influence of meniscus curvature on slippage

Proceedings of the 3rd European Conference on Microfluidics - Microfluidics 2012 - Heidelberg, December 3-5, 2012

µFLU12-100

Molecular Dynamics of TIP4P water over OTS-SAM patterned surfaces: influence of meniscus curvature on slippage. Daniele Gentili1 , Guido Bolognesi2, Mauro Chinappi3 , Alberto Giacomello1 , Carlo Massimo Casciola1 1 Dept. Mechanical and Aerospace Engineering, Sapienza University of Rome, Via Eudossiana 18, 00184 Rome, Italy [email protected], [email protected], [email protected] 2 Dept. of Chemistry, Imperial College London, South Kensington Campus, SW7 2AZ London, UK [email protected] 3 Dept. of Physics, Sapienza University of Rome, Via Eudossiana 18, 00184 Rome, Italy [email protected]

KEY WORDS Nanofluidics, liquid slippage, superhydrophobicity

ABSTRACT Hydrophobic micro-nano patterned surfaces can show remarkable properties when they are in contact with a liquid. Under certain thermodynamic conditions, the liquid at the interface does not fill the pattern of the surface due to the hydrophobicity of the wall and a composite liquid-gas interface is formed (Cassie state). This is the case of the socalled superhydrophobic surfaces. Superhydrophobic surfaces have been shown to induce a significant slippage for water flows inside micro-nanofluidics devices. In the most recent literature several scaling laws have been proposed to relate the slippage to the geometry of the patterns. In most studies the liquid-gas interface is assumed flat and only recently the actual geometry of the meniscus is taken into account. In the present work we focus our attention on the evaluation of the slippage of liquid water flowing on a nano-patterned hydrophobic surface via all-atoms Molecular Dynamics (MD) simulations. A Couette flow of TIP4P/2005 water is simulated on a solid surface endowed with a periodic pattern of nanometer-sized holes and coated with a Self Assembled Monolayer (SAM) of Octadecyltrichlorosilane (OTS). Particularly, we explore how the filling of the nano-corrugation influences the wall slippage showing preliminary results on the effect of meniscus curvature.

1. INTRODUCTION In the last two decades much work has been devoted to the comprehension of flows at the micro and nanoscale [1]. It is nowadays clear that in micro and nanoconfined environment the usual no-slip boundary condition for a liquid flowing on a solid surface fails to reproduce the experimental observations [2]. At these scales the so-called Navier boundary condition (also referred as partial slippage) is considered more appropriate. This reads vw = Ls dv(z)/dz, where v(z) is the velocity profile with z the wall-normal coordinate, vw is the slip velocity and Ls is the slip-length. In the linear regime, the slip length can be interpreted as the extrapolated distance below the wall where the fluid velocity would vanish. When Ls is small as compared to the characteristic scale of the flow, the usual no-slip boundary condition is recovered. The presence of a slip velocity has important consequences for liquid flows inside micro-nanochannels in terms of drag reduction and flow pattern. Slippage may be classified into two broad classes: intrinsic (or molecular) slip, and apparent slip [3]. In the first case, a non-vanishing velocity at the smooth wall results from the sliding of the first few liquid molecular layers on the solid surface. In the second case (apparent slip) the slip velocity vw is due to the alternation of different local properties of the interfaces. Such a non-vanishing average velocity at the nominal wall is an effective feature which belongs to an intermediate scale, since the microscopic details of the actual solid surface have been averaged out. The typical situation is that of air (or vapour) bubbles trapped in the

