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Surface Remobilization of Gas Bubbles in Polymer Solutions Containing Surfactants
Journal of Colloid and Interface Science 256, 249–255 (2002) doi:10.1006/jcis.2002.8578
Surface Remobilization of Gas Bubbles in Polymer Solutions Containing Surfactants Denis Rodrigue1 and Jean-Fran¸cois Blanchet Department of Chemical Engineering (CERSIM), Laval University, Quebec City, Quebec G1K 7P4, Canada Received February 25, 2002; accepted July 9, 2002
In a previous study, Carreau et al. [Rheol. Acta 13, 477 (1974)] did not observe a discontinuity in their graphical representation of velocity as a function of volume (0.1–10 cm3 ) for air bubbles using a 0.5 wt% polyacrylamide solution in a 20/80 glycerin/water mixture. In this study, we report that such discontinuity is occurring when smaller volumes are used. Furthermore, we also show that, for concentrations of an ionic surfactant (sodium dodecyl sulfate) above the critical association concentration of the system, polymer– surfactant association can lead to possible surface remobilization and to the elimination of such discontinuity. C 2002 Elsevier Science (USA) Key Words: bubble; velocity; jump discontinuity; viscoelastic fluids; surfactants; surface remobilization.
INTRODUCTION
For many years now, an impressive amount of work has been performed on the differences between Newtonian and nonNewtonian fluid mechanics. Even a simple physical case like the rising of a single gas bubble in an infinite viscoelastic liquid presents major differences that are still not completely understood. One of these observations is the possible discontinuity in the graphical representation of the velocity as a function of volume (or radius). Since its first report in 1965 by Astarita and Apuzzo (1), a doubt persisted because some investigators did not report such discontinuity for several non-Newtonian liquids. Reviews on the subject have been published recently (2, 3). From all the information gathered over the last 35 years or so, it is most likely that the discontinuity is the result of some kind of instability between elastic and surface tension gradient forces. The jump is believed to represent an abrupt change from a “mostly rigid” interface due to surface-active molecules in solutions (impurities, surfactants, and polymers) to an “almost free” interface (4) produced by normal forces acting to extract molecules from the interface at a certain critical volume or deformation rate (5). If we accept this explanation for the origin of the jump discontinuity, the present investigation has for objectives to answer two important questions: can we produce a jump and can we
eliminate it? To do so, a complete experimental study on the rise velocity of air bubbles in a viscoelastic fluid was performed. To answer the first question, the same polymer solution as that of Carreau et al. (6) was chosen for which no discontinuity was reported. To answer the second question, we based our research on a series of two papers by Stebe and co-workers (7, 8), where they studied the possibility of surface remobilization when surfaceactive agent concentrations are greater than the critical association concentration (CAC) or the critical micelle concentration (CMC) of the system. MATERIALS AND METHODS
All the experiments were performed at a controlled temperature of 23.5◦ C. In each case, distilled water was used for the preparation of the solutions. The base viscoelastic liquid is similar to that used by Carreau, Devic, and Kapellas (6) (hereafter denoted as CDK). It is a 0.5 wt% polyacrylamide (PAA, Separan MG-700) solution in a 20/80 glycerin/water mixture. In the original paper, no velocity discontinuity was observed for air bubbles with volumes between 0.1 and 10 cm3 . To study the effect of surface tension variation, sodium dodecyl sulfate (SDS) was used in concentrations between 0 and 3000 ppm (0.3 wt%). Surface tension was measured by a Fisher Autotensiomat using the Du Nouy ring technique. The density of all solutions was measured using a constant volume pycnometer and found to be around 1047 kg/m3 . The shear viscosities (η) and primary normal stress differences (N1 ) were measured using a Bohlin CVO HR 120 constant stress rheometer. Finally, velocity–volume data were obtained using the same experimental setup as previously used for similar studies (9). The velocity of air bubbles in PAA/SDS solutions was experimentally obtained and photographs of the shapes were simultaneously taken by a digital camera. The range of volumes tested was between 5 and 500 µL, which is much lower than the volumes used by CDK, but for which most of the discontinuities were observed in the past (20–120 µL) (10). RESULTS
Figure 1 shows the variation of surface tension as a function of SDS concentration in distilled water, 20/80 distilled
1 To whom correspondence should be addressed. Fax: 418-656-5993. E-mail: [email protected].
249
0021-9797/02 $35.00
C 2002 Elsevier Science (USA)
All rights reserved.
250
RODRIGUE AND BLANCHET
FIG. 1. Surface tension as a function of SDS. () water; () 20/80 glycerin/water; () 20/80 glycerin/water + 0.5 wt% PAA.
FIG. 3. Primary normal stress difference data for a 0.5 wt% PAA in 20/80 glycerin/water solution. () 0; () 100; () 300; () 1000; () 3000 ppm SDS. The line represents the data of Ref. (6).
water/glycerin, and 0.5 wt% PAA in 20/80 distilled water/ glycerin. As described later, critical concentrations for each fluid are observed as breaks in the general trend of the data (slope changes). Figures 2 and 3 present the rheological data of the solutions where the lines represent the original data of CDK. A very small decrease in viscosity is observed for SDS concentrations less than 300 ppm. Above this point, viscosity decreases substantially due to polymer–surfactant interactions (11, 12). This decrease in viscosity is attributed to the shrinking of the polymer coils via interchain hydrophobic interactions created by polymer–surfactant aggregates. On the other hand, no real distinction can be made for N1 values. The viscosity curves were
fitted to a three-parameter Carreau model: η = η0 [1 + (λγ˙ )2 ](n−1)/2 .
