Modelling of CaCO<SUB>3</SUB> nanoparticle formation during overbasing of lubricating oil additives

A framework of population balance equations has been developed to model formation of CaCO3 nanoparticles during overbasing of lubricating oil. The process involves carbonation of a reverse micellar solution containing lime, present both in the micelles and as a suspension of lime particles in the oil. The mechanism leading to CaCO3 nanoparticles in this setup consists of a number of elementary events such as CO2 transport from gas to reverse micelles through the organic phase, reaction in the reverse micellar core, nucleation of CaCO3, particle growth, and Brownian collisions leading to material exchange, both among reverse micellar drops and between drops and lime particles. A time scale analysis of these steps permits simplification and enables us to divide the whole process into two stages. The first consists of reaction of existing lime in micelles and a burst of nucleation of very short duration, wherein some reverse micelles beget a single nucleus each. The number of such nucleated reverse micelles depends on the relative rates of mass transfer, nucleation, and growth by intermicellar Brownian collisions. This is followed by a slow growth phase of these initial particles through Brownian collisions between nucleated reverse micelles and lime particles. The model predicts the data of Roman et al. (J. Colloid Interface Sci. 1991, 144, 324.), where on average only 10 initial reverse micelles contribute to form a CaCO3 nanoparticle. A simplified version of the model, obtained in the limit of instantaneous gas transfer, is also able to approximately predict the results of Kandori et al. (J. Colloid Interface Sci. 1988, 122, 78.) where, in contrast, a huge number of 108 reverse micelles contribute to form one particle. The model is quite general and can be used for other gas-liquid micellar precipitation systems wherein similar relative orders of time scales are involved