1

c

SHF 2012


micro-nano hollows of the solid surfaces. In this case the liquid experiences a composite interface made of solid-liquid zones alternated by liquid-vapour zones. In a continuum framework, this translates in mixed local boundary conditions, namely no-slip for the solid-liquid interfaces and free-slip for the liquid-vapour ones. In the past decade several authors studied the slip properties of liquid flows past both smooth and patterned surfaces. Experimental techniques include micro Particle Image Velocimetry (µPIV) [4, 5, 6], Surface Force Apparatus (SFA) [7, 8], Total Internal Reflection Velocimetry (TIRV) [9, 10, 11], and rheological methods [12, 13]. A review on the experimental techniques, in particular for apparent slippage on superhydrophobic surfaces, is given, for example, in [14]. Concerning numerical approaches, atomistic simulations have been applied to investigate both molecular and apparent slippage at the nanoscale [15, 16, 17, 18], while different continuum models have been employed to study apparent slippage on rough surfaces [19, 20, 21]. Analytical solutions of the Stokes equation with patterned boundary conditions, alternating free-slip and no-slip patches [22], show that, in the case of a flat meniscus and grooved geometry, the slip depends on the solid fraction φS of the contact area and on the length scale of the groove. Scaling laws for the slip length are also discussed in literature [23], and confirm this picture for generic geometries. The results of these studies show that slip lengths on the order of nanometers, or few decades of nanometers at most [7, 24], are obtained for intrinsic slippage. Large slippage can be achieved instead when air (or vapour) bubbles are trapped within the wall hollows. In this case the liquid is said to be in the superhydrophobic Cassie state [25]. On the contrary, the liquid is in Wenzel state [26] when it completely wets the solid surface. For a more detailed description of Cassie and Wenzel states, their relation with superhydrophobicity, and the related fabrication strategies, the interested readers are referred to [27, 28, 29] and the review paper [30]. For our present purposes, it is worth noting that the stable Cassie state may not correspond to a flat meniscus. Indeed free energy MD simulations show a stable Cassie-like state with high meniscus curvature protruding toward the liquid for an equilibrium liquid-vapour interface [31]. In principle, the curved meniscus may drastically change the flow field, affecting the apparent slip length. For example, a surface with micron scale holes filled with gas bubbles protruding into the liquid may produce negative slip lengths [32], as also expected from a theory based on the continuum picture [33].

The aim of the present paper is to analyse the flow of liquid water on a surface presenting nanohollows, in order to separate the effects of meniscus curvature and water penetration on the resulting liquid-surface slippage. To this purpose we use Molecular Dynamics (MD) simulations of a realistic system where T IP 4P/2005 water [34] is in contact with a surface functionalized with a model Octadecyltrichlorosilane (OTS) coating, which is a very common hydrophobic functionalization [35, 15, 7]. The water model reproduces the actual water viscosity and surface tension. The preliminary results indicate that the depth of water penetration and the meniscus protrusion have a large influence on liquid slippage.

2. SIMULATION SET-UP The system consists of two parallel walls that confine a water slab. A patterned hydrophobic coating covers the bottom wall. The solid walls are composed by a face centred cubic (fcc) crystal of Lennard Jones (LJ) atoms. ˚ −2 on The parameters σwall and ǫwall of the LJ model are chosen to achieve the atom density ρ111 = 1/22 A the 111 plane (here parallel to the Oxy plane) at temperature T = 300 K and pressure P = 1 bar, which is the same surface density of the OTS layer. The hydrophobicity is induced by a coating of alkyl chains with the terminal methyl group exposed to water. Their anchoring to the solid substrate is achieved by a terminal Lennard-Jones atom in direct topological connection with the rest of the alkyl chain [16]. Two molecules characterised by a different number of backbone carbons (S11 molecules with 10 carbons and S29 with 28 carbons, where the first carbon is replaced by a LJ atoms connected with the solid wall as explained above) ˚ with S11 were used to pattern the surfaces (Fig.1). They are arranged to shape a hole of radius r = 29A, molecules forming the hollow. The T IP 4P/2005 water model we use is a rigid four site model which consists of three fixed point charges and one Lennard Jones centre [34]. The T IP 4P/2005 water model is preferred over other less computationally demanding alternatives since it better reproduces viscosity and surface tension of actual liquid water [37]. The top-wall upper layer and the bottom-wall lower layer atoms are constrained in ˚ z the current position of the z direction by a harmonic potential 1/2k(z − z0 )2 with k ≃ 10 kcal/(molA), the atom, and z0 its equilibrium position. The height h of the overall solid-liquid-vapour system is defined as the difference between the equilibrium positions z0 of top and bottom walls. It is hence possible to control the

2

c

SHF 2012


Figure 1: Left) Four periodical images of the simulation cell surface: the solid LJ wall (green) is covered by OTS-SAM (cyan) of two different heights arranged to pattern a nano-hole on the surface. The solid-fraction is φs ≃ 0.72. Right) Set up of the non equilibrium simulations for the generation of the Couette flow. All snapshots are visualised by VMD, Visual Molecular Dynamics software [36].

