Unstable Spreading of Aqueous Anionic SurfactantSolutions on Liquid Films. 2. Highly Soluble Surfactant

Abia B. Afsar-Siddiqui, Paul F. Luckham, and Omar K. Matar*

Department of Chemical Engineering & Chemical Technology, Imperial College of Science,Technology & Medicine, London, SW7 2BY, UK

Received April 20, 2002. In Final Form: September 20, 2002

The spreading of a surfactant solution across a thin water film may be accompanied by a fingeringinstability developing behind the spreading front. In this paper, the role of solubility on this instabilityis investigated by conducting spreading experiments using highly soluble surfactant solutions of sodiumdodecyl sulfate over a range of concentrations on water films ranging from 25 to 100 µm in thickness. Itis found that at surfactant concentrations up to and around the critical micelle concentration, spreadingis largely accompanied by fingers upstream of the spreading front. In comparison with sparingly solubleAOT solutions (sodium di-2-ethylhexyl sulfosuccinate) studied in part 1 of this series, the instability becomesapparent sooner and the fingers are more pronounced and branched. Above the cmc, the instability takesa different form on thinner films, which was not noted in the sparingly soluble case, while fingeringdevelops on thicker films.

1. Introduction

There has been a significant amount of both theoreticaland experimental work carried out in the field of spreadingsurfactant solutions on thin liquid films, which is moti-vated because of applications including drug delivery1 andcoating flows.2 When surfactant solution comes intocontact with a thin homogeneous liquid film, the surfactantadsorbs at the air-liquid interface, leaving a surfactant-rich region surrounded by relatively uncontaminatedliquid. The Marangoni stresses induced at the surfactant-liquid junction advance both the surfactant and theunderlying liquid toward regions of higher surface tension,thus causing deformations in the liquid layer. The strengthof the Marangoni stresses is proportional to the surfacetension gradient and the thickness of the underlying film.

In part 1 of this study,3 the case of insoluble/sparinglysoluble surfactant was considered, where a shocklikestructure develops at the leading edge of the spreadingfront of height almost twice the undisturbed film thicknesstogether with corresponding thinning upstream.4,5 If thesurfactant is soluble and the solid wall beneath the liquidlayer absorbs surfactant, then the continual absorptionof surfactant at the wall reduces surfactant concentrationgradients at the interface, thus reducing spreading ratesand decreasing deformations. At long times there may bea region in which backflow arises even with negligiblegravitational effects.6 In the case of a soluble surfactantwhere the solid wall is impermeable to surfactant (thecase considered in this paper), the surfactant will desorbfrom the surface to the bulk until both bulk and surfaceconcentrations are in local equilibrium.

The behavior of the front is dependent on the sorptionkinetics. If desorption is rapid, then an advancing pulseof fluid develops instead of the shocklike structure of the

insoluble case. The height of this pulse can be in excessof 3 or 4 times the undisturbed film thickness.7 As thesolubility of the surfactant increases, the upstream slopeof the pulse becomes increasingly steep. The downstreamslope is less affected by solubility and so is similar to theinsoluble case. If desorption is slow, initially the surfactantwill spread as in the insoluble case. Then as desorptionbegins to occur preferentially from regions of high surfacesurfactant concentration to the bulk, spreading rates arereduced. Once surface and bulk concentrations are in localequilibrium, the pulse of fluid develops.7 Surface defor-mations in the soluble case are therefore more severe thanin the insoluble case, as shown in Figure 1. While solubilityaffects the flow patterns, it does not have a significanteffect on the spreading exponent and the t1/4 predictionmade by Grotberg and co-workers from the insoluble studyremains valid.7,8

Marmur and Lelah9 first reported seeing the fingeringpatterns accompanying the spreading of various aqueoussurfactant solutions on what they believed to be dry glass.Their observations on the anionic surfactant SDBS showeduniform circular spreading at concentrations below thecritical micelle concentration (cmc). However, spreadingat concentrations above the cmc was accompanied by“fingers” of surfactant originating near the point of originaldeposition, which appeared to branch as they developed.Frank and Garoff10 observed the development of fingerswhen a negatively charged silicon oxide substrate wasbrought into contact with a reservoir of sodium dodecylsulfate (SDS) solution in a vertical geometry. The fingerswere seen to propagate several millimeters up the samplebefore stopping. When spreading the same solution on apositively charged sapphire substrate, however, auto-phobing occurred. Cazabat and co-workers11-14 studiedthe nonionic CnEm surfactants in ethylene and diethylene

