most simple and lated by Huggins a. More rigorous calculation of com- fundamental theory of flexible polymer solutions. The binatory entropy based on the self-avoiding and mutually theory uses the mean field approximation that the avoiding walk has been done by Kurata 9 using the concentration of solution is uniform over the entire range pseudoconfiguration method, and by Bawendi and of solution, for calculating the number of configurations Freed 1°-~2 in the lattice spin-field theory, and by Monte of polymer and solvent molecules and for evaluating the Carlo simulation I a, 14. enthalpic term by a van Laar model 1. The division of This work discusses the correlation between the the free energy into entropic and enthalpic terms made the combinatory entropy in polymer solution and the ideal Flory-Huggins theory even more simple and useful. liquid mixture based on the lattice theories of Flory, Flory also stressed a division of the solution into Huggins, Guggenheim and Miller and a simple and useful intermolecular and intramolecular factors, and stated equation is derived for the entropy of mixing and that thermodynamic properties of a flexible polymer chemical potential in binary solutions of semiflexible and solution are independent of the structure and flexibility rod-like polymers in solvents. Division of the Gibbs free of the macromolecules, and are dependent on only one energy of mixing into the entropic and enthalpic term in parameter, the number of segments per polymer, r. polymer solution is also discussed. Guggenheim 2, Miller 3, and Huggins4 derived essen- tially the same equations for the entropy of mixing in polymer solutions where a parameter Z coordination FUNDAMENTAL RELATION IN THE number is included in addition to the parameter r. CALCULATION OF COMBINATORY ENTROPY Investigation of the combinatory entropy for non-flexible It is well known that the configurational partition polymers such as semiflexible and rod-like polymer has function of ideal, simple liquid mixtures (SM) of N 1 been made by Flory 5'6, where new parameters are molecules of component 1 and N 2 molecules of com- introduced: a fraction f of the bonds bent out of the ponent 2 is given by: co-linear direction of the preceding segment in the semiflexible polymer, and the angle of inclination of the nSM= (N1 + N2)!/(N, !N2 !) (1) particle to the domain axis in the rod-like polymer. Many ~X~'X~ N~ (2) different approaches to the calculation of combinatory entropy and free energy in polymer solutions have been where X i = Ni/(N ~ + N2) is the mole fraction of i and the made. The effects of orientation on combinatory entropy Stirling approximation that N~!'~(N~/e) N' is used. It is 0032-3861/93/0714444)5 © 1993 Butterworth-Heinemann Ltd. 1444 POLYMER, 1993, Volume 34, Number 7 Study of combinatory entropy in solutions: S. Saeki interesting to calculate the number of configurations of and the partial entropy of solvent (1) ASM,1 defined by the mixture under the condition that nt molecules of 1 AS~, 1 = k(t~AIn ~/dN 1 )T,P,N2 and A In f~ = In ~'~M-- In f~N,= 0 are fixed at n~ different positions in the lattice, which is --lnGN2=0 is given by: designated as f~F and is given by: ASM,1/k= -In ~bl-~b 2 (rN 2 fixed) (11) -- ~'~SM x I (3) On the other hand, ASM,1 for the Flory-Huggins theory is: where X~ ~ is the probability that n~ different positions are occupied by n~ molecules of 1 simultaneously, or in ASM, ~/k = -In q~1 - 42 -[- ~)2/r (12) other words n 1 molecules are fixed at nl different The term dpE/rin equation (12) comes from the freedom positions in the lattice. It is obvious that if N1 molecules with which N 2 separated molecules of 2 can move over of 1 and N2 molecules of 2 are fixed at positions of the restricted lattice. A formulation of the combinatory (N~ + N 2), then the number of configuration is one, since entropy in polymer solutions for various theories has ~-~SMXXl vNI ""yN2_ 2 --~,1 from equation (2). It is also interesting been done by the same procedure as mentioned above, to calculate the number of configuration when n 1 which is given by: molecules of 1 are removed or disappear from the __ N2 (r- 1)N2 mixture, which is designated as DnR, and is given by: ~)comb-- q2 ~')id©al~2 f(N1,N2,r,Z) (13) where f is a function of N 1, N2, r and coordination D,R, = (N 1 + N 2 -- n~ )!