Universal Framework for Decomposing Ionic Transport into Interpretable Mechanisms

Abstract

Understanding mechanisms of ion transport in bulk materials is central to designing next-generation ion conductors for energy storage devices, yet studies employing all-atom molecular dynamics (MD) have largely been limited to reporting overall transport coefficients without a quantitative, spatiotemporally resolved breakdown of \emph{how} charge is carried. We present a computational framework that analyzes MD trajectories to quantitatively interpret macroscopic transport by decomposing it into additive contributions from physically motivated events. They are defined either through heuristically identified microscopic transitions, capturing events such as single-ion hops, multi-ion hops, and vehicular motion, or through transitions between chemically interpretable coordination macrostates. The construction guarantees that attributed contributions sum exactly to the Onsager transport coefficients estimated via the Green-Kubo/Einstein formalism, while scanning the sampling window exposes characteristic temporal scales at which distinct transport mechanisms emerge and dominate. Applied across three prototypical electrolytes-inorganic crystals, liquids, and polymers-the framework quantitatively resolves long-standing debates (e.g., the role of concerted motion and exchange), identifies dominant mechanisms and rate-limiting steps, quantifies their frequencies and effectiveness, and extracts activation energies for distinct transport modes, thereby distilling design rules for fast conduction. This general and reproducible analysis tool turns MD trajectories into quantitative mechanism maps, enabling the ion-conductor community to adjudicate mechanistic hypotheses and accelerate discovery.

Figures (11)

