From diffusion optics to photocatalytic rates in multiply scattering porous slabs: finite-slab Green's function, optical-to-kinetic mapping, and application to core-shell aerogels with embedded anatase nanoparticles

We derive an end-to-end framework connecting steady-state photon transport in highly scattering porous slabs to surface-bound photocatalytic rates. Starting from the diffusion approximation (DA) with extrapolated (partial-current) boundary conditions, we solve the finite-slab Green's function with an isotropic internal plane source that rigorously maps external illumination to internal fluence. Using photon units for the fluence rate, Phi(z,lambda) [photons m-2 s-1], the local absorption is qa = mu_a * Phi and the primary carrier/radical generation is G = phi_int * qa. Site-limited and Langmuir-Hinshelwood kinetics are driven by G and closed via an accessible surface per macroscopic volume, S_accessible, yielding intrinsic volumetric (per-volume) and areal (per-area) rate constants. We then specialize to air-core/silica-shell monoliths whose shells host anatase TiO2 nanoparticles (NPs): shell refractive indices are calculated by Maxwell-Garnett/Bruggeman mixing; scattering is obtained from Mie theory (PyMieScatt); absorption follows mu_a = n_NP * sigma_abs with n_NP set by NP packing fraction in the shell. The exact slab solution reduces to a compact, design-useful predictor - our Eq. (14) - in a controlled asymptotic that clarifies when the rate scales as l*l/L (strongly diffusive, l << L) versus proportional to l (optically thin, l >> L). We detail measurement and validation protocols (diffuse R/T inversion of mu_s' and mu_a on the same slab, boundary extrapolation accuracy, lateral-loss control) drawing on prior art, and provide a turnkey recipe to compute rates for NP packing-fraction series. The result is a reproducible workflow that preserves design simplicity while resting on a rigorous finite-slab solution.