ARTICLE pubs.acs.org/Langmuir
Strain-Induced Yielding in Bubble Clusters Anne-Laure Biance,*,† Adelaide Calbry-Muzyka,‡ Reinhard H€ohler,§,‡ and Sylvie Cohen-Addad§,‡ †
Laboratoire de Physique de la Matiere Condensee et Nanostructures (LPMCN), Universite de Lyon, Universite Claude Bernard Lyon 1, Centre National de la Recherche Scientifique (CNRS), Unite Mixte de Recherche (UMR) 5586, Domaine Scientifique de la Doua, F-69622 Villeurbanne Cedex, France ‡ Laboratoire de Physique des Materiaux Divises et des Interfaces (LPMDI), Universite Paris-Est, FRE 3300 CNRS, 5 Bd Descartes, Champs-sur-Marne, 77454 Marne-la-Vallee Cedex 2, France § Institut des NanoSciences de Paris (INSP), Universite Paris 6, UMR 7588, CNRS-Universite Pierre et Marie Curie (UPMC), 4 Place Jussieu, 75005 Paris, France ABSTRACT: We study how shearing clusters of two or four bubbles induces bubble separation or topological rearrangement. The critical deformation at which this yielding occurs is measured as a function of shear rate, liquid composition, and liquid content in the cluster. We establish a geometrical yield criterion in the quasistatic case on the basis of these experimental data as well as simulations. In the dynamic regime, the deformation where the cluster yields increases with the strain rate, and we derive a scaling law describing this phenomenon based on the dynamical inertial rupture of the liquid meniscus linking the two bubbles. Our experiments show that the same scaling law applies to two- and four-bubble clusters.
’ INTRODUCTION Capillary collapse and pinch-off of liquid ligaments occur in a wide variety of natural phenomena and applications.1,2 Among them is the separation of two gas bubbles that are initially in contact and that are pulled apart. Local bubble interactions have recently been considered to model the macroscopic rheological behavior of liquid foams, which are concentrated bubble suspensions in a soap solution.3,4 Although foams are constituted only of fluids, they respond to a small applied strain by a reversible elastic deformation, like a solid. Dynamic adhesion of two bubbles has been studied in the case of such small deformations,5 and it has been linked to the macroscopic viscoelastic foam response.6 Local interactions are also relevant for bubble coalescence,7 bubble breakup,8 or rearrangements of the bubble packing that set in as soon as the foam structure can relax into a new minimum of the interfacial energy.3,4 These latter events accompany the aging process called coarsening, which is due to the diffusive exchange of gas between neighboring bubbles driven by differences in the Laplace pressure. Bubble rearrangements can also be triggered by a large applied strain that induces shear flow. Local bubble interactions have been predicted to govern the dissipation that accompanies such a deformation as bubbles collide and separate upon rearrangements of the packing.3,4,911 A shear flow is called quasistatic if the evolution of the imposed strain on the time scale of the rearrangements is negligible. However, for large strain rates, this evolution can be significant. As a consequence, the yielding behavior is strain-rate-dependent, in agreement with experiments12 and simulations.13,14 So far, no quantitative model explaining how film and bubble dynamics are linked to macroscopic yielding is available. To check to what r 2011 American Chemical Society
extent the macroscopic foam response can be understood on the basis of binary bubble interactions, a comparison with the behavior of larger bubble clusters is needed. Since rearrangements in 3D foams are complex and difficult to observe, several recent studies have focused on rearrangements in 2D foams15 or in clusters containing a number of bubbles that is small but larger than two.1620 Quasistatic shear experiments with such model systems have revealed that the dilatational viscosity of the surfactant-covered gasliquid interfaces governs rearrangement durations in foams of small liquid content called dry foams.15,17 To provide a basis for future models attempting to explain macroscopic yielding of foams in terms of local bubble interactions, we report a study of the dynamic yielding in two- or fourbubble clusters for large deformations, up to the point where the bubbles lose contact. We show that this separation process involves the rupture of a liquid ligament which is analyzed on the basis of previous studies on liquid jet rupture.1 The paper is organized as follows. In the next part, we describe the experimental setup and protocol. The compositions and physicochemical properties of the surfactant solutions are also given. In the third part, the results of quasistatic yielding experiments with two-bubble clusters are shown and compared with Surface Evolver simulations. In the last part, dynamic yielding of a two-bubble cluster is studied as a function of strain rate, liquid content, and chemistry of the solution. Then we discuss our results and compare them with the behavior of four-bubble clusters. Received: July 21, 2011 Revised: October 21, 2011 Published: November 14, 2011 111
