Suspension hard-sphere

Altliough tire behaviour of colloidal suspensions does in general depend on temperature, a more important control parameter in practice tends to be tire particle concentration, often expressed as tire volume fraction ((). In fact, for hard- sphere suspensions tire phase behaviour is detennined by ( ) only. For spherical particles... 

As shown in section C2.6.6.2, hard-sphere suspensions already show a rich phase behaviour. This is even more the case when binary mixtures of hard spheres are considered. First, we will mention tire case of moderate size ratios, around 0.6. At low concentrations tliese fonn a mixed fluid phase. On increasing tire overall concentration of mixtures, however, binary crystals of type AB2 and AB were observed (where A represents tire larger spheres), in addition to pure A or B crystals [105, 106]. An example of an AB2 stmcture is shown in figure C2.6.11. Computer simulations confinned tire tliennodynamic stability of tire stmctures tliat were observed [107, 1081. 

M. Tokuyama and I. Oppenheim, On the theory of concentrated hard-sphere suspensions, Physica A 216, 85 (1995). 

The shear rate range over which the shear thinning of hard sphere suspensions occurs can be determined from the equations due to Krieger26 or Cross27 in conjunction with the reduced stress or Peclet number respectively ... 

Fundamental theories of transport properties for systems of finite concentration are still rather tentative (24). The difficulties are accentuated by the still uncertain effects of concentration on equilibrium properties such as coil dimensions and the distribution of molecular centers. Such problems are by no means limited to polymer solutions however. Even for the supposedly simpler case of hard sphere suspensions the theories of concentration dependence for the viscosity are far from settled (119,120). 

N J. Wagner, G.G. Fuller, and W.B. Russel, The dichroism and birefringence of a hard-sphere suspension under shear, J. Chem. Phys., 89,1580 (1988). 

FIGURE 12.10 Reduced torque versus strain following flow reversal for hard sphere suspension of various concentrations. Polystyrene spheres 45 pm in diameter in silicone oil. Taken from Gadala-Maria and Acrivos [34]. 

Koelman and Hoogerbrugge (1993) have developed a particle-based method that combines features from molecular dynamics (MD) and lattice-gas automata (LGA) to simulate the dynamics of hard sphere suspensions. A similar approach has been followed by Ge and Li (1996) who used a pseudo-particle approach to study the hydrodynamics of gas-solid two-phase flow. In both studies, instead of the Navier-Stokes equations, fictitious gas particles were used to represent and model the flow behavior of the interstial fluid while collisional particle-particle interactions were also accounted for. The power of these approaches is given by the fact that both particle-particle interactions (i.e., collisions) and hydrodynamic interactions in the particle assembly are taken into account. Moreover, these modeling approaches do not require the specification of closure laws for the interphase momentum transfer between the particles and the interstitial fluid. Although these types of models cannot yet be applied to macroscopic systems of interest to the chemical engineer they can provide detailed information which can subsequently be used in (continuum) models which are suited for simulation of macroscopic systems. In this context improved rheological models and boundary condition descriptions can be mentioned as examples. 

Koelman, J. M. V. A., and Hoogerbrugge, P. J., Dynamic simulations of hard-sphere suspensions under steady shear. EuroPhys. Lett 21(3), 363 (1993). 

Dimensional analysis implies that for a given value of , all monodisperse hard-sphere suspensions ought to show an onset of shear thickening at a universal value of the Peclet number Pe, or reduced stress Or. Thus, the critical shear rate Yc foi shear thickening ought... 

The linear viscoelastic properties of hard-sphere suspensions have been measured by de Kruif, Mellema, and coworkers (van der Werff et al. 1989) and by Shikata and Pearson (1994). Figure 6-9 shows G and G" — r a>aT measured as a function of reduced frequency... 

Problems and Worked Examples 6.1 through 6.5, at the end of this chapter, will sharpen your skills in obtaining simple, practical estimations of the viscosity, modulus, and relaxation time of hard-sphere suspensions. 

