Atomic friction coefficient

A stochastic dynamics simulation requires a value to be assigned to the collision frequency friction coefficient 7. For simple particles such as spheres this can be related to the diffusion constant in the fluid. For the simulation of a rigid molecule it may be possible to derive 7 via the diffusion coefficient from a standard molecular dynamics situation. In the more general case we require the friction coefficient of each atom. For simple molecules such as butane the friction coefficient can be considered to be the same for all atoms. The optimal value for 7 can be determined by trial and error, performing a stochastic dynamics simulation for different values of 7 and comparing the results with those from experiment (where available) or from standard molecular dynamics simulations. For large molecules the atomic friction coefficient is considered to depend upon the degree to which each atom is in contact with the solvent and is usually taken to be proportional to the accessible surface area of the atom (as defined in Section 1.5). 

In addition to the intramolecular forces, the chain is subjected to a Brownian force g n,t) due to random collisions with other chains, and an intermolecular frictional force, x(n, t) where < is an atomic friction coefficient. Defining the following Fourier transforms... 

It can be shown that a trajectory generated by integrating the Langevin equation of motion (with at least one non-vanishing atomic friction coefficient y,) maps (at constant volume) a canonical distribution of microstates at temperature To. 

This expression shows diat if die detuning Acuj is negative (i.e. red detuned from resonance), dieii die cooling force will oppose die motion and be proportional to die atomic velocity. The one-diniensional motion of die atom, subject to an opposing force proportional to its velocity, is described by a damped haniionic oscillator. The Doppler damping or friction coefficient is die proportionality factor. 

Equation (Cl.4.35) yields two remarkable predictions first, tliat tire sub-Doppler friction coefficient can be a big number compared to since at far detuning Aj /T is a big number and second, tliat a p is independent of tire applied field intensity. This last result contrasts sharjDly witli tire Doppler friction coefficient which is proportional to field intensity up to saturation (see equation (C1.4.24). However, even tliough a p looks impressive, tire range of atomic velocities over which is can operate are restricted by tire condition tliat T lcv. The ratio of tire capture velocities for Doppler versus sub-Doppler cooling is tlierefore only uipi/uj 2 Figure Cl. 4.6 illustrates... 

The two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

Here, y is the friction coefficient of the solvent, in units of ps and Rj is the random force imparted to the solute atoms by the solvent. The friction coefficient is related to the diffusion constant D of the solvent by Einstein s relation y = kgT/mD. The random force is calculated as a random number, taken from a Gaussian distribu-... 

When the friction coefficient is set to zero, HyperChem performs regular molecular dynamics, and one should use a time step that is appropriate for a molecular dynamics run. With larger values of the friction coefficient, larger time steps can be used. This is because the solution to the Langevin equation in effect separates the motions of the atoms into two time scales the short-time (fast) motions, like bond stretches, which are approximated, and longtime (slow) motions, such as torsional motions, which are accurately evaluated. As one increases the friction coefficient, the short-time motions become more approximate, and thus it is less important to have a small timestep. 

The friction coefficient determines the strength of the viscous drag felt by atoms as they move through the medium its magnitude is related to the diffusion coefficient, D, through the relation Y= kgT/mD. Because the value of y is related to the rate of decay of velocity correlations in the medium, its numerical value determines the relative importance of the systematic dynamic and stochastic elements of the Langevin equation. At low values of the friction coefficient, the dynamical aspects dominate and Newtonian mechanics is recovered as y —> 0. At high values of y, the random collisions dominate and the motion is diffusion-like. 

Friction and Adhesion. The coefficient of friction p. is the constant of proportionality between the normal force P between two materials in contact and the perpendicular force F required to move one of the materials relative to the other. Macroscopic friction occurs from the contact of asperities on opposing surfaces as they sHde past each other. On the atomic level friction occurs from the formation of bonds between adjacent atoms as they sHde past one another. Friction coefficients are usually measured using a sliding pin on a disk arrangement. Friction coefficients for ceramic fibers in a matrix have been measured using fiber pushout tests (53). For various material combinations (43) ... 

