Film thickness at rupture

Capillary (disjoining pressure) and foam film thickness at rupture or at CBF/NBF transition [171)... 

Estimation of the initial film thickness ho is not critical, since initial thinning is fast. After a short time, h hg, allowing evaluation of the drainage rate constant 04, from precise measurements of film thickness versus time. Estimates for the film thickness at rupture from 25 to 500 A have been reported. Studies involving mass transfer from drops show that in the presence of mass transfer, coalescence times are much shorter. 

There are many theories of how actual rupture occurs and at what thickness it happens. For example, a hypothesis by Vrij (1966) suggests that as hydrodynamic thinning proceeds, a point is reached where van der Waals attractive forces dominate over surface (interfacial tension) forces. Therefore, surface waves develop and become unstable, creating a hole where the film is thinnest. Film thickness at rupture was estimated to be in excess of 100 A. 

If a critical film thickness is not reached during film drainage, the drops separate from each other. Conversely, if the critical film thickness is reached, the film ruptures—as a result of van der Waals forces—and the drops coalesce. This generally occurs at thin spots, because van der Waals forces are inversely proportional to h (Verwey and Overbeek, 1948). The value of bent can be determined by setting the van der Waals forces equal to the driving force for film drainage, giving (Verwey and Overbeek, 1948)... 

When a drop (water) falls to a flat interface (benzene-water) the entire drop does not always join the pool (water). Sometimes a small droplet is left behind and the entire process, called partial coalescence, is repeated. This can happen several times in succession. High-speed motion pictures, taken at about 2000 frames per second, have revealed the details of the action (W3). The film (benzene) ruptures at the critical film thickness and the hole expands rapidly. Surface and gravitational forces then tend to drag the drop into the main pool (water). But the inertia of the high column of incompressible liquid above the drop tends to resist this pull. The result is a horizontal contraction of the drop into a pillar of liquid above the interface. Further pull will cause the column to be pinched through, leaving a small droplet behind. Charles and Mason (C2) have observed that two pinches and two droplets occurred in a few cases. The entire series of events required about 0.20 sec. for aniline drops at an aniline-water interface (C2, W3). 

Coalescence must occur through the rupture of a separating film at a critical film thickness. Why can we not predict this thickness more accurately ... 

In order to understand the basis for the prevention of bubble coalescence and hence the formation of foams, let us examine the mechanical process involved in the initial stage of bubble coalescence. The relatively low Laplace pressure inside bubbles of reasonable size, say over 1 mm for air bubbles in water, means that the force required to drain the water between the approaching bubbles is sufficient to deform the bubbles as illustrated in Figure 8.2. The process which now occurs in the thin draining film is interesting and has been carefully studied. In water, it appears that the film ruptures, joining the two bubbles, when the film is still relatively thick, at about lOOnm thickness. However, van der Waals forces, which are attractive in this system (i.e. of air/water/air), are effectively insignificant at these film thicknesses. 

Tween 20 was considerably more effective at reducing the stability of foams of a-la than was the case with /3-lg. There was a significant decrease in a-la foam stability in the presence of Tween, at R values as low as 0.05. Minimal foam stability was observed at R = 0.15. There was no observed change in film drainage behavior or onset of surface diffusion in the adsorbed protein layer up to this R value. The only observed change was a progressive decrease in film thickness. Therefore, it is likely that disruption of adsorbed multilayers is responsible for a reduction in the structural integrity of the adsorbed protein layer and that this increases the probability of film rupture. 

Experiments on the stability of water/surfactant films at various pressures were performed by Exerowa et al.2,3 For a dilute aqueous solution of a nonionic surfactant,3 tetraoxyethylene decyl ether (D(EO>4,5 x 10-4 mol/dm3) or eicosaoxyethylene nonylphenol ether (NP(EO)2o, 1 x 10-5 mol/dm3), and electrolyte (KC1), thick films (with thicknesses of the order of 100 A) were observed at low electrolyte concentrations. With an increase of the electrolyte concentration, the film thickness first decreased, which suggests that the repulsion was caused by the double layer. This repulsive force was generated because of the different adsorptions of the two species of ions on the water/ surfactant interface. At a critical electrolyte concentration, a black film was formed, and the further addition of electrolyte did not. modify its thickness, which became almost independent of the external pressure, until a critical pressure was reached, at which it ruptured. While for NP(EO)2o only one metastable equilibrium thickness was found at each electrolyte concentration, in the case of D(EO)4 a hysteresis of the film thickness with increasing and decreasing pressure (i.e., two metastable minima) was observed in the range 5 x 10 4 to 3 x 10 mol/dm3 KC1. The maximum pressure used in these experiments was relatively low, 5 x 104 N/m2, and the Newton black films did not rupture in the range of pressures employed. 

It was first believed that the dimple in foam films decreases and even disappears at small film thickness. Later, experimental investigations of NaDoS aqueous films proved that the rate of thinning is practically equal in both thin and thick film domains, i.e. the difference by thickness between the thinnest and the thickest domains does not decrease up to the critical thickness of rupture. This leads to an increase in the non-uniformity by thickness (Fig. 3.4). 