volume available to the liquid water by suitably changing the height h. Periodic boundary conditions are applied ˚ and to all the three directions. The wall parallel dimensions of the basic computation cell are Lx ≃ 105.84 A ˚ along the x and y axis, respectively. A clearance between each z-image is also introduced, as Ly ≃ 91.66 A

Figure 2: Upper panels. Snapshots of the constant volume equilibrium simulation reporting the instantaneous ˚ atom configurations in a thin wall-normal slice through the hole centre. From left to right, case a) h = 182 A, ˚ c) h = 172 A, ˚ and d) h = 164 A. ˚ Lower panels. Corresponding average meniscus conb) h = 175 A, figurations visualised by three different density isolines, ρ = 400 kg/m3 (blue), ρ = 600 kg/m3 (green), and ρ = 800 kg/m3 (red). The solid line overlapping the blue symbols is the circular arc best-fitting the ρ = 400 kg/m3 isoline.

3

c

SHF 2012


in [16]. The resulting system consists of 32442 water molecules. All simulations have been performed using NAMD software [38]. Four different cases, corresponding to different heights are here addressed. In the snapshots of Fig. 2, the volume available to water decreases from left to right, passing from a configuration with negative meniscus curvature (case a) to a fully wet condition (case d, Wenzel state). A first issue is characterising the shape of the liquid-vapour interface. To this purpose, we exploit the liquid density field. The density is first computed by splitting the three-dimensional domain in small boxes and by measuring the average number of oxygen atoms in each box at equilibrium. After extracting the density isolines on a wall-normal plane passing through the hole centre, the density isoline corresponding to ρ(x) = 400Kg/m3 is eventually fitted to a circle. The fitted menisci are shown in the bottom panels of Fig. 2 with the curvature radii R of the corresponding circular arcs provided in Table 1. In case d the water completely fills the hollow, corresponding to the Wenzel state. In such a case the contiguous S11 molecules’ layer on the bottom surface is no longer compact and liquid water molecules penetrate the surface coating. This event is indeed compatible with the coating molecule flexibility and it is not expected to have significant impact on slippage.

3. RESULTS AND DISCUSSION

0.4

0.3

1200

0.2

800

0.1

400

0.3 0.2 0.1

2

3

4

5

−20 0 20 40 60 80 100 120

0.4

1600

0.4

0.3

1200

0.2

800

0.1

400

0.3 0.2 0.1

−0.1

0

〈V’y,wall〉 = 0.238 ± 0.007 Å/ps

0

1

2

3 t [ns]

0

−400

0.5

0

10

−5

−20

0

20

40 60 zmin [Å]

80 100 120

−20

0

20

40 60 zmin [Å]

80 100 120

z [Å]

Vy(z) [Å/ps]

V’y,wall (t) [Å/ps]

t [ns]

15 5

0

4

5

−0.1

0 −20 0 20 40 60 80 100 120

z [Å]

−400

30 25 20 Ls [Å]

1

20

3

0

−0.1

25

ρ [kg/m ]

−0.1

0

〈V’y,wall〉 = 0.288 ± 0.007 Å/ps

0

30

Ls [Å]

1600

3

0.4

ρ [kg/m ]

0.5

Vy(z) [Å/ps]

V’y,wall (t) [Å/ps]

For each examined case, the last configuration of the equilibration run has been used as initial condition for the non-equilibrium simulation aimed at measuring the slip length Ls . In addition to the constraints acting in the wall normal direction to keep the distance between the two walls, a force in the y direction is applied to all the atoms of the upper wall (see right panel of Fig. 1). The force intensity is chosen to produce a wall sliding p velocity one order of magnitude below the water thermal speed which is defined as Vt = kB T /mH2 O , where mH2 O is the molecular mass of water. Accordingly, we get Vt ≃ 372.25 m/s at T = 300 K. The module of the force is sufficiently high to accurately extract the motion of the wall from the background thermal noise. For instance in the case b a force Fatom ≃ 8 · 10−2 pN is applied on each atom of the upper wall resulting in a total force F ≃ 176 pN and a total shear τ ≃ 1.82 M P a. In this way, the velocity reached by the upper wall at a stationary state is Vwall ≃ 25.8 m/s which is enough for our purposes. The lower wall is thermostated at T = 300 K to avoid interference with the water velocity field. After the wall velocity reaches the stationary state (see the leftmost panels of Fig. 3), positions and velocities of water oxygen atoms are sampled to evaluate

15 10 5 0 −5

Figure 3: Leftmost panels. Time evolution of the upper wall velocities for case a (top) and case c (bottom). The average value (red line) of the wall velocity is calculated for t > teq (solid vertical line). Centre panels. Average velocity (dashed line) and density (dotted line) profiles for case a (top) and case c (bottom). The position of the nominal upper and bottom walls are reported as vertical solid lines. Rightmost panels. Slip length Ls and its sensitivity to the lower bound zmin of the fitting interval for case a (top) and case c (bottom). The other end of the fitting interval, namely zmax , is kept fixed (solid line).

