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Marginal Regeneration in Thin Vertical Liquid Films
JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.
207, 209 –217 (1998)
CS985646
Marginal Regeneration in Thin Vertical Liquid Films Vincent Adriaan Nierstrasz1 and Gert Frens Laboratory of Physical Chemistry, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands Received October 29, 1997; revised April 28, 1998; accepted May 8, 1998
Marginal regeneration is the rate-determining drainage mechanism in mobile vertical liquid films stabilized with surfactants. Mysels, Frankel, and Shinoda explained this process from (thermal) thickness fluctuations, like capillary waves. The Laplace underpressure in the Plateau border would exert a larger force on a thick, rather than on a thinner film element. This force unbalance would make film elements of different thicknesses move in opposite directions so that they are exchanged at the border. However, experiments and simulations prove that marginal regeneration cannot be the result of thickness fluctuations. Our alternative view is, that marginal regeneration is due to surface tension gradients between the film and its borders. Drainage of film elements into the lower Plateau border causes a local excess of surfactant, and thereby local differences in surface tension. This causes film elements to flow and generates the thickness differences between the absorbed and emerging film elements. The rates of the Marangoni flows reflect the surface dilational properties. This Marangoni effect is a consequence of the compression of the film surface when a film element flows into the lower Plateau border. Marginal regeneration is then a mechanism which returns the surfactant back into the film. © 1998 Academic Press Key Words: marginal regeneration; Marangoni effect; thin vertical liquid films; dynamic surface tension; surface dilational properties.
INTRODUCTION
In dilute foams the gas bubbles are present in the dispersion as spheres, and in aged, concentrated foams there are polyhedric gas volumes, separated by thin liquid films which connect by Plateau borders (Fig. 1). Studies on single isolated foam films are used to model drainage from concentrated foams (1). Some authors investigated the drainage of horizontal foam films: in the absence of gravity, capillary suction is the driving force for drainage (Joye (2), Sharma (3), and Ruckenstein (4)). Others studied vertical films (Mysels (1), Hudales et al. (5–7), Stein (8, 9), and Baets et al. (10)), where gravity and capillarity together determine the drainage rate. Our investigation belongs to the latter type. A stable, vertical film in a gravity field develops a vertical 1
To whom correspondence should be addressed. Present address: Food and Bioprocess Engineering Group, Department of Food Science, Wageningen Agricultural University, P.O. Box 8129, 6700 EV Wageningen, The Netherlands.
surface tension gradient to carry the weight of the film (J. W. Gibbs, 1878) (8, 11). This surface tension gradient causes the surfaces of the film to be immobile. The flow of the liquid can therefore be described as a laminar flow between two fixed walls under the influence of a pressure gradient. This is a very slow process that can be observed in mobile and rigid soap films when there is no marginal regeneration because of the shape of the film or because its surface elasticity, respectively. However, in a vertical soap film with mobile surfaces there is marginal regeneration; rapid turbulent motion is visible at the vertical borders and at the lower horizontal border of the soap film. In these films marginal regeneration is the most important drainage mechanism. Because of marginal regeneration these thin films drain much faster than would be expected on the basis of Poiseuille flow. So much faster, that the effect of the Poiseuille flow is negligible in the drainage of soap films with mobile surfaces. Mysels et al. (1) sought the origin in thermal or other thickness fluctuations in the film. They described the effect as exchange of thick and thin film material along the borders (“margins”) of the film, and therefore gave the process its name. In marginal regeneration complete film elements, including the surfaces and the intralamellar liquid, are rapidly sucked out of the film in exchange for thinner film material, of equal area, which are drawn out of the surrounding Plateau borders. Mysels inferred that the exchange of film patches is only effective for the draining film at the vertical borders. He observed that the rate of drainage is independent of the vertical position and decreases with increasing film width. This implies that at a given height there is no influence on the drainage rate of the amount of film higher up. Assuming that the fluctuations are constant and homogeneous along the vertical borders, this suggests a process in which thin and thick film materials (Fig. 2) of about equal area are exchanged with a rate that can only depend on local conditions, e.g. the local film thickness. The proposed mechanism did not consider the dynamics of the film itself and, especially, not of the surfaces of the exchanging film patches. For example, the possibility of surface tension gradients in the film or near the border, which could cause expansion, shrinkage, and flow of film elements, were not taken into account. The onset of marginal regeneration was explained from (thermal) thickness fluctuations in the film, as may be caused by capillary waves. The Laplace underpressure in the Plateau
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vertical Plateau border near the film, compensated by downwards flows in the border further away from the film, and it explains their observations of a reduced radius of curvature of the border at lower heights. dg h film 5 rg dx 2
FIG. 1. Schematic representation of a concentrated foam. The detail shows the film and the Plateau border.
border, which is equal to the hydrostatic pressure at the height concerned, would exert a larger force on a thick film part than on a thinner film part [1]. As a result the thin and the thick film parts near the same border will move in opposite directions, and their two surfaces and the intralamellar solution are exchanged at the border. F 5 DP Laplace z h film 5
g zh R film
[1]
More recently, Joye et al. (2), Prins et al. (12), Hudales et al. (6), Stein (8, 9), and Baets et al. (10) have studied marginal regeneration. Joye pointed out that surface tension and film thicknesses in horizontal, dimpling, films are not constant. In the barrier ring of such a dimpling film these surface tension differences and thickness differences may, under circumstances, create exchange processes in the film and the border through Mysels’ mechanism. Prins showed that marginal regeneration can be stopped by increasing the surface dilational elasticity, so that rigid soap films do not have marginal regeneration and drain very slowly indeed, but he did not introduce this observation in a more elaborate model for marginal regeneration. Baets studied how liquid viscosity and particles in the bulk liquid affect marginal regeneration and the drainage rate for films with low surface dilational elasticity. Hudales (6) and Stein (8, 9) suggested that the different drainage behaviour of horizontal and of vertical films relate to a surface tension gradient in the vertical Plateau borders. They proposed that during the process of marginal regeneration the vertical surface tension gradient in the film is carried over into the vertical Plateau borders via the film elements that are absorbed. This causes upward flows because of Marangoni effects in the
[2]
However, since it is not the flow inside the Plateau borders which determines the drainage rate but rather the exchange of film elements between the film and those borders, this cannot explain why vertical films drain more rapidly. Marginal regeneration itself, which is the rate-determining drainage mechanism, is in Hudales’ model still considered to be caused by thickness fluctuations in the film, as proposed by Mysels. Then, whereas the film patches which are sucked out of the film by the border have the (higher) surface tension at that particular height, the thinner patches which enter into the film have the (lower) surface tension of the surfactant solution. Consequently, the entering parts would expand in the film and make the fluctuations self-sustaining. In this model it is unclear how film elements that are absorbed by the Plateau border could retain their original surface tension during (rapid) compression in the process. In this paper we will try to show that surface tension gradients in the direction from a film to its borders are the cause, rather than the effect, of marginal regeneration. This new insight is important for the understanding of foam stability in relation to the physical chemistry of the bulk and the surface of the solution. The existence of surface tension gradients near the Plateau borders and their relation to marginal regeneration shows how the dynamics of adsorbed surfactant layers can stimulate liquid flow at interfaces in processes of mass transfer, like dissolution of solids or laundry of soiled textiles, which involve flow in thin liquid films and surface layers.
FIG. 2.
Marginal regeneration along the vertical Plateau borders.
