Friction model

The propordonalitN factor (the friction coefficient) denotes interaction between the solute and the polymer (pore wall). 

Using linear relationships between the fluxes and forces in accordance with the concept of irreversible thermodynamics and assuming isothermal conditions the forces can be described as the gradient of the chemical potential, i.e. 

However, other (external) forces acting on component i, such as the frictional force, must also be included. Thus equation V-64 becomes 

The diffusive solute flux can be written as the product of the mobility, concentration and driving force. The mobility m may be defined as 

If we define a parameter b that relating the fricdonal coefficientfj (between the soluteand the membrane) to (between the solute and water), then 

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Zhu S-B, Lee J, Robinson G W and Lin S H 1988 A microscopic form of the extended Kramers equation. A simple friction model for cis-trans isomerization reactions Chem. Phys. Lett. 148 164-8... 

Fig. 6.8. The dependence of rj2 on x) by the Ivanov model (I) and friction model (F) in comparison with predictions of the extended. /-diffusion (ED) and Langevin (L) models for linear molecules. The line (H) corresponds to the Hubbard inverse proportionality between xgj and xj at very high densities. Experimental data from [81] are in rectangles around line G with the length of their vertical and horizontal sides being equal, correspondingly, to the experimental errors in x el and rj measurements. Experimental data from [270] (J) are shown both in original position and shifted down by a factor of four (broken line).

Fig. 5.27 Model-predicted pressure drops normalized with experimentally measured pressure drops for a circular test section (Triplett et al. 1999b). Model predictions represent the homogeneous wall friction model. Reprinted from Triplett et al. (1999b) with permission...

The FK model accounts for the effects that have been ignored in the Tomlinson model, resulting from the interactions of neighboring atoms. For a more realistic friction model of solid bodies in relative sliding, the particles in the harmonic chain have to be connected to a substrate. This motivates the idea of combining the two models into a new system, as schematically shown in Fig. 24, which is known as the Frenkel-Kontorova-Tomlinson model. Static and dynamic behavior of the combined system can be studied through a similar approach presented in this section. 

In the case of supported membranes also, the support can play an important role in the separation performance of the membrane in the gas as well as in the Hquid phase [101-103]. Transport in these support pores can be accurately described by the Dusty Gas Model [100, 104] although it is put forward by Kerkhof and Geboers that their Binary Friction Model is physically more correct [105]. 

Meares, P. (Ed.), Membrane Separation Processes, Elsevier, Amsterdam, 1976. Meares, P., J. F. Thain, and D. G. Dawson, Transport across ion-exchange membranes The frictional model of transport, in Membranes—A Series of Advances (Ed. G. Eisenman), Vol. 1, p. 55, M. Dekker, New York, 1972. Schlogl, R., see page 415. 

Figure 3.41. Photoyield Y for H2 and D2 associative desorption from Ru(0001)(l x 1)H and Ru(0001)(lxl)D as a function of absorbed 800 nm 130fs laser pulse fluence F. (a) experimental results with circles for H2 desorption and squares for D2 desorption. Solid lines are fits to a ID friction model and dashed lines are fits to power law expressions, Y oc F28 for H2 and Y oc F3 2 for D2. From Ref. [413]. (b) Equivalent photoyields for associative desorption from 3D first principles molecular dynamics with electronic frictions. From Ref. [101].

A. H. Zewail I have a question for Prof. Marcus concerning the fact that, in the bulk solvation problem, there are two regimes for the description of solvation, the continuum model and the detailed molecular dynamics. Do you expect that in clusters the friction model will change as the number of solvent molecules changes from small to large ... 

A number of analogous compounds to BA have been reported, including 5,5 -dibenzo-[a]-pyrenyl (BBPY) [116]. These compounds exhibit emission spectra similar to BA. It would be interesting to explore the ultrafast dynamics of BBPY in order to test the generality of the GLE model. It would also be interesting to study the femtosecond dynamics of BA as a function of applied pressure. Static experiments on the emission of BA, reported by Hara et al. [123], demonstrate that in low viscosity solvents an increase of pressure affects the emission similarly to an increase of solvent polarity. As the pressure is increased, however, the LE/CT interconversion is slowed down. It would be interesting to measure C(r) in these environments and compare the solvation dynamics with LE/CT dynamics, in order to test the generality of the GLE dielectric friction model. 