[1]
Table 1 gives the values of the parameters η0 , λ, and n. The primary normal stress differences (N1 ) were represented by a power-law model (13): N1 = a γ˙ b .
[2]
For the available experimental range (3–1500 s−1 ), a single set of parameters was determined for all PAA/SDS solutions, giving a = 2.82 Pa · sb and b = 0.98. Both sets of data are in reasonable agreement with those from CDK, which were taken in a more limited range of shear rates. Finally, the measured velocity of air bubbles in the PAA/SDS solutions are presented in Figs. 4 and 5 for SDS concentrations below and above 300 ppm with the results of CDK. Figure 4 shows good agreement between both sets of data in the range where bubble volumes overlap. Jump discontinuities are observed for volumes around 40–60 µL. On the other hand, Fig. 5 shows that for SDS concentrations above 300 ppm no discontinuity is observed, pointing to the possible remobilization of the interface. TABLE 1 Carreau Model Parameters for the PAA Solutions
FIG. 2. Viscosity data for a 0.5 wt% PAA in 20/80 glycerin/water solution. () 0; () 100; () 300; () 1000; () 3000 ppm SDS. The line represents the data of Ref. (6).
SDS (ppm)
η0 (Pas)
λ (s)
n
0 100 300 1000 3000
41.2 40.2 35.8 23.9 17.2
70.2 74.6 71.1 39.7 37.1
0.480 0.488 0.500 0.470 0.516
251
SURFACE REMOBILIZATION OF GAS BUBBLES
TABLE 2 Surface Tension as a Function of Polyacrylamide Concentration
FIG. 4. Bubble velocity as a function of volume for a 0.5 wt% PAA in 20/80 glycerin/water solution. () 0; () 10; () 30; () 100; () 300 ppm SDS. () Data taken from Ref. (6); (—) Eq. [3]; (− · · − · · −) Eq. [4]; (− − −) Eq. [5].
DISCUSSION
I. Surface Tension When surfactants are added to a liquid solution without polymer, surface tension decreases monotonically up to a certain concentration called the critical micelle concentration (CMC) and remains constant thereafter (14). Figure 1 clearly shows that this is the case for SDS in distilled water. The measured value for the CMC of SDS in water is in good agreement with the literature; around 2400 ppm (8.3 × 10−3 mol/L) (15). The CMC is not affected much for a solution of 20/80 glycerin/water. On the other hand, surface tension is greatly affected by the presence of polyacrylamide molecules due to the associative nature of the molecules. The surface tension of polymer–surfactant
PAA (wt%)
0
0.05
0.10
0.15
0.20
0.50
σ (mN/m)
71.2
71.1
62.2
60.4
53.5
40.5
systems can be divided into three regions depending on their concentrations: a monomer region, a polymer–surfactant complex region, and a micellization region (14). Surfactant molecules are thus partitioned between free surfactant, polymer-bound surfactant, and micellized surfactant. In our case, the surface tension curve for PAA/SDS shows a minimum for a concentration around 350 ppm of SDS. This kind of minimum is in agreement with early measurements for SDS (16). From Fig. 1, we find that PAA is also acting as a surfaceactive molecule. Table 2 shows the variation of surface tension as a function of PAA in 20/80 glycerin/water solutions. Similar results were obtained by Rodrigue et al. (9) for Separan AP-273 in a 50/50 glycerin/water mixture. The base solvent (concentration of glycerin) is different in both cases, leading to a surface tension reduction which is more important in the present study. Similar results were also obtained by Touhami et al. (14) for poly(ethylene glycol)–SDS systems. Thus, PAA/SDS interactions have an effect on surface tension that is reflected in the bubble velocity curves. II. Velocity Discontinuity As seen in Fig. 4, discontinuities were obtained in the range 40–60 µL for SDS concentrations lower than the CAC (350 ppm). As expected and already discussed by Rodrigue et al. (5), the volumes of CDK were simply too high to observe a discontinuity. Their lowest volume was 100 µL. Not much difference is observed between each set, with the exception of the 300 ppm solution, which is close to the CAC and for which the discontinuity occurs at a lower volume. For concentrations between 0 and 100 ppm of SDS, the jump occurs between 55 and 60 µL, while it is shifted to volumes between 40 and 45 µL at 300 ppm SDS. A comparison between theoretical calculations and experimental data shows similar behavior to the results obtained by Zana and Leal (17) for a different 0.5 wt% aqueous polyacrylamide (Separan AP-30) solution. As presented by Rodrigue et al. (5), the drag coefficient of free surface bubbles or rigid surface spheres moving in liquids represented by the power-law viscosity equation can be related to the generalized Reynolds number and a correction factor for laminar flows. From these models, the velocity of a free surface bubble (Ub ) is given by (5)
ρg D n+1 Ub = 12mY FIG. 5. Bubble velocity as a function of volume for a 0.5 wt% PAA in 20/80 glycerin/water solution. () 0; () 300; () 1000; () 3000 ppm SDS. () Data taken from Ref. (6).
