Apr 20, 2011 - (a) shows the impaled state as a typical situation in the present simulations. In the analytical model (b), the droplet is assumed to be pinned at ...
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Stability and dynamics of droplets on patterned substrates: insights from experiments and lattice Boltzmann simulations
This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2011 J. Phys.: Condens. Matter 23 184112 (http://iopscience.iop.org/0953-8984/23/18/184112) View the table of contents for this issue, or go to the journal homepage for more
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IOP PUBLISHING
JOURNAL OF PHYSICS: CONDENSED MATTER
J. Phys.: Condens. Matter 23 (2011) 184112 (13pp)
doi:10.1088/0953-8984/23/18/184112
Stability and dynamics of droplets on patterned substrates: insights from experiments and lattice Boltzmann simulations F Varnik1,2, M Gross1,2 , N Moradi1 , G Zikos1 , P Uhlmann3 , 4 ¨ , D Magerl4 , D Raabe2 , I Steinbach1 and P Muller-Buschbaum 3 M Stamm 1
Interdisciplinary Center for Advanced Materials Simulation (ICAMS), Ruhr University Bochum, Stiepeler Straße 129, 44780 Bochum, Germany 2 Max-Planck-Institut f¨ur Eisenforschung, Max-Planck Straße 1, 40237 D¨usseldorf, Germany 3 Leibniz-Institut f¨ur Polymerforschung Dresden e.V., Hohe Straße 6, 01069 Dresden, Germany 4 Physik-Department, Technische Universit¨at M¨unchen, LS E13, James-Franck-Straße 1, 85748 Garching, Germany
Received 3 June 2010 Published 20 April 2011 Online at stacks.iop.org/JPhysCM/23/184112 Abstract The stability and dynamics of droplets on solid substrates are studied both theoretically and via experiments. Focusing on our recent achievements within the DFG-priority program 1164 (Nano- and Microfluidics), we first consider the case of (large) droplets on the so-called gradient substrates. Here the term gradient refers to both a change of wettability (chemical gradient) or topography (roughness gradient). While the motion of a droplet on a perfectly flat substrate upon the action of a chemical gradient appears to be a natural consequence of the considered situation, we show that the behavior of a droplet on a gradient of topography is less obvious. Nevertheless, if care is taken in the choice of the topographic patterns (in order to reduce hysteresis effects), a motion may be observed. Interestingly, in this case, simple scaling arguments adequately account for the dependence of the droplet velocity on the roughness gradient (Moradi et al 2010 Europhys. Lett. 89 26006). Another issue addressed in this paper is the behavior of droplets on hydrophobic substrates with a periodic arrangement of square shaped pillars. Here, it is possible to propose an analytically solvable model for the case where the droplet size becomes comparable to the roughness scale (Gross et al 2009 Europhys. Lett. 88 26002). Two important predictions of the model are highlighted here. (i) There exists a state with a finite penetration depth, distinct from the full wetting (Wenzel) and suspended (Cassie–Baxter, CB) states. (ii) Upon quasi-static evaporation, a droplet initially on the top of the pillars (CB state) undergoes a transition to this new state with a finite penetration depth but then (upon further evaporation) climbs up the pillars and goes back to the CB state again. These predictions are confirmed via independent numerical simulations. Moreover, we also address the fundamental issue of the internal droplet dynamics and the terminal center of mass velocity on a flat substrate. (Some figures in this article are in colour only in the electronic version)
includes using droplets as a transport medium for medical agents (e.g. drug delivery and lab-on-chip medical devices) as well as industrial processes such as infiltration, coating and printing. In inkjet printing, for example, tiny droplets (ink) are
1. Introduction The wetting behavior and dynamics of liquid droplets on solid substrates play a fundamental role for many applications. This 0953-8984/11/184112+13$33.00
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J. Phys.: Condens. Matter 23 (2011) 184112
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introductions to the general foundations of the lattice Boltzmann method can be found in recent monographs [29–31] as well as in review articles [32, 33]. Within a lattice Boltzmann method, one basically iterates two simple steps generally referred to as (1) relaxation (collision) and (2) free propagation (streaming),
deposited onto a solid substrate at specified locations. Here, a safe control of the droplet volume and the extent of the droplet spread (contact or base area) on the substrate are of fundamental importance. On an ideally flat and chemically homogeneous substrate, the contact area is larger on hydrophilic as compared to hydrophobic substrates. Real surfaces, however, are neither perfectly flat nor chemically homogeneous. This makes an experimental study of wetting phenomena a challenging task (see e.g. [1] and references therein). In this context, computer simulations may provide a useful and complementary tool. This is particularly interesting since in a computer simulation one can easily focus on ideal cases, such as the behavior of a droplet on a perfectly flat substrate with a well-defined chemical gradient. Conversely, one can study the droplet dynamics on a chemically homogeneous substrate containing a regular topographic pattern. Additional features such as a spatially variable roughness (roughness gradient) or a combination of topographic and chemical patterning may then be introduced into the model in order to study more complex situations. Furthermore, computer simulations often provide access to a number of mutually related quantities allowing us to shed light onto the problem from different perspectives. An example is the possibility to obtain the center of mass velocity of the droplet via two independent approaches, namely (i) from a sequence of images of the droplet during its motion or (ii) by direct averaging over the velocity field within the droplet. While the first approach is the usual way of determining the droplet velocity in real experiments (see e.g. figure 13), the three-dimensional velocity field inside the droplet is hard to access in an experiment. In this paper, we provide some examples on the above mentioned issues. This includes the behavior of liquids on chemically patterned substrates, the motion of droplets driven by a spatial change of roughness density and the behavior of small droplets on a hydrophobic substrate with a regular topographic pattern. In the latter case, an analytic model is proposed, capable of adequate description of the full stability phase diagram. In the case of roughness gradient driven dynamics, on the other hand, simple scaling arguments are found to provide a qualitative understanding of the relation between the average droplet velocity and roughness gradient. Furthermore, we also investigate the dynamics of a droplet on a flat and chemically homogeneous substrate and compare our results to recent molecular dynamics simulations [2] as well as experimental findings [3].
