Light propagation model of titanium dioxide suspensions in water using the radiative transfer equation

Abstract

Constructions of numerical schemes for solving the radiative transfer equation (RTE) are crucial to evaluate light propagation inside photocatalytic systems. We develop accurate and efficient schemes of the three-dimensional and time-dependent RTE for numerical phantoms modeling aqueous titanium dioxide suspensions, in which the anisotropy of the forward-directed scattering varies and the strength of absorption is comparable to that of scattering. To improve the accuracy and efficiency of the numerical solutions, the forward-directed phase function is renormalized in the zeroth or first order with a small number of discrete angular directions. Then, we investigate the influences of the forward-directed scattering on the numerical solutions by comparing with the analytical solutions. The investigation shows that with the anisotropy factor less than approximately 0.7 corresponding to the moderate forward-directed scattering, the numerical solutions of the RTE using the both of the zeroth and first order renormalization approaches are accurate due to the reductions of the numerical errors of the phase function. With the anisotropy factor more than approximately 0.7 corresponding to the highly forward-directed scattering, the first order renormalization approach still provides the accurate results, while the zeroth order approach does not due to the large errors of the phase function. These results suggest that the developed scheme using the first order renormalization can provide accurate and efficient calculations of light propagation in photocatalytic systems.