Multi-angle Dynamic Light Scattering (MADLS) offers improvements in particle sizing resolution, sensitivity and accuracy compared to traditional single angle Dynamic Light Scattering (DLS). By combining measurement data, sequentially collected, from three measurement angles it can provide improvements in resolution, typically from 3:1 to 2:1. It also provides improvements in accuracy by reducing signal to noise ratio, allowing lower smoothing coefficients to be used, and by reducing angular dependence of the final particle size result, as illustrated in the following Mie solution plot for a single protein macromolecule measured at 45, 90, and 173 degree scattering angles.
This white paper provides a technical discussion on how MADLS is implemented in the Malvern Panalytical Zetasizer Ultra system and the advantages of combining data from multiple measurement angles as well as the practical considerations when employing MADLS on your nanomaterial and biomolecular samples.
There are three advantages that can be derived from the introduction of multi angle dynamic light scattering (MADLS) in the new Zetasizer Ultra system;
Given the significance of these improvements compared to historical single angle DLS limitations, an explanation of the justifications for each is warranted.
When a dispersion of particles is illuminated by a coherent light source, a small fraction of the light will be scattered by the particles within the sample, with the magnitude of the scattered intensity being a function of the size, refractive index, and concentration of the particles, as well as the incident light intensity and the observation angle. The influence of these parameters on the scattering profile is well described using Mie’s solution to Maxwell’s electromagnetic equations, which form the fundamental basis of light scattering techniques such as laser diffraction (LD) and static light scattering (SLS), with SLS utilizing a limiting form of the Mie solution called the Rayleigh-Debye-Gans approximation. See for example Figure 1, which shows the normalized scattering per particle as a function of particle size and observation angle for an aqueous protein, under an illuminating wavelength of 633 nm, with the vertical dashed lines indicating the 1st Mie maxima in the scattering profiles for each angle.
Figure 1: Comparison of the size dependent scattering from a single protein macromolecule at 45, 90, and 173 degree scattering angles, as predicted by the Mie solution.
The Mie solution addresses the ‘magnitude’ of scattered light. Particles in solution will move according to Brownian motion. As a consequence, the magnitude of light scattered from a dispersion of particles will fluctuate with time due to constructive and destructive wave interference, as the relative particle positions within the dispersion change. Temporal analysis of these intensity fluctuations is the basis of a technique called dynamic light scattering (DLS).
In DLS, fluctuations in the scattering intensity are statistically analyzed using a technique called correlation, with the measured correlogram representing the lifetime of the ‘similarity in the average’ scattering intensity. The lifetime is influenced by dispersion properties such as; the temperature, which increases Brownian motion; the viscosity, which decreases Brownian motion; and most importantly, the particle size, with smaller particles diffusing faster than larger particles and generating a measured intensity correlogram with a shorter lifetime.
See for example Figure 2, which shows the measured intensity correlation functions for dispersions of 6 nm ovalbumin and 95 nm silicon dioxide.
The DLS measured correlogram contains all the information regarding the motion of all the particles within the dispersion. Deconvolution of the correlogram, using a non-negative least squares (NNLS) fitting algorithm, provides the intensity weighted particle size distribution of particles within the solution.
Figure 2: Dynamic light scattering measured correlograms of 6 nm ovalbumin and 95 nm silicon dioxide in phosphate buffered saline.
The deconvolution of the DLS measured correlogram into a particle size distribution is performed using a NNLS fitting technique according to the vector expression shown below, where g is the autocorrelation coefficient measured at each delay time (τ), x(d) is the particle size distribution, which is intrinsically weighted by the intensity of the light scattered by each size component of the sample, and K(τ, d) is the τ and d dependent scattering matrix, τ is the autocorrelation lagtime and d is the particle diameter.
Each element in the scattering matrix, K(τ, d) is calculated according to the following expression, where g1 is the field autocorrelation function, q is the scattering vector, and Dt is the translational diffusion coefficient which relates to the particle size via the Stokes-Einstein equation.
