The Disjoining Pressure

The disjoining pressure is a useful concept to describe surface forces in general. It is particularly suitable for thin liquid films. Therefore, we introduce it here. 

We consider a medium between two planar, parallel surfaces separated by a distance x. The medium is supposed to be in contact with a large reservoir and we neglect edge effects. In the interfadal zones dose to the surfaces, the properties of the medium can be different from its bulk properties. For example, surface charges change the ion concentration of water dose to a charged surface. Van der Waals forces can impose a dielectric property different from the bulk. 

The sign of II is positive for repulsive surface forces. It is negative for attractive surface forces. 

Box filled with liqiiid at pressure of the bulk phase 

The term disjoining pressure was introduced in 1936 by Derjaguin [134], The disjoining pressure II is equal to the difference between the pressure within a film between two surfaces and the pressure in the bulk phase (Fig. 6.6). It is defined as the change in Gibbs free energy with distance and per unit area at constant cross-sectional area, temperature, and volume  

The concept of disjoining pressure is not in contradiction to the formalism of surface forces. It is sometimes more useful to think in terms of disjoining pressure. For example, if 

Just as with interaction energies, II can be regarded as the sum of several components. These include Ilm due to dispersion interaction, Ilf due to electrostatic interactions between charged surfaces, 11 due to overlapping adsorbed layers of neutral 

Some mention should be made of perhaps the major topic of conversation among surface and colloid chemists during the period 1966-1973. Some initial observations were made by Shereshefsky and co-workers on the vapor pressure of water in small capillaries (anomalously low) [119] but especially by Fedyakin in 1962, followed closely by a series of papers by I eijaguin and co-workers (see Ref. 120 for a detailed bibliography up to 1970-1971). 

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One more experimental result, which is important for PT is as follows. Only polar liquids fill conical capillaries from both sides. We used various penetrants to fill conical defects Pion , LZh-6A , LZhT , LUM-9 etc. It was established that only the penetrants containing polar liquid as the basic liquid component (various alcohols, water and others) manifest two-side filling phenomenon. This result gives one more confirmation of the physical mechanism of the phenomenon, based on liquid film flow, because the disjoining pressure strongly depends just on the polarity of a liquid. 

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). 

Consider the situation illustrated in Fig. VI-9, in which two air bubbles, formed in a liquid, are pressed against each other so that a liquid film is present between them. Relate the disjoining pressure of the film to the Laplace pressure P in the air bubbles. 

Deijaguin and Zorin report that at 25°C, water at 0.98 of the saturation vapor pressure adsorbs on quartz to give a film 40 A thick. Calculate the value of the disjoining pressure of this film and give its sign. 

Fig. 2. Effective interface potential (left) and corresponding disjoining pressure (right) vs film thickness as predicted by DLVO theory for an aqueous soap film containing 1 mM of 1 1 electrolyte. The local minimum in H(f), marked by °, gives the equiHbrium film thickness in the absence of appHed pressure as 130 nm the disjoining pressure 11 = —(dV/di vanishes at this minimum. The minimum is extremely shallow compared with the stabilizing energy barrier.

In response to a hydrostatic pressure, the film thickness thus adjusts itself so that the disjoining pressure balances the appHed pressure and mechanical equiHbrium is restored. 

Theoretically, the diffusion coefficient can be described as a function of the disjoining pressure 77, the effective viscosity of lubricant, 77, and the friction between lubricant and solid surfaces. In relatively thick films, an expression derived from hydrodynamics applies to the diffusion coefficients. 

The disjoining pressure characterizes the wetting properties at short ranges. S and II are related by ... 

In a real liqnid, the nature of the disjoining pressure can be a complicated function of the distance, due to the simultaneous coutributiou of several types of forces [1,4-6], For two bodies with flat surfaces separated by a distance z, the van der Waals interaction, which varies as z (or z, if one considers retardation) for single atoms and molecules, gives rise to a power law of the form ... 

Althongh van der Waals forces are present in every system, they dominate the disjoining pressnre in only a few simple cases, such as interactions of nonpolar and inert atoms and molecnles. It is common for surfaces to be charged, particularly when exposed to water or a liquid with a high dielectric constant, due to the dissociation of surface ionic groups or adsorption of ions from solution, hi these cases, repulsive double-layer forces originating from electrostatic and entropic interactions may dominate the disjoining pressure. These forces decay exponentially [5,6] ... 

From Eq. (9), the following relation between the effective contact angle and the disjoining pressure under constant volume can be obtained ... 

FIG. 7 Effective contact angle of the aqueous KOH droplets on HOPG and mica as a function of droplet height. Solid lines correspond to fits obtained using the disjoining pressure given by Eq. (18). 

In this case, the hydrophobic interaction is very weak compared to that of aq.KOH-graphite system. In spite of this, it still dominates the disjoining pressure. 

It is instructive to compare the data emanating from different force measurement techniques. This requires a conversion of the disjoining pressure in energy per unit area. By integration over the thickness of the disjoining pressure, one obtains the corresponding energy per unit area, E h), between two infinite planes ... 

The disjoining pressure vs. thickness isotherms of thin liquid films (TFB) were measured between hexadecane droplets stabilized by 0.1 wt% of -casein. The profiles obey classical electrostatic behavior. Figure 2.20a shows the experimentally obtained rt(/i) isotherm (dots) and the best fit using electrostatic standard equations. The Debye length was calculated from the electrolyte concentration using Eq. (2.11). The only free parameter was the surface potential, which was found to be —30 mV. It agrees fairly well with the surface potential deduced from electrophoretic measurements for jS-casein-covered particles (—30 to —36 mV). 

The disjoining pressure was transformed into force in the same manner as in the case of /3-casein and compared with results from MCT and SFA (Fig. 2.23). The data from MCT and TFB show reasonably good agreement. The results from SFA are only qualitatively similar to both MCT and TFB data. The reason is essentially the same as in the case of /3-casein-stabilized films, that is, the difference in either... 

We have considered the case of a fluid wedge that can deform under the action of the disjoining pressure. Our simulations show that the extent of deformation of the meniscus (or fluid interface) increases with increase in the volume fraction of nanoparticles/micelles, when a decrease in the diameter of micelles and with a decrease in the capillary pressure resisting the deformation is smaller. The resulting deformation of the meniscus causes the contact line to move so that it displaces the fluid that does not contain the micelles (oil) in favor of the fluid that contains it (aqueous surfactant solution). 

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