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Effect of Confinement on Droplet Breakup in Sheared Emulsions Anja Vananroye, Peter Van Puyvelde,* and Paula Moldenaers Katholieke UniVersiteit LeuVen, Department of Chemical Engineering, W. de Croylaan 46, B-3001 LeuVen (HeVerlee), Belgium ReceiVed February 15, 2006. In Final Form: March 2, 2006 The breakup of Newtonian droplets in a Newtonian matrix during shear flow is investigated in a counterrotating parallel plate device. For bulk conditions, the critical capillary number for breakup is known to be only determined by the viscosity ratio. Here, we show that the critical capillary number is also affected by the degree of confinement: for low viscosity ratios, confinement suppresses breakup, whereas for high viscosity ratios, confinement promotes breakup. This way, above a critical value for the degree of confinement, even droplets with a viscosity ratio larger than 4, which are unbreakable by shear in a bulk situation, can be broken in a simple shear flow field.
Introduction Microscale technologies are rapidly developing in the process industry.1 This has already lead to the development of interesting microfluidic devices and applications.2,3 The processing of geometrically confined blends with a droplet-matrix structure has attracted attention, because the size of the droplet phase can become comparable to the dimensions of the flow geometry. Although a complete physical picture of these geometrically confined blends is still lacking, some remarkable phenomena have already been reported. It was shown, for instance, that in confined emulsions the steady-state droplet size and the relaxation of individual droplets are still in agreement with theories developed for a bulklike environment.4 However, on a larger scale, droplets organize in so-called superstructures, such as pearl necklaces and strings.4 Migler and co-workers5-7 mapped out the transition of droplets to stringlike structures, when concentrated blends with a viscosity ratio near unity become confined between shearing surfaces, a phenomenon that was attributed to coalescence. A morphology diagram in the parameter space of shear rate and mass fraction was constructed, describing the different possible structures. Nevertheless, a clear understanding of the morphological dynamics is still missing. In this paper, the effect of confinement on the breakup of single Newtonian droplets in a Newtonian matrix is considered. To the author’s knowledge, such experiments have not been reported before. Since only single droplet dynamics is discussed, coalescence cannot interfere with the observed results. From the pioneering work of Taylor,8,9 it is known that two dimensionless numbers govern the behavior of a very dilute emulsion consisting of Newtonian components: the capillary number Ca (Ca ) ηmRγ˘ )/R where ηm, R, γ˘ , and R denote respectively the matrix viscosity, the droplet radius, the shear rate, and the interfacial tension) and the viscosity ratio p (p ) * Corresponding author. E-mail:
[email protected]. Tel: +32-16-32.23.57. Fax: +32-16-32.29.91. (1) Stone, H. A.; Stroock, A. D.; Ajdari, A. Annu. ReV. Fluid Mech. 2004, 36, 381-411. (2) Link, D. R.; Anna, S. L.; Weitz, D. A.; Stone, H. A. Phys. ReV. Lett. 2004, 92, 0545031-0545034. (3) Utada, A. S.; Lorenceau, E.; Link, D. R.; Kaplan, P. D.; Stone, H. A.; Weitz, D. A. Science 2005, 308, 537-541. (4) Vananroye, A.; Van Puyvelde, P.; Moldenaers, P. Langmuir 2006, 22, 2273-2280. (5) Migler, K. B. Phys. ReV. Lett. 2001, 86, 1023-1026. (6) Pathak, J. A.; Davis, M. C.; Hudson, S. D.; Migler, K. B. J. Colloid Interface Sci. 2002, 255, 391-402. (7) Pathak, J. A.; Migler, K. B. Langmuir 2003, 19, 8667-8674. (8) Taylor, G. I. Proc. R. Soc. London, A 1932, 138, 41-48. (9) Taylor, G. I. Proc. R. Soc. London, A 1934, 146, 501-523.
Figure 1. Effect of viscosity ratio on droplet breakup in shear flow. Experimental data are taken from Grace.10
ηd/ηm with ηd the droplet viscosity). Above a critical capillary number Cacrit, which only depends on p, a droplet will break up. This relationship, Cacrit ) f(p), was carefully mapped out experimentally by Grace,10 both for simple shear and extensional flow, and the results for shear flow were fitted by a correlation proposed by de Bruijn,11 as shown in Figure 1. These results show that Cacrit reaches a minimum for p between 0.1 and 1. For p above 1, an increase in Cacrit is reported until, for p around 4, breakup is prevented due to the presence of rotational components in the shear flow field. For p > 4, droplets will tumble during start up of flow until an ellipsoidal droplet is obtained, aligned in the flow direction.12 In this research, the effect of confinement on Cacrit is systematically investigated for various values of p. Materials and Methods Poly(isobutylene) (PIB, parapol 1300), with a viscosity of 110 Pa‚s at room temperature (23 °C), is chosen as the matrix phase, whereas various grades of poly(dimethylsiloxane) (PDMS, Rhodorsil/ Silbione) with viscosities ranging from 30 to 1100 Pa‚s have been selected as the droplet phase. It was shown that PIB is slightly soluble in PDMS, but the solubility of PDMS in PIB is very limited.13 Therefore, with PDMS as the droplet phase, no droplet shrinkage is seen, and hence, the system can be considered to be completely immiscible. The interfacial tension R of PDMS/PIB blends is 2.8 mN/m14 and is independent of the molecular weight (Mw), for grades with relatively high Mw, as is the case here.15 (10) Grace, H. P. Chem. Eng. Commun. 1982, 14, 225-277. (11) De Bruijn, R. A. Ph.D. Thesis, Eindhoven University of Technology, 1989. (12) Rumscheidt F. D.; Mason, S. G. J. Colloid Sci. 1961, 16, 210-261. (13) Guido, S.; Villone, M. J. Rheol. 1998, 42, 395-415.
