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

自组装是一种过程，在这种过程中，无序的组件系统仅作为组件相互作用的结果形成有组织的、结构化的模式，在本质上是无处不在的，对许多设计师材料的合成也是重要的。许多表现出自组装和图案形成的系统可以被视为能量驱动的；因此，人们可以通过最小化定义在所有可能的构型上的一些通常假设的自由能来成功地捕捉和分析这些现象。由此产生的变分问题涉及到高度非平凡的能量景观，在数学和计算上都是非常具有挑战性的。在这个方案中，我们将从数学分析和计算的角度研究三个这样的变分模型(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，即那些视野超越对临界点的局部研究和对其稳定性的分析的技术。 第一个变分模型是库仑型对著名的Ginzburg-Landau/Cahn-Hilliard自由能的非局域微扰。这是两嵌段共聚物自组装的最简单模型。丰富而复杂的能量格局，特别是在三维空间中，有许多亚稳态，关于极小化结构的严格分析结果很少。我们将结合关于泛函的分析信息和数值方法来洞察能量格局，并访问较低能量的状态。例如，我们将开发评估特定计算的亚稳态是否接近全局极小值的方法。 第二个变分模型是一种纯几何和有限维的自组装范例，由此可以将点生成器的质心Voronoi镶嵌的概念推广到一般形状。利用水平集公式，我们先验地确定构件结构的几何形状，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，以产生一个简单的算法来填充一般形状。 第三个模型基于定义在密度和度量上的泛函，其中包含一个相互作用核，该核在短距离是排斥的，但在长距离是吸引的。这些泛函与最近备受关注的一类自组装/聚集模型直接相关。我们将探讨全局和局部极小点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.