4

c

SHF 2012


Case a Case b Case c Case d

˚ h[A] 182 175 172 164

˚ Vwall [A/ps] 0.288 0.258 0.238 0.228

˚ Ls [A] 19.21 ± 1.31 17.14 ± 1.06 7.90 ± 0.89 3.12 ± 0.75

˚ R [A] 71.08 271.17 72.62 /

θ 36.2◦ ± 4.12◦ −− ◦ 35.0 ± 4.63◦ −−

Table 1: Height of the channel h, velocity of the upper wall Vwall , slip length Ls , meniscus curvature radius R, and protrusion angle θ in the four cases shown in Fig. 2. Case d) correspond to the completely wet Wenzel state.

the average velocity profile (Fig. 3 centre). Ls is estimated by fitting the profile in the bulk, namely in the ˚ away from the nominal wall which is defined by the average position of the methyl group carbon region 10 A atom of the longest coating molecules (S29). In order to check the robustness of the results to the somewhat arbitrary definition of the bulk region, the estimate is repeated changing its lower boundary zmin . The results are reported in the rightmost panels of Fig. 3. The resulting slip length Ls is provided in Table 1 for the four cases considered. Several considerations follow: i) in cases a and b, the slippage is significantly larger than the ˚ as reported intrinsic slippage measured on a flat surface with a similar simulation protocol (namely, ≃ 5 − 10 A in [15, 16]), ii) once the liquid penetrates the hollow the slippage reduces to the intrinsic slip regardless of the penetration depth, iii) when the meniscus points towards the bulk (case a) the value of Ls is larger than the one observed for the flat interface (case b). Considering issue i), the slippage enhancement with respect to a plain smooth surface [15] due to surface roughness can be compared to the continuum model predictions reported in literature. Indeed the available estimates for slippage on a composite surface provide expressions for Ls in terms of the solid fraction φs (i.e. the ratio between the solid-liquid interface area and the whole nominal area) and intrinsic slippage Ls,in at the solid-liquid interface for different patterns. In our case the closest model is the one analysed by Ng et al. [19] where a Stokes flow is solved on a periodic pattern for both holes and pillars geometries. In the case of holes they propose the following expression Ls,in (1) Ls = Ls,0 + φs where Ls,0 = A (−0.134 ln(φs ) − 0.023) is the estimate for the slip length when the no-slip condition is assumed at pthe solid-liquid interface [19]. The parameter A is the characteristic size of the cell estimated here ˚ as A = Lx Ly and Ls,in is the intrinsic slippage on the solid-liquid interfaces. Assuming Ls,in = 10 A 2 ˚ (see [15]) and φs = (Lx Ly − πr )/Lx Ly ≃ 0.73, we obtain Ls ≃ 16 A. In order to account for the actual geometry here examined, we also implemented and adapted the Ng and Wang [19] computational scheme to our ˚ which is very similar to the one obtained system. The slip length resulting from such an analysis is Ls ≃ 17 A, through Eq.(1). This value is in very good agreement with the one we observe in case b with MD simulations. It is interesting to note how the effect of slippage enhancement due to surface roughness is captured by a very simplified continuum approach, confirming the usefulness of the continuum description in nanofluidics also in the case of multiphase systems. Consideration ii) follows directly from the comparison of cases c and d, with cases a and b. Particularly, as opposed to cases a and b, where the hollow is empty, the case c (hollow partially filled) and case d (hollow completely filled) result in very similar slip lengths which are indeed as small as those corresponding to a perfectly planar, hollow-free surface. This suggests that the surface roughness can effectively enhance the liquid/surface slippage (i.e. issue i)) only if the roughness cavities remain completely empty. In this scenario, the penetration depth of liquid water inside the cavities would not be a sensitive parameter for the resulting slip length. In this respect, the detrimental effects of the liquid penetration in terms of reduced slippage have been already highlighted by [39] via continuum simulations of a micro-grooved surfaces. In that study, it is observed that the liquid penetration progressively reduces the liquid/surface slippage. In our case, we cannot observe a similar smooth transition and the slippage behaviours of an empty cavity and a partially filled one are quite different. However, we remind that the study of [39] differs from ours in both the geometry and the typical length scale of the surface roughness and this might justify the slightly different results. Finally, we conclude with the much more intriguing issue iii), related to curvature effect on surface slippage. In the recent literature, the influence of the meniscus curvature was studied as a function of the so called protrusion angle θ, namely, the angle that the liquid meniscus forms with the nominal wall where it meets the

