The bulk pre-polymerization (first of the three main stages of manufacturing process) is carried out with intense agitation, producing free PS and a graft copolymer (PS-g-PB). The reacting system is homogeneous only at very low conversion, since the incompatibility between the PS and the PB chains forces it to undergo a phase separation mechanism, by which a dispersed, PS-rich phase is formed at the bulk of a PB-rich continuous phase. St monomer is almost evenly distributed between both phases3. As the polymerization proceeds, more PS is produced and the dispersed phase eventually becomes the continuous phase, through a phase inversion (PI) process. The desired morphology is developed at this crucial stage, characterized by a sudden drop of the mixture’s apparent viscosity4.
The PI process is affected by several variables, such as phase viscosity ratio, phase volume ratio, rubber cis/trans content, stirring speed, grafting efficiency, reaction temperature, solvent content, PS and PB molecular weights, etc5. Given that the strong point of HIPS is its enhanced mechanical properties, and that these are the result of the in-situ morphology development during the PI stage, then the understating of this phenomenon and of the relative effect of each operating variable becomes a full chemical engineering challenge. The optimization of the polymerization recipes that provide desired material properties may be achieved by fully understanding the PI phenomenon and correctly predicting the morphology therein developed. This holds a significant interest both from academic and industrial standpoints.
Currently, very few mathematical models have been developed to describe morphology formation by continuous phase entrapment, even in non-reacting systems. Therefore, and with the motivation discussed previously, this work presents a population balance model (PBM) capable of predicting the size distribution of droplets in an agitated dispersion where continuous phase entrapment may occur. The proposed framework extends classical PBM formulations by incorporating a new source term accounting for droplet morphology evolution due to phase entrapment, thus generating “drop-in-drop” structures.
Population balance modeling has been extensively used to describe the evolution of dispersed phases in multiphase systems, primarily through birth and death mechanisms associated with breakage and coalescence6,7. However, in systems with high surfactant concentrations and very low interfacial tension, such as multiple emulsions, coalescence events can trap droplets of the continuous phase inside the dispersed phase. This results in a hierarchical size distribution -droplets within droplets- complicating predictive modeling efforts. Traditional PBM formulations do not account for this phenomenon, leading to discrepancies between predicted and experimentally observed morphologies.
To address this gap, we introduce an extended PBM incorporating additional internal states representing the number and average size of occluded droplets. If one considers a maximum number of N occlusions, then this yields a system of N+1 partial differential equations that must be solved.
Breakage probabilities are computed via direct numerical simulation, capturing the stochastic nature of inner droplet rupture, assuming spherical geometry. Coalescence terms are modified to include morphological constraints governing phase entrapment, employing a stability criterion based on film drainage dynamics. Continuous phase entrapment occurs when the dimple formed during coalescence cannot drain before rim closure, leading to a new occluded morphology8. The critical droplet radius beyond which entrapment is likely, is derived from hydrodynamic film drainage models9.
The model is solved using the Higher-Order Moment-Conserving Method of Classes (HMMC) method, an advanced reconstruction technique preserving key moments of the number density function10. The solution domain is discretized into a two-dimensional grid spanning droplet size and occlusion size. Discretized breakage and coalescence kernels are integrated into a moment-conserving framework, ensuring accurate reconstruction of the morphology distribution.
Simulations of an agitated oil-in-water emulsion system show that continuous phase entrapment significantly affects the droplet size distribution. Compared to classical PBM predictions, our extended model captures the emergence of multimodal distributions corresponding to occluded morphologies. The framework successfully predicts experimentally observed trends in complex emulsification processes11.
This work presents a novel PBM framework that explicitly incorporates continuous phase entrapment, improving morphology predictions in dispersed systems. Future extensions will focus on experimental validation and the incorporation of additional physical effects such as surfactant dynamics and rheological constraints. This approach has broad applications in materials science, in particular for HIPS industrial process, where controlled particle morphology is essential.
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