Abstract
Defects are a ubiquitous feature of ordered media. They have certain universal features, independent of the underlying physical system, reflecting their topological origins. While the topological properties of defects are robust, they appear as ‘unphysical’ singularities, with non-integrable energy densities in coarse-grained macroscopic models. We develop a principled approach for enriching coarse-grained theories with enough of the ‘micro-physics’ to obtain thermodynamically consistent, well-set models that allow for the investigations of dynamics and interactions of defects in extended systems. We also develop associated numerical methods that are applicable to computing energy driven behaviors of defects across the amorphous-soft-crystalline materials spectrum. Our methods can handle order parameters that have a head-tail symmetry, i.e. director fields, in systems with a continuous translation symmetry, as in nematic liquid crystals, and in systems where the translation symmetry is broken, as in smectics and convection patterns. We illustrate our methods with explicit computations.
Highlights
Macroscopic physical systems consist of large numbers of interacting parts and are described by thermodynamic principles
We develop a framework for this procedure by enriching the macroscopic theory to include additional physical fields, that reflect some of the microscopic physics and describe the interaction/dynamics of defects [Kle89]
Patterns and defects are ubiquitous in extended systems. They arise from the interplay between two “universal” mechanisms, the tendency of systems towards order as their energy/temperature is lowered, and the tendency towards disorder from entropic considerations and the likelihood of “getting stuck” in “local” metastable states, that precludes perfect ordering. These defects play a big role in the properties of extended systems and understanding the birth, disappearance and dynamics of defects is critical in explaining a range of phenomena from plasticity, solid-solid phase transitions, fracture, convective transport, and complex fluids
Summary
Macroscopic physical systems consist of large numbers of interacting (microscopic) parts and are described by thermodynamic principles. These additional fields take the form of smooth representations of ‘singular’ parts of higher gradients of the macroscopic fields, since they naturally encode defect behavior Using these two principles, and building on earlier work [ZAWB15, ZZA+16], we develop a general thermodynamic framework to obtain numerically tractable models for defects in extended systems. We enrich the underlying macroscopic theory by adding new physics, involving smooth ‘rehabilitated’ representations of these singular fields and their thermodynamically conjugate variables such that the relevant energy density and stresses are locally integrable everywhere
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