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3640 Phys. Chem. Chem. Phys., 2012, 14, 3640–3650 This journal is c the Owner Societies 2012
Cite this: Phys. Chem. Chem. Phys., 2012, 14, 3640–3650
Determination of the distance-dependent viscosity of mixtures in parallel
slabs using non-equilibrium molecular dynamics
Stanislav Parezaand Milan Predota*
b
Received 30th June 2011, Accepted 9th January 2012
DOI: 10.1039/c2cp22136e
We generalize a technique for determination of the shear viscosity of mixtures in planar slabs
using non-equilibrium computer simulations by applying an external force parallel to the surface
generating Poiseuille flow. The distance-dependent viscosity of the mixture, given as a function of
the distance from the surface, is determined by analysis of the resulting velocity profiles of all
species. We present results for a highly non-ideal water + methanol mixture in the whole
concentration range between rutile (TiO2) walls. The bulk results are compared to the existing
equilibrium molecular dynamics and experimental data while the inhomogeneous viscosity profiles
at the interface are interpreted using the structural data and information on hydrogen bonding.
1. Introduction
The growing interest in nanostructured materials requires
detailed knowledge of the properties of the fluids in nano-
confinement for understanding processes on this scale, leading
to better design of nanodevices.1 While there is significant
advance in the capabilities of experimental techniques in
probing the interfacial properties of fluids adsorbed on solid
materials, computer simulations offer an alternative approach
to provide pieces of information on the structure and dynamics
of such interfaces. The structural information of the inhomo-
geneous region formed at the solid–liquid interface, including
both relaxation of the solid and structure of interfacial liquid,
is more readily available e.g. from X-ray diffraction,2,3
nonlinear optics,4 and neutron scattering.5–7 The pieces of
information on the dynamics of the interfacial molecules
typically include the residence times of ions or molecules in
adsorption layers,8–10 translational diffusivity (either averaged
over the whole volume of the nanopore11 or calculated
bin-wise to yield the distance-dependent diffusivity,12–14
streaming velocity15) or more rarely rotation diffusivity and/or
orientation relaxation times.12
Motivated by the applications in microfluidic devices, mole-
cular sieves and flow through porous and nanostructured
materials in general, we explore in detail a method for
determination of distance-dependent shear viscosity of mixtures
in parallel slabs. While our simulations employ, for simplicity
of both the derivation and simulations, planar 2D-periodic
systems, the key message is the information on the viscosity
profile of a mixture as a function of the distance from the
surface, as the properties of the planar interface represent a
limiting case of surface of (infinitely) a large sphere or cylinder
or generally any surface with small enough curvature.
Methods for determination of shear viscosity from simula-
tions employ equilibrium molecular dynamics16–19 (EMD) as
well as non-equilibrium molecular dynamics (NEMD). The
latter use various approaches, such as e.g. 3D-periodic simula-
tions using oscillatory elongational flow,20–24 planar shear flow
described by SLLOD equations,25–27,29 momentum impulse
relaxation28 or Poiseuille flow between immobile surfaces.14,30,31
Obviously, only the latter example features interactions with
surfaces and thus leads to inhomogeneous profiles of structural
and dynamic properties as a function of the distance from the
surface. The potential models used in homogeneous simulations
range from simple atomic potentials, such as e.g. short ranged
WCA potential,21 argon studied using the Barker–Fisher–Watts
and three-body Axilrod–Teller potentials,29 to molecular systems
such as e.g. n-alkanes17 or ionic liquids.24
The determination of distance-dependent viscosity has been
implemented mostly for atomic fluids based on Lennard-Jones
potential31 or its modifications, e.g. short ranged WCA
potential.30,32 Dynamics of two-site and four-site chain
WCA molecules undergoing planar Poiseuille flow was also
studied,33 including velocity and shear profiles, but viscosity
was not determined. Travis et al.34 determined shear viscosity
by the Poiseuille flow of either atoms or rigid diatomic
molecules between two atomistic walls.
