Flocculation volume fractions

The crucial question is at what value of <)> is the attraction high enough to induce phase separation De Hek and Vrij (6) assume that the critical flocculation concentration is equivalent to the phase separation condition defined by the spinodal point. From the pair potential between two hard spheres in a polymer solution they calculate the second virial coefficient B2 for the particles, and derive from the spinodal condition that if B2 = 1/2 (where is the volume fraction of particles in the dispersion) phase separation occurs. For a system in thermodynamic equilibrium, two phases coexist if the chemical potential of the hard spheres is the same in the dispersion and in the floe phase (i.e., the binodal condition). 

The stability of these dispersions has been investigated. A strong dependence of critical flocculation conditions (temperature or volume fraction of added non-solvent) on particle concentration was found. Moreover, there seems to be little or no correlation between the critical flocculation conditions and the corresponding theta-conditions for the stabilising polymer chains, as proposed by Napper. Although a detailed explanation is difficult to give a tentative explanation for this unexpected behaviour is suggested in terms of the weak flocculation theory of Vincent et al. 

The stability of the various dispersions was assessed and compared by determining the critical flocculation conditions (temperature or volume fraction of added non-solvent for the grafted polymer), as a function of particle concentration. 

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 ...

One of the characteristics of weak flocculation is that the system is reversible. At low volume fractions the system will form some clusters and some single particles. The clusters can be easily disrupted by gentle shaking. As the concentration is increased the system will reach a percolation threshold . The number of nearest neighbours around any test particle reaches 3 at about (p — 0.25 and the attractive forces between... 

The first observation of depletion flocculation by surfactant micelles was reported by Aronson [3]. Bibette et al. [4] have studied the behavior of silicone-in-water emulsions stabilized by sodium dodecyl sulfate (SDS). They have exploited the attractive depletion interaction to size fractionate a crude polydisperse emulsion [5]. Because the surfactant volume fraction necessary to induce flocculation is always lower than 5%, the micelle osmotic pressure can be taken to be the ideal-gas value ... 

Rgure 3.8. Volume fraction of n-alkane at the onset of flocculation as a function of n-aUcane chain length (C H2 +2). The continuous phase is made of a mixture of n-aUcane, dodecane, and SMO (1 wt%). Glycerol droplets (5% in volume) have a diameter of 0.38 pm. T = 65°C. (Adapted from [13].)... 

Evidence for the flocculation of emulsion droplets is commonly derived from a combination of rheological and creaming stability experiments. For instance, a marked increase in both the viscoelasticity of emulsions of moderately high oil volume fraction (40 vol%) and the rapid serum... 

The sedimentation velocity of the flocculated particles is determined by plotting the height of the sedimentation layer (H) as a function of time. Three types of plots exist, depending on the volume fraction of the floes (cp), as shown in Figure 4.37 ... 

Colloidal interactions between emulsion droplets play a primary role in determining emulsion rheology. If attractions predominate over repulsive forces, flocculation can occur, which leads to an increase in the effective volume fraction of the dispersed phase and thus increases viscosity (McCle-ments, 1999). Clustering of milk fat globules due to cold agglutination increases the effective volume fraction of the milk fat globules, thereby increasing viscosity (Prentice, 1992). 

Given the volume fraction of the polymer inside the gel, we are now able to propose a quantitative model of bridging flocculation. Because Crawford et al. [4] studied the contraction of the interlayer spacing as a function of uniaxial stress for the same system without any added polymer, we are able to convert the observed d-values to effective uniaxial pressures caused by the bridging polymers. If we assume that we have one polymer bridge when the end-to-end polymer distance l (calculated according to Equation 12.1) exactly matches the d-value with the... 

As flocculation continues, particle pairs, or doublets, become triplets, and so on, so that floes containing many particles appear. This growth process has been studied by light scattering in dilute or modestly concentrated suspensions, with particle volume fractions less than 0.10. These studies (Dimon et al 1986 Schaefer et al. 1984 Aubert and CanneU 1986) show that the radius ak of a floe containing k particles scales as a power law with k that is. 

A similar, and even more dramatic, viscosity enhancement was observed by Buscall et al. (1993) for dispersions of 157-nm acrylate particles in white spirit (a mixture of high-boiling hydrocarbons). These particles were stabilized by an adsorbed polymer layer, and then they were depletion-flocculated by addition of a nonadsorbing polyisobutylene polymer. Figure 7-9 shows curves of the relative viscosity versus shear stress for several concentrations of polymer at a particle volume fraction of 0 = 0.40. Note that a polymer concentration of 0.1 % by weight is too low to produce flocculation, and the viscosity is only modestly elevated from that of the solvent. For weight percentages of 0.4-1.0%, however, there is a 3-6 decade increase in the zero-shear viscosity ... 

Figure 7.12 Shear-stress dependence of the relative viscosity for dispersions in water of charged polystyrene particles of radius a = 115 nm with nonadsorbing Dextran T-500 polymer (synthesized from glucose) added as a depletion flocculant. The polymer molecular weight is 298,(HX), and its radius of gyration Rg is 15.8 nm. Volume fractions and polymer concentrations are

The modulus of strongly flocculated gels tends to be highly strain-dependent, with linear behavior confined to very low strain amplitudes (Buscall et al. 1988). Figure 7-16 shows the low-frequency modulus in creep versus normalized strain amplitude for strongly flocculated polystyrene particles with volume fractions (j> between 0.1 and 0.25. In... 

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