Coalescence in Surfactant-Stabilized Emulsions ... - ACS Publications

Langmuir 2001, 17, 2647-2655

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Coalescence in Surfactant-Stabilized Emulsions Subjected to Shear Flow Arup Nandi, D. V. Khakhar, and Anurag Mehra* Department of Chemical Engineering, Indian Institute of TechnologysBombay, Powai, Bombay-400076, India Received October 20, 2000. In Final Form: January 29, 2001 Experiments were carried out to study the coalescence dynamics of a neutrally buoyant liquid-liquid emulsion subjected to a simple shear flow in a Couette device. The effect of shear rate and dispersed phase holdup were studied. The surfactant-stabilized emulsions were prepared in a stirred tank at high shear rates which were applied for long times so as to obtain reproducible equilibrium drop size distributions. Low shear rates were used in the Couette device to prevent drop breakup. Sufficiently large drops and density matching ensured that coalescence due to Brownian motion and creaming was negligible and thus the change in drop size distribution with time was entirely due to shear-induced coalescence. The evolving volume density distributions were measured using optical microscopy and image analysis. A population balance model based on Smoluchowski’s result for the rate of shear-induced coalescence and an empirical form for the coalescence efficiency was used to describe the system. Shearing results in stabilization of the emulsion, and the coalescence rate reduces with increasing shear rates. The stabilizing effect due to shearing is greater for higher holdups. The drop size distribution for high values of the holdup becomes bimodal at long times. The coalescence efficiency, back-calculated from experimental data, is independent of drop size for low holdups but is size dependent for higher holdups. This work demonstrates a simple experimental method for evaluating the coalescence efficiency for emulsions with high holdups where theoretical models are not available.

Introduction Liquid-liquid emulsions constitute a wide variety of products and are formed in many processes of industrial relevance. Most lubricants, paints, and cosmetics and many food products are emulsions.1 Processes such as liquid-liquid extraction, petroleum refining, and many areas of polymer processing involve handling and conveying of emulsions.1-3 During these operations the emulsion drops are subjected to flow fields which may cause them to break or coalesce. The simultaneous occurrence of the two phenomena controls the drop size distribution, which in turn has a significant effect on the processing conditions and characteristics of the product. Most emulsions contain surface active impurities or added surfactants, which act as emulsion stabilizers. Thus, a fundamental understanding of the phenomena of drop coalescence and breakage in surfactant-stabilized emulsions is of wide interest. This work explores the effect of shear rate and dispersed phase holdup on the coalescence kinetics of a surfactantstabilized emulsion subjected to a simple shear flow. The shear rates used are small enough such that no drop breakup occurs.4,5 The drops in the emulsions used are large enough so that coalescence due to Brownian motion is negligible. Also, the densities of the drops and the continuous phase are matched so that creaming-induced coalescence is insignificant. Thus, the evolution of the drop size distribution in the system occurs due to shear-induced coalescence alone. Consider the physical processes during the interaction of a pair of drops in a shear flow. The shear flow causes * Corresponding author. E-mail: [email protected]. (1) Venugopal, B. V.; Wasan, D. T. In Encyclopedia of Emulsion Technology; Becher, P., Ed.; Marcel Dekker: New York, 1983; Vol. 2. (2) Mana-Zloczower, I. In Mixing and Compounding of PolymersTheory and Practice; Mana-Zloczower, I., Tadmor, Z., Eds.; Hanser Publishers: Munich, 1994; p 5. (3) Ottino, J. M.; et al. Adv. Chem. Eng. 1999, 25, 105. (4) Grace, H. P. Chem. Eng. Commun. 1982, 14, 225. (5) Khakhar, D. V.; Ottino, J. M. J. Fluid Mech. 1986, 166, 265.

the drops to “collide” (interfaces approach within some specified small length) as a result of the velocity gradient (Figure 1). The drops form a doublet and rotate in a plane that is coplanar to the drop relative velocity vector (u) and the line joining their centers (e). The interfacial film between the drops thins initially, and it may deform due to the pressure within the film. The process of film thinning continues until the following condition is satisfied

u‚e e 0

(1)

Beyond this time the drops move apart and the film begins to thicken. If the film has achieved the critical thickness within this time, it may rupture, leading to coalescence. Otherwise, the drops continue their respective motions with the bulk flow (Figure 1). A review of previous studies related to this process is given below. The simplest model for coalescence of drops in an emulsion subjected to a simple shear flow is based on Smoluchowski’s theory,6 with the coalescence rate given by

γ Cs(ν,ν′) ) [ν1/3 + ν′1/3]3n(ν) n(ν′) π

(2)

where ν and ν′ are the volumes of the coalescing drops, γ is the shear rate, and n(ν) is the number density of the drops. Smoluchowski’s equation is based on the assumption that the drops move along the flow streamlines and hence does not account for hydrodynamic interactions. The effects of physicochemical parameters affecting interdrop film stability are also unaccounted for in the theory. Inclusion of such effects generally results in a coalescence rate that is lower than the Smoluchowski collision rate and is typically expressed as