Studies on distance-dependent viscosity of water in contact
with surfaces are also scarce.14,31,35 As a pioneering work
in this direction we identify the work of Freund,35 who
described the dynamic properties of SPC/E water containing
Cl� ions adjacent to a smooth, positively charged wall of
generic Lennard-Jones atoms. We have later applied a similar
a Institute of Chemical Process Fundamentals, Academy of Sciences ofthe Czech Republic, 165 02 Prague, Czech Republic
b Faculty of Science, University of South Bohemia, Branisovska 31,Ceske Budejovice, 370 05, Czech Republic.E-mail: [email protected]
PCCP Dynamic Article Links
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This journal is c the Owner Societies 2012 Phys. Chem. Chem. Phys., 2012, 14, 3640–3650 3641
approach to study viscosity of pure water in contact with
atomistic rutile (TiO2) surfaces.14 Here we extend our former
work14 to a detailed study of the viscosity profiles of highly
non-ideal aqueous methanol mixtures in contact with the rutile
surfaces. Doing so we also elaborate the determination
of distance-dependent viscosity of mixtures in much more
detail than given by Freund.35 Particularly, we generalize the
formulas for viscosity calculation to an arbitrary number of
mixture species and arbitrary combination of external forces
acting on molecules.
Regarding the simulation results of the components of the
water + methanol system studied here, the viscosity of SPC/E
water was found to be about 10–15% lower than experimental
viscosity of real water36 earlier,25,35 in part also in ref. 16.
Nonequilibrium simulations using Lees–Edwards boundary
conditions of viscosity of water + methanol mixtures (and
also of mixtures with acetone) were carried out by Wheeler
and Rowley25 with SPC/E water, methanol model due to van
Leeuwen and Smit37 and applying a hybrid mixing rule for
cross-interactions. Wensink et al.22 used a periodic perturbation
method employing sinusoidal external forces for NEMD
simulations of diffusion and viscosity of OPLS mixtures of
methanol, ethanol or 1-propanol with TIP4P water using
OLPS potentials for alcohols. However, the agreement of
the viscosities of water + alcohol mixtures with experimental
data was only qualitative.22 Viscosity of methanol + ethanol
mixtures, employing the same model of methanol38 as in
this study, was studied by EMD.18 The recent results of
Guevara-Carrion et al.19 explore in detail the diffusivity and
viscosity, as well as other properties, of aqueous methanol
mixtures using SPC, SPC/E, TIP4P and TIP4P/2005 models
for water and represent a direct EMD benchmark for our
NEMD simulations. Moreover, the TIP4P/2005 water model
and the methanol model used in this study19 offer a qualita-
tively very good match to experimental data without further
fitting of binary parameters.
2. Simulations
Models
Rigid nonpolarizable models based on Lennard-Jones (LJ)
and point charge Coulombic interactions were used for both
methanol and water. For methanol, a united-atom model by
Schnabel et al.38 was adopted for its very good agreement with
experimental data. This model has two LJ sites, one for the
methyl group and one for the oxygen atom. In addition, it
contains three point charges, two are located at positions of LJ
centers and the third is at hydroxyl hydrogen. The interaction
parameters of this molecule are summarized in Table 1
(adopted from Schnabel et al.38). Geometry of methanol is
characterized by bond lengths |CH3–O| = 1.4246 A, |O–H| =
0.9451 A and the angle +CH3–O–H = 108.531. For water,
two very common models in molecular simulations, the SPC/E39
and the TIP4P/2005,40 were employed. The SPC/E model was
chosen to continue our series of papers on the rutile–aqueous
solution interface properties.13,14,41,42 The TIP4P/2005 was
included for its superior dynamic properties in very good
agreement with experiment. Finally, the viscosity of water +
methanol mixtures from EMD simulations, employing the
very same three models (Schnabel model of methanol, SPC/E
and TIP4/2005 for water), has been recently published,19 and
offers thus possibility to benchmark our NEMD results both
in terms of accuracy and efficiency.
Surfaces
In our simulations the liquid was confined between two planar
surfaces (slab geometry). Here we report results for slabs of
sufficiently large separation between the two surfaces, so that a
bulk phase develops in the center of a slab. In all the simula-
tions the distance between walls was higher than 70 A resulting
in the width of the bulk liquid phase ca. 40 A and about 15 A
wide inhomogeneous interfacial regions next to each of the
two surfaces. However, the same approach can be used in
narrow slabs i.e. confinement where all molecules are directly
affected by the presence of the surfaces. The surfaces were
simulated using a rigid atomistic model of a nonhydroxylated
rutile (110) surface.13 Positions of wall atoms were obtained
from our previous work on this surface, as described by
Predota et al.13 Each surface consists of four TiO2 layers.
