10768 DOI: 10.1021/la901314b Langmuir 2009, 25(18), 10768–10781Published on Web 07/10/2009

pubs.acs.org/Langmuir

© 2009 American Chemical Society

Interfacial Water at Hydrophobic and Hydrophilic Surfaces:

Slip, Viscosity, and Diffusion

Christian Sendner,*,† Dominik Horinek,† Lyderic Bocquet,‡ and Roland R. Netz*,†

†Physik Department, Technische Universit€at M€unchen, 85748 Garching, Germany, and‡LPMCN, University Lyon 1 and CNRS UMR 5586, University of Lyon, 69622 Villeurbanne, France

Received April 14, 2009. Revised Manuscript Received June 11, 2009

The dynamics and structure of water at hydrophobic and hydrophilic diamond surfaces is examined via non-equilibrium Molecular Dynamics simulations. For hydrophobic surfaces under shearing conditions, the generalhydrodynamic boundary condition involves a finite surface slip. The value of the slip length depends sensitively onthe surface water interaction strength and the surface roughness; heuristic scaling relations between slip length,contact angle, and depletion layer thickness are proposed. Inert gas in the aqueous phase exhibits pronouncedsurface activity but only mildly increases the slip length. On polar hydrophilic surfaces, in contrast, slip is absent,but the water viscosity is found to be increased within a thin surface layer. The viscosity and the thickness of thissurface layer depend on the density of polar surface groups. The dynamics of single water molecules in the surfacelayer exhibits a similar distinction: on hydrophobic surfaces the dynamics is purely diffusive, while close to ahydrophilic surface transient binding or trapping of water molecules over times of the order of hundreds ofpicoseconds occurs. We also discuss in detail the effect of the Lennard-Jones cutoff length on the interfacialproperties.

Introduction

For many applications in microfluidic technology and almostall situations in the biological domain, the behavior of interfacialwater is of prime importance. The geometric constraint of a solidsurface, as well as the interactions between water molecules andthe substrate, lead to structural changes of water compared to itsbulk properties. Surfaces can be divided into two classes accord-ing to their affinity to water: hydrophilic, water attracting, andhydrophobic, water repellent. Since water molecules interactfavorably with surface charges, surfaces which bear electriccharges or polar groups typically are hydrophilic. In contrast,non polar surfaces are generally hydrophobic since the watermolecules suffer a loss of hydrogen bonding partners at theinterface. This classification into hydrophobic and hydrophilicsurfaces has profound bearing on diverse situations such asinteracting colloidal particles, protein folding, and adsorptionof molecules or more complex structures on surfaces.

Over the past years it has become increasingly clear that the no-slip boundary condition, that is, the condition of zero interfacialfluid velocity, does not necessarily hold at nanoscopic lengthscales.1-3 In fact, the hydrodynamic boundary condition at theliquid/solid interface is of particular importance for microfluidicapplications4,5 or biological nanoscale scenarios, such as thetransport through membrane channels.6 But even in seeminglyunrelated areas such as the automobile industry the importance ofsurface slip effects is acknowledged. As an example, mechanical

components which are coated with hydrophobic, diamond-likecarbon exhibit favorable friction properties,7,8 which could be putto good use in ball bearings and gear boxes to increase efficiency.Likewise, surface slippage amplifies the flow rate for pressuredriven flow, which enhances fluid transport in narrow channels.Clearly, a noticeable increase of fluid transport is only obtained ifthe slip length is comparable to the channel dimension. Forelectrically driven flow, on the other hand, even small slip lengthsin the nanometer range lead to a considerable increase in flow.9,10

All these examples demonstrate that a profound and microscopicunderstanding of the flow boundary condition at surfaces isnecessary.

In this paper we analyze the structure and dynamics ofinterfacial water via molecular dynamic (MD) simulations. Toset the stage, we first consider equilibrium properties of water incontact with a hydrophobic diamond-like surface. By variation ofthe surface water affinity, the hydrophobicity of the surface iscontrolled and quantified by the contact angle. Here we alsodiscuss the influence of the Lennard-Jones cutoff length, aparameter that in simulations of interfaces plays a non-negligiblerole.11,12 Next we study the hydrodynamic boundary condition atthese surfaces by non-equilibrium shear flow simulations. The sliplength at the hydrophobic diamond surface is found to be in thenanometer range. The slip length exhibits a quasi-universalrelation with the contact angle and with the depletion thickness,for which we advance some simple scaling ideas.13 We alsoconsider the effect of varying themagnitude of surface-roughness.Recent experiments yield slip lengths in the range of tens of

*To whom correspondence should be addressed. E-mail: csendner@ph.tum.de (C.S.), netz@ph.tum.de (R.R.N.).(1) Thompson, P. A.; Robbins, M. O. Phys. Rev. A 1990, 41, 6830–6837.(2) Lauga, E.; Brenner, M. P.; Stone, H. A. Handbook of Experimental Fluid

Dynamics; Springer: New York, 2007; Vol. Chapter 19.(3) Bocquet, L.; Barrat, J. L. Soft Matter 2007, 3, 685–693.(4) Stone, H. A.; Stroock, A. D.; Ajdari, A. Annu. Rev. Fluid Mech. 2004, 36,

381.(5) Squires, T. M.; Quake, S. R. Rev. Mod. Phys. 2005, 77, 977.(6) de Groot, B. L.; Grubmuller, H. Science 2001, 294, 2353.

(7) De Barros Bouchet, M.; Matta, C.; Le-Mogne, T.; Michel Martin, J.;Sagawa, T.; Okuda, S.; Kano, M. Tribol. Mater. Surf. Interfaces 2007, 1, 28.

(8) Kano, M. Tribol. Int. 2006, 39, 1682.(9) Joly, L.; Ybert, C.; Trizac, E.; Bocquet, L. Phys. Rev. Lett. 2004, 93, 257805.(10) Bouzigues, C.; Tabeling, P.; Bocquet, L. Phys. Rev. Lett. 2008, 101, 114503.(11) Ismail, A. E.; Grest, G. S.; Stevens, M. J. J. Chem. Phys. 2006, 125, 014702.(12) in ’t Veld, P. J.; Ismail, A. E.; Grest, G. S. J. Chem. Phys. 2007, 127, 144711.(13) Huang, D. M.; Sendner, C.; Horinek, D.; Netz, R. R.; Bocquet, L. Phys.

Rev. Lett. 2008, 101, 226101-4.

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Sendner et al. Article

nanometers for water,14-18 but in the past values up to micro-meters have been reported.19,20 A collection of experimental andtheoretical results can be found in ref 2. As a possible explanationfor the large slip lengths in experiments the presence of a thin gaslayer at the surface was considered.21-23 However, in simulationsof a Lennard-Jones liquid it was found that the slip length is onlymoderately increased in the presence of dissolved gas.24Using ourrealistic water model and gas parameters, we also observe only amodest enhancement of the slippage in the presence of surfaceadsorbed inert gas. The analysis is then extended to polar,hydrophilic surfaces. From the velocity profile in the interfacialregion we obtain zero slip and extract the surface viscosity, whichis increased by a factor of 2 to 4 (depending on the density of polarsurface groups) compared to the bulk value within a layer of a fewangstroms.We do not find evidence for a layer of frozen water orfor an increase of the interfacial viscosity of several orders ofmagnitude at hydrophilic surfaces, as was seen for ultrathin filmsof water on hydrophilic surfaces25,26 and for highly confinedwater between hydrophilic surfaces27-29 both experimentally andtheoretically. In equilibrium simulations we analyze the diffusionof single watermolecules by calculating autocorrelation functionsin slabs. The single water molecule dynamics shows strikinglydifferent behavior on hydrophobic and hydrophilic surfaces: onhydrophobic surfaces, water dynamics is purely diffusive over thefull time range between 1 and 1000 ps. Close to a hydrophilicsurface, on the other hand, diffusion is slowed down by transientbinding or trapping of watermolecules on polar surface sites. Thedecay time of this transient water arrest is of the order of hundredsof picoseconds.

Molecular Dynamics Simulations

In MD simulations, the system is modeled on an atomisticscale, and the trajectories of all constituent particles are calculatedusing Newton’s equations of motion. Particles interact via bond-ing interactions, typically harmonic springs and cosine torsionalpotentials,30 and non-bonded Coulomb and dispersion interac-tions. Dispersion interactions between particle species A and Bwith distance r are described by a 6-12Lennard-Jones potential,

uðrÞ ¼ 4εABσAB

r

� �12

-σAB

r

� �6" #

ð1Þ

Inmost of our simulations, all Lennard-Jones interactions aretruncated at a radiusR0=0.8 nm, as is customarily done to speedup the simulations. The effect of varying values ofR0 on the watercontact angle is discussed in detail below. Long ranged electro-static interactions are calculated with the Particle-Mesh Ewald

(PME)method.31,32 For all simulations,weuse the SPC/E33watermodel. In this three site model the water molecule has partialcharges qO=-0.8476e and qH=0.4238e at the positions of theoxygenandhydrogennuclei. TheOHbond length is 0.1 nmwith atetrahedral bond angle of θ=109.5� at the oxygen position. TheLennard-Jones potential is centered at the oxygen position withparameters given in Table 1. Periodic boundary conditions areapplied in all three spatial directions. The simulations are per-formed in the NAPzT ensemble, that is, at fixed particle numberN, surface area A, temperature T, and vertical pressure Pz, whilethe height of the box is fluctuating. Thewhole system is coupled toa heat bath at 300 K and to a pressure of 1 bar via the Berendsenalgorithm34with coupling constants τT=0.4 ps (temperature) andτp=1.0 ps (pressure). All bonds including hydrogen atoms areconstraint via theLINCS35 algorithm.The simulations are carriedout with the GROMACS36 package.

