Marginal Regeneration and the Marangoni Effect

Marginal Regeneration and the Marangoni Effect

Journal of Colloid and Interface Science 215, 28 –35 (1999) Article ID jcis.1999.6226, available online at http://www.idealibrary.com on Marginal Reg...

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Journal of Colloid and Interface Science 215, 28 –35 (1999) Article ID jcis.1999.6226, available online at http://www.idealibrary.com on

Marginal Regeneration and the Marangoni Effect Vincent Adriaan Nierstrasz 1 and Gert Frens Laboratory of Physical Chemistry, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands Received October 29, 1997; accepted March 22, 1999

Drainage On the basis of experimental observations described earlier, we have proposed that marginal regeneration is caused by surface tension gradients at the borders of mobile foam films. Marginal regeneration is the rate-determining mechanism in the drainage of such films, and, as such, a determining factor in the persistence (or long-term stability) of foams. Marangoni flows are caused by surface tension gradients, and these set off the exchange of thicker for thin film elements along the borders, while the total film area remains the same. In this paper we present simulations of the drainage of liquid in a vertical soap film, and show that it is realistic to expect large surface tension gradients along the lower border of the film under the conditions which lead to marginal regeneration. © 1999 Academic Press Key Words: marginal regeneration; Marangoni effect; foam films; surface tension gradients.

The horizontal bands of uniform color (Fig. 1) in a vertical soap film with mobile surfaces are the result of this interplay between gravity and surface forces. The pattern of horizontal interference colors reveals a vertical thickness profile. The film is thin at the top and thick near the bottom. Film elements at the same horizontal level have the same film thickness h and the same surface tension g. However, in the vertical direction there must be balance of the surface and gravity forces to achieve mechanical equilibrium. The liquid material is kept in place by a vertical surface tension gradient. At every level the local surface tension lifts the weight of the lamella below it, and, therefore, g is higher at the top than at the bottom (2). In spite of this mechanical equilibrium there remains some slow drainage of the viscous liquid from the core of the film. The horizontal lines of equal film thickness are seen to move downward, and as a whole the film is thinning. In a film with rigid, immobilized surfaces the only remaining drainage mechanism of the liquid would be Poiseuille flow in the core of the lamella. For thin films, i.e., at the correspondingly low Reynolds numbers, this drainage very slow. However, in a mobile film the surfaces are only immobilized by the surface tension gradient, but they are not rigid. In a soap film with mobile surfaces there exists a more rapid way of thinning, which is called marginal regeneration (1–7). Thinning of the lamellae is an important property in a foam. As a lamella becomes thinner it also becomes less stable. Therefore, thinning mechanisms and the thinning rates in vertical films are of technological importance in their relation to the persistent stability of foams.

INTRODUCTION

The surfaces of the lamellae in a foam are often stabilized with a “mobile” layer of surfactant molecules (1). Because these surfaces are capable of developing and sustaining gradients in the surface coverage (G), and therefore in the surface tension (g), the weight of the liquid in a vertical film can be carried by the surface tension forces. A balance of forces immobilizes the film elements in the lamellae, so that a foam can survive as a (meta)stable structure on top of a bulk liquid. Without the surface tension gradients it will collapse immediately, which is the reason foams of pure liquids, albeit with low surface tensions, have no persistent stability. The term “mobile layer of surfactant molecules” indicates a familiar type of surfactant surface. An element in a mobile film, which has, locally, acquired a higher surface tension than its surroundings, will tend to contract. By reducing the surface area it lowers the surface tension, until mechanical equilibrium with the rest of the film is established. This effect is responsible for the characteristic rapid, swirling motions that are often seen in foam lamellae.

Marginal Regeneration In a vertical film with mobile surfaces the mechanism which determines the thinning rate is marginal regeneration. Along the “margins” of the film, where it connects with the bulk liquid or with adjacent lamellae of the foam, there are the “Plateau borders”. The Plateau borders (1, 2, 4, 5) around a film have curved surfaces (Fig. 2). Inside the Plateau border there is a lowered Laplace pressure. Capillary suction draws liquid from the film into the border. The liquid drains out of the foam through the interconnected ducts of the Plateau border structure. In marginal regeneration patches of thinner film material are pulled out of the border in exchange for thicker film elements (1). In vertical films these thinner patches tend to

1 To whom correspondence should be addressed. Present address: Textile Technology Group, Department of Chemical Technology, University of Twente, 7500 AE Enschede, The Netherlands.

