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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
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Publication Date (Web): July 10, 2009 | doi: 10.1021/la901314b
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 nonequilibrium Molecular Dynamics simulations. For hydrophobic surfaces under shearing conditions, the general
hydrodynamic boundary condition involves a finite surface slip. The value of the slip length depends sensitively on
the 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 pronounced
surface 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 this
surface layer depend on the density of polar surface groups. The dynamics of single water molecules in the surface
layer exhibits a similar distinction: on hydrophobic surfaces the dynamics is purely diffusive, while close to a
hydrophilic surface transient binding or trapping of water molecules over times of the order of hundreds of
picoseconds occurs. We also discuss in detail the effect of the Lennard-Jones cutoff length on the interfacial
properties.
Introduction
For many applications in microfluidic technology and almost
all situations in the biological domain, the behavior of interfacial
water is of prime importance. The geometric constraint of a solid
surface, as well as the interactions between water molecules and
the substrate, lead to structural changes of water compared to its
bulk properties. Surfaces can be divided into two classes according to their affinity to water: hydrophilic, water attracting, and
hydrophobic, water repellent. Since water molecules interact
favorably with surface charges, surfaces which bear electric
charges or polar groups typically are hydrophilic. In contrast,
non polar surfaces are generally hydrophobic since the water
molecules suffer a loss of hydrogen bonding partners at the
interface. This classification into hydrophobic and hydrophilic
surfaces has profound bearing on diverse situations such as
interacting colloidal particles, protein folding, and adsorption
of molecules or more complex structures on surfaces.
Over the past years it has become increasingly clear that the noslip boundary condition, that is, the condition of zero interfacial
fluid velocity, does not necessarily hold at nanoscopic length
scales.1-3 In fact, the hydrodynamic boundary condition at the
liquid/solid interface is of particular importance for microfluidic
applications4,5 or biological nanoscale scenarios, such as the
transport through membrane channels.6 But even in seemingly
unrelated areas such as the automobile industry the importance of
surface slip effects is acknowledged. As an example, mechanical
*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.
10768 DOI: 10.1021/la901314b
components which are coated with hydrophobic, diamond-like
carbon exhibit favorable friction properties,7,8 which could be put
to good use in ball bearings and gear boxes to increase efficiency.
Likewise, surface slippage amplifies the flow rate for pressure
driven flow, which enhances fluid transport in narrow channels.
Clearly, a noticeable increase of fluid transport is only obtained if
the slip length is comparable to the channel dimension. For
electrically driven flow, on the other hand, even small slip lengths
in the nanometer range lead to a considerable increase in flow.9,10
All these examples demonstrate that a profound and microscopic
understanding of the flow boundary condition at surfaces is
necessary.
In this paper we analyze the structure and dynamics of
interfacial water via molecular dynamic (MD) simulations. To
set the stage, we first consider equilibrium properties of water in
contact with a hydrophobic diamond-like surface. By variation of
the surface water affinity, the hydrophobicity of the surface is
controlled and quantified by the contact angle. Here we also
discuss the influence of the Lennard-Jones cutoff length, a
parameter that in simulations of interfaces plays a non-negligible
role.11,12 Next we study the hydrodynamic boundary condition at
these surfaces by non-equilibrium shear flow simulations. The slip
length at the hydrophobic diamond surface is found to be in the
nanometer range. The slip length exhibits a quasi-universal
relation with the contact angle and with the depletion thickness,
for which we advance some simple scaling ideas.13 We also
consider the effect of varying the magnitude of surface-roughness.
Recent experiments yield slip lengths in the range of tens of
(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 micrometers have been reported.19,20 A collection of experimental and
theoretical results can be found in ref 2. As a possible explanation
for the large slip lengths in experiments the presence of a thin gas
layer at the surface was considered.21-23 However, in simulations
of a Lennard-Jones liquid it was found that the slip length is only
moderately increased in the presence of dissolved gas.24 Using our
realistic water model and gas parameters, we also observe only a
modest enhancement of the slippage in the presence of surface
adsorbed inert gas. The analysis is then extended to polar,
hydrophilic surfaces. From the velocity profile in the interfacial
region we obtain zero slip and extract the surface viscosity, which
is increased by a factor of 2 to 4 (depending on the density of polar
surface groups) compared to the bulk value within a layer of a few
angstroms. We do not find evidence for a layer of frozen water or
for an increase of the interfacial viscosity of several orders of
magnitude at hydrophilic surfaces, as was seen for ultrathin films
of water on hydrophilic surfaces25,26 and for highly confined
water between hydrophilic surfaces27-29 both experimentally and
theoretically. In equilibrium simulations we analyze the diffusion
of single water molecules by calculating autocorrelation functions
in slabs. The single water molecule dynamics shows strikingly
different behavior on hydrophobic and hydrophilic surfaces: on
hydrophobic surfaces, water dynamics is purely diffusive over the
full time range between 1 and 1000 ps. Close to a hydrophilic
surface, on the other hand, diffusion is slowed down by transient
binding or trapping of water molecules on polar surface sites. The
decay time of this transient water arrest is of the order of hundreds
of picoseconds.
Molecular Dynamics Simulations
In MD simulations, the system is modeled on an atomistic
scale, and the trajectories of all constituent particles are calculated
using Newton’s equations of motion. Particles interact via bonding interactions, typically harmonic springs and cosine torsional
potentials,30 and non-bonded Coulomb and dispersion interactions. Dispersion interactions between particle species A and B
with distance r are described by a 6-12 Lennard-Jones potential,
"
#
σ AB 12
σ AB 6
uðrÞ ¼ 4εAB
r
r
ð1Þ
In most of our simulations, all Lennard-Jones interactions are
truncated at a radius R0=0.8 nm, as is customarily done to speed
up the simulations. The effect of varying values of R0 on the water
contact angle is discussed in detail below. Long ranged electrostatic interactions are calculated with the Particle-Mesh Ewald
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Acad. Sci. U.S.A. 2005, 102, 9458–9462.
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Chem. 2004, 25, 1656–1676.
Langmuir 2009, 25(18), 10768–10781
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
C-C
O1-O1
C-O1
0.0
0.3581
0.3166
0.3367
0.0
0.2774
0.6502
0.4247
Ne-Ne
Ne-O1
Ne-C
0.3136
0.3293
0.3351
0.6398
0.4951
0.4213
Ar-Ar
Ar-O1
Ar-C
0.3410
0.3285
0.3494
0.9964
0.8049
0.5257
Og-Og
Og-O1
Og-C
0.3030
0.3098
0.3306
0.4016
0.5110
0.3338
O-O
0.2760
1.2791
O-C
0.3144
0.5957
0.3017
0.8070
O-O1
a
The parameters for gaseous oxygen Og are taken from ref 37.
Ol denotes aqueous oxygen while O denotes the COH group of a
hydrophilic surface.
(PME) method.31,32 For all simulations, we use the SPC/E33 water
model. In this three site model the water molecule has partial
charges qO =-0.8476e and qH =0.4238e at the positions of the
oxygen and hydrogen nuclei. The OH bond length is 0.1 nm with a
tetrahedral bond angle of θ=109.5 at the oxygen position. The
Lennard-Jones potential is centered at the oxygen position with
parameters given in Table 1. Periodic boundary conditions are
applied in all three spatial directions. The simulations are performed in the NAPzT ensemble, that is, at fixed particle number
N, surface area A, temperature T, and vertical pressure Pz, while
the height of the box is fluctuating. The whole system is coupled to
a heat bath at 300 K and to a pressure of 1 bar via the Berendsen
algorithm34 with coupling constants τT=0.4 ps (temperature) and
τp = 1.0 ps (pressure). All bonds including hydrogen atoms are
constraint via the LINCS35 algorithm. The simulations are carried
out with the GROMACS36 package.
Hydrophobic Surface. We first consider a hydrophobic,
hydrogen terminated diamond surface. The diamond slab is
modeled by 2323 carbon atoms, arranged in a double facecentered-cubic lattice with lattice constant a = 3.567 Å. The
surface normal of the (100) plane points in the e^z direction. The
lateral extension of the slab is 3.0 3.0 nm2, its thickness is 1.5 nm.
Carbon atoms are connected by harmonic bonds and are subject
to angular and torsional potentials of the GROMOS96 version
53A6 force field.38 The surface layer of the diamond is reconstructed and terminated by H atoms. Carbon atoms and water
molecules interact via the Lennard-Jones potential in eq 1 with
interaction 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
(31) Darden, T.; York, D.; Pedersen, L. J. Chem. Phys. 1993, 98, 10089–10092.
