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# Shape evolution of drops on surfaces of different wettability gradients

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## Abstract

Passive droplet manipulation on open surfaces can be achieved by creating a wettability gradient on surfaces, which is essential in the fabrication of low cost biological and biochemical chips. We performed 3D numerical simulations to analyze the droplet motion on a broad range of wettability gradient surfaces. We found that the droplet shape evolves with time to maintain a minimum energy state, and the surface energy of the droplet is identical at a particular non-dimensional time (t*) for different wettability gradient surfaces. Although the droplet is at various locations at a fixed t* , the shape of the droplet is found to be identical. The physics behind this interesting phenomenon of identical droplet shape formation is explored. A co-relation for t* is proposed to get the dependency of t* on various geometrical parameters and fluid properties. Three distinct regimes of the droplet identical shape on different wettability gradient surfaces are shown using a regime plot. Along with the identical droplet shape phenomena, the detailed understanding of the dynamics of the droplet shape evolution on different wettability gradient surfaces gives an insight for better open surface passive manipulation.

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... number [15]. Thus, we used an automated time-stepping procedure for solving the equations. ...
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Droplet wetting on solid surfaces is a ubiquitous phenomenon in nature and applications. The wetting behavior of droplets on homogeneous surfaces has been accurately elucidated by the quintessential Young's law. However, on heterogeneous substrates, due to the energy barriers and contact line pinning effect, more than one equilibrated droplet pattern exists, which is more close to reality. Here, we propose a concise mathematical-physical model to delineate the droplet patterns on chemically patterned surfaces: stripe, “chocolate,” and “chessboard.” The present concept is capable of predicting the number as well as the morphologies of the equilibrated droplets on chemically patterned surfaces. We anticipate that the current work can be applied to fabricate programmable surfaces involving droplet manipulation in integrated circuits, biochips, and smart microelectronics.
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Droplets can be used as carrier vehicles for the transportation of biological and chemical reagents. Manipulation of water- and oil- based ferromagnetic droplets in presence of magnetic field have been well-studied. Here, we elucidate the transport of a sessile aqueous (diamagnetic) droplet placed over spikes of oil-based ferrofluid (FF) in presence of a non-uniform magnetic field. An oil-based FF droplet, dispensed over a rigid oleophilic surface, interacts with a magnetic field to get transformed into an array of spikes which then act as a carrier for the transportation of the aqueous droplet. Our study reveals that transportation phenomena is governed by the interplay of three different forces – magnetic force F_m, frictional force F_f and interfacial tension force F_i, which is expressed in terms of the magnetic Laplace number 〖(La〗_m) and magnetic Bond number (〖Bo〗_m) as 〖La〗_m^(-1)=(F_f1⁄F_(m,x) ) and 〖Bo〗_m 〖La〗_m^(-1)=(F_f2⁄F_i ). Based on the values of the dimensionless numbers, three different regimes – steady droplet transport, spike extraction and magnet disengagement, are identified. It is found that steady droplet transport is observed for 〖La〗_m^(-1)≤1 and 〖〖Bo〗_m La〗_m^(-1)≤1, whereas extraction of spikes is observed for 〖La〗_m^(-1)≤1 and 〖〖Bo〗_m La〗_m^(-1)>1, and magnet disengagement is observed for 〖La〗_m^(-1)>1. In the steady droplet transport regime, velocity of the aqueous droplet U_ds was found to be dependent on the volumes of the aqueous droplet V_w and FF droplet V_FF following U_ds~V_w^(-0.19) V_FF^0.36. A simple model is presented that accurately predicts the aqueous droplet velocity U_ds within 5% of the corresponding experimental data. In the spike extraction regime, the spike extraction distance L_se was found to vary with V_w, V_FF and the magnet velocity U_ms following L_se~V_w^(-1.75) V_FF^0.75 U_ms^(-1.56).
