Creating macroscale effective interfaces encapsulating microstructural physics

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This proposal seeks funding for a comprehensive three year researchprogram into methodologies for modeling microscale interfacial phenomena on the macroscale level. A fundamental question stretching across many disciplines is: Given a microstructured interface can it be replaced by an effective ``averaged'' boundary condition entirely posed upon a macroscale. If so, can it accurately reproduce the physical effects created by the microstructure? Can this effective boundary condition be derived rigorously, rather than in some ad-hoc fashion, and what are the limitations in so doing? The proposal aims to answer these questions, with the goal of being able to accurately and efficiently predict complex physical behaviour in three apparently unconnected fields: in wave propagation for surface Rayleigh-Bloch waves and for the reflection of waves from designer structured surfaces, in the statistical mechanics of phase transitions on micropatterned surfaces, and in modeling diffusions through structured domains. These fields all share a complex structured interface and the generic overarching Mathematical approach we propose will lead to effective boundary conditions encapsulating the dominant microscale Physics; this will represent a considerable advance in each of these areas. The primary Mathematical approach will be based around Homogenization theory utilizing the discrepancy in lengthscales to create asymptotics from multiple scales analysis. Homogenization is conventionally used when the bulk material has short-scale fluctuations and the solution varies on a long-scale, its use for interfaces is much less well explored. Importantly we also aim to enhance the range of validity of homogenization theory away from long-wave, quasi-static, regimes to ones that can vary on the same scale as the microstructure. This analytical work will be complemented by detailed numerical simulations that will act to verify the efficacy of the developed interfacial models. The work will be undertaken by a team from the Mathematics Department at Imperial College London with complementary skills and strengths: Pavliotis (Homogenization theory, stochastic processes), Parry (Statistical mechanics, phase transitions) and Craster (Wave propagation, homogenization, fluid mechanics).