Friction-velocity dependence

Because Newton s equations of motion are conservative, the natural ensemble is NVE (micro-canonical), Aat is one in which the internal energy rather than the temperature is held constant. This is inconvenient if one wishes to compare with experiment where it is the temperature that is generally controlled. In order to perform molecular dynamics in the canonical ensemble, a thermostat must be applied to the system. This is accomplished by constructing a pseudo-Lagrangian. Many forms for temperature-conserving Lagrangians have been proposed, most of which can be written in a form that adds a frictional (velocity-dependent) term to the equations of motion (Allen and Tildesley 1989). Physically, the thermostat can be thought of as a heat bath to which the system is coupled. In the NPT ensemble, in which the pressure is held constant, the cell size and shape fluctuates. The choice of dynamical variables is critical. If the lattice parameters are chosen as in the method of Parrinello and Rahman (1981), the time evolution may depend on the chosen size or shape of the supercell. This difficulty is... 

When a coarse grid is used, wall functions are used for imposing boundary conditions near the walls (Section 11.2.3.3). The nondimensional wall distance should be 30 < y < ]Q0, where y = u,y/p. We cannot compute the friction velocity u. before doing the CFD simulation, because the friction velocity is dependent on the flow. However, we would like to have an estimation of y" to be able to locate the first grid node near the wall at 30 < y < 100. If we can estimate the maximum velocity in the boundary layer, the friction velocity can be estimated as n, — 0.04rj, . . After the computation has been carried out, we can verify that 30 nodes adjacent to the walls. 

It has been recognized that the behavior of atomic friction, such as stick-slip, creep, and velocity dependence, can be understood in terms of the energy structure of multistable states and noise activated motion. Noises like thermal activities may cause the atom to jump even before AUq becomes zero, but the time when the atom is activated depends on sliding velocity in such a way that for a given energy barrier, AI/q the probability of activation increases with decreasing velocity. It has been demonstrated [14] that the mechanism of noise activation leads to "the velocity... 

In electric-field driven separations an electric field causes ions to travel through a matrix, such as a gas, liquid, or gel. The movement is retarded by frictional forces from interaction with the matrix and the ions almost instantly reach a steady-state velocity. This velocity depends on properties of both the sample molecules and the surrounding matrix. The two main types of electric-field driven separations are ion mobility spectrometry where the matrix is a gas and electrophoresis where the matrix is a liquid or gel. 

Once temperature comes into play, the jumps of atoms between minima may be invoked prematurely, i.e., before the formation of instabilities, via thermal fluctuations. These thermally activated jumps decrease the force that is required to pull the surface atom, which leads to a decrease in the kinetic friction. The probability that a jump will be thermally activated is exponentially related to the energetic barrier of the associated process, which can be understood in terms of Eyring theory. In general, the energetic barriers are lower when the system is not at its thermal equilibrium position, which is a scenario that is more prominent at higher sliding velocities. Overall, this renders Fk rate or velocity dependent, typically in the following form ... 

An example of the velocity dependence of friction is given in Figure 5 for a boundary lubricant confined between two incommensurate surfaces.25 For the given choice of normal pressure and temperature, one finds four decades in sliding velocity for which Eq. [7] provides a reasonably accurate description. 

Figure 5 Typical velocity relationship of kinetic friction for a sliding contact in which friction is from adsorbed layers confined between two incommensurate walls. The kinetic friction F is normalized by the static friction Fs. At extremely small velocities v, the confined layer is close to thermal equilibrium and, consequently, F is linear in v, as to be expected from linear response theory. In an intermediate velocity regime, the velocity dependence of F is logarithmic. Instabilities or pops of the atoms can be thermally activated. At large velocities, the surface moves too quickly for thermal effects to play a role. Time-temperature superposition could be applied. All data were scaled to one reference temperature. Reprinted with permission from Ref. 25. 

These equations are not as hard to understand as they might appear at first glance. The first and second equations are the same as we saw in Eq. (9.5) apart from the addition of a friction term in the second equation that either increases or decreases the velocity, depending on the sign of The third equation controls the sign and magnitude of The meaning of this equation is clearer if we rewrite Eq. (9.7) as... 

If relief is via a bursting disc, the flow capacity of the relief system will normally depend on friction and choke points in the relief system. The only exception is where friction is not important (LE/D less than about 40), where equation (A6.4) can be used.) Where friction is significant, an isometric sketch of the route of the relief system will be required to determine the capacity. If the system is to be of constant diameter, then using the sketch, the total equivalent length, LE, of the route, including the frictional resistance of bends and fittings can be determined111. This can also be expressed in terms of total frictional velocity head loss, K ... 

The failure of the Rouse theory was attributed to the pathological nature of medium motions in entangled systems, and not any special defect in the Rouse representation of the polymer chain itself. For Rouse chains in a deforming continuous medium, the frictional force depends on the systematic velocity of the bead relative to the medium. The frictional force on a bead is therefore a smootly... 

The relation between friction and viscosity goes beyond the Stokes relation. The Navier-Stokes hydrodynamics has been generalized by Zwanzig and Bixon [23] to include the viscoelastic response of the medium. This generalization provides an elegant expression for the frequency-dependent friction which depends among other things on the frequency-dependent bulk and shear viscosities and sound velocity. 

Fig. 4 Dependency of the deposition rate (k) on the particle diameter and friction velocity...

To avoid stick-slip, one should try to make the spring constant high enough (using stiff materials and stable constructions). It can be shown, that stick-slip may also arise from the velocity dependence of the friction coefficient [460], When the friction coefficient decreases with sliding velocity, stick-slip is amplified. When the friction coefficient increases with velocity, stick-slip is damped out. The former is usually the case at low speeds, certainly for the transition from static to dynamic friction, whereas the latter prevails usually at high velocity. 

The friction velocity at the minimum transport condition may be related to the system configuration and the operating condition by a two-step correlation method, i.e., first, to obtain the velocity at the minimum transport condition under infinite dilutions and, second, to correct for the concentration dependence [Thomas, 1962]. The functional dependence of Uf to the solids concentration is given by... 

The most popular way to control the temperature in the CP MD simulation was introduced by Nose and Hoover.23,24 This approach includes an extra friction term (velocity dependent) into the Car-Parinello equations of motions (cf. Eq. 3) ... 

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