At the end of last century, a near frictionless carbon (NFC) coating was reported, which is practically hydrogen contained DLC film grown on steel and sapphire substrates using a plasma enhanced chemical vapor deposition (PECVD) system [50]. By using a ball on a disk tribo-meter, a super low friction coefficient of 0.001-0.003 between the films coated on both the ball and the disk was achieved [50]. A mechanistic model was proposed that carbon atoms on the surface are partially di-hydrogenated, resulting in the chemical inertness of the surface. Consequently, adhesive interaction becomes weak and super low friction is achieved [22],... 

Therefore, for a rough surface, although it is smooth macro-scopically, the friction coefficient is greatly affected by its surface morphology. For a smooth surface on atomic scale, 0 can be neglected and the friction coefficient is only related to the twist angle of the cantilever. 

The work on carbon nitride solids is strongly related to research on diamondlike carbon (DLC) materials [5, 6]. DLC materials are thin film amorphous metastable carbon-based solids, pure or alloyed with hydrogen, which have properties similar to that of crystalline diamond (high hardness, low friction coefficient, high resistance to wear and chemical attack). This resemblance to diamond is due to the DLC structure, which is characterized by a high fraction of highly cross-linked sp -hybridized carbon atoms. To obtain this diamond-like structure... 

Relatively few works considered the tribological properties of plasma-deposited a-C(N) H films. First, considering the coefficient of friction, nitrogen incorporation was not found to strongly modify it. Prioli et al. [93] studied the friction coefficient of a-C(N) H films against a Si3N4 probe in an atomic force microscope, in ambient air. Nitrogen incorporation up to about 10 at.% into the... 

Here, 7 is the friction coefficient and Si is a Gaussian random force uncorrelated in time satisfying the fluctuation dissipation theorem, (Si(0)S (t)) = 2mrykBT6(t) [21], where 6(t) is the Dirac delta function. The random force is thought to stem from fast and uncorrelated collisions of the particle with solvent atoms. The above equation of motion, often used to describe the dynamics of particles immersed in a solvent, can be solved numerically in small time steps, a procedure called Brownian dynamics [22], Each Brownian dynamics step consists of a deterministic part depending on the force derived from the potential energy and a random displacement SqR caused by the integrated effect of the random force... 

Thus, the strong C—F bond, the special arrangement of atoms in macromolecule, and low surface energy impart some unique physical properties to PTFE and other fluoropolymers high chemical and thermal resistance, nonstick character, low friction coefficient, and low wettability. This combination of properties... 

Figure 21 Friction coefficient for differently oriented Ni(100)/Ni(100) interfaces. Rough surfaces have a 0.8 A rms variation in roughness added to the atomically smooth surfaces. Reproduced with permission from Ref. 85.

Figure 22 Snapshots from the simulations leading to the friction coefficients shown in Figure 21. From left to right Atomically flat commensurate, atomically flat incommensurate, rough commensurate, and rough incommensurate geometries. Only the flat incommensurate surfaces remain undamaged, resulting in abnormally small friction coefficients. Reproduced with permission from Ref. 85.

The average DNA helix diameter used in modeling applications such as the ones described here includes the diameter of the atomic-scale B-DNA structure and— approximately—the thickness of the hydration shell and ion layer closest to the double helix [18]. Both for the calculation of the electrostatic potential and the hydrodynamic properties of DNA (i.e., the friction coefficient of the helix for viscous drag) a helix diameter of 2.4 nm describes the chain best [19-22]. The choice of this parameter was supported by the results of chain knotting [23] or catenation [24], as well as light scattering [25] and neutron scattering [26] experiments. 

Hynes et al. [298] and later Schell et al. [272] have developed a numerical simulation method for the recombination of iodine atoms in solution. The motions of iodine atoms was governed by a Langevin equation, though spatially dependent friction coefficients could be introduced to increase solvent structure. The force acting on iodine atoms was obtained from the mutual potential energy of interaction, represented by a Morse potential and the solvent static potential of mean force. The solvent and iodine atoms were regarded as hard spheres. The probability of reaction was calculated by following many trajectories until reaction had occurred or was most improbable. The importance of the potential of... 

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