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Shear Rupturing of Droplets in Complex Fluids T. G. Mason*,† and J. Bibette‡ Johns Hopkins University, Department of Chemical Engineering, 221 Maryland Hall, 3400 N. Charles Saint, Baltimore, Maryland 21218, and Centre de Recherche Paul Pascal, Ave. A. Schweitzer, F-33600 Pessac, France Received January 21, 1997. In Final Form: May 22, 1997X We have experimentally studied the shear-induced rupturing of viscous droplets in viscoelastic complex fluids. Remarkably, a premixed emulsion of large, polydisperse droplets can be ruptured into monodisperse emulsions of uniform colloidal droplets. The monodispersity becomes most pronounced when the premixed emulsion is viscoelastic and has a shear-thinning viscosity. Since viscoelastic materials may fracture, we reduce the gap of our shear cell to ensure a spatially uniform strain rate for rupturing. We observe monodispersity whether the viscoelasticity arises from the suspending fluid (e.g., concentrated surfactant solution) or droplet deformation as in compressed emulsions. Our observations suggest that the monodispersity results from droplet rupturing alone and that the capillary instability is inhibited by the partial elasticity of the complex fluid. We use the monodispersity to study how the droplet size depends upon the shear rate and composition.
I. Introduction The rupturing of an isolated droplet of one viscous fluid in another immiscible viscous fluid by a shear flow is a classic problem in fluid mechanics that lies at the foundation of dispersion science and multiphase flow. The way in which the shear stresses induced by the flow overcome the interfacial tension between the two fluids to rupture the droplet has been addressed in increasing levels of detail1-8 since the time of Taylor,9 who recognized the importance of the problem in emulsification. The detailed understanding of how the droplet’s free boundary can deform under shear has yielded a rich phenomenology of rupturing scenarios.8 However, this apparent richness explains only a limiting case of the much more general problem of droplet rupturing in a complex fluid,10 which, due to its supramolecular structures, may exhibit viscoelasticity and plastic flow, both of which indicate partial energy storage rather than a simple viscous dissipation.11,12 In this paper, we present the first evidence that the viscoelasticity of the complex fluid used in emulsification can lead to a dramatic alteration of the rupturing phenomena, including uniform rupturing to monodisperse droplets. Moreover, since a concentrated emulsion of many interacting droplets is a complex fluid itself, we are able to go beyond the description for isolated droplets to study the role of the droplet volume fraction, φ, in the rupturing process. In emulsification, work is done by the applied shear stresses against the interfacial tension, σ, to elongate and * E-mail address:
[email protected]. † Johns Hopkins University. ‡ Centre de Recherche Paul Pascal. X Abstract published in Advance ACS Abstracts, August 1, 1997. (1) Bartok, W.; Mason, S. G. J. Colloid Sci. 1959, 14, 13. (2) Chandrasekhar, S. Hydrodynamic and Hydromagnetic Stability; Oxford: London, 1961. (3) Grace, H. P. Chem. Eng. Commun. 1982, 14, 225. (4) Hinch, E. J.; Acrivos, A. J. Fluid Mech. 1980, 98, 305. (5) Khakhar, D. V.; Ottino, J. M. Int. J. Multiphase Flow 1987, 13, 71. (6) Mikami, T.; Cox, R. G.; Mason, S. G. Int. J. Multiphase Flow 1975, 2, 113. (7) Rumscheidt, F. D.; Mason, S. G. J. Colloid Sci. 1961, 16, 238. (8) Rallison, J. M. Annu. Rev. Fluid Mech. 1984, 16, 45. (9) Taylor, G. I. Proc. R. Soc. A 1934, 146, 501. (10) Mason, T. G.; Bibette, J. Phys. Rev. Lett. 1996, 77, 3481. (11) Bird, R. B.; Armstrong, R. C.; Hassager, O. Dynamics of Polymeric Liquids; John Wiley and Sons: New York, 1977. (12) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; John Wiley & Sons: New York, 1980.
S0743-7463(97)00058-9 CCC: $14.00
rupture larger droplets into smaller ones, thereby creating additional surface area between the two fluids while conserving the total droplet volume. This provides a mechanism for increasing the energy stored in the droplet interfaces, since the energy is proportional to the surface area. After mixing, the droplets would prefer to recombine, or coalesce, when they encounter each other to lower this interfacial energy. However, a surfactant, which coats the surfaces of the droplets and provides a short-ranged stabilizing repulsion between the droplet interfaces, can be added to the emulsion to inhibit this coalescence. By choosing a surfactant that provides a strong inhibition of coalescence, it is possible to make emulsions kinetically stable for many years, even when the droplets are densely packed together and deformed as in a compressed emulsion. This strong kinetic interfacial stability is the key to storage life of many products such as mayonnaise and skin lotion; it also differentiates emulsions from thermodynamically stable microemulsion phases in which droplets are formed spontaneously when the fluids and surfactants are placed together in contact.13 Emulsions can be made in many ways, ranging from the biological process of lactation of fat globules for milk production to the enormous industrial mixing processes that produce asphalt emulsions for building roads. We can classify many of the industrial emulsification processes into several broad categories. Shear mixing while adding one fluid to another, as found in many common kitchen recipes, is one broad class. A second is the injection of one fluid into the other through a solid tube or orifice so that droplets are formed; these include membrane emulsification methods. Impinging two high-velocity jets of the fluids can be used to create cavitation that ruptures the droplets, and ultrasonic vibration can likewise serve this purpose. Due to the variety of methods and a fundamental lack of scientific understanding of many of them, the vast majority of industrial emulsion products are made using recipes tailored by trial-and-error to available production devices. The goal of this paper is to provide insight into the process of emulsification through the first broad class: shear mixing. Rather than restrict our attention to the idealized problem of isolated droplet rupturing in viscous fluid subjected to a shear flow, we address how the (13) Kunieda, H.; Shinoda, K. J. Dispersion Sci. Technol. 1982, 3, 233.
© 1997 American Chemical Society
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composition of a real emulsion can affect the rupturing of droplets during emulsification. Emulsions are complex fluids that can have rheological properties far from those of a simple molecular fluid having a Newtonian viscosity. When the droplets are sufficiently concentrated and packed together at high volume fractions, φ, emulsions are known to exhibit plastic behavior; they are elastic for small shear deformations, and they yield and flow for larger deformations. Large surfactant concentrations, C, in the continuous fluid phase can also give the emulsion non-Newtonian rheological properties that may also affect the rupturing, even at low φ. By varying the emulsion’s composition, controlling the shearing conditions, and probing the resulting structure using optical microscopy and light scattering, we demonstrate that shear rupturing alone is sufficient to produce monodisperse emulsions of a highly controlled uniform size. The remainder of this paper is organized as follows. A theory section reviews the shear rupturing of isolated droplets of one viscous fluid in another and also the nonNewtonian rheological properties of emulsions. We then propose two new scaling laws to explain fracture flow in emulsions and how the elasticity of the continuous phase may modify the ordinary capillary instability for viscous fluids. An experimental section describes our sample preparation, characterization, shearing methods, and observation techniques. Our experimental results of the droplet size and monodispersity as a function of the emulsion composition and shearing conditions follow. We discuss these results, which provide clues as to the origin of the monodispersity, and we conclude by illustrating the adaptation of these results for practical, large scale production of monodisperse emulsions. II. Theory II.A. Rupturing of Isolated Droplets. Despite the widespread use of emulsification, the fundamental understanding is essentially limited to the rupturing of a single droplet of viscosity, ηi, suspended in a fluid having viscosity, ηe, at low Reynold’s numbers.8 For droplet deformation to occur, the viscous shear stress of the external phase, ηeγ˘ , must overcome the characteristic Laplace pressure, σ/a, where a is the droplet’s radius, provided ηi can be ignored. For rupturing to occur, the capillary number, Ca ) ηeγ˘ a/σ, defined as the ratio of the shear stress to the Laplace pressure,2 must exceed a critical value of order unity; this implies the droplet has been elongated by the viscous shear before rupturing. If it is assumed that the critical capillary number, Cac, is unity and ηi can be neglected, the average radius of the ruptured droplets is
a≈
σ ηeγ˘
(1)
where geometrical factors (of order unity) accounting for the details of the rupturing have been left out. This important scaling form has been originally derived by Taylor.9 While the Taylor form captures much of the essential physics underlying droplet rupturing, a complete description of the deformation and bursting of isolated droplets under simple shear is complicated and depends on the capillary number, the viscosity ratio, ηi/ηe, and the type and history of the shear flow.8 Experiments have roughly identified three broad classes of rupturing scenarios that may occur when the shear is initiated very gradually. The first is called “tip streaming” because the droplet elongates and rotates under the shear, developing pointed ends that
eject tiny droplets of the dispersed phase.14 In the second scenario, the droplet is ruptured into two droplets of almost equal volume (sometimes accompanied by much smaller “satellite” droplets created during the neckdown).1 In the third scenario, the droplet is stretched into an extremely elongated “liquid thread” that undergoes a capillary (Rayleigh) instability and breaks into a chain of many droplets.5,6 In this case, the critical capillary number can become much greater than unity. Such elongated droplets resemble liquid cylinders that are susceptible to a capillary instability in which the surface tension drives the rupturing of the cylinder into many droplets having less total surface area.2 The fastest-growing wavelength of the capillary instability, determined by the mode of maximum instability, is typically λ ∼ r, where r is the radius of the cylinder, so the resulting droplet radius is also a ≈ r. These three qualitatively different regimes have been identified for shears applied quasi-statically; only qualitative information exists for how rapid transients in the shear rate modify these boundaries.14 II.B. Rupturing of Many Interacting Droplets. Given the complexity of isolated droplet rupturing, theoretical treatments of many-droplet rupturing scenarios in real emulsions have been largely avoided. Any proposed theory must account for the deformations of the free boundaries of the droplets self-consistently with the modification of the flow fields of the continuous and dispersed phase that these boundary changes must introduce. Moreover, surfactant behavior at the interface between the two immiscible phases may alter the flow boundary conditions between the two fluids during the rupturing; this introduces yet further complexity. Given these difficulties, attempts to understand the influence of the emulsion’s composition on droplet rupturing have treated the emulsion as an effective medium, thereby ignoring the precise microscopic dynamics accompanying the rupturing. In particular, the ruptured droplet size in emulsions has been proposed15 to follow eq 1, albeit using an elevated effective viscosity, ηeff, which reflects the role of φ:
a≈
σ ηeffγ˘
(2)
According to this principle, the overall effective viscosity of the emulsion, and not that of the continuous phase alone, determines the shear stress governing the rupturing and, therefore, a. For a compressed emulsion at high φ, ηeff can be many times ηe, so the ruptured droplet sizes can be much smaller than eq 1 would predict. Equation 2 is sufficient to explain some measurements of the ruptured droplet sizes for emulsions where φ plays an important role,15 but it is at odds with the absence of rupturing observed for compressed emulsions having small continuous phase viscosities, yet large effective viscosities.16 II.C. Fracturing in Concentrated Emulsions. To understand how these two different observations can be reconciled, we review the non-Newtonian rheological properties of concentrated emulsions,16,17 including fracturing. As droplets are packed together and deform under an applied osmotic pressure, the emulsion’s flow properties increasingly reflect the role of elastic energy storage through droplet interfacial deformation. Highly concen(14) Torza, S.; Cox, R. G.; Mason, S. G. J. Colloid Interface Sci. 1972, 38, 395. (15) Aronson, M. P. Langmuir 1989, 5, 494. (16) Mason, T. G.; Bibette, J.; Weitz, D. A. J. Colloid Interface Sci. 1996, 179, 439. (17) Princen, H. M. J. Colloid Interface Sci. 1989, 128, 176.
