Friction coefficient phenomenological

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). 

In comparison with the qualitative description of diffusion in a binary system as embodied by Eqs. (11), (12) or (14), the thermodynamic factors are now represented by the quantities a, b, c, and d and the dynamic factors by the phenomenological coefficients which are complex functions of the binary frictional coefficients. Experimental measurements of Dy in a ternary system, made on the basis of the knowledge of the concentration gradients of each component and by use of Eqs. (21) and (22), have been reviewed 35). Another method, which has been used recently36), requires the evaluation of py from thermodynamic measurements such as osmotic pressure and evaluation of all fy from diffusion measurements and substitution of these terms into Eqs. (23)—(26). 

Thirion (239a) has suggested that the plateau and terminal regions are the result of diffuse interchain interactions in a viscoelastic medium. He obtains a modified Rouse spectrum by replacing the subchain frictional coefficient by a time dependent micro-memory function. The theory is partly phenomenological since the memory function is not specified. However, reasonable choices lead to forms for G (co) and G"(o>) which are similar to those observed experimentally. 

Spiegler (164) followed another way. Instead of the Lik s, he introduces the i2iJt s, which have the character of frictional coefficients and are defined by the phenomenological coefficients in the equations, which now express the forces as linear functions of the fluxes. It can be demonstrated that equations (1) maybe written as ... 

Xi = generalised force 0j = flux of species i Llk = phenomenological coefficient E = electrical potential difference A P= pressure difference Le = electrical conductivity of one square cm of membrane n = number of components of a system ut = velocity of component i i3j j = friction coefficients 7, - = electrical transport number lt = reduced transport number or transference number z( = charge of ion i vt = partial volume of ion i per gmol vD — partial volume of the solvent per gmol... 

In this equation, just as in Newton s law adapted for friction, the reciprocal of the phenomenological coefficient Ln has been introduced and acts as a friction coefficient, a resistance. Recalling the relations... 

The internal viscosity force is defined phenomenologically by equations (2.26) formulated above. Various internal-friction mechanisms, discussed in a number of studies (Adelman and Freed 1977 Dasbach et al. 1992 Gennes 1977 Kuhn and Kuhn 1945 Maclnnes 1977a, 1977b Peterlin 1972 Rabin and Ottinger 1990) are possible. Investigation of various models should lead to the determination of matrices Ca/3 and Ga and the dependence of the internal friction coefficients on the chain length and on the parameters of the macromolecule. 

In this formulae, ( is a friction coefficient of a particle in a monomer liquid, while non-dimensional phenomenological quantities B and E are measures of the increase in the external and internal friction coefficients due to the neighbouring macromolecules. 

All viscoelastic functions may be expressed in terms of a single reptation parameter (for example the plateau modulus or tube diameter) and the monomeric friction coefficient (or mobility factor in our terminology), in agreement with the above phenomenological presentation. 

The friction coefficients Cay defined by this equation may be expressed as functions of the phenomenological coefficients. ... 

Kinetic friction differs phenomenologically from static friction in that rather than being in a state of impending or beginning motion, the sliding body is moving overtly with respect to the stationary body. Me write the following expression for the coefficient of kinetic friction... 

During sedimentation under gravity, particle velocity is given by the ratio of force, Fg, to friction coefficient, B. For spherical particles B is given by the Stokes equation, B=6m r, and hence the sedimentation flux and corresponding phenomenological coefficient, an, are given by... 

N) as extension (or eompression) of springs during sliding. The phenomenological measure of friction is in terms of a friction coefficient given by... 

This "phenomenological / should not be confused with the "molecular friction coefficient which is defined as a force needed to have a molecule translate steadily at unit velocity. The latter appears below when the self-diffusion coefficient is introduced. 

The other terms present in the flux equations (eqs. EH - 36 and III - 37) are phenomenological coefficients, and these must also be considered with respect to membrane formation. Also these coefficients are mostly concentration dependent. There are two ways of expressing the phenomenological coefficients when the relationships for the chemical potentials are known i) In diffusion coefficients and ii) in friction coefficients. 

From a purely theoretical point of view, both approaches can be followed. However from a more practical point of view it is preferable to transform ternary parameters into binary parameters. The latter are much more readily measured. For this reason, it is preferable to relate the phenomenological coefficients to binary friction coefficients. 

In formulae (44) and (45), is the friction coefficient of a particle in a monomer liquid, while B and E are phenomenological parameters which will be discussed below. The correlation time r can be interpreted as relaxation time of the mechanical (viscoelastic) reaction of neighbouring macromolecules, i.e. the system as a whole. This quantity will be calculated later (see Sect 6), and the self-consistency of the theory will be demonstrated. Of course, such a choice of memory functions is eventually justified by empirical facts in later Sections, so we consider the memory functions (45) to be empirical, but to give a rather good description for the case

[Pg.165]

In the original tube theo for polymer melts, and the various modifications and refinements, the monomeric friction coefficient and tube diameter are phenomenological parameters. It is important therefore to have a molecular level understanding of these parameters. In the present paper, we will concentrate on the tube diameter in polymer melts and solutions. Our main aim is to provide a quantitative description of the tube diameter in terms of the polymer properties. 

First, we note that in order to predict a phase diagram as a function of shear rate, we must account for the variation with temperature of the molecular dynamics. It is well established that the terminal relaxation time changes rapidly with temperature, due to changes in the monomeric friction coefficient, and a suitable description of the behaviour is provided by the phenomenological WLF formula [68],... 

The most important interaction present in a dilute suspension of a chain is the hydrodynamic interaction. In addition, the excluded volume interaction may be present depending on the nature of the solvent, polymer and temperature this interaction vanishes at the 6 temperature, so there is already an important problem when all interactions are ignored. However the interachain entanglement effects corresponding to the uncrossability of different portions of the same chain are always present. The simplest model which cannot actually be realized physically is to both ignore interactions and hydrodynamic effects and assume ad hoc that the solvent attributes a phenomenological friction coefficient for every monomer. This model is called the Rouse model. 

These bead-spring models of Rouse and Kirkwood-Riseman-Zimm suffer from the artificiality of the beads and springs. The bead friction coefficient is an ad hoc phenomenological coefficient. This should arise naturally from the frictional forces coupling the polymer and solvent directly with the continuous version of the chain without beads and springs. 

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