Dynamic 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.
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.
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 or destructively.
This process is repeated at short time intervals and the resulting set of speckle patterns
are 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 geometry, where the the second polarizer allows light through
that is in the same direction as the primary polarizer.
In vertical/horizontal geometry the second polarizer allows light not in same direction
as the incident light.
Description When light hits small particles, the light
scatters in all directions as long as the particles are small compared to the wavelength.
If the light source is a laser, and thus is monochromatic and coherent, the scattering
intensity fluctuates over time.
This fluctuation is 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.
The angular brackets <> denote the expected value operator, which in some texts is denoted
by a capital E. 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, 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 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.
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.
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 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 determine '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.
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.
Thus there is an optimum angle of detection for each particle size.
A high quality analysis should always be performed at several scattering angles.
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.
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 than by transmission electron microscopy.
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 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
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.
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, 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.
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.
Scattering of non-spherical particles If the particle in question is not spherical,
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
Where is the ratio of the two relaxation modes, 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 due to the fact that 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 size of various
particles including proteins, polymers, micelles, carbohydrates, and nanoparticles.
If the system is monodisperse, 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 CONTIN analysis must
be applied.
For more than two populations CONTIN analysis at several scattering angles is required.
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.
Additionally, in certain DLS machines, stability depending on temperature can be analyzed by
controlling the temperature in situ.
See also
References
External links DLS to determine the radius of small beads
in Brownian motion in a solution Particle sizing using DLS