The elusive fluid-and-crystal coexistence state in simulations of monodisperse, hard-sphere colloids

Abstract

Monodisperse, purely repulsive, hard spheres (MPRHS) are an important model system for mechanistically exploring phase behavior in atomic systems and colloids. Since the 1940s, phase transitions in these systems have been obtained via simulation, theory, and experiments. But there is a gap in this literature: despite decades of reports of phase transition from one pure state to another, no computational studies report spontaneous phase separation into coexisting domains of liquid and crystal regions. This gap owes its origin to the underlying mechanism of entropically-driven phase separation in MPRHS - the competition between short-range entropy and long-range entropy. Frenkel proposed that spontaneous phase separation in simulations of up to 1,000,000 particles would require more than 317,000,000 years to sample enough microstates to converge to phase separated macrostate. Some brute-force simulations do show brief spontaneous coexistence but a metastable crystal or fluid subsequently overtakes the system. To bypass these difficulties, many studies use seeding, gravity, or direct construction of liquid and solid phases to study interfacial energy and nucleation rates of MPRHS systems. WCA potentials have also been used to bypass metastability, where softness provides free volume, lowers osmotic pressure and the energy barrier. It is argued that the transition path taken in bypassing metastability is mechanistically the same as without triggers. But as acknowledged by Alder & Wainwright, explicit observation of spontaneous coexistence is central to computational prediction of first-order transition. Such observation would provide satisfying demonstration of Frenkel's entropy exchange mechanism. We explore literature revealing these interesting behaviors and conclude that computational demonstration of Frenkel's mechanism for MPRHS awaits large systems with careful hardness perturbations.

Figures (5)

• Figure 1: Foundational atomic, colloidal, and rheological studies. Phase transitions in systems of purely-repulsive hard spheres were established beginning with theory in the 1940s kirkwood1940theorykirkwood1941statistical, which matched experiments for argon eisenstein1940. Alder and Wainwright's seminal event-driven simulations produced a fluid line and a crystal line alder1957phasealder1960studies as well as corresponding images (bottom insets) alder1959studies and drove improvement of Wood and Jacobson’s Monte Carlo simulations wood1957preliminary. Hoover and Ree combined lattice-constraint modeling with theory to obtain the phase lines and also used thermodynamic theory to deduce a tie line to put forth the hallmark melting and freezing points for monodisperse hard spheres hr-68. Subsequent virial-expansion theory provided refinements cs-69hall1972anotherwertheim-63mcquarrie-76. Pusey and van Megen's landmark experiments with colloids showed explicit formation of coexisting fluid and crystal domains as shown in top inset pvM-86. Russel and co-workers subsequently deduced colloidal pressure via x-ray measurements of packing, to produce phase lines and infer a coexistence line crossing gravitational layers prczcdo-96.
• Figure 2: Confocal microscopy images from hw-09 showing the interface between colloidal liquid and crystal regions in a suspension of PMMA particles in cyclohexylbromide and decalin. Particles colors indicated structure, from dark blue for crystals to red for liquid. With permission from Proc. Nat. Acad. Sci.
• Figure 3: Methods for triggering crystal nucleation in simulations. (a) The "direct coexistence" method constructs an equilibrated face-centered cubic phase (right half of simulation cell, 2,548 particles) and a separate equilibrated fluid phase (left half of cell, 2,548 particles), pressing them into contact as the initial configuration. Figure from espinosa2013fluid, with permission from AIP/ Journal of Chemical Physics. (b) Biased-sampling Monte Carlo method: snapshot of transient critical nucleus at $\phi=0.5207$ (yellow), surrounded by fluid-phase particles (blue). The nucleus is pre-constructed using biased Monte Carlo simulations to speed the nucleation process. Figure from auer2001prediction, with permission from Nature. (c) Seeding method: snapshot of molecular dynamics simulation illustrating a $5 \times 5$ square seed promoting surrounding crystal growth. From hermes2011nucleation, with permission from Royal Society of Chemistry, Soft Matter.
• Figure 4: Pressure $P$ as a function of number density $\rho$ spanning fluid, coexistence and solid regions from pieprzyk2019thermodynamic. Pressure scaled with $kT/\sigma^3$, where $kT$ is the thermal energy and $\sigma$ is particle diameter. Number density is related to volume fraction as $\phi = \rho \pi \sigma^3/6$. Coexistence tie line obtained by equal-pressure and equal-chemical-potential conditions, giving $\phi_F=0.492$ and $\phi_M=0.543$. With permission from the Royal Society of Chemistry, Physical Chemistry Chemical Physics.
• Figure 5: Recent large-scale Brownian dynamics simulations from wangInprepDemonstration. Far left: simulation cell of 2,000,000 colloids, replicated periodically into an infinite domain in LAMMPS thompson2022lammps. Second and third images: same system at 2x and 5x magnification. Colors correspond to local order, ranging from red for structureless to deep blue for perfect crystal structure. With permission, J. Chem. Phys.