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My current research interests concern the effects of surfactants on droplet interactions in viscous flows. Surfactants are nearly ubiquitous, with both traditional applications in areas such as detergent and the processing of many materials (Schwartz & Perry, 1949) and newer applications in growing research areas such as microfluidics (Stone, Stroock & Ajdari, 2004; Rosslee & Abbott, 2000), ‘smart' drug delivery (Israelachvili, 1994; Karsa, 2000), and liquid crystals (Shah & Abbott, 2001).

The objective of the current research is to develop a broad theory of the role of surfactants on drop interactions in viscous flows, such as buoyancy, thermocapillary motion due to an applied temperature gradient and shear flow, that predicts coalescence, breakup and stability in dilute and nondilute dispersions of spherical and deformable drops, so that emulsions can be made with given characteristics, notably their drop-size distribution and rheological properties.

The impact of surfactants on the settling velocity of an isolated drop in gravity has been well studied. In addition, much attention has been given to how surfactants modify the deformation and breakup of a single drop, particularly in a linear extensional flow. Clearly, such work is necessary to gain a fundamental understanding of the effects of surfactants on drop behavior. Because a major area of practical significance is in the mixing of fluids, as in the preparation of emulsions and in polymer blending, it is crucial that the single-drop studies be extended to two or more drops, especially since surfactants are either used to stabilize drops against coalescence or are nearly always present as impurities. Nonetheless, with the exception of a few investigations, the presence of surfactants for binary and multiple drop interactions has to this point been neglected.

Now that so much progress has been made in understanding the role of surfactants in the motion and deformation of one drop, it is natural to extend present knowledge to theoretical analysis of two or more contaminated drops in shear flow under all conditions. Complicated problems, such as two slightly deformable drops with arbitrary surface coverage, and multiple deformable drops with surfactant in shear flow, remain to be treated. Study of such problems in the presence of surfactants is vital to making sense of emulsion processing and rheology. Similarly, parallel experiments for two or more spherical or deformable drops with surface-active agents need to be conducted.

Theory for Binary Spherical Drop Interactions - Two cases are being considered for binary spherical-drop interactions: arbitrary surface coverage for non-diffusing and diffusing surfactant. In both cases, the goal is to calculate collision efficiencies by a trajectory analysis for use in population dynamics simulations to predict the behavior of dilute dispersions. Such an analysis has been made for surfactant-free drops in linear flows (Wang, Zinchenko & Davis, 1994), in which the problem is linear and can be solved by decoupling motion normal and parallel to the drops' line of centers. For the case of arbitrary surface coverage, however, the coupling between the velocity field and the surfactant equation makes the problem non-linear and series expansions, similar to those of Blawzdziewicz et al. (2000), must be used.

As a starting point, a simple linear model for the dependence of the interfacial tension on surfactant concentration is assumed. The ideal gas equation of state is valid at low surfactant surface concentrations, but this regime is important because even small amounts of surfactant are sufficient to immobilize the drop interfaces (Chesters & Bazhkelov, 2000). The linear model also holds for nondilute systems with small deviations (Blawzdziewicz, Wajnryb & Loewenberg, 1999). Higher surfactant concentrations can be incorporated into the study, but with small amounts of adsorbed surfactant, it is possible to avoid the complications of surface viscosity (Valkovska et al. , 1999, 2000; Pozrikidis, 1994; Li, 1996).

The scaling analysis of Blawzdziewicz et al. (2000) indicates that significant redistribution of bulk-insoluble surfactant can take place on spherical drops if the concentration of adsorbed surfactant is low. When the surfactant concentration is small, the ideal gas equation of state is appropriate, and surface viscosity can be neglected. Although their results are confined to effects on the effective viscosity and first and second normal stress differences in dilute dispersions, we anticipate that even small amounts of adsorbed surfactant can have important consequences for collisions between spherical drops, as well. In particular, we expect that at low drop-to-medium viscosity ratios, surfactant effects will be most evident, since at higher viscosity ratios, the mobility of the drop interfaces is already decreased, even in the absence of contaminants.

As in previous surfactant-free studies, it will be necessary to determine relative trajectories for two drops. In an arbitrary trajectory in gravitational flow, as in Fig. 1, the drops have an initial horizontal offset when well separated, and when one drop catches up to the other one, the drops will either collide and coalesce, or eventually separate. The collision efficiency is determined through the critical horizontal offset demarcating trajectories which lead to coalescence and separation. In Fig. 1, surfactant molecules are indicated by blue lines, and the red lines mark the external flow of liquid around the drops. Competing effects between the bulk flow of diplaced liquid and the squeeze flow between the drops determine the distribution of surfactant along the drop interfaces.

Figure 1. Two surfactant covered drops interacting in buoyancy-driven motion.

To calculate relative trajectories, we propose expanding the surfactant concentration in spherical harmonics and using Lamb's singular series for the velocity field. Substituting into the transport equation and boundary conditions will lead to a series of non-linear ordinary differential equations that can be solved numerically, with fast-convergent, conjugate-gradient iterations at each time step for the velocity field. Close approach will be handled by an asymptotic method.

