pubs.acs.org/Langmuir © 2009 American Chemical Society
Microconfined Shear Deformation of a Droplet in an Equiviscous Non-Newtonian Immiscible Fluid: Experiments and Modeling Mario Minale,*,† Sergio Caserta,‡ and Stefano Guido‡ †
Seconda Universit a di Napoli, Department of Aerospace and Mechanical Engineering Real Casa dell’Annunziata, via Roma 29, 81031 Aversa (CE), Italy, and ‡Universit a Federico II di Napoli, Department of Chemical Engineering Laboratori Gianni Astarita, P.le Tecchio 80, 80125 Napoli, Italy Received June 17, 2009. Revised Manuscript Received July 31, 2009 In this work, the microconfined shear deformation of a droplet in an equiviscous non-Newtonian immiscible fluid is investigated by modeling and experiments. A phenomenological model based on the assumption of ellipsoidal shape and taking into account wall effects is proposed for systems made of non-Newtonian second-order fluids. The model, without any adjustable parameters, is tested by comparison with experiments under simple shear flow performed in a sliding plate apparatus, where the ratio between the distance between the confining walls and the droplet radius can be varied. The agreement between model predictions and experimental data is good both in steady state shear and in transient drop retraction upon cessation of flow. The results obtained in this work are relevant for microfluidics applications where non-Newtonian fluids are used.
Introduction Polymer blending is considered to be a nice alternative to synthesizing new polymers to obtain materials with specified mechanical, electrical, or thermal properties. Indeed, all the properties of the final blend depend not only on the properties of the constituents, but also on the morphology that develops during processing. The topic has, thus, attracted a lot of attention in the literature as reviewed by Tucker and Moldenaers.1 The morphology that develops during flow varies with the concentration starting from the globular kind, in dilute and semidilute conditions, passing through a cocontinuous phase, at a concentration of about 50%, and eventually going back to the globular morphology as concentration further increases. A popular model system for blends with globular morphology consists of a single drop immersed in a infinite fluid subjected to an imposed flow field. Thus, the dynamics of a single drop received much attention in the field of fluid mechanics. First, the “Newtonian systems”, i.e., systems made of a Newtonian drop immersed in a Newtonian matrix, have been widely studied, as reviewed by several authors.2-4 Successively, in recent years, attention has also been paid to study the deformation of a single non-Newtonian drop immersed in a non-Newtonian matrix subjected to a generic flow field: Greco5 developed a theory for small drop deformation, whereas, for instance, Minale6 developed a phenomenological model; several experimental investigations have also been carried out.7,8 In real processing conditions, the fluids are very often nonNewtonian, thus justifying the interest in the literature for the non-Newtonian systems. In addition, confined channel flow is also encountered in important applications, such as mixing. *To whom correspondence should be addressed. Tel.: þ39 081 5010292. Fax: þ39 081 5010204. E-mail:
[email protected].
(1) Tucker, C. L.; Moldenaers, P. Ann. Rev. Fluid Mech. 2002, 34, 177–210. (2) Rallison, J. M. Ann. Rev. Fluid Mech. 1984, 16, 45–66. (3) Stone, H. A. Ann. Rev. Fluid Mech. 1994, 26, 65–102. (4) Guido, S.; Greco, F. Rheol. Rev. 2004, 99–142. (5) Greco, F. J. Non-Newtonian Fluid Mech. 2002, 107, 111–131. (6) Minale, M. J. Non-Newtonian Fluid Mech. 2004, 123, 151–160. (7) Guido, S.; Simone, M.; Greco, F. J. Non-Newtonian Fluid Mech. 2003, 114, 65–82. (8) Verhulst, K.; Moldenaers, P.; Minale, M. J. Rheol. 2007, 51, 261–273.
