Simulated surface diffusion in nanoporous gold and its dependence on surface curvature

TL;DR

This work probes how surface diffusion driving nanoporous gold coarsening depends on surface curvature by performing molecular dynamics on constant-mean-curvature Au surfaces and fitting displacement distributions to extract $D_S$, $D_B$, and $n_S$. Using five curved geometries and four crystal orientations with a MEAM potential, the authors quantify Arrhenius behaviors and activation energies, finding a Curvature-weak dependence for $D_S$ ( activation ~0.74 eV) and a weak curvature impact on $n_S$ (via $E_F\approx0.27$ eV), resulting in a nearly curvature-independent mass-transport rate with total activation ~1.01 eV. They argue that finite lifetimes of mobile surface atoms—exchange with bulk—produce the observed curvature effects and could bias activation energy estimates by ~0.27 eV if not accounted for. The findings provide a more robust basis for predicting np-Au coarsening and emphasize the importance of mobile-atom lifetimes in atomistic diffusion analyses. Overall, the study reconciles curvature-related observations with diffusion theory and offers guidance for interpreting MD-based diffusion measurements in nanoporous metals.

Abstract

The morphological evolution of nanoporous gold is generally believed to be governed by surface diffusion. This work specifically explores the dependence of mass transport by surface diffusion on the curvature of a gold surface. The surface diffusivity is estimated by molecular dynamics simulations for a variety of surfaces of constant mean curvature, eliminating any chemical potential gradients and allowing the possible dependence of the surface diffusivity on mean curvature to be isolated. The apparent surface diffusivity is found to have an activation energy of ~0.74 eV with a weak dependence on curvature, but is consistent with the values reported in the literature. The apparent concentration of mobile surface atoms is found to be highly variable, having an Arrhenius dependence on temperature with an activation energy that also has a weak curvature dependence. These activation energies depend on curvature in such a way that the rate of mass transport by surface diffusion is nearly independent of curvature, but with a higher activation energy of ~1.01 eV. The curvature dependencies of the apparent surface diffusivity and concentration of mobile surface atoms is believed to be related to the expected lifetime of a mobile surface atom, and has the practical consequence that a simulation study that does not account for this finite lifetime could underestimate the activation energy for mass transport via surface diffusion by ~0.27 eV.

Figures (12)

• Figure 1: The simulation cell geometries used to extract surface diffusivity on concave ($\kappa_1<0$) and convex ($\kappa_1>0$) surfaces. $\kappa_2$ is along $\hat{\boldsymbol{z}}$ and $\kappa_1$ along $\hat{\boldsymbol{\theta}}$. The flat plate used as a zero mean curvature surface is not shown.
• Figure 2: The four different crystal orientations used in our simulations. The respective crystallographic directions were oriented along the $x$-, $y$-, and $z$-axes of the simulation cell (the cylinder axis was always along the $z$-axis). For example, the [$211$] [$\bar{1}42$] [$\bar{2}13$] orientation aligns the crystallographic direction [$211$] with the $x$-axis of the simulation cell, [$\bar{1}42$] with the $y$-axis, and [$\bar{2}13$] with the $z$-axis.
• Figure 3: Distributions of atomic displacements in the $z$ direction after $10\ \mathrm{ns}$ at $900\ \mathrm{K}$ for cylinders of radius $34\ \text{\AA}$; the vertical red lines indicate the interatomic distance. Displacements on a cylinder with [$001$] along the $z$ direction (top) generally occur at integer multiples of a lattice translation vector. Combining the displacement data from simulations using the four orientations shown in Fig. \ref{['fig:orientations']} reduces the strength of these peaks.
• Figure 4: The $z$ components (top) and magnitudes (bottom) of atomic displacements on cylinders of radius $30\ \text{\AA}$ during a $10$ ns simulation at $900$ K. The displacement magnitude and the number of displaced atoms visibly depends on the crystallographic orientation (left and right).
• Figure 5: Atomic displacements during $1$ ns as simulated using the EAM potential (left) or the MEAM potential (right). Both simulations started with identical atomic coordinates for a cylinder of radius $30\ \text{\AA}$ and a [$1\bar{7}4$] [$\bar{1}12$] [$311$] orientation and were relaxed for $10$ ns at $1000$ K.