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Phil. Trans. R. Soc. A (2011) 369, 2537–2545
doi:10.1098/rsta.2011.0088
Tilting angle and water slippage over
hydrophobic coatings
BY M. CHINAPPI1, *, F. GALA2 , G. ZOLLO2
AND
C. M. CASCIOLA3
1 Department
of Biochemical Sciences A. Rossi Fanelli,
Sapienza University of Rome, Piazz.le Aldo Moro 5, 00185 Roma, Italy
2 Department of Fundamental and Applied Science for Engineering,
Section of Physics, Sapienza University of Rome, Via A. Scarpa 14,
00161 Roma, Italy
3 Department of Mechanical and Aerospace Engineering, Sapienza,
University of Rome Via Eudossiana 18, 00184 Roma, Italy
Hydrophobic coatings, such as octadecyltrichlorosilane or n-alkyl monolayers, enhance
the slippage of liquids on solid walls. For a given alkyl chain length, the main structural
parameter for homogeneous coatings is the tilt angle between coating molecules and the
surface normal. In this paper, ab initio calculations are used to calculate the equilibrium
configuration of coating molecules, showing that the tilt angle easily changes from 0◦
to 30◦ depending on the specific head group binding the solid substrate. These values
are used to set up classical molecular dynamics of water slippage over the coatings
using different water models (Transferable Intermolecular Potential 3 point (TIP3P),
Transferable Intermolecular Potential 4 point (TIP4P) and TIP4P/2005). The slippage
is found to be robust with respect to the coating tilting, while a slight dependence on
the water model is observed.
Keywords: slip length; hydrophobic coating; ab initio simulation; molecular dynamics
1. Introduction
The recent progress in micro and nano fluidic device fabrication demands a
better comprehension of fluid–solid interactions at the nanoscale where the usual
description in terms of Navier–Stokes equations with no-slip boundary conditions
at the solid wall is unsuitable. There are two possible reasons for this failure,
namely the breakdown of the continuum assumption and fluid slippage at the
wall. For simple liquids, the continuum assumption is rather robust [1,2], and
departures from continuum model predictions are observed only under strong
confinement, when the device dimension approaches the molecular size. For
instance, the so-called single file motion regime sets in for flows through nanopores
only when the pore becomes of the order of three to four molecule diameters
( 1 nm) [1]. Hence, at least for simple liquids and with the exception of subnanometre size, failure of the classical description should be ascribed to the
violation of the no-slip boundary condition at the solid wall.
*Author for correspondence ([email protected]).
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M. Chinappi et al.
Wall slippage is observed for both smooth and patterned surfaces, see [3] for
a review, and it is usually described in terms of the Navier boundary condition
vw = Ls dv/dz, where v(z) is the velocity profile with z being the wall-normal
coordinate and vw the slip velocity, i.e. the relative fluid–solid velocity at the
interface. The parameter Ls is indicated as the slip length. The slip length
encompasses the relevant rheological properties of the interface needed to model
the fluid as a continuum, e.g. [4] where liquid polymer melts flowing over polymer
brushes are addressed by molecular dynamics (MD).
In this paper, we focus our attention on smooth surfaces coated with
self-assembled monolayers (SAMs) of either octadecyltrichlorosilane (OTS)
or 18-alkyl chains, as typical examples of promising technologies for wall
functionalization. In principle, structural properties of the SAM, such as
the tilt angle, could affect the slippage. The study and characterization
of SAMs obtained by deposition of alkyl-silanes (such as OTS) or nalkyl chains on Si or SiO2 surfaces have been undertaken mainly from
the experimental point of view by using several techniques to evaluate
both the film quality (coverage, surface roughness and temperature stability)
and its structure (self-assembling properties, thickness, chain orientation and
hydrophobic termination groups). The literature provides experimental data
concerning structural properties of the deposited films obtained by, e.g.
ellipsometry, infrared spectroscopy, X-ray photoelectron spectroscopy (XPS),
reflectivity and absorption fine structure (near-edge X-ray absorption fine
structure; NEXAFS) [5–9]. It is found that the structural properties of SAM
films depend dramatically on the deposition method and chemical environment
used to deposit it.
