Letter pubs.acs.org/JPCL
Self-Pinning on a Liquid Surface C. Antoine,† J. Irvoas,‡ K. Schwarzenberger,§ K. Eckert,§ F. Wodlei,‡ and V. Pimienta*,‡ †
Laboratoire de Physique Théorique de la Matière Condensée, Université Pierre et Marie Curie, 4 place Jussieu, 75005 Paris, France Laboratoire des Interactions Moléculaires et de la Réactivité Chimique et Photochimique, Université Paul Sabatier de Toulouse, 118 route de Narbonne, 31062 Toulouse Cedex 9, France § Technische Universität Dresden, Institute of Fluid Mechanics, D-01062 Dresden, Germany ‡
S Supporting Information *
ABSTRACT: We report on the first experimental evidence of a self-pinning liquid drop on a liquid surface. This particular regime is observed for a miscible heavier oil drop (dichloromethane) deposited on an aqueous solution laden by an ionic surfactant (hexadecyltrimethylammonium bromide). Experimental characterization of the drop shape evolution coupled to particle image velocimetry points to the correlation between the drop profile and the accompanying flow field. A simple model shows that the observed pinned stage is the result of a subtle competition between oil dissolution and surfactant adsorption. domain for which the application field is vast: oil in water microemulsions. Indeed, CH2Cl2-containing swollen micelles are the final stage of this system.11 The mechanism of formation of such nanoobjects is still an open question in the literature, and the present approach offers a different view described here from the donating phase perspective. The objective of this Letter is to unravel the mechanism behind this unexpected evolution, thereby delivering a key to understand the reasons of the ensuing instability. We first experimentally characterize the shape evolution of the drop and its accompanying flow fields before devising a simple analytical model that accounts for both the organic phase dissolution and surfactant adsorption. CH2Cl2 drops were placed on top of an aqueous 0.5 mM surfactant solution inside a 2.5 cm square cuvette, not hermetically covered by a glass plate. CH2Cl2 is highly volatile (bp = 39.6 °C), denser (ρ = 1.32 g·cm−3) than water and partially soluble in water (solubility = 0.15 M) and its liquid/air surface tension, γ0 = 26.5 mN·m−1, gives it a high surface activity when adsorbed on a water surface. CTAB is hydrophobic; that is, its partition equilibrium is toward the organic phase and shows a strong affinity to both water/air and water/dichloromethane interfaces.11 We measured the contact angle, height, and diameter as a function of time for drops of increasing volume from 5 to 30 μL. As the drop is much denser than water, it is mainly immersed in the aqueous phase and, in the following, we will consider only the part below the water surface.
C
an a dissolving drop be pinned on a liquid surface similarly to what is known for evaporating drops on solid surfaces?1−3 Although dissolution and evaporation seem to act symmetrically at the respective interfaces, no surface heterogeneities can be invoked to explain the pinning on a liquid substrate, contrary to what is generally assumed when describing the coffee ring effect.3−7 Notwithstanding this difference, we also found pinning for the shape evolution of an organic drop (dichloromethane, CH2Cl2) deposited on a surfactant aqueous solution during the induction period that precedes the observation of highly ordered patterns.8,9 These patterns comprise a succession of clear-cut different shapes and dynamic regimes observed by varying the concentration of the ionic surfactant (hexadecyltrimethylammonium bromide, CTAB) in the aqueous phase. Below the critical micellar concentration, these regimes, related to spreading and dewetting effects, give rise to the periodic expansion of the drop radius coupled to the radial ejection of smaller droplets in the form of perfect expanding rings.10 Above the CMC, the drop shows an elongated shape with two tips and self-rotates. At higher CTAB concentration, polygonal structures are observed. In the absence of surfactant, the high surface tension difference between the two liquids induces fast and turbulent spreading of the oil.12 A 25 μL drop disappears in ∼2 s. In the presence of CTAB, the water/air surface tension is reduced, leading to a negative value of the initial spreading coefficient and so to the formation of a lens on the water surface (see SI for experimental conditions). Before the instabilities setup, the drop preserves its lens shape during an induction period and shows signs of selfpinning. This observation shows the complexity of possible interactions arising from coupled buoyant dissolution and surfactant adsorption. This is, in fact, of a broader interest in a © XXXX American Chemical Society
Received: December 8, 2015 Accepted: January 20, 2016
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DOI: 10.1021/acs.jpclett.5b02724 J. Phys. Chem. Lett. 2016, 7, 520−524
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Figure 1. (a) Contact angle, (b) height, and (c) diameter of CH2Cl2 drops of increasing initial volume (V0 = 5, 10, 15, 20, 25, and 30 μL from left to right) deposited on a 0.5 mM CTAB aqueous solution. Vertical dotted lines point time t1 and tinst for the 25 μL drop. Top row: experiments, bottom row: numerical resolution of the system of eqs 2−5. Inset: 10 μL drop diameter in appropriate time scale and higher resolution. Two pulsations are shown for illustration of the instability.
Figure 2. (a) Velocity field obtained by PIV and (b) x component (erratic black curve) of the velocity field at the position of the circle in panel a and diameter evolution (blue dots) of a 25 μL drop on a 0.5 mM CTAB solution. For better readability, the drop contour is made visible by a dashed line.
