Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2022 / Paper 2 / Section A(iii) / 18 / iv

With identical A-selective boundaries, stress-free parallel stacks favour approximately integer multiples of $L_0$. An arbitrary imposed thickness generally cannot preserve both the bulk period and those terminations everywhere.

A small mismatch can be accommodated by compressing or stretching the lamellae. A larger mismatch can favour terraces, islands or holes with locally commensurate heights, separated by steps of roughly $L_0$. For example, a mean height $\overline{h}=2.35L_0$ can be assembled from regions of heights $2L_0$ and $3L_0$, with area fraction $f=0.35$ at the larger height:The cartoon shows this volume-conserving alternative. Which response wins depends on elastic stretching energy, surface energies, lateral constraints, thickness mismatch and equilibration time. If the substrate and processing permit it, dewetting or a different domain orientation can also occur; an arbitrary thickness does not prove that a single specific terrace pattern must form.

Neutralizing the top boundary removes its preference for A, but the substrate still prefers A. The original integer-period constraint from two A-selective interfaces is therefore relaxed. The film can adjust its top termination without paying the same surface penalty, and thickness frustration can be reduced.

A possible morphology is an A-rich wetting layer near the substrate with lamellae turning towards a more perpendicular arrangement in the upper film, as schematically illustrated. A predominantly parallel stack is also possible if the substrate anchoring and thickness commensurability dominate. Neutrality of the top alone does not uniquely determine the orientation.

If both boundaries were neutral, perpendicular lamellae could expose comparable A and B areas without a selective-surface penalty and would avoid the same parallel-stack thickness constraint. That is a different boundary condition from the one given here. A correct sketch must retain the A preference at the bottom rather than silently replacing it with a second neutral interface.