# A Novel Approach to Grain Shape Factor in 3D Hexagonal Cellular Automaton

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Model and Calculation

_{grain}is the surface area of the actual grain, S

_{sphere}is the surface area of the sphere with the same volume as the grain, and S

_{chain}is the surface area of the chain with the same volume as the grain. S

_{sphere}and S

_{chain}can be calculated by Equations (3) and (4), respectively. N

_{grain_surf}is the number of surface cells of the grain (blue spheres in Figure 2c or Figure 2d), N

_{sphere}is the number of surface cells of the sphere (Figure 2a), and N

_{chain}is the number of surface cells of the chain (Figure 2b). S

_{cell}is the area of a cell’s great circle on its circumscribed sphere. This value can be ignored because it cancels out. Therefore, Equation (2) turns into:

_{sphere}and N

_{chain}in Equation (5), we need to build models of spheres and chains with the same volume as the grain. The volume is N

_{volume}, which is the number of cells in the grain. We use a step-by-step growth CA method to add cells to the models. For spheres, we choose the cell that minimizes the surface-area increase. For chains, we choose the cell that maximizes it. This way, we can simulate different grain morphologies with different sphericity degrees. The spheres have a high sphericity degree close to 0, while the chains have a low sphericity degree close to 1. The step-by-step growth CA method allows us to control the size and shape of the grains by adjusting the number and location of the cells added. Then, we count N

_{sphere}and N

_{chain}for each model with different N

_{volume}values (from 1 to 5000). We fit curves for N

_{sphere}and N

_{chain}as functions of N

_{volume}by least squares and plot them in Figure 3 with the actual values [30].

_{chain}and N

_{volume}have a linear relationship in the chain model, implying that each N

_{volume}adds six units of surface area. In contrast, N

_{sphere}has a more complex relationship with N

_{volume}in the sphere model, which can be approximated by Equation (7). To assess the quality of this approximation, we use residual analysis (Equation (8)), where R is the average residual, and a good fit is expected to have R < 0.05 [31]. N

_{i}and ${\widehat{N}}_{i}$ are the actual and fitted values of N

_{sphere}when N

_{volume}= i. However, Equation (7) has a high R value of 0.115, indicating a poor fit, especially when N

_{volume}< 200 (see red dotted curves in Figure 4a,b).

_{volume}, the functions for each segment, and the breakpoints between segments. Then, we can use various methods to estimate the parameters of each function and minimize the error between the fitted values and the actual values.

_{sphere}for any given N

_{volume}. Therefore, with Equations (5) and (9), we can compute the shape factor A of any grain in 3D-HEX using its N

_{volume}and N

_{grain}

_{_surf}, which are easy and efficient to obtain.

## 3. Model Validation

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Schematic view of (

**a**) volume-equivalent-sphere, (

**b**) volume-equivalent-chain, and (

**c**,

**d**) two grains with volume of 500.

**Figure 3.**Relationships between surface cell number of volume-equivalent-sphere (red curve and blue dots), volume-equivalent-chain (black curve and green dots), and their corresponding grain volume number.

**Figure 4.**Comparison of fitting effect between Equation (9) and Equation (7) with volume number within a range of (

**a**) 1~79, (

**b**) 80~199, (

**c**) 200~599, (

**d**) 600~1199, (

**e**) 1200~1799, and (

**f**) 1800~5000.

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**MDPI and ACS Style**

Bao, L.; Shi, J.
A Novel Approach to Grain Shape Factor in 3D Hexagonal Cellular Automaton. *Crystals* **2023**, *13*, 544.
https://doi.org/10.3390/cryst13030544

**AMA Style**

Bao L, Shi J.
A Novel Approach to Grain Shape Factor in 3D Hexagonal Cellular Automaton. *Crystals*. 2023; 13(3):544.
https://doi.org/10.3390/cryst13030544

**Chicago/Turabian Style**

Bao, Lei, and Jun Shi.
2023. "A Novel Approach to Grain Shape Factor in 3D Hexagonal Cellular Automaton" *Crystals* 13, no. 3: 544.
https://doi.org/10.3390/cryst13030544