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

自组装，一个过程，其中一个无序的系统的组件形成一个有组织的，结构化的模式，仅仅作为一个结果的组件相互作用，是无处不在的性质和重要的合成许多设计材料。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配置上的一些通常假设的自由能来成功地捕获和分析这些现象。由此产生的变分问题涉及高度非平凡的能源景观，在数学和计算上非常具有挑战性。在这个建议中，我们将研究三个这样的变分模型（以及它们之间的联系）从数学分析和计算的角度来看。一个共同的主题将是对非局部问题和全球技术的关注，即那些视野超越临界点的局部研究和稳定性分析的人。* 第一个变分模型是对著名的金兹伯格-朗道/卡恩-希利亚德自由能的库仑型非局部扰动。它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，有许多亚稳态，严格的分析结果的结构极小是稀缺的。我们将结合联合收割机的功能与数值方法的分析信息，以深入了解能源景观，并访问较低的能量状态。例如，我们将开发用于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个纯粹的几何和有限维范式的自组装，其中一个概括的概念质心Voronoi镶嵌点生成器一般形状。使用水平集公式，我们先验地固定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，用于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性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，用于一般形状的包装。* 第三个模型是基于定义在密度和措施的相互作用内核，这是在短距离排斥，但在长距离吸引的泛函。这些泛函直接连接到一类自组装/聚集模型，最近受到了很大的关注。我们将探讨全局极小和局部极小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.**