Sphere 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.

Wagner (1914) gave an approximate treatment of the important practical case where a very highly insulating dielectric suffers from inclusions of conductive impurities. Taking the model where the impurity (relative permittivity e2, conductivity a2) exists as a sparse distribution of small spheres (volume fraction f) in the dielectric matrix (relative permittivity e, negligible conductivity), he derived equations for the components of the complex relative permittivity of the composite ... 

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. 

Figure 3 presents the frequency dependence of the electro-optical effects for suspensions of different volume fractions of spherical particles. The stock solution (8% sphere volume fraction) is highly deionized, and dilution is carried out with deionized water. Thus the highly dilute samples are in the liquid-phase state. Modulated responses are detected at low frequencies —- for highly dilute samples... 

FIG. 3 Frequency dependence of the ELS responses from deionized suspensions of CS81 (field intensity 200V/cm) for different sphere volume fractions. 

Figure 5 shows typical responses for this particular sample (0.5% sphere volume fraction). A basic difference in the electro-optical behavior of spherical compared to anisometric particles is the linear response at low field intensity. The modulated signal follows the frequency of the applied field with a phase difference of —90°. With an increase in frequency the ac component relaxes, changing in both amplitude and phase and passing to double frequency, while the dc component changes... 

The electro-optical responses are strongly dependent on the phase state of the colloidal system. The transition of the same sample from the liquid to the crystalline state is accompanied by a considerable increase in the cutoff frequency of the ac component and a displacement of the dc curve to more negative values (Fig. 6). Similar variations can be observed by increasing of the sphere volume fraction, as demonstrated in Fig. 3, which shows that the effective volume fraction determines the process. In parallel, additional modes are observed to appear in the crystalline state. They introduce resonance peaks at frequencies depending on the sphere volume fraction (two volume fractions are presented in Fig. 6). 

The sum of the weights divided by the number of neutrons tracked is the Dancoff factor. The code was used to calculate Dancoff factors for sphere volume fractions from O.OOS to... 

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) ...

Jones, et al. examined 50 nm silica spheres coated with covalently-bound stearyl alcohol dissolved in Shellsol T(55). Viscosities were determined with Ubbelohde viscometers and with three different cone and plate instruments. Sphere volume fractions were taken as high as 0.635, corresponding to T]r as large as 9.2 10 . Shear thinning was apparent at concentrations above 0.4. Systems with (p > 0.64 could not be taken into the low-shear limit in which 17 (/c) becomes independent from /c, so the low-shear rj remains indeterminate at these very large concentrations. 

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