Stochastic approaches: modeling the probability of encounters between H$_{2}$-molecules and metallic atomic clusters in a cubic box

TL;DR

This paper develops a stochastic model to calculate the encounter probability of hydrogen molecules with metallic clusters in a confined space, providing an analytical expression validated against simulations across various conditions.

Contribution

It introduces an original analytical approach based on mathematical rebounds to precisely determine encounter probabilities in a stochastic system involving metallic clusters and hydrogen molecules.

Findings

Encounter probability depends strongly on the number of molecules and the box size.

Temperature has negligible effect on encounter probability.

The analytical model aligns well with distribution-based simulations.

Abstract

In recent years the advance of chemical synthesis has made it possible to obtain \textquotedblleft naked\textquotedblright clusters of different transition metals. It is well known that cluster experiments allow studying the fundamental reactive behavior of catalytic materials in an environment that avoids the complications present in extended solid-phase research. In physicochemical terms, the question that arises is the chemical reduction of metallic clusters could be affected by the presence of H$_{_{2}}$ molecules, that is, by the probability of encounter that these small metal atomic agglomerates can have with these reducing species. Therefore, we consider the stochastic movement of $N$ molecules of hydrogen in a cubic box containing $M$ metallic atomic clusters in a confined region of the box. We use a Wiener process to simulate the stochastic process, with $\sigma$ given by the Maxwell-Boltzmann relationships, which enabled us to obtain an analytical expression for the probability density function. This expression is an exact expression, obtained under an original proposal outlined in this work, \textit{i.e.} obtained from considerations of \textit{mathematical rebounds}. On this basis, we obtained the probability of encounter for three different volumes, $0.1^{^{3}}$, $ 0.2^{^{3}} $ and $ 0.4^{^{3}} $ m$^{^{3}}$, at three different temperatures in each case, $293$, $373$ and $473$ K, for $ 10^{^{1}}\leq N \leq10^{^{10}}$, comparing the results with those obtained considering the distribution of the position as a Truncated Normal Distribution. Finally, we observe that the probability is significantly affected by the number $N$ of molecules and by the size of the box, not by the temperature.