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Unstable Spreading of Aqueous Anionic Surfactant Solutions on Liquid Films. Part 1. Sparingly 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 Received April 20, 2002. In Final Form: September 20, 2002 Experimental work has shown that the spreading of aqueous surfactant solutions on a thin liquid film may be accompanied by a fingering instability, the characteristics of which are dependent on the initial film thickness and surfactant concentration. Experimental results are presented showing how these parameters affect the fingering instability for a sparingly soluble anionic surfactant, sodium di-2-ethylhexyl sulfosuccinate. It is found that at surfactant concentrations well below the critical micelle concentration (cmc) stable spreading occurs but at concentrations in the vicinity of the cmc fingering develops behind the spreading front on water films as thick as 100 µm. At surfactant concentrations greater than the cmc, there is stable spreading with the formation of a central “disk” at lower thicknesses while fingering is observed at a higher thickness. Increasing the initial film thickness results in greater instability onset times and wider fingers, whereas increasing surfactant concentration gives rise to shorter onset times and narrower fingers.

1. Introduction The spreading of surfactant on a thin liquid film of higher surface tension is a process important to many household, industrial, and biomedical applications including coating flows1, ink-jet printing,2,3 and aerosol delivery of medication.4 The spreading process has received considerable attention in the literature. The experimental studies have dealt with either insoluble neat surfactants,5,6 for example, oleic acid, or surfactant solutions on both thick7 and thin8-12 liquid substrates. In this paper, the focus is on the latter case. When a surfactant solution comes into contact with a thin homogeneous liquid film, the surfactant adsorbs at the air-liquid interface creating a surfactant-rich cap surrounded by relatively uncontaminated liquid. Surface tension gradients induce shear or Marangoni stresses at the surfactant-liquid junction, which spontaneously and rapidly advance both the surfactant and the liquid toward regions of higher surface tension. Initial theoretical studies of the spreading behavior of an insoluble surfactant monolayer on a thin liquid film show that in the limit of weak surface diffusion of surfactant, a thickened rim of liquid is formed at the * To whom correspondence should be addressed. E-mail: [email protected]. (1) Schwartz, L. W.; Weidner, D. E.; Eley, R. R. Proceedings of the ACS Division of Polymeric Materials Science and Engineering 1995, 73, 490. (2) Patzer, J.; Fuchs, J.; Hoffer, E. P. Proc. SPIEsInt. Soc. Opt. Eng. 1995, 2413, 167. (3) Le, H. P. J. Imaging Sci. Technol. 1998, 42 (1), 49. (4) Shapiro, D. L. In Surfactant Replacement Therapy; AR Liss: New York, 1989. (5) Ahmad, J.; Hansen, R. S. J. Colloid Interface Sci. 1972, 38, 601. (6) Gaver, D. P.; Grotberg, J. B. J. Fluid Mech. 1992, 235, 399. (7) Joos, P.; Van Hunsel, J. J. Colloid Interface Sci. 1985, 106, 161. (8) Marmur, A.; Lelah, M. D. Chem. Eng. Commun. 1981, 13, 133. (9) Bardon, S.; Cachile, M.; Cazabat, A. M.; Fanton, X.; Valignat, M. P.; Villette, S. Faraday Discuss. 1996, 104, 10. (10) Troian, S. M.; Wu, X. L.; Safran, S. A. Phys. Rev. Lett. 1989, 62, 1496. (11) Stoebe, T.; Lin, Z.; Hill, R. M.; Ward, M. D.; Davis, H. T. Langmuir 1997, 13, 7270. (12) Frank, B.; Garoff, S. Langmuir 1995, 11, 87.

