Mesoscopic Theories in Materials Science

在提出的研究中，我们讨论了描述材料和复杂流体中图案形成的介观方程的建模和分析。我们主要关注两个范例，地面过程和现场响应流体。介观模型是粗粒度偏微分方程或随机偏微分方程，直接准确地从微观相互作用的粒子系统中推导出来，包括详细的原子/分子信息。我们试图回答的主要问题是微观分子间的作用力如何在更大的长度尺度上影响图案的形成和进化。本文提出的分析方法包括非线性偏微分方程、变分法和随机过程。在第一个项目中，我们研究在多种可能相互竞争的机制（如表面扩散、反应和吸附/解吸）的影响下，表面过程中图案的形成和演化。在这里，我们采用伽马收敛技术来理解平衡状态下的模式，以及它们的动态对应物的粘度解和变量。在第二个项目中，我们专注于描述流体中颗粒悬浮物的分子动力学和相关介观模型，特别是对场响应流体的介观PDE的推导和分析。在这里，我们采用质量传递和相对熵方法结合Riesz变换估计来证明解的存在性以及平衡的松弛性。大量的实验证据表明，原子间和分子间的作用力决定了物质的宏观性质，并决定了图案和纹理的形成和选择。值得注意的例子出现在聚合物共混物、合金、催化、先进材料的外延生长和生物介质中。分子动力学和蒙特卡罗算法，在量子/统计力学框架中开发，提供了这些现象的详细，定量的动态和平衡描述；然而，它们仅限于短的空间/时间长度尺度，而实验观察到的形态涉及更大的尺度。计算和实验之间的这种差异强调了开发更大尺度的模型（PDE，随机PDE）的必要性，这些模型考虑了微观细节。我们在数值和分析上开发和研究的“介观”模型正朝着这个方向发展，系统地结合了(a)微观相互作用和(b)未解析的微观尺度波动。所开发的模型和分析方法可以更直接地理解宏观动态和平衡形态行为，并且还可以与通常涉及比微观建模和模拟更大长度尺度的实验进行比较。

In the proposed research we address the modeling and analysis of mesoscopic equations describing pattern formation in materials and complex fluids. We focus mainly on two paradigms, surface processes and field-responsive fluids. Mesoscopic models are coarse grained PDE or Stochastic PDE, derived directly and exactly from microscopic interacting particle systems and include detailed atomistic/molecular information. The principal question we attempt to answer is how microscopic intermolecular forces affect pattern formation and evolution at much largerlength scales. The analysis proposed here draws techniques from nonlinear PDE, calculus of variations and stochastic processes. In the first project we study pattern formation and evolution in surface processes under the influence of multiple and possibly competing mechanisms such as surface diffusion, reaction and adsorption/desorption. Here we employ Gamma-convergence techniques in order to understand patterning at equilibrium, and viscosity solutions and varifolds for their dynamic counterparts. In a second project we focus on molecular dynamics andrelated mesoscopic models describing particle suspensions in fluids and in particular on the derivation and analysis of mesoscopic PDE for field-responsive fluids. Here we employmass transport and relative entropy methods combined with Riesz Transform estimatesto show existence of solutions as well as relaxation to equilibrium.Ample experimental evidence indicates that interatomic and intermolecular forces dictate macroscopic properties of matter and determine formation and selection of patterns and textures.Notable examples arise in polymer blends, alloys, catalysis, epitaxial growth of advancedmaterials and biological media. Molecular dynamics and Monte Carlo algorithms, developed in a Quantum/Statistical Mechanics framework, provide detailed, quantitative dynamic and equilibriumdescriptions of these phenomena; however they are limited to short space/time length scales, while experimentally observed morphologies involve much larger scales. This disparity between computations and experiments underscores the need to develop models (PDE, Stochastic PDE)for larger scales, which take in consideration microscopic details. The "mesoscopic" models we develop and study numerically and analytically are geared towards this direction, incorporating systematically, (a) microscopic interactions, and (b) underresolved microscopic scales fluctuations. The developed models and analysis methods can allow for a more direct comprehension of macroscopic dynamic and equilibrium morphological behaviors and also provide comparisons to experiments which typically involve larger length scales than the ones arising in microscopic modeling and simulation.