Modified Kubelka–Munk model for calculation of the reflectance of coatings with optically-rough surfaces

The Kubelka–Munk two-flux radiative transfer model is strictly applicable only to the case of diffuse illumination but is often applied in the case of collimated illumination. Here, the application of the Kubelka–Munk two-flux model to the collimated illumination of optically-rough surfaces is investigated. Expressions for the reflectance from such surfaces are obtained. A relatively simple treatment of reflection from surfaces of arbitrary roughness is developed that takes into account the characteristics of the spectrophotometer used to measure reflectance. The modified Kubelka–Munk model is tested in the case of an optically-rough rutile titanium dioxide coating on a titanium substrate and found to give good agreement with experiment, even for negligible scattering within the coating. It is expected that if the surface is sufficiently rough to ensure that the light transmitted into the coating is diffuse, the modified Kubelka–Munk model will be applicable irrespective of the magnitude of the absorption and scattering coefficients of the coating material.