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MARGINAL REGENERATION
THEORY
EXPERIMENTAL
If marginal regeneration in a film with immobile surfaces, due to the vertical surface tension gradient that carries the weight of the film, is caused by local thickness fluctuations it is possible to describe these thickness fluctuations (13–16) in terms of the “squeezing mode” of (thermally excited) waves in thin films. Such modes have two characteristic features: their wavelength and their amplitude; and if they are the cause of marginal regeneration these characteristics must, somehow, be reflected in the resulting pattern of turbulence. Under some circumstances the squeezing mode in a thin liquid film can become a growing thickness fluctuation which eventually reaches a critical size and ruptures the film. Vrij (17, 18) and Lucassen et al. (13) showed that the fastest growing wave has a wavelength of
Vertical films were drawn from different sodiumdodecylsulphate (SDS, Merck, reinst) solutions, with and without added NaCl (Baker, zur analyse); N-dodecylpyridiniumchloride (DPCl, Merck, zur synthese) solutions, with and without NaBr added (Merck, reinst); azelaic acid solutions (Fluka, purris.); sodiumdodecylbenzenesulfonate (SDBS, Fluka, tech.) solutions, with and without added CsCl (Merck, reinst) or LiCl (Baker, zur analyse); or decyl-b-D-maltoside (DM, Fluka, purris.) solutions. The technique for the observation of marginal regeneration was, in principle, identical to Mysels’ (1). The films were made by vertically pulling a II-shaped glass frame out of the surfactant solution. The pulling mechanism was placed on top of a glass box in which an all-glass vessel containing the sample solution was placed. The glass box had optically polished windows. Behind the frame, black objects formed a light trap to eliminate spurious reflections. The bottom of the glass box was always covered with a layer of the solution so that the gas atmosphere was kept saturated. All solutions were left in the glass box for a day, to equilibrate at ambient temperatures between 20 and 23°C, before experiments were started. The whole apparatus stood on a vibration-free table. All glass material used, except the enveloping glass box, was cleaned between experiments by immersing it in 62% HNO3 at 75°C for at least 30 min and thoroughly rinsed afterwards with double distilled water. Films were observed in the reflected white light produced by a 50-W halogen lamp. By using an optical glass filter (Schott, SKF 11) the film could also be observed in reflected green light of wavelength 552 nm. The reflected light was monitored by a colour video camera (Sony DXC-151P) and recorded by a video cassette recorder player (Sony EVO-9500P). The advantage of this setup is that recorded experiments on marginal regeneration could be followed in slow motion, reversed in time, or observed in a frame by frame mode. Film thicknesses were obtained from the recorded interference colours. For white light, data tables are available for this purpose (20). When monochromatic light is used, film thicknesses could be obtained with help of Eq. [5], by determination of the interference order, p ( p is an integer). In the case of destructive interference, a dark minimum will occur. In the case of constructive interference, a green maximum will be observed. In the last video frames recorded at the final stage of drainage, the first dark line below the black film at the top is of order p 5 1, the second is of order p 5 2, etc. The local thickness for any element of film is then obtained as
l fastest 5 h 20 Î8 gp 3/A.
[3]
Here k is the wavenumber 2 p / l , A is the Hamaker constant, and h 0 is the mean film thickness. It would be expected that if thickness fluctuations at the border are the cause of marginal regeneration, the fastest growing wavelength for the squeezing mode at a given film thickness will be the most common one. Therefore, one might expect that marginal regeneration at the lower Plateau border is related to the characteristics of the fastest growing squeezing mode thickness fluctuations under the conditions at the border. The Hamaker constant and the surface tension are constants within one experiment, and the film thickness is well defined at the lower border. Therefore, during drainage, the wavelength for the disturbances in thickness at the lower border lexperimental should be proportional with h 20 , which can be verified experimentally. The amplitudes of thickness waves depend on the growth coefficient of the fluctuations. If the initial thickness deviation is of thermal origin, the amplitude is not a function of the film thickness, but only of the surface tension and the temperature (4, 15, 18, 19). If we assume the wavelength lfastest, we can calculate the growth coefficient for the perturbation. For a wave of length lfastest, the growth coefficient (2b) will be (13, 17, 18), 2bfastest 5
A2 96 p 2hg h 50
[4]
which implies that the (thermal) thickness fluctuation which eventually causes the instability would grow or decay at a rate proportional with h 25 . The growth rate is negligible for thickness fluctuations in thick films (h . 1 mm) and increases as the film becomes thinner. Therefore it seems unlikely that thermal fluctuations in film thickness should be the cause for marginal regeneration. It is an experimental fact that marginal regeneration is more active in thicker films and that, unlike the squeezing mode thickness waves, it disappears in films of thicknesses below 0.2 mm.
h green 5
p 2 1/ 2 , 2n cos u
h dark 5
p 2n cos u
[5]
with h 5 the film thickness, n 5 the refractive index the liquid, u 5 the angle of refraction, and p 5 the order of interference.
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FIG. 3.
Photo of marginal regeneration in a vertical liquid film.
RESULTS AND DISCUSSION
If the growth of a squeezing mode thickness fluctuation would eventually cause marginal regeneration it would be expected that the spatial disturbance of the effect (Fig. 3) would reflect this original cause; i.e. that the wavelength of marginal regeneration is related to lfastest of the squeezing mode at an original film thickness h 0 at the lower border. For determining the periodicity or the wavelength of marginal regeneration at the lower border, we counted the number of thin “feathers” per unit film width which were under formation at the same time, i.e. in a single frame of the recorded reflection image of the film. At that time the original thickness of the film h 0 at the border could also be found from the same picture. This procedure was repeated for many frames per solution and at many stages during the drainage process. The measured wavelength was averaged for the frames where h 0 at the border was the same. By this procedure we obtained a value for the wavelength l which is accurate within 10%. This wavelength of the marginal regeneration at the lower Plateau border is plotted in Fig. 4 as a function of h 20 for four different SDS solutions. From these results it is clear that the wavelength of marginal regeneration does not show the linear dependence with h 20 that should be expected if the thickness waves would cause marginal regeneration. Similar behaviour was seen when soap films of other surfactants were used, or when electrolyte was added to the solutions (data not shown). The proposition that the fastest growing squeezing mode wave in the spectrum of thermal thickness fluctuations would cause marginal regeneration at the Plateau borders of a vertical mobile surfactantcovered liquid film appears in conflict with the experimental observations.
Apart from thermal fluctuations, there can be other temporary thickness differences in a liquid film. These would tend to be damped by the Gibbs elasticity which makes the film stable. Capillary suction would then, when strong enough, again, lead to the exchange of thicker and thinner areas when they happened to be side by side along the borders of the film. Because the total surface of the film is constant this would cause equal surface areas of film material to be absorbed by and drawn from the border. However, in our recordings of draining films and marginal regeneration no equivalent area of extra thick film parts was seen to accelerate towards the lower border in exchange for the thinner feathers that are drawn into the film. Only a small “bow wave” was discernible around these rising and expanding patches of thin film material (9), but rather than being drawn in the opposite direction, these thicker film elements moved upwards with the sharply delineated thinner patches until the
FIG. 4. Wavelength of marginal regeneration vs squared film thickness for different SDS concentrations: ■, 0.5 mM; Œ, 7.5 mM; ✚, 10 mM; and }, 15 mM.
MARGINAL REGENERATION
FIG. 5. Inflow thickness vs original film thickness at lower SDS concentrations: ■, 2.5 mM; Œ, 0.8 mM; ✚, 0.6 mM; }, 0.4 mM; and ✖, 0.2 mM.
