# Graph-theoretic analyses of saturation fraction of repulsive dopants in solid solutions

## TL;DR

This paper introduces a mathematical model to predict the maximum fraction of repelling dopants in alloys based on their lattice structure.

## Contribution

A universal model using random graph theory to predict the saturation fraction of repulsive dopants in solid solutions.

## Key findings

- The saturation fraction of repulsive dopants can be universally described using lattice structure parameters.
- Stochastic simulations and mathematical analysis confirmed the model's ability to reproduce saturation trends.
- The model provides insights for designing compositions in multi-principal element alloys.

## Abstract

Short-range order (SRO) of dopant atoms in alloys or solid solutions is one of the most essential factors for materials design. In various alloy materials, dopant atoms repel each other, which causes a non-neighboring SRO and results in a substantial effect on their material properties. The fraction of repelling dopants should have an upper bound to satisfy the non-neighboring placement, where dopants are, as it were, saturated. Such “saturation fraction” is expected to play an important role in composition design for alloys. However, no comprehensive understanding has been established thus far for the saturation fraction of repulsive dopant elements despite its practical importance. Here we show that the saturation fraction of repulsive dopant can be described universally by several simple parameters regarding the lattice structure. We conducted a series of stochastic simulation and mathematical analysis for random packing of repulsive dopant in lattice systems for the purpose of predicting the saturation fraction. The mathematical model, which is based on random graph, successfully reproduced the basic trend of the saturation fraction for a variety of lattice structures. The present analyses can provide new insights into composition design for various kinds of alloys such as multi-principal element alloys.