Equilibrium liquid films

Non-equilibrium liquid films formed in the process of spreading have been considered in some early works, especially in the test of the theory of interfacial tension and the rule of Antonov [204], A review on the rule of Antonov and its interpretation on the basis of isotherms of disjoining pressure in wetting films is presented in [532]. However, these works do not deal with precise measurement of film thickness and the studies confined only the kinetics of spreading and lens formation. 

The situation is more complicated when the aqueous solution contains surfactant micelles, which is a common experimental and practical situation. In such a case the disjoining pressure isotherm H(/j) can exhibit multiple decaying oscillations, whose period is close to the diameter of the micelles (Fig. 8b) (for details see, e.g., Ref 78). The condition for equilibrium liquid film, Eq. (42), can be satisfied at several points, denoted by Hq, h, / 2, and in Fig. 8b the corresponding films contain 0,1,2, and 3 layers of micelles, respectively. The transitions between these multiple equilibrium states represent the phenomenon stratification (see Fig. 9 and Refs 78-91). The presence of dis-... 

A sharp transition from the liquid droplet/ meniscus to the liquid film is impossible (arrow in Fig. 1) otherwise, the capillary pressure will be infinite. Hence, there is a smooth transition from the flat equilibrium liquid film on the solid surfaces to the spherical droplet, as shown in Fig. 1. 

Disjoining Pressure, Rgure 1 Transition zone from a flat equilibrium liquid film on a solid surface to a liquid droplet The arrow shows the point where to the right the liquid protile is concave and to the left the profile is convex... 

The excess free energy per unit area of a flat equilibrium liquid film of thickness on a solid substrate at equilibrium with the vapor in the surrounding air is ... 

Equation 2.3 determines the thickness of the equilibrium liquid film, via disjoining pressure isotherm. Equation 2.4 gives the well-known stability condition of flat equilibrium liquid films [1]. 

This condition shows that the three-phase contact line should be determined at the intersection of the liquid profile with the equilibrium liquid film of thickness h, and not at the intersection with the solid substrate as usually assumed. This further results in... 

As already mentioned in Chapter 1, aU solid surfaces in contact with a volatile or nonvolatile liquid at equilibrium are covered by a thin liquid film. The thickness of this equilibrium film is determined by the action of surface forces (disjoining pressure isotherm). That is, the choice of the reference state is uniquely determined in order to consider the vicinity of the three-phase contact line at the equilibrium state of a bulk liquid in contact with a solid substrate the reference state is the state of solid substrate covered with the equilibrium liquid film That is why a reference state that has a plane parallel film with the lowest possible equilibrium thickness (that is, a-flhns introduced in Section 2.1), which corresponds to the vapor pressure p in the ambient air, is selected. In this section, two-dimensional equilibrium menisci in a flat chamber with a half-width H or two-dimensional equilibrium liquid drops are considered for simplicity. Extension of the derivation, in the following text, to axial symmetry is briefly discussed at the end of this section. 

FIGURE 2.13 Profile of a meniscus in a flat capillary. 1 — a spherical part of the meniscus of curvature p, 2 — transition zone between the spherical meniscus and flat films in front, 3 — flat equilibrium liquid film of thickness h. Further in the text, the liquid profile inside the transition zone will be considered in more detail. 

Note that in all cases under consideration, there is no real three-phase contact line at the equilibrium because the whole solid surface is covered by a flat equilibrium liquid film. This is the reason why we refer to it as an apparent contact line. The transversality condition (4 of Section 2.2) at the apparent three-phase contact line results in Equation 2.21 (Section 2.2) ... 

A region of flat equilibrium liquid film in front of the drop... 

Let ns now consider the case of thin equilibrium liquid films on rough solid substrates when there is complete wetting. In such a case, it is possible to measure only the mean thickness of the film, h. It appears that the mean thickness, h, of wetted films on rongh solid surfaces is bigger than the corresponding thickness of a flat film, h, on a smooth substrate. It also appears that the mean thickness, h, approaches the valne, h, at high and low film thicknesses when the latter are... 

Unfortunately, the disjoining pressure in the form given by Equation 2.230 does not allow any equilibrium liquid films at low thickness (a-films). To overcome this problem, we introduce a cutoff thickness, (Figure 2.39). 

The microscopic contour of a meniscus or a drop is a matter that presents some mathematical problems even with the simplifying assumption of a uniform, rigid solid. Since bulk liquid is present, the system must be in equilibrium with the local vapor pressure so that an equilibrium adsorbed film must also be present. The likely picture for the case of a nonwetting drop on a flat surface is... 

