嵌段共聚物简化模型中的位阻模式
Hexagonal patterns in a simplified model for block copolymers
摘要：
在本文中，我们研究了一个新的模式在两个维度上，灵感来自于二嵌段共聚物熔体的主导相。该模型是足够简单，不仅是服从数值，但也分析，但复杂到足以重现六角形包装结构，类似于在嵌段共聚物实验中观察到的圆柱体图案。从一个尖锐的界面连续模型，一个非局部的能量泛函涉及Wasserstein成本，我们推导出新的模型使用伽玛收敛的限制，其中一个阶段的体积分数趋于零。极限能量是根据原子的尺度来定义的;在三维空间中，原子代表少数相的小球形斑点，在二维空间中，它们代表少数相的薄圆柱体。然后我们研究极限能量的极小化。数值最小化是在两个维度上进行重铸的问题，作为一个计算几何问题，涉及功率图。数值计算结果表明，当粒子数趋于无穷大时，少数相的小粒子倾向于排列在三角形晶格上。这是证明在配套文件伯恩，Peletier和Theil和同意在嵌段共聚物实验中观察到的模式。这是一个罕见的例子，一个非局部的能量驱动的图案形成问题，在二维中，它可以证明，最佳模式是周期性的。
英文摘要：
In this paper we study a new model for patterns in two dimensions, inspired by diblock copolymer melts with a dominant phase. The model is simple enough to be amenable not only to numerics but also to analysis, yet sophisticated enough to reproduce hexagonally packed structures that resemble the cylinder patterns observed in block copolymer experiments. Starting from a sharp-interface continuum model, a nonlocal energy functional involving a Wasserstein cost, we derive the new model using Gamma-convergence in a limit where the volume fraction of one phase tends to zero. The limit energy is defined on atomic measures; in three dimensions the atoms represent small spherical blobs of the minority phase, in two dimensions they represent thin cylinders of the minority phase. We then study minimisers of the limit energy. Numerical minimisation is performed in two dimensions by recasting the problem as a computational geometry problem involving power diagrams. The numerical results suggest that the small particles of the minority phase tend to arrange themselves on a triangular lattice as the number of particles goes to infinity. This is proved in the companion paper Bourne, Peletier & Theil and agrees with patterns observed in block copolymer experiments. This is a rare example of a nonlocal energy-driven pattern formation problem in two dimensions where it can be proved that the optimal pattern is periodic.
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