# Dynamic light scattering

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Dynamic light scattering
Hypothetical Dynamic light scattering of two samples: Larger particles on the top and smaller particle on the bottom

Dynamic light scattering (also known as photon correlation spectroscopy or quasi-elastic light scattering) is a technique in physics, which can be used to determine the size distribution profile of small particles in suspension or polymers in solution.[1] 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 scattering) so 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:

$g^2(q;\tau) = \frac{\langle I(t)I(t+\tau)\rangle}{\langle I(t)\rangle^2}$

where g2(q;τ) is the autocorrelation function at a particular wave vector, q, and delay time, τ, and I 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 g1(q;τ) as follows:

$g^2(q;\tau)= 1+\beta\left[g^1(q;\tau)\right]^2$

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.

## 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. Particularly 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,[2] 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.[3][4] The same method can also be used to correct static light scattering data for multiple scattering contributions.[5] 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 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.

$\ g^1(q;\tau)= \exp(-\Gamma\tau) \,$

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.

$\ \Gamma=q^2D_t\,$

with

$\ q = \frac{4\pi n_0}{\lambda}\sin\left(\frac{\theta}{2}\right)$

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.[6] 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.

$g^1(q;\tau)= \sum_{i=1}^n G_i(\Gamma_i)\exp(-\Gamma_i\tau) = \int G(\Gamma)\exp(-\Gamma\tau)\,d\Gamma.$

It is tempting to obtain data for g1(q;τ) 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 cumulant method,[7][8] from which in addition to the sum of the exponentials above, more information can be derived about the variance of the system as follows:

$\ g^1(q,\tau) = \exp\left(-\bar{\Gamma}\tau\right) \left(1 + \frac{\mu_2}{2!}\tau^2 - \frac{\mu_3}{3!}\tau^3 + \cdots\right)$

where $\scriptstyle \bar{\Gamma}$ is the average decay rate and $\scriptstyle \mu_2/\bar{\Gamma}^2$ is the second order polydispersity index (or an indication of the variance). A third order polydispersity index may also be derived but this is only necessary 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.

$\ \bar{\Gamma}=q^2D_z\,$

One must note that the cumulant method is valid for small $\ \tau$ and sufficiently narrow G(Γ).[9] One should seldom use parameters beyond µ3, because overfitting data with many parameters in a power-series expansion will render all the parameters including $\scriptstyle \bar{\Gamma}$ and µ2, less precise.[10]

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.[11][12] The CONTIN analysis is ideal for heterodisperse, polydisperse and multimodal systems which 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 reducing the χ2 of the fitted data.

## References

1. ^ Berne, B.J.; Pecora, R. Dynamic Light Scattering; Wiley: New York, 1976
2. ^ Schatztel, K. Suppression of multiple-scattering by photon cross-correlation techniques. J. Mod. Opt. 1991, 38, 1849
3. ^ Urban, C.; Schurtenberger, P. Characterization of turbid colloidal suspensions using light scattering techniques combined with cross-correlation methods. J. Colloid Interface Sci. Nov 1 1998, 207 (1), 150–158.
4. ^ Block, I.; Scheffold, F. (2010). "Modulated 3D cross-correlation light scattering: Improving turbid sample characterization". Rev. Sci. Instruments 81 (12): 123107. Bibcode 2010RScI...81l3107B. doi:10.1063/1.3518961.
5. ^ Pusey, P.N. Suppression of multiple scattering by photon cross-correlation techniques. Curr. Opin. Colloid Interface Sci 1999, 4, 177
6. ^ Gohy, Jean-François; Varshney, Sunil K.; Jérôme, Robert (2001). "Water-Soluble Complexes Formed by Poly(2-vinylpyridinium)-block-poly(ethylene oxide) and Poly(sodium methacrylate)-block-poly(ethylene oxide) Copolymers". Macromolecules 34 (10): 3361. Bibcode 2001MaMol..34.3361G. doi:10.1021/ma0020483.
7. ^ Koppel, Dennis E. (1972). "Analysis of Macromolecular Polydispersity in Intensity Correlation Spectroscopy: The Method of Cumulants". The Journal of Chemical Physics 57 (11): 4814. Bibcode 1972JChPh..57.4814K. doi:10.1063/1.1678153.
8. ^ Frisken, Barbara J. (2001). "Revisiting the Method of Cumulants for the Analysis of Dynamic Light-Scattering Data". Applied Optics 40 (24): 4087–91. Bibcode 2001ApOpt..40.4087F. doi:10.1364/AO.40.004087. PMID 18360445.
9. ^ Hassan, Pa; Kulshreshtha, Sk (Aug 2006). "Modification to the cumulant analysis of polydispersity in quasielastic light scattering data". Journal of colloid and interface science 300 (2): 744–8. doi:10.1016/j.jcis.2006.04.013. ISSN 0021-9797. PMID 16790246.
10. ^ Chu, B (1992). Laser Light scattering: Basic Principles and Practice. Academic Press. ISBN 0121745511.
11. ^ Provencher, S (1982). "CONTIN: A general purpose constrained regularization program for inverting noisy linear algebraic and integral equations". Computer Physics Communications 27 (3): 229. Bibcode 1982CoPhC..27..229P. doi:10.1016/0010-4655(82)90174-6.
12. ^ Provencher, S. W. (1982). "A constrained regularization method for inverting data represented by linear algebraic or integral equations". Comp. Phys. Commun. 27 (3): 213–227. Bibcode 1982CoPhC..27..213P. doi:10.1016/0010-4655(82)90173-4.

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