0021-9797/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.

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MARGINAL REGENERATION

29

FIG. 1. A vertical liquid film with mobile surfaces.

expand and move upward (6, 7). This produces a “surplus” of surface area higher up in the film (6 –9). The extra room at the top allows the whole film, including its surfaces, to descend into the lower Plateau border below like a falling curtain (9). In

FIG. 2. Three lamellae in a foam meet at a Plateau border. The films are essentially flat, but the surfaces in the border are curved. This causes a Laplace pressure difference between the Plateau border and the adjacent parts of the film.

thinning vertical films with mobile surfaces the lowest film elements enter into the lower Plateau border with a velocity on the order of millimeters per second (6 – 8). Marginal regeneration. Mysels (1) proposed that marginal regeneration is a consequence of the pressure difference between a film and its borders. He considered (thermal) thickness fluctuations in the film and noticed that the difference in the pulling force on a thin and on a thicker part of film would cause their exchange along the borders. However, our observations (5, 6), on the periodicity of marginal regeneration, on its dependence on the film thickness, and on the rate of drainage, point to an alternative explanation. It seems that marginal regeneration, rather than reflecting differences in the film thickness, is a surface tension related effect. It indicates local deviations of the surface tension from its equilibrium value. Along the perimeter of a soap film with mobile surfaces the surface tension is lower, when a film element is absorbed by its Plateau borders. Compression of surface. In marginal regeneration a flux of film element falls into the lower Plateau border. When a film element falls into the border its thickness increases and its surface area shrinks (Fig. 3). The reduced surface area of a film element becomes crowded with surfactant molecules. Relaxation of the excess of surfactant would be possible by dissolution of the surplus of surfactant into the core solution. But, a foam film is only a thin layer of liquid. The film thickness is of the order of 1 mm, whereas the surface

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NIERSTRASZ AND FRENS

THEORY

FIG. 3. As a film element (shaded area) sinks into the border its surface area shrinks inversely proportional with its thickness.

coverage G with the surface active molecules is high. After doubling the thickness of a film element, restoring equilibrium in the surface by desorption would, typically, lead to an increase of the surfactant concentration inside the core by several orders of magnitude. Such a concentration change entails complex and, compared to marginal regeneration, slow processes like the formation of micelles or of liquid crystalline mesophases, before the surface tension can return to equilibrium. An alternative for the dissolution of the surfactant in the film could be that the saturated surface layer along the border becomes unstable. We have suggested that this leads to a disproportionation of the surface layer into (periodic) areas of higher and of normal surface coverage (7). Some film elements along the border then acquire a higher surface coverage, i.e., a lower surface tension (11), than their surroundings. These parts of the border still have mobile surfaces. They will be expanded in response to the higher surface tension of the film and are pulled back toward the film. But, as the film element expands until its surface coverage is in balance with the surrounding film, the thickness diminishes below the original value. A model like this would explain why there is a constant ratio between the thickness of the undisturbed film at the border and that of the thin patches which are drawn into the film (6, 7, 11). This ratio is independent of the film thickness, but it represents a fixed percentage of surface expansion for all film thicknesses. Therefore, marginal regeneration must be a surface tension effect that is caused by local gradients in surface tension, which allow the stretching of film elements from the border and into the film. The existing interpretation of marginal regeneration is that it determines the rate of thinning for foam films by exchanging thicker for thinner film elements. A more important effect of marginal regeneration on the thinning rate is that it restores the balance of surfactant in the film. The mechanism brings the lost surfactant molecules back into the film, after a peripheral film element, including its surface coverage, has been absorbed into the lower Plateau border. Marginal regeneration is then explained from the expansion of saturated border elements in the direction of the film until the element regains its equilibrium surface tension, which in turn allows the original film to descend faster into the lower Plateau border (6 –9).

