Dynamic light-scattering

Dynamic light scattering (DLS, also known as photon correlation spectroscopy, PCS) can be used in order to determine the diffusion coefficient (D) and thus the hydrodynamic size of particles. 

Making use of the Stokes-Einstein relation, the diffusion coefficient can be converted into the hydrodynamic diameter  

Besides as the hydrodynamic diameter and D as the diffusion coefficient, the equation contains Boltzmaim s constants k, the temperature T, and the solvent viscosity tj. 

In contrast to SLS, DLS allows analysis of smaller particles. For the characterization of mixtures with different components, for example, single 

2 Pulsed Field Gradient and Electrophoretic Nuclear Magnetic Resonance 

Dynamic light scattering (also known as photon correlation spectroscopy) is a measurement technique sensitive to the motions of particles in solution. As we learned previously in this chapter, all suspended particles in a colloidal solution are constantly subject to Brownian motion. Bombardment by solvent molecules leads to rotational, translational, and even more complicated conformational motions (in structured molecules such as polymer chains or proteins, for example). If we probe the solution with visible light, this constant particle motion will result in a time-varying fluctuation of the scattered intensity 7(0). Dynamic 

Imagine a dilute solution of suspended particles. These particles will diffuse a mean-squared distance f in time t according to the diffusion equation 

g t) is the correlation function, A and B are constants, and t is a characteristic time for the exponential decay function, also known as the correlation time or the relaxation time. This characteristic time t can 

Another way of thinking about the correlation function is as a function that describes how the dynamic particle system evolves over time. Imagine taking a snapshot of all particles in the solution at some start time. A short time later, the particles will all have moved slightly in random directions, but their new positions will still be correlated somewhat with the original positions. Later, there will be very little correlation between the original positions as the particles move around randomly. T is a time constant for the exponential decay of positional correlation. 

We can relate this behavior to measurements from a scattering experiment. The scattered intensity from a solution detected at a fixed angle 0 will fluctuate as a function of time because of particle motion, but if two intensity measurements are taken within a very short time interval, there 

The static light-scattering measurements described in the previous section provide a measure of the time-averaged scattered intensity. On the contrary, dynamic light scattering follows the random motion of molecules in solution called Browiuan 

FIGURE 9.7 A typical Zimm plot showing the double extrapolation technique, where - O-represents the experimental points and - - represents the extrapolated points. 

Critical phenomena of gels have been studied mainly by dynamic light scattering technique, which is one of the most well-established methods to study these phenomena [18-20]. Recently, the critical phenomena of gels were also studied by friction measurement [85, 86] and by calorimetry [55, 56]. In the case of these methods, the divergence of the specific heat or dissipation of the friction coefficient could be monitored as a function of an external intensive variable, such as temperature. These phenomena might be more plausible to some readers than the divergence of the scattered intensity since they can observe the critical phenomena in terms of a macroscopic physical parameter. 

The critical phenomena are described in terms of the critical exponents [47]. 

Here we describe briefly the recent studies of the critical phenomena of gels by dynamic light scattering, friction coefficient measurement, and calorimetry. Some of the latest results by neutron scattering are also given. 

Tanaka et aL found the divergence of the scattered intensity, I(q), and diminishing of the relaxation rate, T, at — 17°C when they studied acrylamide gel in 

The critical behaviors of E, and 1(0) was well explained by the mean field theory, 

1993 Chu 1991 Schmitz 1990). For example, the dynamic version of the diffusing wave spectroscopy described in Vignette V is a form of DLS, although in diffusing wave spectroscopy the method of analysis is different in view of multiple scattering. Most of the advanced developments are beyond the scope of this book. However, DLS is currently a routine laboratory technique for measuring diffusion coefficients, particle size, and particle size distributions in colloidal dispersions, and our objective in this section is to present the most essential ideas behind the method and show how they are used for particle size and size distribution measurements. 

16 Schematic illustration of intensity measurement and the corresponding autocorrelation function in dynamic light scattering (a) variation of the intensity of the scattered light with time (b) the variation of the autocorrelation function C(s,td) with the delay time td. 