Hydrophobic Surface. We first consider a hydrophobic,hydrogen terminated diamond surface. The diamond slab ismodeled by 2323 carbon atoms, arranged in a double face-centered-cubic lattice with lattice constant a = 3.567 A. Thesurface normal of the (100) plane points in the ez direction. Thelateral extensionof the slab is 3.0� 3.0 nm2, its thickness is 1.5 nm.Carbon atoms are connected by harmonic bonds and are subjectto angular and torsional potentials of the GROMOS96 version53A6 force field.38 The surface layer of the diamond is recon-structed and terminated by H atoms. Carbon atoms and watermolecules interact via the Lennard-Jones potential in eq 1 withinteraction parameters given in Table 1. Except for the Ne-Ol

and O-Ol interactions and for the interactions involving O2 gas,the combination rules εAB=(εAAεBB)

1/2 and σAB=(σAAσBB)1/2

are used. For the simulations with the diatomic oxygen gas

Table 1. Lennard-Jones GROMOS Forcefield Parameters for

Carbon, SPC/E Water and Noble Gases as Defined in Equation 1a

atom types σ [nm] ε [kJ/mol]

H-H 0.0 0.0C-C 0.3581 0.2774O1-O1 0.3166 0.6502C-O1 0.3367 0.4247

Ne-Ne 0.3136 0.6398Ne-O1 0.3293 0.4951Ne-C 0.3351 0.4213

Ar-Ar 0.3410 0.9964Ar-O1 0.3285 0.8049Ar-C 0.3494 0.5257

Og-Og 0.3030 0.4016Og-O1 0.3098 0.5110Og-C 0.3306 0.3338

O-O 0.2760 1.2791O-C 0.3144 0.5957O-O1 0.3017 0.8070

aThe parameters for gaseous oxygen Og are taken from ref 37.Ol denotes aqueous oxygen while O denotes the COH group of ahydrophilic surface.

(14) Vinogradova, O. I.; Yakubov, G. E. Langmuir 2003, 19, 1227–1234.(15) Joseph, P.; Tabeling, P. Phys. Rev. E 2005, 71, 035303.(16) Maali, A.; Cohen-Bouhacina, T.; Kellay, H. Appl. Phys. Lett. 2008, 92,

053101-2.(17) Cottin-Bizonne, C.; Cross, B.; Steinberger, A.; Charlaix, E. Phys. Rev. Lett.

2005, 94, 056102.(18) Joly, L.; Ybert, C.; Bocquet, L. Phys. Rev. Lett. 2006, 96, 046101.(19) Zhu, Y.; Granick, S. Phys. Rev. Lett. 2001, 87, 096105.(20) Tretheway, D. C.; Meinhart, C. D. Phys. Fluids 2002, 14, L9–L12.(21) Vinogradova, O. I. Langmuir 1995, 1, 2213.(22) de Gennes, P. Langmuir 2002, 18, 3413–3414.(23) Doshi, D. A.; Watkins, E. B.; Israelachvili, J. N.; Majewski, J. Proc. Natl.

Acad. Sci. U.S.A. 2005, 102, 9458–9462.(24) Dammer, S. M.; Lohse, D. Phys. Rev. Lett. 2006, 96, 206101.(25) Odelius, M.; Bernasconi, M.; Parrinello, M.Phys. Rev. Lett. 1997, 78, 2855.(26) Cantrell, W. C.; Ewing, G. E. J. Phys. Chem. B 2001, 105, 5434–5439.(27) Zhu, Y.; Granick, S. Phys. Rev. Lett. 2001, 87, 096104.(28) Zangi, R.; Mark, A. E. Phys. Rev. Lett. 2003, 91, 025502.(29) Li, T. D.; Gao, J. P.; Szoszkiewicz, R.; Landman,U.; Riedo, E.Phys. Rev. B

2007, 75, 115415.(30) Oostenbrink, C.; Villa, A.; Mark, A. E.; Van Gunsteren, W. F. J. Comput.

Chem. 2004, 25, 1656–1676.

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L. G. J. Chem. Phys. 1995, 103, 8577–8593.(33) Berendsen, H. J. C.; Grigera, J. R.; Straatsma, T. P. J. Phys. Chem. 1987, 91,

6269–6271.(34) Berendsen, H. J. C.; Postma, J. P. M.; van Gunsteren, W. F.; DiNola, A.;

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Chem. 1997, 18, 1463–1472.(36) Lindahl, E.; Hess, B.; van der Spoel, D. J. Mol. Modeling 2001, 7, 306–317.(37) Bratko, D.; Luzar, A. Langmuir 2008, 24, 1247–1253.(38) Scott, W. R. P.; Hunenberger, P. H.; Tironi, I. G.; Mark, A. E.; Billeter, S.

R.; Fennen, J.; Torda, A. E.; Huber, T.; Kruger, P.; van Gunsteren, W. F. J. Phys.Chem. A. 1999, 103, 3596–3607.

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10770 DOI: 10.1021/la901314b Langmuir 2009, 25(18), 10768–10781

Article Sendner et al.

the modified combination rule σAB = 0.5(σAA þ σBB) for theLennard-Jones diameter is used.37 The Ne-Ol and O-Ol para-meters are explicitly given in Table 1.

There are different methods to change the hydrophobicity of asurface in simulations, for example, by adding hydrophilic groupsor by rescaling the surface polarity.39-42 Within a certain range,the hydrophobicity of the surface can also be tuned with theliquid/solid interaction energy εCO between surface atoms andwater molecules, which we varied in the range between εCO=0.11to 0.72 kJ/mol, while keeping the Lennard-Jones diameter σCOconstant. Decreasing the liquid/solid interaction energy increasesthe hydrophobicity since water molecules are less attracted by thesolid and the water density close to the interface goes down, asshown in Figure 1a. These density profiles are obtained insimulations of the diamond surface in contact with 1850 watermolecules in a 3.0 � 3.0 � 8.0 nm3 simulation box in the NAPzTensemble. For not too low values of εCO, water layering is foundand the density profile is non-monotonic. For the standardGROMOS value εCO=0.42 kJ/mol, listed in Table 1, the waterdensity in the first layer is roughly twice the bulk density. Low-ering the interaction energy decreases the height of the densitymaximum. The density profile for the lowest interaction energyhas no maximum at all and is similar to the density profile of theair/liquid interface.43 Therefore, by tuning the interaction energy,a smooth transition from a liquid/solid to a liquid/vapor-likeinterface is obtained.

We also considered surfaces with different degrees of nano-roughness, shown inFigure 1. SurfaceR1 is constructedby erasingevery third pair of rows of surface atoms, and surface R2 isobtained by the deletion of every second pair of rows. For theconstruction of R3, every second single row of carbon atoms isdeleted. The roughest surface, R4, is generated by removingcarbon atoms down to the fourth surface layer. One should notethat these nanorough surfaces are very different from so-calledsuperhydrophobic surfaces, for which the surface structuringoccurs on a much larger length scale. From the density profilesin Figure 1b) it is seen that on the rough surfaces water moleculesfill the voids left by the deleted surface atoms.

Hydrophilic Surface. Hydrophilic surfaces are constructedbased on the smooth diamond surface by substituting everyfourth (xOH=1/4) or eighth (xOH=1/8) surface carbon atom bya C-O-H group with standard angular and torsional forcepotentials. The bond angle at the oxygen atom is 108� and thepartial charges are set to qO=-0.674e, qH=0.408e and qC=0.266e, while the standard GROMOS value εCO=0.42 kJ/mol isused. These parameters are taken from the GROMOS96 force-field for serine, the Lennard-Jones parameters are those of theGROMOS96 version 53A6 forcefield, see Table 1. Figure 3 showsthe top view of the two studied hydrophilic surfaces. Torsionalrotations around the C-O bond are in principle possible butseverely inhibited by the large activation energies, so the OHrotational degrees of freedom are effectively quenched.

As can be seen in Figure 1a, the water is closer to the hydro-philic surface compared to the hydrophobic interface, and the firstwater peak is more pronounced and has a smaller width. Onlyminor differences are present between the two hydrophilic sur-faces (data not shown): For the larger OH density, the first waterpeak is slightly higher and closer to the interface.

Contact Angle. One experimentally easily accessible para-meter characterizing the surface hydrophobicity is the contactangle which ranges from 180� (for a hypothetical substrate withthe same water affinity as vapor) down to 0� for a hydrophilicsurface. On smooth hydrophobic surfaces, contact angles up to130� are experimentally observed.44 Even higher contact anglescan be reached with patterned surfaces. In MD simulations, thecontact angle can be determined either by the simulation of ananodroplet on the surface45 or by the calculation of the surfacetension of a flat interface (as used in this study). Via Young’slaw,46 the contact angle is given by

cos θ ¼ -γls - γsv

γlvð2Þ

and depends on the surface tensions of the liquid/solid (ls), solid/vapor (sv), and liquid/vapor (lv) interfaces. In the simulations, thesurface tension is obtained from the diagonal components of thevirial tensor,47 γ=(1/(2A))[2Πzz - (ΠxxþΠyy)], where A denotesthe surface area and the virial tensor is given asΠμν=Æ

Pi=1N ri

νFiμæ

and depends on the positions ri and forcesFi of all single particles.For the calculation of the virial tensor, the surface atoms arefrozen and all interactions between the surface atoms are switchedoff, which directly yields the difference γls- γsv. Freezing the sur-face atoms also avoids subtle problems with substrate stresscontributions.47 To calculate the surface tension of the air-waterinterface, a water slab of up to 1807 molecules is simulated in a3.0� 3.0� 12.0 nm3 box in theNVT-ensemble, yielding a surfacetension of γlv=0.0543 ( 0.002 N/m. The substantial deviationfrom the experimental value of γlv=0.072 N/m is a well-knowndeficiency of the SPC/E water model11 and means that one has tobe careful when comparing interfacial energies with experiments.The systems first were equilibrated for at least 200 ps withsubsequent production runs of 5 ns.

