## 3 Monodispersed Nanorods: Bimodal Distributions due to Translation and Rotation
### Basic Analysis of Particle Suspensions: Hydrodynamic Radius and Poyldispersity Index  
For monodispersed spherical particles, the autocorrelation function decays exponentially. The decay parameter is proportional to the translational diffusion coefficient \(\mathrm{D_{t}}\). The Stokes-Einstein equation describes the (translational) diffusion coefficient \(\mathrm{D_{t}}\) in relation to the hydrodynamic particle radius \(\mathrm{R_{Ht}}\), expressed by:  
\[\mathrm{D_{t}}=\mathrm{kT/6\pi\eta}\mathrm{R_{Ht}}, \tag{1}\]  
where T is the absolute temperature, k the Boltzmann constant, and \(\eta\) the dynamic viscosity of the liquid. If k, T, and \(\eta\) are known, the hydrodynamic particle radius can thus be determined from Equation (1) [7]. Inversely, if \(\mathrm{R_{Ht}}\) is known, it becomes possible to determine the viscosity of the liquid from DLS experiments using well defined monodispersed spherical tracer particles [13; 14]. Such viscosity measurements of BSA protein suspensions have recently been demonstrated using modulated 3D cross-correlation DLS [14]. Their analysis based on microviscosity measurements using DLS agrees well with results of macroviscosity measurements and shows that viscosity measurements can be performed with small amounts of liquid. The diffusion coefficient, \(\mathrm{D_{t}}\), is extracted from the autocorrelation function of the detected scattered light intensity as measured by DLS. In case of spherical particles, analysis is based on an exponential fit of the autocorrelation function, and generally referred to as a Cumulants analysis, and defined in ISO 13321 and ISO 22412. This analysis also gives information on the polydispersity index PDI, which reflects the width of the size distribution. Although widely used as standard, Cumulants analysis has significant disadvantages in case of multi-dispersed particle distributions. This becomes evident, e.g., when considering a suspension with a bimodal distribution of monodispersed spherical particle sizes, which obviously are physically governed by two different diffusion coefficients as smaller particles diffuse faster than larger ones. Therefore, further methods are used to determine diffusion coefficients, and thus hydrodynamic radii, from the measured autocorrelation function of an DLS experiment in the case of multi-dispersed samples. DLS Measurement of Multi-Dispersed Particle Suspensions and Analysis of the Autocorrelation Function  
For multi-dispersed particle suspensions, such as mixtures of nanoparticles of several sizes or e.g., mixtures of nanoparticles and proteins, particles have different velocity leading to different time scales of the correlation decay. As a consequence, the resulting decay of the autocorrelation function cannot any longer be modeled by a single exponential. If a suspension contains fast and slow moving particles, the experimental autocorrelation function of the scattered intensity, g(2)(q,t), can e.g., be modeled by two exponential decay functions as expressed by Equation (2):  
\[\mathrm{g^{(2)}(q,t)}=1+\beta^{2}[\mathrm{A_{f}}\exp(-\Gamma_{f}t)+\mathrm{A_ {s}}\exp(-\Gamma_{s}t)]^{2} \tag{2}\]  
where \(\Gamma_{f}\) and \(\Gamma_{s}\) are the decay rates related to the fast and slow moving particles, and \(\mathrm{A_{f}}\) and \(\mathrm{A_{s}}\) the corresponding amplitudes [15]. Therefore, further algorithms exist to take into account a composition of several decay time scales in the analysis of the autocorrelation function [16]. The CONTIN algorithm is a widely used example [4]. A comparative study of several algorithms has recently been performed showing the non-equivalence between methods as well as the robustness of the Cumulants method [16]. For polydisperse particle distributions, the CONTIN algorithm is performant and widely used, however, it has also limitations, as it is e.g., known to be sensitive to signal noise at large lag times, and can in some cases also lead to phantom peaks. Methods have thus been developed to improve its performance [5].