Much less is known about the actual effects that the SAM structure may
have on the slippage properties of the functionalized wall. The purpose of the
present paper is to describe an investigation recently undertaken in this direction.
We first describe ab initio calculation simulations aimed at extracting the
correct structural parameters. These parameters are then used in a classical MD
simulation of a water flow over the SAM-coated surfaces to address the slippage
properties of the functionalized wall. A careful examination of the experimental
literature on SAM-functionalized Si provides indications concerning the tilt angle
qtilt formed by the carbon chain axis and the normal to the Si surface. More
specifically, from infrared spectroscopy, NEXAFS or equivalent techniques, the
tilt angle for the OTS SAM on (111) Si substrate has been measured in the range
[0◦ ≤ qtilt ≤ 10◦ ], while the carbon chain axis of n-alkyl SAM on (111) Si surface
forms a qtilt 25◦ tilt angle [9]. A clear picture is, however, rather difficult to
extract from the variety of experimental conditions and measurement techniques.
Alternatively, we decided to rely on ab initio simulations. So far, to the author’s
knowledge, no reliable atomistic model of OTS or n-alkyl SAM deposited on Si
or SiO2 substrates has appeared in the literature, which mostly describes only
in qualitative terms the surface chemical bonding and the OTS/n-alkyl chain
structure. Here, preliminary results on reliable atomistic modelling of OTS and
the n-alkyl-isolated molecule deposited on the hydrogenated (111) Si (H : Si)
substrate by means of total energy ab initio calculations are discussed. These
data are then used to set up classical MD simulations in order to estimate the
slip length Ls . The measurement of Ls is performed following the procedure
described in previous work [10]. Both cases of OTS and 18-alkyl are considered
Phil. Trans. R. Soc. A (2011)
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Water slippage on hydrophobic SAMs
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in order to assess the effect of the SAM orientation on slippage. To evaluate the
robustness of the results to the model details, different classical water models
are considered.
The paper is organized as follows. In §2, the ab initio calculations are
presented. Section 3 reports a brief description of the classical MD set up for slip
measurement, and the results for the different cases are considered. A comparison
with experimental and numerical studies in the literature is discussed in §4.
2. Tilting angle: ab initio simulations
In this section, we report some preliminary results on reliable atomistic modelling
concerning the configuration of two typical coating molecules. We deal with a
single molecule of the SAM, either OTS or 18-alkyl, deposited on a hydrogenated
(111) Si substrate by means of total energy ab initio calculations.
The study of the structural stability of SAMs of CH3 (CH2 )17 SiCl3 (OTS) on
Si (111) surface is performed by first-principles calculations based on density
functional theory (DFT; [11]). The silicon surface has been modelled as a threelayer slab of 96 atoms with 32 Si atoms per layer. Periodic boundary conditions
(PBCs) have been employed, and the computational cell is completed with a
vacuum region of approximately 43 Å, corresponding to 18 Si atomic layers. All
the dangling bonds of the Si slab have been passivated with H atoms. The
total zero-temperature energy of the system is E[n(r), Ri ], and the ground
state to be determined minimizes E with respect to both the electron density
n(r) and the ion positions Ri . The equilibrium configuration is found by a
two-step iterative procedure. The first step consists of minimizing the energy
with respect to the electron density n(r) for given positions of the ions. In the
second step, the ion positions are upgraded, accounting for the local energy
gradients vE[n(r), Ri ]/vRi , the so-called Hellmann–Feynman forces. In more
detail, concerning the first step, the total energy of the system is treated, in the
frame of the adiabatic approximation, as a functional of n(r) and is variational
with respect the electron density itself at the ground state
(2.1)
E0 ≤ E[n(r), Ri ] = T [n(r)] + Vee [n(r)] + vext (r, Ri )n(r) dr,
where vext (r, Ri ), T [n(r)] and Vee [n(r)] are the external potential owing to the
ions, the electron kinetic energy and the electron–electron interaction energy
containing the Coulomb repulsive energy and the exchange-correlation energy.