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’ EXPERIMENTS
Once the two bubbles are formed, we reduce the vertical distance H until they touch. At their contact, a flat circular liquid film of radius Rc appears. It is surrounded by a ring of liquid (cf. Figure 1b) that can be compared to a Plateau border in a dry 3D foam. Plateau borders are liquid-filled channels connecting three films.4 Figure 2 illustrates their typical shape, characterized by the radius of curvature RPb. To adjust this parameter, we connect the Plateau border via a syringe needle to a column filled with the surfactant solution, as shown in Figure 1a. Under these conditions, RPb is set by the vertical distance Δh between the liquid level in the column and the position of the contact film, via the balance between hydrostatic and Laplace pressures: σ/RPb = σ/Rc + FgΔh. By analogy with 3D foams, we characterize the liquid content of a cluster by the dimensionless quantity ε = c(RPb/Rb)2, where c is a constant on the order of 0.3.3,4 The studied clusters have liquid contents ranging from 0.001 to 0.02, suggesting that these clusters could be compared to dry 3D foams. Once the desired Plateau border radius of curvature RPb has been established, we retract the syringe needle and translate the bottom tube with the motorized platform at a fixed velocity v chosen in the range between 0.15 and 1500 mm 3 s1 so that the bubbles are sheared in a way that imitates the bubble deformation in a sheared 3D foam. The film at the contact between the two bubbles shrinks progressively, and finally, the bubbles separate, as illustrated in Figure 3a. We use a high-speed camera as described above to study the detachment dynamics of the two bubbles. When the film joining the bubbles disappears, the connection is established only by a liquid ligament (Figure 3c, picture 4). It finally ruptures near its contacts with the bubbles (Figure 3c, pictures 5 and 6), and one or several satellite droplets are detached (Figure 3c, pictures 7 and 8). This is similar to the pinch-off of a rupturing meniscus falling from a faucet,22 to a meniscus elongated between two plates,2,23 or to a soap film streched between two rings.24,25 The horizontal displacement of the mobile tube at the instant of rupture (Figure 3c, picture 4) is denoted Lr. Moreover, we define the diagonal of rupture Dr illustrated in Figure 4 as Dr = (Lr2 + H2)1/2. In the following, we study the variations of Dr with the entrainment velocity v and we identify two cases: a quasistatic one at low velocity and a dynamic one that is strain-rate-dependent.
The setup, shown in Figure 1a, consists of two vertical cylindrical tubes (outside diameter 3.2 mm) facing each other at a vertical distance H which can be adjusted. The top tube is immobile, whereas the bottom tube is fixed on a motorized platform that can be translated horizontally at a controlled velocity v. At the beginning of each experiment, a soap film is deposited at the end of each tube. Then two bubbles of identical volume are inflated by injecting a controlled amount of air through the tubes using a syringe pusher. The obtained bubble radius Rb is measured by videoscopy. Sequences of images are recorded by a high-speed camera (1000 frames per second), on which a macro objective is mounted. The light of a halogen lamp is projected on a diffusive screen to obtain a homogeneous background. We use four surfactant solutions of different liquid and interfacial viscosities. Their physicochemical characteristics are specified in Table 1. All measurements are performed at a temperature of 23 ( 2 °C.
Figure 1. (a) Experimental setup for the two-bubble cluster. (b) Cluster of two bubbles in contact as described in the text.
Figure 2. Schematic view of the Plateau border separating the two bubbles.
