Mass-Spring Model of a Self-Pulsating Drop - Langmuir (ACS

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Mass-Spring Model of a Self-Pulsating Drop Charles Antoine*,† and Véronique 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

‡

ABSTRACT: Self-pulsating sessile drops are a striking example of the richness of far-from-equilibrium liquid/liquid systems. The complex dynamics of such systems is still not fully understood, and simple models are required to grasp the mechanisms at stake. In this article, we present a simple mass-spring mechanical model of the highly regular drop pulsations observed in 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. We introduce an effective timedependent spreading coefficient that sums up all of the forces (due to evaporation, solubilization, surfactant transfer, coffee ring effect, solutal and thermal Marangoni flows, drop elasticity, etc.) that pull or push the edge of a dichloromethane liquid lens, and we show how to account for the periodic rim breakup. The model is examined and compared against experimental observations. The spreading parts of the pulsations are very rapid and cannot be explained by a constant positive spreading coefficient or superspreading.

1. INTRODUCTION

The article is organized as follows. Experimental results are given in section 2, followed by the derivation of the mass-spring model of the drop pulsations in section 3. Interpretations and concluding remarks are given in sections 4 and 5.

Spreading, dewetting, self-motion, and deformations of liquid droplets and sheets attract attention because of their continuously growing technological importance. Essentially limited to liquid/solid systems (i.e., to droplets deposited on solid substrates), this field of research has known, for the last 10 years, a strong expansion toward liquid/liquid systems that present a greater richness and complexity.2−11 Evaporation, solubilization, surfactant transfer, buoyancy effects, and coupling between convection flows inside and below the drop have led to new spatiotemporal dissipative structures, among which regular self-beating and rotating movements are the most striking.1,12,13 The source of energy of these artificial cells is still an open problem but seems to result from a chemical to mechanical energy conversion through various hydrodynamic instabilities. In particular, the combination of evaporation, solubilization, and surfactant transfer appears to be crucial in accounting for the dynamics of these far-from-equilibrium systems. In this article, we focus on the regular pulsating (or beating) mechanism that has been experimentally demonstrated by two teams with a volatile oil drop placed on a water surface, with surfactant either in the drop12 or in the substrate.1 We do not intend to give a comprehensive analytical explanation of the processes at stake, as it is outlined in ref 14 for example, but rather we want to show that the highly regular pulsations in ref 1 can be recovered by a simple mass-spring mechanical model of the drop. It is shown, in particular, that the opposite of the spring modulus can be interpreted as an effective spreading coefficient that oscillates in time in a nontrivial way. © 2013 American Chemical Society

2. EXPERIMENTAL RESULTS The experiment, which is fully described in ref 1 (see also the Supporting Information in ref 1), consists of placing a drop of dichloromethane (CH2Cl2, 25 μL, oil phase, denser than water [ρo = 1.33 g mL−1], highly volatile [boiling point = 39.6 °C], and partially soluble in water) on an aqueous solution of cetylltrimethylammonium bromide (CTAB, surfactant, nonvolatile, partitioning in favor of the oil phase). After the deposition and a short induction period, several remarkable dynamic regimes are spontaneously set up: emulsification at the drop edge, regular lateral oscillations, regular concentric pulsations, a stable rotating structure with tip dropping, and polygonal shapes with moving tips. The transition between these regimes is experimentally controlled by C 0 = [CTAB], the surfactant concentration of the aqueous solution. For C0 < 0.25 mmol L−1, the drop does not present any noteworthy dynamic regime: it spreads out and solubilizes in a few seconds. Between 0.25 and 1.0 mmol L−1 (close to the critical micellar concentration [cmc] equal to 0.9 mmol L−1), the oscillating and pulsating regimes progressively appear and set up. During these regimes, the rim of the drop breaks up, partially or as a whole, and small droplets are ejected (through a Rayleigh-like instability). A stable, rotating structure is observed above the cmc and becomes particularly stable at C0 = 10 mmol L−1. In this dynamic regime, the drop loses its circular symmetry and transforms into an elongated structure with two tips from where small droplets are Received: September 25, 2013 Revised: November 6, 2013 Published: November 7, 2013 14935

dx.doi.org/10.1021/la403678r | Langmuir 2013, 29, 14935−14946

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Figure 1. Ten successive direct shadowgraphy images (top view, time step of 67 ms) during a drop pulsation cycle. The drop radius varies between 3 and 4 mm.

Figure 2. Typical drop radius evolution during the pulsating regime. The drop radius lies between two almost parallel, linearly decreasing functions (blue lines) associated with the minimum and maximum radii (red dots), respectively. regularly ejected. For higher concentrations, the rim of the drop presents a quasi-polygonal structure with some marked tips that move erratically and collide, leading to the ejection of small droplets. For all CTAB concentrations, the final stage of the experiment consists of the complete disappearance of the oil drop by solubilization and evaporation. In this article, we focus on the pulsating regime at C0 = 0.5 mmol L−1. At this surfactant concentration, it is by far the most observed regime and, in most cases, the only regime observed all along the instability. In this regime, different typical stages happen before the complete disappearance of the oil drop. Right after the oil deposit, we clearly observe the rapid formation of a mesoscopic oil film that surrounds the drop. Then starts the induction phase during which we observe a large oil transfer in the water subphase as well as rapid convection rolls inside the drop. This period, during which the drop is stationary with a circular rim, typically lasts 10 to 20 s. Then, the rim becomes unstable and the drop slowly starts spreading. Partial breakups of the rim are observed, which lead to the recoil of the drop in the opposite direction of the oil ejection. These small ejections seem to be linked to interfacial waves that cross and break the upper part of the rim (like a liquid spilling from the top of a container15). Then, one observes a transition to a global ejection regime where the whole rim is involved in the instability and breaking up. In that case, a self-sustained pulsating regime may be maintained for several tens of seconds. Each pulsation consists of a rapid spreading of the drop rim. (For a better

