Effects of Soluble Surfactants on the Deformation and Breakup of

Surfactants are routinely used to control the breakup of drops and jets in many applications such as inkjet printing, crop spraying, and DNA or protei...
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Effects of Soluble Surfactants on the Deformation and Breakup of Stretching Liquid Bridges Ying-Chih Liao, Hariprasad J. Subramani, Elias I. Franses, and Osman A. Basaran* School of Chemical Engineering, Purdue University, 480 Stadium Mall Drive, West Lafayette, Indiana 47907-2100 Received May 14, 2004. In Final Form: August 24, 2004 Surfactants are routinely used to control the breakup of drops and jets in many applications such as inkjet printing, crop spraying, and DNA or protein microarraying. The breakup of surfactant-free drops and jets has been extensively studied. By contrast, little is known about the closely related problem of interface rupture when surfactants are present. Solutions of a nonionic surfactant, pentaethylene glycol monododecyl ether, or C12E5, in water and in 90 wt % glycerol/water are used to show the effects of surfactant and viscosity on the deformation and breakup dynamics of stretching liquid bridges. Equilibrium surface tensions for both solutions can be fitted with the Langmuir-Szyskowski equation. All experiments have been done at 24 °C. The critical micelle concentrations for C12E5 are 0.04 and 0.4 mM in water and the glycerol/water solution, respectively. With high-speed imaging, the dynamic shapes of bridges held captive between two rods of 3.15 mm diameter are captured and analyzed with a time resolution of 0.1-1 ms. The bridge lengths are 3.15 mm initially and about 5-7 mm at pinch-off. Breakup occurs after stretching for about 0.2-0.3 s, depending on the solution viscosity and the surfactant concentration. When the liquid bridges break up, the volume of the sessile drop left on the bottom rod is about 3 times larger than that of the pendant drop left on the top rod. This asymmetry is due to gravity and is influenced by the equilibrium surface tensions. Surfactant-containing low-viscosity water bridges are shown to break up faster than surfactant-free ones because of the effect of gravity. With or without surfactant, water bridges form satellite drops. Surfactant-containing high-viscosity glycerol/water bridges break up more slowly than surfactantfree ones because of strong viscous effects. Moreover, the shapes of the sessile drops close to breakup exhibit a “pear-like” tip; whether a satellite forms depends on the surface age of the bridge before stretching commences. These unexpected effects arising from the addition of surfactants are due to the capillary pressure reduction and Marangoni flows linked to dynamic surface tension.

1. Introduction The stability and breakup of cylindrical liquid columns and threads are important in many established and emerging industrial applications involving drop or jet breakup,1 such as inkjet printing,2 crop spraying,3 and DNA and protein microarrays.4 Liquid bridges, which are geometrically simpler than drops and jets, are commonly used for studying the stability and breakup dynamics of liquid threads.5 A liquid bridge is a liquid column held captive between two coaxial solid disks. When one or both disks are set into motion away from each other, the liquid column is stretched until it finally breaks up, creating a sessile drop on the bottom disk and a pendant drop on the top disk. The volume ratio of the resulting drops depends on many parameters, such as the liquid’s viscosity, its equilibrium and dynamic surface tension, the stretching speed, and geometrical factors. Furthermore, during the rupture process, satellite drops, which are undesired in inkjet printing and microarraying applications, may be formed.6 The breakup dynamics of surfactant-free bridges has been studied experimentally and theoretically for several * Corresponding author. E-mail: [email protected]. Tel: 765-494-4061. Fax: 765-494-0805. (1) Basaran, O. A. AIChE J. 2002, 48 (9), 1842. (2) Kuhn, L.; Myers, R. A. Sci. Am. 1979, 240 (4), 162. (3) Wirth, W.; Strop, S.; Jacobsen, W. Pestic. Sci. 1991, 33, 411. (4) Goldman, T.; Gonzalez, J. S. J. Biochem. Biophys. Methods 2000, 43 (3), 105. (5) Zhang, X.; Padgett, R. S.; Basaran, O. A. J. Fluid Mech. 1996, 329, 207. (6) Notz, P. K.; Chen, A. U.; Basaran, O. A. Phys. Fluids 2001, 13, 549.

