Cavity function

At higher densities, the effect of indirect interactions is represented by the cavity function > (r,p,7), which multiplies tire Boltzirramr distribution... 

Only the first temi in this expansion is shown. It is identical to the last temi shown in the equation for> 2(/ ), which is the coefficient of in the expansion of tlie cavity function y(r). 

The function y r) is called the cavity function or background function. If u r) is discontinuous, g r) is discontinuous for the same values of r. Flowever, y r) is continuous. Eq. (12) is more correctly written as... 

If the cavity function does not change fast in the region of interest, then the derivative Viy(ri3) is small and thus the integral involved may be neglected. In this approximation on integrating Eq. (27), we obtain the integral equation... 

Alternative integral equations for the cavity functions of hard spheres can be derived [61,62] using geometrical and physical arguments. Theories and results for hard sphere systems based on geometric approaches include the scaled particle theory [63,64], and related theories [65,66], and approaches based on zero-separation theorems [67,68]. These geometric theories have been reviewed by Stell [69]. 

When the last equation is used in conjunction with Eq. (51), we obtain the final equation for the pair cavity function... 

Now, we would like to comment on some general features of the solutions of integral equations for the local density. We use superscripts H and P to abbreviate the solutions of the HNCl and PYl equations (6) and (7), respectively. By considering the limiting behavior of the cavity functions inside the solid one obtains... 

For simplicity, we restrict ourselves to a one-component dimerizing fluid, o + o 02, and define the one-particle cavity functions / = 0,1, for bonded and unbonded species as follows... 

Presland, R. B., and Dale, B. A. (2000). Epithelial structural proteins of the skin and oral cavity Function in health and disease. Crit. Rev. Oral Biol. Med. 11, 383-408. 

Thus, another approach consists in selecting some boundary conditions and properties. It is obvious that all exact correlation functions must satisfy and incorporate them in the closure expressions at the outset, so that the resulting correlations and properties are consistent with these criteria. These criteria have to include the class of Zero-Separation Theorems (ZSTs) [71,72] on the cavity function v(r), the indirect correlation function y(r) and the bridge function B(r) at zero separation (r = 0). As will be seen, this concept is necessary to treat various problems for open systems, such as phase equilibria. For example, the calculation of the excess chemical potential fi(iex is much more difficult to achieve than the calculation of usual thermodynamic properties since one of the constraints it has to satisfy is the Gibbs-Duhem relation... 

Here, we report some basic results that are necessary for further developments in this presentation. The merging process of a test particle is based on the concept of cavity function (first adopted to interpret the pair correlation function of a hard-sphere system [75]), and on the potential distribution theorem (PDT) used to determine the excess chemical potential of uniform and nonuniform fluids [73, 74]. The obtaining of the PDT is done with the test-particle method for nonuniform systems assuming that the presence of a test particle is equivalent to placing the fluid in an external field [36]. 

This relation is the potential distribution theorem [73, 74], which gives a physical interpretation of the cavity function in terms of the chemical potential, and the excess interaction generated by the test particle, Y j>2 u(rv)> yia the ensemble average of its Boltzmann factor. In numerical simulation, the use of such a test-particle insertion method is of prime importance in calculating the cavity function at small distances and particularly at zero separation. Note that if the particle labeled 1 approaches the particle labeled 2, a dumbbell particle [41] is created with a bond length L = r2 n corresponding to a dimer at infinite... 

This function is called the cavity function or background function. At low densities... 

A first contribution to J-p is caused by the bonding between two atoms which constitute one molecule. This direct intra-molecular interaction between the atoms is taken into considerations as a f -bond. In addition there are intra-molecular interactions of indirect nature between the both atoms of a molecule. These atoms affect each other indirectly by n point interactions with all remaining atoms and combinations of atoms. The so-called intra-molecular pair cavity function y (ryw) expresses the ensemble of all indirect interactions which appear between the atoms of a molecule in f-bonds [13] and establishes the searched correlation function for all indirect interactions between the atoms inside a molecule. TTie molecular DFT approach evaluates the cluster expansion to calculate y (rjvr) using TPT. This approximation takes into account only presentations with vertices n <— 2, for what reason it is called the single chain approximation (TPT1)[12]. [7,8]... 

This shows that the dielectric constant e of a polar solvent is related to the cavity function for two ions at large separations. One could extend this concept to define a local dielectric constant s(r) for the interaction between two ions at small separations. 

It is now possible to prepare cavities of a specific shape Furthermore there is a method to introduce into the cavity functional groups in a defined topology and some binding sites are available which meet the requirements for substrate binding. 

We have recently shown that lipoxygenases from plants can act on substrate lipids emulsified by proteins or bile salts [42], This means that the interfacial activation is not essential for lipoxygenase reaction in emulsion systems. The data of the three-dimensional structure of lipoxygenases have shown the absence of a lid but the presence of the cavity functioning as a path for a substrate lipid from the exterior of the enzyme molecule to the catalytic site [43]. It is thought that lipoxygenase uses this cavity for the uptake of substrate lipid from the oil droplet surface into the interior catalytic site without substantial conformational change, i.e., interfacial activation. 

Zhou Y, SteH G Chemical association in simple modek of molecular and ionic fluids. III. The cavity function, J Chem Phys 96(2) 1507-1515, 1992. 

Nolle, Rowan, and coworkers prepared cavity-functionalized porphyrin catalysts with diameter of about 9 A. The catalytic macrocycles strongly bound viologen derivatives in their own cavities through electrostatic and n-n stacking interactions K = 10 to 10 M ). The catalytic macrocycle with manganese porphyrin strongly binds a polymer substrate (polybutadiene) that is included in the cavity, and the macrocycle oxidizes polymer substrate complexes. 

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