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Therefore, the dynamics of drops dispersed in polymer blend is often affected by wall presence, either because microfluidics problems are encountered, or because the drops flow in the vicinity of a device wall. Confinement effects in polymer blends have been investigated in the very recent past focusing the attention on Newtonian systems. The transition from droplet morphology to stringlike structures, which occurs when concentrated blends with a viscosity ratio of about 1 become confined between shearing planes, has been mapped experimentally,9-11 whereas Vananroye et al.12 have experimentally shown that concentration affects the blend morphology much more in confined flows than in unconfined ones. The single drop problem received some attention in the literature also for confined flows, and in 2006 Sibillo et al.13 tracked the evolution of a single drop immersed in an equiviscous sheared fluid as a function of the _ degree of confinement and of the Capillary number (Ca=μmRγ/ Γ, where μm is the matrix viscosity, R is the undistorted drop radius, γ_ is the applied shear rate, and Γ is the interfacial tension). An analogous work has been done14 not limiting the attention to equiviscous systems but extending the analysis to systems with several viscosity ratios λ (= μd/μm, where μd is the drop fluid viscosity), and it was concluded that wall effects enhance drop deformation at any viscosity ratio. Drop break-up has been also investigated15 showing that it is inhibited for systems with viscosity ratio less than one, while it is enhanced for systems with λ > 1. The authors also observed that drops with λ > 4 can breakup in confined simple shear flow, while they can not in the same unbounded flow. (9) Migler, K. B. Phys. Rev. Lett. 2001, 86, 1023–1026. (10) Pathak, J. A.; Davis, M. C.; Hudson, S. D.; Migler, K. B. J. Colloid Interface Sci. 2002, 255, 391–402. (11) Pathak, J. A.; Migler, K. B. Langmuir 2003, 19, 8667–8674. (12) Vananroye, A. P.; Van Puyvelde, P.; Moldenaers, P. Langmuir 2006, 22, 2273–2280. (13) Sibillo, V.; Pasquariello, G.; Simeone, M.; Cristini, V.; Guido, S. Phys. Rev. Lett. 2006, 97, 054502. (14) Vananroye, A. P.; Van Puyvelde, P.; Moldenaers, P. J. Rheol. 2007, 51, 139– 153. (15) Vananroye, A. P.; Van Puyvelde, P.; Moldenaers, P. Langmuir 2006, 22, 3972–3974.
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The problem has been studied also theoretically. Shapira and Haber16 derived both the steady drop shape and the full hydrodynamics in the limit of low Reynolds number for a single Newtonian drop immersed in a confined Newtonian fluid undergoing simple shear flow. They used the Lorentz’s reflection method17 stopping to the third reflection, and thus only mild degree of confinement can be accounted for; they predicted an enhancement of drop deformation due to confinement, as experimentally verified later on.13,14 To overcome some limitations of the Shapira and Haber theory,16 Minale18 formulated a phenomenological model based on the assumption that the drop deforms into an ellipsoid, preserving its volume. Minale modified the Maffettone-Minale model19 that is based on the same assumptions and was developed to describe the dynamics of a Newtonian drop immersed in a Newtonian unconfined matrix subjected to a generic flow field. Minale’s model18 recovers the analytical theoretical limit of Shapira and Haber16 for small deformation in simple shear flow and degenerates into the Maffettone and Minale model19 as the drop becomes unconfined. The model predictions are in good agreement with both steady state data18 and transients ones.20 Recently, numerical studies appeared on the topics,21,22 and a review paper has been also published.23 In this paper, we investigate the effects of confinement on a single non-Newtonian drop immersed in a non-Newtonian matrix. We modify Minale’s model6 for non-Newtonian systems to account for confinement, and we compare the model predictions with experiments performed on a Newtonian drop immersed in an equiviscous confined non-Newtonian fluid. The non-Newtonian model was developed to recover analytical limits such as Greco’s theory.5 In the latter, the non-Newtonian effects are accounted for by considering second-order fluids, which show constant viscosity and first normal stress difference (a property associated with fluid elasticity) proportional to the shear rate squared. The second-order fluids are the asymptotic limit to which any non-Newtonian fluid tends at vanishing shear rate. Thus, the model presented in this work catches the major features of any non-Newtonian fluids. Experimentally, a model system approaching the second order fluid behavior is the so-called Boger fluid24 obtained by mixing a small amount of high-molecularweight polymer with a larger amount of a the same polymer with a sensible smaller molecular weight. The latter fluid must show a Newtonian behavior. In this way, the Boger fluid shows a constant viscosity and