Data Treatment for Estimating the Particle Size Distribution from Ensemble Optical Measurements

GLORIA LIA FRONTINI

MEASUREMENT OF PARTICLE SIZE DISTRIBUTION OF POLYMER LATEXES

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Lugar: Kerala, India; Año: 2009;

Light scattering experiments performed on systems of particles may provide all kind of information on the morphological characteristics of the particles [1,2]. The particle size distribution (PSD) in the surrounding media 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. We consider, for instance, Photon Correlation Spectroscopy (PCS) also called 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 light scattered by the latex sample is registered utilizing different experimental setups. In all cases, the mathematical formulation between the obtained data and the unknown function, the Particle Size Distribution leads to an integral equation. The determination of the PSD gives rise to ill-conditioned inverse problems of data reduction. To solve an inverse linear problem usually requires the selection of a regularization parameter and the determination of a function that optimizes a least square problem. Regularization is the approximation of an ill-posed inverse problem by a family of closed well-posed problems. The theory of regularization methods is well developed and the properties of the regularized solutions are established rigorously for linear problems. In this article 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. We consider also the case in which an additional unknown parameter appears 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. The inverse problem established is non-linear in this case: a Fredholm integral equation of the first kind with an unknown parameter in its kernel. We describe an iterative procedure that consists in solving sequentially a linear problem and a non-linear one, in order to estimate the unknown function and the unknown parameter in the kernel, respectively, simultaneously with the selection of a proper regularization parameter.  Through several simulated examples taken from the literature we give some general conclusions about the performance of the discussed methods to retrieve latex PSD.