Immiscible surfactant droplets on thin liquid films: Spreading dynamics, subphase expulsion and oscillatory instabilities

Journal of Colloid and Interface Science 364 (2011) 519–529

Contents lists available at SciVerse ScienceDirect

Journal of Colloid and Interface Science www.elsevier.com/locate/jcis

Immiscible surfactant droplets on thin liquid films: Spreading dynamics, subphase expulsion and oscillatory instabilities David K.N. Sinz, Myroslava Hanyak, Anton A. Darhuber ⇑ Mesoscopic Transport Phenomena Group, Department of Applied Physics, Eindhoven University of Technology, Postbus 513, 5600 MB Eindhoven, The Netherlands

a r t i c l e

i n f o

Article history: Received 19 May 2011 Accepted 21 August 2011 Available online 31 August 2011 Keywords: Surfactant spreading Immiscible liquids Thin liquid films Oscillatory instability Marangoni stresses

a b s t r a c t After deposition of immiscible, surface-active liquids on thin liquid films of higher surface tension, Marangoni stresses thin the liquid film around the surfactant droplet and induce a radially outward flow. We observed an oscillatory instability, caused by temporary trapping and subsequent release of subphase liquid from underneath the surfactant droplet. Height profiles of the thin liquid films were monitored using optical interferometry and fluorescence microscopy, both in the vicinity of the deposited surfactant droplet and at larger distances. Numerical calculations based on the lubrication approximation are compared to the experimental results. Good agreement between the experimental and calculated far-field dynamics and values of the spreading exponents was found. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction Surfactants are used in a large number of environmental or technological processes ranging from pulmonary drug delivery [1], printing and coating processes [2,3], to enhanced oil recovery [4]. The spreading of surface-active liquids on deep liquid layers has been studied intensively, mainly in the context of the remediation of oil spills [5–19]. In contrast to deep liquid layers, surfactant spreading on thin liquid films induces pronounced modulations in the height profile of the subphase film [20–47]. Troian et al. [22] as well as Grotberg and co-workers [21,24,25] reported the formation and propagation of a rim near the surfactant leading edge as well as substantial film thinning in the proximity of the deposited surfactant droplet. Based on linearized sorption kinetics and fast vertical diffusion across the film thickness, Jensen and Grotberg introduced a model for the spreading of soluble surfactants and identified qualitative differences in the flow patterns induced by differences in surfactant solubility [27]. Similarity solutions of surfactant driven flow problems were discussed by Jensen [28]. A fingering instability is often observed near the perimeter of spreading droplets and fronts of surfactant solutions [22,23,34– 48]. A mathematical analogy to the Saffman–Taylor instability was described by Troian et al. who performed a linear stability analysis assuming insolubility [23]. Furthermore a transient growth model for the fingering instability was presented by Matar

⇑ Corresponding author. E-mail address: [email protected] (A.A. Darhuber). 0021-9797/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2011.08.055

and Troian [36]. Cachile and Cazabat [38] reported a profound influence of relative humidity and surface heterogeneities on the instability dynamics. The prime origin of the instability was identified, by Warner et al., as the presence of adverse mobility gradient regions at the leading edge of the spreading drop [44]. A comprehensive review of surfactant spreading dynamics was recently given by Matar and Craster [58]. Previous investigations primarily focused either on soluble surfactants or on the influence of Marangoni-stresses on the evolution of thin liquid films distant from the location of the initial surfactant deposition. In this manuscript, we elucidate the ‘near field’ dynamics and the two-phase flow character at the interface of an immiscible surfactant droplet and a thin subphase film. We provide evidence that, in the case of immiscible surfactants, the fingering instability [22,23,34–48] is preceded by the temporary trapping of subphase liquid and subsequent release from underneath the surfactant droplet, which induces a pronounced increase of the observed far-field spreading rate. Using fluorescence microscopy we observed an oscillatory instability of the three-phase contact line during this subphase expulsion process, which advances in two distinct stages. In an initial ‘global’ phase, liquid is expelled in a concerted fashion around a large fraction of the perimeter of the surfactant droplet. In a second ‘local’ stage, the expulsion proceeds primarily in one or few distinct locations breaking the approximate azimuthal isotropy of the initial stage. In Section 2 we describe the sample preparation, the experimental setup and procedures. Experimental results for stable and unstable spreading dynamics, and the influence of the subphase expulsion on the spreading dynamics are presented in Sections 3.1–3.3. Sections 3.4–3.6 focus on the dynamics in the vicinity of

520

D.K.N. Sinz et al. / Journal of Colloid and Interface Science 364 (2011) 519–529

the three-phase contact line of the surfactant droplet. In Section 4 we outline the theoretical model for the spreading dynamics of insoluble surfactants on thin liquid films along with its numerical implementation. Numerical results are presented in Sections 4.2 and 4.3. 2. Experimental setup Fig. 1a shows a schematic of the spreading process. At time t = 0 a droplet of surfactant (denoted ‘A’) is deposited on a uniformly flat liquid film (labeled ‘Subphase B’), with surface tension c0. Since the surface tension of the subphase covered with surfactant cm is lower than c0, Marangoni stresses (indicated by the arrows in Fig. 1b) are induced, which drive away the subphase liquid from the vicinity of the surfactant droplet. We conducted two types of experiments. First we studied the ‘far-field’ deformation and displacement of the subphase liquid after

ho

2R o

A

(a)

Subphase B Substrate

z r

A

(b)

B

Undisturbed film B

(c)

1 mm

Rim

surfactant deposition using highly reflective Si substrates and interference microscopy. Subsequently we focused on the ‘near-field’ subphase dynamics underneath and in the immediate vicinity of the deposited surfactant droplet using transparent glass substrates and fluorescence microscopy. Details of the two experimental configurations are described below. All experiments were conducted at room temperature T  20 °C under ambient conditions.

2.1. Interference microscopy studies of subphase displacement dynamics Liquid (‘subphase’) films with a thickness between 0.2 and 10 lm were deposited on Si wafers with a diameter of 4 in. by means of spin-coating, which ensures a flat and uniform height profile of the films. The wafers were cleaned by rinsing with trichloroethylene, acetone and isopropanol as well as immersion in a mixture of hydrogen peroxide and sulfuric acid at a temperature of 75 °C prior to the experiments. Two sets of subphase and surfactant materials were used, the properties of which are summarized in Table 1. An insoluble material combination was oleic acid (Fluka, 99% purity, product number 75090) on anhydrous glycerol (Fluka, >99.5% purity). The second set consisted of polydimethylsiloxane (PDMS, Fluka) on pentaphenyltrimethyltrisiloxane (PPTMTS, Dow Corning). After spin-coating, small droplets of surfactant were deposited on the liquid films by using either a piece of Al wire with diameter 300 lm as a dip-pen or a Microdrop droplet-on-demand inkjet system with a capillary nozzle diameter of 70 lm. The droplet volumes ranged from 0.2 to 100 nl. The time evolution of the subphase surface profiles was measured by interference microscopy with an upright Zeiss AxiotechVario microscope and a bandpass filter centered around a wavelength k = 645 nm. A typical snapshot is shown in Fig. 1c. The experiments have been performed in a laminar flowhood in order to eliminate dust contamination.

