Scattered light, correlation functions

Figure 4 Scattered light correlation functions for the AOT system at different distances from the critical point, from 0.07° (top curve) to 2.55° (bottom curve). The curves are fits with the droplet model (see Appendix). The corresponding values of the first cumulant T are given after AT F can be calculated by taking the t = 0 limit of the logarithmic derivative of C(t). (Data from Refs. 31 and 32.)...

Most macromolecules when dissolved in salt solutions acquire charges that are shielded by an atmosphere of counterions. This ion atmosphere affects the diffusion coefficient of the macromolecule and hence the light-scattering time-correlation function. Electrolyte solutions are discussed in Chapters 9 and 13. Recent measurements of diffusion coefficients have been made by several groups. Lee and Schurr (1974) have studied poly-L-lysine-HBr. Schleich and Yeh (1973) have performed similar studies on poly-L-proline. Raj and Flygare (1974) have studied bovine serum albumin (BSA) and find that at high ionic strength and low pH the diffusion constant decreases. This they attribute to the expansion of the molecule. 

Maeda and Saito (1973) have calculated the spectrum of light scattered from optically anisotropic rigid rods whose length is >q J. This calculation is even more complex than that for the integrated intensity (zero-time scattered-field correlation function) for the same model given in Appendix 8.B, and will, therefore, not be given here. The interested reader should consult the article by Maeda and Saito. 

Radiation probes such as neutrons, x-rays and visible light are used to see the structure of physical systems tlirough elastic scattering experunents. Inelastic scattering experiments measure both the structural and dynamical correlations that exist in a physical system. For a system which is in thennodynamic equilibrium, the molecular dynamics create spatio-temporal correlations which are the manifestation of themial fluctuations around the equilibrium state. For a condensed phase system, dynamical correlations are intimately linked to its structure. For systems in equilibrium, linear response tiieory is an appropriate framework to use to inquire on the spatio-temporal correlations resulting from thennodynamic fluctuations. Appropriate response and correlation functions emerge naturally in this framework, and the role of theory is to understand these correlation fiinctions from first principles. This is the subject of section A3.3.2. 

The correlator (6) is of the utmost importance because its generating function enters into an expression which describes the angular dependence of intensity of scattering of light or neutrons [3]. It is natural to extend expression (6) for the two-point chemical correlation function by introducing the w-point correlator ya1... (kl...,kn l) which equals the joint probability of finding in a macromolecule n monomeric units Maj.Ma> divided by (n-1) arbitrary sequences... 

Mw = 2.1 x 106g/mol) in water, which is denoted Cw(t) in the original work [44]. The subscript indicates that both the incoming beam and the scattered light are vertically polarized. The correlation function was recorded for a solution with a concentration of c = 0.005 g/L at a scattering vector of q = 8.31 x 106m-1. The inset shows the distribution function of the relaxation times determined by an inverse Laplace transformation. 

Photon correlation spectroscopy (PCS) has been used extensively for the sizing of submicrometer particles and is now the accepted technique in most sizing determinations. PCS is based on the Brownian motion that colloidal particles undergo, where they are in constant, random motion due to the bombardment of solvent (or gas) molecules surrounding them. The time dependence of the fluctuations in intensity of scattered light from particles undergoing Brownian motion is a function of the size of the particles. Smaller particles move more rapidly than larger ones and the amount of movement is defined by the diffusion coefficient or translational diffusion coefficient, which can be related to size by the Stokes-Einstein equation, as described by... 

This effective Q,t-range overlaps with that of DLS. DLS measures the dynamics of density or concentration fluctuations by autocorrelation of the scattered laser light intensity in time. The intensity fluctuations result from a change of the random interference pattern (speckle) from a small observation volume. The size of the observation volume and the width of the detector opening determine the contrast factor C of the fluctuations (coherence factor). The normalized intensity autocorrelation function g Q,t) relates to the field amplitude correlation function g (Q,t) in a simple way g t)=l+C g t) if Gaussian statistics holds [30]. g Q,t) represents the correlation function of the fluctuat-... 

With the availability of lasers, Brillouin scattering can now be used more confidently to study electron-phonon interactions and to probe the energy, damping and relative weight of the various hydro-dynamic collective modes in anharmonic insulating crystals.The connection between the intensity and spectral distribution of scattered light and the nuclear displacement-displacement correlation function has been extensively discussed by Griffin 236). 

Photon correlation light scattering was carried out by methods described elsewhere (6) to give the (unnormalized) correlation function G( )(q,T). Data were obtained with vertically polarized incident light for G 5)(q,T) and Gjj5)(q,T) obtained, respectively, with the vertical and horizontal components of the scattered light. 

CORRELATION FUNCTION LIGAND DIFFUSION TO RECEPTOR LIGHT SCATTERING (DYNAMIC)... 

The spectrum of scattered light contains dynamical information related to translational and internal motions of polymer chains. In the self-beating mode, the intensity-intensity time correlation function can be expressed (ID) as... 

EXAMPLE 5.5 Determination of the Effective Diameter of an Enzyme Using Dynamic Light Scattering. DLS analysis of a dilute solution of the enzyme phosphofructokinase in water at T = 293K leads to the following data for the correlation function g2(s,td) ... 

As a new subject we have considered the effect of the frequency-dependence of the elastic moduli on dynamic light scattering. The resultant nonexponential decay of the time-correlation function seems to be observable ubiquitously if gels are sufficiently compliant. Furthermore, even if the frequency-dependent parts of the moduli are very small, the effect can be important near the spinodal point. The origin of the complex decay is ascribed to the dynamic coupling between the diffusion and the network stress relaxation [76], Further scattering experiments based on the general formula (6.34) should be very informative. 

Time-dependent correlation functions are now widely used to provide concise statements of the miscroscopic meaning of a variety of experimental results. These connections between microscopically defined time-dependent correlation functions and macroscopic experiments are usually expressed through spectral densities, which are the Fourier transforms of correlation functions. For example, transport coefficients1 of electrical conductivity, diffusion, viscosity, and heat conductivity can be written as spectral densities of appropriate correlation functions. Likewise, spectral line shapes in absorption, Raman light scattering, neutron scattering, and nuclear jmagnetic resonance are related to appropriate microscopic spectral densities.2... 

Both Pecora (16) and Komarov and Fisher (17) adapted van Hove s space-time correlation function approach for neutron scattering (18) to the light-scattering problem to calculate the spectral distribution of the light scattered from a solution. Using a molecular analysis, Pecora assumed the scattering particles to be undergoing Brownian motion, and predicted a Lorentzian line shape for the spectral distribution of the... 

In dynamic or quasi-elastic light scattering, a time dependent correlation function (i (0) i (t)) = G2 (t) is measured, where i (0) is the scattering intensity at the beginning of the experiment, and i (t) that at a certain time later. Under the conditions of dilute solution (independent fluctuation of different small volume elements), the intensity correlation function can be expressed in terms of the electric field correlation function gi (t)... 

To examine in detail some of the time-correlation functions that enter into the theories of transport, light absorption, and light scattering and neutron scattering. 

Fig. 2.49 Dynamic structure factor at two temperatures for a nearly symmetric PEP-PDMS diblock (V = 1110) determined using dynamic light scattering in the VV geometry at a fixed wavevector q = 2.5 X 10s cm-1 (Anastasiadis et al. 1993a). The inverse Laplace transform of the correlation function for the 90 °C data is shown in the inset. Three dynamic modes (cluster, heterogeneity and internal) are evident with increasing relaxation times.

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