Population balance models and Monte Carlo simulation for nanoparticle formation in water-in-oil microemulsions: implications for CdS synthesis

We address controlled CdS nanoparticle formation by tuning experimental synthesis conditions. To this end, a bivariate population balance equation (PBE) model has been developed based on time scale analysis, to explain the mechanism of nanoparticle formation in self-assembled templates. It addresses the process of mixing two water-in-oil (w/o) microemulsions, each containing a pre-dissolved reactant in the microemulsion drops. Brownian collision and coalescence of two water drops of nanometer size results in mixing and exchange of reactant molecules, leading to chemical reaction. The water insoluble reaction product nucleates to form a nanoparticle in an individual drop, which subsequently grows internally by consuming the excess product and by coalescence-exchange with other drops. Finite rates of nucleation and coalescence-exchange are accounted for in the PBE, while the rates of reaction and internal growth of nanoparticles are found to be instantaneous. Experimentally proven binomial redistribution of reactant and product molecules upon drop coalescence is implemented in the present work. This results in a very good prediction of experimental data of the mean aggregate number (MAN) and hence size of CdS nanoparticles. Both our model and Monte Carlo (MC) simulation quantitatively capture the reported variation of MAN with molar excess of Cd2+ concentration and microemulsion drop size. Our results together with previous experimental data establish that usage of stoichiometrically five times or more of excess Cd2+ concentration can cause surface adsorption and desirable enhanced emission intensity of CdS nanoparticles, without altering particle size. We also propose a simplified and computationally efficient univariate PBE model. The univariate model gives very fast (in minutes) and accurate estimates (for low reactant concentrations) of the number and mean size of CdS nanoparticles. Time-scale analysis offers a good a priori choice of the appropriate model based on range of reactant concentrations.