The Billiard Ball Model

This illustration shows an atom as John Dalton (1766-1844) imagined it. Many reference materials refer to Dalton s concept of the atom as the "billiard ball model. Dalton, however, was an avid lawn bowler. His concept of the atom was almost certainly influenced by the smooth, solid bowling balls used in the game. 

Physical chemists are well aware of the usefulness of models. An understanding of the fundamental properties of matter can hardly be gained from watching reality, requiring instead the posing of if-then questions that can be answered only by models. The nature of pressure or temperature of a gas as a collective property of its individual atomic or molecular constituents became obvious only through the billiard ball models of Clausius, Maxwell, and Boltzmann, despite our later insights that true atoms or molecules have quantized motion. 

The billiard ball model is unique in that 9 depends only on b, and not on g. 

The billiard ball model is unique in that 9 depends only on b, and not on g. From the previous result, the quantity b db can be determined ... 

Fig. 8.2 An example of a Partitioning CA reversible rule, /, mapping (2x 2)-blocks of two-valued states to (2 X 2-blocks / (2 X 2) —> (2 X 2). Note that this rule conserves the total number of I s (indicated by a solid circle) and O s (indicated by an empty square). The system that evolves under this rule is in fact a universal CA (see Billiard Ball Model, later in this section).

If we take the billiard-ball, or hard-sphere, model literally, we can calculate the excluded volume constant, b, from the diameter of the molecular billiard balls, ct. The centers of two billiard balls, each of radius a, can come no closer than r = a. Therefore, we can consider that around each molecule there is a... 

It is therefore remarkable that 100 years or so before the laws of thermodynamics were formulated, Daniel Bernoulli developed a billiard ball model of a gas that gave a molecular interpretation to pressure and was later extended to give an understanding of temperature. This is truly a wonderful thing, because all it starts with is the assumption that the atoms or molecules of a gas can be treated as if they behave like perfectly elastic hard spheres—minute and perfect billiard balls. Then Newton s laws of motion are applied and all the gas laws follow, together with a molecular interpretation of temperature and absolute zero. You have no doubt... 

We have been using the word, particle, to indicate that we are still referring to a billiard ball model of atoms or molecules. Molecules also have internal motion, the vibrations of their bonds, etc., so in that case, the velocity refers to the motion of the center of mass of the molecule treated as a whole. 

Since such a molecule has rotational energy, various authors have used the com-pletely smooth billiard ball model which is incapable of changing its rotational energy on collisions (i.e., it slips), while others have used a rough billiard ball with a coefficient aR (0 < aR < 1) to describe the extent to which its rotational energy is involved in a collision. 

Adopting the molecular billiard ball model considering molecules that are smooth and symmetric rigid elastic spheres not surrounded by fields of force, the molecules affect each other motion only at contact. 

To deduce the formula for the dense gas collision frequency a modified relation for the volume of the collision cylinder is required. As mentioned in chap 2, it is customary to consider the motion of particles 2 relative to the center of particles 1 (see Fig 2.2). For a binary molecular collision to occur the center of particle 2 must lie on the sphere of influence with radius di2 about the center of particle 1, see Fig 2.7. The radius of the sphere of influence is defined by (2.152). Besides, since the solid angle dk centered about the apse line k is conveniently used in these calculations in which the billiard ball molecular model is adopted, it is also necessary to specify the direction of the line connecting the centers of the two particles at the instant of contact [86]. The two angles 6 and 4> are required for this purpose. Moreover, when the direction of the apse line lies in the range of 0, 4> and 6 - - dO, 4> + d4>, at the instant of collision, the center of particle 2 must lie on the small rectangle da cut out on the sphere of influence of particle 1 by the angles dO and d< >. The area of this rectangle is ... 

In the discussion of dilute gases in sect 2.4.2 the corresponding surface area element is determined by the product da = bdbd(f>, as illustrated in Fig 2.10. For the billiard ball molecular model the link between the two surface element formulas when centered about the apse line is defined analogous to (2.159). 

In accordance with the ideas presented in sect 2.4.3, the corresponding dilute gas collision operator can be expressed analogous to (2.185). However, the operator is reformulated and defined in terms of k because the billiard ball molecular model is adopted. The details of the transformation is explained in sect. 2.11. The result is ... 

To illustrate this fact, we may adopt the simplest model fluid - the billiard-ball gas -and refer to the simple situation shown in Fig. 2 1. Here we consider a fluid made up of two species namely, black billiard balls and white billiard balls, which are identical apart from their color. By billiard-ball gas we mean that the molecules are modeled as hard spheres that interact only when they colhde. The motion of each billiard ball (or molecule) is stochastic and thus time dependent, but we assume that there is a nonzero, steady macroscopic velocity field u. At an initial moment in time, we imagine a configuration in which the two species are separated by a surface in the fluid that is defined to be locally... 

Onsager Dupuis (i960) and Jaccard (1959, 1964, 1965). The essence of the theory can be seen from a simple billiard-ball model of the molecular processes involved, while a detailed consideration of some of the mechanisms requires, as we shall... 

Such a billiard-ball model of a simple liquid, initiated by J.D. Bernal, has dominated theoretical studies/approaches towards the liquid state of matter for half a century. Such a billiard-ball model, and its generalization to include non-spherical shapes, works not only for liquid argon and krypton, but also for many liquids such as methane, ethane, and carbon tetrachloride, to name a few. However,... 

As electronic excitation is not considered within kinetic sputtering, the collisions can be likened to an atomic-scale billiard ball game that is initiated on primary ion impact. The valence electron shells of the atoms/ions involved would thus represent the billiard ball s surfaces. The linear cascade model, which describes the most prevalent form of ion-induced sputtering, at least that from atomic ions and ions comprising small molecules (common examples used in SIMS include 0 , 02 , Cs, Ar" ", Xe" ", and Ga ), assumes a specific form of kinetic sputtering in which a full isotropic collision cascade is produced close to the surface. This is one form of knock-on sputtering. 

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