Nucleation and droplet growth as a stochastic process (original) (raw)

Droplet nucleation and Smoluchowski’s equation with growth and injection of particles

Physical Review E, 1998

We show that models for homogeneous and heterogeneous nucleation of D-dimensional droplets in a d-dimensional medium are described in mean-field by a modified Smoluchowski equation for the distribution N (s, t) of droplets masses s, with additional terms accounting for exogenous growth from vapor absorption, and injection of small droplets when the model allows renucleation. The corresponding collision kernel is derived in both cases. For a generic collision kernel K, the equation describes a clustering process with clusters of mass s growing between collision withṡ ∝ s β , and injection of monomers at a rate I. General properties of this equation are studied. The gel criterion is determined. Without injection, exact solutions are found with a constant kernel, exhibiting unusual scaling behavior. For a general kernel, under the scaling assumption N (s, t) ∼ Y (t) −1 f (s/S(t)), we determine the asymptotics of S(t) and Y (t), and derive the scaling equation. Depending on β and K, a great diversity of behaviors is found. For constant injection, there is an asymptotic steady state with N (s, t = ∞) ∝ s −τ and τ is determined. The case of a constant mass injection rate is related to homogeneous nucleation and is studied. Finally, we show how these results shed some new light on heterogeneous nucleation with d = D. For d = D = 2 (discs on a plane), numerical simulations are performed, in good agreement with the mean-field results.

Diffusion-limited droplet coalescence

Physica A: Statistical Mechanics and its Applications, 1990

Simulations of diffusion limited (Brownian) droplet coalescence have been carried out in which the droplet diffusion coefficients are related to their sizes (masses), s, by @(s) -s ~. For the case D = d where D is the droplet dimensionality and d is the dimensionality of the substrate, the exponents z and z' describing the algebraic growth of the mean droplet size, S, and the decrease in the number of droplets, N, are given by z = z' = 1/(2/d -3,) with no logarithmic corrections for d =2. If D > d, then for d =2 the growth of S is given by

Kinetics of droplet growth processes: Simulations, theory, and experiments

1989

The formation of a distribution of various size droplets is a characteristic feature of many systems from thin films and breath figures to fog and clouds. In this paper we present the results of our investigations of the kinetics of droplet growth and coalescence. In general, droplet formation occurs either by spontaneous nucleation or by growth from heterogeneously distributed nucleation centers, such as impurities. We have introduced two models to describe these two types of processes. In the homogeneous nucleation model droplets can form and grow anywhere in the system. The results of the simulations of the model are presented and it is shown that the droplet size distribution has a bimodal structure consisting of a monodispersed distribution of large droplets superimposed on a polydispersed distribution of smaller droplets. A scaling description for the evolution of the timedependent droplet size distribution and its moments is presented and it is found that the scaling predictions are in excellent agreement with the simulations. A rate-equation similar to the Smoluchowski equation is also introduced for describing the kinetics of homogeneous droplet growth. The results of the simulations of the homogeneous nucleation model are also compared with the experiments on droplet growth in thin films obtained by vapor deposition of tin on sapphire substrate. It appears that this model captures the essential features of the distribution of droplets in the vapor deposition experiments. We also introduce a heterogeneous nucleation model for studying processes in which droplets only form and grow at certain nucleation centers which are initially chosen at random. Simulations, scaling theory, and a kinetic equation approach for describing the heterogeneously nucleated droplet growth model are also presented. The theoretical predictions are found to be in excellent agreement with the simulations.

Corrections to a mean number of droplets in nucleation

2008

Corrections to a mean number of droplets appeared in the process of nucleation have been analyzed. The two stage model with a fixed boundary can not lead to a write result. The multi stage generalization of this model also can not give essential changes to the two stage model. The role of several first droplets have been investigated and it is shown that an account of only first droplet with further appearance in frame of the theory based on the averaged characteristics can lead to a suitable results. Both decay of metastable phase and smooth variations of external conditions have been investigated. 1

Numerical simulation of diffusion-controlled droplet growth: Dynamical correlation effects

Diffusion-controlled coarsening (Ostwald ripening) of precipitated solutions is studied by numerical simulation. An algorithm is devised which exploits the screening of solute concentration fields, thereby removing the restriction to small systems of previous work. Simulation of the coarsening of 5000 droplets at 10'k volume fraction reveals long-ranged dynamical correlations which broaden the droplet size-distribution function and increase the coarsening-rate coefficient.

Dispersion of nucleation under the smooth variation of external conditions

2004

A simple method to calculate dispersion of the total number of droplets appeared in the process of nucleation caused by the smooth variation of external conditions has been presented. The analytical result for dispersion is compared with results of numerical simulations and the coincidence has been observed. The role of stochastic appearance of several first droplets in formation of dispersion has been analyzed.