Phase-field model for diffusion-induced grain boundary migration: An application to battery electrodes

Diffusion-induced grain boundary migration is a phenomenon in which a grain boundary moves in response to the driving forces generated by diffusing solute species. For example, diffusing solute species change the atomic volume in the host material, either by filling a vacancy with a misfitting solute atom or by expanding host lattices through interstitial diffusion. These volume changes are inhomogeneous and are stored as elastic energy in the material that drives grain boundaries. In this paper, we introduce our previously developed Cahn-Hilliard–phase-field-crystal model (CH-PFC) as a computational tool to investigate diffusion-induced grain boundary migration in crystalline materials. This multiscale phase-field model couples the composition field of a diffusing species with the crystallographic texture of a host material. We apply the CH-PFC model to battery electrodes and investigate whether interstitial solute diffusion induces grain growth in ${\mathrm{FePO}}_4/{\mathrm{LiFePO}}_4$ electrodes. To this end, we compute grain growth in 60 ${\mathrm{FePO}}_4$ electrodes by conducting two parallel trials: In the first trial, we cycle the electrode and calculate diffusion-induced grain growth. In the second trial, we do not cycle the electrode and calculate curvature-driven grain growth. We find a statistically significant grain growth in the cycled electrodes and negligible grain growth in the noncycled electrodes. Overall, we show that the CH-PFC model not only predicts electrode microstructures as a function of the Li composition, but also predicts the crystallographic features of an electrode during battery operation.

Figure 1

Schematic of interstitial solute diffusion into a polycrystalline material. Here, the lattice parameters of the material vary with the solute concentration [i.e., $a\left(c_S\right)$ and $a\left(c_0\right)$ in the presence/absence of an interstitial solute]. This gradient in the lattice parameter produces stresses at the proximity of a grain boundary that induces its movement.

Figure 2

(a) CH-PFC simulation of a partially lithiated ${\mathrm{FePO}}_4$ electrode particle that is surrounded by an amorphous Li reservoir. The color bar shows the Li composition in ${\mathrm{FePO}}_4$ (blue; $c_{ij}=0$) and ${\mathrm{LiFePO}}_4$ (red; $c_{ij}=1$) phases. The small black box shows a diffuse phase boundary (yellow; $0<c_{ij}<1$) separating the two phases along AA. The width of this diffuse interface (highlighted by dashed vertical lines) is numerically calibrated to span approximately four lattice points. (b) Enlarged image of the electrode lattice symmetry is shown in the box labeled BB. In the electrode region, the peak density field ${\psi }_{ij}$ has a waveform where the coordination symmetry of ellipsoidal peaks represents a ${\mathrm{LiFePO}}_4$ lattice. In the Li-reservoir region, ${\psi }_{ij}$ is a constant and describes an amorphous state. At the electrode-reservoir interface, ${\psi }_{ij}$ changes as a function of $r_{ij}$ according to Eq. (A1). (c) “Peak marker image” illustrating the crystallographic features in the polycrystalline electrode: coherent interface, A; triple junction, B; chain of defects, enclosed in green circles, which forms a high-angle grain boundary $(>{15}^{\circ })$, ${\mathrm{C}}^{\prime }$, and an edge-dislocation defect, D.

Figure 3

Lithiation of the polycrystalline ${\mathrm{FePO}}_4$ electrode particle. Starting from an initial ${\mathrm{FePO}}_4$ phase (a), Li ions intercalate into the electrode particle (b, c). A ${\mathrm{LiFePO}}_4$ phase is formed at the end of lithiation (d). The plots in the top row show the temporal evolution of the Li composition $c_{ij}$ in the electrode particle. The plots in the middle row show the structural transformation of lattices in the host-electrode during lithiation. Labels A–D highlight representative crystallographic features in the electrode particle. The plots in the bottom row illustrate the distortion maps ${\delta }_{ij}$ corresponding to each stage of lithiation. We interpret ${\delta }_{ij}$ to show coarse-grained lattice distortions induced from Li intercalation.

Figure 4

Lattice rearrangements in the noncycled polycrystalline electrode particle. The electrode-reservoir system is modeled with a homogeneous Li composition $c_{ij}=0$ and computed for the same time $\left({\tau }_{\mathrm{total}}\right)$ and temperature as in Fig. 3. Host-electrode lattice arrangements (a) before, $\tau /{\tau }_{\mathrm{total}}=0$, and (b) after, $\tau /{\tau }_{\mathrm{total}}=1$, the noncycled grain growth. (c) The lattice distortion map ${\delta }_{ij}$ calculated as the absolute difference in centroid positions between the before and the after stages of the noncycled electrode. The distortion map illustrates coarse-grained lattice rearrangements near grain boundaries.

Figure 5

Grain sizes in electrode particles $d=50{\delta }_0$ (inset: $d=100{\delta }_0)$ in the (a) intervention-cycled, (b) paired-noncycled, and (c) unpaired-noncycled sets. Axes indicate the grain size in electrodes before ($g^0)$ and after $\left(g^{\tau }\right)$ CH-PFC computations. The dotted line is a reference line along which there is zero grain growth in electrodes. Solid lines show the linear model fit to the data points, and gray-shaded areas correspond to the $95\%$ confidence interval. There is statistically significant grain growth in all three sets $(p<0.05)$, however, the estimated mean grain growth in the intervention-cycled $({\overline{G}}_{\mathrm{IC}}=0.459)$ set is greater than the estimated mean grain growth in the paired-noncycled $({\overline{G}}_{\mathrm{IC}}=0.121)$ and unpaired-noncycled $({\overline{G}}_{\mathrm{IC}}=0.131)$ sets.

Figure 6

Schematic of the coupling between lattice symmetry and composition field in the CH-PFC model. The transformation matrix $\mathbf{A}\left(c\right)$ is described as a function of the composition field $c$. For $c=0$, $\mathbf{A}\left(c\right)$ describes a rectangular ${\mathrm{FePO}}_4$ lattice (blue box). For $c=1$, $\mathbf{A}\left(c\right)$ describes the lattice motif of ${\mathrm{LiFePO}}_4$ (red box). For $0<c<1$, intermediate lattice geometries of the interphase ${\mathrm{Li}}_X{\mathrm{FePO}}_4$ (dashed black box) are modeled.