visualization of the process, see the 10 successive pictures in Figure 1.) Then the upper part of the rim breaks up as a whole and progresses at the water surface as a radially expanding torus. After a very short time (a few tens of milliseconds), this toroidal oil filament undergoes a Rayleigh-like instability and breaks up into small droplets that move radially away from the drop (over a distance of 5 to 6 mm) and rapidly disappear by evaporation and dissolution. It is not clear, however, whether the rim is already corrugated before leaving the drop. Right after the rim release, the drop strongly contracts itself and a dewetting process accompanies the new rim formation. The rim regains its initial shape as it recedes, until the drop is at rest. Then, a new spreading− ejection event starts and a regular beating regime is progressively set up, with a mean period of 1 s. Between 5 and 40 regular pulsations are commonly observed before the drop collapses. When the drop is too small, the spreading of the rim leads to the nucleation of a hole in the center of the drop, and all of the oil regroups itself in the expanding rim before being dispersed at the water surface through a Rayleigh-like instability (like an explosion). The last step in the experiment consists of the evaporation and dissolution of the surface oil film, although one still observes the presence of a microscopic oil film on the water surface after several minutes. As is well known, the spreading of an oil drop on a water substrate is mainly controlled by one parameter, the spreading coefficient S16

S = γw/a − γo/a − γo/w 14936

(1)


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where γw/a, γo/a, and γo/w are the surface tensions at the water/air, oil/ air, and oil/water interfaces, respectively. For an aqueous surfactant concentration equal to 0.5 mmol L−1 (i.e., close to the critical micellar concentration), the surfactant adsorption at the water surface leads to a significant decrease in the surface tension: γw/a = 43 mN m−1.17 For the initial CH2Cl2 drop, the surface tensions are supposed to be γo/a = 26.5 mN m−1 and γo/w = 27 mN m−1. However, the latter strongly depends on the surfactant concentration. If we assume a quasiinstantaneous surfactant adsorption at the oil/water interface, it almost immediately has the following low value of γo/w = 1.5 mN m−1.17 So right after the deposition of the drop, the spreading coefficient is expected to be positive, Sinitial = +15 mN m−1. This means that dichloromethane tends to wet the aqueous substrate completely,16 the expected equilibrium state consisting of a uniform, macroscopic layer of oil floating on the aqueous solution. In the presence of evaporation and dissolution, however, the wetting can be slowed down, or frustrated, and a macroscopic drop may coexist with a mesoscopic film. In the experiment described in ref 1, this film has been observed to be of great importance to the drop instability. For example, a strong outward water flow has been monitored below the film. These convection rolls do not have a clear origin for now, but they are associated with a net force pulling both the film and the drop rim outward. Actually, pseudopartial wetting is observed for a positive spreading coefficient and a positive Hamaker constant AH.9,16,19,20 In Hamaker’s approach, the microscopic interaction energy is pairwise additive, and AH takes a simple form: AH ≃ Aoo − Aos ≃ (Aoo)1/2((Aoo)1/2 − (Ass)1/2), where Aoo and Ass are the Hamaker constants of dichloromethane- and surfactant-laden aqueous solution, respectively. Hamaker constants are proportional to the corresponding surface tensions. For a nonpolar liquid such as dichloromethane, one has simply Aoo = 24πh02γo/a ≃ 5.4 × 10−20 J, where h0 = 0.165 nm is the average distance between liquid molecules. As for Ass, it can be obtained through the value of the Hamaker constant of pure water (Apure water = 3.7 × 10−20 J) corrected by the ratio of surface tensions: Ass = Apure water(γw/a/γpure water) ≃ 2.2 × 10−20 J < Aoo, with γpure water = 72.8 mN m−1. In the following section, we will focus on the central drop and will not include the microscopic surface interactions (disjunction pressure due to the van der Waals contribution, for example) in the force balance describing the drop rim evolution. A typical drop radius evolution is given in Figure 2. The first observation is that the rim movement is quasi-periodic, with the period slightly increasing from 0.6 to 1.5 s. Second, one clearly sees the presence of rebounds for both the outward and inward rim movements. They seem to be linked to the fact that it is the upper part of the rim that mainly undergoes spreading, breaking up, and recoil. The lower part of the rim remains attached to the drop and follows the movement of the upper part but with a delay, and friction effects may dampen the bouncing oscillations.21 Third, the pulsations are clearly not time-symmetric: the ascending, spreading parts are slightly longer (∼0.6 s) than the descending, receding parts (∼0.4 s). This time asymmetry increases significantly in the course of time. Fourth, a careful observation reveals that in almost all cases, whenever the minimum of the drop radius is at its lowest, the following maximum is above the maximum average. This mechanical correlation seems to appear every two or three pulsating periods, meaning that such regular superexpansions following supercrunches might be the result of subtle nonequilibrium effects implying subharmonic frequencies. However, the difficulty of accurately defining the maximum drop radii prevents us from analyzing and interpreting these effects in a rigorous and relevant manner. Finally, one observes that the period of the pulsations is much larger than the natural oscillation period of the drop21,22 (equal to (γ/ (ρoV))1/2 ≃ 0.01 s, where V and ρo are the drop volume and density, respectively, and where γ denotes a typical surface tension of the drop, see below), meaning that these pulsations do not correspond to natural or forced mechanical oscillations of the whole drop.