latter merged into the background of original film of the same thickness. At the lower Plateau border, most film material has the normal “original” thickness when it is absorbed into the border. There is no evidence in our observations for the equivalence in the exchange of thicker and thinner film areas, as in Mysels’ original model, moving in opposite directions because of the suction of the Laplace pressure differences. We must conclude that the role of thickness waves as the cause for marginal regeneration has been overestimated. Indeed, the net effect of marginal regeneration is the enhanced drainage of liquid from the film, but this is not caused by the exchange of thicker and thinner film elements. In videographs of marginal regeneration at the lower horizontal Plateau border it is seen that patches of thin film are drawn out of the border in a typical shape (Fig. 3). In between are other film elements, of average thickness, which are part of the overall vertical thickness profile of the draining film. The thickness of the thin, rising portions of film was measured, and compared with the “original” film thickness of the film at the lower border when the patch was originally formed there (Fig. 5). A straight line is obtained, with a slope of 0.8. Remarkably, neither the addition of NaCl (up to 340 mM), nor using other surfactants or surfactant concentrations altered this unexpected experimental result. It is a new observation about marginal regeneration that there is a constant ratio of the “new” and the “original” film thicknesses. During the drainage process of a vertical film the thickness at the lower border diminishes, but the constant ratio of newly formed and the existing film thicknesses persists. This is independent of the value of the film thickness at the lower border. Amplitudes of the thickness waves at the onset of the exchange would not be expected to vary in proportion with the film thickness itself. Both the growth coefficient of thermal thickness fluctuations in thin films and their spatial frequency would have to vary with the film thickness as shown above and in the literature (3, 8, 13). The experimental observations clearly indicate that marginal regeneration cannot be explained from thickness variations. The constant ratio rather suggests that marginal regeneration is caused by a surface tension effect. Thickness change in an
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element of a thin film is coupled to the expansion of the surface area of that film element, since the volume must remain constant. A constant ratio between the thicknesses of “old” and “new” patches of film is then an indication that the degree of surface expansion of film elements was constant, irrespective of the local film thickness itself. This would imply a constant difference in local surface coverage with surfactant molecules between the Plateau border and the film. The expansion of the parts with a high surface coverage equalises the surface tension differences. When elements of a film sink into the lower border the film becomes thicker and its surface shrinks. These surfaces carry a layer of surfactant and, as a result the surfaces of the elements, tend to become supersaturated with surfactant (Fig. 6). If the drainage into the border is fast, relative to relaxation of the (dynamic) surface tension through desorption, surface diffusion, and micelle formation, the supersaturation, as a result of compression of the surface, increases linearly with the thickness in the profile of the lower Plateau border. This supersaturation leads to disproportionation of the surfactant layer in areas with a higher and lower surface coverage, and thereby creates a mechanically unstable situation right at the transition between film and border. Border elements with their excess surface coverage have a lower surface tension. They will expand (and therefore become thinner) in the direction of the film. The surface tension gradient in the film pulls the thinner patches upward. The distance between the thin film patches which are seen rising in the film reflects the periodicity of the disproportionation process. Experimentally, we have made three new experimental observations on marginal regeneration at the lower horizontal Plateau border: —the nonlinear relation of the wavelength of marginal regeneration with the film thickness squared, —the absence of extra thick film parts in exchange for the feathers, —the constant thickness ratio between the original film and new film parts, which point to an alternative explanation of marginal regeneration as the rate-determining factor in the drainage in vertical, mobile soap films. When film elements are absorbed by a Plateau border, they carry their surface coverage with them. However, the total area of the film remains constant and
FIG. 6. Compression of a film element at the lower Plateau border (schematic).
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FIG. 7. Surface tension gradient profile at different times: ■, 0.05 ms; Œ, 1 ms; ✚, 2 ms.
therefore the loss has to be compensated by a flow of surfactant back into the film. This causes elements of normal thickness, but higher surface coverage to expand into the film, which necessarily makes them thinner. The expanding surface area of the new elements augments in the film and makes it possible for the original film to descend faster. In mechanical equilibrium at the transition from the lower Plateau border to the film, the surface tension changes rapidly from a bulk value in the border to a film value which compensates the capillary suction and the height above the liquid surface. This delicate local balance is easily disturbed when film elements become compressed as they drain into the lower Plateau border. The resulting gradients in surface tension will make the transition between border and film unstable. Eventually the differences in surface tension cause Marangoni flow of material from the border into the film. The thickness ratio of 0.8 is then reflecting a surface expansion of the supersaturated film elements along the border of 1.25. Gradients in surface tension between the border and the expanding new feathers of thinner film also explain why there are “tails” which connect the expanding thinner patches with the border. These “tails” are not the wake of ordinary flow, but Marangoni flows from the border into the film element. They are a consequence of the suction caused by the increasing surface tension during the expansion of the new film elements. Mysels, and later on Stein and Hudales, have shown that the drainage rate of a vertical film is reversibly proportional with its width. They conclude from this observation that marginal regeneration at the lower border is not important for film drainage and that the two vertical borders in a rectangular frame carry the rate-determining mechanism. Our alternative reasoning is that the expansion of new film creates extra surface, which must be absorbed by the surrounding Plateau borders. Near the vertical borders, the recordings show that newly formed film elements from the lower border move towards the vertical border while they rise. These elements expand more than those in the middle and they have a higher rising velocity. As they expand, these elements allow the film material between themselves and the vertical border to shrink and to be absorbed by the vertical border nearby. Marginal regeneration,
unlike in Mysels’ reasoning, is thus not constant and homogeneous along the vertical border. It is triggered by the disturbance below through a local compression of the film surface. As they absorb the adjacent film material the vertical borders become themselves supersaturated and act as a secondary source of thinner, expanding film elements. The role of the vertical borders then becomes the upwards transport of the surplus surface because of the process of marginal regeneration. This mechanism makes it possible for the film to descend faster into the lower Plateau border and make the process self-sustaining. The process continues as long as the rate of descent is fast enough to keep the lower borders destabilized. This explains the measured effect of film height and film width on the drainage rate (1). The expansion of fresh film elements into the original area is rate determining for the drainage of film into its borders. The drainage rate of a vertical film is fast enough to create differences in surface tension between the border and the film by supersaturating the film surface at the beginning of the transition between them. Experimentally such local gradients are difficult to prove. Therefore simulations of the draining film into the lower Plateau border were performed to obtain more information. The simulations (Fig. 7) show that, for representative values of draining rates, surface coverages, and surface relaxation mechanisms, large surface tension gradients are indeed present at the transition between the film and the lower Plateau border. Lower drainage rates and higher values for the surface dilational viscosities produce smaller surface tension gradients in the simulations. These computations (for details on the calculations see (21, 22)) show the relevance of our model which does relate marginal regeneration to the magnitude of such surface tension gradients. The surface area of a vertical film in a fixed frame must remain constant. The intensity of marginal regeneration at each moment must then reflect the drainage velocity of the film that is falling into the lower border. We measured the amount of surface area descending in the film from the velocity of descent (dx/dt) of a line of constant thickness h( x) as a function of the film thickness just above the lower Plateau border h 0 , and we related this to the amount of regenerated surface issuing from
FIG. 8. Amount of exchanging surface based on marginal regeneration: ■, 0.1 mM; ✚, 0.6 mM; ✖, 0.8 mM; and on the rate of descent in the film: Œ, 0.1 mM; }, 0.6 mM; and F, 0.8 mM.
MARGINAL REGENERATION
FIG. 9. Rate of descent ✚ compared to the upward frame velocity ■ for a 15-mM SDS film as a function of the film thickness.
the lower Plateau border during the same time interval. The amount of regenerated surface issuing from the lower Plateau border was determined from the recorded video pictures. The number of feathers and their size at the film thickness h 0 were determined. One sees that the surface area lost at the lower border because of drainage is exactly compensated for by the formation of new, thinner film elements (Fig. 8). This shows that at high drainage velocities (dx/dt) is a good estimate for the velocity of film elements of thickness h and, therefore, for the local surface velocity. However, to keep the film stable, the surface coverage must also remain constant. We see the thinner patches expanding as they rise from the lower border. Eventually they acquire the same surface tension and the same thickness as the adjacent parts of the original film. During the process it is unlikely that there is a lateral transport of surfactant from the surroundings into the expanding patch, opposite in direction from the motion of the liquid in the core of the film. Therefore, the thinner parts of the film which are created by marginal regeneration have a higher surface coverage with surfactant and a lower surface tension than their surroundings as long as they do expand. A related observation is that marginal regeneration can be prevented both at the lower and the vertical Plateau borders when the drainage rate of the vertical film is compensated by pulling the frame upwards. When the upward expansion of the film equals the rate of compression at thickness h 0 there will be no surplus of surfactant at the borders and marginal regeneration should therefore be absent. This is indeed seen in experiments (Fig. 9). Increasing the drainage velocity (i.e. the rate of descent into the lower border), either by decreasing the viscosity of the film liquid or by simply pushing the frame back into the bulk solution, or by using a smaller frame width, should result in a different wavelength, frequency, or size of the newly formed film elements in order to keep the balance of the surfactant in equilibrium. The wider the vertical frame, the lower is the lower drainage rate in a vertical, mobile liquid film of equal composition. Rate determining is the marginal regeneration. In Fig. 10 the drainage rate is seen to decrease with increasing frame width for an 15 mM SDS solution with 12.5% glycerol, which is consistent with earlier investigations (1, 6). Figure 11 gives the wavelength of marginal regeneration as a
FIG. 10.