The two deepest layers maintain a strictly periodic bulk crystal
structure while the two layers closer to the interface are
ab initio relaxed.13,43 The surface is terminated with rows of
bridging oxygens protruding out of the interface layer towards
the aqueous phase.
The interaction parameters between the rutile surface and
liquid are based on the ab initio derived potentials13,43 with
SPC/E water. These potentials describe the Ti–O(water) in
terms of Buckingham potential while the O(surface)–O(water)
interaction is given by LJ potential. To be able to model
interactions of surface Ti and O atoms with TIP4P/2005 water
and methanol LJ sites, we fitted the original Ti–O(water)
Buckingham potential by LJ potential and using Lorentz–
Berthelot combining rules derived the corresponding LJ para-
meters of Ti. The resulting parameters of surface atoms are
given in Table 1.
All the used models, including surfaces, then adopt pair-
wise interaction giving the potential energy between two
molecules as
uijðrijabÞ ¼Xna¼1
Xmb¼1
4eabsabrijab
� �12
� sabrijab
� �6" #
þ qaqb
4pe0rijab
where a denotes an interaction site on molecule i, b a site on
molecule j, and n and m are the numbers of sites on molecules i
and j, respectively. qa and qb are the point charges of sites a and b.
Table 1 Point charges and Lennard-Jones parameters of molecularmodels for methanol and surface atoms
Site q [e] s [A] e [kBK]
MethanolCH3 0.24746 3.7543 120.592O �0.67874 3.0300 87.879H 0.43128Rutile (surface)Ti 2.196 3.423 3.525O �1.098 3.166 78.200
3642 Phys. Chem. Chem. Phys., 2012, 14, 3640–3650 This journal is c the Owner Societies 2012
The LJ parameters sab and eab are obtained by Lorentz–
Berthelot combining rules, in agreement with ref. 19,
sab ¼saa þ sbb
2and eab ¼
ffiffiffiffiffiffiffiffiffiffiffieaaebbp
:
Methodology
We applied the theory for determination of shear viscosity14,35,44
and generalize the existing one component formula to fluid
mixtures. Let us adopt a coordinate system in which the xy
plane is parallel to surfaces and the z-axis measures the
distance from the center of the slab towards surfaces (see
Fig. 1). To calculate viscosity from NEMD, velocity gradient
(usually referred to as a shear rate or simply shear) is needed.
For this purpose, an external force was applied along the
x-axis, in the direction parallel to the surfaces, generating
Poiseuille flow in the slab. The resulting shear rate g(z) is
related to the viscosity Z(z) through the equation:
ZðzÞ ¼ �PxzðzÞgðzÞ ð1Þ
where Pxz(z) is the off-diagonal component of the external
pressure tensor (usually called shear stress) implied by the
external force—to distinguish it from the internal contributions
to pressure tensor from intermolecular forces and velocities.
For one-component fluid, calculation of the shear stress and
shear rate using density and streaming velocity profiles is well
established.14,30,31,44 Since the shear stress has the meaning
of the momentum flux density, we have derived the shear stress
for mixtures as a sum of all components’ contributions
(cf. analogous eqn (7) of Thompson31 for cylindrical geometry)
PxzðzÞ ¼Xa
Fax
Zz0
raðz0Þdz0 ð2Þ
where ra(z) is a local number density of a component a and Fax
is the external force acting on this component. Note that for
mixtures, independent forces acting on different components
can be specified. To derive the shear rate, one has to define a
streaming velocity for a mixture. This is possible in terms
of momentum density px and mass density rmass, i.e. vx(z) =
px(z)/rmass(z), since both densities are sums of one-component
contributions. This yields the resulting shear rate:
gðzÞ ¼ @
@z
PamaraðzÞvaxðzÞPamaraðzÞ
24
35 ¼ @vCOM
x ðzÞ@z
ð3Þ
Hence, the shear viscosity of a mixture can be calculated in
the same fashion as the one of a pure fluid provided the shear
stress is a sum of all components’ shear stresses and the
streaming velocity of the given layer of the fluid is character-
ized by the center of mass velocity of the layer. While the latter
result might seem intuitive, the works we cite as leading in the
field of inhomogeneous viscosity determination31,35 do not
make clear, how the streaming velocity was calculated. Parti-
cularly, it is not clear (i) if the COM velocity or only that of
solvent was considered35 and (ii) if differences between streaming
velocities of individual components were taken into account.