/[(N 1 - n I )!N 2 !] (4) number Z and is dependent on the model. It is derived = DsM[N, (N,- 1)... (NI -hi + 1)]/ that f~rod for a mixture of N 2 parallel oriented rigid-rod [(N, +N2)(N x +N 2 -- 1)... (Na +N 2 -n 1 + 1)] (5) polymers with axis ratio r and N~ solvent is equal to The value of f~ is equal to the number of configurations ~rod=(NI+N2)!/(NI!N2 !) and, using equation (4), is given by: in which nl molecules of 1 are combined with the rest of the molecules (Nl-nl) of 1 to make (Nl-na) big or ~'~rod=~)ideal{[N2r(N2 r- 1)... (N 2 T 1)]/ long molecules. [Nt(N,- 1)... (N 1 + N2 + 1)]} (14) where CORRELATION BETWEEN THE ~'~ideal = Nt!/[Nx !(rN2)!] (15) COMBINATORY ENTROPY IN POLYMER AND The f~rod corresponds to the number of configurations of IDEAL LIQUID SOLUTIONS ideal mixture under the restriction that N2(r- 1) separated It is instructive to express the combinatory partition molecules of 2 are removed from the system or combined function in the Flory-Huggins theory by using the with the rest of 2 to make N 2 long rod molecules. The previous relation. The Flory-Huggins equation was ASM,1 for the rod polymer solution is expressed using the originally expressed byl: volume fraction in equation (9) by: ~"~F--H= (N2!)- 1 [Z(Z - 1 )r - 2]N2[NI1 -r)N2] ASM,1/k= - In q~l + ln[1 - (1 - r- 1)q~2] (16) X r(N2r)[(nt/r)!(nt/r-- N2)!] r (6) ~ -In ~bl- (1-r-1)~b2 th2~0 (17) where N i is thc number of solvent molecules (i = 1) and where equation (17) shows that ASM,1 for the rod mixture polymer (i = 2), r is the number of segments per polymer, approaches that of the Flory-Huggins theory at the limit Z is the coordination number and Nt=N~ +rN2 is the of ~b2 = 0. The partial entropy of mixing of polymer in the total number of lattices. Equation (6) reduces to: Flory-Huggins theory is given by: Y~F-H= q~(Nt!/[N1 !(rN2)!]}~b2N:('- 1) (7) ASM,2/k = q~l(r- D-In ~b2 (F-H) (18) where and that for the rod polymer solution by: q2 = Z(Z-- 1)'- 2tier- 1 (8) ASM,2/k=-ln(a2+ln[l+(a~(r-1)] (Rod) (19) and -ln~b2 + ~bl(r- 1) ~bl(r- 1)~0 (20) (b2 = rN2/(N1 + rN2) (9) Therefore ASM,2 for the rod polymer solution at the limit of q~l =0 is equal to that of the Flory-Huggins theory In the Flory-Huggins theory, Dv-n in equation (7) is for a flexible polymer solution. divided into the intramolecular factor q2 for polymer chain and the intermolecular factors, which are expressed by the ideal mixing term Nt!/ENI!(rN2)! ] for rN 2 DERIVATION OF COMBINATORY ENTROPY IN disconnected and separated molecules of 2 from N 2 SOLUTION OF SEMIFLEXIBLE AND ROD-LIKE polymers with r segments per polymer and N 1 solvent POLYMERS IN SOLVENT molecules, and the probability term that N2(r-1 ) separated molecules are fixed at NE(r-1 ) positions in The configurational partition function f2MiX for solutions of semiflexible and rod-like polymers in solvent which the lattice. In other words, equation (7) gives the number of configurations of ideal mixing of N~ of 1 and rN 2 of can predict cases of both flexible polymer solutions and parallel oriented rod solution is given by: 2 under the restriction that (r- 1)N 2 separated molecules __ ,TNL,TN2('I ~N2(r- l)--x are fixed and only N 2 molecules of 2 and N 1 molecules Mix--~/1 1"/2 ~ideal'W2 of 1 are free to move in the lattice with (r-1)N 2 fixed x {[(N 2 +x)(N 2 +x- 1)..