leading edge of the surfactant of height almost twice the undisturbed liquid thickness.13 To accommodate this elevation, the liquid film thins near the point of deposition further upstream14 giving rise to the liquid height profile shown in Figure 1. The spreading velocity and shape of the interfacial profile are strongly dependent on the magnitude of the initial shear stress and also on the film thickness. Gravitational effects, if significant, can cause backflow thus stabilizing the film.14 A theoretical analysis of the surfactant front with negligible gravitational and diffusion effects14,15 predicts that the spreading radius advances in time as t1/4. To validate the modeling prediction, experiments were performed using an oleic acid monolayer on glycerol films of thickness 0.4-2 mm injected with dye streaks, which verified the aforementioned theory. Film thinning was observed in the smaller film thicknesses, whereas reverse flow was observed in the thicker films where gravitational effects become significant. The spreading rate of t1/4 was confirmed.6 The fingering instability was first observed by Marmur and Lelah8 when they spread various aqueous surfactant solutions on what they assumed to be dry glass. Their observations on the sparingly soluble anionic surfactant, AOT, showed uniform, circular spreading at concentrations below the critical micelle concentration (cmc). However, spreading at concentrations above the cmc was accompanied by “fingers” of surfactant originating near the point of drop deposition. Similar experiments were conducted by Troian et al.10 using aqueous AOT solutions on a water film ranging in thickness from 0.1 to 1.0 µm in a closed environment to control evaporation effects. They found that no instability developed on a completely dry substrate and that fingering occurred both above and below the cmc with a dependence of the pattern wavelength on the initial film thickness. The authors concluded that because an underlying film and surface tension gradients were prerequisites for the fingering instability to occur (13) Borgas, M. E.; Grotberg, J. B. J. Fluid Mech. 1988, 193, 151. (14) Gaver, D. P.; Grotberg, J. B. J. Fluid Mech. 1990, 213, 127. (15) Jensen, O. E.; Grotberg, J. B. J. Fluid Mech. 1992, 240, 259.

10.1021/la0258502 CCC: $25.00 © 2003 American Chemical Society Published on Web 01/08/2003

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Figure 1. Schematic diagram of the liquid height profile, H, for a spreading surfactant solution where Ho is the undisturbed film thickness and R is the instantaneous extent of surfactant spreading. The region labeled “cap” denotes a surfactant-rich region corresponding to the original surfactant deposition.

then Marangoni stresses were an essential driving mechanism for the instability. Frank & Garoff12,16 reported seeing “dendrites” when spreading ionic and nonionic surfactant solutions on microscopically thin films in a vertical geometry. They further confirmed12 that fingered spreading cannot occur without the presence of both surface tension gradients and thin water films ahead of the advancing surfactant front, although the presence of a preexisting film is not necessary. They carried out their experiments at ambient humidity and the amount of water adsorbed onto their solid substrate was of the order of a monolayer. Cazabat and co-workers have studied the nonionic CnEm in ethylene and diethylene glycol at various relative humidities and found fingers developing on both noncoated9,17,18 and precoated19 silicon wafers. Some fingering has been reported by Stoebe et al.11 when spreading a variety of surfactant solutions on gold-covered quartz disks deposited with an organosulfur monolayer. Neat surfactant deposited on a thin liquid support can also give rise to the fingering instability. Narrow branching fingers were observed when He and Ketterson20 spread a monolayer of the insoluble ring-shaped surfactant, valinomycin, on a 1 µm water film on a vertical glass slide. Highly ramified fingering patterns were also seen when Fischer et al.21 placed a wire coated with a film of oleic acid on a 10µm layer of glycerol on a silicon wafer. They reported seeing “streamlets” appearing in the thin region ahead of the oleic acid drop. Troian et al.22 provided the first attempt at modeling the fingering instability because of spreading surfactant solutions in order to uncover the underlying mechanism responsible. They exploited the similarity between patterns produced by spreading surfactant and those in HeleShaw flow.23 Their stability analysis, conducted in the large wavenumber limit for Marangoni driven spreading at late times, allowing variation in surfactant concentration but not in film thickness, predicted that the flow was unstable. This suggested that the basic mechanism of fingering was due, as with Hele-Shaw flow, to an adverse mobility gradient. The greater layer height of the drop means that it is more mobile than the adjacent thin regions. When the more mobile drop advances into less mobile areas with constant concentration gradient, infinitesimal protrusions advance faster than neighboring regions of the front thus resulting in an unstable interface. A more rigorous stability analysis, allowing disturbances in both film thickness and surfactant concentration, revealed, however, that the flow was stable to disturbances.24,25 Inclusion of weaker capillary and diffusion forces confirmed the same result.26 A transient growth (16) Frank, B.; Garoff, S. Colloids Surf. A 1996, 116, 31. (17) Cachile, M.; Cazabat, A. M. Langmuir 1999, 15, 1515. (18) Cachile, M.; Cazabat, A. M.; Bardon, S.; Valignat, M. P.; Vadenbrouck, F. Colloids Surf. A 1999, 159, 47. (19) Cachile, M.; Schneemilch, M.; Hamraoui, A.; Cazabat, A. M. Adv. Colloid Interface Sci. 2002, 96, 59. (20) He, S.; Ketterson, J. B. Phys. Fluids 1995, 7, 11. (21) Fischer, B. J.; Darhuber, A. A.; Troian, S. M. Phys. Fluids 2001, 13, 9. (22) Troian, S. M.; Herbolzheimer, E.; Safran, S. A. Phys. Rev. Lett. 1990, 65, 3. (23) Saffman, P. G.; Taylor, G. Proc. R. Soc. London A 1958, 245, 312.