While DLS has a wide working size range from approximately 1 nm to 10 mm, it is considered a low resolution sizing technique. The rule of thumb resolution limit for single angle DLS is 3 to 1 on the particle size, meaning that in order to resolve near-in-size components using DLS, they must differ in size by a factor of 3. The underlying reason for this limitation is the inability to distinguish random noise in the correlogram from real effects associated with weakly scattering particles, such as low concentration protein oligomers or particles in the region of a Mie minima (see Figure 1). Traditional NNLS algorithms compensate for this by blurring the peaks in the particle size distribution – hence the low resolution description.
The challenge of distinguishing correlogram noise from real particle effects is particularly problematic for subsequent transforms of the DLS intensity distribution to a mass or even a number distribution. Given that the scattering intensity scales with the 6th power of the size, DLS intensity distributions appear skewed, with small amounts of larger particles dominating the distribution. While this sensitivity to large particles makes the technique ideal for detecting aggregates, a mass distribution is often the desired result for researchers utilizing DLS. To this end, the Mie solution, along with the optical properties of the particles, can be used to transform the intensity distribution into a mass or number distribution.
The fundamental problem with these transforms, is the assumption that every size bin in the intensity size distribution represents a real particle family, with the band broadening arising from the inability to distinguish correlogram noise from real particle effects leading to significant error in the transformed mass and number distributions. Consider Figure 3 for example, which shows the intensity and volume distributions measured for a 60 nm latex size standard and an oligomeric protein (BSA) mixture.
The intensity distributions in Figure 3 are shown in histogram format, with the bins representing the true sizes of the peak components shown in the darker blue and red for BSA and latex, respectively. The volume distributions are represented by the gray (BSA) and black (latex) curves, and the color coded dashed lines represent the mean values for the intensity and volume weighted distributions.
Note that the latex peak has only a single size component at approximately 30 nm, whereas the BSA peak is polydisperse with monomers, dimers, and trimers at circa 3.5, 4.7, and 5.7 nm radii respectively, confirmed by size exclusion chromatography.
Figure 3: Dynamic light scattering measured intensity (histogram) and volume (line) distributions for a 60 nm size standard and a protein (BSA), using the Zetasizer General Purpose NNLS algorithm.
In the Mie transform, the intensity distribution is considered error free, with every bin in the distribution histogram assumed to represent a real particle size family. As noted in Figure 3 however, there is an inherent degree of polydispersity associated with the DLS measured particle size distribution, because unfortunately the transform algorithm cannot distinguish inherent polydispersity (correlogram noise) from true polydispersity (real particles).
Because of this uncertainty, the net result of the volume transform, when considered in the context of a monodisperse (single particle size family) intensity weighted peak, is a bias or skewing of the peak toward smaller sizes in the volume weighted distribution, as indicated in the 60 nm latex standard result shown above. For polydisperse peaks composed of multiple unresolved size components, such as the mixture of BSA oligomers shown in Figure 3, the inherent component of the polydispersity tends to be less pronounced on a relative basis, due to the increased number of size bins occupied by real particles.
For these types of samples, the volume distribution is more representative of the actual distribution, as indicated in the BSA results in Figure 3. Note however, that more than half of the mass or volume in each distribution is attributed to imaginary particles within the inherently polydisperse collection of size bins. It is for this reason, that mass and number transforms, and therefore subsequent concentration calculations, are problematic when using traditional single angle DLS.
The influence of small amounts of correlogram noise on the resultant DLS particle size distribution is due to the correlogram deconvolution being an ill-posed problem, meaning that small amounts of error in the initial data set can lead to significant variations in the resultant solution. Attempts to negate the correlogram noise and improve DLS resolution are most often focused on increasing the aggressiveness of the NNLS algorithm.
