Graph analysis of proton conduction pathways in scandium-doped barium zirconate

Revised Centrality Measures Tell a Robust Story of Ion Conduction in Solids

“…At trapped sites, ions never go far from the initial site, while on periodic highways, ions are moving at high speeds and thus return to the original sites through periodic boundary conditions quickly. Hence, return-flow centrality highlights both traps and long-range conduction highways, as seen in previous works. , …”

“…Periodic pathways can be linked together to create long-range conduction pathways, which are characteristic of real system conduction if the simulation cell is sufficiently large . Real physical system graphs are much larger and KMC motion through the graph exhibits significant random diffusion through sites, as seen in our earlier work. , To consider conducting an ion–ion correlation or how one conduction ion influences the pathway of another, the edges to the occupied vertex are effectively removed from the graph, as seen in Figure b. Both scenarios depicted in Figure will be considered.…”

“…Flow-through centrality will be calculated with two path sets: (1) all possible periodic paths of length N , which traverse the full simulation box without interior cycles (simplified long-range conduction paths) and (2) all possible paths. Our earlier work , has used a dynamic programming approach for finding the set (1) and sorting paths by probability. Flow-through centrality for site i for these long-range periodic paths is defined here as the sum over the probabilities of all of the paths going through site i .…”

“…Trajectories of ion conduction in solids may include many conducting ion diffusion steps near traps and a few escapes to fast conduction regions intermingled with smaller movements of other ions. One time-based centrality measure has been shown to highlight traps and conduction highways in a single picture, giving an overview of the ion conduction landscape. , However, network theory has developed many centrality measures based on number of steps to highlight different network features. − This paper shows how the three most common types of step-based centrality measures can be converted to time-based centrality measures to probe significant features of ion conduction in solids and their sensitivity to noisy data. Taken as viewpoints of a whole, these measures provide a more robust story of ion conduction, both for single-ion motion in an ionic solid and for ion–ion correlated motion.…”

“…An edge connects two vertices if the minima can be connected via a single transition state. Site probabilities indicate the likelihood of finding the system in the particular minimum or binding site structure, while the rate constant for transition between sites ( k i , j ) characterizes the rate of motion between sites. ,,− Figure a shows a sample conduction graph for an ergodic system, where all vertices are available and accessible from any other vertex. To characterize large systems without unusual edges, these graphs need to include the same periodic boundary conditions common to large system simulations .…”