Transport of a passive scalar in wide channels with surface topography: An asymptotic theory.

We generalize classical dispersion theory for a passive scalar to derive an asymptotic long-time convection-diffusion equation for a solute suspended in a wide, structured channel and subject to a steady low-Reynolds-number shear flow. Our asymptotic theory relies on a domain perturbation approach for small roughness amplitudes of the channel and holds for general surface shapes expandable as a Fourier series. We determine an anisotropic dispersion tensor, which depends on the characteristic wavelengths and amplitude of the surface structure. For surfaces whose corrugations are tilted with respect to the applied flow direction, we find that dispersion along the principal direction (i.e. the principal eigenvector of the dispersion tensor) is at an angle to the main flow direction and becomes enhanced relative to classical Taylor dispersion. In contrast, dispersion perpendicular to it can decrease compared to the short-time diffusivity of the particles. Furthermore, for an arbitrary surface shape represented in terms of a Fourier decomposition, we find that each Fourier mode contributes at leading order a linearly-independent correction to the classical Taylor dispersion diffusion tensor.

From dynamic self-organization to avalanching instabilities in soft-granular threads.

We study the dynamics of threads of monodisperse droplets, including droplet chains and multi-chains, in which the droplets are interconnected by capillary bridges of another immiscible liquid phase. This system represents wet soft-granular matter - a class of granular materials in which the grains are soft and wetted by thin fluid films-with other examples including wet granular hydrogels or foams. In contrast to wet granular matter with rigid grains (e.g., wet sand), studied previously, the deformability of the grains raises the number of available metastable states and facilitates rearrangements which allow for reorganization and self-assembly of the system under external drive, e.g., applied via viscous forces. We use a co-flow configuration to generate a variety of unique low-dimensional regular granular patterns, intermediate between 1D and 2D, ranging from linear chains and chains with periodically occurring folds to multi-chains and segmented structures including chains of finite length. In particular, we observe that the partially folded chains self-organize via limit cycle of displacements and rearrangements occurring at a frequency self-adapted to the rate of build-up of compressive strain in the chain induced by the viscous forces. Upon weakening of the capillary arrest of the droplets, we observe spontaneous fluidization of the quasi-solid structures and avalanches of rearrangements. We identify two types of fluidization-induced instabilities and rationalize them in terms of a competition between advection and propagation. While we use aqueous droplets as the grains we demonstrate that the reported mechanisms of adaptive self-assembly apply to other types of soft granular systems including foams and microgels. We discuss possible application of the reported quasi-1D compartmentalized structures in tissue engineering, bioprinting and materials science.

Rapid Spreading of a Droplet on a Thin Soap Film.

We study the spreading of a droplet of surfactant solution on a thin suspended soap film as a function of dynamic surface tension and volume of the droplet. Radial growth of the leading edge (R) shows power-law dependence on time with exponents ranging roughly from 0.1 to 1 for different surface tension differences (Δσ) between the film and the droplet. When the surface tension of the droplet is lower than the surface tension of the film (Δσ > 0), we observe rapid spreading of the droplet with R ≈ tα, where α (0.4 < α < 1) is highly dependent on Δσ. Balance arguments assuming the spreading process is driven by Marangoni stresses versus inertial stresses yield α = 2/3. When the surface tension difference does not favor spreading (Δσ < 0), spreading still occurs but is slow with 0.1 < α < 0.2. This phenomenon could be used for stretching droplets in 2D and modifying thin suspended films.

Dewetting of Thin Liquid Films Surrounding Air Bubbles in Microchannels.

As an air bubble translates in a microchannel, a thin film of liquid is formed on the bounding walls. In a microchannel with a rectangular cross-section, the liquid in the film leaks toward the low-pressure corners of the geometry, which leads to the appearance of local minima in the film thickness in the cross-sectional plane. In such a configuration, theory suggests that the minimum film thickness scales with Ca and Ca4/3 depending on the distance from the nose of the bubble, where Ca = µUb/γ is the flow capillary number based on the bubble velocity Ub, liquid viscosity µ, and surface tension γ, and Ca ≪ 1. We show that the film of a partially wetting liquid dewets on the channel wall at the sites of the local minima in the film thickness as it acquires thicknesses around and below 100 nm. Our experiments show that the distance Lw between the nose of the bubble and the initial dewetting location is a function of Ca and surface wettability. For channels of different wettability, Lw always scales proportional to Caα, where 1.7 < α < 2 for the range of 10-5 < Ca < 10-2. Moreover, Lw increases up to 10 times by enhancing the wettability of the surface at a given Ca. Our present measurements of Lw provide a design constraint on the lengths of bubbles to maintain a liquid wet channel without dry patches on the wall.

