Complete thermodynamically consistent kinetic model of particle nucleation and growth: Numerical study of the applicability of the classical theory of homogeneous nucleation

Abstract

A complete thermodynamically consistent elementary reaction kinetic model of particle nucleation and growth from supersaturated vapor was developed and numerically evaluated to determine the conditions for the steady-state regime. The model treats all processes recognized in the aerosol science (such as nucleation, condensation, evaporation, agglomerationcoagulation, etc.) as reversible elementary reactions. It includes all possible forward reactions (i.e., of monomers, dimers, trimers, etc.) together with the thermodynamically consistent reverse processes. The model is built based on the Kelvin approximation, and has two dimensionless parameters: S0-the initial supersaturation and Theta-the dimensionless surface tension. The time evolution of the size distribution function was obtained over the ranges of parameters S0 and Theta. At low initial supersaturations, S0, the steady state is established after a delay, and the steady-state distribution function corresponds to the predictions of the classical nucleation theory. At high initial supersaturations, the depletion of monomers due to condensation on large clusters starts before the establishing of the steady state. The steady state is never reached, and the classical nucleation theory is not applicable. The boundary that separates these two regimes in the two dimensionless parameter space, S0 and Theta, was determined. The model was applied to several experiments on water nucleation in an expansion chamber [J. Wolk and R. Strey, J. Phys. Chem. B 105, 11683 (2001)] and in Laval nozzle [Y. J. Kim et al., J. Phys. Chem. A 108, 4365 (2004)]. The conditions of the experiments performed using Laval nozzle (S0=40-120) were found to be close to the boundary of the non-steady-state regime. Additional calculations have shown that in the non-steady-state regime the nucleation rate is sensitive to the rate constants of the initial steps of the nucleation process, such as the monomer-monomer, monomer-dimer, etc., reactions. This conclusion is particularly important for nucleation from supersaturated water vapor, since these processes for water molecules at and below the atmospheric pressure are in the low pressure limit, and the rate constants can be several orders of magnitude lower than the gas kinetic. In addition, the impact of the thermodynamic inconsistency of the previously developed partially reversible kinetic numerical models was assessed. At typical experimental conditions for water nucleation, S0=10 and Theta=10 (T=250 K), the error in the particle nucleation rate introduced by the thermodynamic inconsistency exceeds one order of magnitude.

Comparison between solutions of the general dynamic equation and the kinetic equation for nucleation and droplet growth

“…Usually, the length of a bin is taken proportional to n, 5,19,20 leading to a constant number of bins per order of magnitude of the cluster radius r. Instead, we chose the bin lengths such that the bin edges are equidistant in r space, and the bin length is then approximately proportional to n 2/3 . Our choice is better suited to the problem we are solving, for two reasons.…”

“…This method is most often used to solve large systems of kinetic differential equations and is accurate if the number of bins is large enough. 5 When the division into bins is made, the average cluster concentration f in a bin depends only on the fluxes J n at the bin edges. These fluxes are estimated using a linear interpolation of average f between two adjacent bins.…”

“…However, the boundary condition of the continuous GDE was chosen such that it matched the solution of the discretecontinuous GDE. Chesnokov and Krasnoperov 5 recently compared an extensive kinetic model with a more limited kinetic model and did not make a comparison with a continuous GDE.…”

“…The kinetic model, based on this notion, consists of the kinetic master equations, which describe the concentration of clusters with discrete sizes. This model can be regarded as the most rigorous representation of a system with nucleation and has been in use 4,5 since the groundbreaking work of Courtney 6 and Abraham. 7 The GDE differs from the kinetic equation in three ways.…”

Atmospheric Cluster Dynamics Code: a flexible method for solution of the birth-death equations

“…where the monomer concentration is depleted during the course of the event (Kožíšek et al, 2004;Kožíšek and Demo, 2005;Kožíšek et al, 2006). Efforts have also been made to solve the BDE for various systems using approximate methods instead of explicit solution of the system of differential equations (see, for example, Rao and McMurry (1989); Girshick and Chiu (1990); Koutzenogii et al (1996); Chesnokov and Krasnoperov (2007) and references therein), in order to reduce the total number of equations to something manageable. In addition, some researchers have transformed the water/sulfuric acid and water/acid/ammonia system to a quasi-unary system, which also has the effect of reducing the total number of equations; these equations were then solved explicitly (Yu, 2005(Yu, , 2006Kazil and Lovejoy, 2007).…”