Hard spheres high concentration

From the preliminary work done so far, we can conclude that, as far as the thermodynamic properties are concerned, the MSA approximation gives excellent results for low surface charge density systems both for symmetric and asymmetric electrolytes. In high surface charge density systems, the MSA can lead to serious discrepancies. Finally we want to summarize the results of calculations done using the MSA as an empirical theory, and fitting the experimental results (converted to the MMM systems) to a model of charged hard spheres, with concentration dependent diameters. The diameters... 

As pointed out earlier, the contributions of the hard cores to the thennodynamic properties of the solution at high concentrations are not negligible. Using the CS equation of state, the osmotic coefficient of an uncharged hard sphere solute (in a continuum solvent) is given by... 

We will focus on one experimental study here. Monovoukas and Cast studied polystyrene particles witli a = 61 nm in potassium chloride solutions [86]. They obtained a very good agreement between tlieir observations and tire predicted Yukawa phase diagram (see figure C2.6.9). In order to make tire comparison tliey rescaled the particle charges according to Alexander et al [43] (see also [82]). At high electrolyte concentrations, tire particle interactions tend to hard-sphere behaviour (see section C2.6.4) and tire phase transition shifts to volume fractions around 0.5 [88]. 

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). 

Vincent et al.(3) used a simplified configurational entropy term Ass = -k ln(4>f /4>.). For a dilute dispersion, the In 4>d term is probably correct, but for the floe phase, with of the order of 0.5, a term In 4>f certainly can overestimate the entropy in the floe, because hard spheres with finite volume have at high concentration much less translational freedom than (volumeless) point... 

This is strong evidence for assuming that dispersions of ideal hard spheres would be expected to show a transition in the viscous behaviour between

short time selfdiffusion coefficient Z)s. This still shows a significant value after the order-disorder transition. The problem faced by the rheologist in interpreting hard sphere systems is that at high concentrations there is... 

Fig. 4.8 Micelle volume fraction () versus polymer concentration at different temperatures for solutions of PEO26PPO39PEO26 in D20 (Mortensen 1993a). 4> was obtained from fits of the hard sphere Percus-Yevick model to neutron scattering profiles (see Fig. 3.9). At high concentration the asymptote = for hard sphere crystallization is reached.

Hard Spheres at High Concentration Because viscous forces perturb the microstructure against the restoring effect of Brownian... 

Precrosslinked poly(organosiloxane) particles are composed of crosslinking trifunctional and linear difunctional siloxane units (T and D units, respectively) [5]. The molar ratios of D and T units can be varied without restrictions thus, hard spheres (fillers) as well as soft, elastic silicone particles are accessible. In this study, the siloxane particles were synthesized in emulsion. The particle size was controlled by emulsifier concentration and crosslink density highly crosslinked particles were obtained with particle diameters ranging from 20-50 nm the size of elastic particles could be varied between 70 and 150 nm. The composition of precrosslinked poly(organosiloxane) particles is summarized in Scheme 1 further, organic radicals R which can be incorporated into the partieles are listed [6,7]. 

Problem 6.4(a) Estimate the high-frequency modulus of a suspension of hard spheres of radius 1 /xm at a concentration 0 of 0.4, at room temperature in a solvent with viscosity 1 P. 

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