Figure 4a shows the contact angle for a hydrophobic diamondsurface as a function of the water-surface interaction strength εCOusing the pressure tensormethod (circles), compared to the resultsobtained from the simulation of a nanodroplet on the surface(triangles). The data fall on a straight line that reaches θ=180� forvanishing interaction strength. Whereas the limiting behavior asεCO f 0 is as expected, the linear behavior 180� - θ ∼ εCO is notobvious and will be later discussed in connection with a depletionlayer occurring at the interface. The data for the droplet method

Figure 1. Water density profiles for different liquid/solid interac-tion energies and surface structures. The center of the topmostcarbon atoms is located at z= 0. Panel a shows the water densityprofiles at smooth diamond surfaces for different liquid/solidinteraction energies εCO given in kJ/mol and for one hydro-philic surface with OH-group surface fraction xOH = 1/4 andεCO=0.42 kJ/mol. In panel b, the density profiles for roughhydrophobic surfaces, shown in Figure 2, are plotted for thestandard GROMOS value εCO = 0.42 kJ/mol.

(39) Giovambattista, N.; Debenedetti, P. G.; Rossky, P. J. J. Phys. Chem. C2007, 111, 1323.(40) Giovambattista, N.; Debenedetti, P. G.; Rossky, P. J. J. Phys. Chem. B

2007, 111, 9581.(41) Janecek, J.; Netz, R. R. Langmuir 2007, 23, 8417–8429.(42) Castrill�on, S. R.-V.; Giovambattista, N. J. Phys. Chem. B 2009, 113, 1438–

1446.(43) Sedlmeier, F.; Janecek, J.; Sendner, C.; Bocquet, L.; Netz, R. R.; Horinek,

D. Biointerphases 2008, 3, FC23–FC39.

(44) Genzer, J.; Efimenko, K. Science 2000, 290, 2130–2133.(45) Werder, T.; Walther, J.; Jaffe, R.; Halicioglu, T.; Koumoutsakos, P. J.

Phys. Chem. 2003, 107, 1345–1352.(46) Rowlinson, J.; Widom, B. Molecular theory of capillarity; Oxford Uni-

versity Press: Oxford, 1982.(47) Nijmeijer,M. J. P.; Bruin, C.; Bakker, A. F.; Leeuwen, J.M. J. V.Phys. Rev.

A 1990, 42, 6052–6059.

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Sendner et al. Article

are taken from ref 43 and are in excellent agreement with themethod used in this work except at large contact angles where linetension and cutoff effects influence the droplet data. The contactangle at the diamond surface with the standard GROMOSforcefield, εCO=0.42 kJ/mol, is 106�. In Figure 4b, we plot theresults for the contact angle as 1 þ cos(θ) as a function of theliquid/solid interaction energy εCO. Here again a reasonabledescription is obtained with a linear function that however isshifted by an offset of the order of εCO ≈ 0.14 kJ/mol. Note thatthe two linear laws in Figure 4, panels a and b, are incompatiblewith each other, since 1þcos(θ) ∼ (180� - θ)2 as θ f 180�, andindeed a slight deviation of the data from the straight line forsmall values of εCO is observed in Figure 4b. The offset in εCO as

1þ cos(θ) f 0 in Figure 4b is especially troublesome, since iterroneously points to a drying transition at a finite value of thesurface water Lennard-Jones attraction, which is unreasonable,as will be discussed further below.

The linear dependence 1þ cos(θ) = εCO on the other handfollows from a simplified calculation of the surface tension.Although in this derivation one assumes constant liquid and soliddensities and neglects the presence of a depletion layer, which playan important role as will be discussed later, the derivation whichwe will now sketch is quite useful conceptually. The surfacetension of the liquid/solid interface is related to the workH12 per surface area which is necessary to separate the liquidand solid slabs, defined as46

H12 ¼ γsvþγlv- γls ð3ÞIt can be easily calculated approximately assuming homogeneoussolid and liquid densities Fs and Fl and neglecting any electrostaticor interfacial entropy contributions. For that we define theinteraction energy of a single liquid molecule at a distance z fromthe interface with the solid phase as

ulsðzÞ ¼ 2πFs

Z R0

z

dr

Z 1

z=r

dðcos θÞ r2uðrÞ

¼ 2πFs

Z R0

z

dr ðr2 - zrÞuðrÞ ð4Þ

where the intermolecular liquid/solid interaction potential isu(r) and R0 denotes the upper cutoff. Considering the Len-nard-Jones potential in eq 1, which has a short ranged repulsiveand a long ranged attractive part, the water film will exhibit athin depletion layer of width z* that is defined by the conditionuls(z*)=0 and is (in the limit R0 f ¥) explicitly given by

z/ ¼ σ2

15

� �1=6

ð5Þ

The work term H12 contains the interactions of all water mole-cules with the surface and is given by

H12 ¼ -Fl

Z ¥

z/dz ulsðzÞ ¼ -πFlFs

Z R0

z/dz zðz-z/Þ2uðzÞ ð6Þ

With the linear dependence of the Lennard-Jones potential in eq1 on εCO,H12 turns out to be a linear function of thewater-surfaceinteraction energy εCO as well. From Young’s equation, eq 2, itfollows that the cosine of the contact angle should be linearlydependent on the interaction energy,

1þcos θ ¼ γsvþγlv-γlsγlv

¼ H12

γlv∼εCO ð7Þ

This linear relation gives a fair description of the simulationresults in Figure 4b but, as we have noted before, would suggest avanishing of the expression 1 þ cos θ at a finite liquid solidinteraction energy εCO. Such a drying transition would not beexpected on theoretical grounds since the entropic repulsion of aconfined interface that is governed by surface tension is very

Figure 2. Hydrophobicdiamond-like surfaces.The flat standard surface isdenotedasdiamond.The rough surfaces aredenotedasRnandareconstructed by selectively deleting rows of surface atoms.

Figure 3. Topviewof the hydrophilic diamond surface surfacewithsurface fraction of OH groups xOH = 1/4 (a) and xOH = 1/8 (b).

Figure 4. Contact angle θ of water in contact with the hydropho-bic diamond surface determined via the virial tensor as a functionof the interaction energy εCO. (a)Thedata (spheres) are shown tobeconsistent with the scaling 180� - θ∼ εCO (solid line). In additiondata for the simulation of a nano droplet in contact with thediamond surface (triangles) are presented.43 (b) The data arecompared with the scaling form 1 þ cos θ ∼ εCO including aconstant shift (broken line). Data for different rough surfaces areincluded.Note that the scaling laws shown inpanels a andb as linesare mutually incompatible, see text.

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10772 DOI: 10.1021/la901314b Langmuir 2009, 25(18), 10768–10781

Article Sendner et al.

short-ranged, and the attraction between watermolecules and thesubstrate should keep the depletion layer at a finite width. Thisfollows from theories that incorporate the intricate couplingbetween interfacial fluctuations and interactions between sub-strate and liquid, according to which interfacial fluctuations (thatgive rise to a fluctuation pressure that decays exponentially withdepletion layer thickness) are dominated by the attraction be-tween the water film and the substrate as long as the decay ispower-law-like (which is the case for van-der-Waals type interac-tions).48-50 However, the situation is complicated by the fact thatinMDsimulations all interactions are cut off at a lengthR0 whichon the other hand makes the attraction between water andsubstrate also very short-ranged (an issue that will be consideredin the next section). So our conclusion regarding the validity ofeq 7 is that although at first sight the fit to the simulation data inFigure 4b seems reasonable, we argue that depletion effects anddensity profile variations not included in the derivation of eq 7rather lead to a scaling 180� - θ ∼ εCO, as shown in Figure 4a,which is at odds with eq 7 but in fact is more physically sound, aswill be discussed further below considering the additional effectsof a depletion layer.

Cutoff Dependence. In MD simulations, all Lennard-Jonesinteractions are usually truncated at an upper cutoff R0. In theabove simulations, the cut off radius was set to R0=0.8 nm. Itturns out that the value of the cutoff sensitively influences theresulting contact angle, an issue only recently considered insimulations.11,12 In Figure 5a, the contact angle at a hydrophobicdiamond surface is shown as a function of the liquid/solidinteraction energy εCO and for different values of R0. Theliquid/vapor surface tension was similarly calculated for differentcutoff radii in simulations of a water film of 1807 watermoleculesin the NVT ensemble in a 3 � 3 � 12 nm3 simulation box. Theliquid-vapor surface tension slightly increases with increasingcutoff radiusR0 and shows no statistically significant dependence

on the number of water molecules, see Table 2, in agreement withprevious simulations.11,12 In fact, the main contribution to theinterfacial tension stems from Coulomb interactions which arefully taken into account via Ewald summation.

As seen inFigure 4a, for themost hydrophobic surfaces, that is,large contact angles, a variation in the cutoff radius only leads tominor changes of the contact angle. This is an important fact, as itshows that the spurious drying transition suggestedwhen plottingthe data as in Figure 4b is not caused by too small a cutoff radiusR0. In contrast, for the less hydrophobic surfaces, that is, for smallcontact angles, the cutoff radius substantially influences thecontact angle. Increasing the cutoff radius leads to smaller contactangles. This effect can be understood from the density profiles, seeFigure 5, panels b and c. As R0 increases, the interfacial waterdensity goes up, thus decreasing the substrate-water interfacialtension. This effect is more pronounced for large values of theinteraction strength εCO.

The dependenceof the contact angle on the cutoff radiusR0 canbe rationalized by a simple model calculation. Again assumingconstant liquid and solid densities and neglecting interfacialentropy, the interfacial work H12 as defined in eq 6 can for largeR0 be written as

H12 ¼ 2πεCOFsFlσ4 1

8

15

2

� �1=3

-σ

R0

� �2

þOσ

R0

� �3" #

ð8Þ

where we have used the intermolecular potential eq 1 and the resultfor the depletion layer width eq 5. It is seen that the smaller R0

becomes, the weaker the interfacial attraction and thus the contactangle increases. Figure 5d shows the contact angle as a function ofthe cutoff radius for a diamond surface with fixed εCO=0.57 kJ/molfromsimulations (circles) and fromeq8 (triangles).This is the largestvalueof εCOconsideredbyus, forwhich the cutoff dependenceof thecontact angle is very dramatic. In the calculation, the bulkwater anddiamond densities are used for Fs,l. The trend in the simulations iswell captured; considering the very approximate character of thescaling Ansatz leading to eq 8, the agreement is surprisingly good.Forall subsequent simulations,weusea cutoff radiusofR0=0.8nm.It is important tokeep inmind that, especially for large surfacewaterinteraction strengths, the contact angle and thus all other interfacialproperties, exhibit a pronounced dependence on the value of thecutoff radius R0 employed in the simulations. It follows thatwhenever simulations of interfacial systems are performed, notonly must the interaction strength be reported, but also the cutoffradius. This is important also since the water force-field parametershave been optimized for a certain finite Lennard-Jones cutoffand deviations from the desired water bulk properties will occur ifthe cutoff length deviates drastically from that value. Ideally, oneshould always determine the contact angle, which is the onlyexperimentally meaningful parameter that characterizes the surfacehydrophobicity.