(32) Essmann, U.; Perera, L.; Berkowitz, M. L.; Darden, T.; Lee, H.; Pedersen,
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.;
Haak, J. R. J. Chem. Phys. 1984, 81, 3684–3690.
(35) Hess, B.; Bekker, H.; Berendsen, H. J. C.; Fraaije, J. G. E. M. J. Comput.
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.
DOI: 10.1021/la901314b
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Article
Figure 1. Water density profiles for different liquid/solid interaction energies and surface structures. The center of the topmost
carbon atoms is located at z = 0. Panel a shows the water density
profiles at smooth diamond surfaces for different liquid/solid
interaction energies εCO given in kJ/mol and for one hydrophilic surface with OH-group surface fraction xOH = 1/4 and
εCO = 0.42 kJ/mol. In panel b, the density profiles for rough
hydrophobic surfaces, shown in Figure 2, are plotted for the
standard GROMOS value εCO = 0.42 kJ/mol.
the modified combination rule σAB = 0.5(σAA þ σBB) for the
Lennard-Jones diameter is used.37 The Ne-Ol and O-Ol parameters are explicitly given in Table 1.
There are different methods to change the hydrophobicity of a
surface in simulations, for example, by adding hydrophilic groups
or by rescaling the surface polarity.39-42 Within a certain range,
the hydrophobicity of the surface can also be tuned with the
liquid/solid interaction energy εCO between surface atoms and
water molecules, which we varied in the range between εCO=0.11
to 0.72 kJ/mol, while keeping the Lennard-Jones diameter σCO
constant. Decreasing the liquid/solid interaction energy increases
the hydrophobicity since water molecules are less attracted by the
solid and the water density close to the interface goes down, as
shown in Figure 1a. These density profiles are obtained in
simulations of the diamond surface in contact with 1850 water
molecules in a 3.0 3.0 8.0 nm3 simulation box in the NAPzT
ensemble. For not too low values of εCO, water layering is found
and the density profile is non-monotonic. For the standard
GROMOS value εCO =0.42 kJ/mol, listed in Table 1, the water
density in the first layer is roughly twice the bulk density. Lowering the interaction energy decreases the height of the density
maximum. The density profile for the lowest interaction energy
has no maximum at all and is similar to the density profile of the
air/liquid interface.43 Therefore, by tuning the interaction energy,
a smooth transition from a liquid/solid to a liquid/vapor-like
interface is obtained.
We also considered surfaces with different degrees of nanoroughness, shown in Figure 1. Surface R1 is constructed by erasing
every third pair of rows of surface atoms, and surface R2 is
obtained by the deletion of every second pair of rows. For the
construction of R3, every second single row of carbon atoms is
deleted. The roughest surface, R4, is generated by removing
carbon atoms down to the fourth surface layer. One should note
that these nanorough surfaces are very different from so-called
superhydrophobic surfaces, for which the surface structuring
occurs on a much larger length scale. From the density profiles
in Figure 1b) it is seen that on the rough surfaces water molecules
fill the voids left by the deleted surface atoms.
(39) Giovambattista, N.; Debenedetti, P. G.; Rossky, P. J. J. Phys. Chem. C
2007, 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) Castrillon, 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.
10770 DOI: 10.1021/la901314b
Sendner et al.
Hydrophilic Surface. Hydrophilic surfaces are constructed
based on the smooth diamond surface by substituting every
fourth (xOH=1/4) or eighth (xOH=1/8) surface carbon atom by
a C-O-H group with standard angular and torsional force
potentials. The bond angle at the oxygen atom is 108 and the
partial charges are set to qO = -0.674e, qH = 0.408e and qC =
0.266e, while the standard GROMOS value εCO=0.42 kJ/mol is
used. These parameters are taken from the GROMOS96 forcefield for serine, the Lennard-Jones parameters are those of the
GROMOS96 version 53A6 forcefield, see Table 1. Figure 3 shows
the top view of the two studied hydrophilic surfaces. Torsional
rotations around the C-O bond are in principle possible but
severely inhibited by the large activation energies, so the OH
rotational degrees of freedom are effectively quenched.
As can be seen in Figure 1a, the water is closer to the hydrophilic surface compared to the hydrophobic interface, and the first
water peak is more pronounced and has a smaller width. Only
minor differences are present between the two hydrophilic surfaces (data not shown): For the larger OH density, the first water
peak is slightly higher and closer to the interface.
Contact Angle. One experimentally easily accessible parameter characterizing the surface hydrophobicity is the contact
angle which ranges from 180 (for a hypothetical substrate with
the same water affinity as vapor) down to 0 for a hydrophilic
surface. On smooth hydrophobic surfaces, contact angles up to
130 are experimentally observed.44 Even higher contact angles
can be reached with patterned surfaces. In MD simulations, the
contact angle can be determined either by the simulation of a
nanodroplet on the surface45 or by the calculation of the surface
tension of a flat interface (as used in this study). Via Young’s
law,46 the contact angle is given by
cos θ ¼ -
γls - γsv
γlv
ð2Þ
and depends on the surface tensions of the liquid/solid (ls), solid/
vapor (sv), and liquid/vapor (lv) interfaces. In the simulations, the
surface tension is obtained from the diagonal components of the
virial tensor,47 γ=(1/(2A))[2Πzz - (ΠxxþΠyy)], whereP
A denotes
ν μ
the surface area and the virial tensor is given as Πμν=Æ N
i=1ri Fi æ
and depends on the positions ri and forces Fi of all single particles.
For the calculation of the virial tensor, the surface atoms are
frozen and all interactions between the surface atoms are switched
off, which directly yields the difference γls - γsv. Freezing the surface atoms also avoids subtle problems with substrate stress
contributions.47 To calculate the surface tension of the air-water
interface, a water slab of up to 1807 molecules is simulated in a
3.0 3.0 12.0 nm3 box in the NVT-ensemble, yielding a surface
tension of γlv = 0.0543 ( 0.002 N/m. The substantial deviation
from the experimental value of γlv =0.072 N/m is a well-known
deficiency of the SPC/E water model11 and means that one has to
be careful when comparing interfacial energies with experiments.
The systems first were equilibrated for at least 200 ps with
subsequent production runs of 5 ns.
Figure 4a shows the contact angle for a hydrophobic diamond
surface as a function of the water-surface interaction strength εCO
using the pressure tensor method (circles), compared to the results
obtained from the simulation of a nanodroplet on the surface
(triangles). The data fall on a straight line that reaches θ=180 for
vanishing interaction strength. Whereas the limiting behavior as
εCO f 0 is as expected, the linear behavior 180 - θ ∼ εCO is not
obvious and will be later discussed in connection with a depletion
layer occurring at the interface. The data for the droplet method
(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 University 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.
Langmuir 2009, 25(18), 10768–10781
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Article
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Figure 2. Hydrophobic diamond-like surfaces. The flat standard surface is denoted as diamond. The rough surfaces are denoted as Rn and are
constructed by selectively deleting rows of surface atoms.
Figure 3. Top view of the hydrophilic diamond surface surface with
surface fraction of OH groups xOH = 1/4 (a) and xOH = 1/8 (b).
1 þ cos(θ) f 0 in Figure 4b is especially troublesome, since it
erroneously points to a drying transition at a finite value of the
surface water Lennard-Jones attraction, which is unreasonable,
as will be discussed further below.
The linear dependence 1 þ cos(θ) = εCO on the other hand
follows from a simplified calculation of the surface tension.
