Molar frictional coefficient

Bueche et al. (33) determined chain dimensions indirectly, through measurements of the diffusion coefficient of C1 Magged polymers in concentrated solutions and melts.The self-diffusion coefficient is related to the molar frictional coefficient JVa 0 through the Einstein equation ... 

The diffusion constant is independently evaluated by observing the changes in concentration of a solution in a stationary tube as the solute diffuses into pure solvent with which the solution forms a hquid boundary. From this can be calculated the molar frictional coefficient (/). 

Frictional coefficients Generally, these are introduced in hydrodynamics in order to ascribe a Newtonian friction to particles or molecules. A molar frictional coefficient... 

Frictional Ratios The molar frictional coefficient, /sp, of an unsolvated spherical molecule may be computed by the formula based on Stokes law ... 

In few cases, only the molar frictional coefficient /q, rather than/o//sp, was quoted in the literature. These values are inserted in the same column as the values for the frictional ratio. 

Molar frictional coefficient at zero concentration Concaitration Eq. (A2)... 

Diffusion coefficient, cm /sec Molecular frictional factor Molar frictional factor... 

The dependence of the friction coefficient of the poly(acrylamide) gel on the total concentration of polymers of the gel, including both the main constituent and cross-linker, has been studied under the experimental conditions of a constant molar concentration of the cross-linker at 1 mol%. The total concentration was changed from 400 mM to 2.8 M which corresponds to a concentration from about 3 to 20 wt.%. The temperature was fixed at 20 0.1 °C. 

K above their glass transitions. All linear polymer melts have viscosity proportional to molar mass (ry M) for sufficiently short chains, when the data are determined at a constant friction coefficient as opposed to isothermal data. Longer chains have entanglement effects (discussed in Chapter 9) and have The full chain length dependence of... 

Equation (8.136) is tested in Fig. 8.17 (solid curves) and found to describe the molar mass dependence of constant friction coefficient viscosity data for all three of these linear polymers. The critical molar mass Me for entanglement effects in viscosity is always a factor of 2-4 larger than the entanglement molar mass Mg. that was defined in Eq. (7.47). 

Viscosity data for three linear polymers corrected to the friction coefficient of high molar mass polymer at roughly 7 g+ 120K, fit to Eq. (8.137) (curves). Open circles are polyisobutylene (r=50°C) with Mc = 14000 gmoP. filled squares are polybutadiene (7 =25°C) with A/< = 6700gmor open triangles are free radically prepared polystyrene T =217°C), and filled triangles are anionically prepared polystyrene T=2[1 C) with... 

Classical thermod5mamics and theories of state equilibrium show an admirable flexibility with regard to the choice of components primarily only the number of these is essential. This advantage has been taken over by the diffusion theory with the thermodynamic factor as an intermediate link. As a consequence, it must be admitted that the molar frictions contained in the theory ( /c<) do not necessarily correspond to the frictional coefficients of the special molecular species contained in the mixture. So that the latter shall be the case, a component must consist of only one kind of molecule, and (in calculating the molar properties contained in the theory) the molecular weight of the component must be chosen according to the actual molecular species. 

In Fig. 5.1.2 the Perrin factor is plotted as a function of the axial ratio, defined as the ratio of the long semiaxis to short semiaxis, equal to p for the prolate spheroid and p for the oblate spheroid. As seen, the Perrin factor is always greater than unity, which may have been anticipated, since for equal volumes the surface area of the spheroid will be greater than that of a sphere of the same volume, so the friction coefficients will be greater. Because rhe volume of a molecule is proportional to its molar mass, then, for a constant mass, the more a molecule deviates from a spherical shape, the larger will be its mean friction coefficient. 

The quantities KsOy Ks, and Kd are accessible through independent measurements and are independent of the molar mass. They are consequently called physical constants. On the other hand, A sd, A sr, and A dv are model constants, since they are based on certain assumptions. If, for example, the frictional coefficients from sedimentation and diffusion are of equal magnitude (see Chapter 9), XhttiAso = 1. The model constants can, of course, influence the numerical value of the molar mass, but they have no effect on the composition of the average from the various individual molecular species contributions. Consequently, model constants can always be assumed to have a value of unity until evidence to the contrary is obtained. 

Thus, 5 values alone are not a measure of the molar mass, since the molar mass also depends on the frictional coefficient fs and the buoyancy term (1 — i>2 pi) (see Table 9-2). Frictional coefficients are determined by the shape and degree of solvation of the particles. 

Table 9-2. Sedimentation Coefficient s and Frictional Coefficient fs (as the ratio fsj/sphere) of Macromolecule with Molar Mass Mi...

This equation defines a sedimentation coefficient, s, which may be thought of as the intrinsic speed of the particle and which depends on the molar mass and on the frictional coefficient. The sedimentation coefficient is expressed in Svedberg units, which equal 10 s. Since it varies with temperature and solvent viscosity, it is often expressed as S20,wj that is, the equivalent in 20°C water. The quantity ni — vp) is called the buoyant molecular weight, and corresponds to the mass of the particle less than that of the displaced medium. 

The sedimentation rate (a) is given by the following relations u=V.glf(p-p ) = m.glf (l-p lp)=M.glNf. f (l-p lp) =2r. g 9r) (p-Po). where V = volume of sedimentation particles, p = their density, p = density of dispersion medium, /= friction coefficient, M=molar weigh of particles, = Avogadro s number, rj = dynamic viscosity of dispersion medium. 

For the dependence of the Newtonian viscosity on the molar mass, again the two models (Rouse and reptation) diverge. As established in Chapter 6, the Rouse theory predicts that the viscosity of a fluid is the product of the friction coefficient (Iseg) times a factor F (see Chapter 6)... 

The sedimentation coefficient s°, or its normalized form 5 0 w function of the conformation and flexibility of a macromolecule (via its translational frictional property) and its mass. So if we are going to obtain conformation and flexibility information we need to know the molecular weight (molar mass)... 

We have carried out a wide range of studies concerned with the dextran concentration dependence of the transport of the linear flexible polymers and have varied both molecular mass and chemical composition of this component. Moreover, we have studied the effect of the variation of the molar mass of the dextran on the transport of the flexible polymers 51). In general, the transport of these polymers in dextran solutions may be described on common ground. At low dextran concentrations the transport coefficients of the polymers are close to their values in the absence of the dextran and may even exhibit a lower value. This concentration range has been discussed in terms of normal time-independent diffusional processes in which frictional interactions predominate. We have been able to identify critical dextran concentrations associated with the onset of rapid transport of the flexible polymers. These critical concentrations, defined as C, are summarized in Table 1. They are... 

The Svedberg equation relates 5 to the molar mass of the particles, M2, their translational diffusion coefficient, D (related to the frictional force exerted on the particle), and to (dp/dc2)kl ... 

Regardless of its complex architecture, any polymer relaxing with no topological constraints and no hydrodynamic interactions is well-represented by the Rouse model, with friction proportional to molar mass. To estimate the terminal response of randomly branched polymers, we apply this reasoning to the characteristic polymers, with size consisting of N monomers. The diffusion coefficient of these chains is given by the... 

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