The contact angle, φ, height, h, and diameter, D, (Figure 1) follow similar trends for all initial volumes and show three distinct stages:
constant. At the time t1 an abrupt, synchronous change occurs in the slope of h(t) and φ(t), marking the transition to stage (ii). (ii) A pinned stage t1 < t < tinst during which D(t) is almost constant while both φ(t) and h(t) decrease. The shape
(i) A shape-preserving stage 0 < t < t1 during which h(t) and D(t) show a quasi-linear decrease while φ remains 521
DOI: 10.1021/acs.jpclett.5b02724 J. Phys. Chem. Lett. 2016, 7, 520−524
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The Journal of Physical Chemistry Letters evolution during this stage is actually comparable to the behavior of a “pinned” drop evaporating on a solid substrate. (iii) The instability stage for t > tinst. For drop volumes between 5 and 20 μL a peak in D(t) appears just before the instability starts. This peak is followed by a rapid decrease in the mean diameter (not shown in Figure 1), also observable for the highest volumes. Particle image velocimetry (PIV) was performed to study the flow field during the shape evolution. For this purpose, polystyrene beads (9.66 μm) were added to the aqueous phase, now placed in a 2 cm cuboid cuvette, and their motion was recorded (25 fps) in a light sheet optics. The obtained velocity field likewise displayed three clear stages; see Figure 2a,b. To check comparability, we measured φ(t), h(t), and D(t) again (Figure 2b). Just after the deposition of the drop, a vertical down-flow appears at its bottom (Figure 2a, left). This flow can also be visualized by adding a dye to the drop (see SI). It is due to the buoyant convection induced by dichloromethane dissolution.13 As a result, the oil/water interface of the drop is continuously swept counterclockwise into the aqueous bulk phase. By continuity, the buoyant stream of dichloromethane-rich fluid gives rise to a large-scale toroidal flow around the drop. Its radial inflow is visualized by the x-component of the velocity,14 determined at a distance of 0.5 mm from the initial position of the triple line (circle in Figure 2a). The corresponding negative values in Figure 2b are maintained during the whole induction period; however, two substages appear. The velocity magnitude, fluctuating strongly ∼0.5 mm/s in the beginning, decreases to a quieter period afterward. If we correlate these data to the diameter evolution (Figure 2b, blue dots), it turns out that the decreased activity corresponds to the pinned phase. At the time the instability starts the direction of the subsurface flow is suddenly reversed. The x-component becomes positive and reaches a mean value of 1 mm/s. On the velocity field plot, this corresponds to the formation of a clockwise toroidal flow around the drop. If we compare the evolution obtained in the 2.5 and 2 cm square cuvettes, the only difference is that the initial diameter of the drop is seen to be smaller and the induction period longer. The drop is, in fact, observed to be surrounded by an oily film whose confinement makes the drop more compact because the thicker the dichloromethane film the smaller the contribution of the water/air surface tension exerted on the contact line; however, in larger containers, the drop evolution becomes quantitatively independent of the available aqueous surface and gas volume (Figure 3), unambiguously showing that the film confinement is not at the origin of the shape evolution reported here. Data reported in the literature for evaporating lenses of pure liquids on water show a continuous decrease in the diameter followed, at the very end of the process, by an accelerated shrinkage.15−17 In the present case, evaporation is coupled to dissolution and transfer/adsorption of the surfactant at the oil/ water interface. Mass loss evolution shows that 20% is lost by evaporation. The evaporation rate varies only slightly with the initial drop volume and remains approximately constant in time showing no singularity coinciding with the pinned stage. During the first stage, dissolution drives an intense gravitydriven flow. The volume decrease operates without affecting the shape of the drop (φ(t) = const), suggesting that the capillary
Figure 3. Diameter evolution of a 25 μL drop on a 1 mM solution in Petri dishes of increasing diameter: 11 (red squares), 14.6 (blue circles), and 18.5 cm (green diamonds) diameter, with the liquid height being 5 mm in all cases.
forces exerted at the contact line remain unchanged. Hence, during the first stage, adsorption of the surfactant at the freshly formed oil/water interface appears to be hindered by the renewal of the oil surface due to dissolution. In the second stage, the dissolution flow is highly reduced due to the partial dichloromethane saturation of the immediate surroundings of the drop. This effect now favors adsorption of the surfactant at the water/ oil interface, decreasing the interfacial tension γo/w and hence the contact angle φ(t).18 The downflow decays further by the compression of the surfactant layer, leading to a less mobile interface. During its evolution, the drop is actually submitted to two antagonistic effects: (i) the dissolution volume loss, which tends to decrease D(t) but gradually weakens, allowing (ii) the CTAB adsorption which reduces γo/w(t) and favors a D(t) increase. From this point of view, the “pinned” stage we observe before the instability corresponds to a minimum in the D(t) evolution. The existence of two clearly separated regimes calls for analytical modeling to provide a basic understanding of the underlying physics. For simplicity, the drop is modeled as a timedependent spherical cap with homogeneous surface tensions.4 Taking the diameter D(t) and height h(t) as independent parameters, for example, all other geometric features such as contact angle φ(t), volume V(t), and subsurface area S(t) can be readily expressed analytically. The choice of D(t) and h(t) is motivated by the fact that these two parameters are less sensitive than φ(t) to the departure from the spherical cap geometry. In the following, the model provides the evolution of D(t) and h(t), with φ(t) being calculated afterward by numerically solving the Young−Laplace equations (see SI).19−21 The dissolution process is described by the rate of volume change V [C − C(t )] d V (t ) S(t ) = − m sat dt 9D
(1)
in terms of the transfer area S(t) and driving concentration difference C sat − C(t), where C sat and C(t) are the (homogeneous) saturation and transient concentrations of CH2Cl2 in the aqueous solution,22,23 averaged over a thin layer of fluid below the drop. Vm refers to the CH2Cl2 molar volume. The mass transfer resistance 9D is mainly due to the CH2Cl2 diffusion close to the drop subsurface. It is reduced by natural convection,23 usually described by means of the Sherwood number Sh. For a liquid drop submitted to laminar natural 522
DOI: 10.1021/acs.jpclett.5b02724 J. Phys. Chem. Lett. 2016, 7, 520−524
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m2·s−1 is the CTAB diffusion coefficient in water (calculated using the Wilke and Chang empirical correlation method30). The width lB is the only almost-free parameter of the model, for which we assume physically sound values around the characteristic adsorption length ΓmKL ≃0.23 mm.28,31,32 Equal to ΓmKL for the smallest drop, lB is observed to increase linearly with t1 and V0 (to reach 0.4 mm for the biggest drop, see SI). This correlation makes sense if we consider that the depletion of surfactant in the aqueous solution close to the oil/water interface increases with time. The lB value, which corresponds to the effective boundary layer at the time t1, is thus expected to increase with the duration of the shape preserving stage and so with the initial volume of the drop. Once all of the model inputs are known, eq 1 is readily solved in the shape-preserving stage (φ(t) = φ0) for t < t1. We obtain D(t < t1) = D0 + b0τ(e−t/τ − 1), that is, a linear decrease in the drop diameter at initial times, t ≪ τ, contrary to the square-root decrease expected for a purely diffusion-driven dissolution.33
convection, Cussler24 prescribes a 30% level accuracy correlation
( 2Dh ).
