structures at different surface separations. At H = 9σ, 10.6σ and 13σ, phases are
observed and at H = 15σ, the L4 phase is observed. At H = 9σ and 15σ the lamella
are compressed and at H = 13σ the lamella are stretched than their bulk equilibrium
period, Db- Figure 5.10 shows the excess surface free energy of the symmetric lamellar
phases as a function of the effective film thickness of the copolymer, Hejγ. The free
energy is minimum at the integer values of Heff∙ This minimum free energy is lower
than the earlier case where the two surfaces have comparatively weaker preference for
the A block of the copolymer, ew∕kT = 0.1.
Figure 5.11 shows the anti-symmetric segment density profiles of the two blocks of
the copolymer at different film thickness. At both H = 14σ and 16.4σ, the bɜ phases
are observed. However, the lamella are compressed for H = 14σ and stretched for H
= 16.4σ. The excess surface free energy of this bɜ phase is calculated and compared
with the free energies for the symmetric phases in fig. 5.12. Even the minimum free
energy of this phase (at Heyγ = 1~) is higher than the free energy for the L4 phase.
Hence only the symmetric phases are stable. This is due to the high energy penalty
of having the B block at one of the surfaces in case of anti-symmetric phases. To
quantify this, the free energy penalty of having the B block rather than the A block
near the surface is calculated (at Hey∕ = 1) as ∕w = ʃj1 pA{z)(V^ct(z) — v^[z))dz,
where h is width of the half-lamellae of A next to the surface. For ew∕kT = 0.3,
= 0.6392. This should be equal to the difference in the minimum excess surface
free energy of the anti-symmetric and symmetric phases. From the calculations,
136