Kinetic theory of emulsions with matter supply

TL;DR

This paper develops a kinetic theory for emulsions with continuous matter supply, analyzing how droplet growth and size distribution evolve under different supply and transport limitations.

Contribution

It extends the Lifshitz-Slyozov-Wagner theory to include matter supply, providing new insights into droplet size dynamics under various conditions.

Findings

Droplet size distribution narrows in diffusion-limited regimes with maintained supersaturation.

A universal coarsening law is derived for interface-resistance-limited growth with constant supply.

The average droplet radius follows a power law independent of matter supply.

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.