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trated emulsions exhibit a low-frequency elastic shear modulus, G′p, when perturbative deformations are applied, yet they flow like plastic materials at larger strains beyond the linear regime. The transition from linear to nonlinear rheology can be characterized by a yield strain, γy, and a yield stress, τy. Previous measurements on fractionated monodisperse emulsions stabilized by 10 mM sodium dodecyl sulfate (SDS)16 have shown that both G′p and τy increase dramatically above a critical volume fraction, φc, at which the droplets pack, and both are proportional to the characteristic elastic Laplace pressure scale, σ/a, reflecting droplet deformation. The measured φc corresponds to disordered random close packing (RCP) of monodisperse spheres, φc ≈ 0.64, after accounting for the screened electrostatic repulsion that preserves the stability of the thin films of water between the droplet interfaces. The measured increases in G′p(φ) are consistent with a recent simulation that accounts for the disorder.18 By contrast, for polydisperse emulsions, not all droplets have the same Laplace pressure; moreover, smaller droplets can fit in the spaces between larger droplets without deforming, so φc cannot be so easily identified due to its sensitivity to the complete droplet size distribution.17 Despite the difference in φc and average Laplace pressure, polydisperse emulsions also exhibit plateau moduli and yield stresses. Beyond the linear and yield regimes, the corresponding φ dependence of the steady shear rheology of SDSstabilized monodisperse emulsions has also been investigated. At high φ f 1, the shear stress is nearly independent of γ˘ , so the emulsion’s effective viscosity is shear-thinning, i.e., decreasing as γ˘ increases, indicative of plastic flow. At low φ near φc, the shear-thinning behavior is not as pronounced, ηeff(γ˘ ) decreases less rapidly, as at high φ near unity where the apparent shear-thinning behavior resembles that of an ideal plastic material, ηeff ) τy/γ˘ . However, at high φ, spatially inhomogeneous flow (fracturing) created by the shear has been observed by watching the deformation of a painted stripe on the emulsion’s surface, making the apparent strain rate much smaller than the actual strain rate within the fracture surface. Microscopic observations of the emulsion after shearing have not revealed any noticeable population of smaller droplets, so planes of droplets flow past each other by lubricated slipping that does not induce droplet rupturing. This presents a puzzle, since the effective viscosity of a disordered concentrated emulsion near φ ≈ 1 is quite large and should have produced droplet rupturing at the strain rates used, according to eq 2, yet the droplets are not ruptured. Evidently, the effective medium approach implied by eq 2 can fail. To understand how fracturing may prevent efficient and uniform rupturing in emulsification, we consider shearing a compressed emulsion between two solid plates having a uniform gap spacing, d; with no-slip boundary conditions at the plate walls. The upper plate is moved at a velocity, v, relative to the lower plate. Assuming the emulsion behaves as an ideal plastic, it resists flow elastically until the applied stress exceeds τy and flows with a simple viscosity, ηp, within a fracture plane of thickness, l, which develops between unyielded sections of the material in contact with the plates, as shown in Figure 1. Outside the fracture plane, the strain rate is zero, while inside the fracture plane, it is really γ˘ ) v/l, much larger than the apparent applied strain rate. At either edge of the fracture plane, the viscous stress, ηpγ˘ , of the flowing material must balance the limiting elastic (18) Lacasse, M.-D.; Grest, G. S.; Levine, D.; Mason, T. G.; Weitz, D. A. Phys. Rev. Lett. 1996, 76, 3448.
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Figure 1. Schematic illustration of fracturing in a yield stress material. The spatial gradient in velocity is concentrated within the fracture plane. The viscous stress within the fracture plane balances the yield stress of the static material outside it.
stress of the unyielded material, τy. Solving for the fracture plane thickness, we find
l ) ηpv/τy
(3)
This length scale, while derived in the context of fracturing of emulsions, may be relevant for other dispersions that exhibit a yield stress, yet have a weak ηp reflecting the solvent. To estimate the yield plane thickness for concentrated emulsions, we assume τy ≈ σ/a, and we take ηp to be the external continuous phase viscosity, ηp ≈ ηe. This last approximation ignores the φ dependence of ηp for l . a and underestimates ηp for a dense emulsion. Substituting, we find the fracture length scales as
l ∼ ηev/(σ/a)
(4)
It is proportional to viscosity and inversely proportional to the Laplace pressure. For a concentrated emulsion with little surfactant, a ) 1 µm, σ ) 10 dynes/cm, v ) 10 cm/s, and ηe ≈ 10-2 P, we estimate l ≈ 10-6 cm, much smaller than the droplets themselves. This implies that a thin lubricating film of water can separate one plane of packed droplets as it slips by another plane. This prediction is in accord with our observations of lubricated droplet slip even when the emulsion is highly concentrated and has a yield stress. However, if the viscosity of the continuous phase is elevated to ηe ≈ 102 P, for instance by adding extra surfactant, the thickness increases to l ≈ 10-2 cm, much larger than the droplet size. This distance is sufficiently large that the gap spacing can be made less than l, thereby forcing a homogeneous flow so that all droplets can be sheared and ruptured at the same γ˘ . II.D. Capillary Instability of a Viscous Cylinder in a Yield Stress Material. Since one way of increasing the continuous phase viscosity is to use concentrated surfactant solutions, which themselves can be viscoelastic and shear-thinning, we also consider the rupturing of an isolated droplet in a complex fluid possessing a yield stress. We imagine that, when suspended in a yield stress material and sheared, oil droplets can form highly elongated tubelike liquid threads. We hypothesize that this thread can neckdown and break into many tiny droplets when it is sufficiently stretched so that its radius becomes small enough that the Laplace pressure driving the capillary instability, σ/r, exceeds the yield stress, so the characteristic wavelength of the instability is λ ≈ σ/τy. Thus, the ordinary capillary instability is inhibited until the Laplace pressure of the stretched droplet causes local yielding and flow of the bulk continuous phase surrounding the droplet. The droplets are then ruptured to a radius reflecting this selected wavelength:
a ≈ σ/τy
(5)
We obtain the same result by making the assumption
Shear Rupturing of Droplets
that the emulsion’s effective viscosity can be perfectly shear-thinning, ηeff ) τy/γ˘ , and substituting into the scaling form in eq 2. We find a unique droplet radius that depends only on the surface tension and the yield stress, a ∼ σ/τy; moreover, this radius is independent of the shear rate and the initial droplet volume. If the material is partially shear-thinning with ηeff ∼ γ˘ -R, where R is the shearthinning exponent, the dependence of the droplet radius on the shear rate vanishes when R approaches unity for a perfect yield stress material, while it depends inversely on γ˘ as in eq 2 when R approaches zero for a Newtonian liquid. This weaker size selectivity implies that inhomogeneities in γ˘ can lead to a broader final size distribution for Newtonian fluids than for plastic yield stress fluids. II.E. Origins of Droplet Size Polydispersity. The sharpness of the droplet size distribution after rupturing can be quantified by the polydispersity. Assuming the distribution is monomodal, its full breadth can be characterized using either the polydispersity by droplet
x1 - 〈V2〉/〈V〉2, or the polydispersity by radius, Pa ≡ x1 - 〈a2〉/〈a〉2, where the angled
volume, PV ≡
droplet brackets denote averages from zero to infinity over the volumetric probability density, pV(V), and the radial probability density, pa(a), respectively.19 Normalization requires ∫∞0 pV(V) dV ) 1 and ∫∞0 pa(a) da ) 1, so that pV has units of inverse volume and pa has units of inverse length. Physically, the probability of finding a droplet in the emulsion having a volume between V and V + dV is given by pV(V) dV, whereas the probability of finding a droplet having a radius between a and a + da is given by pa(a) da. For the same emulsion, these two probabilities must be equal, so the radial probability density can be calculated from the volumetric one (measured by light scattering): pa(a) ) (dV/da)pV(V). Since the droplet volume is V ) 4πa3/3, the radial probablity density is pa(a) ) 4πa2pV(4πa3/3). The radial probability density appears more narrow than the volumetric probability density because the tail of pV(V) toward high volumes is effectively compressed into a smaller range of a. Lacking theoretical predictions of the complete droplet size distribution for different rupturing conditions, we estimate an upper limit of the polydispersity one may expect by successively splitting-in-half larger droplets from an initially polydisperse emulsion, for instance, by applying incrementally higher shear rates. Here, we consider that the initial droplet sizes may be nonintegral multiples of the final droplet size; this can be a source of residual polydispersity in the final emulsion. When a droplet is ruptured into two equal volumes, the difference in radius between the two daughter droplets and the parent droplet is δamax ≈ (21/3 - 1)a ≈ 0.26a. This difference is already small, indicating that any desired droplet size much smaller than the initial average size can be obtained in a nearly continuous way from successive rupturing events, regardless of the polydispersity of the initial droplet size distribution, provided no droplets smaller than the final desired size were present in the initial distribution. From this difference, we can estimate the maximum polydispersity in radius, Pa e 0.26; since the width of the distribution must be smaller than this limit, the real polydispersity, where δa represents the true standard deviation, must also be smaller. Polydispersity arising from successive rupturing may be observed when the strain rate is gradually raised from zero until it reaches a fixed γ˘ . The droplets will be successively broken in half, provided Cac ≈ 1 at every (19) Helstrom, C. W. Probability and Stochastic Processes for Engineers; Macmillan: New York, 1984.
