Frictional function

Some years ago, on the basis of the excluded-volume interaction of chains, Hess [49] presented a generalized Rouse model in order to treat consistently the dynamics of entangled polymeric liquids. The theory treats a generalized Langevin equation where the entanglement friction function appears as a kernel... 

Equations (35) and (36) define the entanglement friction function in the generalized Rouse equation (34) which now can be solved by Fourier transformation, yielding the frequency-dependent correlators . In order to calculate the dynamic structure factor following Eq. (32), the time-dependent correlators are needed. 

These include the Rayleigh quotient method" and variational transition state theory (VTST).46 9 xhg 0 called PGH turnover theory and its semiclassical analog/ which presents an explicit expression for the rate of reaction for almost arbitrary values of the friction function is reviewed in Section IV. Quantum rate theories are discussed in Section V and the review ends with a Discussion of some open questions and problems. 

The continuum limit of the Hamiltonian representation is obtained as follows. One notes that if the friction function y(t) appearing in the GLE is a periodic function with period T then Eq. 4 is just the cosine Fourier expansion of the friction function. The frequencies coj are integer multiples of the fundamental frequency and the coefficients Cj are the Fourier expansion coefficients. In practice, the friction function y(t) appearing in the GLE is a decaying function. It may be used to construct the periodic function y(t T) = Y(t TiT)0(t-... 

The friction function (Eq. 4) is then the cosine Fourier transform of the spectral density. 

This extends the previous work (I ) In which the Lennard-Jones type surface potential function and the frictional function representing the Interfaclal forces working on the solute molecule from the membrane pore wall were combined with solute and solvent transport through a pore to calculate data on membrane performance such as those on solute separation and the ratio of product rate to pure water permeation rate in reverse osmosis. In the previous work (1 ) parameters Involved in the Lennard-Jones type and frictional functions were determined by a trial and error method so that the solutions in terms of solute separation and (product rate/pure water permeation rate) ratio fit the experimental data. In this paper the potential function is generated by using the experimental high performance liquid chromatography (HPLC) data in which the retention time represents the adsorption and desorption equilibrium of the solute at the solvent-polymer interface. 

The frictional force is expressed by a function of the ratio of a distance associated with sterlc repulsion at the Interface, to the pore radius. The frictional function Increases steeply with increase in the latter ratio. The method of calculating reverse osmosis separation data by using the surface potential function and the frictional function so generated, in conjunction with the transport equation is illustrated by examples involving cellulose acetate membranes of different porosities and AO nonionized organic solutes in single solute aqueous solution systems. 

In the earlier work (1 ) transport equations were developed on the basis of surface force-pore flow model in which a surface potential function and a frictional function are incorporated. The results can be briefly summarized as follows ... 

Eq 32 implies that the frictional function includes also the effect of tortuosity. 

A in this work). The pore radius used in the Polseuille equation should be, therefore, the smaller radius designated by R]. On the other hand, the interaction force expressed by a potential function such as eq 13 or the frictional function such as eq 39 are exerted throughout the larger pore radius R2. 

The frictional function which was described as a function of the ratio of the distance associated with the steric repulsion at the interface to the pore radius, however, is still an approximation at most, though it is convenient to use, due to its simplified form. A more appropriate functional form including both steric repulsion and interfacial affinity effects on the restricted motion of the solute molecule in the membrane pore is yet to be developed. A further research effort in this direction is called for. 

The friction function that fulfills this requirement may be represented by a polynomial or by the exponential expression... 

The potential of mean force will typically have two wells, corresponding to reactants and products, separated by a barrier. To set the notation, we denote the location of the reactant well, the barrier, and the product well by qa, q, and qb, respectively. One usually expects that the dynamics will be governed by the behavior of the system around the barrier top. Thus the standard procedure (48,49) for generating a GLE is to restrict the system to the barrier top q = q and determine the force autocorrelation function of all other degrees of freedom. The force is just VV, so by using molecular dynamics constrained to the barrier top one can compute the force autocorrelation function (VV(f) W(0)). One then models the true dynamics in terms of a GLE in which the time-dependent friction function is determined through the fluctuation dissipation relation, Eq. (5). 

This procedure has been criticized by Berne and co-workers (50-52). They have shown that the friction function generated in this manner at values of the reaction coordinate different from the barrier top may be quite different from that generated at the barrier top. If the GLE description were accurate, the friction function should be independent of the reaction coordinate. This deficiency may be compensated by using the coordinate dependence of the force correlation function to generate the STGLE (29). One first evaluates the coordinate-dependent friction function... 

To date this procedure has not yet been implemented for a realistic simulation, but one would expect it to give a more accurate representation of the true dynamics. Even this construction is not foolproof the true bath is not Gaussian as implied by the STGLE. There is also some arbitrariness in the prescription since, in principle, one could use any time t and not necessarily the t = 0 value of the friction function t)(f q) in Eq. (23). Only if the system were exactly described by the STGLE would this arbitrariness be removed. 

In practice, the friction function y(t) appearing in the STGLE is not periodic but a decaying function. However, one may use it to construct the periodic function y(t r) = x "y(/ — m)6(t — nT)0[(n + 1)t — r] where 0(x) is the unit step function. The continuum limit is obtained when the period t goes to o°. In any numerical discretization of the STGLE care must always be taken not to extend the dynamics beyond the chosen value of the period t, as beyond this time one is following the dynamics of a system which is considerably different from the continuum STGLE. 

This allows one to define a normal mode friction function (12) (note that the definition here differs from that of Refs. 12 and 67 by a factor of n ,) ... 

The Laplace transform of the friction function K(t) is known in the continuum limit (cf. Eq. (48)) so that one may now directly relate the spectral density of the normal modes /(A) to the spectral density 7(w). One finds... 

The ohmic normal mode friction function is found to be... 

Note that in contrast to the ohmic friction, the normal mode friction function has memory. The corresponding spectral density of normal modes /(X) (cf. Eq. (49)) decays as X4, in a sense it is much better behaved than the ohmic spectral density J(o>), which increases without bound with u>. Finally, the collective bath mode frequency, fl (cf. Eq. (59)),... 

To summarize, using the definition of the coupling coefficients c, and bath frequencies o) as defined from Eqs. (122)-(125) one can construct the time-dependent friction function ... 

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