In theory, NNLS-based DLS deconvolution algorithms could fit the measured correlogram exactly. To do so however would be a mistake, in that underestimation of noise in the signal leads to a very spiky result, with non-real peaks in the intensity size distribution. To avoid this, a parameter defining the degree of smoothness in the distribution result is integrated into the NNLS algorithm. This parameter is called the alpha parameter, and it is integrated into the least squares correlogram fitting expression as shown in the equation below, where g1 is the normalized field autocorrelation function, D is the translational diffusion coefficient, q is the angle dependent scattering vector, τ is the correlation delay time, στ is a weighting factor placing more emphasis on the strongly correlated rather than the weakly correlated (and noisy) data points, and Ai is the area under the curve for each exponential contribution and represents the relative weight for that particular ith exponential function within the sum.
Large alpha values (0.1) limit the spikiness of the solution, leading to smooth distributions. Small alpha values (0.0001) decrease the weighting or importance of the derivative term, subsequently generating more spiky distributions. The alpha parameter then, can be loosely described as representative of the expected level of noise in the measured correlogram.
Figure 4 shows an example of the dangers associated with utilizing an over aggressive alpha parameter. The results shown are those for 0.3 mg/ml lysozyme in PBS. In PBS, lysozyme is known to exist as a mixture of oligomers – monomer, dimer, trimer, etc. As evident in the results shown below, the measured average size of the protein distribution is independent of the a parameter selected. If the General Purpose algorithm is selected, with an alpha value (α) of 0.01, the algorithm generates an accurate intensity weighted mean size of the unresolved oligomeric distribution, with the peak width being representative of the expected DLS polydispersity for a mix of protein oligomers. The more aggressive algorithms, with α < 0.01, artificially squeeze the peak, losing information regarding the unresolved components, plus generate a phantom peak at circa 2 nm. This phantom peak is a consequence of the reduced signal to noise ratio inherent to dilute protein measurements, along with an underestimate of that noise through the use of an aggressive alpha parameter. It was in fact DLS users working with dilute proteins who coined the phrase ‘ghost peaks’ to describe phantom peaks derived from the use of an overly aggressive NNLS algorithm.
Figure 4: Influence of alpha parameter on DLS measured particle size distributions for 0.3 mg/ml lysozyme in PBS at pH 6.8.
In comparison to historical single angle DLS measurements, there are 4 fundamental improvements to the MADLS and particle concentration data collection method that is integrated into the new Zetasizer Ultra system.
Adaptive Correlation – reduces noise in the measured correlogram, by measuring for shorter runs times and separating correlograms that are not statistically identical.
Combining back-scatter (BS), side-scatter (SS), and forward-scatter (FS) Correlograms – increases the sensitivity to particles in single angle Mie minima regions, by coupling correlograms measured in backscatter, 90-degree scattering, and forward scattering modes.
High Resolution NNLS Algorithm – increases the number of size bins utilized in the NNLS fitting and utilizes a multi-angle fitting method to facilitate the extraction of more particle size information from unresolved components within DLS peaks.
Laser Intensity Calibration – eliminates the need to know or measure the incident laser intensity required to calculate concentration, by calibrating the system with a Rayleigh ratio standard (toluene).
In combination, these data collection and analysis improvements lead to more highly resolved and robust DLS particle size measurements by reducing the noise in the correlogram and subsequent inherent polydispersity in the size peak, facilitating the extraction of more information from unresolved near in size components and increasing confidence in subsequent concentration calculations.
The MADLS approach [1, 2] is an extension of the single-angle method but with the addition of multiple measurement angles to the matrix transform. The autocorrelation functions are concatenated as well as the scattering matrices to produce the following system of linear equations, 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 the Mie solution and requires knowledge of the material and dispersant refractive index properties. Without the correct optical properties, the MADLS algorithm will fail to converge on the true solution.
In deriving the solution, the NNLS algorithm aims to reduce the magnitude of the residual (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 procedures utilized in a MADLS and particle concentration measurements are as follows:
1. Measure the correlograms for the sample at BS, SS, & FS, using adaptive correlation.
2. Deconvolute the multi-angle system of linear equations to derive an intensity weighted particle size distribution, using the enhanced resolution MADLS NNLS algorithm;
a) Calculate the expected BS, SS and FS correlograms using known particle refractive index values and the Mie solution.
b) Calculate the fit deviations for all correlograms.
c) Using an iterative least squares process, change the particle size distribution and recalculate the expected correlograms and fit deviations, repeating until the sum of deviations is minimized.
