Correlation function of concentr. fluctuations

Before using Eq. (13.7.8) to compute the correlation functions of concentration fluctuations it is useful to review its more standard applications. 

To study microphase separation transition (MST), we next consider the correlation function of concentration fluctuations, whose Fourier components give the intensity of scattered waves. We have polydisperse clusters whose polydispersity is controlled by the temperature T and the composition 4>. 

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-... 

Measurements of static light or neutron scattering and of the turbidity of liquid mixtures provide information on the osmotic compressibility x and the correlation length of the critical fluctuations and, thus, on the exponents y and v. Owing to the exponent equality y = v(2 — ti) a 2v, data about y and v are essentially equivalent. In the classical case, y = 2v holds exactly. Dynamic light scattering yields the time correlation function of the concentration fluctuations which decays as exp(—Dk t), where k is the wave vector and D is the diffusion coefficient. Kawasaki s theory [103] then allows us to extract the correlation length, and hence the exponent v. 

Our chief emphasis will be on electrolyte solutions in which one of the ionic species is a polyelectrolyte and the remaining species are small ions. The spectrum will then be proportional to the time-correlation function of the polyelectrolyte concentration fluctuation alone. 

The correlation functions for thermodynamic quantities fluctuations will be specially analyzed at the end of Chapter 2, after a detailed discussion of the special forms of the correlation functions of density and concentration fluctuations in solutions near the critical point, or rather, near the spinodal curve. 

As the spinodal is approached, the applicability of Equations 7, 8, 10, 11 disappear (Rk,9o 00 on the spinodal) due to the correlation of concentration fluctuations, so in this vicinity the formalism of correlation functions has to be employed to describe critical opalescence. 

A semidilute solution is characterized by the correlation length of density fluctuations. This correlation length is independent of the degree of polymerization N and decreases as a function of polymer concentration as... 

One of the most important principles of linear response theory relates the system s response to an externally imposed perturbation, which causes it to depart from equilibrium, to its equilibrium fluctuations. Indeed, the system response to a small perturbation should not depend on whether this perturbation is a result of some external force, or whether it is just a random thermal fluctuation. Spontaneous concentration fluctuations, for instance, occur all the time in equilibrium systems at finite temperatures. If the concentration c at some point of a liquid at time zero is (c) + 3c(r, t), where (c) is an average concentration, concentration values at time t + 8t dXt and other points in its vicinity will be affected by this. The relaxation of the spontaneous concentration fluctuation is governed by the same diffusion equation that describes the evolution of concentration in response to the external imposition of a compositional heterogeneity. The relationship between kinetic coefficients and correlations of the fluctuations is derived in the framework of linear response theory. In general, a kinetic coefficient is related to the integral of the time correlation function of some relevant microscopic quantity. 

Probe diffusion was determined using quasi-elastic light scattering spectroscopy. QELSS monitors the temporal evolution of concentration fluctuations by measuring the intensity I(q,t) of the light scattered at time t, and calculating the intensity-intensity correlation function... 

The existence of a critical point is observed when the correlation length associated with the correlation function of the order parameter diverges. By definition, a mean field theory ignores the fluctuations of the order parameter and affords satisfactory results only far away from the critical point in addition to the calculation of the entropy and enthalpy of mixing, de Gennes criticized this main point in the classical theories and observed that the variation of g(a) with the concentration and the distance (a) considered cannot be overlooked or neglected ... 

Another example is that of the self-diffusion and Brownian motion of chains in solution which results in a fluctuation of their concentration on a time scale of about 10 s. As mentioned in Chapter 6, quasi-elastic light scattering is the ideal technique to determine the diffusion coefficient D from the self-correlation function of the scattered intensity. 

In agreement with the DRS measurements of Wettonetol. (154,155], the distribution of correlation times from NMR was found to be broader in the blend compared to the pure polymers. The 2D patterns could not be described by either a single-exponential correlation function or a single stretched exponential a multimodal distribution (bimodal or trimodal) was found more appropriate where the relative weighting of the components was correlated with local concentration fluctuations. 

Here, the response function is given by the correlation function of local concentration fluctuations ... 

The correlation frmction g q,T) indicates the temporal decay behavior of concentration fluctuations in the fluid as a function of delay time r. In ideal systems (dilute solution, monodisperse particles, ideal Brownian diffusion only), it exhibits a single exponential decay with decay rate r ... 

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