• Figure 1: Overview of OnsagerDecomposera, OnsagerDecomposer processes equilibrium MD trajectories sampled at a window length $\Delta t$ to yield a mechanism-resolved decomposition of Onsager transport coefficients. b, Coordination microstates $S_i^t$ encode the identities and indices of solvating species within the coordination shell of the ion of interest (Li) with index $i$ at time $t$, determined by hard distance cutoff or a geometric convex hull method for liquids and amorphous materials, and by mapping to unique Wyckoff positions in crystals. c, Mapping transitions. Method 1 (heuristic classification) assigns each microstate transition $S_i^t\!\to\!S_i^{t{+}\Delta t}$ and its corresponding displacement $\mathbf{d}_i(t,\Delta t)$ to a complete, non-overlapping set of event classes $\{m\}$ (e.g., vehicular vs exchange; intrachain vs interchain; single-ion vs concerted), shown schematically as $\alpha,\beta,\gamma$. Method 2 (macrostate coarse-graining) aggregates high-dimensional microstates to chemically interpretable macrostates, $S_i^t\mapsto\mathbb{S}_i^t$ (e.g., SSIP, CIP, AGG; activated vs non-activated), labeled X, Y, Z. Each macrostate transition and its corresponding displacement $\mathbf{d}_i(t,\Delta t)$ are then mapped into the macrostate transition diagram.
• Figure 2: Exact additive decomposition of Onsager transport coefficients via the event-specific virtual particle construction.a, For each ion $i$ and sampling window $\Delta t$, the event indicator $\chi_i^m(t,\Delta t)$ marks whether the microstate transition $S_i(t)\!\to\!S_i(t{+}\Delta t)$ belongs to event class $m$ (schematically $\alpha,\beta,\gamma$). The event-specific virtual particle displacement illustrates that each virtual particle undergoes displacement only when its parent particle experiences the specific event $m$, such that exactly one class receives the ion’s displacement in each window. In the schematic, white boxes denote $\vec{0}$ and colored boxes denote $\mathbf{d}_i(t,\Delta t)$ for the active event in each window. Aggregating information over all ions $i$ yields the event probabilities. b, Displacement correlations at lag $\tau$ for a given sampling window $\Delta t$. The displacement correlations associated with Li: $C_\mathrm{self}^{\mathrm{LiLi}}$, $C_\mathrm{distinct}^{\mathrm{LiLi}}$, $C_\mathrm{distinct}^{\mathrm{LiAn}}$ (black lines) equals the sum of event-resolved terms (colored lines) at all $\tau$ values, confirming exact additivity. The slopes of $C(\tau; \Delta t)$ with respect to $\tau$ at long times yield the decomposed Onsager coefficients, and thus the additivity of $C$ directly translates to the additivity of these decomposed slopes. c, Schematic illustration of the decomposed Onsager transport matrix for a system with two Li and two anions. Solid lines indicate components of the conventional Onsager matrix, whereas the dotted lines indicate the event-decomposed one. Boxes labeled $\alpha,\beta,\gamma$ denote contributions from individual event types $\alpha,\beta,\gamma$ to $L_{\mathrm{self}}^{\mathrm{LiLi}}$. Analogous decompositions apply to $L_{\mathrm{distinct}}^{\mathrm{LiLi}}$ and $L_{\mathrm{distinct}}^{\mathrm{Li,Anion}}$.
• Figure 3: Mechanistic decomposition of transport in a crystalline solid electrolyte.a, Diffusion network for Li_1+xAl_xTi_2-x(PO4)3 (LATP): nodes are Li Wyckoff sites ($6b$, $18e$, $36f$); edges indicate face-sharing connectivity between polyhedra. Simultaneous hops are detected within a two-edge neighborhood. b, Event classification for transitions between coordination microstates $S_i^t\!\to\!S_i^{t{+}\Delta t}$: (i) local vibration (no hop; no neighbor hop), (ii) local vibration with neighbor hop, (iii) single-ion hop, (iv) aligned concerted hop (at least one neighbor hops with inter-hop angle $\le \theta_{\mathrm{align}}$, here $90^\circ$), and (v) unaligned concerted hop (angle $> \theta_{\mathrm{align}}$). Each window receives exactly one label. c, Arrhenius plot of $D^{\mathrm{Li}}_{\mathrm{self}}$ (black) and its mechanism-resolved components (colors) for $x=1/3$ (Li_1.33Al_0.33Ti_1.67(PO4)3). Colored lines sum to the black line by construction. d, Probability of events $p_m(\Delta t)$ versus sampling window $\Delta t$ at 800 K. e, Contributions of the various microstate transition types to $D^{\mathrm{Li}}_{\mathrm{self}}$ as a function of $\Delta t$. The sum over events at each $\Delta t$ equals the undecomposed $D^{\mathrm{Li}}_{\mathrm{self}}$. f, Effective self-diffusion for each event, defined as $D^{\mathrm{Li}}_{\mathrm{self},m}/p_m(\Delta t)$, as a function of $\Delta t$. Macrostate-based decomposition of transport in stoichiometric and Li-stuffed NASICONs are shown in g-i. g, Event (transition) probabilities between activated (A) and non-activated (non-A) states as a function of sampling window $\Delta t$ for stoichiometric ($x=0$) (left) and Li-stuffed ($x=0.333$) (right) Li_1+xAl_xTi_2-x(PO4)3. h, Contributions of macrostate transitions ($A{\to}A$, $A{\leftrightarrow}\mathrm{non}\text{-}A$, $\mathrm{non}\text{-}A{\to}\mathrm{non}\text{-}A$) to $D^{\mathrm{Li}}_{\mathrm{self}}$. i, Effective $D_{\mathbf{self}}^{\mathrm{Li}}$ for each macrostate transition, defined as its contribution to $D^{\mathrm{Li}}_{\mathrm{self}}$ normalized by the corresponding event probability.
• Figure 4: Mechanistic decomposition of transport in polymer electrolytes.a, Hierarchical event classification for transitions between coordination microstates in polymer electrolytes, yielding six transition types: vehicular (orange), partial intrachain hop (dark green), complete intrachain hop (light green), interchain hop (red), anion exchange (blue), and coupled anion-exchange and interchain hop (purple). These colors are used consistently throughout the figure. b, Event probabilities sampled every 200 ps as a function of Li/EO ratio $r$ in PEO. c, Decomposed $\sigma_{\mathrm{NE}}^{\mathrm{Li}}$ (stacked colored bars) and $\sigma_{\mathrm{NE}}^{\mathrm{anion}}$ (grey bars) as a function of LiTFSI concentration, sampled every 200 ps. d, Effective $\sigma_{\mathrm{NE}}^{\mathrm{Li}}$ (colored points) as a function of $r$. Black squares represent the total $\sigma_{\mathrm{NE}}^{\mathrm{Li}}$, corresponding to the weighted average of the colored points. e--g, Quantitative comparison of ion transport in PEO and PPM polymer electrolytes at $r = 0.08$. e, Event probabilities in PEO (solid bars) and PPM (dotted bars) as a function of sampling interval. f, Decomposed lithium self-diffusion coefficients $D^{\mathrm{Li}}_{\mathrm{self}}$ for PEO (solid bars) and PPM (dotted bars) versus sampling interval. $D^{\mathrm{anion}}_{\mathrm{self}}$ for TFSI is shown in grey bars. g, Comparison of effective $D^{\mathrm{Li}}_{\mathrm{self}}$ in PEO (left) and PPM (right). Black dashed lines indicate the overall $D^{\mathrm{Li}}_{\mathrm{self}}$ in each system, corresponding to the weighted average of the colored points.
• Figure 5: Mechanistic decomposition of lithium transport in liquid electrolytesa, Classification of transitions between coordination microstates for liquid electrolytes. If no constituent solvating molecules change, the transition is labeled as vehicular transport. If only a solvent molecule enters or leaves the solvation shell, the transition is classified as solvent exchange. If only an anion enters or leaves, it is classified as anion exchange. If both solvent and anion entities change within the time window, the transition is classified as a simultaneous solvent–anion exchange event. b, Comparison of decomposed lithium self-diffusion coefficients in EC with 1.0 M LiPF6 and in FAN with 1.3 M LiFSI. Black and gray dashed lines indicate the total self-diffusion coefficients of lithium and anions, respectively. c, Event probabilities as a function of sampling window $\Delta t$. Solid bars correspond to EC with LiPF6, and dotted bars to FAN with LiFSI. d, Effective self-diffusion coefficients for the different transition types in EC with LiPF6 and FAN with LiFSI as a function of $\Delta t$. The black dashed line shows the total lithium self-diffusion coefficient, which is the weighted average of the colored curves.