Correlation direct

The direct correlation fimction c(r) of a homogeneous fluid is related to the pair correlation fimction tiirough the Omstein-Zemike relation... 

The compressibility equation can also be written in tenns of the direct correlation fiinction. Taking the Fourier transfomi of the Omstein-Zemike equation... 

At the critical point p(5 Pldp)j. = 0, and the integral of the direct correlation fiuictioii remains finite, unlike the integral of h r). 

Rearranging this, we have the PY approximation for the direct correlation fimction... 

In the MS approximation, for hard core particles of diameter a, one approximates the direct correlation fiinction by... 

Theories based on the solution to integral equations for the pair correlation fiinctions are now well developed and widely employed in numerical and analytic studies of simple fluids [6]. Furtlier improvements for simple fluids would require better approximations for the bridge fiinctions B(r). It has been suggested that these fiinctions can be scaled to the same fiinctional fomi 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 fiinction c r) between atoms in each molecule and a site-site Omstein-Zemike relation called the reference interaction site... 

In the limit k = (a/i) /i with L < all, the system should consist of dipolar dumb-bells. The asymptotic fonn of the direct correlation fiinction (defined tln-ough the Omstein-Zemike equation) for this system (in the absence of a solvent) is given by... 

The singlet direct correlation function C r) is defined through the relationship C Ur) = + --/t]... 

The PY equation is derived from a Taylor expansion of the direct correlation fimction, and has the fonn... 

Engstrom J R, Goodman D Wand Weinberg W H 1986 Hydrogenolysis of n-butane over the (111) and (110)-(1 2) surfaces of iridium a direct correlation between catalytic selectivity and surface structure J. Am. Chem. Soc. 108 4653... 

Chemical Antioxidant Systems. The antioxidant activity of tea extracts and tea polyphenols have been determined using in vitro model systems which are based on hydroxyl-, peroxyl-, superoxide-, hydrogen peroxide-, and oxygen-induced oxidation reactions (109—113). The effectiveness of purified tea polyphenols and cmde tea extracts as antioxidants against the autoxidation of fats has been studied using the standard Rancimat system, an assay based on air oxidation of fats or oils. A direct correlation between the antioxidant index of a tea extract and the concentration of epigallocatechin gallate in the extract was found (107). 

Crystallization and Melting Point. The abihty of PVA to crystallize is the single most important physical property of PVA as it controls water solubiUty, water sensitivity, tensile strength, oxygen barrier properties, and thermoplastic properties. Thus, this feature has been and continues to be a focal point of academic and industrial research (9—50). The degree of crystallinity as measured by x-ray diffraction can be directly correlated to the density of the material or the swelling characteristic of the insoluble part (Fig. 2). 

Method of Moments The first step in the analysis of chromatographic systems is often a characterization of the column response to sm l pulse injections of a solute under trace conditions in the Henry s law limit. For such conditions, the statistical moments of the response peak are used to characterize the chromatographic behavior. Such an approach is generally preferable to other descriptions of peak properties which are specific to Gaussian behavior, since the statisfical moments are directly correlated to eqmlibrium and dispersion parameters. Useful references are Schneider and Smith [AJChP J., 14, 762 (1968)], Suzuki and Smith [Chem. Eng. ScL, 26, 221 (1971)], and Carbonell et al. [Chem. Eng. Sci., 9, 115 (1975) 16, 221 (1978)]. 

The first step for any structure elucidation is the assignment of the frequencies (chemical shifts) of the protons and other NMR-active nuclei ( C, N). Although the frequencies of the nuclei in the magnetic field depend on the local electronic environment produced by the three-dimensional structure, a direct correlation to structure is very complicated. The application of chemical shift in structure calculation has been limited to final structure refinements, using empirical relations [14,15] for proton and chemical shifts and ab initio calculation for chemical shifts of certain residues [16]. 

Schnell, R., Stamm, M. and Creton, C., Direct correlation between interfacial width and adhesion in glassy polymers. Macromolecules, 31, 2284-2292 (1998). 

In collecting the data care should be taken to avoid data paralysis (see Part 2 Chapter 14). The various quality tools can be used to prioritize the identified problems and corresponding decisions. As with all data collection tasks, you should show a direct correlation between what you are collecting and the goals to be achieved and all conclusions should lead to positive action, otherwise the effort has been futile. 

A common practice is to paraphrase the requirements of ISO/TS 16949 as operational policy statements. Whilst this approach does provide direct correlation with ISO/TS 16949 it does not by itself add any value since users can read the same things by referring to ISO/TS 16949. Operational policies should respond to the requirements, not paraphrase them, and they should provide solutions appropriate to the organization, as given in the following examples ... 

The integrals are over the full two-dimensional volume F. For the classical contribution to the free energy /3/d([p]) the Ramakrishnan-Yussouff functional has been used in the form recently introduced by Ebner et al. [314] which is known to reproduce accurately the phase diagram of the Lennard-Jones system in three dimensions. In the classical part of the free energy functional, as an input the Ornstein-Zernike direct correlation function for the hard disc fluid is required. For the DFT calculations reported, the accurate and convenient analytic form due to Rosenfeld [315] has been used for this quantity. 

Most integral equations are based on the Ornstein-Zernike (OZ) equation [3-5]. The idea behind the OZ equation is to divide the total correlation function h ri2) iiito a direct correlation function (DCF) c r 12) that describes the fact that molecules 1 and 2 can be directly correlated, and an indirect correlation function 7( 12), that describes the correlation of molecule 1 with the other molecules that are also correlated with molecule 2. At low densities, when only direct correlations are possible, 7(r) = 0. At higher densities, where only triplet correlations are possible, we can write... 

For hard spheres, the PY approximation yields an analytic solution [15-17]. The result for the direct correlation function is... 

The equilibrium theory of homogeneous fluids may be constructed by using the hierarchy of the direct correlation functions [48]. This approach has been of much utility for the development of the theory of inhomogeneous simple fluids. The hierarchy of the direct correlation functions is defined by the following relation... 

We need, however, some means to calculate the direct correlation functions F2) of a nonuniform fluid. In order to make further progress, the... 

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