Gas cushion model and hydrodynamic boundary conditions for superhydrophobic textures

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.

Surface Tension and Viscosity Dependence of Slip Length over Irregularly Structured Superhydrophobic Surfaces

A comprehensive understanding of the slip phenomenon on liquid/solid interfaces is essential for multiple real-world applications of superhydrophobic materials, especially those involving drag reduction. In the current contribution, the so-called "slip-length" on an irregularly structured superhydrophobic surface was systematically evaluated, with respect to varying liquid surface tension and viscosity. The superhydrophobic polymer-nanoparticle composite (SPNC) material used exhibits a dual-scale surface roughness and was fabricated via coating a surface with a mixture of polydimethylsiloxane solution and functionalized silica particles. A cone-and-plate rheometric device was employed to quantify the slip length. To independently study the impact of surface tension and viscosity, three types of aqueous solutions were used: sodium dodecyl sulfate, ethanol, and polyethylene glycol. Our experimental results demonstrate that a decreasing surface tension results in a decreasing slip length when the fluid viscosity is held constant. Meanwhile, the slip length is shown to increase with increasing viscosity when the surface tension of the various liquids is matched to isolate effects. The study reveals a linear relationship between slip length and both capillary length and viscosity providing a reference to potentially predict the degree of achievable drag reduction for differing fluids on SPNC surfaces.

A review on slip boundary conditions at the nanoscale: recent development and applications

The slip boundary condition for nanoflows is a key component of nanohydrodynamics theory, and can play a significant role in the design and fabrication of nanofluidic devices. In this review, focused on the slip boundary conditions for nanoconfined liquid flows, we firstly summarize some basic concepts about slip length including its definition and categories. Then, the effects of different interfacial properties on slip length are analyzed. On strong hydrophilic surfaces, a negative slip length exists and varies with the external driving force. In addition, depending on whether there is a true slip length, the amplitude of surface roughness has different influences on the effective slip length. The composition of surface textures, including isotropic and anisotropic textures, can also affect the effective slip length. Finally, potential applications of nanofluidics with a tunable slip length are discussed and future directions related to slip boundary conditions for nanoscale flow systems are addressed.

Influence of Surface Texture on the Variation of Electrokinetic Streaming Potentials

The electrokinetic streaming potential (V_s) obtained through electrolyte flow in a microchannel is shown to be related to the underlying surface pattern. Pillar, mesh, and groove patterns were studied for comparing the relative magnitudes of the V_s with air-/liquid-filled surfaces. A record value of the related figure of merit, in terms of the developed V_s per-unit applied pressure, of ∼0.127 mV/Pa, was observed in a mesh texture liquid-filled surface (LFS) impregnated with an electrolyte-immiscible oil. The study indicated that increasing the solid fraction of the pattern surface decreases the effective slip length while enhancing the overall channel ζ potential. Consequently, maximizing the obtained V_s implies a balancing of the slip with the surface potential, with plausibly more significance of the latter. The work has implications for higher-efficiency electrical voltage sources.

Possibility of Obtaining Two Orders of Magnitude Larger Electrokinetic Streaming Potentials, through Liquid Infiltrated Surfaces

It is shown that the magnitude of the streaming potential (V_s) can be significantly enhanced from ∼0.02 V to as much as ∼1.6 V, in electrokinetic flows through microchannels. This was done through flows on liquid-filled surfaces, where the grooves were filled with oils of viscosity in the range 30-3000 mPa·s. The presence of immiscible oils and the improved slip are both factors that could significantly increase the V_s. The analytical relationship between streaming potential and filled liquid viscosity was derived and verified through corresponding experimental results. The work yields novel insights into complex electrolyte flows and indicates avenues for more efficient energy harvesting.

Flow Structure and Mixing Efficiency of Viscous Fluids in Microchannel with a Striped Superhydrophobic Wall

The peculiarities of a Newtonian fluid flow structure in microchannels with a striped superhydrophobic lower wall texture are studied by means of numerical modeling. In the Cassie-Baxter state, an oblique orientation of such a texture induces helicoidal streamlines with micro spirals. Such a flow structure favors the enhancement of fluid mixing efficiency, which can be quantified using the total root-mean-square deviation of streamlines from the microchannel axis. This characteristic was shown to be a nonmonotonic function of the striped texture tilt angle and to depend strongly on microchannel thickness. The mechanisms of micro and macro helicoidal flow structure formation are investigated, and the mixing quality of miscible fluids is estimated for various Peclet numbers and texture tilt angles. It was found that the striped superhydrophobic wall leads to a notable enhancement in the microchannel mixing efficiency at sufficiently large Peclet numbers.

