Dynamic light scattering
Encyclopedia
Dynamic light scattering (also known as photon correlation spectroscopy or quasi-elastic light scattering) is a technique in physics
that can be used to determine the size distribution profile of small particles in suspension
or polymers in solution
. It can also be used to probe the behavior of complex fluids such as concentrated polymer solutions.
) as long as the particles are small compared to the wavelength (below 250 nm
). If the light source is a laser
, and thus is monochromatic
and coherent
, then one observes a time-dependent fluctuation in the scattering intensity. These fluctuations are due to the fact that the small molecules in solutions are undergoing Brownian motion
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 an autocorrelation of the intensity trace recorded during the experiment. The second order autocorrelation curve is generated from the intensity trace as follows:
where
is the autocorrelation
function at a particular wave vector,
, and delay time, 
, and 
is the intensity. 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 time 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
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:
where the parameter β 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
) from which light is collected. The most important use of the autocorrelation function is its use for size determination.
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.
The simplest approach is to treat the first order autocorrelation function as a single exponential decay. This is appropriate for a monodisperse population.
where Γ is the decay rate. The translational diffusion coefficient Dt may be derived at a single angle or at a range of angles depending on the wave vector q.
with
where λ is the incident laser wavelength, n0 is the refractive index
of the sample and θ is angle at which the detector is located with respect to the sample cell.
Depending on the anisotropy
and polydispersity of the system, a resulting plot of Γ/q2 vs. q2 may or may not show an angular dependence. Small spherical particles will show no angular dependence, hence no anisotropy. A plot of Γ/q2 vs. q2 will result in a horizontal line. Particles with a shape other than a sphere will show anisotropy and thus an angular dependence when plotting of Γ/q2 vs. q2. The intercept will be in any case the Dt.
Dt is often used to calculate the hydrodynamic radius
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
determined by static light 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
with a layer of surfactant will appear larger by dynamic light scattering (which includes the surfactant layer) than by transmission electron microscopy
(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.
It is tempting to obtain data for
and attempt to invert the above to extract G(Γ). Since G(Γ) 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 problem. The methods described below (and others) have been developed to extract as much useful information as possible from an autocorrelation function.
method, from which in addition to the sum of the exponentials above, more information can be derived about the variance
of the system as follows:
where
is the average decay rate and 
is the second order polydispersity index (or an indication of the variance). A third-order polydispersity 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 Dz may be derived at a single angle or at a range of angles depending on the wave vector q.
One must note that the cumulant method is valid for small
and sufficiently narrow G(Γ). One should seldom use parameters beyond µ3, because overfitting data with many parameters in a power-series expansion will render all the parameters including 
and µ2, less precise.
The cumulant method is far less affected by experimental noise than the methods below.
, 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.
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. The maximum entropy method involves a number of iterative steps to minimize the deviation of the fitted data from the experimental data and subsequently reduce the χ2 of the fitted data.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
that can be used to determine the size distribution profile of small particles in suspension
Suspension (chemistry)
In chemistry, a suspension is a heterogeneous fluid containing solid particles that are sufficiently large for sedimentation. Usually they must be larger than 1 micrometer. The internal phase is dispersed throughout the external phase through mechanical agitation, with the use of certain...
or polymers in solution
Solution
In chemistry, a solution is a homogeneous mixture composed of only one phase. In such a mixture, a solute is dissolved in another substance, known as a solvent. The solvent does the dissolving.- Types of solutions :...
. It can also be used to probe the behavior of complex fluids such as concentrated polymer solutions.
Description
When light hits small particles the light scatters in all directions (Rayleigh scatteringRayleigh scattering
Rayleigh scattering, named after the British physicist Lord Rayleigh, is the elastic scattering of light or other electromagnetic radiation by particles much smaller than the wavelength of the light. The particles may be individual atoms or molecules. It can occur when light travels through...
) as long as the particles are small compared to the wavelength (below 250 nm
Nanometre
A nanometre is a unit of length in the metric system, equal to one billionth of a metre. The name combines the SI prefix nano- with the parent unit name metre .The nanometre is often used to express dimensions on the atomic scale: the diameter...
). If the light source is a laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...
, and thus is monochromatic
Monochrome
Monochrome describes paintings, drawings, design, or photographs in one color or shades of one color. A monochromatic object or image has colors in shades of limited colors or hues. Images using only shades of grey are called grayscale or black-and-white...
and coherent
Coherence (physics)
In physics, coherence is a property of waves that enables stationary interference. More generally, coherence describes all properties of the correlation between physical quantities of a wave....
, then one observes a time-dependent fluctuation in the scattering intensity. These fluctuations are due to the fact that the small molecules in solutions are undergoing Brownian motion
Brownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...
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 an autocorrelation of the intensity trace recorded during the experiment. The second order autocorrelation curve is generated from the intensity trace as follows:
where
is the autocorrelationAutocorrelation
Autocorrelation is the cross-correlation of a signal with itself. Informally, it is the similarity between observations as a function of the time separation between them...
function at a particular wave vector,
, and delay time, , and is the intensity. 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 time 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 monodisperseMonodisperse
A collection of objects are called monodisperse, or monosized, if they have the same size and shape when discussing particles, and the same mass when discussing polymers...
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:
where the parameter β 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 pattern
A speckle pattern is a random intensity pattern produced by the mutual interference of a set of wavefronts. This phenomenon has been investigated by scientists since the time of Newton, but speckles have come into prominence since the invention of the laser and have now found a variety of...