analysis was conducted to probe early time dynamics in the presence of Marangoni, capillary, and diffusion forces.27,28 This showed that an explosive growth of disturbances in the film thickness occurred on a time scale of fractions of a second. Disturbances of all wavelengths decay eventually. In the presence of van der Waals forces, the amplification of initially small transverse disturbances was enhanced leading to sustained growth consistent with experimental patterns.29 The precise mechanism for the fingering instability remains unclear and the main aim is to explore possible explanations by experimentally investigating parameters that influence the instability. In this paper, the quantitative effect of varying surfactant concentration and film thickness on the spreading exponent, onset time, and wavelength of the fingers is examined using an aqueous sparingly soluble anionic surfactant on water films up to 100 µm in thickness. This is several orders of magnitude thicker than the films used in previous experimental work on the instability. It is found that below the cmc the fingering instability is more likely to occur on thinner films. However, at concentrations significantly higher than the cmc, the reverse behavior is true. The relative strength of the forces participating in the spreading process is assessed in order to explain the observed behavior, in particular the anomalous behavior above the cmc. 2. Experimental Details 2.1. Materials. The liquid substrate was ultrapure water obtained from a Barnstead NANOpure II filter system with a resistivity of 18 MΩ cm and a surface tension of 72.2 ( 0.5mN/m at 25 °C. This was also used to clean the glassware and syringe as well as to make up the surfactant solutions. The surfactant was AOT (sodium di-2-ethyl-hexyl sulfosuccinate, MW 444.6, 99+%, Aldrich) which is a sparingly soluble anionic surfactant with a cmc of 2.5 × 10-3 M.30 An estimate of the degree of solubility of a surfactant can be obtained from a solubility parameter, β ) ka/kdHo,31 where Ka and Kd are adsorption and desorption coefficients of the surfactant respectively, the values for which were obtained from Chang and Franses.32 For AOT at concentrations well below the cmc, β is of the order of 102 which indicates the low bulk solubility of this surfactant. The surfactant was made up to the desired concentration using ultrapure water. The surface tension of the surfactant solutions was determined using a platinum Wilhelmy plate suspended from a Kruss mi(24) Matar, O. K.; Troian, S. M. In Dynamics in Small Confining Systems III; Drake, J. M., Klafter, J., Kopelman, E. R., Eds.; Materials Research Society, Boston, 1996; Vol. 464, p 237. (25) Matar, O. K.; Troian, S. M. Phys. Fluids 1997, 9, 3645. (26) Matar, O. K., Ph.D. Thesis, Princeton University, 1998. (27) Matar, O. K.; Troian, S. M. Phys. Fluids 1998, 10, 1234. (28) Matar, O. K.; Troian, S. M. Phys. Fluids 1999, 11, 3232. (29) Matar, O. K.; Troian, S. M. Chaos 1999, 9, 141. (30) Mukerjee, P.; Mysels, K. J. National Bureau of Standards, US dept. of Commerce, Washington, DC, 1970. (31) Jensen, O. E.; Grotberg, J. B. Phys. Fluids A 1993, 5, 58. (32) Chang, C. H.; Franses, E. I. Colloids Surf. A 1995, 100, 1.