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Drainage velocity at different frame widths: ■, 1 cm; Œ, 2.5 cm.
function of the film thickness at some different frame widths. From these data it follows that for a film of constant composition the wavelength of the marginal regeneration decreases when the drainage velocity increases. This would be unexpected for a mechanism involving local thermal fluctuations in thickness. But in our explanation of marginal regeneration it is not very surprising. At higher compression rates at the border more surfactant needs to be transported back into the film. Adjustment of the transport back into the film to the rate of absorption of surface by the lower Plateau border is through the wavelength of the instabilities which carry the back-transport mechanism (marginal regeneration) and not only through their size. There is an alternative way to reduce the drainage rate in a vertical soapfilm. Increasing the bulk liquid viscosity slows the drainage process. If the rate of drainage is lower because of an increased viscosity of the surfactant solution, a completely different result is obtained. The drainage rate decreases with increasing viscosity, as expected, but now the wavelength of marginal regeneration also decreases (Figs. 12 and 13). This observation was also made by Baets (10). This observation indicates why the instability in marginal regeneration is of a spatially periodic nature. The film surface becomes unstable at a certain supersaturation with surfactant, which then determines the ratio of 0.8 between the thickness of the new film parts and of the original film thickness. The value of 0.8 for this ratio must be a property of the two-dimensional
FIG. 11. The wavelength of marginal regeneration at different frame widths: ✚, 1 cm; }, 2.5 cm.
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FIG. 12. Drainage rate at different viscosities (mNs/m2) at 2.5-cm frame width: }, 1.0; ■, 1.4; Œ, 1.9; ✚, 4.0.
equation of state that describes the relation of the surface coverage of surfactant with the surface tension in mobile, liquid surfactant stabilized films. The transition zone of the border to the film is supersaturated with surfactant. This results in a pattern of film elements moving into and out of the border, because the supersaturated area disproportionates into regions of lower and of normal surface tension. At a given drainage rate, the amount of exchanged surfactant can be realised by more and smaller or by fewer, larger patches. The dissipation, caused by the flow of core liquid with the surfactant in the surface during the disproportionation and the expansion of the thin film, is seen to lead to the formation of more and smaller patches, instead of less and larger patches, depending on the viscosity of the liquid. At a higher liquid viscosity, the same local Dg between the Plateau border and the film, at an identical surface expansion in the film (d/d0 5 0.8), results in shorter distances between the instabilities at the border. Each feather will then transport less surfactant back into the film. This type of result seems consistent with observations on other Marangoni driven instabilities described in literature (23, 24). Finally, to realize a more rapid compression, and thus surfactant transport to the border, the frame was pushed back into the horizontal lower Plateau border. This results in a decreasing wavelength with an increasing compression velocity (0.2 or 0.9 cm/s), like when the drainage rate is varied by using a different frame width (Fig. 14).
FIG. 13. Wavelength of marginal regeneration at different viscosities (mNs/m2) at 2.5-cm frame width: }, 1.0; Œ, 1.9.
FIG. 14. Wavelength of marginal regeneration at different downwards frame velocities at 2.5-cm frame width: ■, 0.2 cm/s; }, 0.9 cm/s.
Analysis of the data in Fig. 14 shows that l varies linearly with V21/3 z l1/l2 ' (1.2 6 .2)(Vs1/Vs2)21/3. To compare these results s with the experiments with frames of different widths (Figs. 10 and 11), it is necessary to calculate the velocity in both experiments as a function of the film thickness h at the border. The velocity in a free draining film depends on hn. From a log(V)-log(h) plot, n can be determined by linear regression, which gives us the drainage velocity as a function of the film thickness. From this independent combination of experimental data we obtain l1/l2 ' (1.03 6 .1)(Vs1/Vs2)21/3. The error margin in both independent results is mainly due to the wavelength measurement, since l is rather sensitive for the presence of one thin “feather” more or less. But the results seem reasonably consistent, which proves that the drainage rate is the rate-determining cause and that marginal regeneration is the adaptable effect in a self-sustaining, surface tension driven instability. CONCLUSIONS
Our investigations on marginal regeneration lead to the following conclusions: The transition from the film to the lower Plateau border is supersaturated with surfactant. Therefore, marginal regeneration is a Marangoni-driven instability.
Experiments and simulations reveal that marginal regeneration is the result of an instability caused by a surface tension gradient at the transition from the Plateau border to the film. Film elements that are absorbed by the Plateau border when the film which “falls” into the Plateau border are compressed. Since the film elements keep their original surface coverage during compression in the border, the surface tension decreases all along the horizontal borderline. This causes a surface tension gradient opposing gravity. As the surface tension gradient increases with time, the situation becomes unstable, and thin film elements begin to expand and rise in the film as a result of the Marangoni effect. The drainage rate of vertical mobile liquid films is proportional with the rate of marginal regeneration at the lower Plateau border. The intensity of marginal regeneration is determined by the surface compression rate at the transition from the border to the film. Marginal regeneration allows the surface coverage of the film with surfactant of the film to remain constant, while the liquid drains out of the film.
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When the rate of descent of the soap film to the Plateau border is large, there is less time for relaxation mechanisms such as desorption and diffusion. As a result the surface tension gradient will be larger or generated faster. Enhancing the (dilational) viscosity, increasing the frame width, or pulling the film out of its horizontal Plateau border all decrease the flow of surfactant towards the Plateau border. Thus less surfactant needs to be transported back into the film and marginal regeneration is therefore less pronounced. Marginal regeneration can even be stopped this way, if the film becomes rigid or if the upward velocity equals the compression velocity. If, however, the compression velocity is increased, more surfactant must be transported back into the film. Marginal regeneration thus should become more active, and as a result, the wavelength decreases (more “feathers” per unit width). Marginal regeneration is not caused by thickness waves.
The traditional theory explains marginal regeneration from thickness variations in the film, near the vertical borders. Experiments show that the wavelength of marginal regeneration does not show the linear scaling with h 2 , which can be derived for such thickness waves. Experimentally, the thickness of the inflowing film elements is related to the original film thickness by a constant ratio of 0.8. For thickness waves it is expected that the amplitude is constant, when caused by thermal fluctuations. This is also in contradiction with the fact that marginal regeneration is more active in thick than in thin films. The marginal regeneration at the vertical borders is a secondary effect, triggered by the marginal regeneration at the lower horizontal border.
In the existing explanation marginal regeneration is mainly an exchange process at the vertical Plateau borders of the film. This is deduced from the effect of film width on drainage. Experiments show that the “feathers” of marginal regeneration at the lower border near the vertical border are sucked towards these borders. There they disturb the local equilibrium surface tension while they expand and rise and cause secondary marginal regeneration effects along these vertical Plateau borders. Since the compression at a vertical border depends on the expansion of the patches which were generated below, the compression at larger heights is much smaller. Marginal regeneration is therefore not constant and homogenous along the vertical border but decreases with increasing height above the solution. This explains why the rate of drainage in a mobile
vertical film is not dependent on the height above the horizontal border. Marginal regeneration along the vertical borders must be an exchange process of thick and thin film material only. The lower horizontal Plateau border is of much more importance and has been underestimated in the existing explanation. ACKNOWLEDGMENTS An inspiring discussion with Dr. K. J. Mysels was a starting point for some of our ideas on marginal regeneration. 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. Joye, J. L., Hirasaki, G. J., and Miller, C. A., Langmuir 10, 3174 (1994). 3. Sharma, A., and Ruckenstein, E., J. Colloid Interface Sci. 119, 14 (1987). 4. Ruckenstein, E., and Sharma, A., J. Colloid Interface Sci. 119, 1 (1987). 5. Hudales, J. B. M., and Stein, H. N., J. Colloid Interface Sci. 137, 512 (1990). 6. Hudales, J. B. M., and Stein, H. N., J. Colloid Interface Sci. 138, 354 (1990). 7. Hudales, J. B. M., and Stein, H. N., J. Colloid Interface Sci. 140, 307 (1990). 8. Stein, H. N., Adv. Colloid Interface Sci. 34, 175 (1991). 9. Stein, H. N., Colloids Surf. A: Physiochem. Eng. Aspect 79, 71 (1993). 10. Baets, P. J. M., and Stein, H. N., Langmuir 8, 3099 (1992). 11. Prins, A., in “Advances in Food Emulsions and Foams” (E. Dickinson and G. Stainsby, Eds.), p. 91. Elsevier, Amsterdam, 1988. 12. Prins, A., and van Voorst Vader, F., in “Proc. 6th Int. Cong. Surf. Activ. Subst., Zurich, 1972,” p. 441. 13. Lucassen, J., Van den Tempel, M., Vrij, A., and Hesselink, F. Th., Proc. K. Ned. Akad. Wet. Ser. B 73, 109 (1970). 14. Sonin, A. A., Bonfillon, A., and Langevin, D., J. Colloid Interface Sci. 162, 323 (1994). 15. Radoev, B. P., Scheludko, A. D., and Manev, E. D., J. Colloid Interface Sci. 95, 254 (1983). 16. Lucassen-Reynders, E. H., and Lucassen, J., Adv. Colloid Interface Sci. 2, 347 (1969). 17. Vrij, A., Discuss. Faraday Soc. 42, 23 (1966). 18. Vrij, A., and Overbeek, J. Th. G., J. Am. Chem. Soc. 90, 3074 (1968). 19. Sharma, A., and Ruckenstein, E., Langmuir 3, 760 (1987). 20. Isenberg, C., “The Science of Soap Films and Soap Bubbles.” Tieto, Clevedon, 1978. 21. Nierstrasz, V. A., Ph.D. thesis, TU Delft, The Netherlands, 1996. 22. Nierstrasz, V. A., and Frens, G., submitted. 23. Ludviksson, V., and Lightfoot, E. N., AIChE J. 14, 620 (1968). 24. Cazabat, A. M., Heslot, F., Troian, S. M., and Carles, P., Nature 346, 824 (1990).