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stage in the rupturing. After all rupturing has occurred but the shear is still being applied, the emulsion will contain larger droplets that are elongated but cannot be ruptured and less elongated smaller droplets from past rupturing events. For Cac ≈ 1, the larger droplets can be at most twice as large (by volume) as the smallest dropets, making Pa e 0.26. Assuming that the real polydispersity is half of this upper limit, we find that the ruptured droplets can have a highly uniform size. This prediction may likewise apply to emulsions in which continuous rupturing at a fixed γ˘ limits the size of droplets that grow by either discretely coalescencing20 or continuously Ostwald ripening.21 Although this prediction is not based on any assumption of emulsion viscoelasticity, it does assume that extremely small satellite droplets are not created during the neckdown phase of the rupturing. This condition is most likely to be met for ηe . ηi, so that flows inside the droplet can occur quickly enough to prevent stagnation points, and therefore satellite droplets, from appearing. If inhibition of the capillary instability does not occur, we expect the polydispersity to rise when Cac . 1, since the largest elongated droplets that do not rupture in the steady state can be many times bigger than the smallest droplets, or if the strain rate is increased too rapidly, because transients in γ˘ may give rise to one-tomany droplet-rupturing events. To obtain emulsions with polydispersities smaller than the successive rupturing limit, a mechanism of selective rupturing must occur. We propose that a capillary instability of a long cylindrical thread of viscous fluid in viscoelastic complex fluid provides enhanced wavelength selection over the ordinary capillary instability for purely viscous fluids. The mode of maximum instability (and therefore a) is extremely well-defined because it results from the balance of two elastic stresses: the bulk yield stress and the interfacial Laplace pressure; the dispersion relation for the growth modes of the interfacial undulations will be highly peaked around λ -1≈ τy/σ, leading to a sharp peak in pa(a) for the ruptured droplets. A similar wavelength selection will occur for droplets of a viscoelastic complex fluid suspended in a purely viscous continuous phase. If the viscous shear stress is large enough to overcome both the interfacial Laplace pressure and the bulk yield stress within the complex fluid droplet, it will elongate, develop uniform interfacial undulations with λ ≈ σ/τy, and rupture into monodisperse droplets. For the general case of one viscoelastic complex fluid in another immiscible viscoelastic complex fluid, we expect that the larger of the bulk yield stresses will ultimately determine the tube radius at which the interfacial instability can occur and the elongated droplet can rupture. In real emulsification, we expect that the yield stress rather than the linear elastic modulus should be compared with the Laplace pressure since the rupturing of the cylinder requires local strains of order unity that are beyond the linear regime for many complex fluids at the frequencies of interest. Moreover, τy is typically smaller than G′p, allowing rupturing to occur for smaller elongations and larger tube radii. However, in the more idealized Rayleigh problem of the instability of an infinite cylinder of a linear viscoelastic complex fluid having a high-frequency modulus and a low-frequency viscous relaxation, G′p may take the place of τy in setting λ and a. III. Experimental Methods By contrast to standard methods used for rupturing isolated droplets,3,9,14 we have developed a novel approach to emulsifica(20) Bibette, J. Phys. Rev. Lett. 1992, 69, 2439. (21) Sebba, F. Foams and Biliquid Foams-Aphrons; John Wiley and Sons: Chichester, U.K., 1987.
4604 Langmuir, Vol. 13, No. 17, 1997 tion that allows us to probe how the droplet volume fraction affects the rupturing. The essence of our method consists of making a crude premixed emulsion of large droplets having a fixed composition and topology and then applying a spatially homogeneous controlled shear to rupture the droplets. By making a premixed emulsion, the macroscopic φ remains fixed throughout the rupturing; this is precluded for emulsification methods that continuously inject the dispersed phase while shearing. By applying a spatially homogeneous shear, we are also able to uniquely specify one strain rate at which the rupturing occurs. Using optical microscopy and light scattering, we observe the emulsion after all rupturing has occurred. This section is organized as follows: we characterize the constituents of our model oil-in-water emulsion. We then present several different shear cells that are used to rupture the droplets through both oscillatory and steady shears. We conclude by describing our methods for observing the droplet size and polydispersity. III.A. Composition and Characterization of a Model Emulsion System. Our model emulsion composition consists of a dispersed phase of silicone oil (poly(dimethylsiloxane) or PDMS) droplets in an aqueous continuous phase having a mass fraction, C, of the nonionic surfactant Tergitol NP7 (nonyl phenol ethoxy 7). The principal function of the NP7 is to stabilize the droplet interfaces, and its secondary function is to alter the viscosity of the continuous phase. Water-surfactant mixtures can form a variety of phases, including micellar (L1), hexagonal (H), cubic (V), lamellar (LR), and inverse micellar (L2).22,23 We have crudely characterized the phase behavior of the binary NP7water system at ambient temperature, T ) 23 °C, by comparing visual, crossed polarizer, and microscopic observations with those for a similar nonionic surfactant, C12E5, which has a wellcharacterized phase diagram.22 For C < 0.15 but above the critical micelle concentration, aqueous (W) and L1 phases coexist, but for 0.15 < C < 0.45 there is only an L1 phase. At C ≈ 0.45, a hexagonal phase may be present over a very narrow range of concentrations, but we could not precisely determine the boundary. For 0.45 < C < 0.85, the LR phase appears. Above C ≈ 0.85, we find the L2 phase, which cannot stabilize the premixed emulsion against coalescence. Due to our limited methods of observation, these phase boundaries are approximate; a direct measurement of the phase diagram using X-ray or neutron scattering to probe the NP7 surfactant structures remains to be done. Due to its inorganic backbone, silicone oil is a Newtonian fluid that is strictly immiscible in NP7 solutions, eliminating the possibility of spontaneous microemulsion formation. This is important, since light hydrocarbon oils may be partially or completely solubilized in concentrated surfactant solutions, forming lyotropic microemulsion phases.13 For most experiments, we fix ηi ) 350 cP. However, by using silicone oils of different molecular weights, we can vary ηi over a wide range without changing the emulsion’s essential chemical formulation and surface tension, which we have measured to be σ ) 2 dynes/cm at C ) 0.4 using a pendant drop technique. We have used a controlled-strain rheometer equipped with a cone-and-plate geometry to measure ηe of the NP7-water mixtures as a function of strain rate over a wide range of C, as shown in Figure 2. We find that the micellar phase L1 at C ) 0.15 exhibits a Newtonian viscosity more than 2 orders of magnitude larger than that of water (the W phase viscosity is near 1 cP for C ) 0.01). At C ) 0.4, the viscosity of the L1 phase has increased roughly another order of magnitude and remains essentially Newtonian. However, the viscosity becomes strikingly non-Newtonian and shear-thinning above the L1-LR phase boundary, decreasing as a function of γ˘ , for C ) 0.54, and at low strain rates, the viscosity is larger by more than 1 order of magnitude. In the LR phase at C ) 0.73, the shear-thinning behavior persists, and the overall magnitude of the viscosity has increased again by nearly 1 order of magnitude. This shearthinning behavior is qualitatively consistent with that previously observed for lamellar phases.24 Finally, the viscosity of the pure (22) Strey, R.; Schoma¨cher, R.; Roux, D.; Nallet, F.; Olsson, U. J. Chem. Soc., Faraday Trans. 1990, 86, 2253. (23) Ke´kicheff, P.; Grabielle-Madelmont, C.; Ollivon, M. J. Colloid Interface Sci. 1989, 131, 112. (24) Roux, D.; Nallet, F.; Diat, O. Europhys. Lett. 1993, 24, 53.
Mason and Bibette
Figure 2. Viscosity, ηe, of NP7-water mixtures as a function of the shear rate, γ˘ , for a series of NP7 concentrations, C. Solid circles are for C below the L2 phase, while open circles show the pure surfactant viscosity at C ) 1.