Drops with negligible deformations are often met in practical emulsions. At the same time, it would be extremely prohibitive to handle this case by more general (standard boundary-integral or finite-element) methods, due to numerical stiffness and other difficulties; this case requires special tools, like those proposed in this section.

Asymptotic Theory for Small Deformation with Coalescence - Previously, we have determined collision efficiencies for slightly deformable drops in various viscous flows with clean interfaces by combining the solution to the thin-film equations for the region of close approach, where deformation is important, with spherical drop results in the undeformed portion of drop trajectories. By taking an asymptotic approach, we were able to perform systematic calculations for a problem where full boundary-integral calculations are prohibitive. This asymptotic approach, which neglects the effects of pumping (Cristini et al., 2001), was recently validated by multipole-accelerated calculations of Zinchenko & Davis (2005). The solution for shear flow, as opposed to gravitational or uniaxial compressional motion, is particularly interesting because of the dependence on two angles. In Figure 2, the outer quarter circle is the critical cross-section for coalescence for two spherical drops in simple shear in the absence of van der Waals forces. When small deformation and van der Waals forces are taken into account, the critical cross-section becomes irregularly shaped and is reduced in area beyond a certain drop size. As the reduced drop radius increases beyond 0.015 cm for a system of ethyl salicylate (ES) drops in diethylene glycol (DEG), the upstream interception area becomes smaller until at 0.1 cm, the critical cross-section is nearly elliptical with major axes less than 0.4 and 0.19 of the outer circle. Each point on the curves is determined by a single pair of critical angles, and thus many trajectories are required to determine the upstream interception area. The collision efficiency for both contaminated and uncontaminated slightly deformable drops depends on the critical cross-section. Because of the heavy dependence on trajectory calculations, a fast algorithm is required.

Figure 2. The upstream interception area as a function of the reduced drop radius R for an ES/DEG system for drops in simple shear flow with a shear rate of 1 inverse second and size ratio 0.5. (Adapted from Rother & Davis, 2001.)

We propose to modify the explicit algorithm of Chesters & Bazhlekov (2000) for solution of the thin-film equations governing film drainage between contaminated, slightly deformable drops in a manner similar to the semi-implicit scheme of Rother et al. (1997). Using the thin-film results as the inner part of the solution in conjunction with spherical-drop calculations at arbitrary coverage as the outer solution will allow determination of collision efficiencies. We anticipate that, under conditions when the surfactant coverage remains nearly uniform in the outer solution, significant deviation in coverage in the thin inner film may occur at high surface Peclet numbers, resulting in the interfaces becoming immobile (Chesters & Bazhlekov, 2000). Film drainage would then drastically slow and significantly decrease the collision efficiency from spherical-drop results for nearly uniform coverage (Blawzdziewicz, Wajnryb & Loewenberg, 1999)..

Boundary-integral Theory with Moderate-to-Large Deformation - In contrast to small deformations, which inhibit coalescence between two drops, large deformations can enhance coalescence, lead to drop breakup, or result in more complicated outcomes, depending on the flow and flow conditions. Previously, for drops with clean interfaces in buoyancy, by using meshes with fixed topology maintained by "passive" mesh stabilization, we were able to achieve very good convergence for the global shape and even dynamics of the neck cross-section just prior to breakup in 3D calculations by a curvatureless algorithm, with several thousand elements per drop (Zinchenko et al. , 1999; see Fig. 3). In current research, we seek to address the effects of insoluble surfactant on the interactions of two deformable drops in viscous flows, including coalescence and breakup. The idea of using a stretched mesh with fixed connections should succeed, since the curvature term cancels with the z -component contribution of the divergence of the surface velocity in the convective-diffusion equation, when fitting of the surfaces and surfactant concentration is done to local paraboloids in a frame of reference with the z -axis parallel to the outward normal.

A curvatureless boundary-integral formulation and related numerical techniques are being extended for the presence of Marangoni stresses due to surfactant gradients. A difficulty is that the new form is more singular. We are developing an efficient desingularization for the integration surface. Initially, the linear equation of state for surfactant concentration is being employed, where surface viscosity effects can be neglected. Later, the algorithm will be adapted to higher concentrations using the models of Frumkin and Langmuir, which account for maximum packing and methods similar to Pozrikidis (1994) to handle surface viscosity. The surfactant transport equation will be solved by a semi-implicit scheme.

As a long-term goal, continuation of calculations after breakup will be pursued, beginning first with axisymmetric trajectories ( Davis , 1999) and then 3D interactions. For clean drops, the new code will be validated by comparing with convergent calculations we have already made (Fig. 3). We expect that at small surface Peclet numbers, the effect of surfactant will be mainly to lower the interfacial tension, effectively increasing the capillary number, until the drops have begun to stretch, when the increased surface area acts to dilute the surfactant. At higher surface Peclet numbers, there is an interplay between the convection of surfactant and dilution effects, combined with the Marangoni stresses, which oppose the flux of surfactant along the interface.

Figure 3. Stretching and breakup of the smaller drop (the larger one remains compact and is not shown) in buoyancy interaction using a curvatureless algorithm. Adapted from Zinchenko et al. (1999).