While both points can be subtle, namely (i) for low salt
concentration the solvent contribution by far dominates that
of ions and (ii) in many situations the streaming velocities of
individual components are similar and therefore close to the
streaming velocity of COM of the layer, there are examples in
which it is essential to calculate viscosity using generally valid
eqn (3) for the shear rate instead of a solvent velocity
derivative or other one component formula. The particular
example of considerably different streaming velocities of
individual components are electrokinetic or electroosmotic
computer experiments,31,35,48 where the streaming velocities
may have opposite signs as the external force is proportional
to charge of species. However, even applying constant acceleration
on all components, such as e.g. in the gravity driven flow, does
not generally guarantee equal streaming velocities of all
components, owing to unequal mobilities affected by different
strengths of interactions and shapes of molecules. Moreover,
in inhomogeneous environments, different interactions with
Fig. 1 Snapshot of the water + methanol mixture in a slab formed by rutile (TiO2) surfaces.
This journal is c the Owner Societies 2012 Phys. Chem. Chem. Phys., 2012, 14, 3640–3650 3643
the immobile surface (e.g. in aqueous solution of ions) can lead to
different streaming velocities of ions strongly interacting with the
surface and less interacting water with the surface or vice versa.
Simulation details
Starting from a lattice in which methanol and water molecules
were randomly distributed, each system was equilibrated for
about 1 ns with 1 fs timestep. Equilibration was followed with
productive run lasting 10 ns, i.e. 107 productive steps.
Desired pressure was set up applying the Berendsen barostat
in the z direction. That is, the volume of the system was
changed only through altering the distance between the two
solid surfaces and rescaling the centers of mass of fluid
molecules in the z-direction. The pressure was calculated
directly from the z-component of the force of the liquid acting
on the surfaces (P = Fz/S, where S = LxLy is the area of the
periodically replicated box; the pressures acting on both
surfaces have been averaged).
Maintaining the desired temperature in systems under flow
might be complicated due to difficulties in computing the
thermal part of kinetic energy. Consider the translational part
of kinetic energy contributing to thermal motion
K ¼ 1
2
Xi
mi½ðvxi � �vxðziÞÞ2 þ v2yi þ v2zi�
where %vx(zi) is the average streaming velocity in the layer given
by the z coordinate of the molecule i. The problem is that the
average velocity profile is not known until a simulation has
run for sufficiently long time, i.e., to use the above formula,
one would have to iterate the velocity profile, using an improved
value for more accurate thermostatting of the system. We
overcame this problem by using a thermostat (Nose–Hoover)
only in the y direction, perpendicular to the external force,
where the mean velocity %vy is zero. No thermostatting of the
other components of translational velocity or any of the
rotational components of velocity was applied. The thermo-
statting of these velocities was provided only via redistribution
of the energy among all degrees of freedom. Further details on
thermostatting can be found in ref. 14.
3. Results and discussion
The shear viscosity of a water + methanol mixture was
predicted under ambient conditions, 0.1 MPa and 298.15 K.
As mentioned above, two sets of simulations were carried out
differing in the water model applied: in the first set SPC/E, and
in the second set TIP4P/2005. Both sets consist of simulations
of water + methanol mixtures with the total mole fractions of
methanol in the system 0, 0.1, 0.2, 0.3, 0.4, 0.6, 0.8 and 1.
These values were selected to cover the whole concentration
range in a reasonable number of points and with enhanced
sampling close to the viscosity concentration maximum of the
mixture, which is at about 30% methanol.19,45–47 Different
interactions of each component of the mixture with surface
Fig. 2 Numerical density profiles of water (solid lines) and methanol (dashed lines) from water + methanol simulations applying SPC/E (left)
and TIP4P/2005 (right) water models. The coordinate z0 gives the distance from a rutile surface.
3644 Phys. Chem. Chem. Phys., 2012, 14, 3640–3650 This journal is c the Owner Societies 2012
atoms lead to different surface adsorption, as depicted in
Fig. 2. Consequently, the equilibrium bulk concentrations
xMeOH in the center of the slab differ from the total mole
fraction by few percents. The density profiles in the slab were
found to be independent of external forces applied on mole-
cules of the liquid.