2.2. Fluorescence microscopy studies of subphase expulsion dynamics

Crater

A

1 mm

Incline

(d)

Surfactant A

Fig. 1. (a) At time t = 0 a surfactant droplet (‘A’) of radius R0 is deposited on a flat and uniform film of liquid (labeled ‘Subphase B’) of thickness h0 and higher surface tension c0. (b) Due to the surface tension imbalance, Marangoni stresses in the liquid–air interface promote the spreading of a surfactant monolayer, which drives the subphase film radially outwards. (c) Top-view optical interference micrograph of an oleic acid droplet 163 s after deposition on a film of glycerol. Image width 3.6 mm. (d) Fluorescence microscopy image of a glycerol film (h0 = 70 lm) containing fluorescein 12.5 s after deposition of an oleic acid droplet. The image (width 3.48 mm) was recorded through a transparent glass substrate and contrastenhanced.

Rectangular borosilicate glass microscope cover slides (Gold Seal, product number 3334) with dimensions of 48 mm  60 mm and thickness between 0.13 and 0.17 mm, were used as substrates. Prior to deposition of the subphase liquid in each experiment, the substrates were repeatedly cleaned using a solution of hydrogen peroxide (30%, J.T. Baker, product number 7047) and sulfuric acid (95%, J.T. Baker, product number 6057), with a volume ratio of 1:1, and subsequently by use of an ozone cleaner. After the cleaning procedure, liquid films of anhydrous glycerol (purity 99%, Sigma Aldrich, product number 49767) containing 0.5 wt% of fluorescein sodium salt (Sigma Aldrich, product number 46960) were deposited on the substrates. Tensiometric measurements using a Wilhelmy plate showed no detectable change in surface tension due to the addition of fluorescein. The films were deposited via spin coating ensuring uniform and reproducible films of thickness h0 = (70 ± 2.5) lm. Table 1 Density q, viscosity l, surface tension c0 and refractive index n20 D of the subphase materials glycerol [49] [C3H5(OH)3] and pentaphenyltrimethyltrisiloxane (PPTMTS), as well as the surfactants oleic acid [50,51] [cis-9-octadecenoic acid, C18H34O2] and polydimethylsiloxane (PDMS) used in our experiments. The values for glycerol depend somewhat on the water content. Material

q (kg/m3)

l (mPa)

c0 (mN/m)

n20 D

Glycerol Oleic acid PPTMTS PDMS

1261 895 1090 950

1400 27.6 191 1000

63.4 32.5 31.2 21.2

1.474 1.459 1.578 1.404

D.K.N. Sinz et al. / Journal of Colloid and Interface Science 364 (2011) 519–529

After deposition of the subphase liquid, small quantities of the insoluble surfactant cis-9-octadecenoic acid (oleic acid, purity 99%, Sigma Aldrich, product number O1008) were deposited in the center of the substrates using a Hamilton 7000.5SN micro-syringe as a dip-pen. The dynamics following the surfactant deposition were monitored by means of fluorescence microscopy using an Olympus IX71 inverted microscope. The samples were illuminated using a Thorlabs LED light source (product number M455L2-C1)

5

(a)

521

and an Olympus U-MGF-PHQ fluorescence filter cube. The recorded grayscale values correspond to fluorescence intensity, which increases with subphase film thickness. Oleic acid is a surface-active substance that is practically insoluble in glycerol [25]. Gaver and Grotberg [25] have measured the relation between surface tension c and the surfactant surface concentration C for oleic acid on glycerol. The surface tension c drops rapidly from the value for the pure subphase c0 = 63.5 mN/m and asymptotes into cm = 39 mN/m beyond a concentration Cc = 3.5 ll/m2. The maximum spreading pressure Pmax  c0  cm is therefore determined as 24 mN/m. A fit to the reported experimental data according to the function c = cm + Pmaxexp(AC2) results in a fit parameter A = 0.5 m4/ll2.

4

3. Experimental results

ho

3

3.1. Stable surfactant spreading dynamics

s

(b)

1814

2

273 s

t = 17 s

Height h(r,t) [μm]

rmax

1

1 mm 0

0

2

4

6

r [mm] 6

(c)

Deposition area

8.4

Ridge radius rmax [mm]

4

.17

α=0

0.5 mm

(d)

6.1

2

6

0.2

1

rmax ~ tα

6 0.2 α=

ho = 1.3 μm

5

0.2

(e)

0.6 10

Deposition area 1 mm

100

1000

Time t [s] 2

(f)

Ridge radius rmax [mm]

t = 20s

Fig. 1c shows the surface profile of a glycerol film after deposition of a small droplet of oleic acid (labeled ‘A’). The light and dark concentric rings are due to optical interference of the light reflected from the Si wafer and the glycerol–air interface. They trace contour lines of equal thickness, where consecutive dark or light fringes correspond to a vertical spacing Dz = k/2nB  219 nm. In the immediate vicinity of the surfactant droplet, which is marked ‘crater’ region in Fig. 1c, the glycerol film is strongly thinned to a thickness below about 50 nm. At larger radial distances the height profile increases in a ramp-like fashion (labeled ‘incline’ in Fig. 1c) and peaks in a rim, beyond which it decreases and asymptotes into the undisturbed subphase film thickness h0. No detectable instability occurred in the experiment with a very small droplet of oleic acid and a small subphase thickness depicted in Fig. 1c. When a larger oleic acid droplet with a diameter of 2R0 = 1.04 mm was deposited on a thicker glycerol film (h0 = 70 ± 2.5 lm), however, a pronounced finger formation around the surfactant droplet was visible (Fig. 1d). The unstable flow is due to the release of subphase liquid initially located beneath the droplet. The qualitative difference in the observed dynamics is attributed to two factors: (1) the difference in the amount of subphase liquid initially located beneath the droplet and (2) the increased capillary pressure exerted by a smaller droplet onto the underlying film. Fig. 2 presents experimental results for oleic acid spreading on thin films of glycerol. In Fig. 2b, a snapshot of the spreading process is shown. The black dot in the center is the oleic acid droplet, which maintains a finite contact angle and hence does not spread except immediately after deposition. The white ring around the oleic acid is a strongly thinned region, which merges into the incline characterized by the concentric optical interference fringes. In contrast to other groups we never observed any film rupture or dewetting

3 1

√h o

R

0.5 0.7

fit

+

S fit

0.25

~h o

1

10

Film thickness [μm]

Fig. 2. (a) Measured height profiles h(r, t) for oleic acid spreading on a 3 lm thin glycerol layer at times t = 17, 273 and 1814 s. (b) Optical micrograph of oleic acid spreading on glycerol at t = 106 s. Image width 3.05 mm. (c) Time dependence of the rim radius rmax(t) for three different values of h0. The straight lines correspond to power law relations rmax  ta. (d) Optical micrograph of the deposition area for the second data set in (c) (red squares) short after deposition (t = 21 s), the deposited droplet depleted in the course of the experiment causing a reduction in the observed spreading exponent. (e) Optical micrograph of the deposition area and propagating rim for the same data set after droplet depletion (t = 812 s). (f) Rim position rmax(t = 20 s) as a function of h0. Experimentally and numerically obtained values are shown by black and red squares respectively.p The ffiffiffiffiffi black and red solid lines correspond to the function r max ðt ¼ 20 sÞ ¼ Rfit þ Sfit h0 with fit parameters Rfit and Sfit. The green dotted line corresponds to a power law relation 0:25 r max ðt ¼ 20 sÞ  h0 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

