Mesoscopic Theories in Materials Science

材料科学中的介观理论

• 批准号: 0100872
• 负责人: Markos Katsoulakis
• 金额: $ 8.4万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100872&HistoricalAwards=false
• 关键词: Mesoscopic; Theories; Materials; Science

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.

在所提出的研究中，我们解决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。我们主要集中在两个范例，表面过程和场响应流体。介观模型是粗粒度PDE或随机PDE，直接和精确地从微观相互作用粒子系统中导出，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力如何在更大尺度上影响图案的形成和演化。这里提出的分析技术，从非线性偏微分方程，变分法和随机过程。在第一个项目中，我们研究图案的形成和演变的表面过程的影响下，多个和可能的竞争机制，如表面扩散，反应和吸附/脱附。在这里，我们采用伽玛收敛技术，以了解图案在平衡，粘度的解决方案和varifolds为他们的动态同行。在第二个项目中，我们专注于分子动力学和相关的介观模型，描述流体中的颗粒悬浮液，特别是场响应流体介观PDE的推导和分析.在这里，我们采用质量输运和相对熵的方法结合Riesz变换估计显示存在的解决方案，以及松弛到平衡。大量的实验证据表明，原子间和分子间的力量决定了物质的宏观性质，并确定形成和选择的图案和纹理。值得注意的例子出现在聚合物共混物，合金，催化，先进材料和生物介质的外延生长。在量子/统计力学框架中开发的分子动力学和蒙特卡罗算法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模型（PDE，随机PDE）的必要性，这些模型考虑了微观细节。我们开发和研究的“介观”模型的数值和分析是面向这个方向，系统地纳入，（a）微观相互作用，（B）欠解析微观尺度波动。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学行为，并提供与通常涉及比微观建模和模拟中产生的更大长度尺度的实验的比较。