Pair correlation function collision

The rest of this chapter is organized as follows. First, in Section 6.1, we consider the collision term for monodisperse hard-sphere collisions both for elastic and for inelastic particles. We introduce the kinetic closures due to Boltzmann (1872) and Enksog (1921) for the pair correlation function, and then derive the exact source terms for the velocity moments of arbitrary order and then for integer moments. Second, in Section 6.2, we consider the exact source terms for polydisperse hard-sphere collisions, deriving exact expressions for arbitrary and integer-order moments. Next, in Section 6.3, we consider simplified kinetic models for monodisperse and polydisperse systems that are derived from the exact collision source terms, and discuss their properties vis-d-vis the hard-sphere collision models. In Section 6.4, we discuss properties of the moment-transport equations derived from Eq. (6.1) with the hard-sphere collision models. Finally, in Section 6.5 we briefly describe how quadrature-based moment methods are applied to close the collision source terms for the velocity moments. 

Owing to the presence of the pair correlation function, the collision model in Eq. (6.2) is unclosed. Thus, in order to close the kinetic equation (Eq. 6.1), we must provide a closure for written in terms of /. The simplest closure is the Boltzmann Stofizahlansatz (Boltzmann, 1872) ... 

The goal of the next series of steps is to obtain a form of Eq. (6.13) wherein the pair correlation function depends on only one spatial location (i.e. the point where the particles touch during a collision). We start by adding and subtracting the pair distribution function / (x - r/pXi2, vi X, V2) to / (x, vi x + [Pg.219]

The next step is to provide a closure for the pair correlation function appearing in the collision source and collisional-flux terms. For moderately dense flows, the collision frequency for finite-size particles is known to be larger than that found using the Boltzmann Stofizahlansatz (Carnahan Starling, 1969 Enksog, 1921). In order to account for this effect, the pair correlation function can be modeled as the product of two single-particle velocity distribution functions and a radial distribution function ... 

Note that this assumption simply transforms the problem of modeling the pair correlation function into the new problem of modeling o-The usual model for go assumes that the radial distribution function depends neither explicitly on the collision angle (i.e. on X12) nor explicitly on x. The former amounts to assuming that the particle with velocity V2 has no preferential spatial direction relative to the particle with velocity vi. The radial distribution function can then be modeled as a function of the disperse-phase volume fraction. For example, a typical model is (Carnahan Starling, 1969)... 

In general, gap depends on the volume fractions of each particle type and on the particle diameters. However, it can also depend on other moments of the velocity distribution function. For example, if the mean particle velocities Uq. and Vp are very different, one could expect that the collision frequency would be higher on the upstream side of the slower particle type. The unit vector Xi2 denotes the relative positions of the particle centers at collision. If we then consider the direction relative to the mean velocity difference, (Uq, - U ) xi2, we can model the dependence of the pair correlation function on the mean velocity difference as °... 

This question examines a limitation of the pair correlation function. Imagine an experiment in which we excite molecule A in the gas phase into a high-energy state by the absorption of two photons and then transfer that energy to a different molecule B during a collision. For the first step, molecule A must not be next to any other molecules because then the energy of the first photon will be... 

In studying the intense laser interactions with the CS2 molecules, and observing the behaviour of the orientational distribution and pair correlation functions [g(S)J through the time-evolution of the optical field-induced anisotropy, we are emphasizing external field-induced phenomena rather than interaction-induced or collision-induced events. 

These models are still limited by the restriction to rotational tumbling of both the electron and the nucleus. Miiller-Warmuth and coworkers developed combined rotational and translational diffusion models (reviewed in [39]) for dipolar and scalar interactions, assuming independent diffusion of the molecules. They also developed a pulse diffusion model assuming occasional collisions between molecules, described via a Poisson process [39]. Later Hwang et al. [53] used force-free pair correlation functions to account for translational diffusion and ionic interactions for dipolar interactions, leading to the now commonly used equation ... 

D. A. Micha, Atom-polyatomic collisions The role of pair correlation functions, J. Chem. Phys. 70 3165 (1979). 

According to (4.38), the correlation function (5.4) determines the collision integral of the relevant kinetic equation for the distribution of pairs. 

The important feature in the final result (13.45) is the fact that all the needed dynamical information is associated with the time course of a single collision event. To calculate the correlation function that appears here it is sufficient to consider a single binary collision with thermal initial conditions. The result is given in terms of the function B, a two-body property that depends only on the relative position of the particles (the initial configuration for the collision event) and the temperature. The host structure as determined by the V-body force enters only through the configurational average that involves the pair distribution g(r). 

For simple covalent bond breaking reactions, a bound state in the R-BO scheme correlates to a diradical asymptotic state. This latter state represents in the laboratory world a collision pair. In a-space we can define an intermolecular distance. For all values of such distance, the system cannot change its electronic state in an adiabatic process. The asymptotic state must be orthogonal to the bound state. It is therefore necessary that the electronic wave function of the collision pair show one node more than the bounded system. The energy expectation values as a function of the intermolecular distance for the two states would cross above the dissociation energy limit. The corresponding FC factor can hence be different from zero. Experimentally, it is well known that most of the bond-forming processes may have a small barrier (about 1 Kcal/mol) [22]. 

The pair kinetic equation in Section VII.D follows directly from these results if the dynamic memory function " xbs.abs neglected, and the static structural correlations in (D.3) to (D.6) are approximated so that all binary collisions are calculated in the Enskog approximation. [This is the singly independent disconnected (SID) approximation, which is discussed in detail in Ref. 53.] We have also used the static hierarchy to obtain the final form involving the mean force, given in (7.32). This latter reduction involving the static hierarchy is carried out below in the context of a comparison of the singlet and doublet formulations. 

The first pubUshed criticism of the binary collision model was due to Fixman he retained the approximation that the relaxation rate is the product of a collision rate and a transition probabihty, but argued that the transition probability should be density dependent due to the interactions of the colliding pair with surrounding molecules. He took the force on the relaxing molecule to be the sum of the force from the neighbor with which it is undergoing a hard binary collision, and a random force mA t). This latter force was taken to be the random force of Brownian motion theory, with a delta-function time correlation ... 

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