Instability in Ostwald ripening processes

TL;DR

This paper argues that Ostwald ripening in typical settings is governed by a large, stiff dimensionless parameter $\alpha$, which makes the supersaturation dynamics unstable and highly sensitive to counting fluctuations. By formulating dimensionless droplet-growth equations with variables $x$, $y$, and $\tilde{t}$ and analyzing the growth-rate parameter $\tilde{\nu}$, the authors show that fluctuations amplified by $\alpha$ lead to erratic $\nu(t)$ behavior especially when the effective fluctuation scale $\Omega=\alpha x/\sqrt{N}$ is large. Numerical simulations corroborate that, although the mean droplet size $\langle a\rangle$ often tracks Lifshitz-Slyozov predictions, $\nu(t)$ exhibits increasing fluctuations and the asymptotic distribution $p(y)$ deviates from LS, implying the long-time limit is not fully captured by LS theory. A reduced model in the $\alpha\to\infty$ limit confirms that random initial radii induce severe instability, whereas a lattice initialization can suppress counting fluctuations, underscoring the fundamental role of discreteness and fluctuations in Ostwald ripening. Overall, the work indicates that the traditional LS framework may be incomplete for large $\alpha$ and that long-time asymptotics in practical systems require incorporating stochastic and finite-size effects.

Abstract

There is a dimensionless parameter which enters into the equation for the evolution of supersaturation in Ostwald ripening processes. This parameter is typically a large number. Here it is argued that the consequent stiffness of the equation results in the evolution of the supersaturation being unstable. The instability is evident in numerical simulations of Ostwald ripening.

Figures (5)

• Figure 1: Numerical evaluation of $\nu(t)$, for different initial distributions $p_0(a)$ and different values of $\alpha$. In all cases there are initially $N=10^7$ droplets. The long-time simulations also show the running harmonic mean $\mu(t)$ (green), including data for a different random-number seed (blue). The variance of the running weight was $\Delta t=250$ (see equation (\ref{['eq: 3.3']})). The values of $\alpha$ and the choice of initial radius PDF are indicated above each plot: PDF 1, equation (\ref{['eq: 3.4']}) has a distribution of initial radii with an exponential tail, PDF 2, equation (\ref{['eq: 3.5']}) has a distribution of initial volumes with an exponential tail.
• Figure 2: Growth of $\langle a(t)\rangle$ compared with Lifshitz-Slyozov prediction. ( a) Radius distribution has exponential tail (equation (\ref{['eq: 3.4']})). ( b) Volume distribution has exponential tail (equation (\ref{['eq: 3.5']})).
• Figure 3: The same data sets as for figure \ref{['fig: 1']}( e), plotted on shorter intervals, showing fluctuations of $\nu(t)$, comparing them with fluctuations of $\langle y\rangle-1$ and survival probability $P_{\rm s}(t)$ (with the local average over the interval subtracted). ( a) Intermediate time: $t\in[800,825]$. ( b) Late stage: $t\in[8000,8025]$.
• Figure 4: Distribution of the scaled droplet size, $p(y)$, using the same data sets as for figure \ref{['fig: 1']}( e). The PDFs are compared with the Lifshitz-Slyozov distribution, (\ref{['eq: 3.6']}). The PDF was accumulated for two different intervals: ( a), $t\in[500,1000]$, ( b), $t\in [5000,10000]$.
• Figure 5: Evolution of $\nu(t)$ computed using equations (\ref{['eq: 4.5']}), (\ref{['eq: 4.6']}), compared with reference data from figure \ref{['fig: 1']}( e) (initial PDF 2, $\alpha=25$, blue). The simulation of (\ref{['eq: 4.5']}), (\ref{['eq: 4.6']}) with the same random initial (purple) radii show a pronounced scatter almost immediately. For simulations with smooth initial distribution, $\nu$ depends smoothly upon $t$, initially following the reference case quite closely (green).