) from which light is collected. 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. However, 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 nonnegligible contributions from multiple scattering. In particular 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 scatteringStatic 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. Measurement of the scattering intensity at many angles allows calculation of the root mean square...
data for multiple scattering contributions. Alternatively, in the limit of strong multiple scattering, a variant of dynamic light scattering called diffusing-wave spectroscopy
Diffusing-wave spectroscopy
Diffusing-wave spectroscopy is an optical technique derived from dynamic light scattering that studies the dynamics of scattered light in the limit of strong multiple scattering. It has been widely used in the past to study colloidal suspensions, emulsions, foams, gels, biological media and other...
can be applied.
Introduction
Once the autocorrelation data have been generated, different mathematical approaches can be employed to determine 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.The simplest approach is to treat the first order autocorrelation function as a single exponential decay. This is appropriate for a monodisperse population.
where Γ is the decay rate. The translational diffusion coefficient Dt may be derived at a single angle or at a range of angles depending on the wave vector q.
with
where λ is the incident laser wavelength, n0 is the refractive index
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....
of the sample and θ is angle at which the detector is located with respect to the sample cell.
Depending on the anisotropy
Anisotropy
Anisotropy is the property of being directionally dependent, as opposed to isotropy, which implies identical properties in all directions. It can be defined as a difference, when measured along different axes, in a material's physical or mechanical properties An example of anisotropy is the light...
and polydispersity of the system, a resulting plot of Γ/q2 vs. q2 may or may not show an angular dependence. Small spherical particles will show no angular dependence, hence no anisotropy. A plot of Γ/q2 vs. q2 will result in a horizontal line. Particles with a shape other than a sphere will show anisotropy and thus an angular dependence when plotting of Γ/q2 vs. q2. The intercept will be in any case the Dt.
Dt is often used to calculate the hydrodynamic radius
Hydrodynamic radius
The hydrodynamic radius of a macromolecule or colloid particle has two meanings. Some books use it as a synonym for the Stokes radius. Section 6.4 page 290....
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
Radius of gyration or gyradius is the name of several related measures of the size of an object, a surface, or an ensemble of points. It is calculated as the root mean square distance of the objects' parts from either its center of gravity or an axis....
determined by static light scattering
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. Measurement of the scattering intensity at many angles allows calculation of the root mean square...
. 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
Colloidal gold is a suspension of sub-micrometre-sized particles of gold in a fluid — usually water. The liquid is usually either an intense red colour , or a dirty yellowish colour ....
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
Transmission electron microscopy is a microscopy technique whereby a beam of electrons is transmitted through an ultra thin specimen, interacting with the specimen as it passes through...
(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.
It is tempting to obtain data for
and attempt to invert the above to extract G(Γ). Since G(Γ) 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 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 cumulantCumulant
In probability theory and statistics, the cumulants κn of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. The moments determine the cumulants in the sense that any two probability distributions whose moments are identical will have...
method, from which in addition to the sum of the exponentials above, more information can be derived about the variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
of the system as follows:
where
is the average decay rate and is the second order polydispersity index (or an indication of the variance). A third-order polydispersity 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 Dz may be derived at a single angle or at a range of angles depending on the wave vector q.
One must note that the cumulant method is valid for small
and sufficiently narrow G(Γ). One should seldom use parameters beyond µ3, because overfitting data with many parameters in a power-series expansion will render all the parameters including and µ2, less precise.The cumulant method is far less affected by experimental noise than the methods below.
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, polydispersePolydisperse
A collection of objects is called polydisperse or polysized if they have a broad range of size, shape and mass characteristics. A sample of objects that have a uniform size, shape and mass distribution are called monodisperse. In practice, polydisperse collections are common because it is...
, 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 entropyPrinciple of maximum entropy
In Bayesian probability, the principle of maximum entropy is a postulate which states that, subject to known constraints , the probability distribution which best represents the current state of knowledge is the one with largest entropy.Let some testable information about a probability distribution...
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. The maximum entropy method involves a number of iterative steps to minimize the deviation of the fitted data from the experimental data and subsequently reduce the χ2 of the fitted data.
See also
- Nanoparticle Tracking AnalysisNanoparticle tracking analysisNanoparticle Tracking Analysis is a method for visualizing and analyzing particles in liquids that relates the rate of Brownian motion to particle size. The rate of movement is related only to the viscosity and temperature of the liquid, it is not influenced by particle density or refractive index...
- Diffusion coefficient
- Fluorescence correlation spectroscopyFluorescence correlation spectroscopyFluorescence correlation spectroscopy is a correlation analysis of fluctuation of the fluorescence intensity. The analysis provides parameters of the physics under the fluctuations. One of the interesting applications of this is an analysis of the concentration fluctuations of fluorescent...
- Stokes radiusStokes radiusThe Stokes radius, Stokes-Einstein radius, or hydrodynamic radius RH, named after George Gabriel Stokes, is not the effective radius of a hydrated molecule in solution as often mentioned. Rather it is the radius of a hard sphere that diffuses at the same rate as the molecule. The behavior of this...
- Static light scatteringStatic light scatteringStatic 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. Measurement of the scattering intensity at many angles allows calculation of the root mean square...
- Light scatteringLight scatteringLight scattering is a form of scattering in which light is the form of propagating energy which is scattered. Light scattering can be thought of as the deflection of a ray from a straight path, for example by irregularities in the propagation medium, particles, or in the interface between two media...
- Diffusing-wave spectroscopyDiffusing-wave spectroscopyDiffusing-wave spectroscopy is an optical technique derived from dynamic light scattering that studies the dynamics of scattered light in the limit of strong multiple scattering. It has been widely used in the past to study colloidal suspensions, emulsions, foams, gels, biological media and other...
- Protein-protein interactions