Figure 3. Microscope image of a typical premixed emulsion at C ) 0.4 and φ ) 0.7. The premixed emulsion is polydisperse. surfactant in the L2 phase, given by the open symbols, is Newtonian once again, with an overall magnitude much less than that of the lamellar phase. III.B. Preparation of the Premixed Emulsion. Premixing is a necessary part of our method because it renders the emulsion’s composition roughly homogeneous compared to the thickness of the shearing cell. Without premixing, one may imagine that pure oil, water, and surfactant loaded in separate layers between parallel plates may not form droplets at all when sheared. To make our premixed oil-in-water emulsion, we drip oil at a volume rate of V˙ d into a chosen NP7-water mixture of initial volume, Vc, that is gently sheared in a crude mechanical mixer. To avoid inversion to a water-in-oil emulsion, the characteristic rate of volume addition, V˙ d/Vc, is kept much less than the rate of mixing. When the entire volume of oil, Vd, has been added, the volume fraction of the premixed emulsion is φ ) Vd/Vc. An example of a typical premixed emulsion at C ) 0.4 and φ ) 0.7 is shown in the microscope image of Figure 3; it consists of large polydisperse droplets. Premixing is useful since we may independently specify C, φ, ηi, and the droplet topology, whether direct (oil-in-water) or inverse (water-in-oil). Moreover, the existence of the premixed emulsion implies a strong interfacial stability; this allows us to focus on the physics of the rupturing. To induce droplet rupturing, we subject the premixed emulsion to two kinds of controlled shears: oscillatory and steady.25 The oscillatory shear is characterized by two parameters, the shear strain amplitude, γ, and the frequency, ω, whereas the steady shear is characterized only by γ˘ . The oscillatory shear offers the advantage of independently controlling the maximum possible droplet deformation and also γ˘ through ω. (25) Whorlow, R. W. Rheological Techniques; Ellis Horwood Ltd.; Chichester, U.K. 1980.
Shear Rupturing of Droplets
Figure 4. Schematic diagram of an injection couette shear mixer. The premixed emulsion, contained initially in the syringe, is pumped into the thin gap between the outer rotating and inner fixed cylinders. Sealed bearings prevent the sheared emulsion from escaping out the ends and force it to leave through the exit port. III.C. Rupturing Induced by an Oscillatory Shear. For the oscillatory shear experiments, we employ two different geometries. For qualitative observations, we place the premixed emulsion between two glass slides separated by spacers that set the gap thickness between the plates. This thickness is typically kept less than d ) 200 µm to prevent fracturing. The shear is produced by oscillating the upper plate relative to the lower plate (by hand) at peak velocities of v ≈ 10 cm/s. For precise quantitative observations, we control the peak oscillatory shear amplitude and frequency of the rheometer fitted with circular parallel plates having radii R ) 2.5 cm. The premixed emulsion is placed on the base plate and the upper plate is lowered until the gap spacing between the two plates reaches d ) 200 µm; this gap spacing ensures a homogeneous shear deformation of the emulsion along the direction normal to the plate surface. The shear amplitude increases linearly with the radius r from the center of the plate and the angular displacement, θ, of the motor: γ(r,θ) ) rθ/d. After shearing with a fixed peak angular amplitude for 100 cycles to ensure complete rupturing at all γ, we measure the droplet radius by gently separating the two plates, placing a slide in contact with the exposed emulsion, and viewing the slide using a microscope equipped with differential interference contrast. If φ is near RCP, the droplet radius can be measured by counting the number of droplets over a given distance. By marking where the center of the two plates is on the slide’s surface, a(γ) can be found by simply translating the slide relative to this point and viewing microscopically. These measurements are complemented by small angle light scattering using the same translation procedure. For uniform droplets at φ near RCP, a can be determined from the distance between the transmitted beam and the annulus of scattered light. For φ well below RCP, we measure a microscopically, since the droplets are no longer packed and the annulus reflects the interdroplet separation rather than the droplet radius. III.D. Rupturing Induced by a Steady Shear. In the steady shear experiments, we employ two shearing geometries: circular parallel plates and a concentric cylinder couette equipped with a syringe pump for injecting the premixed emulsion, shown in Figure 4. The inner cylinder of radius Ri is stationary and the outer cylinder of radius Ro is attached to a motor that rotates at a fixed angular velocity, ωs. The gap is d ) Ro - Ri, the velocity at the surface of the external cylinder is d ) Roωs, making γ˘ ) ω/(1 - Ri/Ro). For our mixer, the outer radius is Ro ) 2.5 cm and the gap is d ) 200 µm; so for a typical ωs ≈ 10 rad/s we are able to reach γ˘ ≈ 103 s-1. Sealed bearings at the top and bottom of the cylinders maintain the gap spacing to a high degree of precision and prevent the emulsion from flowing out of the ends of the cylinders. The pump forces the premix emulsion up the entrance tube drilled in the internal cylinder, into the region of maximum shear, and out the exit tube for collection as the final product emulsion. The volume flow rate of the injected emulsion is kept small enough that the injection strain rate perpendicular
Langmuir, Vol. 13, No. 17, 1997 4605 to the velocity of the cylinder surfaces is much smaller than γ˘ , ensuring that the transverse γ˘ of the couette is the dominant rate. Although an indication of the droplet polydispersity can be obtained from microscope images and small angle light scattering patterns, we also have measured the droplet size distribution using the angular dependence of the time-averaged intensity, I(q), of light scattered from highly diluted emulsions that have been recovered after shearing. The distribution is extracted by minimizing the error between the measured I(q) and a superposition of weighted scattering form factors for monodisperse spheres of different discrete sizes; the weights determined from this minimization yield the probability per unit volume of encountering a droplet of the given size. From this, pa(a) can be extracted for a large ensemble of droplets. This pa(a) is an unbiased sampling of the droplet size distribution; small droplets below the resolution limit of our microscope and multimodal distributions can be detected. The instrumental resolution of the polydispersity is approximately Pa ≈ 0.1; polydispersities smaller than these cannot be reliably detected.
IV. Experimental Results IV.A. Empirical Map of Rupturing Phenomena. We have mapped out the qualitative phenomenology of the rupturing of premixed silicone oil droplets in NP7-water mixtures over a wide range of φ, C, and γ˘ . We employ the parallel plate geometry with rectangular glass slides to facilitate complementary microscopic and small angle light scattering observations. The oscillations of the upper plate typically induce absolute strains of γ ≈ 103, so that the droplets may become highly stretched. When the plate is oscillated extremely slowly, limiting the peak strain rate to γ˘ < 10 s-1, we find that for all compositions the emulsion remains polydisperse like the premixed emulsion after controlled shear. However, for γ˘ ≈ 10 s-1, microscopic observations reveal that the ruptured droplets are large with a ≈ 10 µm and roughly uniform for C ≈ 0.8 and φ ≈ 0.85, corresponding to the largest ηeff due to both the surfactant ηe and the droplet packing. At lower C and φ, the sheared emulsion remains polydisperse, similar to the premixed emulsion. At larger γ˘ ≈ 102 s-1, this region of monodispersity extends toward lower φ and C, again corresponding to large ηeff, as shown by the darkly shaded region in the C versus φ composition plane of Figure 5(a). The emulsion appears iridescent when illuminated with white light. As ηe decreases, for C and φ below the shaded region, we observe a gradual dropoff in the degree of uniformity and increase in size of the ruptured droplets; these lower boundaries are qualitative guides only. Raising the strain rate to γ˘ ≈ 103 s-1, we find that the previous island of monodispersity with a liquid-like structure found at γ˘ ≈ 102 s-1 has further enlarged toward smaller C and smaller φ, as shown in Figure 5b. The darker shading corresponds with the emulsions having the best monodispersity and also highest effective viscosities, whether due to the surfactant phase at high C or the packing of deformed droplets with a strong surface tension at high φ. The uniform droplet size can be obtained by directly shearing the premixed emulsion or by taking the monodisperse droplets obtained at γ˘ ≈ 102 s-1 and applying a higher shear rate to rupture them to a smaller monodisperse size. The real space microscope image at C ≈ 0.6 and φ ≈ 0.5 within this shaded region is shown in Figure 6. The corresponding small angle light scattering reveals a small ring, as shown in the inset of Figure 6. This ring indicates that there is a uniform distance between the droplets that scatter the light but that the droplet packing structure is disordered. At C ≈ 0.4 and φ ≈ 0.7, the first more intense ring exhibits six spotlike regions of higher intensity having equidistant azimuthal spacing around the ring; this is shown in Figure 7, inset. These spots are evidence of significant correlation of the droplet positions, possible only with highly monodisperse droplets, although the degree of ordering may have been enhanced by the shear. An example of this type of hexagonal ordering of the droplets is shown in the corresponding microscope image of Figure 7. It is striking that this kind of monodispersity can be obtained with a Newtonian ηe (for C in the L1 phase), but an overall shearthinning effective viscosity due to φ. For larger C in the LR phase at dilute φ ≈ 0.1, the ruptured droplets again appear uniform but are found in isolated chains that likely result from the
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a
b
Figure 5. Qualitative maps over φ and C of uniformity of ruptured droplets at shear rates of (a) γ˘ ≈ 102 s-1 and (b) γ˘ ≈ 103 s-1. The more darkly shaded regions correspond to increasing monodispersity in the ruptured droplet size. At high φ and C, coalescence is found, as indicated by the stripes. Fracturing is found at low C and high φ. The solid circles are pointers to microscope images and light scattering patterns in the figures which follow. rupturing of long slender tubes of oil that have been stretched out in the viscoelastic continuous phase by the shear; this is shown in Figure 8. We are able to view these chains because the NP7-water mixture has a strong yield stress; the droplets are unable to rearrange after the shear under entropic driving forces. Light scattering reveals an intense line perpendicular to the direction of shear even at low scattering angles corresponding to the form factor of the long chains of droplets; this scattering pattern is shown in Figure 8, inset. The large difference in refractive indices of the silicone oil and NP7-water mixtures ensures the scattering is dominated by the droplet structures and not defect structures in the lamellar phase itself. Although the positional structures of the ruptured droplets vary, uniformity in the size of the ruptured droplets extends over much of the composition plane. However, at high C, the onset of the L2 phase marks the region where the ruptured droplets are continually coalescencing; an example of the polydispersity that develops seconds after the shear is stopped for φ ≈ 0.7 and C ) 0.87 is shown in Figure 9. In a completely different region, at φ ≈ 0.95 and C ≈ 0.4, a small number of large droplets are found in a sea of smaller monodisperse droplets immediately after shearing, as shown in Figure 10; this is evidence of coalescence of highly compressed emulsions as φ f 1. These general phenomenological trends of monodispersity at large ηeff, polydispersity due to the
Figure 6. Microscope image of a disordered uniform emulsion after shear rupturing at C ≈ 0.6, φ ≈ 0.5, and γ˘ ≈ 103 s-1. Inset: the corresponding small-angle light scattering pattern is a welldefined ring reflecting the uniform separation of the packed droplets.
