Molecular collisions model

Based on the molecular collision model, that describes successfully experimental data for a large number of bimolecular reactions, the rate of the reaction can be calculated as the number of collisions of molecules having energy higher than the required value E [7] ... 

Langevin dynamics simulates the effect of molecular collisions and the resulting dissipation of energy that occur in real solvents, without explicitly including solvent molecules. This is accomplished by adding a random force (to model the effect of collisions) and a frictional force (to model dissipative losses) to each atom at each time step. Mathematically, this is expressed by the Langevin equation of motion (compare to Equation (22) in the previous chapter) ... 

A more sophisticated understanding is linked to an appreciation of the interaction of thermodynamic and kinetic considerations and is likely to be dependent upon the ability to visualise some form of mental model involving molecular collisions and interactions (Gilbert, 2005). This allows the student to see that two reactions are occurring simultaneously ... 

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. 

Multiparticle collision dynamics describes the interactions in a many-body system in terms of effective collisions that occur at discrete time intervals. Although the dynamics is a simplified representation of real dynamics, it conserves mass, momentum, and energy and preserves phase space volumes. Consequently, it retains many of the basic characteristics of classical Newtonian dynamics. The statistical mechanical basis of multiparticle collision dynamics is well established. Starting with the specification of the dynamics and the collision model, one may verify its dynamical properties, derive macroscopic laws, and, perhaps most importantly, obtain expressions for the transport coefficients. These features distinguish MPC dynamics from a number of other mesoscopic schemes. In order to describe solute motion in solution, MPC dynamics may be combined with molecular dynamics to construct hybrid schemes that can be used to explore a variety of phenomena. The fact that hydrodynamic interactions are properly accounted for in hybrid MPC-MD dynamics makes it a useful tool for the investigation of polymer and colloid dynamics. Since it is a particle-based scheme it incorporates fluctuations so that the reactive and nonreactive dynamics in small systems where such effects are important can be studied. 

A. Malevanets and R. Kapral, Mesoscopic multi-particle collision model for fluid flow and molecular dynamics, in Novel Methods in Soft Matter Simulations, M. Karttunen, I. Vattulainen, and A. Lukkarinen (eds.), Springer-Verlag, Berlin, 2003, p. 113. 

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). 

For example, the study of global warming is hindered, in his view, because the intermolecular interactions are not well known. "To properly model the processes, we have to understand the exchange of energy during molecular collisions. If we know them, we can predict an immense amount."... 

A further development is possible by noting that the high frequency shear modulus Goo is related to the mean square particle displacement (m ) of caged fluid particles (monomers) that are transiently localized on time scales ranging between an average molecular collision time and the structural relaxation time r. Specifically, if the viscoelasticity of a supercooled liquid is approximated below Ti by a simple Maxwell model in conjunction with a Langevin model for Brownian motion, then (m ) is given by [188]... 

Based on a molecular kinetic model, Schweikert derives equations which. describe both the process of detonation in a condensed expl and that of the burning of a colloidal powder These processes are shown to differ primarily in the magnitude of the collision efficiency. Relations are derived which relate the max deton vel and pressure with molecular props... 

Content. After a brief overview of molecular collisions and interactions, dipole radiation, and instrumentation (Chapter 2), we consider examples of measured collision-induced spectra, from the simplest systems (rare gas mixtures at low density) to the more complex molecular systems. Chapter 3 reviews the measurements. It is divided into three parts translational, rototranslational and rotovibrational induced spectra. Each of these considers the binary and ternary spectra, and van der Waals molecules we also take a brief look at the spectra of dense systems (liquids and solids). Once the experimental evidence is collected and understood in terms of simple models, a more theoretical approach is chosen for the discussion of induced dipole moments (Chapter 4) and the spectra (Chapters 5 and 6). Chapters 3 through 6 are the backbone of the book. Related topics, such as redistribution of radiation, electronic collision-induced absorption and emission, etc., and applications are considered in Chapter 7. 