* Corresponding author. E-mail: o.matar@ic.ac.uk.(1) Shapiro, D. L. In Surfactant Replacement Therapy; AR Liss: New

York, 1989.(2) La Due, J.; Muller, M. R.; Swangler, M. J. Aircraft 1996, 33, 131.(3) Afsar-Siddiqui, A. B.; Luckham, P. F.; Matar, O. K. Langmuir

2003, 19, 696.(4) Borgas, M. E.; Grotberg, J. B. J. Fluid Mech. 1988, 193, 151.(5) Gaver, D. P.; Grotberg, J. B. J. Fluid Mech. 1990, 213, 127.(6) Halpern, D.; Grotberg, J. B. J. Fluid Mech. 1992, 237, 1.

(7) Jensen, O. E.; Grotberg, J. B. Phys. Fluids 1993, 5, 58.(8) Jensen, O. E.; Grotberg, J. B. J. Fluid Mech. 1992, 240, 259.(9) Marmur, A.; Lelah, M. D. Chem. Eng. Commun. 1981, 13, 133.(10) Frank, B.; Garoff, S. Langmuir 1995, 11, 87.(11) Bardon, S.; Cachile, M.; Cazabat, A. M.; Fanton, X.; Valignat,

M. P.; Villette, S. Faraday Discuss. 1996, 104, 307.(12) Cachile, M.; Cazabat, A. M. Langmuir 1999, 15, 1515.(13) Cachile, M.; Cazabat, A. M.; Bardon, S.; Valignat, M. P.;

Vadenbrouck, F. Colloids Surf. A 1999, 159, 47.

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glycol over a range of relative humidity. They observedprofusely branching fingers at the edge of the surfactantdrop. Other groups who have reported seeing fingeringwith surfactant solutions include Nikolov et al.15 andStoebe et al.,16 who both used aqueous trisiloxane sur-factant solutions on various substrates.

There have been several studies modeling the fingeringinstability in an attempt to understand the drivingmechanism. Most recently, a transient growth analysis ofinsoluble surfactant spreading was conducted to probeearly time dynamics in the presence of Marangoni,capillary, and diffusion forces.17,18 This showed that anexplosive growth of disturbances in the film thicknessoccurred on a time scale of fractions of a second. Distur-bances of all wavelengths decay eventually. In the presenceof van der Waals forces, the amplification of initially smalltransverse disturbances was enhanced, leading to sus-tained growth consistent with experimental patterns.19

Having investigated the effect of varying surfactantconcentration and initial film thickness on the behaviorof a spreading drop of a sparingly soluble anionic sur-factant, AOT,3 this study reports on experimental resultsusing a highly soluble anionic surfactant solution, SDS,to understand the role of solubility. As before, surfactantconcentrations above and below the cmc are deposited onwater films up to 100 µm in thickness, and the effect ofthis on the instability onset time and pattern wavelengthis examined.

2. Experimental Details

2.1. Materials. The liquid substrate was ultrapure waterobtained from a Barnstead NANOpure II filter system with aresistivity of 18 MΩ cm and a surface tension of 72.2 ( 0.5mN/mat 25 °C. This was used to clean the glassware and syringe aswell as to prepare the surfactant solutions.

The surfactant used was SDS (sodium dodecyl sulfate, MW288.4, 99+%, Aldrich), which is a highly soluble anionic surfactantwith a cmc of 8 × 10-3 M.20 The solubility parameter, â ) ka/kdHo,7 where Ka and Kd are adsorption and desorption coefficientsof the surfactant respectively, is on the order 10-2 at a range ofconcentrations below the cmc.21 This indicates the high bulksolubility of SDS. The surface tension of the surfactant solutionswas determined using a platinum Wilhelmy plate suspended

from a Kruss microbalance at a constant temperature of 25 °C.The values obtained were in agreement with published data.20