1 eq [ f i (r , t) − f i (r , t)], τ f i (r + dt ci , t + dt) = f i (r , t),
f i (r , t) = f i (r , t) −
(1) (2)
f i
where we introduced in order to formally separate the relaxation and streaming steps. In equation (1), the quantity τ is the so-called relaxation time which determines the fluid kinematic viscosity, ν = η/ρ (η = viscosity, ρ = fluid density). For the present three-dimensional model with 15 non-zero velocities (D3Q15), one finds [30] ν = (τ − 0.5) dx 2 /(3 dt), where dx is the distance between two neighboring nodes connected along the main coordinate directions and dt is the time step. The physical properties of the system enter the LB eq iteration scheme via the quantity f i equation (1). Obviously, eq the system is ‘pushed’ towards f i with a rate 1/τ . The eq population density f i is, therefore, referred to as the ‘equilibrium distribution’. It is noteworthy that the term ‘equilibrium’ does not refer to a global thermal equilibrium, where no flow exists. Rather, it describes the local velocity distribution in a portion of fluid moving at a velocity u(r ). eq Within the present LB model, one expands fi in powers of the fluid velocity u up to the second order [30]: 1 1 eq f i = ρwi 1 + 2 u · ci + 4 [(u · ci )2 − cs2 u · u] , (3) cs 2c s where cs is the sound speed and wi is a set of weights normalized to unity. For the two-dimensional nine velocity model (D2Q9) used in our studies one finds w0 = 4/9, w1 = w2 = w3 = w4 = 1/9 and w5 = w6 = w7 = w8 = 1/36. Once the discrete populations, f i , are known, the fluid density, ρ(x, t), and velocity, u(x, t), at a given point and time are obtained via ρ= fi (x, t) and ρu = f i (x, t)ci . (4) i
i
The approach described above must be supplemented by additional information accounting for the non-ideal character of a liquid–vapor system. This can be done either by (i) keeping the equilibrium distribution f eq unchanged but introducing density dependent interaction forces [16] or (ii) introducing a Cahn–Hilliard type free energy functional and modifying the equilibrium distribution in such a way as to obtain both the thermodynamic and the irreversible contributions to the pressure tensor correctly [34]. This latter approach has been further modified in order to ensure Galilean invariance [35] as well as to include fluid–solid interactions [36]. We emphasize here that, in this paper, we are not presenting a new numerical method. Rather, we use existing two-phase lattice Boltzmann models to study issues relevant to the stability and dynamics of liquid drops. Unless otherwise stated, we will use the approach proposed in [34] with appropriate modifications. Details of this method can be found e.g. in [20, 27, 36] and references therein.
2. The simulation method We use a simple but powerful computational tool, the lattice Boltzmann (LB) method [4–7], which has proved to be a versatile theoretical tool for the study of a variety of fluid dynamical phenomena such as the flow through porous media [8], roughness effects on inertial flows in narrow channels [9–11], flow of polymer solutions and suspensions [12–15], multiphase flows [16–19], droplets on topographically patterned substrates [20–23] and on gradients of wettability [24–27] and topography [28]. Comprehensive 2
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Figure 1. Creating wettability patterns via fluorosilane treatment. (a) Schematic representation of a substrate, partially covered with a self-organized hydrophobic fluorosilane monolayer. The fluorosilane molecules reach the substrate from the vapor phase and chemically bind to it (CVD technique). The uncovered area is initially coated with a photomask, which prevents fluorosilane from grafting to the substrate. This part of the substrate thus remains hydrophilic. (b) Tiny water drops sprayed on a substrate, prepared as in (a). The contact angle of 106◦ observed here is in accord with table 1. It lies slightly below the average between the maximum values of θA and θR . (c) Another route to chemical patterning: here, the role of the photomask is played by a polymer film, deposited via a commercially available overhead marker (Staedtler™ Lumocolor permanent). Since the polymer film is not chemically bonded to the substrate (only physical adhesion), it can be rinsed after the fluorosilane treatment (again via CVD technique), thus letting free the hydrophilic part of the substrate. (d) Water drops used to visualize a chemical pattern produced via this procedure.
3. Droplets on chemically patterned substrates
Table 1. Results of experimental measurements of the contact angle of water on fluorosilanized glass substrates. θA (θR ) denotes the advancing (receding) contact angle. The number of investigated samples is eleven.
In this section, we briefly address the behavior of droplets on substrates containing spatially variable wettability. The discussion is kept simple by focusing on an example of a substrate with a chess board like chemical pattern. This provides a nice example of how to confine a droplet to a well specified region by the use of wetting gradients. 3.1. Preparation of chemically patterned surfaces
Contact Average angle value (deg)
Standard deviation (deg)
Maximum Minimum (deg) (deg)
θA θR
5.0 4.9
122.6 91.3
113.2 82.8
104.4 74.7
In order to provide a test case between experiments and computer simulations, we used the photomask variant of the silanization technique described above, and coated glass substrates with a (tridecafluoro-1,1,2,2-tetrahydrooctyl)dimethylchlorosilane hydrophobic layer, thereby producing different chemical patterns. One of these patterns is shown in figure 2. The experiments of liquid behavior on these chemically patterned substrates were then paralleled by computer simulation studies of the same situation.
There are various techniques to create hydrophobic coatings of solid surfaces. Here, we briefly sketch two variants of the socalled fluorosilane treatment (figure 1). In the first approach, the parts of the substrate which shall remain hydrophilic are covered by a photomask. The photomask is thus the negative image of the hydrophobic pattern which is to be created on a hydrophilic substrate. The sample is then exposed to the vapor of fluorosilane—(tridecafluoro-1,1,2,2-tetrahydrooctyl)dimethylchlorosilane in our experiments—which deposits on the substrate and chemically grafts to it. Once this chemical vapor deposition (CVD) of fluorosilane has completed, the photomask is removed thus uncovering the hydrophilic part of the substrate. In the second approach, a commercially available overhead marker is used in order to cover parts of the substrate before silanization, which proceeds—similar to the case of the photomask—via the CVD technique. Finally, the polymer film is washed away with a solvent (often ethanol or acetone). While the use of a photomask is well suited to create large structures, the marker-based method allows the design of finer patterns. Within this latter procedure, however, nanoscale hydrophilic domains persist and the contact angles thus reached are significantly lower than those obtained via the use of a photomask [37]. The samples produced via the above described procedures are characterized by contact angle measurements and ellipsometry, the latter yielding the thickness of the hydrophobic coating layer. In the case of glass substrates, however, ellipsometry measurements were not possible and the only route to characterize the samples was via contact angle measurements. The results of these measurements are collected in table 1.