One can wonder if the fast spreading of the drop during the instability can be accounted for by a normal spreading effect due to a constant positive S value. Much experimental and theoretical work has been done on this subject, with all leading to a power law of R(t) ≈ tα with an α exponent close to or less than 1.7,14,23−27 In our case, let us compute the (arithmetic) mean curve of the ascending parts of the regular pulsations of Figure 2 (between 6 and 16 s). Clearly, the left curve of Figure 3 cannot be fitted by a power law with an exponent of

Figure 3. Drop radius vs time: mean spreading (left) and receding (right) parts. The left solid line corresponds to the fit 3.28 + 1.65t2 + 2.70t4, whereas the right solid line corresponds to the fit 3.2 + 1.0 × 10−12.9t. A rebound is clearly visible halfway through the receding part. less than 1. In fact, an accurate fit of this curve requires higher powers of t or an exponential increase (for instance, the fit in Figure 3 is 3.3 + 1.7t2 + 2.7t4), which cannot be consistent with normal drop spreading with a constant positive spreading coefficient (and also is not consistent with superspreading, see the Discussion section). The same is true for the receding parts of the rim evolution curve (right part of Figure 3). The mean curve of the descending parts of the regular pulsations between 6 and 16 s cannot be fitted by a power law, as for usual dewetting processes,19,28 but corresponds to an exponential-like decreasing evolution. (The fit in Figure 3 is equal to 3.2 + 1.0 × 10−12.9 (t−0.6).) To measure the volume ejected at each pulsation, one has to perform two kinds of experimental runs: one with monitoring from the top (to see the rim spreading clearly; see the images of Figure 1) and one with monitoring from the side (to evaluate the volume of the main drop and ejected droplets; see Figure 5). From a numerical analysis of the side images and the Pappus−Guldinus theorem, one can compute the volume of any (axisymmetric) drop and build the curve point by point, giving the drop volume as a function of its radius (inset of Figure 4). For small droplets, this curve is well-defined because the radius−volume relation does not vary from one droplet to another and can be used to calculate the total volume of the droplets ejected at each pulsation. For larger drops (the main drop, for instance), this is no longer the case. Indeed, the drop radius−volume relationship is not the same during the induction period and the pulsating regime. At the beginning of the drop instability, the drop becomes flatter (strong increase in its radius) and takes a pancakelike shape. Because the induction period is not the same for two successive drops deposited on the water surface, the abrupt change in the slope of the radius−volume curve does not appear at the same place in the inset of Figure 4. The periodic evolution of the rim suggests that an oscillator model can be considered in a first approach. This is strengthened by the observation that the pulsation period scales approximately as the square root of the ejected mass of oil.

3. MASS-SPRING MODEL OF THE RIM PULSATIONS By analogy to the mechanical models of a dripping faucet,29,30 a pendant drop,31 and drop fragmentation on impact,32 we 14937


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Figure 4. Lost (red squares) and ejected (blue circles with bars) oil volumes during the pulsating instability. (Inset) Drop volume vs its diameter (experimental data and interpolating curve). The abrupt slope change of the curve at the volume 15 mm3 corresponds to the beginning of the pulsating instability.

simple geometry above, the horizontal projection of their sum is equal to 2πR[(γ′o/a + γ′o/w − γo/w) cos θ − γo/a] and maintains the same form (drop perimeter multiplied by a linear combination of surface tensions) for a more complex geometry. Each of these four surface tensions may depend on time because of various mechanisms. First, as we already underlined, the surfactant distribution at the oil/water interface changes during the course of the experiment. When the drop is deposited on the CTAB aqueous solution, the surfactant molecules rapidly reorganize and adsorb at the drop subsurface (i.e., at the oil/water interface below the drop). Local surface concentration differences may happen during the instability because of ejections, nonuniform evaporation and transfer, surfactant packing, or dilution. They may also induce solutal Marangoni effects on either the film edge, below the film (which has a decreasing width), or below the drop. For example, when the drop is spreading, there is a dilution of surfactant molecules (because the oil/water interface is stretched) and a subsequent increase in γo/w close to the rim. Because of the gradient of γo/w, a solutal Marangoni effect takes place at the drop subsurface, accelerating the outward drop rim movement. Whether the transfer from the aqueous solution to the oil drop is relevant to triggering the instability or sustaining the pulsations is still open. The partitioning of CTAB between the aqueous and CH2Cl2 phases highly favors the organic liquid, but the time scale of the transfer33,34 is longer than a few tens of seconds and, as a consequence, cannot be invoked to be of major importance in the reloading process between two pulsations, although it might be important during the induction period for reaching the instability conditions. The surfactant transfer may nevertheless play a crucial role in the pulsating regime because it is often accompanied by Marangoni-type instabilities. A great deal of work has been recently done on the onset and development of convective Marangoni instabilities due to solute transfer.33−38 When a solute is transferred from the water phase to the oil phase, an orthogonal gradient of

consider a very simple mechanical model of rim pulsations. We describe the drop rim as a toroidal tube of oil located at drop radius position R(t) with a (time-dependent) mass m, for which we assume that we can apply Newton’s second law. Because the pulsating regime starts once there is an oil film surrounding the drop, we consider the geometry of Figure 5.

Figure 5. Cross-sectional view of the drop during the pulsating regime. A mesoscopic oil film surrounds the drop. Green circles mark the position of the rim. Surface tensions and the drop effective weight are shown by red and black arrows. See section 3 for the meaning of the symbols.