Although in this derivation one assumes constant liquid and solid
densities and neglects the presence of a depletion layer, which play
an important role as will be discussed later, the derivation which
we will now sketch is quite useful conceptually. The surface
tension of the liquid/solid interface is related to the work
H12 per surface area which is necessary to separate the liquid
and solid slabs, defined as46
H12 ¼ γsv þ γlv - γls
ð3Þ
It can be easily calculated approximately assuming homogeneous
solid and liquid densities Fs and Fl and neglecting any electrostatic
or interfacial entropy contributions. For that we define the
interaction energy of a single liquid molecule at a distance z from
the interface with the solid phase as
Z
uls ðzÞ ¼ 2πFs
Z
R0
1
dr
Z
¼ 2πFs
dðcos θÞ r2 uðrÞ
z=r
z
R0
dr ðr2 - zrÞuðrÞ
ð4Þ
z
where the intermolecular liquid/solid interaction potential is
u(r) and R0 denotes the upper cutoff. Considering the Lennard-Jones potential in eq 1, which has a short ranged repulsive
and a long ranged attractive part, the water film will exhibit a
thin depletion layer of width z* that is defined by the condition
uls(z*)=0 and is (in the limit R0 f ¥) explicitly given by
Figure 4. Contact angle θ of water in contact with the hydrophobic diamond surface determined via the virial tensor as a function
of the interaction energy εCO. (a) The data (spheres) are shown to be
consistent with the scaling 180 - θ ∼ εCO (solid line). In addition
data for the simulation of a nano droplet in contact with the
diamond surface (triangles) are presented.43 (b) The data are
compared with the scaling form 1 þ cos θ ∼ εCO including a
constant shift (broken line). Data for different rough surfaces are
included. Note that the scaling laws shown in panels a and b as lines
are mutually incompatible, see text.
are taken from ref 43 and are in excellent agreement with the
method used in this work except at large contact angles where line
tension and cutoff effects influence the droplet data. The contact
angle at the diamond surface with the standard GROMOS
forcefield, εCO =0.42 kJ/mol, is 106. In Figure 4b, we plot the
results for the contact angle as 1 þ cos(θ) as a function of the
liquid/solid interaction energy εCO. Here again a reasonable
description is obtained with a linear function that however is
shifted by an offset of the order of εCO ≈ 0.14 kJ/mol. Note that
the two linear laws in Figure 4, panels a and b, are incompatible
with each other, since 1þcos(θ) ∼ (180 - θ)2 as θ f 180, and
indeed a slight deviation of the data from the straight line for
small values of εCO is observed in Figure 4b. The offset in εCO as
Langmuir 2009, 25(18), 10768–10781
1=6
2
z ¼σ
15
/
ð5Þ
The work term H12 contains the interactions of all water molecules with the surface and is given by
Z
H12 ¼ - Fl
¥
z/
Z
dz uls ðzÞ ¼ - πFl Fs
R0
z/
dz zðz - z/ Þ2 uðzÞ ð6Þ
With the linear dependence of the Lennard-Jones potential in eq
1 on εCO, H12 turns out to be a linear function of the water-surface
interaction energy εCO as well. From Young’s equation, eq 2, it
follows that the cosine of the contact angle should be linearly
dependent on the interaction energy,
1 þ cos θ ¼
γsv þ γlv - γls
H12
¼
∼ εCO
γlv
γlv
ð7Þ
This linear relation gives a fair description of the simulation
results in Figure 4b but, as we have noted before, would suggest a
vanishing of the expression 1 þ cos θ at a finite liquid solid
interaction energy εCO. Such a drying transition would not be
expected on theoretical grounds since the entropic repulsion of a
confined interface that is governed by surface tension is very
DOI: 10.1021/la901314b
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Figure 5. (a) Contact angle θ at the hydrophobic diamond surface for different liquid/solid interaction energies εCO and different cutoff radii
R0. (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 surface with εCO = 0.57 kJ/mol as a function of the cutoff radius. The plot shows the contact angles obtained
from 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 is
obtained 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 ( 0.003
1807
0.8
0.0543 ( 0.002
1807
1.0
0.0580 ( 0.002
1807
1.2
0.0588 ( 0.002
1807
1.4
0.0595 ( 0.002
a
The surface tensions are determined via the pressure tensor from
NVT simulations of N water molecules in a 3 3 12 nm3 simulation
box.
short-ranged, and the attraction between water molecules and the
substrate should keep the depletion layer at a finite width. This
follows from theories that incorporate the intricate coupling
between interfacial fluctuations and interactions between substrate and liquid, according to which interfacial fluctuations (that
give rise to a fluctuation pressure that decays exponentially with
depletion layer thickness) are dominated by the attraction between the water film and the substrate as long as the decay is
power-law-like (which is the case for van-der-Waals type interactions).48-50 However, the situation is complicated by the fact that
in MD simulations all interactions are cut off at a length R0 which
on the other hand makes the attraction between water and
substrate also very short-ranged (an issue that will be considered
in the next section). So our conclusion regarding the validity of
eq 7 is that although at first sight the fit to the simulation data in
Figure 4b seems reasonable, we argue that depletion effects and
density profile variations not included in the derivation of eq 7
rather 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, as
will be discussed further below considering the additional effects
of a depletion layer.
Cutoff Dependence. In MD simulations, all Lennard-Jones
interactions are usually truncated at an upper cutoff R0. In the
above simulations, the cut off radius was set to R0 =0.8 nm. It
turns out that the value of the cutoff sensitively influences the
resulting contact angle, an issue only recently considered in
simulations.11,12 In Figure 5a, the contact angle at a hydrophobic
diamond surface is shown as a function of the liquid/solid
interaction energy εCO and for different values of R0. The
liquid/vapor surface tension was similarly calculated for different
cutoff radii in simulations of a water film of 1807 water molecules
in the NVT ensemble in a 3 3 12 nm3 simulation box. The
liquid-vapor surface tension slightly increases with increasing
cutoff radius R0 and shows no statistically significant dependence
(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.
10772 DOI: 10.1021/la901314b
on the number of water molecules, see Table 2, in agreement with
previous simulations.11,12 In fact, the main contribution to the
interfacial tension stems from Coulomb interactions which are
fully taken into account via Ewald summation.
As seen in Figure 4a, for the most hydrophobic surfaces, that is,
large contact angles, a variation in the cutoff radius only leads to
minor changes of the contact angle. This is an important fact, as it
shows that the spurious drying transition suggested when plotting
the data as in Figure 4b is not caused by too small a cutoff radius
R0. In contrast, for the less hydrophobic surfaces, that is, for small
contact angles, the cutoff radius substantially influences the
contact angle. Increasing the cutoff radius leads to smaller contact
angles. This effect can be understood from the density profiles, see
Figure 5, panels b and c. As R0 increases, the interfacial water
density goes up, thus decreasing the substrate-water interfacial
tension. This effect is more pronounced for large values of the
interaction strength εCO.
The dependence of the contact angle on the cutoff radius R0 can
be rationalized by a simple model calculation. Again assuming
constant liquid and solid densities and neglecting interfacial
entropy, the interfacial work H12 as defined in eq 6 can for large
R0 be written as
H12 ¼ 2πεCO Fs Fl σ 4
" 2
3 #
1 15 1=3
σ
σ
þO
8 2
R0
R0
ð8Þ
where we have used the intermolecular potential eq 1 and the result
for the depletion layer width eq 5. It is seen that the smaller R0
becomes, the weaker the interfacial attraction and thus the contact
angle increases. Figure 5d shows the contact angle as a function of
the cutoff radius for a diamond surface with fixed εCO=0.57 kJ/mol
from simulations (circles) and from eq 8 (triangles). This is the largest
value of εCO considered by us, for which the cutoff dependence of the
contact angle is very dramatic. In the calculation, the bulk water and
diamond densities are used for Fs,l. The trend in the simulations is
well captured; considering the very approximate character of the
scaling Ansatz leading to eq 8, the agreement is surprisingly good.
For all subsequent simulations, we use a cutoff radius of R0=0.8 nm.
It is important to keep in mind that, especially for large surface water
interaction strengths, the contact angle and thus all other interfacial
properties, exhibit a pronounced dependence on the value of the
cutoff radius R0 employed in the simulations. It follows that
whenever simulations of interfacial systems are performed, not
only must the interaction strength be reported, but also the cutoff
radius. This is important also since the water force-field parameters
have been optimized for a certain finite Lennard-Jones cutoff
and deviations from the desired water bulk properties will occur if
the cutoff length deviates drastically from that value. Ideally, one
should always determine the contact angle, which is the only
experimentally meaningful parameter that characterizes the surface
hydrophobicity.
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Figure 6. Snapshot of the sheared hydrophobic diamond system
with D = 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 lateral
extension. In the middle, the water density profile is shown. The
right figure shows the solvent velocity profile for diamond velocity
ν0 = 0.02 nm/ps and the definition of the slip length b. The vertical
lines denote the surface velocity ( ν0. The Lennard-Jones parameters are the standard GROMOS values given in Table 1 which
leads to a contact angle of 106 at the diamond surface.
Shear at Hydrophobic Surfaces
At a hydrophobic surface, the partial slip boundary condition
holds, which quantifies the amount of slippage by the slip length
b. This length is defined via the normal gradient of the tangential
fluid velocity field v(z),3
ðv0 -vÞz ¼z0 bðDv=DzÞz ¼z0
ð9Þ
where v0 and z0 denote the velocity and position of the surface, see
Figure 6.