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3. Results

Figure 2. Schematic diagram of experimental set up.

crobalance at a constant temperature of 25 °C. The values obtained were in agreement with published data.33 2.2. Visualization Technique. The experiments were performed in a circular glass Petri dish 15 cm in diameter with an optically flat bottom. This was positioned in a four-point adjustable level stage. The set up was illuminated from above using a fiber optic lamp, and the image was projected onto a tracing paper screen placed beneath the glass dish. A Pulnix CCD progressive scan camera (model TM6710) was used to record the images at a rate of 120 frames per second via a mirror. This entire system rested on an antivibration table to isolate the system from vibrations greater than 1 Hz. A schematic diagram of the set up is shown in Figure 2. 2.3. Experimental Procedure. To ensure complete wettability, the Petri dishes were soaked for at least 12 h in a 2% RBS 50 surfactant solution (Chemical Concentrates Ltd.) before being thoroughly rinsed with ultrapure water. They were then left in an ultrasonic bath for at least 1 h to ensure all traces of detergent were removed from the glass surface. The outside of the Petri dish was then dried as were the inside walls, leaving a continuous water film over the base only. The exact height of this liquid film was calculated from the weight of the water layer. The error in the film thickness was estimated to be no more than 4%, based on film evaporation between weighing and the start of experimentation and the curvature of the Petri dish where the side walls are fused to the base. The glassware used to make up the surfactant solutions was also cleaned in the same way as the Petri dishes. Solutions were used within 24 h to avoid a decrease in surface activity.34 A 20 µL precision Hamilton syringe was used for drop delivery. This was flushed several times with the surfactant solution to be used before the desired amount was drawn, in this case a volume of 6 µL. The Petri dish with the desired film thickness was placed on the adjustable stage. A drop of surfactant was first released from the syringe such that it hung from the tip of the needle. The apex of the drop was then contacted with the water film and was drawn across the water surface by Marangoni stresses. This method proved to be the least disturbing to the water surface. The spreading was followed for about 12s after deposition and the images analyzed using commercially available software. Each spreading run was repeated 3 times to ensure good reproducibility with complete cleaning prior to each run. (33) Williams, E. F.; Woodberry, N.; Dixon, J. K. J. Colloid Interface Sci. 1957, 12, 452. (34) Burcik, E. J.; Vaughn, C. R. J. Coll. Int. Sci. 1951, 6, 522.