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Slow droplet motion on chemically heterogeneous substrates is considered analytically and numerically. We adopt the long-wave approximation which yields a single partial differential equation for the droplet height in time and space. A matched asymptotic analysis in the limit of nearly circular contact lines and vanishingly small slip lengths yields a reduced model consisting of a set of ordinary differential equations for the evolution of the Fourier harmonics of the contact line. The analytical predictions are found, within the domain of their validity, to be in good agreement with the solutions to the governing partial differential equation. The limitations of the reduced model when the contact line undergoes stronger deformations are partially lifted by proposing a hybrid scheme which couples the results of the asymptotic analysis with the boundary integral method. This approach improves the agreement with the governing partial differential equation, but at a computational cost which is significantly lower compared to that required for the full problem.
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The dynamic behavior of compound droplets on the wall of a rectangle channel by the action of an imposed shear flow is simulated using our developed three-dimensional front-tracking method combined with generalized Navier boundary condition. The validity of the present method was confirmed by comparing results of the compound droplet spreading under gravity force with analytical solutions. To determine the physical condition required for detaching/pinching-off the compound droplet, we have performed a large number of simulations with varying capillary numbers of two interfaces and obtained a phase diagram of compound droplets on solid surface in shear flow. The deformation and motion of the compound droplet including its contact line motion are investigated. It is found that the behavior of the compound droplet is controlled by two dimensionless parameters, the capillary numbers of the outer interface and the inner interface. Moreover, we also analyze the deformation and migration of the inner droplet and discuss its effect on the compound droplet. The simulation demonstrate that the lateral migration of the small inner droplet could accelerates the pinch-off process and the large inner droplet could promote the detachment for a moderate capillary number of the outer interface.
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Spontaneous directional transportations of droplets on the solid surfaces driven by the structure gradient have attracted much attention due to its large-scale applications such as, heat transfer, microfluid devices, water collection, separation. It also opens the new sight of theoretical researches between droplets and solid surfaces. This review article summarizes the recent progress in spontaneous directional transportation of droplets on surfaces with structure gradient. Recently, those surfaces with structure gradient can be divided into three types, wedge corner with a gradient opening angle, wedge-shaped surfaces, conical substrate. It focuses on the basic theory, detail transport process, fabricated methods, influence factors and their application development. At last, the perspective of this transportation in future development is proposed.
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In this paper, we experimentally investigated the impact dynamics of different viscous droplets on solid surfaces with diverse wettabilities. We show that the outcome of an impinging droplet is dependent on the physical property of the droplet and the wettability of the surface. Whereas only deposition was observed on lyophilic surfaces, more impact phenomena were identified on lyophobic and superlyophobic surfaces. It was found that none of the existing theoretical models can well describe the maximum spreading factor, revealing the complexity of the droplet impact dynamics and suggesting that more factors need to be considered in the theory. By using the modified capillary-inertial time, which considers the effects of liquid viscosity and surface wettability on droplet spreading, a universal scaling law describing the spreading time was obtained. Finally, we analyzed the post-impact droplet oscillation with the theory for damped harmonic oscillators and interpreted the effects of liquid viscosity and surface wettability on the oscillation by simple scaling analyses.
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We demonstrate the continuous translational invariance of the energy of a capillary surface in contact with reconfigurable solid boundaries. We present a theoretical approach to find the energy-invariant equilibria of spherical capillary surfaces in contact with solid boundaries of arbitrary shape and examine the implications of dynamic frictional forces upon a reconfiguration of the boundaries. Experimentally, we realise our ideas by manipulating the position of a droplet in a wedge geometry using lubricant-impregnated solid surfaces, which eliminate the contact-angle hysteresis and provide a test bed for quantifying dissipative losses out of equilibrium. Our experiments show that dissipative energy losses for an otherwise energy-invariant reconfiguration are relatively small, provided that the actuation timescale is longer than the typical relaxation timescale of the capillary surface. We discuss the wider applicability of our ideas as a pathway for liquid manipulation at no potential energy cost in low-pinning, low-friction situations.
