Mathematical Modeling of Point Defects in Materials Science

Abstract

We survey some recent mathematical works we have contributed to that are related to the modeling of defects in materials science at different scales. We emphasize the similarities (need of a reference, often periodic system; renormalization procedure; etc) shared by models arising in different contexts. Our illustrative examples are taken from electronic structure models, atomistic models, homogenization problems. The exposition is pedagogic and deliberately kept elementary. Both theoretical and numerical aspects are addressed. * Invited article for a special issue of M 3 AS.

Analysis of Boundary Conditions for Crystal Defect Atomistic Simulations

“…This motivates the formulation of a Galerkin-type approximation scheme for (1) (see Sections 2.3 and 3.4 This is a finite dimensional optimisation problem with dim(Ẇ 1,2 N ) ≈ N , and our framework yields a straightforward proof of: supposeū is a strongly stable (cf. (9)) solution to (1) then, for N sufficiently large, there exists a solutionū N to (3) such that…”

“…We see no obstacle to include Lennard-Jones type interactions, but this would require finer estimates and a more complex notation. However, we explicitly exclude Coulomb interactions or any electronic structure model and hence also charged defects (see, for example, [9][10][11]17,27]). …”

“…The mathematical analysis of crystalline defects has traditionally focused on dislocations [1,2,18,20] and on electronic structure models [10,11]; however, see [9] for a comprehensive recent review focused on point defects. The results in the present work, in particular the decay estimates on elastic fields, also have a bearing on this literature since they can be used to establish finer information about equilibrium configurations; see for example, [21].…”

Geometry equilibration of crystalline defects in quantum and atomistic descriptions

“…In the mathematics literature, considerable progress has been made on studying electronic ground states corresponding to local defects in crystals e.g. [3,4,5].…”

Homogenization theory and multiscale numerical approaches for disordered media: some recent contributions

“…The problem (16) in w 1 p is random in nature, but it is in fact easy to see, taking the expectation, that w 1 p = E(w 1 p ) is Q-periodic and solves the deterministic problem…”