Closure approximation spherical

For separations outside the hard core, the direct correlation functions have to be approximated. Classic closure approximations recently applied to QA models axe the Percus-Yevick (PY) closure [301], the mean spherical approximation (MSA) [302], and the hypernetted chain (HNC) closure [30]. None of these relations, when formulated for the replicated system, contains any coupling between different species, and wc can directly proceed to the limit n — 0. The PY closure then implies... 

Equations (130)-(132) are closure-independent results. For explicit results, one needs to evaluate these equations introducing a closure approximation. The mean spherical approximation (MSA) [226] used by Sehweizer and Curro... 

A similar closure approximation is the mean spherical approximation (MSA). The MSA was proposed by Lebowitz and Percus [83] for systems with hard core plus tail potentials. The MSA is expressed as... 

Another linearization of the HNC closure leads to the mean spherical approximation (MSA). For a fluid with a hard core, the MSA is... 

Quite recently, Pini et al. [56] have reported a new, thermodynamically self-consistent approximation to the OZ relation for a fluid of spherical particles for a pair potential given by a hard-core repulsion and a Yukawa attractive tail (Eq. (6)). The closure to the OZ equation they have proposed has the form... 

Once the degree of association is known, the structure of the bulk dimerizing fluid can be determined by implementation of the adequate closures, such as, for example, the extended mean spherical approximation (EMSA)... 

The EMSA requires the degree of dimerization A as an input parameter. This is quite disappointing. However, it ehminates the deficiency of the Percus-Yevick approximation, Eq. (38). The EMSA represents a simpHfied version, to obtain an analytic solution, of a more sophisticated site-site extended mean spherical approximation (SSEMSA) [67-69]. The results of the aforementioned closures can be used as an input for subsequent calculations of the structure of nonuniform associating fluids. 

Fixed-Roof Tanks. The effect of internal pressure on plate structures, including tanks and pressure vessels, is important to tank design. If a flat plate is subjected to pressure on one side, it must be made quite thick to resist bending or deformation. A shallow cone-roof deck on a tank approximates a flat surface and is typically built of 3/ 16-in. (4.76-mm) thick steel (Fig. 4a). This is unable to withstand more than a few inches of water column pressure. The larger the tank, the more severe the effect of pressure on the structure. As pressure increases, the practicality of fabrication practice and costs force the tank builder to use shapes more suitable for internal pressure. The cylinder is an economic and easily fabricated shape for pressure containment. Indeed, almost all large tanks are cylindrical. The problem, however, is that the ends must be closed. The relatively flat roofs and bottoms or closures of tanks do not lend themselves to much internal pressure. As internal pressure increases, tank builders use roof domes or spheres. The spherical tank is the most economic shape for internal pressure storage in terms of required thickness, but it is generally more difficult to fabricate than a dome- or umbrella-roof tank because of its compound curvature. 

From the various possible closures, the mean spherical approximation (MSA) [189] has found particularly wide attention in phase equilibrium calculations of ionic fluids. The Percus-Yevick (PY) closure is unsatisfactory for long-range potentials [173, 187, 190]. The hypemetted chain approximation (HNC), widely used in electrolyte thermodynamics [168, 173], leads to an increasing instability of the numerical algorithm as the phase boundary is approached [191]. There seems to be no decisive relation between the location of this numerical instability and phase transition lines [192-194]. Attempts were made to extrapolate phase transition lines from results far away, where the HNC is soluble [81, 194]. 

This another popular closure [43] deals with spherical particles fluids that interact through an infinite repulsive potential at short range u(r) = +oo for r mean-spherical approximation (MSA) is formulated in terms of an ansatz for the direct correlation function. In this approach, c(r) is supposed to be... 

The simplest closure relation is the so-called mean spherical approximation (MSA). This approximation is defined by the conditions [25,32]... 

Figure 11.25 (a) Shear rate ymax at which the first normal stress difference reaches its positive maximum Nimax. versus reduced concentration U/U predicted by the Smoluchowski equation using an exact spherical harmonic solution and using the Hinch-Leal approximate closure (see Baek et al. 1993b). These predictions use... 

The HMSA closure (10) has been used to solve the OZ equation it is an interpolation between the soft mean spherical approximation at small r and the HNC closure at large r. 

A treatment for polar solvents on the basis of the mean spherical approximation was first given by Wertheim [24, 25]. The closure conditions are based simply on the dipole-dipole interaction energy between the polar molecules in the system. Neglecting molecular polarizability, these conditions are... 

The evaluation of pre.scnts a closure problem because the term /4 ,ph, which is a function of time, is not known over the path of the system. However, i4 pb is closety related to the minimum surface area of the aero.sol—(hat is, the surface area that would be attained if each individual agglomerate particle became spherical. Indeed, the two would be equal if the aerosol were monodi.sperse with size v. When the rate of coalescence is fast compared with the collision rate, the minimum surface area can be approximated by the self-preserving size distribution for coalescing spheres (Chapter 7). 

Whether these requirements can be met depends on the model considered and on the closure relation involved for the calculation of the correlation functions. Examples for which Eq. (7.54) has actually been used pertain to the class of simple QA systems, that is, QA systems with no rotational degree of freedom where the interaction potentials contain a spherical hard-core contribution plus (at most) an attractive perturbation. For such sj stems, the free energy has been calculated on the basis of correlation functions in the mean sphericfxl approximation (or an optimized random-phase approximation) [114, 298). 

Marcus theory 37 Maxwell construction 46, 50, 51 mean spherical approximation (MSA) 8,49-51,171, 172,178-180 -closure 9,49,50,171,180 mechanical instability 51 melittin 122... 

Curro and Schweizer have carried out numerical [60,61] and analytical [23] studies of the symmetric blend using the Mean Spherical Approximation (MSA) closure successfully employed for atomic, colloidal, and small molecule fluids [5,6]. This closure corresponds to the approximation ... 

The simplest molecular closure based on the above ideas is one that builds in the hard core reference behavior and correctly treats the longer ranged attractive potentials in the weak coupling limit. It is called the Reference Molecular Mean Spherical Approximation (RMMSA) and is given in real space for a homopolymer blend by [68-70]... 

Haytcr [ 19] from the mean spherical approximation closure to the Omstcin-Zemike equation [20] ... 

The nuclear models that resulted in the prediction of an island of superheavy nuclei have evolved in response to experimental measurements of the decay properties of the heaviest elements. While the prediction of a spherical magic N = 184 is robust and persists across the models [8], the shell closure associated with Z — 114 is weaker, and different models place it at higher atomic numbers, from Z = 120 to 126 [60-69] or even higher [70] (see Nuclear Structure of Superheavy Elements ). Interpretation of the decay properties of the heaviest elements may support this [71, 72], but the most part decay and reaction data do not conclusively establish the location of the closed proton shell. Because of this, the domain of the superheavy elements can be considered to start at approximately Z = 106 (seaborgium), the point at which the liquid-drop fission barrier has vanished [9]. For our purposes, the transactinide elements (Z > 103) will be considered to be superheavy (see Nuclear Structure of Superheavy Elements ). 

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