Estimation of particle size distribution from Vis/NIR scattering measurements

Emulsions are an important group of products in the food industry characterized by the dispersion of two or more immiscible liquids. To ensure stable and high quality products, quality characteristics should be monitored throughout the production process. Optical measurements such as Vis/NIR spectroscopy have already proven their usefulness in determining the chemical composition of food products based on light absorption characteristics. Furthermore, emulsion quality has also an important physical aspect, namely the size and number of globules present. Since light scattering is related to these physical properties, optical measurements have the potential for estimating the particle size distribution (PSD). Nevertheless, further insight in the relation between particle size distribution and bulk scattering properties is necessary to optimize estimation routines and sensor design.
The estimation of a PSD from a bulk scattering coefficient spectrum is an ill-posed problem. Mathematically, there are several solutions, but not all are suited candidate PSD’s because they are physically impossible or lack consistency with knowledge from previous measurements. In this study, the potential of several inverse estimation methods to estimate the PSD and volume concentration of milk fat globules from measured bulk scattering coefficient spectra are evaluated.
In order to create variable particle sizes, subsamples of the original raw samples were homogenized for 0 - 1200 seconds. The reference PSD of both raw and homogenized samples were determined with laser diffraction measurements. The bulk scattering properties in the 550 - 1800 nm wavelength range were derived from total reflectance, total transmittance and unscattered transmittance measurements by means of an Inverse Adding-Doubling algorithm.
An extension of the Mie solution of Maxwell’s equations was exploited to relate a bulk scattering coefficient spectrum to a specific monodisperse particle size. As a first method to regularize the solution for polydisperse systems, assumptions were made regarding the shape of the PSD. Based on literature, the PSD of milk fat was assumed to follow a lognormal or Weibull distribution. Secondly, more flexible, shape independent estimation methods were examined for the regularization of the inverse estimation. The unknown PSD was approximated by a weighted summation of B-splines. The B-spline weights were obtained by solving a least-squares problem with Tikhonov regularization, or by using a two-step method in which the solution was regularized by the number and position of the B-splines. Finally, a third methodology was tested, were both the bulk scattering coefficient spectrum and PSD were approached as a weighted sum of B-splines. Next, a functional linear model was fitted to allow prediction of PSD spline weights from spline weights fitted to the bulk scattering coefficient spectrum.
It was found that the proposed methodology resulted in relatively accurate estimates for the PSD’s and the particle volume concentrations. Although the procedure which used the predefined probability density function to regulate the PSD was least flexible, it resulted in the most accurate and robust estimations.
The estimation of milk fat globule size distribution presented here can serve as an exploration step for inverse estimation methods to be used for other (more complex) emulsions and suspensions.