The rim is at the boundary between the drop and the film, and we have specified the film data with quotation marks. The spreading coefficient S′ of the drop (considered to be independent from the film) may be defined as ′ + γo/w ′ ) − (γo/a + γo/w ) S′ = (γo/a

(2)

where γ′o/a and γ′o/w are the surface tensions at oil/air and oil/ water interfaces of the film. The film is curved (to compensate for the effective weight of the drop) and makes an angle with the horizontal roughly equal to θ, the angle of the oil/water interface of the drop near the rim. In a first approach, if we consider that γo/a ′ = γo/a and γo/w ′ = γo/w, then S′ = 0 and the net force per unit length of the rim is equal to −γo/a(1 − cos θ) < 0, pulling the rim toward the drop center. Several different pulling and pushing forces act on the rim. Some are proportional to R and lead to springlike forces. Such is the case, for instance, for the surface tension forces. With the 14938


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The higher evaporation rate close to the edge of the drop also leads to convection flow from the center of the drop to its periphery (invoked to explain the famous coffee ring effect15,46,47) so as to compensate for the higher mass loss. The same effect is expected to happen in the surrounding oil film, with a permanent flow to offset the mass loss due to evaporation. This outward flow not only produces a timedependent force proportional to 2πR but also creates the wavelike structure of the rim, with a small bump on the rim (Figure 5) where there is a kind of bottleneck.6,11,14 Solubilization is also of great importance in our case because dichloromethane is partially soluble in water. (Its solubility is equal to 13 g L−1, which amounts to 10 oil drops in the 25 mL of water contained in the Petri dish.) To quantify the relevance of this effect, we compared (Figure 4) the volume of oil that is lost by ejection (at each pulsation) with the total volume lost during the instability. One can see that during the course of time the oil volume loss originates from both the solubilization/ evaporation and ejection processes, with the latter playing an increasingly important role. Let us note that the convection flows, which make up for the oil lost by solubilization, are superimposed on the evaporation-induced convection flows (toward the rim and the film) and contribute to the rim pulling. During the pulsation regime, the nonmonotonic oil volume loss in Figure 4 leads to a monotonic decrease in the minimum radius of the drop (Figure 2). From a mechanical point of view, it suggests that the rim mass has to increase enough, between two successive pulsations, to compensate for the decrease in the drop radius. A global linear increase in the width of the rim (at the instant the drop is fully contracted, i.e., just before the spreading part of a pulsation cycle) can indeed be inferred from Figure 8. As for the maxima of the R(t) curve, they seem to decrease monotonically as well (Figure 2) in a manner similar to that for minima, meaning that the extension of the drop pulsations is constant in time. Finally, to complement this springlike forces list, one can also add the elastic force that may be introduced to account for the elasticity of the surfactant-covered drop, even if this force is hard to quantify. Because the lower surface of the drop is covered with a monolayer of surfactant molecules, this interface is rigidified and the solubilization flow is strongly reduced. The rigidified oil/water interface can be seen as an elastic, permeable membrane that may also play a significant role in the instability.48 In particular, it could help us to understand why the drop becomes progressively flatter and flatter. One can similarly wonder about the effect of the rigidification of the water/air interface when the drop is initially deposited on the solution. The elasticity of the surface monolayer aggregate might intervene in the explanation of the good floatation of the initial drop. (Let us mention that here the triple-line elasticity is not expected to be a fringe elasticity because the drop is large and the wavelength of the rim deformations (of the order of the drop perimeter) is much larger than the capillary lengths of interest.16) Differences in the surface rigidification may also happen without any surfactant influence. A surface that has a greater surface tension in some places (because it is colder, for example) will behave as if it is more rigid locally. In our case, the dichloromethane drop is expected to show non-negligible variations in the surface temperature, with a cold zone in the center of the drop. As a consequence, the center of the drop is naturally more rigid than the external part of the drop, which can be more easily stretched. As observed in the pulsating

solute concentration appears at the interface, which may enhance a local, small (thermal or solutal) Marangoni convection.39 Both the convection flow and concentration gradient grow in time until the instability fades because of convection-induced solute homogenization. Depending on the experimental conditions, the oil−water−solute system can undergo several successive such instabilities and even reach a stable oscillatory regime. When the instability develops in the system, the convective flow causes an abrupt decrease in the interfacial tension γo/w where the surfactant molecules are present in excess. Both the time scale of this effect (on the order of 1 s) and its amplitude (of the order of a few mN m−135) are compatible with the present data (section 5). One of the other obvious phenomena that has a good time scale is evaporation. Dichloromethane is highly volatile, and evaporation has been shown to be responsible for a 30% decrease in the drop mass in 50 s (i.e., the lifetime of the drop). Oil evaporation is important, not only for triggering and sustaining the pulsations but also for making the drop float. Indeed, the drops have been observed to sink when the gas phase is saturated by dichloromethane (by hermetically closing the container, for example), and evaporation is expected to play a role in compensating the (effective) weight of the dichloromethane drop. The heat required for evaporation results in a lower surface temperature.40−43 The evaporation is more intense for curved surfaces and thin liquid layers, but the cooling of the drop surface strongly depends on the thickness of the drop and heat diffusion in the substrate. For the dichloromethane/water system, we expect to observe the drop surface to be cooler, by a few degrees, in its center than on its rim and the water surface to be hotter than the drop surface. Because the surface tension decreases linearly with surface temperature44 as ∼0.1 mN m−1 K−1, one may reasonably think that evaporation can increase the drop and film surface tensions by a few tenths of a mN m−1. Another perimeter proportional force is due to the Marangoni stress caused by the surface tension gradients near the rim. Indeed, on the upper surface of the drop, the temperature difference between the rim and the drop interior leads to a Marangoni flow directed toward the drop center16 and a subsequent viscous constraint (per unit length) equal to − γo/a)/2, where γcenter is the surface tension of the (γcenter o/a o/a (colder) drop center. Because the volume of the drop decreases over time, this constraint is expected to have less and less importance during the pulsating regime, and the direction of this thermal Marangoni flow might even be reversed40,45 when the contact angle becomes small enough. The lower part of the drop may also produce a Marangoni effect if there is nonuniformity in the adsorbed surfactant molecules. However, it is tricky to decide whether there is a well-defined solutal Marangoni flow and, if so, if its direction is inward or outward. It is, however, tempting to say that there is surfactant depletion in the rim after a rim breakup simply because the oil ejected from the rim possesses a high concentration of surfactants (which accumulate because of the various outward convection flows) and is partially replaced by some oil coming from the bulk of the drop (i.e., noncontaminated by surfactant). On the contrary, the convection flows toward the rim and the film tend to pack and accumulate the surfactant molecules in the rim subsurface, leading to a decrease in γo/w. Let us insist on the fact that, as mentioned before, these Marangoni flows may couple to the surfactant transfer to enhance these γo/w variations. 14939


