Models hard sphere

Hard Sphere Model. Here the molecule is assumed to be the equivalent of a billiard ball. That is, the molecule is presented as a rigid sphere of diameter or, mass m (the molecular weight), and the capability 

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The hard sphere model considers each molecule to be an impenetrable sphere of diameter a so that... 

Hard-sphere models lack a characteristic energy scale and, hence, only entropic packing effects can be investigated. A more realistic modelling has to take hard-core-like repulsion at small distances and an attractive interaction at intennediate distances into account. In non-polar liquids the attraction is of the van der Waals type and decays with the sixth power of the interparticle distance r. It can be modelled in the fonn of a Leimard-Jones potential Fj j(r) between segments... 

In the theory of the liquid state, the hard-sphere model plays an important role. For hard spheres, the pair interaction potential V r) = qo for r < J, where d is the particle diameter, whereas V(r) = 0 for r s d. The stmcture of a simple fluid, such as argon, is very similar to that of a hard-sphere fluid. Hard-sphere atoms do, of course, not exist. Certain model colloids, however, come very close to hard-sphere behaviour. These systems have been studied in much detail and some results will be quoted below. 

Anotlier model system consists of polymetliylmetliacrylate (PMMA) latex, stabilized in organic solvents by a comb polymer, consisting of a PMMA backbone witli poly-12-hydroxystearic acid (PHSA) chains attached to it [10]. The PHSA chains fonn a steric stabilization layer at tire surface (see section C2.6.4). Such particles can approach tire hard-sphere model very well [111. 

The van der Waals surface (or the hard sphere model, also known as the scale model or the corresponding space-filling model) is the simplest representation of a molecular surface. It can be determined from the van dcr Waals radii of all... 

The first molecular dynamics simulation of a condensed phase system was performed by Alder and Wainwright in 1957 using a hard-sphere model [Alder and Wainwright 1957]. In this model, the spheres move at constant velocity in straight lines between collisions. All collisions are perfectly elastic and occur when the separation between the centres of... 

The solvent-excluded volume is a molecular volume calculation that finds the volume of space which a given solvent cannot reach. This is done by determining the surface created by running a spherical probe over a hard sphere model of molecule. The size of the probe sphere is based on the size of the solvent molecule. 

First, we would like to eonsider a simple hard sphere model in a hard sphere matrix, similar to the one studied in Refs. 20, 21, 39. However, our foeus is on a very asymmetric hard sphere mixture adsorbed in a disordered matrix. Moreover, having assumed a large asymmetry of diameters of the eomponents and a very large differenee in the eoneentration of eomponents, here we restriet ourselves to the deseription of the struetural properties of the model. Our interest in this model is due, in part, to experimental findings eoneerning the potential of the mean foree aeting between eolloids in a eolloidal dispersion in the presenee of a matrix of obstaeles [12-14]. 

A plot of A versus r, the calibration curve of OTHdC, is shown in Fig. 22.2. The value of constant C depends on whether the solvent/polymer is free draining (totally permeable), a solid sphere (totally nonpermeable), or in between. In the free-draining model by DiMarzio and Guttman (DG model) (3,4), C has a value of approximately 2.7, whereas in the impermeable hard sphere model by Brenner and Gaydos (BG model) (8), its value is approximately 4.89. 

Fig. 1.2 Hard-sphere model of face-centred cubic (f.c.c.) lattice showing various types of sites. Numbers denote Miller indices of atom places and the different shadings correspond to differences in the number of nearest neighbours (courtesy Erlich and Turnbull )...

So far the structure of pure metals has been discussed with reference to bulk characteristics and continuous crystals. However, corrosion is essentially a surface phenomenon and it is necessary to consider how the structure and defects already described interact with free surfaces. At this stage it is convenient to consider only a film-free metal surface, although of course in most corrosion phenomena the presence of surface films is of the utmost importance. Furthermore, it is at free surfaces that the hard sphere model of metals... 

In its most elementary aspects, kinetic theory is developed on the basis of a hard sphere model of the particles (atoms or molecules) making up the gas.1 The assumption is made that the particles are uniformly distributed in space and that all have the same speed, but that there are equal numbers of particles moving parallel to each coordinate axis. This last assumption allows one to take averages over... 

Where, the diffusivity D for the transfer of one gas in another is not known and experimental determination is not practicable, it is necessary to use one of the many predictive procedures. A commonly used method due to Gilliland 6 is based on the Stefan-Maxwell hard sphere model and this takes the form ... 

