Freezing line of polydisperse hard spheres via direct-coexistence simulations

TL;DR

This work tackles the challenge of predicting phase boundaries in polydisperse hard-sphere systems by introducing a direct-coexistence approach within a semi-grand canonical framework. By constraining the fluid phase to a Gaussian size distribution and using a semi-grand ensemble to couple pressure and particle-size chemical potentials, the authors map the fluid-FCC freezing line (cloud point) and identify the corresponding crystal shadow across polydispersities. They find that increasing polydispersity raises the coexisting pressure and shifts the crystal toward larger, less polydisperse particles, with the crystal density eventually dropping below that of the fluid at high $p$, consistent with size-fractionation effects. The study also characterizes the fluid–crystal interface, revealing rising surface stress and pronounced interfacial size-selective absorption, illustrating the method’s utility for exploring complex polydisperse phase behavior beyond monodisperse hard spheres.

Abstract

In experimental systems, colloidal particles are virtually always at least somewhat polydisperse, which can have profound effects on their ability to crystallize. Unfortunately, accurately predicting the effects of polydispersity on phase behavior using computer simulations remains a challenging task. As a result, our understanding of the equilibrium phase behavior of even the simplest colloidal model system, hard spheres, remains limited. Here, we present a new approach to map out the freezing line of polydisperse systems that draws on direct-coexistence simulations in the semi-grand canonical ensemble. We use this new method to map out the conditions where a hard-sphere fluid with a Gaussian size distribution becomes metastable with respect to partial crystallization into a face-centered-cubic crystal. Consistent with past predictions, we find that as the polydispersity of the fluid increases, the coexisting crystal becomes increasingly size-selective, exhibiting a lower polydispersity and larger mean particle size than the fluid phase. Interestingly, for sufficiently high polydispersity, this leads to a crystal phase with a lower number density than that of the coexisting fluid. Finally, we exploit our direct-coexistence simulations to examine the characteristics of the fluid-crystal interface, including the surface stress and interfacial absorption.

Figures (10)

• Figure 1: a) Surface forces $\mathbf{f}_i = - f_{\sigma_i}$ measured for a 16000 particle system with Gaussian particle size distribution and polydispersity $p = 0.06$, for a range of system densities $\rho \bar{\sigma}^3$ (top to bottom: 0.94, 0.95, 0.96, 0.97, and 0.98). Shading corresponds to one standard error and dashed black lines represent the fits $f\left(\sigma, \rho\bar{\sigma}^3\right)$ we employed in subsequent simulations. b) Particle size distribution for a system with polydispersity $p=0.06$. c) Configurational contribution (dashed line), mixing contribution (dotted line), and total chemical potential difference for the above system at density 0.96. d) Total chemical potential difference for the systems shown in the left panel.
• Figure 2: Crystal pressure as a function of crystal density obtained from semi-grand simulations of a bulk crystal of 16384 particles. The imposed chemical potential difference corresponds to that of a trial fluid system with polydispersity $p = 0.06$ at density $\rho_f^\textrm{trial} \bar{\sigma}^3 = 0.9618$. Pressure of the trial fluid is reported on the horizontal dotted line. Standard errors are shown for all points but are typically found to be smaller than the point size. The dashed line corresponds to the quadratic fit employed to determine the crystal density at mechanical equilibrium.
• Figure 3:
• Figure 4: a) Coexistence pressure as a function of the polydispersity of the fluid phase. b) Freezing line (blue) and corresponding shadow curve (red) in the polydispersity-packing fraction plane. c) Freezing line and shadow curve in the polydispersity-density plane. Data presented here corresponds to a fluid in coexistence with a FCC crystal, the fluid-crystal interface corresponds to the square lattice plane of the FCC crystal. Note that error bars are shown on all points but are smaller than the point size (typically on the order of $0.1\%$). All lines correspond to the functional forms reported in the main text. Data for the monodisperse case was obtained from Ref. smallenburg2024simple.
• Figure 5: Mean particle size (a) and polydispersity (b) of the crystal phase as a function of the fluid polydispersity $p$. The dashed lines indicate the corresponding mean size and polydispersity for the fluid phase for comparison.