Configurational osmotic pressure

Strauss et al. [28] has developed a numerical method for the nonlinear Poisson-Boltzmann equation 4 > 25 mV for this spherical particle in a spherical cell geometry. Figure 11.5 is a plot of the osmotic pressure for a suspension of identical particles with 100 mV surface potential and KU = 3.3. In this figure, the configurational osmotic pressure is also given and is much smaller than that of the osmotic pressure due to the double layer. The osmotic pressure increases with increased volume fraction due to the further overlap of the double layers sur-roimding each particle. 

FIGURE IIS Osmotic pressure versus volume fraction for = 100 mV and ko = 3.3 (a) body-centered cubic DLVO, (b) face[Pg.515]

Let us consider the following case of removing an inorganic salt from an aqueous stream. It is desired to reduce the salt content of a 26 m /hr water stream (Qf) whose feed concentration, Cp, of 0.035 kmol/m (approximately 2,000 ppm). The feed osmotic pressure (rrp) is 1.57 atm. A 30 atm (Pp) booster pump is used to pressurize the feed. Sixteen hollow fiber modules are to be employed for separation. The modules are configured in parallel with the feed distributed equally among the units. The following properties are available for the HFRO modules ... 

V, is the molar volume of polymer or solvent, as appropriate, and the concentration is in mass per unit volume. It can be seen from Equation (2.42) that the interaction term changes with the square of the polymer concentration but more importantly for our discussion is the implications of the value of x- When x = 0.5 we are left with the van t Hoff expression which describes the osmotic pressure of an ideal polymer solution. A sol vent/temperature condition that yields this result is known as the 0-condition. For example, the 0-temperature for poly(styrene) in cyclohexane is 311.5 K. At this temperature, the poly(styrene) molecule is at its closest to a random coil configuration because its conformation is unperturbed by specific solvent effects. If x is greater than 0.5 we have a poor solvent for our polymer and the coil will collapse. At x values less than 0.5 we have the polymer in a good solvent and the conformation will be expanded in order to pack as many solvent molecules around each chain segment as possible. A 0-condition is often used when determining the molecular weight of a polymer by measurement of the concentration dependence of viscosity, for example, but solution polymers are invariably used in better than 0-conditions. 

In order to understand the source of this force, consider two particles separated by a distance d as shown in Figure 13.17. The dispersed polymer molecules exert an osmotic pressure force on all sides of the particles when the particles are far apart, that is, when d > Rg. Then, there is no net force between the two particles. However, when d < Rg, there is a depletion of polymer molecules in the region between the particles since otherwise the polymer coils in that region lose configurational entropy. As a consequence, the osmotic pressure forces exerted by the molecules on the external sides of the particles exceed those on the interior (see Fig. 13.17), and there is a net force of attraction between the two particles. The range of this attraction is equal to Rg in our highly simplified model. 

There is no simple, comprehensive theory and steric forces are complex and difficult to describe. Different components contribute to the force, and depending upon the situation, dominate the total force. The most important interaction is repulsive and of entropic origin. It is caused by the reduced configuration entropy of the polymer chains. If the thermal movement of a polymer chain at a surface is limited by the approach of another surface, then the entropy of the individual polymer chain decreases. In addition, the concentration of monomers in the gap increases. This leads to an increased osmotic pressure. 

Because of restrictions on the number of possible configurations, non-adsorbing polymers tend to stay out of a region near the surfaces of the particles, known as the depletion layer. As two particles approach, the polymers in the solution are repelled from the gap between the surfaces of the particles. In effect the polymer concentration in the gap is decreased and is increased in the solution. As a result, an osmotic pressure difference is created which tends to push the particles together. The resulting attractive force is the reason for depletion flocculation. In contrast to this, depletion stabilisation has been mentioned above. 

We have studied the phase and micellization behavior of a series of model surfactant systems using Monte Carlo simulations on cubic lattices of coordination number z = 26. The phase behavior and thermodynamic properties were studied through the use of histogram reweighting methods, and the nanostructure formation was studied through examination ofthe behavior ofthe osmotic pressure as a function of composition and through analysis of configurations. 

Figure 9 summarizes the simulation results for monodisperse packings. It shows a plot of Go, Goo and r, all scaled with , versus the packing fraction. These values of the shear moduli and the osmotic pressure were consistently reproduced over nine sample configurations for each fraction and show little spread, with error bars smaller than the symbols. Up to a packing fraction of around =0.64 (i.e., the close-packing density), there is no contact between particles, and both the moduli... 

We have already discussed confinement effects in the channel flow of colloidal glasses. Such effects are also seen in hard-sphere colloidal crystals sheared between parallel plates. Cohen et al. [103] found that when the plate separation was smaller than 11 particle diameters, commensurability effects became dominant, with the emergence of new crystalline orderings. In particular, the colloids organise into z-buckled" layers which show up in xy slices as one, two or three particle strips separated by fluid bands see Fig. 15. By comparing osmotic pressure and viscous stresses in the different particle configurations, tlie cross-over from buckled to non-buckled states could be accurately predicted. 

The thermodynamic parameters 1/) and K introduced above, pertaining to polymer-solvent interactions in dilute solutions, may be determined from thermodynamic studies of dilute solutions of the polymer, e.g., from osmotic pressure or turbidity measurements at different temperatures. These parameters may also be determined, at least in principle, from viscosity measurements on polymer solutions (see Frictional Propcitics of Polymers). The parameter ij, which is a measure of the entropy of mixing, appears to be related to the spatial or geometrical character of the solvent. For those solvents having cyclic structures, which are relatively compact and symmetrical (e.g., benzene, toluene, and cyclohexane), xp has relatively higher values than for the less symmetrical acyclic solvents capable of assuming a number of different configurations. Cyclic solvents are thus more favorable... 

The question of whether the same term is warranted for chains irreversibly attached to dispersed particles, which are capable of translational motion, is not so easily resolved. The centres of mass of the attached chains possess translational movement by virtue of the motion of the particles. However, the chains attached to one particle undergo correlated movement, quite different from their uncorrelated behaviour in free solution. Hence, even for dispersed particles, this term is unlikely to be correct but it is not immediately obvious how the configurational entropy of the sterically stabilized particles can be properly introduced. Finally, we note parenthetically that the presence of a term associated with the osmotic pressure of the dispersion medium may be appropriate at high compressions. 

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