3.2. Computer simulation of droplets on a chemical pattern In the modeling part of our studies, the surface is assumed to be perfect (no contact angle hysteresis). A survey of table 1, however, reveals the presence of significant hysteresis effects in our experiments. This is best seen by a comparison of the advancing and receding contact angles. Thus, we do not expect a full quantitative agreement between theory and experiments. Rather, we want to examine whether the qualitative features observed in the experiments (e.g., the localization of the droplet to the chemically patterned area) are also found in our computer simulations. As can be seen in figure 2, this is indeed the case. The observed deviations between simulation and experiment, on the other hand, are rather expected. In the present simulation, the system is able to avoid high curvatures at the contact zone of the two hydrophilic parts by slightly expanding towards the hydrophobic domains. In the corresponding experiment, however, such a process is very probably hindered by the contact angle hysteresis. Regarding the dynamics, our studies of droplet spreading on chemical gradients reveal that, despite complications due to the geometry and spatial variation of wetting properties, the capillary time still remains one of the most important 3
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Figure 2. (a) A glass substrate with a chess board like chemical pattern. (b) Liquid deposited on this substrate. (c) Lattice Boltzmann computer simulation of a similar situation. (d) Footprint of the same droplet shown in (c). While the overall confinement of the droplet caused by chemical patterning is well reproduced in the simulations, parts of the hydrophobic domain are also covered by the liquid. This is necessary in order to avoid the high curvature of the liquid–vapor interface at the contact zone of the two hydrophilic domains. The presence of such a high curvature in the experiment is very probably caused by the presence of hysteresis and the related pinning forces. The size of the simulation box is 100 × 100 lattice units.
Figure 3. Quasi-static evaporation of a droplet on patterned (top) and perfectly smooth (bottom) hydrophobic substrates (Young contact angle on the flat substrate = θY = 95◦ ). The comparison of the upper and lower images reveals that the rather large contact angle hysteresis observed here (≈difference between the contact angles in (d) and (a)) largely arises from the presence of topography.
characteristic times of the separation process. Further studies of other related aspects and the governing scaling relations can be found in [27].
Indeed, in these simulations, the contact angle continuously decreases without ever reaching a steady value. This implies that, for the case considered here, the receding contact angle is quite small (if not zero). The advancing contact angle, on the other hand, is larger than 90◦ (see figure 3(a)). A few words are necessary regarding the definition and implementation of the topographic patterns in our simulations. We introduce rectangular posts of linear dimension 4 × 4 × 5 (length × width × height) lattice units. The distance between two neighboring posts is set to eight lattice units along both the x and y directions. Thus, only 50% of the substrate is covered by the posts. At each solid node, two conditions are applied. The first condition is to ensure that the solid acts as an impenetrable barrier with no-slip boundary conditions. The second condition regards the wetting behavior of the substrate. A possible way to take account of this feature is by introducing the contribution of the fluid–solid interaction to the total free energy of the system in terms of the fluid density. Linearizing this free energy [38], one obtains a simple condition for the gradient of the fluid density at the solid surface [27, 36]. However, since this gradient must be computed in the direction normal to the surface, its implementation in the case of an arbitrary orientation of the solid surface becomes rather complicated. This explains why the topographic patterns used in our studies are made of
3.3. Pinning at substrate roughness We already remarked on the rather large difference between the advancing and receding contact angles for our experimental samples. A major reason for this large contact angle hysteresis is the surface roughness. This effect is nicely demonstrated via our lattice Boltzmann simulations, shown in figure 3. The aim of these simulations is to highlight the effect of surface topography on the receding contact angle. Instead of directly subtracting the mass from the liquid, we allow here a low but constant outflow of vapor from the topmost layer (a horizontal plane at L z − 1) of the simulation box. By doing so, we mimic a real situation corresponding to an open system with controlled outflow, without direct modification of the transport processes on the liquid–vapor interface. The term ‘low’ here means that the resulting evaporation process is sufficiently slow so that, at each instant, the shape of the droplet corresponds to the local minimum of the surface free energy (evaporation time capillary time). In this sense, the studied evaporation process is quasi-static. 4
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Figure 4. Left: top view of two step gradient substrates. In the upper panel (referred to as case A), the pillar density to the left side ( x < 50) is φLeft = 0.187 (square posts of length a = b = 3) while it is set to φRight = 0.321 on the right side ( x > 50, rectangular posts of length a = 9 and width b = 3). The spacing distance of the pillars in the x -direction is dx = 5 and in the y -direction is d y = 3 overall on the substrate. The height of the posts is c = 6. The lower panel (case B) is obtained from A by shifting the posts on each second row horizontally by an amount (a + dx )/2 with dx = 5, a = 3 for x < 50 and a = 9 for x > 50. All lengths are given in LB units. Middle: the initial setup and final states of a spherical droplet placed on the substrates of type A (top) and B (bottom). The droplet remains pinned to the roughness gradient zone in case A while it passes over this zone in the case of substrate B. Right: images showing that similar final states are also observed for the case of cylindrical droplets. Adapted from [28].
roughness density, one may expect that a droplet placed at the contact zone will leave this area towards the region of higher roughness density. However, the experimental observation of a roughness gradient induced spontaneous motion is not an easy task [42]. In fact, even though equation (5) predicts a decrease in the effective contact angle upon an increase in the roughness density and hence a driving force along the gradient of φ , the contact angle hysteresis [46] may be strong enough in order to prevent a spontaneous droplet motion [42]. The authors of [41, 42], for example, resort to vertically shaking the substrate in order to overcome the pinning forces. Furthermore, in these and similar experiments, it is often observed that the behavior of droplets is not unique [39–42]. In order to study this issue via computer simulations, we adopt the above described case of a step wise change in the roughness density and design a substrate divided into two regions, each with a constant pillar density. Furthermore, in order to underline the crucial role of pillar arrangement on the behavior of droplets, we consider two different pillar arrangements leading to the same pillar density gradient as shown in the left panels of figure 4. The simulation results shown in figure 4 clearly demonstrate that the behavior of a droplet on substrates patterned by a pillar microstructure with the same pillar density gradient but different pillar arrangement and geometries can be qualitatively different. In particular, in the case of the substrate of type A, the droplet motion is stopped on the gradient zone, while in the case of substrate B it completely reaches the more favorable region of higher φ . A detailed comparison of the effects of arrangements A and B on pinning forces seems quite a complicated and demanding task. Instead, we investigate the dependence of the
pieces of planar surfaces with normal vectors along the lattice directions. Despite this simple geometry, careful book-keeping is still necessary. It is particularly important here to realize and take account of the fact that corners, edges and faces provide different topologies for a computation of density gradients and thus require a separate treatment. A more detailed discussion of this issue can be found in [20].