From observations on marginal regeneration we have deduced (6, 7, 11) that the absorption of peripheral film elements, including their adsorbed, mobile, surface layer of soluble surfactant, into the lower Plateau border creates a local surface tension gradient. To investigate the conditions for which these surface tension gradients might develop, and to show that the relaxation to equilibrium by a Marangoni flow is a faster process than dissolving the surplus of surfactant in the core of the film, the drainage of a mobile soap film is simulated. A Model for a Draining Soap Film A vertical liquid film will be considered. The vertical and horizontal (film thickness 2h) coordinates are x and y, respectively. The film is wide relative to the film thickness: the edges are far from each other in the z-direction and we investigate the lower Plateau border and the vertical film above it. This assumption effectively reduces the dimensionality of the model describing the film (where properties are constant in the zdirection). The origin ( x 5 0, y 5 0) is at the top of the film. The thickness profile of the film, as well as of the Plateau border, is symmetrical around the center-plane of the film ( y 5 0). During drainage this thickness profile remains continuous over the whole length of the film, including the transition zone where the film connects to the Plateau border below, and the curved surfaces of the Plateau border. To keep the problem as simple as possible, it is assumed that —the liquid in the film is an incompressible Newtonian fluid with a viscosity h, —the flow obeys the lubrication approximation in the film, the transition zone and the Plateau border, —the drainage is symmetrical relative to the plane y 5 0. Bulk Fluid Equations of Motion The flow in the film is determined by gravity forces, and if there is no curvature, the pressure gradient ­P/­x is equal to 2rg. This does apply to the situation just after formation of the film. When drainage proceeds, a disjoining pressure may become relevant, at least for the higher, thinner parts of the film. In the thick films with marginal regeneration considered here, disjoining pressure is not relevant. The equations of the fluid motion are ­P ­ 2V x 5h ­x ­y2

[1]

­P 5 0, ­y

[2]

and the continuity equation is ­V x ­V y 1 5 0, ­x ­y

[3]

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where P is the pressure and V x and V y are the components of the velocity vector in the x and y directions, respectively (12, 13). These equations are solved subject to the boundary conditions, Vx 5 Vs

at y 5 6h

[4]

­V x 50 ­y

at y 5 0

[5]

at y 5 6h,

[6]

Vy 5 2

­h ­t

where V s is the velocity of the interface in the x direction. With these boundary conditions, Eqs. [1] to [3] are integrated and rearranged. Vx 5 2 2

1 ­P 2 ~h 2 y 2 ! 1 V s 2h ­ x

S

­h h 3 ­P ­ 5 2 1 V sh ­t ­x 3h ­ x

D

[7]

Surfactant Mass Balance of the Intralamellar Liquid There is the possibility that surfactant molecules will be exchanged between the surfaces and the intralamellar solution. If the diffusivity D bulk in the (incompressible) liquid is constant, the surfactant mass balance in the core liquid becomes

S

D

­c ­ 2c ­ ­ 2c ­ 1 ~cV x ! 1 ~cV y ! 5 D bulk 1 . ­t ­ x ­y ­x2 ­y2

In the above equations all variables are known for a given or measured film profile, except V s . Just after the formation of the film, surface tension gradients will be absent. When drainage proceeds the surface properties will affect the drainage velocity. Unlike in a pure liquid, a surfactant-covered film element changes its surface tension as its surface shrinks or expands. Gradients in surface tension will develop as the film elements are stretched due to the gravitational forces. These gradients will slow down the motion of the liquid and create, eventually, a mechanical equilibrium between gravity and surface forces. The surface velocity V s will therefore depend on the thickness profiles (h( x)), the surface tension gradient ( g ( x)), and the surface excess concentration (G( x)). The developing surface tension gradients, which are affected themselves by the redistribution of surfactant between the intralamellar liquid and the surfaces (at y 5 6h) of each film element, can stop the motion of the surfaces and keep a vertical foam film upright and stable. The gradients in the surface coverage create surface stresses in the film. Neglecting curvature effects and the density of the surface, this can be expressed as ­P ­ g ­ 2V s 5 1 ~ m s 1 m d! , ­x ­x ­x2

[10]

The model parameter a, which describes this local relation, is experimentally obtained from adsorption isotherms and accompanying surface tensions. In the simulations, one combined parameter m for the surface shear and dilational properties will be used, instead of the separate parameters m s and m d . This is a simplified model for a Newtonian mobile interface. More complicated models describing films with viscoelastic interfaces exist (12). This is not necessary since the simplified model is capable, for a study of the transition zone where a falling film is disappearing into the Plateau border, and can identify experimental conditions where large surface tension gradients might be expected.

