Solution frictional coefficient

Exact solution. Friction coefficient. Following Blasius [43], we express the fluid velocity components via the stream function L according to (1.1.6) and substitute them into (1.7.2). Then we seek the stream function in the form... 

Ti-15Mo-5Zr-3Al NaCl solution Friction coefficient 0.82 (P alloy) 3 times that of stainless steel 316L Steinemann et ai, 1993... 

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

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. 

Second, we take account of the frictional drag as the solute molecule moves through the solvent. The frictional force is taken to be proportional to the velocity of the particle, with a proportionality constant called the friction coefficient... 

Fig. 3.8a,b Friction coefficients as a function of solution Reynolds number Reji, in (a) adiabatic flow, and (b) diabatic flow. Reprinted from Hetsroni et al. (2004) with permission... 

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. 

Ni is Avogadro s number, / the particle friction coefficient, u the velocity and v the partial specific volume of the solute [79,80]. Inertial forces are negligible and the balancing of the above forces yields the sedimentation coefficient ... 

Hardness measurements of non-metallic solids are influenced by environmental factors. These have been studied extensively by Westwood (Westwood et al., 1981) and others. However, the evidence is that most, if not all, of the observed effects result from changes in the indenter/specimen friction coefficient caused by adsorption. Under ambient conditions, water vapor is commonly adsorped (Hanneman and Westbrook, 1968). In the presence of various liquids both solvents and solutes are adsorped. Since the effects are not intrinsic to the specimens, they will not be discussed further here. 

The time dependent friction coefficient, per solute mass p, is related to the fluctuating forces exerted by the solvent on the solute coordinate x through their time correlation function ... 

Hydrodynamic properties, such as the translational diffusion coefficient, or the shear viscosity, are very useful in the conformational study of chain molecules, and are routinely employed to characterize different types of polymers [15,20, 21]. One can consider the translational friction coefficient, fi, related to a transport property, the translational diffusion coefficient, D, through the Einstein equation, applicable for infinitely dilute solutions ... 

This result is appHcable to semi-dilute and concentrated solutions [21], and is also useful to check many simulations that do not include HI. For non-draining chains, introducing Gaussian statistics in Eq. (35), and transforming the summations over a large number of units in Eq. (34) into integrals, the translational friction coefficient can finally be written as [ 15]... 

The translational friction coefficient ft of the polymer chain at infinitely dilute solutions is calculated by equating the net force acting on the chain, — and —fR, where R is the net drift velocity of the center of mass of the chain... 

Historically, one of the central research areas in physical chemistry has been the study of transport phenomena in electrolyte solutions. A triumph of nonequilibrium statistical mechanics has been the Debye—Hiickel—Onsager—Falkenhagen theory, where ions are treated as Brownian particles in a continuum dielectric solvent interacting through Cou-lombic forces. Because the ions are under continuous motion, the frictional force on a given ion is proportional to its velocity. The proportionality constant is the friction coefficient and has been intensely studied, both experimentally and theoretically, for almost 100... 

---