Equilibrium Fluid-Solid Coexistence of Hard Spheres

Abstract

We present a tethered Monte Carlo simulation of the crystallization of hard spheres. Our method boosts the traditional umbrella sampling to the point of making practical the study of constrained Gibbs' free energies depending on several crystalline order-parameters. We obtain high-accuracy estimates of the fluid-crystal coexistence pressure for up to 2916 particles (enough to accommodate fluid-solid interfaces). We are able to extrapolate to infinite volume the coexistence pressure (pco = 11.5727(10)kBT /σ 3 ) and the interfacial free energy (γ {100} = 0.636(11)kBT /σ 2 ).PACS numbers: 05.10. Ln, 64.60.My, Crystallization is a vast field of research, where experiments and theory cross-fertilize. Hard-spheres (HS) provide a celebrated example: the numerical finding of a fluid-solid phase transition [1] motivated experiments on colloids [2,3]. Finding an accurate procedure to locate the equilibrium phase boundaries for HS is a crucial step to address the self-assembly of complex molecules [4], as modeled by HS plus non-spherical interactions (e.g patchy [5] and Janus particles [6]).Up to now, numerical simulations of crystallization phase transitions have been well behind their fluid-fluid counterpart (e.g. vapor-liquid equilibria [7]). Actually, HS are the preferred benchmark for numerical approaches to crystallization. Yet, the lack of exact solutions enhances the importance of accurate numerical and/or experimental studies.However, for preexisting numerical methods, a simulation whose starting configuration is a fluid never reaches the equilibrium crystal. Much as in experiments [3], the simulation gets stuck in a metastable crystal, or a defective crystal (or even a glass [8]). The proliferation of metastable states defeats optimized Monte Carlo (MC) methods that overcome free-energy barriers in simpler systems [9][10][11]. Besides, experimental and numerical determinations of the interfacial free energy are plainly inconsistent (maybe due to a small electrical charge in the colloidal particles [12]).Since feasible numerical methods [13] could not form the correct crystalline phase spontaneously, choosing the starting particle configuration became an issue (e.g. crystalline or a carefully crafted mixture of solid and fluid phases). Methods can be classified as equilibrium or nonequilibrium. In the phase switch MC [14], one tries to achieve fluid-crystal equilibrium (only up to N = 500 HS [15] As for the accuracy, in equilibrium computations the coexistence pressure p co was obtained with precisions of ∼ 0.1% (at finite N ). Yet, the N values that can be simulated are rather small. An N → ∞ extrapolation is mandatory, which degrades the final accuracy to ∼ 1% [14,15,19] (results are summarized in Table I). The situation improves by an order of magnitude for the direct-coexistence method. With the exception of [15], the different estimations of p co are compatible, although with widely differing accuracies.The computation of the interfacial free energy, γ, is more involved, since the issue of spatially heterogen...

Bridging and depletion mechanisms in colloid-colloid effective interactions: A reentrant phase diagram

“…16,17,22 The metastable fluidfluid demixing coexistence, responsible for the broadening at η s > 0 of the stable fluid-solid coexistence (0.492 ≤ η l ≤ 0.543) for pure HSs (η s → 0), 23,24 remains always metastable and exists at small enough q in the AHS case, whereas it becomes stable at large q in the AO case, where a triple point appears. Figure 3 sketches (in the plane σ ss /σ s vs τ −1 sl ) the different models referred to above.…”

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