decades.5,7-9 Since surface tension plays an important role in the breakup dynamics of liquid threads, effects of the addition of surfactants are crucial to many industrial processes and need to be elucidated. Fundamental understanding of surfactant-induced flows in the breakup dynamics is lacking because of the experimental difficulties in probing the surface density and bulk concentration profiles directly and in real time on a rapidly moving and deforming interface. Instead of such experiments, theoretical models, which couple the hydrodynamics and interfacial mass transfer, have been developed to help understanding of the mass transfer and dynamic surface tensions indirectly. Dynamic shapes and breakup times of bridges, which are affected by the addition of surfactants, can be used to examine the accuracy and consistency of these models. Recently, models based on a rigorous finite element algorithm have been shown to agree well with dynamic shapes for surfactant-free liquid systems with different geometries.1,10,11 By extension of these models to include interfacial mass transfer effects, the surfactant effects can be predicted accurately.12 In this letter, we report the first detailed data for liquid bridges of water and glycerol/water solutions containing a nonionic surfactant, pentaethylene glycol monododecyl ether. New surface tension data for the glycerol/water (7) Chen, T.-Y.; Tsamopoulos, J. J. Fluid Mech. 1993, 255, 373. (8) Padday, J. F.; Petre, G.; Rusu, C. G.; Gamero, J.; Wozniak, G. J. Fluid Mech. 1997, 352, 177. (9) Yildirim, O. E.; Basaran, O. A. Chem. Eng. Sci. 2001, 56, 211. (10) Chen, A. U.; Notz, P. K.; Basaran, O. A. Phys. Rev. Lett. 2002, 88 (17), 174501. (11) Wilkes, E. D.; Phillips, S. D.; Basaran, O. A. Phys. Fluids 1999, 11, 3577. (12) Liao, Y.-C. Ph.D. Thesis, Purdue University, West Lafayette, IN, 2004.

10.1021/la0487949 CCC: $27.50 © 2004 American Chemical Society Published on Web 09/28/2004

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solutions show typical surfactant adsorption and micellar behavior, qualitatively similar to that in water. Highspeed digital imaging with high spatial resolution allows accurate visualization of surfactant and viscosity effects on liquid bridge deformation and breakup. 2. Experimental Section 2.1. Materials. A nonionic alkyl ethylene oxide surfactant, pentaethylene glycol monododecyl ether, or C12E5, of purity 99+%, was purchased from Fluka (Milwaukee, WI). The water used for all samples was first distilled and then passed through a Millipore four-stage cartridge system, resulting in a water resistivity of 18 MΩ‚cm at the exit port. Glycerol of purity 99+% was purchased from Sigma Chemical Co. (St. Louis, MO). The C12E5/glycerol/ water solutions were prepared by mixing C12E5/water solutions with glycerol. The viscosities of water and glycerol/water (90 wt % glycerol) solutions were 0.98 and 165 mPa‚s, respectively, measured with a RS-150 rheometer from ThermoHaake (Karlsruhe, BW, Germany). The specific gravity of the glycerol/water solutions was 1.23. Addition of surfactant had a negligible effect on the density and viscosity. 2.2. Apparatus and Procedures. A Langmuir trough, from KSV Instruments, Finland, with a roughened platinum Wilhelmy plate, was used to obtain the equilibrium surface tensions of C12E5 in water or glycerol/ water. The solutions were placed in a glass beaker, which has a diameter and height of 5 cm. The temperature was controlled at 24 ( 0.5 °C. The surfactant-free solution had constant and reproducible surface tension. The surfaces of surfactant solutions were first aspirated to ensure chemical purity. The surface tension decreased with time until it became steady for at least 1 h. The steady-state value is defined as the equilibrium surface tension (EST). Since the apparatus for liquid bridge experiments has been described previously,11,13 a brief account is given here. The liquid, of volume V ) 24.5 µL in all experiments here, is vertically confined between two stainless steel rods of 3.15 mm diameter to form a liquid bridge of length L ) 3.15 mm. The bottom rod remains fixed. The upper rod is connected to a computer-controlled Newport (Irvine, CA) translation stage. Initially, the upper rod moves with an initial speed of 0.025 cm/s and a constant acceleration of 10 cm/s2 until it reaches 2 cm/s, after which the socalled stretching speed U is constant. A dual imager setup is used to ensure the axial symmetry of the liquid bridges. The primary camera is a Kodak Motion Corder Analyzer SR Ultra, capable of recording from 30 to 10 000 frames per second. Evaporation is found to be insignificant for glycerol/water (the volume decreased by less than 5% after 5 h) but is more important for water (the volume decreased by 20% after 1 h). After the formation of liquid bridges, about 5 min for water solutions and about 60 min for glycerol/water solutions are allowed for adsorption equilibration. Then, the top rod is set into motion and the shapes of the bridges are photographed with the high-speed camera. All experiments were done at room temperature, around 24 ( 1 °C. 2.3. Results of Surface Tensiometry of Adsorbed C12E5 at Air/Liquid Interfaces. The equilibrium surface tension of aqueous C12E5 decreases with increasing bulk concentration C0 until the critical micelle concentration (cmc) of 0.04 mM, beyond which it remains constant (Figure 1).14 The EST for C12E5 in glycerol/water solution (13) Panditaratne, J. Ph.D. Thesis, Purdue University, West Lafayette, IN, 2003.