The simulation setup consists of two diamond blocks which are
separated by two SPC/E water slabs of typical thickness D ≈ 4 nm
each. This corresponds to roughly 1000 water molecules in each
water slab. In Figure 6, we show a snapshot of half the simulation
system, where only one of the two water slabs is shown. As was
done in previous simulation setups,51 Couette shear flow is
induced by attaching harmonic springs with spring constants
k=1000 kJ mol-1 nm-2 to the upper and lower surface. The upper
spring is pulled with a velocity of typically v0=0.02 nm/ps in the
x-direction and the lower spring is pulled with the same speed in
the opposite direction such that the net momentum input
vanishes. The motion of the diamond blocks induces a linear
velocity profile for the solvent flow, see Figure 6. The shear rate
follows as γ_ 2v0/D where D is the water slab thickness. Using the
definition of a partial slip boundary condition at the position of
the surface in eq 9, the slip length b is obtained by extrapolating
the velocity profile. For that purpose, the velocity profile is fitted
to a linear function. The location of the surface at which the slip
boundary condition is applied is defined by the center of the
topmost layer of surface atoms. The systems are equilibrated for
200 ps and then subsequent production runs of up to 30 ns are
performed. Several simulations with the same parameters are
performed and all trajectories are used for further analysis.
The used Berendsen weak-coupling thermostat is in principle
critical for shear simulations. We checked this issue by performing
two benchmark simulations, where a Berendsen thermostat with
velocity scaling in all Cartesian directions and a Nose-Hoover
thermostat with velocity scaling only in the direction perpendicular
to the shear direction and perpendicular to the surface normal were
applied, during otherwise identical simulations.13 We found no
(51) Thompson, P. A.; Robbins, M. O. Science 1990, 250, 792–794.
Langmuir 2009, 25(18), 10768–10781
Figure 7. Slip length b as a function of shear rate γ_ at a smooth
diamond surface with the standard GROMOS Lennard-Jones
parameter value εCO = 0.42 kJ/mol given in Table 1. The different
symbols correspond to different thicknesses D of the water film
between the two diamond slabs.
difference between these different simulation protocols, meaning
that the type of thermostat does not influence the slip-length results.
Since experimental shear rates are substantially lower than the
rates used in MD simulations, a careful examination of the
influence of the applied pulling velocity on the resultant slip length
is necessary.52 Therefore, shear flow simulations at different shear
rates are performed to rule out artifacts that have to do with
deviations from the linear-response behavior. Up to shear rates of
1010 s-1, the slip length b is almost independent of the shear rate
and independent of the water film thickness which was varied
between D = 2 and 8 nm, see Figure 7. To obtain reliable data
at acceptable computational cost a shear rate of γ_ = 1010 s-1 is
utilized for all subsequent shear simulations, with a water-slab
thickness of D=4 nm.
In Figure 8a we show the slip length as a function of the surface
energy parameter εCO for the four different surface topologies
shown in Figure 2. For decreasing interaction energies, that is, for
more hydrophobic surfaces, the slip length increases. For the
lowest interaction energy, the slip length is of the order of 20 nm.
The slip length for the rough surfaces is always smaller than for
the smooth diamond surface. At the roughest surface R4, the slip
length is smallest. Increasing surface roughness leads to smaller
values for b, since the friction at the liquid/solid interface is
enhanced by the stronger corrugation of the liquid/solid interaction potential. A similar dependence of the slip length on the
surface interaction strength was seen in simulations on polymeric
melt systems53,54 and in simulations of liquid-liquid interfaces.55
To rationalize the dependence of the slip length on static
surface properties, especially on the interaction energy εCO, we
follow an argument of Bocquet and Barrat,56,57 who related the
slip length of a fluid with viscosity η at a surface with surface area
A to the friction force Fx exerted by the fluid on the solid
Fx η
¼ vx ðz ¼ z0 Þ ¼ ηv0 x ðz ¼ z0 Þ
b
A
ð10Þ
The friction coefficient λ is defined as λ=Fx/(vx(z=z0)A) which
together with eq 10 yields b=η/λ. The Green-Kubo relation for
(52)
(53)
(54)
(55)
(56)
(57)
Thompson, P. A.; Troian, S. M. Nature 1997, 389, 360–362.
Barsky, S.; Robbins, M. O. Phys. Rev. E 2002, 65, 021808.
Servantie, J.; M€uller, M. Phys. Rev. Lett. 2008, 101, 026101.
Koplik, J.; Banavar, J. R. Phys. Rev. Lett. 2006, 96, 044505.
Bocquet, L.; Barrat, J.-L. Phys. Rev. E 1994, 49, 3079–3092.
Barrat, J. L.; Bocquet, L. Faraday Discuss. 1999, 112, 119–127.
DOI: 10.1021/la901314b
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Sendner et al.
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 for
the different surface structures. (b) The slip length is plotted versus the contact angle. The broken line shows a fit to b (1 þ cos θ)-2 for all
data points on the smooth diamond surface while the solid line shows a fit according to b (180 - θ)-2. See text for more details.
R
the friction coefficient reads λ= ¥
0 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 meansquared lateral force acting between the substrate and the liquid
phase. For the relaxation time we write τ ∼ σ2/D where σ is a
characteristic length scale and D is the diffusion coefficient of
a fluid particle. The mean squared lateral surface force comes
from force fluctuations, and in the thermodynamic limit scales as
ÆF2xæ ∼ C^ (εCO/σ2)2A where C^ is a geometric factor that accounts
for roughness effects: The rougher the surface, the larger C^.
Putting everything together, we arrive at the scaling expression
b∼
σ2 ηDkB T
C^ ε2CO
ð11Þ
b ¼ δðη0 =ηs -1Þ
which incorporates a number of drastic approximations and
simplifications but on the other hand constitutes an easily testable
relation between slip length and surface energy εCO. In Figure 8a,
the slip length is plotted as a function of 1/ε2CO for the different
roughnesses considered. The agreement with the straight lines
verifies the above scaling considerations. Also, the rougher the
surface, the smaller the slip length, again in agreement with
the scaling in eq 11 if one takes the coefficient C^ as a measure
of the degree of surface roughness.
Together with the linear dependence of the cosine of the contact
angle on the liquid/solid interaction energy, eq 7, a simple relation
between the slip length and contact angle is obtained,13
b ¼ 0:63 nm 3 ð1 þ cos θÞ -2
ð12Þ
where the numerical prefactor is obtained from a fit to the data,
broken line in Figure 8b. We hasten to add that the alternative and
more physically sound relation εCO ∼ 180 - θ, as suggested by
Figure 4a, gives the modified scaling relation
b ¼ 12 μm 3 ð180 - θÞ -2
ð13Þ
which in fact gives a comparable description of the data, solid line
in Figure 8b. The effects of roughness on b on the one hand and on
θ on the other hand partially cancel out: Increasing roughness
leads 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 to
decreasing contact angles, see Figure 4b. The influence of roughness on the scaling plot in Figure 8b is thus reduced. Despite the
10774 DOI: 10.1021/la901314b
rough estimates leading to eq 11, the simulation results follow
nicely the predicted scaling. The dependence of the slip on the
contact angle shows the same dependence also for different
surface structures such as fcc(100) Lennard-Jones surfaces or
alkane chains.13 This quasi-universal relation between contact
angle and slippage is of particular interest, since the contact angle
is 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 existence
of a thin depletion layer between the surface and the water
with thickness δ. The viscosity ηs of this layer is assumed to
be substantially lower than the bulk water viscosity η0, which
gives rise to an apparent slip length21,23,54
ð14Þ
which is linearly dependent on the depletion width. To test this
prediction, we used two different definitions of the depletion layer
width δ: (i) the position where the water density is half its bulk
value, δ0 , and (ii) the integrated density deficit of the water layer41,58
Z ¥
δ ¼
dz ½1 - Fs ðzÞ=Fbs - Fl ðzÞ=F1 b ð15Þ
0
as visualized in Figure 9. Here Fbl,s are the bulk densities of the liquid
and solid phase. Both depletion lengths can be directly calculated
from the density profiles of the simulations. For both definitions,
we do not find a linear dependence of the slip length on the
depletion width, see Figure 10c. Therefore, unless one postulates a
parametric dependence of the depletion-layer viscosity ηs on the
depletion width δ, the data do not support the prediction eq 14 of
the two-viscosity model. In addition, the depletion length is less
than a molecular diameter which makes the definition of an
effective viscosity for such a thin layer awkward.
In Figure 10a, the depletion length is plotted as a function of
the liquid/solid interaction energy, which suggests the scaling
δ -2 ∼ εCO
ð16Þ
Combining this with the relation eq 11, one immediately obtains
a quartic dependence of the slip length on the depletion length,
b ∼ δ4
ð17Þ
(58) Mamatkulov, S. I.; Khabibullaev, P. K.; Netz, R. R. Langmuir 2004, 20,
4756–4763.