[8]

Surface Equations of Motion

h

­G ­ g ­ g ­G 5 5a . ­ x ­G ­ x ­x

[9]

where g is the local surface tension, and m s and m d are the surface shear viscosity and the surface dilational viscosity, respectively. The surface tension gradient is related to the local surface coverage G,

[11]

In this equation, c is the surfactant concentration of the intralamellar solution. The film is very thin, and the velocity gradient perpendicular to the surfaces is relatively small at high thinning velocities. Therefore, the third term on the left-hand side may be neglected. Assuming first-order mass transfer between surface and solution and Langmuir adsorption kinetics, we obtain 2D bulk

S D ­c ­y

5 y56h

S D ­G ­t

S

5 k ads c 1 2 ads

2 k des

G G max

D

G 5 K~G eq 2 G!, G max

[12]

where K5

k ads c eq 1 k des G max

G eq 5

k ads c eq G max , k ads c eq 1 k des

[13] c eq 5

k des G eq . G max 2 G eq k ads

[14]

The ratio k des /k ads is the Langmuir–Szyszkowski constant a. This parameter is known from direct measurements or from the literature. Equations [13] and [14] are obtained with the aid of the Langmuir isotherm for a nonionic surfactant, G5

G maxc . a1c

[15]

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NIERSTRASZ AND FRENS

In the case of an ionic surfactant, the left-hand side of the equation must be multiplied by 2. As a consequence, G and G eq in Eqs. [14] and [12] must in that case also be corrected with the same factor. Integrating Eq. [11] from y 5 2h to y 5 h, and substituting Eq. [7] for V x and Eq. [12] for the bulk-to-surface mass transport, we obtain

S

D

­~ch! ­ ch ­P ­ ~ch! 5 2 chV s 1 D bulk 1 K~G 2 G eq !. ­t ­x 3 h ­ x ­x2 3

2

[16] With this expression the developing surfactant distribution in film elements is described as a function of the hydrodynamic and physical chemical properties. The first and second terms on the right-hand side describe the motion of the liquid and the surface velocity. The third and fourth terms represent the exchange rates of surfactant between bulk and surface by diffusion and adsorption. Surface Surfactant Mass Balance The (local) surfactant concentration at the surfaces of film elements can now be expressed, with the aid of Eq. [12], as ­G ­~GV s ! ­ 2G 1 2 D surf 1 K~G 2 G eq ! 5 0, ­t ­x ­x2

[19]

­c 50 ­x

at x 5 0

[20]

Vs 5 0

at x 5 0 ` x $ x Bulk

[21]

The first boundary condition is due to the fact that the volume of the lower Plateau border is much larger than the volume of the film. When a film element enters the Plateau border its surface will be compressed and its velocity will decrease since the thickness of the lower Plateau border increases rapidly. The lower Plateau border is a large reservoir in contact with the bulk solution and can therefore maintain equilibrium between its liquid core and the surfaces. The other boundary conditions describe the situation at the top of the film. The surfactant concentration gradient at the top of the film should be zero because there is no surfactant entering or leaving the film at the glass frame. At the top, the surface velocity is zero because there should be no slip at the glass frame. The last boundary condition describes the situation at the horizontal bulk solution surface. The surface velocity approaches zero when approaching the horizontal bulk solution surface. During the simulations we varied the x-position of this last boundary condition, but there was no influence on the results. Simulation of the Thinning Soap Film

[17]

where D surf is the surface diffusion coefficient. Usually the rate of surface diffusion is very small compared to Marangoni flow and desorption kinetics. However, we did not neglect surface diffusion at this stage since we have no information about the magnitude of the developing surface tension gradients at the transition from the falling film to the lower Plateau border, yet. Plateau Border, Boundary, and Initial Conditions The description above fits the situation as it develops in the draining flat film. However, in the Plateau border the effect of the curved surfaces needs to be taken into account. Inside the border the net downward flow of liquid is assumed to be independent of the height since the dimensions of the lower Plateau border are much larger than those in the film. This results in ­P 3h 3hVs 5 3 Q 2 ­ x 2h h2

at x $ x PB

c 5 c bulk

[18]

for the influx of film elements with a descent rate V s from a thinning film into the lower Plateau border (1, 14), where Q is the flux over the total thickness of the border (2h). Equations [8], [9], [16], [17], and [18] can be solved numerically with the boundary conditions