for Sh, under which 9D depends only on φ = 2 arctan
The concentration C(t) of CH2Cl2 in the aqueous solution dC(t ) dt −1
close to the drop subsurface obeys 1 dV (t ) V0 dt 0
be expressed as τ ≈
=
= V0 S0
Csat − C(t ) where τ may τ −1 VmCsat
( ) 9 0D
. With this
expression, the final mass transfer equation writes ⎛D 4Vm dD 6h ⎞ dh ⎟· (Csat − C(t )) +⎜ + =− ⎝ ⎠ 9D(h/D) dt 2h D dt
(2)
A second equation is required to determine the two unknown quantities D(t) and h(t). The drop profile is settled by the surface tensions exerted on the contact line. In the framework of the spherical cap approximation, the relation between surface tensions and drop geometry is provided by the approximate Neumann triangle relation, which, expressed in terms of D(t) and height h(t), writes
γo/w(t ) ·
2h(t ) D(t ) 2h(t ) D(t )
2
≃ γa/w − γa/o ≃ constant
a(h , D) =
(3)
(4)
where Rg is the gas constant, T is the temperature, Γm = 2.1.10−6 mol·m−2 is the maximum surface concentration of surfactant, KL = 110 m3·mol−1 is the adsorption parameter,11 and γ0 is the surfactant-free surface tension. CS(t) is the CTAB concentration in water close to the oil/water interface. It is intricately coupled to the various diffusive and convective fluxes at the oil/water interface and depends on the partition coefficient of the surfactant. Particularly hard to grasp is the surfactant dragging induced by oil dissolution, which counterbalances the diffusioncontrolled adsorption25 and delays it by a time t1. Rather than solving the fully coupled transport and transfer equations, we assume here, for simplicity, that the subsurface CTAB concentration in water, CS(t), obeys a first-order process:26−28 CS(t ) = CS0 + (CCAC − CS0)(1 − e−(t − t1)/ τS)
scales as 1/sin2(φ0 /2), the
( 2Dh + 6Dh ) ddht counterbalances the shrinkage due to
the right-hand side of eq 2 and makes the drop expand after a minimum has been reached. Because the dominant term of a(h,D) is inversely proportional to h, this expansion is faster and more intense for smaller drops, explaining the steep increase at the end of the drop diameter evolution. Despite its simplicity, the present model quantitatively accounts for all of the experimental features (see bottom row of Figure 1). The pinned stage of the drop is precisely captured as well as the faster shrinkage of smaller drops that also show a steeper rise in the drop diameter before the pulsating instability sets in. Although this rise is seen here for the small volumes, it is a key finding in all experiments employing Petri dishes, the optimum geometry to observe the instability. We report here for the first time on the self-pinned stage of a dissolving drop on a liquid surface. This particular regime has been observed for a heavier, volatile, miscible oil drop deposited on an aqueous surfactant solution. This effect is due to a subtle interplay between several processes including dissolution, free convection, surfactant adsorption, and transfer. The model shows that the pinned stage is the result of two antagonistic effectsdissolution and surfactant adsorptionwhich lead to a mathematical minimum of the drop diameter. In this way, the model accurately maps all relevant trends in the induction period including the final rise of the diameter. The combined experimental and numerical insights therefore form a sound basis for understanding the ensuing pulsating instability stage described in ref 8, with the final diameter increase appearing as a key ingredient to trigger this Marangoni induced instability because it causes a nonhomogeneous surfactant distribution at the oil/water interface.
