Random-roughness hydrodynamic boundary conditions

Squirmer hydrodynamics near a periodic surface topography

The behaviour of microscopic swimmers has previously been explored near large-scale confining geometries and in the presence of very small-scale surface roughness. Here, we consider an intermediate case of how a simple microswimmer, the tangential spherical squirmer, behaves adjacent to singly and doubly periodic sinusoidal surface topographies that spatially oscillate with an amplitude that is an order of magnitude less than the swimmer size and wavelengths that are also within an order of magnitude of this scale. The nearest neighbour regularised Stokeslet method is used for numerical explorations after validating its accuracy for a spherical tangential squirmer that swims stably near a flat surface. The same squirmer is then introduced to different surface topographies. The key governing factor in the resulting swimming behaviour is the size of the squirmer relative to the surface topography wavelength. For instance, directional guidance is not observed when the squirmer is much larger, or much smaller, than the surface topography wavelength. In contrast, once the squirmer size is on the scale of the topography wavelength, limited guidance is possible, often with local capture in the topography troughs. However, complex dynamics can also emerge, especially when the initial configuration is not close to alignment along topography troughs or above topography crests. In contrast to sensitivity in alignment and topography wavelength, reductions in the amplitude of the surface topography or variations in the shape of the periodic surface topography do not have extensive impacts on the squirmer behaviour. Our findings more generally highlight that the numerical framework provides an essential basis to elucidate how swimmers may be guided by surface topography.

The effect of morphology and particle-wall interaction on colloidal near-wall dynamics

We investigated the near-wall Brownian dynamics of different types of colloidal particles with a typical size in the 100 nm range using evanescent wave dynamic light scattering (EWDLS). In detail we studied dilute suspensions of silica spheres and shells with a smooth surface and silica particles with controlled surface roughness. While the near wall dynamics of the particle with a smooth surface differ only slightly from the theoretical prediction for hard sphere colloids, the rough particles diffuse significantly slower. We analysed the experimental data by comparison with model calculations and suggest that the deviating dynamics of the rough particles are not due to increased hydrodynamic interaction with the wall. Rather, the particle roughness significantly changes their DLVO interaction with the wall, which in turn affects their diffusion.

Effect of a liquid flow on the forces between charged solid surfaces and the non-equilibrium electric double layer

The physical properties of fluids can change as they flow through confined charged solid areas, such as a charged pore or channel, allowing the transport of fluid through the channels to be controlled. The liquid flow is influenced by the electrical double layer (EDL) that is next to the charged surface. The overlap of the EDL of two nearby charged solid surfaces results in the formation of an electrostatic force. A flow will change the EDL from an equilibrium state to a non-equilibrium state, causing the forces to also change from an equilibrium (static) state to a non-equilibrium (dynamic) state. There are numerous studies that have been performed by molecular dynamics (MD) simulations and surface force experiments which concern the equilibrium EDL and the equilibrium surface forces. However, there are significantly less studies concerning the non-equilibrium EDL and non-equilibrium surface forces, including the effect of a liquid flow on the EDL and the surface forces. This review will focus on how a liquid flow changes the EDL and the surface forces of charged hydrophilic solid surfaces in aqueous electrolyte solutions. Results obtained by MD simulations and surface force experiments are discussed in this review. A flow was seen to be able to distort the EDL, causing the surface forces to change. The EDL and surface forces were affected by the surface charge, the structuring ability of the liquid molecules and ions near the surfaces, the ion type and their specificity towards the surface, the ionic concentration, and the rate of flow of the liquid. The physical properties of the system were shown to change with a flow, e.g. the increase in the fluid viscosity next to a charged solid surface that accompanies a flow. The number of counterions adsorbed to a charged solid surface was also seen to affect the direction of flow in an EDL. The surface forces were shown to change with a flow due to changes in hydrodynamic and electrostatic forces. Information on the effect of the liquid flow on the EDL and surface forces will help improve applications that require fluid to be transported in a defined way through a charged solid vessel.