Figure 5. (a) Contact angle θ at the hydrophobic diamond surface for different liquid/solid interaction energies εCO and different cutoff radiiR0. (b) Water density profiles at the diamond surface for εCO = 0.11 kJ/mol and (c) εCO = 0.57 kJ/mol for two different cutoff radii R0.(d) Contact angle for a diamond surfacewith εCO=0.57 kJ/mol as a functionof the cutoff radius. The plot shows the contact angles obtainedfrom simulations (O) and from the calculation of H12 in eq 8 via eq 7 (4). The surface tension of the liquid/vapor interface used in eq 7 isobtained from the simulation of a N= 1807 water film, see Table 2.

Table 2. Liquid Vapor Surface Tension γlv for Different System Sizes

and Different Lennard-Jones Cutoff Values R0a

N R0 [nm] γlv [N/m]

751 0.8 0.0527 ( 0.0031807 0.8 0.0543 ( 0.0021807 1.0 0.0580 ( 0.0021807 1.2 0.0588 ( 0.0021807 1.4 0.0595 ( 0.002

aThe surface tensions are determined via the pressure tensor fromNVT simulations of N water molecules in a 3 � 3 � 12 nm3 simulationbox.

(48) Lipowsky, R.; Fisher, M. E. Phys. Rev. B 1987, 36, 2126.(49) Lipowsky, R. Random Fluctuations and Pattern Growth; Kluwer Akad.

Publ.: Dordrecht, 1988; Vol. Nato ASI Series E, Vol. 157, pages 227-245.(50) Lipowsky, R.; Grotehans, S. Biophys. Chem. 1994, 49, 27.

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Shear at Hydrophobic Surfaces

At a hydrophobic surface, the partial slip boundary conditionholds, which quantifies the amount of slippage by the slip lengthb. This length is defined via the normal gradient of the tangentialfluid velocity field v(z),3

ðv0 -vÞz¼z0� bðDv=DzÞz¼z0

ð9Þ

where v0 and z0 denote the velocity and position of the surface, seeFigure 6.

The simulation setup consists of two diamond blockswhich areseparated by two SPC/Ewater slabs of typical thicknessD≈ 4 nmeach. This corresponds to roughly 1000 water molecules in eachwater slab. In Figure 6, we show a snapshot of half the simulationsystem, where only one of the two water slabs is shown. As wasdone in previous simulation setups,51 Couette shear flow isinduced by attaching harmonic springs with spring constantsk=1000kJmol-1 nm-2 to the upper and lower surface.The upperspring is pulled with a velocity of typically v0=0.02 nm/ps in thex-direction and the lower spring is pulled with the same speed inthe opposite direction such that the net momentum inputvanishes. The motion of the diamond blocks induces a linearvelocity profile for the solvent flow, see Figure 6. The shear ratefollows as _γ 2v0/D where D is the water slab thickness. Using thedefinition of a partial slip boundary condition at the position ofthe surface in eq 9, the slip length b is obtained by extrapolatingthe velocity profile. For that purpose, the velocity profile is fittedto a linear function. The location of the surface at which the slipboundary condition is applied is defined by the center of thetopmost layer of surface atoms. The systems are equilibrated for200 ps and then subsequent production runs of up to 30 ns areperformed. Several simulations with the same parameters areperformed and all trajectories are used for further analysis.

The used Berendsen weak-coupling thermostat is in principlecritical for shear simulations. We checked this issue by performingtwo benchmark simulations, where a Berendsen thermostat withvelocity scaling in all Cartesian directions and a Nose-Hooverthermostat with velocity scaling only in the direction perpendicularto the shear direction and perpendicular to the surface normal wereapplied, during otherwise identical simulations.13 We found no

difference between these different simulation protocols, meaningthat the type of thermostat does not influence the slip-length results.

Since experimental shear rates are substantially lower than therates used in MD simulations, a careful examination of theinfluence of the applied pulling velocity on the resultant slip lengthis necessary.52 Therefore, shear flow simulations at different shearrates are performed to rule out artifacts that have to do withdeviations from the linear-response behavior. Up to shear rates of1010 s-1, the slip length b is almost independent of the shear rateand independent of the water film thickness which was variedbetween D=2 and 8 nm, see Figure 7. To obtain reliable dataat acceptable computational cost a shear rate of _γ=1010 s-1 isutilized for all subsequent shear simulations, with a water-slabthickness of D=4 nm.

In Figure 8awe show the slip length as a function of the surfaceenergy parameter εCO for the four different surface topologiesshown in Figure 2. For decreasing interaction energies, that is, formore hydrophobic surfaces, the slip length increases. For thelowest interaction energy, the slip length is of the order of 20 nm.The slip length for the rough surfaces is always smaller than forthe smooth diamond surface. At the roughest surface R4, the sliplength is smallest. Increasing surface roughness leads to smallervalues for b, since the friction at the liquid/solid interface isenhanced by the stronger corrugation of the liquid/solid interac-tion potential. A similar dependence of the slip length on thesurface interaction strength was seen in simulations on polymericmelt systems53,54 and in simulations of liquid-liquid interfaces.55

To rationalize the dependence of the slip length on staticsurface properties, especially on the interaction energy εCO, wefollow an argument of Bocquet and Barrat,56,57 who related theslip length of a fluid with viscosity η at a surface with surface areaA to the friction force Fx exerted by the fluid on the solid

Fx

A¼ η

bvxðz ¼ z0Þ ¼ ηv0xðz ¼ z0Þ ð10Þ

The friction coefficient λ is defined as λ=Fx/(vx(z=z0)A) whichtogether with eq 10 yields b=η/λ. The Green-Kubo relation for

Figure 6. Snapshot of the sheared hydrophobic diamond systemwithD=4 nm gap size (left). Not shown is the second water film.The diamond slabs are 1.5 nm thick and have 3.0� 3.0 nm2 lateralextension. In the middle, the water density profile is shown. Theright figure shows the solvent velocity profile for diamond velocityν0= 0.02 nm/ps and the definition of the slip length b. The verticallines denote the surface velocity ( ν0. The Lennard-Jones para-meters are the standard GROMOS values given in Table 1 whichleads to a contact angle of 106� at the diamond surface.

Figure 7. Slip length b as a function of shear rate _γ at a smoothdiamond surface with the standard GROMOS Lennard-Jonesparameter value εCO= 0.42 kJ/mol given in Table 1. The differentsymbols correspond to different thicknesses D of the water filmbetween the two diamond slabs.

(51) Thompson, P. A.; Robbins, M. O. Science 1990, 250, 792–794.

(52) Thompson, P. A.; Troian, S. M. Nature 1997, 389, 360–362.(53) Barsky, S.; Robbins, M. O. Phys. Rev. E 2002, 65, 021808.(54) Servantie, J.; M€uller, M. Phys. Rev. Lett. 2008, 101, 026101.(55) Koplik, J.; Banavar, J. R. Phys. Rev. Lett. 2006, 96, 044505.(56) Bocquet, L.; Barrat, J.-L. Phys. Rev. E 1994, 49, 3079–3092.(57) Barrat, J. L.; Bocquet, L. Faraday Discuss. 1999, 112, 119–127.

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the friction coefficient reads λ=R0¥ dt ÆFx(0)Fx(t)æ/(AkBT), which

for many systems can be simplified to λ = τ ÆFx2æ /(AkBT) where

the relaxation time τ has been defined and ÆFx2æ is the mean-

squared lateral force acting between the substrate and the liquidphase. For the relaxation time we write τ ∼ σ2/D where σ is acharacteristic length scale and D is the diffusion coefficient ofa fluid particle. The mean squared lateral surface force comesfrom force fluctuations, and in the thermodynamic limit scales asÆFx2æ∼C^ (εCO/σ

2)2AwhereC^ is a geometric factor that accountsfor roughness effects: The rougher the surface, the larger C^.Putting everything together, we arrive at the scaling expression

b∼ σ2ηDkBT

C^ε2COð11Þ

which incorporates a number of drastic approximations andsimplifications but on the other hand constitutes an easily testablerelation between slip length and surface energy εCO. In Figure 8a,the slip length is plotted as a function of 1/εCO

2 for the differentroughnesses considered. The agreement with the straight linesverifies the above scaling considerations. Also, the rougher thesurface, the smaller the slip length, again in agreement withthe scaling in eq 11 if one takes the coefficient C^ as a measureof the degree of surface roughness.