4. Spontaneous droplet motion induced by a gradient of texture While droplet behavior on homogeneous roughness has widely been investigated in the literature, only a few works exist dealing with the case of inhomogeneous topography [39–43]. The starting point is the Cassie–Baxter (CB) equation for the effective contact angle of a droplet suspending on the top of roughness tips [44], cos θCB = φ cos θY − (1 − φ) = φ(1 + cos θY ) − 1,
(5)
where the roughness density φ gives the fraction of the droplet’s base area, which is in contact with the solid. It is important to realize that equation (5) does not explicitly take account of three-phase contact line structure. This shortcoming may, however, be neglected as long as the contact area reflects the structure and energetics of the three-phase contact line [45]. Noting that 1 + cos θY > 0 also holds for hydrophobic substrates (θY > 90◦ ), it is easily seen from equation (5) that cos θCB decreases upon a decrease of φ , resulting in a larger effective contact angle. Thus, the lower the pillar density, the higher the effective hydrophobic character of the patterned substrate. If one brings into contact two hydrophobic substrates, each homogeneously decorated with a different 5
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in the viscous regime, we neglect inertial terms in the Navier– Stokes equation and write for the steady state 0 = −∇ p+η u . Here, u is the fluid velocity, p is the hydrostatic pressure and η is the viscosity. The velocity u varies only over a distance of the order of the droplet radius, hence u ∼ u/R 2 . On the other hand, ∇ p ∼ −d pLaplace (θCB )/R = (σ/R 2 ) d R/R , assuming that the driving force originates from the Laplace pressure variation (over a length of the order of R ) within the droplet. For the case of a cylindrical droplet of unit length, the condition of constant droplet volume, = R 2 [θCB − sin(2θCB )/2], equation (5) and some algebra lead to d R/R ∼ (π − θCB ) dφ ∼ φ¯ dφ (the relation π − θCB ∼ φ¯ follows from equation (5) assuming θCB close to π [42] and φ¯ is defined as (φRight + φLeft )/2). Putting all this together, and after a change of notation √ dφ ≡ φ = φRight − φLeft , we arrive at ηu/R 2 ∼ (σ/R 2 ) φ φ . Hence,
Figure 5. The x -component of the center of mass position versus time for a cylindrical droplet using, φLeft = 0.187 and φRight = 0.321, 0.333, 0.35 and 0.375. The two groups of curves belong to two different surface tensions of σ0 = 5.4 × 10−4 (LB units) (right) and 4σ0 (left). Adapted from [28].
u∼ average droplet velocity during its passage over the gradient zone as a function of the pillar density gradient. For this purpose, we create substrate patterns of type B with various values of φ by keeping φLeft unchanged and varying φRight . To simplify the analysis further, we place a cylindrical droplet whose axis is along the y -direction. This allows us to use the periodicity of the pillar arrangement along the y -direction and thus also reduce the computation time. The results on the dynamics of a cylindrical drop on such texture gradient substrates are shown in figure 5. A survey of the center of mass position versus time in figure 5 reveals that the droplet motion is first linear in time until it reaches a constant value. The plateau corresponds to the case where the droplet has completely left the region of lower pillar density. This is in line with the fact that no driving force exists in this final state. Using the linear section of the data in figure 5, we define an average velocity for the motion of the droplet’s center of mass. Importantly, figure 5 reveals the strong effect of the surface tension on droplet dynamics. Both the absolute values of the droplet velocity for a given φ as well as the slope of the data significantly depend upon σ . It is possible to rationalize these observations by simple scaling arguments. First, for the sake of simplicity, we do not consider pinning effects here. Since the flow we consider is
σ η
¯ φ φ.
(6)
It is noteworthy that both equations (6) and equation (5) in [42] predict a linear dependence of the droplet velocity on φ . In [42], the lateral velocity is estimated from the roughness gradient induced asymmetry of the dewetting of a droplet, flattened due to impact. However, the situation we consider is different. There is no impact and hence a related flattening is absent in the present case. Furthermore, the dynamics we study is in the viscous regime whereas the high impact velocity in [42] supports the relevance of inertia. These differences show up in different predictions regarding the dependence of the droplet velocity on surface tension, fluid viscosity and density. While equation (5) in [42] predicts a dependence on the square root of σ , equation (6) suggests that, in our case, a linear dependence on σ should be expected. We therefore examine equation (6) not only with regard to the relation between the droplet velocity u and the difference in roughness density φ (left panel in figure 6), but we also check how u changes upon a variation of the surface tension σ for a fixed φ . The results of this latter test are depicted in the right panel of figure 6, confirming the expected linear dependence of u on σ . It is noteworthy that σ in the right panel of figure 6 varies roughly by a factor of ten so that a square root dependence can definitely be ruled out.
Figure 6. Left: the droplet’s center of mass velocity versus the difference in pillar density, φ = φRight − φLeft , extracted from the linear part of the center of mass motion (see e.g. figure 5). Results for three different liquid–vapor surface tensions are depicted. From top to bottom: 4σ0 , 2σ0 and σ0 , where σ0 = 5.4 × 10−4 (LB units). In all cases, a linear variation is seen in accordance with the simple model, equation (6). Right: a further test of equation (6), where the dependence of droplet velocity on surface tension is shown for a fixed φ .
6
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Figure 7. A small droplet on a hydrophobic pillar array. Here, the term ‘small’ means that the linear dimension of the droplet is of the same order of magnitude as the scale of the topographic pattern. (a) shows the impaled state as a typical situation in the present simulations. In the analytical model (b), the droplet is assumed to be pinned at the edges of the pillars, i.e. it has a fixed base radius a . For a given droplet size, the only way to minimize the overall free energy is thus through a change in the penetration depth p. This also fixes the apparent contact angle θ . Adapted from [23].
The total volume of the droplet shall be fixed, Vtot = const = Vsph (θ ) + Vcyl ( p), with Vsph = 13 πa 3 (2 − 3 cos θ + cos3 θ )/ sin3 θ the volume of the cap and Vcyl = (πa 2 − b 2) p the volume of the penetrating cylinder. In the following, instead of the droplet volume Vtot , we will usually refer to the effective droplet radius Reff which corresponds to a spherical 3 droplet of the same volume (4π/3 Reff = Vtot ). We consider p as the free variable and determine the dependence of θ on p via the fixed volume condition. Since the volume of the drop (and the temperature) is constant, only surface energy contributions play a role for a change in the total free energy. The free energy f ( p) of the model droplet, neglecting gravity and terms associated with the Wenzel transition, and normalizing such that f (0) = 0, is then given by
5. Stability of small droplets The issues discussed above have one common point, namely that the size of the droplet is large compared to the typical roughness scale of the substrate. This separation of length scales often justifies the use of arguments based on average properties (see equation (6)). Here, we would like to consider a situation where the droplet size is comparable to the length scale of the roughness. This situation is not only important for a better understanding of the wetting properties of microscale systems [47], but also in many industrial applications, as, for example, in the production of efficient self-cleaning surfaces [48], robust metal coatings, or in plasma spraying techniques. Moreover, small droplets naturally occur in many condensation or evaporation processes [49, 50].
f ( p) = σLV [Ssph ( p)− Ssph (0)+ Scyl,LV ( p)− 8bp cos θY ]. (7)
5.1. An analytically tractable model
Here, Ssph = 2πa 2 (1 − cos θ )/ sin2 θ is the surface area of the spherical cap Scyl,LV = (2πa − 4b) p is the lateral liquid– vapor surface area of the cylinder and −σLV 8bp cos θY is the (positive, since θY > 90◦ ) free energy associated with the wetting of the (eight) side walls of the pillars.
