Phase separation patterns for diblock copolymers on spherical surfaces: A finite volume method

We explore phase separation on spherical surfaces by solving the Cahn-Hilliard equation modified for diblock copolymers using a finite volume method. The spherical surface is discretized into almost uniform triangles by employing successive dyadic refinements of the spherical icosahedron, a methodology that avoids potential mathematical and numerical problems related to the poles in spherical coordinates. The finite volume method is based on averaging Voronoi cells built from triangular meshes to calculate the Laplace-Beltrami operator on the curved surface, which greatly improves both the accuracy and speed of calculation as compared to the conventional finite difference method. By using this method we simulate the phase separation of diblock copolymers on a spherical surface. It is found that stable and intrinsic defects, which would not occur in a flat space after sufficient annealing, appear in the periodic arrangement of the domains on the curved surface due to the distinct Euler characteristic of the surface.

Figure 1

The schematic of dyadic icosahedral triangulation of a spherical surface.

Figure 2

A vertex with six neighbors and one-ring Voronoi region for evaluating the Laplace-Beltrami operator on a spherical surface.

Figure 3

The evolution of morphologies with time $t$ for $AB$ diblock copolymers with the copolymer composition $\varphi =0.5$ and $\alpha =0.2$.

Figure 4

The time evolution of morphologies for $AB$ diblock copolymers with the copolymer composition $\varphi =0.35$ and $\alpha =0.1$.

Figure 5

Log-log plots of $N_B$ representing the domain size as a function of time $t$ for polymer blends. The guide line is $N_B\sim t^{-1∕3}$.

Figure 6

Lattice points used to calculate $X_{\prime \alpha }=\partial X\left(u\right)∕\partial u^{\alpha }$ and $X_{\prime \alpha \beta }={\partial }^{2}X\left(u\right)∕\partial u^{\alpha }\partial u^{\beta }$ for the vertex with (a) six and (b) five neighbors.