The Marangoni flow of soluble amphiphiles
Matthieu Roché, Zhenzhen Li, Ian M. Griffiths, Sébastien Le Roux, Isabelle Cantat, Arnaud Saint-Jalmes, Howard A. Stone
aa r X i v : . [ phy s i c s . f l u - dyn ] D ec The Marangoni flow of soluble amphiphiles
Matthieu Roch´e, ∗ Zhenzhen Li,
1, †
Ian M. Griffiths, S´ebastien LeRoux, Isabelle Cantat, Arnaud Saint-Jalmes, and Howard A. Stone Department of Mechanical and Aerospace Engineering,Princeton University, Princeton, New Jersey 08544, USA Mathematical Institute, University of Oxford, Oxford, OX2 6GG, England Institut de Physique de Rennes, CNRS UMR 6251, Universit´e Rennes 1, 35042 Rennes, France
Surfactant distribution heterogeneities at a fluid/fluid interface trigger the Marangoni effect, i.e. a bulk flowdue to a surface tension gradient. The influence of surfactant solubility in the bulk on these flows remains in-completely characterized. Here we study Marangoni flows sustained by injection of hydrosoluble surfactants atthe air/water interface. We show that the flow extent increases with a decrease of the critical micelle concen-tration, i.e. the concentration at which these surfactants self-assemble in water. We document the universalityof the surface velocity field and predict scaling laws based on hydrodynamics and surfactant physicochemistrythat capture the flow features.
The release of a drop of water mixed with dishwashing liq-uid on the surface of pure water covered with pepper grainsdemonstrates the Marangoni effect [1]: after the drop touchesthe water surface, pepper grains are transported rapidly tothe edge of the bowl (see movie M1 in Supplementary Ma-terials). The flow results from the difference in surface ten-sion between water at the point of release and clean waterfar away. The occurrence of the Marangoni effect plays animportant role in many natural and industrial processes suchas pulmonary surfactant replacement therapy [2, 3], motionand defence of living organisms [4–6], the stability of emul-sions and foams [7, 8] and many others [9–12]. In these set-tings, surfactants generally have a finite solubility in one ofthe phases, but the effect of interface-bulk mass exchangeon Marangoni flows is still not understood despite its con-sequences on flow (see Movie M2 in Supplementary Mate-rials). Most studies [13–16] have focused on the depositionof droplets of surfactant solutions on thin water films, but thetransient nature of the induced flow and the small size of thefilm prevented the validation of proposed theoretical descrip-tions. Here, we investigate axisymmetric Marangoni flows in-duced by hydrosoluble surfactants on centimeter-thick waterlayers. We show how the flow extent and the associated ve-locity field depend on the surfactant chemical structure, henceon amphiphile thermodynamic properties such as the criticalmicelle concentration (cmc). We propose scaling laws basedon hydrodynamics and physicochemistry for the flow radiusand its velocity that are in excellent agreement with the exper-iments.We characterized the Marangoni flow of water induced byhydrosoluble surfactants using eight surfactants from the alkyltrimethylammonium halides (C n TABr, n =10 to 14; C n TACl, n = 12 and 16) as well as from the sodium alkyl sulfate(C n NaSO , n = 8 to 12) families (purchased from Sigma-Aldrich before each experimental run, purity 99%), whosecritical micelle concentration varies over two orders of mag-nitude [17–23] (See Supplementary Materials). Surfactant so-lutions, seeded with light-scattering 10- m m olive oil droplets,were supplied on the surface of a ultra-pure water layer (Mil-lipore Q, resistivity s = 18.2 M W .cm) using a syringe pump (Harvard Apparatus PHD2000) at a constant surfactant mo-lar flow rate Q a = q Qc , with q = V s / ( V s + V oil ) the volumefraction of surfactant solution in the injected liquid, V s and V oil the volumes of surfactant solution and oil used to pre-pare the injected solution, Q the total volume flow rate and c the surfactant concentration (Fig. 1a). We made sure that theoil droplets acted as passive tracers and did not influence theproperties of the flow (See Supplementary Materials). Theflow is divided into three concentric regions (Fig. 1b andmovie M3 in Supplementary Materials). A zone of signifi-cant light scattering, the source, surrounds the injection point,over distance r s . Further downstream, we observed a regionof radius r t which exhibits little scattered light that we callthe transparent zone. Outside the transparent zone, intenselight scattering is observed again, and vortex pairs similar tothose reported in the case of thermocapillary Marangoni flows[24] grow along the air/water interface and expand outwards.Further from the source, the tracers move only slightly, sug-gesting that surface tension is spatially homogeneous in thisregion and that the Marangoni flow is located in the transpar-ent region. A side view of the experiment reveals the existenceof a three-dimensional recirculating flow in the bulk fluid be-low the transparent zone, which changes direction at r = r t and then follows the bottom of the container back towards thesource.The slow interfacial vortices might be related to thefate of surfactants at the air/water interface in the outer region[25], which does not have a significant influence on the mainflow characteristics relevant to the transparent zone, as we willshow later.The size of the different flow regions depends on the sur-factant molecular structure. We observed that, for a constantmolar flow rate Q a , the radius of the transparent zone r t variesover almost two orders of magnitude when n increases two-fold (Fig. 2a). Also, r t is sensitive to the properties of thesurfactant polar headgroup, in particular to its effective radius r e f f , which takes into account electrostatic and ion-specificeffects in its definition [26, 27]. For example, an increase of r e f f by using C NaSO instead of C TAB, which differ onlyby their polar headgroups, results in a decrease of r t .The radius of the transparent zone r t also varies with time h ~ 10 -2 m surfactanttracer FlowFlow Q a ultra-pure waterair d = 1.8x10 -3 m (a) r t r s C NaSO Side view (b) highlow D r o p l e t s u r f ace concen t r a t i on FIG. 1.
