Dynamic light scattering
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Dynamic light scattering (DLS) is a technique in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
that can be used to determine the size distribution profile of small
particle In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from ...
s in
suspension Suspension or suspended may refer to: Science and engineering * Suspension (topology), in mathematics * Suspension (dynamical systems), in mathematics * Suspension of a ring, in mathematics * Suspension (chemistry), small solid particles suspende ...
or
polymers A polymer (; Greek '' poly-'', "many" + ''-mer'', "part") is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic an ...
in
solution Solution may refer to: * Solution (chemistry), a mixture where one substance is dissolved in another * Solution (equation), in mathematics ** Numerical solution, in numerical analysis, approximate solutions within specified error bounds * Soluti ...
. In the scope of DLS, temporal fluctuations are usually analyzed using the intensity or photon auto-correlation function (also known as photon correlation spectroscopy or quasi-elastic light scattering). In the time domain analysis, the autocorrelation function (ACF) usually decays starting from zero delay time, and faster dynamics due to smaller particles lead to faster decorrelation of scattered intensity trace. It has been shown that the intensity ACF is the Fourier transform of the
power spectrum The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...
, and therefore the DLS measurements can be equally well performed in the spectral domain. DLS can also be used to probe the behavior of complex fluids such as concentrated polymer solutions.


Setup

A monochromatic light source, usually a laser, is shot through a polarizer and into a sample. The scattered light then goes through a second polarizer where it is collected by a photomultiplier and the resulting image is projected onto a screen. This is known as a speckle pattern (Figure 1). All of the molecules in the solution are being hit with the light and all of the molecules diffract the light in all directions. The diffracted light from all of the molecules can either interfere constructively (light regions) or destructively (dark regions). This process is repeated at short time intervals and the resulting set of speckle patterns is analyzed by an autocorrelator that compares the intensity of light at each spot over time. The polarizers can be set up in two geometrical configurations. One is a vertical/vertical (VV) geometry, where the second polarizer allows light through that is in the same direction as the primary polarizer. In vertical/horizontal (VH) geometry the second polarizer allows light that is not in the same direction as the incident light.


Description

When light hits small particles, the light scatters in all directions (
Rayleigh scattering Rayleigh scattering ( ), named after the 19th-century British physicist Lord Rayleigh (John William Strutt), is the predominantly elastic scattering of light or other electromagnetic radiation by particles much smaller than the wavelength of th ...
) as long as the particles are small compared to the wavelength (below 250 nm). Even if the light source is a
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fi ...
, and thus is monochromatic and
coherent Coherence, coherency, or coherent may refer to the following: Physics * Coherence (physics), an ideal property of waves that enables stationary (i.e. temporally and spatially constant) interference * Coherence (units of measurement), a deri ...
, the scattering intensity fluctuates over time. This fluctuation is due to small particles in suspension undergoing
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
, and so the distance between the scatterers in the solution is constantly changing with time. This scattered light then undergoes either constructive or destructive interference by the surrounding particles, and within this intensity fluctuation, information is contained about the time scale of movement of the scatterers. Sample preparation either by filtration or centrifugation is critical to remove dust and artifacts from the solution. The dynamic information of the particles is derived from the autocorrelation of the intensity trace recorded during the experiment. The second order autocorrelation curve is generated from the intensity trace as follows: : g^2(q;\tau) = \frac where is the autocorrelation function at a particular wave vector, , and delay time, , and is the intensity. The angular brackets <> denote the expected value operator, which in some texts is denoted by a capital . At short time delays, the correlation is high because the particles do not have a chance to move to a great extent from the initial state that they were in. The two signals are thus essentially unchanged when compared after only a very short time interval. As the time delays become longer, the correlation decays exponentially, meaning that, after a long period has elapsed, there is no correlation between the scattered intensity of the initial and final states. This exponential decay is related to the motion of the particles, specifically to the diffusion coefficient. To fit the decay (i.e., the autocorrelation function), numerical methods are used, based on calculations of assumed distributions. If the sample is monodisperse (uniform) then the decay is simply a single exponential. The Siegert equation relates the second-order autocorrelation function with the first-order autocorrelation function as follows: : g^2(q;\tau)= 1+\beta\left ^1(q;\tau)\right2 where the first term of the sum is related to the baseline value (≈1) and the parameter is a correction factor that depends on the geometry and alignment of the laser beam in the light scattering setup. It is roughly equal to the inverse of the number of speckle (see
Speckle pattern Speckle, speckle pattern, or speckle noise is a granular noise texture degrading the quality as a consequence of interference among wavefronts in coherent imaging systems, such as radar, synthetic aperture radar (SAR), medical ultrasound and optic ...
) from which light is collected. A smaller focus of the laser beam yields a coarser speckle pattern, a lower number of speckle on the detector, and thus a larger second-order autocorrelation. The most important use of the autocorrelation function is its use for size determination.