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We consider the wetting of water droplets on substrates with different chemical composition and molecular spacing, but with an identical equilibrium contact angle. A combined approach of large-scale molecular dynamics simulations and a continuum phase field model allows us to identify and quantify the influence of the microscopic physics at the contact line on the macroscopic droplet dynamics. We show that the substrate physico-chemistry, in particular hydrogen bonding, can significantly alter the flow. Since the material parameters are systematically derived from the atomistic simulations, our continuum model has only one adjustable parameter, which appears as a friction factor at the contact line. The continuum model approaches the atomistic wetting rate only when we adjust this contact line friction factor. However, the flow appears to be qualitatively different when comparing the atomistic and continuum models, highlighting that non-trivial continuum effects can come into play near the interface of the wetting front.
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A liquid drop moves on a solid surface if it is subjected to a gradient of wettability or temperature. However the pinning defects on the surface manifesting in terms of a wetting hysteresis, or a first order non-linear friction, limits the motion in the sense that a critical size has to be exceeded for a drop to move. The effect of hysteresis can, however, be mitigated by an external vibration that can be either structured or stochastic, thereby creating a directed motion of the drop. Many of the well-known features of rectification, amplification and switching that are generic to electronics can be engineered with such types of movements. A specific case of interest is the random coalescence of drops on a surface that gives rise to a self-generated noise. This noise overcomes the pinning potential diffusively, thereby generating a random motion of the coalesced drops. Randomly-moving coalesced drops themselves exhibit a purely diffusive flux when a boundary is present to eliminate them by absorption. With the presence of a bias, the coalesced drops execute a diffusive drift motion that can have useful application in various water and thermal management technologies.
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We consider the static and dynamic behaviour of two-dimensional droplets on inclined heterogeneous substrates. We utilize an evolution equation for the droplet thickness based on the long-wave approximation of the Stokes equations in the presence of slip. Through a singular perturbation procedure, evolution equations for the location of the two moving fronts are obtained under the assumption of quasi-static dynamics. The deduced equations, which are verified by direct comparisons with numerical solutions to the governing equation, are scrutinized in a variety of dynamic and equilibrium settings. For example, we demonstrate the possibility for stick–slip dynamics, substrate-induced hysteresis, the uphill motion of the droplet for sufficiently strong chemical gradients and the existence of a critical inclination angle beyond which the droplet can no longer be supported at equilibrium. Where possible, analytical expressions are obtained for various quantities of interest, which are also verified by appropriate numerical experiments.
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A level set method for capturing the interface between two fluids is combined with a variable density projection method to allow for computation of two-phase flow where the interface can merge/break and the flow can have a high Reynolds number. A distance function formulation of the level set method enables one to compute flows with large density ratios (1000/1) and flows that are surface tension driven; with no emotional involvement. Recent work has improved the accuracy of the distance function formulation and the accuracy of the advection scheme. We compute flows involving air bubbles and water drops, to name a few. We validate our code against experiments and theory.
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This paper presents results of an experimental investigation of a single drop impact onto a dry, partially wettable substrate and its numerical simulation. Particularly, the drop spreading diameter and the dynamic contact angle are measured at different time instants after impact. Two surfaces, wax (low wettability) and glass (high wettability), are used to study the effect of surface wettability (static contact angle) on the impact dynamics. It is shown that existing empirical models for the dynamic contact angle (e.g., Hoffman-Voinov-Tanner law) do not predict well the change of the dynamic contact angle, especially at high capillary numbers. In addition to the experimental investigations, the drop impact was studied numerically, focusing primarily on the contact angle treatment. The singularity in the neighborhood of the moving contact line is removed from the computational domain and replaced by a local force with some dependence on the instantaneous advancing/receding contact-line velocity. The predicted time dependence of the drop spreading diameter and of the dynamic contact angle agrees well with the experimental data for both the advancing and receding phases of the impact process.