522

D.K.N. Sinz et al. / Journal of Colloid and Interface Science 364 (2011) 519–529

1 mm

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

1 mm

2 mm

Fig. 3. (a–c) Stable spreading of PDMS on PPTMTS. Images recorded (a) 65 s, (b) 165 s and (c) 325 s after droplet deposition. Image width 3.0 mm. (d–f) Unstable spreading of PDMS on PPTMTS. Images recorded (d) 74 s, (e) 254 s and (f) 664 s after droplet deposition. Image width 3.0 mm. (g–i) Unstable spreading of oleic acid on glycerol. Images recorded (g) 32 s, (h) 82 s and (i) 182 s after droplet deposition. Image width 5.8 mm.

phenomena around the surfactant droplet [25,26]. Fig. 2a depicts height profiles extracted from the location of the interference fringes at three different times during an experiment. The vertical arrows in Fig. 2a label the radial position of the rim maximum rmax. The peak height hmax of the advancing rim at early times is about 60% higher than the asymptotic film thickness h0. The rim height markedly decreases and the rim width increases with time. In Fig. 2c, the rim position rmax is plotted as a function of time for three different values of h0. To good approximation, the rim position exhibits power law behavior rmax(t)  ta with exponents around a = 0.25. The second curve (h0 = 6.1 lm) corresponds to an experiment where the deposited droplet of oleic acid was depleted after about 300 s, as indicated by the vertical line. From that moment on, the deposited oleic acid droplet was completely redistributed into a monolayer spreading on the glycerol film. As a consequence, the concentration around r = 0 was no longer constant and the effective spreading exponent decreased, indicating the transition from an effectively infinite to a finite surfactant volume. The spreading following the depletion can be approximated by a power law relation with an exponent of a = 0.17, indicated by the dashed black line. Fig. 2d and e shows interference images of the deposition area before and after surfactant depletion, respectively. The position of the small oleic acid droplet is marked in both images by the red arrows. The data presented in Fig. 2c are somewhat misleading inasmuch as a rather wide range of spreading exponents a between about 0.23 and 0.4 has been measured, which will be elucidated in Section 3.2. Experiments performed not immediately after spin-coating showed a systematic decrease of a by about 4–8% per hour as well as a reduction in the rim height hmax.

Fig. 2f presents the dependence of rmax(t = 20 s) on the initial film thickness h0. The scatter in the experimental data (black squares) is mainly due to variations in the surfactant droplet radius R0. A scaling argument for the dependence of rmax on h0 follows from equating the rim propagation rate with the Marangoni velocity in the region beyond the crater area, r > Rc, where the film thickness is of order h0

drmax h0 Dcrim ðtÞ 1=2  ! rmax ðtÞ  Rc þ Sh0 : dt lðrmax  Rc Þ

ð1Þ

Here, we have assumed that the expansion rate of the crater dRc/dt is much smaller than the rim propagation rate drmax/dt. Here, Dcrim  c(rmax)  c(Rc) is the (time-dependent) surface tension difference between the edge of the crater and the rim position. The function S, which stems from the time-integration of Dcrim/l and is, thus, influenced by the film evolution in the crater region, determines the spreading exponent a. The open symbols in Fig. 2f correspond to the results of numerical simulations (for parameter values R0 = 165 lm, poffset = 0 and C0 ¼ 1), which will be discussed in Section pffiffiffiffiffi 4.3. The solid lines correspond to the function r max ¼ Rfit þ Sfit h0 with fit-parameters Rfit and Sfit. For comparison, the dotted line corresponds to a pow0:25 erlaw rmax  h0 . Both the experimental and the numerical data are very well approximated by the scaling relation Eq. (1) with Rfit = 255 lm, but slightly different prefactors Sfit. This difference can most likely be attributed to the fact that the numerical simulations were based on the nominal viscosity of pure, anhydrous glycerol, whereas the viscosity in the experiments decreased due to water uptake from the humidity of the ambient atmosphere.

523

D.K.N. Sinz et al. / Journal of Colloid and Interface Science 364 (2011) 519–529

3.2. Instability induced by immiscible surfactants A well-known fingering instability occurs during the spreading of surfactant solutions on thin liquid films [22,23,34–48,52]. In Ref. [52] it was observed that the front of the spreading surfactant droplet first steepens and develops a rim, which divides into fingers with an initially well defined wavelength. These fingers undergo tip-splitting, which is the onset for the formation of a rather irregular morphology including branches and multiple unstable spreading fronts. In this manuscript, we do not consider solutions but instead droplets of effectively immiscible surfactants. If the subphase and surfactant are not or only sparingly miscible, the interface between the surfactant droplet and the thin liquid film underneath has to be taken into account. We consider two material systems in this section (1) oleic acid, which maintains a finite contact angle on the subphase glycerol and (2) PDMS on PPTMTS, which wets the subphase completely. Both subphases completely wet the solid substrates used in the experiments. If the surface-active liquid spreads on the subphase film as in the case of PDMS on PPTMTS, a volume of subphase material becomes essentially permanently trapped underneath the spreading surfactant droplet. This phenomenon is clearly visible in Fig. 3a–c due to the large difference in the refractive indices of PDMS and PPTMTS. The PDMS surrounds the trapped PPTMTS volume around its entire perimeter and the spreading process is completely stable. Fig. 3d–f shows unstable spreading of PDMS on PPTMTS. During deposition of the PDMS droplet the dispensing wire was moved laterally by a fraction of a mm, thus exposing the underlying PPTMTS in one location. A finger of subphase liquid formed that subsequently branched into a series of fingers. When the spreading of the PDMS droplet cut off the supply of PPTMTS, the branching stopped and the fingers gradually thinned, widened and vanished. In contrast to PDMS, which appears to completely wet PPTMTS, oleic acid is partially wetting and maintains a finite contact angle on glycerol. Using a holographic interferometer we measured the morphology of oleic acid droplets of varying size below the capillary length on thin films of glycerol, deposited on borosilicate glass microscope cover slides. We found that the liquid–air interface of the droplets maintains the shape of a spherical cap with an effective contact angle of h  9.5 ± 0.5°. An analogous instability to the one described for PDMS on PPTMTS is observed primarily for large surfactant droplets and thick subphase films, where glycerol trapped beneath the oleic acid is driven forward and develops fingers that subsequently tip-split and branch out (see Fig. 3g–i). Due to the finite contact angle, the material supply is not cut off by the spreading of the surfactant droplet and the fingers continue to evolve until the trapped subphase material is depleted. On thin subphase films (h0 < 10 lm) the expulsion often does not affect the entire perimeter of the surfactant droplet (Fig. 3g— i), whereas on thicker films an initial stage of expulsion along the entire droplet perimeter is observed (Fig. 1d and Fig. 4b and c). In both cases, axisymmetry is not maintained. A net force acting on the surfactant droplet results, which can induce shape deformations and swaying. Fig. 4 shows the trajectory of the center of a droplet of oleic acid on a thin film of glycerol. After the trapped glycerol is depleted, the jiggling motion stops and the droplet perimeter assumes a circular shape. 3.3. Expulsion induced modification of the spreading dynamics Fig. 5a illustrates the effect of the afore described subphase expulsion on the spreading dynamics of the surfactant. The ridge position rmax(t) exhibits two distinct phases as a consequence of pronounced subphase expulsion. Initially (t < 150 s) the ridge

0.5 mm

0.2 mm

(b)

(a)

(c)

Fig. 4. (a) Oleic acid droplet deposited on a thin film of glycerol. The white circles denote the trajectory of the center of the surfactant droplet. Neighboring symbols correspond to a time difference of 8 s. Image width 1.06 mm. (b and c) Spontaneous shape deformation of an oleic acid droplet deposited on a thicker film of glycerol (h0 = 30 lm). Illuminating light is passband limited around k = 550 nm. Time difference between frames 0.1 s. Image width 0.49 mm.