Togetherwith the linear dependenceof the cosineof the contactangle on the liquid/solid interaction energy, eq 7, a simple relationbetween the slip length and contact angle is obtained,13

b ¼ 0:63 nm 3 ð1þ cos θÞ-2 ð12Þwhere the numerical prefactor is obtained from a fit to the data,broken line inFigure 8b.Wehasten to add that the alternative andmore physically sound relation εCO ∼ 180� - θ, as suggested byFigure 4a, gives the modified scaling relation

b ¼ 12 μm 3 ð180�- θÞ-2 ð13Þwhich in fact gives a comparable description of the data, solid lineinFigure 8b. The effects of roughness on bon the one handand onθ on the other hand partially cancel out: Increasing roughnessleads to decreasing slip lengths because of the enhanced friction(by increasing the factor C^), see Figure 8a. On the other hand,since the liquid/solid contact area is increased on a rough surface,the liquid/solid interaction energy becomes larger which leads todecreasing contact angles, see Figure 4b. The influence of rough-ness on the scaling plot in Figure 8b is thus reduced. Despite the

rough estimates leading to eq 11, the simulation results follownicely the predicted scaling. The dependence of the slip on thecontact angle shows the same dependence also for differentsurface structures such as fcc(100) Lennard-Jones surfaces oralkane chains.13 This quasi-universal relation between contactangle and slippage is of particular interest, since the contact angleis an experimentally easily accessible quantity.Slippage and Depletion Layer.An alternative theory for the

slip length at the liquid/solid interface is based on the existenceof a thin depletion layer between the surface and the waterwith thickness δ. The viscosity ηs of this layer is assumed tobe substantially lower than the bulk water viscosity η0, whichgives rise to an apparent slip length21,23,54

b ¼ δðη0=ηs -1Þ ð14Þwhich is linearly dependent on the depletion width. To test thisprediction, we used two different definitions of the depletion layerwidth δ: (i) the position where the water density is half its bulkvalue,δ0, and (ii) the integrated density deficit of thewater layer41,58

δ ¼Z ¥

0

dz ½1-FsðzÞ=Fbs -FlðzÞ=F1b� ð15Þ

as visualized inFigure 9.Here Fl,sb are the bulk densities of the liquidand solid phase. Both depletion lengths can be directly calculatedfrom the density profiles of the simulations. For both definitions,we do not find a linear dependence of the slip length on thedepletion width, see Figure 10c. Therefore, unless one postulates aparametric dependence of the depletion-layer viscosity ηs on thedepletion width δ, the data do not support the prediction eq 14 ofthe two-viscosity model. In addition, the depletion length is lessthan a molecular diameter which makes the definition of aneffective viscosity for such a thin layer awkward.

In Figure 10a, the depletion length is plotted as a function ofthe liquid/solid interaction energy, which suggests the scaling

δ-2 ∼ εCO ð16ÞCombining this with the relation eq 11, one immediately obtains

a quartic dependence of the slip length on the depletion length,

b∼ δ4 ð17Þ

Figure 8. (a) Slip length b plotted versus the inverse square of the liquid/solid interaction energy, 1/εCO2. The solid lines show linear fits forthe different surface structures. (b) The slip length is plotted versus the contact angle. The broken line shows a fit to b � (1þ cos θ)-2 for alldata points on the smooth diamond surface while the solid line shows a fit according to b � (180� - θ)-2. See text for more details.

(58) Mamatkulov, S. I.; Khabibullaev, P. K.; Netz, R. R. Langmuir 2004, 20,4756–4763.

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Sendner et al. Article

which is nicely confirmedby thedata inFigure 10c (broken line).Onthe other hand, combining eq 16 with eq 7 one obtains the scalingrelation

δ-2�1þ cos θ ð18Þwhich is compared with the simulation data shown in Figure 10b.The agreement is fair, but note that the line for the smooth diamondsurface extrapolates to a finite depletion length for θf 180�, whichis at odds with the expectation that the interaction between thewater and diamond slabs vanishes only when δ f ¥. This is thesame problem one encounters when applying eq 7 to the simulationdata in Figure 4b.We can in fact bring out the shortcomings of eq 7more clearly: Taking into account the presence of a depletion layerofwidthδbetween the solid and the liquid phase (which amounts toreplacing the lower integration boundary z* in eq 6 by δ), theinterfacial energy is given by

H12 ∼ FlFsσ6εCO=δ

2 ð19ÞNaive application of Young’s equation, eq 2, and using thedefinition ofH12 in eq 3 gives the relation 1þcos θ� εCO/δ

2, whichcannot be simultaneously satisfied with eq 16 and eq 18, as notedbefore.13 A simple way out of this apparent dilemma is to postulatea second contribution to the interfacial energy,

-H12 ∼ a=δm - FlFsσ6εCO=δ

2 ð20Þwith at first an arbitrary exponent m and prefactor a. In fact,the relation eq 16 follows from eq 20 by minimizing-H12 withrespect to δ if one chooses m=4; at that minimum one findsH12 ∼ εCO

2, in contrast to the naive scaling result in eq 7.Indeed, using H12 ∼ εCO

2 together with eqs 2 and 3 yieldsεCO ∼ (1þcos θ)1/2 ∼ 180� - θ which gives an equally sounddescription of the data, as shown in Figure 4a. Finally,combining εCO ∼ 180� - θ with the result eq 11 gives the

modified scaling law b∼ (180�- θ)-2, which describes the datain Figure 8b equally well as the form proposed in eq 12.

We conclude that the quality of the numerical data does notallow to firmly distinguish between the conflicting scaling formsfor the dependence of the contact angle θ on εCO, as suggested bythe straight lines in Figure 4, panels a and b. The scaling relationb∼ εCO

-2 , shown inFigure 8a, and the relation δ-2∼ εCO, shown inFigure 10a, seemmore robust and in fact are fully consistent withthe scaling result eq 11 and the modified interfacial free energyscaling form in eq 20. Taken together, these relations lead to thestriking result b ∼ δ4, which is presented in Figure 10c.

There are a number of effects that could lead to the postulatedrepulsive contribution ∼ a/δm with m=4 in eq 20 with well-studied consequences on the wetting and dewetting behavior.59

One possible candidate, the repulsion due to confinement of theinterface fluctuations, is by itself quite complicated48-50 anddisplays a distance-dependent crossover from being dominatedby interfacial tension (at large length scales) to being dominatedby interfacial bending rigidity (at short length scales).60 On theother hand, the water density profile changes as a function oftheLennard-Jones parameter εCO,which also gives rise to sizablecorrections to H12. Finally, the modification of the electrostaticinteractions between water molecules in the interfacial layerbecause of the presence of the solid substrate will also contributeto the interfacial free energy. At this stage, we can only state that aconsistent description of the data is possible with an ad-hocmodified interfacial free energy as in eq 20, the microscopic originof which is left for future investigation.Slippage and Dissolved Gas. To examine the effect of

dissolved gas on the slip length, shear flow simulations onhydrophobic diamond surfaces are performed with added gasparticles. Inmost simulations, 10 gas particles are inserted in eachof the twowater slabs.We examine different types of gas particles.Species X (mx=12.01 u) interacts equally with all other atomspresent in the system via a purely repulsive potential,

VXðrÞ ¼ 4εXσX

r

� �12

ð21Þ

with σX=0.3581 nm and εX=0.2774 kJ/mol, the Lennard-Jones parameter of carbon. GROMOS force field parametersare used to model the noble gases Ne (mNe=20.18 u) and Ar(mAr=39.95 u). For the diatomic oxygen gas O2 (mO=16.00 u),we use the Lennard-Jones parameters from ref 37 with a bondlength of 1.21 A. The interaction parameters for all consideredgas types are summarized in Table 1. For the simulations withargon and gas type X, we consider variations of the surfacehydrophobicity, while for oxygen and neon only the standarddiamond parameters are used. For the simulations with argon,all interaction energies involving the surface atoms are recal-culated by εCAr=(εCCεArAr)

1/2 and εCO=(εCCεOO)1/2 while for

the simulations with gas type X, only the liquid/solid interac-tion energy εCO is varied, while the gas/solid interaction εX isunchanged and given by eq 21.

Figure 11 shows the density profiles obtained within shearflow simulations, which in fact are indistinguishable fromdensity profiles obtained in equilibrium simulations. Thedensities are averaged over all four interfaces in the simulationcell. As expected, the gas atoms accumulate at the hydro-phobic interface, as was previously observed in MD simula-tions of Lennard-Jones fluids24 and in grand canonical

Figure 9. (a) Density profile of the diamond slab in contact withwater, (b) the integrand f(z) = 1- Fs(z)/Fs - Fl(z)/Fl

b of eq 15, and(c) its running integral g(z)=

R0z dz0 f(z0) together with the defini-

tion of the depletion length. The graphs show data for the hydro-phobic diamond surface with εCO = 0.2550 kJ/mol which corre-sponds to a contact angle of 136�.

(59) Dietrich, S.; Schick, M. Phys. Rev. B 1986, 33, 4952–4962.(60) Netz, R. R.; Lipowsky, R. Europhys. Lett. 1995, 29, 345–350.

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Monte Carlo simulations employing the SPC water model.37

The density of neon, argon, and oxygen gas at the interface isincreased by a factor of roughly 20-60 compared to thedensity in the bulk of the water slab. The gas adsorption isrestricted to one monolayer of gas particles, there is only onenarrow peak present in the density profiles shown in Figure 11.The density of the purely hydrophobic gas type X at theinterface is increased by a factor of more than 150. All X atomsare mostly present on one surface and the formation of a largecluster of gas atoms can be seen in the simulations, seeFigure 12 This strong clustering is reflected in the shoulderof the X-density profile in Figure 11. The other gases do notexhibit such a clustering and they are more or less equallydistributed on the two interfaces.

A variation of the surface hydrophobicity does not lead toqualitative changes in the density profiles. The accumulation ofargon atoms is strongest at the most hydrophobic surface (εCO=0.26 kJ/mol). This effect is also observed for gas type X.Pronounced interfacial gas accumulation is not seen in simula-tions of dissolved gas between the hydrophilic surfaces, as shownfor argon inFigure 11b:Here, the gas density at the surface is onlytwice the bulk value.

Although the gas particles Ne, Ar, and O2 are stronglyattracted to the interface, they frequently desorb from the inter-face as seen in the snapshot, Figure 12a. The gas accumulation atthe interface is not because of the Lennard-Jones attractionbetween the surface and the gas atoms, since also gas typeXwith apurely repulsive potential exhibits an increased density at thesurface.Rather, the hydrogenbonding network is less perturbed ifthe gas particles are at the interfacewhich favors gas adsorption ata hydrophobic surface.