Population Dynamics for Spherical Drops in the Dilute Limit - When the probability of three-drop interactions is small, as in the case of dilute dispersions, population dynamics can be used to describe the evolution of the drop distribution. Population dynamics has been used often to model experiments on dispersions. Figure 4 shows results for a homogeneous dilute dispersion of poly(propylene glycol) (PPG) drops in poly(ethylene glycol) (PEG) at 90 degrees Celsius in shear flow. In Fig. 4a on the left, the volume-averaged diameter is shown as a function of time. The symbols are experimental data, and the solid lines are results using population dynamics. In Fig. 4b, the evolution of the drop-size distribution is shown for a shear rate g = 20 inverse seconds. The dotted-line histograms are from experiment, and the solid lines from theory. The effective volume fraction of the dispersed phase varies from 5 to 18%. In Fig. 4, the volume-averaged diameter increases rapidly while the drops remain spherical. Once small deformation becomes important, however, droplet growth is arrested by the deformation and slow film drainage, and the drop-size distribution becomes unimodal around the radius at which the collision efficiency decreases rapidly toward zero. The final average drop size can be controlled by proper choice of the shear rate.

Figure 4. a) Growth of volume-averaged diameter vs. dimensionless time for a PPG/PEG system at shear rates of 2, 5, 10, 20, and 50 inverse seconds from top to bottom, respectively. The symbols are from experiment and the solid lines from theory. b) Evolution of drop-size distribution for a shear rate of 20 inverse seconds. The figure is adapted from Burkhart et al. (2001).

To assess the effect of surfactants on dilute emulsions, we will incorporate the collision efficiency results for both spherical and slightly deformable drops. In this way, accurate predictions for the evolution of the drop-size distribution can be made and used as a basis for experiments.

Interactions of Two Deformable Drops - To study the effects of bulk-insoluble surfactant on the interactions of two deformable drops in buoyancy, we propose using glycerol/water drops in a matrix of castor oil. This system has been used to investigate binary deformable drop interactions with clean interfaces (Kushner et al. , 2001). By varying the water content in the drop phase, the drop-to-medium viscosity ratio can be changed from 0.001 to 1.5. The Plexiglas tank used in the experiments is 40 cm x 40 cm with a height of 120 cm to minimize wall effects.

The larger drop, which is introduced second, catches up to the leading smaller drop giving rise to several possible outcomes. The interactions depend on the Bond number, initial horizontal offset, drop-to-medium viscosity ratio, and size ratio. Upon coming into close contact, the drops may separate, the drops may coalesce through the larger drop coating the smaller drop or the smaller drop being captured by the larger one, either drop may break up, or they may interact in a more complicated way. At smaller viscosity ratios, coalescence through capture and coating is very important, while at viscosity ratios close to unity, breakup can overwhelm coalescence. One interesting phenomenon observed for drop with clean interfaces is “pass-through,” where the smaller drop may lose some volume through breakup at the tail, subsequently passes through a plume in the larger drop, moves behind the larger drop, and repeats the cycle several times before reaching the bottom of the tank (Fig. 5).

Figure 5. Initial cycle of pass-through where the larger drop moves by the smaller one, the tail of the smaller drop stretches and breaks in the extensional flow behind the larger one, and the head of the smaller drop passes through the larger one. Adapted from Kushner et al. (2001).

As surfactant, we will add a small amount of Triton-X 100. If the concentration is kept small, the surfactant can be modelled as bulk insoluble, since the diffusion flux from the bulk is slow compared to the surface convection flux (Pawar & Stebe, 1996). Experiments can also be performed at higher surfactant concentrations for comparison with theory as it develops. At low surface Peclet numbers, it is expected that the surfactant concentration will be nearly uniform, resulting in a lowering of the interfacial tension. However, once the drops deform significantly. the surfactant will be diluted by the increase in interfacial area, and the interfacial tension will increase. In this limit, the critical Bond number for breakup will increase by an almost predictable factor (Stone & Leal, 1990). In general, however, the effect of surfactant on the drop interactions is complicated and not predictable a priori (Stone and Leal, 1990; Pawar and Stebe, 1996).

Dilute and Concentrated Emulsions with Phase Separation – Wang & Davis (1996) conducted experiments on dilute dispersion of 1,2-propanediol drops in dibutyl sebacate in a cuvette 1 cm x 1cm with a height of 20 cm. Typical droplet sizes of 5 to 30 m m were obtained by vortex mixing, and the settling was videotaped. Drop distributions were obtained by sampling with a syringe, and the phase separation was measured with time from the videotape. Surfactant effects will be examined on the same system by using Triton-X 100 and other surface-active agents. More concentrated emulsions will be considered, and the effect of deformation on coalescence and phase separation will be investigated, as well. Other systems being considered include 10 drop/matrix combinations evaluated by Wang (1994), castor oil in silicon oil (Zhang et al. , 1993), fluorinert drops in glycerol or silicon oil (Subramanian & Balasubramaniam, 2001), and polymer systems with compatibilizer, such as those used by Hu et al. (2000) and Hudson et al. (2003) in shear flow.

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Research Description