# The effects of competition between random sequential nucleation of point-sized seeds and island growth by adsorption of finite-sized grains

## TL;DR

This paper models the competition between seed nucleation and grain growth in a one-dimensional adsorption process, deriving analytical expressions for coverage and validating with simulations.

## Contribution

It introduces a solvable model capturing the interplay of nucleation and growth, providing explicit formulas for coverage and gap distribution.

## Key findings

- Jamming coverage approaches 1 as growth rate dominates.
- Analytical results agree with Monte Carlo simulations.
- Coverage follows a power-law decay in the limit of high growth rate.

## Abstract

We study random sequential adsorption of particles from pool onto a one dimensional substrate following ballistic deposition rules, with separate nucleation and growth processes occurring simultaneously. Nucleation describes the formation of point-sized seeds, and after a seed is sown, it acts as an attractor and grows in size by the addition of grains of a fixed-sized. At each time step either an already-nucleated seed can increase in size, or a new seed may be nucleated. We incorporate a parameter $m$, to describe the relative rates of growth to nucleation. We solve the model analytically to obtain gap size distribution function and a general expression for the jamming coverage as a function of $m$. We show that the jamming coverage $\theta(m)$ reaches its maximum value $\theta(m)=1$ in the limit $m\rightarrow \infty$ following a power-law $\theta(\infty) - \theta(m) \sim m^{-1/2}$. We also perform extensive Monte Carlo simulation and find excellent agreement between analytic and numerical results.