Collision dynamics binary collisions

J.-P. Estrade, H. Carentz, G. Lavergne, Y. Biscos Experimental investigation of dynamic binary collision of ethanol droplets — a model for droplet coalescence and bouncing, Int. J. Heat Huid How 20,486-491 (1999). 

The dynamics of ion surface scattering at energies exceeding several hundred electronvolts can be described by a series of binary collision approximations (BCAs) in which only the interaction of one energetic particle with a solid atom is considered at a time [25]. This model is reasonable because the interaction time for the collision is short compared witii the period of phonon frequencies in solids, and the interaction distance is shorter tlian the interatomic distances in solids. The BCA simplifies the many-body interactions between a projectile and solid atoms to a series of two-body collisions of the projectile and individual solid atoms. This can be described with results from the well known two-body central force problem [26]. 

The volume fraction dependence of /cq/(4>) is plotted in Fig. 21b and shows that it increases strongly with 4>- Recall that this rate coefficient is independent of 4> if simple binary collision dynamics is assumed to govern the boundary layer region. The observed increase arises from the obstacle distribution in the vicinity of the catalytic sphere surface. When obstacles are present, a reactive... 

Binary collision dynamics, reactive hybrid MPC-molecular dynamics,... 

To simulate the particle-particle collision, the hard-sphere model, which is based on the conservation law for linear momentum and angular momentum, is used. Two empirical parameters, a restitution coefficient of 0.9 and a friction coefficient of 0.3, are utilized in the simulation. In this study, collisions between spherical particles are assumed to be binary and quasi-instantaneous. The equations, which follow those of molecular dynamic simulation, are used to locate the minimum flight time of particles before any collision. Compared with the soft-sphere particle-particle collision model, the hard-sphere model accounts for the rotational particle motion in the collision dynamics calculation thus, only the translational motion equation is required to describe the fluid induced particle motion. In addition, the hard-sphere model also permits larger time steps in the calculation therefore, the simulation of a sequence of collisions can be more computationally effective. The details of this approach can be found in the literature (Hoomans et al., 1996 Crowe et al., 1998). 

In the dynamical approach, one attempts to solve directly the quantum-mechanical or classical equations of motion for a system, Such a direct approach is practicable, for example, for treating the binary collisions between molecules in a gas, by either classical or quantum-mechanical methods.3 However, in a dense system such as a liquid, only the classical equations are tractable,4 even with high-speed computers,... 

The friction on a tagged particle is expressed in terms of the time-dependent force-force autocorrelation function. Although the bare, short-time part of the friction that arises from binary collisions can be calculated from kinetic theory, the long-time part needs the knowledge of the solvent and the solute dynamics and the coupling between them. The solvent dynamic quantities... 

It should be noted that although in Eq. (90) only the connected motion of the solute and the solvent is retained, in the argument presented on the time scale it is the disconnected parts which have been considered. This is because in the latter part, for the derivation of the expression of Ci. the solute and the solvent motions are assumed to be disconnected. This assumption is the same as those made in the density functional theory and also in mode coupling theories where a four-point correlation function is approximated as the product of two two-point correlation functions. This approximation when incorporated in Ci. means that after the binary collision takes place, the disturbances in the medium will propagate independently. A more exact calculation would be to consider the whole four-point correlation function, thus considering the dynamics of the solute and the solvent to be correlated even after the binary collision is over. Such a calculation is quite cumbersome and has not been performed yet. 

Note that the above study is performed for a simple system. There exists a large body of literature on the study of diffusion in complex quasi-two-dimensional systems—for example, a collodial suspension. In these systems the diffusion can have a finite value even at long time. Schofield, Marcus, and Rice [17] have recently carried out a mode coupling theory analysis of a quasi-two-dimensional colloids. In this work, equations for the dynamics of the memory functions were derived and solved self-consistently. An important aspect of this work is a detailed calculation of wavenumber- and frequency-dependent viscosity. It was found that the functional form of the dynamics of the suspension is determined principally by the binary collisions, although the mode coupling part has significant effect on the longtime diffusion. 

Simulations. In addition to analytical approaches to describe ion—solid interactions two different types of computer simulations are used Monte Carlo (MC) and molecular dynamics (MD). The Monte Carlo method relies on a binary collision model and molecular dynamics solves the many-body problem of Newtonian mechanics for many interacting particles. As the name Monte Carlo suggests, the results require averaging over many simulated particle trajectories. A review of the computer simulation of ion—solid interactions has been provided (43). 

Ashgriz, N. and Givi, P. (1987). Binary collision dynamics of fuel droplets. Inter. J. Heat Fluid Flow, 8 205-210. 

The isolated binary collision (IBC) theory is one of the earliest dephasing theories developed for liquids (77). Despite criticism of its assumptions (78), it remains popular for the interpretation of experimental data, because it relates the dephasing to easily measurable solvent properties. It assumes that the solvent dynamics are essentially gas-like, consisting of uncorrelated collisions of the vibrating molecule with solvent molecules. The solvent-solute interaction is taken as purely repulsive. The dependence of... 

The reflection of carbon atoms and ions at the limiter or divertor surfaces in the ERO modeling is determined by TRIM [29], However, the binary collision approximation used in the TRIM code is no longer valid at small energies of the incoming particles where chemical effects start to influence the interaction of the particles with the solid. To take this into account reflection coefficients calculated with a molecular dynamic code MolDyn [55] were implemented. 

Two of the most common classes of particle-dynamic simulations are termed hard-particle and soft-particle methods. Hard-particle methods calculate particle trajectories in response to instantaneous, binary collisions between particles and allow particles to travel ballistically between collisions. This class of... 

In a hard sphere approach, particles are assumed to interact through instantaneous binary collisions. This means particle interaction times are much smaller than the free flight time and therefore, hard particle simulations are event (collision) driven. For a comprehensive introduction to this type of simulation, the reader is referred to Allen and Tildesley (1990). Hoomans (2000) used this approach to simulate gas-solid flows in dense as well as fast-fluidized beds. There are three key parameters in such hard sphere models, namely coefficient of restitution, coefficient of dynamic friction and coefficient of tangential restitution. Coefficient of restitution is discussed later in this chapter. Detailed discussion of these three model parameters can be found in Hoomans (2000). 

To calculate the integrals defining the source term and the flux term, appropriate expressions for Aip and ip[ — ipi have to be determined from an analysis of the inelastic binary particle collision dynamics. 

In this section the dynamics of inelastic binary particle collisions are examined. The theory represents a semi-empirical extension of the binary collision theory... 

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). 

Recently, the sapt approach has been applied (32) to compute the interaction-induced polarizability for the helium diatom. The computed polarizability invariants have been analytically fitted, and used in quantum-dynamical calculations of the binary collision-induced Raman spectra. The results of the dynamical calculations are summarized in Fig. 1. 

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