Failure mechanisms of air entrainment in drop impact on lubricated surfaces.

Lubricated surfaces have recently been introduced and studied due to their potential benefit in various configurations and applications. Combining the techniques of total internal reflection microscopy and reflection interference microscopy, we examine the dynamics of an underlying air film upon drop impact on a lubricated substrate where the thin liquid film is immiscible to the drop. In contrast to drop impact on solid surfaces where even the smallest asperities cause random breakup of the entraining air film, we report two air film failure mechanisms on lubricated surfaces. In particular, using ≈5 µm thick liquid films of high viscosity, which should make the substrate nearly atomically smooth, we show that air film rupture shifts from asperity-driven to a controlled event. At low Weber numbers (We < 2, We = ρlU02R/σ, U0 the impact velocity, R the drop radius, and ρl the density and σ the surface tension of the droplet) the droplet bounces. At intermediate We (2 < We < 10), the air film fails at the center as the top surface of the drop crashes downward owing to impact-induced capillary waves; the resulting liquid-liquid contact time is found to be independent of We. In contrast, at high We (We > 10), the air film failure occurs much earlier in time at the first inflection point of the air film shape away from the drop center, where the liquid-liquid van der Waals interactions become important. The predictable failure modes of the air film upon drop impact sheds light on droplet deposition in applications such as lubricant-infused self-cleaning surfaces.

Dripping and jetting in microfluidic multiphase flows applied to particle and fiber synthesis.

Dripping and jetting regimes in microfluidic multiphase flows have been investigated extensively, and this review summarizes the main observations and physical understandings in this field to date for three common device geometries: coaxial, flow-focusing and T-junction. The format of the presentation allows for simple and direct comparison of the different conditions for drop and jet formation, as well as the relative ease and utility of forming either drops or jets among the three geometries. The emphasis is on the use of drops and jets as templates for microparticle and microfiber syntheses, and a description is given of the more common methods of solidification and strategies for achieving complex multicomponent microparticles and microfibers.

Michaelis-Menten kinetics in shear flow: Similarity solutions for multi-step reactions.

Models for chemical reaction kinetics typically assume well-mixed conditions, in which chemical compositions change in time but are uniform in space. In contrast, many biological and microfluidic systems of interest involve non-uniform flows where gradients in flow velocity dynamically alter the effective reaction volume. Here, we present a theoretical framework for characterizing multi-step reactions that occur when an enzyme or enzymatic substrate is released from a flat solid surface into a linear shear flow. Similarity solutions are developed for situations where the reactions are sufficiently slow compared to a convective time scale, allowing a regular perturbation approach to be employed. For the specific case of Michaelis-Menten reactions, we establish that the transversally averaged concentration of product scales with the distance x downstream as x(5/3). We generalize the analysis to n-step reactions, and we discuss the implications for designing new microfluidic kinetic assays to probe the effect of flow on biochemical processes.

Wetting of flexible fibre arrays.

Fibrous media are functional and versatile materials, as demonstrated by their ubiquity both in natural systems such as feathers and adhesive pads and in engineered systems from nanotextured surfaces to textile products, where they offer benefits in filtration, insulation, wetting and colouring. The elasticity and high aspect ratios of the fibres allow deformation under capillary forces, which cause mechanical damage, matting self-assembly or colour changes, with many industrial and ecological consequences. Attempts to understand these systems have mostly focused on the wetting of rigid fibres or on elastocapillary effects in planar geometries and on a fibre brush withdrawn from an infinite bath. Here we consider the frequently encountered case of a liquid drop deposited on a flexible fibre array and show that flexibility, fibre geometry and drop volume are the crucial parameters that are necessary to understand the various observations referred to above. We identify the conditions required for a drop to remain compact with minimal spreading or to cause a pair of elastic fibres to coalesce. We find that there is a critical volume of liquid, and, hence, a critical drop size, above which this coalescence does not occur. We also identify a drop size that maximizes liquid capture. For both wetting and deformation of the substrates, we present rules that are deduced from the geometric and material properties of the fibres and the volume of the drop. These ideas are applicable to a wide range of fibrous materials, as we illustrate with examples for feathers, beetle tarsi, sprays and microfabricated systems.