Figure 1. Equilibrium surface tension of adsorbed C12E5 at 24 °C for (9) water and (b) 90 wt % glycerol/water solution. The dashed lines are the best fit of the data to eq 1; the parameters used are Γm ) 6.21 and 3.45 µmol/m2 and KL ) 330 and 120 m3/mol for water and glycerol/water solutions, respectively. The apparent cmc is about 0.04 mM in water and 0.4 mM in the glycerol/water solution.

(Figure 1) shows a similar trend, but the surface tension remains constant when C0 > 0.4 mM. This indicates that C12E5 acts as a typical surfactant in a glycerol/water solution. Since the surfactant solutions were clear with no visible particles, the concentration “break” of 0.4 mM is probably a cmc. A higher cmc in the glycerol/water solution is not unexpected, since glycerol is less polar than water and might exhibit a less pronounced hydrophobic effect. The time to reach equilibrium for surfactant solutions at C0 ) cmc is about 30 s in water and about 3000 s in glycerol/water solutions due in part to the much higher viscosity of glycerol/water solutions. The pre-cmc ESTs fit fairly well to the standard twoparameter Langmuir-Szyskowski equation (Figure 1):

γ ) γ0 - ΓmRgT ln(1 + KLC0)

(1)

where γ0 ≡ γ(0), Rg is the gas constant, T is the temperature, Γm is the maximum surface density, and KL is the adsorption equilibrium constant. For water, Γm ) 6.2 µmol/m2 and KL ) 330 m3/mol; for glycerol/water solution, Γm ) 3.5 µmol/m2 and KL ) 120 m3/mol. Evidently, the maximum area per molecule is larger for glycerol/ water solution, suggesting that the solvated headgroup is larger in glycerol/water; KL is also smaller in glycerol/ water, indicating that the surfactant shows less surface activity in the solution. 2.4. Results of Stretching Liquid Bridges. To demonstrate the effects of surfactant on the breakup dynamics, we present the bridge shapes close to the breakup with different surfactant concentrations and liquid viscosities. The initial equilibrium shapes of bridges with different surfactant concentrations are shown in the top rows of Figures 2 and 3. Equilibrium shapes of surfactant-containing bridges show thinner “waists” (narrowest parts) with increasing bulk concentration below the cmc (images a-c), due to the lowering of surface tension force relative to gravitational force. The relative importance of these forces is described by the gravitational Bond number G ≡ FgR2/γ, where F is the density, g is the gravitational acceleration, and R is the rod radius of the bridge. Equilibrium shapes of liquid bridges and their stability are governed by three parameters:5,15 G, V/R3, (14) Siddiqui, F. A.; Franses, E. I. AIChE J. 1997, 43, 1569. (15) Coriell, S. R.; Hardy, S. C.; Cordes, M. R. J. Colloid Interface Sci. 1977, 60, 126.

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Letters Table 1. Limiting Lengths and Volumes of Sessile Drops of Bridges fluid water glycerol/water

γ(C0),a mN/m

C 0, mol/m3

Vd,b µL

Ld,c mm

td,d ms

72 39 32 64 34 32 32

0 0.02 0.04 0 0.28 0.40 0.56

18.53 19.38 19.57 18.41 18.50 18.60 18.69

5.48 ( 0.03 5.35 ( 0.03 5.28 ( 0.03 6.19 ( 0.03 6.31 ( 0.03 6.38 ( 0.03 6.45 ( 0.03

217 210 206 252 258 262 265

a Estimated from Figure 1, at 24 °C. b Measured volume of sessile drops, i.e., ones on the bottom rod. c Measured limiting lengths 1 ms before breakup. d Breakup time calculated from Ld.

