Particle volume fractions, effect dispersions

Fig. 7. Effect of volume fraction of dispersed particles in a gel on the shear modulus of the gel under electric fields. The sample used was a silicone gel containing PMACo particles. The particles were dispersed at random in the gel...

The next problem is to find an expression for Asg. This entropy difference is a function of the particle volume fractions in the dispersion ( ) and in the floe (<(> ). As a first approximation, we assume that Ass is independent of the concentration and chain length of free polymer. This assumption is not necessarily true the floe structure, and thus < >f, may depend on the latter parameters because also the solvent chemical potential in the solution (affected by the presence of polymer) should be the same as that in the floe phase (determined by the high particle concentration). However, we assume that these effects will be small, and we take as a constant. 

Increasing the concentration of particles (roughly for a volume fraction approaching 10%), the stress concentration effects of neighboring particles can overlap (Fig. 13.2). Therefore, a large volume fraction of the matrix supports an average load higher than the applied load and can yield. This stress concentration effect increases when the volume fraction of dispersed particles increases or the interparticle distance decreases. 

When the superparamagnetic theory is applied for interpretation of any measured susceptibility line, it means that some model function x(V,a>,T) depending on several material parameters, is processed through averaging like such as Eq. (4.121) or (1.122). In our case the basic set of the material parameters comprises magnetization /, anisotropy energy density K, relaxation time To, and the particle volume fraction tp. Obviously, for nanosize-dispersed systems the effective values of /, K, and t0 do not coincide with those for a bulk material. The size/volume averaging itself introduces two independent... 

When the particle volume fraction / was increased to 0.015, the oscillations of the effective interaction between identical charged particles became larger than those for / = 0.005 at an electrolyte concentration of 10 5 M (Fig. 3). The effective interaction between identical charged particles versus the distance between particles is plotted in Fig. 3 for various values of Z (Z = 300, Z = 600, Z = 1200). As shown in Fig. 4, the colloidal dispersion has a disordered liquidlike structure for Z = 600, but a more ordered structure for Z = 1200. When the electrolyte concentration was increased to 10-4 M, the interaction between identical charged particles became completely screened. As shown in Fig. 5, no oscillations of the effective interaction potential were present for Z = 300,600, and 1200. 

The effective volume fraction can be obtained from the intrinsic viscosity (rj) of the dispersion. According to Einstein s viscosity relation, the product of (ijl and the mass concentration c is proportional to the particle volume fraction, which is bigger for covered particles than for bare ones. The ratio (t 1 / n. where (rjlg is the intrinsic viscosity of the dispersion of bare particles, is therefore equal to the ratio between the volume of a covered pcutlcle and that of a bare one, l.e., it equals (l + d /a) . This method has been used by several authorsespecially in older work. 

In Eulerian-Eulerian (EE) simulations, an effective reaction source term of the form of Eq. (5.32) can be used in species conservation equations for all the participating species. The above comments related to models for local enhancement factors are applicable to the EE approach as well. It must be noted that interfacial area appearing in Eq. (5.32) will be a function of volume fraction of dispersed phase and effective particle diameter. It can be imagined that for turbulent flows, the time-averaged mass transfer source will have additional terms such as correlation of fluctuations in volume fraction of dispersed phase and fluctuations in concentration even in the absence... 

Unlike the case of coagulation without growth, the volume fraction of dispersed material increases with time as a result of gas-to-particle conversion. The total surface area, on the other hand, lends to an approximately constant value. Coagulation tends to reduce surface area, whereas growth tends to increase it. and the two effects in this case almost balanced each other. 

It has been reported (5) that the elastic modulus of ABS resins prepared by either mass or emulsion polymerization can be represented by a single relationship with the dispersed phase volume fraction. This is in agreement with the theory that the modulus of a blend with dispersed spherical particles depends only on the volume fraction and the modulus ratio of particles to matrix phase. Since the modulus of rubber is almost 1000 times smaller than the modulus of the matrix SAN, the rubber particle volume fraction alone is the most important parameter controlling modulus values of ABS resins. Even for rubber particles containing a high occlusion level, as in ABS produced by mass polymerization, the modulus of the composite particle still remains imchanged from pure rubber, suggesting a unique relationship between modulus and dispersed phase volume fraction. Also, the modulus of a material is a small strain elastic property and is independent of particle size in ABS. The effects of rubber content on modulus and on tensile... 

Another method for the determination of electrophoretic mobility which has emerged in recent years is that of the measurement of the electrokinetic sonic amplitude (ESA) for a particle subjected to an alternating current (8). This electroacoustic effect is a result of the oscillation of the particles near the electrodes where a sound wave is produced that can be picked up by a pressure transducer located behind the electrode. The ESA pressure signal is simultaneously proportional to the dynamic mobility of the particle, the particle volume fraction and the density difference between particle and solvent. Thus, the electroacoustic effect is appropriate for concentrated dispersions where conventional electrophoretic methods are inappropriate. However, one disadvantage of the method is that it is not appropriate to systems having low density differences between the particles and suspending liquid. 

Particle surface modification could be characterized with the surface energy measurement, ilow the surface energy of dispersed particle affects the ER efiect was systematically addressed by Hao [95]. A set of oxidized polyacrylonitrile (OP) materials of different surface properties were employed for such a purpose. The five kinds of water-free ER fluids composed of oxidized polyacrylonitriles (OP) particle dispersed in a low viscosity silicone oil were used for correlating the ER effect with the particle surface energy. The powdered OP materials with average particle size 0.1-10 pm were treated at 150 C for 8 h, and then dispersed in silicone oil immediately at the particle volume fraction of 35 vol /o. The surface energy was measured by means of the dynamic wicking method [96]. Water and... 

Special oscillatory structural forces appear in thin films containing small colloidal particles like surfactant micelles, polymer coils, protein macromolecules, and latex or silica particles [268,280-283]. For larger particle volume fractions, these forces are found to stabilize thin films and dispersions, whereas at low particle concentrations, the oscillatory force degenerates into the depletion attraction, which has the opposite effect see Sec. VI.C. 

Although the thermodynamic analysis of weak flocculation and colloidal phase separation, given above, illustrates the basic principles, some of the details are incorrect, in particular for more concentrated dispersions. One missing feature is the prediction of an order/disorder transition in hard sphere dispersions (for which Vmin is 0), where, at equilibrium, a colloidal crystal phase is predicted to coexist with a disordered phase over a narrow range of particle volume fractions (ip), that is, 0.50 < tp < 0.55 (Dickinson, 1983). In molecular hard-sphere fluids this is known as the Kirkwood-Alder transition , and is an entropy-driven effect. 

This response time should be compared to the turbulent eddy lifetime to estimate whether the drops will follow the turbulent flow. The timescale for the large turbulent eddies can be estimated from the turbulent kinetic energy k and the rate of dissipation e, Xc = 30-50 ms, for most chemical reactors. The Stokes number is an estimation of the effect of external flow on the particle movement, St = r /tc. If the Stokes number is above 1, the particles will have some random movement that increases the probability for coalescence. If St 1, the drops move with the turbulent eddies, and the rates of collisions and coalescence are very small. Coalescence will mainly be seen in shear layers at a high volume fraction of the dispersed phase. 

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