C(ν,ν′) ) Cs(ν,ν′)

10.1021/la001473m CCC: $20.00 © 2001 American Chemical Society Published on Web 04/05/2001

(3)

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Figure 1. Schematic view of the stages of the collision between two drops in a shear flow where u is the relative velocity vector and e is the unit vector along the line joining the centers.

where is the coalescence efficiency, defined as the fraction of collisions which result in coalescence. The drainage of the film between closely spaced drops, which may determine coalescence efficiency, has been studied in several works, and a review is given by Chesters.7 Expressions for the coalescence efficiency based on models for the rate of thinning of films for fully mobile, partially mobile, and immobile drop interfaces are derived.7 An assumption of the models is that films below a critical thickness rupture, thereby leading to coalescence. Several more recent works focus on the details of film thinning between drops. Davis et al.8 carried out a lubrication analysis of film thinning for nondeformable drops pressed together by a constant force. Hartland et al.9 considered both the viscous and inertial components for a flat circular interface with partially mobile interfaces. Later work by Jeelani and Hartland10 included the effect of van der Waals forces on the interfacial film thinning rate during the axisymmetric motion of a pair of equal-sized liquid drops pressed together by a constant force. Abid and Chesters11 carried out detailed calculations of the film thinning rate for a pair of deformable drops with partially mobile interfaces. Their results are close to the rates predicted by the model of Chesters7 for the same case. Hartland and Jeelani12 also studied the effect of interfacial tension gradients on the rate of thinning of the intervening film. Computations of the coalescence efficiency, taking into account the detailed hydrodynamics around a pair of drops, have been carried out in a few works. Zeichner and Schowalter13 studied the coagulation of equal-sized rigid particles with immobile interfaces in shear and extensional flow taking into account van der Waals forces. Particles which approach closer than a critical distance form a permanent doublet (coagulate) as a result of van der Waals forces. This work was extended by Feke and Schowalter14 to include the case of particles with different diameters. Wang et al.15 calculated the coalescence efficiency for rigid drops with partially mobile interfaces under the action of a simple shear and uniaxial extensional flow. Their results indicate that the highest probability of coalescence exists for drops which are of the same size. For a given emulsion system, their results predict a critical drop diameter ratio (smaller drop diameter to larger drop diameter) below which there is no coalescence. The value of this critical (6) Smoluchowski, M. V. Z. Phys. Chem. 1917, 92, 129. (7) Chesters, A. K. Trans. Inst. Chem. Eng. 1991, 69, 259. (8) Davis, R. H.; Schonberg, J. A.; Rallison, J. M. Phys. Fluids A 1989, 1, 77. (9) Hartland, S.; Jeelani, S. A. K.; Suter, A. Chem. Eng. Sci. 1989, 44, 387. (10) Jeelani, S. A. K.; Hartland, S. J. Colloid Interface Sci. 1993, 156, 467. (11) Abid, S.; Chesters, A. K. Int. J. Multiphase Flow 1994, 20, 613. (12) Hartland, S.; Jeelani, S. A. K. Colloids Surf., A 1994, 88, 289. (13) Zeichner, G. R.; Schowalter, W. R. AIChE J. 1977, 23, 243. (14) Feke, D. L.; Schowalter, W. R. J. Fluid Mech. 1983, 133, 17. (15) Wang, H.; Zinchenko, A. Z.; Davis, R. H. J. Fluid Mech. 1994, 265, 161.

Nandi et al.

diameter ratio becomes smaller with decreasing viscosity ratio (µd/µc), where µd is the viscosity of the dispersed phase and µc that of the continuous phase. Only relatively few experimental studies of coalescence in emulsions subjected to a laminar shear flow have been previously reported. Hazlett and Schechter16 subjected an emulsion of methyl methacrylate in a mixture of methanol and water to tangential Couette flow. The coalescence efficiency, calculated using a population balance model, was studied at temperatures close to the critical point of the emulsion. Their results indicate that there is no variation of the coalescence efficiency at temperatures near the critical point. Vinckier et al.17 studied shear-induced coalescence in a polymer blend produced at a shear rate higher than that at which the emulsion was broken. Their results, obtained from measured average drop diameters, matched the coalescence efficiency model for partially mobile interfaces given by Chesters.7 Experiments at high dispersed phase holdups gave a coalescence efficiency which increased with dispersed phase holdup. This was explained on the basis of an increased critical film thickness for coalescence at higher holdups. Mishra et al.18 subjected a surfactantfree emulsion of pentadecane in an aqueous solution of NaCl to a simple shear flow using a concentric cylinder apparatus. The evolving drop size was measured using laser Doppler anemometry. Their results follow the hard sphere predictions of Zeichner and Schowalter13 and Feke and Schowalter.14 They also found the drop size distribution to be self-similar, in agreement with the analysis of Swift and Friedlander,19 for systems with a constant coalescence efficiency. The coalescence rate was found to increase with shear rate as well as with electrolyte concentration. Shear-induced coalescence in the presence of surfactant was studied by Mousa and van de Ven.20 An emulsion of silicone oil in water, stabilized by SDS (sodium dodecyl sulfate) was sheared between rotating circular plates. The coalescence efficiency was back-calculated using the population balance equation incorporating an assumed functionality for the coalescence efficiency given by

