A Bayesian Approach for the Estimation of the Particle Size Distribution Combining Static Light Scattering and Scanning Electron Microscopy

In this article, Static Light Scattering (SLS) measurements are processed to estimate the Particle Size Distribution (PSD) of polymeric particle systems incorporating prior information obtained from an alternative experimental technique: Scanning Electron Microscopy (SEM). SEM is a direct experimental technique for particle sizing but it uses just a small fraction of the particles so it has a considerable statistical error. The inverse problem stated based on SLS, is solved using a Bayesian approach implemented through the MetropolisHastings (MH) algorithm, following two different schemes for the representation of the PSD. In the first one, the PSD is represented by a parameterized family of distributions in a fixed-form scheme. In the second one, there is no assumption on the shape of the PSD, i.e. a free-form scheme is used. Both schemes are tested by the analysis of simulated examples of concentrate and semi-concentrate polymeric particles where simulated SLS measurements are generated using a rigurous model (Vrij’s Hard Sphere (HS) model). An approximate model (Pedersen’s Local Monodisperse Approximation (LMA)) is used in the inversion stage. A few priors from SEM micrographs are simulated in a Monte Carlo routine. Results show that a Bayesian approach combining data from two experimental techniques can improve the quality of the estimated parameters as far as an approximate model and reliable additional information are available. Nomenclature ) (R f Particle Size Distribution (PSD) ) R q , (n F p , 2 Form Factor H Regularization matrix ) (q I s Light Scattering Intensity  I Scattered Intensity Measurements K Global model constant M Number of measurements N Number of components of the discretized PSD m n Solvent Refractive Index p n Refractive Index of Particles