Transverse flow in thin superhydrophobic channels

Modeling of Shear Flows over Superhydrophobic Surfaces: From Newtonian to Non-Newtonian Fluids

The design and use of superhydrophobic surfaces have gained special attentions due to their superior performances and advantages in many flow systems, e.g., in achieving specific goals including drag reduction and flow/droplet handling and manipulation. In this work, we conduct a brief review of shear flows over superhydrophobic surfaces, covering the classic and recent studies/trends for both Newtonian and non-Newtonian fluids. The aim is to mainly review the relevant mathematical and numerical modeling approaches developed during the past 20 years. Considering the wide ranges of applications of superhydrophobic surfaces in Newtonian fluid flows, we attempt to show how the developed studies for the Newtonian shear flows over superhydrophobic surfaces have been evolved, through highlighting the major breakthroughs. Despite the fact that, in many practical applications, flows over superhydrophobic surfaces may show complex non-Newtonian rheology, interactions between the non-Newtonian rheology and superhydrophobicity have not yet been well understood. Therefore, in this Review, we also highlight emerging recent studies addressing the shear flows of shear-thinning and yield stress fluids in superhydrophobic channels. We focus on reviewing the models developed to handle the intricate interaction between the formed liquid/air interface on superhydrophobic surfaces and the overlying flow. Such an intricate interaction will be more complex when the overlying flow shows nonlinear non-Newtonian rheology. We conclude that, although our understanding on the Newtonian shear flows over superhydrophobic surfaces has been well expanded via analyzing various aspects of such flows, the non-Newtonian counterpart is in its early stages. This could be associated with either the early applications mainly concerning Newtonian fluids or new complexities added to an already complex problem by the nonlinear non-Newtonian rheology. Finally, we discuss the possible directions for development of models that can address complex non-Newtonian shear flows over superhydrophobic surfaces.

Analytical model for drag reduction on liquid-infused structured non-wetting surfaces

Liquid-infused structured non-wetting surfaces provide alternating no-slip and partial slip boundary conditions to the fluid flow, resulting in reduced friction at the interface. In this paper, an analytical model is developed for the evaluation of effective slip and, in turn, friction factor and drag reduction on liquid-infused structured non-wetting surfaces. By considering the entire range of anisotropy and heterogeneity of the surface structures as well as the full range of partial slip offered by the infusion liquid, the present model eliminates empirical fitting or correlations that are inherent in previous studies. Based on the effective slip length, drag reduction and skin friction coefficient values for Newtonian flow between two infinite parallel plates and flow in round tubes are presented. Extension of Moody charts for non-wetting surfaces and design maps of surface meso/micro/nano texturing for achieving desired drag reduction are presented for a broad range of engineering applications. The article further presents independent validation of the model across experimental and computational data from the literature and brings together several previous studies in a unified manner.

Advective superdiffusion in superhydrophobic microchannels

We consider pressure-driven flows in wide microchannels, and discuss how a transverse shear, generated by misaligned superhydrophobic walls, impacts cross-sectional spreading of Brownian particles. We show that such a transverse shear can induce an advective superdiffusion, which strongly enhances dispersion of particles compared to a normal diffusion, and that maximal cross-sectional spreading corresponds to a crossover between its subballistic and superballistic regimes. This allows us to argue that an advective superdiffusion can be used for boosting dispersion of particles at smaller Péclet numbers compared to known concepts of passive microfluidic mixing. This implies that our superdiffusion scenario allows one efficient mixing of much smaller particles or using much thinner microchannels than methods, which are currently being exploited.

Probing effective slippage on superhydrophobic stripes by atomic force microscopy

While the effective slippage of water past superhydrophobic surfaces has been studied over a decade, theoretical predictions have never been properly confirmed by experiments. Here we measure a drag force on a sphere approaching a plane decorated by superhydrophobic grooves and compare the results with the predictions of semi-analytical theory developed here, which employs the gas cushion model to calculate the local slip length at the gas sectors. We demonstrate that at intermediate and large (compared to a texture period) separations the half-sum of longitudinal and transverse effective slip lengths can be deduced from the force-distance curve by using the known analytical theory of hydrodynamic interaction of a sphere with a homogeneous slipping plane. This half-sum is shown to depend on the fraction of gas sectors and its value is in excellent agreement with theoretical predictions. At small distances the half-sum of effective longitudinal and transverse slip lengths becomes separation-dependent, and is in quantitative agreement with the predictions of our semi-analytical theory.