a significant amount of elasticity, as measured by the first normal stress difference. The advantage of the use of such a fluid is that the shear thinning non-Newtonian effect (i.e., the decrease of the viscosity with the shear rate) is decoupled from the elastic response. Some experimental data on a (16) Shapira, M.; Haber, S. Int. J. Multiphase Flow 1990, 16, 305–321. (17) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; Martinus Nijhoff Publishers: Dordrecht, The Netherlands, 1983. (18) Minale, M. Rheol. Acta 2008, 47, 667–675. (19) (a) Maffettone, P. L.; Minale, M. J. Non-Newtonian Fluid Mech. 1998, 78, 227–241. Erratum J. Non-Newtonian Fluid Mech. 1999, 84, 105–106. (20) Vananroye, A.; Cardinaels, R.; Van Puyvelde, P.; Moldenaers, P. J. Rheol. 2008, 52, 1459–1475. (21) Janssen, P. J. A.; Anderson, P. D. Phys. Fluids 2007, 19, 043602/1–043602/ 11. (22) Vananroye, A.; Janssen, P. J. A.; Anderson, P. D.; Van Puyvelde, P.; Moldenaers, P. Phys. Fluids 2008, 20, 013101/1–013101/10. (23) Van Puyvelde, P.; Vananroye, A.; Cardinaels, R.; Moldenaers, P. Polymer 2008, 49, 5363–5372. (24) Boger, D. V. J. Non-Newtonian Fluid Mech. 1977, 3, 87–91. (25) Verhulst, K..; Cardinaels, R.; Renardy, Y.; Moldenaers, P. Proceedings of the XVth International Congress on Rheology, Monterey, CA, Aug 3-8, 2008; Society of Rheology: Orono, ME, 2008; pp 1000-1002 (26) Verhulst, K.; Cardinaels, R.; Moldenaers, P. Polymer Processing Society Europe/Africa Regional Meeting, G€oteborg, Sweden, Aug 28-30, 2007; Polymer Processing Society: Melville, NY, 2007; CD-ROM.
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similar model system at a different viscosity ratio between the drop and the external fluid have been recently presented.25-28 Here, we focus on the equiviscous case with the aim of studying the effects of the capillary number and the degree of confinement in a non-Newtonian matrix, under both steady-state shear and transient relaxation upon cessation of flow.
The Model The model is based on the following hypotheses: (A) the drop deforms keeping the ellipsoidal shape and preserving its volume; (B) the non-Newtonian effects appear at the second order in the Capillary number as in the unconfined non-Newtonian case.5 With these hypotheses at O(Ca) the system behaves as the confined Newtonian system and thus the model must recover the analytical limit of Shapira and Haber16 for small steady drop deformation. We then hypothesize: (C) as predicted by Shapira and Haber for the Newtonian case, also in the non-Newtonian case, the effects of confinement scale with the degree of confinement to the power three, i.e., as (R/H)3, where H is the distance between the device walls (e.g., the gap in a shear cell.) Because Minale’s model for non-Newtonian systems6 already satisfies hypotheses A and B, we here modify this model to account for confinement effects so to fulfill also hypothesis C. Because the drop is assumed to be ellipsoidal it is described by S-1:rr = 1, where S is a second rank, symmetric, positive definite tensor made dimensionless with the squared undistorted drop radius R2, and r is the position vector made nondimensional with R. The eigenvalues of S represent the squared dimensionless drop semiaxes. The nondimensional drop evolution equation proposed by Minale is ( DS - CaðΩ 3 S - S 3 ΩÞ ¼ - f1 ðS - gIÞ þ Ca f2 ðD 3 S þ S 3 DÞ þ Dt ) S:I ð1Þ f3 ðD 3 S 3 S þ S 3 S 3 DÞ - ðD 3 S þ S 3 DÞ 3 where t is the time made dimensionless with the so-called emulsion time τ = μmR/Γ, Ω = (rv - rvT)/2 (where rv is the velocity gradient tensor) is the vorticity tensor made dimensionless with γ_ and it represents the rotational contribution of the flow field, D= (rvþrvT)/2 is the rate of deformation tensor also made dimensionless with γ_ and it represents the stretching contribution of the flow field, and I is the unit tensor. In eq 1, the LHS is the frame invariant Jaumann time derivative,29 whereas the RHS is the driving force for drop deformation that results from the competing actions of the interfacial tension, which drives the drop to recover the undistorted spherical shape (the unit tensor I), and the viscous drag that tends to deform the drop (the term multiplied by Ca). The function g is plugged into the interfacial term of eq 1 to preserve the drop volume, i.e., the third scalar invariant of S and is given by f3 IIIS ð2Þ g ¼ 3 - 2Ca ISD f1 IIS where ISD is the first scalar invariant (the trace) of the tensor S 3 D, IIIS is the third scalar invariant (the determinant) of the tensor S, (27) Cardinaels, R.; Verhulst, K.; Moldenaers, P. 9th European Symposium on Polymer Blends, Palermo, Italy, Sept 9-12, 2007; University of Palermo: Palermo, Italy, 2007; CD-ROM. (28) Cardinaels, R.; Vananroye, A.; Verhulst, K.; Moldenaers, P. Polymer Processing 24th Annual Meeting, Salerno, Italy, June 15-19, 2008; Polymer Processing Society: Melville, NY, 2008; CD-ROM. (29) Astarita, G.; Marrucci, G. Principles of Non-Newtonian Fluid Mechanics; McGraw-Hill: New York, 1974.
