Tuning Nucleation Kinetics via Nonequilibrium Chemical Reactions

Unlike fluids at thermal equilibrium, biomolecular mixtures in living systems can sustain nonequilibrium steady states, in which active processes modify the conformational states of the constituent molecules. Despite qualitative similarities between liquid-liquid phase separation in these systems, the extent to which the phase-separation kinetics differ remains unclear. Here we show that inhomogeneous chemical reactions can alter the nucleation kinetics of liquid-liquid phase separation in a manner that is consistent with classical nucleation theory, but can only be rationalized by introducing a nonequilibrium interfacial tension. We identify conditions under which nucleation can be accelerated without changing the energetics or supersaturation, thus breaking the correlation between fast nucleation and strong driving forces that is typical of phase separation and self-assembly at thermal equilibrium.

Figure 1

Simulating driven chemical reactions at a phase-separated NESS. (a) Schematic of an open system with inhomogeneous chemical reactions. The effective internal free-energy differences between the $B$ and $I$ states in the liquid and vapor phases are $\mathrm{\Delta }{f}_l$ and $\mathrm{\Delta }{f}_v$, respectively. (b) An example steady-state distribution in an inhomogeneous model. (c) Phase diagram for equilibrium (green), nonequilibrium homogeneous (blue), and inhomogeneous (orange) models, and (d) quantification of the inhomogeneous reactions assuming $\beta \epsilon =-2.95$, $k^{\circ }={10}^{-1}$, and ${\rho }_v=0.05$ at coexistence. The shaded (unshaded) region indicates where liquid (vapor) is stable for all models in (c) and for the inhomogeneous model only in (d).

Figure 2

Nucleation kinetics at a NESS obey classical nucleation theory (CNT) with modified interfacial properties. (a) A schematic illustration of diffusion on an (equilibrium) free-energy landscape, $F(n)$. (b) Tests of CNT and the nucleation theorem (inset) for nonequilibrium homogeneous (blue) and inhomogeneous (orange) models under far-from-equilibrium conditions (at $\beta \mathrm{\Delta }{\mu }_{\mathrm{coex}}=1.87$ using the same parameters as Figs. 1 and 1. Solid and dashed curves show the equilibrium prediction and a fit of the inhomogeneous results to the CNT rate equation, respectively.

Figure 3

Inhomogeneous reactions at a NESS alter the interfacial tension, which strongly affects the nucleation kinetics. (a) Deviation of the nonequilibrium line tension, $\mathrm{\Delta }\sigma \equiv \sigma -{\sigma }_{\mathrm{eq}}$, as determined from nucleation rate calculations, with respect to $\beta \mathrm{\Delta }{\mu }_{\mathrm{coex}}$ and $k^{\circ }$ (inset). The simulation parameters are the same as in Figs. 1 and 1. (b) Comparison of nonequilibrium nucleation rate densities, $J$, to corresponding equilibrium rate densities, $J_{\mathrm{eq}}$, at constant supersaturation, $S=1.27$ (see text). Orange and blue colors indicate nonequilibrium inhomogeneous and homogeneous models, respectively. Symbols report FFS results, and lines show theoretical predictions.

Figure 4

Kinetic scheme of particle exchange and internal chemical reactions. In our simulations of an open system, each lattice site stochastically transitions between being unoccupied ($E$) or being occupied by either a bonding ($B$) or inert ($I$) particle with the specified transition rates.

Figure 5

FLEX schematic of a single-layer configuration at a liquid-vapor interface. The effective bonding energy at the interface is obtained from the steady-state distribution at the tagged site under a fixed local configuration. Colors correspond to the same lattice-site states as in Fig. 4.