Morphology of soft and rough contact via fluid drainage

The dynamic of contact formation between soft materials immersed in a fluid is accompanied by fluid drainage and elastic deformation. As a result, controlling the coupling between lubrication pressure and elasticity provides strategies to design materials with reversible and dynamic adhesion to wet or flooded surfaces. We characterize the elastic deformation of a soft coating with nanometer-scale roughness as it approaches and contacts a rigid surface in a fluid environment. The lubrication pressure during the approach causes elastic deformation and prevents contact formation. We observe deformation profiles that are drastically different from those observed for elastic half-space when the thickness of the soft coating is comparable to the hydrodynamic radius. In contrast, we show that surface roughness favors fluid drainage without altering the elastic deformation. As a result, the coupling between elasticity and slip (caused by surface roughness) can lead to trapped fluid pockets in the contact region.

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.

From Dot to Ring: The Role of Friction in the Deposition Pattern of a Drying Colloidal Suspension Droplet

The deposition of particles on a substrate by drying a colloidal suspension droplet is at the core of applications ranging from traditional printing on paper to printable electronics or photovoltaic devices. The self-pinning induced by the accumulation of particles at the contact line plays an important role in the formation of a deposit. In this article, we investigate, both numerically and theoretically, the effect of friction between the particles and the substrate on the deposition pattern. Without friction, the contact line shows a stick-slip behavior and a dotlike deposit is left after the droplet is evaporated. By increasing the friction force, we observe a transition from a dotlike to a ringlike deposit. We propose a theoretical model to predict the effective radius of the particle deposit as a function of the friction force. Our theoretical model predicts a critical friction force when self-pinning happens and the effective radius of deposit increases with increasing friction force, confirmed by our simulation results. Our results can find implications for developing active control strategies for the deposition of drying droplets.

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.

Rolling Spheres on Bioinspired Microstructured Surfaces

Microstructured surfaces, such as those inspired by nature, mediate surface interactions and are actively sought after to control wetting, adhesion, and friction. In particular, the rolling motion of spheres on microstructured surfaces in fluid environments is important for the transport of particles in microfluidic devices or in tribology. Here, we characterize the motion of smooth silicon nitride spheres (diameters 3-5 mm) as they roll down inclined planes decorated with hexagonal arrays of microwells and micropillars. For both types of patterned surfaces, we vary the area fraction of the micropatterned features from 0.04 to 0.96. We measure directly and independently the rotational and translational velocities of the spheres as they roll down planes with inclination angles that vary between 5 and 30°. For a given area fraction, we find that spheres have a higher translational and rotational velocity on surfaces with microwells than on micropillars. We rely on the model of Smart and Leighton [Phys. Fluids A 5, 13 (1993)] to obtain an effective gap width and coefficient of friction for all microstructured surfaces investigated. We find that the coefficient of friction is significantly higher for a surface with micropillars than that for one with microwells, likely due to the presence of interconnected drainage channels that provide additional paths for the fluid flow and favor solid-solid contact on the surface with micropillars. We find that while the effective gap width at a very low solid fraction is equal to the height of the patterned features, the effective separation decreases exponentially as the surface coverage of microstructures increases, with little measured differences between the two geometries. Superposition of resistance functions is used to relate the rapid decrease in the effective gap height with increase in the surface coverage observed in experiments.

Concurrent Modeling of Hydrodynamics and Interaction Forces Improves Particle Deposition Predictions

It is widely believed that media surface roughness enhances particle deposition-numerous, but inconsistent, examples of this effect have been reported. Here, a new mathematical framework describing the effects of hydrodynamics and interaction forces on particle deposition on rough spherical collectors in absence of an energy barrier was developed and validated. In addition to quantifying DLVO force, the model includes improved descriptions of flow field profiles and hydrodynamic retardation functions. This work demonstrates that hydrodynamic effects can significantly alter particle deposition relative to expectations when only the DLVO force is considered. Moreover, the combined effects of hydrodynamics and interaction forces on particle deposition on rough, spherical media are not additive, but synergistic. Notably, the developed model's particle deposition predictions are in closer agreement with experimental observations than those from current models, demonstrating the importance of inclusion of roughness impacts in particle deposition description/simulation. Consideration of hydrodynamic contributions to particle deposition may help to explain discrepancies between model-based expectations and experimental outcomes and improve descriptions of particle deposition during physicochemical filtration in systems with nonsmooth collector surfaces.

