Electronic friction theory

From the point of view of associative desorption, this reaction is an early barrier reaction. That is, the transition state resembles the reactants.46 Early barrier reactions are well known to channel large amounts of the reaction exoergicity into product vibration. For example, the famous chemical-laser reaction, F + H2 — HF(u) + H, is such a reaction producing a highly inverted HF vibrational distribution.47-50 Luntz and co-workers carried out classical trajectory calculation on the Born-Oppenheimer potential energy surface of Fig. 3(c) and found indeed that the properties of this early barrier reaction do include an inverted N2 vibrational distribution that peaks near v = 6 and extends to v = 11 (see Fig. 3(a)). In marked contrast to these theoretical predictions, the experimentally observed N2 vibrational distribution shown in Fig. 3(d) is skewed towards low values of v. The authors of Ref. 44 also employed the electronic friction theory of Tully and Head-Gordon35 in an attempt to model electronically nonadiabatic influences to the reaction. The results of these calculations are shown in... 

Fig. 3. Vibrational population distributions of N2 formed in associative desorption of N-atoms from ruthenium, (a) Predictions of a classical trajectory based theory adhering to the Born-Oppenheimer approximation, (b) Predictions of a molecular dynamics with electron friction theory taking into account interactions of the reacting molecule with the electron bath, (c) Born—Oppenheimer potential energy surface, (d) Experimentally-observed distribution. The qualitative failure of the electronically adiabatic approach provides some of the best available evidence that chemical reactions at metal surfaces are subject to strong electronically nonadiabatic influences. (See Refs. 44 and 45.)...

The developed theory of two-phase laminar flow with a distinct interface which is based on a one-dimensional approximation, takes into account the major features of the process the inertia, gravity, surface tension and friction forces and leads to the physically realistic pattern of a laminar flow in a heated micro-channel. This allows one to use the present theory to study the regimes of flow as well as optimizing a cooling system of electronic devices with high power densities. 

Figure 2.5 Electron transfer rate as a function of the electronic interaction A. The full line is the prediction of first-order perturbation theory. The upper points correspond to a solvent with a low friction the lower points to a high friction. The data have been taken from Schmickler and Mohr [2002].

In a free solution, the electrophoretic mobility (i.e., peiec, the particle velocity per unit applied electric field) is a function of the net charge, the hydrodynamic drag on a molecule, and the properties of the solutions (viscosity present ions—their concentration and mobility). It can be expressed as the ratio of its electric charge Z (Z = q-e, with e the charge if an electron and q the valance) to its electrophoretic friction coefficient. Different predictive models have been demonstrated involving the size, flexibility, and permeability of the molecules or particles. Henry s theoretical model of pdcc for colloids (Henry, 1931) can be combined with the Debye-Hiickel theory predicting a linear relation between mobility and the charge Z ... 

The differences in antiwear properties of disulfides are related to their ability to be physisorbed about 100 to 1000 times faster than monosulfide on metal surfaces. The differences can be explained in terms of the lower energy needed for the formation of the same number of RS" ions from disulfides (Kajdas,1994). The exposed metal surface is extremely reactive to lubricant components, especially antiwear and extreme-pressure additives resulting the formation of a film on the contact surface. The reaction of emitted electrons of low energy (1 to 4 eV) with molecules of oil additives adsorbed on the friction surface may lead to formation of negative ions and negative ion radicals. The investigator (Kajdas, 1994 and 1985) pointed out the indispensability of the metal oxide film on the rubbing surface from the viewpoint of the theory of sulfide film formation. 