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Figure 9. (a) Density profile of the diamond slab in contact with
water, (b) the integrand f(z) =R1 - Fs(z)/Fs - Fl(z)/Fbl of eq 15, and
(c) its running integral g(z) = z0 dz0 f(z0 ) together with the definition of the depletion length. The graphs show data for the hydrophobic diamond surface with εCO = 0.2550 kJ/mol which corresponds to a contact angle of 136.
which is nicely confirmed by the data in Figure 10c (broken line). On
the other hand, combining eq 16 with eq 7 one obtains the scaling
relation
δ -2 1 þ cos θ
ð18Þ
which is compared with the simulation data shown in Figure 10b.
The agreement is fair, but note that the line for the smooth diamond
surface extrapolates to a finite depletion length for θ f 180, which
is at odds with the expectation that the interaction between the
water and diamond slabs vanishes only when δ f ¥. This is the
same problem one encounters when applying eq 7 to the simulation
data in Figure 4b. We can in fact bring out the shortcomings of eq 7
more clearly: Taking into account the presence of a depletion layer
of width δ between the solid and the liquid phase (which amounts to
replacing the lower integration boundary z* in eq 6 by δ), the
interfacial energy is given by
H12 ∼ Fl Fs σ6 εCO =δ2
ð19Þ
Naive application of Young’s equation, eq 2, and using the
definition of H12 in eq 3 gives the relation 1þcos θ εCO/δ2, which
cannot be simultaneously satisfied with eq 16 and eq 18, as noted
before.13 A simple way out of this apparent dilemma is to postulate
a second contribution to the interfacial energy,
-H12 ∼ a=δm - Fl Fs σ 6 εCO =δ2
ð20Þ
with at first an arbitrary exponent m and prefactor a. In fact,
the relation eq 16 follows from eq 20 by minimizing -H12 with
respect to δ if one chooses m=4; at that minimum one finds
H12 ∼ εCO2, in contrast to the naive scaling result in eq 7.
Indeed, using H12 ∼ ε2CO together with eqs 2 and 3 yields
εCO ∼ (1þcos θ)1/2 ∼ 180 - θ which gives an equally sound
description of the data, as shown in Figure 4a. Finally,
combining εCO ∼ 180 - θ with the result eq 11 gives the
Langmuir 2009, 25(18), 10768–10781
modified scaling law b ∼ (180 - θ)-2, which describes the data
in Figure 8b equally well as the form proposed in eq 12.
We conclude that the quality of the numerical data does not
allow to firmly distinguish between the conflicting scaling forms
for the dependence of the contact angle θ on εCO, as suggested by
the straight lines in Figure 4, panels a and b. The scaling relation
-2
∼ εCO, shown in
b ∼ ε-2
CO, shown in Figure 8a, and the relation δ
Figure 10a, seem more robust and in fact are fully consistent with
the scaling result eq 11 and the modified interfacial free energy
scaling form in eq 20. Taken together, these relations lead to the
striking result b ∼ δ4, which is presented in Figure 10c.
There are a number of effects that could lead to the postulated
repulsive contribution ∼ a/δm with m = 4 in eq 20 with wellstudied consequences on the wetting and dewetting behavior.59
One possible candidate, the repulsion due to confinement of the
interface fluctuations, is by itself quite complicated48-50 and
displays a distance-dependent crossover from being dominated
by interfacial tension (at large length scales) to being dominated
by interfacial bending rigidity (at short length scales).60 On the
other hand, the water density profile changes as a function of
the Lennard-Jones parameter εCO, which also gives rise to sizable
corrections to H12. Finally, the modification of the electrostatic
interactions between water molecules in the interfacial layer
because of the presence of the solid substrate will also contribute
to the interfacial free energy. At this stage, we can only state that a
consistent description of the data is possible with an ad-hoc
modified interfacial free energy as in eq 20, the microscopic origin
of which is left for future investigation.
Slippage and Dissolved Gas. To examine the effect of
dissolved gas on the slip length, shear flow simulations on
hydrophobic diamond surfaces are performed with added gas
particles. In most simulations, 10 gas particles are inserted in each
of the two water slabs. We examine different types of gas particles.
Species X (mx = 12.01 u) interacts equally with all other atoms
present in the system via a purely repulsive potential,
VX ðrÞ ¼ 4εX
σX
r
12
ð21Þ
with σX = 0.3581 nm and εX = 0.2774 kJ/mol, the LennardJones parameter of carbon. GROMOS force field parameters
are 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 bond
length of 1.21 Å. The interaction parameters for all considered
gas types are summarized in Table 1. For the simulations with
argon and gas type X, we consider variations of the surface
hydrophobicity, while for oxygen and neon only the standard
diamond parameters are used. For the simulations with argon,
all interaction energies involving the surface atoms are recalculated by εCAr=(εCCεArAr)1/2 and εCO=(εCCεOO)1/2 while for
the simulations with gas type X, only the liquid/solid interaction energy εCO is varied, while the gas/solid interaction εX is
unchanged and given by eq 21.
Figure 11 shows the density profiles obtained within shear
flow simulations, which in fact are indistinguishable from
density profiles obtained in equilibrium simulations. The
densities are averaged over all four interfaces in the simulation
cell. As expected, the gas atoms accumulate at the hydrophobic interface, as was previously observed in MD simulations of Lennard-Jones fluids24 and in grand canonical
(59) Dietrich, S.; Schick, M. Phys. Rev. B 1986, 33, 4952–4962.
(60) Netz, R. R.; Lipowsky, R. Europhys. Lett. 1995, 29, 345–350.
DOI: 10.1021/la901314b
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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 the
contact 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 depletion
length δ for smooth and rough surfaces. The dashed line shows a fit for the diamond surface to b δ4. The depletion length δ0 in the inset (note
the different scale of the x-axis) is defined by the position where the water density is half its bulk value.
Figure 12. Snapshots of simulations with dissolved gas. (a) Simulation 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.
Figure 11. Density profiles for different gas types at the smooth
hydrophobic diamond surface. The gas densities are scaled by their
bulk concentrations (for gas type X the density is additionally
divided by a factor of 2). The center of the topmost carbon surface
atoms is located at z = 0. The total amount of dissolved gas is
10 gas particles per water slab, except for one simulation for
O2 with only 4 oxygen molecules per water slab. In panel a, data
are 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 shown
for different surface hydrophobicities, (i) εCO = 0.255 kJ/mol,
(ii) εCO = 0.425 kJ/mol, and (iii) εCO = 0.570 kJ/mol, for the gas
type X and Ar. Additionally, data for the simulation of dissolved
argon gas between two hydrophilic surfaces (xOH = 1/4) is shown.
Monte Carlo simulations employing the SPC water model.37
The density of neon, argon, and oxygen gas at the interface is
increased by a factor of roughly 20-60 compared to the
density in the bulk of the water slab. The gas adsorption is
restricted to one monolayer of gas particles, there is only one
narrow peak present in the density profiles shown in Figure 11.
The density of the purely hydrophobic gas type X at the
interface is increased by a factor of more than 150. All X atoms
are mostly present on one surface and the formation of a large
cluster of gas atoms can be seen in the simulations, see
Figure 12 This strong clustering is reflected in the shoulder
of the X-density profile in Figure 11. The other gases do not
exhibit such a clustering and they are more or less equally
distributed on the two interfaces.
10776 DOI: 10.1021/la901314b
A variation of the surface hydrophobicity does not lead to
qualitative changes in the density profiles. The accumulation of
argon 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 simulations of dissolved gas between the hydrophilic surfaces, as shown
for argon in Figure 11b: Here, the gas density at the surface is only
twice the bulk value.
Although the gas particles Ne, Ar, and O2 are strongly
attracted to the interface, they frequently desorb from the interface as seen in the snapshot, Figure 12a. The gas accumulation at
the interface is not because of the Lennard-Jones attraction
between the surface and the gas atoms, since also gas type X with a
purely repulsive potential exhibits an increased density at the
surface. Rather, the hydrogen bonding network is less perturbed if
the gas particles are at the interface which favors gas adsorption at
a 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 of
1 atm and T=298.15 K.61 In the simulations, the total pressure is
fixed at 1 bar ≈ 1 atm. Since the partial pressure of water vapor at
room temperature is roughly 0.02 atm, the partial gas pressure in
the simulation is comparable to the experimental situation, pgas ≈
(1 - 0.02) atm ≈ 1 atm. From the density profiles, we deduce that
the mole fraction of dissolved argon gas in the bulk liquid has a
(61) Wilhelm, E.; Battino, R.; Wilcock, R. J. Chem. Rev. 1977, 77, 219.