During thinning thermodynamically unstable films keep their shape in a large range of thicknesses until the critical thickness is approached, at which the film ruptures. This thickness is called critical thickness of rupture hcr. Therefore, the thermodynamic instability is a necessary but not a sufficient condition for film instability. There are other factors determining instability which at thicknesses smaller than the critical cease to act. Two are the possible processes involved in film instability - film thinning with retaining film shape, and film rupture. Which of them is realised when thermodynamic instability is reached, requires analysis of the various mechanisms of film rupture. 

Early studies of rupture of unstable thin films have been performed with macroscopic emulsion films [94] and foam films [53]. Very high values for hcr were obtained (of the order of 10 pm). Systematic investigations with microscopic films [e.g. 29,64,73] have shown that their critical thickness is considerably smaller. The probability character of rupture is illustrated by the curves in Fig. 3.12. As it is seen the most probable critical thickness increases with the increase in film radius. The most probable critical thickness of rupture is 30 nm (r = 0.1 mm). Usually such a thickness is reached by films from aqueous solutions of low molecular fatty alcohols at which the surfactant concentration is chosen so that the surface tension is equal in all cases [29,73]. Aniline films exhibit a higher hcr 42 nm. 

The question of the ( -potential value at the electrolyte solution/air interface in the absence of a surfactant in the solution is very important. It can be considered a priori that it is not possible to obtain a foam film without a surfactant. In the consideration of the kinetics of thinning of microscopic horizontal foam films (Section 3.2) a necessary condition, according to Reynolds relation, is the adsorption of a surfactant at both film surfaces. A unique experiment has been performed [186] in which an equilibrium microscopic horizontal foam film (r = 100 pm) was obtained under very special conditions. A quartz measuring cell was employed. The solutions were prepared in quartz vessels which were purified from surface impurities by a specially developed technique. The strong effect of the surfactant on the rate of thinning and the initial film thickness permitted to control the solution purity with respect to surfactant traces. Hence, an equilibrium thick film with initial thickness of about 120 nm was produced (in the ideal case such a film should be obtained right away). Due to the small film size it was possible to produce thick (100 - 80 nm) equilibrium films without a surfactant. In many cases it ruptured when both surfaces of the biconcave drop contacted. Only very precise procedure led to formation of an equilibrium film. 

At equilibrium film thickness hi the disjoining pressure equals the external (capillary) pressure, n = p This is a common thin film and its equilibrium is described by the DLVO-theory. If h < hcr, at which the film ruptures (see Section 3.2.2), the film is common black (schematically presented in Fig. 3.42). Such a film forms via black spots (local thinnings in the initially thicker non-equilibrium film). The pressure difference nmax - pa is the barrier which hinders the transition to a film of smaller thickness. According to DLVO-theory after nmax the disjoining pressure should decrease infinitely. Results from measurements of some thermodynamic parameters of foam films [e.g. 251,252] show the existence of a second minimum in the 17(6) isotherm (in direction of thickness decrease) after which the disjoining pressure sharply ascends. 

Table 3.8 presents the values of disjoining pressure n and film thickness h at which the film either ruptures or a CBF/NBF transition occurs. In the NaCl concentration range from 10 4 to 0.15 mol dm3, the films rupture in a certain pressure interval which becomes narrower with the rise in electrolyte concentration. 

It has been assumed [30] that if with the decrease in film thickness, the elasticity modulus increases, this leads to extension of the thicker film parts, equalisation of thickness and stabilisation of the film. When the modulus of elasticity decreases with the decrease in film thickness, which is realised at low surfactant concentrations, the thin film parts enlarge during extension. This leads to greater non-uniformity in film thickness and to rupture of the thin parts. The calculations in [30] have shown that, indeed, depending on the surfactant concentration, it is possible to obtain different types of curves for the Ef(h) function and there is a decrease in elasticity with the decrease in film thickness. The poor direct experimental evidence of the relation between the elasticity of a newly formed film and its stability proves qualitatively the assumption that the films become unstable in the range of surfactant concentration where the modulus of elasticity decreases with the diminishing film thickness. 

The study of wet steady-state foams has shown that the foam films at the upper layers rupture at very large thicknesses, i.e. before reaching thicknesses at which specific thermodynamic properties begin to appear [96]. Under these conditions the properties of wet steady-state foams are determined mainly by the effects of Marangoni and Gibbs, which stabilise kinetically the whole system [94-97,116,121,122]. 

In the initial state the film surface is planar until a contact with a solid particle occurs. At the moment of rupture when the menisci tips come into contact, the film thickness becomes h = 2h. The rate of mutual approach of the menisci depends on the volume rate of film thinning and on the excess liquid volume (V + Vy. 

The real surfaces are characterized by a certain extent of roughness. It is assumed that film thickness cannot be lower than the wall roughness value. When film thickness achieves this value, the film ruptured may occur depending on rivulet width, which corresponds to film flow rate at that time. With this statement the system of equations proves to contain the relationship describing the flow rate in a rivulet in the dependence on its length and contact angle. This correlation was obtained by the same way as that for the meniscus flow rate. 

The interface boundary inside the chaimel is shown on figures 4 and 5 as a function of the liquid Reynolds number. At large liquid flow rate the considerable part of the liquid flows in the corners and the film is thinned both on the long and short sides of the channel. This enhances the heat transfer in comparison with uniform film. At small liquid Reynolds numbers the minimum film thickness becomes the same as the wall roughness and film rupture occurs leading to stable rivulet flow. Dry areas exist on the wall, which are not wetted by liquid. This reduces the heat transfer. The... 

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