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The description of physical mechanisms involved in the impact of a drop upon a dry, partially wettable substrate is still a matter of debate. One way to analyze the balance of these mechanisms is the development of an analytical one-dimensional (1D) model based upon the energy equation. The assimilation of the drop to a cylinder allows a reduction of the energy equation to a second-order differential equation. This paper proposes a semi-empirical description of viscous dissipation taking into account the rolling motion near the contact line. The dissipation due to the rolling motion is added to the calculated dissipation in the core of the droplet. We compare our model to previous ones using a large set of literature data covering a wide range of viscosity, velocity impact, and equilibrium contact angle values. The new dissipation description proposed is shown to supersede those described in previous 1D models. Our model closely predicts the maximum spread factor and the time at which it is obtained on the whole range of Ohnesorge and Weber numbers considered. It also distinguishes between deposition with a steady variation in the wetted area from deposition with advancing and receding phases. The main limitations of the model lie in its inability to reproduce the spread factor at the very beginning of the impact and the rebounding observed after a receding phase for very high values of the equilibrium contact angle.
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Microdroplet formation is an emerging area of research due to its wide-ranging applications within microfluidic based lab-on-a-chip devices. Our goal is to understand the dynamics of droplet formation in a microfluidic T-junction in order to optimize the operation of the microfluidic device. Understanding of this process forms the basis of many potential applications: synthesis of new materials, formulation of products in pharmaceutical, cosmetics and food industries. The two-phase level set method, which is ideally suited for tracking the interfaces between two immiscible fluids, has been used to perform numerical simulations of droplet formation in a T-junction. Numerical predictions compare well with experimental observations. The influence of parameters such as flow rate ratio, capillary number, viscosity ratio and the interfacial tension between the two immiscible fluids is known to affect the physical processes of droplet generation. In this study the effects of surface wettability, which can be controlled by altering the contact angle, are investigated systematically. As competitive wetting between liquids in a two-phase flow can give rise to erratic flow patterns, it is often desirable to minimize this phenomenon as it can lead to a disruption of the regular production of uniform droplets. The numerical simulations predicted that wettability effects on droplet length are more prominent when the viscosity ratio λ (the quotient of the viscosity of the dispersed phase with the viscosity of the continuous phase) is O(1), compared to the situation when λ is O(0.1). The droplet size becomes independent of contact angle in the superhydrophobic regime for all capillary numbers. At a given value of interfacial tension, the droplet length is greater when λ is O(1) compared to the case when λ is O(0.1). The increase in droplet length with interfacial tension, σ, is a function of lnσ with the coefficients of the regression curves depending on the viscosity ratio.
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Dynamic transportation of water droplet on microstructured hydrophobic silicon substrate with a contact angle gradient is studied in this article. We propose a new type of substrate designed with microridges on a silicon wafer fabricated by photolithograph and subsequent coating with octadecyltrichlorosilane (OTS). When horizontal vibration is applied on the substrate, the water droplet can move to the direction with larger solid–liquid contact area fraction. It is found that the dynamic contact angle of the water droplet varied with the vibration direction and the speed of the substrate. The contact angle difference at the left and the right edge of the water droplet on the vibrated surface is obviously magnified compared to the contact angle difference of the droplet on the static surfaces, resulting in the increasing driving force. When the vibration amplitude of the exciter source (20Hz) increases from 0.14 to 0.43mm, the average velocity of 10μL water droplet increases from 10 to 23mm/s. The internal flow pattern of the water droplet moving on the microstructured hydrophobic surfaces is also obtained using particle image velocimetry (PIV) and particle tracking velocimetry (PTV) techniques. Both rolling and slipping motions are observed for the water droplet during its movement in the vibrated substrate.
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It is of both fundamental and practical interest to study the flow physics in the manipulation of droplets. In this paper, we investigate complex flow in liquid droplets actuated by a linear gradient of wettability using dissipative particle dynamics simulation. The wetting property of the substrate ranging from hydrophilic to hydrophobic is achieved by adjusting the conservative solid-liquid interactions which results in a variation of solid-liquid surface tension. The internal three-dimensional velocity field with transverse flow in droplet is revealed and analyzed in detail. When the substrate is hydrophobic, it is found that there is slight deformation but strong flow circulation inside the droplet, and the droplet rolling is the dominant mechanism for the movement. However, large deformation of the droplet is generated after the droplet reaches the hydrophilic surface, and a mechanism combining rolling and sliding dominates the transportation of the droplet. Another interesting finding is that the thermal fluctuation can accelerate the spontaneous motion of a liquid droplet under a wetting gradient. The effects of the steepness of wetting gradient and the size of droplet on the translation speed are studied as well.