Experimental observation of the continuous surfactant-induced Marangoni flow. (a) Schematic of the experimental set-up inthe region surrounding the point of injection. (b) Side view of a typical experiment. In this view, regions of high coverage in light-scatteringtracers are white. Surfactant molar flow rate Q a = . × − mol.s − , scale bar: 3 × − m. r s r ( - m ) Q a (10 -6 mol.s -1 ) r t,max -1 -3 -2 -1 C TAB C TAB C NaSO C TAB C TAC C NaSO C TAB C NaSO r t ( - m ) t (s) (b)(a) increasing N C FIG. 2.
Characterization of the transparent zone. (a) Radius ofthe transparent zone r t as a function of time t for different surfac-tants. Surfactant molar flow rate Q a = . × − mol.s − for allexperiments. (b) Maximum radius r t , max (filled symbols) and radiusof the source r s (open symbols) as a function of surfactant molar flowrate Q a . Data collected for the same amount of injected C NaSO . (Fig. 2a). After an initial increase at the onset of injection, r t remains constant at a maximal value r t , max for a time depen-dent on finite-size effects due to the container (see Fig. S2 in Supplementary Materials). Then, r t decreases slowly, beforea sharp decrease is observed at longer times, correspondingto a significant increase of the surfactant concentration in thewater layer (see Fig. S3 in Supplementary Materials).The relationship between the surfactant molar flow rate Q a and r t , max is nonlinear (Fig. 2b) in contrast with the linear de-pendence between r t , max and Q a reported in earlier studies ofthe continuous Marangoni flow of partially miscible fluids onwater [28]. The size of the source r s remains equal to the nee-dle diameter until a threshold flow rate is reached, after which r s increases. The value of the threshold flow rate appears todepend little on the formulation of the injected solution.To understand the physics underlying the observed flows,we reconstructed the surface velocity field by tracking the in-terfacial motion of the tracers in the steady regime r t = r t , max (see movie M4 in Supplementary Materials). The tracersmoved along the radial direction only with a velocity u whoseshape as a function of r is similar for all the surfactants wetested (Fig. 3a). When the tracers leave the source, where u ≈ − m.s − , they accelerate, reach a maximum velocity u max ≈ . − , before decelerating as they travel across thetransparent area. Finally, tracers decelerate abruptly as theyreach r = r t . The magnitude of u max decreases with an in-crease of n and/or r e f f . The injection flow rate has little effecton the velocity field (Fig. 3b). We note that the shape of thevelocity fields is qualitatively similar to those reported in ear-lier studies on the spreading of partially miscible fluids on wa-ter [28–30], though no systematic scaling laws were identifiedin these earlier works and only partial theoretical descriptionswere given.Inspired by the similarity of the velocity profiles obtainedfor surfactants (Fig. 3a), we plotted u / u max versus a rescaledradial coordinate ( r − r s ) / ( r t , max − r s ) . Figure 4a showsthat the velocity fields obtained for different surfactants col-lapse on a nearly universal profile when plotted with therescaled coordinates. The location at which u = u max is ( r − r s ) / ( r t , max − r s ) ≈ . Q a = 0.52 10 -6 mol.s -1 Q a = 0.78 10 -6 mol.s -1 u ( m . s - ) r (10 -3 m) SDeS
DoTAB
SDoS u ( m . s - ) r (10 -3 m) (a)(b) C NaSO C NaSO C TAB
FIG. 3.
