Data treatment for estimating the particle size distribution from ensemble optical measurements

Abstract

Light scattering experiments performed on systems of particles may provide all types of information on the morphological characteristics of the particles. Specifically, the particle size distribution (PSD) can be assessed as far as suitable experiments and models to analyze the data are available. There are a number of techniques in use at the present time in industrial and research laboratories for a wide range of particles. For instance, we can mention Dynamic Light Scattering (DLS), Elastic Light Scattering (ELS), Turbidimetry, Espectroscopy and Fraunhofer diffraction. For all these methods the light source is typically a laser in the visible spectrum; an ensemble of measurements of light scattered by the latex sample is registered using different experimental setups. In all cases, the mathematical formulation between the obtained data and the unknown function, the PSD, leads to an integral equation. The determination of the PSD gives rise to ill conditioned inverse problems of data reduction. In this work we present a brief description of a variety of methods applied successfully to invert light scattering measurements to estimate latex PSD. Truncated singular value decomposition and Phillips-Tikhonov regularization methods are presented in detail. Several techniques to select the regularization parameter, such as Morozov´s discrepancy principle, L-curve and the generalized crossed validation method are introduced as well. Through several simulated examples we give some general conclusions about the performance of the discussed methods. Also, we consider the case in which there is an additional unknown parameter in the integral kernel. Typically, the value of the relative refractive index of the particles to the medium can be uncertain and it should be estimated to accomplish the retrieval of the PSD. This problem, formulated as a non-linear inverse problem, is solved following an iterative procedure that leads to the simultaneous determination of the PSD, the refractive index and the value of the regularization parameter.