**Although several sources of information are available that provide a mathematical description of the terms used in light scattering, these will not usually provide assistance in understanding their use in the practical application of the technique. **

The list of terms given below provides a descriptive definition, with notes on their specific application in the context of Dynamic Light Scattering (DLS).

## Z-Average Size

The Z-Average mean or Z-Average size used in DLS is a parameter also called the cumulants mean. It is the main and most stable parameter yielded by the technique. When employed in a quality control setting, the Z-Average mean is the best value to report because it is defined in ISO 13321 and more recently ISO 22412 which describes this mean as the 'harmonic intensity averaged particle diameter'.

The Z-average size will only be analogous with the size measured by other techniques if the sample is spherical or near spherical in shape, monomodal (i.e. only one peak), and monodisperse (i.e. very narrow width of distribution). The sample is prepared in a suitable dispersant, because the Z-Average mean size can be sensitive to even slight changes in the sample, for example, the presence of a small proportion of aggregates. It is important to note that the Z-average is a hydrodynamic parameter and is hence only relevant to molecules in solution or particles in dispersion.

## Cumulants Analysis

Cumulants analysis is a simple technique to analyze the autocorrelation function produced by a DLS experiment. The calculation is defined in ISO 13321 and ISO 22412. Since it is a moments expansion, it can generate several values, but then only the first two terms are used in practice — a mean value for the size (Z-Average) and a width parameter called the Polydispersity Index (PdI). The Intensity-based calculated value — Z-Average — should never be confused with or directly compared to a Mass or Number mean value produced by other techniques. The calculation is defined in the ISO standards, hence all systems that use this calculation as recommended should provide similar results if the same scattering angle is used.

## Polydispersity Index

This index is a number calculated from a simple 2 parameter fit to the correlation data (the cumulants analysis). The Polydispersity Index is dimensionless and scaled in such a way that values lesser than 0.05 are seldom seen other than with highly monodisperse standards. Values above 0.7 denote that the sample has a very broad size distribution and may not be ideal for the DLS technique. The numerous size distribution algorithms work with data that lies between these two extremes. The calculations for these parameters are defined in the ISO standard document 13321:1996 E and ISO 22412:2008.

## Polydispersity

In light scattering, the term % polydispersity and polydispersity are obtained from the Polydispersity Index — a parameter calculated from Cumulants analysis of the DLS-measured intensity autocorrelation function. In the Cumulants analysis, a single particle size mode is considered and a single exponential fit is applied to the autocorrelation function, and the polydispersity defines the width of the assumed Gaussian distribution. With regards to a protein analysis, a % polydispersity below 20% signifies that the sample is monodisperse.

## Diffusion Coefficient

Molecules and particles in suspension/solution experience Brownian motion. This is the motion caused by the bombardment of solvent molecules that themselves are moving because of their thermal energy. If a laser is used to illuminate the molecules or particles, the scattered light intensity fluctuates at a rate that is dependent upon the particle size because smaller particles are "kicked" further by the solvent molecules and move more quickly.

Analysis of these intensity fluctuations yields the velocity of the Brownian motion and thus the particle size using the Stokes-Einstein relationship. Therefore, the diffusion coefficient describes this Brownian motion of the particle or analyte in that specific solvent environment. The translational diffusion coefficient will be dependent on both the size of the particle "core" and any surface structure, as well as the type and concentration of ions in the medium.

## Hydrodynamic Diameter

The hydrodynamic size measured by DLS is defined as "the size of a hypothetical hard sphere that diffuses in the same manner as that of the particle being measured". However, practically, macromolecules or particles in solution are dynamic (tumbling), non-spherical, and solvated. Due to this, the diameter calculated from the particle’s diffusional properties will indicate the apparent size of the dynamic solvated/hydrated particle. Hence, it is known as the hydrodynamic diameter. Thus, the hydrodynamic diameter, or Stokes diameter, is that of a sphere that has the same translational diffusion coefficient as the particle being measured, imagining a hydration layer surrounding the molecule or particle.

## Correlation Curve - or Correlation Function

The correlation curve is the measured data in a DLS experiment. The curve must be a smooth, single exponential decay function for mono-size particle dispersion. All of the information concerning the diffusion of particles within the sample being measured is embodied within the correlation curve. The diffusion coefficient (D – which is proportional to the lifetime of the exponential decay) can be calculated by fitting the correlation curve to an exponential function. As the diffusion coefficient (D) is now known, the hydrodynamic diameter can be calculated with the help of a variation of the Stokes-Einstein equation. This curve is a sum of exponential decays for a polydisperse sample.

## Y-Intercept or Intercept

In the DLS technique, the Y-Intercept, or more simply Intercept, means the intersection of the correlation curve on the y-axis of the correlogram. The y-intercept can be used to assess the signal-to-noise ratio from a measured sample and hence is often used to judge data quality. It is typically scaled such that an ideal signal will offer a value of 1, and a good system will give intercepts more than 0.6, and above 0.9 for the best systems.

## Deconvolution or Deconvolution Algorithm

This is an algorithm-based method for resolving a mixture of exponentials obtained from a polydisperse sample into several intensity values each related to a discrete size band. The particle size distribution from DLS is derived from a deconvolution of the measured intensity autocorrelation function of the sample. Usually, this is achieved using a non-negatively constrained least squares (NNLS) fitting algorithm, with CONTIN being a common example.

## Count Rate or Photon Count Rate

In DLS, the count rate is simply the number of photons detected and is often stated in a "per second" basis. This is helpful in determining the quality of the sample, 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. The count rate has to be beyond some minimum value to have an adequate signal for analysis, but all detectors possess a maximum count rate where the response continues to be linear, and if the count rate is not adjusted automatically, then it would be best to observe the manufacturer recommendations for adjusting the count rate.