1/n [3]
Y is a correction factor, which is a function of n (power-law index) alone. In the case of a rigid sphere, the velocity (Us ) is
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RODRIGUE AND BLANCHET
given by (5) Us =
ρg D n+1 18m X
1/n ,
[4]
where X is also a function of n alone. For higher volumes (18), the relation of Davies and Taylor, Usc = 1.02 (g R), [5] for a spherical-cap bubble is also included. Equations [3]–[5] are
presented in Fig. 4 for the 0 ppm SDS case (n = 0.444, Y = 1.44, X = 1.91). It is seen that elasticity in the liquid phase tends to decrease the drag coefficient, thus increasing the terminal velocity at small volumes. On the other hand, at higher volumes, elasticity seem to play a negligible role and the data are well represented by Eq. [5]. The evolution of bubble shapes is depicted in Fig. 6. The shapes change from a spheroid to a tear drop, leading finally to the presence of a cusp. As already discussed by Liu et al. (19), a cusp was reported to appear at the rear stagnation point of
FIG. 6. Photographs of bubble shapes for a 0.5 wt% PAA in 20/80 glycerin/water solution. The first and second numbers are the SDS concentration in ppm and the bubble volume in µL, respectively. The jump occurs between 55 and 65 µL for the 0 and 100 ppm SDS solutions and between 40 and 55 µL for the 300 ppm SDS solution.
SURFACE REMOBILIZATION OF GAS BUBBLES
bubbles for volumes around the critical one. Up until now, not much theoretical or numerical work has been performed on problems involving viscoelastic flows over free boundaries, but it is known that elasticity in the continuous phase increases deformation (20). Nevertheless, some numerical simulations were done to predict the shape of a bubble rising in viscoelastic fluids. Noh et al. (21) performed numerical simulations based on a finite difference solution of the conservation equations using an adjustable orthogonal curvilinear coordinate system and tried to explain the deformation of bubbles. As the polymer/solvent mixture approaches the front stagnation point of the bubble, it experiences a sudden biaxial extensional flow. The molecules are therefore extended in a direction tangent to the surface producing non-zero normal stress components. As the fluid proceeds further along the surface, it experiences a rapid extension near the rear stagnation point, where an uniaxial straining local field with a high extension normal to the surface is produced. This added contribution to the normal stress act as to pull out the surface of the bubble against surface tension. In their case, no cusping was observed due to low capillary numbers used for convergence of the calculations (Ca < 0.25). They also showed that the role of inertia is to flatten the bubble and to produce greater normal stresses pulling out the surface near the rear stagnation point. These results were recently confirmed by Pillapakkam (22). With a FENE-CR constitutive equation combined with a finite element numerical method based on the Marchuk–Yanenko operator splitting method, higher capillary numbers could be obtained and cusps were produced. The author also showed that when stresses become larger than a critical value, the surface of the bubble is pulled out near the rear stagnation point and the bubble appears to go through an instability. For certain combinations of capillary and Deborah numbers (critical values not defined in the paper), it was reported that no steady state solution could be obtained, thus leading to a possible jump discontinuity (20). Originally, Liu et al. (19) reported that such cusping is not sufficient to predict a discontinuity and they proposed that the capillary number based on the zero shear viscosity becomes equal to unity at the discontinuity. Unfortunately, this criterion was found to be inappropriate to represent all the available discontinuities encountered in the literature since elasticity, shear thinning, and surface tension gradient are not included. These properties are believed to influence the discontinuity and were introduced recently by Rodrigue et al. (5). The latter proposed that the jump discontinuity is associated with a sudden change at the gas–liquid interface. For a gas bubble rising in a purely Newtonian fluid, viscous, surface tension, and gravity forces control velocity and shape. In the case of viscoelastic fluids, variable viscosity and elastic forces must also be taken into account. When surface-active agents are present, surface tension gradient forces (Marangoni) can also be involved. Therefore, it is believed that the discontinuity is the result of an imbalance or an instability at the gas–liquid interface. The origin of the instability should be related to normal forces that, in certain critical conditions, are extracting both surfactant and polymer
253
molecules from the bubble surface, leaving a zone of interfacial and rheological different character. Based on this hypothesis, Rodrigue et al. (5) proposed that the jump is effectively the result of elastic forces normal to the bubble surface acting to extract surface-active agents from the bubble surface and locally changing the boundary conditions. From Fig. 6, it can be seen that, near the rear stagnation point of the bubble, a high degree of curvature exists, creating high local strains. This leads to the development of substantial normal stresses. In this region, polymer and surfactant molecules are stretched along the fluid streamlines, locally modifying both momentum and mass transfer. Recent flow visualization measurements (birefringence) by Funfschilling and Li (23) seem to confirm this scheme. They showed a very different flow field around bubbles rising in PAA solution vs a Newtonian glycerin solution. For non-Newtonian fluids, three distinct zones were observed: an upward flow in front as in the Newtonian case, a downward flow in the central wake as described earlier as a negative wake, and a conical upward flow just around the central wake. Theses different flow zones with direction reversal could be the origin of the jump discontinuity but further visualization measurements are needed, especially when surfactants are present. III. Surface Remobilization It is known that surface-active molecules adsorbed onto surfaces of moving fluid particles are convected