(b)

(a)

500 μm

(c)

500 μm

Fig. 5. (a) Experimental time evolution of the rim position rmax(t) for h0 = 2.9 lm, revealing two clearly distinguishable spreading stages due to subphase expulsion. Initially the ridge radius increases as rmax  t0.26 (indicated by the black solid line) while for later times (t > 150 s) rmax  t0.40 holds (as indicated by the red dashed line). (b and c) Microscope images of the (b) initial (t = 34 s) and (c) later (t = 194 s) stages of the experiment. In (c) the expelled liquid is clearly visible and undergoes a fingering instability. The solid white lines indicate the position at which the ridge radius is measured, the blue arrows mark the data points corresponding to the images. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

radius increases as rmax  t0.26 (indicated by the fitted black solid line) while for later times (t > 150 s) the increase can be approximated by rmax  t0.40 (indicated by the fitted red dashed line). Fig. 5b and c shows microscope images of the early and late stage respectively. While in Fig. 5b only a very small and locally expelled amount of subphase liquid is visible, in Fig. 5c the expelled liquid is clearly visible and covers most of the crater region around the surfactant droplet. The expelled liquid in the crater region enhances the transport of surfactant from the droplet to the rim and causes the increase in the spreading rate in Fig. 5a at around t = 150 s. The experiment shown in Fig. 5 exhibits two distinct spreading phases due to delayed subphase expulsion. Other experiments exhibited expulsion directly after deposition which resulted in an overall increased exponent as mentioned in Section 3.1 rather than two distinct phases of spreading with different exponents.

524

D.K.N. Sinz et al. / Journal of Colloid and Interface Science 364 (2011) 519–529

3.4. Subphase expulsion dynamics near the three-phase contact line In the experiments reported in the previous sections, rather thin films (h0 < 10 lm) were used; the expulsion of subphase liquid and the associated occurrence of the fingering instability was observed only occasionally. In this section, we consider thicker films with h0  70 lm, for which subphase expulsion and flow instabilities were always prominent. We found that the expulsion process can be divided into two stages. In the first stage the expulsion occurs in a concerted fashion along a large fraction of the perimeter of the surfactant droplet. This ‘global’ expulsion is associated with a discernible motion of the three-phase contact line of the droplet. Fig. 1d shows a fluorescence microscopy image of the global expulsion stage. The ‘global’ expulsion phase appears to be reminiscent of the spontaneous oscillatory deformation of a sessile oil lens containing a volatile surfactant on an aqueous subphase, which has recently been studied by Stocker and Bush [57]. In their system, however, the subphase layer thicknesses were approximately 100–1000 times larger, the subphase viscosity was three orders of magnitude smaller than ours and evaporation of the surfactant was a prerequisite for the occurrence of the oscillations. The material system we study, oleic acid and glycerol, can be considered nonvolatile at our experimental conditions. The sustained motion and pronounced deformations of droplets of immiscible liquids on comparably deep aqueous subphase layers as a consequence of surfactant transport have been investigated by Sumino et al. [59,60] and Chen et al. [61]. In the second or late stage, the expulsion occurs in a localized fashion at one or a few positions around the perimeter. An example is shown in Fig. 6, which also illustrates the associated oscillatory deformation of the three-phase contact line of the surfactant droplet. Videos of both processes are provided as Supplementary Information. In both stages, the spreading of the expelled liquid is unstable and undergoes a fingering instability. The graininess of the images in Fig. 6 is not due to camera noise, but a consequence of an emulsification of glycerol as the dispersed phase inside the surfactant droplet of oleic acid as the continuous phase. Further information on this emulsion formation is provided as Supplementary Material. 3.5. Global expulsion Due to the expulsion of liquid located under the surfactant droplet, the corresponding fluorescence intensity decreases as time

progresses. During the stage of global expulsion, the rate of this decrease depends on the size of the deposited surfactant droplet. This is illustrated in Fig. 7a, where the temporal evolution of the fluorescence intensity I(t) in the center of various deposited surfactant droplets is shown. In each experiment, the fluorescence intensity I(x, y, t) was integrated over a circular region with diameter equal   to the droplet radius r 6 12 R0

Z

IðtÞ ¼

R0 =2

R0 =2

Z pffiffiffiffiffiffiffiffiffiffiffiffiffi R20 =4x2

Iðx; y; tÞdydx: pffiffiffiffiffiffiffiffiffiffiffiffiffi 2



ð2Þ

R0 =4x2

The droplets in the experiments presented in Fig. 7a had diameters of 2R0 = 0.88 mm (red triangles), 2.17 mm (blue circles) and 3.76 mm (black squares). It is evident that the rate of of the intensity decrease is higher for the smaller droplets. At the end of the global expulsion phase t = Dt, the concerted motion of the threephase contact line ceases and the rate of decrease of the integrated fluorescence intensity is reduced as indicated by the green vertical lines. Fig. 7b shows the dependence of Dt on the radius R0 of the deposited surfactant droplet. The straight line corresponds to the scaling relation Dt  R20 , which can be derived by considering the volume of subphase liquid expelled in the global stage Vgl. Let d be the fraction d  Vgl/V0 relative to the quantity initially trapped under the droplet and in an analogous way dI the ratio of the integrated intensity prior to and following the global expulsion stage dI  I(Dt)/I(0). Fig. 7a indicates that dI  0.4 is independent of the droplet radius R0. The initial film height h0 was kept constant in the experiments. Moreover, the film profiles underneath the droplet to good approximation exhibited geometrical similarity in the integration region with respect to different R0. For these two reasons we conclude that d  Vgl/V0, i.e. the fraction of expelled subphase liquid, is essentially independent of R0. We assume no-slip at the solid–liquid interface z = 0 and no stress at the subphase/droplet interface, i.e. l@vr/ @zjz=h  0, which is admissible owing to the large ratio of subphase and drop viscosity l/ldrop  1. The flow velocity in radial direction then follows as v r ¼  @p zð2h  zÞ=2l. Assuming axi@r symmetry, the rate of volume loss of subphase liquid V_ is then given by

V_ ¼

Z

h

Z 2p

0

0

 3 2pR0 h @p v r R0 du dz ¼   3l @r 

:

ð3Þ

r¼R0

200 μm

200 μm

(a)

(b)

Fig. 6. Fluorescence microscopy images of a localized expulsion process (image width 632 lm). The three-phase contact line of the surfactant droplet oscillates forward and backward in the radial direction. (a) Contact line in its most retracted and (b) in its extended position. A corresponding video is provided as Supplementary Material.