Multiple scattering

Dynamic light scattering provides insight into the dynamic properties of soft materials by measuring single scattering events, meaning that each detected photon has been scattered by the sample exactly once. In principle, the DLS measurements can be performed with the detector positioned at any angle. The choice of the best angle depends on the sample properties, such as turbidity and particle size. Back scattering detection (e.g., 173° or 175°) is particularly interesting for turbid and highly concentrated samples, which contain large particles. Side scattering detection (90°) is recommended for weakly scattering samples, including small particles and transparent samples. Finally, forward scattering detection (e.g., 13° or 15°) is suitable for detection of samples containing small particles with few large particles. Some DLS instruments in the market also allow automatic angle selection based on a continuous transmittance measurement. At the lower end of the turbidity range, the Cavity Amplified Scattering Spectroscopy method makes use of an integrating cavity to elongate photon paths through quasi non-scattering samples. As opposed to conventional DLS instruments, this method is angle independent as it probes samples isotropically from all directions. Even though the DLS measurement using a single-angle detection has been the most diffuse technique, the application to many systems of scientific and industrial relevance has been limited due to often-encountered multiple scattering, wherein photons are scattered multiple times by the sample before being detected. Accurate interpretation becomes exceedingly difficult for systems with non-negligible contributions from multiple scattering. Especially for larger particles and those with high refractive index contrast, this limits the technique to very low particle concentrations, and a large variety of systems are, therefore, excluded from investigations with dynamic light scattering. However, as shown by Schaetzel, it is possible to suppress multiple scattering in dynamic light scattering experiments via a cross-correlation approach. The general idea is to isolate singly scattered light and suppress undesired contributions from multiple scattering in a dynamic light scattering experiment. Different implementations of cross-correlation light scattering have been developed and applied. Currently, the most widely used scheme is the so-called 3D-dynamic light scattering method. The same method can also be used to correct
static light scattering Static light scattering is a technique in physical chemistry that measures the intensity of the scattered light to obtain the average molecular weight ''Mw'' of a macromolecule like a polymer or a protein in solution. Measurement of the scattering ...
data for multiple scattering contributions. Alternatively, in the limit of strong multiple scattering, a variant of dynamic light scattering called diffusing-wave spectroscopy can be applied.


