Kinetic theory of emulsions with matter supply

TL;DR

The work extends Lifshitz–Slyozov–Wagner theory to emulsions fed by ongoing matter supply, distinguishing diffusion-limited and interface-resistance-limited growth under constant-supersaturation and constant-matter-supply driving. By deriving a coarse-grained concentration equation and a continuum droplet-size distribution, it uncovers universal coarsening behavior in the interface-resistance-limited regime, with a closed-form distribution and a supply-independent growth law, while demonstrating nonuniversal and initial-condition–dependent coarsening in diffusion-limited cases with constant supply. The constant-supersaturation scenario yields decoupled, time-dependent growth and, depending on regime, either narrowing (diffusion) or drifting, fixed-shape distributions (interface-resistance). The constant-matter-supply case reveals semi- or fully universal coarsening in an interface-resistance-limited regime, and non-universal, supply-dependent behavior in the diffusion-limited regime, with implications for chemically fueled emulsions and biomolecular condensates. Overall, the theory provides a unified framework linking LSW ripening, transport bottlenecks, and sustained matter supply to predict droplet-size distributions and growth laws across contexts.

Abstract

In this work, we propose a theory for the kinetics of emulsions in which a continuous supply of matter feeds droplet growth. We consider cases where growth is either limited by bulk diffusion or the transport through the droplets' interfaces. Our theory extends the Lifshitz-Slyozov-Wagner (LSW) theory by two types of matter supply, where either the supersaturation is maintained or the supply rate is constant. In emulsions with maintained supersaturation, we find a decoupling of droplets at all times, with the droplet size distribution narrowing in the diffusion-limited regime and a drifting distribution of a fixed shape in the interface-resistance-limited case. In emulsions with a constant matter supply, there is a transition between narrowing and broadening in the diffusion-limited regime, and the distribution is non-universal. For the interface-resistance-limited regime, there is no transition to narrowing, and we find a universal law governing coarsening kinetics that is valid for any constant matter supply. The average radius evolves according to a power law that is independent of the matter supply, and we find a closed-form expression for the droplet size distribution function. Our theory is relevant to biological systems, such as biomolecular condensates in living cells, since droplet material is not conserved and the growth of small droplets is proposed to be interface-resistance-limited.

Figures (10)