2.2.VisualizationTechnique. All the spreading experimentswere carried out in a glass Petri dish fitted with an optically flatbottom. This was illuminated from above using a fiber optic lamp,and the resulting image was projected onto a tracing paper screenplaced beneath the Petri dish. The images were recorded usinga 120 Hz CCD progressive scan camera (Pulnix TM6710) via amirror. The setup has been described in greater detail previously3

and is illustrated in Figure 2.2.3. Experimental Procedure. After cleaning the Petri dish

with RBS50 detergent (Chemical Concentrates Ltd.) and thor-ough rinsing with ultrapure water, the water film thickness wasevaluated gravimetrically as previously described.3 A 9-µL dropof aqueous surfactant was contacted with the water film usinga clean 20-µL precision Hamilton syringe. The spreading wasfollowed for about 4 s after deposition, and the images wereanalyzed using commercially available software. Each spreadingrun was repeated three times to ensure good reproducibility withcomplete cleaning prior to each run.

3. Results

The spreading of aqueous SDS solutions over a widerange of concentrations on water films gives rise to threedistinct types of behavior, which are presented in Table 1.

At very low surfactant concentrations, spreading isstable. Increasing the surfactant concentration up to andaround the cmc results in fingered spreading. At still

(14) Cachile, M.; Schneemilch, M.; Hamraoui, A.; Cazabat, A. M.Adv. Colloid Interface Sci. 2002, 96, 59.

(15) Nikolov, A. D.; Wasan, D. T.; Chengara, A.; Koczo, K.; Policello,G. A.; Kolossvary, I. Adv. Colloid Interface Sci. 2002, 96, 59.

(16) Stoebe, T.; Lin, Z.; Hill, R. M.; Ward, M. D.; Davis, H. T.Langmuir1997, 13, 7270.

(17) Matar, O. K.; Troian, S. M. Phys. Fluids 1998, 10, 1234.(18) Matar, O. K.; Troian, S. M. Phys. Fluids 1999, 11, 3232.(19) Matar, O. K.; Troian, S. M. Chaos 1999, 9, 141.(20) Mukerjee, P.; Mysels, K. J. National Bureau of Standards, US

Dept. of Commerce, Washington, DC, 1970.(21) Chang, C. H.; Franses, E. I. Colloids Surf. A 1995, 100, 1.

Figure 1. Schematic diagram of the liquid height profile, H, for (a) an insoluble spreading surfactant solution and (b) a solublespreading surfactant solution where Ho is the undisturbed film thickness and R is the instantaneous extent of surfactant spreading.Note that in the soluble case, the upstream slope is steeper and the leading edge resembles a pulse.

Figure 2. Schematic diagram of the experimental setup.

Table 1. Variation of Spreading Behavior with InitialSurfactant Concentration and Water Film Thickness

spreading behaviorinitialsurfactant

concn (mM) 25 µm 50 µm 100 µm

0.32 (0.04 cmc) stable spreading3.2 (0.4 cmc) fingering stable6.4 (0.8 cmc) fingering9.6 (1.2 cmc) fingering

12.8 (1.6 cmc) “disk” instability fingering22.4 (2.8 cmc) “disk” instability fingering32 (4 cmc) “disk” instability fingering

704 Langmuir, Vol. 19, No. 3, 2003 Afsar-Siddiqui et al.

higher concentrations, above the cmc, two different typesofbehaviorareobserved,dependingonthe initial thicknessof the underlying film. On the thinner films, a “disk” ofliquid remains in the center at the point of originalsurfactant deposition from which protrusions extend.Increasing the film thickness results in the fingeringinstability.

Typically the speed of spreading is about 0.5 cm/s.Spreading rates increase with increasing concentrationup to 1.2 cmc and decrease thereafter. There is no greatvariation in spreading rates with film thickness, despitereduced viscous dissipation effects at the higher thick-nesses. Given that the radius of the spreading front, R,advances with time, t, raised to an exponent, R, theexponents for each of the spreading runs were determinedfrom a logarithmic plot of the radius of the spreading frontagainst time for the same time duration in each case.These results are shown in Figure 3. The spreadingexponents are broadly in agreement with the t1/4 predictionmade by Grotberg and co-workers.5,8

3.1. Evolution and Onset of Fingering Instability.Figure 4 shows the evolution of the spreading drop andthe onset and development of the fingers with time. Almostimmediately after deposition, the thickened front can beseen as a dark ring, which initially spreads uniformly(Figure 4a). At a time which depends on the initialsurfactant concentration and film thickness, a lighter frontbecomes visible behind the thickened rim and it is fromthis thinned region that small protrusions grow anddevelop (Figure 4b). They appear to be pushing the darkfront, which becomes distorted. The fingers grow andbranch (Figure 4c), and at later times, the thin region isbarely discernible (Figure 4d). The bright, white regionbeyond the spreading front appears to be an optical effect.