5

c

SHF 2012


solid surface. If the meniscus is outside the cavity, θ is positive, whereas θ is negative when the meniscus is inside. Recent numerical and experimental studies by Steinberger et al. [32] of liquid slippage over bubble mattresses have shown that meniscus curvature has a negative impact over the slippage: the largest slippage is obtained when the liquid-gas interface is flat (θ = 0). Moreover when the protrusion angle θ exceeds a certain critical value, the slip-length Ls becomes even negative resulting in an enhancement of the apparent friction. This scenario seems to be confirmed by the Lattice-Boltzmann simulations by Hyvaluoma et al. [40] and by the theoretical analysis of Davis and Lauga [41]. Nevertheless, in both these studies a small increase in Ls is observed for low positive values of the protrusion angle, θ ∈ (0, 25◦ ). Similar results are reported by Teo and Khoo[42] via simulations of the Couette flow over transverse grooves. The estimate of the protrusion angle from MD simulation is not straightforward since different procedures are possible and no one of them appears to be more appropriate than the others. We arbitrarily define the protrusion angle as the one formed by the best-fitted circle and the nominal wall. According to such operative definition, we obtain the values reported in Table 1. Case a) is in slight disagreement with the predictions of the above mentioned papers, according to which the value of Ls is expected to be (at that protrusion angle) smaller than the one corresponding to the flat meniscus. Several explanations of this discrepancy are possible: from an improper definition of protrusion angle in the MD simulation to an actual phenomenological difference possibly due to intrinsic slippage or failure of the continuum assumption. Although the results discussed in this proceedings do not allow to discriminate between the different arguments, a few considerations may be attempted. In particular we observe contact line depinning. This results in an actually larger liquid-vapour interface than estimated in the mentioned articles where the triple line always coincides with the corner of the hole. In principle this might justify the observed discrepancy between MD and continuum simulations. Clearly, in order to settle the issue the study needs to be extended to different geometries and systematically compared with the outcomes of the available continuum models.

4. CONCLUSIONS In this paper we performed full atom molecular dynamics simulations aimed at characterising the effect on wall slippage of water filling a surface hollow. Our work represents a first step towards the systematic study of the topic. Particularly, varying the distance between the two solid walls the water slab is confined to, we isolated and analysed four different scenarios: empty hollow with curved meniscus pointing towards the bulk, empty hollow with flat meniscus, partially filled hollow with a curved meniscus pointing outwards and completely filled hollow. The preliminary results indicate that for the investigated geometry the depth of water filling and meniscus protrusion have a large influence on liquid slippage. However, when those results are compared to the continuum theory and to the numerical simulations reported in literature, different behaviours emerge. In particular, in our study a low partial filling of the hollow results in a significant drop of slip length whose value is close to the intrinsic slippage of an ideally flat surface. In view of that, partial filling of the hollow can be as detrimental as complete filling. On the contrary, continuum theoretical models for micro-structured surfaces [39] show a more progressive reduction of slippage with liquid penetration depth. Interestingly, a behaviour specific to our simulations concerns meniscus curvature. In theoretical and numerical studies of continuum models [40, 41], meniscus curvature is recognised to negatively affect wall slippage. In this study, we show that a meniscus pointing towards the bulk may instead result in a larger slippage than for a flat meniscus. At this stage of the study, we cannot assess whether or not we have consistency with the predictions of the continuum theory and with the experimental results on micron-sized systems reported in literature [32]. Indeed, many factors may affect the results, starting from the procedure adopted to compare the outcomes of discrete and continuum models (e.g., the definition of the protrusion angle). In any case, in the specific instance of protruding meniscus, we observed contact line depinning resulting in a larger liquid-vapour interface which might affect the measured wall slippage. The promising results herein reported naturally call for further and systematic investigations of this subject in order to better understand and clarify the mechanics through which the liquid penetration depth and meniscus curvature affect the liquid slippage on nano-structured surfaces.

