A general expression for the statistical error in a diffusion coefficient obtained from a solid‐state molecular‐dynamics simulation

Abstract

Analysis of the mean squared displacement of species k, rk2, as a function of simulation time t constitutes a powerful method for extracting, from a molecular‐dynamics (MD) simulation, the tracer diffusion coefficient, Dk*. The statistical error in Dk* is seldom considered, and when it is done, the error is generally underestimated. In this study, we examined the statistics of rk2t curves generated by solid‐state diffusion by means of kinetic Monte Carlo sampling. Our results indicate that the statistical error in Dk* depends, in a strongly interrelated way, on the simulation time, the cell size, and the number of relevant point defects in the simulation cell. Reducing our results to one key quantity—the number of k particles that have jumped at least once—we derive a closed‐form expression for the relative uncertainty in Dk*. We confirm the accuracy of our expression through comparisons with self‐generated MD diffusion data. With the expression, we formulate a set of simple rules that encourage the efficient use of computational resources for MD simulations.

Oxygen grain-boundary diffusion in (La,Sr)FeO3−δ perovskite-oxides probed by molecular-dynamics simulations

“…The relative error in D* O was calculated from the expression of Usler et al 55 In order to compare our results with literature data, we calculated from a tracer diffusion coefficient D* O the oxygenvacancy diffusion coefficient D v , since our MD simulations were performed at constant oxygen deficiency, whereas experiment refers to constant oxygen partial pressure, and thus varying oxygen deficiency.…”

Oxygen Diffusion in Brownmillerite Sr2Fe2O5 is Two-Dimensional: Results from a Molecular Dynamics Study

“…For the BM phase, we therefore considered ⟨ r O 2 ⟩ = ⟨ x O 2 ⟩ + ⟨ z O 2 ⟩ and for the PV phase, ⟨ r O 2 ⟩ = ⟨ x O 2 ⟩ + ⟨ y O 2 ⟩ + ⟨ z O 2 ⟩. The corresponding relative error in D O * was obtained from the expression by Usler et al u normalr normale normall ( D normalO * ) = 2 d · N normalO · true[ 1 − exp ( − ⟨ r normalO 2 ⟩ normalf s O 2 ) true] where N O is the number of oxygen tracer particles, false⟨ r O 2 false⟩ f is the value of the mean-squared displacement at the end of a simulation run, and s O is the jump distance. Although this expression is strictly valid for only vacancy and interstitialcy mechanisms, we also used it for the combined interstitial–interstitialcy mechanism.…”

A comprehensive review on potential of diffusion length enhancement to upraise perovskite solar cell performance

“…Petersen et al minimized the errors in diffusion coefficient measurement using spherical cell immobilization metrices [78]. Recently, for statistical analysis of error in diffusion coefficient, User et al designed a general expression from solid-state molecular-dynamics simulation software [79]. The mean squared displacement in the tracer diffusion coefficient was obtained using kinetic Monte Carlo sampling.…”