Experimentally, a mole fraction of 0.25 � 10-4 Ar-molecules(0.23� 10-4 for O2) is soluble in water at a partial gas pressure of1 atm and T=298.15 K.61 In the simulations, the total pressure isfixed at 1 bar≈ 1 atm. Since the partial pressure of water vapor atroom temperature is roughly 0.02 atm, the partial gas pressure inthe simulation is comparable to the experimental situation, pgas≈(1- 0.02) atm≈ 1 atm. From the density profiles, we deduce thatthe mole fraction of dissolved argon gas in the bulk liquid has a

Figure 10. Plot of the inverse square of the depletion length δ versus the liquid/solid interaction energy εCO (a) and versus the cosine of thecontact angle (b). The depletion length δ is defined in eq 15. The broken lines show linear fits. (c) Slip length b as a function of the depletionlengthδ for smooth and rough surfaces. The dashed line shows a fit for the diamond surface tob�δ4. The depletion lengthδ0 in the inset (notethe different scale of the x-axis) is defined by the position where the water density is half its bulk value.

Figure 11. Density profiles for different gas types at the smoothhydrophobic diamond surface. The gas densities are scaled by theirbulk concentrations (for gas type X the density is additionallydivided by a factor of 2). The center of the topmost carbon surfaceatoms is located at z = 0. The total amount of dissolved gas is10 gas particles per water slab, except for one simulation forO2 with only 4 oxygen molecules per water slab. In panel a, dataare shown for one liquid solid interaction energy εCO = 0.425 kJ/mol. For comparison, the water density profile (not normalized)without dissolved gas is also shown. In panel b, results are shownfor different surface hydrophobicities, (i) εCO = 0.255 kJ/mol,(ii) εCO = 0.425 kJ/mol, and (iii) εCO = 0.570 kJ/mol, for the gastype X and Ar. Additionally, data for the simulation of dissolvedargon gas between two hydrophilic surfaces (xOH=1/4) is shown.

Figure 12. Snapshots of simulations with dissolved gas. (a) Simu-lation with dissolved argon gas at the diamond surface (εCO =0.425 kJ/mol). (b) Simulation with dissolved gas type X at εCO =0.570 kJ/mol.

(61) Wilhelm, E.; Battino, R.; Wilcock, R. J. Chem. Rev. 1977, 77, 219.

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value of about 5� 10-3 (similarly, for oxygen themole fraction inthe bulk is about 4 � 10-3) and is thus much higher than themaximally soluble mole fraction (the gas bulk concentration isdetermined from all gas molecules that have a surface separationof at least 1 nm). However, we do not observe phase separationof argon or oxygen in the simulations, which is related to thevery large nucleation barrier for gas bubble formation in suchsmall systems. We nevertheless checked whether the unphysicallylarge gas concentration in the simulations leads to spuriouseffects. For this we performed one simulation with only fouroxygen molecules per water slab. In the normalized densityprofiles in Figure 11a, no difference is seen between the twodifferent oxygen concentrations; we conclude that for all gasesexcept the artificial gas type X (for which the solubility presum-ably is even lower than for the other gases considered) thesimulation results approximately reflect the situation of a satu-rated solution of gas in water.

The surface adsorption of gas particles enhances the slip lengthonly slightly, see Table 3 and Figure 13. The largest relativechange in slip length b because of gas adsorption is seen for thehighest liquid/solid interaction energy (i.e., for the smallestcontact angle). There, the slip length roughly doubles when, forexample, Ar atoms are added to the water. Dissolved gas onlymoderately amplifies the slip length for large contact angles,which remain in the range of only a few nanometers. Thus, weconclude that large slip lengths as measured experimentally arenot caused by surface adsorbed thin gas layers. However, experi-mental measurements suggest the formation of gas-nanobubblesat the liquid/solid interface.62-66 The lateral dimension of thesebubbles is in the order of 100 nm, with a height of severalnanometers which could significantly enlarge the slip length.These gas cavities are much larger than simulation box sizesone can possibly study and are therefore out of reach for presentcomputer simulations.

Shear Flow Simulations at Hydrophilic Surfaces

We now examine the hydrodynamic boundary condition andthe viscosity of water close to polar, hydrophilic surfaces. Experi-ments report on viscosities for confined water between micasurfaces at room temperature, which are comparable to the bulkviscosity.67,68 Sub-nanometer confinement was shown to lead toan increase in viscosity of approximately 80 times the bulk

viscosity in simulations.69 Other experiments and simulationstudies report on a strong increase in viscosity of several ordersof magnitude for highly confined water films.27,29,70 Besides fromthese conflicting results for the viscosity in confined water layers,also the structure of water at hydrophilic surfaces is under debate.Spectroscopy experiments and computer simulations find anice-like structure of thin water films.25,26,28,71-73 These crystallike structures are identified via a sharp drop in the diffusionconstant, a substantially increased shear viscosity, and can sustainshear stress. At hydrophobic interfaces, in contrast, this ice likewater structure is not observed. A critical summary of simulationwork on confined water has recently been published.74

To clarify the structure and properties of water at hydrophilicsurfaces, we perform NEMD simulations of water at polar,hydrophilic surfaces with relative large water slab thickness. Inthe non equilibrium shear flow simulations, the shear viscosityprofile of interfacial water is obtained from the fluid velocityprofiles. Via this method, we are able to locally probe the shearviscosity of water close to interfaces without introducing addi-tional effects due to confinement. The velocity profiles, Figure 14,differ qualitatively from those obtained for the hydrophobicsurfaces: close to the surface, the gradient of the velocity profileis smaller than in the middle of the water slab. In contrast, thegradient of the velocity profiles at the hydrophobic surfaces isconstant, even close to the interface, see Figure 6. At thehydrophilic surface, the water molecules at the interface aredragged along with the surface. The region over which thisdragging of water molecules occurs roughly corresponds to theextent of the first water layer. This feature is due to the stronginteraction between the hydroxyl surface groups and the watermolecules. These findings are in agreement with velocity profilesobtained from simulations of sheared water films between micasurfaces.75

Since the shear viscosity is linearly related to the inverse of thevelocity gradient, it is possible to define two different viscosities inthe system. One in the surface region, ηs, and the other one in thebulk region, η0. To obtain the shear rates in the two regions, the

Table 3. Slip lengths for Shear Flow Simulations with Dissolved Gas,

bgas, for the Hydrophobic Diamond Surfacea

gas εCO [kJ/mol] bgas [nm] b [nm]

Ar 0.255 8.92 ( 2.12 7.54 ( 0.76X 0.425 2.69 ( 0.18 2.17 ( 0.15Ne 0.425 2.53 ( 0.10 2.17 ( 0.15Ar 0.425 2.77 ( 0.55 2.17 ( 0.15O2(10) 0.425 2.61 ( 0.10 2.17 ( 0.15O2(4) 0.425 2.57 ( 0.24 2.17 ( 0.15X 0.570 1.19 ( 0.19 0.75 ( 0.10Ar 0.570 1.42 ( 0.04 0.75 ( 0.10

aFor comparison, also the results for the simulations without gas areshown, b.

Figure 13. Slip length with and without dissolved argon gas at thehydrophobic smooth diamond surface plotted versus the contactangle. In the simulationswe add 10 argon atoms to eachwater slab,which corresponds to amolar gas fraction of about 5� 10-3 in thebulk.

(62) Simonsen, A. C.; Hansen, P. L.; Klosgen, B. J. Colloid Interface Sci. 2004,273, 291–299.(63) Tyrrell, J. W. G.; Attard, P. Phys. Rev. Lett. 2001, 87, 176104.(64) Schwendel, D.; Hayashi, T.; Dahint, R.; Pertsin, A.; Grunze, M.; Steitz, R.;

Schreiber, F. Langmuir 2003, 19, 2284–2293.(65) Switkes, M.; Ruberti, J. W. Appl. Phys. Lett. 2004, 84, 4759–4761.(66) Holmberg, M.; Kuhle, A.; Garnas, J.; Morch, K. A.; Boisen, A. Langmuir

2003, 19, 10510–10513.(67) Raviv, U.; Klein, J. Science 2002, 297, 1540–1543.(68) Raviv, U.; Laurat, P.; Klein, J. Nature 2001, 413, 51–54.

(69) Leng, Y.; Cummings, P. T. Phys. Rev. Lett. 2005, 94, 026101-4.(70) Sakuma, H.; Otsuki, K.; Kurihara, K. Phys. Rev. Lett. 2006, 96, 046104-4.(71) Leng, Y.; Cummings, P. T. J. Chem. Phys. 2006, 124, 074711.(72) Pertsin, A.; Grunze, M. Langmuir 2008, 24, 135–141.(73) Pertsin, A.; Grunze, M. Langmuir 2008, 24, 4750–4755.(74) Lane, J. M. D.; Chandross, M.; Stevens, M.; Grest, G. Langmuir 2008, 24,

5209.(75) Leng, Y.; Cummings, P. T. J. Chem. Phys. 2006, 125, 104701.

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velocity profile is separately fitted to a linear function in thesurface and bulk regions. The fit region for the velocity profile ofthe interfacial layer starts at the positionwhere thewater density isfor the first time equal to the density at the first density minimumand ends at the position of the first water density minimum,shown as the gray shaded areas in Figure 14. The ratio of the twoviscosities is then determined by the ratio of the velocity gradientsin the interfacial, _γs, and bulk, _γ0, region.

In Table 4, the numerical values of the interfacial viscosities areshown. For the more hydrophilic surface (xOH=1/4), the inter-facial viscosity is larger by a factor of 4 compared to the bulkviscosity. The less hydrophilic surface exhibits an interfacialviscosity roughly twice the bulk value. For both surfaces, thechange in viscosity is moderate and does not support an ice-likeinterfacial water structure.

As for the hydrophobic surfaces, eq 9 can be used to define aslip length b. Therefore, the velocity profile is fitted in the bulkregion to a linear function and extrapolated. By definition, thebulk region ranges from the first to the last minimum of the waterdensity profile, which is depicted as the white region in betweenthe gray shaded areas inFigure 14a. The slip length is definedwithrespect to the center of the oxygen atoms of the OH surfacegroups. This procedure leads to a negative slip length of roughlyb ≈ -0.3 nm for both OH surface concentrations, see Table 4.