While the final section is devoted to distance-dependent
viscosity profiles, we first discuss the bulk values of viscosity,
which can be directly compared with the corresponding data at
the same xMeOH obtained from experiment as well as EMD
simulations. The viscosity in the bulk is determined on the
basis of eqn (1)–(3) using densities ra and streaming velocity
vCOMx (z) only in the bulk region, i.e. in the z-range where the
densities are constant. In this range the streaming velocity was
fitted with a parabola a + bz2, in agreement with a solution of
Navier–Stokes equation for liquids with constant density and
viscosity.
Verification of the independence of the NEMD method on
applied external forces
First we verified that the NEMD method is correct and robust
as follows. The relations (1)–(3) for computing viscosity of
mixtures must be valid for any combination of external forces
(within the linear regime, see below) acting on molecules of
individual species. To verify this, we carried out three inde-
pendent simulations (denoted by I, II, and III in Table 2) at a
bulk concentration xMeOH = 0.31 (total concentration 0.3) in
which an external force Fax was applied to one component, to
the other, or to both of them, respectively. Furthermore, two
extra simulations (IV and V) of pure water were run to test
independence of viscosity on the magnitude of the
external force.
The independence of viscosity on external forces holds only
in the linear regime for sufficiently small forces, when the
liquid behaves as Newtonian, i.e. streaming velocity is propor-
tional to the applied force and viscosity does not depend on
magnitude of shear. Therefore, the values Fw and FMeOH were
selected so that (i) the streaming velocity was large enough to
give satisfactory signal to noise ratio and, at the same time,
(ii) the maximum shear in the system was smaller than a threshold
beyond which the liquid would deviate from Newtonian.
According to eqn (1) and (2), if one ignores the viscosity and
density variations, the largest shear occurs close to the soli-
d–liquid interface gmax ¼ �PxzðzÞZðzÞ
���z¼Lz=2
. This leads, as a rule of
thumb, to a recommended maximum external force
Fmax ¼ 2gmaxZrLz
. Applying values gmax E 1010 s�1, Z E 10�3 Pa
s, r E 1028 m�3 and Lz E 10�9 m results in Fmax E 10�12 N.
Substituting this force in the simple homogeneous formula for
streaming velocity of Poiseuille flow vxðzÞ ¼ Fr2Z ½ð
Lz2Þ2 � z2�
leads to a rough estimate of the maximum streaming velocity
in the center of the slab vmax = gmaxLz/4, i.e. vmax E 101 m s�1
in our case, which is significantly less than root mean square
velocity (371 m s�1 for water and 278 m s�1 for methanol).
The maximum shear rate achieved in our simulations was
indeed of the order of 1010 s�1, which corresponds to the
dimensionless shear rate g� ¼ gsffiffiffiffiffiffiffiffim=e
pof about 0.02, which is
a small value compared to magnitudes of g* of about 1, where
a strong effect of the shear rate on not only viscosity, but also
configuration energy and pressure was observed.27,29 For
comparison, the smallest force applied, 1.38 � 10�13 N,
corresponds to the Coulomb force between two electrons
separated by approximately 400 A or to the Lennard-Jones
attraction between two oxygen atoms in two SPC/E water
molecules at distance 2.6 sSPC/E.Resulting viscosities from simulations I–V are listed in
Table 2. Obviously, viscosities from simulations which were
carried out under the same thermodynamical conditions (but
different external forces) agree within their uncertainties. To
demonstrate that selected pairs of forces Fw and FMeOH
produce different dynamics of the liquid, we show streaming
Table 2 Simulations verifying the independence of results on externalforces. TIP4P/2005 model was used for water
xMeOH
[mol mol�1]Fw
[10�13 N]FMeOH
[10�13 N]Z[10�4 Pa s]
Forces acting on different componentsSimulation I 0.31 1.38 2.07 14.5 � 0.3Simulation II 0.31 5.52 0.00 14.3 � 0.4Simulation III 0.31 0.00 9.32 14.3 � 0.3Scaling of forcesSimulation IV 0.00 1.38 0.00 9.4 � 0.3Simulation V 0.00 4.14 0.00 9.5 � 0.1
Fig. 3 Streaming velocity profiles for simulations verifying independence of the present method on a combination of external forces (left) as well
as on the magnitude of the forces (right). For details see Table 2.