In general, one might start a simulation from any arbitrary thickness profile, like from a film of constant thickness, and predict the evolution of the profile in time. Our focus is, however, on the possibility that a surface tension gradient develops at the transition between the film and the lower Plateau border, and on the influence of the surface and bulk properties of the surfactant solution on this surface tension gradient. This surface tension gradient can explain marginal regeneration along the lower border Plateau. It is then more convenient and less time consuming to start the simulation with an experimental thickness profile of a vertical soap film with mobile surfaces. For this film profile we calculated the equilibrium shape of the attached Plateau border for zero surface velocity (15, 16). The surface tension in the film is assumed to equal the bulk value at the beginning of the calculations. The surfactant concentrations in the intralamellar solution and at the surface are calculated from the surface tension with the Langmuir– Szyszkowski equation [22], which is only valid at concentrations below the cmc, and with Eq. [14],

S

g 0 2 g 5 2RTG maxln 1 2

D

G . G max

[22]

Here, g 0 is the surface tension of the solvent. In our simulations h , m s , m d , a , g , G max, k ads , k des , D bulk , D surf , and r are parameters which can be chosen depending on the solution studied, and can be obtained from the literature.

MARGINAL REGENERATION

33

FIG. 4. (a) Velocity profile of the interface at different times, for m 5 1 mNs/m: ■, 0.05 msec; Œ, 1 msec; ✚, 2 msec. (b) Velocity profile of the interface at different times, for m 5 10 mNs/m: ■, 0.05 msec; Œ, 1 msec; ✚, 2 msec.

RESULTS

Two characteristic situations are described by the simulation results. These are shown in the Figs. 4a, 5a, and 6a for mobile films and Figs. 4b, 5b, and 6b for rigid films. These figures describe the initial stages of thinning for a vertical soap film. The data for the parameters used in these simulations are rendered in Table 1. In combination with these values, two different values for the combined surface dilational and surface shear viscosity (m 5 1 mNs/m and m 5 10 mNs/m) were used in the simulations shown in Figures 4 – 6. These differ by an order of magnitude. A surface dilational and shear viscosity m 5 10 mNs/m is typical for the surface properties in an (almost) rigid film (17). For mobile films the (lower) surface dilational viscosity of m 5 1 mNs/m is a typical value (17, 18). In test runs we found (6), as was to be expected, that the drainage rates slow down when the liquid viscosity h is increased (18, 19) and when the film thickness is reduced. A more interesting result is that a larger surface dilational or shear viscosity will also result in a lower rate of descent of the film elements toward the Plateau border. For the chosen values

m 5 1 mNs/m and m 5 10 mNs/m the computed velocity V s of the film elements which are entering the lower Plateau border (at x 5 0, 3 cm) is of the order of 10 23 m/s and 10 24 m/s, respectively (Figs. 4a and 4b), which is comparable with experiments (6 – 8, 11). The velocity profiles in the Plateau border resemble those found in the literature (14). The slow drainage of rigid films with a large surface dilational viscosity is no surprise. But in real experiments with vertical mobile liquid films the measured thinning rates are of the same order as in the simulations. This is surprising, since we know that in the real experiments, contrary to the simulation, the thinning is controlled by marginal regeneration as the rate limiting mechanism. An experimenter could make marginal regeneration less effective without affecting the conditions at the Plateau border below the film, e.g., by changing the width (1, 4) or the shape of the film. It is then rather surprising to find that when he allows marginal regeneration, the thinning rate can maintain itself more or less at the initial level. In a two-dimensional simulation, when marginal regeneration is absent, this would not be possible. Then, the thinning rate

FIG. 5. (a) Surface excess concentration profile at different times, for m 5 1 mNs/m: ■, 0.05 msec; Œ, 1 msec; ✚, 2 msec. (b) Surface excess concentration profile at different times, for m 5 10 mNs/m: ■, 0.05 msec; Œ, 1 msec; ✚, 2 msec.

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NIERSTRASZ AND FRENS

FIG. 6. (a) Surface tension gradient profile at different times, for m 5 1 mNs/m: ■, 0.05 msec; Œ, 1 msec; ✚, 2 msec. (b) Surface tension gradient profile at different times, for m 5 10 mNs/m: ■, 0.05 msec; Œ, 1 msec; ✚, 2 msec.

