# Characterization of Magnetic-Silica Magnetic Fluids by Laser Scattering
## 2 Materials and Methods
### DLS-Based Original Experimental Technique  
In dynamic light scattering, time dependences of the scattering intensity fluctuations caused by the Brownian motion of macromolecules in solution are measured. That allows one to measure the diffusion coefficients of different kinds: translational and rotational movements and internal dynamics [24]. Scattering from independent particles in solution can be represented as a sum of contributions from individual scatterers. When irradiation with polarized light is used, the scattered radiation will consist of polarized (polarization vector is perpendicular to the scattering plane: \(VV\)) and depolarized (polarization vector in the scattering plane: \(VH\)) components [25]:  
\[E_{s}{}^{VV}=\sum_{j}E_{j}=\sum_{j}A^{VV}{}_{j}e^{i\varphi_{j}}e^{-i\omega_{0}t}, \tag{1}\]  
\[E_{s}^{VH}=\sum_{j}E_{j}^{VH}=\sum_{j}A_{j}^{VH}e^{i\varphi_{j}}e^{-i\omega_{0}t},\]  
where \(A_{j}^{VV}\) and \(A_{j}^{VH}\) are the amplitudes of the polarized and depolarized components of the scattered light from the \(j\)th scattering element, and \(\varphi_{j}\) are the corresponding phases. If we take \(\varphi=0\) at the origin, then  
\[\varphi_{j}=(\mathbf{k}_{0}-\mathbf{k}_{s})\cdot\mathbf{r}_{j}=\mathbf{q} \cdot\mathbf{r}_{j}, \tag{2}\]  
where \(\mathbf{q}\) is the scattering wave vector, \(\mathbf{k}_{0}\) and \(\mathbf{k}_{s}\) are the wave vectors of the incident and scattered waves (Figure 1), and \(\mathbf{r}_{j}\) is the scatterer position. Since the frequency of scattered radiation is shifted relative to the original frequency insignificantly, then \(|\,\mathbf{k}_{0}\,|\approx|\,\mathbf{k}_{s}\,|\), and the magnitude of the scattered light can be written as [26]  
\[|\mathbf{q}|=|\mathbf{k}_{0}-\mathbf{k}_{s}|\approx 2|\mathbf{k}_{0}|\sin \left(\frac{\theta}{2}\right)=\frac{4\pi n_{0}}{\lambda_{0}}\sin\left(\frac{ \theta}{2}\right), \tag{3}\]  
where \(\theta\) is the scattering angle, \(\lambda_{0}\) is the light wavelength in the medium, and \(n_{0}\) is the refractive index of the medium (in this case, the solvent). Thus, the polarized and depolarized components of the scattered radiation can be rewritten as  
\[E_{s}{}^{VV}=\sum_{j}A^{VV}{}_{j}e^{i\mathbf{qr}_{j}(t)}e^{-i\omega_{0}t}, \tag{4}\]  
\[E_{s}^{VH}=\sum_{j}A_{j}^{VH}e^{i\mathbf{qr}_{j}(t)}e^{-i\omega_{0}t},\]  
When isotropic particles in the classical DLS method are considered, only the polarized component \(E_{s}\) is measured and analyzed. To process the scattering signal, we calculate an autocorrelation function  
\[G^{(1)}(\tau)=\,<E_{S}^{*}(t)E_{s}(t+\tau)>. \tag{5}\]  
\(G^{(1)}(\tau)\) is called an autocorrelation function of the first kind. In real experiments, square-law detectors (photoelectron multipliers) are typically used. These detectors capture fluctuations of the scattered radiation intensity rather than of the field. In this case, an autocorrelation function of the second kind is calculated as  
\[G^{(2)}(\tau)=\,<E_{S}^{*}(t)E_{S}(t)E_{S}^{*}(t+\tau)E_{s}(t+\tau)>. \tag{6}\]In the case when the scattered light is a Gaussian random process, the autocorrelation function of the second kind will be related to the autocorrelation function of the first kind by the Siegert relation [26]  
\[G^{(2)}(\tau)=\Bigr{|}G^{(1)}(\tau)\Bigr{|}_{2}+1. \tag{7}\]  
This relationship makes it easy to go from the measured \(G^{(2)}(\tau)\) to \(G^{(1)}(\tau)\) and continue to work with it.