where the air/water and air/oil surface tensions may reasonably be assumed to remain constant during the induction period. Hence, taking D(0) and h(0) from the experiments and calculating γo/w(0) by numerically solving the Young−Laplace equation, the set of eqs 1 and 2 can be solved providing the γo/w(t) value. Although the system is out of equilibrium, it is convenient to consider local equilibrium at the oil/water interface expressed by the Langmuir adsorption isotherm and the Szyszkowski equation of state γo/w(t ) = γ0 − 2R gT Γm·ln(1 + KLCS(t ))
4 cotan(φ0 / 2) VmCsat 2 + cos(φ0) 9D(φ0)
flatter (smaller) is the drop, the stronger is its shrinkage, in accordance with experimental data. In the pinned stage, for t > t1, the eqs 2 and 3 are solved numerically. The (negative) coefficient
2
( ) 1+( ) 1−
Because b0 =
(5)
for times greater than t1, for which a linear scaling with the initial drop volume V0 according to t1/V0 ≈ constant was experimentally found. In this equation, CS0 is the value of CS(t) between t = 0 and t1 and corresponds to the initial value γo/w(0), CCAC is the surfactant aggregation concentration in the presence of dichloromethane,11 CCAC = 0.1 mmol L−1, and τS ≈ l2B/DS gives the time scale of CTAB diffusion over an “effective” finite boundary layer,27,29 lB, which is assumed to sum up all the previously quoted effects (adsorption controlled by diffusion and dragging/ desorption induced by transfer and dissolution). DS ≈ 2.10−10
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.5b02724. Further information about the experimental conditions, the confinement effect on the film surrounding the droplet, the role of evaporation, the time and volume 523
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(19) Pujado, P.; Scriven, L. Sessile lenticular configurations: translationally and rotationally symmetric lenses. J. Colloid Interface Sci. 1972, 40, 82−98. (20) Kriegsmann, J. J.; Miksis, M. J. Steady motion of a drop along a liquid interface. SIAM J. Appl. Math. 2003, 64, 18−40. (21) Burton, J.; Huisman, F.; Alison, P.; Rogerson, D.; Taborek, P. Experimental and numerical investigation of the equilibrium geometry of liquid lenses. Langmuir 2010, 26, 15316−15324. (22) Peña, A. A.; Miller, C. A. Solubilization rates of oils in surfactant solutions and their relationship to mass transport in emulsions. Adv. Colloid Interface Sci. 2006, 123, 241−257. (23) Wegener, M.; Paul, N.; Kraume, M. Fluid dynamics and mass transfer at single droplets in liquid/liquid systems. Int. J. Heat Mass Transfer 2014, 71, 475−495. (24) Cussler, E. L. Diffusion: Mass Transfer in Fluid Systems; Cambridge University Press, 2009. (25) Mucic, N.; Kovalchuk, N.; Pradines, V.; Javadi, A.; Aksenenko, E.; Krägel, J.; Miller, R. Dynamic properties of C n TAB adsorption layers at the water/oil interface. Colloids Surf., A 2014, 441, 825−830. (26) Johannsen, E.; Chung, J.; Chang, C.; Franses, E. Lipid transport to air/water interfaces. Colloids Surf. 1991, 53, 117−134. (27) Chang, C.-H.; Franses, E. I. Adsorption dynamics of surfactants at the air/water interface: a critical review of mathematical models, data, and mechanisms. Colloids Surf., A 1995, 100, 1−45. (28) Moorkanikkara, S. N.; Blankschtein, D. Possible existence of convective currents in surfactant bulk solution in experimental pendantbubble dynamic surface tension measurements. Langmuir 2009, 25, 1434−1444. (29) Alvarez, N. J.; Vogus, D. R.; Walker, L. M.; Anna, S. L. Using bulk convection in a microtensiometer to approach kinetic-limited surfactant dynamics at fluid-fluid interfaces. J. Colloid Interface Sci. 2012, 372, 183− 191. (30) Reid, R. C.; Sherwood, T. K. The Properties of Gases and Liquids; McGraw-Hill: New York, 1958. (31) Chang, C.; Wang, N.-H.; Franses, E. Adsorption dynamics of single and binary surfactants at the air/water interface. Colloids Surf. 1992, 62, 321−332. (32) Chen, L.-H.; Lee, Y.-L. Adsorption behavior of surfactants and mass transfer in single-drop extraction. AIChE J. 2000, 46, 160−168. (33) Epstein, P. S.; Plesset, M. S. On the stability of gas bubbles in liquid-gas solutions. J. Chem. Phys. 1950, 18, 1505−1509.
dependence of the effective boundary layer lB, and some other minor aspects of the modeling.(ZIP)
AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We wish to acknowledge the support of the European Space Agency. V.P. thanks the Centre National d’Etude Spatiales (DAR115349) for financial support. K.E. and K.S. thank DFG (Ec201/2-2) for financial support as part of priority program SPP1506.
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REFERENCES
(1) Cazabat, A.-M.; Guéna, G. Evaporation of macroscopic sessile droplets. Soft Matter 2010, 6, 2591−2612. (2) Sobac, B.; Brutin, D. Triple-line behavior and wettability controlled by nanocoated substrates: influence on sessile drop evaporation. Langmuir 2011, 27, 14999−15007. (3) Weon, B. M.; Je, J. H. Self-pinning by colloids confined at a contact line. Phys. Rev. Lett. 2013, 110, 028303. (4) Larson, R. G. Transport and deposition patterns in drying sessile droplets. AIChE J. 2014, 60, 1538−1571. (5) Lopes, M. C.; Bonaccurso, E. Influence of substrate elasticity on particle deposition patterns from evaporating water-silica suspension droplets. Soft Matter 2013, 9, 7942−7950. (6) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. Capillary flow as the cause of ring stains from dried liquid drops. Nature 1997, 389, 827−829. (7) Sefiane, K. Patterns from drying drops. Adv. Colloid Interface Sci. 2014, 206, 372−381. (8) Pimienta, V.; Brost, M.; Kovalchuk, N.; Bresch, S.; Steinbock, O. Complex shapes and dynamics of dissolving drops of dichloromethane. Angew. Chem., Int. Ed. 2011, 50, 10728−10731. (9) Pimienta, V.; Antoine, C. Self-propulsion on liquid surfaces. Curr. Opin. Colloid Interface Sci. 2014, 19, 290−299. (10) Antoine, C.; Pimienta, V. Mass-spring model of a self-pulsating drop. Langmuir 2013, 29, 14935−14946. (11) Tadmouri, R.; Zedde, C.; Routaboul, C.; Micheau, J.-C.; Pimienta, V. Partition and water/oil adsorption of some surfactants. J. Phys. Chem. B 2008, 112, 12318−12325. (12) Rideal, E. K.; Davies, J. T. Interfacial Phenomena; Academic Press: New York, 1963. (13) Agble, D.; Mendes-Tatsis, M. The effect of surfactants on interfacial mass transfer in binary liquid-liquid systems. Int. J. Heat Mass Transfer 2000, 43, 1025−1034. (14) The point is chosen close to the water/air interface because this way the x-component approximately accounts for the global velocity, with the advantage that the inversion point is clearly detected. Points analyzed all around the drop give similar qualitative behavior. (15) Bentwich, M.; Landau, U.; Sideman, S. Direct contact heat transfer with change of phase: evaporation of discrete volatile films from the surface of a stagnant immiscible liquid. Int. J. Heat Mass Transfer 1970, 13, 945−956. (16) Kaneko, M.; Mori, Y. H. Evaporation of a volatile-liquid lens floating on the surface of a stagnant immiscible liquid. Int. Commun. Heat Mass Transfer 1984, 11, 209−218. (17) Nosoko, T.; Ohyama, T.; Mori, Y. Evaporation of volatile-liquid lenses floating on an immiscible-liquid surface: effects of the surface age and fluid purities in n-pentane/water system. J. Fluid Mech. 1985, 161, 329−346. (18) Phan, C. M. Stability of a Floating Water Droplet on an Oil Surface. Langmuir 2014, 30, 768−773. 524