A third definition of surface mobility is essentially a rheological one it represents the extension to films of the criteria we use for bulk phases and, of course, it is the basis for distinguishing states of films on liquid substrates. Thus as discussed in Chapter IV, solid films should be ordered and should show elastic and yield point behavior liquid films should be coherent and show viscous flow gaseous films should be in rapid equilibrium with all parts of the surface. 

With a reactive solvent, the mass-transfer coefficient may be enhanced by a factor E so that, for instance. Kg is replaced by EKg. Like specific rates of ordinary chemical reactions, such enhancements must be found experimentally. There are no generalized correlations. Some calculations have been made for idealized situations, such as complete reaction in the liquid film. Tables 23-6 and 23-7 show a few spot data. On that basis, a tower for absorption of SO9 with NaOH is smaller than that with pure water by a factor of roughly 0.317/7.0 = 0.045. Table 23-8 lists the main factors that are needed for mathematical representation of KgO in a typical case of the absorption of CO9 by aqueous mouethauolamiue. Figure 23-27 shows some of the complex behaviors of equilibria and mass-transfer coefficients for the absorption of CO9 in solutions of potassium carbonate. Other than Henry s law, p = HC, which holds for some fairly dilute solutions, there is no general form of equilibrium relation. A typically complex equation is that for CO9 in contact with sodium carbonate solutions (Harte, Baker, and Purcell, Ind. Eng. Chem., 25, 528 [1933]), which is... 

Here rj is the viscosity of the dewetting liquid. Note that a relaxational term proportional to a has been added, with fi(j)) being the chemical potential of the vapor. This term alone guarantees that a homogeneous liquid film will relax to its equilibrium value hooip) by evaporation or condensation. For h = hooip) this term vanishes. 

The first part of Eq. (89), proportional to the inverse viscosity r] of the liquid film, describes a creeping motion of a thin film flow on the surface. In the (almost) dry area the contributions of both terms to the total flow and evaporation of material can basically be neglected. Inside the wet area we can, to lowest order, linearize h = hoo[ + u x,y)], where u is now a small deviation from the asymptotic equilibrium value for h p) in the liquid. Since Vh (p) = 0 the only surviving terms are linear in u and its spatial derivatives Vw and Au. Therefore, inside the wet area, the evolution equation for the variable part u of the height variable h becomes... 

The shape of a droplet or of the front end of a film can be determined from the surface energies and interaction forces between the interfaces. These also determine the equilibrium thickness of a liquid film that completely wets a surface. The calculation is done by minimization of the free energy of the total system. In a two-dimensional case the free energy of a cylindrical droplet can be expressed as [5] ... 

The technique used to study dewetting dynamics on materials consists of making a flat, smooth elastomer surface. A hquid puddle is deposited within a 50-mm-diameter ring of 0.1-mm-thick plasticized adhesive paper adhering to the substrate. The adhesive paper acts as a spacer. A microscope slide is drawn over the liquid to obtain a liquid film of ca. 0.1-mm thickness. At this thickness, the liquid film is unstable, being much less than the equilibrium value, of ca. 1.5 mm calculated from Eq. (29). Nucleation of dry patches... 

Overall mass transfer coefficients are only constant when both liquid film coefficients are constant and also when the slope of the equilibrium line is constant. Thus, for a non-linear equilibrium relationship, the overall mass transfer coefficient will vary with concentration. How would you implement this effect into the program ... 

Our "superheated liquid-film concept" stands on the thermodynamic basis of (1) equilibrium shifts due to reactive separation under boiling and refluxing conditions and (2) irreversible processes of heat flows through the catalyst layer as well as bubble formation from the catalyst surface. 

As far as the Prigogine-type coupling between heat- and mass flow is operative from the catalytic active sites to the vigorously generating bubbles, the adsorbed products will be readily taken out and a large amount of vacant sites will be generated stationarily. Therefore, the restriction of equilibrium conversion (AG < 0) even becomes removable as a consequence of AG < AG by adopting the superheated liquid-film catalysis. 

The characteristics of equilibrium shifts and the irreversible nature in the superheated liquid-film concept are summarized in Figure 13.28. 

Reaction between an absorbed solute and a reagent lowers the equilibrium partial pressure of the solute and thus increases the rate of mass transfer. The mass transfer coefficient likewise may be enhanced which contributes further to increased absorption rate. Three modes of contacting gas and liquid phases are possible The gas is dispersed as bubbles in the liquid, the liquid is dispersed as droplets, the two phases are contacted on a thin liquid film deposited over a packing or wall. The choice between these modes is an important practical problem. 

In the steady state the diffusional rates through the gas and liquid films equal the rate of surface reaction. The concentration in the gas phase is Ag, at the interface At and at the surface As. A is the equilibrium value in the liquid.For a reaction of order m,... 

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