Modulation of the Streaming Potential and Slip Characteristics in Electrolyte Flow over Liquid-Filled Surfaces

A significant enhancement in the streaming potential ( V_s) was obtained in experiments considering the flow of electrolyte over liquid-filled surfaces (LFSs), where the grooves in patterned substrates are filled with electrolyte immiscible oils. Such LFSs yield larger V_s (by a factor of 1.5) compared to superhydrophobic surfaces, with air-filled grooves, and offer tunability of electrokinetic flow. It is shown that the density, viscosity, conductivity, as well as the dielectric constant of the filling oil, in the LFS, determine V_s. Relating a hydrodynamic slip length to the obtained V_s offers insight into flow characteristics, as modulated by the liquid interfaces in the LFS.

Ultrafast water harvesting and transport in hierarchical microchannels

Various natural materials have hierarchical microscale and nanoscale structures that allow for directional water transport. Here we report an ultrafast water transport process in the surface of a Sarracenia trichome, whose transport velocity is about three orders of magnitude faster than those measured in cactus spine and spider silk. The high velocity of water transport is attributed to the unique hierarchical microchannel organization of the trichome. Two types of ribs with different height regularly distribute around the trichome cone, where two neighbouring high ribs form a large channel that contains 1-5 low ribs that define smaller base channels. This results in two successive but distinct modes of water transport. Initially, a rapid thin film of water is formed inside the base channels (Mode I), which is followed by ultrafast water sliding on top of that thin film (Mode II). This two-step ultrafast water transport mechanism is modelled and experimentally tested in bio-inspired microchannels, which demonstrates the potential of this hierarchal design for microfluidic applications.

Extending the Lotus Effect: Repairing Superhydrophobic Surfaces after Contamination or Damage by CHic Chemistry

Superhydrophobic surfaces have gained a reputation to show a self-cleaning behavior ("Lotus effect") as drops rolling off the surface take along loosely adhering dust particles. However, this self-cleaning process reaches its limits when such surfaces are brought in contact with sticky contaminants such as oils and smaller particles. Once intimate contact is established between the surface and a small particle, it will be almost impossible to remove it because of strong surface interactions. Such contaminations, however, lead to contact line pinning and destroy the superhydrophobic effect. Because the fragility of the micro- and nanostructures prohibits any mechanical cleaning, the sample is usually doomed. Here, we report a universal method for restoring superhydrophobicity: by simple dip-coating, a conformal ultrathin layer (≈10 nm) of a highly hydrophobic and photoreactive fluoropolymer is deposited. Through short UV irradiation (5 min), this thin layer is cross-linked and chemically attached to the underlying surface by C,H-insertion cross-linking, thus covering the contaminant like a thin veil. We use this "cover-up" strategy of masking the contaminants to restore superhydrophobicity. We demonstrate this principle by deliberately soiling the surface with various model contaminants, such as oily substances and particles, and study the repair process.

Wall slip for complex liquids - Phenomenon and its causes

In this review, we tried to qualify different types and mechanisms of wall slip phenomenon, paying particular attention to the most recent publications and issues. The review covers all type of fluids - homogeneous low molecular weight liquids, polymer solution, multi-component dispersed media, and polymer melts. We focused on two basic concepts - fluid-solid wall interaction and shear-induced fluid-to-solid transitions - which are the dominant mechanisms of wall slip. In the first part of the review, the theoretical and numerical studies of correlation of wetting properties and wall slip of low molecular weight liquids and polymeric fluids are reviewed along with some basic experimental results. The influence of nanobubbles and microcavities on the effectiveness of wall slip is illuminated with regard to the bubble dynamics, as well as their stability at smooth and rough interfaces, including superhydrophobic surfaces. Flow of multi-component matter (microgel pastes, concentrated suspensions of solid particles, compressed emulsions, and colloidal systems) is accompanied by wall slip in two cases. The first one is typical of viscoplastic media which can exist in two different physical states, as solid-like below the yield point and liquid-like at the applied stresses exceeding this threshold. Slip takes place at low stresses. The second case is related to the transition from fluid to solid states at high deformation rates or large deformations caused by the strain-induced glass transition of concentrated dispersions. In the latter case, the wall effects consist of apparent slip due to the formation of a low viscous thin layer of fluid at the wall. The liquid-to-solid transition is also a dominant mechanism in wall slip of polymer melts because liquid polymers are elastic fluids which can be in two relaxation states depending on the strain rate. The realization of these mechanisms is determined by polymer melt interaction with the solid wall.

