On a relationship between grain boundary free energy, grain boundary segregation, and grain boundary diffusion

Abstract

We present a detailed analysis of the universal relationship between grain boundary (GB) free energy and GB self-diffusion coefficient derived by Borisov et al. (1964). This relationship was expressed by a simple equation that was used in many publications to predict GB energies on the basis of experimental diffusion data. Meanwhile, the physical assumptions and approximations underlying the Borisov model are poorly understood. As a result, the Borisov equation was often used outside its intended limits. Here, we re-derive the Borisov equation from ground up, identifying its underlying assumptions, correcting some errors and inconsistencies, and extending the original model to the case of impurity diffusion and diffusion by the interstitialcy and interstitial-dumbbell mechanisms. The meaning of the key assumption of the Borisov model, related to the free energy of the activated complex, is discussed, and ways to test this assumption are proposed.

Figures (3)

• Figure 1: Schematic representation of the atomic free energy as a function of position in a crystalline solid with a point defect undergoing a transition to a new state. The blue oval outlines the activated complex containing $n=3$ atoms. The transition rate $w$ is given by equation (\ref{['eq:132']}).
• Figure 2: Schematic representation of free energies of the activated complexes for vacancy jumps in the lattice ($\hat{g}$), in the GB ($\hat{g}^{\prime}$), and between the GB and the lattice ($\hat{g}_{t}$). The plot shows the free energies of atoms as a function of position in the perfect lattice ($\mu_{0}$), in the GB ($\mu_{0}^{\prime}$), and in the activated complexes. The atomic jump rates are $w$ in the lattice, $w^{\prime}$ in the GB, and $w_{t}$ and $w_{t}^{\prime}$ into and out of the GB, respectively. All free energies are defined relative to the same reference state. The blue ovals symbolize the activated complexes. The orange dashed lines demonstrate that $\hat{g}$, $\hat{g}^{\prime}$, and $\hat{g}_{t}$ could be close to each other but are generally different. An activated complex may contain a single atom or a group of several atoms undergoing a collective rearrangement. The diagram is purely conceptual. In reality, free energies of individual atoms are not well-defined, which is why the equations in the main text are formulated in terms of total free energies of regions containing many atoms.
• Figure 3: Schematic representation of the free energies of interstitial impurity atoms during diffusion by the direct interstitial mechanism. The plot shows the free energies of atoms as a function of position. The host atoms are represented by white circles, and the interstitial atoms before their jumps and in the activated state are represented by green and red circles, respectively. The blue ovals symbolize the activated complexes, whose free energies are $\hat{g}$ and $\hat{g}^{\prime}$. $\tilde{\mu}_{I}$ and $\tilde{\mu}_{I}^{\prime}$ are the non-configurational parts of the chemical potentials of the impurity atoms. $g^{*}$ and $g^{*\prime}$ are the activation barriers of the interstitial jumps. Note that $\tilde{\mu}_{I}^{\prime}<\tilde{\mu}_{I}$, leading to the negative segregation energy $g_{s}=\tilde{\mu}_{I}^{\prime}-\tilde{\mu}_{I}<0$ and impurity segregation at the GB. Assuming that $\hat{g}^{\prime}\approx\hat{g}$, the activation free energy of GB diffusion is higher than that of lattice diffusion: $g^{*\prime}>g^{*}$. The diagram is purely conceptual. In reality, free energies of individual atoms are not well-defined.