Mesoscopic Theories in Materials Science

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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.