The hard-sphere model

In the first models, due to Born and Madelung, the ions are described as hard spheres, put together in the most compact way (Kittel, 1990). 

Ionic radii The radii of these spheres are estimated from inter-atomic equilibrium distances measured in the bulk compounds. Typical values of 

The determinations of ionic radii have been constantly refined. The most recent compilations give values, not only as a function of the charge state of the atom but also as a function of its coordination number Z (Shannon, 1976). An increase in Z is always associated with an increase of the ionic radius e.g. r = 0.99 A if Z = 4, r = 1.12 A if Z = l,r = 1.24 A if Z = 9 and r = 1.39 A if Z = 12, for Na+. 

Madelung energies The lattice electrostatic energy involves a sum of elementary coulomb interactions between pairs of ions (ij), bearing charges Qi and Qj, at a distance Rij  

This is called the Madelung energy. For a binary crystal containing N formula units, in which the charges are equal to —Q and -I- nQ (e.g. Ti02 6 = 2, n = 2), m may be written  

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

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. 

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... 

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... 

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... 

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.

To simulate the particle-particle collision, the hard-sphere model, which is based on the conservation law for linear momentum and angular momentum, is used. Two empirical parameters, a restitution coefficient of 0.9 and a friction coefficient of 0.3, are utilized in the simulation. In this study, collisions between spherical particles are assumed to be binary and quasi-instantaneous. The equations, which follow those of molecular dynamic simulation, are used to locate the minimum flight time of particles before any collision. Compared with the soft-sphere particle-particle collision model, the hard-sphere model accounts for the rotational particle motion in the collision dynamics calculation thus, only the translational motion equation is required to describe the fluid induced particle motion. In addition, the hard-sphere model also permits larger time steps in the calculation therefore, the simulation of a sequence of collisions can be more computationally effective. The details of this approach can be found in the literature (Hoomans et al., 1996 Crowe et al., 1998). 

In a hard-sphere system, the trajectories of particles are determined by momentum conserving binary collisions. The interactions between particles are assumed to be pair-wise additive and instantaneous. In the simulation, the collisions are processed one by one according to the order in which the events occur. For not too dense systems, the hard-sphere models are considerably faster than the soft-sphere models. Note that the occurrence of multiple collisions at the same instant cannot be taken into account. 

In the following, we focus on the soft-sphere method since this really is the workhorse of the DPMs. The reason is that it can in principle handle any situation (dense regimes, multiple contacts), and also additional interaction forces—such as van der Waals or electrostatic forces—are easily incorporated. The main drawback is that it can be less efficient than the hard-sphere model. 

The conclusion is that the soft-sphere model can be used as an alternative for the hard-sphere model, as far as the calculation of the excess compressibility is concerned. 

Fig. 19. Simulation results for both the soft-sphere model (squares) and the hard-sphere model (the crosses), compared with the Carnahan-Starling equation (solid-line). At the start of the simulation, the particles are arranged in a FCC configuration. Spring stiffness is K = 70,000, granular temperature is 9 = 1.0, and coefficient of normal restitution is e = 1.0. The system is driven by rescaling.

These three structures are the predominant structures of metals, the exceptions being found mainly in such heavy metals as plutonium. Table 6.1 shows the structure in a sequence of the Periodic Groups, and gives a value of the distance of closest approach of two atoms in the metal. This latter may be viewed as representing the atomic size if the atoms are treated as hard spheres. Alternatively it may be treated as an inter-nuclear distance which is determined by the electronic structure of the metal atoms. In the free-electron model of metals, the structure is described as an ordered array of metallic ions immersed in a continuum of free or unbound electrons. A comparison of the ionic radius with the inter-nuclear distance shows that some metals, such as the alkali metals are empty i.e. the ions are small compared with the hard sphere model, while some such as copper are full with the ionic radius being close to the inter-nuclear distance in the metal. A consideration of ionic radii will be made later in the ionic structures of oxides. 

For the hard sphere model, when energy collision leads to reaction, the rate constant is given by... 

The interfacial solution layer contains h3 ated ions and dipoles of water molecules. According to the hard sphere model or the mean sphere approximation of aqueous solution, the plane of the center of mass of the excess ionic charge, o,(x), is given at the distance x. from the jellium metal edge in Eqn. 5-31 ... 

Fig. 6-21. Charge distribution profile across a metal/aqueous solution interface (M/S) (a) the hard sphere model of aqueous solution and the jellium model of metal (the jellium-sphere model), (b) the effective image plane (IMP) and the effective excess charge plane x, (c) reduction in distance /lxd,p to the closest approach of water molecules due to electrostatic pressure, o, = differential excess charge on the solution side og = total excess charge on the solution side Oy = total excess charge on the metal side.

The interfacial electric double layer is represented by a combination of the improved jeUium model on the metal side and the hard sphere model of ions and dipoles on the aqueous solution side. Then, the electric capacity, Ch, of the compact layer is given by Eqn. 5-34 ... 

Fig. 6-26. For the hard sphere model on metal electrodes (a) interfacial dipole induced by adsorbed water molecules and (b) interfadal dipole induced by contact adsorption of partially ionized bromine atoms. - 6 = charge number of adsoihed particle (z ). [From Schmickler, 1993.]...

Air at room temperature and pressure consists of 99.9% void and 0.1% molecules of nitrogen and oxygen. In such a dilute gas, each individual molecule is free to travel at great speed without interference, except during brief moments when it undertakes a collision with another molecule or with the container walls. The intermolecular attractive and repulsive forces are assumed in the hard sphere model to be zero when two molecules are not in contact, but they rise to infinite repulsion upon contact. This model is applicable when the gas density is low, encountered at low pressure and high temperature. This model predicts that, even at very low temperature and high pressure, the ideal gas does not condense into a liquid and eventually a solid. 

The ideal gas law is consistent with the assumption that molecules are point masses with no volume. However, the hard sphere model assumes that molecules are spheres with a finite diameters a and introduces a central concept of the mean free path k, which is the mean distance traveled by a molecule before collision with another molecule. It is given by... 

The coordination number of a cation depends on the number of anions or ligand atoms that can be fit around it in three dimensions (Fig. 2). In the hard-sphere model the coordination number is determined by the ratio of the radius of the cation to that of the anion... 

Mass Diffusivity in Liquid Metais and Ailoys. The hard-sphere model of gases works relatively well for self-diffusion in monatomic liquid metals. Several models based on hard-sphere theory exist for predicting the self-diffusivity in liquid metals. One such model utilizes the hard-sphere packing fraction, PF, to determine D (in cm /s) ... 

In this lab, the students determine the compression factor, (9) Z = PV/nRT, for Argon using the hard sphere model, the soft sphere model, and the Lennard-Jones model and compare those results to the compression factor calculated using the van der Waals equation of state and experimental data obtained from the NIST (70) web site. Figure 3 shows representative results from these experiments. The numerical accuracy of the Virtual Substance program is reflected by the mapping of the Lennard-Jones simulation data exactly onto the NIST data as seen in Figure 3. 

The parameters R and Rj in equation (5.32) are the radii of the equivalent hard spheres representing biopolymers i and y, respectively (where i = j for interactions between the same macroions). The equivalent hard sphere corresponds to the space occupied in the aqueous medium by a single biopolymer molecule (or particle) which is completely inaccessible to other biopolymers. In practice, the hard sphere model is a highly satisfactory description for many globular proteins. 

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