Ceramics-Silikáty 57 (2) 151-161 (2013)

A CLOSED-FORM EXPRESSION APPROXIMATING THE MIE SOLUTION FOR THE REAL-IN-LINE TRANSMISSION OF CERAMICS WITH SPHERICAL INCLUSIONS OR PORES

A new closed-form expression is presented for estimating the real-in-line transmission of ceramics consisting of non-absorbing phases in dependence of the inclusion or pore size. The classic approximations to the exact Mie solution of the scattering problem for spheres are recalled (Rayleigh, Fraunhofer, Rayleigh-Gans-Debye/RGD, van de Hulst), and it is recalled that the large-size variant of the RGD approximation is the basis of the Apetz-van-Bruggen approach. All approximations and our closed-form expression are compared mutually and vis-a-vis the exact Mie solution. A parametric study is performed for monochromatic light in the visible range (600 nm) for two model systems corresponding to composites of yttrium aluminum garnet (YAG, refractive index 1.832) with spherical alumina inclusions (refractive index 1.767), and to porous YAG ceramics with spherical pores (refractive index 1). It is shown that for the YAG-alumina composites to achieve maximum transmission with inclusion volume fractions of 1 % (and slab thickness 1 mm), inclusion sizes of up to 100 nm can be tolerated, while pore sizes of 100 nm will be completely detrimental for porosities as low as 0.1 %. While the van-de-Hulst approximation is excellent for small phase contrast and low concentration of inclusions, it fails for principal reasons for small inclusion or pore sizes. Our closed-form expression, while less precise in the aforementioned special case, is always the safer choice and performs better in most cases of practical interest, including high phase contrasts and high concentrations of inclusions or pores.

Keywords: Optical properties, YAG ceramics (porosity, pore size), YAG-alumina composites, Real-in-line transmission (RIT), Mie solutions and approximations (Rayleigh, Fraunhofer, van de Hulst), Apetz-van-Bruggen approach