Table 1. Foaming Solutions and Their Characteristicsa solution
TTAB concn(g/g)
glycerol concn (g/g)
A B
0.003 0.003
0.1 0.4
C
0.003
0.6
D
0.003
0.1
LOH concn (g/g)
0.0004
σ (mN/m)
η (mPa 3 s)
F (kg/m3)
k (mPa 3 m 3 s)
32 ( 1 29 ( 1
1.3 3.6
1030 1100
0.095 0.13
28 ( 1
10.7
1160
23 ( 1
1.3
1030
0.25 40
a
The foaming solutions used in the experiments contain the following chemicals: TTAB (tetradecyltrimethylammonium bromide; Sigma, 99%, cat. no. T4762), LOH (1-dodecanol; Fluka, purum, 97%, GC, cat. no. 44110), glycerol (Fluka, anhydrous p.a., >99.5%, GC, cat. no. 49770). The surfactants are used as received and dissolved in mixtures of water (Millipore Milli-Q) and glycerol by stirring and heating at 60 °C overnight. The concentrations are defined as the mass of the added chemical divided by the total mass of the solution. The viscosity η has been determined with a rheometer in a plateplate geometry (Anton Paar), and the surface tension σ has been measured by the pendant drop method (TECLIS Tracker). The surface dilatational viscosity k has been measured with a precision of 10% using the oscillating bubble technique (TECLIS Tracker) at a frequency of 0.1 Hz and a dilatation amplitude of 2.5% for a soap bubble radius of 1 mm. The density F of each surfactant solution is supposed to be equal to that of a waterglycerol mixture with the same concentration (wt %) of glycerol published in the literature.21 112
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Figure 5. Diagonal of rupture Dr versus bubble size Rb. The plate velocity v = 0.15 mm 3 s1, RPb = 0.5 mm, and the vertical distance between the tubes H is in the range of 6.88.8 mm. Solution A is represented by solid circles, solution C by open squares, and solution D by open circles. Surface Evolver simulations with no liquid in the Plateau border are represented by open triangles. The continuous line is a linear fit to Surface Evolver simulation results (Dr = 5.25 mm + 5.57Rb), and the dashed line has the equation Dr = 4.45 mm + 5.57Rb. Simulation results with liquid in the Plateau border are represented by open tilted squares for liquid contents ε increasing from bottom to top between 0.006 and 0.06.
Figure 3. (a) Shape of a bubble cluster made with solution A (RPb = 0.5 mm), strained up to the bubble separation. Pictures are taken at times t = 0.00, 1.49, 8.09, and 9.02 s. The speed of the lower tube is v = 1.25 mm/s. (b) The experiment illustrated in (a) is simulated using the Surface Evolver software. (c) Snapshots of the bubble separation process, taken at time intervals of 0.55 ms (RPb = 0.5 mm). The solution and velocity v are identical to those in (a). Picture 1 is taken at t = 8.28 s.
determined as a function of both of these parameters. We perform this calculation numerically, since the interfacial area and energy are difficult to deduce from our video observations. Surface Evolver Simulation. Cluster Geometry and Diagonal of Rupture. To simulate the bubble shapes observed under quasistatic conditions, we use the Surface Evolver software.26 The interfaces are approximated as unions of triangular facets which are adjusted iteratively to converge toward a configuration of minimal total interfacial area under given boundary conditions. The two bubbles are simulated as two connected bodies of fixed volume that are each pinned on circular boundaries, representing the tubes. The hypothesis of constant volume is in good agreement with our experiments because the dead volume of gas in the experimental setup is small. The shear deformation is implemented by alternating interfacial area minimization and small horizontal displacements of the lower circular boundary. Finally, under the effect of the increasing imposed shear, the film connecting the two bodies shrinks to a point, indicating that rupture is imminent. The bodies representing the bubbles are then disconnected, and each one converges to a spherical shape. A comparison between parts a and b of Figure 3 shows that this simulated evolution of the cluster geometry closely matches the experimental observations. Moreover, Figure 5 shows that the observed and simulated dependencies of the diagonal of rupture Dr on the bubble radius are similar. In the range of the simulation results, the diagonal of rupture varies also linearly with the bubble radius Rb. The data can be fitted by the linear relationship Dr = 5.25 mm + 5.57Rb. However, the simulated values differ from the measured ones by a small offset. Two origins of this discrepancy can be identified: the finite liquid content of the Plateau border around the central film and the liquid menisci at the junctions between one bubble and its tube. The effect of liquid in the central Plateau border is explored using Surface Evolver simulations. To do so, a third liquid-filled body is introduced around the bubble contact face. As shown in Figure 5, the offset is reduced as the liquid content increases. The effect of liquid in the junction between one bubble and its tube can be taken into account by adding to Dr the lengths of these two liquid menisci, which are estimated at 0.4 mm each. The simulation
Figure 4. Schematic view of the bubble cluster illustrating the diagonal of rupture Dr discussed in the text.