3.1. Evolution of Spreading and Onset of Fingering. When a drop of aqueous AOT solution is placed on the water layer, surface deformations are apparent almost immediately. Figure 3 shows the evolution of the spreading surfactant with time. Soon after deposition a light central region is visible which corresponds to the spreading cap of surfactant, whereas the outer dark circle is the thickened rim. The dark edge of the inner circle is due to the curvature of the cap. An inferred liquid height profile is also given in Figure 3a. The thickened front continues to advance uniformly with time (Figure 3b). At a point in time, determined by the initial film thickness and surface tension gradient, fingers may start to become apparent behind the thickened front (Figure 3c). These continue to grow with time for as long as the spreading is followed. In the vicinity of the fingers, interference colors were also observed. For the chosen experimental conditions, two distinct types of spreading characteristics were noted. Either the spreading was stable or fingers developed behind the spreading front; no transition behavior was observed. Table 1 shows a summary of the observed behavior corresponding to the experimental conditions investigated in this study. This reveals that below the cmc of the surfactant, the spreading develops as a uniform, circular disk with the exception of the 0.8 cmc drop on the 25 µm film. Just above the cmc (1 < cmc < 2), spreading develops with fingering on all of the chosen thicknesses. However, at concentrations significantly above the cmc (4 cmc), uniform radial spreading is observed at 25 and 50 µm, which differs slightly from the spreading seen below the cmc. Here, only part of the deposited surfactant seems to participate in the spreading while the rest remains as a distinct surfactant-covered “disk” of liquid in the center, which retracts at late times. Interestingly, however, spreading of a 4 cmc surfactant solution on a 100 µm thick film results in fingered spreading. To investigate further, experiments were conducted on a water film thickness of 200 µm, but it was found that the finger instability no longer occurs at this thickness for any surfactant concentration. Typically, the speed of spreading is about 0.5 cm/s with the rate increasing with increasing film thickness because of reduced viscous dissipation effects. The rate also increases with increasing surfactant concentration up to 1.6 cmc and then shows a decrease with further increase in surfactant concentration. Stoebe et al.35 also reported that the spreading rate of aqueous AOT on gold-covered quartz increases with increasing concentration up to 0.5 wt % (cmc ) 0.03 wt %) and decreases with increasing concentration thereafter. Given that the radius of the spreading front, R, advances with time, t, raised to an exponent, R, the exponents for each of the spreading runs were determined from a logarithmic plot of the radius of the spreading disk against time. These results are shown in Figure 4. It is found that the spreading exponents are largely in agreement with the t1/4 theoretical prediction.14,15 Some of the exponents for the thicker films and the concentrations around the cmc seem to be closer to 0.3. This effect arises because the spreading radius became larger than the field of view in these cases because of the large spreading rates associated with this range of parameters. Thus, the complete spreading process could not be analyzed resulting in a slightly higher exponent. (35) Stoebe, T.; Hill, R. M.; Ward, M. D.; Davis, H. T. Langmuir 1997, 13, 7276.

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Figure 3. 6 µL drop of 1.6 cmc (4mM) AOT deposited on a 25 µm thick water film (a) 0.05s after deposition with inferred liquid height profile. The black line is the needle of the syringe; (b) 0.4s after deposition; and (c) 5s after deposition. In panel c) only a portion of the spreading front has been shown for clarity.

Figure 4. Variation in the spreading exponent with initial surfactant concentration and water film thickness. These points represent the average of the 3 runs, and the error bars arise from the uncertainty in the measurement of the radius of the spreading front. (b) 25, (9) 50, and (2) 100 µm. Table 1. Variation of Spreading Behavior with Initial Surfactant Concentration and Water Film Thickness initial film thickness initial surfactant concentration 0.1 mM (0.04 cmc) 1 mM (0.4 cmc) 2 mM (0.8 cmc) 3 mM (1.2 cmc) 4 mM (1.6 cmc) 10 mM (4 cmc)

25 µm

50 µm

100 µm

stable spreading stable spreading fingering stable spreading fingering fingering stable with disk fingering

Table 2. Variation in Fingering Instability Onset Time with Surfactant Concentration and Water Film Thickness initial surfactant concentration 0.1 mM (0.04 cmc) 1 mM (0.4 cmc) 2 mM (0.8 cmc) 3 mM (1.2 cmc) 4 mM (1.6 cmc) 10 mM (4 cmc)

25 µm

4.5 s 4s 3.5 s

initial film thickness 50 µm 100 µm

6s 5s

10 s 8s 6s

3.2. Characteristics of the Fingering Pattern. The fingering patterns observed are faint and difficult to image because the instability began to develop at relatively late times and at a radius of a few centimeters. Figure 2c shows that the fingers are straight and round-tipped with very little branching. This is in contrast to the spreading of AOT on very thin films as reported by Troian et al.10 where the fingers appear almost immediately on deposition at the periphery of the deposited drop and branch profusely as they grow. The onset times of the fingering instability observed in this study are presented in Table 2. These values show that the onset time decreases with increasing surfactant concentration and increases with increasing initial film thickness.