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Figure 6. Mean spreading part of the drop radius: experimental data (red dots) and various solutions of eq 4 for different linear and nonlinear fits of Seff(t). The curves correspond to Seff(t) = 0 (dotted−dashed black curve), meaning that the spreading is caused by the hydrostatic force only; Seff(t) = constant > 0 without the hydrostatic term (tiny dashed black curve); Seff(t) < 0, linearly increasing (short dashed green curve); Seff(t) < 0, quadratically increasing (very long dashed purple curve); Seff(t) < 0, increasing as t1.5 (dashed blue curve); and Seff(t) < 0, increasing as t1.75 (orange curve).

include the energy loss due to the flows near the contact line, at the rim core and inside the drop. In accordance with the experiments, the drop is also supposed to be far from the wall of the Petri dish so that one can totally neglect any meniscus effect that may alter the forces listed above. Finally, one may write the dynamics of the rim as

regime, the drop rim seems to exhibit dynamic behavior that is nearly independent of the central part of the drop. Two other forces may be important for describing the observed phenomena. First, because the drops that we consider are not small compared to the smallest capillary length −1 = κo/w

γo/w (ρo − ρw )g

≃ 0.7 mm (3)

R 4 d ⎛⎜ dR ⎞ dR m(t ) ⎟ = −k(t )R + k 0 03 − bR ⎝ ⎠ dt dt dt R

we cannot neglect the effect of gravity. (The small drops cannot lead to a stable pulsating regime because they explode at the first pulsation in a way similar to the main drop at the end of its lifetime.) The hydrostatic force acting on a unit length of the rim scales as (ρo − ρw)gh2/2, where h is the height of the drop, g is the gravitational acceleration, and the density difference is positive for a heavy dichloromethane drop. Because the drops are relatively flat and almost completely immersed in the aqueous solution, it is not worth taking into account a more precise drop structure. Within this approximation, the height of the pancake-shaped drop can be related to its radius as h ≈ V/ πR2, where V is the drop volume, and the total hydrostatic force can be expressed as (ρo − ρw)gV2/(πR3), scaling as R−3. The last force of our overview also depends on rim perimeter 2πR: it is the damping force caused by the friction of the (half)toroidal rim when it moves in the water/oil medium. This frictional force is proportional to (4πη/|log(LH/l)|)(2πR) dR/ dt according to Brenner’s formula,19,49 with quantities LH and l being the smallest viscous characteristic length of the problem and the rim width, respectively, and η ≃ 10−3 Pa s being the dynamic viscosity of water at 20 °C. Using the Stokes length η/ (ρov0)49,50 for LH, where v0 characterizes the rim velocity, one can write the frictional force as −bR dR/dt with b = 8π2η/| log(1/Re)| and Re = ρov0l/η being the Reynolds number. Let us mention that although the viscous friction of a liquid− liquid system is not dominated by hydrodynamic effects inside the rim, as for a rim moving on a solid substrate, for example, a better, comprehensive modeling of the pulsations should

(4)

with the help of a time-dependent spring parameter k(t) that accounts for all of the springlike forces acting on the rim (surface tensions, Marangoni stress, coffee ring effect, and drop surface elasticity due to the surfactant monolayer) and can be seen as the opposite of an effective spreading coefficient Seff(t) = −k(t)/2π. Note that the hydrostatic term has been rewritten as k0R04/R3 so as to obtain R = R0 when the rim is at rest. The exact time dependence of m(t) and k(t) is not known, but we intend to estimate it through a numerical fit to the observed values of R(t). This does not encourage us to define a particular scaling, and we use the commonly used units for expressing the experimental data in the following section. The experiments prompt us to proceed in three steps. First, we model the mean ascending part of R(t) (Figure 3) with a constant mass m0 and a time-dependent (decreasing) spring parameter k(t). Second, we model the mean receding part (Figure 3) with a constant spring parameter k0 and a timedependent (increasing) mass m(t). Finally, we put these two models together and show that we can simulate the whole curve of R(t) (i.e., the succession of pulsations of Figure 2) by choosing a simple rim breakup criterion and by assuming that an oil annulus is ejected at each pulsation (in line with the minimum drop radius curve of Figure 2). A further refinement of the model consists of considering a slight time increase in the rim mass over all of the pulsations. 14940


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Figure 7. Modeling of the mean receding part of the drop rim pulsations. Experimental data (in red) and various solutions of eq 4: for m = m0 and b = b0 (in green); for m = m0 and b ≃ 1.45b0 (in blue); and for m(t) and b ≃ 1.45b0 (in black). (Inset) The same curves close to the initial time, where we see how much better the modeling is when a time-dependent mass is considered.

