Kinetic model collision

Bashford D Weaver D. L. and Karplus M. Diffusion-collision model for the folding kinetics of the lambda-repressor operatorbinding domain. J. Biomol. Str. Dyn. (1984) 1 1243-1255. 

In ISS, ions such as H, He and Ar are scattered off a surface and their energy distribution is observed. During the scattering process, the ions lose energy to the surface atoms. The collision process is usually so rapid (with kinetic energies of the order of 1 keV to 1 MeV) that a binary collision model is a good description of the situation. 

Kinetic theory of gases (collision model or Monte Carlo approach)... 

In the condensed phase the AC permanently interacts with its neighbors, therefore a change in the local phase composition (as were demonstrated on Figs. 8.1 and 8.2) affects the activation barrier level (Fig. 8.6). Historically the first model used for surface processes is the analogy of the collision model (CM) [23,48,57]. This model uses the molecular-kinetic gas theory [54]. It will be necessary to count the number of the active collisions between the reagents on the assumption that the molecules represent solid spheres with no interaction potential between them. Then the rate constant can be written down as follows (instead of Eq. (6)) ... 

In the review information only about the first steps of MC simulation is given as today this method is dominant by comparison with the kinetic theory. The calculations based on the dynamic MC methods for the lattice-gas model are carried out using the master equation (24). The calculation results depend appreciably on the way of assigning the probabilities of transitions Wa. This was repeatedly pointed out in applying both the cluster methods (Section 3) and the MC method (see, e.g. Ref. [269]). Nevertheless, practically in all the papers of Section 7 the expressions (29) and (30) do not take into account the interaction between AC and its neighbors (i.e., the collision model was used). It means s (r) = 0, whereas analysis of the cluster simulations demonstrated important influence of the parameter s (r) (that restricts obtained MC results). 

The kinetic molecular theory of gases predicts that an increase in temperature increases molecular velocities and so increases the frequency of in-termolecular collisions. This agrees with the observation that reaction rates are greater at higher temperatures. Thus there is qualitative agreement between the collision model and experimental observations. However, it is found that the rate of reaction is much smaller than the calculated collision frequency in a given collection of gas particles. This must mean that only a small fraction of the collisions produces a reaction. Why ... 

If the incident ion has a mass of Mi and kinetic energy of Eq, and the ion backscattered at an angle 0 (relative to the direction of the incident ion) has an energy of Ei, the two-collision model allows the determination of the surface elemental composition based on the following equation ... 

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) ... 

At equilibrium, where the yelocity distribution is Maxwellian, it is straightforward to show that < > = 4 J p/7T, where 0p is the granular temperature. We should note that Eq. (6.109) corresponds to an inelastic Maxwell particle (Maxwell, 1879), and, most importantly, it still contains the exact dependence on tu = (1 + e)/2. We will therefore refer to this kinetic model as the inelastic Maxwell collision model. 

In this section, we look at a kinetic equation for the velocity NDF n t, x, v), where v = ( , v) is a two-component velocity vector (i.e. the velocity phase space is two-dimensional). In order to show the dynamics for different amounts of particle-particle collisions, we will use the BGK collision model. (See Chapter 6 for more details on collision models.) The inhomogeneous kinetic equation for this case is... 

Chapter 6 is devoted to the topic of hard-sphere collision models (and related simpler kinetic models) in the context of QBMM. In particular, the exact source terms for integer moments due to collisions are derived in the case of inelastic binary collisions between two particles with different diameters/masses, and the use of QBMM to overcome the closure problem is illustrated. 

Applications of this kinetic equation for isomerization dynamics have been carried out by considering the motion of a particle in the external force corresponding to the double minimum potential in Fig. 3.2. Since this model treats the free streaming in the potential correctly and specifies a not unreasonable model for the collisions, which provide the energy dissipation, interesting results for the dynamics of the reaction can be obtained. Other more complex collision models, which contain solute and solvent molecule mass effects explicitly, have also been studied. We discuss some of these results in Section Xll. 

Before closing this section, we should remark that although this analysis of velocity relaxation effects has focused on a simple collision model, we expect that the detailed structure of the rate kernel for short times will depend on the precise form of the chemical interactions in the system under consideration. It is clear, however, that a number of fundamental questions need to be answered before more specific calculations can be undertaken form the kinetic theory point of view. 

Model kinetic equation approaches of these types should probably be more thoroughly investigated for such complex systems before more elaborate kinetic theories are constructed. Ultimately, however, difficult problems such as the nature of the friction coefficient or collision frequency associated with an internal coordinate must be solved. What, for instance, is the form of its space and time nonlocality The solution of this problem will involve a more complex calculation than that outlined in Section IX.B for the two-particle friction tensor. 

According to the simple collision model, the fraction of collisions with a kinetic energy sufficient to overcome the energy barrier to reaction increases with increasing temperature. This behaviour largely accounts for the temperature dependence of the theoretical rate constant. 

The simplest model to account for reaction kinetics is the collision model... 

A process is said to be spontaneous if it occurs without outside intervention. Spontaneous processes may be fast or slow. As we will see in this chapter, thermodynamics can tell us the direction in which a process will occur but can say nothing about the speed of the process. As we saw in Chapter 12, the rate of a reaction depends on many factors, such as activation energy, temperature, concentration, and catalysts, and we were able to explain these effects using a simple collision model. In describing a chemical reaction, the discipline of chemical kinetics focuses on the pathway between reactants and products thermodynamics considers only the initial and final states... 

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. 

We consider the orientational dynamics only and ignore the spatial coordinates of interacting rods (an analog of the Maxwell model of binary collisions in kinetic theory of gases, see e.g. [15]). Since the motor residence time on microtubules (about 10 sec) is much smaller than the characteristic time of pattern formation (10 min or more), we model molecular motor - microtubule inelastic interaction as an instantaneous colUsion in which two rods change... 

---