Figure 2. Digital images of liquid bridges before stretching (top row, images a-c) and close to pinch-off (bottom row, images a′-c′), at 24 °C. In each bridge, 24.5 µL of water is held between two stainless steel rods with a diameter of 3.15 mm. The initial surfactant concentrations C0 are (a,a′) 0.0 mM (no surfactant), (b,b′) 0.02 mM, and (c,c′) 0.04 mM. The arrows in each picture indicate the locations of the contact lines where the bridge liquid, the solid rod, and air meet. The bright stripes in the center of the liquid bridges arise from the transmitted light.

and L/R. For a bridge of a fixed V/R3, the bottom to top asymmetry of the bridge increases and the maximum allowable value of L/R before a static bridge becomes unstable decreases as G increases. When C0 > cmc, the initial shapes are the same as at C0 ) cmc because of the same EST. The dynamic bridge shapes, from a few milliseconds before the initiation of stretching, were recorded for 2 s with the high-speed camera at 1000 frames per second. It took about 0.2 s for the water bridges and about 0.26 s for glycerol/water bridges to break (Table 1). The dynamic shapes in the bottom rows of Figures 2 and 3 were taken about 1 ms before the breakup. The volumes of the sessile drops Vd were calculated from these drop shapes. The amount of liquid left on the bottom rod after the breakup, Vd, increases as surfactant concentration C0 increases. Since γ falls whereas G rises as C0 rises, more liquid is left on the bottom rod as C0 rises because of the increasing importance of gravitational force to surface tension force.15 The bridge breakup dynamics depends not only on forces due to gravity and EST but also on ones due to stretching speed, dynamic surface tension (DST), and viscosity. In a thinning filament of radius h, surface tension gives rise

Figure 3. Same as Figure 2 but for glycerol/water solutions. The surfactant concentration C0 is (a,a′) 0.0 (no surfactant), (b,b′) 0.28 mM, (c,c′) 0.4 mM, and (d,d′) 0.56 mM. Images a′-d′ are the corresponding photographs of the bridges in images a-d after being stretched to 1 ms before breakup, where a thin thread can still be seen. In images a′ and b′, the enlarged images (4×) of thin threads show that only a surfactant-free glycerol/water bridge has a “bulge” at the center.

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Figure 4. Images of liquid bridges before stretching (first column), close to breakup (second column), and after breakup (third column). The bridges are composed of 24.5 µL of 0.38 mM C12E5 in the glycerol/water solution. Stretching started at (a) 1, (b) 30, or (c) 60 min after the bridge formation to test the DST effects. In cases a and b, satellites form, as shown in the enlarged (10×) pictures. In case c, either no satellite forms or the satellite is too small (less than about 5 µm) to be visible.

to a capillary pressure pc due to surface curvature that scales as γ/h. Since pc increases as h decreases, the axial velocity of the liquid evacuating the waist increases as the filament radius decreases. Since the thinning is so fast in the narrowest part of the bridge, surface expansion is also largest there. Therefore, liquid flow out of the waist and surface dilatation in the vicinity of the waist work synergistically to produce smaller surface densities or larger surface tensions there than away from it. The result is a surface tension gradient, or a Marangoni stress, which exerts a surface force toward the waist and delays the pinching process. It has been shown that the limiting length of a surfactant-free liquid bridge increases as the stretching speed U increases.5,9 This “stabilizing” effect is due to the increasing importance of inertial force relative to surface tension force, which can be estimated from the Weber number, We ≡ FU2R/γ. Moreover, when the stretching speed is so fast that We . 1, the breakup shapes will shift from a “bottom breaks first” pattern to “top breaks first” pattern.5 In the current experimental setup, the stretching speed is kept at 2 cm/s, for which We ) 0.01, and the breakup shapes exhibit the former pattern. The relative importance of viscous force to surface tension force can be described by the Ohnesorge number Oh ≡ µ/(FRγ)1/2, where µ is the liquid viscosity. The higher the viscosity, the more slowly the capillary pressure can

“pinch” or “squeeze” the bridge, leading to a longer “limiting length” (the length of the bridge at the incipience of breakup) Ld. Moreover, as the viscosity gets higher, the Marangoni stress, which drives the fluid back into the pinching region, can affect the flow further away from the interface because of more efficient momentum transfer from the interface into the bulk. For water bridges, Oh is ca. 0.003. Thus, viscous force is negligible in the breakup dynamics except at the incipience of pinch-off,16 and breakup is essentially determined by the gravitational and surface tension forces (G ) 0.5 for pure water). The limiting lengths of surfactant/ water bridges decrease with increasing surfactant concentration (Table 1). This decrease is reproducible and is due to an increase in the effective gravitational Bond number due to a decrease in surface tension with increasing concentration. In glycerol/water bridges, both viscous force and Marangoni stress are important. In the present experiments, Oh ) 0.5 and the Marangoni stress is roughly 2 orders of magnitude larger in a glycerol/water bridge than in a water bridge. Thus, limiting lengths of glycerol/water bridges are longer than those of water bridges (Table 1) and increase with bulk concentration, by contrast to water bridges. This increase is even larger when C0 > cmc (16) Eggers, J. Phys. Rev. Lett. 1993, 71 (21), 3458.

