Substrate-Mediated Evaporation and Stochastic Evolution of Supported Au Nanoparticles

Abstract

We use in situ transmission electron microscopy with automated tracking to study supported gold nanoparticles (NPs) during high-temperature vacuum annealing. \rev{The average mass loss per NP is governed by a flat, nearly size-independent substrate-mediated evaporation profile.} On top of \rev{this mean shrinkage}, individual NPs show significant fluctuations in apparent growth or shrinkage, and NP volume follows a \rev{random-walk-like trajectory. To rationalize both the ensemble-mean behavior and the particle-resolved variability, we develop a self-consistent theory that couples substrate-mediated evaporation to collective 2D Ostwald-type mass exchange through a shared adatom field, described in terms of a renormalized screening length and background concentration. In the experimentally relevant regime, the theory predicts an approximately size-independent mean shrinkage rate and clarifies how net mass loss suppresses classical coarsening.} \rev{Superimposed on this deterministic drift, we quantify stochastic volume trajectories and capture their fluctuation spectrum with a minimal Langevin description consistent with intermittent adatom attachment and detachment events.} In addition, we characterize the lateral diffusive motion of NPs, which is responsible for their coalescence. Altogether, our results highlight that stochasticity is intrinsic at the nanoscale \rev{and that predicting the evolution of supported NPs at early and intermediate times requires a unified framework combining substrate-mediated evaporation, collective mass exchange, and stochastic fluctuations.

Figures (10)

• Figure 1: (a) Bright-field TEM image of Au nanoparticles supported on an amorphous Si$_3$N$_4$ membrane during annealing at $950\,^{\circ}\mathrm{C}$ in the microscope column. (b) Binary image generated from (a) by the automatic pipeline used for quantification. Extracted particle footprints (white) are tracked frame-to-frame for size and position statistics.
• Figure 2: Volume-conserving coalescence. Evolution of three neighboring Au NPs undergoing two consecutive merger events. The total footprint area, $S_1+S_2+S_3$, is not conserved. In contrast, the total volume proxy parameter, $S_1^{3/2}+S_2^{3/2}+S_3^{3/2}$, stays approximately constant.$S$ and $S^{3/2}$ are normalized so that both sum to $1$ at $t=7$ min .
• Figure 3: Ensemble evolution during annealing.(a) Total NP count $N(t)$ versus time $t$ shows a monotonic decrease due to coalescence and evaporation. (b) Normalized total volume proxy $\sum_i S_i^{3/2}(t)/N_{\max}$ (green squares) and ensemble average volume proxy per particle, $\langle S^{3/2}\rangle(t)=N^{-1}(t)\sum_i S_i^{3/2}(t)$ (red circles). While $N(t)$ decreases, $\langle S^{3/2}\rangle$ increases, which is consistent with net coarsening. The total volume proxy, $\sum_i S_i^{3/2}$ shows a gradual decrease attributable to sublimation.
• Figure 4: Bimodal distribution of NP size increments. a) Scatter plot of the volume proxy change, $\Delta S^{3/2}$, versus $S^{3/2}$. Each point corresponds to a NP size change over a time interval $\Delta t=10$ min. Two distinct clusters emerge: large positive $\Delta S^{3/2}$ values indicate coalescence events (black crosses), while a narrow band in the vicinity of zero reflects sublimation-dominated size change (red dots). The solid line shows the mean sublimation rate of binned data as a function of $S^{3/2}$, with whiskers representing the standard deviation within each bin. b) Correlation between the number of NPs exhibiting large size increase ($\Delta S^{3/2}>50 nm^3$) (blue circles), and the decrease of the total particle count (orange diamonds). Both changes are taken between the two consecutive measurements. This supports our interpretation of large size increases as coalescence events.
• Figure 5: Schematic of substrate-mediated evaporation. Supported NP of radius $R$ on a substrate with adatom diffusion length $\xi$ and interparticle spacing $L$. When $L\gtrsim \xi$, particles behave quasi-independently; evaporation includes (i) direct sublimation from the NP surface and (ii) desorption of adatoms generated within a substrate ring of order $\xi$ around each NP. The latter yields an approximately size-independent contribution to the per-particle mass loss.