Linear algebraic characterization of particle concentration and size distribution

Abstract

Particle counting and sizing are essential in various fields of science and engineering. Most existing particle characterization methods rely on a rigorous analysis of nonlinear particle–light interactions. Here, we present a novel algebraic approach for counting and sizing colloidal particles. We construct a mathematical vector space in which the scattered signal distributions from the colloidal dispersions form vectors. These vectors are expanded using the basis vectors corresponding to the scattered signal distributions from particles of known sizes. We then determine the expansion coefficients that yield the number concentration as a function of particle size via mathematical optimization. Further, we experimentally evaluate the algebraic optics and find that the formalism accurately recovers the particle size and concentration. Thus, this algebraic method provides a means of quantifying particulate matter in fluids that are highly concentrated and difficult to measure individually and entirely.