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and IIs = (I2s - IS2)/2 is the second scalar invariant of S. The functions fi’s are determined so to recover all the available analytical limits; for non-Newtonian unconfined systems, the model is then forced to recover Greco’s second order theory5 thus obtaining the fiM’s functions reported in the Appendix, as slightly modified by Verhulst et al.8 to better predict some experimental data at high Ca values. The fi’s depend on the nondimensional parameters governing the system, which for systems made of second-order fluids of Coleman and Noll30 are the viscosity ratio λ; the capillary number Ca; the ratio between the elastic relaxation time of the matrix fluid and the interfacial relaxation time (the emulsion time), De = Ψ1mΓ/2Rμ2m (where Ψ1m is the first normal stress coefficient of the matrix fluid); the ratio between the first and the second normal stress differences of the matrix fluid, Ψ = -N2m/N1m; the ratio between the elastic relaxation time of the droplet fluid and the interfacial relaxation time, Ded = Ψ1dΓ/2Rμ2d (where Ψ1d is the first normal stress coefficient of the droplet fluid); the ratio between the first and the second normal stress differences of the droplet fluid,Ψd=-N2d/ N1d. In the limit of Newtonian fluids, for which De=Ded=0, the model degenerates to all the analytical limits for small deformation of the Newtonian systems. In agreement with Minale,18 to take into account the confinement and to fulfill hypothesis C, it is enough to modify the fi’s functions as follows f1 ¼ f1M =½1 þ Cs ðR=HÞ3 f1c f2 ¼ f2M ½1 þ Cs ðR=HÞ3 f2c f3 ¼ f3M ½1 þ Cs ðR=HÞ3 f3c
ð3Þ
44 þ 64λ - 13λ2 2ð1 þ λÞð12 þ λÞ 10 - 9λ ¼ 12 þ λ
f2c
ð4Þ
With this choice, whatever the choice of f3c could be, Shapira and Haber theory is recovered by the model; indeed, the terms that multiply f3c give their first contribution only at O(Ca2). The function Cs is numerically given by Shapira and Haber. Notice that in a confined problem a new length scale is introduced: The shear cell gap H, and consequently, a new dimensionless parameter, the degree of confinement R/H, appears in the problem. To completely specify the model the function f3c must be determined. In analogy with the other fi’s functions up to now determined, we assume f3c to be a nondiverging rational function, the simplest possible choice being (aþbλ)/(cþλ). It must be underlined, however, that the model is not very sensitive to f3c, at least up to moderate Ca values. Nonetheless, we have determined the three parameters, a, b, and c, from a best fit of the available steady state data on the deformation of a confined Newtonian drop immersed in a Newtonian matrix. The data have been taken from Sibillo et al.13 and from Vananroye et al.14 Only (30) Truesdell, C. A.; Noll, W. The Non-Linear Field Theories of Mechanics; Flugge, S., Ed.; Encyclopedia of Physics; Springer: Berlin, 1965; Vol. III/3.
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the deformation data have been used, because the orientation angle is independent of f3c; we propose the following f3c ¼ -
20 - 5λ 11 þ λ
ð5Þ
With eq 5, the model governed by eqs 1 and 2 with eqs 3 and 4 is fully specified and has no adjustable parameters to describe the dynamics of a confined non-Newtonian drop immersed in a nonNewtonian matrix.