Figure 7. Microscope image of an ordered uniform emulsion after shear rupturing at C ≈ 0.4, φ ≈ 0.7, and γ˘ ≈ 103 s-1. Inset: the light scattering pattern is a diffuse ring with six bright spots reflecting hexagonal spatial ordering of the droplets. absence of rupturing for low ηeff at dilute φ and small C, fracturing at large φ but small C, and coalescence at very high C or φ are also seen at higher γ˘ ≈ 5000 s-1. IV.B. Rupturing by an Oscillatory Shear. Having characterized the qualitative experimental phenomena of droplet rupturing, we focus on quantitative experiments that may lead to a more precise understanding of the rupturing to a uniform droplet size. We have chosen φ ) 0.7 to ensure that the droplets are packed and C ) 0.4 so that the interfacial stability of the droplets is strong, and ηe is Newtonian but much larger than the viscosity of water to prevent fracturing. We employ the controlled strain rheometer rather than the crude glass slides. Since the amplitude of the strain may play an important role in the deformation of droplets imbedded in partially elastic materials, we have characterized a(γ) at fixed frequencies. The dependencies for three fixed frequencies, ω ) 1 rad/s, ω ) 10 rad/s, and ω ) 100 rad/s are shown in Figure 11; the solid symbols represent the unique radius of a monodisperse emulsion, while the open symbols represent the average radius of a polydisperse emulsion. For small deformations below a threshold γc ≈ 6, as shown by the vertical dashed line on the plot, the resulting emulsion is not monodisperse, regardless of ω. At deformations immediately above γc, it is possible to obtain monodisperse
Shear Rupturing of Droplets
Figure 8. Microscope image of chains of uniform droplets after shear rupturing at C ≈ 0.7, φ ≈ 0.1, and γ˘ ≈ 103 s-1. Inset: the light scattering pattern is a line reflecting the form factor of the chains.
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Figure 10. Microscope image of large droplets in a sea of monodisperse droplets found in a highly compressed emulsion at C ) 0.4 and φ ≈ 0.95 immediately after shearing at γ˘ ≈ 103 s-1. By contrast with Figure 9, visible coalescence is absent at γ˘ ≈ 0.
Figure 11. Dependence of the droplet radius, a, on the strain amplitude, γ, at fixed frequencies of ω ) 1 rad/s (circles), 10 rad/s (triangles), and 100 rad/s (diamonds). The open symbols represent average radii of polydisperse emulsions, while the solid symbols represent unique monodisperse radii. To the left of the vertical dashed line, the emulsion is polydisperse. Figure 9. Microscope image of a continually coalescing emulsion at C ≈ 0.87, φ ≈ 0.7, and γ˘ ≈ 103 s-1 approximately 10 s after the shear has been stopped. Large droplets have grown in a sea of initially monodisperse droplets. The polydispersity continues to increases after the image is taken. emulsions for fast oscillations, ω g 10 rad/s. By contrast, for ω , 1 rad/s, we cannot obtain monodisperse emulsions even at the largest accessible γ. For a fixed frequency, the monodisperse radius becomes smaller for larger strain amplitudes, becoming nearly independent of the strain amplitude at large γ. For a fixed strain amplitude, the radius becomes smaller for larger frequencies; by gradually increasing ω for γ > γc, we observe that the uniform droplet radius becomes successively smaller nearly continuously. For the C we have chosen, the droplet interfacial stability is strong; once ruptured, the droplets do not become reversibly larger for all ω less than the maximum shear rate applied. Thus, the droplet radius reflects the shear history, since a depends upon the maximum past ω and not necessarily its present value. Evidently, the maximum past shear rate, γ˘ ) ωγ, sets the radius, provided γ has exceeded the critical value for rupturing. IV.C. Rupturing by a Steady Shear. To test this idea, we explore how the final droplet radius of the emulsion depends on its viscoelasticity and γ˘ for steady shearing conditions. Starting from low shear rates where the premixed emulsion droplets are not ruptured, we have measured the effective viscosity of the premixed emulsion as a function of shear rate, as shown by the points in the inset of Figure 12. The emulsion is shear-thinning,
Figure 12. Dependence of the droplet radius, a, on the steady shear rate, γ˘ (points) for C ) 0.4 and φ ) 0.7. The solid line is a fit using the scaling form, eq 2, using the measured γ˘ -dependence of the effective viscosity, ηeff, shown by the points in the inset. The solid line in the inset is an empirical power law fit, ηeff ≈ 210 γ˘ -1/2 P, which is extrapolated to higher γ˘ for use in eq 2. exhibiting a smaller effective viscosity at higher shear rates following an empirical power law behavior of ηeff ∼ 210 γ˘ -1/2 P, shown by the solid line in the inset, up to a maximum shear rate of γ˘ ) 102 s-1, limited by the maximum torque of the rheometer’s transducer. This effective viscosity is more than 10 times larger than the viscosity of the continuous phase alone, reflecting the influence of φ. We apply larger shear rates to induce droplet rupturing and measure the radius as a function of γ˘ , as shown by the points in Figure 12. Above γ˘ ≈ 102 s-1, the radius drops
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Figure 13. Dependence of the droplet radius, a, on the volume fraction, φ, for fixed C ) 0.4 and γ˘ ≈ 103 s-1.
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Figure 15. Dependence of the droplet radius, a, on the oil viscosity, ηi, for fixed C ) 0.4, φ ) 0.7, and γ˘ ≈ 103 s-1. The dashed line on the plot represents the emulsion’s ηeff at γ˘ ≈ 103 s-1 extrapolated from Figure 12, inset.
Figure 14. Dependence of the droplet radius, a, on the NP7 surfactant composition, C, for fixed φ ) 0.7 and γ˘ ≈ 103 s-1. sharply and saturates at higher γ˘ ; this behavior is reminiscent of the data in Figure 11 for a fixed ω and strong shear amplitudes. Since the effective viscosity can be strongly modified by the volume fraction, we investigate the dependence of the droplet radius on φ, using the same continuous phase composition and fixing the shear rate at γ˘ ) 103 s-1, as shown in Figure 13. At low φ, the ηeff is set primarily by the continuous phase, resulting in the φ-independent radius. However, above φ ≈ 0.55, the droplet radius begins to decrease, qualitatively consistent with the large increase in the elasticity and yield stress when the droplets form a colloidal glass.16 At yet higher φ > 0.63 (RCP), the radius continues to decrease as φ f 1 due to droplet deformation.16,26 Since the concentration of NP7 can also have a strong effect on ηeff, we have measured a(C) at fixed φ and γ˘ , as shown in Figure 14 for φ ) 0.7 and γ˘ ) 103 s-1. Toward lower C the droplet radius increases sharply and the degree of monodispersity becomes smaller. This reflects the weakening viscosity of the L1 phase as the micelles become diluted (see Figure 2) and also the onset of inhomogeneous flow at the lowest C. Toward higher C, the droplets facilitate rupturing to a smaller size, reflecting the higher effective viscosity of the surfactant solutions. The onset of the inverted micellar phase, which permits rapid droplet coalescence, precludes formation of the premixed emulsion and, therefore, size measurements near C ≈ 1. As ηi becomes larger than ηeff, we expect it to play an important role in the rupturing since the droplets viscously resist distortion. We have measured a(ηi) for fixed C ) 0.4, φ ) 0.7, and γ˘ ) 103 s-1, as shown in Figure 15. For small ηi < 103 cP, the droplet radius is insensitive to ηi; the best monodispersity is found here. However, when the internal phase viscosity becomes much larger, the distribution becomes less uniform and a increases, indicating that the strong dissipative dynamics of deforming and breaking the internal phase are becoming important. IV.D. Role of Droplet Coalescence. We supplement these measurements of a(φ,C,γ˘ ,ηi) with a qualitative experiment to determine if droplet growth through coalescence during the shear is necessary to obtain monodispersity. We make a premix emulsion of equal volumes of high viscosity (ηi ) 5 × 104 cP) and low viscosity (ηi ) 12 cP) oil droplets at C ) 0.4. After shearing at γ˘ ≈ 103 s-1, we have obtained a binary alloy of emulsion droplets having two discrete monodisperse radii. If significant coalescence had been present during the shear, then only one droplet size would have been found, corresponding to the average viscosity (26) Mason, T. G.; Bibette, J.; Weitz, D. A. Phys. Rev. Lett. 1995, 75, 2051.
Figure 16. Radial probability density, pa(a), of the emulsion made using the injection couette mixer determined from angledependent light scattering measurements. The solid line is a Gaussian fit using eq 6, yielding an average radius 〈a〉 ) 0.56 µm and a standard deviation δa ) 0.09 µm. if the two oils had been mixed together before making the premixed emulsion. However, we find two discrete droplet sizes, supporting the absence of coalescence and the conclusion that the monodispersity can be obtained by droplet rupturing alone in a highly controlled shear. However, this does not preclude the possibility that monodispersity may also arise when coalescence does occur. To directly investigate the consequences of coalescence on the rupturing, we have prepared a premixed emulsion at φ ≈ 0.7 and C ) 0.87, barely within the L2 phase. The quiescent premixed emulsion is unstable, but the rate of coalescence between the droplets is slow enough that the emulsion can persist for several days without completely phase separating into pure oil and pure surfactant solution. This coalescence is microscopically visible in the absence of shear. After shearing the emulsion between two glass plates at γ˘ ≈ 103 s-1, we find that the droplets are still ruptured to a monodisperse size, rather than being forced to phase separate through shear-induced coalescence. However, the droplets begin coalescing immediately after the shear is stopped. This leads to larger droplets in a sea of monodisperse droplets and, therefore, an increasing polydispersity after the shearing is stopped. IV.E. Large Scale Production: Couette Injection Mixer. Finally, to demonstrate the feasibility of scaling-up droplet rupturing between thin plates to a continuous industrial production of monodisperse emulsions where large volumes are required, we have used our newly developed injection couette mixer. We pump a premixed emulsion of C ) 0.4 and φ ≈ 0.9 at a volume rate of 10 mL/min into the couette operating at γ˘ ≈ 1500 s-1. The premixed emulsion is ruptured to a monodisperse emulsion by the shear, and we collect it from the exit tube. We then measure the droplet size distribution using the angular dependence of the time-averaged intensity of light scattered from a highly diluted suspension of droplets taken from the final emulsion; the radial probability density based on this measurement is shown as the points in Figure 16. This distribution is highly peaked around a ≈ 0.56 µm.
