Friction collisional

Johnson, P C., Frictional-collisional equations of motion for particulate flows with applications to chutes and shear cells, Ph.D. dissertation, Princeton University, 1987. 

Johnson, PC. and Jackson, R. (1987), Frictional collisional constitutive relations for granular materials with application to plain shearing, J. Fluid Meek, 176, 67. 

For instance, this is the compact form for the symmetrized time evolution MFPK operator for two Brownian particles (or rotators see below) coupled via a potential V and a frictional (collisional) matrix oi". Although both the terms acting on the momentum space and the positional space are written, for the sake of simplicity, as formal raising and lowering operators, actually only the properties of the operators will be used in the following. Note that we have not specified the nature of the gradient operators in x, so they could be a set of rotational coordinates (in this case we should include a precession-like term in but we shall see in the next section that the presence of the precession operator is irrelevant). We define n) as the direct product of the eigenfunctions of... 

P. C. Johnson and R. Jackson. "Frictional-collisional constitutive relations for granular materials, with application to plane shearing," /. Fluid Mech., 176, 67-93, 1987. 

P.C. Johnson, P. Nott and R. Jackson, Frictional-collisional Equations of Motion for Particulate Flows and their Application to Chutes, Journal of Fluid Mechanics 210 501-535 (1990)... 

Johnson, P. C., P. Nott, and R. Jackson. 1990. Frictional-collisional equations of motion for particulate flow and their applications to chutes.. Fluid Mech. 210, 501-535. 

G. R. Fleming We have not yet looked at the issue raised by Prof. Rice. We understand much more about dielectric friction than about collisional friction at short times. The calculation you suggest would be very interesting. 

The reaction of trimethylene biradical was successfully treated by means of dynamics simulations by two groups with different PESs as described above.11 15 The success led one of the groups to extend the study to analyze the collisional and frictional effects in the trimethylene decomposition in an argon bath.16 A mixed QM/MM direct dynamics trajectory method was used with argon as buffer medium. Trimethylene intramolecular potential was treated by AM1-SRP fitted to CASSCF as before, and intermolecular forces were determined from Lennard-Jones 12-6 potential energy functions. 

The second term on the left is the heat transport VQ with Q = f d3v yw2n/, due to convection, viscous heat transport and conduction, respectively. On the right-hand side one has the heating due to the work done by the electric field, the collisional frictional force due to the flow relative to the other species (here electrons) and from the collisional heating due to collisions with other species (electrons) ... 

III. Single-Particle Motion—A Stochastic Approach A. Stochastic Formulation of Friction Self-Diffusion in Hard-Sphere Fluids Collisional Friction... 

Quantum Effects in Collisional Friction The Tlansition Regime of Aerodynamics Hydrodynamic and Viscoelastic Effects Mass Dependence of Self-Diffusion Pair Dissociation—A Stochastic Approach A. Diatomic Dissociation on a J-Averaged Potential Adiabatic Effects in Diatomic Dissociation... 

For a large particle in a fluid at liquid densities, there are collective hydro-dynamic contributions to the solvent viscosity r, such that the Stokes-Einstein friction at zero frequency is In Section III.E the model is extended to yield the frequency-dependent friction. At high bath densities the model gives the results in terms of the force power spectrum of two and three center interactions and the frequency-dependent flux across the transition state, and at low bath densities the binary collisional friction discussed in Section III C and D is recovered. However, at sufficiently high frequencies, the binary collisional friction term is recovered. In Section III G the mass dependence of diffusion is studied, and the encounter theory at high density exhibits the weak mass dependence. 

In Section IV A the model is applied to diatomic dissociation on a Morse potential in the low-friction limit. This models diatomic dissociation in a low-density gas in which the collisional excitation is impulsive, being modeled by a zero-frequency friction, that is, the duration of the collision is assumed to be small relative to all other time scales. The objective of the section is to test whether the reduction to a one-dimensional effective potential P (l ) leads to an accurate formula for the dissociation rate. This is done by comparison... 