toward their rear ends where they accumulate and create interfacial tension gradients (Marangoni effect). These gradients are directed in the opposite direction of convection, thus retarding interfacial motion. In a series of two papers, Stebe and co-workers (7, 8) discussed that the movement of the interface can remain unhindered if, relative to the convective rate along the surface, desorption of the surface-active molecules is fast and the bulk concentration is high enough so that diffusion away from the particle is fast. In this way, a uniform surface concentration of surfactant is maintained and surface tension gradients disappear, restoring interfacial motion. Their argument, which was proved experimentally using pressure drop measurements of oil/water/surfactant mixtures inside tubes, is the following. A surface-active molecule kinetically rapidly exchanging at a concentration above its CMC can be used to control the surface mobility. In this situation, micelles, acting as monomer reservoirs, keep a sublayer concentration uniform at the CMC value. The surface concentration, in equilibrium with this sublayer, is made uniform and Marangoni stresses are removed, thus creating a remobilization along the interface. This kind of bubble surface remobilization was tested for our PAA/SDS solutions. For this, SDS concentrations higher than the CAC value (around 350 ppm) were used. The resulting bubble velocities are shown in Fig. 5. As the amount of SDS increases, an increase in velocity at constant volume is observed, which is related to a lower viscosity of the liquid as shown in Fig. 2. Furthermore, a gradual transition is produced from a jump
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RODRIGUE AND BLANCHET
to a no-jump situation between 300 and 1000 ppm SDS. Crossing the CAC value gives support to the interfacial remobilization model of Stebe and co-workers. Elimination of Marangoni stresses seems to eliminate at the same time the discontinuity, giving further support to the origin of the jump as stated by Rodrigue et al. (5), for which the phenomenon is related to surface-active molecules at the bubble surface. Figure 7 shows pictures of the bubbles for SDS concentrations above the CMC. It can be seen that a cusp appears in these cases even if no velocity discontinuity is observed. This result is in
agreement with the statement of Liu et al. (19) that the presence of a cusp is not a sufficient condition for a jump to appear. It is simply related to the viscoelastic character of the liquid phase, as predicted by the numerical simulations reported by Pillapakkam (22) and the experimental measurements of Funfschilling and Li (23). The different and complex hydrodynamic–mass transfer phenomena occurring at the bubble interface are believed to be responsible for this jump–no jump change in the bubble behavior: retarded interface vs surface remobilization. For the moment no further experimental data or explanation is available. A future report will focus on the special conditions leading to this new observation. SUMMARY
Using a polymer solution similar to one used in the work of Carreau et al. (6), we showed that a discontinuity in the velocity– volume plot of air bubbles rising in viscoelastic fluids containing surface-active molecules can be observed if sufficiently small bubble volumes are used. From all the observations gathered so far, the origin of the jump is most likely related to a change in interfacial condition due to an imbalance in surface tension gradient and elastic forces at the gas–liquid interface. We also showed that the jump can be eliminated by using surfactant concentrations above their critical association concentration. Combined with special conditions related to mass transfer of molecules at the bubble surface, it was shown that a surface remobilization is possible, leading to a jump–no jump transition. Due to the complex coupling of the hydrodynamics– mass transfer character of this situation, the criteria previously proposed for the prediction of the jump do not apply. For this reason, further experimental, numerical, and theoretical studies are needed to explain completely the physics behind this new observation. Finally, care should be given in the future to the possible effect of extensional viscosity of polymer/surfactant solutions. This property of viscoelastic fluids is believed to play an important role for this phenomenon to occur due to the different character of the flow around a bubble rising in non-Newtonian vs Newtonian fluid. ACKNOWLEDGMENTS The authors wish to acknowledge financial support from the National Sciences and Engineering Research Council of Canada (NSERC) and from the Imperial Oil University Research Grant (IO159LAV).
REFERENCES
FIG. 7. Photographs of bubble shapes for a 0.5 wt% PAA in 20/80 glycerin/water solution. The first and second numbers are the SDS concentration in ppm and the bubble volume in µL, respectively.
1. Astarita, G., and Apuzzo, G., AIChE J. 11, 815 (1965). 2. Rodrigue, D., and De Kee, D., “Proceedings 13th International Congress on Rheology, Cambridge, Aug. 20–25, 2000.” Vol. 2, p. 241. British Society of Rheology, Glascow, 2000. 3. Rodrigue, D., and De Kee, D., in “Transport Processes in Bubbles, Drops, and Particles” (D. De Kee and R. P. Chhabra, Eds.), 2nd ed. “Recent
SURFACE REMOBILIZATION OF GAS BUBBLES
4. 5. 6. 7. 8. 9. 10.
11. 12.
Development in the bubble jump discontinuity,” Chap. 4, p. 79–101. Taylor & Francis, New York, 2002. Rodrigue, D., and De Kee, D., Rheol. Acta 38, 177 (1999). Rodrigue, D., De Kee, D., and Chan Man Fong, C. F., J. Non-Newt. Fluid Mech. 79, 45 (1998). Carreau, P. J., Devic, M., and Kapellas, M., Rheol. Acta 13, 477 (1974). Stebe, K. J., Lin, S. Y., and Maldarelli, C., Phys. Fluids A 3, 3 (1991). Stebe, K. J., and Maldarelli, C., J. Colloid Interface Sci. 163, 177 (1994). Rodrigue, D., De Kee, D., and Chan Man Fong, C. F., J. Non-Newt. Fluid Mech. 66, 213 (1996). De Kee, D., Chhabra, R. P., and Rodrigue, D., in “Handbook of Applied Polymer Processing Technology” (N. P. and P. N. Cheremisinoff, Eds.), pp. 87–123. Dekker, New York, 1996. Mylonas, Y., Karayanni, K., and Staikos, G., Langmuir 14, 6320 (1998). Methemitis, C., Morcellet, M., Sabbadin, J., and Francois, J., Eur. Polym. J. 22, 619 (1986).
