A New Analytical Solution for Solving the Smoluchowski Equation Due to Nanoparticle Brownian Coagulation for Non-Self-Preserving System

Mingzhou Yu,Martin Seipenbusch,Hanhui Jin,Jinghua Yang

doi:10.4209/aaqr.2014.03.0059

Abstract

The Smoluchowski equation has become a fundamental equation in nanoparticle processes since it was proposed in 1917, whereas the achievement of its analytical solution remains a challenging issue. In this work, a new analytical solution, which is absolutely different from the conventional asymptotic solutions, is first proposed and verified for nonself-preserving nanoparticle systems in the free molecular regime. The Smoluchowski equation is first converted to the form of moment ordinary differential equations by the performance of Taylor expansion method of moments and subsequently resolved by the separate variable technique. In the derivative, a novel variable, g = m0m2/m1 2 , where m0, m1 and m2 are the first three moments, is first revealed which can be treated as constant. Three specific models are proposed, two with a constant g (an Analytical Model with Constant g (AMC), and a Modified Analytical Model with Constant g (MAMC)), and another with varying g (a finite Analytical Model with Varying g (AMV)). The AMC model yields significant errors, while its modified version, i.e., the MAMC model, is able to produce highly reliable results. The AMV is verified to have the capability to solve the Smoluchowski equation with the same precision as the numerical method, but an iterative procedure has to be employed in the calculation.

Highlights

Brownian coagulation is regarded as the most important inter-particle mechanism modifying the size distribution of particles in processes involving nanoparticle synthesis, polymerization, aerosol, emulsification and flocculation (Fox, 2008; Anand et al, 2012; Buesser and Pratsinis, 2012; Menz et al, 2014)
In the free molecular regime, the successful achievement of an analytical solution for the Smoluchowski equation (SE) is attributed to the novel dynamical property of particles that the geometric standard deviation (GSD) of the size distribution can be treated as a constant (Lee et al, 1984; Vemury et al, 1994; Friedlander and Wang, 1966)
The newly proposed analytical solution is absolutely different from the existing asymptotic solutions, and it is expected to extend beyond the asymptotic solution in its scope of application

Summary

INTRODUCTION

Brownian coagulation is regarded as the most important inter-particle mechanism modifying the size distribution of particles in processes involving nanoparticle synthesis, polymerization, aerosol, emulsification and flocculation (Fox, 2008; Anand et al, 2012; Buesser and Pratsinis, 2012; Menz et al, 2014). Lee and Chen (1984), Lee et al (1984) and Park et al (1999) found that it is feasible to assume the size distribution to follow the lognormal distribution, and based on this assumption, his group successfully developed a series of analytical models for solving SE due to Brownian coagulation at different size regimes As they acknowledged in their works, the formulas for obtaining total particle number concentration, geometric standard deviation or geometric mean diameter have to be coupled and dependent of each other, iterative techniques have to solve the coupled equations. It needs to emphasize the TEMOM ODEs with three order Taylor expansion series is the best selection for deriving the analytical solution due to its both simple mathematical structure and high precision, which has been discussed in (Yu et al, 2008).

AND DISCUSSION

CONCLUSIONS