Kinetic friction coefficient

Static Friction Coefficient, Kinetic Friction Coefficient, u... 

Predicting the solvent or density dependence of rate constants by equation (A3.6.29) or equation (A3.6.31) requires the same ingredients as the calculation of TST rate constants plus an estimate of and a suitable model for the friction coefficient y and its density dependence. While in the framework of molecular dynamics simulations it may be worthwhile to numerically calculate friction coefficients from the average of the relevant time correlation fiinctions, for practical purposes in the analysis of kinetic data it is much more convenient and instructive to use experimentally detemiined macroscopic solvent parameters. 

Here the friction coefficient is completely detemiined by the instantaneous values of the coordhiates and momenta. It is easy to see that the kinetic energy jc = Y.ia constant of the motion ... 

The friction and wear properties of fullerene LB fihns have been investigated. The coefficient of kinetic friction was measured using a steel ball-on-glass disk method, with the LB films deposited onto the glass disk [326,327]. The friction coefficient dropped from 0.8... 

The various physical methods in use at present involve measurements, respectively, of osmotic pressure, light scattering, sedimentation equilibrium, sedimentation velocity in conjunction with diffusion, or solution viscosity. All except the last mentioned are absolute methods. Each requires extrapolation to infinite dilution for rigorous fulfillment of the requirements of theory. These various physical methods depend basically on evaluation of the thermodynamic properties of the solution (i.e., the change in free energy due to the presence of polymer molecules) or of the kinetic behavior (i.e., frictional coefficient or viscosity increment), or of a combination of the two. Polymer solutions usually exhibit deviations from their limiting infinite dilution behavior at remarkably low concentrations. Hence one is obliged not only to conduct the experiments at low concentrations but also to extrapolate to infinite dilution from measurements made at the lowest experimentally feasible concentrations. 

The friction coefficient of a large B particle with radius ct in a fluid with viscosity r is well known and is given by the Stokes law, Q, = 67tT CT for stick boundary conditions or ( = 4jit ct for slip boundary conditions. For smaller particles, kinetic and mode coupling theories, as well as considerations based on microscopic boundary layers, show that the friction coefficient can be written approximately in terms of microscopic and hydrodynamic contributions as ( 1 = (,(H 1 + (,/( 1. The physical basis of this form can be understood as follows for a B particle with radius ct a hydrodynamic description of the solvent should... 

Figure 18 Coverage dependence of the kinetic friction coefficient pk of a system containing 0.25-2.5 monolayers of a simple fluid. Commensurate systems (c) are denoted with open symbols, and incommensurate systems (ic) are designated with closed symbols. Reproduced with permission from Ref. 82.

In their study, Park et al.100 investigated the frictional properties of fluorine-terminated alkanethiol SAMs grafted to gold surfaces. The frictional properties of the system were investigated by sliding two SAMs past one another at velocities in the stick-slip regime under various external loads. The simulations yield the shear stress as and the kinetic friction coefficient pk can be estimated from the slope of a plot of as versus load, using the relationships contained in Eqs. [4] and [5]. 

Cartesian kinetic SDEs with unprojected, geometrically projected, and inertially projected random forces require the same correction forces in certain special cases. Inertial and geometric projections are completely equivalent for models with an equal bead mass m for all beads, for which the mass tensor m v = is proportional to the identity. Unprojected and geometrically projected random forces require identical correction forces in the case of local, isotropic friction with an equal friction coefficient for all beads, as in the Rouse or Kramers model, for which the friction tensor ... 

In other words, it is assumed here that the particles are surrounded by a isotropic viscous (not viscoelastic) liquid, and is a friction coefficient of the particle in viscous liquid. The second term represents the elastic force due to the nearest Brownian particles along the chain, and the third term is the direct short-ranged interaction (excluded volume effects, see Section 1.5) between all the Brownian particles. The last term represents the random thermal force defined through multiple interparticle interactions. The hydrodynamic interaction and intramolecular friction forces (internal viscosity or kinetic stiffness), which arise when the macromolecular coil is deformed (see Sections 2.2 and 2.4), are omitted here. 

The friction coefficient can be measured in two ways the static friction coefficient Qus) and the dynamic or kinetic friction coefficient (fikX The static friction coefficient is defined as the ratio of the force required to initiate relative movement and the normal force between the surfaces the dynamic or kinetic friction coefficient is defined as the ratio of the friction force to the normal force when the two surfaces are moving relative to each other. For simplicity, much of the research has focused on the dynamic friction coefficients wherein the two surfaces move at a relative constant velocity. Most of the friction studies on skin have dealt with the dynamic friction coefficient and the subscript k is usually dropped. This overview references the dynamic coefficient of friction unless otherwise noted. 

FIGURE 33.1 Variation of the friction coefficient versus time during the recording of an experiment on the forearm. The two curves correspond to two different loads applied on the sliding pad. The maximum of the curves corresponds to the static friction coefficient and the asymptotic value to the kinetic coefficient.3... 

If the static friction is greater than the kinetic friction, slip-stick motion may be the result. In rigid plastics the kinetic friction coefficient is normally lower than the static coefficient, in elastomers the reverse applies. At high velocities it is sometimes difficult to separate the effects of velocity and temperature. 

In contrast, the motion in the electric field of kinetically rigid molecules for which tq is less than should be governed by the mechanism of the orientation of the molecule as a whole and the rate of this process should be determined by the rotational friction coefficient W or rotatory diffusion coefficient Dj (Eq. (53)). Hence, the study of the orientation kinetics of kinetically rigid macromolecules in the electric field permits the determination of their rotational mobility (i.e. W and Dj). 

