Evidence for the existence of an effective interfacial tension between miscible fluids. 2. Dodecyl acrylate-poly(dodecyl acrylate) in a spinning drop tensiometer

Drop Behavior on Heterogeneous Ratchet-Structured Substrates Harmonically Vibrated in Lateral Direction

We analyze numerically a new ratchet system: a liquid drop is sitting on a heterogeneous ratchet-structured solid plate. The coated plate is subject to a lateral harmonic oscillation. The systematic investigation performed in the frame of a phase field model shows the possibility of realizing a long-distance net-driven motion for isolated domains of the forcing parameters. The studied problem might be of considerable interest for controlled motion in micro- and nanofluidics.

Trade Off between Hydrodynamic and Thermodynamic Forces at the Liquid-Liquid Interface

Viscous fingering (VF) instability has been investigated in the case of a partially miscible binary system by nonlinear numerical simulations. Partially miscible fluid systems offer the possibility of phase separation coupled with VF instability. The thermodynamics of such systems are governed by the Margules parameter (interaction parameter) as well as the fluid concentrations. Kinetics of the decomposition is also influenced by dynamical parameters such as the viscosity of the fluid, which incidentally also affects the hydrodynamic forces. Here, we explore the effects of concentration and Margules parameter in order to ascertain the trade-offs incurred between hydrodynamic and thermodynamic effects at the interface as well as the thermodynamics of the bulk. Based on the Gibb's free energy versus concentration curve, we select concentrations (i) outside spinodal and binodal regions, (ii) within binodal but outside the spinodal, and (iii) within the spinodal curve. We solve the modified Cahn-Hilliard-Hele-Shaw equation employing the COMSOL Multiphysics software. Applying high-resolution numerical simulations, we show a strong dependence of the thermodynamic forces on the concentration of the mixtures. Rapid phase separation and hence a faster rate of droplet formation have been found when the concentration lies inside the spinodal region. Further, we have investigated the correlation between the fractal dimension and dynamics of the system. The spatiotemporal studies presented in this work clearly illustrate the competition between hydrodynamic and thermodynamic forces and provide insights on the kinetics of decomposition and growth of interfacial instabilities.

Durably Self-Sustained Droplet on a Fully Miscible Liquid Film

Droplets impacting onto a solid or liquid surface inducing wetting, floatation, splash, coalescence, etc. is ubiquitous in nature and industrial processes. Here, we report that liquid droplets exhibit spherical caps upon contact with a fully miscible liquid film of lower surface tension, despite the spontaneous mixing of the two liquids. Such a spherical cap on a continuous liquid surface sustains a long lifespan up to minutes before ultimately merging into the film. Benefiting from large viscous forces in a thin film as a result of spatial confinement, the surface flow is substantially suppressed. Therefore, the surface tension gradient responsible for this phenomenon is maintained because the normal diffusion of film liquid into the droplet can timely dilute film liquid supplied by uphill Marangoni flow at the droplet surface. The present finding removes the conventional cognition that droplet coalescence is prompt on fully miscible continuous liquid surfaces, thus benefiting design of new types of microfluidic devices.

Anomalous patterns of Saffman-Taylor fingering instability during a metastable phase separation

Phase separation is important in biology, biochemistry, industry, and other areas and is divided into two types: a spinodal decomposition type and a nucleation and growth type. The spinodal decomposition type phase separation occurs under the thermodynamically unstable conditions, and the nucleation and growth type phase separation occurs under thermodynamically metastable conditions. On the other hand, when a less viscous fluid displaces a more viscous one in porous media, the interface of the two fluids becomes hydrodynamically unstable and forms a finger-like pattern. The coupling of the hydrodynamic instability with the thermodynamic instability has been studied. It is reported that the hydrodynamic instability under thermodynamically unstable conditions, where spinodal decomposition type phase separation occurs, creates multiple moving droplets with a radius of 3-4 mm because of the spontaneous convection induced by the Korteweg force, which is driven by a compositional gradient during phase separation. However, the hydrodynamic instability under metastable conditions, where the phase separation of nucleation and growth type occurs, is still unrevealed. In this study, we applied fingering instability (hydrodynamic instability) under the metastable conditions, where the patterns are changed from fingering or droplets to anomalous patterns such as tip-widening or needle-like (top-pointed) fingering patterns when the initial concentration is metastable, which is considered near a binodal curve. These patterns are ubiquitous in nature, similar to dendrite crystals (snowflakes) or our body's cells. Thus, the patterns created can be controlled through hydrodynamic conditions such as the injection flow and thermodynamic conditions such as spinodal decomposition (thermodynamically unstable conditions) and metastable conditions.