255
13. Barnes, H. A., Hutton, J. F., and Walters, K., “An Introduction to Rheology”. Elsevier, Amsterdam, 1989. 14. Touhami, T., Rana, D., Neale, G. H., and Hornof, V., Colloid Polym. Sci. 279, 297 (2001). 15. Goddard, E. D., and Ananthapadmanabhan, K. P., in “Interactions of surfactants with polymers and proteins,” Chap. 1. CRC Press, Boca Raton, FL, 1993. 16. Powney, J., and Addison, C. C., Trans. Faraday Soc. 33, 1243 (1937). 17. Zana, E., and Leal, L. G., Int. J. Multiphase Flow 4, 237 (1978). 18. Haberman, W. L., and Morton, R. K., Trans. ASCE 121, 227 (1956). 19. Liu, Y. J., Liao, T. Y., and Joseph, D. D., J. Fluid Mech. 304, 321 (1995). 20. Ramaswamy, S., and Leal, L. G., J. Non-Newt. Fluid Mech. 88, 149 (1999). 21. Noh, D. S., Kang, I. S., and Leal, L. G., Phys. Fluids A 5, 1315 (1993). 22. Pillapakkam, S. B., Proc. ASME FED-250, 169 (1999). 23. Funfschilling, D., and Li, H. Z., Chem. Eng. Sci. 56, 1137 (2001).
Surface Remobilization of Gas Bubbles in Polymer Solutions Containing Surfactants Denis Rodrigue1 and Jean-Fran¸cois Blanchet Department of Chemical Engineering (CERSIM), Laval University, Quebec City, Quebec G1K 7P4, Canada Received February 25, 2002; accepted July 9, 2002
In a previous study, Carreau et al. [Rheol. Acta 13, 477 (1974)] did not observe a discontinuity in their graphical representation of velocity as a function of volume (0.1–10 cm3 ) for air bubbles using a 0.5 wt% polyacrylamide solution in a 20/80 glycerin/water mixture. In this study, we report that such discontinuity is occurring when smaller volumes are used. Furthermore, we also show that, for concentrations of an ionic surfactant (sodium dodecyl sulfate) above the critical association concentration of the system, polymer– surfactant association can lead to possible surface remobilization and to the elimination of such discontinuity. C 2002 Elsevier Science (USA) Key Words: bubble; velocity; jump discontinuity; viscoelastic fluids; surfactants; surface remobilization.
INTRODUCTION
For many years now, an impressive amount of work has been performed on the differences between Newtonian and nonNewtonian fluid mechanics. Even a simple physical case like the rising of a single gas bubble in an infinite viscoelastic liquid presents major differences that are still not completely understood. One of these observations is the possible discontinuity in the graphical representation of the velocity as a function of volume (or radius). Since its first report in 1965 by Astarita and Apuzzo (1), a doubt persisted because some investigators did not report such discontinuity for several non-Newtonian liquids. Reviews on the subject have been published recently (2, 3). From all the information gathered over the last 35 years or so, it is most likely that the discontinuity is the result of some kind of instability between elastic and surface tension gradient forces. The jump is believed to represent an abrupt change from a “mostly rigid” interface due to surface-active molecules in solutions (impurities, surfactants, and polymers) to an “almost free” interface (4) produced by normal forces acting to extract molecules from the interface at a certain critical volume or deformation rate (5). If we accept this explanation for the origin of the jump discontinuity, the present investigation has for objectives to answer two important questions: can we produce a jump and can we
eliminate it? To do so, a complete experimental study on the rise velocity of air bubbles in a viscoelastic fluid was performed. To answer the first question, the same polymer solution as that of Carreau et al. (6) was chosen for which no discontinuity was reported. To answer the second question, we based our research on a series of two papers by Stebe and co-workers (7, 8), where they studied the possibility of surface remobilization when surfaceactive agent concentrations are greater than the critical association concentration (CAC) or the critical micelle concentration (CMC) of the system. MATERIALS AND METHODS
All the experiments were performed at a controlled temperature of 23.5◦ C. In each case, distilled water was used for the preparation of the solutions. The base viscoelastic liquid is similar to that used by Carreau, Devic, and Kapellas (6) (hereafter denoted as CDK). It is a 0.5 wt% polyacrylamide (PAA, Separan MG-700) solution in a 20/80 glycerin/water mixture. In the original paper, no velocity discontinuity was observed for air bubbles with volumes between 0.1 and 10 cm3 . To study the effect of surface tension variation, sodium dodecyl sulfate (SDS) was used in concentrations between 0 and 3000 ppm (0.3 wt%). Surface tension was measured by a Fisher Autotensiomat using the Du Nouy ring technique. The density of all solutions was measured using a constant volume pycnometer and found to be around 1047 kg/m3 . The shear viscosities (η) and primary normal stress differences (N1 ) were measured using a Bohlin CVO HR 120 constant stress rheometer. Finally, velocity–volume data were obtained using the same experimental setup as previously used for similar studies (9). The velocity of air bubbles in PAA/SDS solutions was experimentally obtained and photographs of the shapes were simultaneously taken by a digital camera. The range of volumes tested was between 5 and 500 µL, which is much lower than the volumes used by CDK, but for which most of the discontinuities were observed in the past (20–120 µL) (10). RESULTS
Figure 1 shows the variation of surface tension as a function of SDS concentration in distilled water, 20/80 distilled
1 To whom correspondence should be addressed. Fax: 418-656-5993. E-mail: [email protected].
249
0021-9797/02 $35.00
C 2002 Elsevier Science (USA)
All rights reserved.