Spiegler has used the friction model to describe a system consisting of sodium ions (1), chloride ions (2), water (3) and a charged matrix (4). He neglects the interaction of the sodium ions with the chloride ions. Then five independent measurements are needed to calculate the friction coefficients. Spiegler chose to be measured the self-diffusion coefficient... 

In the Kramers approach the friction models collisions between the particle and the surrounding medium, and it is assumed that the collisions occur instantaneously. There is a time-scale separation between the reactive mode and its thermal bath. The dynamics are described by the Langevin equation (4.141). The situation where the collisions do not occur instantaneously but take place on a time scale characterizing the interactions between the particle and its surrounding can be described by a generalized Langevin equation (GLE),158,187... 

For the most important intermediate pressure range from 10 2 to 0.5 mbar exist only empirical friction models. In [25] an extension of the shock wave model is proposed, which is valid from 10 2 to 0.5 mbar at lower target-to-substrate distance from 20 to 50 mm. 

Inertial regime. For low-viscosity liquids like molten metals, the use of the viscous friction model is questionable because inertial forces can become very important. For a sessile drop of mass ma, the inertial force is given by ... 

Here,yjm is obtained from the binary friction model. 

As an alternative to the DGM, the Binary Friction Model (BFM) is of interest. This principally new model has been developed recently [22]. The BFM flux equation, contending with Equation 3.20, is ... 

Reprinted from Chemical Engineering Journal, 64, P.J.A.M. Kerkhof, A modified Maxwell-Stefan model for transport through inert membranes the binary friction model, 319-344,1996, with kind permission from Elsevier Science S.A., P.O. Box 564,1001 Lausanne, Switzerland. 

Chapter 3 described a new model for transport through porous media, developed recently by Kerkhof [5] and called the binary friction model (BFM). It is of interest to see how this model can be applied to the description of available experiments and to compare the results with those of the dusty gas model (DGM). Kerkhof [5] took the experimental data of Evans et al. [6,7] for the permeation of He and Ar through a low-permeability porous graphite septum. The experimental set-up, similar to the Wicke-Kallenbach diffusion cell, is sketched in Figure 9.7. Of interest are the steady... 

Figure 9.11 Net pressure difference in the counterdiffusion of N2 and CjH4. Symbols show the experimental data of Waldmann and Schmitt [11] drawn line show simulation with the binary friction model. Highest pressure on the nitrogen side. The DGM predicts no pressure difference (from Kerkhof [5]). Reprinted from Chemical Engineering Journal, 64, PJ.A.M. Kerkhof, A modified Maxwell-Stefan model for transport through inert membranes the binary friction model, 319-344,1996, with kind permission from Elsevier Science S.A., RO. Box 564,1001 Lausanne, Switzerland.

Based on the lower desorption probability and the shape of the 2PC spectrum (not shown), the friction model gives a three times smaller electron-coupling time (0.025 ps) for the step sites. The absolute error is significant, but the relative value (compared to terrace NO) can be determined reliably. A similar analysis of the CO data reveals an electron-coupling time of 0.3 ps and 0.1 ps for CO adsorbed on the terraces and the steps, respectively. Thus for both CO and NO the electron coupling time is three times faster at the step sites than at the terrace sites [37]. 

This sequence of events, excitation of the substrate electrons, energy transfer to the frustrated translation of the CO molecnles and the associated changes in the C-0 stretch vibration, can be described again with the friction model [50], the result of which is shown as black lines in Fig. 10.6. To reproduce the data in Fig. 10.6, coupling times of Tj=2.5 0.5 ps and 4 0.5 ps for terrace and step, respectively, are required. The simple one-dimensional model fully reproduces the time-dependent width and central frequency. 

It should be noted that the extracted electron coupling times for NO are smaller than the inverse mode frequency of the low-frequency modes. This is unphysical, as energy transfer into the low-frequency modes cannot occur faster than the motion associated with the modes. Although the absolute values for the friction coefficient obtained with this simple one-dimensional friction model may have limited meaning, the relative difference between step and terrace coefficient clearly indicates a 3-fold stronger couphng of the laser-heated electrons to the adsorbate at the steps relative to the terraces... 

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