(92a) Effect of Drop Size Distribution Shape Upon Dispersion Stability

A discretized population balance has been developed utilizing the moving pivot technique in order to understand effects of the droplet distribution shape upon the emulsion stability. Specifically, we examine the effects of distribution shape upon coalescence in simple shear flow with no inter-droplet interactions. Two facets of the distribution shape are considered: (1) the shape of the initial drop size distribution and (2) the shape of the underlying distribution as resolved by the measurement technique. The effect of the shape of the initial drop size distribution is explored by monitoring the short and long-time evolution of the drop size distribution for initial distribution shapes that include monodisperse, lognormal, bimodal, multidisperse, multimodal and step distributions. The effect of the drop size measurement technique upon the observed rate of coalescence is explored by examining the effects of drop size measurement range and drop size measurement resolution upon the ?observed' rate of coalescence and mean drop size. For the study of the initial distribution shape, simulations show that the rate of shear-induced coalescence is independent of average drop size, and depends only upon the shape of the distribution. Narrow distributions are characterized by faster rates of coalescence than broad distributions with monodisperse systems showing the fastest rate of coalescence. All other distributions show a decrease in the rate of coalescence with an increase in the width of the distribution. As the width of the distribution is increased, step distributions show up to a 25% decrease in the rate of coalescence and lognormal distributions show up to a 75% decrease in the rate of coalescence from the monodisperse condition. Other distributions show intermediate decreases in the rate of coalescence. These differences can be understood by examining the shape of the large-drop tail of the initial drop size distributions. Holding the width of each respective distribution constant, it is seen that distributions whose shapes contain a large fraction of droplets at large drop sizes, coalesce at faster rates than distributions with a smaller fraction of large droplets. Asymptotic theories in the long time limit predict that, regardless of the shape of the initial distribution, systems undergoing coalescence in simple shear approach time-independent, self-similar distributions. This asymptotic distribution is a monotonically decreasing function of droplet size, that shows an algebraic decease in the number of drops as drop size is increased at small and intermediate drop sizes, and an exponential decrease in the number of drops as drop size is increased at large drop sizes. In our simulations, only the monodisperse distribution shows a full approach to the asymptotic distribution at all drop sizes. All other distributions retain ?fingerprints' of the initial distribution at small drop sizes, even as the rest of the distribution approaches the long-time asymptotic behavior. Overall, the rate of coalescence is bounded by the initial rate of coalescence, which is a function of the shape of the initial distribution, and the final rate of coalescence, which is a function of the shape of the asymptotic distribution. Furthermore, since the rate of coalescence is a function of the shape of the drop size distribution, changes in the ?measured' shape of the underlying distribution due to measurement techniques can significantly alter the ?measured' coalescence behavior. Experimental techniques used to measure droplet size data such as nuclear magnetic resonance (NMR), phase doppler anemometry (PDA), and dynamic light scattering, are constrained by the range of drop sizes they can measure. Not surprisingly, the population balance results show that the average drop size is strongly under-predicted if the entire range of drop sizes is not captured. A much less intuitive result is that failure of the measurement technique to capture drops at the upper end of the size range can also seriously underestimate the coalescence rate. This is the case even when the majority of droplets are still within the measurement window, and can be understood in light of the results for the different initial distribution shapes above, in that the presence of large drops greatly increases the rate of coalescence. Measurement techniques also often assume the shape of the drop-size distribution a priori in order to measure drop sizes (NMR) or analyze the coalescence mechanism. This can have a significant impact on evolution of the ?measured' drop size versus the ?true' drop size. Similarly, improving the resolution with which the drop size distribution is measured can significantly alter the ?measured' rate of coalescence versus the ?true' rate of coalescence. For instance, narrowly distributed emulsions are often represented as monodisperse. This approximation can have a significant impact on the ?measured' versus the ?true' rate of coalescence, since the true polydisperse distribution coalesces at a slower rate than the idealized monodisperse condition. This difference in coalescence rate would lead to an underprediction of the coalescence efficiency (i.e. not all collisions between drops result in coalescence) for droplets represented by a monodisperse model. In conclusion, the shape of the drop size distribution can have a significant impact upon the rate of coalescence within emulsions. Experimental measurements of drop-size distributions are constrained by the degree of resolution with which drop sizes can be measured, the range of droplet sizes that can be measured and also often involve a priori assumptions of drop size distribution shape. These constraints should be recognized during the interpretation of results, since they can significantly affect the explanation of the underlying mechanism. These insights have potential to be combined with recent experimental advances in the development of emulsions with controlled initial distributions in order to elucidate potential coalescence mechanisms.