Grain Boundary Solute Drag Model in Regular Solution Alloys

We present a grain boundary (GB) solute drag model in regular solution alloys. The model accounts for solute-solute interactions in both the bulk and GBs and captures effects such as monolayer, multilayer, and asymmetrical segregation. Our analysis shows that deviations from ideal solution thermodynamics play a paramount role, in which solute drag is shown to scale with solute-solute interaction parameters. Further, it is found that the asymmetry in GB segregation introduces an additional component to solute drag. A universal solute drag-GB velocity relation is proposed and used to explain recent experimental observations of sluggish grain growth in a wide range of engineering alloys.

Figure 1

The four $\mathbf{\Upsilon }$ functions used to describe GB-solute interactions. The Gaussian (green), smoothed boxcar (black), left-(blue), and right-skewed (red) functions.

Figure 2

Solute concentration profiles across the GB for $\overline{\mathcal{V}}=0$, 4, and 12 for the four $\mathbf{\Upsilon }$ functions used in this work. Here, we set $(c_{\infty },{\overline{\mathrm{\Omega }}}_b,{\overline{\mathrm{\Omega }}}_{*},{\overline{G}}_{*})=(0.1,0.75,-5.75,5.75)$.

Figure 3

For the four $\mathbf{\Upsilon }$ functions used in this work: (a) A surface plot depicting solute drag ${\overline{P}}_d$ as a function of ${\overline{\mathrm{\Omega }}}_{*}$ and $\overline{\mathcal{V}}$. (b) Slices of the ${\overline{P}}_d$ surface for various ${\overline{\mathrm{\Omega }}}_{*}$ values depicting the shift in the maximum solute drag to larger $\overline{\mathcal{V}}$ with decreasing ${\overline{\mathrm{\Omega }}}_{*}$. In both panels, $(c_{\infty },{\overline{\mathrm{\Omega }}}_b)=(0.1,0.75)$.

Figure 4

(a) A plot of the proposed form for $G_{\mathrm{ref}}$ [Eq. (9)] along with the collapsed solute drag-velocity curves (gray lines) for alloys with $c_{\infty }=0.05$, 0.1, and 0.2 and using the four $\mathbf{\Upsilon }$ functions employed in this work. Using Eq. (8), a plot of (b) ${\overline{P}}^{*}={\overline{P}}^{*}(\overline{{\mathrm{\Omega }}_{*}})$, and (c) ${\overline{\mathcal{V}}}^{*}={\overline{\mathcal{V}}}^{*}({\overline{\mathrm{\Omega }}}_{\text{GB}},{\overline{\mathrm{\Omega }}}_b)$ for the alloy with $c_{\infty }=0.1$ for various values of the bulk heat of mixing ${\overline{\mathrm{\Omega }}}_b$.