Langmuir 2009, 25(18), 10768–10781
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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]
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Ar
0.255
8.92 ( 2.12
7.54 ( 0.76
X
0.425
2.69 ( 0.18
2.17 ( 0.15
Ne
0.425
2.53 ( 0.10
2.17 ( 0.15
Ar
0.425
2.77 ( 0.55
2.17 ( 0.15
0.425
2.61 ( 0.10
2.17 ( 0.15
O2(10)
0.425
2.57 ( 0.24
2.17 ( 0.15
O2(4)
X
0.570
1.19 ( 0.19
0.75 ( 0.10
Ar
0.570
1.42 ( 0.04
0.75 ( 0.10
a
For comparison, also the results for the simulations without gas are
shown, b.
value of about 5 10-3 (similarly, for oxygen the mole fraction in
the bulk is about 4 10-3) and is thus much higher than the
maximally soluble mole fraction (the gas bulk concentration is
determined from all gas molecules that have a surface separation
of at least 1 nm). However, we do not observe phase separation
of argon or oxygen in the simulations, which is related to the
very large nucleation barrier for gas bubble formation in such
small systems. We nevertheless checked whether the unphysically
large gas concentration in the simulations leads to spurious
effects. For this we performed one simulation with only four
oxygen molecules per water slab. In the normalized density
profiles in Figure 11a, no difference is seen between the two
different oxygen concentrations; we conclude that for all gases
except the artificial gas type X (for which the solubility presumably is even lower than for the other gases considered) the
simulation results approximately reflect the situation of a saturated solution of gas in water.
The surface adsorption of gas particles enhances the slip length
only slightly, see Table 3 and Figure 13. The largest relative
change in slip length b because of gas adsorption is seen for the
highest liquid/solid interaction energy (i.e., for the smallest
contact angle). There, the slip length roughly doubles when, for
example, Ar atoms are added to the water. Dissolved gas only
moderately amplifies the slip length for large contact angles,
which remain in the range of only a few nanometers. Thus, we
conclude that large slip lengths as measured experimentally are
not caused by surface adsorbed thin gas layers. However, experimental measurements suggest the formation of gas-nanobubbles
at the liquid/solid interface.62-66 The lateral dimension of these
bubbles is in the order of 100 nm, with a height of several
nanometers which could significantly enlarge the slip length.
These gas cavities are much larger than simulation box sizes
one can possibly study and are therefore out of reach for present
computer simulations.
Shear Flow Simulations at Hydrophilic Surfaces
We now examine the hydrodynamic boundary condition and
the viscosity of water close to polar, hydrophilic surfaces. Experiments report on viscosities for confined water between mica
surfaces at room temperature, which are comparable to the bulk
viscosity.67,68 Sub-nanometer confinement was shown to lead to
an increase in viscosity of approximately 80 times the bulk
(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.
Langmuir 2009, 25(18), 10768–10781
Figure 13. Slip length with and without dissolved argon gas at the
hydrophobic smooth diamond surface plotted versus the contact
angle. In the simulations we add 10 argon atoms to each water slab,
which corresponds to a molar gas fraction of about 5 10-3 in the
bulk.
viscosity in simulations.69 Other experiments and simulation
studies report on a strong increase in viscosity of several orders
of magnitude for highly confined water films.27,29,70 Besides from
these 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 an
ice-like structure of thin water films.25,26,28,71-73 These crystal
like structures are identified via a sharp drop in the diffusion
constant, a substantially increased shear viscosity, and can sustain
shear stress. At hydrophobic interfaces, in contrast, this ice like
water structure is not observed. A critical summary of simulation
work on confined water has recently been published.74
To clarify the structure and properties of water at hydrophilic
surfaces, we perform NEMD simulations of water at polar,
hydrophilic surfaces with relative large water slab thickness. In
the non equilibrium shear flow simulations, the shear viscosity
profile of interfacial water is obtained from the fluid velocity
profiles. Via this method, we are able to locally probe the shear
viscosity of water close to interfaces without introducing additional effects due to confinement. The velocity profiles, Figure 14,
differ qualitatively from those obtained for the hydrophobic
surfaces: close to the surface, the gradient of the velocity profile
is smaller than in the middle of the water slab. In contrast, the
gradient of the velocity profiles at the hydrophobic surfaces is
constant, even close to the interface, see Figure 6. At the
hydrophilic surface, the water molecules at the interface are
dragged along with the surface. The region over which this
dragging of water molecules occurs roughly corresponds to the
extent of the first water layer. This feature is due to the strong
interaction between the hydroxyl surface groups and the water
molecules. These findings are in agreement with velocity profiles
obtained from simulations of sheared water films between mica
surfaces.75
Since the shear viscosity is linearly related to the inverse of the
velocity gradient, it is possible to define two different viscosities in
the system. One in the surface region, ηs, and the other one in the
bulk region, η0. To obtain the shear rates in the two regions, the
(69)
(70)
(71)
(72)
(73)
(74)
5209.
(75)
Leng, Y.; Cummings, P. T. Phys. Rev. Lett. 2005, 94, 026101-4.
Sakuma, H.; Otsuki, K.; Kurihara, K. Phys. Rev. Lett. 2006, 96, 046104-4.
Leng, Y.; Cummings, P. T. J. Chem. Phys. 2006, 124, 074711.
Pertsin, A.; Grunze, M. Langmuir 2008, 24, 135–141.
Pertsin, A.; Grunze, M. Langmuir 2008, 24, 4750–4755.
Lane, J. M. D.; Chandross, M.; Stevens, M.; Grest, G. Langmuir 2008, 24,
Leng, Y.; Cummings, P. T. J. Chem. Phys. 2006, 125, 104701.
DOI: 10.1021/la901314b
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Sendner et al.
Table 4. Shear Rate and Viscosities for the Bulk (γ_ 0,η0) and
Interfacial Region (γ_ s,ηs) and the Corresponding Slip Length,
Obtained from the Velocity Profiles of the Non-Equilibrium MD
Simulationsa
xOH
γ_ 0 [1/ns]
γ_ s [1/ns]
ηs/η0
slip length [nm]
13.4
3.6 ( 0.8
3.7 ( 0.9
-0.32 ( 0.01
12.8
5.9 ( 1.0
2.2 ( 0.4
-0.29 ( 0.01
a
The slip length b is defined with respect to the layer of oxygen atoms
of the surface C-O-H groups.
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1/4
1/8
Figure 14. (a) Water density profiles (top row) and velocity profiles (bottom row) for the hydrophilic surfaces with xOH = 1/8 (left)
and xOH = 1/4 (right). The topmost layer of carbon atoms defines
the origin z = 0. (b) shows the density profiles and velocity profiles
(þ) close to the surface. The fit functions for the velocity profile in
the peak region and in the bulk region are shown as solid and
broken lines. The gray shaded areas denote the extent of the peak
region used in the fitting.
velocity profile is separately fitted to a linear function in the
surface and bulk regions. The fit region for the velocity profile of
the interfacial layer starts at the position where the water density is
for the first time equal to the density at the first density minimum
and ends at the position of the first water density minimum,
shown as the gray shaded areas in Figure 14. The ratio of the two
viscosities is then determined by the ratio of the velocity gradients
in the interfacial, γ_ s, and bulk, γ_ 0, region.
In Table 4, the numerical values of the interfacial viscosities are
shown. For the more hydrophilic surface (xOH =1/4), the interfacial viscosity is larger by a factor of 4 compared to the bulk
viscosity. The less hydrophilic surface exhibits an interfacial
viscosity roughly twice the bulk value. For both surfaces, the
change in viscosity is moderate and does not support an ice-like
interfacial water structure.
As for the hydrophobic surfaces, eq 9 can be used to define a
slip length b. Therefore, the velocity profile is fitted in the bulk
region to a linear function and extrapolated. By definition, the
bulk region ranges from the first to the last minimum of the water
density profile, which is depicted as the white region in between
the gray shaded areas in Figure 14a. The slip length is defined with
respect to the center of the oxygen atoms of the OH surface
groups. This procedure leads to a negative slip length of roughly
b ≈ -0.3 nm for both OH surface concentrations, see Table 4.
From the velocity profiles, only the ratio between the interfacial
and the bulk viscosity can be calculated. For the explicit determination of the viscosities, the force acting on the diamond slabs is
10778 DOI: 10.1021/la901314b
needed. In the simulation, the two solid slabs are attached to
harmonic springs which are pulled with constant velocities (ν0.
