A new approach for deriving $c^2$-bounds on the effective conductivity of random dispersions

A new variational procedure for evaluating the effective conductivity of a dilute random dispersion of spheres is proposed. The classical variational principles are employed, in which a class of trial fields in the form of suitable truncated factorial series is introduced. In general, this class leads to a rigorous formula for the effective conductivity, which is correct to the order "square of sphere fraction", and makes use of the disturbance to the temperature field in an unbounded matrix, generated by two spherical inhomogeneities. The basic idea in the present study consists in replacing this "two-sphere" field by a superposition of disturbances, generated by the same two spheres, but considered as single already, together with the disturbance due to another single sphere, centered between them and radially inhomogeneous. In this way new variational bounds on the effective conductivity are derived and discussed in more detail for a special choice of the middle sphere's properties. The obtained bounds improve, in particular, on the known three-point bounds on the effective conductivity of the dispersion.