Molecular friction coefficient

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. 

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. 

In this equation, % is a proportionality factor known as the bead-solvent friction coefficient which purports to account in some kind of average way for the complex molecular interactions as the polymer segments (schematized by the bead) move about in the solvent. Following Stokes law of drag resistance, this friction coefficient is usually given as = 67trisa, with a equal to the bead radius. 

In the past decade, effects of an EEF on the properties of lubrication and wear have attracted significant attention. Many experimental results indicate that the friction coefficient changes with the intensity of the EEF on tribo-pairs. These phenomena are thought to be that the EEF can enhance the electrochemical reaction between lubricants and the surfaces of tribo-pairs, change the tropism of polar lubricant molecules, or help the formation of ordered lubricant molecular layers [51,73-77]. An instrument for measuring lubricant film thickness with a technique of the relative optical interference intensity (ROII) has been developed by Luo et al. [4,48,51,78] to capture such real-time interference fringes and to study the phenomenon when an EEF is applied, which is helpful to the understanding of the mechanism of thin film lubrication under the action of the EEF. 

Phenomenal studies were made to observe the frictional behavior of L-B films and SAMs and its dependence on applied load and sliding velocity, which has been summarized in a review article by Zhang [33]. It has been confirmed that in comparison to the bare surface of the substrates, the friction on molecular films is significantly reduced, with friction coefficients in a range of 0.05-0.1. Friction forces are found... 

The shape of the maser curve not only depends on the rubber compound, but also on the surface on which it slides. On dry, clean polished glass the friction master curve for gum rubbers rises from very small values at low log ajv to a maximum which may reach friction coefficients of more than 3 and falls at high log ajv to values which are normally associated with hard materials, i.e., 0.3 shown for an ABR gum compound in Figure 26.2. If the position of the maximum on the log a-fV axis for different gum rubbers is compared with that of their maximum log E frequency curves, a constant length A = 6 X 10 m results which is of molecular dimension, indicating that this is an adhesion process [10]. 

Rothe, GM, Determination of Molecular Mass, Stoke radius. Frictional Coefficient and Isomer-Type of Non-denatured Proteins by Time-Dependent Pore Gradient Gel Electrophoresis, Electrophoresis 9, 307, 1988. 

Of major importance is the fact that the specific character of polymer chains of a given type enters the relationship (17) only through the effective size of one of its beads as indicated by the ratio f/970. Even the effect of this factor vanishes when the total internal resistance to flow is sufficiently large. Hence, in this limit, which will include nearly all actual cases of interest (see Sec. 4), the molecular frictional coefficient should depend only on the size /s and not otherwise on the nature of the polymer. Accordingly, we choose to let... 

In the canonical ensemble (P2) = 3kBTM and p M. In the microcanonical ensemble (P2) = 3kgT i = 3kBTMNm/(M + Nm) [49]. If the limit M —> oo is first taken in the calculation of the force autocorrelation function, then p = Nm and the projected and unprojected force correlations are the same in the thermodynamic limit. Since MD simulations are carried out at finite N, the study of the N (and M) dependence of (u(t) and the estimate of the friction coefficient from either the decay of the momentum or force correlation functions is of interest. Molecular dynamics simulations of the momentum and force autocorrelation functions as a function of N have been carried out [49, 50]. 

F. Ould-Kaddour and D. Levesque, Determination of the friction coefficient of a Brownian particle by molecular-dynamics simulation, J. Chem. Phys. 118, 7888 (2003). 

Fig. 8. Self-diffusion coefficients of polyethylene chains as a function of molecular mass. The measurements were carried out at the same value of the monomeric friction coefficient. (Reprinted with permission from [48]. Copyright 1987 American Chemical Society, Washington)...

In addition to the Rouse model, the Hess theory contains two further parameters the critical monomer number Nc and the relative strength of the entanglement friction A (0)/ . Furthermore, the change in the monomeric friction coefficient with molecular mass has to be taken into account. Using results for (M) from viscosity data [47], Fig. 16 displays the results of the data fitting, varying only the two model parameters Nc and A (0)/ for the samples with the molecular masses Mw = 3600 and Mw = 6500 g/mol. 

To simulate the particle-particle collision, the hard-sphere model, which is based on the conservation law for linear momentum and angular momentum, is used. Two empirical parameters, a restitution coefficient of 0.9 and a friction coefficient of 0.3, are utilized in the simulation. In this study, collisions between spherical particles are assumed to be binary and quasi-instantaneous. The equations, which follow those of molecular dynamic simulation, are used to locate the minimum flight time of particles before any collision. Compared with the soft-sphere particle-particle collision model, the hard-sphere model accounts for the rotational particle motion in the collision dynamics calculation thus, only the translational motion equation is required to describe the fluid induced particle motion. In addition, the hard-sphere model also permits larger time steps in the calculation therefore, the simulation of a sequence of collisions can be more computationally effective. The details of this approach can be found in the literature (Hoomans et al., 1996 Crowe et al., 1998). 

Figure 5.22 A schematic of the log of the viscosity multiplied by the ratio of friction coefficients versus the log of the molecular weight...

We can consider the friction coefficient to be independent of the molecular weight. At times less than this or at a frequency greater than its reciprocal we expect the elasticity to have a frequency dependence similar to that of a Rouse chain in the high frequency limit. So for example for the storage modulus we get... 

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