Field-theoretic simulations: An emerging tool for probing soft material assembly
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particles to fields Standard texts on statistical mechanics discuss building equilibrium models of classical assemblies of molecules in various ensembles.1 These descriptions generally invoke pair-wise interactions among atoms or atomic sites on molecules, sometimes correcting the classical force field with three-body terms. A less-known fact is that techniques exist for exactly converting such a many-body problem involving interacting particle degrees of freedom into a statistical field-theory description, wherein fluctuating fields become the relevant degrees of freedom.2 Why would re-expressing the partition function of an ensemble of molecules in terms of functional integrals over fields, rather than ordinary integrals over atomic coordinates, be a useful thing to do? First, from the standpoint of numerical simulations, fields can be represented in many different ways (e.g., spectrally or in real space), and on uniform or nonuniform grids. This gives field-based simulations a significant computational advantage, as the degrees of freedom can be easily adapted to the nano-, meso-, and microstructure of the material investigated. For example, interfaces in a multiphase material can be tracked at high resolution, while
nearly structureless bulk domains are resolved on a coarse grid. In contrast, a molecular simulation based on a model with ∼106 atomic coordinates is forced to track all of those coordinates throughout the simulation trajectory or risk losing vital information. Second, field-theory descriptions are more natural for multiscale and coarse-graining strategies, which aim to extend computationally accessible length scales by eliminating degrees of freedom in a thermodynamically consistent way, than particle models. In coarse-graining particle models, one must decide among various complex procedures3,4 for eliminating atomic sites in favor of fewer lumped “molecular” coordinates. With fields, simple block averaging, spectral filtering, or interpolation operations affect the desired up- or downscaling. A third, but less obvious, advantage of field-theory descriptions is that statistical field theories admit mean-field solutions, similar to Hartree–Fock theory in quantum chemistry, which uses a single Slater determinant ansatz for the manybody wave function to develop a self-consistent mean-field theory. The availability of mean-field approaches can sometimes permit drastically less expensive simulation options. In the case of interacting polymers, for example, the mean-field
Glenn H. Fredrickson, Mitsubishi Chemical Center for Advanced Materials, University of California, Santa Barbara, USA; [email protected] Kris T. Delaney, Materials Research Laboratory, University of California, Santa Barbara, USA; [email protected] doi:10.1557/mrs.2018.97
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