Rigorous convergence of molecular models to continuum mechanics beyond perfect periodic zero-temperature crystals: defects, free-surface boundary layers, and temperature

In a 2001 Comptes Rendus note, Blanc, Le Bris and Lions rigorously derived continuum hyperelastic energies as asymptotic limits of molecular models (pair-potential and Thomas-Fermi) as the interatomic distance tends to zero, under four simplifying assumptions: a perfect periodic lattice without defects, periodicity up to free boundaries (no surface relaxation), one common deformation field at microscopic and macroscopic scales, and zero temperature. The note's conclusion poses the extension beyond these assumptions as the open challenge. The scale-dependent-deformation restriction has since been methodologically absorbed by the discrete-to-continuum Gamma-convergence literature, and substantial rigorous results now exist for dilute point defects, two-dimensional edge dislocations, Peierls-Nabarro-type dislocation models, dilute three-dimensional line-tension limits for dislocations, one-dimensional boundary layers, interface energies, and low-temperature regimes. The general problem remains open: rigorous molecular-to-continuum derivations for crystals with dislocation networks and grain boundaries, free-surface geometry relaxation in two and three dimensions, and positive temperature beyond the low-temperature, dilute windows covered by existing theorems.

Background

Continuum mechanics describes an elastic solid through a stored-energy functional $\mathcal{E}(u)=\int_\Omega E(\nabla u(x))\,dx$, where $u$ is the macroscopic deformation of a reference domain $\Omega$ and $E$ is the elastic energy density. In hyperelasticity this density is normally postulated on symmetry grounds: frame indifference, $E(RM)=E(M)$ for rotations $R$, and invariance under the crystal's lattice symmetry group. Whether such continuum energies can be derived rigorously from atomic-scale physics, rather than postulated, is a foundational question of mathematical materials science. The Cauchy-Born rule is the classical heuristic: under a deformation $u$, each atom at a lattice site $i$ is assumed to move to $u(i)$, so the continuum density is read off from the energy of the homogeneously deformed cell obtained by applying the gradient $\nabla u$ to the lattice. For the microscopic description one may use pair potentials between atoms, or simplified quantum models such as Thomas-Fermi theory, in which the electrons are replaced by a self-consistent charge density interacting Coulombically with fixed nuclei - crude compared to the full Schroedinger equation, but tractable for rigorous analysis.

Blanc, Le Bris and Lions (C. R. Acad. Sci. Paris, 2001; extended in Arch. Ration. Mech. Anal., 2002) made this derivation rigorous in the following framework: identical atoms occupy the sites of a perfect periodic lattice with spacing $\varepsilon$, interact through a potential of range $\delta$ (or through a Thomas-Fermi-type energy), and the atomistic energy is evaluated on the configuration generated by a fixed smooth deformation $u$. As $\varepsilon\to0$ (and $\delta\to0$) under various scalings of $\varepsilon$ against $\delta$, they identify the limit of the energy per unit volume: explicit continuum densities $E(\nabla u)$ that retain the memory of the lattice geometry, an $O(\varepsilon)$ surface correction, and an $O(\varepsilon^2)$ volume correction of strain-gradient type, i.e. a correction depending on the second derivatives of the deformation. The scheme is of Gamma-convergence type, Gamma-convergence being the standard notion under which minimizers of a sequence of variational problems converge to minimizers of a limit functional.

The derivation rests on four simplifying assumptions, each of which is violated by the very solids the resulting continuum models are used for. First, the crystal is perfectly periodic: real crystals contain vacancies (missing atoms), dislocations (line defects where the lattice registry is broken, and the microscopic carriers of plasticity) and grain boundaries (interfaces between differently oriented crystallites), which govern plasticity and fracture. Second, the lattice is treated as periodic up to the boundary, whereas near a free surface the atoms relax into a boundary layer whose geometry differs from that of the bulk, so that the surface contribution to the energy is not captured by simply truncating the periodic expansion. Third, the same smooth deformation $u$ is imposed at the atomic scale, although the physically relevant atomistic equilibria are $\varepsilon$-dependent configurations $u_\varepsilon$ that should merely converge, in a suitable sense, to the macroscopic $u$; relaxing this turns the computation into a genuine relaxation problem, since the lattice energies are non-convex and produce fine-scale microstructure, with relaxed limits governed by quasiconvexity - the weakened convexity notion, replacing convexity for vector-valued deformations, under which variational problems of elasticity are well-posed. Fourth, the temperature is zero, while real crystals exhibit thermal expansion, phase transformations and temperature-dependent plasticity, which require a statistical-mechanical, free-energy description. The 2001 note's conclusion states these four extensions as the open challenge, referring to geometry-optimization approaches for surfaces and to the $\varepsilon$-dependent-deformation route in the spirit of the thin-film passage of Friesecke and James.

Problem Statement

Extend the rigorous asymptotic derivation of continuum mechanics from molecular models established by Blanc, Le Bris and Lions for perfect periodic zero-temperature crystals with a single imposed deformation, by removing its simplifying restrictions while preserving the character of the result: an explicitly identified limiting continuum energy or model, together with a proven convergence statement as the interatomic distance $\varepsilon$ tends to zero. The setting, model classes and assumptions are those of the original derivation (three-dimensional crystalline solids, periodic lattice of spacing $\varepsilon$, two-body/pair-potential or Thomas-Fermi-type energies as treated in the cited papers), with the continuum limit taken as $\varepsilon$ tends to zero. The scale-dependent-deformation restriction is methodologically covered by the discrete-to-continuum Gamma-convergence literature. Concretely, for molecular (pair-potential or Thomas-Fermi-type quantum) models of crystalline solids, establish with full rigor, in at least the three-dimensional setting of the original derivation:

1. Defects: the derivation for crystals containing vacancies, dislocations and grain boundaries, delivering an explicitly identified limit and a proven convergence statement.
2. Free surfaces: the derivation when the crystal has a free boundary at which atomic positions are allowed to relax (geometry optimisation), capturing the boundary layer, its geometry, and its contribution to the energy expansion beyond the order-$\varepsilon$ surface term already derived under the periodicity-up-to-the-boundary assumption.
3. Temperature: the derivation at positive temperature, where the object of study becomes a free energy defined by statistical mechanics of the thermal fluctuations, in regimes going beyond the low-temperature, dilute-defect windows of existing results.

The three items share one objective - a continuum mechanics of real (defective, bounded, finite-temperature) crystalline solids rigorously derived from molecular models - in the sense of the original derivation: an explicitly identified limit together with a proven convergence theorem.

References

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