All the information in a dynamic light scattering (DLS) measurement on Prometheus Panta is calculated from the autocorrelation function (ACF) of the measured light intensity. The decay of the ACF is related to the diffusion coefficient of the particles in solution: The ACF will decay earlier for small particles diffusing at a higher rate and decay later for large particles diffusing at a lower rate. Furthermore, the gradient of the decay contains information on the heterogeneity of samples, which is expressed in form of a polydispersity index. To extract the diffusion coefficient and the polydispersity index, the ACF is fit with appropriate algorithms.

The intensity fluctuations of the scattering signal of a sample are correlated for the duration of a DLS acquisition. Each acquisition is then evaluated separately and its result, the ACF along with fitting results and quality parameters (see below), can be accessed via the acquisition details tab of a measurement.

Figure: Illustration of the autocorrelation function (ACF) of a DLS acquisition recorded with PR.Panta Control. The autocorrelation g² (t) is plotted over the time increments t. The time when the function starts to decay is related to the mean diffusion coefficient, while the gradient of the decay indicates the polydispersity/heterogeneity of the sample. The amplitude and the baseline of the ACF are marked with orange lines.

You do not need to inspect all the individual ACFs of a measurement. However, the ACF along with the fitting results can help you to understand where inconsistencies in DLS data come from. PR.Panta Control also provides guidance in DLS data analysis in the form of automatic quality checks.

Quality parameters of the autocorrelation function

Prometheus Panta software reports three different parameters that describe the quality of an ACF, which help to decide if the results of a particular acquisition should be included in the analysis: the amplitude, the signal-to-noise ratio, and the baseline.

Theoretically, the maximum amplitude of the ACF is 1. However, the amplitude is influenced by several factors including the concentration of the particle of interest, the level of dispersity of the sample, the static background signal, and optical setup. On Prometheus Panta, most measurements that can be categorized as good or acceptable have ACF amplitudes ranging from 0.9 to 0.2, with the majority falling in the range of 0.5 to 0.4.

The amplitude is just one parameter among several to judge ACF quality. The signal-to-noise ratio was introduced to help evaluate the quality of the measured autocorrelation function. This parameter not only takes the amplitude but also the noise at each t into account. Good acquisitions show values of 125 or higher. The signal-to-noise ratio is used in the ACF rating of the quality checks.

An optimal baseline displays a value of 1. However, the baseline can be influenced by aggregates or dust, by absorption/fluorescence, or number fluctuations due to low particle concentration.

Mathematical formalism of the autocorrelation function

The autocorrelation function g² (t) of the measured light intensity can be described by:

where t is the delay time, b is an instrument factor, and g¹ the electric field correlation function.

For monodisperse samples, g¹ can be expressed by a single exponential decay dependent on the decay rate G:

while for polydisperse samples, g¹ is expressed by an intensity-weighted distribution function of decay rates:

The decay rate G is proportional to the translational diffusion coefficient D:

where q² is the scattering vector:

with h being the solvent refractive index, l the wavelength of the incident light (405 nm), and q the scattering angle. PR.Panta Control uses a refractive index of 1.335 (water) to calculate the scattering vector.

A more detailed derivation can be found in the literature, for example Stetefeld et al.: Dynamic light scattering: a practical guide and applications in biomedical sciences. Biphys Rev. 2016.