The above variational problem is usually recast into an independent electron
eigenvalue problem [11], and the corresponding so-called Kohn–Sham equations
can be solved self-consistently to find the ground-state electron density and
total energy.
The key factor affecting the accuracy of ab initio calculations is the exchangecorrelation functional that is unknown a priori. Here, we have employed the
generalized gradient approximation (GGA), using the Perdew–Burke–Ernzerhof
(PBE) formula for the exchange-correlation potential and energy functionals
[12]. The usual choice for the exchange-correlation potential for semiconductor
bulk systems or surfaces (such as the Si surface employed here) is the local
density approximation (LDA) that has been derived from a uniform electron gas
Phil. Trans. R. Soc. A (2011)
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2540
M. Chinappi et al.
(a)
(b)
(c)
total energy per atom (Ry)
–7824
–7826
33°
–7828
–7830
–7832
10
15
20
25
30
energy cut-off (Ry)
35
Figure 1. (a) Total energy of the Si slab as a function of the energy cut-off for different k-point MP
grids. (b) OTS and (c) 18-alkyl stable configurations on (111) H : Si surface. Circles with dashed
lines, 1 × 1 × 1; squares with dashed lines, 2 × 2 × 1; diamonds with solid lines, 3 × 3 × 1.
adapted locally. This approximation, despite being quite rough, is satisfactory
for semiconductors, but is not satisfactory for molecules and for systems where
long-range interactions are involved. Moreover, it is a general opinion that the
reaction energy calculated using LDA is quite distant from the experimental
values, especially for molecules. Thus, several GGA schemes have been developed
and checked, even though they have not always improved the LDA results. Among
all these schemes, the PBE density functional is a good choice to obtain fair
accuracy for systems ranging from molecules to solids; hence, it is fit for the
present case study. Concerning the external potential, norm-conserving pseudopotentials have been constructed following the Troullier–Martins scheme [13] that
allows the inclusion of oxygen 3-d electrons in a soft way, thus not affecting
the computational cost too much. The Kohn–Sham equations are solved using a
plane-wave basis set with a cut-off of 30 Ry for the wave functions and a 120 Ry
cut-off for the electronic charge density, while a (3 × 3 × 1) Mokhorst–Pack (MP)
k-point grid [14] has been used for Brillouin zone sampling and integration.
The above computational parameters have been chosen after several accurate
convergence tests on the total energy of the Si surface for different k-point MP
grids as reported in figure 1; subsequently, the energy cut-off has been resized to
include the presence of an alkyl-silane molecule. Finally, the Broyden–Fletcher–
Goldfarb–Shanno (BFGS; [15]) algorithm has been used to optimize the atomostic
configurations. All the calculations have been performed by using the QUANTUM
ESPRESSO package [16]. The convergence threshold for the total force has been
set to 0.001 (atomic units) and a PBC dipole correction is used to handle the
dipole electric field [17].
The above specified ab initio theoretical model shows that the OTS stable
configuration is nearly normal to the Si surface (figure 1b), whereas the carbon
chain of an 18-alkyl molecule adhering on a hydrogenated Si (111) surface is about
30◦ off the surface normal (figure 1c).
Phil. Trans. R. Soc. A (2011)
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Water slippage on hydrophobic SAMs
2541
3. Water slippage: classical molecular dynamics
Concerning slippage, the main relevant structural property of the coating is the
chemical nature of the terminal group exposed to the liquid. However, for a given
terminal group, certain geometrical features may, in principle, affect the liquid–
solid interface. Beyond chain length, for uniform coatings, the tilt angle q is
expected to be the most important. The results of §2 suggest that q strongly
depends on the specific system, and it is extremely difficult to determine from
experiments. The zero-temperature ab initio simulations discussed so far may not
correspond to the conditions encountered at finite temperature. Nevertheless,
their indications can be used as starting point to investigate the effect of q
on slippage.