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Droplet motion on solid substrates has been widely studied not only because of its importance in fundamental research but also because of its promising potentials in droplet-based devices developed for various applications in chemistry, biology, and industry. In this paper, we investigate the motion of an evaporating droplet in one-component fluids on a solid substrate with a wettability gradient. As is well known, there are two major difficulties in the continuum description of fluid flows and heat fluxes near the contact line of droplets on solid substrates, namely, the hydrodynamic (stress) singularity and thermal singularity. To model the droplet motion, we use the dynamic van der Waals theory [ Phys. Rev. E 75 036304 (2007)] for the hydrodynamic equations in the bulk region, supplemented with the boundary conditions at the fluid-solid interface. In this continuum hydrodynamic model, various physical processes involved in the droplet motion can be taken into account simultaneously, e.g., phase transitions (evaporation or condensation), capillary flows, fluid velocity slip, and substrate cooling or heating. Due to the use of the phase field method (diffuse interface method), the hydrodynamic and thermal singularities are resolved automatically. Furthermore, in the dynamic van der Waals theory, the evaporation or condensation rate at the liquid-gas interface is an outcome of the calculation rather than a prerequisite as in most of the other models proposed for evaporating droplets. Numerical results show that the droplet migrates in the direction of increasing wettability on the solid substrates. The migration velocity of the droplet is found to be proportional to the wettability gradients as predicted by Brochard [ Langmuir 5 432 (1989)]. The proportionality coefficient is found to be linearly dependent on the ratio of slip length to initial droplet radius. These results indicate that the steady migration of the droplets results from the balance between the (conservative) driving force due to the wettability gradient and the (dissipative) viscous drag force. In addition, we study the motion of droplets on cooled or heated solid substrates with wettability gradients. The fast temperature variations from the solid to the fluid can be accurately described in the present approach. It is observed that accompanying the droplet migration, the contact lines move through phase transition and boundary velocity slip with their relative contributions mostly determined by the slip length. The results presented in this paper may lead to a more complete understanding of the droplet motion driven by wettability gradients with a detailed picture of the fluid flows and phase transitions in the vicinity of the moving contact line.
Article
We simulate the moving contact line in two-dimensional chemically patterned channels using a diffuse-interface model with the generalized Navier boundary condition. The motion of the fluid–fluid interface in confined immiscible two-phase flows is modulated by the chemical pattern on the top and bottom surfaces, leading to a stick–slip behaviour of the contact line. The extra dissipation induced by this oscillatory contact-line motion is significant and increases rapidly with the wettability contrast of the pattern. A critical value of the wettability contrast is identified above which the effect of diffusion becomes important, leading to the interesting behaviour of fluid–fluid interface breaking, with the transport of the non-wetting fluid being assisted and mediated by rapid diffusion through the wetting fluid. Near the critical value, the time-averaged extra dissipation scales as U, the displacement velocity. By decreasing the period of the pattern, we show the solid surface to be characterized by an effective contact angle whose value depends on the material characteristics and composition of the patterned surfaces.