250
RODRIGUE AND BLANCHET
FIG. 1. Surface tension as a function of SDS. () water; () 20/80 glycerin/water; () 20/80 glycerin/water + 0.5 wt% PAA.
FIG. 3. Primary normal stress difference data for a 0.5 wt% PAA in 20/80 glycerin/water solution. () 0; () 100; () 300; () 1000; () 3000 ppm SDS. The line represents the data of Ref. (6).
water/glycerin, and 0.5 wt% PAA in 20/80 distilled water/ glycerin. As described later, critical concentrations for each fluid are observed as breaks in the general trend of the data (slope changes). Figures 2 and 3 present the rheological data of the solutions where the lines represent the original data of CDK. A very small decrease in viscosity is observed for SDS concentrations less than 300 ppm. Above this point, viscosity decreases substantially due to polymer–surfactant interactions (11, 12). This decrease in viscosity is attributed to the shrinking of the polymer coils via interchain hydrophobic interactions created by polymer–surfactant aggregates. On the other hand, no real distinction can be made for N1 values. The viscosity curves were
fitted to a three-parameter Carreau model: η = η0 [1 + (λγ˙ )2 ](n−1)/2 .
[1]
Table 1 gives the values of the parameters η0 , λ, and n. The primary normal stress differences (N1 ) were represented by a power-law model (13): N1 = a γ˙ b .
[2]
For the available experimental range (3–1500 s−1 ), a single set of parameters was determined for all PAA/SDS solutions, giving a = 2.82 Pa · sb and b = 0.98. Both sets of data are in reasonable agreement with those from CDK, which were taken in a more limited range of shear rates. Finally, the measured velocity of air bubbles in the PAA/SDS solutions are presented in Figs. 4 and 5 for SDS concentrations below and above 300 ppm with the results of CDK. Figure 4 shows good agreement between both sets of data in the range where bubble volumes overlap. Jump discontinuities are observed for volumes around 40–60 µL. On the other hand, Fig. 5 shows that for SDS concentrations above 300 ppm no discontinuity is observed, pointing to the possible remobilization of the interface. TABLE 1 Carreau Model Parameters for the PAA Solutions
FIG. 2. Viscosity data for a 0.5 wt% PAA in 20/80 glycerin/water solution. () 0; () 100; () 300; () 1000; () 3000 ppm SDS. The line represents the data of Ref. (6).
SDS (ppm)
η0 (Pas)
λ (s)
n
0 100 300 1000 3000
41.2 40.2 35.8 23.9 17.2
70.2 74.6 71.1 39.7 37.1
0.480 0.488 0.500 0.470 0.516
251
SURFACE REMOBILIZATION OF GAS BUBBLES
TABLE 2 Surface Tension as a Function of Polyacrylamide Concentration
FIG. 4. Bubble velocity as a function of volume for a 0.5 wt% PAA in 20/80 glycerin/water solution. () 0; () 10; () 30; () 100; () 300 ppm SDS. () Data taken from Ref. (6); (—) Eq. [3]; (− · · − · · −) Eq. [4]; (− − −) Eq. [5].
DISCUSSION
I. Surface Tension When surfactants are added to a liquid solution without polymer, surface tension decreases monotonically up to a certain concentration called the critical micelle concentration (CMC) and remains constant thereafter (14). Figure 1 clearly shows that this is the case for SDS in distilled water. The measured value for the CMC of SDS in water is in good agreement with the literature; around 2400 ppm (8.3 × 10−3 mol/L) (15). The CMC is not affected much for a solution of 20/80 glycerin/water. On the other hand, surface tension is greatly affected by the presence of polyacrylamide molecules due to the associative nature of the molecules. The surface tension of polymer–surfactant
PAA (wt%)
0
0.05
0.10
0.15
0.20
0.50
σ (mN/m)
71.2
71.1
62.2
60.4
53.5
40.5
systems can be divided into three regions depending on their concentrations: a monomer region, a polymer–surfactant complex region, and a micellization region (14). Surfactant molecules are thus partitioned between free surfactant, polymer-bound surfactant, and micellized surfactant. In our case, the surface tension curve for PAA/SDS shows a minimum for a concentration around 350 ppm of SDS. This kind of minimum is in agreement with early measurements for SDS (16). From Fig. 1, we find that PAA is also acting as a surfaceactive molecule. Table 2 shows the variation of surface tension as a function of PAA in 20/80 glycerin/water solutions. Similar results were obtained by Rodrigue et al. (9) for Separan AP-273 in a 50/50 glycerin/water mixture. The base solvent (concentration of glycerin) is different in both cases, leading to a surface tension reduction which is more important in the present study. Similar results were also obtained by Touhami et al. (14) for poly(ethylene glycol)–SDS systems. Thus, PAA/SDS interactions have an effect on surface tension that is reflected in the bubble velocity curves. II. Velocity Discontinuity As seen in Fig. 4, discontinuities were obtained in the range 40–60 µL for SDS concentrations lower than the CAC (350 ppm). As expected and already discussed by Rodrigue et al. (5), the volumes of CDK were simply too high to observe a discontinuity. Their lowest volume was 100 µL. Not much difference is observed between each set, with the exception of the 300 ppm solution, which is close to the CAC and for which the discontinuity occurs at a lower volume. For concentrations between 0 and 100 ppm of SDS, the jump occurs between 55 and 60 µL, while it is shifted to volumes between 40 and 45 µL at 300 ppm SDS. A comparison between theoretical calculations and experimental data shows similar behavior to the results obtained by Zana and Leal (17) for a different 0.5 wt% aqueous polyacrylamide (Separan AP-30) solution. As presented by Rodrigue et al. (5), the drag coefficient of free surface bubbles or rigid surface spheres moving in liquids represented by the power-law viscosity equation can be related to the generalized Reynolds number and a correction factor for laminar flows. From these models, the velocity of a free surface bubble (Ub ) is given by (5)
ρg D n+1 Ub = 12mY FIG. 5. Bubble velocity as a function of volume for a 0.5 wt% PAA in 20/80 glycerin/water solution. () 0; () 300; () 1000; () 3000 ppm SDS. () Data taken from Ref. (6).
