# Cell-size distribution and scaling in a one-dimensional KJMA lattice model with continuous nucleation

## TL;DR

This paper presents a mean-field approach to the 1D KJMA lattice model with continuous nucleation, deriving an analytical cell-size distribution that aligns with known scaling laws and fits Weibull distribution, validated by simulations.

## Contribution

It introduces a simple analytical method for the 1D KJMA model that accurately predicts cell-size distributions and scaling behavior.

## Key findings

- The derived cell-size distribution follows Weibull distribution.
- The approach reproduces established KJMA scaling laws.
- Simulation data confirms the analytical predictions.

## Abstract

The Kolmogrov-Johnson-Mehl-Avrami (KJMA) growth model is considered on a one-dimensional (1D) lattice. Cells can growth with constant speed and continuously nucleate on the empty sites. We offer an alternative, mean-field like approach for describing theoretically the dynamics and derive an analytical cell-size distribution function. Our method reproduces the same scaling laws as the KJMA theory and has the advantage that it leads to a simple closed form for the cell-size distribution function. It is shown that a Weibull distribution is appropriate for describing the final cell-size distribution. The results are discussed in comparison with Monte Carlo simulation data.