The time taken for the fingers to become visible as afunction of the surfactant concentration and initial filmthickness is presented in Table 2. The fingering instabilityonset time decreases with increasing surfactant concen-tration and with decreasing film thickness. Note that theonset times for SDS are an order of magnitude faster thanthose observed for AOT.3

In addition to the instability onset time, the averagewidth of the fingers has been measured using imageanalysis software on developed profiles. The fingers onthe thinner films were found to be broader at the base andnarrower at the tips. The measurement taken is that atthe tip. The variation in finger width with initial filmthickness and surfactant concentration is presented inFigure 5. There is an increase in finger width withdecreasing surfactant concentration and increasing filmthickness.

Figure 6 shows qualitatively the difference in spreadingpatterns that arise at different film thickness when asurfactant solution of 1.2 cmc SDS is deposited on a waterfilm. With increasing film thickness, the fingers becomeincreasingly round-tipped, shorter, straighter, and wider.The distance between the tips of the fingers and thethickened front is smaller and the front itself is morecircular and less corrugated. A similar effect is seen withdecreasing surfactant concentration, as shown in Figure7.

3.2. Evolution of “Disk” Instability. Surfactantdeposition at concentrations above the cmc (1.6, 2.8, and4 cmc) on 25 and 50 µm films gives a different type ofspreading behavior. Figures 8 and 9 show the evolutionof this instability for 1.6 cmc on a 25 µm water film and2.8 cmc on a 50 µm water film, respectively. Figure 10shows side views for the same parameters as those usedto generate Figure 9.

In the fingering case described in section 3.1, thereremains no remnant of the original surfactant deposition.For the cases depicted in Figures 8 and 9, however, acentral “disk” or cap remains in the center, which is clearlyshown by the side views in Figure 10. Protrusions thenappear to extend from the edge of this disk, the shape andonset time of which seem to be determined by thesurfactant concentration and film thickness. These pro-trusions are long and straight on a thinner film, whilethey appear sooner and much shorter on the thicker film.In this respect, the protrusions behave in a fashion similarto the fingers. However, growth of fingers leads tocorrugation of the thickened front (see Figure 4b), whereasthe protrusions do not affect the spreading front, whichremains circular (see Figures 9 and 10). The disk itselfdoes not grow appreciably with time; the spreading front,however, does grow. The early time pattern in Figure 8abears some resemblance to patterns obtained by Fournierand Cazabat,22 who observed forklike dendrites at thebulk edge of a water-ethanol mixture, which gave rise toa solutal Marangoni effect. There is a darker zone at thebulk edge that is interpreted as a depression in the liquidsurface. The side views in Figure 10 do not readily indicatethat there is a depression; it is possible that the darkerzone results from the curvature at the edge of the lens.

4. Discussion

The spreading exponents shown in Figure 3 are in goodagreement with the t1/4 prediction.5,8 This indicates thatspreading is driven primarily by Marangoni convectionat all film thicknesses and surfactant concentrations.However, the qualitative difference in behavior over therange of concentrations and thickness is due to thestrength of Marangoni stresses in relation to other forces.

The experimental results show that at very low sur-factant concentrations spreading is stable. This is becausethesurface tensiongradient is too lowtogeneratesufficientMarangoni flows to initiate the fingering instability. Athigher concentrations up to the cmc, Marangoni stresses

(22) Fournier, J. B.; Cazabat, A. M. Europhys. Lett. 1992, 20, 517.

Figure 3. Variation in the spreading exponent with initialsurfactant concentration and water film thickness: (b) 25 µm,(9) 50 µm, (2) 100 µm. These points represent the average ofthe three runs, and the error bars arise from the uncertaintyin the measurement of the radius of the spread drop.

