Quantifying Acoustophoretic Separation of Microparticle Populations by Mean-and-Covariance Dynamics for Gaussians in Mixture Models

Abstract

A method for the quantification of acoustophoretic separation and dispersion for microparticle populations featuring continuously distributed physical parameters is presented. The derivation of the method starts by (i)~considering the equation of motion for a particle ensemble in the coordinate+parameter space, (ii)~performing moment analysis on the transport equation for the probability density function (PDF), and (iii)~expanding up to the first-order the drift (and the diffusion coefficient) around the mean of the PDF. Following these steps, a system of ordinary differential equations for the evolution of the mean and the covariance in the coordinate+parameter space is derived. These differential equations enable for the approximation of the acoustophoretic separation dynamics of particle ensembles by using a gaussian mixture for which the mean and the covariance of each gaussian evolve according to the mean-and-covariance dynamics. The approximation property of this method is shown by comparison with direct numerical simulations of particle ensembles in the cases of prototypical models of acoustophoretic and free-flow acoustophoretic separations for which the particle populations are distributed according to the radius. Furthermore, the indicators for quantifying free-flow acoustophoretic separation performance are introduced, and a method for the inference of particle-histogram parameters is illustrated.