From the average displacement of the slabs with respect to the
minimum of the spring potential, the average force F acting on the
slabs is determined. The bulk and surface viscosities are given by
η0=F/(2Aγ_ 0) and ηs=F/(2Aγ_ s) with surface area A (the factor 2
in the expressions for the shear viscosities arises because each
diamond slab is connected to two water slabs). From the shear
rates γ_ 0 and γ_ s, obtained from the velocity profiles, and the
average force F measured in the simulations, the viscosities given
in Table 5 are thus obtained. The bulk viscosity η0 in the inner
region of the water film is similar for the hydrophilic and
hydrophobic surfaces. The moderate confinement of 4 nm does
not lead to strong deviations from the bulk viscosities, that is, our
estimates for the bulk viscosity of the water film are in good
agreement with the literature value η=0.642 cP76 for the SPC/E
water model, obtained in non equilibrium simulations at 300 K
(note that the SPC/E water model considerably underestimates
the viscosity of real water, η=0.851 cP77).
Since the used shear rates γ_ ∼ 1010 s-1 are much larger than
those typically used in experiments, it is crucial to check if the
linear response regime is reached for the hydrophilic surface. We
account for this issue by performing two benchmark simulations
with double and half of the diamond pulling speed. Since neither
the measured bulk viscosities nor the slip length changed with
different pulling speed, see Table 5, we infer that the system is still
in the linear response regime.
Diffusion
The viscosity η of a liquid directly affects the diffusion constant
for a particle with radius a, which is given by the Stokes-Einstein
relation D=kBT/6πηa for the no-slip case or D=kBT/4πηa for the
perfect-slip case. In this section we seek to understand the relation
between the increased surface-viscosity at a hydrophilic surface,
and the diffusivity of water molecules themselves. Not much is
known for water dynamics at a single surface. Recent polarization-resolved pump-probe spectroscopy studies showed that
water around small hydrophobic methyl groups is effectively
immobilized,78 but the dynamics on planar surfaces is less clear.
Simulations studied the diffusion of water molecules at various
surfaces.79-82 In the present work, we shall be mainly interested in
the difference of the diffusivity perpendicular to hydrophobic and
hydrophilic surfaces. In simulations, diffusion constants are
typically determined from the mean square displacement or from
the velocity autocorrelation function. This leads to difficulties for
(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.
Langmuir 2009, 25(18), 10768–10781
Sendner et al.
Article
the determination of a local diffusion constant. Another route to
obtain diffusion constants is to consider the time-correlation of a
function f(z) which is unity if the particle is inside a certain layer
and zero otherwise,
(
1 if
0
f ðzÞ ¼
z0 < z < z0 þ Δz
else
ð22Þ
CðtÞ N
Z
Z
z0 þΔz
z0 þΔz
dz
z0
dz0 pðz, z0 , tÞ f ðzÞ f ðz0 Þ Fðz0 Þ
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Publication Date (Web): July 10, 2009 | doi: 10.1021/la901314b
ð23Þ
where the function p(z,z0 ,t) is the probability that a particle is at z
at time t, given that it was at z0 at t=0 and the liquid density is
given by F(z). The normalization constant N assures that C(0)=1.
p(z,z0 ,t) is the solution of the one-dimensional diffusion equation
for a particle in an external potential W(z),
(
Dt pðz, z , tÞ ¼ Dz
)
1
0
ðDz WðzÞÞ þ Dz pðz, z , tÞ
DðzÞ
kB T
pðz, z0 , 0Þ ¼ δðz -z0 Þ
ð24Þ
ð25Þ
For bulk water with zero external potential and position independent diffusion constant D, the solution is given by
0
pðz, z , tÞ ¼ ð4πDtÞ
-1=2
ðz - z0 Þ2
exp 4Dt
!
ð26Þ
with the boundary condition p(z,z0 ,t) f 0 for |z-z0 | f ¥. Since for
bulk water, the density F(z)Ris constant,
the normalization conR
stant in eq 23 is given by N= dz dz0 δ(z-z0 ) f(z) f(z0 ) F(z)=(Δz)
F0. The integrals in eq 23 can be performed and the autocorrelation function is given by
Z
CðtÞ ¼
Z
z0 þΔz
z0 þΔz
dz
z0
0
2 -1=2
dz ½4πDtðΔzÞ z0
ðz - z0 Þ2
exp
4Dt
!
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffi
2
4Dt
¼
ðe -ðΔzÞ =ð4DtÞ -1Þ þ erfðΔz= 4DtÞ
2
πðΔzÞ
ð27Þ
With the asymptotic behavior of the Error function, the asymptotics are C(t) = Δz/(4πDt)1/2 for long times t . (Δz)2/D and
C(t) = 1 - (2/Δz)(Dt/π)1/2 for short times t , (Δz)2/D.
We first calculate autocorrelation functions in bulk (in the
absence of a substrate) from simulations for an NAPzT ensemble
of 6000 water molecules in a 3 3 20.5 nm3 box. As for the other
simulations, the system is coupled to a temperature of 300 K. The
obtained autocorrelation functions are shown in Figure 15 together
with the exact expression eq 27 for four different values of the
correlation-slab thickness Δz. The only free parameter is the
diffusion constant, for which we fit the value D = 2.70 nm2/ns.
This value is in good agreement with previous publications for
(83) Mark, P.; Nilsson, L. J. Phys. Chem. 2001, 105, 9954–9960.
Langmuir 2009, 25(18), 10768–10781
ν0 [nm/ps]
η0 [ 10-3 N s /m2 ]
ηs [ 10-3 N s /m2 ]
1/4
1/8
0
0.02
0.02
0.02
0.73 ( 0.03
0.71 ( 0.01
0.66 ( 0.06
2.7 ( 0.7
1.5 ( 0.3
ν0 [nm/ps]
η0 [ 10-3 N s /m2 ]
b [nm]
0.01
0.71 ( 0.03
-0.31 ( 0.01
0.04
0.722 ( 0.005
-0.317 ( 0.004
a
The errors of the viscosities are calculated from the uncertainties of
the measured spring force and of the shear rate. Also the results of two
benchmark simulations at half and double pulling speed v0 are shown
(for which the interfacial viscosity is not calculated because of shorter
simulation runs and the considerable statistical fluctuation in the
velocity profile).
1/4
1/4
z0
0
xOH
xOH
The autocorrelation function (ACF) of f(z) is given by
-1
Table 5. Simulation Results for the Bulk and Surface Viscosities
Calculated According to η0 = F/(2Aγ_ 0) and ηs = F/(2Aγ_ s)a
the SPC/E water model, where the diffusion constant was determined to be D = 2.70-2.79 nm2/ns.83 Experiments on the selfdiffusion 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 way
from tens of femtoseconds to nanoseconds. This means, not
unexpectedly, that in bulk the water dynamics is purely diffusive.
The upper panel shows the data on a logarithmic time scale. The
lower panel shows the data on a linear time scale; here some
deviations are observed in the time window below one picosecond:
in fact, water seems to diffuse faster than predicted by its long-time
diffusion constant. The microscopic reason for this interesting
behavior 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 short
times, from which one could extract a time constant of the order of
500 fs (with a spurious dependence on the correlation-slab thickness Δz). Only the comparison with the exact expression eq 27
shows that this is a non-sensical procedure, as the behavior is purely
diffusive down to the smallest time scales with no typical time
constant present in the data.
For water molecules close to the solid/liquid interface, the
solution of the diffusion equation, eq 24, is much more
complicated. Generally, the liquid particles experience a
non-zero surface potential W(z) and in addition the mobility
of a single water molecule depends on its distance from the
surface because of hydrodynamic boundary effects. In fact,
the diffusion tensor even becomes anisotropic with different
components for motion laterally and normal to the surface:
simulations have shown that the water molecules are
more mobile along the lateral direction.79 More sophisticated
models are necessary to calculate the diffusion constant
in the interfacial layer under the presence of a surface
potential.80 As a crude approximation, we consider the
solution of the diffusion equation at an interface assuming
a position-independent diffusion constant and vanishing
external potential. Since there is no flux of particles across
the interface, the first derivative of the probability distribution must be zero at the location of the surface, z = 0. The
solution of eq 24 that satisfies this no-flux boundary condition is given by
2
!
!3
ðz - z0 Þ2
ðz þ z0 Þ2 5
-1=2 4
0
þexp ps ðz, z , tÞ ¼ ð4πDtÞ
exp 4Dt
4Dt
ð28Þ
(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.
DOI: 10.1021/la901314b
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Sendner et al.
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Figure 16. Autocorrelation functions for water inside a Δz =
0.1 nm wide region, centered at the first water peak, at the
hydrophobic surface (bottom data set), and at the hydrophilic
surfaces with OH fraction of xOH = 1/8 (middle data set) and
xOH = 1/4 (upper data set). On the hydrophobic surface the
dynamics is purely diffusive and no exponential decay is discerned.
