Nigel Goldenfeld's Group:
Pattern formation on multiple scales







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The Problem

Why is it so hard to predict the properties of real materials?  Unlike simple crystalline solids, real materials, produced by a wide range of processing conditions, contain defects and multiple grains that strongly impact mechanical, thermal, electrical response, and give rise to such important phenomena as plasticity, hysteresis, work hardening and glassy relaxation.  Moreover, it is frequently the case that a faithful description of materials processing requires simultaneous treatment of dynamics at scales ranging from the nanoscale up to the macroscopic.  Another part of the problem is that in a polycrystalline material, different grains collide and so one must keep track of the crystallographic orientation in order to properly model grain boundaries and dislocation arrays.

Despite these obstacles, progress in rational material design requires a complete and fundamental understanding of the way in which useful properties emerge as the mesoscale is approached.  Questions that must be addressed include: What is the collective behavior of assemblies of nanoscale objects?  How best can target mesoscale properties be achieved from nanoscale constituents? And how can the properties at nanoscale, mesoscale and intermediate scales simultaneously be captured in a quantitative and predictive fashion?

Renormalization group approach to multiscale modeling

Our approach is to start, not with a molecular dynamics model at the nanoscale, but with a density functional description (known in this context as the phase field crystal model), whose equilibrium solutions are periodic density modulations.  The phase field crystal model has all sorts of nice properties: for example, it automatically represents linear and non-linear elasticity!  And it is a representation of the appropriate diffusive dynamics on a time scale 6 orders of magnitude greater than can be accessed through conventional methods based on brute force molecular dynamics.

A system that is periodic at the nanoscale can be parameterized in terms of a uniform phase and an amplitude: the amplitude describes the maximum variations in the density of the system through the unit cell, while the phase describes uniform spatial translations.  A system that has an underlying periodicity, but which experiences defects or other nanostructure, is equivalent to a density wave whose amplitude is practically constant, or at least slowly varying on the nanoscale, and a phase that is essentially uniform everywhere, except near a defect. This observation suggests that the phase of the density is the appropriate dynamical variable to use for describing spatially-modulated nanoscale structure in a mesoscopic system, and in the vicinity of a defect it must be supplemented by the amplitude.

How can we obtain the correct dynamical equations for the amplitude and phase, assuming that we have a good description of the dynamics of the density?  We are using insights developed from the study of hydrodynamic instabilities in convection and other fluid pattern formation systems, in conjunction with our own renormalization group methods for extracting universal long-wavelength behaviour from partial differential equations.

This renormalization procedure is more general than real-space renormalization, which has been attempted in a non-systematic way in related contexts.  In particular, our technique directly focuses on the important dynamical instabilities that characterize the dynamics, which may not have simple real-space interpretations.  Once the phase is determined, the actual structure at the nanoscale on up to the mesoscale can be reconstructed by using the relation between the density and the phase.  Thus, because the phase equation describes solutions that are slowly-varying everywhere, except near a defect, adaptive mesh refinement can be used to solve the phase equation.

And we already pioneered the use of adaptive mesh refinement to simulate dendritic growth using phase field models ...

In summary, the nanoscale-to-mesoscale simulation problem can be solved by the combination of three techniques: (1) phase-field crystal formulation; (2) renormalization group theory; (3) adaptive mesh refinement.

Does it work?  Next page ...

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