**There are various sources of information that provide a mathematical explanation of the terms used in light scattering. However, these will not generally provide help in understanding their application in the practical use of the method. (Ref, 1, 2, 3, 4, 5, 6)**

This article provides a descriptive definition with notes on their particular use in the context of Dynamic Light Scattering.

## Z-Average Size

The Z-Average mean or Z-Average size is a parameter used in dynamic light scattering and is also known as the cumulants mean. It is the most stable and primary parameter produced by the method. When used in a quality control setting, the Z-Average mean is the best value to report as it is described in ISO 13321 and more recently ISO 22412 which defines the Z-Average mean as the 'harmonic intensity averaged particle diameter'.

The Z-average size can be compared with the size measured by other methods only if the sample is monodisperse (i.e. very narrow width of distribution), near-spherical or spherical in shape, monomodal (i.e. only one peak), and the sample is prepared in an appropriate dispersant, because the Z-Average mean size can be susceptible to even slight changes in the sample, for example the presence of a small amount of aggregates. It must to be noted that the Z-average mean is a hydrodynamic parameter and hence relevant to molecules in a solution and particles in a dispersion.

## Cumulants Analysis

This is a simple technique for analyzing the autocorrelation function produced by a DLS test. The calculation is defined in ISO 22412 and ISO 13321. Since it is a moments expansion, it can generate a number of values; however, in practice, only the first two terms are used — a width parameter called the Polydispersity Index (PdI) and a mean value for the size (Z-Average). Considered as an Intensity-based calculated value, the Z-Average should not be confused with or compared directly to a Mass or Number mean value generated by other techniques. The calculation is defined in the ISO standards; therefore, all systems using this calculation as suggested, should provide similar results if the same scattering angle is used.

## Polydispersity Index

The Polydispersity Index is a number analyzed from a simple 2 parameter fit to the correlation data (the cumulants analysis). This index is dimensionless and scaled in such a way that values smaller than 0.05 are hardly seen other than with monodisperse standards. However, values more than 0.7 show that the sample has an extremely broad size distribution and is perhaps not appropriate for the dynamic light scattering (DLS) technique. The different size distribution algorithms function with data that falls between these two limits. The calculations for these types of parameters are defined in ISO 22412:2008 and the ISO standard document 13321:1996 E.

## Polydispersity

The term polydispersity and % polydispersity in light scattering are derived from the Polydispersity Index, a parameter calculated from a Cumulants analysis of the DLS-measured intensity autocorrelation function. During the Cumulants analysis, a single particle size method is assumed and a single exponential fit is used to autocorrelation function, and the width of the assumed Gaussian distribution is described by the polydispersity. With regard to a protein analysis, a % polydispersity below 20% points out that the sample is monodisperse.

## Diffusion Coefficient

Molecules and particles in suspension/solution undergo Brownian motion, which is produced by the bombardment by solvent molecules that move due to their thermal energy. Illuminating the molecules or particles with a laser will cause the scattered light intensity to fluctuate at a rate based upon the size of the particles, because smaller particles are "kicked" further by the solvent molecules and they move more quickly.

Analysis of these intensity fluctuation produces the velocity of the Brownian motion and so the particle size using the Stokes-Einstein equation. Therefore, the diffusion coefficient defines this Brownian motion of the particle or analyte in that particular solvent environment. This translational diffusion coefficient will depend on the size of the particle "core" and on any surface structure, as well as the type and concentration of ions in the medium.

## Hydrodynamic Diameter

The hydrodynamic size is determined by Dynamic Light Scattering (DLS). It is defined as "the size of a hypothetical hard sphere that diffuses in the same fashion as that of the particle being measured". Though in practice, macromolecules or particles in solution are solvated, dynamic (tumbling) and non-spherical. Due to this, the diameter calculated from the particle’s diffusional properties will signify the apparent size of the solvated/dynamic hydrated particle. Hence the terminology used is Hydrodynamic diameter. Therefore, the Stokes diameter, or hydrodynamic diameter, is that of a sphere that has the similar translational diffusion coefficient as the particle that is being calculated, assuming a hydration layer surrounding the molecule or particle.

## Correlation Curve or Correlation Function

The data measured in a dynamic light scattering (DLS) test is the correlation curve which must be a smooth, single exponential decay function for a mono-size particle dispersion. All of the data regarding the diffusion of particles within the sample that is being measured is embodied within the correlation curve.