Scaling Hydrodynamic Boundary Conditions of Microstructured Surfaces in the Thin Channel Limit

Despite its importance in many applications and processes, a complete and unified view on how nano- and microscale asperities influence hydrodynamic interactions has yet to be reached. In particular, the effects of surface structure can be expected to become more dominant when the length scale of the asperities or textures becomes comparable to that of the fluid flow. Here we analyze the hydrodynamic drainage of a viscous silicone oil squeezed between a smooth plane and a surface decorated with hexagonal arrays of lyophilic microsized cylindrical posts. For all micropost arrays studied, the periodicity of the structures was much larger than the separation range of our measurements. In this thin channel geometry, we find the hydrodynamic drainage and separation forces for the micropost arrays cannot be fully described by existing boundary condition models for interfacial slip or a no-slip shifted plane. Instead, our results show that the influence of the microposts on the hydrodynamic drag exhibits three distinct regimes as a function of separation. For large separations, a no slip boundary condition (Reynolds theory) is observed for all surfaces until a critical (intermediate) separation, below which the position of the no-slip plane scales with surface separation until reaching a maximum, just before contact. Below this separation, a sharp decrease in the no-slip plane position then suggests that a boundary condition of a smooth surface is recovered at contact.

Non-linear, non-monotonic effect of nano-scale roughness on particle deposition in absence of an energy barrier: Experiments and modeling

Deposition of colloidal- and nano-scale particles on surfaces is critical to numerous natural and engineered environmental, health, and industrial applications ranging from drinking water treatment to semi-conductor manufacturing. Nano-scale surface roughness-induced hydrodynamic impacts on particle deposition were evaluated in the absence of an energy barrier to deposition in a parallel plate system. A non-linear, non-monotonic relationship between deposition surface roughness and particle deposition flux was observed and a critical roughness size associated with minimum deposition flux or “sag effect” was identified. This effect was more significant for nanoparticles (<1 μm) than for colloids and was numerically simulated using a Convective-Diffusion model and experimentally validated. Inclusion of flow field and hydrodynamic retardation effects explained particle deposition profiles better than when only the Derjaguin-Landau-Verwey-Overbeek (DLVO) force was considered. This work provides 1) a first comprehensive framework for describing the hydrodynamic impacts of nano-scale surface roughness on particle deposition by unifying hydrodynamic forces (using the most current approaches for describing flow field profiles and hydrodynamic retardation effects) with appropriately modified expressions for DLVO interaction energies, and gravity forces in one model and 2) a foundation for further describing the impacts of more complicated scales of deposition surface roughness on particle deposition.

Interfacial slip on rough, patterned and soft surfaces: a review of experiments and simulations

Advancements in the fabrication of microfluidic and nanofluidic devices and the study of liquids in confined geometries rely on understanding the boundary conditions for the flow of liquids at solid surfaces. Over the past ten years, a large number of research groups have turned to investigating flow boundary conditions, and the occurrence of interfacial slip has become increasingly well-accepted and understood. While the dependence of slip on surface wettability is fairly well understood, the effect of other surface modifications that affect surface roughness, structure and compliance, on interfacial slip is still under intense investigation. In this paper we review investigations published in the past ten years on boundary conditions for flow on complex surfaces, by which we mean rough and structured surfaces, surfaces decorated with chemical patterns, grafted with polymer layers, with adsorbed nanobubbles, and superhydrophobic surfaces. The review is divided in two interconnected parts, the first dedicated to physical experiments and the second to computational experiments on interfacial slip of simple (Newtonian) liquids on these complex surfaces. Our work is intended as an entry-level review for researchers moving into the field of interfacial slip, and as an indication of outstanding problems that need to be addressed for the field to reach full maturity.

