Hard spheres volume fraction

In practice, tliere are various ways by which ( ) can be detennined for a given sample, and tire results may be (slightly) different. In particular, for sterically stabilized particles, tire effective hard-sphere volume fraction will be different from tire value obtained from tire total solid content. 

Fig. 37. The ratio of the equivalent hard sphere volume fraction based on the measured intrinsic viscosity as a function of for polyfmethyl methacrylate) spheres with grafted poly( 12-hydroxy stearic add) layers such that a/L = 4.7 (Mewis et ai, 1989). Open and closed circles correspond to the low and high shear limits of suspension viscosity.

In this equation, hs is the hard sphere volume fraction which is about 14% larger in o/w-droplet microemulsions of non-ionic surfactant than the dispersed volume fraction. This is caused by the water penetration in the surfactant layer [64]. S(q) approaches unity for q values smaller than the minimum of I(q). This behaviour occurs even for fairly high volume fractions in non-ionic surfactant systems (see for example Fig. 8 in Ref. [64]). Seeing that the value of the radius is fixed by the position of the minimum of I(q), the approximation of S(q) 1 in Eq. (2.12) does not lead to a significant error in the determination of Rq if the low q part of the experimental curve is not taken into... 

Here, is the hard-sphere volume fraction. We use (p (volume fraction of copolymer) instead of for the calculation of Do because the exact volume of micelles is unknown. The concentration of Sii4C3EO is = 2 vol% so that the calculated values. Do, differs from D by -5% and this difference is noted as an error bar. 

In addition, the droplets have a hydrodynamic radius, th, which is obtained from the diffusion coefficient extrapolated to infinite dilution. As will be shown below, the droplet interactions can, to a very good approximation, be described in terms of hard spheres. A third characteristic radius, the hard-sphere radius, ths, then enters to describe the interactions. Associated with the three radii, there are three different characteristic droplet volumes, and therefore three different characteristic droplet volume fractions. If 0 = 0 -t- 0o denotes the total volume fraction of surfactant and oil, the hard sphere volume fraction, 0hs, can be written as follows ... 

In Figure 17.19, which mainly highlights the concentrated regime, we saw that the relevant parameter for describing the concentration was the hard-sphere volume fraction 0hs- At high dilution, on the other hand, the viscosity is governed by the hydrodynamic volume... 

Figure 17.19. Variation of the normalized low shear viscosity, with the hard-sphere volume fraction, 0hs- Samples from the microemulsion (data taken from ref. (22)) were measured in a capillary ( ) or in a cone and plate rheometer (A). Open symbols show the data obtained for different radii of coated silica spheres in oil, taken from ref. (23), reproduced by permission of society of Rheology. The continuous line shows the prediction of equation (17.17)...

Figure 17.21. Variation of the normalized collective Dc/Do) (A) and long-time self-diffusion (D /A)) ( ) coefficients with the hard-sphere volume fraction 0hs- All filled symbols refer to microemulsion data (taken from ref. (17)). The A/Do data, shown as open triangles correspond to silica spheres, taken from ref. (26), while the T>s/A) data, shown as open circles, correspond to the self-diffusion of traces of silica spheres in a dispersion of poly(methyl methacrylate) spheres (data taken from ref. (27)). The dashed line represents the equation, Dc/Dq = 1 + 1.30HS, while the continuous line represents the relationship. A/A) = (1 — 0hs/O-63) ...

Figure 17.22. Plots of the normalized self-diffusion coefficient (Z)s/A)) the inverse normalized low shear viscosity rj/ijo as a function of the hard-sphere volume fraction 0hs- The open circles correspond to rj/rjo and the filled circles to (A/A) while the continuous line represents the Quemada function, i.e. (1 — 0hs/O-63)...

The low shear viscosity, rj, was measured using capillary and, at higher concentrations, a cone-plate rheometer [4], The two techniques gave equivalent results in the overlapping concentration range. The variation of the normalised low shear viscosity rj/rjo, where rjo is the water solvent viscosity, with the hard-sphere volume fraction Hs is shown in Fig. 5. For comparison, we have also plotted data from van der Werff and de Kruif [18] for hard-sphere silica dispersions of three different sizes. As can be seen, there is a perfect agreement between the microemulsion and silica data. The solid line in Fig. 5 shows the Quemada expression [19]... 

Fig. 8 Plot of the inverse normalized self-diffusion coefficient (DJDo) and the normalized low shear viscosity rf/rio as a function of hard-sphere volume fraction pas- Open circles correspond to rj/rjo and filled circles to (DJDo). The solid line is the Quemada function (1 — (f)Hs/0.63) ...

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