# The Role of Grain Boundary Diffusion in the Solute Drag Effect

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

## 3. Results and Discussion

#### 3.1. Grain Migration in Pure Cu

#### 3.2. Grain Boundary Migration in the Random Alloy

#### 3.3. The Alloying Effect on GB Mobility

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Schematic force–velocity diagram according to the classical model of GB solute drag [29,48,49]. The maximum of the drag force separates two kinetic regimes, with the GB dragging the segregation atmosphere at low velocities and breaking away from it at high velocities. (

**b**) Driving force for the planar $\Sigma 17\left(530\right)\left[001\right]$ GB as a function of GB velocity at different alloy compositions is indicated in the key [50]. The GB is driven by an applied shear stress.

**Figure 2.**Cu bicrystal with a half-loop GB studied in this work. The GB crystallography is $\Sigma $17 [001] with symmetrical tilt inclinations along the vertical portions and at the tip point and asymmetrical tilt inclinations along other curved surfaces. The kite-shaped structural units of the GB structure are outlined. The atoms of the outer and inner grains are shown in blue and green, respectively, with the GB atoms shown in red.

**Figure 3.**GB dynamics in MD simulations of pure Cu. (

**a**) The grain height as a function of time at several temperatures. (

**b**) Arrhenius diagram of GB mobility. The points in (

**b**) correspond to the temperatures indicated in (

**a**), with the straight line representing the linear fit to determine the effective activation energy as GB migration.

**Figure 4.**GB dynamics in MD simulations of Cu–Ag alloys at the temperatures of (

**a**) 950 K and (

**b**) 1100 K.

**Figure 5.**Displacement–time curve for a GB moving in the 1.25 at.%Ag alloy at the temperature of 1000 K (left column) in comparison with GB images at the moments of time indicated by the vertical red line. (

**a**) Initial GB. (

**b**) GB after 50 ns of simulation. (

**c**) GB at the end of the simulation. Atoms with FCC environment are invisible. The GB atoms are shown in gray and the Ag clusters containing five or more atoms are shown in red. The oval indicates the group of clusters responsible for the plateau in the displacement curve. Note how, in (

**b**), the GB bows out toward these clusters.

**Figure 6.**Interaction of the moving GB with solute clusters. Cu atoms of the outer and inner grains are shown in blue and green, respectively, with GB atoms shown in red and Ag atoms in yellow. The GB is moving towards the lower left corner of the images. In (

**a**), the GB has just separated from the group of atoms (cluster) A but is still pinned by the cluster B. In (

**b**), the GB broke away from the cluster B. Panels (

**c**,

**d**) are zoomed in views of the two clusters. The cyan arrows point to the clusters.

**Figure 7.**(

**a**,

**b**) GB position in the beginning of the MD simulation (

**a**) and closer to the end (

**b**). Cu atoms of the outer and inner grains are shown in blue and green, respectively. The GB atoms are shown in red and Ag atoms in yellow. The purple rectangular region is selected to show the SRO formation and is swept by the GB motion. The orange region lies completely in the outer grain and is unaffected by the GB motion. The graphs below compare the Ag–Ag RDFs in the two states shown (

**a**,

**b**) for the outer (

**c**) and inner (

**d**) regions, respectively. Note the significant rise in the first peak in the region traversed by the moving boundary.

**Figure 8.**Change in the RDF’s first peak height (measure of the SRO) in the inner and outer rectangular regions shown in Figure 7 during the shrinkage of the inner grain. The results are shown for three alloy compositions indicated in the key at the temperature of 1000 K. Note that the SRO in the outer region changes very little, while the SRO in the inner region increases as the GB sweeps through its volume and levels out after the GB exits the region.

**Figure 9.**GB velocity as a function of temperature and alloy composition. The two zero-velocity points at 1000 K represent the GB arrest at concentrations above 1.5 at.%Ag.

**Figure 10.**Composition and temperature dependencies of the equilibrium GB free energy in the Cu–Ag solid solution. (

**a**) Comparison of the exact free energy obtained by thermodynamic integration [54] (lines) with the dilute solution approximation by Equation (3) (points). (

**b**) Calculations from Equation (3) for several temperatures and alloy compositions.

**Figure 11.**The capillary driving force for GB migration as a function of temperature and alloy composition.

**Figure 12.**(

**a**) The upper and lower bounds of the GB mobility coefficient as a function of temperature and alloy composition. (

**b**) The upper and lower bounds of the activation energy of GB mobility as a function of alloy composition.

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Koju, R.K.; Mishin, Y.
The Role of Grain Boundary Diffusion in the Solute Drag Effect. *Nanomaterials* **2021**, *11*, 2348.
https://doi.org/10.3390/nano11092348

**AMA Style**

Koju RK, Mishin Y.
The Role of Grain Boundary Diffusion in the Solute Drag Effect. *Nanomaterials*. 2021; 11(9):2348.
https://doi.org/10.3390/nano11092348

**Chicago/Turabian Style**

Koju, R. K., and Y. Mishin.
2021. "The Role of Grain Boundary Diffusion in the Solute Drag Effect" *Nanomaterials* 11, no. 9: 2348.
https://doi.org/10.3390/nano11092348