## 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 fraction or family. For biological polymers or materials, the particle scattering intensity is proportional to the square of the molecular weight. As such, the intensity distribution can be misleading to some extent, because a small amount of agglomeration/aggregation or presence of a larger particle species can dominate the distribution. However, this distribution can be used as a sensitive detector to detect the presence of large material in the sample.

## Volume Distribution

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

When an intensity distribution is being converted to a volume/mass distribution, there are four assumptions that should be accepted.

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

An understanding of these assumptions is particularly important because the DLS technique itself generates distributions with inherent peak broadening, and hence there will always be some error in the representation of the intensity distribution. As a result, volume and number distributions that are derived from these intensity distributions are best used for comparative purposes, or for estimating the relative proportions where there are multiple peaks, or modes, and should never be considered absolute. Therefore, it is a good practice to report the size of the peak depending on an intensity analysis and report the relative percentages only (not size) from a Volume distribution analysis.

## Adaptive Correlation

The amount of light scattered by diffusive particles is monitored by DLS instruments. The size of the particles considerably influences the intensity of the scattered light. For instance, with regards to isotropic scatterers, the intensity is proportional to the 6^{th} power of the particle diameter. In the Zetasizer Nano range, 50% of the sub runs with the highest count rates are discarded to reduce the effect of erratic data induced by sample contaminants (larger particles = higher count rates). The new Zetasizer Range uses a new statistical method - each sub run is individually analyzed and depending on how much they vary statistically from the other sub runs, they can be categorized either as steady-state or transient data.

### Steady State Data

Steady-state data sets describe particles that are constantly part of the measurement volume and hence are characteristic of the entire sample being analyzed. The key parameter for this data classification is the polydispersity index (PDI) of each sub run. The logic behind this strategy is that PDI is specifically susceptible to the presence of larger populations and also to other effects (noise) in the correlation function. Using statistic models, the statistical relevance of PDI can be determined, and sub runs can be classified as either representative of the sample or simply as transient events.

### Transient Data

On the other hand, transient data are typically particles that are not representative of the detection volume or the bulk of the sample (i.e. dust, aggregates, and other contaminants). The effect (or frequency) of data labeled as transient events can be validated in the “Run Retention” parameter, which reveals the percentage of runs that have been employed for the steady-state analysis, and consequently the percentage of runs that have been excluded. It should be noted that transient data is never deleted from the analysis, and it can be shown by itself, or how it would have impacted the original sample by showing the unfiltered results. In this way, the analyst can monitor how adaptive correlation enhances the results.

The ability to further decrease the time of analysis is another advantage of adaptive correlation. It has been found that the length of shorter sub-runs yields more reliable results by restricting the overall impact of transient events in the analysis. It has been shown that 10x one-second long sub-runs produce more repeatable results when compared to ten-second long sub-runs. In the new Zetasizer, the number of sub-runs as well as their length is established until a point where adding more data would not considerably improve the confidence in the result, thus giving a final result with enhanced reproducibility.

## MADLS

Particles larger than 1/10 of the λlaser exhibit an angular dependency on the intensity of light scattered. Moreover, this effect becomes exponentially more important with increasing particle sizes, up to a point where the scattering of particles becomes a complex function of minima and maxima based on the detection angle.

With its significant distortion towards forward angles of detection, it has been suggested that when looking for the presence of aggregates, the 13° detector should be used. A dual angle measurement feature in the Zetasizer Nano software** **enables carrying out two individual measurements at forward scattering and backscattering angles of detection which allows for a more complete picture to be obtained. However, analysts would be presented with two entirely different results instead of just one plot representative of their whole sample. In the new Zetasizer Ultra, three detectors are placed at various angles (back, side, and forward), instead of just two. These detectors can be used to obtain a single higher resolution result - Multi-Angle Dynamic Light Scattering (MADLS^{®}).

As a technique, DLS is known to have restrictions in resolving different size populations within the same sample. The angular dependency of the scattered light is used by MADLS to enhance the resolution of the technique by integrating the information obtained at the different angles and providing a single, higher resolution size distribution. It should be noted that the range of concentrations that can be used in this type of measurement is more restricted than those of backscattering (NIBS^{®}) analysis. This is because some effects often present at forward and side scattering measurements may also be detected (for example, multiple scattering, number fluctuations, and so on). MADLS results are mainly displayed as volume weighted particle size distributions, but they can also be converted to intensity (backscattered weighted) and number particle size distributions thereby enabling the extraction of even more information.

## Particle Concentration

The Zetasizer Ultra can offer information on the number of particles per mL of solution by measuring the particle size and the angular dependent intensity of the scattered light — from which the buffer scattered intensity (background), is subtracted. Furthermore, if different populations are present in a sample, it can also produce a reliable concentration of particles for each mode present, because it uses the same principles of a MADLS measurement (higher resolution size determination). Particle concentration results can be reported as distributed particle concentration, cumulative particle concentration plots, or as a total particle concentration value. Similar to a MADLS measurement, the range of concentrations that can be used is more restricted than when performing a NIBS^{®} measurement.

Although in the ZS XPLORER software, the particle concentration is displayed as a stand-alone measurement, it is an extension of a multi-angle DLS measurement, and therefore a MADLS result is also obtained.

## References

- International Standard ISO13321 Methods for Determination of Particle Size Distribution Part 8: Photon Correlation Spectroscopy, International Organization for Standardization (ISO) 1996.
- International Standard ISO22412 Particle Size Analysis - Dynamic Light Scattering, International Organization for Standardization (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 Micromeritics Instrument Corporation.

For more information on this source, please visit Micromeritics Instrument Corporation.