Self-Consistent Field Theory and Its Applications by M. W. Matsen

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1.7 Polymer Blends 43 Note that the value of χ is inversely proportional to the segment density, ρ 0 , and thus is not an invariant quantity; one must be careful of this fact when referring to literature values of χ. Equation (1.192) is the simplest and by far most common expression for the interactions, but SCFT is perfectly capable of accommodating more elaborate treatments (Matsen 2002a), if so desired. 1.7.1 SCFT for a Polymer Blend The partition function for a polymer blend is hardly any more involved than that for the polymer brush in Section 1.6.4. In this case, it becomes ∫ nA ( 1 ∏ ∏n B Z ∝ ˜Dr A,α ˜Dr B,β exp − U[ ˆφ A , ˆφ ) B ] δ[1 − n A !n B ! k B T ˆφ A − ˆφ B ] (1.194) α=1 β=1 As before, there is a functional integration over the configuration of each polymer and a Dirac delta functional is used to enforce the incompressibility condition. Of course, the delta functions constraining the chain ends are gone, and there is now a Boltzmann weight accounting for the energy of the segment interactions. Also present are factors of n A ! and n B ! to account for the indistinguishability of the A- and B-polymers, respectively. This time, we start by inserting a functional integral over δ[Φ A − ˆφ A ], allowing the operator, ˆφ A (r), to be swapped with the ordinary function, Φ A (r). This transforms δ[1 − ˆφ A − ˆφ B ] into δ[1 − Φ A − ˆφ B ], which, in turn, allows the exchange of ˆφ B (r) with 1 − Φ A (r). The result of this manipulation is Z ∝ 1 n A !n B ! ∫ DΦ A n A ∏ α=1 ˜Dr A,α n B ∏ β=1 ( ˜Dr B,β exp − U[Φ ) A, 1 − Φ A ] × k B T δ[Φ A − ˆφ A ]δ[1 − Φ A − ˆφ B ] (1.195) As with the brush, the two delta functionals are replaced with integral representations analogous to that in Eq. (1.142), which allows us to complete the integrals over the polymer configurations, leading to the result, ∫ 1 ( ρ0 ) Z ∝ DΦ A DW A DW B n A !n B ! N Q nA ( ρ0 ) A[W A ] N Q nB B[W B ] × ( exp − U[Φ A, 1 − Φ A ] + ρ ∫ ) 0 dr[W A Φ A + W B (1 − Φ A )] (1.196) k B T N where ∫ ( ∫ 1 ) Q γ [W γ ] ∝ ˜Dr γ,α exp − ds W γ (r γ,α (s)) (1.197) 0 is the partition function of a single γ-type polymer in the external field, W γ (r). (For the purpose of future simplification, a factor of (ρ 0 /N ) n has been extracted from the unspecified proportionality constant in Eq. (1.196).) Next, the factorials are replaced by the usual Stirling approximations (e.g., ln(n γ !) ≈ n γ ln(n γ ) − n γ ), giving ∫ ( Z ∝ DΦ A DW A DW B exp − F [Φ ) A,W A ,W B ] (1.198) k B T
44 1 Self-consistent field theory and its applications where [ ( F φV nk B T = φ ln ∫ 1 V Q A [W A ] ) ] [ ( ) ] (1 − φ)V − 1 +(1− φ) ln − 1 + Q B [W B ] dr[χNΦ A (1 − Φ A ) − W A Φ A − W B (1 − Φ A )] (1.199) As before with the brush, all the steps above are exact. The partition functions, Q A [W A ] and Q B [W B ], can still be calculated by the method outlined in Section 1.2, and thus F [Φ A ,W A ,W B ] can be evaluated without difficulty. It is the functional integration of exp(−F/k B T ) in Eq. (1.198) that poses a problem. So once again, the saddle-point approximation is implemented, which requires us to locate the extremum denoted by the lower-case functions, φ A , w A , and w B . Setting to zero the functional derivative of F [Φ A ,W A ,W B ] with respect to W A gives the condition, φ A (r) =−Vφ D ln(Q A[w A ]) (1.200) Dw A (r) Equation (1.32) identifies φ A (r) as the average concentration of n A polymers in the external field, w A (r), which is evaluated by where φ A (r) =− Vφ Q A [w A ] ∫ Q A [w A ]= ∫ 1 0 ds q A (r,s)q † A (r,s) (1.201) dr q A (r,s)q † A (r,s) (1.202) Here, the partial partition functions, q A (r,s) and q † A (r,s), obey the previous diffusion Eqs. (1.24) and (1.28), but with the field, w A (r). Differentiation of F [Φ A ,W A ,W B ] with respect to W B leads to the incompressibility requirement, φ A (r)+φ B (r) =1 (1.203) where φ B (r) ≡−Vφ D ln(Q B[w B ]) Dw B (r) (1.204) is the average concentration of n B polymers in the external field, w B (r), evaluated in the same manner as φ A (r). Following arguments 〈 already presented in Section 1.6.4, φ A (r) and φ B (r) are the SCFT approximations for ˆφA (r)〉 and 〈ˆφB (r)〉 , respectively. The last remaining differentiation of F [Φ A ,W A ,W B ] with respect to Φ A gives the self-consistent field condition, w A (r) − w B (r) =χN(1 − 2φ A (r)) (1.205) In principle, the two Eqs. (1.203) and (1.205) should be enough to determine the two functions, w A (r) and w B (r), but this is not exactly the case. With similar arguments to those
Page 1 and 2: Self-Consistent Field Theory and It
Page 3 and 4: 2 Contents Index 83
Page 5 and 6: 4 1 Self-consistent field theory an
Page 11 and 12: 10 1 Self-consistent field theory a
Page 43: 42 1 Self-consistent field theory a
Page 80 and 81: Bibliography Almdal, K., Rosedale,
Page 82 and 83: BIBLIOGRAPHY 81 Matsen, M. W. and G
Page 84 and 85: Index action 17 binodal point 46 bl

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