Diffusive Growth of a Single Droplet with Three Different Boundary Conditions

TL;DR

This paper investigates the diffusive growth of a single droplet on a 2D substrate under various boundary conditions, revealing different early-time growth exponents and a universal long-time growth rate independent of boundary conditions.

Contribution

It introduces and solves models with different boundary conditions, including adsorption, radiation, and phenomenological, providing new insights into droplet growth dynamics.

Findings

Early-time growth exponents: 1/4, 1/2, 3/4 for different boundary conditions.

Universal long-time growth rate: (t/ln t)^{1/3} for all models.

Power law growth with exponent 1/3 for constant flux boundary condition.

Abstract

We study a single, motionless three-dimensional droplet growing by adsorption of diffusing monomers on a 2D substrate. The diffusing monomers are adsorbed at the aggregate perimeter of the droplet with different boundary conditions. Models with both an adsorption boundary condition and a radiation boundary condition, as well as a phenomenological model, are considered and solved in a quasistatic approximation. The latter two models allow particle detachment. In the short time limit, the droplet radius grows as a power of the time with exponents of 1/4, 1/2 and 3/4 for the models with adsorption, radiation and phenomenological boundary conditions, respectively. In the long time limit a universal growth rate as $[t/\ln(t)]^{1/3}$ is observed for the radius of the droplet for all models independent of the boundary conditions. This asymptotic behaviour was obtained by Krapivsky \cite{krapquasi} where a similarity variable approach was used to treat the growth of a droplet with an adsorption boundary condition based on a quasistatic approximation. Another boundary condition with a constant flux of monomers at the aggregate perimeter is also examined. The results exhibit a power law growth rate with an exponent of 1/3 for all times.