The Mechanics of Nucleation and Growth and the Surface Tensions of Active Matter

TL;DR

This work develops a purely mechanical framework for homogeneous nucleation in active matter, enabling determination of the critical nucleus solely from dynamics without invoking equilibrium thermodynamics. It introduces three surface tensions—$gamma_mech$, $gamma_cw$, and $gamma_ost$—and a nonlocal term $K$ in the dynamic stress to define the generalized Young-Laplace relation and nonequilibrium coexistence criteria. By mapping Active Brownian Particle (ABP) behavior to this framework, it recovers Gibbs-Thomson-like relations and derives Langevin growth dynamics for nucleus radius, yielding an active nucleation barrier higher for droplets than for bubbles. It further combines fluctuating hydrodynamics with classical coarsening theories (LSW, Voorhees-Glicksman) to reveal $t^{1/3}$ coarsening and to extend to finite minority fractions, highlighting how nonequilibrium stresses shape nucleation and growth in active fluids.

Abstract

Homogeneous nucleation, a textbook transition path for phase transitions, is typically understood on thermodynamic grounds through the prism of classical nucleation theory. However, recent studies have suggested the applicability of classical nucleation theory to systems far from equilibrium. In this Article, we formulate a purely mechanical perspective of homogeneous nucleation and growth, elucidating the criteria for the properties of a critical nucleus without appealing to equilibrium notions. Applying this theory to active fluids undergoing motility-induced phase separation, we find that nucleation proceeds in a qualitatively similar fashion to equilibrium systems, with concepts such as the Gibbs-Thomson effect and nucleation barriers remaining valid. We further demonstrate that the recovery of such concepts allows us to extend classical theories of nucleation rates and coarsening dynamics to active systems upon using the mechanically-derived definitions of the nucleation barrier and surface tensions. Three distinct surface tensions -- the mechanical, capillary, and Ostwald tensions -- play a central role in our theory. While these three surface tensions are identical in equilibrium, our work highlights the distinctive role of each tension in the stability of active interfaces and the nucleation and growth of motility-induced phases.

Figures (5)

• Figure 1: Density profiles of critical bubbles and droplets at $\ell_o/d_{\rm hs} = 150$. All profiles have been shifted by their corresponding critical radius, given in Fig. \ref{['fig:figure2']}. Here, the volume fraction is defined as the number density multiplied by the effective volume of a particle $\rho \pi d_{\rm hs}^3 / 6$.
• Figure 2: Critical radius and relative change in nucleus density (${\nu^{\rm in} \equiv |\rho^{\rm in} - \rho^{\rm gas, \rm liq}| / \rho^{\rm gas, \rm liq}}$) as a function of supersaturation (${\nu^{\rm sat} \equiv |\rho^{\rm sat} - \rho^{\rm gas, \rm liq}| / \rho^{\rm gas, \rm liq}}$), as calculated from Eq. \ref{['eq:coexistrestate']} using the numerical procedure outlined in Section 1.3 of the SM. Each data point requires self-consistent solution of the corresponding full spatial profile, e.g. those shown Fig. \ref{['fig:figure1']}. Linear dependence of $\nu^{\rm in}$ on $\nu^{\rm sat}$ and inverse dependence of $R_C/d_{\rm hs}$ on $\nu^{\rm sat}$ implies a Gibbs-Thomson relation with $\nu^{\rm {in}}\propto 1/R$.
• Figure 3: Barrier $U(R_C)$ relative to active energy scale as a function of activity and supersaturation. The distinct white contour lines distinguish values of $U(R_C)/k_BT^{\rm act}$ separated by orders of magnitude.
• Figure 4: Surface tensions as defined by $\upgamma_{\rm mech}$, $k\to 0$ limit of $\upgamma_{\rm cw}$, and $R\to\infty$ limit of $\upgamma_{\rm ost}$ for both bubbles and droplets as a function of run length.
• Figure 5: Average radius growth as a function of time for nuclei initialized with an LSW distribution and average radius $750 d_{\rm hs}$. Radius growth curves across different activities, as well as between bubbles and droplets, collapse onto one another when scaled by prefactor ${K\equiv \left(\rho^{\rm sat}\upgamma_{\rm ost}d_{\rm hs}/\zeta U_o\Delta\rho^2\right)^{1/3}}$.