The fact that we deal with droplets of a size comparable to the roughness scale allows the use of a simple analytical model. The roughness of a surface is modeled by a regular array of cuboidal pillars with width b , height h and spacing d (figure 7). The intrinsic hydrophobicity of the flat parts of the surface is described by the Young contact angle θY . In this work, we only focus on the case θY > 90◦ , which is also a necessary condition for a (meta-) stable Cassie–Baxter state. Gravity will be neglected throughout, as we only consider droplets that are smaller than the capillary length (∼2.7 mm for water). In the analytical model, we assume the part of the droplet located above the pillars to be a spherical cap with base a = b/2 + d = R sin θ (with R being the radius of the cap and θ the apparent contact angle). The impaled part is approximated as a cylindrical liquid column with radius a and height p (penetration depth), surrounding the central pillar. The macroscopic contact line of the droplet is assumed to remain pinned at the edges of the outer pillars. Note that in this model the fully penetrated state (Wenzel [51]) would correspond to p = h and the suspended (CB) state to p = 0. Since the mechanism of the Wenzel transition has been discussed in detail by many authors in previous publications, we will ignore the Wenzel state completely, and, for the rest of this work, assume the pillars to be so tall that no contact between the liquid and the bottom of the grooves is possible.
5.2. Analytical results Figures 8(a) and (b) show the dependence of the free energy on the penetration depth p for varying Young contact angles and droplet sizes. Several interesting observations can be made: firstly, as also found in the case of droplets which are large compared to the roughness scale [52–56], the stability of the CB state, determined by the slope of f at p = 0, depends not only on the contact angle but also on the size of the droplet. A novel feature is the appearance of a local minimum of the free energy at large penetration depths, existing in addition to the possible minimum associated with the CB and Wenzel states. From the condition d f /d p = 0, which is easily evaluated with the help of the fixed volume constraint, a necessary condition for the existence of a minimum of the free energy arises, namely, θY < arccos(−1/2 + b/(4a)), with a = b/2 + d being the base radius of the spherical cap. Interestingly, this condition does not depend on the droplet size. 7
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Figure 8. Predictions of the analytical model. (a) Dependence of the free energy f /4πa 2 σLV on the penetration depth p and the Young contact angle θY for a droplet of fixed size Reff = a . (The inset shows a magnification of the curve for θY = 108◦ .) (b) Dependence of the free energy f /4πa 2 σLV on the penetration depth p and the droplet size Reff for a fixed contact angle of θY = 100◦ . (c) Contributions to the free energy f /4πa 2 σLV for Reff = a and θY = 100◦ . f sph is the surface free energy of the liquid–vapor interface of the spherical cap, f cap is the free energy due to the wetting of the ‘capillary’ constituted by the pillars and f = f sph + f cap is the total free energy. The insets sketch the droplet configuration for different p according to the analytical model. All curves in (a)–(c) are given for b/d = 1 and plotted up to a value of p where no further volume is left in the spherical cap. Adapted from [23].
Figure 9. Left: regions of stability for the CB state. We fix the pillar width b and plot the set of points (Reff , θY ) fulfilling d f /d p(Reff , θY )| p=0 = 0, for varying values of the mutual pillar distance d (solid curves). The CB state is stable to the right hand side of each solid curve. The dotted lines mark the limiting value of θY for the existence of a local minimum of the free energy. Right: reentrant transition. Given a moderate contact angle, a large droplet is predicted by our analytical model to assume a metastable CB state (stage 1). On reduction of its size the CB state becomes unstable and the droplet penetrates into the grooves (stage 2). If the height of the pillars prohibits a Wenzel transition, the droplet eventually enters the CB state again upon further reduction of its size (stage 3). The boundary curve of the CB stability region is shown for the case b = d . Adapted from [58].
The origin of this new state with a finite penetration depth can be understood by imagining the pillars to represent a (partly open) hydrophobic capillary tube, wetted by a small droplet that is placed at its entrance. In this situation, the equilibrium state of the droplet is a consequence of the balance between the Laplace pressure within the spherical cap (pushing the droplet into the capillary) and an opposing capillary force due to the hydrophobicity of the substrate. To illustrate this idea, we split the free energy (equation (7)) into the contributions of the spherical cap and the remaining ‘capillary’ part. As shown in (figure 8(c)), an increase of the droplet penetration p leads to a linear increase of capillary free energy, while the free energy associated with the spherical cap decreases in a nonlinear fashion. As a result, the total free energy f may exhibit a local minimum. This simple reasoning suggests that the intermediate minimum constitutes a generic equilibrium state of a droplet, occurring in any situation of filling hydrophobic capillaries by a spherical liquid reservoir. Indeed, further simulations using various
surface geometries (e.g. omitting the central pillar) clearly underline this assertion [57]. These results also show under which conditions we are allowed not to consider the Wenzel state in the first place: a transition to this state can be inhibited, if the pillar height h is larger than the penetration depth p of a droplet at the local minimum, plus a small correction of the order of d 2 /Reff [1] that accounts for the curvature of the lower droplet interface. As becomes clear from figure 9, the shape of the stability region of the CB state is largely independent of the geometry, and its characteristic shape offers the interesting possibility of a reentrant transition: for a given Young contact angle, imagine a droplet that is very large and initially deposited at the top of the pillar array. According to our analytical model, the droplet will adopt a CB state (stage 1 in figure 9). If the droplet is now reduced in size (e.g. through evaporation) and the Young contact angle is not too large, the droplet will enter the region of CB instability (stage 2). Depending on the tallness of the pillars, the droplet will now either go over into the Wenzel state 8
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Figure 10. Reentrant transition through evaporation of a droplet. In the simulation, the droplet is initially (t = 0) prepared in a (meta-)stable (partially impaled) Cassie state (a). Evaporation is then switched on and proceeds by reducing, at a sufficiently low rate satisfying quasi-static equilibrium, the mass of the vapor phase across a x y -plane close to the top of the simulation box. In the course of the evaporation process, the Cassie state becomes unstable and adopts an impaled state (b), from where it gradually climbs up the pillars (c) until it again reaches a Cassie state (d). (e) shows the penetration depth of the lower droplet interface as it is observed in the simulation ( ) and predicted by the analytical model (——). Simulation parameters: contact angle θY = 103◦ , initial droplet size Reff = 1.2a , b = d = 12 LB units, evaporation rate 5 × 10−7 /LB time steps. All lengths are given in units of the LB grid spacing. Adapted from [23].
