Tang Ping, Qiu Feng, Zhang Hongdong, Yang Yuliang
The Key Laboratory of Molecular Engineering of Polymers, Ministry of Education, China.
Phys Rev E Stat Nonlin Soft Matter Phys. 2005 Jul;72(1 Pt 2):016710. doi: 10.1103/PhysRevE.72.016710. Epub 2005 Jul 18.
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.
我们通过使用有限体积法求解针对二嵌段共聚物修改后的Cahn-Hilliard方程，来探索球面上的相分离。通过对球形二十面体进行连续的二元细化，将球面离散为几乎均匀的三角形，这种方法避免了与球坐标中的极点相关的潜在数学和数值问题。有限体积法基于对由三角形网格构建的Voronoi单元进行平均，以计算曲面上的拉普拉斯-贝尔特拉米算子，与传统的有限差分法相比，这大大提高了计算的准确性和速度。通过使用这种方法，我们模拟了二嵌段共聚物在球面上的相分离。结果发现，由于曲面独特的欧拉特征，在充分退火后在平坦空间中不会出现的稳定且内在的缺陷，出现在曲面上畴的周期性排列中。