Table 2. Variation in Fingering Instability Onset Timewith Surfactant Concentration and Initial Film

Thickness

onset time (s)initialsurfactant

concn (mM) 25 µm 50 µm 100 µm

0.32 (0.04 cmc)3.2 (0.4 cmc) 0.125 0.176.4 (0.8 cmc) 0.10 0.14 0.159.6 (1.2 cmc) 0.075 0.12 0.13

12.8 (1.6 cmc) 0.1222.4 (2.8 cmc) 0.1132 (4 cmc) 0.11

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are strong enough to cause fingering, which is reflectedby the shorter instability onset times and narrower, morebranched fingers. With increasing film thickness, thefingers are broader and straighter and have a longer onsettime. This can be explained by considering the effect ofsorption kinetics.3 Although sorption kinetics constantsdo not change with concentration up to the cmc (both kaand kd increase at the same rate21), the solubilityparameter, â ) ka/kdHo (see section 2.1), decreases withincreasing film thickness, implying that the time scalefor adsorption is larger than that for desorption. The

increased rate of desorption, therefore, diminishes themagnitude of the Marangoni driving force on thicker filmsand exerts a stabilizing influence on the spreading process.This may counteract the decrease in viscous drag con-tributing to the observed weak dependence of the spread-ing rates on the film thickness.

At concentrations above the cmc, the reverse trendoccurs, with fingering occurring on the thicker films anda different instability developing on the thinner films. Atthese concentrations the sorption kinetics are different,because of the large number of micelles in the bulk. This

Figure 4. The development of the fingering pattern when a 9-µL drop of 1.2 cmc (9.6 mM) SDS is deposited on a 25 µm waterfilm: (a) formation of the thickened front at 0.067 s, (b) onset of fingering from the thinned region at 0.14 s, (c) development ofthe fingers at 0.31 s, and (d) fully developed fingers at 2 s. Corresponding side views are shown in parts e, f, g, and h, respectively.

Figure 5. Variation in finger width with variation in surfactantconcentration and water film thickness: ([) 1.2 cmc, (9) 1.6cmc. Each value is the average of 30 measurements over threeruns. The solid lines are a power law fit to the data. Theexponents are ([) 0.61 and (9) 0.63. The regression coefficientsare 0.99 for both sets of data.

Figure 6. Fingering patterns produced after 0.5 s when a 9-µLdrop of 1.2 cmc (9.6 mM) SDS is deposited on (a) a 25 µm waterfilm and (b) a 100 µm water film.

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results in a higher rate of surfactant adsorption from thebulk to the interface. This reduces the magnitude of thesurface tension gradients, which is reflected in the lowerspreading rates observed above the cmc. However, onthicker films the surface cannot be replenished rapidlyenough to suppress Marangoni flow, even at these highconcentrations, so fingering occurs at this thickness. Thereasons for the instability at the edge of the “disk” are notyet understood.

The surface and bulk Peclet numbers, showing therelative strength of Marangoni stresses to surface or bulkdiffusion, are on the order of 106 and 103 respectively.

While confirming the dominance of Marangoni forces,these also indicate that bulk diffusive transport isrelatively more significant than surface diffusion (moreso on thinner films).

Gravitational forces are characterized by the Bondnumber, which relates hydrostatic pressure to spreadingpressure. This is on the order of 10-3 for the thinner films,rising to ∼0.2 on the 100 µm film at the lower surfactantconcentrations, accounting for the elevations in filmthickness and the relaxation of the concentration gradi-ents. The fact that significant variation in spreading rateswas not found with film thickness, despite reduced viscousretardation on thicker films, could also be brought aboutby gravitational forces, which reinforce the effects ofsurfactant desorption in counterbalancing the reducedviscous effects on thicker films. Moreover, spreadingexponents for surfactant concentrations up to the cmc onthe 100 µm film were found to be closer to 0.2. The exponenttakes this value when gravitational effects become sig-nificant (derived in part 1 of this study3), thus furtherindicating that these effects may be significant on thethicker films, particularly at the lower concentrations.

The dimensionless Hamaker constant, which gives therelative strength of van der Waals forces to Marangoniconvection, achieves values on the order of 10-11. Whilethis does not appear to be significant, it does not accountfor the substantial degree of thinning that can occur inthe film. It was previously demonstrated3 that in the caseof dominant Marangoni forces, the finger width, λ, isrelated to the initial film thickness, Ho, by the followingsimple relation derived using scaling analysis: λ ∼ Ho

2/3.In the case of significant van der Waals forces, this relationbecomes λ ∼ Ho

2. The results shown in Figure 5 show goodagreement with the 2/3 scaling law, indicating that thespreading is Marangoni-driven. However, this does notrule out the possibility of van der Waals interactions.