DOI: 10.1021/acs.jpclett.5b02724 J. Phys. Chem. Lett. 2016, 7, 520−524
Self-Pinning on a Liquid Surface C. Antoine,† J. Irvoas,‡ K. Schwarzenberger,§ K. Eckert,§ F. Wodlei,‡ and V. Pimienta*,‡ †
Laboratoire de Physique Théorique de la Matière Condensée, Université Pierre et Marie Curie, 4 place Jussieu, 75005 Paris, France Laboratoire des Interactions Moléculaires et de la Réactivité Chimique et Photochimique, Université Paul Sabatier de Toulouse, 118 route de Narbonne, 31062 Toulouse Cedex 9, France § Technische Universität Dresden, Institute of Fluid Mechanics, D-01062 Dresden, Germany ‡
S Supporting Information *
ABSTRACT: We report on the first experimental evidence of a self-pinning liquid drop on a liquid surface. This particular regime is observed for a miscible heavier oil drop (dichloromethane) deposited on an aqueous solution laden by an ionic surfactant (hexadecyltrimethylammonium bromide). Experimental characterization of the drop shape evolution coupled to particle image velocimetry points to the correlation between the drop profile and the accompanying flow field. A simple model shows that the observed pinned stage is the result of a subtle competition between oil dissolution and surfactant adsorption. domain for which the application field is vast: oil in water microemulsions. Indeed, CH2Cl2-containing swollen micelles are the final stage of this system.11 The mechanism of formation of such nanoobjects is still an open question in the literature, and the present approach offers a different view described here from the donating phase perspective. The objective of this Letter is to unravel the mechanism behind this unexpected evolution, thereby delivering a key to understand the reasons of the ensuing instability. We first experimentally characterize the shape evolution of the drop and its accompanying flow fields before devising a simple analytical model that accounts for both the organic phase dissolution and surfactant adsorption. CH2Cl2 drops were placed on top of an aqueous 0.5 mM surfactant solution inside a 2.5 cm square cuvette, not hermetically covered by a glass plate. CH2Cl2 is highly volatile (bp = 39.6 °C), denser (ρ = 1.32 g·cm−3) than water and partially soluble in water (solubility = 0.15 M) and its liquid/air surface tension, γ0 = 26.5 mN·m−1, gives it a high surface activity when adsorbed on a water surface. CTAB is hydrophobic; that is, its partition equilibrium is toward the organic phase and shows a strong affinity to both water/air and water/dichloromethane interfaces.11 We measured the contact angle, height, and diameter as a function of time for drops of increasing volume from 5 to 30 μL. As the drop is much denser than water, it is mainly immersed in the aqueous phase and, in the following, we will consider only the part below the water surface.
C
an a dissolving drop be pinned on a liquid surface similarly to what is known for evaporating drops on solid surfaces?1−3 Although dissolution and evaporation seem to act symmetrically at the respective interfaces, no surface heterogeneities can be invoked to explain the pinning on a liquid substrate, contrary to what is generally assumed when describing the coffee ring effect.3−7 Notwithstanding this difference, we also found pinning for the shape evolution of an organic drop (dichloromethane, CH2Cl2) deposited on a surfactant aqueous solution during the induction period that precedes the observation of highly ordered patterns.8,9 These patterns comprise a succession of clear-cut different shapes and dynamic regimes observed by varying the concentration of the ionic surfactant (hexadecyltrimethylammonium bromide, CTAB) in the aqueous phase. Below the critical micellar concentration, these regimes, related to spreading and dewetting effects, give rise to the periodic expansion of the drop radius coupled to the radial ejection of smaller droplets in the form of perfect expanding rings.10 Above the CMC, the drop shows an elongated shape with two tips and self-rotates. At higher CTAB concentration, polygonal structures are observed. In the absence of surfactant, the high surface tension difference between the two liquids induces fast and turbulent spreading of the oil.12 A 25 μL drop disappears in ∼2 s. In the presence of CTAB, the water/air surface tension is reduced, leading to a negative value of the initial spreading coefficient and so to the formation of a lens on the water surface (see SI for experimental conditions). Before the instabilities setup, the drop preserves its lens shape during an induction period and shows signs of selfpinning. This observation shows the complexity of possible interactions arising from coupled buoyant dissolution and surfactant adsorption. This is, in fact, of a broader interest in a © XXXX American Chemical Society
Received: December 8, 2015 Accepted: January 20, 2016
520
DOI: 10.1021/acs.jpclett.5b02724 J. Phys. Chem. Lett. 2016, 7, 520−524
Letter
The Journal of Physical Chemistry Letters
Figure 1. (a) Contact angle, (b) height, and (c) diameter of CH2Cl2 drops of increasing initial volume (V0 = 5, 10, 15, 20, 25, and 30 μL from left to right) deposited on a 0.5 mM CTAB aqueous solution. Vertical dotted lines point time t1 and tinst for the 25 μL drop. Top row: experiments, bottom row: numerical resolution of the system of eqs 2−5. Inset: 10 μL drop diameter in appropriate time scale and higher resolution. Two pulsations are shown for illustration of the instability.
