Small Volume Fraction Limit of the Diblock Copolymer Problem: I. Sharp-Interface Functional
Small Volume Fraction Limit of the Diblock Copolymer Problem: I. Sharp-Interface Functional
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二嵌段共聚物问题的小体积分数极限：I. 锐界面泛函
DOI：
10.1137/090764888
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发表时间：
2009
期刊：
影响因子：
--
通讯作者：
M. Peletier
中科院分区：
文献类型：
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作者：
Rustum Choksi;M. Peletier
我们提出的第二个两篇文章的小体积分数限制的非本地Cahn-Hilliard功能引入模型微相分离的二嵌段共聚物。在建立了尖界面版本的泛函（arXiv：0907.2224）的结果之后，我们在这里考虑全扩散界面泛函，并解决了其中的ε和体积分数趋于零但少数相（称为粒子）的数量保持为O（1）的极限。使用语言的伽玛收敛，我们专注于两个层次的这种收敛，并获得一阶和二阶有效能量，其能量景观更简单，更透明。这些极限能量仅在δ函数的加权和上是有限的，对应于质量集中到"点粒子"中。在最高水平上，有效能量完全是局部的，包含关于每个粒子的大小的信息，但没有关于它们的空间分布的信息。在下一个层次上，我们遇到了粒子之间的库仑式相互作用，这是图案形成的原因。我们提出的结果在三维和评论他们的二维类似物。
We present the second of two articles on the small volume fraction limit of a nonlocal Cahn-Hilliard functional introduced to model microphase separation of diblock copolymers. After having established the results for the sharp-interface version of the functional (arXiv:0907.2224), we consider here the full diffuse-interface functional and address the limit in which epsilon and the volume fraction tend to zero but the number of minority phases (called particles) remains O(1). Using the language of Gamma-convergence, we focus on two levels of this convergence, and derive first- and second-order effective energies, whose energy landscapes are simpler and more transparent. These limiting energies are only finite on weighted sums of delta functions, corresponding to the concentration of mass into `point particles'. At the highest level, the effective energy is entirely local and contains information about the size of each particle but no information about their spatial distribution. At the next level we encounter a Coulomb-like interaction between the particles, which is responsible for the pattern formation. We present the results in three dimensions and comment on their two-dimensional analogues.