3. Transform the best fit intensity weighted particle size distribution into mass/volume and number distributions using the known refractive index of the particles and the Mie solution.
4. Use the intensity distribution to determine the fraction of the total scattering intensity attributed to each size component and calculate the concentration of each component, by comparison of component scattering to the toluene scattering reference.
The MADLS solution to low DLS resolution (stemming from noise in the correlogram) is the integration of adaptive correlation. In adaptive correlation, the longer run times employed in historical DLS measurements are replaced with short runs. Noise is minimized by separating correlograms that are not statistically identical, yielding a high quality low noise average correlogram from which the particle size distribution can be calculated. This reduction in noise facilitates use of an enhanced resolution NNLS algorithm, with observed resolution limits being around 2 to 1 on the size, rather than 3 to 1 seen historically.
The use of multi-angle correlogram collection in the new MADLS method enhances detection of low concentration weakly scattering particles in the Mie minima size regions. Particles difficult to detect in the backscatter mode, are easily observed at the other angles, particularly the forward angle with no Mie minima until the particle size exceeds ~ 900 nm. Multi angle correlogram collection also leads to a reduction in correlogram nose (or uncertainty) and subsequent inherent peak polydispersity, further justifying the use the enhanced resolution MADLS NNLS algorithm.
In comparison to traditional single angle DLS NNLS algorithms, the MADLS enhanced resolution algorithm utilizes more (smaller) size bins. The increase in the number of size bins is justified by the reduction in the correlogram noise arising from the integration of adaptive correlation and multi angle correlogram collection, providing extraction of more highly resolved size information, in the absence of inherent peak polydispersity or any false squeezing of the peak arising from an overly aggressive alpha parameter.
Since each correlogram is an independent measurement of the sample, angle-specific noise does not manifest in the result since it is not common to all angles. The net result is a higher resolution intensity size distribution and a number transform that is more representative of the actual distribution of particles in the sample.
As a consequence of correlogram noise and uncertainty in traditional single angle DLS measurements, use of the total intensity and DLS measured intensity size distribution to calculate particle concentration is problematic. Correlogram noise compounds into inherent peak polydispersity, which compounds into artificial skewing of the number distribution toward imaginary size components, which is ultimately manifested as calculated concentration values with order of magnitude levels of uncertainty or bias. Given the reduction in correlogram noise and higher resolution distributions derived from the MADLS technique however, calculation of the particle concentration from the DLS measured scattering intensity and the derived intensity weighted size distribution is no longer problematic. But it does require knowledge of the incident laser intensity.
The concentration of particles within a solution can be calculated from the scattering intensity using the expression shown below, where K1 is a known geometric constant, Io is the incident intensity, IA is the sample analyte scattering intensity (sample scattering – buffer scattering), d is the particle diameter, ño is the dispersant refractive index, ñP is the particle refractive index, λ is the incident wavelength, q is the observation or scattering angle relative to the transmitted incident beam, C is the concentration in units of particles per unit volume, and the function f represents the Mie solution to Maxwell’s equations.
For a fixed optical configuration and particles of known optical properties, all of the parameters in the above expression are known, with the exception of the incident intensity (Io). Since direct measurement of Io is problematic, it is typically determined using a scattering standard of know Rayleigh ratio (Rθ), such as toluene, using the expression shown here, where IT is the scattering intensity of toluene.
Once Io is known, calculating the particle concentration within the sample is straightforward, with the intensity size distribution defining the percentage of the total scattering intensity attributed to each size bin within the distribution, and the previous concentration expression, along with the particle refractive index and Mie solution, used to calculate the concentration of each particle family within the sample.