Figure 2. (a) Velocity field obtained by PIV and (b) x component (erratic black curve) of the velocity field at the position of the circle in panel a and diameter evolution (blue dots) of a 25 μL drop on a 0.5 mM CTAB solution. For better readability, the drop contour is made visible by a dashed line.
The contact angle, φ, height, h, and diameter, D, (Figure 1) follow similar trends for all initial volumes and show three distinct stages:
constant. At the time t1 an abrupt, synchronous change occurs in the slope of h(t) and φ(t), marking the transition to stage (ii). (ii) A pinned stage t1 < t < tinst during which D(t) is almost constant while both φ(t) and h(t) decrease. The shape
(i) A shape-preserving stage 0 < t < t1 during which h(t) and D(t) show a quasi-linear decrease while φ remains 521
DOI: 10.1021/acs.jpclett.5b02724 J. Phys. Chem. Lett. 2016, 7, 520−524
Letter
The Journal of Physical Chemistry Letters evolution during this stage is actually comparable to the behavior of a “pinned” drop evaporating on a solid substrate. (iii) The instability stage for t > tinst. For drop volumes between 5 and 20 μL a peak in D(t) appears just before the instability starts. This peak is followed by a rapid decrease in the mean diameter (not shown in Figure 1), also observable for the highest volumes. Particle image velocimetry (PIV) was performed to study the flow field during the shape evolution. For this purpose, polystyrene beads (9.66 μm) were added to the aqueous phase, now placed in a 2 cm cuboid cuvette, and their motion was recorded (25 fps) in a light sheet optics. The obtained velocity field likewise displayed three clear stages; see Figure 2a,b. To check comparability, we measured φ(t), h(t), and D(t) again (Figure 2b). Just after the deposition of the drop, a vertical down-flow appears at its bottom (Figure 2a, left). This flow can also be visualized by adding a dye to the drop (see SI). It is due to the buoyant convection induced by dichloromethane dissolution.13 As a result, the oil/water interface of the drop is continuously swept counterclockwise into the aqueous bulk phase. By continuity, the buoyant stream of dichloromethane-rich fluid gives rise to a large-scale toroidal flow around the drop. Its radial inflow is visualized by the x-component of the velocity,14 determined at a distance of 0.5 mm from the initial position of the triple line (circle in Figure 2a). The corresponding negative values in Figure 2b are maintained during the whole induction period; however, two substages appear. The velocity magnitude, fluctuating strongly ∼0.5 mm/s in the beginning, decreases to a quieter period afterward. If we correlate these data to the diameter evolution (Figure 2b, blue dots), it turns out that the decreased activity corresponds to the pinned phase. At the time the instability starts the direction of the subsurface flow is suddenly reversed. The x-component becomes positive and reaches a mean value of 1 mm/s. On the velocity field plot, this corresponds to the formation of a clockwise toroidal flow around the drop. If we compare the evolution obtained in the 2.5 and 2 cm square cuvettes, the only difference is that the initial diameter of the drop is seen to be smaller and the induction period longer. The drop is, in fact, observed to be surrounded by an oily film whose confinement makes the drop more compact because the thicker the dichloromethane film the smaller the contribution of the water/air surface tension exerted on the contact line; however, in larger containers, the drop evolution becomes quantitatively independent of the available aqueous surface and gas volume (Figure 3), unambiguously showing that the film confinement is not at the origin of the shape evolution reported here. Data reported in the literature for evaporating lenses of pure liquids on water show a continuous decrease in the diameter followed, at the very end of the process, by an accelerated shrinkage.15−17 In the present case, evaporation is coupled to dissolution and transfer/adsorption of the surfactant at the oil/ water interface. Mass loss evolution shows that 20% is lost by evaporation. The evaporation rate varies only slightly with the initial drop volume and remains approximately constant in time showing no singularity coinciding with the pinned stage. During the first stage, dissolution drives an intense gravitydriven flow. The volume decrease operates without affecting the shape of the drop (φ(t) = const), suggesting that the capillary
Figure 3. Diameter evolution of a 25 μL drop on a 1 mM solution in Petri dishes of increasing diameter: 11 (red squares), 14.6 (blue circles), and 18.5 cm (green diamonds) diameter, with the liquid height being 5 mm in all cases.
