# Dynamic Light Scattering: Measuring the Particle Size Distribution

Dynamic Light Scattering (DLS, also known as Photon Correlation Spectroscopy or Quasi-Elastic Light Scattering) is one of the most popular light scattering techniques because it allows particle sizing down to 1 nm diameter. Typical applications are emulsions, micelles, polymers, proteins, nanoparticles or colloids. The basic principle is simple: The sample is illuminated by a laser beam and the fluctuations of the scattered light are detected at a known scattering angle θ by a fast photon detector.

Simple DLS instruments that measure at a fixed angle can determine the mean particle size in a limited size range. More elaborated multi-angle instruments can determine the full particle size distribution.

From a microscopic point of view the particles scatter the light and thereby imprint information about their motion. Analysis of the fluctuation of the scattered light thus yields information about the particles. Experimentally one characterizes intensity fluctuations by computing the intensity correlation function  g2(t), whose analysis provides the diffusion coefficient of the particles (also known as diffusion constant).

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DLS Correlator

The diffusion coefficient D is then related to the radius R of the particles by means of the Stokes-Einstein Equation:

$$D=\frac{k_B T}{6 \pi \eta R }$$

Where kB is the Boltzmann-Konstant, T the temperature and η the viscosity.

The correlation of the intensity can be performed by electronic hardware (digital correlator) or software analysis of the photon statistics. Because fluctuation are typically in the range of nanoseconds to milliseonds, electronic hardware is typically faster and more reliable at this job.

## Data Analysis

Cumulant Method

To obtain the diffusion coefficient the intensity correlation function must be analyzed. The standard procedure for this is the application of the cumulant method. By fitting a polynomial of third degree to the logarithm of the intensity correlation function, the decay rate Γ is obtained (1. cumulant).

The decay rate is directly related to the diffusion coefficient D:

$$\Gamma =q^{2}D$$

Where is q is the wave vector, which is dependend of the scattering angle.

Higher orders of the fitting result (2. and 3. cumulant) give the polydispersity index of the sample. Modern dynamic light scattering instruments perform cumulant analysis automatically. The quality of the result however depends significantly on the quality of the data and the constraint settings of the fitting procedure. The cumulant analysis can only determine the particle size distribution of a Gaussian distribution around on mean particle size. For more bi- or polymodal particle size distributions more complex analysis methods such as the Contin method are required.

## Quality of measurement

The quality of a DLS measurement depends on several factors. Some obvious, such as the quality of the components (the laser, the detector, the correlator...), other factors are not as straightforward but may influence the measurement significantly. Some important points to be considered are listed below.

The scattering angle

The decay rate depends on the wave vector and thus the scattering angle.  Particles of different sizes scatter with different intensities in dependence of the scattering angle. Thus there is an optimum angle of detection for each particle size. A high quality analysis should always be performed at several scattering angles (multiangle DLS). This becomes even more important in case of polydisperse samples with unknown particle size distribution since at certain angles, the scattering intensity of some particles will completely overwhelm the weak scattering signal of other particles, thus making them invisible to the data analysis at this angle.
DLS instruments working exclusively at a fixed angle can only deliver good results for some particles. Therefore, special attention should be paid while considering a precision of an advertised DLS instrument. For these fixed angle instruments such indications are only ever true for certain particles.

Multiple scattering

The theory of Dynamic Light Scattering is only valid for single scattered light. Like all scattering methods the interpretation becomes exceedingly difficult for systems with non-negligible contributions from multiple scattering. Already small contributions of multiple scattering can result in large analysis errors. Particularly for larger particles with high scattering contrast, this limits the technique to very low particle concentrations. A large variety of systems are therefore excluded from investigations with conventional dynamic light scattering. However, it is possible to suppress multiple scattering in DLS via the cross-correlation approach. The general idea is to isolate singly scattered light and suppress undesired contributions from multiple scattering in a DLS experiment. Different implementations of cross-correlation light scattering have been developed and applied. Currently the most successful scheme is the so called 3D cross-correlation method. The same method can also be used to correct Static Light Scattering (SLS) data for multiple scattering contributions. Alternatively, in the limit of strong multiple scattering, a variant of dynamic light scattering called Diffusing Wave Spectroscopy (DWS) can be applied.