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such situations adequately, the present model must be extended appropriately. Such a study presents an interesting topic for future work. Second, it is interesting to address the range of length scales for the validity of the present predictions. For this purpose, we recall that solid deformability, thermal fluctuations and gravity are neglected in the investigated analytic model. In other words, only surface free energies are considered in the model. This makes the model scale invariant since, in this case, f (αa, αb, αd, αp, α R) = α 2 f (a, b, d, p, R), where α is an arbitrary scale factor. As a consequence, all the predictions made within our model are scale invariant as long as the above approximations hold. While the first two of the above mentioned approximations set a lower bound to the available length scales (of the order of microns, the exact value depending on the substrate elastic modulus and the surface tension), the third one implies that the droplet size should be less than the capillary length (≈2.7 mm for water). Thus, when experimentally testing the present predictions, it is a priori not necessary to use micro-pillars with correspondingly small micro-droplets (which are well known to be prone to strong evaporation). Rather, experiments on sufficiently large droplets with diameters, d , below the capillary length (e.g., d 1 mm) would also be adequate for such a purpose.
or the impaled state. In the latter case, further reduction of the droplet’s size brings it back again into the CB state (stage 3). This remarkable behavior is due to the peculiar dependence of the position of the intermediate minimum of the free energy on droplet size (figure 10(b)). Figure 10 provides evidence from simulations for the existence of the reentrant transition. This is achieved through a quasi-static evaporation process of a relatively large droplet. In the beginning (figure 10(a)), the droplet resides on the top of the pillars with a tiny penetration depth. However, after reaching a critical size, the droplet suddenly penetrates into the pillar grooves and goes over to the intermediate minimum of the free energy (figure 10(b)). During the impaled phase (figures 10(b) and (c)) the droplet gradually climbs up the pillars again, still residing in the local minimum. Note that its penetration depth is in nice agreement with the analytical predictions (figure 10(e)). In the end of the process, the droplet again attains a stable Cassie–Baxter state (figure 10(d)), as expected from the phase diagram, figure 9. However, we remark here that this transition actually happens for a slightly larger radius than predicted analytically, since the pinning condition cannot be maintained if the penetration depth is below the thickness of the liquid–vapor interface, which is of the order of 3–4 lattice units in the present case. According to the common understanding of self-cleaning, impalement is considered unfavorable and the droplet cleans the surface by rolling over the top of the texture. Interestingly, the existence of a reentrant transition suggests the possibility of a qualitatively new self-cleaning mechanism, since the droplet not only touches the top of the substrate, but also its inner parts. The existence of a reentrant transition can also explain some recent experimental observations, which found that small evaporating droplets indeed tend to remain close to the top of the substrate [55, 59] and not get trapped inside of the texture. We close this section with two important remarks. First, in the present analytical model as well as in the accompanying lattice Boltzmann simulations, the pillars are assumed to be perfectly rigid. In the case of sufficiently tall micron-sized pillars, however, the elasticity of the pillars and the resulting deformation may become important [60]. In order to describe
6. Droplet dynamics on a flat substrate Here, we employ a recent variant of the two-phase lattice Boltzmann method in order to investigate the detailed dynamics of fluid drops on solid substrates. The method combines a free energy approach to the liquid–vapor coexistence with a numerical scheme allowing us to literally eliminate the so-called parasitic currents [61]. It is important to realize that the elimination of the spurious currents is an important step towards a reliable description of fluid dynamics inside a droplet. Particular attention has also been paid to the correct implementation of the forcing term [62] in order to minimize error terms arising from the inhomogeneous nature 9
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Figure 11. (a) Fluid momentum in a channel containing a cylindrical droplet which moves under the action of a force proportional to the fluid density. Obviously, there is hardly momentum in the vapor phase. (b) Depiction of the same data viewed from a frame of reference comoving with the drop’s center of mass. The dark (bright) color indicates large (small) density. All quantities are in LB units.
Figure 12. (a) Shear stress across the channel visualized via a color code (dark = high stress). The arrows show the velocity field in the lab frame. The strongest shear stresses occur at the three-phase contact line and decay towards the middle of the droplet. Furthermore, the shear stress is significantly higher close to the substrate than in the center of the drop. In particular, it is practically negligible at the liquid–vapor interface. (b) Velocity profiles along vertical lines across the droplet (i.e. u x versus y ) at different lateral positions x as indicated in the panel (a). (c) A plot similar to (b) but using the momentum data (ρu x (y)). (d) Terminal drop velocity versus drop volume for different choices of kinematic viscosities νliquid and νvapor. Viscous liquid: νliquid > νvapor. Viscous vapor: νliquid < νvapor. Equal viscosities: νliquid = νvapor . All quantities are in LB units.
small. However, as the plot clearly shows, the momentum contained in the vapor phase is negligible as compared to the liquid momentum. This also reflects itself in the significant decrease of the shear stress as the vapor phase is approached (see figure 12). Shear stress plays a central role in theoretical models aimed at a description of the droplet or liquid film dynamics. It is often assumed that shear stress vanishes at the liquid–vapor interface (see e.g. [63]). It is, therefore, important to examine how this quantity varies across the channel in the present case. For this purpose we show in figure 12(a) the shear stress field in the entire channel using a color code. Obviously, the strongest shear stresses occur at the three-phase contact line and close to the liquid–solid contact. Away from the solid, the shear stress decays rapidly. In the center of the droplet, where rotational flow is observed in the comoving frame (figure 11(b)), the
of the mean field type interactions, the latter being essential for the two-phase behavior of the system. Figure 11 shows the results of LB simulations on the dynamics of a two-dimensional droplet under the action of a gravity like force (equivalent to the case of a cylindrical droplet in the 3D studies of [2]). In the left panel, the fluid momentum field is shown in the laboratory frame of reference, while the right panel depicts the same data within a reference frame, which moves with the droplet’s center of mass. While the no-slip boundary condition is evident from the left panel (fluid momentum is zero close to the substrate) an observer moving with the droplet’s center of mass will confirm the presence of a well established rotational flow inside the droplet. Interestingly, similar rotational flows are also observed in molecular dynamics simulations of polymeric liquids [2]. It is noteworthy that the vapor velocity is not necessarily 10
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Figure 13. (a) Selected optical microscopy snap-shots of a toluene droplet (volume 0.3 μl) moving down an incline (from top to bottom, left to right) on a HDMS covered Si substrate. The time between individual pictures is 0.34 s and in total 5.1 s were measured. Each image shows an area of 4.3 × 3.2 mm2 . (b) Terminal droplet velocity v plotted as a function of the contact area A for toluene droplets of different volumes on basic cleaned Si. The solid line is a fit to the data with a model as explained in the text.