525

D.K.N. Sinz et al. / Journal of Colloid and Interface Science 364 (2011) 519–529

deposited droplet. If – consistent with the experimental finding in Fig. 7a – also d is presumed independent of R0, then the scaling Dt  R20 results, which agrees well with the experimental data shown in Fig. 7b. 3.6. Local expulsion

(a)

(b)

Fig. 7. (a) Temporal evolution of the fluorescence intensity integrated over the region r 6 0.5R0 underneath the deposited surfactant droplet for 2R0 = 0.88 mm (red triangles), 2.17 mm (blue circles) and 3.76 mm (black squares). Vertical lines indicate the end of the global expulsion phase. (b) Duration of the global expulsion stage Dt as a function of surfactant droplet radius R0 as determined by image analysis. The solid line corresponds to the scaling relation Dt  R20 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

In the later stages of the experiments, localized expulsion phenomena are frequently observed, where liquid is visibly expelled from underneath the surfactant droplet only at a certain position along the droplet perimeter. This goes along with an oscillatory deformation of the droplet three-phase contact line as shown in the fluorescence microscopy images in Fig. 6. The conformation of the contact line in its most retracted position is shown in Fig. 6a and in Fig. 6b for its most extended state. Fig. 8a shows a fluorescence microscopy image of a local expulsion zone. The rectangle with length 100 lm indicates the zone over which the intensity was averaged over 10 pixels normal to the n-direction as well as the zero position of the n-axis. The averaged fluorescence intensity hIi(n, t) is presented in Fig. 8b. Each profile has been normalized with the saturation intensity of the camera and progressively shifted along the ordinate axis by a constant amount for clarity. The local maxima in the hIi(n, t) profiles, approximately in the range n = 80–100 lm, are caused by emulsion droplets inside the drop of oleic acid. The dotted red line traces a pronounced sequence of local maxima in hIi(n, t), as illustrated by the lightblue arrow in Fig. 8a. We ascribe them to a meniscus of subphase liquid forming at the position of the three-phase contact line ncl. Fig. 8c and d shows that ncl undergoes an oscillatory motion, where the retraction phase occurs much faster than the extension phase, as evident from the asymmetric peak shapes in Fig. 8d. The gaps between the individual parts of the curve correspond to times in which no distinct maximum is detectable. We consistently observed the receding motion (positive slope in Fig. 8c and d) to be faster than the advancing motion (negative slope in Fig. 8c and d). 4. Numerical model for subphase redistribution induced by insoluble surfactants 4.1. Governing equations and boundary conditions

Integrating the volume flux V_ over the duration of the global expulsion stage Dt yields the scaling relation for the expelled volume

V gl ¼ dpR20 h0 ¼

Z 0

Dt

3

_  2ph0 R0 @p ðR0 ÞDt: Vdt @r 3l

ð4Þ

The expulsion duration Dt scales therefore as

Dt

3l 3

2ph0 R0 @p ðR0 Þ @r

dV 0 ¼

3l 3

2ph0 R0 @p ðR0 Þ @r

dpR20 h0 :

ð5Þ

Subphase liquid underneath the surfactant droplet is subject to the capillary pressure of the surfactant droplet, since the curvature of the liquid–liquid interface is negligible compared to the curvature of the liquid–air interface for thin subphase films h0  R0. For small R0 < ‘c this pressure is large compared to hydrostatic pressure contributions. Assuming that the relevant length scale for the pressure gradient at the three-phase contact line of the droplet is independent of the droplet diameter 2R0, it scales as @p  cROA0 , with cOA denoting the liquid–air interfacial tension of the @r

We performed model calculations for the axisymmetric spreading of insoluble surfactants on thin viscous films by coupling the lubrication equation for thin film flow with the equation for surfactant surface transport by convection and diffusion [54,55] and an equation of state (EOS). The latter provides information on the equilibrium relation between the surface tension c and the surfactant surface concentration C. This equilibrium relation is expected to be valid for the dynamic surface tension as well, i.e. c only depends on the local surfactant concentration, but not e.g. the velocity profile or previous system configurations. We introduce the   h=h0 ; following non-dimensionalized variables: r  r=R0 ; h C  C=Cc , as well as

t  t=t M  t h0 Pmax lR20

 and p

pR20 h0 Pmax

;

ð6Þ

where p is the augmented pressure and Cc is introduced below. The timescale tM is chosen as the ratio of R0 and the Marangoni velocity h0s/l  h0Pmax/(lR0). Using the scales introduced above, the following system of coupled equations can be derived [22–24]

526

D.K.N. Sinz et al. / Journal of Colloid and Interface Science 364 (2011) 519–529

(a)

ξ = 0 µm

100 µm

" #  1 @ r h 2 @ c  @ C 2 3 @ p  @h ¼ 0; þ  r h @r 3 @t r @r 2 @ C @r    2  2 @ p  r @ C @C 1 @  @c rhC  r Ch þ  ¼ 0; @r Pes @r @r 2 @t r @r !   @ @h c r  ¼ Bo h ; p  r @r @r h  i. c c ¼ ¼ cm þ Pmax exp AC2 Pmax ;

ð7Þ ð8Þ ð9Þ ð10Þ

Pmax

Contact line

(b)

where   h0 =R0 ; Bo  qgR20 =Pmax ; A  AC2c is a constant, Pes  h0 Pmax/(lDs) is the Peclet number for surface transport and Ds is the surface diffusion constant. The EOS Eq. (10) was fitted to experimental data of Gaver and Grotberg [25], which yielded the fit parameters cm ¼ 39 mN=m; A ¼ 6:125 and Cc = 3.5 ll/m2. The boundary conditions are

  @h @p ð0; tÞ ¼ 0; ð0; tÞ ¼ 0; @r @r  r r ; tÞ ¼ 1; p ðr r ; tÞ ¼ 0; hð

Normalized intensity (a.u.)

@C  ð0; tÞ ¼ 0; @r Cðr r ; tÞ ¼ Cd ;

ð11Þ ð12Þ

where r r is the right boundary of the computational domain ½0; rr . The initial conditions for the surfactant distribution used in the numerical simulations are sketched in Fig. 9a. Two cases are considered

(

Cðr; t ¼ 0Þ ¼

f ðr Þ þ Cd 6f ðr Þ þ Cd

ðcase IÞ ðcase IIÞ

ð13Þ

;

where

f ðr Þ 

ξ

ð14Þ

is used to provide a smooth and continuous variation of C at r ¼ 1. The parameter Cd accounts for a pre-existing homogeneous contamination of the subphase with a surface-active material. For simplicity, we assume that the EOS and surface diffusion constant of the contaminant are identical to that of oleic acid. The initial condi r; t ¼ 0Þ ¼ 1 in both tion for the height profile was a flat film hð cases.

ξ

ξ

(c)

1 ðtanh ½10ð1  r Þ þ 1Þ; 2

(d)

(b)

(a)

Fig. 8. (a) Fluorescence microscopy image of a local expulsion zone. The rectangle (length 100 lm) indicates the position n = 0 and the region over which the intensity was integrated normal to the n-direction. (b) Averaged fluorescence intensity h Ii(n) measured perpendicular to the droplet contact line. Individual lines corresponding to time increments of 0.6 s are shifted vertically for clarity. The dashed lines represent the baselines (hIi = 0) for each curve. The dotted red line traces the local intensity maxima that we interpret as the location of the three-phase contact line. (c) Contact line position ncl as a function of time. (d) Zoom of (c). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 9. Model calculations for axisymmetric spreading of oleic acid on thin glycerol films assuming Pes = 50,000,  = 0.01 and Cd ¼ 0 (a) Initial surfactant distributions: Cðr ¼ 0; t ¼ 0Þ ¼ 1 (case I) and Cð0; 0Þ ¼ 6 (case II). (b) Time evolution of the rim position r max ðtÞ. The dotted line labeled ’linear’ corresponds to case I with a hypothetical linear EOS c ¼ cðC ¼ 0Þ  Pmax C, which yields a = 0.25.