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CS985646
Marginal Regeneration in Thin Vertical Liquid Films Vincent Adriaan Nierstrasz1 and Gert Frens Laboratory of Physical Chemistry, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands Received October 29, 1997; revised April 28, 1998; accepted May 8, 1998
Marginal regeneration is the rate-determining drainage mechanism in mobile vertical liquid films stabilized with surfactants. Mysels, Frankel, and Shinoda explained this process from (thermal) thickness fluctuations, like capillary waves. The Laplace underpressure in the Plateau border would exert a larger force on a thick, rather than on a thinner film element. This force unbalance would make film elements of different thicknesses move in opposite directions so that they are exchanged at the border. However, experiments and simulations prove that marginal regeneration cannot be the result of thickness fluctuations. Our alternative view is, that marginal regeneration is due to surface tension gradients between the film and its borders. Drainage of film elements into the lower Plateau border causes a local excess of surfactant, and thereby local differences in surface tension. This causes film elements to flow and generates the thickness differences between the absorbed and emerging film elements. The rates of the Marangoni flows reflect the surface dilational properties. This Marangoni effect is a consequence of the compression of the film surface when a film element flows into the lower Plateau border. Marginal regeneration is then a mechanism which returns the surfactant back into the film. © 1998 Academic Press Key Words: marginal regeneration; Marangoni effect; thin vertical liquid films; dynamic surface tension; surface dilational properties.
INTRODUCTION
In dilute foams the gas bubbles are present in the dispersion as spheres, and in aged, concentrated foams there are polyhedric gas volumes, separated by thin liquid films which connect by Plateau borders (Fig. 1). Studies on single isolated foam films are used to model drainage from concentrated foams (1). Some authors investigated the drainage of horizontal foam films: in the absence of gravity, capillary suction is the driving force for drainage (Joye (2), Sharma (3), and Ruckenstein (4)). Others studied vertical films (Mysels (1), Hudales et al. (5–7), Stein (8, 9), and Baets et al. (10)), where gravity and capillarity together determine the drainage rate. Our investigation belongs to the latter type. A stable, vertical film in a gravity field develops a vertical 1
To whom correspondence should be addressed. Present address: Food and Bioprocess Engineering Group, Department of Food Science, Wageningen Agricultural University, P.O. Box 8129, 6700 EV Wageningen, The Netherlands.
surface tension gradient to carry the weight of the film (J. W. Gibbs, 1878) (8, 11). This surface tension gradient causes the surfaces of the film to be immobile. The flow of the liquid can therefore be described as a laminar flow between two fixed walls under the influence of a pressure gradient. This is a very slow process that can be observed in mobile and rigid soap films when there is no marginal regeneration because of the shape of the film or because its surface elasticity, respectively. However, in a vertical soap film with mobile surfaces there is marginal regeneration; rapid turbulent motion is visible at the vertical borders and at the lower horizontal border of the soap film. In these films marginal regeneration is the most important drainage mechanism. Because of marginal regeneration these thin films drain much faster than would be expected on the basis of Poiseuille flow. So much faster, that the effect of the Poiseuille flow is negligible in the drainage of soap films with mobile surfaces. Mysels et al. (1) sought the origin in thermal or other thickness fluctuations in the film. They described the effect as exchange of thick and thin film material along the borders (“margins”) of the film, and therefore gave the process its name. In marginal regeneration complete film elements, including the surfaces and the intralamellar liquid, are rapidly sucked out of the film in exchange for thinner film material, of equal area, which are drawn out of the surrounding Plateau borders. Mysels inferred that the exchange of film patches is only effective for the draining film at the vertical borders. He observed that the rate of drainage is independent of the vertical position and decreases with increasing film width. This implies that at a given height there is no influence on the drainage rate of the amount of film higher up. Assuming that the fluctuations are constant and homogeneous along the vertical borders, this suggests a process in which thin and thick film materials (Fig. 2) of about equal area are exchanged with a rate that can only depend on local conditions, e.g. the local film thickness. The proposed mechanism did not consider the dynamics of the film itself and, especially, not of the surfaces of the exchanging film patches. For example, the possibility of surface tension gradients in the film or near the border, which could cause expansion, shrinkage, and flow of film elements, were not taken into account. The onset of marginal regeneration was explained from (thermal) thickness fluctuations in the film, as may be caused by capillary waves. The Laplace underpressure in the Plateau
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vertical Plateau border near the film, compensated by downwards flows in the border further away from the film, and it explains their observations of a reduced radius of curvature of the border at lower heights. dg h film 5 rg dx 2
FIG. 1. Schematic representation of a concentrated foam. The detail shows the film and the Plateau border.
border, which is equal to the hydrostatic pressure at the height concerned, would exert a larger force on a thick film part than on a thinner film part [1]. As a result the thin and the thick film parts near the same border will move in opposite directions, and their two surfaces and the intralamellar solution are exchanged at the border. F 5 DP Laplace z h film 5
g zh R film
[1]
More recently, Joye et al. (2), Prins et al. (12), Hudales et al. (6), Stein (8, 9), and Baets et al. (10) have studied marginal regeneration. Joye pointed out that surface tension and film thicknesses in horizontal, dimpling, films are not constant. In the barrier ring of such a dimpling film these surface tension differences and thickness differences may, under circumstances, create exchange processes in the film and the border through Mysels’ mechanism. Prins showed that marginal regeneration can be stopped by increasing the surface dilational elasticity, so that rigid soap films do not have marginal regeneration and drain very slowly indeed, but he did not introduce this observation in a more elaborate model for marginal regeneration. Baets studied how liquid viscosity and particles in the bulk liquid affect marginal regeneration and the drainage rate for films with low surface dilational elasticity. Hudales (6) and Stein (8, 9) suggested that the different drainage behaviour of horizontal and of vertical films relate to a surface tension gradient in the vertical Plateau borders. They proposed that during the process of marginal regeneration the vertical surface tension gradient in the film is carried over into the vertical Plateau borders via the film elements that are absorbed. This causes upward flows because of Marangoni effects in the
[2]
However, since it is not the flow inside the Plateau borders which determines the drainage rate but rather the exchange of film elements between the film and those borders, this cannot explain why vertical films drain more rapidly. Marginal regeneration itself, which is the rate-determining drainage mechanism, is in Hudales’ model still considered to be caused by thickness fluctuations in the film, as proposed by Mysels. Then, whereas the film patches which are sucked out of the film by the border have the (higher) surface tension at that particular height, the thinner patches which enter into the film have the (lower) surface tension of the surfactant solution. Consequently, the entering parts would expand in the film and make the fluctuations self-sustaining. In this model it is unclear how film elements that are absorbed by the Plateau border could retain their original surface tension during (rapid) compression in the process. In this paper we will try to show that surface tension gradients in the direction from a film to its borders are the cause, rather than the effect, of marginal regeneration. This new insight is important for the understanding of foam stability in relation to the physical chemistry of the bulk and the surface of the solution. The existence of surface tension gradients near the Plateau borders and their relation to marginal regeneration shows how the dynamics of adsorbed surfactant layers can stimulate liquid flow at interfaces in processes of mass transfer, like dissolution of solids or laundry of soiled textiles, which involve flow in thin liquid films and surface layers.