’ SMALL STRAIN RATES: QUASISTATIC YIELDING Experimental Results. We measure the diagonal of rupture Dr as a function of the bubble radius Rb for the lowest plate velocity accessible with our setup (v = 0.15 mm 3 s1), a vertical distance between the tubes H in the range from 6.8 to 8.8 mm, and a Plateau border radius of curvature RPb in the range from 0.4 to 0.6 mm. The results obtained for foaming solutions A, C, and D shown in Figure 5 collapse on a single curve, indicating that the solution composition has no impact on the results. This is expected in a quasistatic experiment where variations of surface and bulk viscosities should not matter. Between A and C, the liquid viscosity η is multiplied by a factor of 8, the surface viscosity k being almost constant. Between A and D, η is kept constant while k is increased by a factor of 400. Complementary experiments show that the diagonal of rupture Dr does not depend on the Plateau border radius of curvature in the range between 0.15 and 0.5 mm. All these data show a consistent almost linear increase of the diagonal of rupture with the bubble radius. We measure the deformation of the cluster with respect to a reference state where the elastic energy is minimal. The interfacial energy is a function not only of the shear displacement Lr, but also of the distance H, which controls whether the two bubbles are compressed or elongated in the vertical direction. Therefore, the interfacial energy and the reference state must be 113
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Figure 6. (a) Normalized energy of the bubble cluster simulated using the Surface Evolver software versus the height H. The energy is normalized by the surface energy of a single spherical bubble, i.e., σ4πRb2. The simulations are performed without taking into account the liquid in the Plateau border for a bubble radius Rb = 2.7 mm. (b) The distance H0 corresponding to the energy minimum is represented versus the bubble radius Rb, and it is fitted by the expression H0 = 3.82Rb 2.70 mm.
linear fit shifted by 0.8 mm is in very good agreement with our experimental data as shown in Figure 5. Rupture Criterion. Figure 6a shows the interfacial energy of the bubble cluster before shearing as a function of the tube separation H. When the tube separation is large, the bubbles are stretched, which results in an energy increase. When, on the contrary, the tubes are close to each other, the bubbles are squeezed, which also results in an energy increase. Between these two cases, there is an energy minimum, characterized by a tube distance H = H0, which is a linear function of Rb. This behavior can be qualitatively understood by a simple geometrical argument, assuming that the bubbles are truncated spheres with a central contact film. The radius of each sphere is imposed by volume conservation and by the separation distance H. Under these hypotheses, the total surface of the cluster is found to be minimal for a distance H0 = 25/3Rb. This linear relationship is consistent with the simulations, but it slightly overestimates H0 because the assumed spherical bubble geometry is too schematic. We use the characteristic length H0, deduced from the simulation, to construct a dimensionless diagonal of rupture, Dr/H0. Experimentally, in the quasistatic regime, all observed ruptures occur for the same value of (Dr/H0)QS, equal to 1.80 ( 0.04. This provides a rupture criterion. Alternatively, it can also be expressed in terms of a yield strain γc, but this parameter depends on the vertical stretching or squeezing, described by the ratio H/H0. For H/H0 = 1, we find γc = Lr/Ho = 1.5. This value is larger than the yield strain of dry foams, typically found in the range of 0.10.3,10,12 presumably due to the geometrical constraints in a 3D bubble packing. To summarize, we have shown that, in the quasistatic regime, the shape of the two-bubble cluster is entirely determined by surface minimization up to the yielding point. This must be distinguished from previous studies on 2D foams15 or fourbubble clusters17 focused on the duration of the rearrangement following the yielding point. In the quasistatic regime, the dynamics of the rearrangement are independent of the entrainment velocity and driven by the interplay between surface viscoelasticity and surface tension. In the next part, we study how the shape of the cluster at the rupture point varies for large entrainment velocities.
Figure 7. Diagonal of rupture divided by the reference height Dr/H0 versus v/H0 for the different solutions described in Table 1: A (O), B (]), C (b), D (4). RPb is 0.5 mm, and Rb varies between 2.1 and 2.7 mm. The continuous line corresponds to the empirical law in eq 1 with the fitted parameters given in the text. The dashed line represents eq 7 with T = 5 ms.