Experimental Method
where Cs is a coefficient that depends on the distance between the drop center and the closest device wall (h) made dimensionless with H, and fic’s are functions of the viscosity ratio that must be determined. We want the model to recover Shapira and Haber analytical limit16 for small deformations of a confined Newtonian system and because at O(Ca) the model coincides with the Maffettone and Minale model modified to account for confinement,18 we keep f1c and f2c the same as in Minale18 f1c ¼
Figure 1. Rheological characterization of the elastic matrix, T=
23 °C.
The rheo-optical apparatus used in this work7 essentially consists of a shear cell coupled with an optical microscope equipped with a monochromatic CCD video camera and a motorized focus system. In all the experiments, observations were performed in bright field, using long working distance optics. The drops are observed along the vorticity direction of shear flow by using as parallel plates a couple of glass bars of square section (100 mm 50 mm 50 mm). Parallelism between the two plates is adjusted by a set of micrometric rotary and tilting stages (the residual error is around 10 μm over a length of 10 cm). Shear flow at a constant shear rate γ_ is obtained by translating one of the plates at a constant speed with respect to the other through a computer-controlled motorized translating stage with micrometric precision setting. The microscope itself is also mounted on a separate motorized translating stage, which is used to keep the deforming drop within the field of view during motion. All the experiments are performed at room temperature, 23 °C ( 0.5 °C, in a thermostated room. Isolated drops are injected in the continuous phase, preliminarily loaded between the parallel plates, by a tiny glass capillary, fixed to a micromanipulator for precise positioning. Buoyancy effects on the injected drop are found to be negligible on the time scale of the experiments because of the high viscosities and the small density difference between the two liquids. The continuous phase is a constant-viscosity, elastic polymer (Boger fluid) prepared7 by mixing a Newtonian polyisobutylene (PIB) sample (Indopol 100, Innovene) with a small amount (0.5%) of a high molecular weight grade (Mv = 470 000) of the same polymer (Aldrich), preliminarily dissolved in kerosene at the concentration of 4 wt %. After the preparation kerosene was eliminated by evaporation. Rheological data were obtained by using a constant-stress rheometer equipped with a normal stress transducer (Bohlin, CVO 120), in the cone-and-plate configuration. As shown in Figure 1, the Boger fluid viscosity is essentially constant, μm = 28.7 Pa s, in the range of shear rate of interest, presenting an initial thinning for γ_ > 0.1 s-1, and rather large values of the first normal stress difference are found, with a first normal stress coefficient Ψ1m=90 Pa s2. Langmuir 2010, 26(1), 126–132
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Figure 2. Sketch of a deformed drop in shear flow as observed along the vorticity direction z. The axis x is the flow direction, and y the velocity gradient direction. Observable quantities measured in this work, i.e., drop axes and orientation, the gap between the parallel plates H, and the distance between the drop center and the closest device wall h, are also defined here.
Figure 3. Comparison of drop shapes during steady state flow at Ca=0.33 and R/H=0.26 (a) and 0.37 (b). Both images are acquired along the vorticity direction of shear flow. An elliptical contour is superimposed to each drop image as done in data analysis. Standard image processing routines have been applied for background subtraction and contrast enhancement. The drop phase is prepared by mixing Newtonian silicone oils of different molecular weight in such proportions to match the matrix viscosity, obtaining a viscosity ratio λ= 1.0. The interfacial tension Γ of the pair of liquids is measured by a standard experimental procedure based on fitting the Shapira and Haber16 predictions accounting for wall confinement to the experimental values of the deformation parameter D (defined below) at small deformations, and is 1.7 mN/m. In fact, though Shapira and Haber16 theory is valid for Newtonian systems, it can also be applied to non-Newtonian systems as long as the capillary number is small enough (indeed, non-Newtonian effects show up at the second order in Ca). The drop shape parameters, as shown in Figure 2, are the semiaxes L0 and B0 in the shear plane, the deformation parameter D = (L0 - B0 )/(L0 þ B0 ), and the orientation angle θ between the major semiaxis L0 and the flow direction x. Such shape parameters are measured by image analysis techniques (fitting an ellipse to the drop contour). The experiments are started with an initial setting of the nondimensional gap R/H, between the confining plates, high enough that no significant wall effects are anticipated. Several shear flow runs are then carried out at increasing values of γ_ (from one run to the other, the drop is allowed to retract back to the spherical shape at rest). This shear rate sweep is iterated at progressively lower gap settings by reducing the distance between the parallel plates through a micrometric translating stage. The degree of confinement R/H investigated in this work ranges from 0.17 to 0.37. Drop shapes from two representative runs at Ca=0.33 and R/H=0.26 and 0.37 are shown in images a and b in Figure 3, respectively.
