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Soft matter is assuming an ever growing role in fundamental science and in technological applications. Soft materials can rarely be understood on one length scale and time scale alone. A minute change of molecular interaction may lead to a massive change of the system's observed macroscopic properties. Over the last decade, multiscale modeling has therefore attracted rising interest in soft matter science and engineering (Peter 2010). Multiscale modeling has also a long-standing tradition in applied mathematics (E 2003, Kevrekidis 2003, E 2007), and this community is also beginning to take interest in the modeling of complex materials (Ren 2005, Barrett 2012, Krumscheid 2013). Emergency of the development of systematic modeling and simulation approaches for multiscale problems has been reflected in founding a new journal of the Society for Applied and Industrial Mathematics “Multiscale Modelling and Simulation” in 2003. This highly internationally recognized mathematical journal focuses on fundamental modeling and computational principles appearing in various multiscale problems arising in science and engineering. Nevertheless, the development of multiscale methods has in general proceeded independently in the soft matter community and the applied mathematics community, and with few exceptions, the exchange between communities has been rather limited so far. The proposed workshop will bring together scientists from these communities to discuss the open problems in the field from different perspectives.
Multiscale simulation techniques have been developed for many classes of soft-matter systems and they have made a large contribution to the understanding of structure and properties of these systems. A systematic mapping between models of different resolution has been achieved (Reith 2003, Izvekov 2004, Shell 2008, Lyubartsev 2009, Karimi-Varzaneh 2012), and in many cases the fundamental connections between structural variations on one scale and the materials properties on another have been established. Contributions have been made to the multiscale behavior of systems as diverse as polymers, polyelectrolytes, lipids, amphiphiles, liquid crystals, and many others. Methods in the soft matter community range from quantum chemistry and ab-initio molecular dynamics (Senn 2009) via coarse-grained molecular dynamics to a battery of mesoscale methods (Müller 2005, Dünweg 2009, Gompper 2009, Pagonabarraga 2010, Smiatek 2012) that can be used to study properties of complex fluids on mesoscopic length and time scales where microscopic details can be omitted to some extent, but thermal fluctuations are still considered to be important. In contrast, the applied mathematics community is traditionally interested in “macroscopic” continuum models. Bridging the enormous range of dynamically coupled scales is a fundamental challenge and has shown to be a driving force in the development of new mathematical tools and techniques, from homogenization to averaging, renormalization and a large variety of coupling multiscale algorithm. For the latter, let us mention, for example, the Fourier and general multiple scale analysis, various computational methods such as multigrid, adaptive multiresolution techniques, wavelets or space/time adaptive finite element/finite volume methods and the heterogeneous multiscale methods (E 2003, E 2007, Kevrekidis 2003). Furthermore, mathematics provides tools to analyze stability and convergence of numerical schemes and the numerical pitfalls of inverse problems, as typically arise in the context of coarse-graining (Wang 2012).
Among the many challenges in the field of multiscale modeling methods, we will focus on the following issues in the workshop:
Transferability and representability: Coarse-grained parameters (e.g., potentials) are typically derived at one particular set of thermodynamic parameters, and they cannot necessarily be transferred to nearby state points. This severely restricts the applicability of such coarse-grained models for studying heterogeneous situations or nonequilibrium systems. Strategies must be developed to overcome this problem (Villa 2010, Mukherjee 2012).
Multiresolution coupling schemes: In many applications, it is desirable to use different levels of coarse-graining in one simulation model, e.g., to study a small functional part of a system in high detail and coarse-grain the less interesting rest of the system. This motivates the development of hybrid models, e.g., mixed quantum mechanical/classical schemes (QM/MM; Senn 2009), mixed fine grained/coarse grained particle models (AdResS) (Praprotnik 2008, Potestio 2013), mixed particle/continuum models (Delgado-Buscalioni, 2008). In particular, adaptive simulation schemes where fine-grained regions can dynamically adjust to the current state of the system, are still at the beginning.
Particle-continuum gap. There are numerous mapping schemes between models of different resolution within the class of particle-based models, and within the class of continuum models. In the particle language, this coarse-graining corresponds to collecting several detailed particles into one simplified particle. In the continuum world, such reductions of complexity are known as homogenization, whereby detailed-resolved domains with locally varying properties are collected into a larger representative volume element. In contrast, there exist only few successful attempts to combine particle models with continuum models in a nontrival fashion (Daoulas 2006, Ren 2007, Milano 2009, Donev 2010, E 2007, Kevrekidis 2003). The particle-continuum gap for complex molecular systems is the missing link in the chain of hierarchical models to describe soft materials on all their length and time scales.
Dynamics. Coarse-graining dynamical properties in a systematic manner is arguably the biggest and most pressing current challenge in the field. Several aspects need to be considered. First, coarse-grained energy landscapes are projections and thus smoother than the original landscapes. This speeds up the dynamics, reduces viscosities, and moves time scales closer to each other (Louis 2010, Depa 2011). Second, the fact that degrees of freedom are integrated out upon coarse-graining introduces sources of friction, noise and possibly memory, in the dynamics of the coarse-grained system. So far, systematic coarse-graining methods for dynamical properties are still in their infancy (Langeloth 2013). This problem is particularly severe in the context of multiresolution methods.