Figure 5. Variation in finger width with variation in surfactant concentration and water film thickness. ([) 1.2 and (9) 1.6 cmc. Each value is the average of 30 measurements over 3 runs. The solid lines are a power law fit to the data. The exponents are ([) 0.59 and (9) 0.72. The regression coefficients are ([) 0.96 and (9) 0.99.

Figure 5 shows the variation in finger width with initial film thickness and surfactant concentration. This figure shows a clear increase in finger width with decreasing surfactant concentration and increasing water film thickness. Previous work for the spreading of AOT on thin water films10 has also shown that increasing the film thickness resulted in broader fingering, although in that study the effect was less significant. 4. Discussion Spreading occurs because of the interplay of a variety of effects, which include Marangoni convection, viscous retardation, long-range intermolecular forces, sorption kinetics, surface and bulk diffusion, and gravitational forces. The relative strength of these is determined in order to explain the observed behavior summarized in Table 1. Grotberg and co-workers14,15 predicted that the radius of a surfactant monolayer, driven by Marangoni convection with negligible gravitational and diffusion effects, advances in time as t1/4. The spreading exponents in Figure 4 show values which are largely in agreement with this figure, thus strongly suggesting that Marangoni convection is the main driving force for the spreading at all film thicknesses and surfactant concentrations. The difference in behavior over a range of film thickness and surfactant concentration can be explained by comparing the strength of Marangoni convection in relation to other forces. The Peclet number is a measure of the relative strength of Marangoni stresses to either surface or bulk diffusive forces. The surface Peclet number, Pes, is defined as Pes ) UR/Ds where U ) (SHo/µR) is the characteristic Marangoni velocity, S is the spreading pressure, Ho is the initial film thickness, µ is the film viscosity and Ds is the surface diffusivity. The experimental parameters in this case give Pes values of the order of 106, showing that Marangoni stresses dominate over surface diffusive effects. The bulk Peclet number, Peb, is defined as Peb ) UHo/Db ) (SHo/µDb) where Db is the bulk diffusivity and  )

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(Ho/R) ,1. This is of the order of 104 suggesting that, although Marangoni convection dominates, bulk diffusive transport is more significant than surface diffusion especially on thinner films. The ratio of Marangoni to bulk diffusive time scales is given by tm/tDb ) (µR2/SHo)/ (Ho2/Db) ) 1/(Peb), where R is the radial extent of spreading. This achieves values in the range of 1-10-3 for film thicknesses in the range of 10-100 µm. In addition to bulk diffusion, sorption kinetics also play a significant role in determining the strength of the surface tension gradient at the air-water interface. The solubility parameter, β ) ka/kdHo, first defined in section 2.1, gives a measure of the relative significance of adsorption and desorption. Time scales for these can be defined as ta ) Ho/ka for adsorption and td ) 1/kd for desorption; thus, β ) td/ta. As the film thickness increases, β becomes smaller implying that the time scale for adsorption is larger than that for desorption. The surface tension gradient at the air-water interface is reduced and so is the Marangoni driving force. With increasing surfactant concentration both ka and kd increase. However, the increase in ka is greater,33 so with increasing concentration up to the vicinity of the cmc, the rate of desorption is reduced in relation to that of adsorption, allowing high concentration gradients to be maintained at the interface. These considerations are reflected in the experimental results shown in Table 1. As the surfactant concentration increases up to 1.6 cmc, fingering is more likely to occur as td/ta increases and the surface tension gradients necessary to drive Marangoni convection are maintained. As the film thickness increases, the tendency for fingering decreases as reflected by the longer instability onset times and broader fingers. Here, td/ta decreases with increasing film thickness, thus diminishing the surfactant concentration at the interface. Marangoni forces are reduced despite reduced viscous dissipation and bulk diffusion effects. However, there is an anomaly at a surfactant concentration of 4 cmc, which exhibits stable spreading with the presence of a surfactant-coated “disk” in the center on the thinner films that retracts at late times. This may be due to the fact that micelles present in the bulk at this high concentration act as monomer sources that continually replenish the interface thus eliminating surface tension gradients. The consequent reduction in Marangoni stresses may not be sufficiently strong to pull the entire surfactant cap across the liquid surface resulting in the formation of a central disk. This retracts at late times following the dilution of surfactant, which is accompanied by an increase in the local surface tension that causes the drop to retract. Troian et al.10 also found that a 10 mM (4 cmc) AOT solution did not finger on water films up to 1 µm and proposed that transport of excess surfactant from the bulk to the surface may be inhibiting the formation of surface tension gradients large enough to give rise to unstable flow. They did not, however, observe the presence of a central disk. For a thicker 100 µm film, the air-liquid interface cannot be replenished with surfactant from the bulk at a rate fast enough to suppress surface tension gradients. These gradients are therefore sustained giving rise to the fingering observed at this thickness. Matar and Troian30 proposed that van der Waals forces become significant in regions of the film which have been thinned to at least 100 nm, which give rise to the fingering instability. The dimensionless Hamaker constant, AD, which is defined as AD ) (A/6πHo2S),15 where A is the Hamaker constant, gives the relative strength of the van der Waals forces as compared to Marangoni convection . Values of the order 10-11 are obtained for the film

