Let us prove that the RPA condition (5.63) reduces to the spinodal condition (5.46) if the wavenumber is allowed to go to zero. To simplify the notation we definen ≡ nA+ nB
for the sumof the DP of both species and writenA≡ na,nB≡ nb with a +b = 1.
For q=0 we have Jij=1, and Alm(0)=(nal)2,Blm(0)=(nbm)2, andClm=n2ablm.
Hence we haveSAA◦ = n2a2 l2 νS,SBB◦ = n2b2 m2 νS, andSAB◦ = n2ab lm νS, where a bracket ··· abbreviates the number-weighted average ··· n. By definition we obtain
S(0)
W(0)= (al +bm)2
(nab)2( l2 m2 − lm 2)νS.
To express theκ-functions in terms of the average quantities, we take the derivative of the two relations (5.27a) and (5.27b) with respect toφ, and find
na[xνx+x(νxxx+νxyy)] = 1, (5.71a) nb[yνy+y(νyxx+νyyy)] = −1, (5.71b)
where a prime indicates the derivative with respect to φ, and νxy, etc., are the par-tial derivatives of νS. By the use of the identities x2νxx=
l(l − 1)νl,m= ( l2 − l )νS,y2νyy=( m2 − m )νS, andxyνxy= lm νS, we eliminate the partial derivatives
in favor of the number-averages. The relations(5.71a) are transformed into l2
l κA−a b
lm m κB= 1, b
a lm
l κA− m2
m κB= −1, which, when solved with respect toκ, give
κA(φ) = a lm +b m2 b( l2 m2 − lm 2) l , κB(φ) = a l2 +b lm
a( l2 m2 − lm 2) m .
Theκ-functions have thus been expressed in terms of the average cluster sizes and their fluctuations. Substituting the result into (5.46), we confirmthat it is equivalent to (5.63) withq =0. The RPA scattering function has thus been most generally proved to give the lattice-theoretical spinodals in the limit of vanishing wavenumber.
References
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[4] Prigogine, I.; Bellemans, A.; Mathot, V., The Molecular Theory of Solutions. North-Holland:
Amsterdam, 1957.
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[6] Hirschfelder, J.; Stevenson, D.; Eyring, H., J. Chem. Phys. 5, 896 (1937).
[7] Flory, P. J., J. Chem. Phys. 12, 425 (1944).
[8] Huggins, M. L., J. Chem. Phys. 46, 151 (1942).
[9] Flory, P. J., Principles of Polymer Chemistry. Cornell University Press: Ithaca, NY, 1953.
[10] Koningsveld, R.; Stockmayer, W. H., Nies, E., Polymer Phase Diagrams–A Text Book. Oxford University Press: Oxford, 2001.
[11] Clark, A. H.; Ross-Murphy, S. B., Adv. Polym. Sci. 83, 57 (1987).
[12] Russo, R. S., in Reversible Polymeric Gels and Related Systems, Russo, R. S. (ed.). American Chemical Society: New York, 1987.
[13] Kramer, O., Biological and Synthetic Polymer Networks. Elsevier: London and New York, 1988.
[14] Guenet, J. M., Thermoreversible Gelation of Polymers and Biopolymers. Academic Press:
London, 1992.
[15] te Nijenhuis, K., Adv. Polym. Sci. 130, 1 (1997).
[16] Tanaka, F., Macromolecules 22, 1988 (1989).
[17] Tanaka, F.; Matsuyama, A., Phys. Rev. Lett. 62, 2759 (1989).
[18] Tanaka, F., Macromolecules 23, 3784; 3790 (1990).
[19] Tanaka, F.; Koga, T., Bull. Chem. Soc. Jpn 74, 201 (2001).
[20] Tanaka, F., Polym. J. 34, 479 (2002).
[21] Solc, K., Macromolecules 3, 665 (1970).
[22] de Gennes, P. G., J. Phys. (Paris) 31, 235 (1970).
[23] de Gennes, P. G., Faraday Disc. Roy. Soc. Chem. 68, 96 (1979).
[24] de Gennes, P. G., Scaling Concepts in Polymer Physics. Cornell University Press: Ithaca, NY, 1979.
[25] Leibler, L., Macromolecules 13, 1602 (1980).
[26] Bates, F. S., Science 251, 898 (1991).
[27] Tanaka, F.; Ishida, M.; Matsuyama, A., Macromolecules 24, 5582 (1991).
[28] For example see (i) Burchard, W., In Light Scattering from Polymers; Springer-Verlag: Hei-delberg, 1983; p.1 (ii) Picot, Cl. in Static and Dynamic Properties of the Polymeric Solid State Pethrick, R. A. Richards, R. W., (eds.). Reidel: New York, 1982, p. 127.
This chapter presents some important nongelling binary associating mixtures. Throughout this chapter, we assume the pairwise association of reactive groups, the strength of which can be expressed in terms of the three association constants for A·A, B·B, and A·B association. We apply the general theory presented in Chapter5to specific systems, such as dimerization, linear association, side-chain association, hydration, etc. The main results are summarized in the form of phase diagrams.
