Viscosity coefficient hard spheres

There are not a great number of studies on the viscoelastic behaviour of quasi-hard spheres. The studies of Mellema and coworkers13 shown in Figure 5.5 indicate the real and imaginary parts of the viscosity in a high-frequency oscillation experiment. Their data can be normalised to a characteristic time based on the diffusion coefficient given above. 

The physical properties of these polymeric dendrimers have been studied to some extent. Intrinsic viscosity measurements combined with MW afford values of according to Eq. (5). Alternatively, the translational diffusion coefficient leads to Rh according to Eq. (6). These equations may well be applicable, since it is observed that Rn and Rh scale with the 1/3 power of MW in support of the equal density hard-sphere assumption [88]. 

It should be noted again that the numerical coefficient above has units of mole. According to this equation, the gas viscosity coefficient should be independent of pressure and should increase with the square root of the absolute temperature. The viscosities of gases are in fact found to be substantially independent of pressure over a wide range. The temperature dependence generally differs to some extent from because the effective molecular diameter is dependent on how hard the molecules collide and therefore depends somewhat on temperature. Deviation from hard-sphere behavior in the case of air (diatomic molecules, N2 and O2) is demonstrated by Eq. (4-19). 

The direct calculation of the collective contribution DJDs to the self-diffusion coefficient is complicated by the inadequate temperature dependence of the shear viscosity in ref. [ ]. Indeed, it is easy to verify that the ratio r / r g for the model argon increases with temperature on isochors. From the physical viewpoint, this result is inadequate. It is worth noting that for ( ) < 0.4 the values of r from ref. f ] and those determined on the basis of the Enskog theory for hard spheres diameter of which coincides with the effective diameter... 

Similar results, which showed behavior of the viscosity according to a hard-sphere model, have been observed for a variety of O/W microemulsion systems such as Brij 96-butanol-hexadecane-water [52], Tween 60-Span 80-glycerol-paraffin-water [53], and Brij 96-pentanol-hexadecane-water [54]. Again the extrapolated intrinsic viscosity is found to be about 60% greater than expected according to Einstein s equation and the Huggins coefficient kn [cf Eq. (13)] to be about 1.8 [52]. This shows that such behavior will quite generally be observed for droplet-type microemulsions irrespective of the inter-... 

Figure 37 Relative zero-shear viscosity (normalized to the solvent tis) as a function of the effective volume fraction

Here is the appropriate microscopic current and tlim denotes the thermodynamic limit. (For reviews of this theory, see Zwanzig and Steele. ) We will here focus our attention on the self-diffusion process, both because it is the simplest case from a pedagogical standpoint and because it is the one that has been most extensively studied by means of molecular dynamics calculations. Calculations of the coefficients of viscosity and thermal conductivity and the associated time-correlation functions have been reported by Alder et for hard spheres. 

The simplest way to determine the effective hard-sphere diameter d of the fluid molecules at temperature T and the corresponding value of x ts to fit exp( the experimental value for the coefficient of shear viscosity at low density, to the theoretical formula for hard-sphere molecules. That is, d is determined by using the first Sonine polynomial approximation to 17 for a gas of hard spheres [Eq. (115a)],... 

Yoon and Ohr derived an expression for the compressibility of hard-sphere fluids in terms of the radial free space distribution function, (r), which is the probability of acceptance of a (radial) displacement, r, in a Monte Carlo simulation.Alemany et al. performed MD simulations of liquid 50 and calculated its diffusion coefficient and shear viscosity. They simulated a system of 1372 Ceo molecules interacting through Girifalco s potential. 

Fig. 2.8. Reciprocal of the apparent value of as determined from viscosity (—) and self-diffusion coefficients (- -) using hard-sphere transport theory (Table 2.2). Transport coefficient data from H. H. Landolt and R. Bomstein, Zahlenwerte and Funktionen, 6th ed. (Berlin Springer, 1969), Vol. Ill, Part 5a, pp. 3, 516.

Since experimental values of diffusion coefficients are known in relatively few instances, it is often convenient to express them in terms of the viscosity (rj) of the medium, which is known for many solvents and is easy to measure. Application of classical hydrodynamic theory to the hard-sphere model leads, as was shown in Section 1.3.2, to the Stokes-Einstein equation (1.3) for the relative diffusion coefficient Dab, which may be... 

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