)

{

[

R0 0,

]

4q (1 + q)2

5.0

,

if ν and ν′ < νc if ν or ν′ > νc

(4)

In the above equation, q ) (ν/ν′)1/3 is the ratio of the diameter of the colliding drops, and νc is the maximum volume a drop can attain due to coalescence with other drops. The index value (5.0) was selected on the basis of phenomenological arguments. Experiments were carried out at a dispersed-phase holdup of 0.2%, and the volumeaveraged diameters of the coalescing emulsion were found from the transmitted light intensity measurements. Their results indicate that for a surfactant-free emulsion the coalescence efficiency term (R0) first decreased with shear rate and then showed an abrupt increase beyond a certain value of the shear rate. A similar qualitative behavior was reported earlier in the theoretical study by van de Ven and Mason.21 The addition of salts such as KCl and AlCl3 resulted in a rise in the coefficient R0 with shear (16) Hazlett, R. D.; Schechter, R. S. Colloid Surf. 1988, 29, 71. (17) Vinckier, I.; et al. AIChE J. 1998, 44, 951. (18) Mishra, V.; Kresta, S. M.; Masliyah, J. H. J. Colloid Interface Sci. 1998, 197, 57. (19) Swift, D. L.; Friedlander, S. K. J. Colloid Sci. 1964, 19, 621. (20) Mousa, H.; van de Ven, T. G. M. Colloids Surf., A 1991, 60, 39. (21) van de Ven, T. G. M.; Mason, S. G. Colloid Polym. Sci. 1977, 255, 468.

Coalescence in Surfactant-Stabilized Emulsions

rate and then a drop beyond a critical value. The addition of sodium dodecyl sulfate (SDS) to the emulsion gave a similar result. Schokker and Dalgleish22 reported flocculation of drops in an oil-in-water emulsion stabilized by sodium casseinate. Their experiments were conducted in a tangential Couette device at very high shear rates (670740 s-1). They showed that the coalescence rate was strongly influenced by the storage time as well as by the apparatus used for making the emulsion. We have recently studied coalescence in a neutrally buoyant, surfactant-stabilized emulsion subjected to a simple shear flow in a Couette device.23 The experiments were done for a low dispersed phase holdup (1%) at which multidrop collisions are negligibly small. The coalescence efficiency was determined by fitting the theoretically obtained drop size distribution profile to the experimental data and found to be independent of colliding drop sizes. The model of Chesters7 was found to overpredict the rate of coalescence by 2 orders of magnitude. A model for the computation of coalescence efficiency, based on the stochastic nature of film rupture, gave good agreement with experimental data. The stability of the film between flat interfaces was studied by Nikolov and Wasan;24 they found that the coalescence time in the presence of a surfactant was much higher than that for the corresponding surfactant-free system as well as the value predicted by models for film thinning. Coalescence times for identical drops were distributed over a range of values, and the stability of the interface was found to be more important than the fluid mechanics governing the thinning of the film. Dreher et al.25 conducted a similar study in a surfactant-free system and found that the coalescence time was log-normally distributed. Ghosh26 has demonstrated similar results for surfactant loaded systems using different surfactants. The objective of the current work is to study the effect of shear rate on surfactant-stabilized emulsions at high dispersed-phase holdups. Despite the practical importance of emulsions with high holdup, no theoretical or experimental work for the case of well-defined laminar flows has been previously reported. We present here the results of an experimental study of the evolution of the drop size distribution in a neutrally buoyant, surfactant-stabilized emulsion subjected to a shear flow. The evolving drop size distributions obtained from a population balance model simulation are fitted to the experimentally obtained drop size distributions to back-calculate the coalescence efficiency. Experimental details are given in the next section followed by a description of the model and computational procedures used. Experimental Section Materials and Method. Monochlorobenzene (analytical grade, Glaxo Chemicals, India) was dispersed in a mixture of glycerol (LR grade, Merck, India), double-distilled water, and sodium chloride (Loba Chemie, India), using TWEEN-80 (spectroscopic grade, SD Fine Chemicals, India) as the surfactant. Emulsions were prepared in a 1 L beaker fitted with stainless steel baffles. A shrouded turbine impeller was used for emulsification. The beaker was covered in order to avoid dust contamination as well as to minimize the evaporation of chlorobenzene. To avoid gravity-driven motion of the drops, the specific gravity of the continuous phase was adjusted, by varying (22) Schokker, E. P.; Dalgleish, D. G. Colloids Surf., A 1998, 145, 61. (23) Nandi, A.; Mehra, A.; Khakhar, D. V. Phys. Rev. Lett. 1999, 83, 2461. (24) Nikolov, A. D.; Wasan, D. T. Ind. Eng. Chem. Res. 1996, 34, 3653. (25) Dreher, T. M.; et al. AIChE J. 1999, 45, 1182. (26) Ghosh. Ph.D. Thesis, I. I. T.sBombay.