’ DYNAMIC YIELDING Experimental Results. In this section we show how the yielding of a bubble cluster depends on the shear velocity v. Experimental results for solutions AD specified in Table 1 whose bulk and surface viscosities vary over a wide range are reported in Figure 7. The shear velocity is normalized by H0 to obtain a quantity comparable to a shear rate. Remarkably, the data follow the same empirical law for all of the investigated solutions:
Dr ¼ H0
Dr H0
þ QS
Tv H0
α
ð1Þ
with (Dr/H0)QS = 1.8 as measured in quasistatic experiments in the previous section. The fitted parameters are T = 4.3 ( 0.5 ms and α = 0.45 ( 0.10. This relationship is sublinear with respect to the plate velocity v. The characteristic time T observed here is of the same order of magnitude as the pinch-off time of a millimetric 114
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pinch-off tr, the decrease of the ligament radius due to this effect is given by the following relation, where a is a dimensionless constant equal to 0.11 ( 0.04:24,27,31 1=3 σ rðtÞ = a ðtr tÞ2=3 ð3Þ F Since the two mentioned mechanisms are combined, we compare in Figure 8 the thinning due to stretching (eq 2) to the thinning due to the inertial instability (eq 3) for typical values of Ω, Rb, and tr, assuming that the bubble remains spherical. At short times, we expect stretching to be the dominant mechanism since it is the faster one. However, prior to pinch-off, the thinning due to Rayleigh instability should drive the dynamics. Within this simplified picture, the crossover between the two cases occurs when the rates of thinning are equal. This happens at a time tc and for a radius rc where the curves described by eqs 2 and 3 meet with a common tangent (cf. Figure 8). By combining eqs 2 and 3 and their time derivatives, tc and rc read 3=7 2=7 Ω F ð4Þ tc ≈ v σ
Figure 8. Evolution of the meniscus radius versus time. The thick red line corresponds to volume conservation (eq 2) with vD = 0.4 m/s and Ω = 1010 m3 corresponding to a Plateau border radius of 0.21 mm and a bubble radius of 2.7 mm. The thin blue lines illustrate how the radius locally shrinks to zero due to a Rayleigh instability, according to eq 3 with a = 0.1131 and σ = 29 mN/m, for different tr values (0.2, 0.3, and 0.4 ms). The second of these curves matches the red line at a time denoted tc. The inset is a sketch of the Plateau border (dark gray) between the bubbles (light gray) just before separation.
liquid cylinder.27 We now discuss the increase of Dr/H0 with v/H0 that sets in at higher velocities. Analysis and Discussion. To predict Dr as a function of shear velocity, the dynamics of the bubble separation must be analyzed. As described above, a liquid ligament is formed between the bubbles and stretched up to its pinch-off (cf. pictures 1 and 4 in Figure 3c). This is reminiscent of the rupture dynamics of a liquid meniscus connecting two solid surfaces, which has been studied experimentally2,28 and theoretically.23,24,29 In all these cases, the rupture dynamics are governed by a balance between surface tension, which is the driving force, and a resistive force that can be due to bulk viscous friction, surface viscous friction, or inertia. Since in our experiments a variation of the bulk viscosity by an order of magnitude does not significantly affect the measured diagonal of rupture for a given shear velocity, we deduce that bulk viscous friction cannot be the dominant resistive force. The same remark applies to surface viscous friction since a variation by 2 orders of magnitude of the dilatational interfacial viscosity does not have any significant effect either. We therefore consider that the rupture dynamics are driven by a balance between inertial forces and surface tension. For a cylindrical jet of radius r, rupture via such a Rayleigh instability arises on a time scale τ = (Fr3/σ)1/2.24,30 If r is on the order of 0.2 mm, which corresponds to the meniscus radii that we observe just before pinch-off, this yields τ ≈ 0.5 ms, consistent with the time interval between snapshots 3 and 4 of Figure 3c. In our experiments, the ligament thins down as it is stretched due to liquid volume conservation. At the same time, the Rayleigh instability develops. To determine when it becomes predominant, we now analyze these two thinning dynamics using a scaling approach. We describe the ligament as a liquid cylinder of radius r and of height h. As h increases at a constant velocity vD (cf. the inset of Figure 8), conservation of the volume Ω = πr2h yields sffiffiffiffiffiffiffiffiffi Ω rðtÞ ¼ ð2Þ πvD t
