Estimation of mixing index and contact number by spot sampling

Abstract

Most of the available definitions of the mixing index, which specifies homogeneity or distribution of the composition in a solids mixture, are based on the variance of the concentration of a certain component among spot samples. However, for a solid—solid chemical reaction or any process involving contact between different solid phases, its rate is proportional to the contact points or area among particles of the different phases. Thus a definition of a microscopic and geometric mixing index based on the number of contact points appears to be of practical significance. The contact number is the number of contact points between two different types of particles for one key particles, a particle species which is selected as a reference. In this paper, the estimation of the mean contact number from spot samples is considered. An expression for estimating the contact number from spot samples is derived. Expressions for the expected value (population mean contact number) and the variance of this mean contact number are also derived. To verify these expression, random numbers with a uniform distribution are generated to simulate a binary component mixture in the completely mixed state. Results of the simulation are in reasonably good agreement with the derived expressions. The mixing index based on the mean contact number is able to indicate the homogeneity of a mixture with regular packing arrangement. In such a mixture, particles are packed either cubically or hexagonally in each layer, and therefore it is difficult to estimate the homogeneity of the mixture from the sample variance.