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Figure 2. Initial volume density distributions for all three holdups. Table 1. Time of Stirring (ts) and Rotation Speed (ωs) for Making the Emulsion and Resulting Volume-Average Diameters (D h ) for the Starting Emulsions for the Three Holdups (O), Together with the Interfacial Tension Values (σ) φ

ts (h)

ωs (rpm)

D h (µm)

σ (mN/m)

1% 5% 15%

2.5 5.5 8.5

2090 ( 40 2200 ( 50 2450 ( 40

8.41 ( 0.09 8.07 ( 0.24 8.83 ( 0.28

16.99 ( 0.01 16.95 ( 0.05 16.92 ( 0.12

the ratio of glycerol to water, to match that of the dispersed phase to an accuracy of (0.005 g/cm3. All experiments were conducted at room temperature, which was maintained at 25 ( 1 °C. The continuous-phase constituted 0.525 L of glycerol and 10 g of sodium chloride (0.1131 M) dissolved in 1 L of doubledistilled water. The addition of sodium chloride masks the trace ionic impurities and suppresses the electric double layer, resulting in a characteristic Debye-Huckel length of 3.35 × 10-2 µm.27 The surfactant was initially dissolved in the dispersed phase, and a surfactant concentration of 0.85% (w/w) of the dispersed phase was used in all the emulsions. Preparation of Emulsion. Emulsions were stirred for long times to achieve a dynamic equilibrium between drop breakup and coalescence and thus to obtain a steady state for the drop size distribution. This ensured reproducibility of the starting drop size distributions in the shear flow experiments. Further, the impeller rotation speed was adjusted to obtain nearly similar drop size distributions for the different cases. The emulsionmaking conditions for the various holdups are summarized in Table 1, where φ is the volume fraction of the dispersed phase, ts is the time of stirring, and ωs is the impeller rotational speed. The interfacial tension (σ) between the two phases was measured using a drop volume tensiometer after equilibrating the two phases (oil and aqueous) for 36 h. The values of the interfacial tensions are quite close to each other (Table 1) for the three holdups. The initial volume average diameters (D h ) 〈D3〉1/3) are highly reproducible and are quite close for all three holdups (Table 1). Figure 2 shows the initial drop size distributions for the three holdups, which are very similar. The distributions are shown on a volume basis with (πD3/6)f(D) dD being the volume fraction of drops in the size range (D, D + dD). The distributions are normalized by the total volume fraction (φ) for comparison. The error bars, which indicate the standard deviation over a minimum of three runs, are small and show the reproducibility of the initial distributions. (27) Stokes, R. J.; Evans, D. F. Advances in Interfacial Engineering: Wiley-VCH: New York, 1997.

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Figure 3. Schematic diagram of the tangential Couette flow assembly.

Figure 4. Plot of the volume-average diameter versus the number of drops counted.

Figure 5. Schematic diagram showing a rectangular cavity made using cover slips for viewing the emulsion sample in the microscope. The typical height of the rectangular cavity varied from 270 to 300 µm, which was significantly higher than the diameter of the largest drop in the experiments.

Shear Flow Apparatus. The emulsion was sheared in a tangential Couette flow apparatus with the inner cylinder rotating. A schematic view of the apparatus along with dimensions is shown in Figure 3. The rotation of the inner cylinder generates a shear flow in the gap between the cylinders which is small compared to the diameter of the cylinders to ensure a uniform shear rate. The ratio of the width of the gap to the length of the cylinder was >20, which resulted in minimization of longitudinal instabilities28 and negligible end effects. The cylinders were made of brass to obtain a smooth surface and to prevent corrosion. The bottom surface of the lower cylinder was cut in the form of a cone with an angle of 6° to ensure a uniform shear rate at the bottom. The inner cylinder was powered by an ac motor, through a set of gears with a high reduction ratio, so that small fluctuations in the shaft speed resulted in minimal changes of the rotation speed of the inner cylinder ((0.5 rpm). The critical angular speed of the inner cylinder for transition from simple shear flow to Taylor vortices is given by29

[

Ωc ) ν

]

3390(d22 - d12) 2

4d1 (d2 - d1)

4

0.5

(5)

where d1 and d2 are the diameters of the outer and the inner cylinders, respectively (Figure 3). The critical angular velocity calculated using the kinematic viscosity (ν) of the continuous phase is 5.0 rad s-1 (48 rpm). Measurement of Drop Size Distribution. The drop size distribution was obtained by using an Image analysis system consisting of a CCD camera (SONY 94C) mounted on an optical microscope (Olympus BX60). Various combinations of objective lenses were used to accurately measure drops of diameter between 2 and 200 µm. The emulsion samples were diluted to prevent overlap of drops during viewing in the microscope and to control agglomeration as well as coalescence of the drops during microscopy. The diluent used was the pure continuous phase, equilibrated with chlorobenzene, and containing 1% (w/w of the continuous phase) surfactant. The large quantity of surfactant present prevented the drops from coalescing during storage on the slide. A drop of the diluted emulsion was placed on a glass slide with spacers, and a cover slip was carefully placed over it. Thus, the emulsion sample was enclosed in a rectangular cavity, as shown in Figure 4. This procedure ensures that the top cover slip does not press the drops and deform them. Nor does the emulsion flow out due to the weight of the cover slip. To ensure that all drops are within the same plane of focus, the specific (28) Snyder, H. D. Phys. Fluids 1968, 11, 1599. (29) Chandrashekhar, S. In Hydrodynamic and hydromagnetic stability; Dover publications: New York, 1981; pp 272-339.