According to Vitanov et a/.,61,151 C,- varies in the order Ag(100) < Ag(lll), i.e., in the reverse order with respect to that of Valette and Hamelin.24 63 67 150 383-390 The order of electrolytically grown planes clashes with the results of quantum-chemical calculations,436 439 as well as with the results of the jellium/hard sphere model for the metal/electro-lyte interface.428 429 435 A comparison of C, values for quasi-perfect Ag planes with the data of real Ag planes shows that for quasi-perfect Ag planes, the values of Cf 0 are remarkably higher than those for real Ag planes. A definite difference between real and quasi-perfect Ag electrodes may be the higher number of defects expected for a real Ag crystal. 15 32 i25 401407 10-416-422 since the defects seem to be the sites of stronger adsorption, one would expect that quasi-perfect surfaces would have a smaller surface activity toward H20 molecules and so lower Cf"0 values. The influence of the surface defects on H20 adsorption at Ag from a gas phase has been demonstrated by Klaua and Madey.445... 

The C, values for Sb faces are noticeably lower than those for Bi. Just as for Bi, the closest-packed faces show the lowest values of C, [except Bi(lll) and Sb(lll)].28,152,153 This result is in good agreement with the theory428,429 based on the jellium model for the metal and the simple hard sphere model for the electrolyte solution. The adsorption of organic compounds at Sb and Bi single-crystal face electrodes28,152,726 shows that the surface activity of Bi(lll) and Sb(lll) is lower than for the other planes. Thus the anomalous position of Sb(lll) as well as Bi(lll) is probably caused by a more pronounced influence of the capacitance of the metal phase compared with other Sb and Bi faces28... 

Instead of the hard-sphere model, the Lennard-Jones (LJ) interaction pair potential can be used to describe soft-core repulsion and dispersion forces. The LJ interaction potential is... 

More realistic treatment of the electrostatic interactions of the solvent can be made. The dipolar hard-sphere model is a simple representation of the polar nature of the solvent and has been adopted in studies of bulk electrolyte and electrolyte interfaces [35-39], Recently, it was found that this model gives rise to phase behavior that does not exist in experiments [40,41] and that the Stockmeyer potential [41,42] with soft cores should be better to avoid artifacts. Representation of higher-order multipoles are given in several popular models of water, namely, the simple point charge (SPC) model [43] and its extension (SPC/E) [44], the transferable interaction potential (T1PS)[45], and other central force models [46-48], Models have also been proposed to treat the polarizability of water [49],... 

Figure 5.8. Left STM picture of the so-called clock reconstruction that occurs when 0.5 ML of carbon is adsorbed on Ni(lOO). Right Hard sphere model of the reconstruction. Dotted rings indicate the unreconstructed positions of...

Based on the molecular collision cross-section, a particle might undergo a collision with another particle in the same cell. In a probabilistic process collision partners are determined and velocity vectors are updated according to the collision cross-section. Typically, simple parametrizations of the cross-section such as the hard-sphere model for monoatomic gases are used. 

Interestingly, it has been shown that some adatoms can be selectively deposited on step sites, taking advantage of the enhanced reactivity of these sites. Figure 7.5 shows the voltammogram of a Pt(775) surface in 0.5 M H2SO4. The hard sphere model for... 

Figure 7.5 Cyclic voltammogram of a Pt(775) electrode in 0.5 M H2SO4 solution and a hard sphere model of this surface. Sweep rate 50 mV/s. In the hard sphere model, four atoms forming the (110) step site have been identified in black.

Whereas the Mg atoms are in contact with each other and the Cu atoms are in contact with each other, the Cu partial structure floats inside the Mg partial structure. The hard sphere model proves to be insufficient to account for the real situation atoms are not really hard. The principle of the most efficient filling space should rather be stated as the principle of achieving the highest possible density. Indeed, this shows up in the actual densities of the Laves phases they are greater than the densities of the components (in some cases up to 50 % more). For example, the density of MgCu2 is 5.75 g cm-3, which is 1% more than the mean density of 5.37 g cm-3 for 1 mole Mg + 2 moles Cu. Therefore,... 

Figure 5.7 The variation of the potential energy as two non-bonded atoms approach each other curve a, the hard sphere model curve b, a potential of the form V = C/r12.

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