• Figure 1: Emulsion supplied by droplet matter. (a) Illustration of an emulsion coupled to a reservoir (red) supplying the system with matter, leading to growing droplets. (b) The matter supply $J$ is switched on at time $t^*$. Supply cases: (i) To keep the supersaturation constant, the matter supply $J\propto t^2$ increases since droplets grow in time. (ii) Constant matter supply (orange).
• Figure 2: Single droplet in a system coupled to a material reservoir. (a) Illustration of a droplet of radius $R$ in a system of size $R_\text{sys} \gg R$. Far away from the droplet interface, at the system's boundary, the matter is supplied from the reservoir. Due to fast diffusion, we assume no spatial gradients of the droplet material concentration at the system's boundary. (b) Illustration of the concentration profile in the diffusion-limited regime. The concentration is constant inside the droplet, while outside it relaxes to $\bar{c}$ with a spatial dependence $\propto R/r$. (c) Illustration of the concentration profile in the interface-resistance-limited regime. Due to fast diffusion, the concentration outside relaxes instantaneously to $\bar{c}$. (d) Growth of the droplet in the diffusion-limited regime for the three supply cases. In the presence of a matter supply, the radius grows indefinitely, while in the passive case, the growth ceases close to equilibrium. $\tau = \ell_\gamma^2/D^\text{out}$ in the log-log plot; $\ell_\gamma$ denotes the capillary length. (e) Growth speed of the droplet radius $R$ as a function of time in the log-log plot. In the case of constant supersaturation, there is a regime of accelerated growth that decreases and follows the $R^{-1}$ law at late times. The same slowdown is valid for a constant matter supply, and it starts when the droplet is large enough such that the $R^{-1}$ term in the growth law dominates the $R^{-2}$ term. For the passive case, the growth speed decreases as $R^{-2}$. Results shown in panels (d,e) were obtained solving Eqs. \ref{['eq:growth1-conc1']}, for the diffusion-limited regime with the parameters given in Table \ref{['tab-param1']}.
• Figure 3: Dynamics of two droplets in the diffusion-limited regime. Phase portrait of two droplets with radii $(R_1, R_2)$ in the (a) passive, (b) constant supersaturation, and (c, d) the case with constant matter supply. Black lines and arrows indicate the flow field lines $(\dot{R}_1,\dot{R}_2)$, and colored solid lines the separatrices. A separatrix splits the phase portraits into domains that differ in the asymptotic behaviour at long times. Solid disks represent stable fixed points, and open circles are unstable fixed points. (a) In the passive case, there are three domains of dynamics. Droplets with radii smaller than the critical radius always shrink and dissolve, and there are two domains within which either $R_1$ or $R_2$ will not dissolve but reach the steady state. (b) For the constant supersaturation, there are three domains where one or both droplets shrink. The domain in the upper right corner describes the growth of both droplets. This growth is indefinite, and there is no stable fixed point. (c) At early times and for the constant supply, the dynamics are similar to those of the constant supersaturation. This similarity arises because droplet material is in excess, and the small droplets grow initially indefinitely. (d) At a late time and for the constant supply, the dynamics are similar to the passive case since droplets grew to large sizes where the constant matter supply is growth-limiting. As a result, there are two stable fixed points and a single droplet persisting over large times. Results were obtained solving Eqs. \ref{['eq:cons-law-discrete2-c']}, \ref{['eq:cons-law-integralc']} for the diffusion-limited regime with the parameters given in Table \ref{['tab-param2']}.
• Figure 4: Coarsening kinetics in an emulsion with constant supersaturation. (a) Average radius as a function of time $t = t/\tau$, $\tau = \ell_\gamma^2/D^\text{out}$. The solid lines correspond to different initial choices of $\beta(\langle R(0)\rangle)$. Depending on the initial choice, the growth of emulsion is determined by diffusion- or interface-resistance-limited transport, with the average radius following $t^{1/2}$ or $t$ power law, respectively. The star marks the crossover between the two regimes for a particular initial condition. (b) The crossover from the interface-resistance to diffusion-limited regime happens when $\beta(\langle R \rangle) \geq 1$, indicated by the star. (c) After all droplets smaller than the critical radius have dissolved, the droplet number density saturates at a constant value. (d) After an initial regime of droplet dissolution, the standard deviation decreases. (e) In the diffusion-limited regime, the droplet size distribution function narrows towards a delta peak and is not universal. (f) In the interface-resistance-limited regime, the droplet size distribution function drifts with increasing velocity in space, maintaining a fixed shape and a constant standard deviation. Results were obtained solving Eqs. \ref{['eq:em-cont']}, \ref{['eqC3:dotR_cconst']} for constant supersaturation $\varepsilon$, with the parameters given in Table \ref{['tab-param']}.
• Figure 5: Non-universal coarsening kinetics in an emulsion with constant matter supply in the diffusion-limited regime. (a) Average radius for different values of constant supply density $J$. The supply is switched on at $t^*$, with time rescaled as $t \to t/\tau$, and $\tau = \ell_\gamma^2/D^\text{out}$. (b) The average radius rescaled by the critical radius for different values of supply. The droplet size distribution function narrows for $\kappa > 2$ (red dashed line). For the supply such that $\kappa < 3/2$, the coarsening converges to a broadening distribution function, for which $\kappa$ relaxes to $\kappa = 3/2$vollmer_ripening_2014. Since, in this limit, the asymptotic solution of the distribution function depends on the number density at the time the supply starts, we term it semi-universal. For $\kappa > 3/2$, its value depends on the matter supply $J/n(t^*) = 4\pi \ell_\gamma c^{(0)} D^\text{out}(\kappa -1)$. (c) Droplet number density converges to a constant value (red dashed line). (d) Standard deviation crosses from the broadening to the narrowing regime with the increasing supply density $J$. The color code for the supply strength is indicated in (a). (e) The droplet size distribution function collapsed for a supply in the broadening regime via rescaling of the spatial coordinate by the critical radius. The choice of supply strength corresponds to the broadening regime. The color code represents different times for measuring the distribution function, while the line style corresponds to the strength of the supply density. (f) The droplet size distribution function narrows towards a delta peak for a supply in the narrowing regime. The color code represents the measurement times, with yellow indicating early times and blue indicating late times. Results were obtained solving Eqs. \ref{['eq:em-cont']}, \ref{['eq:model-both']} for different values of constant matter supply density $J$, with the parameters given in Table \ref{['tab-param']}.