Values of 0◦ for OTS chains and 30◦ for 18-alkyl chains are used to set up
classical MD simulations of a Couette flow in the nano-channel between two
planar walls (figure 2a). The lower wall is coated, while the upper one is kept
in motion by a constant force. The slip length is obtained from the steadystate velocity profile. The simulations are performed at 300 K and 1 atm. For
the level of detail needed here, a precise description of the head group (the
one towards the solid wall), of the specific binding mechanism and of the solid
crystal is not necessary. The relevant information is the hexagonal structure
of the SAM and its surface density. Hence, the sample is modelled as a (111)
Lennard-Jones (LJ) solid surface parallel to the Oxy plane. Previous studies
showed that, after equilibration, the molecules of the coating spontaneously
achieve a tilt angle qtilt 30◦ [10]. This fact is exploited here to reproduce
the inclination of the ab initio simulation for the 18-alkyl chain. The case
with 0◦ inclination is obtained by fixing the chain carbon atoms. Sensitivity
tests to the water model were also performed, using Transferable Intermolecular
Potential 3 Point (TIP3P), Transferable Intermolecular Potential 4 Point (TIP4P)
[18] and TIP4P/2005 [19] models for water, see Chinappi & Casciola [10] for
further details.
The mean velocity profile is shown in figure 2b for the TIP4P/2005 water
model with 30◦ tilt angle of the alkyl chains. For each water model, the slope
is consistent with the respective viscosity, with TIP4P/2005 (m = 0.855 m Pa s−1 )
getting closest to the experimental value of m = 0.896 m Pa s−1 [20]. Defining the
wall surface as the average position of the methyl group carbon atom of the chains,
Ls is obtained from the position where the velocity profile extrapolates to zero.
The results are summarized in table 1. No difference is observed by changing
forcing direction and intensity (data not shown). Figure 2c shows the number
density profile within the fluid (rf , z > 0) and the solid (rs , z < 0), formed either by
the LJ crystal (uncoated case, solid line) or by the SAM (coated surface, qtilt = 30,
dashed line). The hydrophobic behaviour of the LJ crystal appears from the first
strong peak in the water density (z 3.5 Å), which is instead much weaker for
the hydrophilic SAM coating. Denoting the bulk values by a superscript ‘b’, this
behaviour is quantified by the depletion layer thickness (last line of table 1),
d=
Phil. Trans. R. Soc. A (2011)
∞ −∞
rf
rs
1− b − b
rs
rf
dz.
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2542
M. Chinappi et al.
(a)
(b)
constrained
Vx(Å ps–1)
forced
0.3
0.2
0.05
0.1
0
−20 −10
0
−25
0
25
−10
−5
z (Å)
50
z (Å)
0
75
100
5
10
(c)
ρ (Å–3)
0.09
0.06
0.03
0
thermostated
fixed
−20
−15
0
Figure 2. (a) Snapshot of a Couette flow simulation showing the tilted SAM at the lower wall.
(b) Velocity profile for the Couette flow simulation for the TIP4P/2005 water model and tilted
SAM. (c) Number density profiles for the uncoated surface (solid line) and the tilted SAM coating
(dashed line). The nominal wall is at z = 0, the liquid is in the z > 0 region.
