Hard-sphere collisions

Figure A3.1.9. Hard sphere collision geometry in the plane of the collision. Here a is the diameter of the spheres.

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... 

We are concerned with bimolecular reactions between reactants A and B. It is evident that the two reactants must approach each other rather closely on a molecular scale before significant interaction between them can take place. The simplest situation is that of two spherical reactants having radii Ta and tb, reaction being possible only if these two particles collide, which we take to mean that the distance between their centers is equal to the sum of their radii. This is the basis of the hard-sphere collision theory of kinetics. We therefore wish to find the frequency of such bimolecular collisions. For this purpose we consider the relatively simple case of dilute gases. 

If the transition state theory is applied to the reaction of two hard spheres, the result is identical with that of simple collision theory. - pp Because transition state theory is an equilibrium theory, it can be inferred that collision theory is also an equilibrium theory. 

In general, the distribution function changes in time because of the underlying motion of the hard-spheres. Consider first the nonphysical case where there are no collisions. Phase-space conservation, or Louiville s Theorem [bal75], assures us that... 

Now let us add the possibility of collisions. Before we proceed, we make the following two assumptions (1) only binary collisions occur, i.e. we rule out situations in which three or more hard-spheres simultaneously come together (which is a physically reasonable assumption provided that the gas is sufficiently dilute), and (2) Boltzman s Stosszahlansatz, or his molecular chaos assumption that the motion of the hard-spheres is effectively pairwise uncorrelated i.e. that the pair-distribution function is the product of individual distribution functions ... 

Now, equation 9.27 must of course be modified in the presence of collisions. Since, by definition, the rate of change of f x, v, t) (Px(Pv is equal to the increase in hard-spheres in d xd v as a result of collisions, we have, in general, that... 

Consider a sphere-sphere collision. Let the incoming and outgoing distributions be given by /i, /2 and /i, /2, respectively. Also, assume that the pre- and postcollision velocity-space regions occupied by hard-spheres 1 and 2 are given by cPv, cPv and cPv2, cPv2, respectively. 

Boltzman s H-Theorem Let us consider a binary elastic collision of two hard-spheres in more detail. Using the same notation as above, so that v, V2 represent the velocities of the incoming spheres and v, V2 represent the velocities of the outgoing spheres, we have from momentum and energy conservation that... 

Fig. 9.6 Collision of two hard-spheres shown in a reference frame in which sphere 1 is at rest n is a unit vector along the line of the two centers of the colliding spheres.

Suppose that k — k(x, v) is some quantity associated with a hard-sphere such that in any collision between spheres that takes place at position x, we have that... 

Figure 7-3 shows an experimental study of the collision of hard spheres. The experimenter imparts energy of motion to the white ball (see Figure 7-3A, 7-3B). He does so by doing work by striking the ball with the end of a cylindrical stick (a cue). The amount of energy of motion (kinetic energy) received by the ball is fixed by the amount of work done. If the ball is struck softly (little work being done), it moves slowly. If the ball is struck hard (much work being done), it moves rapidly. The kinetic energy of the white ball appears because work was done—the amount of work, IV, determines and equals the amount of kinetic energy, (KE),. In symbols,...

For hard sphere collisions, v(v) would be proportional to v9 and the mean free path independent of v A(v) is an equivalent mean free path for a- general force law. Cf. S. Chapman and T. G. Cowling, The Mathematical Theory of Non- Uniform Oases, pp. 91 and 348, Cambridge University Press, 1958. 

This is an indication of the collective nature of the effect. Although collisions between hard spheres are instantaneous the model itself is not binary. Very careful analysis of the free-path distribution has been undertaken in an excellent old work [74], It showed quite definite although small deviations from Poissonian statistics not only in solids, but also in a liquid hard-sphere system. The mean free-path X is used as a scaling length to make a dimensionless free-path distribution, Xp, as a function of a free-path length r/X. In the zero-density limit this is an ideal exponential function (Ap)o- In a one-dimensional system this is an exact result, i.e., Xp/(Xp)0 = 1 at any density. In two dimensions the dense-fluid scaled free-path distributions agree quite well with each other, but not so well with the zero-density scaled distribution, which is represented by a horizontal line (Fig. 1.21(a)). The maximum deviation is about... 

Einwohner T., Alder B. J. Molecular dynamics. VI. Free-path distributions and collision rates for hard-sphere and square-well molecules, J. Chem. Phys. 49, 1458-73 (1968). 

As described above, the magnitude of Knudsen number, Kn, or inverse Knudsen number, D, is of great significance for gas lubrication. From the definition of Kn in Eq (2), the local Knudsen number depends on the local mean free path of gas molecules,, and the local characteristic length, L, which is usually taken as the local gap width, h, in analysis of gas lubrication problems. From basic kinetic theory we know that the mean free path represents the average travel distance of a particle between two successive collisions, and if the gas is assumed to be consisted of hard sphere particles, the mean free path can be expressed as... 

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 ... 

In what follows, unless specified otherwise the breaking and joining parameters, Pb and J, will be assigned the neutral values Pb = TO and J = 1.0 appropriate to hard-sphere (billiard ball) collisions. In some cases it will be of interest to depart from this simple model and to alter these values to find the influences of intermolecular attractions and repulsions on the results. 

If we assume that molecules can be considered as billiard balls (hard spheres) without internal degrees of freedom, then the probability of reaction between, say, A and B depends on how often a molecule of A meets a molecule of B, and also if during this collision sufficient energy is available to cross the energy barrier that separates the reactants, A and B, from the product, AB. Hence, we need to calculate the collision frequency for molecules A and B. 

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. 

The second generalization is the reinterpretation of the excluded volume per particle V(). Realizing that only binary collisions are likely in a low-density gas, van der Waals suggested the value Ina /I for hard spheres of diameter a and for particles which were modeled as hard spheres with attractive tails. Thus, for the Lennard-Jones fluid where the pair potential actually is... 

Given that a collision takes place, the nature of the momentum transfer between the cells must be specified. This should be done in such a way that the total momentum and kinetic energy on the double cell are conserved. There are many ways to do this. A multiparticle collision event may be carried out on all particles in the pair of cells. Alternatively, a hard sphere collision can be mimicked by exchanging the component of the mean velocities of the two cells along da,... 

Lattice gas models are simple to construct, but the gross approximations that they involve mean that their predictions must be treated with care. There are no long-range interactions in the model, which is unrealistic for real molecules the short-range interactions are effectively hard-sphere, and the assumption that collisions lead to a 90° deflection in the direction of movement of both particles is very drastic. At the level of the individual molecule then, such a simulation can probably tell us nothing. However, at the macroscopic level such models have value, especially if a triangular or hexagonal lattice is used so that three-body collisions are allowed. 

Termolecular Reactions. If one attempts to extend the collision theory from the treatment of bimolecular gas phase reactions to termolecular processes, the problem of how to define a termolecular collision immediately arises. If such a collision is defined as the simultaneous contact of the spherical surfaces of all three molecules, one must recognize that two hard spheres will be in contact for only a very short time and that the probability that a third molecule would strike the other two during this period is vanishingly small. 

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