On the hydrophilic surfaces, on the other hand, exponentially
decaying surface binding with time constants τ = 210 ps (for
xOH=1/8) and τ=340 ps (for xOH=1/4) is present. The lines are
the theoretical prediction from eq 30 with fit parameters in Table 6.
Figure 15. Autocorrelation functions for bulk water for different
correlation-slab thicknesses Δz = 0.1,0.2,0.4,1 nm (from bottom
to top) compared with the expression eq 27 in a log-log and in a
log-lin plot. The fitted diffusion constant is D = 2.70 nm2/ns.
Assuming a constant density F0 inside the layer, the surfaceautocorrelation function can be evaluated exactly as
Cs ðtÞ ¼ N -1
Z
Z
z0 þΔz
z0 þΔz
dz
z0
z0
dz0 ps ðz, z0 , tÞF0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
Dt
2
ð2e -ðΔzÞ =ð4DtÞ -2e -ð2z0 þΔzÞ =ð4DtÞ -2 þ e -z0 =ðDtÞ
¼
2
πðΔzÞ
pffiffiffiffiffiffiffiffi
2
þe -ðz0 þΔzÞ =ðDtÞ ÞþerfðΔz= 4DtÞ
pffiffiffiffiffiffiffiffi z0
pffiffiffiffiffiffi
2z0 þ Δz
erfðð2z0 þ ΔzÞ= 4DtÞþ erfðz0 = DtÞ
Δz
Δz
pffiffiffiffiffiffi
z0 þ Δz
þ
ð29Þ
erfððz0 þ ΔzÞ= DtÞ
Δz
The surface autocorrelation function decays with the inverse
square root of time as Cs(t) = Δz/(πDt)1/2 for long times t . (Δz)2/
Dt. In contrast to the bulk solution in eq 27, the prefactor is larger
by a factor of 2, which is caused by the reflecting boundary
condition. The autocorrelation functions, shown in Figure 16, are
obtained from the simulation of one solid surface in contact with
1830 water molecules in a 3.0 3.0 8.0 nm box in the NPzT
ensemble. The region of the water slab considered for the
estimation of the diffusion constant is depicted in Table 6 and
corresponds to a Δz = 0.1 nm wide layer centered around the
maximum of the density in the first water peak. For the hydrophobic surface, the lowest curve in Figure 16, the expression eq 29
gives a good description of the data over the complete time range,
provided the diffusion constant is fitted to a value D=0.85 nm2/ns,
which is reduced by a factor of about three compared to the bulk
10780 DOI: 10.1021/la901314b
case (note that D is the diffusion coefficient normal to the surface).
This decrease is not surprising, as the hydrodynamic self-mobility
of a particle at a surface is reduced compared to the bulk case. The
correlation-slab width is taken as Δz = 0.1 nm as used in the
simulations, the correlation-slab position is fitted as z0 =0.03 nm
which is in rough agreement with the actual distance between the
correlation slab and the point where the water density approaches
zero (note that the actual position of the reflecting boundary is not
well-defined within the simulation, which means that this is
actually a free parameter in the theoretical description). For the
hydrophilic surface data shown in Figure 16 (middle and upper
curves), the expression eq 29 cannot fit the data, in fact, the
functional form of the data suggests an exponential process that
has to do with transient binding or trapping of water at polar
surface groups. The simplest model for such a trapping process is
one where initially a fraction φ of water molecules is bound to the
surface, then exponentially released, and only after release subject
to diffusion. Neglecting rebinding of water molecules to surface
sites, the autocorrelation function that allows for trapping thus
reads
"
Cs ðtÞ
¼ ð1 - φÞCs ðtÞþφ e -t=τ þ
Z
0
t
dt0
0
Cs ðt - t0 Þe -t =τ
τ
#
ð30Þ
Keeping the same diffusion constant D as on the hydrophobic
surface for both hydrophilic surfaces, we can successfully fit the
data with two fitting parameters, namely, binding fraction φ and
binding time τ, see Table 6. That the diffusion constant is at least
very similar for all three hydrophobic and hydrophilic considered
surfaces is suggested by the fact that for long times beyond one
nanosecond the curves are seen to converge. The most interesting
property is the time scale τ over which a water molecule is bound
to the polar surface groups, which turns out to be τ=210 ps for the
xOH=1/8 surface and τ=340 ps for the xOH=1/4 surface. These
are long times on simulation time scales, which means that
equilibrating polar surfaces takes a long time. The fact that the
time scale on the xOH=1/4 surface is larger than on the xOH=1/8
surface points to the presence of cooperative effects, meaning that
Langmuir 2009, 25(18), 10768–10781
Sendner et al.
Article
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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. c Here, xOH is the surface fraction of OH groups, Δz is the correlation slab thickness, and z0 its distance from the
hydrodynamic boundary (see graph on the right which shows the water density at a hydrophobic substrate with standard εCO parameter and arbitrarily
shifted 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 characteristic
release time of those bound water molecules.
on the xOH =1/4 surface a water molecule presumably binds to
two OH surface groups simultaneously or into a network of other
surface-bound water molecules. The fraction of bound water
molecules, φ=0.33 and φ=0.51 on the two surfaces, is rather high
and close to saturation. The fact that the xOH=1/4 surface does
not bind twice as many water molecules as the xOH=1/8 surface
again points to some cooperative (i.e., saturation) effects. The fact
that our decay times are rather sensitive to the surface hydrophilicity is in contrast to the results in ref 79, where the residence
time is insensitive to the surface hydrophilicity. The longer
residence time at the hydrophilic surface is due to the fact that
the water molecules are strongly attracted to the polar surface
groups, whereas on the hydrophobic surface no binding is
observed at all but purely diffusive behavior is seen when
compared with the exact prediction from the diffusion equation.
Note that water density fluctuations close to hydrophobic spherical cavities display a somewhat different behavior that suggests
short-time exponential decay.82 In bulk water, see Figure 15, the
behavior is also purely diffusive down to time scales as short as
20 fs, which is shorter than the lifetime of a single hydrogen
bond τH ∼ 1 ps.86
Care has to be taken with the interpretation of the obtained
(normal) surface diffusion constant because of our crude assumptions. The neglect of a surface potential and of a position
dependent mobility and the assumption of a constant water
density in the surface region most likely lead to changes in the
fitting of the surface diffusion constant, but recent results on
Lennard-Jones fluids suggest that the changes are small.87
Conclusion
The structure and dynamics of liquid water close to interfaces is
examined. We use nonpolar hydrophobic, as well as polar
hydrophilic, substrates. In shear flow simulations we find slip
lengths of only a few nanometers at hydrophobic surfaces with
realistic contact angles. Via variation of the surface hydrophobicity, we obtain a heuristic scaling of the slip length with the
(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.
Langmuir 2009, 25(18), 10768–10781
contact angle of the surface, which is backed up by some simple
scaling arguments. We also find a strong dependence of the slip
length on the water depletion thickness at the hydrophobic
surfaces. Dissolved gas is found to strongly adsorb to the
hydrophobic surfaces but only moderately increases the slip
length. In contrast, at the hydrophilic surfaces, no major gas
accumulation at the interface is observed. From the velocity
profile obtained from shear flow simulations at the polar, hydrophilic substrates, we infer that the viscosity of water in the
interfacial region is increased by a factor of 2 to 4, compared to
the bulk viscosity, depending on the surface fraction of hydroxyl
groups. From the decay of the autocorrelation function, the
dynamics of water molecules at hydrophobic and hydrophilic
surfaces is obtained. On hydrophobic surfaces, the dynamics is
found to be purely diffusive, with no sign of surface binding or
trapping. On hydrophilic surfaces, on the other hand, exponentially decaying surface trapping with typical decay times of
hundreds of picoseconds is found. It is this transient water binding
that most likely causes the effectively increased surface viscosity
on hydrophilic surfaces in the shear flow geometry.
Even for contact angles as large as θ = 140, the slip length
obtained in our simulations is only of the order of b=18 nm, see
Figure 8b, whereas recent experiments consistently yield larger
slip lengths, of the order of 20 nm at a contact angle of 105.17,18
The reason for this disagreement is at present not clear. However,
our simulations and scaling arguments give evidence for a strong
dependence b ∼ δ4 of the slip length b on the depletion length δ.
Thus, a small deviation in depletion length between experiment
and theory would lead to a much amplified disagreement between
slip length in experiment and theory. Indeed, compared to
experiments for silanized surfaces, the simulations underestimate
the 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 is
acknowledged from the DFG via SPP 1164 (Nano- and Microfluidics) and via the Excellence Cluster Nano-Initiative Munich,
and from the Elitenetzwerk Bayern in the framework of CompInt.
L.B. acknowledges support from the von Humboldt foundation.
DOI: 10.1021/la901314b
10781