Results We start describing the steady-state results by comparing model predictions and experimental data. In all the model calculations, the parameter Ψ is set to 0.1 as already done in the literature.8 As a reference for the unconfined non-Newtonian Langmuir 2010, 26(1), 126–132
Figure 4. Comparison of experimental data with theoretical predictions: Model presented here (solid line), non-Newtonian unconfined6 (dashed), Newtonian confined18 (dotted), Shapira and Haber16 (dash-dotted). λ=1.0, De=1.1, R/H=0.26, h/H=0.49. (a) Deformation parameter plotted vs the capillary number; (b) three dimensionless semiaxes L, B, W. The experimental values of W are calculated from volume conservation.
and the confined Newtonian cases, only comparisons with models from the literature, which have already been validated,6,8,18 are given for the sake of clarity. Indeed, the unconfined non-Newtonian model showed to be in good agreement with the experimental data up to Ca ≈ 0.3 both in steady state and transient flow,6,8 whereas for the confined Newtonian model, the agreement extends up to Ca ≈ 0.4.18 Drop deformation was investigated at several degrees of confinement. In Figure 4a the deformation parameter D=(L B)/(L þ B), where L and B are the semiaxes L0 and B0 , respectively, made dimensionless with the undistorted drop radius R, is plotted vs the capillary number Ca. The experimental data are compared with the model presented here (solid line), the reference non-Newtonian unconfined case (dashed) according to the Minale’s model,6 the reference Newtonian confined case18 (dotted), and finally Shapira and Haber predictions16 (dashdotted). The calculations are based on the following experimental values: λ = 1.0, De = 1.1, R/H = 0.26, h/H = 0.49. The nonNewtonian confined model (solid line) well compares to the data in all the experimental range of Ca investigated. At this degree of confinement the deviations from the non-Newtonian unbounded predictions (dashed) are minor. However, a significant difference is found with respect to the Newtonian case at increasing capillary number, as calculated both from the Shapira and Haber16 (dashdotted) and the Minale18 (dotted) model. At vanishing values of Ca, the present model recovers the analytical Newtonian limit of Shapira and Haber, in agreement with the data. Each data point is the average of several values (between 10 and 100) at steady state within the same run, and the error bars, calculated as the data standard deviation, fall within the symbol size. In Figure 4b, the nondimensional semiaxes in the shear plane L and B DOI: 10.1021/la902187a
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Figure 5. Orientation angle vs capillary number. The predictions of the model presented here (solid line), the non-Newtonian unconfined one (dashed), and the Newtonian confined one (dotted) are compared with experimental measurements. λ=1.0, De=1.1, R/H=0.26, h/H=0.49.
(see Figure 1), and that along the vorticity axis W are shown for the same system as a function of the capillary number. The experimental values of W are calculated from volume conservation by assuming ellipsoidal shape.31 The comparison of predictions to these experimental quantities is a more severe test for the models since the semiaxes provide a full description of the drop shape as opposed to the lumped deformation parameter D, which is less sensitive to shape details. In fact, it can be noticed that the Shapira and Haber model, while providing a rather accurate prediction for the major nondimensional semiaxis L, fails in predicting quantitatively the minor semiaxis B and even qualitatively the third semiaxis W, which is indeed not constant with Ca. In Figure 5, the drop orientation angle θ (see Figure 2) is shown as a function of Ca. As before, the solid line corresponds to the present model, the dashed line is the unconfined non-Newtonian case, and the dotted line is the unconfined Newtonian case (Shapira and Haber model predicts a constant angle of 45°). In agreement with the measured angle, the predictions of the present model show that the drop is more oriented than in both the unconfined non-Newtonian and the confined Newtonian case. Indeed, a more significant deviation between the present model and the reference ones is found here as compared to the semiaxes shown in Figure 4. Data and predictions for a more confined situation are shown in Figure 6 for the deformation parameter (a) and the drop semiaxes (b). For the sake of clarity, Shapira and Haber predictions are omitted. The calculations are based on the following experimental values: λ=1.0, De=1.1, R/H=0.37, h/H=0.48. Also in this case, the data are well-described by the present model, and significant deviations from the reference models are now seen. At low values of Ca, the Newtonian behavior is again recovered, whereas the unconfined non-Newtonian model starts with a wrong, less-pronounced slope. With increasing Ca, data and predictions of the present model deviate from the reference confined Newtonian case, going toward the unconfined non-Newtonian reference model. Similar considerations apply to the nondimensional semiaxes shown as a function of Ca in Figure 6b. In Figure 7, the drop orientation angle is plotted as a function of Ca. The agreement between data and predictions of the present model is once again very good. It can be noticed that in this more confined situation the drop is more oriented and there is a stronger departure from the unconfined non-Newtonian case, (31) Guido, S.; Villone, M. J. Rheol. 1992, 42, 395–415.