V. Discussion V.A. Empirical Map of Rupturing Phenomena. Our empirical map of the dependence of rupturing
Shear Rupturing of Droplets
phenomena on γ˘ , φ, and C provides a general guide for understanding how the rheology of the surfactant-water mixtures and the droplet packing may affect the final droplet size and polydispersity. This map is not a phase diagram, nor is it a nonequilibrium phase diagram, since emulsions are not phases and their kinetically-stable interfacial structure depends completely on their past shear history. Instead, it represents a concise snapshot of qualitatively different regimes of rupturing and the resulting droplet structure. Our novel approaches of using a premixed emulsion with a well-defined composition and narrow shear geometries which help eliminate fracture flow have enabled us to build this map. Using it, we qualitatively interpret the rupturing phenomena and resulting droplet structure by considering C in terms of the surfactant phase behavior and ηe, φ in terms of the droplet packing, and γ˘ relative to (σ/a)/ηeff. This permits us to specify different origins of polydispersity. We first focus on the polydisperse regions of the map, pointing out general features regarding the rupturing at low and high C and φ, at fixed γ˘ . We then address the monodisperse regions and compare our quantitative results for a and Pa with the theoretical predictions. For low C in the dilute L1 and W phases, lubrication of the droplets by the weakly viscous continuous phase precludes rupturing by a homogeneous shear. When the viscosity of the surfactant phase falls below ηc ≈ 1 P, the thickness of the fracture plane can be estimated as l < 1 µm for γ˘ < 103 s-1, much less than the premixed emulsion’s droplet size and the separation between our plates. This implies that the flow will be inhomogeneous, despite our effort to use a very narrow gap separation. This is why emulsification with weakly viscous continuous phases usually yields polydisperse emulsions, even at high φ where the droplets can pack and the emulsion exhibits a considerable yield stress. As C is increased, the L1 phase becomes viscous enough that fracturing can be eliminated, and the droplets can be uniformly ruptured to one size, provided φ plays a significant role in ηeff. At yet larger C, in the LR phase, ηe becomes shear-thinning,24 so ηeff is also shear-thinning even at dilute φ, and the size selectivity of the ruptured droplets becomes even more pronounced, according to the selection mechanism of the modified capillary instability for viscoelastic complex fluids. For extremely large C in the L2 phase, droplet coalescence either prevents the formation of a premixed emulsion (near C ≈ 1) or it causes the polydispersity of the sheared emulsion to increase rapidly after the shear is stopped (near the LR boundary at C ≈ 0.87), independent of φ or γ˘ . While this coalescence formally limits our after-shear description to be polydisperse, near the LR boundary for a slow coalescence rate, the emulsion under steady shear can be maintained as monodisperse by continual rupturing of droplets that have grown to be larger than the critical size through coalescence. While the phase behavior of the surfactant plays a critical role in understanding the C dependence of the rupturing phenomena, the packing of droplets plays a corresponding role in the φ dependence. As φ f 0, the droplets are well-separated and their effect on ηeff is negligible, yet it is possible to rupture droplets to a uniform size for concentrated C where ηe sets ηeff. At low C, this corresponds to the classical regime of rupturing in a viscous fluid, which has been extensively studied. However, the degree of uniformity we observe noticeably improves above C ≈ 0.45, corresponding to a pronounced elastic component in the complex shear modulus and a shear-thinning effective viscosity. At yet higher C, the suspending fluid exhibits a yield stress which restricts the Brownian motion of the ruptured droplets. Our microscopic observations
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of linear chains of uniform droplets oriented along the shear direction show that the Laplace pressure of extremely elongated liquid threads is sufficient to overcome the yield stress and cause rupturing. The signature of these threads in the light scattering is a line perpendicular to the direction of shear, reflecting the form factor of these rodlike chains when oriented perpendicular to the incident beam. Both the observations of the chains of droplets and the uniform ruptured droplet size supports our interpretation that the capillary instability for highly stretched threads is inhibited until the Laplace pressure driving the instability exceeds the characteristic bulk yield stress required for flow. When φ approaches and exceeds RCP, the emulsion’s ηeff increases sharply and becomes shear-thinning, reflecting the onset of yield behavior due to the elasticity of the droplet’s deformable interfaces. This leads to an extension of the monodisperse region in Figure 5b for concentrated emulsions with φ > 0.6 in weaker continuous phases at lower C; the droplets can exhibit enhanced monodispersity for Newtonian ηe in the L1 phase since the shear-thinning ηeff is due to φ. While the relationship between packing and φ throughout the rupturing process from the polydisperse premixed emulsion remains obscure, it is clear that in the later stages of rupturing to monodisperse droplets, the droplet packing can be clearly understood in terms of packings of monodisperse spheres. If sufficiently monodisperse, hard sphere suspensions27 and emulsions28 can entropically crystallize to increase their translational free volume if concentrated to high φ. For crystallization to occur, the radial polydispersity must be less than Pa ≈ 0.15, consistent with experiments29 and simulations.30 A signature of the ordering is the appearance of Bragg spots in the light scattering pattern, rather than a simple ring characteristic of disorder. Under entropic driving forces, the crystals may take a very long time to form; by contrast, a driving shear may induce or enhance ordering after the droplets are ruptured,31 leading to pronounced Bragg spots. In our experiments, shearinduced ordering in the packed droplets for φ immediately below and above RCP give rise to Bragg spots in the light scattering pattern; these spots degrade to a liquid-like ring for φ far below RCP. Even above RCP, the positional ordering persists, and the relatively even intensities of the spots suggest that the droplets have formed randomly stacked hexagonally close-packed (RHCP) planes32 to reduce the overall degree of droplet deformation. Near the top of the composition plane as φ f 1 for C below the L2 phase, our observations of large droplets in a sea of monodisperse droplets may indicate that some coalescence is occurring while the shear is driving rupturing in highly compressed emulsions. Assuming that coalescence between two droplets will occur when the pressure between them exceeds a critical disjoining pressure, Π*,20 we imagine that the effective viscous stress, ηeffγ˘ , may set the scale for the impinging pressure between adjacent droplets. When this pressure, or effective normal stress, exceeds Π*, coalescence can occur. This implies that the rate of coalescence may increase with the shear rate to reduce the emulsion’s osmotic pressure by increasing the droplet packing efficiency and the thickness of the films between the remaining droplets. This speculation (27) Pusey, P. N.; vanMegen, W. Nature 1986, 320, 340. (28) Mason, T. G.; Krall, A. H.; Gang, H.; Bibette, J.; Weitz, D. A. in Encyclopedia of Emulsion Technology; Becher, P., Ed.; Marcel Dekker, Inc.: New York, 1996; Vol. 4, p 299. (29) Pusey, P. N. J. Phys. 1987, 48, 709. (30) Barrat, J. L.; Hansen, J. P. J. Phys. 1986, 47, 1547. (31) Ackerson, B. J.; Clark, N. A. Phys. Rev. A 1984, 30, 906. (32) Chaikin, P. M.; Lubensky, T. Principles of Condensed Matter Physics; Cambridge University Press: Cambridge, U.K., 1995.