Since DFT calculations are in principle only applicable for the electronic ground state, they cannot be used in order to describe electronic excitations. Still it is possible to treat electronic exciations from first principles by either using quantum chemistry methods [114] or time-dependent density-functional theory (TDDFT) [115,116], First attempts have been done in order to calculate the chemicurrent created by an atom incident on a metal surface based on time-dependent density functional theory [117, 118]. In this approach, three independent steps are preformed. First, a conventional Kohn-Sham DFT calculation is performed in order to evaluate the ground state potential energy surface. Then, the resulting Kohn-Sham states are used in the framework of time-dependent DFT in order to obtain a position dependent friction coefficient. Finally, this friction coefficient is used in a forced oscillator model in which the probability density of electron-hole pair excitations caused by the classical motion of the incident atom is estimated. 

In the case of the equivalent niobium compounds, the same electronic effects are not present. He postulated that in pure stoichiometric niobium disulphide this results in poor lubrication. When good lubrication behaviour is observed, it is probably caused by additional niobium atoms intercalated between the lamellae, which contribute non-bonding electrons. On the basis of this theory, non-bonded atoms intercalated between the lamellae can increase the inter-lamellar spacing, whereas bonded intercalated atoms increase the resistance to inter-lamellar shear, and therefore the friction. However, an alternative interpretation is that certain intercalated atoms alter the interaction between the niobium atoms, allowing rearrangement to the 2H structure of molybdenum disulphide, and it is the favourable structure which provides good lubrication performance. 

Next consider the motion of these electrons. It was already mentioned that in addition to their density, metallic electrons are characterized, at this level of theory, by a relaxation time t. In the Drude theory this enters via a simple friction force by assuming that under a given force f (i) the electron moves according to... 

The Marcus theory, as described above, is a transition state theory (TST, see Section 14.3) by which the rate of an electron transfer process (in both the adiabatic and nonadiabatic limits) is assumed to be determined by the probability to reach a subset of solvent configurations defined by a certain value of the reaction coordinate. The rate expressions (16.50) for adiabatic, and (16.59) or (16.51) for nonadiabatic electron transfer were obtained by making the TST assumptions that (1) the probability to reach transition state configuration(s) is thermal, and (2) once the reaction coordinate reaches its transition state value, the electron transfer reaction proceeds to completion. Both assumptions rely on the supposition that the overall reaction is slow relative to the thermal relaxation of the nuclear environment. We have seen in Sections 14.4.2 and 14.4.4 that the breakdown of this picture leads to dynamic solvent effects, that in the Markovian limit can be characterized by a friction coefficient y The rate is proportional to y in the low friction, y 0, limit where assumption (1) breaks down, and varies like y when y oo and assumption (2) does. What stands in common to these situations is that in these opposing limits the solvent affects dynamically the reaction rate. Solvent effects in TST appear only through its effect on the free energy surface of the reactant subspace. 

The hot-spot theory, expounded by Bowden and Yoffe in their earlier work [2], is the foundation for most of their interpretations of initiation experiments. Heat (certainly), shock, friction, and intense electron beams were aU described as causing localized heating, such that if the region grows beyond a critical size the rate of heat generation exceeds the rate of heat loss and explosion occurs. 

The Warden theory is noteworthy for its heavy reliance on examples and analogies that have a scientific connotation, such as equations (i.e.. Physical x Moral = Outcome) the enemy system as a human body, or solar system reference to electrons and atoms reliance of architectural metaphors and the use of the concept of centers of gravity. This theory was not the first time in history that such reference had been made. " For example, Clausewitz s theories of friction in war and centers of gravity were identified as being borrowed from the physics of the day. Airpower theory, and the targeting assessments that it spawned, have long had an association, however, tenuous, with scientific terminology. 

Generality of the conduction modei. As explained in case study All Reactive Chemical Species in Chapter 4, the classical approach in kinetics is based on the transition state theory (Laidler and King 1998). The Formal Graph approach is based on a simpler theory of conduction which is much more general as it works in all energy varieties. For instance, the same theory is able to model electrons and holes in a p-n junction (Shockley diode) as well as molecules or enzymes involved in chemical or electrochemical reactions. Mechanical friction or viscous fluids may also be modeled with this transverse approach. 

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