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Theory
Understanding Dynamic Light Scattering

Brownian motion

When in solution, macromolecules are buffeted by the solvent molecules. This leads to a random motion of the molecules called Brownian motion. For example, consider this movie of 2 micrometer diameter particles in pure water. As can be seen, each particle is constantly moving, and its motion is uncorrelated with the other particles. (Movie courtesy of Dr. Eric R. Weeks, Physics Department, Emory University.)

As light scatters from the moving macromolecules, this motion imparts a randomness to the phase of the scattered light, such that when the scattered light from two or more particles is added together, there will be a changing destructive or constructive interference. This leads to time-dependent fluctuations in the intensity of the scattered light.

In Dynamic Light Scattering (DLS), a.k.a. Quasi-Elastic Light Scattering (QELS), the time-dependent fluctuations in the scattered light are measured by a fast photon counter. The fluctuations are directly related to the rate of diffusion of the molecule through the solvent, which is related in turn to the particles’ hydrodynamic radii. DLS is employed by the DynaPro NanoStar, the DynaPro™ Plate Reader II, the Möbius™ and the WyattQELS™ Dynamic Light Scattering Module for MALS detectors to determine the effective particle size.

The fluctuations are quantified via a second order correlation function given by:

(1)

where I(t) is the intensity of the scattered light at time t, and the brackets indicate averaging over all t. The correlation function depends on the delay t, that is, the amount that a duplicate intensity trace is shifted from the original before the averaging is performed. A typical correlation function for a monodisperse sample is shown in Fig. 1

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Fig. 1: Model correlation function for a 20 nm rh particle and the WyattQELS multi-tau correlator.

As described in various light scattering texts [cf. B. Chu, Laser Light Scattering: Basic Principles and Practice, (Academic Press, Boston, 1991)], the correlation function for a monodisperse sample can be analyzed by the equation:

(2)

where B is the baseline of the correlation function at infinite delay, b is the correlation function amplitude at zero delay, and G is the decay rate.

The ASTRA software uses a nonlinear least squares fitting algorithm to fit the measured correlation function to equation 2 to retrieve the correlation function decay rate G. From this point, G can be converted to the diffusion constant D for the particle via the relation:

(3)

Here, q is the magnitude of the scattering vector, and is given by

(4)

where n0 is the solvent index of refraction, l0 is the vacuum wavelength of the incident light, and q is the scattering angle.

Finally, the diffusion constant can be interpreted as the hydrodynamic radius rh of a diffusing sphere via the Stokes-Einstein equation:

(5)

where k is Boltzmann's constant, T is the temperature in K, and h is the solvent viscosity.