Understanding clay minerals with fuzzy mathematics

A long-existing geochemical problem is the lack of a means of chemically defining non-stoichiometric complex minerals such as clays, which have no distinct composition and no clear compositional boundaries. We propose here a novel approach for describing the chemical nature of clay minerals using fuzzy logic. This non-conventional mathematical approach allows us to quantify compositional vagueness in such systems. We show that a clay mineral can be described in terms of how compositionally representative it is, of its own type ("belonging-ness") and to what extent it resembles other types (compositional overlap). Many clay minerals are seen to be far from the ideal, and most minerals are also to a lesser (and sometimes greater) extent, close to types other than their own. This has provided a means of grading such minerals, evaluating how "good" a sample is, and defining the extent of transition to other phases. We have derived here the fundamental methodology for such computations, which could be a framework for the analysis of other complex chemical systems, especially in the fields of geology and metallurgy.