From the velocity profiles, only the ratio between the interfacialand the bulk viscosity can be calculated. For the explicit determi-nation of the viscosities, the force acting on the diamond slabs is

needed. In the simulation, the two solid slabs are attached toharmonic springs which are pulled with constant velocities (ν0.From the average displacement of the slabs with respect to theminimumof the spring potential, the average forceF acting on theslabs is determined. The bulk and surface viscosities are given byη0=F/(2A _γ0) and ηs=F/(2A _γs) with surface area A (the factor 2in the expressions for the shear viscosities arises because eachdiamond slab is connected to two water slabs). From the shearrates _γ0 and _γs, obtained from the velocity profiles, and theaverage force Fmeasured in the simulations, the viscosities givenin Table 5 are thus obtained. The bulk viscosity η0 in the innerregion of the water film is similar for the hydrophilic andhydrophobic surfaces. The moderate confinement of 4 nm doesnot lead to strong deviations from the bulk viscosities, that is, ourestimates for the bulk viscosity of the water film are in goodagreement with the literature value η=0.642 cP76 for the SPC/Ewater model, obtained in non equilibrium simulations at 300 K(note that the SPC/E water model considerably underestimatesthe viscosity of real water, η=0.851 cP77).

Since the used shear rates _γ ∼ 1010 s-1 are much larger thanthose typically used in experiments, it is crucial to check if thelinear response regime is reached for the hydrophilic surface. Weaccount for this issue by performing two benchmark simulationswith double and half of the diamond pulling speed. Since neitherthe measured bulk viscosities nor the slip length changed withdifferent pulling speed, see Table 5, we infer that the system is stillin the linear response regime.

Diffusion

The viscosity η of a liquid directly affects the diffusion constantfor a particle with radius a, which is given by the Stokes-EinsteinrelationD=kBT/6πηa for the no-slip case orD=kBT/4πηa for theperfect-slip case. In this section we seek to understand the relationbetween the increased surface-viscosity at a hydrophilic surface,and the diffusivity of water molecules themselves. Not much isknown for water dynamics at a single surface. Recent polariza-tion-resolved pump-probe spectroscopy studies showed thatwater around small hydrophobic methyl groups is effectivelyimmobilized,78 but the dynamics on planar surfaces is less clear.Simulations studied the diffusion of water molecules at varioussurfaces.79-82 In the present work,we shall bemainly interested inthe difference of the diffusivity perpendicular to hydrophobic andhydrophilic surfaces. In simulations, diffusion constants aretypically determined from the mean square displacement or fromthe velocity autocorrelation function. This leads to difficulties for

Figure 14. (a) Water density profiles (top row) and velocity pro-files (bottomrow) for the hydrophilic surfaceswithxOH=1/8 (left)and xOH= 1/4 (right). The topmost layer of carbon atoms definesthe origin z=0. (b) shows the density profiles and velocity profiles(þ) close to the surface. The fit functions for the velocity profile inthe peak region and in the bulk region are shown as solid andbroken lines. The gray shaded areas denote the extent of the peakregion used in the fitting.

Table 4. Shear Rate and Viscosities for the Bulk ( _γ0,η0) and

Interfacial Region ( _γs,ηs) and the Corresponding Slip Length,

Obtained from the Velocity Profiles of the Non-Equilibrium MD

Simulationsa

xOH _γ0 [1/ns] _γs [1/ns] ηs/η0 slip length [nm]

1/4 13.4 3.6 ( 0.8 3.7 ( 0.9 -0.32 ( 0.011/8 12.8 5.9 ( 1.0 2.2 ( 0.4 -0.29 ( 0.01

aThe slip length b is defined with respect to the layer of oxygen atomsof the surface C-O-H groups.

(76) Hess, B. J. Chem. Phys. 2002, 116, 209.(77) Weast, R. C. CRC Handbook of Chemistry and Physics; CRC Press: Boca

Raton, FL, 1986.(78) Rezus, Y. L. A.; Bakker, H. J. Phys. Rev. Lett. 2007, 99, 148301.(79) Lee, S. H.; Rossky, P. J. J. Chem. Phys. 1994, 100, 3334–3345.(80) Liu, P.; Harder, E.; Berne, B. J. J. Phys. Chem. B 2004, 108, 6595–6602.(81) Liu, P.; Harder, E.; Berne, B. J. J. Phys. Chem. B 2005, 109, 2949–2955.(82) Mittal, J.; Hummer, G. Proc. Natl. Acad. Sci. U.S.A. 2008, 105, 20130–

20135.

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Sendner et al. Article

the determination of a local diffusion constant. Another route toobtain diffusion constants is to consider the time-correlation of afunction f(z) which is unity if the particle is inside a certain layerand zero otherwise,

f ðzÞ ¼ 1 if z0 < z < z0 þ Δz0 else

(ð22Þ

The autocorrelation function (ACF) of f(z) is given by

CðtÞ � N -1

Z z0þΔz

z0

dz

Z z0þΔz

z0

dz0 pðz, z0, tÞ f ðzÞ f ðz0Þ Fðz0Þ

ð23Þwhere the function p(z,z0,t) is the probability that a particle is at zat time t, given that it was at z0 at t=0 and the liquid density isgiven by F(z). The normalization constantN assures thatC(0)=1.p(z,z0,t) is the solution of the one-dimensional diffusion equationfor a particle in an external potential W(z),

Dtpðz, z0, tÞ ¼ Dz DðzÞ 1

kBTðDzWðzÞÞ þ Dz

� �pðz, z0, tÞ

( )ð24Þ

pðz, z0, 0Þ ¼ δðz-z0Þ ð25Þ

For bulk water with zero external potential and position inde-pendent diffusion constant D, the solution is given by

pðz, z0, tÞ ¼ ð4πDtÞ-1=2 exp -ðz-z0Þ24Dt

!ð26Þ

with the boundary condition p(z,z0,t)f 0 for |z-z0|f¥. Since forbulk water, the density F(z) is constant, the normalization con-stant in eq 23 is given byN=

RdzRdz0 δ(z-z0) f(z) f(z0) F(z)=(Δz)

F0. The integrals in eq 23 can be performed and the autocorrela-tion function is given by

CðtÞ ¼Z z0þΔz

z0

dz

Z z0þΔz

z0

dz0 ½4πDtðΔzÞ2�-1=2 expðz- z0Þ24Dt

!

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4Dt

πðΔzÞ2s

ðe-ðΔzÞ2=ð4DtÞ -1ÞþerfðΔz=ffiffiffiffiffiffiffiffi4Dt

pÞ

ð27Þ

With the asymptotic behavior of the Error function, the asymp-totics are C(t) = Δz/(4πDt)1/2 for long times t . (Δz)2/D andC(t) = 1 - (2/Δz)(Dt/π)1/2 for short times t , (Δz)2/D.

We first calculate autocorrelation functions in bulk (in theabsence of a substrate) from simulations for an NAPzT ensembleof 6000watermolecules in a 3� 3� 20.5 nm3 box.As for the othersimulations, the system is coupled to a temperature of 300 K. Theobtained autocorrelation functions are shown inFigure 15 togetherwith the exact expression eq 27 for four different values of thecorrelation-slab thickness Δz. The only free parameter is thediffusion constant, for which we fit the value D=2.70 nm2/ns.This value is in good agreement with previous publications for

the SPC/E water model, where the diffusion constant was deter-mined to be D=2.70-2.79 nm2/ns.83 Experiments on the self-diffusion of real water yield a value of 2.30 nm2/ns at 298 K.84,85

The agreement between the data and eq 27 is excellent all the wayfrom tens of femtoseconds to nanoseconds. This means, notunexpectedly, that in bulk the water dynamics is purely diffusive.The upper panel shows the data on a logarithmic time scale. Thelower panel shows the data on a linear time scale; here somedeviations are observed in the time window below one picosecond:in fact, water seems to diffuse faster than predicted by its long-timediffusion constant. The microscopic reason for this interestingbehavior is at present unclear.We also note that in the lower panel,one is tempted to erroneously fit a straight line to the data for shorttimes, fromwhich one could extract a time constant of the order of500 fs (with a spurious dependence on the correlation-slab thick-ness Δz). Only the comparison with the exact expression eq 27shows that this is a non-sensical procedure, as the behavior is purelydiffusive down to the smallest time scales with no typical timeconstant present in the data.

For water molecules close to the solid/liquid interface, thesolution of the diffusion equation, eq 24, is much morecomplicated. Generally, the liquid particles experience anon-zero surface potential W(z) and in addition the mobilityof a single water molecule depends on its distance from thesurface because of hydrodynamic boundary effects. In fact,the diffusion tensor even becomes anisotropic with differentcomponents for motion laterally and normal to the surface:simulations have shown that the water molecules aremore mobile along the lateral direction.79 More sophisticatedmodels are necessary to calculate the diffusion constantin the interfacial layer under the presence of a surfacepotential.80 As a crude approximation, we consider thesolution of the diffusion equation at an interface assuminga position-independent diffusion constant and vanishingexternal potential. Since there is no flux of particles acrossthe interface, the first derivative of the probability distribu-tion must be zero at the location of the surface, z=0. Thesolution of eq 24 that satisfies this no-flux boundary condi-tion is given by

psðz, z0, tÞ ¼ ð4πDtÞ-1=2 exp -ðz-z0Þ24Dt

!þexp -

ðzþ z0Þ24Dt

!24

35

ð28Þ

Table 5. Simulation Results for the Bulk and Surface Viscosities

Calculated According to η0 = F/(2A _γ0) and ηs = F/(2A _γs)a

xOH ν0 [nm/ps] η0 [ 10-3 N s /m2 ] ηs [ 10

-3 N s /m2 ]

1/4 0.02 0.73 ( 0.03 2.7 ( 0.71/8 0.02 0.71 ( 0.01 1.5 ( 0.30 0.02 0.66 ( 0.06

xOH ν0 [nm/ps] η0 [ 10-3 N s /m2 ] b [nm]

1/4 0.01 0.71 ( 0.03 -0.31 ( 0.011/4 0.04 0.722 ( 0.005 -0.317 ( 0.004

aThe errors of the viscosities are calculated from the uncertainties ofthe measured spring force and of the shear rate. Also the results of twobenchmark simulations at half and double pulling speed v0 are shown(for which the interfacial viscosity is not calculated because of shortersimulation runs and the considerable statistical fluctuation in thevelocity profile).