Principles of transverse flow fractionation of microparticles in superhydrophobic channels

We propose a concept of fractionation of micron-sized particles in a microfluidic device with a bottom wall decorated by superhydrophobic stripes. The stripes are oriented at an angle α to the direction of a driving force, G, which generally includes an applied pressure gradient and gravity. Separation relies on the initial sedimentation of particles under gravity in the main forward flow, and their subsequent lateral deflection near a superhydrophobic wall due to generation of a secondary flow transverse to G. We provide some theoretical arguments allowing us to quantify the transverse displacement of particles in the microfluidic channel, and confirm the validity of theoretical predictions in test experiments with monodisperse fractions of microparticles. Our results can guide the design of superhydrophobic microfluidic devices for efficient sorting of microparticles with a relatively small difference in size and density.

Flows and mixing in channels with misaligned superhydrophobic walls

Aligned superhydrophobic surfaces with the same texture orientation reduce drag in the channel and generate secondary flows transverse to the direction of the applied pressure gradient. Here we show that a transverse shear can be easily generated by using superhydrophobic channels with misaligned textured surfaces. We propose a general theoretical approach to quantify this transverse flow by introducing the concept of an effective shear tensor. To illustrate its use, we present approximate theoretical solutions and Dissipative Particle Dynamics simulations for striped superhydrophobic channels. Our results demonstrate that the transverse shear leads to complex flow patterns, which provide a new mechanism of a passive vertical mixing at the scale of a texture period. Depending on the value of Reynolds number two different scenarios occur. At relatively low Reynolds number the flow represents a transverse shear superimposed with two corotating vortices. For larger Reynolds number these vortices become isolated, by suppressing fluid transport in the transverse direction.

Electrokinetics over charge-modulated surfaces in the presence of patterned wettability: role of the anisotropic streaming potential

In the present study, we focus on evaluating the induced streaming electric field along the orthogonal directions in a narrow fluidic confinement in the presence of patterned surface wettability and modulated surface charges. We attempt to assess the implications of such modulations on the related important quantities and pinpoint the regimes of improved induced streaming potential field and the resulting anisotropy in the induced potential. Our results reveal that for certain combinations of the parameters characterizing the modulated slip, a significant amount of augmentation in the streaming electric field might be obtained, whereas in other cases the effects may lead to adverse consequences. We further demonstrate that the presence of anisotropic modulations on the channel walls give rise to considerable off-diagonal effects, which makes the streaming potential "disoriented" with the applied pressure gradient, when the same is not applied along one of the orthogonal directions. Our analysis also shows that one can remove such "mis-orientations" by finely tuning several relevant flow and geometric parameters, which may bear immense scientific and technological consequences towards an improved design of miniaturized energy conversion devices.

Flow past superhydrophobic surfaces with cosine variation in local slip length

Anisotropic superhydrophobic surfaces have the potential to greatly reduce drag and enhance mixing phenomena in microfluidic devices. Recent work has focused mostly on cases of superhydrophobic stripes. Here, we analyze a relevant situation of cosine variation of the local slip length. We derive approximate formulas for maximal (longitudinal) and minimal (transverse) directional effective slip lengths that are in good agreement with the exact numerical solution and lattice-Boltzmann simulations. Compared to the case of superhydrophobic stripes, the cosine texture can provide a very large effective slip. However, the difference between eigenvalues of the slip-length tensor is smaller, indicating that the flow is less anisotropic.

Effective hydrodynamic boundary conditions for microtextured surfaces

Understanding the influence of topographic heterogeneities on liquid flows has become an important issue with the development of microfluidic systems, and more generally for the manipulation of liquids at the small scale. Most studies of the boundary flow past such surfaces have concerned poorly wetting liquids for which the topography acts to generate superhydrophobic slip. Here we focus on topographically patterned but chemically homogeneous surfaces, and measure a drag force on a sphere approaching a plane decorated with lyophilic microscopic grooves. A significant decrease in the force compared with predicted even for a superhydrophobic surface is observed. To quantify the force we use the effective no-slip boundary condition, which is applied at the imaginary smooth homogeneous isotropic surface located at an intermediate position between the top and bottom of grooves. We relate its location to a surface topology by a simple, but accurate analytical formula. Since grooves represent the most anisotropic surface, our conclusions are valid for any texture, and suggest rules for the rational design of topographically patterned surfaces to generate desired drag.

Effects of surface morphology and anisotropy on the tangential-momentum accommodation coefficient between Pt(100) and Ar

In this paper, we study the influence of platinum (100) surface morphology on the tangential-momentum accommodation coefficient with argon using a molecular dynamics method. The coefficient is computed directly by beaming Ar atoms onto the surfaces and measuring the relative momentum changes. The wall is maintained at a constant temperature and its interaction with the gas atoms is governed by the Kulginov potential. To capture correctly the surface effect of the walls and the atoms' trajectories, the quantum Sutton-Chen multibody potential is employed between the Pt atoms. The effects of wall surface morphology, incident direction, and temperature are considered in this work and provide full information on the gas-wall interaction.

