Droplets nucleation, and Smoluchowski's equation with growth and injection of particles

Stephane Cueille; Clement Sire

doi:10.1103/PhysRevE.57.881

paper

Droplets nucleation, and Smoluchowski's equation with growth and injection of particles

arXiv:cond-mat/9706208 · doi:10.1103/PhysRevE.57.881

Abstract

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 with ds/dt=As^β $, an d inj ec t i o n o f m o n o m er s a t a r a t e I (t) . G e n er a l p r o p er t i eso f t hi se q u a t i o na r es t u d i e d . T h e g e l cr i t er i o ni s d e t er min e d . W i t h o u t inj ec t i o n, e x a c t so l u t i o n s a r e f o u n d w i t ha co n s t an t k er n e l, e x hibi t in g u n u s u a l sc a l in g b e ha v i or . F or a g e n er a l k er n e l, u n d er t h esc a l in g a ss u m pt i o n N (s, t) \sim Y (t)^{- 1} f (s / S (t)), w e d e t er min e t h e a sy m pt o t i cso f S (t) an d Y (t), an dd er i v e t h esc a l in g e q u a t i o n . D e p e n d in g o n β an d K, a g r e a t d i v er s i t y o f b e ha v i or s i s f o u n d . F or co n s t an t inj ec t i o n, t h er e i s ana sy m pt o t i cs t e a d y s t a t e w i t h N (s, t = \infty) \propto s^{- τ} an d τ$ 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.

Revtex, 25 pages (multicol.sty), 13 eps figures (uses epsfig)

Droplets nucleation, and Smoluchowski's equation with growth and injection of particles

Abstract

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