Particle volume fraction

Charged particles in polar solvents have soft-repulsive interactions (see section C2.6.4). Just as hard spheres, such particles also undergo an ordering transition. Important differences, however, are that tire transition takes place at (much) lower particle volume fractions, and at low ionic strengtli (low k) tire solid phase may be body centred cubic (bee), ratlier tlian tire more compact fee stmcture (see [69, 73, 84]). For tire interactions, a Yukawa potential (equation (C2.6.11)1 is often used. The phase diagram for the Yukawa potential was calculated using computer simulations by Robbins et al [851. 

Rheology. Flow properties of latices are important during processing and in many latex appHcations such as dipped goods, paint, inks (qv), and fabric coatings. For dilute, nonionic latices, the relative latex viscosity is a power—law expansion of the particle volume fraction. The terms in the expansion account for flow around the particles and particle—particle interactions. For ionic latices, electrostatic contributions to the flow around the diffuse double layer and enhanced particle—particle interactions must be considered (92). A relative viscosity relationship for concentrated latices was first presented in 1972 (93). A review of empirical relative viscosity models is available (92). In practice, latex viscosity measurements are carried out with rotational viscometers (see Rpleologicalmeasurement). 

Hindered Settling When particle concentration increases, particle settling velocities decrease oecause of hydrodynamic interaction between particles and the upward motion of displaced liquid. The suspension viscosity increases. Hindered setthng is normally encountered in sedimentation and transport of concentrated slurries. Below 0.1 percent volumetric particle concentration, there is less than a 1 percent reduction in settling velocity. Several expressions have been given to estimate the effect of particle volume fraction on settling velocity. Maude and Whitmore Br. J. Appl. Fhys., 9, 477—482 [1958]) give, for uniformly sized spheres,... 

Figure 2 Variation of relative viscosity with particle volume fraction.

Particle volume fraction 0m = Maximum particle volume fraction... 

In order to define the r), - and r 2-exponents, it is necessary to dispose a second equation, besides relation (31) for the evaluation of r.-radius, and relation (27) for the definition of the difference (T t —ri2). For this purpose we used the values of the composite moduli evaluated for various particle-volume fractions of iron-epoxy particulates determined experimentally and given in Ref.I4>. 

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. 

Figure 3. Critical flocculation temperature (T) versus log (particle volume fraction ) for the two Si02 g PDMS dispersions in bromocyclohexane O, S15/PDMS5 x, S15/ PDHS3.

Figure 4. Critical flocculation solvent composition toluene + n-hexane (v = volume fraction of toluene), versus log (particle volume fraction, (J>) for various SiC -g-PS systems at 24 1°C V, S12/PS13c 0, S12/PS13a ...

The second term on the RHS of this equation shows that the compressibility of the solid phase is taken directly into account in the estimation of the new particle volume fractions. Furthermore, the expression for the derivatives of the velocities to the solid pressure can be obtained by combination with the x-momentum equation for the gas phase that results in... 

Fig. 29. Snapshots of particle volume fraction fields obtained while solving a kinetic theory-based TFM. 75 pm fluid catalytic particles in ambient air. Simulations were done over a 16 x 32 cm periodic domain. The average particle volume fraction in the domain is 0.05. Dark (light) color indicates regions of high (low) particle volume fractions. (See Refs. Agrawal et al., 2001 Andrews et al., 2005) for other parameter values.) Source Andrews and Sundaresan (2005).

Strictly speaking, one should use 2D bins involving particle volume fraction and a Reynolds number based on slip velocity to classify the filtered drag coefficient however in these simulations, the Reynolds number effect was found to be weak and hence the data were collapsed to just volume fraction bins. 

Andrews and Sundaresan (2005) have also extracted the filtered particle-phase viscosity from these simulations and found that at low particle volume fractions (0.0-0.25), the filtered viscosity varies nearly linearly with particle volume, and that it increases monotonically (and nearly linearly) with filter size. 

Fig. 33. Comparison of the domain-average slip velocity (in cm/s) determined by solving a microscopic TFM and the corresponding filtered TFM. 16 x 32cm periodic domain. Domain-average particle volume fraction = 0.05. Number of grids in the vertical direction is twice that in the lateral direction. Source Andrews and Sundaresan (2005).

Figure 3.10. Phase diagrams of attractive monodisperse dispersions. Uc is the contact pair potential and (j) is the particle volume fraction. For udk T = 0, the only accessible one-phase transition is the hard sphere transition. If Uc/hgT 0, two distinct scenarios are possible according to the value of the ratio (range of the pair potential over particle radius). For < 0.3 (a), only fluid-solid equilibrium is predicted. For % > 0.3 (b), in addition to fluid-solid equilibrium, a fluid-fluid (liquid-gas) coexistence is predicted with a critical point (C) and a triple point (T).

In suspensions of particles with an aspect ratio (length to diameter) greater than 1, particle rotation during flow results in a large effective hydrodynamic volume, and Kh > 2.5 (see Figure 4.7). At particle volume fractions above about 5-10%, interaction between particles during flow causes the viscosity relationship to deviate from the Einstein equation. In such instances, the reduced viscosity is better described by the following relationship ... 

A.D. Brailsford and P. Wynblatt. The dependence of Ostwald ripening kinetics on particle volume fraction. Acta Metall., 27(3) 489—497, 1979. 

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