Hard-sphere collision models

By using the hard sphere collision model we can compute a collision frequency for three molecules A, B, and C by first computing the stationary concentration of the three possible binary complexes AB, BC, and CA. If we call tab, tbc, and tca the mean lifetime of these binary complexes/ their stationary concentrations are approximately given by... 

FIGURE 29 Drift-time distribution recorded for LaQo with injection energy of 400 eV. The curve shows the drift-time distribution calculated from the transport equation for ions in the drift tube. The arrows show the expected drift times for endohedral La( Qo and exohedral La(Qo) complexes. The mobility was calculated using a simple hard-sphere collision model. 

Hard-sphere collision model, 143 Hard template assisted synthesis, 449 Heterobimetallic precursor, 400 Heterometallic rare-earth oxide, 387... 

Many disperse-phase systems involve collisions between particles, and the archetypical example is hard-sphere collisions. Thus, Chapter 6 is devoted to the topic of hard-sphere collision models in the context of QBMM. In particular, because the moment-transport equations for a GBPE with hard-sphere collisions contain a source term for the rate of change of the NDF during a collision, it is necessary to derive analytical expressions for these source terms (Fox Vedula, 2010). In Chapter 6, the exact source terms are derived... 

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. 

For inelastic collisions, the hard-sphere collision model is given by (Fox Vedula, 2010 Jenkins Mancini, 1989 Jenkins Richman, 1985)... 

Thus, at least qualitatively, a hard-sphere collision model results in an equation which has a firm basis in experiment. This suggests that the Arrhenius parameters, which were treated as purely experimental parameters in Section 6, can be given a physical interpretation, at least for bimolecular processes. 

At first glance it seems paradoxical to treat unimolecular reactions, in which a single molecule is apparently involved in reaction, in terms of a collision theory based on pairwise interactions. Indeed, we have developed a rather specific picture of a chemical reaction from the hard-sphere collision model, in which bonds are formed rather than broken and in which the energetics of reaction are represented in terms of relative kinetic energy. 

A listing of typical estimates of this sort is given in Table 2.4. It can be seen that as the reaction becomes more complex in relation to the hard-sphere collision model, the steric factor decreases. In practice, then, one might be able to use the magnitude of experimentally determined steric factors as the basis for more detailed hypotheses concerning the nature of a given reaction. 

The colUsion cross section as a function of Ejp is determined from the increase of linewidth with pressure. At higher Ejp the hard-sphere collision model was found to be a good approximation. The main contribution to the cross section of ions in their parent gases is from charge transfer 35). 

Langridge, D. Giles, K. Hoyes, J.B. Simulation of ion motion in a travelling wave mobility separator using a hard-sphere collision model. Proc. 56th ASMS Conference on Mass Spectrometry and Allied Topics, Denver, CO, June 1-5, 2008, WP 064. 

To represent the true environment of the QIT, a hard-sphere collision model (HSl coded by Manura) [97] was implemented with a user program in SIMION. Ion trajectories were computed individually from an initial g =0.70 for m/z 491. The RF voltage was scanned at a rate equivalent to 5500 Th s, allowing approximately 2 ms prior to ion ejection for the ions to equilibrate with the helium collision gas held at a pressure of 1 mTorr. 

---