FIG. 2.
Marginal regeneration along the vertical Plateau borders.
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THEORY
EXPERIMENTAL
If marginal regeneration in a film with immobile surfaces, due to the vertical surface tension gradient that carries the weight of the film, is caused by local thickness fluctuations it is possible to describe these thickness fluctuations (13–16) in terms of the “squeezing mode” of (thermally excited) waves in thin films. Such modes have two characteristic features: their wavelength and their amplitude; and if they are the cause of marginal regeneration these characteristics must, somehow, be reflected in the resulting pattern of turbulence. Under some circumstances the squeezing mode in a thin liquid film can become a growing thickness fluctuation which eventually reaches a critical size and ruptures the film. Vrij (17, 18) and Lucassen et al. (13) showed that the fastest growing wave has a wavelength of
Vertical films were drawn from different sodiumdodecylsulphate (SDS, Merck, reinst) solutions, with and without added NaCl (Baker, zur analyse); N-dodecylpyridiniumchloride (DPCl, Merck, zur synthese) solutions, with and without NaBr added (Merck, reinst); azelaic acid solutions (Fluka, purris.); sodiumdodecylbenzenesulfonate (SDBS, Fluka, tech.) solutions, with and without added CsCl (Merck, reinst) or LiCl (Baker, zur analyse); or decyl-b-D-maltoside (DM, Fluka, purris.) solutions. The technique for the observation of marginal regeneration was, in principle, identical to Mysels’ (1). The films were made by vertically pulling a II-shaped glass frame out of the surfactant solution. The pulling mechanism was placed on top of a glass box in which an all-glass vessel containing the sample solution was placed. The glass box had optically polished windows. Behind the frame, black objects formed a light trap to eliminate spurious reflections. The bottom of the glass box was always covered with a layer of the solution so that the gas atmosphere was kept saturated. All solutions were left in the glass box for a day, to equilibrate at ambient temperatures between 20 and 23°C, before experiments were started. The whole apparatus stood on a vibration-free table. All glass material used, except the enveloping glass box, was cleaned between experiments by immersing it in 62% HNO3 at 75°C for at least 30 min and thoroughly rinsed afterwards with double distilled water. Films were observed in the reflected white light produced by a 50-W halogen lamp. By using an optical glass filter (Schott, SKF 11) the film could also be observed in reflected green light of wavelength 552 nm. The reflected light was monitored by a colour video camera (Sony DXC-151P) and recorded by a video cassette recorder player (Sony EVO-9500P). The advantage of this setup is that recorded experiments on marginal regeneration could be followed in slow motion, reversed in time, or observed in a frame by frame mode. Film thicknesses were obtained from the recorded interference colours. For white light, data tables are available for this purpose (20). When monochromatic light is used, film thicknesses could be obtained with help of Eq. [5], by determination of the interference order, p ( p is an integer). In the case of destructive interference, a dark minimum will occur. In the case of constructive interference, a green maximum will be observed. In the last video frames recorded at the final stage of drainage, the first dark line below the black film at the top is of order p 5 1, the second is of order p 5 2, etc. The local thickness for any element of film is then obtained as
l fastest 5 h 20 Î8 gp 3/A.
[3]
Here k is the wavenumber 2 p / l , A is the Hamaker constant, and h 0 is the mean film thickness. It would be expected that if thickness fluctuations at the border are the cause of marginal regeneration, the fastest growing wavelength for the squeezing mode at a given film thickness will be the most common one. Therefore, one might expect that marginal regeneration at the lower Plateau border is related to the characteristics of the fastest growing squeezing mode thickness fluctuations under the conditions at the border. The Hamaker constant and the surface tension are constants within one experiment, and the film thickness is well defined at the lower border. Therefore, during drainage, the wavelength for the disturbances in thickness at the lower border lexperimental should be proportional with h 20 , which can be verified experimentally. The amplitudes of thickness waves depend on the growth coefficient of the fluctuations. If the initial thickness deviation is of thermal origin, the amplitude is not a function of the film thickness, but only of the surface tension and the temperature (4, 15, 18, 19). If we assume the wavelength lfastest, we can calculate the growth coefficient for the perturbation. For a wave of length lfastest, the growth coefficient (2b) will be (13, 17, 18), 2bfastest 5
A2 96 p 2hg h 50
[4]
which implies that the (thermal) thickness fluctuation which eventually causes the instability would grow or decay at a rate proportional with h 25 . The growth rate is negligible for thickness fluctuations in thick films (h . 1 mm) and increases as the film becomes thinner. Therefore it seems unlikely that thermal fluctuations in film thickness should be the cause for marginal regeneration. It is an experimental fact that marginal regeneration is more active in thicker films and that, unlike the squeezing mode thickness waves, it disappears in films of thicknesses below 0.2 mm.
h green 5
p 2 1/ 2 , 2n cos u
h dark 5
p 2n cos u
[5]
with h 5 the film thickness, n 5 the refractive index the liquid, u 5 the angle of refraction, and p 5 the order of interference.
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FIG. 3.
Photo of marginal regeneration in a vertical liquid film.
RESULTS AND DISCUSSION
If the growth of a squeezing mode thickness fluctuation would eventually cause marginal regeneration it would be expected that the spatial disturbance of the effect (Fig. 3) would reflect this original cause; i.e. that the wavelength of marginal regeneration is related to lfastest of the squeezing mode at an original film thickness h 0 at the lower border. For determining the periodicity or the wavelength of marginal regeneration at the lower border, we counted the number of thin “feathers” per unit film width which were under formation at the same time, i.e. in a single frame of the recorded reflection image of the film. At that time the original thickness of the film h 0 at the border could also be found from the same picture. This procedure was repeated for many frames per solution and at many stages during the drainage process. The measured wavelength was averaged for the frames where h 0 at the border was the same. By this procedure we obtained a value for the wavelength l which is accurate within 10%. This wavelength of the marginal regeneration at the lower Plateau border is plotted in Fig. 4 as a function of h 20 for four different SDS solutions. From these results it is clear that the wavelength of marginal regeneration does not show the linear dependence with h 20 that should be expected if the thickness waves would cause marginal regeneration. Similar behaviour was seen when soap films of other surfactants were used, or when electrolyte was added to the solutions (data not shown). The proposition that the fastest growing squeezing mode wave in the spectrum of thermal thickness fluctuations would cause marginal regeneration at the Plateau borders of a vertical mobile surfactantcovered liquid film appears in conflict with the experimental observations.
Apart from thermal fluctuations, there can be other temporary thickness differences in a liquid film. These would tend to be damped by the Gibbs elasticity which makes the film stable. Capillary suction would then, when strong enough, again, lead to the exchange of thicker and thinner areas when they happened to be side by side along the borders of the film. Because the total surface of the film is constant this would cause equal surface areas of film material to be absorbed by and drawn from the border. However, in our recordings of draining films and marginal regeneration no equivalent area of extra thick film parts was seen to accelerate towards the lower border in exchange for the thinner feathers that are drawn into the film. Only a small “bow wave” was discernible around these rising and expanding patches of thin film material (9), but rather than being drawn in the opposite direction, these thicker film elements moved upwards with the sharply delineated thinner patches until the
FIG. 4. Wavelength of marginal regeneration vs squared film thickness for different SDS concentrations: ■, 0.5 mM; Œ, 7.5 mM; ✚, 10 mM; and }, 15 mM.