527

D.K.N. Sinz et al. / Journal of Colloid and Interface Science 364 (2011) 519–529

4.2. Numerical results – depleting surfactant source The initial condition Cðr ¼ 0; t ¼ 0Þ > 1 can be viewed as a simple model for a highly compressed surfactant monolayer or an ultrathin layer of oleic acid with a thickness on order of 6 molecules. It increases the amount of surfactant present in the deposition region compared to the case Cðr ¼ 0; t ¼ 0Þ ¼ 1 and can, thus, reproduce experiments with a very small surfactant source that depletes during the course of the experiment. Fig. 9b shows the rim position rmax as a function of dimensionless time for Pes = 5  104,  = 0.01, and the two initial conditions (Eqs. (13) and (14)) considered. For case II and t < 200 a power law behavior r max  t a with exponent a  0.28 is observed, which is in good agreement with the experimental power law exponents shown in Fig. 2c. After t  200; Cðr < 1Þ falls below unity, the slope of rmax ðtÞ decreases and is comparable for both cases considered. The situation corresponds exactly to the second curve in Fig. 2c, where the surfactant droplet is depleted at t  350 s. The value of the predicted spreading exponent a = 0.16 agrees very well with the experimental value 0.17 shown in Fig. 2c. The dotted line in Fig. 9b corresponds to case I and a hypothet=@ C ¼ 1 ically linear EOS c = [cm + Pmax(1  C/Cc)], for which @ c is constant. An inaccurate spreading exponent of 1/4 results, which indicates that a realistic EOS is crucial to properly reproduce the experimentally observed spreading rates.

(a)

(b)

500 μm

(c)

4.3. Numerical results – enhanced spreading due to subphase expulsion In an attempt to model the effect of subphase expulsion described in Section 3.3, we consider solutions of Eqs. (7)–(10) for which the surfactant concentration is kept constant for r < 1

Cðr 6 1Þ ¼ C0 ¼ const:

ð15Þ

Moreover, we added a pressure offset to Eq. (9) in the region r 6 1

offset p

2c R20  OA ; R1 Pmax h0

(d)

ð16Þ

where R1  R0/sin h. Eq. (16) accounts for the capillary pressure that is exerted by a droplet of surfactant placed on top of the thin film of subphase liquid. Hydrostatic contributions can be neglected for the small droplets considered here. These modifications are intended to more closely account for the presence of a macroscopic surfactant droplet deposited on the liquid film. They do not, however, faithfully represent local effects caused by the existence of the threephase contact line at the edge of the droplet.  r; tÞ for Fig. 10a presents a time series of the height profile hð offset ¼ 0 and Cðr 6 1; tÞ ¼ 1. For t K 15 h0 ¼ 3 lm; R0 ¼ 250 lm; p  r; tÞ is observed at r  1, which subsequently moves ina ridge in hð wards and disappears under the influence of capillary pressure. At t  400 a small foot in the height profile appears at r  1, which develops into a thin film that is expelled from underneath the surfactant droplet. A magnified view of the height profile in the crater region is presented in Fig. 10c. Fischer and Troian [53] considered the linear stability of thin film flows for the case of a step-like increase in the Marangoni stress s1 ? s2 with and without additionally assuming an initial step-like change in the height profile h1 ? h2. They found that the flow was unstable if a thicker film (h1 > h2) flows into a region of reduced thickness h2 but increased Marangoni stress s2. They predict that the lateral wavelength corresponding to the most unstable mode scales as k  h0(s2/s1)1/2. Warner et al. [44], Jensen and Naire [45], Craster and Matar [46] and Edmonstone et al. [47] considered the full problem including surfactant bulk- and surface

 r; tÞ for h = 3 lm, R = 250 lm, Fig. 10. (a) Time evolution of the height profile hð 0 0 Pes = 2.5  104, and Cd ¼ 0:002. Labels (1–7) correspond to non-dimensional times 10, 79, 500, 800, 1500, 3000 and 7000. (b) Unstable expulsion of glycerol trapped below an oleic acid droplet with R0 = 465 lm. The thickness of the expelled film is below 100 nm and increases from the perimeter of the oleic acid droplet towards the fingers. (c) Zoom of (a) in the crater region. For clarity, curves corresponding to different times were shifted consecutively by 0.05 along the ordinate. Dotted horizontal lines denote the solid surface z ¼ 0 for each solution. The dashed curve is the dimensionless concentration Cðt ¼ 7000Þ. (d) Rim position r max and height  rmax Þ extracted from the simulations in (a) [curve (1)] and equivalent ones for hð Cðr 6 1; tÞ ¼ 0:5 [curve (2)] and Cðr 6 1; tÞ ¼ 0:25 [curve (3)]. Solid lines correspond to powerlaws rmax  t a with a = 0.22 and 0.32. Curves (4) and (5) track the rim height for the data in (a) [curve (5)] and a smaller value of Cd ¼ 0:001 with all other parameters unchanged [curve (4)]. Curve illustrates rmax(t) for Ds = 0 and a boundary condition Cðr ¼ 0; tÞ ¼ 1 representing surfactant supply only from a point source. No expulsion is observed. The resulting spreading exponent a = 0.21 remains constant in time.