and 2=7 1=7 Ω σ rc ≈ v F
ð5Þ
To calculate the rupture time tr, we substitute eqs 4 and 5 into eq 3. This yields 3=7 2=7 Ω F ð6Þ tr ≈ v σ tr must be proportional to tc since only one characteristic time can be constructed with these physical hypotheses. The length at which the ligament pinches off is given by hr = vDtr. vD is directly related to the velocity of the plate v by vD = v, which is confirmed experimentally. Since Dr = Dr(v = 0) + hr, the diagonal length where pinch-off occurs reads 4=7 Dr Dr vT ¼ þ ð7Þ H0 H0 H0 QS with T≈
Ω H0
3=4 1=2 F σ
ð8Þ
The characteristic time T depends on the liquid surface tension and density, on the liquid content in the cluster, and on the bubble radius. This prediction is in good agreement with experimental data shown in Figure 7. Indeed, the predicted exponent 4/7 is close to the value α = 0.45 of the empirical law in eq 1. Moreover, for the results shown in Figure 7, eq 8 predicts T to be on the order of 1 ms. This is the same order of magnitude as the value T = 5 ms deduced by fitting eq 7 to the data. Pinch-off lengths and times have previously been discussed in the literature but not in the case of ligaments between bubbles. Some authors report no dependence on the pinch-off time with the deformation velocity,2 but the range of velocities remains small in these studies. On the contrary, ligaments stretched between two solid spheres show a length of rupture
Besides this thinning, the cylinder is also subject to the Rayleigh instability. For time t close to the instant of the final 115
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Figure 9. Characteristic time T versus Plateau border radius RPb. The symbols correspond to data obtained with solution D (cf. Table 1). The line corresponds to eq 10 with T = (1.7 ( 0.1)(Rb/H0)3/4(F/σ)1/2RPb3/2. The bubble radius Rb = 2.6 ( 0.1 mm, and H0 is 7.2 mm.
Figure 10. (a) Experimental setup for the four-bubble cluster as described in ref 17. (b) Structure obtained after a topological rearrangement where bubbles 1 and 4 lose contact.
that scales with the square root of the velocity,28 which is close to our result (eq 7). Effect of the Liquid Content. The liquid content is set at the beginning of each experiment by injecting/withdrawing liquid from the Plateau border between the two bubbles. A geometrical calculation relates Ω to Rc and Rb: pffiffiffi π ð9Þ Rc RPb 2 Ω = 2π 3 2 In addition, Surface Evolver simulations performed on the twobubble cluster without a Plateau border show that the initial contact radius Rc is proportional to the initial bubble radius: Rc = 0.82Rb. For a finite Plateau border volume, this relation still holds provided that RPb , Rb. Consequently, Ω can be determined directly from a measurement of Rb for a given RPb. Equation 6 implies that the quantity of liquid in the Plateau border Ω has an impact on the time T defined in eq 7. If Ω is expressed in terms of Rb and RPb, using eqs 9 and 8, T reads
Rb T≈ H0
3=4 rffiffiffi F RPb 3=2 σ
Figure 11. (a) Image sequence showing the yielding in a four-bubble cluster. A ligament rupture is clearly visible in the encircled region. The picture height is 1.5 mm. (b) Enlarged view of the ligament. (c) Dynamic yield strain versus strain rate for a two-bubble cluster () with RPb = 0.5 mm and for a four-bubble cluster (0) with RPb = 0.4 mm. The solution used is D (cf. Table 1). The straight line has a slope of 4/7, predicted by eq 7.
