Omstein-Zemike form

If, being the Fourier transform of u x). From Eqs. (4.3) and (4.10), the structure factor for the fluctuation is of the usual Omstein-Zemike form. 

There exist pre-transition effects in the isotropic phase heralding the I-N phase transition. Such pre-transition effects, which are consistent with the weakly first-order nature of the I-N transition, can be attributed to the development of short-range orientational order, which can be characterized by a position-dependent local orientational order parameter Q(r), where all component indices have been omitted [2]. In the Landau approximation, the spatial correlation function < G(0)G(r) > has the Omstein-Zemike form < G(0)G(r) exp(—r/ )/r, where is the coherence length or the second-rank orientational correlation length. The coherence length is temperature-dependent and the Landau-de Gennes theory predicts... 

Usually, h r) represented by eq 2.36 is called the Omstein-Zemike form. It shows that h r) decays faster with r as becomes smaller. Thus, may be taken as a measure of the range of r in which density fluctuations are effectively correlated. It is called the correlation length for density fluctuation [17]. In general, eq 2.33 holds only for small k so that h r) for r < is no longer the Zemike-Omstein form. This means that characterizes the decay rate of h r) in the tail region (r > (). To calculate h r) valid over the entire range of r (hence S k) or H k) valid for all k) we have to face a very difficult problem. An approximate solution by Benoit and Benmouna is described in Section 2.4. 

H) Near the spinodal, the pair correlation g r) becomes of long range and is well approximated by an Omstein-Zemike form ... 

The limiting behavior of the correlations at large distances is given by an Omstein-Zemike form (introduced in Section X.1.4). Inverting the Laplace transformation [eq. (X.39)], it is then possible to fmd the asymptotic form of or of [Pg.279]

Several studies have been directed at elucidating the limits of applicability of the mean-field approximation for mixtures of polymers. At small scattering wave vectors (0 0) Eqs. (7.25) and (7.26) reduce to the well-known Omstein-Zemike form [115],... 

From the many tools provided by statistical mechanics for determining the EOS [36, 173, 186-188] we consider first integral equation theories for the pair correlation function gxp(ra,rp) of spherical ions which relates the density of ion / at location rp to that of a at ra. In most theories gafi(ra,rp) enters in the form of the total correlation function hxp(rx,rp) = gxp(rx,rp) — 1. The Omstein-Zemike (OZ) equation splits up hap(rx,rp) into the direct correlation function cap(ra, rp) for pair interactions plus an indirect term that reflects these interactions mediated by all other particles y ... 

Theories based on the solution to integral equations for the pair correlation functions are now well developed and widely employed in numerical and analytic studies of simple fluids [6]. Further improvements for simple fluids would require better approximations for the bridge functions B r). It has been suggested that these functions can be scaled to the same functional form for different potentials. The extension of integral equation theories to molecular fluids was first accomplished by Chandler and Andersen [30] through the introduction of the site-site direct correlation function c r) between atoms in each molecule and a site-site Omstein-Zemike relation called the reference interaction site... 

In the limit X = (a/L) /r with L < a/2, the system should consist of dipolar dumb-bells. The as>miptotic form of the direct correlation function (defined through the Omstein-Zemike equation) for this system (in the absence of a solvent) is given by... 

The limit x > 1 corresponds to small correlated regions (small ) in this region a simple Omstein-Zemike picture becomes valid, and the spatial decay of correlations has the form 1/r exp (— r/f) (for d = 3). The limit jc = 0 corresponds to t = (f infinite) at t = Tc, the correlation decays slowly, like a power law We return to the region of small x in the... 

In combination with Eq. (12) the above relation can be put in form similar to the Omstein-Zemike equation,... 

As stated in the introduction, in this work we have used the PM in order to include the ion size in the EDL description. To this end, it is necessary to know the structure formed by counterions and coions around the macroions. Consequently, the Omstein-Zemike (OZ) formalism applied to isotropic systems could be a starting point for its description... 

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