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Figure 6. Model presented here (solid line), the non-Newtonian unconfined one (dashed), and the Newtonian confined one (dotted) are compared with experimental data for a higher confinement: λ=1.0, De=1.1, R/H=0.37, h/H=0.48. (a) Deformation parameter plotted vs the capillary number; (b) three dimensionless semiaxes L, B, and W. The experimental values of W are calculated from volume conservation.
Figure 7. Orientation angle vs capillary number. The predictions of the model presented here (solid line), the non-Newtonian unconfined one (dashed), and the Newtonian confined one (dotted) are compared with experimental measurements. λ=1.0, De=1.1, R/H=0.37, h/H=0.48.
whereas the discrepancy with the confined Newtonian case remains more or less the same as in Figure 5. In this work, experiments at two other degrees of confinement were performed, and are not shown in detail for the sake of brevity. The overall results are summarized in Figures 8 and 9 by plotting the deformation parameter and the orientation angle as a function of the degree of confinement R/H, respectively. Because in the experiments, the drop center was not always located in the middle of the gap (h/H=0.5), we have rescaled the ratio R/H such that the parameter Cs(R/H)3, as calculated for the drop in the middle of the gap, corresponds to the actual measured value. The plots refer to four values of the capillary number as a parameter in Langmuir 2010, 26(1), 126–132
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Figure 8. Deformation parameter vs degree of confinement: comparison of models predictions (model presented here, solid line; Newtonian confined model, dotted line) with data. Data series are parametric in Ca: 0.08 (open circles), 0.16 (filled triangles), 0.23 (open squares), 0.4 (filled diamonds).
Figure 10. Drop relaxation upon cessation of flow. (a) Deformation parameter D is compared with predictions of model presented here (solid line), the non-Newtonian unconfined model (dashed line), and the Newtonian confined one (dotted line). (b) Three dimensionless semiaxes are plotted vs the dimensionless time, t and only the model presented here is shown. The experimental values of W are calculated from volume preservation. R/H = 0.37, h/H = 0.48, De=1.1, λ=1.0.
Figure 9. Orientation angle vs degree of confinement: comparison of models predictions (model presented here, solid line; Newtonian confined model, dotted line) with data. Data series are parametric in Ca: 0.08 (open circles), 0.16 (filled triangles), 0.23 (open squares), 0.4 (filled diamonds).
the range 0.08-0.4 (when data were not available at those exact values, they were calculated by linear interpolation). In Figure 8, the deformation parameter is well-described by the present model and shows a much slighter increase with R/H as compared to the confined Newtonian case (in fact, D is almost constant at any Ca). Even at the highest Ca value of 0.4, where a significant difference with the Newtonian confined prediction is appreciated, the agreement with data is still satisfactory. In Figure 9, the orientation angle, as calculated according to the present model, starts from a plateau at vanishing degree of confinement, and decreases with R/H. Model predictions are in good agreement with the experiments (more data scatter is due to higher experimental error in measuring the angle), and here deviations from the Newtonian case can be already appreciated at the smallest Ca values, where the predictions of the two models for the deformation parameter are indistinguishable. A crude estimate of the experimental error can be inferred from the distance between data points at almost the same degree of confinement (e.g., (0.03 for D at Ca=0.16 in Figure 8). Such experimental error does not affect our results. A further test of the model predictions was made by comparison with data taken during drop retraction upon cessation of flow at Ca=0.45, R/H=0.37, h/H=0.48, De=1.1, λ=1.0. In Figure 10, the three drop semiaxes, L, B, and W (the latter calculated from volume preservation as before), are shown vs time Langmuir 2010, 26(1), 126–132
made nondimensional with respect to the emulsion time τ=μmR/ Γ. Because there is a slight difference between steady-state model predictions and data at such a high value of Ca, to start from the same initial condition the model has been fitted to the initial steady state experimental drop semiaxes (this fit was found for Ca= 0.41). In this way, the relaxation process is investigated without the possible error coming from the steady state. In Figure 10a, data compare very well with model predictions, with a significant difference with respect to both reference cases, which show a faster relaxation. It should be noticed that experimental data of the unconfined non-Newtonian case from the literature,8,32,33 show two characteristic relaxation times that are not accounted for by the models considered here. The longer relaxation tail found in the experiments is already apparent at nondimensional time around 9, as shown by several authors.8,32,33 In our confined situation, on the contrary, a single relaxation time is found in the investigated experimental range (up to t ≈ 20) in agreement with the present model. The results of the three nondimensional semiaxes presented in Figure 10b confirm the good model performance in describing transient drop shape.