In order to be able to use the fluctuation of the intensity around the average value, we need to find a way to represent the fluctuations in a convenient manner. In Section 5.3b in our discussion of Rayleigh scattering applied to solutions, we came across the concept of fluctuations of polarizabilities and concentration of scatterers and the role they play in light scattering experiments. In the present section, what we are interested in is the time dependence of such fluctuations. In general, it is not convenient to deal with detailed records of the fluctuations of a measured quantity as a function of time. Instead, one reduces the details of the fluctuations to what is known as the autocorrelation function C(s,td), as defined below  

The Siegert relation is valid except in the case of scattering volumes with a very small number of scatterers or when the motion of the scatterers is limited. We ignore the exceptions, which are rare in common uses of DLS, and consider only autocorrelations of the type shown in Equation (104). As mentioned above, modern DLS instruments use computer-controlled correlators to calculate the intensity autocorrelation function automatically and to obtain the results in terms of the function gi(s,/rf) therefore we only need to concern ourselves here with the interpretation of gi(s,td). 

1 Monosize Spherical Particles Measuring Diffusion Coefficient and Particle Size 

In equation (4.51) of Chapter 4, the equal time structure factor, C (x), was defined. For light scattered from fluctuations in the dielectric properties of a material, it was shown that the light intensity was proportional to this quantity. In the problem of total intensity light scattering discussed in that chapter, the measurement is integrated over time and time-dependent fluctuations are not directly observed. When time-dependent fluctuations are 

The fringe pattern is predicted to translate in space with a speed proportional to co 1. In 

The amplitude factor will vary on a timescale comparable to the transit time of a scattering element as it convects through the scattering volume. If this volume is characterized by a length scale, L, that timescale is 

Calculation of the autocorrelation function proceeds by noting that the single particle scattering function, F- (q, t), is simply the Fourier transform of G- (x ), the Van- 

Hove self space-time correlation function. [8] In other words, 

The diffusional or Brownian motion of molecules in a liquid or gas gives rise to fluctuations in density or concentration that can be observed by optical methods. 

Analysis of the shape and frequency of this flickering pattern gives the autocorrelation time (t), which is related to the diffusion constant (/)) of the molecules. This information can be used to determine the relative molecular masses and heterogeneities of macromolecular samples. 

1 General Concepts Determination of Particle Sizes in Dilute Solutions 

Although homodyne is the most used method in PCS, we describe shortly also the heterodyne method, which is widely used for Doppler velocimetry experiments or when the Siegert relation is not applicable. Heterodyning means that we mix in the detector the scattered light with a strong nonscattered signal (named commonly as the local oscillator), that is. 

Equation 18.52 becomes the dynamic equivalent of Equation 18.35 where D is the probability of 

We have omitted all the front factors since they are not relevant for the time dependence of the scattering intensity. 

Let us now consider the simplest case, a system consisting of diluted suspension of spheres, such as a latex particle suspension. As with any rigid body, there will be only two dynamic modes a translation of the center of mass and the rotation around the center of mass. For a sphere, the only mode that participates in the fluctuations in concentration will be the translational one since a rotation around the center of mass will have no effect on mass transfer in a scattering volume element. This means that in the case of a sphere, we can factorize t) as 

The regime kRg 1. In this regime, only the overall translational motion of the polymer can be seen because g k, t) is written as 

The underlined terms may be put at zero since they are of the order of kRg, which is much less than unity. Thus 

If t is large, the distribution of i o(r) -/ g(0) becomes Gaussianf with the variance 2Dat, hence 

Thus the decay of g(k, t) for a long time region is written as 

DLS is a widely used experimental method for examining microgel size and shape and has been used to monitor the temperature-induced swelling/de-swelling process. It is therefore a particularly useful technique to monitor the conformational behaviour of microgels in different solvent environments. 

This well-established technique measures the diffusion coefficient of particles through the decay of the intensity of the correlation function. To convert the measured diffusion coefficient (D) into particle diameter, the Stokes-Einstein relationship is used (equation (9.1))  

T is the temperature, k the Boltzmann constant, rj the viscosity of the solvent, and a is the diameter of a hydrodynamicaUy equivalent sphere. For an ideal dilute solution of non-interacting particles D corresponds to the self-diffusion coefficient [45]. 

There are numerous reports in the literature where DLS has been used to follow the collapse of the colloidal microgel particles across the VPT [45,47]. By comparing the hydro-dynamic diameter before and after the transition it is possible to determine the swelling ratio of the microgel. 

