Soap films equilibrium film

As it stands, eqn. (7.7) contains too many unknowns. But there is one additional piece of information that we can use. The interfacial energies, Ysl> Yes 7cl ct as surface tensions in just the way that a soap film has both a surface energy and a surface tension. This means that the mechanical equilibrium around the edge of the nucleus can be described by the triangle of forces... 

It is possible that solid ejected films would return to the surface phase if sufficient time were permitted to re-establish equilibrium. On the other hand the polar groups which have been removed forcibly fi om the water surface may be heavily solvated and the structure of the ejected material may approximate to that of a partly crystallised soap film possessing but little tendency towards adhesion with a water surface. 

The equilibrium thickness of a (meta-)stable soap film will depend on the strength and range of the repulsive forces in the film. Electrostatic forces are long-range in water and hence give rise to thick (0.2 micron) films, which are highly coloured due to the interference of visible light... 

In this situation, the equilibrium thickness at any given height h is determined by the balance between the hydrostatic pressure in the liquid (hpg) and the repulsive pressure in the film, that is n = hpg. Cyril Isenberg gives many beautiful pictures of soap films of different geometries in his book The Science of Soap Films and Soap Bubbles (1992). Sir Isaac Newton published his observations of the colours of soap bubbles in Opticks (1730). This experimental set-up has been used to measure the interaction force between surfactant surfaces, as a function of separation distance or film thickness. These forces are important in stabilizing surfactant lamellar phases and in cell-cell interactions, as well as in colloidal interactions generally. 

If a soap film is sufficiently thin, its equilibrium thickness is the result of the double-layer repulsion, given by Equation (82), and van der Waals attraction, given by... 

At the surface of the solution the cationic soap constitutes a positive layer of organic cations neutralized by Cl" and PoCl62" ions 210Po emits a-rays. The surface density as well as the kinetics of the adsorption of PoCl62" are determined by measuring the radioactivity above the surface of the solution. An analogous technique is used to determine the density of the soap adsorbed at the air-solution interface, when the soap is labelled with 14C. Therefore, these measurements allow a direct analysis of the composition of the adsorbed soap films at equilibrium and during their formation. 

Equilibrium of Adsorption, (a) The Soap Films. The isotherms of adsorption of the organic cations of the soaps are reproduced on the Figure 2. Classical methods (20) are used to obtain the surface densities of the soaps I and IV. 

Curvature relates to the local change in interface area when an interface moves. The energy change per unit volume swept out by the interface is equal to the product of k and the interfacial energy per unit area 7. Normally, for fluids, 7 is independent of the interface inclination h in this case, the interface is isotropic. For example, a soap bubble has isotropic interface tension. If perturbed, a floating individual soap bubble will quickly re-establish its equilibrium form—a sphere of fixed volume. Such a soap bubble will also shrink slowly—the gas will diffuse out of the bubble because of a pressure difference across the soap film (AP = jk = /Rc). Thus,... 

The composition of the soap solution used has a great influence on the stability and properties of the films. For good results very highly purified oleic acid must be used and the best results cannot apparently be obtained without the use of a trace of ammonia or an amine. Excess of alkali is said to be fatal this points to the hydrolytic equilibrium between acid and neutral soap being of great importance. A 5 per cent, solution of ammonium oleate in 50 per cent, glycerine makes a good solution for ordinary work details of this may be found in Lawrence s Soap Films. Perrin, however, used a 2 per cent, solution of ordinary soap ... 

Thickness is one of the main parameters of a foam film. The most widely employed technique for its determination is interferometry. It is based on the comparison between the intensities of the light falling on the thin film and that reflected from it. This technique permits to evaluate the thickness of equilibrium as well as thinning films. It has been used by Perrin [48] and Wells [49] with soap films. The intensity of the reflected light was measured with an interferometer comparing the two parts of the visual field (in a microscope). Thus the film thickness was determined with an accuracy of about 0.5 nm. 