528

D.K.N. Sinz et al. / Journal of Colloid and Interface Science 364 (2011) 519–529

transport; however, due to its increased complexity, a simple expression for the instability wavelength could not be given. The dashed curve in Fig. 10c corresponds to the concentration profile CðrÞ for the solution at time t 7 ¼ 7000. As can be seen, Cðr Þ has a very pronounced kink in the thinned region just ahead of the rim of the expelled film. The Marangoni stress in this region is approximately a factor of 20 higher than behind the ridge and the film thickness decreases by more than a factor of 10. Therefore, the conditions of the instability criterion in the model of Fischer and Troian [53] are met and the film is unstable, consistent with Figs. 10b and 3g–i. offset – 0 according If we consider a value of the offset pressure p to Eq. (16), the dashed curve in Fig. 10a results at time t ¼ 7000. It offset ¼ 0 (solid differs very little from the solution obtained for p line in Fig. 10a), which is also the case in the crater region and underneath the surfactant droplet. This indicates that the subphase transport is dominated by the Marangoni stresses and that the pressure exerted by the surfactant droplet does not substantially contribute for the values of  considered. As illustrated by the dashed curve in Fig. 10c, the surfactant concentration is significantly higher on top of the expelled film than in the region ahead of it. This implies that the expulsion process of the ultrathin film is very efficient in convecting ‘‘additional’’ surfactant across the crater region. It is therefore not surprising that this effect leads to an increase in both the slope of max as shown in Fig. 10d. rmax ðtÞ and the height of the rim h The non-dimensional rim height increases from about 1.66 to 1.71 around t ¼ 800 for curve (4). At the same time r max ðtÞ, which is well described by a powerlaw r max  ta with a = 0.22 over the interval 20 < t < 200, changes exponent towards a = 0.32 after t  700. Also shown in Fig. 10d is the rim position rmax ðtÞ for Cðr ¼ 0; tÞ ¼ 1 and Ds = 0 [curve (6)]. For this system the expulsion, shown in Fig. 10c, was absent and the rim position could be approximated by r max  ta with a = 0.21 for the entire range shown (illustrated by the solid line). Experimentally we found that the expulsion process can increase the spreading exponent a by up to about 0.15. The instability can be weakened or suppressed by reducing the surfactant droplet volume, i.e. by decreasing R0. Analogous simulations for Cðr < 1; tÞ ¼ 0:25 and 0.5 exhibit the same pre-expulsion powerlaw exponents of 0.22 and a corresponding jump in the rim height at t ¼ 800. For the boundary condition Cðr < 1; tÞ ¼ 0:25, the highest Marangoni coefficient @ c/@ C occurs at the boundary value of the surfactant concentration, i.e. at r ¼ 1. On the other hand, for Cðr < 1; tÞ ¼ 0:5 and 1, the stress maximum is assumed at a lower concentration of approximately C ¼ 0:285 for A ¼ 6:125, which occurs ahead of the boundary in the crater region. Consequently, the fact that the shape evolution proceeds in a qualitatively identical fashion indicates that the precise functional form of the EOS is not crucial for the occurrence of the expulsion process and the ensuing fingering instability. In Fig. 11a we present the dimensionless onset time t o of the expulsion process as a function of . The parameter t o is defined as the time when the rate of increase of the local film thickness  tÞðr ¼ 1Þ reaches a maxiat the rim of the surfactant droplet ð@ h=@ mum. The solid line in Fig. 11a corresponds to a power law relation 2:6 to  1:6 , which translates into to  R3:6 0 =h0 . This scaling is reminiscent of the timescale for capillary leveling [56] of infinitesimal disturbances tOrchard  L4/hhi3, where L is the disturbance wavelength and hhi is the average film thickness. As exemplified in Fig. 10a, the expulsion is indeed preceded by the profile of the trapped film evolving from initially flat h(r,t = 0) = h0 to approximately a spherical cap shape. Fig. 11b shows the time evolution of the dimensionless center c  hð   height h x ¼ 0Þ and the subphase volume V s

(a)

(b)

Fig. 11. (a) Dimensionless onset t o and expulsion times te as a function of . The solid and dashed lines correspond to power law relations with exponents 1.6 and   3, respectively. (b) Dimensionless height hð x ¼ 0Þ (open symbols) and subphase volume V s underneath the surfactant droplet for 0 <  x < 1 (solid symbols) for different values of . The solid line corresponds to a power law relation with exponent 0.25. The onset times to for the different aspect ratios are indicated by the vertical line segments.

V s  2p

Z

1

 dr ; r h

ð17Þ

0

underneath the surfactant droplet for different values of . The initial volume located underneath the surfactant droplet is V 0 ¼ pR20 h0 or equivalently V s ðt ¼ 0Þ ¼ p. After the center-height reaches a maximum, the rate of volume loss shows a rapid increase. The vertical line segments indicate the onset times t o , which precede the  c ðtÞ resemble power law behavior regimes where V s ðtÞ and h  t 0:25 as indicated by the solid line. Corresponding curves of  c ðtÞ and V s ðtÞ follow the same power law behavior, since the volh ume of a spherical cap of small aspect ratio is to good approximation V cap ¼ pR20 hc =2, where R0 is the base radius and hc the center height; i.e. the subphase volume V s is proportional to the center c for droplet diameters below the capillary length. height h We define the expulsion time t e as the instant when the subphase volume reaches a fraction of 1/e  37% of its initial value. In Fig. 11a te is plotted as a function of aspect ratio  (open symbols). The dashed line corresponds to a power law te  3 . The experimentally observed behavior, described in Section 3.5, deviates from the results of our axisymmetric model in Fig. 11. According to Fig. 7 only a certain fraction of the trapped liquid is expelled in the global phase, which can be considered as an approximately axisymmetric process. The model, therefore, cannot capture the pronounced transition in the volume depletion rate from global towards local expulsion occurring at time Dt. Furthermore, the normalized intensity in Fig. 7a remains below a value of

D.K.N. Sinz et al. / Journal of Colloid and Interface Science 364 (2011) 519–529

1, whereas an overshoot, i.e. a value I/I0 > 1, would be expected from the numerical model calculations based on the center height evolution shown in Fig. 11b. Furthermore the numerical model does not faithfully capture the effects induced by the presence of the liquid–liquid interface between the droplet and the subphase film nor the (dynamic) deformation of the surfactant droplet in the vicinity of the three-phase contact line.

529

acknowledge that this research is supported partially by the Dutch Technology Foundation STW, applied science division of NWO and the Technology Program of the Ministry of Economic Affairs. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.jcis.2011.08.055.

4.4. Implications of subphase expulsion for the spreading dynamics References If the expulsion initiates at an earlier stage, the film thickness in the crater region has not thinned and its extension has not increased as much. Consequently for smaller R0, the crater region constitutes less of a flow obstacle for the expelled film. This has a noticeable influence on the spreading exponents, which were determined as a = 0.24 before and 0.29 after expulsion for  = 0.04 and a = 0.27 for  = 0.1 after the expulsion phase (data not shown). The results for  = 0.012 in Fig. 10d were a = 0.32 after and a = 0.22 before expulsion. This range of values matches very well with the range of measured exponents between 0.23 and 0.34 mentioned in Section 2 as well as the scatter in the data presented in Fig. 2d. Ahmad and Hansen [20] studied the spreading of oleic acid on thicker films (320–1210 lm) of glycerol and reported exponents a of about 0.5 for spreading in a 2 cm wide, open channel filled with glycerol. This large value of a appeared inconsistent with previous experiments and model calculations. Gaver and Grotberg suggested that their results might be obscured by the additional flow resistance or a possible surface activity of the talk marker particles that Ahmad and Hansen had used [24] or by film rupture [25]. For the latter there appears to be no indication in Ref. [20]. Ahmad and Hansen do not report the volumes of the oleic acid droplets they deposited. If we assume that their size is comparable to the channel width and if we repeat the calculations outlined in Section 3.2 for a planar, Cartesian geometry, the resulting spreading exponents range around 0.43. It is therefore possible that the described expulsion process also explains Ahmad and Hansen’s data in a quantitative fashion. 5. Summary We have investigated the spreading dynamics of effectively immiscible, surface-active liquids on thin liquid films. Fluorescence microscopy and optical interferometry were applied to monitor the subphase morphology in the vicinity and far from the deposited surfactant droplet. Numerical simulations of the far-field spreading dynamics of an insoluble surfactant compare very favorably with the experimental results. A fingering instability was observed similar in appearance to the case of soluble surfactants, which is induced by the temporary entrapment of subphase liquid beneath the deposited surfactant droplet and its subsequent release. Two distinct phases of this expulsion process were identified: a ‘global’ regime that was observed primarily for thicker films and early times after surfactant deposition, where subphase liquid is expelled along most of the three-phase contact line. The global was followed by a ‘local’ expulsion stage, where flow of subphase liquid is restricted to one or few locations along the droplet perimeter. Furthermore, we observed a pronounced oscillatory instability of the three-phase contact line, which temporally modulates the subphase flow. Acknowledgments The authors would like to thank Steffen Berg and Axel Makurat from Shell International Exploration and Production (Rijswijk, The Netherlands) for the inspiring collaboration. The authors gratefully