3.1. Mean Spreading Part (m0, k(t)). We assume that k(t = 0) = k0 so that initially the rim is at rest: R(t = 0) = R0. Then we estimate the numerical values of m0, k0, R0, and b from the experimental data. We choose, in this subsection, the data corresponding to the mean ascending part given in Figure 3. The (arithmetic) mean minimum drop radius (Figure 3) is equal to R0 = 3.3 mm, whereas its height is h ≃ 1 mm, giving a value of k0 ≃ 10 mN m−1 when balanced with the hydrostatic force per unit length (ρo − ρw)gh2/2 ≃ 65 mN m−1. Let us pay attention to the fact that the mean initial drop radius value of 3.3 mm has no real meaning: it comes from an arithmetic average over the regular pulsations of the curve in Figure 2. The rim mass is deduced from its apparent volume observed by direct shadowgraphy and is estimated to mo ≃ 5 mg. As for the friction coefficient, one obtains bo ≃ 0.5 Pa s and a Reynolds number very close to 1. This Reynolds number value prevents us simplifying the eq 4 by omitting either the inertial term or the viscous effect. The different dynamics obtained by numerically integrating eq 4 are given in Figure 6. They correspond to different linear and nonlinear increases in the effective spreading coefficient Seff(t) = −k(t)/2π. If a constant, positive spreading coefficient (dashed curve) or a linear increase in Seff(t) < 0 (orange curve) give back the qualitatively good tendency, then it is also clear that they cannot quantitatively explain the strong spreading of the drop radius. Fits of Seff(t) in tδ with δ > 1 are much better. The best fit is obtained for δ close to 7/4. In this case, the total increase in Seff(t) over the pulsation spreading time is equal to 1.3 mN m−1. This is a relatively low value (compared to the various surface tensions at stake) that is compatible with the physical processes highlighted above (Discussion section). 3.2. Mean Receding Part (m(t), k0). As mentioned before, the increase in the pulling force acting on the rim during drop spreading is controlled by various mechanisms: evaporation and dissolution of the film surrounding the drop, outward convection flow due to the positive global spreading parameter, surfactant inhomogeneities and Marangoni flows, and more intense evaporation of the drop close to the rim. Even if we do not know what is the dominant effect, we observe that the rim breakup causes a strong contraction of the drop. There is, of course, a simple mechanical effect due to the momentum

conservation of the rim that breaks into two parts: the torusshaped upper part that progresses away in water and a smaller part that remains attached to the receding drop. However, the main effect seems to be linked to the oil that is released in water. This oil is different from the oil contained in the film that surrounds the drop and that has been submitted to evaporation, dissolution, and transfer for 1 s (since the last ejection). Actually, the ejected oil may be considered to be similar to the oil that was surrounding the drop before the beginning of the present pulsation. As a consequence, this oil “jet” brings the whole system back to the initial state, and the spring parameter k(t) is expected to come back to its initial value k0 after each ejection. We assume furthermore that this value does not change for the whole drop contraction (because the film immediately next to the receding rim is “clean”). As for the other parameters in our model (mass and friction coefficients), we could, in a first approach, consider them to be constant and equal to the previous ones. This is a crude approximation because the rim mass is reduced after the rim breakup and because the viscous effects are a little bit different during the receding stage. Indeed, the receding rim is still a torus of oil moving in water but it goes toward the oil drop and penetrates it, and we should, strictly speaking, consider the damping of strong oscillations of the drop.21,51 Actually, this does not alter very much the expression of the effective viscous friction force in eq 4, but we expect that parameter b is slightly larger than in the previous spreading stage. For the effective rim mass, it is hard to quantify its change due to the rim breakup. What we observe is that a new rim is formed approximately halfway to the receding time, essentially as a result of dewetting processes (like a sea wave after lapping a beach). According to the dewetting theory of liquid sheets, the rim mass growth is expected to be closely linked to the drop radius evolution.16,19 As a consequence, the exponential decrease in the receding parts of the drop pulsations implies an exponential increase in the rim mass during approximately half of the receding time. This increase is then followed by a plateau, the rim returning to its initial mass. In accordance with the previous subsection, we choose the data corresponding to the mean receding part given in Figure 3. The results are given in Figure 7. The red curve (with dots) 14941


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Figure 8. Normalized values of the rim width (green rhombus symbols), the volume ejected by the rim breakup (red squares), and the pulsation period (blue circles) during the regular part of the drop pulsations.

enough, the rim area behaves as the edge of an expanding liquid sheet of width R(t) − R0 (similar to what happens for splashing or strongly oscillating drops). One can then roughly say that the rim breaks up when the size of the nearest pinch point exceeds the capillary length, which amounts to writing the rim breakup condition as Rmax − R0 ≈ κ−1 o/w. As a consequence, we assume that this deformation criterion is valid for our data. It is similar to the contact angle criterion used by the authors of ref 14 but differs from the criterion used in ref 31, which is based on an a priori time modulation of the surface tension coefficient. One has also to estimate the quantity of oil that is ejected at each pulsation so as to determinate the (sequential) evolution of the minimum drop radius. Actually, as illustrated in Figure 4, the ejected part of the oil loss is only a part of the whole volume decrease and cannot be inferred from a single wellidentified mechanism. Figure 2 shows that the minimum radius of the pancake-shaped drop linearly decreases over time, meaning that the volume lost at each pulsation corresponds to a ring of width δR. In the most regular part of Figure 2 (between 6 and 11 s, when the pulsation has a quasi-constant period, see Figure 8), δR is quasi-constant and a rough estimate can be computed: δR = δR0 ≃ 0.09 mm. Outside of this regular band, δR(t) is observed to increase slowly along with the pulsation period T(t). In the present model, the evolution law of δR(t) does not result from a fit but strictly follows the linear decrease of Rmin(t) in Figure 2. Our rather simple mechanical model of the drop pulsations works quite well but it implies, up to this point, a constant period that is not in line with the experimental data. As it is clearly shown in Figure 8, this period broadening is concomitant with the increase in the mass of oil that is driven and ejected at each pulsation. Two main mechanisms help to explain this effect. First, the progressive mass reduction of the drop (because of the successive ejections of oil due to the rim breakup) leads to better floatation of the drop, which means that a larger part of the drop rim emerges from the water.52 As explained above, it is this part of the rim that is ejected at each pulsation. Let us note also that the hydrostatic driving force is all the more reduced