Nonequilibrium Capillary Pressure of a Miscible Meniscus

We examine the dynamics of a miscible displacement in a capillary, calculating the nonequilibrium capillary pressure of a moving (and slowly diffusing) miscible meniscus. During the displacement, the capillary pressure varies with time following stretching and smearing of a miscible interface. The capillary pressure remains different from zero for a long time (on a diffusion time scale), slowing the displacement. This capillary pressure is however completely ignored by all theories currently available for practical modeling of miscible displacements in capillaries and porous matrices.

Spinning elastic beads: a route for simultaneous measurements of the shear modulus and the interfacial energy of soft materials

Large deformations of soft elastic beads spinning at high angular velocity in a denser background fluid are investigated theoretically, numerically, and experimentally using millimeter-size polyacrylamide hydrogel particles introduced in a spinning drop tensiometer. We determine the equilibrium shapes of the beads from the competition between the centrifugal force and the restoring elastic and surface forces. Considering the beads as neo-Hookean up to large deformations, we show that their elastic modulus and interfacial energy constant can be simultaneously deduced from their equilibrium shape. Also, our results provide further support to the scenario in which interfacial energy and interfacial tension coincide for amorphous polymer gels.

Spinning Drop Dynamics in Miscible and Immiscible Environments

We report on the extensional dynamics of spinning drops in miscible and immiscible background fluids following a rotational speed jump. Two radically different behaviors are observed. Drops in immiscible environments relax exponentially to their equilibrium shape, with a relaxation time that does not depend on the centrifugal force. We find an excellent quantitative agreement with the relaxation time predicted for quasi-spherical drops by Stone and Bush (Q. Appl. Math. 1996, 54, 551), while other models proposed in the literature fail to capture our data. By contrast, drops immersed in a miscible background fluid do not relax to a steady shape: they elongate indefinitely, their length following a power-law l(t)∼t2/5 in very good agreement with the dynamics predicted by Lister and Stone (J. Fluid Mech. 1996, 317, 275) for inviscid drops. Our results strongly suggest that low compositional gradients in miscible fluids do not give rise to an effective interfacial tension measurable by spinning drop tensiometry.

Drop Behavior Influenced by the Correlation Length on Noisy Surfaces

We investigate numerically the role of the correlation length in drop behavior on noisy surfaces. To this aim, a phase field tool has been used. Theoretical results are confirmed by experiments of distilled water drops sitting on stainless steel and silicon surfaces textured by laser-induced periodic self-organized structures: an increase of the noise amplitude results in an amplification of the original behavior (i.e., hydrophobic is getting more hydrophobic, hydrophilic is getting more hydrophilic). Furthermore, computer simulations in two and three spatial dimensions allow for predictions of drop behavior on noisy sloped substrates under a gravitational force, a problem of large interest in controlled motion in micro- and nanofluidics.

Evolution and Disappearance of Solvent Drops on Miscible Polymer Subphases

Traditionally, an interface is defined as a boundary between immiscible phases. However, previous work has shown that even when two fluids are completely miscible, they maintain a detectable "effective interface" for long times. Miscible interfaces have been studied in various systems of two fluids with a single boundary between them. However, this work has not extended to the three-phase system of a fluid droplet placed on top of a miscible pool. We show that these three-phase systems obey the same wetting conditions as immiscible systems, and that their drop shapes obey the Augmented Young-Laplace Equation. Over time, the miscible interface diffuses and the shape of the drop evolves. We place 2-microliter drops of water atop miscible poly(acrylamide) solutions. The drop is completely wetted by the subphase, and then remains detectable beneath the surface for many minutes. An initial effective interfacial tension can be approximated to be on the order of 0.5 mN/m using the capillary number. Water and poly(acrylamide) are completely miscible in all concentrations, and yet, when viewed from the side, the drop maintains a capillary shape. Study of this behavior is important to the understanding of effective interfaces between miscible polymer phases, which are pervasive in nature.