MARGINAL REGENERATION
FIG. 5. Inflow thickness vs original film thickness at lower SDS concentrations: ■, 2.5 mM; Œ, 0.8 mM; ✚, 0.6 mM; }, 0.4 mM; and ✖, 0.2 mM.
latter merged into the background of original film of the same thickness. At the lower Plateau border, most film material has the normal “original” thickness when it is absorbed into the border. There is no evidence in our observations for the equivalence in the exchange of thicker and thinner film areas, as in Mysels’ original model, moving in opposite directions because of the suction of the Laplace pressure differences. We must conclude that the role of thickness waves as the cause for marginal regeneration has been overestimated. Indeed, the net effect of marginal regeneration is the enhanced drainage of liquid from the film, but this is not caused by the exchange of thicker and thinner film elements. In videographs of marginal regeneration at the lower horizontal Plateau border it is seen that patches of thin film are drawn out of the border in a typical shape (Fig. 3). In between are other film elements, of average thickness, which are part of the overall vertical thickness profile of the draining film. The thickness of the thin, rising portions of film was measured, and compared with the “original” film thickness of the film at the lower border when the patch was originally formed there (Fig. 5). A straight line is obtained, with a slope of 0.8. Remarkably, neither the addition of NaCl (up to 340 mM), nor using other surfactants or surfactant concentrations altered this unexpected experimental result. It is a new observation about marginal regeneration that there is a constant ratio of the “new” and the “original” film thicknesses. During the drainage process of a vertical film the thickness at the lower border diminishes, but the constant ratio of newly formed and the existing film thicknesses persists. This is independent of the value of the film thickness at the lower border. Amplitudes of the thickness waves at the onset of the exchange would not be expected to vary in proportion with the film thickness itself. Both the growth coefficient of thermal thickness fluctuations in thin films and their spatial frequency would have to vary with the film thickness as shown above and in the literature (3, 8, 13). The experimental observations clearly indicate that marginal regeneration cannot be explained from thickness variations. The constant ratio rather suggests that marginal regeneration is caused by a surface tension effect. Thickness change in an
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element of a thin film is coupled to the expansion of the surface area of that film element, since the volume must remain constant. A constant ratio between the thicknesses of “old” and “new” patches of film is then an indication that the degree of surface expansion of film elements was constant, irrespective of the local film thickness itself. This would imply a constant difference in local surface coverage with surfactant molecules between the Plateau border and the film. The expansion of the parts with a high surface coverage equalises the surface tension differences. When elements of a film sink into the lower border the film becomes thicker and its surface shrinks. These surfaces carry a layer of surfactant and, as a result the surfaces of the elements, tend to become supersaturated with surfactant (Fig. 6). If the drainage into the border is fast, relative to relaxation of the (dynamic) surface tension through desorption, surface diffusion, and micelle formation, the supersaturation, as a result of compression of the surface, increases linearly with the thickness in the profile of the lower Plateau border. This supersaturation leads to disproportionation of the surfactant layer in areas with a higher and lower surface coverage, and thereby creates a mechanically unstable situation right at the transition between film and border. Border elements with their excess surface coverage have a lower surface tension. They will expand (and therefore become thinner) in the direction of the film. The surface tension gradient in the film pulls the thinner patches upward. The distance between the thin film patches which are seen rising in the film reflects the periodicity of the disproportionation process. Experimentally, we have made three new experimental observations on marginal regeneration at the lower horizontal Plateau border: —the nonlinear relation of the wavelength of marginal regeneration with the film thickness squared, —the absence of extra thick film parts in exchange for the feathers, —the constant thickness ratio between the original film and new film parts, which point to an alternative explanation of marginal regeneration as the rate-determining factor in the drainage in vertical, mobile soap films. When film elements are absorbed by a Plateau border, they carry their surface coverage with them. However, the total area of the film remains constant and
FIG. 6. Compression of a film element at the lower Plateau border (schematic).
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FIG. 7. Surface tension gradient profile at different times: ■, 0.05 ms; Œ, 1 ms; ✚, 2 ms.
therefore the loss has to be compensated by a flow of surfactant back into the film. This causes elements of normal thickness, but higher surface coverage to expand into the film, which necessarily makes them thinner. The expanding surface area of the new elements augments in the film and makes it possible for the original film to descend faster. In mechanical equilibrium at the transition from the lower Plateau border to the film, the surface tension changes rapidly from a bulk value in the border to a film value which compensates the capillary suction and the height above the liquid surface. This delicate local balance is easily disturbed when film elements become compressed as they drain into the lower Plateau border. The resulting gradients in surface tension will make the transition between border and film unstable. Eventually the differences in surface tension cause Marangoni flow of material from the border into the film. The thickness ratio of 0.8 is then reflecting a surface expansion of the supersaturated film elements along the border of 1.25. Gradients in surface tension between the border and the expanding new feathers of thinner film also explain why there are “tails” which connect the expanding thinner patches with the border. These “tails” are not the wake of ordinary flow, but Marangoni flows from the border into the film element. They are a consequence of the suction caused by the increasing surface tension during the expansion of the new film elements. Mysels, and later on Stein and Hudales, have shown that the drainage rate of a vertical film is reversibly proportional with its width. They conclude from this observation that marginal regeneration at the lower border is not important for film drainage and that the two vertical borders in a rectangular frame carry the rate-determining mechanism. Our alternative reasoning is that the expansion of new film creates extra surface, which must be absorbed by the surrounding Plateau borders. Near the vertical borders, the recordings show that newly formed film elements from the lower border move towards the vertical border while they rise. These elements expand more than those in the middle and they have a higher rising velocity. As they expand, these elements allow the film material between themselves and the vertical border to shrink and to be absorbed by the vertical border nearby. Marginal regeneration,
unlike in Mysels’ reasoning, is thus not constant and homogeneous along the vertical border. It is triggered by the disturbance below through a local compression of the film surface. As they absorb the adjacent film material the vertical borders become themselves supersaturated and act as a secondary source of thinner, expanding film elements. The role of the vertical borders then becomes the upwards transport of the surplus surface because of the process of marginal regeneration. This mechanism makes it possible for the film to descend faster into the lower Plateau border and make the process self-sustaining. The process continues as long as the rate of descent is fast enough to keep the lower borders destabilized. This explains the measured effect of film height and film width on the drainage rate (1). The expansion of fresh film elements into the original area is rate determining for the drainage of film into its borders. The drainage rate of a vertical film is fast enough to create differences in surface tension between the border and the film by supersaturating the film surface at the beginning of the transition between them. Experimentally such local gradients are difficult to prove. Therefore simulations of the draining film into the lower Plateau border were performed to obtain more information. The simulations (Fig. 7) show that, for representative values of draining rates, surface coverages, and surface relaxation mechanisms, large surface tension gradients are indeed present at the transition between the film and the lower Plateau border. Lower drainage rates and higher values for the surface dilational viscosities produce smaller surface tension gradients in the simulations. These computations (for details on the calculations see (21, 22)) show the relevance of our model which does relate marginal regeneration to the magnitude of such surface tension gradients. The surface area of a vertical film in a fixed frame must remain constant. The intensity of marginal regeneration at each moment must then reflect the drainage velocity of the film that is falling into the lower border. We measured the amount of surface area descending in the film from the velocity of descent (dx/dt) of a line of constant thickness h( x) as a function of the film thickness just above the lower Plateau border h 0 , and we related this to the amount of regenerated surface issuing from
FIG. 8. Amount of exchanging surface based on marginal regeneration: ■, 0.1 mM; ✚, 0.6 mM; ✖, 0.8 mM; and on the rate of descent in the film: Œ, 0.1 mM; }, 0.6 mM; and F, 0.8 mM.
MARGINAL REGENERATION
FIG. 9. Rate of descent ✚ compared to the upward frame velocity ■ for a 15-mM SDS film as a function of the film thickness.
the lower Plateau border during the same time interval. The amount of regenerated surface issuing from the lower Plateau border was determined from the recorded video pictures. The number of feathers and their size at the film thickness h 0 were determined. One sees that the surface area lost at the lower border because of drainage is exactly compensated for by the formation of new, thinner film elements (Fig. 8). This shows that at high drainage velocities (dx/dt) is a good estimate for the velocity of film elements of thickness h and, therefore, for the local surface velocity. However, to keep the film stable, the surface coverage must also remain constant. We see the thinner patches expanding as they rise from the lower border. Eventually they acquire the same surface tension and the same thickness as the adjacent parts of the original film. During the process it is unlikely that there is a lateral transport of surfactant from the surroundings into the expanding patch, opposite in direction from the motion of the liquid in the core of the film. Therefore, the thinner parts of the film which are created by marginal regeneration have a higher surface coverage with surfactant and a lower surface tension than their surroundings as long as they do expand. A related observation is that marginal regeneration can be prevented both at the lower and the vertical Plateau borders when the drainage rate of the vertical film is compensated by pulling the frame upwards. When the upward expansion of the film equals the rate of compression at thickness h 0 there will be no surplus of surfactant at the borders and marginal regeneration should therefore be absent. This is indeed seen in experiments (Fig. 9). Increasing the drainage velocity (i.e. the rate of descent into the lower border), either by decreasing the viscosity of the film liquid or by simply pushing the frame back into the bulk solution, or by using a smaller frame width, should result in a different wavelength, frequency, or size of the newly formed film elements in order to keep the balance of the surfactant in equilibrium. The wider the vertical frame, the lower is the lower drainage rate in a vertical, mobile liquid film of equal composition. Rate determining is the marginal regeneration. In Fig. 10 the drainage rate is seen to decrease with increasing frame width for an 15 mM SDS solution with 12.5% glycerol, which is consistent with earlier investigations (1, 6). Figure 11 gives the wavelength of marginal regeneration as a
FIG. 10.