(83) Mark, P.; Nilsson, L. J. Phys. Chem. 2001, 105, 9954–9960.(84) Mills, R. J. Phys. Chem. 1973, 77, 685–688.(85) Price, W. S.; Ide, H.; Arata, Y. J. Phys. Chem. A 1999, 103, 448–450.

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Assuming a constant density F0 inside the layer, the surface-autocorrelation function can be evaluated exactly as

CsðtÞ ¼ N -1

Z z0þΔz

z0

dz

Z z0þΔz

z0

dz0psðz, z0, tÞF0

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDt

πðΔzÞ2s

ð2e-ðΔzÞ2=ð4DtÞ -2e-ð2z0þΔzÞ2=ð4DtÞ -2þe-z20=ðDtÞ

þe-ðz0þΔzÞ2=ðDtÞÞþerfðΔz=ffiffiffiffiffiffiffiffi4Dt

pÞ

-2z0þΔz

Δzerfðð2z0þΔzÞ=

ffiffiffiffiffiffiffiffi4Dt

pÞþ z0

Δzerfðz0=

ffiffiffiffiffiffiDt

pÞ

þ z0þΔz

Δzerfððz0þΔzÞ=

ffiffiffiffiffiffiDt

pÞ ð29Þ

The surface autocorrelation function decays with the inversesquare root of time asCs(t)=Δz/(πDt)1/2 for long times t. (Δz)2/Dt. In contrast to the bulk solution in eq 27, the prefactor is largerby a factor of 2, which is caused by the reflecting boundarycondition. The autocorrelation functions, shown in Figure 16, areobtained from the simulation of one solid surface in contact with1830 water molecules in a 3.0 � 3.0 � 8.0 nm box in the NPzTensemble. The region of the water slab considered for theestimation of the diffusion constant is depicted in Table 6 andcorresponds to a Δz=0.1 nm wide layer centered around themaximum of the density in the first water peak. For the hydro-phobic surface, the lowest curve in Figure 16, the expression eq 29gives a good description of the data over the complete time range,provided the diffusion constant is fitted to a valueD=0.85 nm2/ns,which is reduced by a factor of about three compared to the bulk

case (note thatD is the diffusion coefficient normal to the surface).This decrease is not surprising, as the hydrodynamic self-mobilityof a particle at a surface is reduced compared to the bulk case. Thecorrelation-slab width is taken as Δz=0.1 nm as used in thesimulations, the correlation-slab position is fitted as z0=0.03 nmwhich is in rough agreement with the actual distance between thecorrelation slab and the point where the water density approacheszero (note that the actual position of the reflecting boundary is notwell-defined within the simulation, which means that this isactually a free parameter in the theoretical description). For thehydrophilic surface data shown in Figure 16 (middle and uppercurves), the expression eq 29 cannot fit the data, in fact, thefunctional form of the data suggests an exponential process thathas to do with transient binding or trapping of water at polarsurface groups. The simplest model for such a trapping process isone where initially a fraction φ of water molecules is bound to thesurface, then exponentially released, and only after release subjectto diffusion. Neglecting rebinding of water molecules to surfacesites, the autocorrelation function that allows for trapping thusreads

C�s ðtÞ ¼ ð1-φÞCsðtÞþφ e-t=τþ

Z t

0

dt0

τCsðt- t0Þe-t0=τ

" #ð30Þ

Keeping the same diffusion constantD as on the hydrophobicsurface for both hydrophilic surfaces, we can successfully fit thedata with two fitting parameters, namely, binding fraction φ andbinding time τ, see Table 6. That the diffusion constant is at leastvery similar for all three hydrophobic and hydrophilic consideredsurfaces is suggested by the fact that for long times beyond onenanosecond the curves are seen to converge. The most interestingproperty is the time scale τ over which a water molecule is boundto the polar surface groups,which turns out tobe τ=210 ps for thexOH=1/8 surface and τ=340 ps for the xOH=1/4 surface. Theseare long times on simulation time scales, which means thatequilibrating polar surfaces takes a long time. The fact that thetime scale on the xOH=1/4 surface is larger than on the xOH=1/8surface points to the presence of cooperative effects,meaning that

Figure 15. Autocorrelation functions for bulk water for differentcorrelation-slab thicknesses Δz = 0.1,0.2,0.4,1 nm (from bottomto top) compared with the expression eq 27 in a log-log and in alog-lin plot. The fitted diffusion constant isD= 2.70 nm2/ns.

Figure 16. Autocorrelation functions for water inside a Δz =0.1 nm wide region, centered at the first water peak, at thehydrophobic surface (bottom data set), and at the hydrophilicsurfaces with OH fraction of xOH= 1/8 (middle data set) andxOH = 1/4 (upper data set). On the hydrophobic surface thedynamics is purely diffusive and no exponential decay is discerned.On the hydrophilic surfaces, on the other hand, exponentiallydecaying surface binding with time constants τ = 210 ps (forxOH=1/8) and τ=340 ps (for xOH=1/4) is present. The lines arethe theoretical prediction fromeq 30with fit parameters inTable 6.

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Sendner et al. Article

on the xOH=1/4 surface a water molecule presumably binds totwoOH surface groups simultaneously or into a network of othersurface-bound water molecules. The fraction of bound watermolecules,φ=0.33 and φ=0.51 on the two surfaces, is rather highand close to saturation. The fact that the xOH=1/4 surface doesnot bind twice as many water molecules as the xOH=1/8 surfaceagain points to some cooperative (i.e., saturation) effects. The factthat our decay times are rather sensitive to the surface hydro-philicity is in contrast to the results in ref 79, where the residencetime is insensitive to the surface hydrophilicity. The longerresidence time at the hydrophilic surface is due to the fact thatthe water molecules are strongly attracted to the polar surfacegroups, whereas on the hydrophobic surface no binding isobserved at all but purely diffusive behavior is seen whencompared with the exact prediction from the diffusion equation.Note that water density fluctuations close to hydrophobic sphe-rical cavities display a somewhat different behavior that suggestsshort-time exponential decay.82 In bulk water, see Figure 15, thebehavior is also purely diffusive down to time scales as short as20 fs, which is shorter than the lifetime of a single hydrogenbond τH ∼ 1 ps.86

Care has to be taken with the interpretation of the obtained(normal) surface diffusion constant because of our crude assump-tions. The neglect of a surface potential and of a positiondependent mobility and the assumption of a constant waterdensity in the surface region most likely lead to changes in thefitting of the surface diffusion constant, but recent results onLennard-Jones fluids suggest that the changes are small.87

Conclusion

The structure and dynamics of liquidwater close to interfaces isexamined. We use nonpolar hydrophobic, as well as polarhydrophilic, substrates. In shear flow simulations we find sliplengths of only a few nanometers at hydrophobic surfaces withrealistic contact angles. Via variation of the surface hydropho-bicity, we obtain a heuristic scaling of the slip length with the

contact angle of the surface, which is backed up by some simplescaling arguments. We also find a strong dependence of the sliplength on the water depletion thickness at the hydrophobicsurfaces. Dissolved gas is found to strongly adsorb to thehydrophobic surfaces but only moderately increases the sliplength. In contrast, at the hydrophilic surfaces, no major gasaccumulation at the interface is observed. From the velocityprofile obtained from shear flow simulations at the polar, hydro-philic substrates, we infer that the viscosity of water in theinterfacial region is increased by a factor of 2 to 4, compared tothe bulk viscosity, depending on the surface fraction of hydroxylgroups. From the decay of the autocorrelation function, thedynamics of water molecules at hydrophobic and hydrophilicsurfaces is obtained. On hydrophobic surfaces, the dynamics isfound to be purely diffusive, with no sign of surface binding ortrapping. On hydrophilic surfaces, on the other hand, exponen-tially decaying surface trapping with typical decay times ofhundreds ofpicoseconds is found. It is this transientwater bindingthat most likely causes the effectively increased surface viscosityon hydrophilic surfaces in the shear flow geometry.

Even for contact angles as large as θ=140�, the slip lengthobtained in our simulations is only of the order of b=18 nm, seeFigure 8b, whereas recent experiments consistently yield largerslip lengths, of the order of 20 nm at a contact angle of 105�.17,18The reason for this disagreement is at present not clear. However,our simulations and scaling arguments give evidence for a strongdependence b ∼ δ4 of the slip length b on the depletion length δ.Thus, a small deviation in depletion length between experimentand theory would lead to amuch amplified disagreement betweenslip length in experiment and theory. Indeed, compared toexperiments for silanized surfaces, the simulations underestimatethe depletion length typically by a few angstr€oms.23

Acknowledgment.We thank S.Dietrich, D.Huang,D. Lohse,F. Sedlmeier, and L. Schimmele for discussions. Funding isacknowledged from the DFG via SPP 1164 (Nano- and Micro-fluidics) and via the Excellence Cluster Nano-Initiative Munich,and from theElitenetzwerkBayern in the framework ofCompInt.L.B. acknowledges support from the von Humboldt foundation.

Table 6. Parameters Obtained from Fitting the Autocorrelation Function in Bulka to the Analytic Expression Equation 27 and the Autocorrelation

Function on the Hydrophobic and Hydrophilic Surfacesb to the Analytic Expression Equation 29 without Surface Trapping (for the Hydrophobic

Surface) and to the Expression Equation 30 with Surface Trapping for the Hydrophilic Surfacesc

a See Figure 15. b See Figure 16. cHere, xOH is the surface fraction of OH groups, Δz is the correlation slab thickness, and z0 its distance from thehydrodynamic boundary (see graph on the right which shows the water density at a hydrophobic substrate with standard εCO parameter and arbitrarilyshifted origin of the z-axis), D is the fitted diffusion constant, φ is the fraction of initially surface-trapped water molecules, and τ is the characteristicrelease time of those bound water molecules.

(86) Ball, P. Chem. Rev. 2008, 108, 74–108.(87) Mittal, J.; M., T. T.; Errington, J. R.; Hummer, G. Phys. Rev. Lett. 2008,

100, 145901.

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