Frenkel's entropy-exchange mechanism in monodisperse, nearly hard-sphere colloids: minimal perturbations to access fluid-crystal coexistence

TL;DR

The paper investigates kinetic access to fluid–solid coexistence in monodisperse, purely repulsive hard spheres by studying nearly hard-sphere colloids under Brownian dynamics. It implements minimal, quantified perturbations: small global softness (via a Morse potential with $B_2^*$ near 1) and 2–4% distributed crystal seeds, to reveal Frenkel's long-range/short-range entropy-exchange mechanism within the theoretical coexistence region. Results show explicit phase separation along the tie line on finite timescales, with phase envelopes approaching hard-sphere predictions as hardness increases; spontaneous coexistence can occur even without seeds at sufficient hardness. The findings connect Frenkel’s entropic mechanism to realistic colloidal kinetics and demonstrate how slight perturbations convert a kinetically inaccessible hard-sphere transition into observable phase separation, with practical implications for simulating and interpreting entropy-driven crystallization in colloidal suspensions.

Abstract

Entropically driven fluid-solid transitions in monodisperse, purely repulsive hard spheres (MPRHS) are well established in theory, simulation, and experiment for atomic and colloidal systems. For MPRHS, however, coexistence is usually located via bulk free-energy calculations; the underlying microscopic balance between configurational and vibrational entropy is left implicit. Frenkel clarified this mechanism explicitly as an exchange of long-range configurational entropy for short-range vibrational entropy, but in the pristine MPRHS limit the nucleation barrier near coexistence is so high that phase separation is predicted only on astronomical time scales. Consistent with this, even unbiased simulations do not show spontaneous, equilibrium fluid-crystal coexistence; transient mixtures are mostly overtaken by a single phase; observed coexistence is still algorithmically-driven. Nearly hard-sphere colloid experiments do observe fluid-crystal coexistence, but always in the presence of unavoidable triggers such as gravity, walls, and polydispersity. We treat the hard-sphere phase diagram as settled and ask how the entropic exchange mechanism can be revealed in nearly hard-sphere colloidal simulations. We probe the mechanism on finite time scales by introducing minimal perturbations that trigger phase separation: small reductions in hardness that increase locally accessible free volume (and thus gently increase vibrational entropy), and 2-4% distributed crystal seeds. These perturbations produce coexisting fluid and crystal domains with crystal fraction, phase envelope and osmotic pressure that, with systematically increasing particle hardness, approach the hard-sphere limit. These results demonstrate that slight enhancements to vibrational entropy provide a dynamically accessible route to realizing the long-range/short-range entropy exchange required for phase separation.

Figures (13)

• Figure 1: Comparison of potentials used to represent hard-sphere colloids in simulations, plotted as a function of particle center-to-center distance, where values smaller than unity indicate 'overlap'. The purely repulsive Morse potential with $\kappa a=30$ (solid lines) is shown for varying hardness values as indicated in the legend. A commonly-used Weeks-Chandler-Anderson potential is also shown (black dashed line). Truly hard-sphere interaction is a Heaviside function at unity.
• Figure 2: Snapshots of our Brownian dynamics simulations of the phase behavior of solvent-suspended colloids. 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. Figure from Wang et al.wangInreviewElusive, with permission.
• Figure 3: Simulation images from present study showing particle arrangements for a range of volume fraction $\phi$ and crystal fraction $\zeta$. Particles are colored according to $6^{th}$ order average local-order parameter $\bar{q}_6$. Particles surrounded by amorphous structure ($\bar{q}_6<0.29$) are colored pink and made translucent for visibility. Red particles are surrounded by marginally crystalline structure ($\bar{q}_6\approx0.3$); green particles are surrounded by substantially crystalline structure ($\bar{q}_6\approx0.4$); and blue particles ($\bar{q}_6\geq0.5$) are surrounded by very crystalline structure. Particle hardness set as $V_0=6kT$ and $\kappa a=30$. All images from samples initially close to the theoretical metastable fluid line (all using the slow melting protocol, except for $\phi=0.505$ and $\phi=0.51$ that used the quasi-equilibrium melting protocol).
• Figure 4: Extent of crystal and fluid-like structure at 12 volume fractions as shown. Total crystal fraction $\zeta$ shown in each plot. The probability P($\bar{q}_6$) is plotted as a function of the $6^{th}$ order average local-order parameter $\bar{q}_6$, calculated for each of the 2,000,000 particles. Measurement taken at $2,000a^2/D$ after achieving target volume fraction. Dotted vertical line marks the boundary between fluid-like structure ($\bar{q}_6 < 0.29$) and crystalline structure ($\bar{q}_6 \ge 0.29$). Particle hardness $V_0=6kT$, $\kappa a=30$ ($B_2^*=0.985$), cf Figure \ref{['fig:fig_potential']}.
• Figure 5: Crystal fraction as a function of volume fraction. Fast freezing ($\triangle$), slow freezing ($\blacktriangle$), fast melt ($\square$), slow melt ($\blacksquare$), and quasi-equilibrium melting (${\medblackdiamond}$) are shown (see Methods for rates), with the initial crystal seeding fractions also shown in the legend. A linear fit predicts freezing at $\phi=0.503$ and melting at $\phi=0.547$. Particle hardness parameters $V_0=6kT$ and $\kappa a=30$ ($B_2^*=0.985$).