ACKNOWLEDGEMENTS Computing resources were made available by CASPUR under HPC Grant 2011 and 2012.

6

c

SHF 2012


REFERENCES AND CITATIONS [1] T.M. Squires and S.R. Quake. Microfluidics: Fluid physics at the nanoliter scale. Reviews of Modern Physics, 77(3):977, 2005. [2] L. Bocquet and J.L. Barrat. Flow boundary conditions from nano- to micro-scales. Soft Matter, 3(6):685– 693, 2007. [3] E. Lauga, M.P. Brenner, and H.A. Stone. Microfluidics: The no-slip boundary condition, in Handbook of Experimental Fluid Dynamics, Chap 15. Springer, Berlin, 2005. [4] P. Joseph and P. Tabeling. Direct measurement of the apparent slip length. Physical Review E, 71(3):035303, 2005. [5] P. Tsai, A.M. Peters, C. Pirat, M. Wessling, R.G.H. Lammertink, and D. Lohse. Quantifying effective slip length over micropatterned hydrophobic surfaces. Physics of Fluids, 21:112002, 2009. [6] G. Bolognesi. Optical studies of micron-scale flows. Lap Lambert Academic Publishing, 2012. [7] C. Cottin-Bizonne, A. Steinberger, B. Cross, O. Raccurt, and E. Charlaix. Nanohydrodynamics: The intrinsic flow boundary condition on smooth surfaces. Langmuir, 24(4):1165–1172, 2008. [8] L. Zhu, C. Neto, and P. Attard. Reconciling slip measurements in symmetric and asymmetric systems. Langmuir, 28(20):7768–7774, 2012. [9] S. Jin, P. Huang, J. Park, JY Yoo, and KS Breuer. Near-surface velocimetry using evanescent wave illumination. Experiments in fluids, 37(6):825–833, 2004. [10] P. Huang, J.S. Guasto, and K.S. Breuer. Direct measurement of slip velocities using three-dimensional total internal reflection velocimetry. Journal of Fluid Mechanics, 566(447-464):5, 2006. [11] H. Li and M. Yoda. An experimental study of slip considering the effects of non-uniform colloidal tracer distributions. Journal of Fluid Mechanics, 662, 2010. [12] C. Lee and CJ Kim. Influence of surface hierarchy of superhydrophobic surfaces on liquid slip. Langmuir: the ACS journal of surfaces and colloids, 27(7):4243, 2011. [13] Z. Ming, L. Jian, W. Chunxia, Z. Xiaokang, and C. Lan. Fluid drag reduction on superhydrophobic surfaces coated with carbon nanotube forests (cnts). Soft Matter, 7(9):4391–4396, 2011. [14] A. Maali and B. Bhushan. Measurement of slip length on superhydrophobic surfaces. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 370(1967):2304–2320, 2012. [15] M. Chinappi, F. Gala, G. Zollo, and CM Casciola. Tilting angle and water slippage over hydrophobic coatings. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 369(1945):2537–2545, 2011. [16] M. Chinappi and CM Casciola. Intrinsic slip on hydrophobic self-assembled monolayer coatings. Physics of Fluids, 22:042003, 2010. [17] C. Cottin-Bizonne, J.L. Barrat, L. Bocquet, and E. Charlaix. Low-friction flows of liquid at nanopatterned interfaces. Nature Materials, 2(4):237–240, 2003. [18] H. Zhang, Z. Zhang, and H. Ye. Molecular dynamics-based prediction of boundary slip of fluids in nanochannels. Microfluidics and Nanofluidics, pages 1–9, 2012. [19] C.O. Ng and CY Wang. Apparent slip arising from stokes shear flow over a bidimensional patterned surface. Microfluidics and Nanofluidics, 8(3):361–371, 2010. [20] R. Benzi, L. Biferale, M. Sbragaglia, S. Succi, and F. Toschi. Mesoscopic two-phase model for describing apparent slip in micro-channel flows. EPL (Europhysics Letters), 74:651, 2006. ´ Charlaix, L. Bocquet, and J.L. Barrat. Dynamics of simple liquids at [21] C. Cottin-Bizonne, C. Barentin, E. heterogeneous surfaces: Molecular-dynamics simulations and hydrodynamic description. The European Physical Journal E: Soft Matter and Biological Physics, 15(4):427–438, 2004. [22] E. Lauga and H.A. Stone. Effective slip in pressure-driven stokes flow. Journal of Fluid Mechanics, 489(1):55–77, 2003. [23] C. Ybert, C. Barentin, C. Cottin-Bizonne, P. Joseph, and L. Bocquet. Achieving large slip with superhydrophobic surfaces: Scaling laws for generic geometries. Physics of fluids, 19:123601, 2007. [24] CI Bouzigues, P. Tabeling, and L. Bocquet. Nanofluidics in the debye layer at hydrophilic and hydrophobic surfaces. Physical review letters, 101(11):114503, 2008. [25] ABD Cassie and S. Baxter. Wettability of porous surfaces. Trans. Faraday Soc., 40(0):546–551, 1944. [26] R.N. Wenzel. Resistance of solid surfaces to wetting by water. Industrial & Engineering Chemistry, 28(8):988–994, 1936. [27] L. Afferrante and G. Carbone. Microstructured superhydrorepellent surfaces: effect of drop pressure on