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because the adsorption is faster.17 Moreover, the axial positions at which the sessile drops of glycerol/water bridges break increase as C0 increases, as shown in Figure 3. The small satellite drop (Figure 3a′) formed in the surfactant-free viscous bridge is not observed in surfactant-containing viscous bridges. The enlarged picture (Figure 3b′) shows that the shape in the pinching region for a surfactant-containing bridge is a long thin thread, instead of a bulge in the center that is connected by two thin threads to the sessile and pendant drops (Figure 3a′). This bulge then quickly forms a satellite drop after the breakup. Formation of satellite drops is not observed in surfactant-containing glycerol/water bridges (C0 > 0.28 mM) if adsorption equilibrium is reached before bridge stretching. The breakup dynamics, which depends strongly on the interfacial transport, is affected by the age of the surface te (the time since the bridge formation). For a 0.38 mM solution, when the bridge is stretched 1 min after its formation, a small satellite (≈70 µm diameter) is formed (Figure 4a), and the bridge shape (or limiting length) is close to that of a pure glycerol/water solution, indicating that surfactant adsorbs little. For te ) 30 min (Figure 4b), the initial shape has a thinner waist (about 2 pixels thinner than for te ) 1 min) due to lower surface tension. When this bridge is stretched, the sessile drop also has a wider maximum radius than the one stretched after te ) 1 min. Satellites are also observed with a diameter varying from 70 to 90 µm (the maximum is shown here) depending on the initial surface density. The bridge has a still thinner waist (Figure 4c) for te ) 60 min than for te ) 30 min. No satellite is observed, and the tip of the sessile drop at the incipience of pinch-off is higher than in the other two cases. Similar shapes are observed for bridges kept for te g 60 min, and again no satellite is formed. A similar sequence of phenomena is observed in the 0.56 mM solution (Figure 5), but the critical te for the adsorption equilibrium decreases to 10 min. 3. Conclusions Dynamic shapes of surfactant-containing liquid bridges have been visualized by high-speed imaging and shown to depend strongly on liquid viscosity and surface tension. The equilibrium surface tension data of a nonionic surfactant, C12E5, fit well to the Langmuir-Szyskowski equation. At 24 °C, the cmc in a glycerol/water solution, which is 0.4 mM, is higher than that in water, which is 0.04 mM. Values of the maximum surface density Γm and the adsorption equilibrium constant KL are lower in a glycerol/water solution compared to those in water. Liquid bridges containing surfactants have equilibrium shapes with thinner waists compared to surfactant-free ones, and the former bridges differ significantly from the latter ones with respect to both limiting lengths and breakup shapes. It takes about 0.21 s for the water bridges to break and about 0.26 s for the glycerol/water bridges. Volumes of sessile drops that result upon breakup increase as surfactant concentration C0 increases because surface tension γ falls as C0 rises. The limiting length of the surfactant-containing water bridges is shorter, however, than that of the pure water bridge, 5.3 versus 5.5 mm, because the breakup dynamics is determined virtually by (17) Liao, Y.-C.; Basaran, O. A.; Franses, E. I. AIChE J. 2003, 49, 3229.

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Figure 5. Same as Figure 4 but for 0.56 mM C12E5 in the glycerol/water solution. Here the long time asymptotic behavior is reached at about 10 min because the surface tension equilibrates faster.

the competition between surface tension and gravitational forces. By contrast, at the higher solution viscosity (165 vs 1 cP), the limiting length of the surfactant-containing glycerol/water bridges is longer than that of the surfactantfree bridge, 6.4 versus 6.2 mm. Moreover, no satellite is formed in a surfactant-containing glycerol/water bridge, if it is stretched after adsorption equilibration. The time after which satellite drop formation is not observed is about 60 min for glycerol/water bridges below the cmc and is about 10 min above the cmc. These results can be important for printing and microarraying applications where accurate control of drop volume is desired. Acknowledgment. This research was supported in part by grants from the National Science Foundation (Grant CTS 0135317) to Elias I. Franses and grants from the BES Program of the U.S. Department of Energy and the Purdue Research Foundation to Osman A. Basaran. Supporting Information Available: Figure 1: Images of liquid bridges before stretching and close to pinch-off. Figure 2: Images of liquid bridges before stretching, close to breakup, and after breakup. This material is available free of charge via the Internet at http://pubs.acs.org. LA0487949