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thicknesses considered in this work, which appear too small to be of consequence. This is not to be wholly reflective of the significance of van der Waals forces over the entire domain because the Ho values inserted are for the initial film thickness, whereas substantial thinning of the film does occur,15 the extent of which could not be measured here. However, interference colors have been observed in the vicinity of the fingering patterns. This is a preliminary indication that film thicknesses of the order of 100 nm may be achieved in which case van der Waals interactions could possibly be significant. The relationship between finger width, λ, and the initial film thickness, Ho, can be used to provide further insight into the forces which dominate the fingering event. Two relationships have been derived using a simple scaling analysis; one for the case of dominant Marangoni forces and the other for the case of significant van der Waals interactions. If there exists a balance between Marangoni and capillary forces, then h2Γy ∼ σh3hyyy, where h denotes the local film thickness, Γ the surfactant surface concentration, σ the surface tension, and y the transverse coordinate. Assuming that variations in the local concentration gradient are small and that y ∼ λ and h ∼ Ho, where λ denotes the wavelength of a finger, then λ ∼ Ho2/3. However, if there is a balance between van der Waals and capillary forces, then A/h3 ∼ σhyy. This balance yields λ ∼ (σ/A)1/2 h2, hence λ ∼ Ho2. The results in Figure 5 show good agreement with the 2/3 scaling law, suggesting that Marangoni forces dominate the spreading. However, van der Waal forces may be significant in areas of local thinning. Gravitational forces do not appear to be significant at the film thicknesses used here. The Bond number, Bo, is defined as Bo ) (FgHo2)/S, where F is the liquid film density and g is the gravitational acceleration, and relates hydrostatic pressure to spreading pressure. This is of the order of 10-2 for a 100 µm film accounting for elevations in the film thickness and a relaxation of the concentration gradients. 5. Conclusions In an attempt to understand the mechanisms that give rise to the fingering instability, a study has been conducted to investigate quantitatively the effects of varying surfactant concentration and film thickness on the characteristics of the fingering pattern for film thicknesses from 25 to 100 µm. The fingering patterns for the sparingly soluble anionic surfactant solution, AOT, are faint with round-tipped, straight fingers. It is found that the fingering instability is more likely to occur with decreasing film thickness and increasing concentration up to the vicinity of the cmc, in which case the fingers are narrower and longer with shorter instability onset times. At very low surfactant concentrations, the surface tension gradient is not sufficiently large to generate Marangoni flows strong enough to initiate the fingering instability. On thin films and high concentrations (up to and around the cmc), the time scale for adsorption is smaller than that for desorption thus maintaining the surface tension gradients at the air-water interface which give rise to Marangoni forces strong enough to initiate the fingering instability. Anomalous behavior is observed at a surfactant concentration significantly above the cmc. Here, the thinner films show stable spreading, whereas the thicker film develops fingers. It is possible that sorption kinetics are different at this concentration because of the large numbers of micelles present in the bulk. These may

