Effective medium formalism of binary mixture properties involving two substrate variables

Abstract

Recently we made a preliminary attempt [1] to extend the effective medium formalism for dielectric binary mixtures developed in [2]. Essentially, we recognize that the formalism as used in [2] is sufficiently general that we expect its validity would not be limited only to dielectric properties of mixtures. However, complications arise as soon as it is attempted to apply it further, for example glass transition temperatures of polymer blends or to the prediction of the shear moduli of composite systems, or to other mixture problems whose equations involve one more physical property in addition to the property in question. In [1] we considered the implication of such complications on the original effective medium formalism via the simplest example possible, viz. the simple rule-of-mixtures expressed in terms of the mass concentrations of the constituents. The densities of the constituents then become explicit in the mixture formula, and difficulties are resolved there by allowing the substrate density also to be subjected to effective medium considerations. The effective medium formalism thus modified provides the insight for the more general framework formulated in this letter. The notation for a mixture introduced in [1] is convenient and will be restated, for the present purpose, in more general terms. Suppose in a binary mixture the mixture properties /~ and r are dependent on the corresponding properties x and y of the substrate and on the concentration 0 (either by volume or weight) of the filler, then these mixture properties may be expressed as functions of x, y and 0, say /~(x, y , 0) and ~:(x, y , 0), and the mixture itself may be denoted by (x ,y , 0). This gives #(x, y, 0), for instance, a direct physical interpretation as the property # of the mixture (x, y, 0), as well as its conventional meaning as a multivariable function. With this notation understood, we now proceed with the effective medium formalism. Consider a mixture (x, y , 01) as substrate. This substrate has properties given by #(x, y, 01) and z-(x, y, 01), and therefore when it is used to form a mixture with the further addition of the same filler so that the filler proportion is 02 reckoned with respect to (x, y, 01) as substrate, then the final mixture is one of [/~(x, y , 01) , T(X, y, 01), 02]. This mixture is the same as one of (x, y , 0) as substrate with filler amounting to 01 + 02 0102. Effective medium considerations will then give