Quasi-elastic Light Scattering

QELS allows one to study the dyamics of a system in real time [see, e.g., Berne and Pecora, Dynamic Light Scattering, John Wiley & Sons, 1978]. In this method one sits at a particular scattering angle and measures the temporal autocorrelation function for that particular scattering vector. For example, in a binary mixture of two fluids, under proper conditions the two fluids will phase separate (each phase having a different composition from the original) in a continuous phase transition. Near the transition temperature T_c it does not "cost" much energy for fluctuations to occur, which diverge near T_c. These fluctuations can be spatially Fourier transformed so that at a given scattering angle, one measures a particular momentum transfer Q_scatter for light, given by Q_final - Q_incident. Here the magnitudes of both the incident and final momenta |Q_i,f| of the light are 2/ , where is the wavelength of light in the medium. The directions of these vectors are different, however. (Note that |Q_i,f| depends on the scattering angle as well as the refractive index of the medium).

For smaller scattering angles and therefore smaller |Q_scatter|, the concentration changes less rapidly and is therefore less costly; thus the fluctuations are larger and longer lasting. The larger momentum transfers represent shorter length scales where the fluctuations are smaller and die away more rapidly. By picking out a particular scattering wavevector, one can measure the intensity of scattered light (which corresponds to the susceptibility) and the autocorrelation function (which corresponds to the lifetime of the fluctuation) and observe both change as a function of temperature.

Another very common application of QELS is particle sizing. As particles translationally diffuse, the scattered light intensity fluctuates. [see Berne and Pecora]. One can think of this problem as follows: imagine a large number of statistically independent scatterers. At a given point in time one measures a scattering intensity I(t). At this time they can be thought of as one very powerful superscatterer located at the origin, such that the spatial Fourier components of this delta function all have the same amplitude. A short time later the particles have all translationally diffused in different directions, and their scattering fields will interfere to produce a different intensity. Statistically, this is equivalent to the one intense scatterer spreading out in accordance with the diffusion equation. Thus, instead of a delta function, at this later time the superparticle is washed out a bit in space, and now each spatial Fourier component has a different amplitude. At an even later time the particles will have diffused even more, and the super scatterer will be even more washed out in space. It turns out that the larger Q_scatter components of the spatial Fourier transform of the washed out distribition will have decayed away faster than the samller Q_scatter components. The light scattering geometry picks out just one Q_scatter component, and the intensity-intensity autocorrelation function tells you how that component decays in time. In a homodyning experiment, the decay time = (2DQ²)-1, where D is the diffusion constant given by the Stokes-Einstein relation. By measuring the decay time, one can thus determine the average "hydrodynamic radius" of the particle. In fact, by doing both static and quasi-elastic scattering, the hydrodynamic radius and radius of gyration will give you information about the shape of the particle, if it is not spherical.

In a liquid crystal experiment, the situation is complicated by the fact that the nematic and smectic phases are either uniaxial or biaxial, meaning that the incident and final momenta are determined by the direction of propagation and the polarization of the light. Judicious choice of scattering geometry may facilitate different sorts of measurements on the same system. A classical example is the study of "director fluctuations" in the nematic phase, corresponding to collective orientational modes of the molecular axis. For an appropriate incident momentum and polarization of light, the scattered light may be "depolarized," i.e., the polarization vector may change. Thus, detecting the light with a crossed polarizer would permit only this depolarized component of light to pass, and all other scattering (from both the liquid crystal and artifacts in the oven) would be blocked from the detector. A study of the time evolution of the scattered light then tells us information about the dynamics of the collective modes.

Applications of Light Scattering