*4.3. Models*

## 4.3.1. Optical Properties Predictions

The experimentally measured absorbance, *A*exp, of a colloidal solution can be expressed in terms of predicted extinction cross section, *σ*ext as:

$$A\_{\rm pred} = N \frac{\sigma\_{\rm ext}}{2.303} d\_0 \tag{2}$$

where *N* (m−3) is the number density of the NPs, *d*<sup>0</sup> (cm) is the path length of the spectrometer. For spherical, homogeneous and isotropic NPs, the "Mie scattering theory" [45,50] can be used to compute the exact values of the *Q*ext, absorption (*Q*abs) and scattering (*Q*sca) efficiency as well as the anisotropy factor (*g*) as follows [46]:

$$Q\_{\rm ext} = \frac{2}{k^2} \sum\_{n=1}^{\infty} (2n+1) \text{Re}(a\_n + b\_n) \tag{3a}$$

$$Q\_{\rm sca} = \frac{2}{k^2} \sum\_{n=1}^{\infty} (2n+1) \left[ (|a\_n|^2 + |b\_n|^2) \right] \tag{3b}$$

$$Q\_{\rm abs} = Q\_{\rm ext} - Q\_{\rm sca} \tag{3c}$$

$$g = \frac{4}{k^2 Q\_{\text{sca}}} \sum\_{n=1}^{\infty} \left[ \frac{n(n+2)}{n+1} \operatorname{Re} (a\_n a\_{n+1}^\* + b\_n b\_{n+1}^\*) + \frac{2n+1}{n(n+1)} \operatorname{Re} (a\_n b\_n^\*) \right] \tag{3d}$$

where *k* is the NP size parameter (= 2*πa*/*λ*). *an* and *bn*, the scattering coefficients in terms of the spherical Ricatti-Bessel functions, *ψ<sup>n</sup>* and *ηn*, respectively, are defined as:

$$a\_n = \frac{\psi\_n'(mx)\psi\_n(x) - m\psi\_n(mx)\psi\_n'(x)}{\psi\_n'(mx)\eta\_n(x) - m\psi\_n(mx)\eta\_n'(x)}\tag{4a}$$

$$b\_n = \frac{m\psi\_n'(m\mathbf{x})\psi\_n(\mathbf{x}) - \psi\_n(m\mathbf{x})\psi\_n'(\mathbf{x})}{m\psi\_n'(m\mathbf{x})\eta\_n(\mathbf{x}) - \psi\_n(m\mathbf{x})\eta\_n'(\mathbf{x})} \tag{4b}$$

where *<sup>m</sup>* is the ratio of complex refractive index (*nS* <sup>=</sup> <sup>√</sup>*S*) of the sphere to that of the surrounding medium (*nm*) asterix (∗) and prime (-) indicate complex conjugate and derivative with respect to *x* and *mx*, respectively. The numerical calculations were performed with a python code implementation of the original algorithm published by Wiscombe [51]. The wavelength dependent complex refractive index, *n*(*λ*), was obtained from Ref. [52].

To account for polydispersity, the size range of the nanoparticle was discretized into a varying number of terms (*n*t) and then number-averaged to obtain the ensemble optical properties,

$$
\overline{\sigma\_{\mathbf{k}}} = \frac{1}{n\_{\mathbf{t}}} \sum\_{r=R\_{\mathbf{l}}}^{R\_{\mathbf{u}}} \sigma\_{\mathbf{k}}(r+\mathbf{i}) \quad \mathbf{k} = \text{ext}, \text{ abs, sca}, \quad n\_{\mathbf{t}} = 2, 3, 4, \dots, N \tag{5}
$$

where *Ru* and *Rl* are the upper and lower limits of the NP size range, respectively. *i* is the step size which is calculated as: *i* = *Ru* − *Rl*/(*n* − 1) and *σ*<sup>k</sup> is the mean *k* (i.e., extinction, absorption, scattering) cross sections of the NP.
