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

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1.8 Block Copolymer Melts 63 φ A (z) 1.2 0.8 χN = 50 20 11 0.4 (a) 0.0 -1.6 -0.8 0.0 0.8 1.6 z/aN 1/2 3 (b) D/aN 1/2 2 1 0.7 w I /aN 1/2 0.4 0.2 (b) 1-phase 0.1 (c) 0.07 10 20 50 100 χN Figure 1.17: (a) Segment profiles, φ A(z), from the lamellar phase of symmetric diblock copolymers, f =0.5 plotted for three levels of segregation, χN. (b) Domain spacing, D, versus segregation plotted logarithmically; the dashed curve denotes the SST prediction from Eq. (1.302). (c) Interfacial width, w I, versus segregation plotted logarithmically; the dashed and dotted lines correspond to the SST predictions in Eqs. (1.239) and (1.299), respectively. and the interfacial width, w I ≡ φ A(D/2) − φ A (−D/2) φ ′ A (0) (1.292)
64 1 Self-consistent field theory and its applications narrows as the segregation increases. Plots 1.17(b) and 1.17(c) show these trends on logarithmic scales so as to highlight the well-known scaling behavior that emerges for stronglysegregated microstructures. The dashed line in Fig. 1.17(b) denotes the domain scaling of D ∝ aχ 1/6 N 2/3 , where the proportionality constant will be derived in the next Section using SST. At intermediate segregations, the rate of increase is somewhat more rapid, consistent with the fact that the experimentally measured exponent for N tends to be ∼ 0.8 (Almdal et al. 1990). The dashed and dotted lines in Fig. 1.17(c) show SST predictions for the interfacial width, which will be discussed in the next Section. L C S G PL Figure 1.18: Minority domains from the periodically ordered phases observed in diblock copolymer melts. The first three, lamellar (L), cylindrical (C), and spherical (S), are referred to as the classical phases, and the latter two, gyroid (G) and perforated-lamellar (PL), are denoted as the complex phases. Although the PL phase is often observed, our best experimental evidence indicates that it is only ever metastable (Hajduk et al. 1997). The lamellar phase is only observed for f ≈ 0.5. When the diblock copolymer becomes sufficiently asymmetric, the domains transform into other periodic geometries as depicted in Fig. 1.18. In the cylindrical (C) phase, the shorter minority blocks form cylindrical domains aligned in a hexagonal packing with the longer majority blocks filling the intervening space. In the spherical (S) phase, the minority blocks form spheres that generally pack in a bodycentered cubic (bcc) arrangement. Note, however, that there are occasions where the spherical (S cp ) phase prefers close-packed (e.g., fcc or hcp) arrangements. In addition to these classical phases, two complex phases, gyroid (G) and perforated-lamellar (PL), have also been observed. The minority domains of the G phase (Hajduk et al. 1994; Schulz et al. 1994) form two interweaving networks composed of three-fold coordinated junctions. The PL phase (Hamley et al. 1993) is much like the classical L phase, but the minority-component lamellae develop perforations through which the majority-component layers become connected. The perforations tend to be aligned hexagonally within the layers and staggered between neighboring layers. Fortunately, the only part of the spectral method that changes when considering the nonlamellar morphologies are the basis functions, and these changes are entirely contained within the values of λ i and Γ ijk . Take the C phase for example. Its hexagonal unit cell is shown in Fig. 1.19 with a coordinated system defined such that the cylinders are aligned in the z
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Self-Consistent Field Theory and It
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2 Contents Index 83
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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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Self-Consistent Field Theory and Its Applications by M. W. Matsen

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