[1] J.B. Grotberg, Annu. Rev. Fluid Mech. 26 (1994) 529. [2] P.M. Schweizer, S.F. Kistler, Liquid Film Coating, Chapman & Hall, London, 1997. [3] L.W. Schwartz, D.E. Weidner, R.R. Eley, Langmuir 11 (1995) 3690. [4] L.L. Schramm, Surfactants: Fundamentals and Applications in the Petroleum Industry, Cambridge University Press, Cambridge, 2000. [5] A. Cary, E.K. Rideal, Proc. R. Soc. Ser. A 109 (1925) 309. [6] E. Landt, M. Vollmer, Z. Phys. Chem. 122 (1926) 398. [7] J.A. Fay, D.P. Hoult (Eds.), Oil on the Sea, Plenum, NY, 1969, pp. 55–63. [8] D.P. Hoult, Annu. Rev. Fluid Mech. 4 (1972) 341. [9] C. Huh, M. Inoue, S. Mason, Can. J. Chem. Eng. 53 (1975) 367. [10] M. Foda, R.G. Cox, J. Fluid Mech. 101 (1980) 33. [11] W. Camp, J. Berg, J. Fluid Mech. 184 (1987) 445. [12] J.G. Fraaije, A.M. Cazabat, J. Colloid Interface Sci. 133 (1989) 452. [13] O.E. Jensen, J. Fluid Mech. 293 (1995) 349. [14] A.D. Dussaud, S.M. Troian, Phys. Fluids 10 (1998) 23. [15] A.D. Dussaud, S.M. Troian, Phys. Fluids 10 (1998) 1588. [16] E. Tan, S.T. Thoroddsen, Phys. Fluids 10 (1998) 3038. [17] T.F. Svitova, R.M. Hill, C.J. Radke, Langmuir 15 (1999) 7392. [18] M.J. Vogel, A.H. Hirsa, J. Fluid Mech. 472 (2002) 283. [19] S. Berg, Phys. Fluids 21 (2009) 032105. [20] J. Ahmad, R.S. Hansen, J. Colloid Interface Sci. 38 (1972) 601. [21] M.S. Borgas, J.B. Grotberg, J. Fluid Mech. 193 (1988) 151. [22] S.M. Troian, E. Herbolzheimer, S.A. Safran, Phys. Rev. Lett. 62 (1989) 1496. [23] S.M. Troian, E. Herbolzheimer, S.A. Safran, Phys. Rev. Lett. 65 (1990) 333. [24] D.P. Gaver, J.B. Grotberg, J. Fluid Mech. 213 (1990) 127. [25] D.P. Gaver, J.B. Grotberg, J. Fluid Mech. 235 (1992) 399. [26] O.E. Jensen, J.B. Grotberg, J. Fluid Mech. 240 (1992) 259. [27] O.E. Jensen, J.B. Grotberg, Phys. Fluids A 5 (1993) 58. [28] O.E. Jensen, Phys. Fluids 6 (1994) 1084. [29] V.M. Starov, A. de Ryck, M.G. Velarde, J. Colloid Interface Sci. 190 (1997) 104. [30] J.L. Bull, L.K. Nelson, J.T. Walsh Jr., M.R. Glücksberg, S. Schürch, J.B. Grotberg, J. Biomed. Eng. 121 (1999) 89. [31] P.L. Evans, L.W. Schwartz, R.V. Roy, J. Colloid Interface Sci. 227 (2000) 191. [32] A.D. Dussaud, O.K. Matar, S.M. Troian, J. Fluid Mech. 544 (2005) 23. [33] D.K.N. Sinz, M. Hanyak, J.C.H. Zeegers, A.A. Darhuber, Phys. Chem. Chem. Phys. 13 (2011) 9768. [34] B. Frank, S. Garoff, Langmuir 11 (1995) 87. [35] S. He, J.B. Ketterson, Phys. Fluids 7 (1995) 2640. [36] O.K. Matar, S.M. Troian, Phys. Fluids 10 (1998) 1234. [37] O.K. Matar, S.M. Troian, Phys. Fluids 11 (1999) 3232. [38] M. Cachile, A.M. Cazabat, Langmuir 15 (1999) 1515. [39] M. Cachile, M. Schneemilch, A. Hamraoui, A.M. Cazabat, Adv. Colloid Interface Sci. 96 (2002) 59. [40] B.J. Fischer, S.M. Troian, Phys. Rev. E 67 (2003) 016309. [41] B.J. Fischer, S.M. Troian, Phys. Fluids 15 (2003) 3837. [42] A.B. Afsar-Siddiqui, P.F. Luckham, O.K. Matar, Langmuir 19 (2003). 696 and 703. [43] A.B. Afsar-Siddiqui, P.F. Luckham, O.K. Matar, Adv. Colloid Interface Sci. 106 (2003) 183. [44] M.R.E. Warner, R.V. Craster, O.K. Matar, J. Fluid Mech. 510 (2004) 169. [45] O.E. Jensen, S. Naire, J. Fluid Mech. 554 (2006) 5. [46] R.V. Craster, O.K. Matar, Phys. Fluids 18 (2006) 032103. [47] B.D. Edmonstone, R.V. Craster, O.K. Matar, J. Fluid Mech. 564 (2006) 105. [48] A. Marmur, L.D. Lelah, Chem. Eng. Commun. 13 (1981) 133. [49] A.A. Newman, L.V. Cocks, Glycerol, CRC Press, Cleveland, 1968. [50] H. Noureddini, B.C. Teoh, D.L. Clements, J. Am. Oil Chem. Soc. 69 (1992) 1189. [51] L.D.A. Chumpitaz, L.F. Coutinho, J.A. Meirelles, J. Am. Oil Chem. Soc. 76 (1999) 379. [52] A.A. Darhuber, S.M. Troian, Phys. Fluids 15 (2003) S9. [53] B.J. Fischer, Non-modal Linear Stability Analysis of Thin Film Spreading by Marangoni Stresses, PhD thesis, Princeton University, 2003. [54] A. Oron, S.H. Davis, S.G. Bankoff, Rev. Mod. Phys. 69 (1997) 931. [55] R.V. Craster, O.K. Matar, Rev. Mod. Phys. 81 (2009) 1131. [56] S.E. Orchard, Appl. Sci. Res. 11 (1962) 451. [57] R. Stocker, J.W.M. Bush, J. Fluid Mech. 583 (2007) 465. [58] O.K. Matar, R.V. Craster, Soft Matter 5 (2009) 3801. [59] Y. Sumino, H. Kitahata, H. Seto, S. Nakata, K. Yoshikawa, J. Phys. Chem. B 113 (2009) 15709. [60] Y. Sumino, H. Kitahata, H. Seto, K. Yoshikawa, Soft Matter 7 (2011) 3204. [61] Y.-J. Chen, Y. Nagamine, K. Yoshikawa, Phys. Rev. E 80 (2009) 016303.