Figure 6. Volume density distribution of the same emulsion sample taken separately on two different slides: (A) measurement made 8 min after sampling and again after 2.5 h; (B) measurement made 8 min after sampling from the same emulsion stock from which A was withdrawn. gravity of the diluting phase was maintained around 0.4%-0.6% higher than that of the dispersed phase. This resulted in all the drops rising and resting lightly against the top cover slip, and consequently, all drops were in the plane of focus of the objective lens. Each image typically contained 80-200 drops, and several nonoverlapping images (at least 156) were taken to capture a sufficiently large number of drops. Consistency Tests. Several tests were performed to check the consistency and accuracy of the experimental procedure. The minimum number of drops to be counted in order to obtain a good description of the drop size distribution was examined by counting an increasing number of drops. Figure 5 shows the variation of volume-average diameter with the number of drops counted for an emulsion sample. The value of the volume-average diameter becomes nearly constant beyond 15 000 drops, and beyond this limit reasonably smooth distributions were obtained as well. Hence, in all the reported data, the number of drops counted was in excess of 15 000. The stability of the emulsion, withdrawn for making drop size measurements, after dilution was tested as follows. Two samples (samples A and B) were taken


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on the size ratio. The last two terms account for the fact that there may be a critical drop diameter (Dc) at which the coalescence efficiency is maximum or minimum, depending on the value of m. The prefactor R0 is an overall coalescence efficiency term which is assumed to depend only on the shear rate and holdup. The parameters of the model (R0, C, m, and Dc) are obtained by a regression procedure applied to the experimental drop size distributions as described below. The population balance equation (eq 6) along with the coalescence efficiency (eq 7) was solved using a 4th-order Runge-Kutta method with a variable time step. The experimentally obtained initial drop size distribution was used as the initial condition in the computations. Thereafter, at each integration time step, the error (E) between the theoretical and the experimental distributions was calculated using N

E) Figure 7. Volume density distribution of emulsion samples taken from the top sampling port and the bottom of the Couette apparatus after 2.5 h of shearing at 10 rpm, for holdup φ ) 1%. from the same emulsion stock and then diluted separately. The volume density distribution was measured first for sample A within 8 min of sampling and then again after it had been left standing for 2.5 h, while sample B was assessed for the drop size distribution only once, after 8 min of sampling. The distributions are shown in Figure 6 and are nearly identical. This indicates the consistency of the sampling method as well as the stability of the emulsion after it has been diluted and transferred to the slide. The third test consisted of shearing an emulsion of 1% holdup in the tangential Couette flow apparatus for a period of 2.5 h. Samples were then withdrawn from the sampling port and also from the top and bottom of the apparatus. The volume density distributions for each of the samples were found to be nearly the same (Figure 7). This shows that the shear rate along the length of the cylinder is uniform and that creaming and settling of the drops during the course of the experiment are negligible.

Theoretical Analysis The time evolution of the drop size distribution for a purely coalescing system is given by the population balance equation

∫0νCs(ν,ν-ν′) dν′ - ∫0∞Cs(ν,ν′) dν′

∂n 1 ) ∂t 2

(6)

where n(ν) is the number density of drops of volume ν and Cs is the Smoluchowski coalescence rate given by eq 1 for drops of size ν and ν′. An assumption in the above equation is that only pairwise collisions are considered. If all collisions result in coalescence, ) 1. However, as will be seen below, the experimental results indicate that , 1 and the efficiency values also depend on the sizes of the colliding drops. Here we assume the following expression for the coalescence efficiency, which is a generalization of that proposed by Mousa and van de Ven,20

[

) R0

][ (

4q (1 + q)2

C

1+ 1-

Di Dc

)] [ ( 2 -m

1+ 1-

Dj Dc

)]

2 -m

(7)

The first two terms on the right-hand side of the expression are the same as those assumed by Mousa and van de Ven.20 The second term captures the dependence of the coalescence efficiency on the ratio of the diameters (q ) Di/Dj) of the colliding drops. The value of the exponent C indicates the strength of the dependence of the coalescence efficiency

[(Di3n(Di)|theor - Di3n(Di)|exp)∆Di]2 ∑ i)0

(8)

