Mechanism-enabled population balances and the effects of anisotropies in the complex Ginzburg-Landau equation
Date
2019
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Abstract
This paper considers two problems. The first is a chemical modeling problem which makes use of ordinary differential equations to discover a minimum mechanism capable of matching experimental data in various metal nanoparticle nucleation and growth systems. This research has led to the concept of mechanism-enabled population balance modeling (ME-PBM). This is defined as the use of experimentally established nucleation mechanisms of particle formation to create more rigorous population balance models. ME-PBM achieves the goal of connecting reliable experimental mechanisms with the understanding and control of particle-size distributions. The ME-PBM approach uncovered a new and important 3-step mechanism that provides the best fits to experimentally measured particle-size distributions (PSDs). The three steps of this mechanism are slow, continuous nucleation and two surface growth steps. The importance of the two growth steps is that large particles are allowed to grow more slowly than small particles. This finding of large grow more slowly than small is a paradigm-shift away from the notion of needing nucleation to stop, such as in LaMer burst nucleation, in order to achieve narrow PSDs. The second is a study of the effects of anisotropy on the dynamics of spatially extended systems through the use of the anisotropic Ginzburg-Landau equation (ACGLE) and its associated phase diffusion equations. The anisotropy leads to different types of solutions not seen in the isotropic equation, due to the ability of waves to simultaneously be stable and unstable, including transient spiral defects together with phase chaotic ripples. We create a phase diagram for initial conditions representing both the longwave k = 0 case, and for wavevectors near the circle |k| = μ using the average L² energy.
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Subject
Ginzburg-Landau
nucleation
population balance
nanoparticles
anisotropy
pattern formation