Collision Model. Figure 3 shows an instructive, efficient, and convenient method for treating the case of mixed surface and particle-diffusion control. It is instructive because it is easy to visualize. It is efficient because although it is a molecular approach, it does not require Monte Carlo calculations, which are expensive and should be avoided whenever possible. It is convenient because it leads to a set of differential equations... 

The high-pressure limit of the rate constant Um,oo is readily measured. From the assumptions in the model, molecular collision theory should be adequate to predict the excitation-reaction rate constant ke, using Eq. 10.76 ... 

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). 

D. A. Micha. Dynamics of Molecular Collisions, Vol. IA, chapter Optical models in molecular collision theory, page 81. Plenum Press, New York, 1976. 

The most satisfactory treatment of the reactions of interest in this chapter is in terms of classical trajectories on potential energy surfaces. They provide a detailed consideration of the reactive interaction (for which the kinematic models are limiting cases7), and provide ample scope for the theoretician to apply his intuition in explaining reactive molecular collisions. Reactions are naturally divided into those which take place on a single surface, usually leading to vibrational excitation, and those which involve two or more surfaces, often leading to electronic excitation. 

The rotational relaxation of polyatomic spherical top molecules can be treated approximately on the classical rough sphere model. This has been done for homo-molecular collisions by Wang Chang and Uhlenbeck101. They find a simple expression resembling that obtained by Brout for diatomic molecules... 

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)) ... 

On a microscopic scale, atoms and molecules travel faster and, therefore, have more collisions as the temperature of a system is increased. Since molecular collisions are the driving force for chemical reactions, more collisions give a higher rate of reaction. The kinetic theory of gases suggests an exponential increase in the number of collisions with a rise in temperature. This model fits an extremely large number of chemical reactions and is called an Arrhenius temperature dependency, or Arrhenius law. The general form of this exponential relationship is... 

As already mentioned the present treatment attempts to clarify the connection between the sticking probability and the mutual forces of interaction between particles. The van der Waals attraction and Born repulsion forces are included in the calculation of the rate of collisions between two electrically neutral aerosol particles. The overall interaction potential between two particles is calculated through the integration of the inter-molecular potential, modeled as the Lennard-Jones 6-12 potential, under the assumption of pairwise additivity. The expression for the overall interaction potential in terms of the Hamaker constant and the molecular diameter can be found in Appendix 1. The motion of a particle can no longer be assumed to be... 

It seems, therefore, with the current renewal of theoretical interest in atomic and molecular collision problems, reactive scattering, and predissociation phenomena, that it is worthwhile to examine the VB theory as a useful model that is capable of yielding accurate potential energy surfaces. 

Berend and Benson s classical treatment of V-V transfer [81] employs a two-dimensional collision model of a pair of diatomics with identical Morse interaction potentials between each pair of atoms. The Morse range parameter a was determined from experimental data for the N2-N2 T-V process. In-all, six functions are employed, one between each pair of atoms (Figure 3.5). Molecule CD, oriented at an angle / relative to its velocity vector, collides with molecule AB, with impact parameter b. Molecule AB is taken to be oriented parallel to the velocity vector of CD. The instantaneous angle between the molecular axis of AB and the line joining the centers of mass is denoted t). Cross sections for the reactions... 

The Quantum Mechanical study of molecular collisions and of the chemical reaction is itself an important topic There are several review papers and textbooks - on the subject. Unfortunately, there are no exact quantum results within a realistic model of a chemical reaction yet, not even for the simplest 3-atoms exchange. Thus the comparison is limited to particular cases. 

In [112] the insight and accuracy of the approach has been demonstrated it has been extended afterwards to a series of molecules in view of perspective applications to molecular dynamics modeling of collision induced chirality changing processes. 

G. Duration of a Collision. The hard sphere model is very useful because it permits us to describe molecular collisions in terms of a single, simple, molecular parameter, the collision diameter. It is, however, insufficient to permit a detailed description of a chemical reaction, which is an event that transpires during a collision between two molecules, because the duration in time of a hard sphere collision is precisely zero. 

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