On the Fokker-Planck approximation in the kinetic equation of multicomponent classical nucleation theory

TL;DR

This paper investigates the conditions under which the Fokker-Planck approximation accurately describes multicomponent nucleation kinetics, highlighting when higher-order effects must be considered for precise modeling.

Contribution

It provides a detailed analysis of the parameters affecting the validity of the Fokker-Planck approximation in multicomponent nucleation and suggests necessary modifications for accurate predictions.

Findings

The Fokker-Planck approximation is valid only under certain smallness conditions of parameters.

Higher-order derivatives or nonlinear force coefficients emerge when parameters are not small.

Numerical results indicate the conventional nucleation rate expression may need modification for binary systems.

Abstract

We examine the validity of the Fokker-Planck equation with linear force coefficients as an approximation to the kinetic equation of nucleation in homogeneous isothermal multicomponent condensation. Starting with a discrete equation of balance governing the temporal evolution of the distribution function of an ensemble of multicomponent droplets and reducing it (by means of Taylor series expansions) to the differential form in the vicinity of the saddle point of the free energy surface, we have identified the parameters whereof the smallness is necessary for the resulting kinetic equation to have the form of the Fokker-Planck equation with linear (in droplet variables) force coefficients. The "non-smallness" of these parameters results either in the appearance of the third or higher order partial derivatives of the distribution function in the kinetic equation or in its force coefficients becoming non-linear functions of droplet variables, or both; this would render the conventional kinetic equation of multicomponent nucleation and its predictions inaccurate. As a numerical illustration, we carried out calculations for isothermal condensation in five binary systems of various non-ideality at T=293.15 K: 1-butanol--1-hexanol, water--methanol, water--ethanol, water--1-propanol, water--1-butanol. Our results suggest that under typical experimental conditions the kinetic equation of binary nucleation of classical nucleation theory may require a two-fold modification and, hence, the conventional expression for the steady-state binary nucleation rate may not be adequate for the consistent comparison of theoretical predictions with experimental data.