The density correlation length

Of special interest is the density correlation length defined as... 

It is important to note that in the inhomogeneous gel, the average crosslinking density is not a relevant parameter for determining the frictional pore size of the gel. It is the spatial correlation length of the density fluctuations that determines the bulk frictional behavior of water in the gel. 

A more subtle problem occurs for quantities involving several characteristic length scales, Consider for instance the density correlation function in the limit of large momenta (qi ff)2 > 1 where 1/q defines a length scale of interest, which is much smaller than Rg. In the excluded volume limit simple scaling considerations (cf. Sect. 9.1, Eq. (9.20)) suggest... 

In order to explain the interactions between silica surfaces, the polarization model is adapted to poorly-organized surfaces. To account for the disorder induced in water by the rough surfaces of silica, the dipole correlation length Am, which is the main parameter of the polarization model, is allowed to decrease from Am=14.9A obtained for water perfectly organized in ice-like layers in the vicinity of a surface to smaller values. For Am=4A, good agreement with experiment is obtained for reasonable values of the parameters involved (such as surface dipole and charge densities) [7.9],... 

Figure 16-8. Left two limiting behavior of the electron correlation length free (upper curve) and localized (lower). Right electron correlation lengths in SCW (circles) and SCA (squares) as function of the reduced density. The horizontal arrows indicate die correlation lengths for solvated electrons at die corresponding triple points of die two fluids. Change in slope indicates the onset of localization [28]...

Figure 8.14 The product of the tension of the liquid-vapor surface, cr, and the isothermal compressibility, Kj-, identified by Egelstaff and Widom (1970) as proportional to the spatial correlation length. This combination was suggested as appropriate for the low density of the coexisting vapor phase.

A more subtle problem occurs for quantities involving several characteristic length scales. Consider for instance the density correlation function in the limit of large momenta > 1 where 1/ defines a length scale of... 

Tt o- Taking the short correlation length 2 A, we roughly estimate that 77 6 A, so that the smallest " He cluster will consist of a central atom and its first coordination layer. Thus the threshold size domain for the realization of the lambda transition is TVmin " 5-13. Such a low value of Amin is consistent with the value Amin < 8 for the exhibition of the superfluid density in finite systems [155]. Finally, the threshold size for the appearance of rotons in the elementary excitation spectra of ( He) y clusters [128] is realized for 20 < Amin < 70 (Section l.D). 

Equation 14.29 defines the density correlation function C(r), where p(f) is the density of material at position r, and the brackets represent an ensemble average. In Equation 14.30, A is a normalization constant, D is the fractal dimension of the object, and d is the spatial dimension. Also in Equation 14.30 are the limits of scale invariance, a at the smaller scale defined by the primary or monomeric particle size, and at the larger end of the scale h(rl ) is the cutoff function that governs how the density autocorrelation function (not the density itself) is terminated at the perimeter of the aggregate near the length scale As the structure factor of scattered radiation is the Fourier transform of the density autocorrelation function. Equation 14.30 is important in the development below. 

In the framework of the scaling theory, the corona of a spherical micelle can be envisioned [53-56] as an array of concentric spherical shells of closely packed blobs. The blob size, (r) = r/grows as a function of the radial distance r from the center of the core. Each blob comprises a segment of the chain within the local correlation length of the monomer density fluctuations [57], and corresponds to a contribution to the free energy of steric repulsion between the coronal chains. After calculating the total number of blobs in the micellar corona, one finds fhe free energy (per coronal chain) as ... 

Here = /Lsm/0 Sm is the smectic correlation length in the direction along the smectic normal, Dq is the phase offset, which is due to the uncertainty of the determination of zero surface separation in the AFM force measurements and ao is the smectic period. S o is the amplitude of the smectic order on the surfaces and Lsm is the coefficient of the smectic gradient term in the LdG free energy density expansion. 

Thus, the angular dependence of the scattering intensity of a binary mixture can be analyzed as the Fourier transform of the density correlation function of any of the scattering particles scaled by the square of the difference in scattering lengths of the two components. 

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