6.1 Dimer formation as associated block-copolymers
The first systemwe study is a mixture of R{A1} and R{B1} chains, each carrying a functional group A or B at one end. Diblock copolymers are formed by the end-to-end association (hetero-dimerization) [1,2]. End groups A and B are assumed to be capable of forming pairwise bonds A·B by thermoreversible hetero-association. The hydrogen bond between acid and base pair is the most important example of this category.
For such mixtures, composite diblock copolymers R{A1}-block-R{B1} with a tem-poral junction are formed (Figure6.1). The system is made up of a mixture of diblock copolymers (1,1), and unassociated homopolymers of each species (1,0) and (0,1). It is similar to the mixture of chemically connected diblock copolymers dissolved in their homopolymer counterparts [3,4], but its phase behavior is much richer because the population of the block copolymers varies with both temperature and composition.
Letn≡nA+nBbe the total number of the statistical units on a block copolymer chain, and leta ≡ nA/n (b ≡ nB/n) be the fraction of A-chain (B-chain). The relation a +b = 1 holds by definition.
Our starting free energy is given by
F = ν11+ν10lnφ10+ν01lnφ01+ν11lnφ11+χ(T )φ(1−φ), (6.1)
where
≡ β(µ◦A·B−µ◦A−µ◦B) (6.2) is the free energy of dimer formation. By differentiation, we find the chemical potentials for each component as
βµ10= 1+ln x −nAνS+χnA(1−φ)2, (6.3a)
+
A B A • B
Fig. 6.1 Associated diblock copolymer formed by a pairwise bond between the end groups.
βµ01= 1+ln y −nBνS+χnBφ2, (6.3b)
βµ11= 1++ln z−nνS+χ[nA(1−φ)2+nBφ2], (6.3c) whereνS≡ ν10+ ν01+ ν11is the total number of molecules that possess translational degree of freedom, and the abbreviated notationsx ≡ φ1,0,y ≡ φ0,1,z ≡ φ1,1have been used.
The association equilibriumcondition (5.19) then leads to
z = Kxy (6.4)
for the volume fraction z of the block copolymers, where K ≡ exp(1−) is the temperature-dependent equilibrium constant. Because of the nongelling nature, we have the identity
φS= x +y +Kxy ≡ 1. (6.5)
The number density of clusters is given by νS= ν =1
n
x a+y
b+Kxy
. (6.6)
The coupled equations (5.27a) and (5.27b) take the form
x(1+aKy) = φ, (6.7a)
y(1+bKx) = 1−φ. (6.7b)
The solution is given by
x(φ) = φ −a −K−1+ D(φ)!
/2b, (6.8a)
y(φ) = a −φ −K−1+ D(φ)!
/2a, (6.8b)
whereD(φ) ≡ [a(1−φ)+bφ +K−1]2−4abφ(1−φ). Hence we have z(φ) = 1
2ab
"
a(1−φ)+bφ +K−1− D(φ)#
. (6.9)
The logarithmic derivatives ofx and y yield specific forms of the κ-functions for the dimerization
κA(φ) = 1−az
1−az/φ, κB= 1+bz
1−bz/(1−φ), (6.10)
wherez(φ) is the concentration derivative of z(φ). Explicitly, it is z(φ) = K(y −x)/
[1+K(ay −bx)].
Let us consider the free energy (6.2) of the dimer formation. The conformational free energy appears because the entropy of disorientation is reduced when two chains are combined. If we use the lattice-theoretical entropy of disorientation (2.90), we have
Sdis≡ Sdis(nA+nB)−Sdis(nA)−Sdis(nB) = kBln
σ(ζ −1)2 ζ enab
, (6.11)
for the entropy change. The free energy is given byfconf= −T Sdis. Combining the free energy of bondingf0= −T s, we find that the equilibriumconstant is given in the form
K = λ0e−β , (6.12)
whereλ0≡ σ (ζ −1)2es/kB/ζenab is a temperature-independent constant.
Let us proceed to the calculation of the scattering functions. Simple algebra gives
A10= A11= nA
i,j=1
exp(−κ|i −j|) = (na)2D(aQ), (6.13)
for the AA component of the intramolecular scattering function, where Q ≡ nκ = n(aq)2/6 = (RGq)2is the dimensionless squared wavenumber measured relative to the unperturbed gyration radiusRG≡ a√
n/6 of a diblock copolymer. The entire scattering function depends on Q, T , and φ. The function D(x) is Debye function (1.49). The amplitudeAlmis the same for both A-unimer (1,0) and copolymer (1,1).
Similar calculation leads to
B01= B11= (nb)2D(bQ), (6.14)
for the BB components, and
C11=
for the AB component of a block copolymer. Putting these results together, we find SAA◦ = A10 x and the condition for stability limit (5.63) to be solved is given by
F (Q)−2nχ = 0, (6.17)