# Dynamic Light Scattering: Theory

Dynamic Light Scattering (DLS) refers to an optical technique used for analyzing dynamic properties and size distribution of a broad variety of physical, chemical and biological systems composed of several suspended constituents [1]. These can be colloidal particles, macromolecules, bubbles or droplets. For simplicity we will simply talk of particles in all three cases.

DLS is based on the extraction of spectral information derived from time-dependent fluctuations of the light scattered from a spatially limited volume within the sample. Specifically, when a suspension of  particles is hit by a monochromatic coherent beam of light, generated scattered light waves spread out in all directions. Scattered waves interference in the far field region generates a net scattered light intensity $$I_s(t)$$. Due to the random motion of the suspended particles within the sample the interference can be stochastically either constructive or destructive, hence resulting in a stochastic light intensity signal.

The suspended particles of the colloidal dispersion under investigation undergo Brownian motion. This motion results in fluctuations of the distances between the particles and hence also in fluctuations of the phase relations of the scattered light. Additionally, the number of particles within the scattering volume may vary in time.

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The net result is a fluctuating scattered intensity . The corresponding measured normalized intensity correlation function is written as:

$$g_2(q,\tau)=\frac{\left<I_s(q,t)I_s(q,t+\tau)\right>}{\left<|I_s(q,t)|^2\right>},$$              (1)

with $$\tau$$ being the time lag and  $$q$$ being the scattering vector module. By means of the Siegert relation  can be related to the electric field correlation function $$g_1(q,\tau)$$:

\begin{eqnarray}
g_2(q,\tau)&=& 1+\beta |g_1(q,\tau)|^2 \\
g_1(q,\tau)&=& \frac{\left<E_s(q,t)E_s^*(q,t+\tau)\right>}{\left<|E_s(q,t)|^2\right>}
\end{eqnarray}

$$\beta$$ being the so-called intercept. The field correlation function may be used to determine the diffusion coefficient $$D$$ of the scatterers. For a monodisperse sample,  is fitted to an exponential function

$$g_1(q,\tau)=\exp(-\Gamma \tau)$$

yielding the decay rate, $$\Gamma$$. From its definition:

$$\Gamma=q^2 D$$

one obtains the diffusion coefficient. By using the Stokes-Einstein equation one can obtain the hydrodynamic radius as

$$R = \frac{kT}{6 \pi \eta D}$$

with $$k$$ the Boltzmann constant, $$T$$ the temperature in Kelvins, and $$\eta$$ the viscosity of the suspending medium.

It is important to note that the equation above are only true for the case of single scattered light. If particle concentration is to high, such that the mean free path l becomes smaller than the sample dimension d, mutiple scattering will occur which will render the equation invalid (l < d).

[1] B.J. Berne and R. Pecora. Dynamic light scattering. Dover Publications, Mineola, 2000.