Structure factor long-wavelength limit

The formal description of thermodiffusion in the critical region has been discussed in detail by Luettmer-Strathmann [79], The diffusion coefficient of a critical mixture in the long wavelength limit contains a mobility factor, the Onsager coefficient a = ab + Aa, and a thermodynamic contribution, the static structure factor S(0) [7, 79] ... 

These expressions are formally exact and the first equality in Eq. (123) comes from Euler s theorem stating that the AT potential u3(rn, r23) is a homogeneous function of order -9 of the variables r12, r13, and r23. Note that Eq. (123) is very convenient to realize the thermodynamic consistency of the integral equation, which is based on the equality between both expressions of the isothermal compressibility stemmed, respectively, from the virial pressure, It = 2 (dp/dE).,., and from the long-wavelength limit S 0) of the structure factor, %T = p[.S (0)/p]. The integral in Eq. (123) explicitly contains the tripledipole interaction and the triplet correlation function g (r12, r13, r23) that is unknown and, according to Kirkwood [86], has to be approximated by the superposition approximation, with the result... 

One should notice here, that according to the compressibility equation and Eq. (35), the osmotic compressibility is given by the long wavelength limit (k - 0) of the static structure factor, i.e. 

Dynamic light scattering (DLS) is an effective technique to measure the collective diffusion coefficients by measuring the time correlation of the concentration fluctuations. These experiments allow us to can determine the collective modes of the system that couple to concentration fluctuations. In binary systems this connection is quite useful since there is only one independent concentration variable. Then one can obtain the collective diffusion coefficient theoretically from the dynamic structure factor in the long wavelength limit. 

In discussing the thermodynamics of the kinetic model description we will concentrate only on those properties which directly affect the behavior of S(k,(o). These are the isothermal compressibility xt, the specific heats Cy and Cp, and the adiabatic sound speed c . The compressibility is the long-wavelength limit of the static structure factor Sik), ... 

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