Initiation of immersed granular avalanches

By means of coupled molecular dynamics-computational fluid dynamics simulations, we analyze the initiation of avalanches in a granular bed of spherical particles immersed in a viscous fluid and inclined above its angle of repose. In quantitative agreement with experiments, we find that the bed is unstable for a packing fraction below 0.59 but is stabilized above this packing fraction by negative excess pore pressure induced by the effect of dilatancy. From detailed numerical data, we explore the time evolution of shear strain, packing fraction, excess pore pressures, and granular microstructure in this creeplike pressure redistribution regime, and we show that they scale excellently with a characteristic time extracted from a model based on the balance of granular stresses in the presence of a negative excess pressure and its interplay with dilatancy. The cumulative shear strain at failure is found to be ≃ 0.2, in close agreement with the experiments, irrespective of the initial packing fraction and inclination angle. Remarkably, the avalanche is triggered when dilatancy vanishes instantly as a result of fluctuations while the average dilatancy is still positive (expanding bed) with a packing fraction that declines with the initial packing fraction. Another nontrivial feature of this creeplike regime is that, in contrast to dry granular materials, the internal friction angle of the bed at failure is independent of dilatancy but depends on the inclination angle, leading therefore to a nonlinear dependence of the excess pore pressure on the inclination angle. We show that this behavior may be described in terms of the contact network anisotropy, which increases with a nearly constant connectivity and levels off at a value (critical state) that increases with the inclination angle. These features suggest that the behavior of immersed granular materials is controlled not only directly by hydrodynamic forces acting on the particles but also by the influence of the fluid on the granular microstructure.

Effective slip-length tensor for a flow over weakly slipping stripes

We discuss the flow past a flat heterogeneous solid surface decorated by slipping stripes. The spatially varying slip length, b(y), is assumed to be small compared to the scale of the heterogeneities, L, but finite. For such weakly slipping surfaces, earlier analyses have predicted that the effective slip length is simply given by the surface-averaged slip length, which implies that the effective slip-length tensor becomes isotropic. Here we show that a different scenario is expected if the local slip length has steplike jumps at the edges of slipping heterogeneities. In this case, the next-to-leading term in an expansion of the effective slip-length tensor in powers of max[b(y)/L] becomes comparable to the leading-order term, but anisotropic, even at very small b(y)/L. This leads to an anisotropy of the effective slip and to its significant reduction compared to the surface-averaged value. The asymptotic formulas are tested by numerical solutions and are in agreement with results of dissipative particle dynamics simulations.

Effect of the charge and roughness of surfaces on normal and friction forces measured in aqueous solutions

We used the atomic force microscope (AFM) to determine how the roughness and charge on a surface affect the adhesion and friction when measured against a smooth surface (colloid probe) in an aqueous solution. The effect of roughness was investigated by coating TiO2 crystal substrates with TiO2 nano- or micro-sized particles, where an increase in the particle size increased the RMS roughness of the substrate. The charge of the substrate was varied by changing the pH of the aqueous solution. Force-separation curves and friction-load data were measured for the smooth colloid probe-rough substrate systems. The adhesion and friction between two surfaces in solution were seen to depend on the surface charge and roughness. A noncharged surface gave the greatest adhesion, while a charged surface gave weaker adhesions. Increasing the roughness of the surface resulted in a stronger adhesion. The magnitude and range of the adhesions were not affected by the measuring velocity in the case of a noncharged substrate but decreased with an increasing velocity for charged surfaces. The friction was seen not to depend on roughness in the case of a noncharged surface. However, in the case of a charged surface, the friction decreased with an increased roughness for low loads and then showed no dependence on the surface roughness for high loads. The results of this experiment show that the adhesion and friction of a system can be decreased via the roughness and charge of the substrate and the ion types in the solution.