Comparison of Spreading Behavior of SDS withAOT3 on Water Films. The spreading behavior of boththe highly soluble SDS and the sparingly soluble AOTsolutions on water films shows many similarities in trendswith some notable differences. Upon deposition, a centralcap is visible at the point of deposition at early times forAOT, whereas there is no remnant of surfactant depositionin the case of SDS. Spreading rates are on the same orderfor both surfactants and exhibit a maximum around thecmc. However, while spreading rates increase withincreasing film thickness due to reduced viscous effectsfor AOT, no such variation is seen with SDS. This isattributed to rapid surfactant desorption and relativelysignificant gravitational effects at larger film thicknessesfor SDS, which counterbalance the reduced viscousdissipation.

Fingering occurs over a broader range of concentrationsin the case of SDS, with fingering patterns apparent ata surfactant concentration of 0.4 cmc, while in the caseof AOT fingering is not seen until 0.8 cmc for all valuesof the film thickness considered. Fingers developing as aresult of SDS deposition are more pronounced, with earlier

Figure 7. Fingering patterns produced after 0.5 s when a 9-µLdrop of (a) 0.4 cmc (3.2 mM) and (b) 1.2 cmc (9.6 mM) SDS isdeposited on a 25 µm water film.

Figure 8. Spreading pattern produced after (a) 0.08 s and (b)3.5 s when a 9-µL drop of 1.6 cmc (12.8 mM) SDS is depositedon a 25 µm water film.

Figure 9. Spreading pattern produced after (a) 0.1 s and (b)3.5 s when a 9-µL drop of 2.8 cmc (22.4 mM) SDS is depositedon a 50 µm water film.

Figure 10. Side view of a 9-µL drop of 2.8 cmc (22.4 mM) SDS on a 50 µm water film after (a) 0.07 s and (b) 2 s.

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onset times (<0.2 s), and show branching, while fingersdue to AOT deposition are faint, straight, and show onsettimes of more than 3 s. However, the trends of onset timeand finger width with surfactant concentration and filmthickness are the same in both cases.

Above the cmc on the thinner films, a central “disk” ispresent as a remnant of the surfactant deposition in bothcases, which is attributed to diminished Marangonistresses as a result of surface replenishment of surfactant.At a higher film thickness, fingering occurs in both cases.However, in the case of SDS, protrusion are seen to extendfrom the “disk”, while in the AOT case, the “disk” remainsstable but retracts at long times.

5. ConclusionsTo investigate the role of solubility on the fingering

instability, a series of experiments have been conductedusing the highly soluble surfactant SDS. The effect ofvarying surfactant concentration and film thickness upto 100 µm on the characteristics of the fingering patternshas been investigated, and comparisons have been drawnbetweenthisandapreviousstudyusingasparinglysolublesurfactant solution, AOT.3

The fingeringpattern isverypronouncedwithbranchingfingers, a highly corrugated leading edge, and short onsettimes, more so on thinner films and higher concentrations

up to the cmc. This suggests that fingering is enhancedby a large concentration gradient and an initially thinfilm. Surface diffusion appears to be weak, while bulkdiffusion (on thinner films) and gravitational effects (onthicker films) are nonnegligible. However, Marangonieffects dominate the spreading process.

For the case of spreading at concentrations above thecmc on 25 and 50 µm films, a “disk” remains in the centerafter surfactant deposition. This feature may be due to areduction in the magnitude of surface tension gradients,driven by surface replenishment from the bulk to thesurface. At the edge of the disk an instability develops,the reasons for which remain unclear. For a 100 µm thickfilm, fingering is observed. This may be due to a reductionin the rate of surfactant replenishment for thicker filmsresulting inMarangoni forces,whicharesufficientlystrongto initiate fingering.

Deposition of SDS solution on a water film results inmore unstable spreading in comparison to AOT deposi-tion.

Acknowledgment. We are grateful to Dr. AnneDussaud (Unilever) for invaluable advice concerning ourexperimental setup and to the EPSRC for the funding ofthis project.

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