Boundary conditions for equilibrating incommensurate periodic patterns

Simulation of periodic patterns often suffer from artifacts due to incommensurability of the intrinsic length scale and the system size. We introduce a simple numerical scheme to avoid this problem in finding equilibrium domain morphologies from a Ginzburg-Landau-type free energy. In this scheme, the boundary values are determined only by the local equilibrium condition at the adjacent bulk sites. The scheme is especially advantageous in equilibrating patterns that have two or more characteristic lengths. We demonstrate it using a model of lamellar-lamellar coexistence in block copolymer blends.

Figure 1

A schematic domain morphology of coexisting lamellar phases. Layers with different thicknesses are joined by making a finite angle between them.

Figure 2

Time evolution of the coexisting lamellar patterns. Snapshots at (a) $t=0$, (b) $t=100$, (c) $t=500$, and (d) $t=1000$. The left-hand and right-hand columns show the result for the local equilibrium and periodic boundary condition, respectively.

Figure 3

Decay of the mean-square relaxation rate $H\left(t\right)$ for the local equilibrium and periodic boundary conditions. The solid line (local equilibrium) is always below the dotted line (periodic). Inset, relaxation in the early stage.

Figure 4

(a) Snapshot at $t=1000$ with the local equilibrium boundary condition [same as Fig. 2d in the left-hand column]. (b) Change of order parameter ${\varphi }_2(t=1000)-{\varphi }_2(t=0)$ along cross sections $A\text{−}B$ (third outermost sites) and $C\text{−}D$ (center line) in panel (a).

Figure 5

Domain morphology for initial conditions with a large angle between the $AB$ and $CD$ lamellae. Snapshots at $t=1000$ obtained with the local equilibrium boundary condition. The angle $\gamma$ is (a) 40° with $q_{1x}$ and $q_{2x}$ having the different sign, (b) 90°. (c) Magnified picture of the enclosed area of (b), which shows an “overshoot” in the wider stripe close to the phase boundary.