Constant number Monte Carlo simulation of population balances with multiple growth mechanisms

Robust metric for quantifying the importance of stochastic effects on nanoparticle growth

Comprehensive representation of nanoparticle dynamics is necessary for understanding nucleation and growth phenomena. This is critical in atmospheric physics, as airborne particles formed from vapors have significant but highly uncertain effects on climate. While the vapor–particle mass exchange driving particle growth can be described by a macroscopic, continuous substance for large enough particles, the growth dynamics of the smallest nanoparticles involve stochastic fluctuations in particle size due to discrete molecular collision and decay processes. To date, there have been no generalizable methods for quantifying the particle size regime where the discrete effects become negligible and condensation models can be applied. By discrete simulations of sub-10 nm particle populations, we demonstrate the importance of stochastic effects in the nanometer size range. We derive a novel, theory-based, simple and robust metric for identifying the exact sizes where these effects cannot be omitted for arbitrary molecular systems. The presented metric, based on examining the second- and first-order derivatives of the particle size distribution function, is directly applicable to experimental size distribution data. This tool enables quantifying the onset of condensational growth without prior information on the properties of the vapors and particles, thus allowing robust experimental resolving of nanoparticle formation physics.

Taylor series expansion scheme applied for solving population balance equation

Population balance equations (PBE) are widely applied to describe many physicochemical processes such as nanoparticle synthesis, chemical processes for particulates, colloid gel, aerosol dynamics, and disease progression. The numerical study for solving the PBE, i.e. population balance modeling, is undergoing rapid development. In this review, the application of the Taylor series expansion scheme in solving the PBE was discussed. The theories, implement criteria, and applications are presented here in a universal form for ease of use. The aforementioned method is mathematically economical and applicable to the combination of fine-particle physicochemical processes and can be used to numerically and pseudo-analytically describe the time evolution of statistical parameters governed by the PBE. This article summarizes the principal details of the method and discusses its application to engineering problems. Four key issues relevant to this method, namely, the optimization of type of moment sequence, selection of Taylor series expansion point, optimization of an order of Taylor series expansion, and selection of terms for Taylor series expansion, are emphasized. The possible direction for the development of this method and its advantages and shortcomings are also discussed.

Optimality and adaptation of phenotypically switching cells in fluctuating environments

Stochastic switching between alternative phenotypic states is a common cellular survival strategy during unforeseen environmental fluctuations. Cells can switch between different subpopulations that proliferate at different rates in different environments. Optimal population growth is typically assumed to occur when phenotypic switching rates match environmental switching rates. However, it is not well understood how this optimum behaves as a function of the growth rates of phenotypically different cells. In this study, we use mathematical and computational models to test how the actual parameters associated with optimal population growth differ from those assumed to be optimal. We find that the predicted optimum is practically always valid if the environmental durations are long. However, the regime of validity narrows as environmental durations shorten, especially if subpopulation growth rate differences differ from each other (are asymmetric) in two environments. Furthermore, we study the fate of mutants with switching rates previously predicted to be optimal. We find that mutants which match their phenotypic switching rates with the environmental ones can only sweep the population if the assumed optimum is valid, but not otherwise.