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An experimental study is presented on contact angle dynamics during spreading/recoiling of mm-sized water droplets impacting orthogonally on various surfaces with $\hbox{\it We}\,{=}\,O(0.1)-O(10)$, $Ca\,{=}\,O(0.001)-O(0.01)$, $\hbox{\it Re}\,{=}\,O(100)-O(1000)$, $Oh\,{=}\,O(0.001)$ and $Bo\,{=}\,O(0.1)$. In this impact regime, inertial, viscous and capillary phenomena act in unison to influence contact angle dynamics. The wetting properties of the target surfaces range from wettable to non-wettable. The experiments feature accelerating and decelerating wetting lines, capillary surface waves in the early impact stages, contact angle hysteresis, and droplet rebound under non-wetting conditions. The objective of the work is to provide insight into the dynamic behaviour of the apparent (macroscopic) contact angle $\theta$ and its dependence on contact line velocity $V_{\hbox{\scriptsize{\it CL}}}$ at various degrees of surface wetting. By correlating the temporal behaviours of $\theta$ and $V_{\hbox{\scriptsize{\it CL}}}$, the angle vs. speed relationship is established for each case examined. The results reveal that surface wettability has a critical influence on dynamic contact angle behaviour. The hydrodynamic wetting theory of Cox (J. Fluid Mech. vol. 357, 1998, p. 249) and the molecular-kinetic theory of wetting by Blake & Haynes (J. Colloid Interface Sci.) vol. 30, 1969, p. 421) are implemented to extract values of the corresponding microscopic wetting parameters required to match the experimentally observed $\theta$ vs. $V_{\hbox{\scriptsize{\it CL}}}$ data. Application of hydrodynamic theory indicates that in the slow stage of forced spreading the slip length and the microscopic contact angle should be contact line velocity dependent. The hydrodynamic theory performs well during kinematic (fast) spreading, in which solid/liquid interactions are weak. Application of the molecular kinetic theory yields physically reasonable molecular wetting parameters, which, however, vary with impact conditions. The results indicate that even for a single liquid there is no universal expression to relate contact angle with contact line speed. Finally, analysis of the spreading dynamics on the non-wettable surfaces shows that it conforms to the Cassie-Baxter regime (only partial liquid/solid contact is maintained). The present results offer guidance for numerical or analytical studies, which require careful attention to the implementation of boundary conditions at the moving contact line, including the need to specify the dependence of contact angle on contact line speed.
Article
The equilibrium shape of a liquid droplet, lying on a horizontal solid surface, is controlled by the spreading coefficient S. For partial wetting (S < 0), a drop smaller than the Laplace length κ-1 forms a spherical cap, while a drop much larger than κ-1 forms a thick "pancake". For complete wetting, the liquid spreads as a very thin pancake. If S is nonuniform, all these structures are set into motion. We discuss this in the limit of small gradients (∇S → 0) for two cases: (I) ∇S is obtained via a chemical treatment of the solid surface; here we predict that the droplets, or pancakes, move in the direction of lower surface energies. (II) ∇S is the consequence of a temperature gradient ∇T. The Marangoni effect plays an important role. If the surface tension γ of the liquid is temperature dependent, while all other interfacial tensions are not, we predict motions toward the regions of higher surface energies.
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Super-hydrophobic surfaces, with a water contact angle (CA) greater than 150°, have attracted much interest for both fundamental research and practical applications. Recent studies on lotus and rice leaves reveal that a super-hydrophobic surface with both a large CA and small sliding angle () needs the cooperation of micro- and nanostructures, and the arrangement of the microstructures on this surface can influence the way a water droplet tends to move. These results from the natural world provide a guide for constructing artificial super-hydrophobic surfaces and designing surfaces with controllable wettability. Accordingly, super-hydrophobic surfaces of polymer nanofibers and differently patterned aligned carbon nanotube (ACNT) films have been fabricated.
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In this review we focus on recent developments in applications of bio-inspired special wettable surfaces. We highlight surface materials that in recent years have shown to be the most promising in their respective fields for use in future applications. The selected topics are divided into three groups, applications of superhydrophobic surfaces, surfaces of patterned wettability and integrated multifunctional surfaces and devices. We will present how the bio-inspired wettability has been integrated into traditional materials or devices to improve their performances and to extend their practical applications by developing new functionalities.