Steric effects of ions in the charge-related wetting phenomena.

Steric effects of ions on the charge-related wetting phenomena are studied. Along with a general treatment, three specific problems in two-dimensional system are considered: a droplet on an electrode, a droplet on a charged surface, and an electrowetting phenomenon on a dielectric. For computation of wetting tension, the electromechanical approach is adopted with the principle of mechanical force balance for each phase. The modified Poisson-Boltzmann equation, which was originally proposed by Bikerman [Philos. Mag. 33, 384 (1942)], is adopted for the analysis of the steric effects. It is found that the steric hindrance reduces significantly both the osmotic pressure and the electrical stress near the triple contact line. This reduction results in a considerable decrease in the wetting tension when the ratio of the capacitance per unit area of the electrical double layer to that of the dielectric layer is small.

Non-coalescence of oppositely charged drops.

Electric fields induce motion in many fluid systems, including polymer melts, surfactant micelles and colloidal suspensions. Likewise, electric fields can be used to move liquid drops. Electrically induced droplet motion manifests itself in processes as diverse as storm cloud formation, commercial ink-jet printing, petroleum and vegetable oil dehydration, electrospray ionization for use in mass spectrometry, electrowetting and lab-on-a-chip manipulations. An important issue in practical applications is the tendency for adjacent drops to coalesce, and oppositely charged drops have long been assumed to experience an attractive force that favours their coalescence. Here we report the existence of a critical field strength above which oppositely charged drops do not coalesce. We observe that appropriately positioned and oppositely charged drops migrate towards one another in an applied electric field; but whereas the drops coalesce as expected at low field strengths, they are repelled from one another after contact at higher field strengths. Qualitatively, the drops appear to 'bounce' off one another. We directly image the transient formation of a meniscus bridge between the bouncing drops, and propose that this temporary bridge is unstable with respect to capillary pressure when it forms in an electric field exceeding a critical strength. The observation of oppositely charged drops bouncing rather than coalescing in strong electric fields should affect our understanding of any process involving charged liquid drops, including de-emulsification, electrospray ionization and atmospheric conduction.

Dynamics of wetting: from inertial spreading to viscous imbibition.

We report the influence of the nature of boundaries on the dynamics of wetting. We review some work recently published and highlight new experimental observations. Our paper begins with the spreading of drops on substrates and demonstrates how the exponents of the spreading laws are affected either by the surface chemistry or by the droplet shape. We then discuss the imbibition of completely and partially wetting fluids into channels and over microtextured surfaces. Starting with the one-dimensional imbibition of completely wetting liquids in tubes and surface textures, we show that (i) shape variations of channels change the power-law response of the imbibition and (ii) the geometrical parameters of a surface roughness change the spreading behavior. For partially wetting fluids, we observe directionally dependent spreading: polygonal wetted domains can be obtained. We conclude with a tabular summary of our findings, allowing us to draw connections between the different systems investigated, and shed light on open questions that remain to be addressed.

Bridging kinematics and concentration content in a chaotic micromixer.

We analyze the mixing properties of the microfluidic herringbone configuration introduced to mix scalar substances in a narrow channel at low Reynolds but large Péclet numbers. Because of the grooves sculpted on the channel floor, substantial transverse motions are superimposed onto the usual longitudinal Poiseuille dispersion along the channel, whose impact on both the mixing rate and mixture content is quantified. We demonstrate the direct link between the flow kinematics and the deformation rate of the mixture's concentration distribution, whose overall shape is also determined.

In-fiber semiconductor filament arrays.

We report a novel physical phenomenon in which a cylindrical shell undergoing a scaling process evolves into an ordered array of filaments upon reaching a characteristic thickness. We observe that the tendency to breakup is related to the material viscosity in a manner reminiscent of capillary instability. However, unlike the classical breakup of a fluid cylinder into droplets, the structural evolution in our system occurs exclusively in the cross sectional plane while uniformity is maintained in the axial direction. We propose a fluid front instability mechanism to account for the observed phenomena. The fleeting evolution of fluid breakup from a thin film to a filament array is captured in the frozen state by a thermal drawing process which results in extended lengths of solid sub-100 nm filaments encapsulated within a polymer fiber. Hundreds of glassy semiconductor filament arrays are precisely oriented within a polymer fiber matrix making electrical connections trivial. This approach offers unique opportunities for fabrication of nanometer scale devices of unprecedented lengths allowing simplified access and connectivity.