the shape is no longer constant. On hexamethyldisiloxane (HDMS) covered Si substrates the contact angle is significantly increased which allowed for experiments with moving toluene droplets of constant shape (see figure 13(a)). The experiments were always prepared on freshly prepared HDMS films. Evaporation of the toluene from the droplet was prevented by a surrounding toluene atmosphere. Figure 13 shows the example of a toluene droplet moving down a HDMS covered Si substrate (inclination angle 60◦ ). Very clearly the droplet exhibits a spherical cap shape and its front is well visible. During movement the droplet does not get expanded along the direction of the acting body force, which results in a constant shape. The movement of the droplet front was considered. For each image the position of the droplet front was probed and a distance–time diagram was extracted from the data. After an acceleration phase the slope of the distance–time diagram is constant being the measured droplet velocity. It has to be noted that unavoidable evaporation causes a deceleration of the droplet for long distances measured from the point of droplet deposition. In addition, it turned out to be more reliable to use the contact area of the droplet with the underlying HDMS covered Si substrate instead of the droplet volume. Depending on the actual volume, and in particular for very small sub-microliter droplets, the real deposited volume and the nominal volume showed non-negligible deviations. Thus figure 13(b) shows the droplet velocities as a function of the contact area. The critical contact area, which droplets needed to have in order to exhibit motion on an incline with 60◦ inclination was A = 2.0 mm2 . Smaller droplets did not move. The data suggest a linear dependence of the terminal droplet velocity on the contact area, if the latter exceeds the mentioned threshold value. However, due to a rather strong scatter of droplet volume in these experiments, it is not possible to make a statement on the relation between the terminal drop velocity and its volume. Note that a linear dependence on the contact area would suggest that the main dissipation mechanism arises from the friction between the substrate and the fluid. Following the analysis in [64], surface dissipation is expected to dominate
shear stress is quite small in accordance with the idea that rotational motion minimizes the dissipation loss. Figure 12(b) depicts the fluid velocity profiles at various vertical cuts across the droplet, i.e. u x versus y for various lateral positions, x . Analytical treatments of droplet motion often assume a simple parabolic shape for the fluid velocity inside the droplet [64]. Interestingly, a parabolic envelope for the velocity field has recently been reported in molecular dynamics simulations [2]. Within our lattice Boltzmann simulations and for the chosen set of parameters, however, we see that the velocity profile changes with the lateral position x in a rather non-trivial way. Parts of this profile can nevertheless be approximated by a parabolic profile. Another closely related issue addressed in our studies is the scaling of terminal (steady state) droplet velocity with droplet volume. As shown in figure 12(d), the steady state drop velocity is a linear function of the drop volume. Interestingly, this behavior does not depend on the ratio of kinematic viscosities of the liquid and vapor phases. A linear dependence between the terminal drop velocity and drop volume has been suggested in [2] for the case of large droplets but the observation of this behavior was hampered by the rather small droplet sizes accessible to molecular dynamics simulations [2]. However, as will be discussed below, providing direct experimental evidence for this relation still remains a challenge.
7. Experimental studies of droplet motion on solid substrates In the experimental part of the study, we use optical microscopy to detect the droplet motion on solid substrates. To simplify the experiments and allow for comparison with theory, simple liquids were used. However, it turned out that the movement of liquids such as toluene on simple substrates such as chemically pre-treated silicon (Si) substrates cannot be easily compared with theory due to the droplet shape. Caused by the small contact angle of toluene droplets on Si substrates (e.g. 5◦ for toluene on basic cleaned Si), the droplets get elongated during movement [3]. As a consequence, 11
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distribution of the shear stress. The latter is quite strong in the proximity of the three-phase contact line, whereas it becomes negligibly small at the liquid–vapor interface. These results call for more elaborate studies of droplet dynamics on solid substrates. A particularly important aspect here is the effect of wettability and topographic roughness as well as chemical patterning on the hysteresis and the resulting pinning. Hysteresis makes reproducible experiments a challenging task. A better understanding of this issue could help to eliminate or at least significantly reduce the hysteresis and related undesired effects. It is also highly desirable to build upon the valuable knowledge gained via these studies in order to design new experiments as well as perform well tuned computer simulations of more complex phenomena such as the flow-mediated deposition of ordered structures on top of solid surfaces.
when the slip length becomes large compared to the vertical dimension of the droplet, which is of macroscopic size (1 mm) in our experiments. However, the sticky behavior observed in the present studies does not support such a large slip length. The present experiments thus must be complemented by new ones allowing safe control of the droplet volume so that reliable data on the volume dependence of the terminal velocity can be obtained. Another interesting challenge for new experiments would be the optical visualization of the droplet velocity in order to obtain direct information on the internal droplet dynamics.
8. Conclusion In this report, we present our recent achievements within the DFG-priority program 1164, Nano- and Microfluidics. In particular, we discuss some selected issues relevant for the behavior of liquid drops on chemically or topographically patterned substrates. An example is the effect of a chemical step on droplet dynamics, the motion of droplets driven by a spatial change of roughness density and the behavior of small droplets on a hydrophobic substrate with a regular topographic pattern. When placed on a perfectly flat substrate, the action of a chemical gradient drives the fluid from a less towards a more hydrophilic region. It is shown how this property can be used in order to separate an emulsion into its individual components. As to the behavior of a droplet on a gradient of topography, it is found that contact angle hysteresis often plays a crucial role. Nevertheless, if care is taken in the choice of the topographic patterns (in order to reduce the hysteresis effects), a motion may be observed. Interestingly, in this case, simple scaling arguments adequately account for the dependence of the droplet velocity on the roughness gradient. In the case of hydrophobic substrates with a periodic arrangement of pillars, it is possible to propose an analytically solvable model for the case where the droplet size becomes comparable to the roughness scale. Two important predictions of the model are highlighted and confirmed via independent numerical simulations. (i) There exists a state with a finite penetration depth, distinct from the full wetting (Wenzel) and suspended (CB) states. (ii) Upon quasi-static evaporation, a droplet initially on the top of the pillars (CB state) undergoes a transition to this new state with a finite penetration depth but then climbs up the pillars and goes back to the Cassie–Baxter state again. Furthermore, we also discuss the dynamics of liquid drops on solid substrates and investigate the motion of drops under the action of a gravity like force and its dependence on parameters such as the drop volume and kinematic viscosity of the liquid and vapor phases. Experiments clearly show a strong effect of the droplet volume. This is best seen by the observation that drops with a contact area smaller than ≈14 mm2 did not move at all. For larger drops it is found that the terminal drop velocity scaled linearly with the drop-substrate contact area. Our lattice Boltzmann computer simulations shed light onto very important aspects of the problem such as the velocity field inside the droplet and the
Acknowledgments We gratefully acknowledge stimulating discussions with various colleagues within the DFG-priority program 1164 Nano- and Microfluidics during the past six years: M M¨uller, B Wu, P Truman, T Haraszti, A Wixforth, M F Schneider, O Vinogradova, J Harting, H Zabel, K Jacobs, M Rauscher. Most of the simulations reported here (sections 3–5) were performed using a variant of the D3Q15 non-ideal fluid LB code provided by A Dupuis. Financial support was provided by the DFG in the framework of the priority program 1164 Nano- and Microfluidics (grats MU1487/2, STA324/27 and VA 205/3).