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We systematically investigate droplet movement, coalescence, and splitting on an open hydrophobic surface. These processes are actuated by magnetic beads internalized in an oil-coated aqueous droplet using an external magnet. Results are organized into an 'operating diagram' that describes regions of droplet stable motion, breakage, and release from the magnet. The results are explained theoretically with a simple model that balances magnetic, friction, and capillary-induced drag forces and includes the effects of particle type, droplet size, surrounding oil layer, surface tension, and viscosity. Finally, we discuss the implications of the results for the design of magnet-actuated droplet systems for applications such as nucleic acid purification, immunoassay and drug delivery.
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Droplet behaviors on substrates with wettability controlled both in space and in time are numerically investigated by using the lattice Boltzmann method. Several typical droplet responses are found under different designs of substrate wettability control. Special attention is drawn to the conditions under which rapid transport of droplets can be achieved. It is found that on alternating non-wetting-wetting units with proper non-wetting confining stripes, this objective can be realized when the frequency of wettability switch approximately matches that of the droplet to move across one unit. The variation of the "optimal" frequency with the size of the confining stripe is sought within certain ranges. The various types of droplet movement are analyzed by looking at the three-phase lines on the substrate, as well as the droplet shapes under different conditions. The results may provide useful implications for droplet manipulation in microfluidic devices.
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It is well known that a liquid drop with a low contact angle (approximately 45 degrees ) and low wetting hysteresis moves toward the colder region of a temperature gradient substrate as a result of the thermal Marangoni force. A moderately sized water drop, however, usually does not move on such a surface because of the overwhelming effect of hysteresis. The water drop can, however, be forced to move when it is vibrated on a temperature gradient surface with its velocity exhibiting maxima at the respective Rayleigh frequencies. A simple model is presented that captures the dependence of drop velocity on hysteresis, vibration amplitude, and the forcing and resonance frequencies of vibration.
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We investigate the applicability of a mesoscale modeling approach, lattice Boltzmann simulations, to the problem of contact line motion in one- and two-component two phase fluids. In this, the second of two papers, we consider binary systems. We show that the contact line singularity is overcome by diffusion which is effective over a length scale L about the contact line and derive a scaling form for the dependence of L on system parameters.
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In this Letter, we report on the motion of water droplets on surfaces decorated with molecular gradients comprising semifluorinated (SF) organosilanes. SF molecular gradients deposited on flat silica substrates facilitate faster motion of water droplets relative to the specimens covered with an analogous hydrocarbon gradient. Further increase in the drop speed is achieved by advancing it along porous substrates coated with the SF wettability gradients. The results of our experiments are in quantitative agreement with a simple scaling theory that describes the faster liquid motion in terms of reduced friction at the liquid/substrate interface.
Article
The hydrodynamic force experienced by a spherical-cap drop moving on a solid surface is obtained from two approximate analytical solutions and used to predict the quasi-steady speed of the drop in a wettability gradient. One solution is based on approximation of the shape of the drop as a collection of wedges, and the other is based on lubrication theory. Also, asymptotic results from both approximations for small contact angles, as well as an asymptotic result from lubrication theory that is good when the length scale of the drop is large compared with the slip length, are given. The results for the hydrodynamic force also can be used to predict the quasi-steady speed of a drop sliding down an incline.
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Results from experiments performed on the motion of drops of tetraethylene glycol in a wettability gradient present on a silicon surface are reported and compared with predictions from a recently developed theoretical model. The gradient in wettability was formed by exposing strips cut from a silicon wafer to dodecyltrichlorosilane vapors. Video images of the drops captured during the experiments were subsequently analyzed for drop size and velocity as functions of position along the gradient. In separate experiments on the same strips, the static contact angle formed by small drops was measured and used to obtain the local wettability gradient to which a drop is subjected. The velocity of the drops was found to be a strong function of position along the gradient. A quasi-steady theoretical model that balances the local hydrodynamic resistance with the local driving force generally describes the observations; possible reasons for the remaining discrepancies are discussed. It is shown that a model in which the driving force is reduced to accommodate the hysteresis effect inferred from the data is able to remove most of the discrepancy between the observed and predicted velocities.