would gradually be reduced when, at longer simulation times, surface tension gradients develop and immobilize the surfaces of the vertical film. For the mobile film a gradient in the surface coverage (G) is seen to develop, in the film and in the lower Plateau border, during the initial stages of thinning (Figs. 5a and 5b). In the slower, rigid film this gradient is hardly noticeable, but it is clearly seen in the “mobile” case. The gradient is clearly asymmetric around the transition zone. Going from the film into the Plateau border, the film elements entering the lower border are quickly compressed and hardly any relaxation takes place. Further away from the film, the excess concentration tends to relax due to desorption. Expressed in terms of surface tension gradients, which bring the film to the mechanical equilibrium eventually, this is still a rather small effect. More important differences between a rigid and a mobile film become apparent near the transition from the film to the lower Plateau border. A high velocity V s of the surfactant-covered film elements creates a considerable surplus of surfactant. This lowers the surface tension in the transition zone between a mobile film and the Plateau border into which it is draining. This aspect is highlighted in Figs. 6a and 6b. The development of the surface tension gradients dg/dx in a rigid and a mobile film during the simulation is shown. In the rigid film the TABLE 1 Model Parameters h x5x_PB h a g G max k ads k des D bulk D surf T r

2.14 mm 1.0E-3 Ns/m 2 1.0E4 (N/m)/(mol/m 2) 45.0E-3 N/m 6.0E-6 mol/m 2 1.3E-3 m/s 1.3E-3 mol/(m 2s) 1.0E-9 m 2/s 1.0E-11 m 2/s 298 K 1000 kg/m 3

surface velocity is very low, and the system remains close to equilibrium. In the mobile film, however, a large surface tension gradient is built up at the transition between the border and the draining film, which is magnitudes larger than those in the film above the Plateau border. These large gradients develop where film elements enter into the Plateau border with a velocity on the order of 1 mm/s, that is, with a surface velocity which is typical for mobile foam films of mm thickness. Also, this result is obtained with quite normal values assigned to the surface properties and to relaxation rates by diffusion and adsorption. And, the initial thickness profile of the film was already derived from experiments with mobile films. Therefore, the motions and the surface tension gradients in the simulation are typical for what must be expected at the lower Plateau border of a real film with mobile surfaces. The higher surface tensions in the immediate surroundings of the transition line will, indeed, be unbalanced and apply strong forces on the saturated film elements of the border. These will tend to stretch their mobile surfaces, until the local surface tensions balance into a new equilibrium. DISCUSSION AND CONCLUSIONS

The simulation produces a new view into in the reasons why marginal regeneration at the borders of a vertical mobile foam film determines the rate of thinning. Immediately after its formation a film begins to fall into the Plateau border below (8). Expansion of film elements moves a vertical film in the direction of mechanical equilibrium, and this effect reduces the rate of thinning in the same way we found in the simulation. The velocity of the film elements which become part of the borders is of the order of 1 mm/s, depending on the surface dilational and shear viscosity of the stabilizing surfactant layers. With a low surface dilational and shear viscosity, like in a mobile film, the rate of descent V s of the film elements at the transition from film to border can be large enough for developing a considerable gradient in the surface tension in the transition zone.

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MARGINAL REGENERATION

These gradients determine the presence or absence of marginal regeneration, and therefore the mode of thinning. If the surface tension gradient is large enough to make the transition zone unstable, it causes marginal regeneration (6, 20). If the gradient remains small, because the rate of descent is too slow or because the relaxation to the equilibrium surface tension is too fast, the rate of thinning will continue to slow down. Eventually, after the equilibrium between surface tension and gravity forces has been established, film thinning is reduced to the slow Poiseuille flow of the viscous core liquid between immobilized film surfaces, like in a rigid film. In the case of the mobile film, the quicker relaxation mechanism for the saturated area along the border is through marginal regeneration. The surface tension gradient becomes unstable and the most saturated film areas along the border line expand and rise into the film. These film elements expand and become thinner while they rise. The expansion of these film elements increases their surface tension until it approaches the value of the surrounding film. Expanding film elements produce extra film surface (8, 9), but the total area of the film must remain approximately constant. The expansion of film elements, therefore, allows the original film to continue the downward motion by which it disappears into the borders, with essentially the velocity which is necessary to deliver an equal amount of surfactant to the lower Plateau border as is generated by the expansion of new film elements in marginal regeneration (7). This is a self-sustaining mechanism. In the steady state the rate of thinning is determined by a balance of fluxes of the surfactant into the lower Plateau border and back into the film, into the transition zone by the descent of film elements, and out of it by the expansion of film elements. It is not surprising then that a fast rate of thinning can be maintained in a vertical mobile film. This mechanism will work as long as the thinning is fast enough and creates surface tension gradients which can destabilize the lower Plateau border. Experimental observations support this model. Marginal regeneration diminishes as the film thickness h decreases (6, 7). A smaller film thickness leads to a lower velocity of the surface at the lower Plateau border. This reduces the influx of surfactant into the transition zone, even when the surface tension and the surface coverage of the descending film elements remain the same. As a consequence of the smaller flux the surface tension gradient in the lower Plateau border also diminishes, as does the regeneration of film surface at the Plateau border which compensates this flux. By altering the thinning rate without changing the properties of the foaming solution (6, 7), we could relate the value of V s to the wavelength of marginal regeneration (6, 7, 20). Of course, such an experiment would not lead to a result if thermal thickness fluctuations were the basis of the exchange of thin and thick film elements along the Plateau borders. Independent of the chosen thinning rate, the amount of surface which disappears when film

