## 4 Theoretical Background and Simulations
### From Microscopic to Macroscopic Quantities: Ab Initio Calculations
The distribution can be determined via dynamic light scattering, atomic force microscopy, electron microscopy or differential sedimentation [155; 156; 157; 158]. A typical particle size distribution is shown in Figure 15.  
Owing to a significant contribution to diffuse scattering originating from small particles (cf. Figure 13), it is of utmost importance to resolve the size distribution down to low nanometer-levels. In this regard, disc centrifuges (e.g., by CPS Instruments, Inc. or Brookhaven Instruments Corp. [159; 160]) have proven to be a valuable asset in accurate size determination. The particle size distributions \(q_{0}\) and \(q_{3}\) are mutually dependent on each other and can be transformed as follows:  
\[q_{3}(d)=\frac{q_{0}(d)\cdot d^{3}}{\int_{0}^{\infty}q_{0}(d)\cdot d^{3}\ \mathrm{d}d}\enspace. \tag{10}\]  
The denominator of Equation (10) is required for normalization, i.e., the cumulative undersize should approach 100% for \(d\rightarrow\infty\). Unless otherwise defined, we henceforth use the particle size distribution by number denoted by \(q_{0}\) for simulations. For each particle size \(d_{i}\), \(Q_{\mathrm{sca}}\), \(Q_{\mathrm{abs}}\), \(S_{1}\) and \(S_{2}\) are calculated. Afterwards, their weighted average is determined and used in subsequent calculations. This weighing is performed by using \(q_{0}(d_{i})\). For increased accuracy, especially when broad size distributions occur, a considerable amount of particle sizes should be sampled. In general, any polydispersity will wash out sharp Mie resonances at a single particle size. Provided with an initial size distribution and a complex-valued refractive index \(m=m^{\prime}+im^{\prime\prime}\), one may estimate the diffuse reflectance and vice versa, calculate \(m^{\prime\prime}\) from \(R_{\infty}\), as shown in Figure 16.  
Figure 15: Particle size distribution (blue) and cumulative undersize (red) according to number (solid lines) and volume density (dash-dotted) for a TiO\({}_{2}\) nanopowder. If the diffuse reflectance spectrum is known, further optimization may be carried out by modifying both \(m\) and \(q_{0}(d)\) until a sufficient overlap between experiment and simulation is achieved. Because \(m\) and \(q_{0}\) both have significant influence on macroscopic quantities, it is viable to affix one in order to calculate the other. For example, by setting \(q_{0}\) immutable due to a preliminary size characterization, \(m\) alone is varied. Based on this approach, the absorption coefficient [162]  
\[\alpha=\frac{4\pi}{\lambda}m^{\prime\prime} \tag{11}\]  
can be determined. This finding is of crucial importance, considering that it grants access to a quantity that traditionally could only be concluded from transmissive measurements which are impossible to conduct for many industrially important materials as they only exist as powders. While it is possible to directly calculate \(\alpha\) from Mie as indicated in reference [163], the additional step via K-M theory helps relating predicted diffuse reflectances to measured ones for further parameter refinement and extraction of intermediary quantities such as \(K\) or \(S\). Figure 16: Calculation steps as outlined in Figure 14. (**a**) Measured diffuse reflectance spectrum \(R_{\infty}(\lambda)\) (blue) and calculated values (circles) and the derived absorption coefficient \(\alpha\) (red line); for better visibility, simulated values are shown only for selected positions; full calculations take the entire experimental dataset into account. The absorption-dip at 700 nm results from experimental artifacts that is discussed in Section 3.2. (**b**) Kubelka–Munk absorption and scattering coefficient \(K\) and \(S\) from which \(R_{\infty}\) is calculated according to Equation (9). (**c**) Microscopic scattering and absorption efficiencies \(Q_{\text{sca}}\) and \(Q_{\text{abs}}\) calculated from Mie theory [154] used for deriving \(K\) and \(S\). (**d**) Complex-valued refractive index (left); the real part is adapted from ref.