Employing global optimization techniques to establish an empirical model for ultrasound emulsification: Predicting the droplet size of various O/ W-Emulsions

Abstract

Current theoretical correlations, based on Hinze's and Taylor's foundational theories, predominantly attribute droplet deformation in turbulent regimes during ultrasound emulsification (USE) to hydrodynamic pressure fluctuations and viscous stresses at the droplet interface, generated by turbulent eddies. However, these existing theoretical models lack comprehensive validation with experimental data and broader applicability. Thus, this paper introduces an empirically validated model for USE, applicable to a wide range of O/W-emulsion systems by establishing and fitting an emulsification dataset of three distinct O/W-emulsions produced through batch and flow USE. This dataset was modelled by implementing AI-tools in a fitting-algorithm, assuming power-law relationships between several dimensionless groups which characterize the droplet formation process. This study compares the performance of three Global Optimization Techniques-Bayesian Optimization, Particle Swarm Optimization, and Differential Evolution-with the latter exhibiting the most optimal performance. In the end, an empirical correlation was obtained achieving an adjusted coefficient of determination (R adj ²) of 0.81 and a Relative Root Mean Squared Error (RRMSE) of 22 %, ensuring droplet size prediction accuracy within ±61 nm at a 95 % confidence level, corresponding to an prediction accuracy of 37 % relative to the average droplet size. Furthermore, this empirical correlation was validated using literature emulsification data of a wide range of O/ W-emulsions, yielding an R adj ²-coefficient of 0.89 and a RRMSE of 15 %, underscoring its broader applicability. Introduction Emulsions have applications in the food-, pharmaceutical-and cosmetic industry due to their favourable optical, physical and chemical properties which are in a large part influenced by the size and poly-dispersity of the droplets [1-3]. Nano-emulsions, characterized by droplet sizes between 50 and 500 nm, are not thermodynamically stable, meaning that their formation is a non-spontaneous process [3-5]. Nonetheless, their small size makes them less prone to destabilization mechanisms such as creaming, sedimentation and flocculation, therefore creating kinetically stable dispersions [6]. Nano-emulsions have previously been produced using high-energy methods which apply high pressures or shear stresses to deform the oil/water-interphase [7]. Ultrasound emulsification (USE) is the most prominent method, due to its superior energy efficiency compared to high shear mixing, high pressure homogenization and microfluidization [8-10]. The ultrasound waves travelling through the continuous phase cause compression and expansion of vapour bubbles [11]. These bubbles grow over a number of cycles until a certain size is exceeded leading to an implosion called cavitation [11-14]. This cavitation generates microjets and eddies (turbulent vortices) which disrupt the oil/water-interphase and deform the emulsion droplets, decreasing their size [15]. Despite the application of external stresses, emulsion droplets resist deformation and breakup due to an increasing internal pressure as a result of surface tension γ (N/m), which is defined as the Laplace pressure ΔP l (Pa), represented in Eq. (1) [16-18]. In case of nano-droplets, they can be approached as spheres meaning R 1 = R 2 hence the equation can be further simplified. To cause droplet breakup, the relative pressure difference inside and outside of the droplet has to be overcome by either inertial or viscous stresses [18]. ΔP l = γ⋅ (1 R 1 + 1 R 2) = 2γ R (1) The critical Weber number (We crit) quantifies the relation between