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is merely one possibility; one may likewise imagine that the droplets are truly monodisperse while the shear is applied (since any large droplets that have grown by coalescence would be ruptured) but that they become polydisperse due to coalescence immediately after the shear is stopped. Since we cannot measure the droplet size distribution in our concentrated emulsion while the shear is applied, we cannot differentiate between these two possibilities. However, we believe that the second explanation is more consistent with the apparent absence of coalescence at lower φ and our observations of the binary emulsion using oils having two different viscosities. While coalescence is not a requirement for monodispersity, it is also possible to create monodisperse emulsions through a combination of rupturing and coalescence, as demonstrated at surfactant concentrations in the L2 phase near the LR boundary. Indeed, coalescence is expected to play an important role in emulsification in the following limits: at extremes in the surfactant concentration, either C f 0, where the interfaces are depleted of surfactant, or C f 1, where the pure surfactant is likewise incapable of stabilizing the interfaces; as φ f 1, where the osmotic pressure of the emulsion exceeds the critical disjoining pressure; as γ˘ f ∞, where the effective viscous stress exceeds the critical disjoining pressure. The last limit offers the advantage that the coalescence ceases as the strain rate is reduced, thereby preserving the monodispersity of the ruptured emulsion. In all cases where rupturing is the dominant effect, the droplet size is set by a balance of the interfacial Laplace pressure and effective viscous stresses. If the coalescence rate is independent of γ˘ and greatly exceeds the rupturing rate, the emulsion will eventually phase separate, precluding monodispersity. However, it may be possible that the rate of coalescence can become negligible as the droplet size becomes larger; for instance when Π* > σ/a; the unique droplet size would reflect Π* and not ηeffγ˘ . V.B. Dependence of the Ruptured Droplet Size on Composition and Shear. By contrast with droplet rupturing in viscous fluids, our results from oscillatory measurements indicate that the specification of Cac (or equivalently γ˘ ) is insufficient for determining the rupturing of droplets in viscoelastic media. Instead, a and Pa may also depend on the strain amplitude. This is natural since the energy stored in partially elastic materials depends on the strain. For instance, one may imagine that the degree of deformation of a weakly viscous droplet in a completely elastic material is determined only by the strain, provided the elasticity remains much greater than the Laplace pressure of the deformed droplet. Thus, our observations of a large critical rupture strain are a reflection of the partially elastic behavior of the effective medium in which the droplets are ruptured. The value we find for the one composition we explored, γc ≈ 6, is significantly larger than unity, indicating that the droplets must be stretched to at least a similarly large aspect ratio before rupturing. However, we know of no predictions of this quantity as a function of the effective viscoelasticity of the medium surrounding the droplet; we expect γc > 1, since the droplets must elongate before rupturing can occur, but the specific value of γc may vary with the viscoelasticity of the medium. Once the emulsion flows with γ > γc, we find eq 2 can be used to predict a. To determine if the shear-thinning effective viscosity of the emulsion sets the ruptured droplet size, we fit the data for a(γ˘ ) using the scaling form of eq 2. We describe ηeff by the empirical power law, ηeff ) 210 γ˘ -1/2 P, as shown by the solid line in Figure 12, inset, and we extrapolate this to higher shear rates where rupturing occurs. We obtain excellent agreement between the measured and
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predicted droplet sizes when the dimensionless coefficient of the scaling form is 0.45. Since this is of order unity, the fit confirms the perspective that this emulsion can be treated as an effective medium with a shear-thinning viscosity that governs the rupturing. However, we do not expect this scaling form for a(γ˘ ) to be universal; the region of its validity and that of the effective medium approach in the composition plane remain imprecise. We also expect this scaling form to be limited to low ηi, since high ηi . ηeff has been shown to critically affect the ruptured droplet size. Given the complexity in understanding the droplet structure and packing fraction during the shear, it is difficult to quantitatively interpret the dependence of a(φ). However, its behavior can be divided into two qualitatively different regions at low and high φ. At low φ, corresponding to isolated or weakly-interacting droplets, ηe essentially sets ηeff, so the droplet size remains roughly constant. However, at large φ > 0.55, the elastic deformation of the strongly-interacting droplet interfaces greatly elevates ηeff, which may become shear-thinning despite the Newtonian ηe. This causes the observed decrease in a and also accentuates the monodispersity. Our observations of a(C) corroborate this qualitative interpretation using ηeff. For low C in the L1 phase, ηeff is weak and Newtonian, so the droplets are ruptured to a larger size. However, when the effective viscosity of the surfactant phase is larger and shear-thinning, such as in the LR phase, the droplet size rapidly decreases and then saturates as the coalescence region of the L2 phase is approached. In general, we find that for C near phase boundaries, which may depend sensitively on the temperature, or for C in multiphase coexistence regions, which may have locally inhomogeneous viscoelasticities, controlling the droplet size can be difficult. The increase in droplet radius at higher ηi shown in Figure 15 may occur when the internal phase viscosity becomes greater than the emulsion’s effective viscosity at the strain rate used for rupturing. To make this comparison, we assume that the effective viscosities of all the emulsions are independent of the internal viscosities. We determine the effective viscosity by extrapolating ηeff (γ˘ ) shown for ηi ) 350 cP in Figure 12, inset, up to γ˘ ) 103 s-1, where we find ηeff ≈ 500 cP, as indicated by the dashed vertical line on the plot. For ηi > ηeff at this strain rate, we find an increase in the droplet radius by a factor of 4 over two decades in ηi. For one decade below ηi ) ηeff at this strain rate, we find the droplet size is constant. These results are qualitatively consistent with the internal viscosity dependence of isolated droplets ruptured in a purely viscous fluid. 3 V.C. Polydispersity of the Droplet Size Distribution after Rupturing. To determine the polydispersity one may find in a typical production device, we fit the raw size distribution directly measured using light scattering for our sealed couette using eq 6, as shown by the solid line in Figure 16. The measured distribution exhibits an asymmetric skew that the Gaussian clearly does not capture. In principle, we could calculate the moments of this distribution to determine the average droplet size and polydispersity, but we find it instructive to fit these data with a two-parameter normalized Gaussian distribution:19 2
pa(a) )
2
e-[(a-〈a〉) /2δa ]
x2πδa2
(6)
where 〈a〉 is the mean droplet size and δa is the standard deviation. We obtain 〈a〉 ) 0.56 µm and δa ) 0.09 from
Shear Rupturing of Droplets
Figure 17. Radial probability density, pa(a), of the emulsion shown in Figure 7 determined directly from the microscope image (points). The solid line is a fit using eq 6, yielding an average radius 〈a〉 ) 2.31 µm and a standard deviation δa ) 0.12 µm.
the fit, shown by the solid line in Figure 16, yielding a polydispersity of Pa ) 0.16. This monodispersity is equivalent to that obtained with four or five size fractionations of an initially polydisperse emulsion,28,33 and rivals that presently available through filtration and porous glass membrane emulsification methods.34,35 To illustrate the monodispersity one may obtain in optimized laboratory conditions, we have measured the droplet size distribution of the emulsion shown in Figure 7; our results are shown in Figure 17. The distribution is highly peaked. The fit, shown by the solid line in Figure 17, is in good agreement with our distribution, although it cannot account for a slight skew toward larger droplet radii. We extract 〈a〉 ) 2.31 µm, slightly higher than that measured for precisely controlled γ˘ , and δa ) 0.12 µm, making the residual polydispersity Pa ) δa/〈a〉 ) 0.05. This is impressively small, well below that required for entropically-driven crystallization, and is consistent with the positional ordering in the image and the Bragg spots we find in the light scattering. While the larger polydispersity of the injection couette emulsion is comparable to our estimate from successive droplet rupturing, the very small polydispersity of the microscope image is well below our estimate. This suggests that a different mechanism such as wavelength selection through the viscoelastically-modified capillary instability is enhancing the uniformity of the ruptured droplets. While speculative, these two simple ideas may capture much of the essential physics governing the polydispersity from the limit of viscous fluids to the limit of highly elastic complex fluids. A precise analysis of the dispersion relation for the unstable growth modes for these two different limits may lead to quantitative predictions for pa(a) that can be directly compared with the measurements. VI. Practical Applications Our shear emulsification method offers the distinct advantage of a high production throughput since it is naturally suited for concentrated φ, rather than dilute φ required by other methods.33,34 The low measured polydispersity of the emulsion produced with the injection couette mixer demonstrates the feasibility of scaling-up our discovery for commercial use. Successful adaptation of our methods for a specific application hinges upon two features: the design of the mixer and the desired composition of the emulsion. This section covers some practical guidelines for adapting our technique to make emulsions with an arbitrary composition and droplet size, (33) Bibette, J. J. Colloid Interface Sci. 1991, 147, 474. (34) Omi, S. Colloids Surf. 1996, 109, 97. (35) Omi, S.; Katami, K. I.; Taguchi, T.; Kaneko, K.; Iso, M. J. Appl. Polym. Sci. 1995, 57, 1013.