Hydrodynamic instabilities in miscible fluids

Hydrodynamic instabilities in miscible fluids are ubiquitous, from natural phenomena up to geological scales, to industrial and technological applications, where they represent the only way to control and promote mixing at low Reynolds numbers, well below the transition from laminar to turbulent flow. As for immiscible fluids, the onset of hydrodynamic instabilities in miscible fluids is directly related to the physics of their interfaces. The focus of this review is therefore on the general mechanisms driving the growth of disturbances at the boundary between miscible fluids, under a variety of forcing conditions. In the absence of a regularizing mechanism, these disturbances would grow indefinitely. For immiscible fluids, interfacial tension provides such a regularizing mechanism, because of the energy cost associated to the creation of new interface by a growing disturbance. For miscible fluids, however, the very existence of interfacial stresses that mimic an effective surface tension is debated. Other mechanisms, however, may also be relevant, such as viscous dissipation. We shall review the stabilizing mechanisms that control the most common hydrodynamic instabilities, highlighting those cases for which the lack of an effective interfacial tension poses deep conceptual problems in the mathematical formulation of a linear stability analysis. Finally, we provide a short overview on the ongoing research on the effective, out of equilibrium interfacial tension between miscible fluids.

Effect of pseudo-gravitational acceleration on the dissolution rate of miscible drops

The effect of pseudo-gravitational acceleration on the dissolution process of two phase miscible systems has been investigated at high acceleration values using a spinning drop tensiometer with three systems: 1-butanol/water, isobutyric acid/water, and triethylamine/water. We concluded that the dissolution process involves at least three different transport phenomena: diffusion, barodiffusion, and gravitational (buoyancy-driven) convection. The last two phenomena are significantly affected by the centrifugal acceleration acting at the interface between the two fluids, and the coupling with the geometry of the dissolving drop leads to a change of the mass flux during the course of the dissolution process.

Off-equilibrium surface tension in miscible fluids

The interfacial tension between immiscible fluids is responsible for a wealth of every-day phenomena, from the spherical shape of small drops and bubbles to the ability to walk on water of many insects. More than a century ago, physicist and mathematician D. Korteweg postulated the existence of an effective interface tension for miscible fluids, whenever a composition gradient exists, as encountered, e.g., in many flow geometries. In this mini-review, we discuss experimental work performed in the last decades that demonstrates the existence of a positive effective interface tension in a variety of systems, from molecular, near-critical liquids to complex fluids such as polymer solutions and colloidal suspensions. The various experimental strategies that have been deployed are discussed, together with their advantages and limitations. Finally, some of the key theoretical questions still open are outlined.

Self-Propelled Vesicles Induced by the Mixing of Two Polymeric Aqueous Solutions through a Vesicle Membrane Far from Equilibrium

This study describes the development of self-propelled vesicles using transient interfacial energy in an aqueous two-phase system composed of polyethylene glycol (PEG), dextran (DEX), and water. The transient interfacial energy was generated at the mixing boundary between the PEG and DEX solutions when the two miscible liquids were in contact with each other far from equilibrium. Vesicles encapsulating 20 wt % DEX solution traveled spontaneously when the PEG concentration in the environmental media was >15 wt %. The motility of the vesicles varied with the permeability of the vesicle membrane. The permeability increased significantly when the concentration of PEG was >15 wt %. PEG had a profound effect not only on mass transfer through the membrane but also on the motility of the vesicles.

Bulk and interfacial stresses in suspensions of soft and hard colloids

We explore the influence of particle softness and internal structure on both the bulk and interfacial rheological properties of colloidal suspensions. We probe bulk stresses by conventional rheology, by measuring the flow curves, shear stress versus strain rate, for suspensions of soft, deformable microgel particles and suspensions of near hard-sphere-like silica particles. A similar behaviour is seen for both kinds of particles in suspensions at concentrations up to the random close packing volume fraction, in agreement with recent theoretical predictions for sub-micron colloids. Transient interfacial stresses are measured by analyzing the patterns formed by the interface between the suspensions and their solvent, due to a generalized Saffman-Taylor hydrodynamic instability. At odds with the bulk behaviour, we find that microgels and hard particle suspensions exhibit vastly different interfacial stress properties. We propose that this surprising behaviour results mainly from the difference in particle internal structure (polymeric network for microgels versus compact solid for the silica particles), rather than softness alone.