1/n [3]
Y is a correction factor, which is a function of n (power-law index) alone. In the case of a rigid sphere, the velocity (Us ) is
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RODRIGUE AND BLANCHET
given by (5) Us =
ρg D n+1 18m X
1/n ,
[4]
where X is also a function of n alone. For higher volumes (18), the relation of Davies and Taylor, Usc = 1.02 (g R), [5] for a spherical-cap bubble is also included. Equations [3]–[5] are
presented in Fig. 4 for the 0 ppm SDS case (n = 0.444, Y = 1.44, X = 1.91). It is seen that elasticity in the liquid phase tends to decrease the drag coefficient, thus increasing the terminal velocity at small volumes. On the other hand, at higher volumes, elasticity seem to play a negligible role and the data are well represented by Eq. [5]. The evolution of bubble shapes is depicted in Fig. 6. The shapes change from a spheroid to a tear drop, leading finally to the presence of a cusp. As already discussed by Liu et al. (19), a cusp was reported to appear at the rear stagnation point of
FIG. 6. Photographs of bubble shapes for a 0.5 wt% PAA in 20/80 glycerin/water solution. The first and second numbers are the SDS concentration in ppm and the bubble volume in µL, respectively. The jump occurs between 55 and 65 µL for the 0 and 100 ppm SDS solutions and between 40 and 55 µL for the 300 ppm SDS solution.
SURFACE REMOBILIZATION OF GAS BUBBLES
bubbles for volumes around the critical one. Up until now, not much theoretical or numerical work has been performed on problems involving viscoelastic flows over free boundaries, but it is known that elasticity in the continuous phase increases deformation (20). Nevertheless, some numerical simulations were done to predict the shape of a bubble rising in viscoelastic fluids. Noh et al. (21) performed numerical simulations based on a finite difference solution of the conservation equations using an adjustable orthogonal curvilinear coordinate system and tried to explain the deformation of bubbles. As the polymer/solvent mixture approaches the front stagnation point of the bubble, it experiences a sudden biaxial extensional flow. The molecules are therefore extended in a direction tangent to the surface producing non-zero normal stress components. As the fluid proceeds further along the surface, it experiences a rapid extension near the rear stagnation point, where an uniaxial straining local field with a high extension normal to the surface is produced. This added contribution to the normal stress act as to pull out the surface of the bubble against surface tension. In their case, no cusping was observed due to low capillary numbers used for convergence of the calculations (Ca < 0.25). They also showed that the role of inertia is to flatten the bubble and to produce greater normal stresses pulling out the surface near the rear stagnation point. These results were recently confirmed by Pillapakkam (22). With a FENE-CR constitutive equation combined with a finite element numerical method based on the Marchuk–Yanenko operator splitting method, higher capillary numbers could be obtained and cusps were produced. The author also showed that when stresses become larger than a critical value, the surface of the bubble is pulled out near the rear stagnation point and the bubble appears to go through an instability. For certain combinations of capillary and Deborah numbers (critical values not defined in the paper), it was reported that no steady state solution could be obtained, thus leading to a possible jump discontinuity (20). Originally, Liu et al. (19) reported that such cusping is not sufficient to predict a discontinuity and they proposed that the capillary number based on the zero shear viscosity becomes equal to unity at the discontinuity. Unfortunately, this criterion was found to be inappropriate to represent all the available discontinuities encountered in the literature since elasticity, shear thinning, and surface tension gradient are not included. These properties are believed to influence the discontinuity and were introduced recently by Rodrigue et al. (5). The latter proposed that the jump discontinuity is associated with a sudden change at the gas–liquid interface. For a gas bubble rising in a purely Newtonian fluid, viscous, surface tension, and gravity forces control velocity and shape. In the case of viscoelastic fluids, variable viscosity and elastic forces must also be taken into account. When surface-active agents are present, surface tension gradient forces (Marangoni) can also be involved. Therefore, it is believed that the discontinuity is the result of an imbalance or an instability at the gas–liquid interface. The origin of the instability should be related to normal forces that, in certain critical conditions, are extracting both surfactant and polymer
253
molecules from the bubble surface, leaving a zone of interfacial and rheological different character. Based on this hypothesis, Rodrigue et al. (5) proposed that the jump is effectively the result of elastic forces normal to the bubble surface acting to extract surface-active agents from the bubble surface and locally changing the boundary conditions. From Fig. 6, it can be seen that, near the rear stagnation point of the bubble, a high degree of curvature exists, creating high local strains. This leads to the development of substantial normal stresses. In this region, polymer and surfactant molecules are stretched along the fluid streamlines, locally modifying both momentum and mass transfer. Recent flow visualization measurements (birefringence) by Funfschilling and Li (23) seem to confirm this scheme. They showed a very different flow field around bubbles