The decay of g r) with increasing r carries information relating to the rate of movement of the solute molecules. For a monodisperse solute 

For polydisperse solutes such as polymers, g r) approximates to a weighted sum of exponentials 

The translational diffusion coefficient of a molecule is related to its frictional coefficient, / , by the Einstein diffusion equation 

Photon correlation spectroscopy (PCS) is particularly suitable for studies of the hydrodynamic behaviour of polymers, and for studies of the effects of solvency conditions (e.g. solvent, temperature) and skeletal structure (e.g. linear, branched, cyclic) upon chain dimensions. The development of PCS has been greatly assisted by developments in multi-bit processors (which have enabled g r) to be determined with ever greater accuracy) and in the analytical methods by which information is extracted from 

Nowadays, most light scattering instruments are designed so that they are capable of performing both static (i.e. total intensity ) and dynamic light scattering measurements, the latter by PCS. 

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The scattering techniques, dynamic light scattering or photon correlation spectroscopy involve measurement of the fluctuations in light intensity due to density fluctuations in the sample, in this case from the capillary wave motion. The light scattered from thermal capillary waves contains two observables. The Doppler-shifted peak propagates at a rate such that its frequency follows Eq. IV-28 and... 

B. J. Berne and R. Pecora, Dynamic Light Scattering, Wiley, New York, 1976. 

Foam rheology has been a challenging area of research of interest for the yield behavior and stick-slip flow behavior (see the review by Kraynik [229]). Recent studies by Durian and co-workers combine simulations [230] and a dynamic light scattering technique suited to turbid systems [231], diffusing wave spectroscopy (DWS), to characterize coarsening and shear-induced rearrangements in foams. The dynamics follow stick-slip behavior similar to that found in earthquake faults and friction (see Section XU-2D). 

The dynamics of polymers at surfaces can be studied via dynamic light scattering (DLS), as described in Section IV-3C. A modification of surface DLS using an evanescent wave to probe the solution in a region near the interface has... 

Berne B J and Pecora R 1976 Dynamic Light Scattering (New York Wiley) ch 10... 

Schmitz K S 1990 An introduction to Dynamic Light Scattering by Macromoiecuies (New York Academic)... 

Pusey P N and Tough R A 1985 Dynamic Light Scattering ed R Pecora (New York Plenum) ch 4... 

Photon Correlation Spectroscopy. Photon correlation spectroscopy (pcs), also commonly referred to as quasi-elastic light scattering (qels) or dynamic light scattering (dls), is a technique in which the size of submicrometer particles dispersed in a Hquid medium is deduced from the random movement caused by Brownian diffusion motion. This technique has been used for a wide variety of materials (60—62). 

Ferre-D Amare, A.R., Burley, S.K. Use of dynamic light scattering to assess crystallizability of macromolecules and macromolecular assemblies. Structure 2 357-359,... 

Direct measurements of equilibrium stress-strain isotherms of SAH are complicated by the gel softness. Nevertheless, a number of experiments on compression and tension of the gels has been reported (see, for example, Refs. [18, 21, 42]). The method of dynamic light scattering is free from such inconveniences... 

Burchard, W. Static and Dynamic Light Scattering from Branched Polymers and Biopolymers. Vol. 48, pp. 1—24. 

Dynamic light scattering (DLS) Translational diffusion coefficient, hydrodynamic or Stokes radius branching information (when Rh used with Rg) Fixed 90° angle instruments not suitable for polysaccharides. Multi-angle instrument necessary. [3]... 

G(S) and G(X) have been estimated by quantifying the effect on molecular size distributions inferred from sedimentation velocity, gel permeation chromatography, and dynamic light-scattering measurements [58]. 

FIGURE 1 Effect of (sequential) extrusion of MLV dispersions through polycarbonate membrane filters (Unipore) with pore sizes of 1.0, 0.6, 0.4, 0.2, and 0.1 ym on the mean liposome diameter. DXR-containing MLV (phosphatidylcholine/phosphatidylserine/ cholesterol 10 1 4) mean diameter of nonextruded dispersion about 2 ym pH 4. Mean particle size determined by dynamic Light scattering (Nanosizer, Coulter Electronics). (From Crommelin and Storm, 1987.)... 

An alternative approach is the use of pH-sensitive fluorophores (Lichtenberg and Barenholz, lOSS). These probes are located at the lipid-water interface and their fluorescence behavior reflects the local surface pH, which is a function of the surface potential at the interface. This indirect approach allows the use of vesicles independent of their particle size. Recently, techniques to measure the C potential of Liposome dispersions on the basis of dynamic light scattering became commercially available (Muller et al., 1986). 

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