corresponds to the mean experimental data. The green curve is the solution of eq 4 with the same (constant) mass m0 and friction coefficient b0 as for the spreading part. The blue curve is obtained similarly but with a slightly increased friction coefficient (b ≃ 1.5b0 ≃ 0.7 Pa s). The black curve is obtained with a time-dependent mass. m(t) is chosen in accordance with the observations described in the previous paragraph (exponential increase followed by a plateau), and we determined that the exact form of the curve of m(t) is not important: it hardly changes the initial slope of R(t). The main conclusions of these numerical solutions are the following. First, one sees that the reset of the spring parameter k(t) → k0 right after the rim breakup is sufficient to simulate qualitatively the receding parts of the R(t) pulsations curve. Second, and as mentioned above, the friction coefficient has to be increased to account for the damping effect of the contracting drop. With b ≃ 1.45b0, one obtains a very good fit to the experimental data. Third, the initial steep slope of R(t) comes from the strong loss of mass of the rim during its breakup. The closer to zero this mass, the larger the acceleration experienced by the rim. With a simple model of the rim mass evolution, one can get back the steep slope of R(t) (inset of Figure 7). 3.3. Rim Breakup Criterion and Global Model of the Pulsations. To model the whole pulsation curve of Figure 2 or at least a sequence of its regular pulsations, one has to add to the two previous models (of, respectively, the spreading and receding parts) a rim breakup condition. Force, energy, and deformation thresholds are the most common criteria in the literature. By differentiating the experimental curve of Figure 2, we observed that the acceleration and velocity of the rim at the instant of breakup fluctuate a lot, which forced us to conclude that there is no identif iable constant force or energy limit beyond which the rim is ejected. By contrast, Figure 2 shows that the difference between the maximum and minimum drop radii remains approximately constant (equal to 0.94 mm, on the order of the capillary length κ−1 o/w defined in eq ), meaning that the rim breaks up when it is stretched by this length. Such an elongating condition is not surprising for a pancake-shaped drop with a quasi-autonomous rim. When the drop spreads 14942


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Figure 9. Global model of the drop pulsations. The dashed line corresponds to the experimental data whereas the solid red line is the curve resulting from the model detailed in the main text. (Inset) Time modulation of the rim mass (upper red curve) and effective spreading parameter Seff(t) = −k(t)/2π (lower black curve).

spreading parameter Seff(t) and the mass m(t) of the rim are given in the inset of this figure. The modeling curve agrees strikingly well with almost all of the experimental data. As said before, the pulsation period almost doubles over time, with longer and longer spreading parts, and this period drift is also well captured by the model. As for Seff(t), we clearly see that it increases more and more and eventually becomes positive for the last pulsations (i.e., when the drop is too small for the hydrostatic pressure term k0R04/R3 in eq 4 to be strong enough to contribute significantly to drop spreading). In the next section, we discuss the validity of the numbers used in the model, and we see how to improve the model.

when the drop volume is low. At the end of the instability, this pushing force is almost negligible with respect to surface tension forces, and the spreading time is increased. Let us mention that this effect might be simply understood by a dimensional analysis of the drop when it is considered as a whole. In this case, the pulsation period T is expected to scale as (γ/(ρV))1/2, where ρ, V, and γ denote, respectively, a typical density, volume, and surface tension of the system. If we assume that V is the drop volume, then we expect a 2-fold increase in T when the drop volume is 4-fold decreased, which roughly corresponds to the observations between t = 6 and 25 s (Figure 2 and the inset of Figure 4). Even if the quantity (γ/ (ρV))1/2, which is similar to the natural oscillation period of the drop, is 2 orders of magnitude smaller than the observed pulsation period T, this 1/(V)1/2 scaling of T suggests that an alternate model might be relevant for describing the drop pulsations. The second mechanism that may significantly contribute to an increase in the rim mass is the periodic oil loss itself, which leads to a monotonic increase in the surfactant concentration of the drop subsurface, especially near the drop rim. According to the Laplace law, the curvature of this region increases so as to compensate the surface tension γo/w reduction, and a larger rim is created after each ejection. As a consequence, we consider that the initial mass of the rim is not the same during each pulsation cycle. Thus, it should be noted that now the rim mass has two different time dependencies: a strong, exponential one during the receding parts of the pulsations (subsection 3.2) and a soft one on the scale of all of the pulsations. According to Figure 8, one can roughly say that the rim width increases linearly over time during the whole instability, meaning that the rim mass increases quadratically over time (because the volume of a cylinder scales as the square of its diameter), pulsation by pulsation. We can infer the global evolution curve of the rim mass from the curve of Figure 8, but the bad quality of the latter prompts us to boil down to a fit (quadratic in time) the only one of our global model. The modeling of the drop radius evolution of Figure 2 is given in Figure 9. The time modulation of the effective

4. DISCUSSION As seen above, the global variation of Seff(t) over a pulsation period is on the order of 1.5 mN m−1 s−1 and reaches 2 mN m−1 s−1 during the last pulsations. It is interesting to assess the various contributions to this change and determine if the physical mechanisms considered in our modeling are compatible with such a variation. Let us examine, for example, the importance of the oil/water surface tension modification due to surfactant inhomogeneities under the drop and the film surrounding the drop. In the presence of dichloromethane, equilibrium measurements have shown that the aggregates form an oil-in-water microemulsion and that these aggregates appear at a critical aggregation concentration (cac) that is lowered to 0.1 mmol L−1 (instead of 0.9 mmol L−1 in pure water17). Under our experimental conditions, the initial CTAB concentration (0.5 mmol L−1) is higher than the cac, which means that no significant variation of the interfacial tension is expected because of a small variation of the surfactant bulk concentration. This reasoning omits, however, the masstransfer contribution. As already mentioned, the partition coefficient of CTAB largely favors the organic phase. As a result, solute mass transfer leads to the formation of concentration gradients normal to the interface. For simply structured surfactants such as CTAB, the kinetics of adsorption is diffusion-controlled. A local equilibrium can be assumed 14943