Conclusions In this work, the shape of a single Newtonian droplet immersed in an equiviscous non-Newtonian Boger fluid subjected to simple shear flow is investigated by modeling and experiments. A novel phenomenological model is proposed on the basis of the assumption of drop ellipsoidal shape. The model accounts for wall effects (32) Sibillo, V.; Simeone, M.; Guido, S.; Greco, F.; Maffettone, P. L. J. NonNewtonian Fluid Mech. 2006, 134, 27–32. (33) Verhulst, K.; Cardinaels, R.; Moldenaers, P.; Afkhamib, S.; Renardy, Y. J. Non-Newtonian Fluid Mech. 2009, 156, 44–57.
DOI: 10.1021/la902187a
131
Article
and non-Newtonian fluid components. Model parameters are chosen to ensure recovery of all the analytical small deformation limits and best fit with the Newtonian confined case. Therefore, no adjustable parameter is needed for the non-Newtonian system investigated here. We have focussed on a fixed viscosity ratio of one and have studied the effects of the capillary number and the degree of confinement. Experiments are carried out in a parallel plate apparatus, allowing us to generate simple shear flow and to adjust the distance between the plates (thus varying the degree of confinement). Good agreement between model predictions and experimental data is found both in steady state and transient drop retraction upon cessation of flow. The effect of confinement on drop deformation is less pronounced with respect to the Newtonian case. On the contrary, drop orientation along the flow direction shows a significant increase as a function of both the non-Newtonian character of the matrix fluid and the degree of confinement. Concerning the drop retraction upon cessation of flow, the data show, in agreement with model predictions, essentially a single relaxation time in contrast to the unbounded case, where a slower second relaxation mechanism appeared. Potential applications of this work are in the field of microfluidics, where droplets are subjected to significant wall effects. Acknowledgment. The authors thank the Italian Ministry of Research for partial support to this work through the PRIN 2006 program. The authors thank Angelo Pommella for his valuable work on the experimental side.
132 DOI: 10.1021/la902187a
Minale et al.
Appendix 120ð1 þ λÞð16 þ 19λÞ ; q 2 15ð16 þ 19λÞ 3Ca2 þ 0:75 f2M ¼ ; q 2 þ 6Ca2 q þ 4Deq 16De q 1 2 d 3 f3M ¼ 84ð1 þ λÞq f1M ¼
20335 þ 6Deð5613 - 9106ΨÞ - 8Ded ð1359 - 848Ψd Þ 3675 þ 1496De þ 360Ded 325Ca ð1 þ 26CaÞð199 þ 51λÞ
ðA1Þ
where q ¼ 3ð3 þ 2λÞð16 þ 19λÞ2 þ 8Deð176 þ 436λ þ 323λ2 Þ þ 360Ded λ2 ð3 þ 2λÞ q1 ¼ ð16 þ 19λÞð880 þ 12849λ þ 10673λ2 Þ q2 ¼ ð67040 þ 141668λ þ 114796λ2 þ 30115λ3 Þ 2Ψð41840 þ 107508λ þ 105276λ2 þ 32215λ3 Þ q3 ¼ λ2 ½9ð1445 þ 1726λÞ - 16Ψd ð440 þ 673λÞ
ðA2Þ
Langmuir 2010, 26(1), 126–132