Computations were carried out for fixed values of m, C, and Dc, and the value of R0 was obtained at which the error was minimum for an experimental drop size distribution corresponding to a particular time. The parameters m, C, and Dc were varied in the ranges C ∈ [-4.0,4.0] in steps of 0.5, m ∈ [-1.0,2.5] in steps of 0.5, and Dc ∈ [3.0,81] µm in steps of 3.0 µm, and the set of values at which the error is a minimum was obtained for the distribution. This was repeated for each experimental distribution in a run and for all runs corresponding to various shear rates (for a fixed holdup). The arithmetic averages of the values obtained for all samples for a particular holdup were used as the final values of C, m, and Dc. Thereafter, the computation was repeated with these average values of m, C, and Dc and the value of R0 was recalculated. For a given shear rate, the average R0 was the mean of the values obtained (for constant m, C, and Dc corresponding to that particular holdup) calculated over the four samples for that particular shear rate and holdup. Results and Discussions We first present the experimental results and discuss the significant qualitative features of the data. The model predictions and their implications are discussed next. The time evolution of the drop size distributions for the three holdups studied are shown in Figures 8-10, and the volume-averaged drop diameters for all three holdups are shown in Figure 11. Each figure shows data for three different rotational speeds used. Some trends are common to all the cases, and we list these first. The peak in the drop size distributions falls and the tail lengthens with increasing time of shearing. The rate of coalescence decreases with increasing shear rate, and the coalescence rate is highest at the lowest shear rate. Both features are evident from the extent of change of the distributions with time for the different shear rates as well as from the slopes of the average diameter versus time curves. There are also significant differences in the results for the different volume holdups studied. From Figure 10 it can be seen that the most significant difference is the formation of the second peak at larger drop sizes in the high-holdup case (φ ) 15%) whereas at φ ) 5% (Figure 9) just a shoulder appears. At, φ ) 1% there is simply a smooth tail. The rate of coalescence also increases with holdup, and this is most clearly evident from the time variation of the average diameters. However, the rate of

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Figure 8. Time variation of the volume density distributions for various shear rates at 1% holdup: (symbols) experimental data, error bars show standard deviation over 3 runs; (solid lines) predictions of the population balance model using parameters given in Table 2 and Figure 14.


increase is not proportional to φ2 (as predicted by Smoluchowski6 theory) but slower. Thus, the coalescence efficiency decreases with increasing holdup. Finally, from Figure 11 it can be observed that the nature of the average diameter versus time curves are different. For φ ) 1%, the variation is almost linear with time, and this is indicative of a coalescence efficiency that is independent of the sizes of the colliding drops, as shown in Nandi et al.23 However, at φ ) 5%, the rate initially rises with time and then slows down toward the end. For the dispersedphase holdup φ ) 15%, the rate of coalescence goes up with time for the entire period of the experiment. This is evident from the increasing slope of the plot showing the nondimensionalized volume-average diameter with time. These two cases indicate a dependence of the coalescence efficiency upon the diameters of the colliding drops. A reduction in the coalescence rate at later times (as seen from the 5% holdup) was also observed by Mishra et al.18 in their experimental results with a surfactant-free emulsion system. The slowing was attributed to the flattening of the surfaces of the colliding drops, resulting in a slow rate of film thinning.7 This does not occur during the initial stages of the process when drops are small, since interfacial tension prevents significant flattening. A similar effect could be active in the present system as well. Possible reasons why this effect is not seen in the lower and higher holdup systems are as follows. In the 1% holdup system the drops never become large enough. In the 15% holdup case, chaining of drops due to shear flow may provide the extra force to overcome the additional resistance to thinning of the film due to flattening of the larger drops. More detailed drop-level studies, however, are required to confirm these explanations. The drop size distributions for the case with φ ) 1%

were scaled using the analysis of Swift and Friedlander.19 A plot of the cumulative oversize number fraction (1 Nνi/NT) versus the normalized drop volume (νi/〈ν〉) for φ ) 1% and for three rotation speeds is shown in Figure 12 along with that for φ ) 15% for the lowest shear rate. i ∞ nk, NT ) ∑i)1 ni, and 〈ν〉 ) NT/φ. For the Here, Nν ) ∑k)1 lowest holdup (φ ) 1%) the scaled distributions at different times of shearing superimpose (Figure 12a-c), showing that the distributions are self-similar and that the coalescence efficiency is independent of drop size. However, the curves for φ ) 15% and γ ) 10.8 s-1 (Figure 12d) do not scale, indicating that the coalescence efficiency in this case is dependent on drop size. The solid lines in Figures 8-10 are the model predictions obtained using the fitted parameters. The predictions match the experimental distributions within the limits of experimental error. Note that the development of the shoulder for the 5% emulsions and the formation of the second peak for the 15% holdup are well-described by the model. The prediction of the variation of the volumeaverage drop diameter with time also matches well with experimental data for all cases except for the 5% holdup at long times. The values of the fitted parameters (C, DC, and m) along with the standard deviations for each value of the dispersed-phase holdup are given in Table 2. The values of these parameters obtained from fitting the model to the experimental distributions at different time steps (during any single run) were nearly the same. The values were also found to be nearly independent of shear rate at a particular holdup. The parameter values provide an insight into the coalescence at the scale of individual drops. At the lowest volume holdup, C ) 0 and m ) 0, implying that the coalescence efficiency is independent of the sizes



