Friction coefficient

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. 

Btamp/e Conformations of molecules like n-decane can be globally characterized by the end-to-end distance, R. In a comparison of single-molecule Brownian (Langevin) dynamics to molecular dynamics, the average end-to-end distance for n-decane from a 600 ps single-molecule Langevin dynamics run was almost identical to results from 19 ps of a 27-molecule molecular dynamics run. Both simulations were at 481K the time step and friction coeffi- 

The friction coefficient between fibers also increases with increase in temperature. As a matter of fact temperature effect is one of the eanses for the increased friction coefficients at high sliding speeds. At very high speeds, the temperature of fibers may increase, and this could contribute to the increased frietion coefficients. 

The friction of fibers on other materials also is important. For example, to convert fibers to final products, they often need pass around cyhnder surfaces, such as 

The Capstan equation can be rewritten to give the fiiction coefficient of fibers on other materials  

Source Buckle, H., et. al., Journal of the Textile Institute Transactions, 39, T199-T210, 1948.  

This is the Slokes-Einstein expression for the coefficient of diffusion, ft relates D to the properties of the fluid and the particle through the friction coefficient discussed in the next section, 

A careful experimental test of the theory was carried out by Perrin (1910). Emulsions composed of droplets about 0.4 jum in diameter were observed under an optical microscope and the positions of the particles were noted at regular time intervals. The Stokes-Einstein equation was checked by writing it in the form 

The friction coefficient is a quantity fundamental to most particle transport processes. The Stokes law form. / = 3jt fidp, holds for a rigid sphere that moves through a fluid at constant velocity with a Reynolds number. dpUjv, much less than unity. Here U is the velocity, and V is the kinematic viscosity. The particle must be many diameters away from any surfaces and much larger than the mean free path of the gas molecules, ip, which is about 0.065 jum at 25°C. 

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 accommodation coefficient or represents the fraction of the gas molecules that leave the surface in equilibrium with the surface. The fraction I — cr is specularly reflected such that the velocity normal to the surface is reversed. As in the case of Stokes law, the drag is proportional to the velocity of the spheres. However, for the free molecule range, the friction coefficient is proportional to dj whereas in the continuum regime dp ip), it is proportional to dp. The coefficient a must, in general, be evaluated experimentally but is usually near 0.9 for momentum transfer (values differ for heat and mass transfer). The friction coefficient calculated from (2.19) is only 1% of that from Stokes law for a 20-A particle. 

For t the dynamics of entangled polymer melts are described by the Rouse model. Owing to the fact that x, Eq. (11.33) can be written as 

ilogCr(0/dlogt =-1/2 at short times moreover, since iV, Eq. (11.45) suggests that G(t) and are independent of molecular weight. 

The friction coefficient is customarily obtained from either the relaxation or retardation spectrum, H x) or L x), respectively. At short times, i.e., on the transition from the glassy-like to the rubbery plateau, the viscoelastic processes obey Rouse dynamics, and the relaxation modulus is given by Eq. (11.45). Since H x) = —dG/d nx t, one obtains 

The coefficient of friction can alternatively be obtained from the retardation spectrum. In this case, 

A double logarithmic plot of L(x) versus x gives a straight line with slope 1/2 at short times. Values of can be obtained from Eq. (11.48) by using the procedures described above to evaluate this parameter from the relaxation spectrum. 

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Fig. 7 shows the torque necessary to obtain the specified body force under construction conditions and in tbe state when removed from the bridge. It can well be seen that the change of the friction coefficient causes a very big scattering, and the necessary torque is much bigger than specified. The distribution of the results of a measurement performed on 1,127 bolts is presented in Figure 8. An average of 80% of nominal body force was found by the new method. The traditional method found the nuts could be swivelled much further than specified on 42 bolts, these bolts were found to have 40 - 60 % body force by the new method. 

The traditional method for investigating the forces originating in the body of the bolt, which is based on measuring the torque of the nut, can detect only the bolts with a very great lack of body force since tbe friction coefficient worsens with time. 

The above described MBN measuring method investigates the stress state, so the obtained results are independent from the friction coefficient. 

Thus the average velocity decays exponentially to zero on a time scale detennined by the friction coefficient and the mass of the particle. This average behaviour is not very interesting, because it corresponds to tlie average of a quantity that may take values in all directions, due to the noise and friction, and so the decay of the average value tells us little about the details of the motion of the Brownian particle. A more interesting... 

Thus, the requirement that the Brownian particle becomes equilibrated with the surrounding fluid fixes the unknown value of, and provides an expression for it in tenns of the friction coefficient, the thennodynamic temperature of the fluid, and the mass of the Brownian particle. Equation (A3.1.63) is the simplest and best known example of a fluctuation-dissipation theorem, obtained by using an equilibrium condition to relate the strengtii of the fluctuations to the frictional forces acting on the particle [22]. 

This result is often called the Stokes-Einstein formula for the difflision of a Brownian particle, and the Stokes law friction coefficient 6iiq is used for... 

Another, purely experimental possibility to obtain a better estimate of the friction coefficient for rotational motion in chemical reactions consists of measuring rotational relaxation times of reactants and calculating it according to equation (A3,6,35) as y. =... 

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. 

In liquid solution. Brownian motion theory provides the relation between diffiision and friction coefficient... 

In the Smoluchowski limit, one usually assumes that the Stokes-Einstein relation (Dq//r7)a = C holds, which fonns the basis of taking the solvent viscosity as a measure for the zero-frequency friction coefficient appearing in Kramers expressions. Here C is a constant whose exact value depends on the type of boundary conditions used in deriving Stokes law. It follows that the diffiision coefficient ratio is given by ... 

Multidimensionality may also manifest itself in the rate coefficient as a consequence of anisotropy of the friction coefficient [M]- Weak friction transverse to the minimum energy reaction path causes a significant reduction of the effective friction and leads to a much weaker dependence of the rate constant on solvent viscosity. These conclusions based on two-dimensional models also have been shown to hold for the general multidimensional case [M, 59, and 61]. 

To conclude this section it should be pointed out again that the friction coefficient has been considered to be frequency independent as implied in assuming a Markov process, and that zero-frequency friction as represented by solvent viscosity is an adequate parameter to describe the effect of friction on observed reaction rates. 

Here is a friction coefficient which is allowed to vary in time 2 is a thennal inertia parameter, which may be replaced by v.j., a relaxation rate for thennal fluctuations g 3Ais the number of degrees of freedom. 

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 ... 

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... 

Figure C 1.4.6. Comparison of capture velocity for Doppler cooling and Tin-periD-lin sub-Doppler cooling. Notice tliat tire slope of tire curves, proportional to tire friction coefficient, is much steeper for tire sub-Doppler mechanism. (After [17].)...

We assume that the unbinding reaction takes place on a time scale long ( ompared to the relaxation times of all other degrees of freedom of the system, so that the friction coefficient can be considered independent of time. This condition is difficult to satisfy on the time scales achievable in MD simulations. It is, however, the most favorable case for the reconstruction of energy landscapes without the assumption of thermodynamic reversibility, which is central in the majority of established methods for calculating free energies from simulations (McCammon and Harvey, 1987 Elber, 1996) (for applications and discussion of free energy calculation methods see also the chapters by Helms and McCammon, Hermans et al., and Mark et al. in this volume). 

The effective moment of inertia / and the friction coefficient / could easily be estimated. The force constant k associated with the relative motion of the lobes was determined from an empirical energy function. To do so, the molecule was opened in a step-wise fashion by manipulating the hinge region and each resulting structure was energy minimized. Then, the interaction energy between the two domains was measured, and plotted versus 0. 

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). 

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