10.1021/la060442+ CCC: $33.50 © 2006 American Chemical Society Published on Web 03/23/2006
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Figure 2. Scheme of counterrotating parallel plate device. The experiments were performed with a counterrotating parallel plate shearing device (Paar-Physica), schematically shown in Figure 2. It consists of two parallel quartz plates with a diameter of 50 mm. A transparent glass cup is fixed around the bottom plate, serving as a container for the matrix material. In the present work, a gap spacing of 1 mm was chosen. The motion of both plates can be separately controlled; as it is possible to rotate the plates in opposite directions, a stagnation plane can be created. To study the effects of confinement, expressed as the ratio of the droplet diameter 2R to the gap spacing d, and viscosity ratio, droplets with radii R between 55 and 455 µm and viscosity ratios p between 0.31 and 12, are injected in the PIB matrix using a homemade injection device. After injection, the threedimensional position of the droplet, as well as the droplet diameter are determined. Droplets that are situated close to one of the two walls are rejected for further analysis. From the radial position of the droplet, appropriate shear rates can be generated. The droplets, located in the stagnation plane, are observed with optical microscopy. Images can be captured both in the vorticity-velocity plane, as well as in the velocity-velocity gradient plane by simply repositioning the microscope and camera. From this, it is possible to extract information about the deformation and orientation of droplets during flow. A constant, low shear rate is initially applied to the system. When steady state is reached without the occurrence of breakup, the flow is temporarily stopped. After retraction of the droplet, the shear rate is slightly increased. This procedure is repeated until eventually breakup is observed. From the shear rate at which this occurs, the critical capillary number (Cacrit) can be calculated.
Experimental Results and Discussion In Figure 3, the experimentally determined critical capillary numbers of several droplets are depicted as a function of the degree of confinement (2R/d). Figure 3a shows the results for p ∼ 0.31 (b) and p ∼ 1 (0). For systems with a viscosity ratio near unity, the capillary number at which droplet breakup occurs remains comparable to the bulk results of Grace (Figure 1) and to the fitted value of de Bruijn (0.48), for all values of 2R/d. Hence, it appears that breakup of single droplets is not influenced by the confining walls when the viscosity ratio is near unity, at least for 2R/d values up to 0.9. This result is in line with recent data presented by Sibillo et al., who investigated droplet breakup in confined geometries at a single viscosity ratio of 1.16 At a viscosity ratio of 0.31, good agreement between the experimental critical capillary number and the bulk critical capillary number of de Bruijn (0.48) is only obtained for low values of 2R/d. For more confined droplets, Cacrit increases, indicating that, at a viscosity ratio below 1, breakup of individual droplets can be suppressed. As can be seen in Figure 3a, the ratio of droplet diameter to gap spacing at which this deviation starts to appear is approximately 0.3. (14) Sigillo, I.; di Santo, L.; Guido, S.; Grizzuti, N. Polym. Eng. Sci. 1997, 37, 1540-1549. (15) Kobayashi, H.; Owen, M. J. Trends Polym. Sci. 1995, 3, 330-335. (16) Sibillo, V.; Pasquariello, G.; Simeone, M.; Guido, S. Presentation giVen at the 77th annual meeting of the Society of Rheology; Vancouver 2005.
Figure 3. Effect of degree of confinement on the critical capillary number Ca; (a) p ∼ 0.31 (b) and p ∼ 1 (0). The dashed line represents the bulk critical capillary number of Grace;10 (b) p ∼ 5 (b) and p ∼ 11.5 (0) (dotted lines to guide the eye).
