Finite-Size Effects on Liquid-Solid Phase Coexistence and the Estimation of Crystal Nucleation Barriers

Antonia Statt, Peter Virnau, and Kurt Binder

Phys. Rev. Lett. 114, 026101 – Published 13 January 2015

Abstract

A fluid in equilibrium in a finite volume $V$ with particle number $N$ at a density $ρ = N / V$ exceeding the onset density $ρ_{f}$ of freezing may exhibit phase coexistence between a crystalline nucleus and surrounding fluid. Using a method suitable for the estimation of the chemical potential of dense fluids, we obtain the excess free energy due to the surface of the crystalline nucleus. There is neither a need to precisely locate the interface nor to compute the (anisotropic) interfacial tension. As a test case, a soft version of the Asakura-Oosawa model for colloid-polymer mixtures is treated. While our analysis is appropriate for crystal nuclei of arbitrary shape, we find the nucleation barrier to be compatible with a spherical shape and consistent with classical nucleation theory.

Received 15 October 2014

DOI:https://doi.org/10.1103/PhysRevLett.114.026101

Authors & Affiliations

Antonia Statt^1,2,*, Peter Virnau¹, and Kurt Binder¹

¹Institut für Physik, Johannes Gutenberg-Universität Mainz, Staudinger Weg 9, 55128 Mainz, Germany
²Graduate School of Excellence Materials Science in Mainz, Staudinger Weg 9, 55128 Mainz, Germany

^*Corresponding author. statt@uni-mainz.de

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Vol. 114, Iss. 2 — 16 January 2015

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Images

Figure 1
Schematic plot of the chemical potential $μ$ vs density $ρ$ for a system undergoing a liquid-solid transition in a finite box volume $V_{box}$ with periodic boundary conditions. (A plot of pressure $p$ vs $ρ$ would qualitatively look just the same). Because of interfacial effects, non-negligible in finite systems, the isotherm deviates from $p = p_{coex}$ in the two-phase coexistence region, $ρ_{f} < ρ < ρ_{m}$ . The features in the curve (kinks in reality are rounded due to fluctuations) are due to transitions between the different states shown in the figure via snapshots of the simulated generalized Asakura-Oosawa model. Only the part where the solid phase is the minority phase is discussed. For further explanations, see the text.
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Figure 2
Normalized pressure $\tilde{p} = p σ_{c}^{3} / k_{B} T$ plotted vs packing fraction $η \equiv ρ π σ_{c}^{3} / 6$ of the colloids, for the Eff AO model (asterisks) and its soft version (soft Eff AO, squares). Curves are a guide to the eye only. These data were obtained from simulations of homogeneous liquid and solid (fcc) phases, while the pressure where two-phase coexistence occurs was found from the “interface velocity method” [37], namely, $\tilde{p} = 8.44 \pm 0.04$ (soft Eff AO) and $\tilde{p} = 8.06 \pm 0.06$ (Eff AO). The coexistence packing fractions are $η_{f} = 0.495 (1)$ and $η_{m} = 0.636 (1)$ for the soft Eff AO case. The inset compares the potentials of the Eff AO (which is singular at $r = σ_{c} = 1$ ) and soft Eff AO models.
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Figure 3
(a) Illustration of the method to compute the chemical potential of a very dense fluid, using an $L \times L \times L_{z}$ slab geometry, with a soft wall at $z = 0$ and a hard wall at $z = L_{z} = 30$ (lengths being measured in units of $σ_{c}$ , $L = 7$ , and four choices of $N$ are used, $N = 750$ , 950, 1100 and 1250, respectively). Inset shows $μ$ in units of $k_{B} T$ as a function of $z$ , for the four choices shown, over the regions of $z$ where particle insertion works. (b) Chemical potential $μ$ (in units of $k_{B} T$ ) plotted vs $η$ , for different choices of $L$ and $L_{z}$ , as indicated, to show that finite-size effects are negligible. The data labeled by “Widom” at not so large $η$ values are obtained by the standard particle insertion method for homogeneous bulk systems. Arrows on abscissa and ordinate indicate $η_{f}$ and $μ_{c} (p_{coex}) / k_{B} T$ , respectively.
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Figure 4
(a) Pressure difference $Δ p = p_{l} - p_{coex}$ between the pressure $p_{l}$ in a fluid surrounding a crystal nucleus of finite size and the coexistence pressure, plotted vs the average packing fraction $η$ in the simulation box, for particle number $N = 6000$ , 8000, and 10 000 (symbols, from bottom to top). Curves show the formula $Δ p = (2 γ / R^{*}) c / (η_{m} / η_{f} - 1)$ with $c = 1.07$ , extracting $R^{*}$ from the assumption of a spherical nucleus $(V^{*} = 4 π R^{* 3} / 3$ ) and taking $γ \approx {\tilde{γ}}_{111} \approx 1.013$ [45]. (b) $Δ F^{*}$ computed from $p_{l}, p_{c}$ and $V^{*}$ [using Eqs. (2), (5), and (6)] plotted vs $V^{* 2 / 3}$ ; straight line is Eq. (3) with $A = A_{iso}$ and $\tilde{γ} = γ_{111}$ . The dotted line is a fit illustrating $Δ F^{*} \propto {V^{*}}^{2 / 3}$ .
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