The diffusion coefficient (D) can be calculated, by fitting the correlation curve to an exponential function. (D is proportional to the lifetime of the exponential decay). Once the diffusion coefficient (D) is known, the hydrodynamic diameter can be measured by using a variation of the Stokes-Einstein equation. This curve is a sum of exponential decays for a polydisperse sample.

## Y-Intercept or Intercept

In DLS the Y-Intercept, or just Intercept, indicates the intersection of the correlation curve on the y-axis of the correlogram. The y-intercept can be employed to assess the signal-to-noise ratio from a quantified sample and thus is normally employed to judge data quality. It is often scaled such that a perfect signal will give a value of 1, and a good system will provide intercepts greater than 0.6 and the best systems will give greater than 0.9.

## Deconvolution or Deconvolution Algorithm

An algorithm-based approach to resolving a mixture of exponentials derived from a polydisperse sample into numerous intensity values, each related to a discrete size band. The particle size distribution from dynamic light scattering (DLS) is derived from a deconvolution of the measured intensity autocorrelation function of the sample. Usually, a non-negatively constrained least squares (NNLS) fitting algorithm, for instance CONTIN, is used to accomplish this.

## Count Rate or Photon Count Rate

In DLS, this refers to merely the number of photons identified and is typically stated in a "per second" basis. This is helpful for determining the sample quality, by monitoring its stability as a function of time, and is also employed to set instrument parameters such as the attenuator setting and at times analysis duration. In order to have sufficient signals for analysis, the count rate should be above some minimum value. Conversely, all detectors have a maximum count rate where the response continues to remain linear. If the count rate is not automatically adjusted, the manufacturer recommendations for adjusting the count rate must be observed.

## Intensity Distribution

The first order result from a DLS experiment is an intensity distribution of particle sizes. The intensity distribution is naturally weighted based on the scattering intensity of each particle family or fraction. The particle scattering intensity is proportional to the square of the molecular weight in the case of biological polymers or materials. The intensity distribution, by itself, can be ambiguous to an extent, in that the distribution is dominated by a small amount of agglomeration/aggregation or the presence of a larger particle species. This distribution, however, can be employed as a sensitive detector for the presence of large material in the sample.

## Volume Distribution

Although the fundamental size distribution generated by DLS is an intensity distribution, this can be converted, using Mie theory, to a volume distribution or a distribution describing the relative proportion of multiple components in the sample based on their mass or volume rather than based on their scattering (Intensity).

Four assumptions must be accepted when an intensity distribution is converted to a volume/mass distribution:

- There is no error in the intensity distribution
- The optical properties of the particles are known, i.e. the imaginary and real components of the refractive index
- All particles are homogeneous
- All particles are spherical

A basic insight about these assumptions is very important as the DLS technique itself generates distributions with inbuilt peak broadening, and thus some error will always be present in the representation of the intensity distribution. As such, number and volume distributions derived from this intensity distribution are optimally employed for comparative purposes, or for predicting the relative proportions where there are several peaks or modes, and should never be considered absolute. Therefore, reporting the size of the peak according to an intensity analysis and reporting the relative percentages alone (not size) from a volume distribution analysis is considered to be a good practice.

**References**

- International Standard ISO13321 Methods for Determination of Particle Size Distribution Part 8: Photon Correlation Spectroscopy, International Organisation for Standardisation (ISO) 1996.
- International Standard ISO22412 Particle Size Analysis - Dynamic Light Scattering, International Organisation for Standardisation (ISO) 2008.
- Dahneke, B.E. (ed) Measurement of Suspended Particles by Quasi-elastic Light Scattering, Wiley, 1983.
- Pecora, R. Dynamic Light Scattering: Applications of Photon Correlation Spectroscopy, Plenum Press, 1985.
- Washington, C. Particle Size Analysis In Pharmaceutics And Other Industries: Theory And Practice, Ellis Horwood, England, 1992.
- Johnson, C.S. Jr. and Gabriel, D.A. Laser Light Scattering, Dover Publications, Inc., New York 1981

This information has been sourced, reviewed and adapted from materials provided by Malvern Panalytical.

For more information on this source, please visit Malvern Panalytical.

Why do not we make standard solutions of polydispersed?