Effect of Péclet number on miscible rectilinear displacement in a Hele-Shaw cell

The influence of fluid dispersion on the Saffman-Taylor instability in miscible fluids has been investigated in both the linear and the nonlinear regimes. The convective characteristic scales are used for the dimensionless formulation that incorporates the Péclet number (Pe) into the governing equations as a measure for the fluid dispersion. A linear stability analysis (LSA) is performed in a similarity transformation domain using the quasi-steady-state approximation. LSA results confirm that a flow with a large Pe has a higher growth rate than a flow with a small Pe. The critical Péclet number (Pec) for the onset of instability for all possible wave numbers and also a power-law relation of the onset time and most unstable wave number with Pe are observed. Unlike the radial source flow, Pec is found to vary with t0. A Fourier spectral method is used for direct numerical simulations (DNS) of the fully nonlinear system. The power-law dependence of the onset of instability ton on Pe is obtained from the DNS and found to be inversely proportional to Pe and it is in good agreement with that obtained from the LSA. The influence of the anisotropic dispersion is analyzed in both the linear and the nonlinear regimes. The results obtained confirm that for a weak transverse dispersion merging happens slowly and hence the small wave perturbations become unstable. We also observ that the onset of instability sets in early when the transverse dispersion is weak and varies with the anisotropic dispersion coefficient, ε, as ∼√[ε], in compliance with the LSA. The combined effect of the Korteweg stress and Pe in the linear regime is pursued. It is observed that, depending on various flow parameters, a fluid system with a larger Pe exhibits a lower instantaneous growth rate than a system with a smaller Pe, which is contrary to the results when such stresses are absent.

Can vibrations control drop motion?

We discuss a mechanism for controlled motion of drops with applications for microfluidics and microgravity. The mechanism is the following: a solid plate supporting a liquid droplet is simultaneously subject to lateral and vertical harmonic oscillations. In this way the symmetry of the back-and-forth droplet movement along the substrate under inertial effects is broken and thus will induce a net driven motion of the drop. We study the dependency of the traveled distance on the oscillation parameters (forcing amplitude, frequency, and phase shift between the two perpendicular oscillations) via phase field simulations. The internal flow structure inside the droplet is also investigated. We make predictions on resonance frequencies for drops on a substrate with a varying wettability.

Off-equilibrium surface tension in colloidal suspensions

We study the fingering instability of the interface between two miscible fluids, a colloidal suspension and its own solvent. The temporal evolution of the interface in a Hele-Shaw cell is found to be governed by the competition between the nonlinear viscosity of the suspension and an off-equilibrium, effective surface tension Γe. By studying suspensions in a wide range of volume fractions, ΦC, we show that Γe∼ΦC2, in agreement with Korteweg's theory for miscible fluids. The surface tension exhibits an anomalous increase with particle size, which we account for using entropy arguments.

All-aqueous multiphase microfluidics

Immiscible aqueous phases, formed by dissolving incompatible solutes in water, have been used in green chemical synthesis, molecular extraction and mimicking of cellular cytoplasm. Recently, a microfluidic approach has been introduced to generate all-aqueous emulsions and jets based on these immiscible aqueous phases; due to their biocompatibility, these all-aqueous structures have shown great promises as templates for fabricating biomaterials. The physico-chemical nature of interfaces between two immiscible aqueous phases leads to unique interfacial properties, such as an ultra-low interfacial tension. Strategies to manipulate components and direct their assembly at these interfaces needs to be explored. In this paper, we review progress on the topic over the past few years, with a focus on the fabrication and stabilization of all-aqueous structures in a multiphase microfluidic platform. We also discuss future efforts needed from the perspectives of fluidic physics, materials engineering, and biology for fulfilling potential applications ranging from materials fabrication to biomedical engineering.

Partial coalescence of sessile drops with different miscible liquids

Using computer simulations in three spatial dimensions, we examine the interaction between two deformable drops consisting of two perfectly miscible liquids sitting on a solid substrate under a given contact angle. Driven by capillarity and assisted by Marangoni effects at the droplet interfaces, several distinct coalescence regimes are achieved after the droplets' collision.

