HS model

FIG. IS Series of snapshot configurations obtained using lattices of side L = 1024 during the stationary regime of the HS model. (A) Cellular structures, (B) target patterns, (C) double spirals and (D) turbulence. (From Ref. 15.)... 

The HS model exhibits a rich variety of spatio-temporal patterns. During the oscillatory behavior, if the simulation starts with an empty grid in the hexagonal phase the only possible event is CO adsorption. Consequently, when a certain CO coverage is reached, the surface starts to convert into the 1 X 1 phase. Oxygen cannot adsorb yet, due to the lack of empty sites. 

When bounding walls exist, the particles confined within them not only collide with each other, but also collide with the walls. With the decrease of wall spacing, the frequency of particle-particle collisions will decrease, while the particle-wall collision frequency will increase. This can be demonstrated by calculation of collisions of particles in two parallel plates with the DSMC method. In Fig. 5 the result of such a simulation is shown. In the simulation [18], 2,000 representative nitrogen gas molecules with 50 cells were employed. Other parameters used here were viscosity /r= 1.656 X 10 Pa-s, molecular mass m =4.65 X 10 kg, and the ambient temperature 7 ref=273 K. Instead of the hard-sphere (HS) model, the variable hard-sphere (VHS) model was adopted in the simulation, which gives a better prediction of the viscosity-temperature dependence than the HS model. For the VHS model, the mean free path becomes ... 

Inspection of Eq. 7 reveals that the molecular interference function, s(x), can be derived from the ratio of the total cross-section to the fitted IAM function, when the first square bracketed factor has been accounted for. A widely used model of the liquid state assumes that the molecules in liquids and amorphous materials may be described by a hard-sphere (HS) radial distribution function (RDF). This correctly predicts the exclusion property of the intermolecular force at intermolecular separations below some critical dimension, identified with the sphere diameter in the HS model. The packing fraction, 17, is proportional for a monatomic species to the bulk density, p. The variation of r(x) on 17 is reproduced in Fig. 14, taken from the work of Pavlyukhin [29],... 

The present relations differ from the KM approximation since the factor 3 is replaced by the bridge function at zero separation. This feature does not seem to be unreasonable because, from diagrammatic expansions, B (r) = B r)/3 is supposed to be accurate only at very low densities. Eq. (112) presents two advantages at high density i) it provides a closed-form expression for Bother fluids than the HS model and ii) it allows to ensure a consistent calculation of the excess chemical potential by requiring only the use of the pressure consistency condition (the Gibbs-Duhem constraint, no longer required, is nevertheless implicitly satisfied within 1%). 

Houk and coworkers investigated the diastereofacial selectivity in radical additions of substituted cyclohexyl radicals to alkenes [13]. In this work, the force field developed by Spellmeyer and Houk was applied to intermolecular homolytic addition with success and demonstrated the added versatility of the HS model over the BS procedure which is limited to intramolecular systems. Extraordinarily accurate predictions of diastereoselectivity were made. For example, acrylonitrile is predicted to react with the 4-ter -butyl-2-methylcyclohexyl radical 28 to alford the products... 

Myers also found that the HS model provided accurate predictions of the regio-and stereochemical outcomes of ring closures aimed at the synthesis of cembranoid natural products [15]. Calculations for the cyclization of macrocyclic radicals 35... 

The reader is referred to numerous other examples in which these modeling methods have been employed with great success [16]. While these examples, and those provided above, demonstrate the versatility of both the BS and HS modeling methods and provide examples of strong performance, the reader should be aware that examples exist in which these methods perform poorly. Most notably, these are cases where considerable electronic influences exist in the radicals and transition states in question or where conformational factors lead to modeling difficulties. One early example involves the 6-methylenecyclodecyl radical (41) which was incorrectly predicted by the BS model to cyclize to afford the ra s -decalinylmethyl radical 43 (Scheme 7) [17]. Other examples include systems expected to be influenced by the anomeric effect, as in the case of allyloxytetrahydropyranyl radicals [18]. 

Potential 1 is extremely simple the only parameter is the radius of the sphere. In spite of this simplicity, an impressive number of physical results have been obtained using the HS potential on the whole range of densities and aggregations. Also, mixtures of liquids have been successfully treated, introducing in the computational machinery the desired number of spheres with appropriate radii. The HS model is at the basis of the Scaled Particle Theory (SPT), which still constitutes a basic element of modem solvation methods (see later for more details). 

The values of the dimensionless parameters 2 and CO for the most classic collision models are given in Table 1. The Maxwell molecules (MM) model assumes a linear relationship between viscosity and temperature, although for the hard sphere (HS) model, the viscosity is proportional to the square root of the temperature. These models could be roughly considered as limits for the real behavior of gases, and the variable hard sphere (VHS) model proposed by Bird [2] is much more accurate. Another sophistication has been proposed by Koura and Matsumoto who developed the variable soft... 

A corrective coefficient has been calculated by different authors [6] who have shown that a better prediction of the flow out of the Knudsen layer would be obtained with this corrective coefficient slightly different from unity, unlike as initially proposed by Maxwell. Its value depends on the collision model for example, = 1.11 for a HS model [7]. Equation 12 is called first-order slip boundary condition, because it involves the Knudsen number (9(Kn)) and the first derivative du Jdn ) . 

Thompson KE, Fogler HS. Modeling flow in disordered packed beds from pore-scale fluid mechanics. AIChE Journal 1997 43 1377-1389. 

For representing the repulsive nature of two-body interaction, HS model is widely adopted in hquid state theories. As depicted in Fig. 6, the HSs are defined simply as impenetrable spheres which cannot overlap with each other in space. 

Here x = 0A5 gives a value A(0) = 0.025. Kj-, in this approximation, is then a function of x T), T, and n T). McAlister and Crozier (1981) have recently measured the velocity of sound, v, in liquid La, Ce, Pr, and Yb, and in Ce-10% Yb alloy, and have obtained a measure of Kj from these data. (See table 3.) These data agree rather well with calculations of Kj based on Ahs(O) the agreement is improved further by incorporation of nearly free electron effects into the HS model. The anomalous sign of (dvJdT)p for liquid Ce may be associated with increasing delocalization of the 4f electron with increasing temperature this will be discussed in more detail later (see sections 3.2. and 3.3.). 

Yokoyama et al. (1988) have recently calculated S and along similar lines for the liquid R s, but have used a one-component plasma (OCP) model instead of the HS model. The results of their calculations of c are shown in table 8, and differ only slightly from results based on the HS model. One would wish to use eq. (34) to calculate Cp for a comparison with experiment, but a and jSj- are not available for these liquids. Instead, we note that for a wide variety of liquid metals, c /c is in the range of 1.1 to 1.3. (See Khanna 1981, Hafner 1977, Faber 1972.) Taking Cp/c = 1.2 as a reasonable average for these liquid R s, we obtain Cp in table 8. Comparison with the experimental values is moderately good, and... 

Ko GH, Ryou HS Modeling of droplet collision-induced breakup process, Int J Multiphase Flow 31 723-738, 2005. 

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