Effective conductivity bounds by inserting adiabatic or isothermal surfaces

Structure-dependent bounds of the effective thermal conductivity ETC in polyphasic materials, with a regular inclusion array, are proposed. The bounds account for the contact resistance and are based on the enhancement of the ETC by insertion of any isothermal surface (for the upper bounds) and on the decrease of the ETC by insertion of any adiabatic surface (for the lower bounds). The method is above all able to introduce bounds for the high contrast filled composite polymers, whose inclusions are much more conductive than the polymer matrix. Computations for the ETC of a simple cubic array of spheres, which simulate well the heat conduction features in filled polymers, show that two such lower bounds and an upper one are easily computable. For all ETC higher than 3 or 4 times the matrix thermal conductivity, the bounds by insertion of isothermal or adiabatic surfaces are tighter than the other known structure-dependent bounds which account for the imperfection of the perfect contact - the 3rd order variational bounds of Torquato and Rintoul [S. Torquato, M. Rintoul, Effect of the interface on the properties of composite media, Phys. Rev. Lett. 75 (1995) 4067-4070].