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superspreading54−56 cannot be invoked to explain the strong, quasi-exponential expansion of the drop radius. Only the viscous inertial regime of drop impact rapid spreading57,58 bears some similarities to the spreading observed here, without, however, making possible a direct analogy because there is no external energy suddenly provided to the drop in our case. Nevertheless, the rebounds encountered during drop impacts prompt us to reflect on the possible connection between these problems and the experiment presented here. As previously mentioned, the regular oil loss leads to a continuous increase in the surfactant concentration of the drop subsurface, especially near the drop rim. The resulting rigidification of the rim may dampen the drop oscillations and create more stringent conditions for the spreading and breakup of the drop rim. Hysteresis may be introduced to account for such an effect. It is overcome by interfacial waves and Marangoni and other convection flows. Let us eventually mention the torque exerted on the rim (resulting from the increase in the outward pulling force on its top and the decrease in the inward pulling force on its bottom) that helps the rim break and spilling out. Several other effects can be considered to improve the massspring model: by accounting for the disjunction pressure in the film or the ionic nature of the surfactant, for example, or by introducing a plasticity threshold of the rim. To reproduce the rebounds observed in Figure 2, one may also divide the rim into two (upper and lower) parts, with the upper part alone being involved in the oil-ejection process. Finally, one may also take into account the damping effects due to the proper elasticity of the drop. The presence of small surfactant aggregates might also be important in explaining the instability (because the experiment is realized above the cac).

between the surface layer and the subsurface on the aqueous side. In a two-layer system, the mean interfacial tension measured during mass transfer from an aqueous phase of initial concentration 1 mmol L−1 has been measured to be 12 mN m−1 (i.e., far from the value obtained at the cac, 1.5 mN m−1 34). In this domain, dγo/a/dC0 reaches 300 mN m2 mol−1, leading to a huge sensitivity of the interfacial tension to bulk concentration variations. For instance, a local fluctuation reaching 1% of C0 leads to an interfacial tension variation of 1.5 mN m−1. Such a variation may happen in different regions of the experiment at stake. Under the drop rim, for example, four main causes have been identified: (1) that due to the outward flow (Marangoni and coffee ring effects, hydrostatic pressure, and outward convection in the water layer under the drop), which pushes and packs the surfactant molecules in the rim area (leading to a decrease in γo/w); (2) that due to drop stretching (increase in γo/w) during spreading and shrinkage (decrease in γo/w) during the receding parts of the pulsations; (3) that due to the higher evaporation rate close to the rim, which leads to a higher surfactant concentration Cdrop inside the rim (decrease in γo/w); and (4) that due to oil ejections that precisely lower this surfactant concentration (increase in γo/w). Similar effects are also possible in the film surrounding the drop, adding that the ejected oil that spills on the film is strongly laden with surfactant. The oil evaporation, in the drop or in the film, and the surface cooling that follows may also lead to significant changes in γo/a and Seff(t) over a pulsation period. For dichloromethane, one has44 dγo/a dTo/a

= −0.13 mN m−1 K−1 (5)

5. CONCLUSIONS We have shown in this article how to simulate the spontaneous, regular self-pulsations in ref 1 by a simple mass-spring mechanical model of the sessile oil drop. The opposite of the spring modulus can be interpreted as an effective spreading coefficient Seff(t) that oscillates in time with an amplitude equal to 1.5 mN m−1. The time dependence of Seff(t) is compatible neither with the usual spreading (with a constant, positive spreading coefficient) nor with the superspreading of drops deposited on a liquid surface. It has been found that the time increase in the drop radius is quasi-exponential and that Seff(t) should scale as t1.75 to account for this rapid spreading. The model accounts for the regular rim breakup and reformation of the rim mass through a dewetting process. Several improvements in the model have been contemplated and already implemented (a slow increase in the rim mass, for example, to simulate the extension of the pulsation period), but they do not provide a substantially better understanding of the mechanisms at stake. The reasons that Seff(t) may vary with time by 1.5 mN m−1 in one period are numerous: evaporation/ dissolution of both the drop and the film surrounding the drop, outward convection flow due to the positive global spreading parameter, coffee ring effect, dependence of interfacial tension on the dichloromethane concentration, temperature, and surfactant inhomogeneities producing solutal and thermal Marangoni effects. The rigidity and elasticity of the surfactant-covered drop subsurface have been also underlined as playing a non-negligible role. Of course, even if the present model works very well, a deeper and finer modeling of the observed pulsations would

which means that a few degrees of modification of the oil surface temperature To/a can lead to a variation of γo/a on the order of 0.5 mN m−1. Another closely linked effect is the Marangoni friction force that acts on the drop rim. As mentioned above, this inward force appears in the Seff(t) rim center expression as (γcenter − γrim are the o/a o/a)/2, where γo/a and γo/a surface tensions of the (hotter) rim and (colder) drop center, respectively. Usual IR movies of volatile oil drops15,40−43,46,47 show a time variation of almost 2°/s between these two regions, which means a variation in Seff(t) of a few tenths of a mN m−1 that could be invoked to complement the previously explained γo/w variation. As for the γw/a variations, a 0.3 mN m−1 s−1 decrease over the whole instability has been observed far from the drop and the film, which implies a variation of a few tenths of a mN m−1 during one pulsation period. This variation appears to be linked to the slow increase in the number of dichloromethane molecules that are present at the water surface (mainly due to oil ejection). A local increase in γw/a has also to be contemplated because of the evaporation and dissolution of the film surrounding the drop (which pulls on the water surface). The way in which all of these surface tensions depend on time is not clear for now. A linear time increase is expected for such small changes of a few mN m−1, and the particular scaling of Seff(t) as t1.75 is not reproducible by usual evolution laws of To/a(t) (through the equation describing the surface temperature of an evaporating fluid layer53) and of C0(t) (through the kinetics of surfactant adsorption and transfer34). Even drop 14944


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require the solution of the full hydrodynamic equations. Because the particular geometry of the problem allows us to consider the lubrication approximation and axisymmetric variables, the problem should be greatly simplified.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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Mass-Spring Model of a Self-Pulsating Drop - Langmuir (ACS

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