elements fall into the border must always be precisely compensated by the regeneration of the extra surface area of thinner film which spreads from the borders into the film above. The two compensating areas are the same, their surface coverage is equal, and this equality is persistent during thinning, independent of the film thickness or the thinning rate. Marginal regeneration, or the generation of extra film area at the borders, is the effect rather than the cause of a high thinning rate. The regeneration process stops when the thinning rate becomes too slow. In fast thinning, like in the drainage of mobile, vertical liquid films, the borders become unstable because a surplus of surfactant creates large gradients in the surface tension. These are the conditions which trigger marginal regeneration. The film maintains the fast thinning rates because it acts as an essential link in the chain of balanced fluxes in and out of the thinning film. From that position marginal regeneration controls the rate of the overall thinning process. It is the mechanism which continuously compensates for loss of surfactant from the film while it drains into the borders. ACKNOWLEDGMENTS IR-TNO, Delft, The Netherlands, Household and Personal Care Research, the VEMATEX, the NVW, and the Unilever Research Laboratory, Vlaardingen, The Netherlands, are kindly acknowledged for their financial support.

REFERENCES 1. Mysels, K. J., Shinoda, K., and Frankel, S., “Soap Films: Studies of their Thinning and a Bibliography.” Pergamon, New York, 1959. 2. Stein, H. N., Adv. Colloid Interface Sci. 34, 175 (1991). 3. Hudales, J. B. M., and Stein, H. N., J. Colloid Interface Sci. 137, 512 (1990). 4. Hudales, J. B. M., and Stein, H. N., J. Colloid Interface Sci. 138, 354 (1990). 5. Hudales, J. B. M., and Stein, H. N., J. Colloid Interface Sci. 140, 307 (1990). 6. Nierstrasz, V. A., Ph.D. Thesis, TU Delft, The Netherlands, 1996. 7. Nierstrasz, V. A., and Frens, G., J. Colloid Interface Sci. 207, 209 (1998). 8. Sandor, N., Gelade, E., van Marris, R., and Racz, G., J. Appl. Polym. Sci. 67, 1739 (1998). 9. Mysels, K. J., “Forum on Micro Fluid Mechanics, Am. Soc. Mechanical Engineers: New York” (L. M. Trefethen, Ed.) FED 113, p. 36, 1991. 10. Frankel, S. P., and Mysels, K. J., J. Phys. Chem. 73, 3038 (1969). 11. Nierstrasz, V. A., and Frens, G., “Proceedings, 37th International WFK Det. Conference, Krefeld,” p. 70, 1996. 12. Tambe, D. E., and Sharma, M. M., J. Colloid Interface Sci. 124, 137 (1991). 13. Sonin, A. A., Bonfillon, A., and Langevin, D., J. Colloid Interface Sci. 162, 323 (1994). 14. Barber, A. D., and Hartland, S., Can. J. Chem. Eng. 54, 279 (1976). 15. Princen, H. M., in “Surface and Colloid Science” (E. Matijevic, Ed.) Vol. 7, p. 1. Plenum, New York, 1974. 16. Hudales, J. B. M., Ph.D. Thesis, TU Eindhoven, The Netherlands, 1989. 17. Prins, A., and van Voorst Vader, F., “Proceedings, 6th International Congress Surface Active Substances, Zurich,” p. 441, 1972. 18. Baets, P. J. M., Ph.D. Thesis, TU Eindhoven, The Netherlands, 1993. 19. Baets, P. J. M., and Stein, H. N., Langmuir 8, 3099 (1992). 20. Nierstrasz, V. A., and Frens, G., to be published.