7

c

SHF 2012


fakir-state stability and apparent contact angles. Journal of Physics: Condensed Matter, 22:325107, 2010. [28] Neelesh A. Patankar. Consolidation of hydrophobic transition criteria by using an approximate energy minimization approach. Langmuir, 26:8941–8945, 2010. [29] E. S. Savoy and F. A. Escobedo. Molecular simulations of wetting of a rough surface by an oily fluid: Effect of topology, chemistry and droplet size on wetting transition rates. Langmuir, 28:3412–3419, 2012. [30] Xi Zhang, Feng Shi, Jia Niu, Yugui Jiang, and Zhiqiang Wang. Superhydrophobic Surfaces: From Structural Control to Functional Application. J. Mater. Chem., 18:621, 2008. [31] A. Giacomello, S. Meloni, M. Chinappi, and C.M. Casciola. Cassie–baxter and wenzel states on a nanostructured surface: Phase diagram, metastabilities, and transition mechanism by atomistic free energy calculations. Langmuir, 28(29):10764–10772, 2012. [32] A. Steinberger, C. Cottin-Bizonne, P. Kleimann, and E. Charlaix. High friction on a bubble mattress. Nature materials, 6(9):665–668, 2007. [33] M. Sbragaglia and A. Prosperetti. A note on the effective slip properties for microchannel flows with ultrahydrophobic surfaces. Physics of Fluids, 19:043603, 2007. [34] JLF Abascal and C. Vega. A general purpose model for the condensed phases of water: Tip4p/2005. The Journal of chemical physics, 123:234505, 2005. [35] F. Gala and G. Zollo. Functionalization of hydrogenated (111) silicon surface with hydrophobic polymer chains. Physical Review B, 84(19):195323, 2011. [36] W. Humphrey, A. Dalke, K. Schulten, et al. Vmd: visual molecular dynamics. Journal of molecular graphics, 14(1):33–38, 1996. [37] C. Vega and E. de Miguel. Surface tension of the most popular models of water by using the test-area simulation method. Journal of Chemical Physics, 126(15):4707, 2007. [38] J.C. Phillips, R. Braun, W. Wang, J. Gumbart, E. Tajkhorshid, E. Villa, C. Chipot, R.D. Skeel, L. Kale, and K. Schulten. Scalable molecular dynamics with namd. Journal of computational chemistry, 26(16):1781– 1802, 2005. [39] C.O. Ng and CY Wang. Stokes shear flow over a grating: Implications for superhydrophobic slip. Physics of Fluids, 21:013602, 2009. [40] J. Hyv¨aluoma, C. Kunert, and J. Harting. Simulations of slip flow on nanobubble-laden surfaces. Journal of Physics: Condensed Matter, 23:184106, 2011. [41] A.M.J. Davis and E. Lauga. Geometric transition in friction for flow over a bubble mattress. Physics of Fluids, 21:011701, 2009. [42] CJ Teo and BC Khoo. Flow past superhydrophobic surfaces containing longitudinal grooves: effects of interface curvature. Microfluidics and Nanofluidics, 9(2):499–511, 2010.

8

c

SHF 2012