Modeling of Coalescence Cascades in Liquid-Liquid dispersions

Abstract

Agitated dispersions of one liquid in another immiscible liquid invert at a critical value of dispersed phase fraction, nearly catastrophically. The dispersed phase becomes continuous and vice versa. This phenomenon has been studied over the past several decades, but a mechanistic understanding of this phenomenon still eludes us. Population balance-based approaches with realistic kernels for first-order drop breakup and second-order drop coalescence processes fail to predict the onset of an inversion-like instability. If during the process of breakup or coalescence of drops, the new drop being born comes in contact with another drop due to the volume constraints, the two can coalesce. The newly born drop can come in contact with yet another drop and follow the same process, giving rise to what we call a coalescence cascade. The present work develops two approaches to capture the effect of coalescence cascades on the dynamics of dispersions—Monte Carlo simulations (MC) and mean-field models. MC simulations of an agitated liquid–liquid dispersion with breakup and random movement of drops as stochastic processes capture the increasingly dominant role of coalescence cascades as the dispersed phase fraction is increased. The simulations predict the steady-state average drop size to be independent of the size of the simulation box at dispersed phase fractions below a critical value. At larger values, the average drop size increases unboundedly with an increase in system size. It leads to the prediction that at the critical value, the average drop size jumps from a finite value to infinity for system size approaching infinity. This instability is taken to mimic phase inversion. The mechanism responsible for this behavior is the inability of the drop breakup process to decrease the average drop size at critical dispersed phase fraction, as all the newly born daughter drops come in collisional contact with neighboring drops. The model captures several hitherto unexplained experimental findings. The same MC model predicts the existence of a new kinetic phase transition for pure breakup of sticky particles (which coalesce with other particles on contact). Below a critical value of dispersed phase fraction, sticky particles continue to break indefinitely. At dispersed phase fractions larger than the critical value, breakup of sticky particles leads to a steady-state size distribution comprising a small population of drops, independent of the system size taken. A new simple power-law scaling of critical volume fraction with dimensionality of space is revealed. A new population balance-based mean-field description of concentrated dispersions in which coalescence cascades of order up to infinity can be triggered by first-order breakup and second-order coalescence of drops is developed in this work. The new mean-field model successfully captures all the features brought out by the MC simulations for the two cases discussed above. It captures the system size-independent and system size-dependent behavior of the model system at dispersed phase fraction smaller and larger than the critical value, respectively. The mean-field description also predicts phase inversion-like transition at a critical value of dispersed phase fraction, and the new scaling observed for the variation of critical volume fraction with the dimensionality of space for the breakup of sticky particles. The mean-field model captures the sub-critical and the super-critical dynamics of the model system very well. The new mean-field description was next tested for coalescence of drops when they move relative to each other in a defined manner as in (i) shear flow and (ii) buoyancy-induced relative motion among drops of different sizes. The predictions of the mean-field model with coalescence cascade are verified by event-driven simulations (analogous to molecular dynamics-type simulations). The predictions of the event-driven simulations and mean-field model, which are quite different from the predictions of the classical mean-field theory which does not account for coalescence cascades in it, are in good agreement. The present work also shows that the addition of commercial-grade silica particles in size range of tens of microns at extremely small concentrations (as small as 0.05% by volume) can destabilize agitated liquid–liquid dispersions. The dispersed phase fraction at which w/o dispersions for toluene–water system inverts can be reduced from 0.54 to 0.40 by the addition of trace amounts of hydrophilic silica particles. The dispersed phase fraction for inversion of the o/w dispersion for the same system is decreased from 0.66 to 0.50 by the addition of the same silica particles after their silanation. High-speed conductivity measurements of agitated water-in-oil dispersions with and without the presence of hydrophilic particles reveal that at extremely low surface coverage itself, particles promote coalescence of drops substantially, which leads to the formation of large connected domains of water, precursor to phase inversion, at a substantially reduced dispersed phase fraction.