forces exerted at the contact line remain unchanged. Hence, during the first stage, adsorption of the surfactant at the freshly formed oil/water interface appears to be hindered by the renewal of the oil surface due to dissolution. In the second stage, the dissolution flow is highly reduced due to the partial dichloromethane saturation of the immediate surroundings of the drop. This effect now favors adsorption of the surfactant at the water/ oil interface, decreasing the interfacial tension γo/w and hence the contact angle φ(t).18 The downflow decays further by the compression of the surfactant layer, leading to a less mobile interface. During its evolution, the drop is actually submitted to two antagonistic effects: (i) the dissolution volume loss, which tends to decrease D(t) but gradually weakens, allowing (ii) the CTAB adsorption which reduces γo/w(t) and favors a D(t) increase. From this point of view, the “pinned” stage we observe before the instability corresponds to a minimum in the D(t) evolution. The existence of two clearly separated regimes calls for analytical modeling to provide a basic understanding of the underlying physics. For simplicity, the drop is modeled as a timedependent spherical cap with homogeneous surface tensions.4 Taking the diameter D(t) and height h(t) as independent parameters, for example, all other geometric features such as contact angle φ(t), volume V(t), and subsurface area S(t) can be readily expressed analytically. The choice of D(t) and h(t) is motivated by the fact that these two parameters are less sensitive than φ(t) to the departure from the spherical cap geometry. In the following, the model provides the evolution of D(t) and h(t), with φ(t) being calculated afterward by numerically solving the Young−Laplace equations (see SI).19−21 The dissolution process is described by the rate of volume change V [C − C(t )] d V (t ) S(t ) = − m sat dt 9D
(1)
in terms of the transfer area S(t) and driving concentration difference C sat − C(t), where C sat and C(t) are the (homogeneous) saturation and transient concentrations of CH2Cl2 in the aqueous solution,22,23 averaged over a thin layer of fluid below the drop. Vm refers to the CH2Cl2 molar volume. The mass transfer resistance 9D is mainly due to the CH2Cl2 diffusion close to the drop subsurface. It is reduced by natural convection,23 usually described by means of the Sherwood number Sh. For a liquid drop submitted to laminar natural 522
DOI: 10.1021/acs.jpclett.5b02724 J. Phys. Chem. Lett. 2016, 7, 520−524
Letter
The Journal of Physical Chemistry Letters
m2·s−1 is the CTAB diffusion coefficient in water (calculated using the Wilke and Chang empirical correlation method30). The width lB is the only almost-free parameter of the model, for which we assume physically sound values around the characteristic adsorption length ΓmKL ≃0.23 mm.28,31,32 Equal to ΓmKL for the smallest drop, lB is observed to increase linearly with t1 and V0 (to reach 0.4 mm for the biggest drop, see SI). This correlation makes sense if we consider that the depletion of surfactant in the aqueous solution close to the oil/water interface increases with time. The lB value, which corresponds to the effective boundary layer at the time t1, is thus expected to increase with the duration of the shape preserving stage and so with the initial volume of the drop. Once all of the model inputs are known, eq 1 is readily solved in the shape-preserving stage (φ(t) = φ0) for t < t1. We obtain D(t < t1) = D0 + b0τ(e−t/τ − 1), that is, a linear decrease in the drop diameter at initial times, t ≪ τ, contrary to the square-root decrease expected for a purely diffusion-driven dissolution.33
convection, Cussler24 prescribes a 30% level accuracy correlation
( 2Dh ).
for Sh, under which 9D depends only on φ = 2 arctan
The concentration C(t) of CH2Cl2 in the aqueous solution dC(t ) dt −1
close to the drop subsurface obeys 1 dV (t ) V0 dt 0
be expressed as τ ≈
=
= V0 S0
Csat − C(t ) where τ may τ −1 VmCsat
( ) 9 0D
. With this
expression, the final mass transfer equation writes ⎛D 4Vm dD 6h ⎞ dh ⎟· (Csat − C(t )) +⎜ + =− ⎝ ⎠ 9D(h/D) dt 2h D dt
(2)
A second equation is required to determine the two unknown quantities D(t) and h(t). The drop profile is settled by the surface tensions exerted on the contact line. In the framework of the spherical cap approximation, the relation between surface tensions and drop geometry is provided by the approximate Neumann triangle relation, which, expressed in terms of D(t) and height h(t), writes
γo/w(t ) ·
2h(t ) D(t ) 2h(t ) D(t )
2
≃ γa/w − γa/o ≃ constant
a(h , D) =
(3)
(4)
where Rg is the gas constant, T is the temperature, Γm = 2.1.10−6 mol·m−2 is the maximum surface concentration of surfactant, KL = 110 m3·mol−1 is the adsorption parameter,11 and γ0 is the surfactant-free surface tension. CS(t) is the CTAB concentration in water close to the oil/water interface. It is intricately coupled to the various diffusive and convective fluxes at the oil/water interface and depends on the partition coefficient of the surfactant. Particularly hard to grasp is the surfactant dragging induced by oil dissolution, which counterbalances the diffusioncontrolled adsorption25 and delays it by a time t1. Rather than solving the fully coupled transport and transfer equations, we assume here, for simplicity, that the subsurface CTAB concentration in water, CS(t), obeys a first-order process:26−28 CS(t ) = CS0 + (CCAC − CS0)(1 − e−(t − t1)/ τS)
scales as 1/sin2(φ0 /2), the
( 2Dh + 6Dh ) ddht counterbalances the shrinkage due to
the right-hand side of eq 2 and makes the drop expand after a minimum has been reached. Because the dominant term of a(h,D) is inversely proportional to h, this expansion is faster and more intense for smaller drops, explaining the steep increase at the end of the drop diameter evolution. Despite its simplicity, the present model quantitatively accounts for all of the experimental features (see bottom row of Figure 1). The pinned stage of the drop is precisely captured as well as the faster shrinkage of smaller drops that also show a steeper rise in the drop diameter before the pulsating instability sets in. Although this rise is seen here for the small volumes, it is a key finding in all experiments employing Petri dishes, the optimum geometry to observe the instability. We report here for the first time on the self-pinned stage of a dissolving drop on a liquid surface. This particular regime has been observed for a heavier, volatile, miscible oil drop deposited on an aqueous surfactant solution. This effect is due to a subtle interplay between several processes including dissolution, free convection, surfactant adsorption, and transfer. The model shows that the pinned stage is the result of two antagonistic effectsdissolution and surfactant adsorptionwhich lead to a mathematical minimum of the drop diameter. In this way, the model accurately maps all relevant trends in the induction period including the final rise of the diameter. The combined experimental and numerical insights therefore form a sound basis for understanding the ensuing pulsating instability stage described in ref 8, with the final diameter increase appearing as a key ingredient to trigger this Marangoni induced instability because it causes a nonhomogeneous surfactant distribution at the oil/water interface.