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using our results for the silicone oil/NP7 + water composition as a foundation. The minimum requirements for an effective mixer design are a uniformly thin gap to prevent fracture flow; a controllable, uniform strain rate; and an injection system that forces all the premixed emulsion through the region of maximum shear. The injection system is the key to overcoming the limitation of the thin gap, facilitating high throughput and continuous production. Our injection couette mixer is only one example suitable for large-scale continuous production of monodisperse emulsions. In general, the shearing geometry need not be a rotational couette; rotational cone-plate, rotational cone-cone, or oscillatory plate-plate geometries could also be used. Sealed bearings are useful for forcing the emulsion out of the region of shear through an exit port for collection or further processing. For continuous operation, a pump can be used to draw the premixed emulsion from a crudelycontrolled feed mixer into the controlled mixer. The injection volumetric flow rate can be converted to an injection shear rate using the dimensions of the geometry; this injection shear rate should be kept much less than γ˘ . Since the ruptured droplet radius is predicted to be insensitive to the shear rate for a perfectly shear-thinning ηeff, according to eq 5, it may be possible to obtain monodisperse emulsions even in a poorly controlled mixer in which γ˘ varies spatially, provided the entire premixed emulsion is pumped through the region of maximum γ˘ where no fracturing occurs. Since in industrial emulsion production, the dispersed phase composition of the emulsion is set by the desired use, it is important to demonstrate that the mechanisms leading to monodispersity are independent of the specific chemical formulation. For different emulsion compositions, we find that the phenomena of droplet rupturing to a monodisperse size in viscoelastic media is general. We have made monodisperse emulsions of alkanes such as mineral oil and hexadecane and immiscible thermotropic liquid crystals. Even inhomogeneous oils that contain nanometric particles have been monodispersely emulsified. Regarding the continuous phase, concentrated ionic surfactant solutions (e.g., SDS phases) that are viscoelastic also yield monodispersity. Viscoelastic polymer solutions, rather than surfactant phases can also be used. For instance, a continuous phase of 105 MW Dextran at 20% by mass with only 1% SDS to stabilize the droplet interfaces also leads to monodisperse emulsions. Since both silicone oil and the Dextran-water mixture are polymeric liquids, we have effectively shown that monodisperse emulsions of homopolymer droplets in an immiscible viscoelastic homopolymer matrix can be formed. Moreover, this demonstrates that the expense of a concentrated surfactant phase can be avoided through the use of standard polymeric viscosity modifiers. While monodispersity can be obtained for many different compositions, the boundaries of the regions in the compositional map do depend on the specific formulation primarily through the effective viscosity. This is especially true for concentrated surfactants, which have phase boundaries, and therefore effective viscosities, that may be very different from that of our example, NP7. A simple characterization of ηeff(γ˘ ) for the composition in question would provide the designer sufficient information to select a shear rate consistent with the desired droplet size using eq 2. While the polydispersity cannot yet be predicted with equal ease, it should be minimal for a strongly shearthinning ηeff, provided that inhomogeneous fracturing is avoided. If the continuous phase can be altered to achieve this desired ηeff(γ˘ ) (e.g., through the addition of polymers or surfactants) without sacrificing product performance,
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the monodispersity may be improved by design. Once the composition is selected, a fast way to screen if a monodisperse emulsion can be made is to shear a small quantity of premixed emulsion between glass slides and observe the droplets using a microscope. Since attractive droplet interactions due to depletion or solvent forces can create large contact angles between droplets visible at low φ, the boundaries of monodispersity may change from those of the repulsive case (e.g., NP7). Such attractive droplets are nonspherical and do not pack except at extremely high φ; this modifies ηeff. When inadequately packed at lower φ, they form aggregates or flocs that have locally inhomogeneous effective viscosities which can preclude uniform rupturing. We have observed this effect for invert emulsions of water-in-oil stabilized by oil-soluble surfactants such as SPAN 80 that exhibit monodispersity only above φ ≈ 0.90. Presently, we are able to control the uniform droplet size to be between 0.1 µm < a < 10 µm by tuning γ˘ and ηeff through φ and C. The lower limit is essentially set by the yield stress of the most elastic concentrated surfactant phase we have used (a hexagonal phase of SDS with τy ≈ 106 dynes/cm2). The upper size is limited by contamination of the final droplet distribution with smaller droplets that were originally present in the premixed emulsion. Greater care in preparing the premix should allow larger droplet sizes to be obtained, and the use of surfactant or polymeric continuous phases that have larger yield stresses promise to yield droplet sizes smaller than those we have obtained. VII. Perspective Our study of droplet rupturing in complex fluids represents a generalization of the simpler problem of droplet rupturing in a purely viscous fluid. We have shown that the partial elasticity of the complex fluid may inhibit rupturing of droplets deformed by a shear until the droplet becomes sufficiently stretched that its Laplace pressure can induce the complex fluid to flow plastically. While this may not be so surprising when the structures giving rise to the elasticity within the complex fluid are much smaller than the droplet size, it is surprising that the same idea apparently holds when the structures are the droplets themselves, as in concentrated emulsions. We find that the concentrated emulsion can be considered as an effective medium throughout the rupturing, provided that fracturing through the lubricated slip of planes of droplets can be avoided. In this limit, each droplet can be considered as isolated as it ruptures in a homogeneous medium, which reflects the viscoelasticity arising from all sources: the surface tension of packed droplets and the continuous phase through surfactant phases or polymer networks. The gross simplification of using an effective viscosity to predict the ruptured droplet size breaks down when the viscosity of the continuous phase is small and inhomogeneous fracture flow occurs. To go beyond the simple scaling form of eq 2 and predict its numerical prefactor, the details of the colloidal structures around the rupturing droplet must be considered. Once fracturing is eliminated, we find that several different scenarios can give rise to monodisperse distributions through shear rupturing. The first is the modification of the capillary instability for isolated droplets in a yield stress fluid. This equivalently can be viewed as the γ˘ -independent prediction for the droplet size for a perfectly shear-thinning ηeff. Second, fluids exhibiting a partially shear-thinning ηeff can be successively ruptured to smaller and smaller monodisperse sizes by successively applying incrementally higher shear rates. Our observations suggest that the polydispersity increases as the emulsion becomes less shear-thinning. Finally, a balance between
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rupturing and coalescence rates can lead to an equilibrium monodisperse size that can depend on coalescence criteria that balance the composition-dependent critical film rupture pressure Π* with the γ˘ -dependent effective viscous stress. This complements an earlier finding of monodispersity for rupturing that limits the continuous growth of nucleated droplets when a binary liquid mixture is quenched into the two-phase region.36 We emphasize that regardless of whether the droplet growth mechanisms are discrete (coalescence) or continuous (ripening), ultimately, the limiting steady-state monodispersity is determined by the physics of the rupturing alone. The mechanism we propose for the modification of the capillary instability by the bulk yield stress of the complex fluid is similar to that found for pearling behavior of cylindrical lipid membranes.37 When excited by a laser, these membranes form monodisperse pearls. The pearls do not rupture into vesicles, but remain connected by tiny cylindrical tubes. An explanation has been proposed in which the curvature energy of the membrane modifies the surface tension-driven capillary instability induced by the laser excitation; a balance between the curvature energy density and Laplace pressure yields the pearl size. In spirit, this resembles the characteristic wavelength selection in the undulations of a spherical droplet which has a surface tension that competes with the curvature energy,38 although the cylindrical and spherical geometries are very different. Our proposed mechanism of inhibition of the capillary instability differs from both of these since we compare a bulk rather than an interfacial elastic stress with σ/a. Moreover, once a droplet is sufficiently elongated that the instability occurs, it does completely rupture into many discrete uniform droplets. The formation of monodisperse multilamellar vesicles (spherulites) by shearing a lyotropic lamellar phase39 exhibits some similarities with and also some differences from our observations for emulsions. The lamellar phase’s structure under shear consists of: smectic planes with defects at low γ˘ , then monodisperse spherulite structures with periodic defects at higher γ˘ , after which smectic layers nearly free from defects at the highest γ˘ . These observations are analogous to ours in that droplets cannot be ruptured at low γ˘ , and they can be ruptured to a monodisperse size at higher γ˘ , and that shear-induced droplet coalescence may occur at extremely high γ˘ for weak Π*. In the monodisperse region, the average spherulite radius has been observed to scale as a(γ˘ ) ∼ γ˘ -1/2.39 It is surprising that this scaling is similar to what we observe in our concentrated emulsion in Figure 12. We believe that this similarity is merely coincidental since the shearthinning power law for ηeff for concentrated monodisperse emulsions at φ ≈ 0.7 for very dilute surfactant concentrations far outside of the lamellar phase is also near 1/2.16 Moreover, the monodispersity of the spherulites is likely to arise from a combination of coalescence and rupturing, since when the shear rate is reduced, the spherulites become larger reversibly. From this, we may conclude that coalescence is always occurring, even after the shear is stopped, so that the spherulites are unstable in a quiescent state and will eventually fuse to relieve local stresses retained in the defect structure. This is comparable with the small region in our compositional map (for C in the L2 phase near the LR boundary) where coalescence is always present. By contrast, when we ensure strong interfacial stability, we have shown that (36) Min, K. Y.; Goldburg, W. I. Phys. Rev. Lett. 1993, 70, 469. (37) Bar-Ziv, R.; Moses, E. Phys. Rev. Lett. 1994, 73, 1392 . (38) Granek, R.; Ball, R. C.; Cates, M. E. J. Phys. II Fr. 1993, 3, 829 . (39) Diat, O.; Roux, D.; Nallet, F. J. Phys. II Fr. 1993, 3, 1427 .
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the monodispersity of emulsions can be obtained from droplet rupturing alone in the absence of coalescence. The droplet size is set irreversibly by the highest shear rate, and the droplets are completely stable against coalescence when the shear is stopped. VIII. Conclusion This study represents the first systematic exploration of emulsification through the shear-rupturing of droplets in complex fluids. Our approach is a generalization of the classic Taylor problem of the rupturing of a viscous droplet in a viscous fluid to the rupturing of a viscous droplet in a viscoelastic complex fluid. In the course of mapping the qualitative rupturing behavior as a function of C, φ, and γ˘ , we have discovered many interesting phenomena, including rupturing to a monodisperse size. This monodispersity has enabled us to measure the droplet radius as a function of γ˘ , C, φ, and ηi, fixing the other parameters. We have shown that the effective viscosity of the emulsion plays a key role in determining the droplet size when fracturing is eliminated and leads to a high degree of monodispersity when ηeff is shear-thinning. We believe that the ordinary capillary instability is modified due to the partial elasticity of the complex fluid; this elasticity leads to a highly selected wavelength in the instability, resulting in enhanced monodispersity. The general description of isolated droplet rupturing in a viscoelastic material remains an interesting theoretical problem to be explored. The solution of the full threedimensional free boundary problem for the droplet shape throughout the rupturing in an arbitrary viscoelastic medium is likely to be quite complicated. Yet, this problem is probably simple in comparison to describing real emulsification which involves the effects of φ and droplet interfaces pressing against each other during rupturing. Ultimately, we hope our measurements and scaling arguments will motivate theoretical predictions or simulations of the polydispersity and complete droplet size distribution over the complete map, and establish precise
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boundaries where the effective medium approach works or fails. In future experiments, measurements of pa(a) using light scattering over all γ˘ , C, and φ will be made to quantify the complete distribution over the entire map. Our method of observing the emulsion before and after rupturing neglects the intricate interfacial dynamics while the rupturing is occurring; this would also be interesting to explore. Time-resolved microscopic observations of an isolated droplet’s interface during all stages of the elongation, neckdown, and pinching off could provide direct verification of our hypothesis of strong wavelength selection in the modified capillary instability. At high φ, the boundaries for shear-induced ordering of droplets may be quite different from those found for hard sphere dispersions since the droplets can deform; performing small angle light scattering while shearing the emulsion may give insight into the changing droplet positional structure. The direct production of monodisperse emulsions through droplet rupturing in viscoelastic complex fluids represents a new and well-controlled way of making colloidal fluid dispersions. The power of this emulsification method lies in its simplicity, throughput, control, and applicability to a broad range of chemical compositions of the fluids, making it a good candidate for commercial emulsion production.40 The monodispersity of the droplets should simplify the design of emulsion products; the final droplet size and rheological behavior can be selected through the composition and the shear rate using available rheological data as a guideline.16,26 Many exciting fundamental experimental studies lie ahead to more fully quantify the basic phenomenology we have presented in this work. Acknowledgment. We have greatly benefited from discussions with David Weitz and Didier Roux. LA9700580 (40) Bibette, J.; Mason, T. G.; French Patent No. 96 04736, 1996.