Influence of substrate conductivity on circulation reversal in evaporating drops.

Nonuniform evaporation from sessile droplets induces radial convection within the drop, which produces the well-known "coffee-ring" effect. The evaporation also induces a gradient in temperature and consequently a gradient in surface tension, generating a Marangoni flow. Here we investigate theoretically and experimentally the thermal Marangoni flow and establish criteria to gauge its influence. An asymptotic analysis indicates that the direction of the flow depends on the relative thermal conductivities of the substrate and liquid, k_{R} identical withk_{S}/k_{L}, reversing direction at a critical contact angle over the range 1.45

Coalescence of spreading droplets on a wettable substrate.

We investigate experimentally and theoretically the coalescence dynamics of two spreading droplets on a highly wettable substrate. Upon contact, surface tension drives a rapid motion perpendicular to the line of centers that joins the drops and lowers the total surface area. We find that the width of the growing meniscus bridge between the two droplets exhibits power-law behavior, growing at early times as t1/2. Moreover, the growth rate is highly sensitive to both the radii and heights of the droplets at contact, scaling as ho3/2/Ro. This size dependence differs significantly from the behavior of freely suspended droplets, in which the coalescence growth rate depends only weakly on the droplet size. We demonstrate that the scaling behavior is consistent with a model in which the growth of the meniscus bridge is governed by the viscously hindered flux from the droplets.

Monodisperse double emulsions generated from a microcapillary device.

Double emulsions are highly structured fluids consisting of emulsion drops that contain smaller droplets inside. Although double emulsions are potentially of commercial value, traditional fabrication by means of two emulsification steps leads to very ill-controlled structuring. Using a microcapillary device, we fabricated double emulsions that contained a single internal droplet in a core-shell geometry. We show that the droplet size can be quantitatively predicted from the flow profiles of the fluids. The double emulsions were used to generate encapsulation structures by manipulating the properties of the fluid that makes up the shell. The high degree of control afforded by this method and the completely separate fluid streams make this a flexible and promising technique.

Foam drainage on the microscale II. Imaging flow through single Plateau borders.

The liquid in foam forms an interconnected network, which is composed of Plateau borders, nodes, and films. One of the dominant pathways for foam drainage is flow through Plateau borders, and we use confocal microscopy to obtain experimental results for the flow fields inside individual Plateau borders. For three types of surfactants detailed comparisons are made with a model based upon the influence of surface viscosity at free boundaries between the gas in the bubbles and the liquid in the Plateau borders. The model describes the flows well, and we find good agreement between the surface viscosity predicted by this model and representative values found in the literature. We also give a qualitative description of the flow in the nodes.

Foam drainage on the microscale I. Modeling flow through single Plateau borders.

The drainage of liquid through a foam involves flow in channels, also called Plateau borders, which generally are long and slender. We model this flow by assuming the flow is unidirectional, the shear is transverse to the flow direction, and the liquid/gas interfaces are mobile and characterized by a Newtonian surface viscosity, which does not depend on the shear rate. Numerical finite difference simulations are performed, and analytical approximations for the velocity fields inside the channels and the films that separate the bubbles are given. We compare the liquid flow rates through interior channels, exterior channels (i.e., channels contacting container walls) and films. We find that when the number of exterior channels is comparable to the number of interior channels, i.e., narrow container geometries, the exterior channels can significantly affect the dynamics of the drainage process. Even for highly mobile interfaces, the films do not significantly contribute to the drainage process, unless the amount of liquid in the films is within a factor of ten of the amount of liquid in the channels.

Geometrically mediated breakup of drops in microfluidic devices.

Microfluidic technology offers capabilities for the precise handling of small fluid volumes dispersed as droplets. To fully exploit this potential requires simultaneous generation of multiple size droplets. We demonstrate two methods for passively breaking larger drops into precisely controlled daughter drops using pressure-driven flow in simple microfluidic configurations: (i) a T junction and (ii) flow past isolated obstacles. We quantify conditions for breakup at a T junction and illustrate sequential breakup at T junctions for making small drops at high dispersed phase volume fractions.