where the air/water and air/oil surface tensions may reasonably be assumed to remain constant during the induction period. Hence, taking D(0) and h(0) from the experiments and calculating γo/w(0) by numerically solving the Young−Laplace equation, the set of eqs 1 and 2 can be solved providing the γo/w(t) value. Although the system is out of equilibrium, it is convenient to consider local equilibrium at the oil/water interface expressed by the Langmuir adsorption isotherm and the Szyszkowski equation of state γo/w(t ) = γ0 − 2R gT Γm·ln(1 + KLCS(t ))
4 cotan(φ0 / 2) VmCsat 2 + cos(φ0) 9D(φ0)
flatter (smaller) is the drop, the stronger is its shrinkage, in accordance with experimental data. In the pinned stage, for t > t1, the eqs 2 and 3 are solved numerically. The (negative) coefficient
2
( ) 1+( ) 1−
Because b0 =
(5)
for times greater than t1, for which a linear scaling with the initial drop volume V0 according to t1/V0 ≈ constant was experimentally found. In this equation, CS0 is the value of CS(t) between t = 0 and t1 and corresponds to the initial value γo/w(0), CCAC is the surfactant aggregation concentration in the presence of dichloromethane,11 CCAC = 0.1 mmol L−1, and τS ≈ l2B/DS gives the time scale of CTAB diffusion over an “effective” finite boundary layer,27,29 lB, which is assumed to sum up all the previously quoted effects (adsorption controlled by diffusion and dragging/ desorption induced by transfer and dissolution). DS ≈ 2.10−10
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.5b02724. Further information about the experimental conditions, the confinement effect on the film surrounding the droplet, the role of evaporation, the time and volume 523
DOI: 10.1021/acs.jpclett.5b02724 J. Phys. Chem. Lett. 2016, 7, 520−524
Letter
The Journal of Physical Chemistry Letters
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(19) Pujado, P.; Scriven, L. Sessile lenticular configurations: translationally and rotationally symmetric lenses. J. Colloid Interface Sci. 1972, 40, 82−98. (20) Kriegsmann, J. J.; Miksis, M. J. Steady motion of a drop along a liquid interface. SIAM J. Appl. Math. 2003, 64, 18−40. (21) Burton, J.; Huisman, F.; Alison, P.; Rogerson, D.; Taborek, P. Experimental and numerical investigation of the equilibrium geometry of liquid lenses. Langmuir 2010, 26, 15316−15324. (22) Peña, A. A.; Miller, C. A. Solubilization rates of oils in surfactant solutions and their relationship to mass transport in emulsions. Adv. Colloid Interface Sci. 2006, 123, 241−257. (23) Wegener, M.; Paul, N.; Kraume, M. Fluid dynamics and mass transfer at single droplets in liquid/liquid systems. Int. J. Heat Mass Transfer 2014, 71, 475−495. (24) Cussler, E. L. Diffusion: Mass Transfer in Fluid Systems; Cambridge University Press, 2009. (25) Mucic, N.; Kovalchuk, N.; Pradines, V.; Javadi, A.; Aksenenko, E.; Krägel, J.; Miller, R. Dynamic properties of C n TAB adsorption layers at the water/oil interface. Colloids Surf., A 2014, 441, 825−830. (26) Johannsen, E.; Chung, J.; Chang, C.; Franses, E. Lipid transport to air/water interfaces. Colloids Surf. 1991, 53, 117−134. (27) Chang, C.-H.; Franses, E. I. Adsorption dynamics of surfactants at the air/water interface: a critical review of mathematical models, data, and mechanisms. Colloids Surf., A 1995, 100, 1−45. (28) Moorkanikkara, S. N.; Blankschtein, D. Possible existence of convective currents in surfactant bulk solution in experimental pendantbubble dynamic surface tension measurements. Langmuir 2009, 25, 1434−1444. (29) Alvarez, N. J.; Vogus, D. R.; Walker, L. M.; Anna, S. L. Using bulk convection in a microtensiometer to approach kinetic-limited surfactant dynamics at fluid-fluid interfaces. J. Colloid Interface Sci. 2012, 372, 183− 191. (30) Reid, R. C.; Sherwood, T. K. The Properties of Gases and Liquids; McGraw-Hill: New York, 1958. (31) Chang, C.; Wang, N.-H.; Franses, E. Adsorption dynamics of single and binary surfactants at the air/water interface. Colloids Surf. 1992, 62, 321−332. (32) Chen, L.-H.; Lee, Y.-L. Adsorption behavior of surfactants and mass transfer in single-drop extraction. AIChE J. 2000, 46, 160−168. (33) Epstein, P. S.; Plesset, M. S. On the stability of gas bubbles in liquid-gas solutions. J. Chem. Phys. 1950, 18, 1505−1509.
dependence of the effective boundary layer lB, and some other minor aspects of the modeling.(ZIP)
AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We wish to acknowledge the support of the European Space Agency. V.P. thanks the Centre National d’Etude Spatiales (DAR115349) for financial support. K.E. and K.S. thank DFG (Ec201/2-2) for financial support as part of priority program SPP1506.
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DOI: 10.1021/acs.jpclett.5b02724 J. Phys. Chem. Lett. 2016, 7, 520−524
