# Effect of random terms on the shape of dendrites analyzed by the phase-field model

## TL;DR

This paper studies how random terms affect dendrite shapes in simulations, finding that variance has the biggest impact.

## Contribution

The study introduces and compares two random terms (χ and ∇·q) in phase-field models to understand their effect on dendrite morphology.

## Key findings

- The variance of random terms has the greatest effect on dendrite shape.
- Uniform distribution random numbers reduce calculation time by more than twice compared to Gaussian distribution.
- The probability density function of random numbers does not influence dendrite shape.

## Abstract

In this study, we focus on random terms with a significant effect on the calculation time and the sidebranch structure of dendrites. We quantitatively evaluate the effect of the random terms on dendrites by changing the term introduced into the evaluation equation, the distribution of the random term, and variance. We introduced the terms: χ and ∇·q. In the phase-field equation, we use χ to represent interfacial noise. In the heat conduction equation, we use ∇·q to represent thermal noise. We compare the results of calculation using only χ with those using only ∇·q and analyze the effect of each random term introduced on the shape of dendrites. In terms of the probability distribution for generating random numbers, the uniform and Gaussian distributions are used for comparison. The magnitude of the noise Fu, which controls the variance of noise, is used with 3 or 4 patterns of values to the extent that the calculation results do not diverge. Each value of Fu differs by a factor of 10. Sidebranch length, contour length, and area are used as indices for evaluating dendrites. The succinonitril is used for comparing the simulation and experimental results for dendrites to design the optimal random number model. On the basis of the introduced terms, we clarify that each term requires a different order of the variance. The probability density function of random numbers does not affect the shape of dendrites. On the other hand, the calculation time which is used the random numbers following the uniform distribution is more than twice as fast as when using random numbers that follow a Gaussian distribution. The variance of a random term has the greatest effect on the shape of dendrites.