of the colliding drops and depends only on the shear rate, as reported earlier.23 With increasing holdup, the parameter C decreases while m increases. The critical drop diameter is found to be about 18 µm for both 5% and 15% holdup. A lower value of C indicates a higher probability of coalescence for drops of unequal sizes. The coalescence efficiency attains a maximum (since m is positive) at the critical diameter (Dc). However, coalescence efficiency is a product of these two terms, and the variation of the coalescence efficiency with the diameters of the colliding drops is shown in Figure 13 for the highest dispersedphase holdup. This figure shows that coalescence is most likely when one of the pair of colliding drops is close to the critical diameter and the other is small (∼2.0 µm). While the existence of an optimal drop size for coalescence has been proposed earlier,30 it is not clear whether the same effects are applicable here. We speculate that the observed maximum could be the result of long-lived clusters of one large and several small drops which eventually coalesce. The clusters comprising 18 µm and several 2 µm drops are perhaps the most stable under shear flow. We note that a bimodal distribution is not produced if m ) 0 (which corresponds to the model of Mousa and van de Ven20), for any value of the exponent C. Thus, the existence of a critical diameter seems essential to explain this feature of the results. The shear rate dependence of the coalescence rate is given by the product of R0 with the shear rate (γ) (eq 3). Figure 14 shows that the coalescence rate decreases monotonically with shear rate, indicating that coalescence efficiency decreases faster than the inverse of the shear (30) Kumar, S.; Kumar, R.; Gandhi, K. S. Chem. Eng. Sci. 1993, 48, 2025.

Langmuir, Vol. 17, No. 9, 2001 2653

Figure 11. Variation of the nondimensionalized volumeaveraged diameters with time for various shear rates (corresponding to each holdup as shown in Figures 8-10).

rate. Further, R0γ decreases with increasing holdup, indicating that the probability of coalescence is lower in the case of higher holdups. Thus, the coalescence rate increases with holdup but at a rate that is lower than the φ2 proportionality, as suggested by Smoluchowski’s theory.6 The magnitude of coalescence efficiency obtained is 2 orders of magnitude lower than that predicted by Chesters’7 model. Experiments were also performed in which the emulsion was kept standing in a beaker and samples were withdrawn at specified times and the size distributions analyzed. For the case of 1% holdup, the extent of increase of the volume-averaged diameter was slightly higher (D h /D h0 ) 1.337 after standing for 4.5 h) than that at the lowest shear rate studied. For the holdup of 5%, the dimensionless volume-averaged diameter increased to 3.7 after standing for 4.5 h, which is much larger than the result for the lowest shear rate. In the 15% holdup case, the emulsion separated into two distinct layers on standing for 0.5 h, the bottom layer being the dispersed phase, as its specific gravity was higher than that of the continuous phase by 0.003 g/cm3. The high coalescence rate in standing emulsions is due to the formation of long-lived aggregates of drops. The thin films in these aggregates have sufficient time to rupture. Weak Brownian motion and small flow currents generated due to handling cause the drops to collide and form aggregates which can be seen under the microscope if a sample is carefully taken. Formation of such aggregates was reported by Bibette et al.31 The reduction in the coalescence efficiency is primarily due to the fact that, with higher shear rates, the interdrop contact time (tc) goes down approximately as the inverse of the shear rate.7 As a result of this, the time available for the film between two drops to rupture is reduced as (31) Bibette, J.; et al. Phys. Rev. Lett. 1992, 69, 981.

2654


Nandi et al.

Figure 14. Variation of R0γ with shear rate (γ) for different holdup emulsions: (symbols) values back-calculated from experimental distributions; (solid line) prediction of eq 10. Table 3. Back-calculated Dimensionless Contact Times (γtc) for Colliding Drops Based on the Results Given in Figure 14

Figure 12. Scaled cumulative oversize distribution for different shear rates and holdups. The distributions are self-similar for the emulsion having φ ) 1% .

φ

γ ) 10.8 s-1

γ ) 21.5 s-1

γ ) 43.0 s-1

1% 5% 15%

1.00 0.61 0.54

1.00 0.75 0.64

1.00 1.04 0.83

be stochastically distributed, with a log-normal distribution25,26

P(ln tb) )

( [

])

1 1 ln tb - µb exp 2 σb σx2π

2

(9)

Assuming that the interdrop contact time is tc ) 1/γ7 for dilute emulsions (φ ) 1%), the coalescence efficiency is given by

R0 )

Figure 13. Variation of the normalized coalescence efficiency (/R0) with diameters of the colliding drops (Di,Dj) obtained from eq 7 for the 15% emulsion (Table 2). Table 2. Fitted Values of the Parameters C, m and Dc for the Three Holdups Studied (Refer to Eq 7) and Standard Deviations of Each of the Parameters over the Three Shear Rates for Each Holdup φ

C

M

Dc (µm)

