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

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1.7 Polymer Blends 47 -0.92 -0.96 (a) χN = 3 F h /nk B T -1.00 -1.04 F s φ (1) φ (2) -1.08 5 4 2-phase χN 3 2 (b) 1-phase 1 0.0 0.2 0.4 0.6 0.8 1.0 φ Figure 1.10: (a) Free energy, F h , of a homogeneous blend as a function of composition, φ, calculated for χN =3. The dashed line denotes the free energy, F s, for a macrophase-separated blend with compositions, φ (1) =0.3 and φ (2) =0.6, and the dotted line is the double-tangent construction locating the binodal points denoted by triangles. The two diamonds denote spinodal points, where the curvature of the free energy curve switches sign. (b) Phase diagram showing the binodal (solid) and spinodal (dotted) curves as a function of χN. The triangular and diamond symbols correspond to those in (a), and the solid dot denotes a critical point. 1.7.3 Scattering Function for a Homogeneous Blend A beam of radiation (e.g., x-rays or neutrons) will pass undeflected through a perfectly homogeneous blend, ˆφ A (r) =φ, assuming the wavelength, λ, is above the atomic resolution and that absorption is negligible. However, there are always thermal functuations that disturb the uniform composition, which in turn scatter the radiation by various angles, θ. To predict the
48 1 Self-consistent field theory and its applications resulting pattern of radiation, I(θ), we must first calculate the free energy cost, F [Φ A ], of producing a specified composition fluctuation, ˆφ A (r) =Φ A (r). The fact that we will continue to implement mean-field 〈 theory 〉 implies that, in reality, the free energy we calculate will be for a fixed φ A (r) ≡ ˆφA (r) as opposed to a fixed ˆφ A (r); this limitation will be discussed further in Section 1.10. 3 φ = 0.5 S(k)/N 2 1 1.0 1.5 χN=1.8 0 0 1 2 3 kaN 1/2 Figure 1.11: Scattering function, S(k), for binary blends of symmetric composition, φ =0.5, plotted for several degrees of segregation, χN. To perform the calculation, we use the Fourier representation developed in Section 1.5.4. In terms of the Fourier transforms of the composition, φ A (k) and φ B (k), the free energy is approximated by F nk B T = F h nk B T + ∫ 1 2φ(2π) 3 dk φ A(−k)φ A (k) V k≠0 g(x, 1) ∫ 1 2(1 − φ)(2π) 3 dk φ B(−k)φ B (k) + V k≠0 g(x, 1) ∫ χN (2π) 3 dkφ A (−k)φ B (k) (1.216) V k≠0 The first term is the free energy of a perfectly homogeneous state, Eq. (1.208), the following two integrals represent the loss in configurational entropy, Eq. (1.105), experienced by the A- and B-type polymers, and the final integral accounts for the increased number of A-B contacts, Eq. (1.192) transformed by Eq. (1.82). The incompressibility condition is enforced by setting, φ B (k) =−φ A (k), which simplifies the free energy to F nk B T = F h nk B T + ∫ N 2(2π) 3 dkS −1 (k)φ A (−k)φ A (k) (1.217) V k≠0 +
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 47: 46 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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