ð10Þ
In Figure 9, we compare this prediction to our experimental data: The characteristic time T obtained as T = (Dr/H0 (Dr/H0)QS)7/4/(v/H0) is plotted versus RPb. Two regions can be identified. At low liquid content or for RPb e 0.4 mm, T diverges sharply. In this region, ligament rupture is in competition with bubble breakup or with bubble coalescence,7 another mechanism whose discussion is beyond the scope of this paper. At larger liquid contents, an increase of T is observed in qualitative agreement with eq 10. A refined calculation should take into account the nonspherical shape of the bubbles. Note that the minimal Plateau border radius of curvature RPb = 0.4 mm where the range of validity of our model begins corresponds to a very dry cluster that would be comparable to a dry 3D foam with liquid content ε = 0.003. To summarize, we have shown that dynamic yielding is characterized by the shape of the two-bubble cluster that strongly depends on the entrainment velocity. In this regime, the diagonal length at the rupture point increases with the velocity following a power law which results from a balance between surface tension and inertia. In the next paragraph, we study the shape of
four-bubble clusters and evidence similarities with the twobubble cluster case. Four-Bubble Clusters. We have performed shear experiments with four-bubble clusters. The bubbles are attached to two groups of two cylinders facing each other, and the cluster is strained to induce a topological rearrangement where bubbles exchange neighbors, as described in ref 17 and sketched in Figure 10. We study the dynamic regime where the deformation of the cluster reached at the yielding point depends on the entrainment velocity. This also corresponds to the regime where the duration of the rearrangement itself increases with the entrainment velocity.17 The evolution of the cluster is different from that observed in the two-bubble case since the bubbles are not entirely disconnected. Instead, they switch neighbors within the cluster, and a new film is formed. However, parts a and b of Figure 11 reveal that the bubbles remain linked via a liquid ligament inside the freshly formed film. This ligament finally pinches off at its two ends, much as in the two-bubble case. 116
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In analogy with this case, we define a yield strain γc based on this instant of ligament pinch-off and a strain rate γ_ for four-bubble clusters. We choose the value of H0 defined in the two-bubble case as the reference state for the four-bubble cluster. γc is determined as the diagonal joining two initially opposite tubes at the instant of pinch-off divided by H0. γc0 is defined as γc in the quasistatic limit. The strain rate γ_ is the entrainment speed v divided by H0. The data presented in Figure 11c show good agreement between the yielding behavior of two- and fourbubble clusters of the same bubble radius, which both follow the scaling predicted by eq 7. This finding raises the question of whether macroscopic foams with comparable bubble sizes and liquid contents would show the same behavior, but such data are so far not available in the literature.
(7) Biance, A.-L.; Delbos, A.; Pitois, O. How topological rearrangements and liquid fraction control liquid foam stability. Phys. Rev. Lett. 2011, 106 (6), 068301. (8) Golemanov, K.; Tcholakova, S.; Denkov, N. D.; Ananthapadmanabhan, K. P.; Lips, A. Breakup of bubbles and drops in steadily sheared foams and concentrated emulsions. Phys. Rev. E 2008, 78 (5), 051405. (9) Tcholakova, S.; Denkov, N. D.; Golemanov, K.; Ananthapadmanabhan, K. P.; Lips, A. Theoretical model of viscous friction inside steadily sheared foams and concentrated emulsions. Phys. Rev. E 2008, 78 (7), 011405. (10) H€ohler, R.; Cohen-Addad, S. Rheology of liquid foam. J. Phys.: Condens. Matter 2005, 17 (41), R1041–R1069. (11) Cohen-Addad, S.; H€ohler, R. Bubble dynamics relaxation in aqueous foam probed by multispeckle diffusing-wave spectroscopy. Phys. Rev. Lett. 2001, 86 (20), 4700–4703. (12) Rouyer, F.; Cohen-Addad, S.; Vignes-Adler, M.; H€ ohler, R. Dynamics of yielding observed in a three-dimensional aqueous dry foam. Phys. Rev. E 2003, 67 (2), 021405–7. (13) Kraynik, A. M.; Hansen, M. G. Foam rheology: A model of viscous phenomena. J. Rheol. 