Characterization of the velocity field. (a) Radial com-ponent u of the interfacial velocity field in the transparent zoneas a function of the radial position r . Surfactant molar flow rate Q a = . × − mol.s − . (b) Radial component u of the interfa-cial velocity field for a flow induced by C NaSO at different molarflow rates Q a . profiles have a similar slope during the deceleration stage for ( r − r s ) / ( r t , max − r s ) > . ℓ n ≈ r n r ∗ u ∗ , (1)with n = hr the kinematic viscosity, h and r respectively thedynamic viscosity and the density of the fluid in the layer, u ∗ acharacteristic velocity at the interface and r ∗ the distance over which radial velocity gradients are established, i.e. the size ofthe flow we want to determine. We assume that surface ten-sion gradients in regions extending to r > r ∗ are much smallerthan in the area defined by r < r ∗ , an assumption supported byprevious work on continuous Marangoni flows [25].The fluid moving along the interface advects surfactants.As there is no surfactant far from the interface, surfactantsdesorb and diffuse towards the bulk. We assume that adsorp-tion/desorption processes occur on timescales much shorterthan the surfactant diffusion in bulk water. Interface-bulkmass exchange is thus diffusion-limited, and a mass transferboundary layer grows, whose thickness scales as: ℓ c ≈ r Dr ∗ u ∗ = Sc − / ℓ n , (2)with Sc = n D the Schmidt number, which compares the kine-matic viscosity n , i.e momentum diffusion constant, to thesurfactant bulk diffusion constant D . Equation 2 is valid if theviscous boundary layer is much larger than the mass transferboundary layer, i.e. if Sc ≫
1, a condition that is fulfilled inour case as, for a diffusion coefficient D = − m .s − and n = − m .s − for water, Sc ≈ . The bulk concentrationthus varies from a high value just below the interface to zeroat the bottom of the mass boundary layer. We choose the cmcof the surfactants as the concentration scale relevant to the de-scription of surfactant transport because of the dependence ofthe radius of the Marangoni flow on the properties of both thehydrophobic tail and the polar headgroup of the surfactants(Fig. 2a), which are key elements in the thermodynamic defi-nition of the cmc [27, 31, 32].Our rationale is based on the assumption that the Marangoniflow stops when surfactants injected at the source at a molarflow rate Q a have all desorbed from the interface. Hence, thesurfactant mass balance can be expressed as: Q a (cid:181) r ∗ D c ∗ ℓ c , (3)with c ∗ the critical micelle concentration. Replacing ℓ c by Eq.2, we find: Q a (cid:181) r ∗ / ( Du ∗ ) / c ∗ . (4)The continuity of stress at the air/water interface writes: h u ∗ ℓ n ≈ g w − g s r ∗ , (5)with g w the surface tension of ultra-pure water and g s the sur-face tension of the surfactant solution. From this condition,we obtain an expression for the velocity: u ∗ = A (cid:18) ( g w − g s ) hr r ∗ (cid:19) / , (6)and by replacing u ∗ in Eq. 4 with Eq. 6, we obtain: r ∗ = B (cid:18) hr ( g w − g s ) D (cid:19) / (cid:18) Q a c ∗ (cid:19) / , (7) -2 -1 C TAB
SDS
SDeS r t, m a x -r s ( - m ) Q a (10 -6 mol.s -1 ) -2 -1 -1 Q a = 0.17 10 -6 mol.s -1 Q a = 0.52 10 -6 mol.s -1 Q a = 0.85 10 -6 mol.s -1 r t, m a x -r s ( - m ) cmc -1 (m .mol -1 ) 3 -2 -1 Q a = 0.17 10 -6 mol.s -1 Q a = 0.52 10 -6 mol.s -1 Q a = 0.85 10 -6 mol.s -1 (r t, m a x -r s ) / Q a3 / ( S . I. ) cmc -1 (m .mol -1 ) SDoS C TAB C TAC
SDeS u / u m a x (r-r s )/(r t -r s ) (a) Experimental data
Equation 1 u m a x ( m . s - ) r t,max -r s (10 -3 m) (c) (b)(d) NaSO C NaSO C TABC TACC NaSO C NaSO C TAB (r-r s )/(r t,max -r s ) u / u m a x FIG. 4.
Universality of the velocity field in the transparent zone in steady state and scaling laws. (a) Rescaled velocity profiles u / u max as a function of the rescaled radial coordinate ( r − r s ) / ( r t − r s ) for identical amounts of injected surfactants. (b) Comparison between Eq.6 and experimental data. (c) Comparison between Eq. 7 and experimental data for the maximal size of the transparent zone r t , max − r s as afunction of the surfactant molar flow rate Q a . (d) Comparison between Eq. 7 and experimental data for the maximal size of the transparentzone r t , max − r s as a function of the inverse of the critical micellar concentration c ∗ . Inset: collapse of the experimental data for r t , max − r s when values are rescaled by Q / a as a function of cmc − . All data points were measured for the same surfactant amount injected in the layer, n s = Q a t = . × − mol. where A and B are two dimensionless prefactors. We estimatethe values predicted for u ∗ and r ∗ with typical values of thedifferent parameters involved in Eqs. 6 and 7 while assumingthat c ∗ = − M, g w − g s is constant for all experiments andequal to 33 mN.m − , a realistic value for the surfactant solu-tions we used. Setting both A and B to unity, we find u ∗ ≈ − and r ∗ ≈ × − m, which compare very well withour experimental findings for the maximum velocity (Fig. 3).We compare Eq. 6 to the experimental data by taking u ∗ = u max and r ∗ = r t , max − r s . Figure 4b shows that Eq. 6 cap-tures the experimental measurements very well, with a prefac-tor A ≈