The black spots on soap films, which are not more than 10 to 20 molecules thick, can remain for weeks in equilibrium with the thicker, coloured parts of the film,4 and hence it is assumed that they have the same vapour pressure as the normal liquid, and that Thomson s formula can be applied for a radius of curvature of 200 x 10 cm. or less. Bakker<5 gave reasons for supposing that the surface tension is independent of the radius of curvature of the capillary layer, although he recognised that in very thin films it has abnormal values, and he calculated that the maximum ascent of a liquid occurs in a tube of 2 5 m[jL radius. Woodland and Mack found no change of surface tension in a tube of 6 7 [I radius. 

In our first simple example the electrostatic potential set up by CsCl is almost but not quite a minimal surface [10]. The reason is that the Coulomb electrostatic energy is only a part of the whole electromagnetic field. Two body, three and higher order, non-additive van der Waals interactions contribute to the complete field, distributed within the crystal. This leads one to expect that the condition that the stress tensor of the field is zero, as for soap films, yields the condition for equilibrium of the crystal. Precisely that condition is that for the existence of a minimal surface. Strictly speaking the minimal surface might be defined by the condition that the electromagnetic stress tensor is zero. But in any event, we see in this manner that the occurrence of minimal surfaces, should be a consequence of equilibrium (cf. Chapter 3,3.2.4). Indeed a statement of equilibrium may well be equivalent to quantum statistical mechanics. 

We consider the simple example of a soap film supported on a rectangular frame and held in equilibrium by a force exerted on a moveable edge (Figure 12.1). The work done in stretching the film, and increasing its area by dA, is... 

Figure 12-1 Soap film on a rectangular frame with one moveable side held in equilibrium by a face f f /l = 2,.

Grain boundaries are internal interfaces and behave much like external surfaces, but now we have to be concerned with two crystal orientations, not one. Just as for surfaces, we have a pressure difference associated with the GB curvature and a driving force that tends to lead to an overall increase in grain size whenever possible. Grain morphology and GB topology are two aspects of the same topic. It is instructive to think of the model of soap foams a soap film is flat when in equilibrium and it has a finite thickness. Three soap films meet along a line—a TJ. If you blow on a soap film (apply a pressure) it bows out until the surface tension balances the applied pressure. 

A vertical soap film can be in mechanical equilibrium only if the force of gravity acting on each film element is balanced by a gradient in the surface tension on both surfaces. Calculate the necessary surface tension gradient for a film with a thickness of 100 nm, given a soap solution density of 1 g cm (g = 980 cm s ). 

To illustrate the design of a fundamental function, consider a process in which the surface tension is fixed and the area can vary. For example, area a can be varied at constant surface tension y by pushing with constant force sideways on a floating bar in a dish pan (called a Langmuir trough) to keep a soap film corralled on one side. You want to find a function whose extremum identifies the state of equilibrium for constant T, p, N, and y. Begin by expressing U = U(5, V,N,a). The total differential dU is... 

On an intuitive level, the concept of surface tension becomes apparent in the description of Duprd s original experiment, in which one observes the stretching of a soap film, either in a bubble or on a frame (Figure 1.1). Equilibrium is established when a force, F = 2da, is applied to the moving boundary, where d is the frame width, a is the surface tension, and the numerical coefficient 2 implies that the film has two sides. 

In the latter half of the nineteenth century Josiah Willard Gibbs, who is well known for his theoretical contributions to the study of statistical mechanics and thermodynamics, observed the draining and thinning of soap films. Some of these observations are reported in his paper entitled Equilibrium of Heterogeneous Substances. -... 

The two conditions, (a) and (b), are satisfied by most fluids. For a soap film, however, it is possible that (b) will only be satisfied approximately. For example consider the equilibrium of a section of a vertical soap film of thickness t, width /, and height h (Fig. 1.8). Let the surface tension at the bottom of the section of film be ao and that at height h, ah. The vertical force at the top of the film is 2/(7, the factor of 2 arises because the film has two surfaces. This force balances the force at the bottom of the film, 2/ao, plus the weight of the film mg, where m is the mass of the film. If p is the density of the fluid in the film then m = tlhp. For equilibrium. 

