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

More documents

Recommendations

Info

1.8 Block Copolymer Melts 65 end-view of the cylinders y=D/ 3 y y x x=D/2 x x=R hexagonal unit cell unit cell approximation Figure 1.19: End-view showing the hexagonal arrangement of the minority domains of the cylinder (C) phase. Below to the left is the proper hexagonal unit cell, and to the right is a circular unit-cell approximation (UCA), where the radius R is set by the equal area condition, πR 2 = √ 3D 2 /2. direction. The orthonormal basis functions for this symmetry start off as f 0 (r) = 1 (1.293) f 1 (r) = √ 2/3[cos(2Y )+2cos(X)cos(Y )] (1.294) f 2 (r) = √ 2/3[cos(2X)+2cos(X)cos(3Y )] (1.295) f 3 (r) = √ 2/3[cos(4Y )+2cos(2X)cos(2Y )] (1.296) f 4 (r) = √ 4/3[cos(3X)cos(Y )+cos(2X)cos(4Y )+cos(X)cos(5Y )] (1.297) where X ≡ 2πx/D and Y ≡ 2πy/ √ 3D. (The general formula is provided by Henry and Lonsdale (1969) under the space-group symmetry of p6mm.) The corresponding eigenvalues of the Laplacian are λ 0 =0, λ 1 =16π 2 /3, λ 2 =16π 2 , λ 3 =64π 2 /3, λ 4 = 112π 2 /3, and so on. Although there are simply too many elements of Γ ijk to begin enumerating them, it is a trivial matter to generate them by computer. Once they have been evaluated and stored away
66 1 Self-consistent field theory and its applications in a file, the free energy of the C phase is calculated using the exact same expressions as for the L phase. The same goes for the S and G phases, where their basis functions are catalogued by Henry and Lonsdale (1969) under the space-group symmetries, Im ¯3m and Ia¯3d, respectively. For the S cp and PL phases, there are near degeneracies (Matsen and Bates 1996a) in the arrangement of spheres and the stacking of layers, respectively. As suggested by Matsen and Bates (1996a), this is likely to result in irregular non-periodic arrangements, which is indeed consistent with recent experimental observations (Sakamoto et al. 1997; Zhu et al. 2001). Nevertheless, for the purpose of calculating phase boundaries, it is sufficient to use the symmetries Fm¯3m (i.e., fcc packing) and R¯3m (i.e., ABCABC... stacking), respectively. The PL phase is also different in that it has two distinct periodicities, which implies that the eigenvalue Eq. (1.165) for the basis functions must be generalized to ) ( λ‖,i ∇ 2 f i (r) =− D‖ 2 + λ ⊥,i D⊥ 2 f i (r) (1.298) where D ‖ is the in-plane spacing between perforations and D ⊥ is the out-of-plane lamellar spacing. However, this distinction only enters in an obvious extension to Eqs. (1.284) and (1.285), and in the need to minimize F with respect to both D ‖ and D ⊥ . 40 30 S C L C S χN 20 S cp G S cp 10 disordered 0 0.0 0.2 0.4 0.6 0.8 1.0 f Figure 1.20: Theoretical phase diagram for diblock copolymer melts showing where the various morphologies in Fig. 1.18 are stable. All phase transitions are discontinuous except for the mean-field critical point denoted by the solid dot. Along part of the order-disorder transition (ODT) is a narrow region, S cp, where spherical domains prefer a close-packed arrangement over the usual bcc one. In the strong-segregation limit, the G region pinches off at χN ≈ 60, the ODT approaches the two edges of the diagram, and the C/S and L/C boundaries approach the fixed compositions estimated by the arrows at the top of the diagram. Comparing the free energies of the disordered phase and all the ordered phases of Fig. 1.19 generates the phase diagram in Fig. 1.20, which maps the phase of lowest free energy as
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 7 and 8:
Page 9 and 10:
Page 11 and 12:
10 1 Self-consistent field theory a
Page 13 and 14:
12 1 Self-consistent field theory a
Page 15 and 16: 14 1 Self-consistent field theory a
Page 65: 64 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
show all

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

Create successful ePaper yourself

Delete template?

Save as template?