Collision-induced vibrational excitation and relaxation by the bath molecules are the fundamental processes that characterize dissociation and recombination at low bath densities. The close relationship between the frequency-dep>endent friction and vibrational relaxation is discussed in Section V A. The frequency-dependent collisional friction of Section III C is used to estimate the average energy transfer jjer collision, and this is compared with the results from one-dimensional simulations for the Morse potential in Section V B. A comparison with molecular dynamics simulations of iodine in thermal equilibrium with a bath of argon atoms is carried out in Section V C. The nonequilibrium situation of a diatomic poised near the dissociation limit is studied in Section VD where comparisons of the stochastic model with molecular dynamics simulations of bromine in argon are made. The role of solvent packing and hydrodynamic contributions to vibrational relaxation are also studied in this section. 

At high frequencies or low densities. 4(co) -> 0. In terms of the Fourier variable (0, the collisional friction depends only on the power spectrum < Fp(oj) > of a binary collision so that... 

The impact of bath molecules on the atoms of the molecule of interest cannot be treated as impulsive because the strong binding forces of chemical bonds places a significant fraction of the vibrational sjjectrum of the molecule above the collisional bandwidth, broadly defined as the reciprocal of the duration of a collision. Thus collisional vibrational relaxation and excitation are inefficient relative to rotational relaxation. Binary collision theory is well develojjed at the microcanonical level because of the its importance in chemical reactions. The relationship to the friction is of interest, " primarily because stochastic treatments have the potential of bridging the gas-phase limit of resolved binary collisions and the liquid phase where collective phenomena of the solvent can preclude interpretation in terms of binary collisions. 

Elimination of the solvent molecule coordinates leads to stochastic forces acting on each atom of the molecule, and to a first approximation these forces on different atoms can be treated as uncorrelated and the coupling to the bath linear in the atomic displacement. In order to implement such a model, a realistic atom-atom collisional friction needs to be determined, and this will be done in this section. 

The concept of collisional friction arises from an analysis of molecular dynamics simulations, which shows that the short-time collisional contri-... 

Figure S. Collisional friction kernel B at) for classical Morse potential (-and its asymptotic...

Figure 6, Collisional friction kernel B((b) for classical Morse potential (--) and its asymptotic limit (---versus reduced frequency for several values of reduced Morse vibrational...

Figure 7. Quantum collisional friction kernel B(c5) for exponential atom-atom potential (----)...

In the regime of aerodynamics the size of the test particle, of radius R, and the mean free path 2 of the bath particles are both large compared with the size of bath particles. The only important size parameter is the Knudsen number In the free molecule limit Kn - oo and the diffusion coefficient of the large test particle is derived from the hard-sphere low-density limit of the collisional friction Eq. (3.4), namely,... 

In the previous sections a model of the frequency-dependent collisional friction has been derived. Because the zero-frequency friction for a spherical particle in a dense fluid is well modeled by the Stokes-Einstein result, even for particles of similar size as the bath particles, there has been considerable interest in generalizing the hydrodynamic approach used to derive this result into the frequency domain in order to derive a frequency-dependent friction that takes into account collective bath motions. The theory of Zwanzig and Bixon, corrected by Metiu, Oxtoby, and Freed, has been invoked to explain deviation from the Kramers theory for unimolec-ular chemical reactions. The hydrodynamic friction can be used as input in the Grote-Hynes theory [Eq. (2.35)] to determine the reactive frequency and hence the barrier crossing rate of the molecular reaction. However, the use of sharp boundary conditions leads to an unphysical nonzero high-frequency limit to Ib(s). which compromises its utility. 

The discussion presented above applies only for impulsive collisions for which Pc 1. that is, m. This is not a realistic formulation, except near the threshold for dissociation. In most cases of interest the vibrational frequencies are too high relative to the collision bandwidth cuq defined in Section III C. For a single degree of freedom (n = 1) it is essential to include the frequency dejjendence of the collisional friction discussed in that section. In this case the energy diffusion coefficient D(E) is modeled by Eq. (4.9) which, in the limit of constant friction, is in accord with Eq. (5.1), deduced from the density of states. At low bath pressures, where the collisions are resolved, D(E) is the second moment of P(E, ), ... 

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