1987, 31 (2), 175–205. (14) Cantat, I. Gibbs elasticity effect in foam shear flows: A non quasi-static 2d numerical simulation. Soft Matter 2010, 7, 448–455. (15) Durand, M.; Stone, H. A. Relaxation time of the topological T1 process in a two-dimensional foam. Phys. Rev. Lett. 2006, 97 (22), 226101. (16) Hutzler, S.; Saadatfar, M.; van der Net, A.; Weaire, D.; Cox, S. J. The dynamics of a topological change in a system of soap films. Colloids Surf., A 2008, 323 (13), 123–131. (17) Biance, A.-L.; Cohen-Addad, S.; H€ ohler, R. Topological transition dynamics in a strained bubble clusters. Soft Matter 2009, 5, 4672–4679. (18) Bohn, S. Bubbles under stress. Eur. Phys. J. E 2003, 11 (2), 177–189. (19) Cox, S. J.; Weaire, D.; Vaz, M. F. The transition from twodimensional to three-dimensional foam structures. Eur. Phys. J. E 2002, 7 (4), 311–315. (20) Rosa, M. E.; Fortes, M. A. Uniaxial deformation tests of aqueous foams: Rate effects and topological changes. Philos. Mag. 2006, 86 (32), 4997–5007. (21) Dorsey, N. E. Properties of Ordinary Water Substance; Reinhold: New York, 1940. (22) Shi, X. D.; Brenner, M. P.; Nagel, S. R. A cascade of structure in a drop falling from a faucet. Science 1994, 265 (5169), 219–222. (23) Eggers, J. Universal pinching of 3d axisymmetric free-surface flow. Phys. Rev. Lett. 1993, 71, 3458–3460. (24) Chen, Y. J.; Steen, P. H. Dynamics of inviscid capillary breakup: Collapse and pinchoff of a film bridge. J. Fluid Mech. 1997, 341, 245–267. (25) Robinson, N. D.; Steen, P. H. Observations of singularity formation during capillary collapse and bubble pinch-off of a soap-film bridge. J. Colloids Interface Sci. 2001, 241 (6), 448–458. (26) Brakke, K. A. The Surface Evolver. Exp. Math. 1992, 1, 141. (27) Lord Rayleigh. On the instability of jets. Proc. London Math. Soc. 1879, 10, 4–13. (28) Pitois, O.; Moucheront, P.; Chateau, X. Liquid bridge between two moving spheres: An experimental study of viscosity effects. J. Colloid Interface Sci. 2000, 231, 26–31. (29) Frankel, I.; Weihs, D. Stability of a capillary jet with linearly increasing axial velocity (with application to shaped charges). J. Fluid Mech. 1985, 155, 289–307. (30) de Gennes, P. G.; Brochard, F.; Quere, D. Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves; Springer: New York, 2004. (31) Clanet, C.; Lasheras, J. C. Transition from dripping to jetting. J. Fluid Mech. 1999, 383, 307–326.
’ CONCLUSION We report a study of dynamic yielding of strained clusters of two bubbles. These experiments evidence a critical deformation at which the bubbles separate. At low strain rates, we establish on the basis of experiments and simulations a geometrical criterion for this critical deformation that can be expressed as a yield strain. With increasing strain rate, this yield strain rises and we identify a regime of large liquid contents and bubble sizes where dynamic yielding is dominated by the interplay between surface tension and inertial forces. On the basis of a simple physical model, we derive a scaling law that successfully describes our experimental findings. This law also describes the separation of bubbles in fourbubble clusters undergoing a strain-induced topological rearrangement, suggesting that similar dynamics might be relevant for the yielding of 3D foams. To make further progress toward a full understanding of yielding in bubble clusters and foams, it will be necessary to determine the conditions in which separation mechanisms based either on inertia or on viscous friction prevail or shear-induced bubble rupture occurs. ’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT We thank H. Suzin and J. Laurent for technical support. We acknowledge financial support from the French Space Agency (Convention CNES/CNRS Number 103954) and the European Space Agency (Contracts MAP AO 99-108 and 14914/02/NL/SH). ’ REFERENCES (1) Eggers, J.; Villermaux, E. Physics of liquid jets. Rep. Prog. Phys. 2008, 71, 036601. (2) Marmottant, P.; Villermaux, E. Fragmentation of stretched liquid ligaments. Phys. Fluids 2004, 16 (8), 2732–2741. (3) Cantat, I.; Cohen-Addad, S.; Elias, F.; Graner, F.; H€ohler, R.; Pitois, O.; Rouyer, F.; Saint-Jalmes, A. Les Mousses. Structure et Dynamique; Belin (Echelles): Paris, 2010. (4) Weaire, D.; Hutzler, S. The Physics of Foams; Clarendon Press: Oxford, U.K., 1999. (5) Besson, S.; Debregeas, G. Statics and dynamics of adhesion between two soap bubbles. Eur. Phys. J. E 2007, 24 (2), 109–117. (6) Besson, S.; Debregeas, G.; Cohen-Addad, S.; H€ohler, R. Dissipation in a sheared foam: From bubble adhesion to foam rheology. Phys. Rev. Lett. 2008, 101 (21), 214504. 117
dx.doi.org/10.1021/la202817t |Langmuir 2012, 28, 111–117