"Nickel Nanoflowers" with Surface-Attached Fluoropolymer Networks by C,H Insertion for the Generation of Metallic Superhydrophobic Surfaces

Metallic superhydrophobic surfaces (SHSs) combine the attractive properties of metals, such as ductility, hardness, and conductivity, with the favorable wetting properties of nanostructured surfaces. Moreover, they promise additional benefits with respect to corrosion protection. For the modification of the intrinsically polar and hydrophilic surfaces of metals, a new method has been developed to deposit a long-term stable, highly hydrophobic coating, using nanostructured Ni surfaces as an example. Such substrates were chosen because the deposition of a thin Ni layer is a common choice for enhancing corrosion resistance of other metals. As the hydrophobic coating, we propose a thin film of an extremely hydrophobic fluoropolymer network. To form this network, a thin layer of a fluoropolymer precursor is deposited on the Ni substrate which includes a comonomer that is capable of C,H insertion cross-linking (CHic). Upon UV irradiation or heating, the cross-linker units become activated and the thin glassy film of the precursor is transformed into a polymer network that coats the surface conformally and permanently, as shown by extensive extraction experiments. To achieve an even higher stability, the same precursor film can also be transformed into a chemically surface-attached network by depositing a self-assembled monolayer of an alkane phosphonic acid on the Ni before coating with the precursor. During cross-linking, by the same chemical process, the growing polymer network will simultaneously attach to the alkane phosphonic acid layer at the surface of the metal. This strategy has been used to turn fractal Ni "nanoflower" surfaces grown by anisotropic electroplating into SHSs. The wetting characteristics of the obtained nanostructured metallic surfaces are studied. Additionally, the corrosion protection effect and the significant mechanical durability are demonstrated.

Apparent slip of shear thinning fluid in a microchannel with a superhydrophobic wall

The peculiarities of simple shear flow of shear thinning fluids over a superhydrophobic wall consisting of a set of parallel gas-filled grooves and solid stripes (domains with slip and stick boundary conditions) are studied numerically. The Carreau-Yasuda model is used to provide further insight into the problem of the slip behavior of non-Newtonian fluids having a decreasing viscosity with a shear rate increase. This feature is demonstrated to cause a nonlinear velocity profile leading to the apparent slip. The corresponding transverse and longitudinal apparent slip lengths of a striped texture are found to be noticeably larger than the respective effective slip lengths of Newtonian liquids in microchannels of various thicknesses and surface fractions of the slip domains. The viscosity distribution of the shear thinning fluid over the superhydrophobic wall is carefully investigated to describe the mechanism of the apparent slip. Nonmonotonic behavior of the apparent slip length as a function of the applied shear rate is revealed. This important property of shear thinning fluids is considered to be sensitive to the steepness of the viscosity flow curve, thus providing a way to decrease considerably the flow resistance in microchannels.

Molting Materials: Restoring Superhydrophobicity after Severe Damage via Snakeskin-like Shedding

The nanostructures that are required to generate superhydrophobic surfaces are always sensitive to shear and are easily damaged, especially by scratching with sharp objects. As a result of this destruction, the water repellency will be lost. We introduce a novel approach to restoring the original surface properties after mechanical damage. In this approach, the damaged layer is shed like the skin of a snake. This is demonstrated with a three-layer stack as a proof-of-principle system: when the original, superhydrophobic surface layer is damaged, this leads to the dissolution of a sacrificial layer below it. Thus, the damaged layer is shed, a new unscathed surface is uncovered, and superhydrophobicity can easily be restored after a short washing.

Effective slip for flow in a rotating channel bounded by stick-slip walls

This paper aims to look into how system rotation may modify the role played by boundary slip in controlling flow through a rotating channel bounded by stick-slip walls. A semianalytical model is developed for pressure-driven flow in a slit channel that rotates about an axis perpendicular to its walls, which are superhydrophobic surfaces patterned with periodic alternating no-shear and no-slip stripes. The cases where the flow is driven by a pressure gradient parallel or normal to the stripes are considered. The effects of the no-shear area fraction on the velocities and effective slip lengths for the primary and secondary flows are investigated as functions of the rotation rate and the channel height. It is mathematically proved that the secondary flow rate is exactly the same in the two cases, irrespective of whether the primary flow is parallel or normal to the wall stripes. For any rotation speed, there is an optimal value of the no-shear area fraction at which the primary flow rate is maximum. This is a consequence of two competing effects: the no-shear part of the wall may serve to reduce the wall resistance, thereby enhancing the flow especially at low rotation, but it also weakens the formation of the near-wall Ekman layer, which is responsible for pumping the flow especially at high rotation. Wall slip in a rotating environment is to affect flow in the Ekman layer, but not flow in the geostrophic core.

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.

Electrohydrodynamics near hydrophobic surfaces

We show that an electro-osmotic flow near the slippery hydrophobic surface depends strongly on the mobility of surface charges, which are balanced by counterions of the electrostatic diffuse layer. For a hydrophobic surface with immobile charges, the fluid transport is considerably amplified by the existence of a hydrodynamic slippage. In contrast, near the hydrophobic surface with mobile adsorbed charges, it is also controlled by an additional electric force, which increases the shear stress at the slipping interface. To account for this, we formulate electrohydrodynamic boundary conditions at the slipping interface, which should be applied to quantify electro-osmotic flows instead of hydrodynamic boundary conditions. Our theoretical predictions are fully supported by dissipative particle dynamics simulations with explicit charges. These results lead to a new interpretation of zeta potential of hydrophobic surfaces.

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.