It has been assumed that the fluid, or soap film, is at constant temperature in thermodynamic equilibrium. Under these conditions it is found that the surface tension, a, of a fluid surface depends only on the temperature. This is known as the static surface tension. The surface tension will differ from the static value if the fluid, or soap film, is not in thermodynamic equilibrium. An example of a common non-equilibrium situation occurs in a jet of water issuing from a pipe (Fig. 1.9). The molecules at the surface of the water are not in thermodynamic equilibrium. The environment of the molecules at, or... 

It has been assumed that a/ remains constant in investigating the behaviour of the bubbles. For soap films with low surfactant concentrations of will not remain constant. As the radius of the bubble decreases more soap ions will be adsorbed into the surface, which will decrease a/. By increasing the radius of the bubble the film will become more water-like, as the surface density of surfactant ions will decrease. Hence surface tension than soap solution. Under these conditions a bubble will be in stable equilibrium if condition (1.19) is satisfied. That is, from (1.15), if... 

It is shown in Appendix IV, for a restricted class of problems, that Eq. (1.27) is the mathematical condition for a surface to have minimum area. This mathematical minimum area property can be shown to hold generally for any surface that has zero excess pressure across it and hence satisfies Eq. (1.27). We can verify this result on physical grounds by considering the energy of a soap film, as in section 1.3, and showing that the condition for stable equilibrium of a soap film requires that the area of the film be minimized providing that of is constant. 

The stability of soap films is determined by the amphipathic ions in the surface. If a soap film is perturbed from equilibrium so that the area of an element of film increases, the surface density of amphipathic ions will decrease. That is, the number of ions per unit area will diminish and consequently the surface will behave more like the surface of water. Hence the surface tension of the surface element will increase because the surface tension of water is greater than that of soap solution. This increased force in the region of increased area will restore the surface to its former equilibrium configuration. This stabilizing effect was first observed by Marangoni and is known as the Marangoni effect. 

Fig. 1.18(a) Interference of light produced by a vertical soap film after withdrawal from a bath of soap solution, (b) The film sometime later, (c) The final equilibrium film. 

The film eventually reaches an equilibrium thickness in which both faces of the film are parallel (Fig. 1.18(c)). In this state there is no variation in intensity over the surface of the film. This is called the common black film. It occurs, typically, at a thickness of 300A (30nm). A further decrease in the film thickness, to another stable equilibrium state with a thickness of about 50A, is often possible and is known as the Newton black film. This film is darker than the first black film, the common black film, as the second of the two split rays, that is refracted into the film, travels through a thinner soap film than in the case of the common black film. Consequently the phase difference between the two split rays of the Newton black film is closer to tt than in the case of the common black film. Some films have only one equilibrium state while others have two or more equilibrium states. 

A freshly formed surface of a soap film reaches its equilibrium shape in the order of seconds. However the thickness of the film reaches its equilibrium value in a time that is orders of magnitude greater than that for the surface. For the fastest draining films, of low surface viscosity, this will be minutes whilst films of high surface viscosity will take hours to drain to their equilibrium thickness. 

In a controlled environment a soap film will thin until its thickness becomes appreciably less than the wavelength of light so that it appears black when viewed by reflected light. Commonly it is found that the film reaches a stable thickness, in thermodynamic equilibrium, of about 300 A. This is known as the common black film. If a small amount of evaporation is allowed to take place the film will often thin to a new equilibrium thickness of about 50 A. This is the Newton black film. The thickness of these two equilibrium states can be obtained by measuring the intensity of the reflected light by using photoelectric detectors. The application of Eq. (2.26) will enable the thickness of the film, t, to be determined. 

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