Bridging Atomistic and Mesoscale Lithium Transport via Machine-Learned Force Fields and Markov State Models

TL;DR

The paper tackles the challenge of connecting atomistic lithium diffusion mechanisms to mesoscale transport in Li-Si systems. It introduces a bottom-up AIMD→MLFF→MSM workflow in which machine-learned force fields trained on first-principles data enable nanosecond-scale simulations that reduce finite-size biases, and Markov state models coarse-grain these trajectories to yield statistically converged jump networks and mechanistic diffusion channels. The approach delivers near-DFT accuracy for migration barriers and diffusion coefficients, reproduces mean-square displacements across long times, and reveals slow relaxation modes and dominant pathways through MSM spectra. This framework is transferable to amorphous, defect-containing silicon anodes and provides a scalable route to couple atomistic transport to electrode-scale models for next-generation solid-state batteries.

Abstract

Lithium diffusion in solid-state battery anodes occurs through thermally activated hops between metastable sites often separated by large energy barriers, making such events rare on ab initio molecular dynamics (AIMD) timescales. Here, we present a bottom-up multiscale workflow that integrates AIMD, machine-learned force fields (MLFFs), and Markov state models (MSMs) to establish a quantitatively consistent link between atomistic hopping mechanisms and mesoscale transport. MLFFs fine-tuned on AIMD reference data retain near-DFT accuracy while enabling large-scale molecular dynamics simulations extending to tens of nanoseconds. These extended trajectories remove the strong finite-size bias present in AIMD and yield diffusion coefficients in excellent agreement with experiment. Furthermore, from these long MLFF trajectories, we obtain statistically converged lithium jump networks and construct MSMs that remain Markovian across more than two orders of magnitude in the lag times used for their construction. The resulting MSMs faithfully reproduce mean-square displacements and recover rare diffusion processes that do not occur on AIMD timescales. In addition to propagating lithium distributions, the MSM transition matrices provide mechanistic insight: their eigenvalues and eigenvectors encode characteristic relaxation timescales and dominant transport pathways. Although demonstrated for defect-free crystalline Li$_x$Si$_y$ phases, the AIMD$\rightarrow$MLFF$\rightarrow$MSM framework is general and provides a transferable approach for describing lithium transport in amorphous materials, defect-mediated diffusion, and next-generation solid-state anodes.

Figures (9)

• Figure 1: (a) Snapshot of the Li$_{12}$Si$_7$ crystal structure. Lithium and silicon atoms are shown in red and teal, respectively. Panels (b)--(d) illustrate comparisons of the radial distribution function $g(r)$ obtained from AIMD (black filled circles), the MACE foundation model (green line), and the fine-tuned model (orange line) for Li--Li, Li--Si, and Si--Si pairs, respectively.
• Figure 2: Representative lithium migration pathways and activation energies. Panels (a) and (c) show selected migration paths in Li$_{12}$Si$_7$ and Li$_{13}$Si$_4$, respectively. Panels (b) and (d) show corresponding NEB energy profiles computed using DFT (blue), the MACE foundation model (red), and the fine-tuned model (orange).
• Figure 3: (a) Mean-square displacement (MSD) from AIMD and MLFF simulations. (b) Convergence of the diffusion coefficient (MSD slope) with respect to system size in the $x$-direction from MLFF simulations. Values 1, 2, 3, 4, and 8 in the legend correspond to a 1-, 2-, 3-, 4-, and 8-fold increase of the supercell dimension in the $x$-direction. Apparent diffusivities computed in AIMD-accessible supercells exhibit a pronounced finite-size bias, whereas larger MLFF supercells approach the asymptotic, size-independent limit.
• Figure 4: $z$-component of the MSD of Li$_{13}$Si$_4$ obtained from 100 ps AIMD, 30 ns MLFF-MD, and Markov state model (MSM) simulations. For the "projected" MSDs, the positions of the Li atoms were mapped in each time step to the nearest lithium lattice sites. This procedure removes the contributions from local fluctuations around crystallographic lattice sites and is implicitly included in the Markov models by construction. Meaningful diffusion coefficients in z-direction can only be obtained from the MLFF and MSM simulations.
• Figure 5: Average Li$^{+}$ jump frequency maps at 500 K. Top row: Li$_{12}$Si$_7$ with (a) AIMD (100 ps), (b) fine-tuned MLFF (1 ns), (c) foundation model (1 ns). Bottom row: Li$_{13}$Si$_4$ with (d) AIMD (100 ps), (e) fine-tuned MLFF (1 ns), (f) foundation model (1 ns). The foundation model overestimates jump rates; fine-tuned MLFFs recover AIMD-observed channels and sample rare events.