Osmotic pressure double layer

In the preceding derivation, the repulsion between overlapping double layers has been described by an increase in the osmotic pressure between the two planes. A closely related but more general concept of the disjoining pressure was introduced by Deijaguin [30]. This is defined as the difference between the thermodynamic equilibrium state pressure applied to surfaces separated by a film and the pressure in the bulk phase with which the film is equilibrated (see section VI-5). 

In support of the association theory, colloid chemists cited non-reproduceable cryoscopic molecular weight determinations (which were eventually shown to be caused by errors in technique) and claimed that the ordinary laws of chemistry were not applicable to matter in the colloid state. The latter claim was based, not completely without merit, on the ascerta-tion that the colloid particles are large aggregates of molecules, and thus not accessible to chemical reactants. After all many natural colloids were shown to form double electrical layers and adsorb ions, thus they were "autoregulative" by action of their "surface field" (29). Furthermore, colloidal solutions were known to have abnormally high solution viscosities and abnormally low osmotic pressures. 

The contribution of double-layer forces to the osmotic pressure of HIPEs was also investigated [98], These forces arise from the repulsion between adjacent droplets in o/w HIPEs stabilised by ionic surfactants. It was observed that double-layer repulsive forces significantly affected jt for systems of small droplet radius, high volume fraction and low ionic strength of the aqueous continuous phase. The discrepancies between osmotic pressure values observed by Bibette [97] and those calculated by Princen [26] were tentatively attributed to this effect. 

It is important to note that the concept of osmotic pressure is more general than suggested by the above experiment. In particular, one does not have to invoke the presence of a membrane (or even a concentration difference) to define osmotic pressure. The osmotic pressure, being a property of a solution, always exists and serves to counteract the tendency of the chemical potentials to equalize. It is not important how the differences in the chemical potential come about. The differences may arise due to other factors such as an electric field or gravity. For example, we see in Chapter 11 (Section 11.7a) how osmotic pressure plays a major role in giving rise to repulsion between electrical double layers here, the variation of the concentration in the electrical double layers arises from the electrostatic interaction between a charged surface and the ions in the solution. In Chapter 13 (Section 13.6b.3), we provide another example of the role of differences in osmotic pressures of a polymer solution in giving rise to an effective attractive force between colloidal particles suspended in the solution. 

Why is it that the force of double-layer interactions for curved surfaces cannot be derived using osmotic pressure arguments as is done in the case of planar double layers ... 

Figure 4.2 illustrates several features of the diffuse electric double layer. The potential decreases exponentially with increasing distance. This decrease becomes steeper with increasing salt concentration. The concentration of co-ions is drastically increased close to the surface. As a result the total concentration of ions at the surface and thus the osmotic pressure is increased. 

The double-layer force p acting between two parallel plane surfaces has two contributions—the Maxwell stress and the osmotic pressure ... 

On the other (copper) electrode the electrolytic solution pressure is lower than the osmotic pressure of the cations in the solution and therefore cupric ions from tho solution are deposited, thus giving the metal a positive charge, while the solution becomes negative due to the excess of anions (SO ). Both kinds of charges Cu++ and SO - are attracted and form again the electrical double layer. In this case, however, the double layer has an opposite effect than at the zinc electrode as it facilitates the transfer of the cupric ions from the electrode to the solution and prevents them being transferred in the opposite direction. Equilibrium will be attained, when the electrostatic forces of the double layer and the solution pressure of copper together will counterbalance the osmotic pressure of the cupric ions in the solution. 

For uniform, infinite surfaces the stress tensor is independent of position on the surface and therefore can be taken outside the area integration. For a symmetric double layer system, the end result can be expressed as the osmotic pressure difference due to the double layer that is, the concentration of electrolyte at the plane of symmetry minus the bulk concentration, all multiplied by kT. For symmetric electrolytes this is [1]... 

The interaction between two spherical colloids can be transformed by the Derjaguin approximation [29] to the interaction between two flat surfaces (see Appendix A). The net osmotic pressure in an electric double layer is the difference between the internal force, F n, and the external or bulk force, Fex, and is related to the force between two colloids Posm = F n — Fex/a, where a is the area. 

The osmotic pressure in an electric double layer can be derived at any plane between the two surfaces in a salt solution. Let us consider the case where two colloids are treated as infinitely large flat surfaces. There are two natural... 

Saturated Hydraulic Conductivity, 393 Diffuse double layer, 367-369 Na-Ioad, 379, 410-416 ESP 379, 410-416 SAR, 197, 412 Dispersion, 414 Swelling, 103-115 Osmotic pressure, 377 Secondary Contaminants, 478, 479 Copper, 479, 488 Iron, 479, 488 Zinc, 479, 488 Foaming Agents, 488 Chloride, 488 Color, 489 Corrosivity, 489 Hardness, 489 Manganese, 489 Odor, 490 pH, 490 Sodium, 490 Sulfate, 490 Taste, 490... 

Osmotic Pressure of the Double Layer in a Colloidal Suspension... 

An aqueous colloidal suspension also has an osmotic pressure associated with both the double layer of the particles in solution and the structure of the particles. The osmotic pressure term for the structure is given in Section 11.6 for both ordered and random close packing. The osmotic pressure associated with the double layer surrounding the ceramic particles in aqueous solution is discussed here. 

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. 

If, further, the particle is charged, the particle charge and electrolyte ions (mainly counterions) form an electrical double layer around the particle, as shown in Chapter 1. The osmotic pressure becomes... 

The electrolyte ions in the electrical double layer thus exert an excess osmotic pressure ATI(r) on the particle, which is given by... 

When two charged colloidal particles approach each other, their electrical double layers overlap so that the concentration of counterions in the region between the particles increases, resulting in electrostatic forces between them (Fig. 8.2). There are two methods for calculating the potential energy of the double-layer interaction between two charged colloidal particles [1,2] In the first method, one directly calculates the interaction force P from the excess osmotic pressure tensor All and... 

FIGURE 8.1 Electrical double layer around a charged particle exert the excess osmotic pressure AH and the Maxwell stress T on the particle. 

As shown in Chapter 8, the interaction force P can be calculated by integrating the excess osmotic pressure AH and the Maxwell stress T over an arbitrary closed surface E enclosing either one of the two interacting plates (Eq. (8.6)). As an arbitrary surface E enclosing plate 1, we choose two planes x= —oo and x = 0, since )J/(x) = d)J//dx = 0 at x=—oo (Eqs. (10.3) and (10.4)) so that the excess osmotic pressure AH and the Maxwell stress T are both zero at x = —oo. Thus, the force P(h) of the double-layer interaction per unit area between plates 1 and 2 can be expressed as... 

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