References [1] Qu´er´e D 2008 Annu. Rev. Mater. Res. 38 71 [2] Servantie J and M¨uller M 2008 J. Chem. Phys. 128 014709 [3] M¨uller-Buschbaum P et al 2011 J. Phys.: Condens. Matter 23 184111 [4] McNamara G and Zanetti G 1988 Phys. Rev. Lett. 61 2332 [5] Higuera F and Jimenez J 1989 Europhys. Lett. 9 663 [6] Benzi R, Succi S and Vergassola M 1992 Phys. Rep. 3 145 [7] Qian Y, d’Humieres D and Lallemand P 1992 Europhys. Lett. 17 479 [8] Gunstensen A and Rothman D 1993 J. Geophys. Res. 98 6431 [9] Varnik F and Raabe D 2006 Modelling Simul. Mater. Sci. Eng. 14 857 [10] Varnik F, Dorner D and Raabe D 2007 J. Fluid Mech. 573 191 [11] Varnik F and Raabe D 2007 Mol. Simul. 33 583 [12] Ahlrichs P and D¨unweg B 1999 J. Chem. Phys. 111 8225 [13] Ladd A and Verberg R 2001 J. Stat. Phys. 104 1191 [14] Dupin M M et al 2007 Phys. Rev. E 75 066707 [15] Kr¨uger T, Varnik F and Raabe D 2011 Comput. Math. Appl. at press [16] Shan X and Chen H 1993 Phys. Rev. E 47 1815 [17] Swift M, Orlandini E, Osborn W and Yeomans J 1996 Phys. Rev. E 54 5041 [18] Luo L S 1998 Phys. Rev. Lett. 81 1618 [19] Benzi R, Chibbaro S and Succi S 2009 Phys. Rev. Lett. 102 026002 [20] Dupuis A and Yeomans J M 2005 Langmuir 21 2624 [21] Dupuis A and Yeomans J M 2006 Europhys. Lett. 75 105 [22] Sbragaglia M and Succi S 2006 Europhys. Lett. 73 370 [23] Gross M, Varnik F and Raabe D 2009 Europhys. Lett. 88 26002
12
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F Varnik et al
[24] Leopoldes J, Dupuis A, Bucknall D G and Yeomans J M 2003 Langmuir 19 9818 [25] Dupuis A and Yeomans J M 2004 Future Gener. Comput. Syst. 20 993 [26] Chang Q and Alexander J 2006 Microfluidics Nanofluidics 2 309 [27] Varnik F et al 2008 Phys. Fluids 20 072104 [28] Moradi N, Varnik F and Steinbach I 2010 Europhys. Lett. 89 26006 [29] Rothman D H and Zaleski S 1997 Lattice-Gas Cellular Automata (Simple Models of Complex Hydrodynamics) (Cambridge: Cambridge University Press) [30] Wolf-Gladrow D 2000 Lattice-Gas Cellular Automata and Lattice Boltzmann Models (Berlin: Springer) [31] Succi S 2001 The Lattice Boltzmann Equation: For Fluid Dynamics and Beyond, Numerical Mathematics and Scientific Computation (Oxford: Oxford University Press) [32] Chen S and Doolen G 1998 Annu. Rev. Fluid Mech. 30 329 [33] Raabe D 2004 Modelling Simul. Mater. Sci. Eng. 12 R13 [34] Swift M, Osborn W and Yeomans J 1995 Phys. Rev. Lett. 75 830 [35] Holdych J-G G D J, Rovas D and Buckius R O 1998 Int. J. Mod. Phys. C 9 1393 [36] Briant A J, Wagner A J and Yeomans J M 2004 Phys. Rev. E 69 031602 [37] Trumann P 2007 PhD Thesis Technical University of Dresden [38] de Gennes P G 1985 Rev. Mod. Phys. 57 827 [39] Zhu L, Feng Y, Ye X and Zhou Z 2006 Sensors Actuators A 130/131 595 [40] Yang J, Chen J, Huang K and Yeh J 2006 J. Microelectromech. Syst. 15 697 [41] Shastry A, Case M and B¨ohringer K 2006 Langmuir 22 6161
[42] Reyssat M, Pardo F and Qu´er´e D 2009 Europhys. Lett. 87 36003 [43] Fang G, Li W, Wang X and Qiao G 2008 Langmuir 24 11651 [44] Cassie A B D and Baxter S 1944 Trans. Faraday Soc. 40 546 [45] Gao L and McCarthy T 2007 Langmuir 23 3762 [46] Kusumaatmaja H and Yeomans J M 2007 Langmuir 23 6019 [47] Seemann R et al 2005 Proc. Natl Acad. Sci. USA 102 1848 [48] Blossey R 2003 Nature Mater. 2 301 [49] Narhe R D and Beysens D A 2004 Phys. Rev. Lett. 93 076103 [50] Dorrer C and R¨uhe J 2007 Langmuir 23 3820 [51] Wenzel R N 1936 Indust. Eng. Chem. 28 988 [52] Lafuma A and Qu´er´e D 2003 Nature Mater. 2 457 [53] McHale G et al 2005 Langmuir 21 11053 [54] Moulinet S and Bartolo D 2007 Eur. Phys. J. E 24 251 [55] Reyssat M, Yeomans J M and Qu´er´e D 2008 Europhys. Lett. 81 26006 [56] Kusumaatmaja H, Blow M L, Dupuis A and Yeomans J M 2007 Europhys. Lett. 81 36003 [57] Moradi N, Varnik F and Steinbach I 2010 in preparation [58] Gross M, Varnik F, Raabe D and Steinbach I 2010 Phys. Rev. E 81 051606 [59] Jung Y C and Bhushan B 2007 Scr. Mater. 57 1057 [60] Br¨ucker C 2011 J. Phys.: Condens. Matter 23 184120 [61] Lee T and Fischer P E 2006 Phys. Rev. E 74 046709 [62] Guo Z, Zheng C and Shi B 2002 Phys. Rev. E 65 04630 [63] Landau L D and Lifschitz E M 1991 Hydrodynamik vol 6 (Berlin: Akademie) [64] de Gennes F B P G and Qu´er´e D 2004 Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves (New York: Springer)
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