# Atomistic Assessment of Solute-Solute Interactions during Grain Boundary Segregation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Thermodynamics of Grain Boundary Segregation

#### 2.1. Free Energy vs. Enthalpy of Segregation

#### 2.2. Classical Segregation Models

#### 2.3. Spectral Segregation Models

## 3. Atomistic Simulation Methods

#### 3.1. Production of Pure Al Polycrystal

^{3}, 60,367 total atoms, and 10 grains of random orientation with an average diameter of 6 nm (Figure 1). The polycrystal was randomly initialized via Voronoi tessellation using the toolkit Atomsk (Version b0.11.1, University of Lille, Villeneuve d’Ascq, France) [54], followed by structural relaxation with conjugate gradient minimization. The polycrystal was then thermally annealed in an isothermal isobaric ensemble with a Nose-Hoover thermostat/barostat, at zero pressure and a temperature of 600 K for 0.5 ns. Finally, the polycrystal was cooled to 0 K over 0.25 ns, followed by a final conjugate gradient minimization.

^{3}polycrystal used in this work, at an average grain size of 6 nm, is significantly smaller than the (15 nm)

^{3}and (36 nm)

^{3}polycrystals used by Wagih and Schuh previously [43,47], with grain sizes of 9 and 12 nm, respectively. However, preliminary work in analyzing the grain size dependence of the segregation energy distribution indicates that changes in the distribution with respect to grain size are due primarily to the increased presence of triple junctions and quadruple nodes at smaller grain sizes. While this effect is non-negligible, for most alloys, including Al-Mg, the effective difference in segregation energy when decreasing the grain size from 12 nm to 6 nm is of at least an order of magnitude less than the effective segregation energy itself.

#### 3.2. Dilute Limit Segregation Energy Distributions

^{3}polycrystal [43].

#### 3.3. The True Equilibrium Segregation State: Hybrid MC/MS

#### 3.4. Grain Boundary Heat of Mixing Distributions

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Visualization of the grain boundary network of the pure Al polycrystal after relaxation and annealing, with dimensions of (10 nm)

^{3}, 10 randomly oriented grains of average diameter 6 nm, and 60,367 total atoms.

**Figure 2.**Dilute limit segregation energy distribution for Al-Mg, calculated from the (10 nm)

^{3}polycrystal, with a fitted skew-normal distribution overlaid.

**Figure 3.**Correlation plot of the average segregation energy of the nearest neighbors of a given grain boundary site vs. the segregation energy of that site for Al-Mg in the (10 nm)

^{3}polycrystal.

**Figure 4.**(

**a**) Al-Mg polycrystal with 5% total solute, equilibrated with hybrid MC/MS at 600 K. (

**b**) Segregation energy distribution with the equilibrium occupied distribution shown in red. Predicted occupied distribution is shown for the dilute case (Equation (9) (blue)). (

**c**) For the (10 nm)

^{3}Al-Mg polycrystal: McLean-style isotherm with effective segregation energy $\Delta {\overline{E}}_{eff}^{seg}$ = −26.5 (Equation (3) (green)), dilute limit spectral isotherm (Equation (9) (blue)), and polycrystal equilibrated via MC/MS, with a fitted linear interaction parameter ${\Omega}^{GB}$ = −22.86 kJ/mol (Equation (11) (red)).

**Figure 5.**Example 2d atomic configurations used to calculate the per-bond parameter ${w}_{ij}^{GB}$ for bond $i-j$, by measuring the per-atom energy of atom I in the fully relaxed polycrystal, ${E}_{ij,IJ}^{GB}$, where atoms I and J can be either solvent A or solute B.

**Figure 6.**For every GB site in the (10 nm)

^{3}Al-Mg polycrystal: (

**a**) Atomic coordination of every GB site. (

**b**) Correlation plot of atomic coordination and per-site parameter ${w}_{i}^{GB}$. (

**c**) Average per-site parameter ${w}_{i}^{GB}$. (

**d**) Average per-site heat of mixing parameter ${\Omega}_{i}^{GB}$.

**Figure 7.**(

**a**) 2D histogram of the dilute limit segregation energy and per-site interaction parameter ${w}_{i}^{GB}$, exhibiting a bivariate skew-normal distribution. (

**b**) Bivariate normal distribution fitted to the data depicted in Figure 7a.

**Figure 8.**(

**a**) For the (10 nm)

^{3}Al-Mg polycrystal: isotherm for the polycrystal equilibrated via MC/MS at 600 K, with a fitted linear interaction parameter ${\Omega}^{GB}$ = −22.86 kJ/mol (Equation (11) (red)), spectral isotherm with the average bulk interaction parameter ${\Omega}^{c}=$ −28.32 kJ/mol and average grain boundary interaction parameter ${\overline{\Omega}}^{GB}=$ −27.10 kJ/mol (Equation (12) (solid black)), and spectral isotherm with fitted bivariate normal distribution (Equation (17) (magenta)). (

**b**) Equilibrium occupied distribution, with predicted occupation distributions using: a fitted linear interaction parameter ${\Omega}^{GB}$ = −22.86 kJ/mol (Equation (11) (red)), average interaction parameters ${\Omega}^{c}=$ −28.32 kJ/mol and ${\overline{\Omega}}^{GB}=$ −27.10 kJ/mol (Equation (12) (black)) and the full fitted bivariate normal distribution (Equation (17) (magenta)).

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Matson, T.P.; Schuh, C.A.
Atomistic Assessment of Solute-Solute Interactions during Grain Boundary Segregation. *Nanomaterials* **2021**, *11*, 2360.
https://doi.org/10.3390/nano11092360

**AMA Style**

Matson TP, Schuh CA.
Atomistic Assessment of Solute-Solute Interactions during Grain Boundary Segregation. *Nanomaterials*. 2021; 11(9):2360.
https://doi.org/10.3390/nano11092360

**Chicago/Turabian Style**

Matson, Thomas P., and Christopher A. Schuh.
2021. "Atomistic Assessment of Solute-Solute Interactions during Grain Boundary Segregation" *Nanomaterials* 11, no. 9: 2360.
https://doi.org/10.3390/nano11092360