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where R denotes the surfactant “activity”. If σ ) σm, then Γ ) Γm, and S can be written as

σo - σm ) RΓm ) S

(5)

Substituting (5) into (2) yields Figure 6. Schematic representation of the initial conditions of the spreading process: a surfactant drop of surface tension, σm, resting on an intially undisturbed film of thickness, Ho, and surface tension σo. Here, r and z represent the radial and vertical coordinates and U a characteristic velocity.

act as monomer sources that replenish the interface with surfactant thus wiping out the surface tension gradients and diminishing Marangoni stresses on thinner films. On thicker films, the replenishment rate is slower; surfactant concentration gradients are maintained so Marangoni stresses are sufficiently strong to give rise to fingers. Bulk diffusion effects were found to be more significant than surface diffusion particularly on thinner films, while gravitational effects appear to be weak. The role of longrange intermolecular forces needs to be investigated further, but preliminary experimental observations suggest that there may be sufficient thinning of the liquid film to allow significant van der Waal interactions. Acknowledgment. We are grateful to Dr. Anne Dussaud (Unilever) for invaluable advice concerning our experimental set up and to the EPSRC for funding of this project. Appendix Derivations of the spreading exponents are presented using simple scaling arguments. It is shown that the radius of a spreading front of insoluble surfactant from a finite source spreads in time as t1/4 for the case of Marangoni dominated spreading. When gravitational effects become more significant, however, this exponent decreases to 1/5. Consider a drop of surfactant on a thin liquid film as illustrated in Figure 6. σm represents the surface tension of the surfactant-covered liquid, whereas σo represents the surface tension of the uncontaminated film. Marangoni Dominated Spreading (see also Jensen and Grotberg15 and Espinosa et al.36). The Marangoni stress balance at the air-liquid interface is given by

∂u ∂σ )µ ∂r ∂z

(1)

where σ is the local surface tension, u is the tangential velocity, and µ is the film viscosity. A simple scaling of (1) gives

U S ∼µ R Ho

(2)

in which R denotes the radial extent of the drop, U a characteristic velocity, and Ho the initial film thickness. The spreading pressure, S, is given by

S ) σ o - σm

(3)

RΓm U ∼µ R H

(6)

The total mass of surfactant in the spreading drop, M, is

M)

∫0∞ 2πrΓ dr ∼ R2Γm

(7)

which yields the following simple relation for Γm

Γm ∼

M R2

(8)

Substituting (8) into (6) yields

U RM ∼µ 3 H R o Now, U )

dR dt

(9) (10)

hence substituting (10) into (9) and rearranging gives

RMHo dR ∼ R3 µ dt

(11)

Integrating (11) gives

4RMHo t ∼ R4 µ

(12)

so that the following power-law relation for Marangonidriven spreading is obtained:

R ∼ t1/4

(13)

Gravitationally Dominated Spreading. The equation of momentum conservation in the lubrication approximation is given by

∂2u ∂p )µ 2 ∂r ∂z

(14)

in which the pressure, p, can approximated as follows in the case of significant gravitational effects:

p ∼ FgHo

(15)

Here, F denotes the density and g the gravitational acceleration. A simple scaling of (14) then yields

µUR Fg

(16)

µUR3 RM

(17)

Ho3 ∼ whereas from (9), we obtain

Ho ∼

Expressing surface tension as a function of surfactant surface concentration, Γ

Substituting (17) into (16) together with U ) dR/dt yields

σ ) σo - RΓ

(36) Espinosa, F. F.; Shapiro, A. H.; Fredberg, J. J.; Kamm, R. D. J. Appl. Physiology 1993, 75, 2028.

(4)

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Afsar-Siddiqui et al.

3

dR (RM) ∼ R4 dt Fgµ2

(18)

hence, the following power-law relation for gravitationally dominated spreading is obtained:

R ∼ t1/5

Integration of (18) yields

R∼

(

)

25 (RM)3 Fgµ2

1/10

t1/5

(19) LA0258502

(20)