Encounter dynamics

B. Elimination of Thennalized Coordinates C Encounter Dynamics of Two Particles... 

A broad range of physical and chemical processes occur as a result of an encounter between pairs of particles. Depending on the physical phenomenon being investigated, the particles may range in size from the atomic scale to colloidal, while the bath density may range from that of a low-pressure gas to a dense liquid. The primary aim of this chapter is to show that a simple model for encounter dynamics can be derived from the Fokker-Planck equation (FPE), which applies, within limits, over the entire range of particle size and bath density. 

In this chapter size effects in encounter and reaction dynamics are evaluated using a stochastic approach. In Section IIA a Hamiltonian formulation of the Fokker-Planck equation (FPE) is develojjed, the form of which is invariant to coordinate transformations. Theories of encounter dynamics have historically concentrated on the case of hard spheres. However, the treatment presented in this chapter is for the more realistic case in which the particles interact via a central potential K(/ ), and it will be shown that for sufficiently strong attractive forces, this actually leads to a simplification of the encounter problem and many useful formulas can be derived. These reduce to those for hard spheres, such as Eqs. (1.1) and (1.2), when appropriate limits are taken. A procedure is presented in Section IIB by which coordinates such as the center of mass and the orientational degrees of freedom, which are often characterized by thermal distributions, can be eliminated. In the case of two particles the problem is reduced to relative motion on the one-dimensional coordinate R, but with an effective potential (1 ) given by K(l ) — 2fcTln R. For sufficiently attractive K(/ ), a transition state appears in (/ X this feature that is exploited throughout the work presented. The steady-state encounter rate, defined by the flux of particles across this transition state, is evaluated in Section IIC. 

Section VI can be applied from an estimate of the volume of the confinement region. However, in some cases of interest this limit is not applicable, and a more careful treatment is required. This is done in Section VIIC, in which reaction in a hemispherical pocket is considered. Another type of confinement occurs in the case of intramolecular reactions of sites on a polymer chain. The maximum separation is dictated by the polymer, and the encounter dynamics can be strongly influenced by the relaxation rates of the polymer configurations. This is discussed in Section VIID. In both of these examples the problem is reduced from three dimensions to one using the effective potential. 

The Langmuir-Hinshelwood picture is essentially that of Fig. XVIII-14. If the process is unimolecular, the species meanders around on the surface until it receives the activation energy to go over to product(s), which then desorb. If the process is bimolecular, two species diffuse around until a reactive encounter occurs. The reaction will be diffusion controlled if it occurs on every encounter (see Ref. 211) the theory of surface diffusional encounters has been treated (see Ref. 212) the subject may also be approached by means of Monte Carlo/molecular dynamics techniques [213]. In the case of activated bimolecular reactions, however, there will in general be many encounters before the reactive one, and the rate law for the surface reaction is generally written by analogy to the mass action law for solutions. That is, for a bimolecular process, the rate is taken to be proportional to the product of the two surface concentrations. It is interesting, however, that essentially the same rate law is obtained if the adsorption is strictly localized and species react only if they happen to adsorb on adjacent sites (note Ref. 214). (The apparent rate law, that is, the rate law in terms of gas pressures, depends on the form of the adsorption isotherm, as discussed in the next section.)... 

Flowever, we have also seen that some of the properties of quantum spectra are mtrinsically non-classical, apart from the discreteness of qiiantnm states and energy levels implied by the very existence of quanta. An example is the splitting of the local mode doublets, which was ascribed to dynamical tiumelling, i.e. processes which classically are forbidden. We can ask if non-classical effects are ubiquitous in spectra and, if so, are there manifestations accessible to observation other than those we have encountered so far If there are such manifestations, it seems likely that they will constitute subtle peculiarities m spectral patterns, whose discennnent and interpretation will be an important challenge. 

As discussed above, the nonlinear material response, P f) is the most connnonly encountered nonlinear tenn since vanishes in an isotropic medium. Because of the special importance of P we will discuss it in some detail. We will now focus on a few examples ofP spectroscopy where just one or two of the 48 double-sided Feymnan diagrams are important, and will stress the dynamical interpretation of the signal. A pictorial interpretation of all the different resonant diagrams in temis of wavepacket dynamics is given in [41]. 

If we wish to know the number of (VpV)-collisions that actually take place in this small time interval, we need to know exactly where each particle is located and then follow the motion of all the particles from time tto time t+ bt. In fact, this is what is done in computer simulated molecular dynamics. We wish to avoid this exact specification of the particle trajectories, and instead carry out a plausible argument for the computation of r To do this, Boltzmann made the following assumption, called the Stosszahlansatz, which we encountered already in the calculation of the mean free path ... 

Most chemically reacting systems tliat we encounter are not tliennodynamically controlled since reactions are often carried out under non-equilibrium conditions where flows of matter or energy prevent tire system from relaxing to equilibrium. Almost all biochemical reactions in living systems are of tliis type as are industrial processes carried out in open chemical reactors. In addition, tire transient dynamics of closed systems may occur on long time scales and resemble tire sustained behaviour of systems in non-equilibrium conditions. A reacting system may behave in unusual ways tliere may be more tlian one stable steady state, tire system may oscillate, sometimes witli a complicated pattern of oscillations, or even show chaotic variations of chemical concentrations. 

Reactive atomic and molecular encounters at collision energies ranging from thermal to several kiloelectron volts (keV) are, at the fundamental level, described by the dynamics of the participating electrons and nuclei moving under the influence of their mutual interactions. Solutions of the time-dependent Schrodinger equation describe the details of such dynamics. The representation of such solutions provide the pictures that aid our understanding of atomic and molecular processes. 

The many approaches to the challenging timestep problem in biomolecular dynamics have achieved success with similar final schemes. However, the individual routes taken to produce these methods — via implicit integration, harmonic approximation, other separating frameworks, and/or force splitting into frequency classes — have been quite different. Each path has encountered different problems along the way which only increased our understanding of the numerical, computational, and accuracy issues involved. This contribution reported on our experiences in this quest. LN has its roots in LIN, which... 

The Reynolds number for flow in a tube is defined by dvpirj, where d is the diameter of the tube, V is the average velocity of the fluid along the tube, p is the density of the fluid, and rj is its dynamic viscosity. At flow velocities corresponding with values of the Reynolds number of greater than 2000, turbulence is encountered. 

In order to treat crystallization systems both dynamically and continuously, a mathematical model has been developed which can correlate the nucleation rate to the level of supersaturation and/or the growth rate. Because the growth rate is more easily determined and because nucleation is sharply nonlinear in the regions normally encountered in industrial crystallization, it has been common to... 

The effect of ozone is complicated in so far as its effect is largely at or near the surface and is of greatest consequence in lightly stressed rubbers. Cracks are formed with an axis perpendicular to the applied stress and the number of cracks increases with the extent of stress. The greatest effect occurs when there are only a few cracks which grow in size without the interference of neighbouring cracks and this may lead to catastrophic failure. Under static conditions of service the use of hydrocarbon waxes which bloom to the surface because of their crystalline nature give some protection but where dynamic conditions are encountered the saturated hydrocarbon waxes are usually used in conjunction with an antiozonant. To date the most effective of these are secondary alkyl-aryl-p-phenylenediamines such as /V-isopropyl-jV-phenyl-p-phenylenediamine (IPPD). 

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