Model based process optimization for nanoparticle precipitation

Abstract

This study introduces a derivative-based optimization framework for the controlled synthesis of nanoparticles, with a focus on achieving target particle size distributions via dynamic manipulation of reactant inflow during precipitation. The key novel aspects of our approach include coupling population balance equations (PBE)-which accounts for nucleation and growth phenomena-with a prior first-order reduction reaction formulated using the exact Method of Moments (eMoM), and implementing a discretize-then-optimize approach for efficient computation of parameter sensitivities, enabling effective optimization of the time-dependent inflow profile. The objective is to minimize the variance in particle size while ensuring a prescribed mean diameter. Numerical case studies explore the role of process constraints and regularization strategies in control performance. The following quantitative results highlight the effectiveness of our approach: first, the optimized time-varying inflow profiles significantly outperform constant-rate strategies. Within the size range μ_D ± 0.1μ_D, the optimal control yields nanoparticles with a purity exceeding 99%, compared with about 77% for the most effective constant control. Furthermore, the framework remains robust under variations in process parameters, sustaining performance despite limits on maximum inflow rate and total synthesis time. This framework offers a systematic and computationally tractable approach to optimizing transient operating conditions in nanoparticle synthesis, with strong potential for industrial application.