1% 5% 15%

0.0 -1.5 ( 0.43 -2.0 ( 0.90

0.0 0.5 ( 0.5 1.75 ( 0.66

18.75 ( 2.54 18.00 ( 3.5

the shear rate is increased and thus the probability of coalescence is also reduced. A model based on this mechanism was presented in ref 23. The time (tb) for which the film between the drops survives is assumed to

∫-∞ln t

c

P(ln tb) d ln tb )

{

}

µb + ln γ 1 erfc 2 x2σb

(10)

The above equation was fitted to the values of R0 obtained for φ ) 1% emulsions and for different shear rates to yield µb ) 4.35 and σb ) 1.99. The solid line in Figure 14 shows that the model adequately describes the experimental data. In the case of high holdups, although film stability may be well-described by the log-normal probability density (eq 9), the average contact time is difficult to estimate independently. Fitting eq 10 to the data for R0 given in Figure 14 for the higher holdups yields estimates of the average contact times (tc). The results obtained for different shear rates are given in Table 3. The increased stabilization (compared to that for the lower holdups) may be due to multidrop collisions which reduce the contact times as shown in Table 3. A quantitative understanding of such phenomena is complicated, and as yet models to calculate parameters such as average contact time are unavailable in the literature. The lack of dependence of the coalescence efficiency on the sizes of the colliding droplets, for the case of a dispersed-phase holdup of 1%, can be explained from the model of Chesters7 for droplets with immobile interfaces. Calculations show that, on reducing the critical thickness


at which the film can rupture, the coalescence efficiency becomes weakly dependent on the sizes of the colliding species. In the case of our emulsion system, the critical thickness for the rupture of the film is low, since the film is stabilized by the presence of surfactant. Further, only “near head-on” collisions give contact times long enough to allow for film rupture and coalescence. The contact time may be divided into two regimes. Initially, the film thins rapidly, and thereafter, the film thinning becomes very slow. The second part may be termed as the “drop rest time”. For near head-on collisions, the period for the drop rest time is significantly higher than the time for which the film thins. The rate of film thinning is dependent on the sizes of the colliding species, however, the drop rest time is not. Thus, for collisions in which the time of thinning is small compared to the drop rest time, the dependence of the coalescence efficiency on the drop sizes becomes insignificant. At higher dispersed-phase holdups, hydrodynamic, multibody interactions are likely to affect the interdroplet contact times and the situation of a size independent coalescence efficiency will therefore not hold, as has been observed in this work. Conclusions A neutrally buoyant emulsion, stabilized by surfactant and containing an electrolyte, was subjected to a tangential Couette flow in the simple shear regime. The drop size distributions were measured using optical microscopy and image analysis. Studies were conducted for varying shear rates and holdups. An increasing shear rate was found to stabilize the emulsion in all the cases. Increasing the dispersed-phase holdup gave rise to a bimodal distribution, unlike in the case of the lowest holdup (1%), where the drop size distribution curves showed an elongating tail with time and a diminishing peak. On scaling the experimental distributions, it was noted that the distributions of samples withdrawn at various times for the case of 1% holdup showed self-similarity, indicating that the coalescence efficiency was independent of drop size. A similar graph for the case with 15% holdup showed a wide difference between the scaled distributions, indicat-

Langmuir, Vol. 17, No. 9, 2001 2655

ing the dependence of the coalescence efficiency on the size of the coalescing droplets. A population balance model was used to describe transient evolution of the experimentally obtained drop size distribution and to back-calculate the coalescence efficiency. The coalescence efficiency was found to decrease with increasing shear rate in all the cases. At low holdup (1%) the coalescence efficiency was found to be independent of drop size, in agreement with the scaling result. At high holdups the coalescence efficiency exhibited dependence on colliding drop sizes and the highest efficiency was obtained for a pair of drops with one drop size being about 18 µm and the other approximately 2 µm. The coalescence efficiency was found to reduce significantly with increased holdup. Thus, the coalescence rate increases at a rate slower than φ2. An estimate of the coalescence efficiency was obtained considering the process of film rupture to be controlling and assuming a log-normal distribution of the rupture time. The analysis gave a good description of the variation of coalescence efficiency with shear rate at low holdups. With increasing holdups, the drop-drop contact times were found to decrease, indicating the importance of interdrop interactions. This work demonstrates an experimental method for back-calculating the shear-induced coalescence efficiency for emulsions, for which quantitative models are as yet unavailable. The experimental results and analysis provide a qualitative insight into drop level processes during shear-induced coalescence. More detailed models and computations including multidrop interactions are perhaps required to obtain a quantitative description of the process. Acknowledgment. The financial support of the BRNS through Project No. 36/7/94/R&D-II is gratefully acknowledged. D.V.K. acknowledges the financial support from DST, India, through the Swarnajayanti fellowship. The authors thank Dr. B. K. Das and Dr. P. D. Sawant of the Alchemie Research Center, Bombay, for the help in measuring the interfacial tension values. LA001473M

Coalescence in Surfactant-Stabilized Emulsions ... - ACS Publications

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