## 4 Theoretical Background and Simulations
### From Microscopic to Macroscopic Quantities: Ab Initio Calculations  
Performing ab initio calculations on particle ensembles to estimate their diffuse reflectance is a challenging endeavor due to the large number of steps and parameters involved that require considerable effort to determine or are straight out unknown. For example, while analytical expressions for individual particles exist, the response in the framework of dependent scattering may differ significantly. In practice, however, Mie calculations with a few corrective terms are typically sufficient [140]. Starting from arbitrarily chosen particle- and material-parameters such as their size distribution \(q_{j}\) (with the most prominent quantities being the size distribution by number \(q_{0}\) and by volume \(q_{3}\)[148]) and complex-valued relative refractive index \(m=n_{\text{particle}}/n_{\text{medium}}=m^{\prime}-im^{\prime\prime}\)[1, 118], Mie theory is employed to calculate the Mie coefficients \(a_{n},b_{n}\) in order to determine microscopic scattering, absorption and extinction efficiencies \(Q_{\text{sca}}\), \(Q_{\text{abs}}\), \(Q_{\text{ext}}\)[119]. The link between these microscopic and their respective macroscopic quantities can be established by calculating the average path-length parameter and the forward-scattering ratio for diffuse radiation [141; 149; 150; 151; 152]. From these calculations, absorption and scattering quantities \(K,S\) are extracted that in turn may be invested into the Kubelka-Munk function (Equation (3)) to yield a theoretical value of the diffuse reflectance (cf. Figure 14). Based on these calculations, the starting parameters may have to be refined to yield a more accurate result in combination with experimental data. In a nutshell, a rough recipe to approach this kind of calculations would be as follows:  
1. Use a Mie model to calculate the (microscopic) scattering and absorption efficiencies \(Q_{\text{sca}}\), \(Q_{\text{abs}}\) as well as the scattering amplitudes \(S_{1}\), \(S_{2}\) from the Mie coefficients \(a_{n}\), \(b_{n}\)[1, 119; 153; 154]. This can be based on classic Mie theory of spheres or advanced models such as stratified spheres, anisotropic particles or arbitrary objects. For simplicity, we exclusively consider isotropic spheres henceforth. 2. The K-M absorption coefficient \(K\) can then be defined as [152] \[K=\frac{3f_{V}}{d}Q_{\text{abs}}\] (7) with the volume filling fraction \(f_{V}\) and the particle diameter \(d\). The K-M scattering coefficient \(S\) is defined in a similar fashion [152]: \[S=\frac{3f_{V}}{d}(1-\sigma_{c})Q_{\text{sca}}\] (8) where \(\sigma_{c}\) is the forward scattering ratio that is derived and given in references [136; 152]. 3. For sufficiently thick samples, Equation (3) can be solved for \(R_{\infty}\), yielding: \[R_{\infty}=\frac{\sqrt{2 KS}-S}{2K-S}\ \.\] (9) In this case, neither particle size \(d\) nor volume filling fraction \(f_{V}\) play any further role as they cancel out in Equation (9). However, for a translucent coating layer, Equation (9) becomes inaccurate and needs to be expanded to expressions including the diffuse reflectance of layers beneath the sample [1, 152]. However, since this case is not covered by classic K-M theory, it will be omitted within this context. This way, the diffuse reflectance of an opaque sample can be approximated by theoretical calculations for a single wavelength and particle size. The aforementioned procedure can be repeated for arbitrary wavelengths, thus generating a reflectance spectrum. Because this model only accounts for single scattering processes, it tends to overestimate the diffuse reflectance, especially where absorption becomes prominent: light is only attenuated once as opposed to potential subsequent scattering (and consequentially absorption) events at neighboring particles. When the investigated sample is polydisperse, knowledge about the particle size distribution in either its differential form \(q_{0,3}\) or integral form \(Q_{0,3}\) (also termed as _cumulative undersize_) [148] is crucial.