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Drainage velocity at different frame widths: ■, 1 cm; Œ, 2.5 cm.
function of the film thickness at some different frame widths. From these data it follows that for a film of constant composition the wavelength of the marginal regeneration decreases when the drainage velocity increases. This would be unexpected for a mechanism involving local thermal fluctuations in thickness. But in our explanation of marginal regeneration it is not very surprising. At higher compression rates at the border more surfactant needs to be transported back into the film. Adjustment of the transport back into the film to the rate of absorption of surface by the lower Plateau border is through the wavelength of the instabilities which carry the back-transport mechanism (marginal regeneration) and not only through their size. There is an alternative way to reduce the drainage rate in a vertical soapfilm. Increasing the bulk liquid viscosity slows the drainage process. If the rate of drainage is lower because of an increased viscosity of the surfactant solution, a completely different result is obtained. The drainage rate decreases with increasing viscosity, as expected, but now the wavelength of marginal regeneration also decreases (Figs. 12 and 13). This observation was also made by Baets (10). This observation indicates why the instability in marginal regeneration is of a spatially periodic nature. The film surface becomes unstable at a certain supersaturation with surfactant, which then determines the ratio of 0.8 between the thickness of the new film parts and of the original film thickness. The value of 0.8 for this ratio must be a property of the two-dimensional
FIG. 11. The wavelength of marginal regeneration at different frame widths: ✚, 1 cm; }, 2.5 cm.
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FIG. 12. Drainage rate at different viscosities (mNs/m2) at 2.5-cm frame width: }, 1.0; ■, 1.4; Œ, 1.9; ✚, 4.0.
equation of state that describes the relation of the surface coverage of surfactant with the surface tension in mobile, liquid surfactant stabilized films. The transition zone of the border to the film is supersaturated with surfactant. This results in a pattern of film elements moving into and out of the border, because the supersaturated area disproportionates into regions of lower and of normal surface tension. At a given drainage rate, the amount of exchanged surfactant can be realised by more and smaller or by fewer, larger patches. The dissipation, caused by the flow of core liquid with the surfactant in the surface during the disproportionation and the expansion of the thin film, is seen to lead to the formation of more and smaller patches, instead of less and larger patches, depending on the viscosity of the liquid. At a higher liquid viscosity, the same local Dg between the Plateau border and the film, at an identical surface expansion in the film (d/d0 5 0.8), results in shorter distances between the instabilities at the border. Each feather will then transport less surfactant back into the film. This type of result seems consistent with observations on other Marangoni driven instabilities described in literature (23, 24). Finally, to realize a more rapid compression, and thus surfactant transport to the border, the frame was pushed back into the horizontal lower Plateau border. This results in a decreasing wavelength with an increasing compression velocity (0.2 or 0.9 cm/s), like when the drainage rate is varied by using a different frame width (Fig. 14).
FIG. 13. Wavelength of marginal regeneration at different viscosities (mNs/m2) at 2.5-cm frame width: }, 1.0; Œ, 1.9.
FIG. 14. Wavelength of marginal regeneration at different downwards frame velocities at 2.5-cm frame width: ■, 0.2 cm/s; }, 0.9 cm/s.
Analysis of the data in Fig. 14 shows that l varies linearly with V21/3 z l1/l2 ' (1.2 6 .2)(Vs1/Vs2)21/3. To compare these results s with the experiments with frames of different widths (Figs. 10 and 11), it is necessary to calculate the velocity in both experiments as a function of the film thickness h at the border. The velocity in a free draining film depends on hn. From a log(V)-log(h) plot, n can be determined by linear regression, which gives us the drainage velocity as a function of the film thickness. From this independent combination of experimental data we obtain l1/l2 ' (1.03 6 .1)(Vs1/Vs2)21/3. The error margin in both independent results is mainly due to the wavelength measurement, since l is rather sensitive for the presence of one thin “feather” more or less. But the results seem reasonably consistent, which proves that the drainage rate is the rate-determining cause and that marginal regeneration is the adaptable effect in a self-sustaining, surface tension driven instability. CONCLUSIONS
Our investigations on marginal regeneration lead to the following conclusions: The transition from the film to the lower Plateau border is supersaturated with surfactant. Therefore, marginal regeneration is a Marangoni-driven instability.
Experiments and simulations reveal that marginal regeneration is the result of an instability caused by a surface tension gradient at the transition from the Plateau border to the film. Film elements that are absorbed by the Plateau border when the film which “falls” into the Plateau border are compressed. Since the film elements keep their original surface coverage during compression in the border, the surface tension decreases all along the horizontal borderline. This causes a surface tension gradient opposing gravity. As the surface tension gradient increases with time, the situation becomes unstable, and thin film elements begin to expand and rise in the film as a result of the Marangoni effect. The drainage rate of vertical mobile liquid films is proportional with the rate of marginal regeneration at the lower Plateau border. The intensity of marginal regeneration is determined by the surface compression rate at the transition from the border to the film. Marginal regeneration allows the surface coverage of the film with surfactant of the film to remain constant, while the liquid drains out of the film.
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MARGINAL REGENERATION
When the rate of descent of the soap film to the Plateau border is large, there is less time for relaxation mechanisms such as desorption and diffusion. As a result the surface tension gradient will be larger or generated faster. Enhancing the (dilational) viscosity, increasing the frame width, or pulling the film out of its horizontal Plateau border all decrease the flow of surfactant towards the Plateau border. Thus less surfactant needs to be transported back into the film and marginal regeneration is therefore less pronounced. Marginal regeneration can even be stopped this way, if the film becomes rigid or if the upward velocity equals the compression velocity. If, however, the compression velocity is increased, more surfactant must be transported back into the film. Marginal regeneration thus should become more active, and as a result, the wavelength decreases (more “feathers” per unit width). Marginal regeneration is not caused by thickness waves.
The traditional theory explains marginal regeneration from thickness variations in the film, near the vertical borders. Experiments show that the wavelength of marginal regeneration does not show the linear scaling with h 2 , which can be derived for such thickness waves. Experimentally, the thickness of the inflowing film elements is related to the original film thickness by a constant ratio of 0.8. For thickness waves it is expected that the amplitude is constant, when caused by thermal fluctuations. This is also in contradiction with the fact that marginal regeneration is more active in thick than in thin films. The marginal regeneration at the vertical borders is a secondary effect, triggered by the marginal regeneration at the lower horizontal border.
In the existing explanation marginal regeneration is mainly an exchange process at the vertical Plateau borders of the film. This is deduced from the effect of film width on drainage. Experiments show that the “feathers” of marginal regeneration at the lower border near the vertical border are sucked towards these borders. There they disturb the local equilibrium surface tension while they expand and rise and cause secondary marginal regeneration effects along these vertical Plateau borders. Since the compression at a vertical border depends on the expansion of the patches which were generated below, the compression at larger heights is much smaller. Marginal regeneration is therefore not constant and homogenous along the vertical border but decreases with increasing height above the solution. This explains why the rate of drainage in a mobile
vertical film is not dependent on the height above the horizontal border. Marginal regeneration along the vertical borders must be an exchange process of thick and thin film material only. The lower horizontal Plateau border is of much more importance and has been underestimated in the existing explanation. ACKNOWLEDGMENTS An inspiring discussion with Dr. K. J. Mysels was a starting point for some of our ideas on marginal regeneration. 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.
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