To demonstrate the resolution improvements in the MADLS technique over traditional single angle DLS, a mixture of polystyrene spheres with diameters of 150 nm and 300 nm and similar scattering intensities were prepared in a 10 mM aqueous solution of NaCl. The single-angle intensity weighted particle size distributions results (backscatter, side scatter and forward scatter) are presented in Figure 5. Only a single population is discernible and the size is variable.
Figure 5: Intensity-weighted particle size distribution of a 150 and 300 nm polystyrene sphere dispersion measured using DLS in backscatter (red), side scatter (green) and forward scatter (blue).
These single angle DLS results can be explained using the background information presented earlier. The 1st Mie minima at a backscatter angle is circa 300 nm. Because of the reduced scattering intensity of the 300 nm component in backscatter, it is unresolvable from random correlogram noise, with the 150 nm latex scattering dominating the intensity PSD results. In the absence of a Mie minima for the side and forward scattering angles, the 300 nm signal dominates the correlograms, as well as the derived particle size distributions. The difference in the widths or polydispersity of the side and forward scatter results is attributable to the known physical polydispersity of the 300 nm latex standard, which contains a small amount of slightly larger latex spheres. The increased scattering of these larger particles at low angle leads to a spreading of the unresolved size peak and a subsequent increase in the peak mean.
The multi-angle MADLS result is presented in Figure 6, which shows a distinct peak corresponding to each component of the mixture, with near baseline resolution achieved for a 2:1 ratio by size.
Figure 6: Volume-weighted particle size distribution of a 150 and 300 nm polystyrene sphere dispersion measured using the Zetasizer Ultra MADLS technique.
While the MADLS technique leads to an improvement in DLS component resolution, there are still limitations to the amount of information that can be extracted, as well as sample suitability.
The concentration calculations are highly sensitive to the particle size. So high accuracy in the intensity weighted particle size distribution is essential. Restricted diffusion effects (sample viscosity) must be accounted for, as they will lead to large errors in the calculated concentrations.
Electrostatic repulsion effects on the apparent particle size cannot be corrected. So samples exhibiting electrostatic repulsion (increased diffusion) are not suitable for concentration measurements, with significant inaccuracy expected within the mass and number distributions.
While the MADLS technique minimizes inherent polydispersity in the intensity size peaks, the limiting assumptions regarding particle optical properties in the intensity to mass transform still apply, with the particles assumed to be spherical with a homogeneous and known refractive index. For core-shell type particles, such as liposome, viruses, coated nanoparticles, etc., the homogeneous particle assumption may still be valid, but would necessitate determination of an average particle refractive index value appropriate to the sample type and size.
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. Samples with large populations (i.e. greater than 500 nm diameter) will only be measurable after specific optimization of the concentration, if at all.
The MADLS technique has been explained for users interested in how the measurement data is collected and analyzed. The fundamental basis of the improved resolution over single-angle measurements is a reduction in correlogram noise, stemming from three mechanisms:
The reduction in correlogram noise facilitates use of an enhanced resolution NNLS algorithm, which utilizes a larger number of size bins (= small bin size) and fits to data from multiple measurement angles simultaneously. This fitting approach facilitates extraction of more highly resolved size information, in the absence of inherent peak polydispersity or any false squeezing of the peak arising from an overly aggressive alpha parameter, with the net result being a higher resolution intensity size distribution and a number transform that is more representative of the actual distribution of particles in the sample.
This MADLS technique allows users of the Zetasizer Ultra to gather additional insight into their sample by providing an angular independent result, bridging the gap between dynamic light scattering and high-resolution techniques such as SEC or TEM.
[1] P.G. Cummins, E.J. Staples, Particle Size Distributions Determined by a “Multiangle” Analysis of Photon Correlation Spectroscopy Data, Langmuir, 1987, 3, 1109-1113
[2] G. Bryant, J.C. Thomas, Improved Particle Size Distribution Measurements Using Multiangle Dynamic Light Scattering, Langmuir, 1995, 11, 2480-2485
