Generating dense packings of hard spheres by soft interaction design

Abstract

Packing spheres efficiently in large dimension d is a particularly difficult optimization problem. In this paper we add an isotropic interaction potential to the pure hard-core repulsion, and show that one can tune it in order to maximize a lower bound on the packing density. Our results suggest that exponentially many (in the number of particles) distinct disordered sphere packings can be efficiently constructed by this method, up to a packing fraction close to 7 d 2 −d . The latter is determined by solving the inverse problem of maximizing the dynamical glass transition over the space of the interaction potentials. Our method crucially exploits a recent exact formulation of the thermodynamics and the dynamics of simple liquids in infinite dimension.

Data From: Postponing the dynamical transition density using competing interactions

“…The maximum accessible liquid density, ϕ * d , is however, not attained with λ 0 = 0.03, but rather with one of λ * 0 ≈ 0.019. Although our result for λ * 0 ≈ 0.019 is in the vicinity of the infinite-dimensional theoretical prediction for this optimization (λ * 0 ≈ 0.029) [13], it is nonetheless significantly different from it. Because the intricate liquid structure of finite-dimensional systems is neglected in the analytical study, this discrepancy is not particularly surprising.…”

“…This nontrivial feature could be a finite dimensional echo of the d → ∞ prediction. The theoretical prediction that further tuning the interaction potential could engender additional (smaller) gains in ϕ * d [13] is nevertheless unlikely to be verifiable in three-dimensional systems.…”

“…in the disordered regime [10][11][12]. A recent theoretical proposal suggests that certain SALR models exhibit unusual dynamical features in the very dense fluid regime [13] as well. Maimbourg et al [13]'s extension of a high-dimensional treatment of the glass transition [14][15][16] suggests that certain models should display a very pronounced dynamical reentrance upon changing temperature.…”

“…A recent theoretical proposal suggests that certain SALR models exhibit unusual dynamical features in the very dense fluid regime [13] as well. Maimbourg et al [13]'s extension of a high-dimensional treatment of the glass transition [14][15][16] suggests that certain models should display a very pronounced dynamical reentrance upon changing temperature. More precisely, the theoretical analysis suggests that a carefully chosen highdensity SALR system that is glassy at low temperature should, upon heating, first melt and then get dynamically arrested once again, all while remaining completely disordered, i.e., without crystallizing.…”

“…As a result liquids with a higher packing fraction than is possible from either mechanism can then be stabilized [35]. Adding a supplementary, longer-ranged repulsion is understood as effectively deepening the well created by the short-range attraction, and thus leads to a slightly more efficient packing of neighboring spheres in the liquid state [13]. In the mean-field description, the nonergodicity transition to a glass phase is then pushed to even higher densities although only over a very narrow temperature window [13].…”