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.

Combined effects of surface roughness and wetting characteristics on the moving contact line in microchannel flows

The present study investigates moving contact lines in microfluidic confinements with rough topographies modeled with random generating functions. Using matched asymptotic expansion, the description of the whole contact line is obtained and the dynamic contact angle is extracted by extrapolating the bulk meniscus to the channel wall. Significant variations are observed in the contact angle because of the heterogeneities of the confining walls of the microfluidic channel. The effects of the surface wetting condition also play a crucial role in altering the description of the contact line bearing particular nontrivial interactions with the topological features of the solid boundaries. In an effort to assess the underlying consequences, two different surface wetting conditions are studied; namely, complete wetting substrate and partial wetting substrate. Our studies reveal that the consequent wetting characteristics are strongly influenced by action of intermolecular forces in presence of surface roughness. The effect of slip, correlation length, and roughness parameters on the dynamic contact angle have been also investigated.

Calculation of effective slip on rough chemically heterogeneous surfaces using a homogenization approach

We present an expression for the effective slip length of a nanoscale rough chemically heterogeneous surface. A heterogeneous surface may be regarded as having an effective slip length generated by extrapolating the uniform velocity profile found in the far field. We consider two-dimensional steady-state Stokes flow over a surface that has periodic roughness and an intrinsic slip length varying over the same period. Using weak convergence methods for partial differential equations, we derive an expression for the effective slip length in terms of the intrinsic slip length and contact area of the surface. The result predicts that roughness causes a significant reduction in effective slip and that slip effects are dominated by the minimum intrinsic slip length of the system.

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.

Molecular transport and flow past hard and soft surfaces: computer simulation of model systems

The equilibrium and flow of polymer films and drops past a surface are characterized by the interface and surface tensions, viscosity, slip length and hydrodynamic boundary position. These parameters of the continuum description are extracted from molecular simulations of coarse-grained models. Hard, corrugated substrates are modelled by a Lennard-Jones solid while polymer brushes are studied as prototypes of soft, deformable surfaces. Four observations are discussed. (i) If the surface becomes strongly attractive or is coated with a brush, the Navier boundary condition fails to describe the effect of the surface independently of the strength and type of the flow. This failure stems from the formation of a boundary layer with an effective, higher viscosity. (ii) In the case of brush-coated surfaces, flow induces a cyclic, tumbling motion of the tethered chain molecules. Their collective motion gives rise to an inversion of the flow in the vicinity of the grafting surfaces and leads to strong, non-Gaussian fluctuations of the molecular orientations. The flow past a polymer brush cannot be described by Brinkman's equation. (iii) The hydrodynamic boundary condition is an important parameter for predicting the motion of polymer droplets on a surface under the influence of an external force. Their steady-state velocity is dictated by a balance between the power that is provided by the external force and the dissipation. If there is slippage at the liquid-solid interface, the friction at the solid-liquid interface and the viscous dissipation of the flow inside the drop will be the dominant dissipation mechanisms; dissipation at the three-phase contact line appears to be less important on a hard surface. (iv) On a soft, deformable substrate like a polymer brush, we observe a lifting-up of the three-phase contact line. Controlling the grafting density and the incompatibility between the brush and the polymer liquid we can independently tune the softness of the surface and the contact angle and thereby identify the parameters for maximizing the deformation at the three-phase contact.

Simulations of slip flow on nanobubble-laden surfaces

On microstructured hydrophobic surfaces, geometrical patterns may lead to the appearance of a superhydrophobic state, where gas bubbles at the surface can have a strong impact on the fluid flow along such surfaces. In particular, they can strongly influence a detected slip at the surface. We present two-phase lattice Boltzmann simulations of a flow over structured surfaces with attached gas bubbles and demonstrate how the detected slip depends on the pattern geometry, the bulk pressure, or the shear rate. Since a large slip leads to reduced friction, our results give assistance in the optimization of microchannel flows for large throughput.

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.