Evolution with time of hydrophobicity and microrelief of a cation-exchange membrane surface and its impact on overlimiting mass transfer

Surface properties were measured together with electrochemical characteristics of a CMX (Neosepta, Tokuyama Corp.) cation-exchange membrane. Relative hydrophobicity was controlled by the contact angle; XPS and SEM were used for characterizing chemical composition and microrelief of the surface, respectively. Voltammetry, chronopotentiometry, and mass transfer rate measurements were made as well. A "fresh" membrane and samples after 10, 25, 100, and 150 h of operation in an electrodialysis cell at an overlimiting current equal to 3 theoretical limiting currents, in a 0.02 M NaCl solution, were characterized. Some electrochemical properties were also measured for a Neosepta cation-exchange membrane, aged 2 years, in an industrial food process. It was found that the hydrophobicity of the CMX membrane has increased after the first 10 h of operation; more and more cavities of the dimension of the order of 1 μm have appeared with time testifying electrochemical erosion of the surface. The limiting current density (i(lim)) and the overlimiting transfer rate through the CMX membrane increased with time of its operation under overlimiting current. In the case of new CMX, i(lim) was very close to the theoretical value i(lim)(theor) calculated by the Lévêque equation. After 10 h of operation, i(lim) increased by 5%, and after 25, 100, and 150 h, the increase was by 30%, 70%, and 100%, respectively. Similarly, the mass transfer rate was found to increase up to 5 times (when desalting 0.005 M NaCl under 3 V) in comparison with the theoretical value. The ensemble of data was explained by the hypothesis that the passage of intensive current produces erosion of the ion-exchange polymer forming a continuous phase in CMX. This erosion results in exposure at the surface of the other constituent of CMX: small (about 100 nm) particles of relatively hydrophobic polyvinylchloride. Increasing surface hydrophobicity facilitates the slip of electroconvective vortexes along the surface. Besides, the geometry of the cavities gives rise to appearing tangential electric force applied to the extended space charge density at cavity's walls. As the local limiting current density within a cavity is lower than at the flat surface, electroconvective vortices arise at current densities lower than i(lim)(theor). With time, the number and the size of cavities increase (apparently, due to paired electroconvective vortices occurring inside them) that seems the main reason for overlimiting transfer increase.

Tensorial slip of superhydrophobic channels

We describe a generalization of the tensorial slip boundary condition, originally justified for a thick (compared to texture period) channel, to any channel thickness. The eigenvalues of the effective slip-length tensor, however, in general case become dependent on the gap and cannot be viewed as a local property of the surface, being a global characteristic of the channel. To illustrate the use of the tensor formalism we develop a semianalytical theory of an effective slip in a parallel-plate channel with one superhydrophobic striped and one hydrophilic surface. Our approach is valid for any local slip at the gas sectors and an arbitrary distance between the plates, ranging from a thick to a thin channel. We then present results of lattice Boltzmann simulations to validate the analysis. Our results may be useful for extracting effective slip tensors from global measurements, such as the permeability of a channel, in experiments or simulations.

Drag force on a sphere moving toward an anisotropic superhydrophobic plane

We analyze theoretically a high-speed drainage of liquid films squeezed between a hydrophilic sphere and a textured superhydrophobic plane that contains trapped gas bubbles. A superhydrophobic wall is characterized by parameters L (texture characteristic length), b1 and b2 (local slip lengths at solid and gas areas), and φ1 and φ2 (fractions of solid and gas areas). Hydrodynamic properties of the plane are fully expressed in terms of the effective slip-length tensor with eigenvalues that depend on texture parameters and H (local separation). The effect of effective slip is predicted to decrease the force as compared with what is expected for two hydrophilic surfaces and described by the Taylor equation. The presence of additional length scales, L, b1, and b2, implies that a film drainage can be much richer than in the case of a sphere moving toward a hydrophilic plane. For a large (compared to L) gap the reduction of the force is small, and for all textures the force is similar to expected when a sphere is moving toward a smooth hydrophilic plane that is shifted down from the superhydrophobic wall. The value of this shift is equal to the average of the eigenvalues of the slip-length tensor. By analyzing striped superhydrophobic surfaces, we then compute the correction to the Taylor equation for an arbitrary gap. We show that at a thinner gap the force reduction becomes more pronounced, and that it depends strongly on the fraction of the gas area and local slip lengths. For small separations we derive an exact equation, which relates a correction for effective slip to texture parameters. Our analysis provides a framework for interpreting recent force measurements in the presence of a superhydrophobic surface.

Wetting, roughness and flow boundary conditions

We discuss how the wettability and roughness of a solid impacts its hydrodynamic properties. We see in particular that hydrophobic slippage can be dramatically affected by the presence of roughness. Owing to the development of refined methods for setting very well controlled micro- or nanotextures on a solid, these effects are being exploited to induce novel hydrodynamic properties, such as giant interfacial slip, superfluidity, mixing and low hydrodynamic drag, that could not be achieved without roughness.