Heat conduction in ceramics: Pores, cracks and splat boundaries

The theory of phonon heat conduction in ceramics is applied to the case of uniform porosity. Each pore is treated as a region of zero conductivity, and a theory of conduction in inhomogeneous media based on minimum entropy production is applied. This is a perturbation theory and has two forms: one is in terms of conductivity and one in terms of resistivity. One chooses the form based on the smaller perturbation. For pores of zero conductivity the first form is preferred. The effective thermal conductivity kappa(e) becomes kappa(e) = kappa(1 - 4p/3)where p is the volume fraction porosity and kappa the conductivity of the matrix. Splat boundaries can be treated as ellipsoidal regions of enhanced porosity. For this either formalism can be used. For cracks, again of zero conductivity, the conductivity formalism is used, making one major axis of the ellipsoid much smaller than the others. This assumes that a local value of the conductivity can be defined. For phonons the mean free path is small enough to do so, except in the immediate vicinity of cracks. At high temperatures the radiative component of thermal conductivity becomes significant. Each pore is a region of differing dielectric constant and all pores reduce the dielectric constant and thus reduce the radiative component similar to the reduction in the lattice conductivity. Sub-micron sized pores can also act as scattering centers. Expressions are obtained for the reductions in conductivity and dielectric constant, and applied to various cases, with emphasis on the overall heat transfer in thermal barrier coatings of stabilized zirconia.