Ornstein Zernicke

Figure 3. Illustration of the resonance peak at the antiferromagnetic wave vector Q = (n, tv) for hole-doped cuprates (after [12]). While the normal-state data (lower curve) may be described by an Ornstein-Zernicke form (see Eq. (3)), in the superconducting state a strong suppression for small frequencies occurs followed by a large peak at uires — 2Ao.

The second term of Eq. (4) (Ornstein-Zernicke function) accounts for the liquid-like contributions from the polymer network and provides the correlation length (mesh size or hydrodynamic blob size) of the PVFA-co-PBVU/silica and PVAm-co-PBVU/silica network meshes [100]. The mesh sizes for PVFA-co-PBVU/silica and PVAm-co-PBVU/silica hydrogel hybrids are listed in Table 2 as a function of temperature. 

A adsorption model for a templated porous material may be posed in terms of seven replica Ornstein-Zernicke (ROZ) integral equations [58-60] that relate the direct and total correlation functions, c, (r) and hij (r), respectively, of the matrix-adsorbate system... 

J. S. H0ye and G. Stell, Ornstein-Zernicke equation with a core condition and direct correlation function of Yukawa form. Mol. Phys. 32, 195-207 (1976). 

If A is the order parameter, assuming L° has no critical anomaly, Ornstein-Zernicke theory gives A°a . Typically, U has no critical anomaly, and we assume... 

The integral equations themselves are obtained by inserting various approximations to the direct correlation function c, (r) into the Ornstein-Zernicke equation. It is common practice to omit the bridge graphs Bij in these approximations. 

Once the integral equation is solved together with the Ornstein-Zernicke equation for a given potential Uijir), equations of, the type of Eqs. (69) and (70) may be used to obtain thermodynamic properties from the resulting g,y(r) functions. At the MM-level, Eq. (69) yields the osmotic pressure H instead of the total pressure p of the solution. 

These conditions together with the Ornstein-Zernicke equation allow the calculation of c, (r) at short distances and gijir) at long distances. This approximation is called... 

The starting point of many modern theories of fluids is the Ornstein Zernicke equation (0Z)(10), which relates the direct correlation function c j(r) and the total correlation function h j(r)... 

The kernel is a well known quantity that appears in several branches of theoretical physics. For example, evaluated for the electron gas, /xc is, up to a factor, the local field correction . To emphasize the correspondence to the effective interaction of Landau s Fermi-liquid theory, to which it reduces in the appropriate limit, /xc plus the bare Coulomb interaction is sometimes called the effective interaction , while in the theory of classical liquids the same quantity is referred to as the Ornstein-Zernicke function. 

It has been assumed that u 0.1 in order to be far from the symmetrical regime. This means that the labeled chains merging at the crosslink are 9 times shorter than the external chains. One can first assume that the quantity S(q) is constant this means that there are short-range order interactions between the copolymer molecules as in classical liquids this is not reasonable and has practically the same effect as changing the value of the X parameter. Therefore, we have to introduce a q dependence of J((j) It can be assumed, as a simple approximation, that J(q) obeys an Ornstein Zernicke type of law and has the form ... 

This intensity is made of two terms one is the scattering by the centers O of the copolymers and the other the contribution of the structure of the copolymer. The separation of these terms is not difficult to perform at large q (q H > 1 ) since the copolymer term vanishes rapidly when q increases. As an example, we have considered in Fig.(5) the case of a four-arms star, plotting ijq) as function of q. The curve 1 corresponds to S(q) =0 it is the contribution of the copolymer term and reaches a plateau. Curve 2 corresponds to an Ornstein Zernicke scattering function for J q) One sees clearly the q tail. This shows that, if there is a fractal exponent, it is theoretically possible to measure it. 

Fig. 5. Intensity scattered by stars with labeled centers as function of qR Curve 1 the centers are randomly distributed. Curve 2 they obey an Ornstein-Zernicke type distribution.

Equation 12 can also be considered as an Ornstein-Zernicke equation describing the degree of thermal composition fluctuations of correlation length The correlation length is evaluated from V 2 S(0) and becomes infinite at the critical point as described by the scaling law The... 

The mode coupling approximation for m (0 yields a set of equations that needs to be solved self-consistently. Hereby the only input to the theory is the static equilibrium structure factor 5, that enters the memory kernel directly and via the direct correlation function that is given by the Ornstein-Zernicke expression = (1 - l/5,)/p, with p being the average density. In MCT, the dynamics of a fluid close to the glass transition is therefore completely determined by equilibrium quantities plus one time scale, here given by the short-time diffusion coefficient. The theory can thus make rather strong predictions as the only input, namely, the equilibrium structure factor, can often be calculated from the particle interactions, or even more directly can be taken from the simulations of the system whose dynamics is studied. 

At low q, the intensity should start to saturate and follow an Ornstein-Zernicke law for q < 1/ ic. 

In SAXS analysis, the Ornstein-Zernicke (OZ) correlation function (Stanley, 1971), jf(r) is introduced by analyzing critical scattering. The correlation function is given by. 

---