Data analysis


Introduction

Once the autocorrelation data have been generated, different mathematical approaches can be employed to obtain 'information' from it. Analysis of the scattering is facilitated when particles do not interact through collisions or electrostatic forces between ions. Particle-particle collisions can be suppressed by dilution, and charge effects are reduced by the use of salts to collapse the
electrical double layer A double layer (DL, also called an electrical double layer, EDL) is a structure that appears on the surface of an object when it is exposed to a fluid. The object might be a solid particle, a gas bubble, a liquid droplet, or a porous body. The D ...
. The simplest approach is to treat the first-order autocorrelation function as a single exponential decay. This is appropriate for a monodisperse population. : \ g^1(q;\tau)= \exp(-\Gamma\tau) \, where is the decay rate. The translational diffusion coefficient may be derived at a single angle or at a range of angles depending on the
wave vector In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
. : \ \Gamma=q^2D_t\, with : \ q = \frac\sin\left(\frac\right) where is the incident laser wavelength, is the solvent
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
and is the angle at which the detector is located with respect to the sample cell. The refractive index of the solvent plays a crucial role in light scattering and is important to calculate the
Stokes radius The Stokes radius or Stokes–Einstein radius of a solute is the radius of a hard sphere that diffuses at the same rate as that solute. Named after George Gabriel Stokes, it is closely related to solute mobility, factoring in not only size but also ...
from the Stokes-Einstein equation. Therefore, previous refractive index data from the scattering medium should be evaluated with dedicated instruments, known as
refractometer A refractometer is a laboratory or field device for the measurement of an Refractive index, index of refraction (refractometry). The index of refraction is calculated from the observed refraction angle using Snell's law. For mixtures, the index ...
s. Alternatively, DLS instruments containing a refractive index measurement module allow a good estimative for this important parameter within ±0.5%, which is the accuracy defined by ISO 22412:2017 for refractive index values required for DLS. Besides the refractive index of the medium, the refractive index of the particles is only necessary when analyzing larger particle size (usually above 100 nm) and volume- or number-weighted size distributions are needed. In these cases, prior knowledge of the refractive index and absorbance of the material is required in order to apply the
Mie scattering The Mie solution to Maxwell's equations (also known as the Lorenz–Mie solution, the Lorenz–Mie–Debye solution or Mie scattering) describes the scattering of an electromagnetic plane wave by a homogeneous sphere. The solution takes the f ...
. Depending on the
anisotropy Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...
and
polydispersity In chemistry, the dispersity is a measure of the heterogeneity of sizes of molecules or particles in a mixture. A collection of objects is called uniform if the objects have the same size, shape, or mass. A sample of objects that have an incons ...
of the system, a resulting plot of vs. may or may not show an angular dependence. Small spherical particles will show no angular dependence, hence no anisotropy. A plot of vs. will result in a horizontal line. Particles with a shape other than a sphere will show anisotropy and thus an angular dependence when plotting vs. . The intercept will be in any case the Dt. Thus, there is an optimum angle of detection for each particle size. A high-quality analysis should always be performed at several scattering angles (multiangle DLS). This becomes even more important in a polydisperse sample with an unknown particle size distribution. At certain angles the scattering intensity of some particles will completely overwhelm the weak scattering signal of other particles, thus making them invisible to the data analysis at this angle. DLS instruments which only work at a fixed angle can only deliver good results for some particles. Thus, the indicated precision of a DLS instrument with only one detection angle is only ever true for certain particles. is often used to calculate the
hydrodynamic radius The hydrodynamic radius of a macromolecule or colloid particle is R_. The macromolecule or colloid particle is a collection of N subparticles. This is done most commonly for polymers; the subparticles would then be the units of the polymer. R_ ...
of a sphere through the Stokes–Einstein equation. It is important to note that the size determined by dynamic light scattering is the size of a sphere that moves in the same manner as the scatterer. So, for example, if the scatterer is a random coil polymer, the determined size is not the same as the
radius of gyration ''Radius of gyration'' or gyradius of a body about the axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the body's actual distribution of mass, if the total mass of the body were concentr ...
determined by
static light scattering Static light scattering is a technique in physical chemistry that measures the intensity of the scattered light to obtain the average molecular weight ''Mw'' of a macromolecule like a polymer or a protein in solution. Measurement of the scattering ...
. It is also useful to point out that the obtained size will include any other molecules or solvent molecules that move with the particle. So, for example,
colloidal gold Colloidal gold is a sol or colloidal suspension of nanoparticles of gold in a fluid, usually water. The colloid is usually either wine-red coloured (for spherical particles less than 100  nm) or blue/purple (for larger spherical particl ...
with a layer of surfactant will appear larger by dynamic light scattering (which includes the surfactant layer) than by
transmission electron microscopy Transmission electron microscopy (TEM) is a microscopy technique in which a beam of electrons is transmitted through a specimen to form an image. The specimen is most often an ultrathin section less than 100 nm thick or a suspension on a g ...
(which does not "see" the layer due to poor contrast). In most cases, samples are polydisperse. Thus, the autocorrelation function is a sum of the exponential decays corresponding to each of the species in the population. : g^1(q;\tau)= \sum_^n G_i(\Gamma_i)\exp(-\Gamma_i\tau) = \int G(\Gamma)\exp(-\Gamma\tau)\,d\Gamma. It is tempting to obtain data for and attempt to invert the above to extract . Since is proportional to the relative scattering from each species, it contains information on the distribution of sizes. However, this is known as an
ill-posed The mathematical term well-posed problem stems from a definition given by 20th-century French mathematician Jacques Hadamard. He believed that mathematical models of physical phenomena should have the properties that: # a solution exists, # the sol ...
problem. The methods described below (and others) have been developed to extract as much useful information as possible from an autocorrelation function.