Despite significant simplification of the speedskate ice friction problem, FAST 1.0 is able to predict kinetic ice friction coefficients that are consistent with the published measurements of de Koning et al., with an error of about 20% or less. 

Under conditions typical of competitive speedskating, frictional melting, squeeze flow and heat conduction into the ice all play an important role in determining skate blade lubrication. Pressure-induced freezing point depression and the quasi-liquid layer are accounted for in the model, but they play only a minor role in determining the kinetic ice friction coefficient. 

Because of the complex nature of the temperature and velocity dependence of the kinetic ice friction coefficient, it is important to measure ice friction under conditions that are as close as possible to those where the data will be applied. It is inadvisable to apply or extrapolate measurements and inferences made at low velocities to skating at much higher velocities. 

Under static conditions, 0 rises linearly with time and the static friction coefficient rises as In t. The steady-state value of 0 during sliding can be found by setting the time derivatives in Eq. (6) to zero, yielding O,. = D /v. Inserting this in Eq. (5) gives p = Po + hi(w/r)o). Thus the steady-state kinetic... 

It might yet be possible to predict trends from Eq. (52). Persson and Tosatti [27] argue that decreases quickly with increasing normal pressure. This has important consequences for tectonic motion for instance, small earthquakes typically do not occur close to the earth s surface. Sokoloff [122] concluded recently that even for his small value of the contribution to kinetic friction due to elastic instabilities that result from a competition between surface roughness and elastic interactions would lead to rather small friction coefficients on the order of 10 provided that no contamination layer or other local friction mechanisms were present in the contacts. 

Figure 22. (a) Loss spectrum Im (ai)]/ ((B) as a function of frequency (i). (b) Kinetic friction coefficient at constant sliding velocity w, (c) Transient behavior of the friction coefficient as a function of time t. At the smallest t. the velocity is abruptly changed from zero velocity to finite velocity. With permission from Ref. 267. Phys. Rev. B 65, 134106 (2002). 

As particle size is decreased to the point where dp tp, the drag for a given velocity becomes less than predicted from Stokes law and continues to decrease with particle size. In the range dp ip, the free molecule range (Chapter 1), an expression for the friction coefficient can be derived from kinetic theory (Epstein, 1924) ... 

The Cooper-Mann theory of monolayer transport was based on the model of a sharply localized interfacial region in which ellipsoidal molecules were constrained to move. The surfactant molecules were assumed to be massive compared with the solvent molecules that made up the substrate and a proportionate part of the interfacial region. It was assumed that the surfactant molecules had many collisions with solvent molecules for each collision between surfactant molecules. A Boltzmann equation for the singlet distribution function of the surfactant molecules was proposed in which the interactions between the massive surfactant molecules and the substrate molecules were included in a Fokker-Planck term that involved a friction coefficient. This two-dimensional Boltzmann equation was solved using the documented techniques of kinetic theory. Surface viscosities were then calculated as a function of the relevant molecular parameters of the surfactant and the friction coefficient. Clearly the formalism considers the effect of collisions on the momentum transport of the surfactant molecules. 

For dilute monolayers, the intermolecular potential between two surfactant molecules was not needed to construct the kinetic formulas. Obviously there are solvent-surfactant interactions since the entire system is a liquid. However, it was possible to incorporate such interactions into a friction coefficient that can be computed in principle or left as a parameter details are given by Cooper and Mann (7). The extension to dense gas monolayers requires a potential function and a radial distribution function for the surfactant molecules. The formulation based on the Rice-Allnatt approach was developed by Cooper and Mann (7). 

The construction of Cooper and Mann (7) for the surface viscosity includes the substrate effect by a model that represents the result of very frequent molecular collisions between the small substrate molecules and the larger molecules of the monolayer. This was done by adding a term to the Boltzmann equation for the 2D singlet distribution function that is equivalent to the friction coefficient term of the Fokker-Planck equation from which Equations 24 and 25 can be constructed. Thus a Brownian motion aspect was introduced into the kinetic theory of surface viscosity. It would be interesting to derive the collision frequency of Equation 19 using the better model (7) and observe how the T/rj variable of Equation 26 emerges. 

For the polyion equivalent conductivity, conditions are different. Here an appreciable concentration dependence is expected even in very dilute solutions. This is partly due to the direct dependence of Ap on a, a quantity that may vary with concentration, and partly due to the concentration dependence of the friction coefficient/p. As in the case of polymeric solutes in general, the friction coefficient depends on the polyion chain conformation, which for flexible polyelectrolytes is strongly concentration dependent. Furthermore, the polyion friction coefficient also includes contributions from the fraction (1 — a) of the counterions, which form a kinetic unit with the polyion. The friction coefficient can therefore be written in the form... 

Kinetic theory can be used to extend this simple description in several ways. First, even if the calculation is carried out at the Langevin equation level, the kinetic theory results for the friction coefficient can be used to go beyond the simple (and possibly inadequate) approximation of a constant friction. Second, a direct solution of the kinetic equation rather than the Langevin equation can be carried out. Both types of calculation should increase our understanding of the microscopic dynamics of these processes. 

Model kinetic equation approaches of these types should probably be more thoroughly investigated for such complex systems before more elaborate kinetic theories are constructed. Ultimately, however, difficult problems such as the nature of the friction coefficient or collision frequency associated with an internal coordinate must be solved. What, for instance, is the form of its space and time nonlocality The solution of this problem will involve a more complex calculation than that outlined in Section IX.B for the two-particle friction tensor. 

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