Figure 3b depicts the results for p ∼ 5 (b) and p ∼ 11.5 (0). As can be seen, droplets with a viscosity ratio of 11.5 and with a small degree of confinement cannot be broken by shear flow, which is in agreement with the data of Grace (Figure 1). However, with increasing degree of confinement, a large decrease in Cacrit is present for both viscosity ratios, though the effect is more pronounced for p ∼ 11.5. When 2R/d > 0.7, the droplets break at very low capillary numbers (Cacrit ∼ 0.5) with respect to the bulk situation. Therefore, it can be concluded that breaking droplets with high viscosity ratios in a simple shear flow field becomes possible at low capillary numbers when the droplets are highly confined. Figure 4 illustrates the behavior of the droplets with a very high viscosity ratio as a function of strain γ () γ˘ t, with t the shearing time), both in unconfined [Figure 4a] and confined [Figure 4b] conditions. The time t ) 0 corresponds to the start up of the shear flow. The viscosity ratios of the two droplets differ slightly due to a difference in experimental temperature. A shear rate corresponding to Ca ) 1 is applied to the droplet in Figure 4a. As can be seen at this low degree of confinement (2R/d ) 0.21), the droplet tumbles during flow due to the rotational components in the flow field, until it eventually aligns in the flow direction. This tumbling was seen for Ca ranging from 0.5 to 50. In Figure 4b, images of a highly confined droplet (2R/d ) 0.75) are shown. The droplet is subjected to a shear rate corresponding to a droplet capillary number of 0.6. The viscosity ratio for this experiment is as high as 11.7. In this case, the confining walls prevent the droplet from tumbling. After starting the flow, the droplet stretches continuously until it eventually breaks, via necking, into two equally sized daughter drops with small satellite droplets in between. Figure 5 shows normalized critical capillary numbers for several viscosity ratios as a function of the degree of confinement.
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Figure 4. Time evolution of droplet deformation and orientation during shear flow; (a) Ca ) 1, 2R/d ) 0.21, p ) 11.2; (b) Ca ) 0.6, 2R/d ) 0.75, p ) 11.7.
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at p ∼ 11.5 [(0) in Figure 3b] will not be added on this normalized graph. For all values of p, the scaled critical capillary number Ca*crit in Figure 5 is near unity for 2R/d < 0.3. As stated before, no effect of confinement is detected for p ) 1. For p < 1, when the degree of confinement exceeds a value of 0.3, the critical capillary number increases with increasing degree of confinement. On the contrary, for viscosity ratios above unity, Ca*crit decreases with increasing degree of confinement. No clear explanation of the reported phenomena is available at present. Confined droplets were observed to be more deformed than their unconfined counterparts, irrespective of the viscosity ratio. In addition, they displayed more blunt ends,17 suggesting more internal flow. The complexity of the internal flow patterns in confined conditions has recently been discussed by Hodges et al.18 The higher degree of deformation is consistent with the analysis of Shapira and Haber, who reported that, for all viscosity ratios and in a Couette flow, droplets close to a wall deform more than those positioned near the center.19 It was also remarkable that the deformation of confined droplets went through a maximum before the actual breakup occurred. This was especially pronounced at low viscosity ratios. Finally, the observed breakup mechanisms in confined geometries also differ from the ones reported in bulk conditions. At low viscosity ratios, breakup occurs through Rayleigh-like instabilities and end pinching. The fact that flow is known to stabilize Raleigh instabilities, might partially explain the suppression of breakup observed under such conditions. At high viscosity ratios, breakup in confined conditions occurs via the usual splitting in two droplets as it was shown in Figure 4b. Increased deformation, together with a shift from rigid body rotation to internal flow, might contribute to the rupture of these droplets
Conclusions
Figure 5. Effect of the degree of confinement on the normalized critical capillary number.
Normalization is performed by scaling the experimentally determined Ca with the bulk critical capillary number of Grace.10 For viscosity ratios of 0.31, 0.96, and 1.9, the values are scaled with the corresponding fitted values of de Bruijn.11 For a viscosity ratio of 5, scaling with the bulk critical capillary number is a priori impossible. However, as a first approximation, a quadratic fitting through the data of Grace in the p-window above p ) 0.4, could be used to estimate a hypothetical critical bulk capillary number. However, such an extrapolation will only be reasonable for p values close to 4. Due to this uncertainty, the data points
The effect of confinement on the breakup of Newtonian droplets in a Newtonian matrix during a steady-state shear flow is investigated. Confined droplets show different breakup behavior than nonconfined ones in simple shear flow: breakup of confined droplets occurs at higher Ca with respect to the bulk critical capillary number for p < 1 and at lower Ca for p > 1. In particular, it has been shown that by applying a shear field to highly confined droplets breakup can be achieved even with p > 4, which is not common in bulk shear flow. Acknowledgment. FWO Vlaanderen (Project G.0523.04) and onderzoeksfonds K.U.Leuven (GOA 03/06) are gratefully acknowledged for financial support. LA060442+ (17) Vananroye, A.; Van Puyvelde, P.; Moldenaers, P. To be submitted. (18) Hodges, S. R.; Jensen, O. E.; Rallison, J. M. J. Fluid Mech. 2004, 501, 279-301. (19) Shapira, M.; Haber, S. Int. J. Multiphase Flow. 1990, 16, 305-321.