On the coalescence of sessile drops with miscible liquids

Sessile drops sitting on highly wettable solid substrates fuse in qualitatively different ways after contact, depending on the surface tension gradients between the mixing droplets. In early time evolution the drop coalescence can be fast or delayed (intermittent). In long time evolution a secondary drop formation can occur. We study numerically droplet dynamics during coalescence in two and three spatial dimensions, within a phase field approach. We discuss criteria to distinguish different coalescence regimes. A comparison with recent experiments will be done.

Delayed coalescence of droplets with miscible liquids: Lubrication and phase field theories

Mixing of droplets with a body of different liquids shows an interesting behavior for small contact angles at solid substrate. The droplets interact with each other, a liquid exchange appears between the approaching drops owing to surface tension gradients at the droplets interface. But the drops remain separated for some seconds (up to minutes), until the merging into a single drop occurs (Langmuir 24, 6395 (2008)). We investigate this phenomenon using lubrication approximation and phase field approach. For both methods, 2D quantitative computer simulations for delayed fusion of perfectly miscible thin liquid films/droplets with low contact angles are reported.

Boussinesq approximation of the Cahn-Hilliard-Navier-Stokes equations

We use the Cahn-Hilliard approach to model the slow dissolution dynamics of binary mixtures. An important peculiarity of the Cahn-Hilliard-Navier-Stokes equations is the necessity to use the full continuity equation even for a binary mixture of two incompressible liquids due to dependence of mixture density on concentration. The quasicompressibility of the governing equations brings a short time-scale (quasiacoustic) process that may not affect the slow dynamics but may significantly complicate the numerical treatment. Using the multiple-scale method we separate the physical processes occurring on different time scales and, ultimately, derive the equations with the filtered-out quasiacoustics. The derived equations represent the Boussinesq approximation of the Cahn-Hilliard-Navier-Stokes equations. This approximation can be further employed as a universal theoretical model for an analysis of slow thermodynamic and hydrodynamic evolution of the multiphase systems with strongly evolving and diffusing interfacial boundaries, i.e., for the processes involving dissolution/nucleation, evaporation/condensation, solidification/melting, polymerization, etc.

Transient surface tension in miscible liquids

Evidence of the existence of a transient surface tension between two miscible fluid phases is given. This is done by making use of a density matched free of gravity perturbations, binary liquid of isobutyric acid and water, which presents a miscibility gap and is studied by light scattering. The experiment is performed very near the critical point of the binary liquid, where the diffusion of phases is extremely slow. The surface tension is deduced from the evolution of the structure factor obtained from low angle light scattering. The latter evolution is successfully analyzed in terms of a local equilibrium diffusive approach that makes explicit how the surface tension decreases with time.

Different behaviors of delayed fusion between drops with miscible liquids

Fusion of sessile droplets with body of different liquids is delayed when the approaching drops are sitting on a highly wettable solid substrate. Owing to the surface tension gradients between the mixing drops, a Marangoni driven flow through the connecting channel appears. Experiments of delayed coalescence were recently reported in [Langmuir 24, 6395 (2008)10.1021/la800630w] for millimeters sized drops. For droplets of submillimeter dimensions, capillary forces dominate. The control of interfacial energies becomes an important strategy for manipulating tiny droplets along the solid surfaces. In this paper we present phase field simulations in two spatial dimensions for microdroplets of perfectly miscible liquids. For drops with a given geometry, systematic investigations were performed for different fluid viscosities. Different behaviors are observed from chasing droplets to droplets repelling.

Radial viscous fingering in miscible Hele-Shaw flows: a numerical study

A modified version of the usual viscous fingering problem in a radial Hele-Shaw cell with immiscible fluids is studied by intensive numerical simulations. We consider the situation in which the fluids involved are miscible, so that the diffusing interface separating them can be driven unstable through the injection or suction of the inner fluid. The system is allowed to rotate in such a way that centrifugal and Coriolis forces come into play, imposing important changes on the morphology of the arising patterns. In order to bridge from miscible to immiscible pattern forming structures, we add the surface tensionlike effects due to Korteweg stresses. Our numerical experiments reveal a variety of interesting fingering behaviors, which depend on the interplay between injection (or suction), diffusive, rotational, and Korteweg stress effects. Whenever possible the features of the simulated miscible fronts are contrasted to existing experiments and other theoretical or numerical studies, usually resulting in close agreements. A number of additional complex morphologies, whose experimental realization is still not available, are predicted and discussed.