rising in PAA solution vs a Newtonian glycerin solution. For non-Newtonian fluids, three distinct zones were observed: an upward flow in front as in the Newtonian case, a downward flow in the central wake as described earlier as a negative wake, and a conical upward flow just around the central wake. Theses different flow zones with direction reversal could be the origin of the jump discontinuity but further visualization measurements are needed, especially when surfactants are present. III. Surface Remobilization It is known that surface-active molecules adsorbed onto surfaces of moving fluid particles are convected toward their rear ends where they accumulate and create interfacial tension gradients (Marangoni effect). These gradients are directed in the opposite direction of convection, thus retarding interfacial motion. In a series of two papers, Stebe and co-workers (7, 8) discussed that the movement of the interface can remain unhindered if, relative to the convective rate along the surface, desorption of the surface-active molecules is fast and the bulk concentration is high enough so that diffusion away from the particle is fast. In this way, a uniform surface concentration of surfactant is maintained and surface tension gradients disappear, restoring interfacial motion. Their argument, which was proved experimentally using pressure drop measurements of oil/water/surfactant mixtures inside tubes, is the following. A surface-active molecule kinetically rapidly exchanging at a concentration above its CMC can be used to control the surface mobility. In this situation, micelles, acting as monomer reservoirs, keep a sublayer concentration uniform at the CMC value. The surface concentration, in equilibrium with this sublayer, is made uniform and Marangoni stresses are removed, thus creating a remobilization along the interface. This kind of bubble surface remobilization was tested for our PAA/SDS solutions. For this, SDS concentrations higher than the CAC value (around 350 ppm) were used. The resulting bubble velocities are shown in Fig. 5. As the amount of SDS increases, an increase in velocity at constant volume is observed, which is related to a lower viscosity of the liquid as shown in Fig. 2. Furthermore, a gradual transition is produced from a jump
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to a no-jump situation between 300 and 1000 ppm SDS. Crossing the CAC value gives support to the interfacial remobilization model of Stebe and co-workers. Elimination of Marangoni stresses seems to eliminate at the same time the discontinuity, giving further support to the origin of the jump as stated by Rodrigue et al. (5), for which the phenomenon is related to surface-active molecules at the bubble surface. Figure 7 shows pictures of the bubbles for SDS concentrations above the CMC. It can be seen that a cusp appears in these cases even if no velocity discontinuity is observed. This result is in
agreement with the statement of Liu et al. (19) that the presence of a cusp is not a sufficient condition for a jump to appear. It is simply related to the viscoelastic character of the liquid phase, as predicted by the numerical simulations reported by Pillapakkam (22) and the experimental measurements of Funfschilling and Li (23). The different and complex hydrodynamic–mass transfer phenomena occurring at the bubble interface are believed to be responsible for this jump–no jump change in the bubble behavior: retarded interface vs surface remobilization. For the moment no further experimental data or explanation is available. A future report will focus on the special conditions leading to this new observation. SUMMARY
Using a polymer solution similar to one used in the work of Carreau et al. (6), we showed that a discontinuity in the velocity– volume plot of air bubbles rising in viscoelastic fluids containing surface-active molecules can be observed if sufficiently small bubble volumes are used. From all the observations gathered so far, the origin of the jump is most likely related to a change in interfacial condition due to an imbalance in surface tension gradient and elastic forces at the gas–liquid interface. We also showed that the jump can be eliminated by using surfactant concentrations above their critical association concentration. Combined with special conditions related to mass transfer of molecules at the bubble surface, it was shown that a surface remobilization is possible, leading to a jump–no jump transition. Due to the complex coupling of the hydrodynamics– mass transfer character of this situation, the criteria previously proposed for the prediction of the jump do not apply. For this reason, further experimental, numerical, and theoretical studies are needed to explain completely the physics behind this new observation. Finally, care should be given in the future to the possible effect of extensional viscosity of polymer/surfactant solutions. This property of viscoelastic fluids is believed to play an important role for this phenomenon to occur due to the different character of the flow around a bubble rising in non-Newtonian vs Newtonian fluid. ACKNOWLEDGMENTS The authors wish to acknowledge financial support from the National Sciences and Engineering Research Council of Canada (NSERC) and from the Imperial Oil University Research Grant (IO159LAV).
REFERENCES
FIG. 7. Photographs of bubble shapes for a 0.5 wt% PAA in 20/80 glycerin/water solution. The first and second numbers are the SDS concentration in ppm and the bubble volume in µL, respectively.
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SURFACE REMOBILIZATION OF GAS BUBBLES
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