Cumulant method

One of the most common methods is the
cumulant In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will ha ...
method, from which in addition to the sum of the exponentials above, more information can be derived about the
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
of the system as follows: : \ g^1(q;\tau) = \exp\left(-\bar\left(\tau - \frac\tau^2 + \frac\tau^3 + \cdots\right)\right) where is the average decay rate and is the second order polydispersity index (or an indication of the variance). A third-order
polydispersity In chemistry, the dispersity is a measure of the heterogeneity of sizes of molecules or particles in a mixture. A collection of objects is called uniform if the objects have the same size, shape, or mass. A sample of objects that have an incons ...
index may also be derived but this is necessary only if the particles of the system are highly polydisperse. The z-averaged translational diffusion coefficient may be derived at a single angle or at a range of angles depending on the wave vector . : \ \bar=q^2D_z\, One must note that the cumulant method is valid for small and sufficiently narrow . One should seldom use parameters beyond µ3, because overfitting data with many parameters in a power-series expansion will render all the parameters, including \scriptstyle \bar and µ2, less precise. The cumulant method is far less affected by experimental noise than the methods below.


Size-distribution function

The particle size distribution can also be obtained using the autocorrelation function. However, polydisperse samples are not well resolved by the cumulant fit analysis. Thus, the combination of non-negative least squares (NNLS) algorithms with regularization methods, such as the
Tikhonov regularization Ridge regression is a method of estimating the coefficients of multiple-regression models in scenarios where the independent variables are highly correlated. It has been used in many fields including econometrics, chemistry, and engineering. Also ...
, can be used to resolve multimodal samples. An important feature of the NNLS optimization is the regularization term used to identify specific solutions and minimize the deviation between the measure data and the fit. There is no ideal regularization term that is suitable for all samples. The shape of this term can determine if the solution will represent a general broad distribution with small number of peaks or if narrow and discrete populations will be fit. Alternatively, the calculation of the particle size distribution is performed using the CONTIN algorithm.


CONTIN algorithm

An alternative method for analyzing the autocorrelation function can be achieved through an inverse Laplace transform known as CONTIN developed by Steven Provencher. The CONTIN analysis is ideal for heterodisperse, polydisperse, and multimodal systems that cannot be resolved with the cumulant method. The resolution for separating two different particle populations is approximately a factor of five or higher and the difference in relative intensities between two different populations should be less than 1:10−5.


Maximum entropy method

The Maximum entropy method is an analysis method that has great developmental potential. The method is also used for the quantification of sedimentation velocity data from
analytical ultracentrifugation Analytical ultracentrifugation is an analytical technique which combines an ultracentrifuge with optical monitoring systems. In an analytical ultracentrifuge (commonly abbreviated as AUC), a sample’s sedimentation profile is monitored in real tim ...
. The maximum entropy method involves several iterative steps to minimize the deviation of the fitted data from the experimental data and subsequently reduce the χ2 of the fitted data.