Table 1. Summary of the simulations. The confidence interval on the Ls is of the order of 2 Å, while
on the d is 0.3 Å. Cases A and B refer to the uncoated surface.
qtilt
H2 O
Ls (Å)
d (Å)
A
B
C
D
E
F
G
—
TIP3P
−1.8
−0.25
—
TIP4P
−1.2
−0.20
30◦
TIP3P
5.8
1.4
30◦
TIP4P
8.6
1.4
30◦
TIP4P/2005
13.1
1.6
0◦
TIP3P
5.9
1.5
0◦
TIP4P/2005
12.1
1.5
4. Discussion
The tilt angle of coating molecule chains presents a wide variability in the
dependence of the specific wall-binding mechanism, §2. Actually, molecules, such
as OTS and 18-alkyl, differing only by the binding group, attain largely different
tilt angles when deposited on a hydrogenated Si (111) surface. Their groundstate structures have been obtained by both fully relaxed BFGS optimization
and constrained static calculations where the tetrahedral coordination of the
sylane group (OTS) and of the entire carbon chain (18-alkyl) have been kept
fixed. Concerning OTS, three –OH groups were supposed to replace the chlorine
atoms originally present in the trichlorosylane group. The substitution reaction
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Water slippage on hydrophobic SAMs
2543
occurs in water solution where the OTS molecules can easily adhere to the Si
surface through an oxygen bridge by eliminating a hydrogen molecule. The 18alkyl ground-state structure has other equivalent orientations with respect to the
(111) Si surface. The relevant transition energy barriers should then be measured
to evaluate the temperature stability of the tilt angle. In addition, in the OTS
case, the –OH group orientation with respect to the chain axis and the H:Si
surface may affect the tilt angle. A deeper analysis of such structural problems
is under way, and will be reported in the near future.
Despite the significant differences between the single molecule, zerotemperature simulations and the conditions found in actual coatings, i.e. finite
temperature and interaction with neighbouring chains, these results suggest that
the tilt angle may change from case to case, potentially affecting the slippage.
In fact, our all-atom classical MD results show no relevant difference in slip
length between cases with normal and tilted chain coatings (compare case C
and F (TIP3P) and E and G (TIP4P/2005) in table 1). This behaviour is
confirmed by addressing the depletion layer thickness, which globally quantifies
the hydrophobicity of the surface, which, within statistical accuracy, is the same
for the two tilt angles. A statistically significant, although slight, variation
is instead observed in the classical simulations when changing the model for
the water molecule (simulations C, D and E). In particular, the recently
developed TIP4P/2005 model gives a slip length Ls 13 Å, significantly larger
than the TIP3P value Ls 6 Å and a slightly larger depletion layer thickness.
As a general conclusion drawn from the present and from previous [10,21]
MD simulations, the intrinsic slip for liquid water over a smooth surface
coated with methyl-terminating SAMs is in the range 0.5–2 nm. This result
is robust under changes in both the SAM structure (tilt angle) and the
water model. It is one order of magnitude smaller than the most credited
experimental value [22]. In our opinion, this discrepancy could hardly be ascribed
to inaccuracies of the classical MD force field or to details of the specific
simulation procedure.
This turns our attention to other possible explanations, like the presence of
gas nanobubbles trapped in nanoscale defects. Several studies on the structure of
OTS coatings suggest that perfectly smooth coatings such as those addressed
in ideal MD simulations do not exist in practice. Cottin-Bizonne et al. [22]
and Joseph & Tabeling [23] provide estimates of a few nanometres for the
height of typical asperities in their coatings. Additionally, spectroscopic variable
angle ellipsometry, neutron reflection and atomic force microscopy carried out
by Styrkas et al. [24] suggest that OTS films often consist of nearby multi-layer
domains of different thickness, ranging from one to three and even four times the
monolayer thickness.
In conclusion, the small intrinsic slip on hydrophobic homogeneous monolayer
coatings provided by classical MD simulations does not emerge as an artefact
of poor/improper modelling. The discrepancy with experimental values is most
probably owing to imperfections in the deposited layer, a conjecture still to
be confirmed by addressing realistic nanoscale defects and possible trapping
of nanobubbles.
Help with some MD simulations by D. Gentili and L. Mammucari is acknowledged. Computing
resources were made available by CASPUR under HPC Grant 2010.
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