Variational Models for Self-Assembly and Pattern Formation: Analysis and Computation

自组装是一种无序的组件系统形成有组织、结构化模式的过程，仅作为组件相互作用的结果，它在自然界中无处不在，对许多设计材料的合成也很重要。许多表现出自组装和模式形成的系统可以看作是能量驱动的；因此，人们可以通过最小化某些（通常是假设的）定义在所有可能构型上的自由能来成功地捕获和分析这种现象。由此产生的变分问题涉及高度非平凡的能量景观，并且在数学和计算上非常具有挑战性。在这个建议中，我们将从数学分析和计算的角度研究三种这样的变分模型（以及它们之间的联系）。一个共同的主题将是对非局部问题和全球技术的关注，即那些超越了对临界点的局部研究和对其稳定性的分析的人。***第一个变分模型是对著名的Ginzburg-Landau/Cahn-Hilliard自由能的库仑比克型非局部摄动。这是二嵌段共聚物自组装的最简单模型。丰富而复杂的能量景观，特别是在三维空间中，具有许多亚稳态，而对最小化器结构的严格分析结果很少。我们将把关于函数的分析信息与数值方法结合起来，以深入了解能量景观，并获得较低能量的状态。例如，我们将开发评估特定计算亚稳态是否接近全局最小值的方法。第二种变分模型是一种纯几何和有限维的自组装范式，人们将点生成器的质心Voronoi镶嵌的概念推广到一般形状。使用水平集公式，我们先验地固定了组件结构的几何形状，并考虑完全由距离函数决定的自组装。我们最近提出了一种新的快速算法来模拟这种刚性形状的自组装（广义质心Voronoi镶嵌），现在将使用这种方法来经验地探索非凸能量景观。我们还将扩展我们的距离函数公式，以生成一个用于一般形状打包的简单算法。***第三种模型是基于密度和度量上定义的泛函，由一个相互作用核组成，它在近距离是排斥的，但在远距离是吸引的。这些函数直接连接到一类最近受到广泛关注的自组装/聚合模型。我们将探讨全局极小值和局部极小值的存在性和性质问题

Self-assembly, a process whereby a disordered system of components forms an organized, structured pattern solely as a consequence of component interactions, is both ubiquitous in nature and important for the synthesis of many designer materials. Many systems which exhibit self-assembly and pattern formation may be viewed as energy-driven; therefore, one can successfully capture and analyze the phenomena via minimization of some, often postulated, free energy defined over all possible configurations. The resulting variational problems involve highly non-trivial energy landscapes and are mathematically and computationally very challenging. In this proposal we will study three such variational models (and connections between them) from the point of view of mathematical analysis and computation. A common theme will be the focus on nonlocal problems and global techniques, i.e. those whose vision extends beyond the local study of critical points and the analysis of their stability. ***The first variational model is a nonlocal perturbation of Coulombic-type to the well-known Ginzburg-Landau/Cahn-Hilliard free energy. It is the simplest model for self-assembly of diblock copolymers. The rich and complex energy landscape, particularly in three space dimensions, has many metastable states, and rigorous analytical results on the structure of minimizers are scarce. We will combine analytical information about the functional with numerical methods to gain insight into the energy landscape, and access states of lower energy. For example, we will develop methods for assessing whether or not a particular computed metastable state is close to a global minimizer. ***The second variational model is a purely geometric and finite-dimensional paradigm for self-assembly whereby one generalizes the notion of centroidal Voronoi tessellations for point generators to general shapes. Using a level set formulation, we a priori fix the geometry for the component structures, and consider self-assembly entirely dictated by distance functions. We have recently produced a novel fast algorithm for the simulation of such self-assemblies of rigid shapes (generalized centroidal Voronoi tessellations) and will now use this method to empirically explore the non-convex energy landscape. We will also extend our distance function formulation to produce a simple algorithm for the packing of general shapes. ***The third model is based upon functionals defined over densities and measures consisting of an interaction kernel which is repulsive at short range but attractive at long ranges. These functionals are directly connected to a class of self-assembly/aggregation models which recently have received much attention. We will explore questions of existence and properties of global and local minimizers.**