The effective volume fraction of particles is therefore 0eff = 0 (< eff/2a). Since a hard-sphere suspension forms a macrocrystalline lattice at a volume fraction of around 0 0.55,... 

Stable particle suspensions exhibit an extraordinarily broad range of rheological behavior. which depends on particle concentration, size, and shape, as well as on the presence and type of stabilizing surface layers or surface charges, and possible viscoelastic properties of the suspending fluid. Some of the properties of suspensions of spheres are now reasonably well understood, such as (a) the concentration-dependence of the zero-shear viscosity of hard-sphere suspensions and (b) the effects of deformability of the steric-stabilization layers on the particles. In addition, qualitative understanding and quantitative empirical equations... 

Disordered solutions of spherical micelles are not particularly viscoelastic, or even viscous, unless the volume fraction of micelles becomes high, greater than 30% by volume. Figure 12-7, for example, shows the relative viscosity (the viscosity divided by the solvent viscosity) as a function of micellar volume fraction for a solution of hydrated micelles of lithium dodecyl sulfate in water. Qualitatively, these data are reminiscent of the viscosity-volume-fraction relationship for suspensions of hard spheres, shown as a dashed line (see Section 6.2.1). The micellar viscosity is higher than that of hard-sphere suspensions because of micellar ellipsoidal shape fluctuations and electrostatic repulsions. 

The flow properties of disordered micellar phases are now reasonably well understood. For spherical micelles the viscosity can be estimated from modified hard-sphere-suspension theories, while for disordered semidilute cylindrical micelles the Cates theory of entangled living polymers provides at least a good starting point, and in some cases nearly quantitative prediction of rheological properties. 

Figure 11 shows the relative-viscosity-concentration behavior for a variety of hard-sphere suspensions of uniform-size glass beads. Even though the particle size was varied substantially (0.1 to 440 xm), the relative viscosity is independent of the particle size. However, when the particle diameter was small ( 1 fJLm), the relative viscosity was calculated at high shear rates, so that the effect of Brownian motion was negligible. Figure 8 shows that becomes independent of the particle size at high shear stress (or shear rate). 

The polyelectrolyte microgels have been established as model soft spheres as, in addition to the above features, their softness and properties can be tuned by altering the physico-chemical environment (pH, ionic strength, degree of ionization) [152-160], The response varies from that of colloidal (polydisperse hard-sphere) suspensions and that of polymer gels and in this respect such microgels fit within the theme of Fig. 1 [157-160],... 

In the presence of size polydispersity, there is an additional incoherent contribution to C(, t) decaying through the self-diffusion coefficient Ds((/)) [42,43,91]. The latter can also be measured for monodisperse hard sphere suspensions at finite concentrations at qR corresponding to the first minimum of S( ), i.e., when the interactions can be ignored [91]. These three different diffusion coefficients exhibit distinctly different dependence on q and 0. From these three transport quantities, Z>cou( ) is absent in monodisperse homopolymers, whereas Ds can hardly be measured in polydisperse homopolymers due to the vanishingly small contrast. 

Fig. 14 Normalized Dcoii/Bo of PDMS coated silica suspension with = 0.3 in a symmetric mixture of toluene and heptane (solid circles) along with hard sphere suspension (open squares) at similar volume fraction. The hydrodynamic interactions expressed in H(q) for the two systems (solid squares for the hard sphere suspension) are shown in the inset [101]. This system is crystallized by sedimentation as seen in the photograph...

The use of confocal microscopy to study concentrated colloidal suspensions was pioneered by van Blaaderen and Wiltzius [83], who showed that the structure of a random-close-packed sediment could be reconstructed at the single-particle level. Confocal microscopy of colloidal suspensions in the absence of flow has been recently reviewed [13-16]. We refer the reader to these reviews for details and references. Here, we simply note that this methodology gives direct access to local processes, such as crystal nucleation [84] and dynamic heterogeneities in hard-sphere suspensions near the glass transition [34,37]. 

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