Scattering of non-spherical particles

If the particle in question is not spherical, the rotational motion must be considered as well because the scattering of the light will be different depending on orientation. According to Pecora, rotational Brownian motion will affect the scattering when a particle fulfills two conditions; they must be both optically and geometrically anisotropic. Rod-shaped molecules fulfill these requirements, so a rotational diffusion coefficient must be considered in addition to a translational diffusion coefficient. In its most succinct form, the equation appears as : \frac = \frac \frac Where is the ratio of the two relaxation modes (translational and rotational), contains information about the axis perpendicular to the central axis of the particle, and contains information about the axis parallel to the central axis. In 2007, Peter R. Lang and his team decided to use dynamic light scattering to determine the particle length and aspect ratio of short gold nanorods. They chose this method since it does not destroy the sample and it has a relatively easy setup. Both relaxation states were observed in VV geometry and the diffusion coefficients of both motions were used to calculate the aspect ratios of the gold nanoparticles.


Applications

DLS is used to characterize the size of various particles including proteins, polymers, micelles, Protein cages and virus-like particles, vesicles, carbohydrates, nanoparticles, biological cells, and gels. If the system is not disperse in size, the mean effective diameter of the particles can be determined. This measurement depends on the size of the particle core, the size of surface structures, particle concentration, and the type of ions in the medium. Since DLS essentially measures fluctuations in scattered light intensity due to diffusing particles, the diffusion coefficient of the particles can be determined. DLS software of commercial instruments typically displays the particle population at different diameters. If the system is monodisperse, there should only be one population, whereas a polydisperse system would show multiple particle populations. If there is more than one size population present in a sample then either the CONTIN analysis should be applied for photon correlation spectroscopy instruments, or the power spectrum method should be applied for Doppler shift instruments. Stability studies can be done conveniently using DLS. Periodical DLS measurements of a sample can show whether the particles aggregate over time by seeing whether the hydrodynamic radius of the particle increases. If particles aggregate, there will be a larger population of particles with a larger radius. In some DLS machines, stability depending on temperature can be analyzed by controlling the temperature ''in situ''.


See also

*
Scanning ion occlusion sensing Tunable Resistive Pulse Sensing (TRPS) is a single-particle technique used to measure the size, concentration and zeta potential of particles as they pass through a size-tunable nanopore. The technique adapts the principle of resistive pulse se ...
* Nanoparticle tracking analysis *
Diffusion coefficient Diffusivity, mass diffusivity or diffusion coefficient is a proportionality constant between the molar flux due to molecular diffusion and the gradient in the concentration of the species (or the driving force for diffusion). Diffusivity is enco ...
*
Fluorescence correlation spectroscopy Fluorescence correlation spectroscopy (FCS) is a statistical analysis, via time correlation, of stationary fluctuations of the fluorescence intensity. Its theoretical underpinning originated from L. Onsager's regression hypothesis. The analysis p ...
*
Stokes radius The Stokes radius or Stokes–Einstein radius of a solute is the radius of a hard sphere that diffuses at the same rate as that solute. Named after George Gabriel Stokes, it is closely related to solute mobility, factoring in not only size but also ...
*
Static light scattering Static light scattering is a technique in physical chemistry that measures the intensity of the scattered light to obtain the average molecular weight ''Mw'' of a macromolecule like a polymer or a protein in solution. Measurement of the scattering ...
*
Light scattering Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including ...
* Diffusing-wave spectroscopy * Protein–protein interactions * Differential dynamic microscopy *
Multi-angle light scattering Multiangle light scattering (MALS) describes a technique for measuring the light scattering, scattered by a sample into a plurality of angles. It is used for determining both the absolute molar mass and the average size of molecules in Solution (c ...
*
Differential Static Light Scatter (DSLS) Differential static light scatter (DSLS) is a term coined to represent the change in total light scatter of a system over time or temperature in a static environment. Static light scattering or SLS and its many types are well outlined in liter ...


References


External links


DLS to determine the radius of small beads in Brownian motion in a solutionParticle sizing using DLSDynamic Light Scattering for particle size characterization of proteins, polymers and colloidal dispersions
{{DEFAULTSORT:Dynamic Light Scattering Scattering, absorption and radiative transfer (optics) Biochemistry methods Physical chemistry Spectroscopy Colloidal chemistry Scientific techniques