Time dependent nucleation

Abstract

Continuum approximations to the discrete birth and death equations for classical nucleation are investigated. The discrete equations are parametrized by rate coefficients αi and βi for a cluster of size i to lose or gain a monomer, respectively. The continuum equations considered for the distribution function f(x,t) of clusters containing x monomers at time t are all of the form of a Fokker–Planck equation: ∂f/∂t=∂/∂x[Bf eq∂(f/f eq)/∂x], where f eq(x) is the equilibrium distribution and B(x) is a diffusion coefficient. The dependence of B(x) on various continuum approximations to the rate coefficients is discussed at length. Three different forms of B(x) are considered; that used by Frenkel [Kinetic Theory of Liquids (Oxford, Oxford, 1946)], that suggested by Goodrich [Proc. R. Soc. London Ser. A 371, 167 (1964)], and a third form proposed here. Steady state distributions and time lags obtained from the continuous and discrete equations are compared. The time-dependent Fokker–Planck equation is solved by an eigenfunction expansion and the eigenfunctions and eigenvalues of the Fokker–Planck operator are compared with those of the birth and death equations. A change of variables transforms the Fokker–Planck equation into a Schrödinger equation and permits the interpretation of the eigenvalues as energy levels in a potential function. Since the potential is approximately quadratic near its minimum, the lower eigenvalues are close to the harmonic oscillator results. The results show that the choice for B(x) suggested by Goodrich and the one proposed in this paper generally give better agreement with values from the discrete equations than the Frenkel form, used previously by most workers.