Multi-angle dynamic light scattering (MADLS®) delivers an angular-independent particle size distribution (PSD) with increased resolution by combining scattering information from multiple angles. The measurement provides improved insight into all particle size populations present in the sample, as populations that may be weakly scattering at one detection angle, are revealed in the other detection angles, and brought into the combined-angle PSD.
A MADLS measurement carried out by the Zetasizer Ultra is an automated series of single-angle measurements: backscatter, side scatter and forward scatter, that are combined to provide a single PSD. As in a single angle measurement, where PSD is derived from a single autocorrelation function, the MADLS measurement derives the PSD by analysing multiple autocorrelation functions recorded at the different angles. The autocorrelation functions are treated as parallel observations, finding a single PSD that is a best fit to all. This technical note describes how the analysis is achieved and the data that it generates, using a measurement example for a sample containing 150 nm and 300 nm diameter polystyrene latex beads.
The analysis of a single-angle DLS measurement is carried out using a least squares minimization algorithm to determine the size distribution solution that best fits the measured autocorrelation function. Typically, the specific algorithm chosen is influenced by constraints that may be placed on the solution, such as smoothness or non-negativity. The predicted instrument response (hereon termed scattering matrix) provides the necessary algebraic transformation from result to measurement:
Where g1(τ) is the autocorrelation coefficient measured at each lag time (τ). The particle size distribution is x(d) and is intrinsically weighted by the intensity of the light scattered by each particle size component, d. Each element in the scattering matrix, K(τ,d), a function of lag time and particle size is calculated according to the following:
Where g1(τ) is the field autocorrelation function, q is the scattering vector and Dt is the translational diffusion coefficient which relates to the particle size (hydrodynamic radius or diameter) via the Stokes-Einstein equation.
The MADLS approach [1, 2] is an extension of the single-angle method but with the addition of multiple measurement angles to the transform. The autocorrelation functions are concatenated as well as the scattering matrices to produce the following system of linear equations, c.f. similarity to eq.1:
Where θ1 denotes the primary angle, θ2 denotes the secondary angle etc.
When combining data from multiple angles, it is important to preserve the weighting inherent in the PSD. Therefore, the scattering matrix for each measurement angle is weighted by the expected scattered intensity. In the Zetasizer Ultra, the PSD can be viewed as a backscatter-equivalent intensity-weighted PSD or as a volume-weighted distribution, which is independent of scattering angle. The scattered intensity (specifically differential scattering cross-section) is calculated using Mie theory and requires knowledge of the material and dispersant optical properties (refractive index and absorption). Without the correct optical properties, the MADLS algorithm will fail to converge on the true solution.
In deriving the solution, the least squares minimisation algorithm aims to reduce the magnitude of the residual (the difference between the measured autocorrelation function and the prediction). Each angular contribution to the residual is weighted separately to account for the result uncertainty expected in that angle.
The MADLS analysis sequence described here is represented schematically in Figure 1. It is important to recognise that the solution PSD is the result of fitting to multi-angle auto-correlation data simultaneously to derive a single solution that satisfies the measurement at all three angles.
Figure 1: Schematic of MADLS analysis sequence.
To demonstrate the MADLS measurement, a mixture of polystyrene spheres with diameter 150 nm and 300 nm and similar scattering intensities were prepared in a 10 mM aqueous solution of NaCl. The single-angle results (backscatter, side scatter and forward scatter) are presented in Figure 2, showing that measurement at just one angle can resolve one only size population and that the size differs dependent on the measurement angle. Therefore, a single result at each angle is difficult to interpret – each is different. This behavior can be inferred from measurement using backscatter alone. As the size is greater than ~60 nm, particles will be non-isotropically scattering and they will scatter at different intensities dependent on the angle of observation and the particle size. Therefore, a small degree of sample polydispersity is likely to complicate interpretation of the results for the user as information from additional angles can seem contradictory.
Figure 2: Intensity weighted particle size distribution of a 150 nm+300 nm polystyrene sphere dispersion measured using DLS in backscatter (red), side scatter (green) and forward scatter (blue).
The multi-angle MADLS result is presented in Figure 3 and shows a distinct peak corresponding to each component of the mixture. The peaks are separated by a 2:1 ratio by size. The user can now see a more complete and realistic particle size distribution, free of angular dependence that is more readily interpretable than three separate size results.
Figure 3: Volume-weighted particle size distribution of a 150nm+300nm polystyrene sphere dispersion measured using MADLS.
A benefit of MADLS is that it is a higher resolution technique than single-angle DLS and can be used to further characterise a sample after initial screening by DLS. In some cases, the use of MADLS can reduce the need of using other techniques, such as TEM, that may have otherwise been employed to complete characterisation at every development stage up to final stage verification.
Improved resolution arises through two mechanisms:
If the sample is an isotropic scatterer, the extra resolution arises because of the multi-variate information that we have available in the analysis. This improves the signal to noise and constrains the solution so that only common components from all autocorrelation functions are incorporated, which in turn rejects non-common components (noise).
If the sample scattering displays angular dependence, improved resolution can arise from detection of mixtures with different relative intensity as a function of the angle. For example, one population that is very weakly scattering in backscatter may be the dominant component in side scatter.
The highest resolution arises when both of these conditions are met. It is possible to resolve both components in a mixture of 150 nm and 300 nm diameter spheres because there is sufficient anisotropy in the scattering by diameters 300 nm and above, i.e. with the resolution is 2:1. It is usually not possible to resolve 100 nm and 200 nm mixtures of polystyrene since both populations scatter with little angular dependence and the resolution is not maximized.
The requirement of MADLS to measure the sample at three separate angles restricts the concentration range that is measurable, since the concentration range must satisfy the requirements of each detection angle. For example, turbid samples are not suitable because such samples require NIBS technology which is possible in backscatter only.
The MADLS technique has been explained for users interested in how the measurement data is analysed. The origins of the improved resolution over a single-angle measurement have been discussed. This powerful technique allows users of the Zetasizer Ultra to gather additional insight into their sample by providing an angular independent result – multiple angles are measured to generate a single, more complete, size distribution allowing users to easily interpret their size data with no need to guess component size populations from multiple results from multiple angles. MADLS’s increased resolution helps to bridge the gap between traditional dynamic light scattering and high-resolution techniques such as SEC or TEM, reducing the requirements to run samples on such time-consuming techniques.
Further information on MADLS can be found here – https://www.malvernpanalytical.com/en/learn/knowledge-center/application-notes/AN180516GoldNanoparticlesZetasizerUltra
 P.G. Cummins, E.J. Staples, Particle Size Distributions Determined by a “Multiangle” Analysis of Photon Correlation Spectroscopy Data, Langmuir, 1987, 3, 1109-1113
 G. Bryant, J.C. Thomas, Improved Particle Size Distribution Measurements Using Multiangle Dynamic Light Scattering, Langmuir, 1995, 11, 2480-2485