This journal is c the Owner Societies 2012 Phys. Chem. Chem. Phys., 2012, 14, 3640–3650 3645
velocity profiles in Fig. 3. Even though the streaming velocity
of simulation V is 3� larger relative to that of simulation IV,
Table 2 confirms that the resulting viscosity is independent of
the magnitude of the external forces. The sharp peaks at the
ends of the velocity profiles are unphysical and unimportant;
they are artifacts of poor statistics at gaps between molecular
layers of liquid close to the surface, where density vanishes
(see Fig. 2). A few rare events, when a molecule appears in a
bin which is normally abandoned, can therefore yield a peak
with a height of the order of thermal velocity.
The uncertainties of viscosities in Table 2 were estimated as
standard deviations of viscosities computed from shorter
sections. Dividing each simulation into N shorter sections of
2 ns length, the standard deviation of the viscosity of the whole
simulation was estimated by
DZ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
NðN � 1ÞXNi¼1ðZi � �ZÞ2
vuut :
The values of viscosities determined from 2 ns sections Zi aresummarized in Table 3. Simulations II–V comprise of N = 5
sections giving the total simulation length 10 ns, which is the
same as the length of simulations used for obtaining final results
in the next section. For simulation I we extended the number of
2 ns-sections up to N = 12 to verify that the values determined
from 5 sections are representative and the deviations deter-
mined from longer runs agree with those from the 10 ns run.
The result of simulation from the first 5 sections is 14.0 � 0.4,
the result from all 12 sections is 14.5 � 0.3, i.e. they agree.
The length of a section, 2 ns, is assumed to be large enough
to exclude self-correlations of the calculated viscosities. To
support this statement we estimated the correlation time of
vCOMx (z), which is the quantity with the longest correlation
length among those entering the viscosity calculation. Fig. 4
shows the standard deviation of vCOMx (0) calculated by a block
averaging method.50 The correlation time can be estimated by
locating the beginning of a plateau, i.e. corresponding to
about 7 block operations. As the streaming velocity was
sampled each 100 fs, we can consider the velocities after each
100 � 27 = 12 800 fs to be independent. This ensured us that
viscosities determined from subsequent 2 ns sections were
statistically independent. We did not perform direct block
averaging of viscosities because that would require multiple
fitting of the velocity profiles obtained from shorter sections.
For the same reason we did not compute viscosity uncertainties
for other concentrations. Since we estimated here the deviation
of viscosity close to its concentration maximum, we can expect
that the precision of the results for other concentrations is not
worse, i.e. DZ r 0.4 � 10�4 Pa s.
Concentration dependence of bulk properties
For studying concentration dependence of viscosity, external
forces Fw and FMeOH were kept constant regardless of concen-
tration, particularly Fw = 1.38 � 10�13 N, FMeOH = 2.07 �10�13 N (the same as in Simulation I in Table 2). Results of the
present NEMD method are shown in Table 4 for both water
models used. Furthermore, in Fig. 5 and 6 our viscosity
concentration profiles are compared to EMD results of
Guevara et al.,19 who used the same molecular models, and
to experimental data45–47 cited there. Our results show a very
good agreement with their simulation data and in the case of
TIP4P/2005 also with experimental data. As concluded by
Guevara et al., the TIP4P/2005 gives the best qualitative
agreement for transport properties among the commonly used
water models. On the other hand they showed that both
models underestimate viscosity of the mixture in the peak
region, especially for lower temperature (278 K). Considering
that the models were fitted to static properties and simple
Lorentz–Berthelot combining rules were used to describe the
interaction between unlike LJ centers, the quality of prediction
of viscosity is still remarkable.
Table 3 Detailed results for simulations described in Table 2. Eachsimulation was divided into five 2 ns long sections yielding the listedaverage values and standard deviations of the average
Section #xMeOH
[mol mol�1]Z[10�4 Pa s] Section #
xMeOH
[mol mol�1]Z[10�4 Pa s]
Simulation I Simulation III1 0.31 14.8 1 0.31 152 0.31 13.2 2 0.31 14.93 0.31 13.8 3 0.31 14.54 0.31 13.0 4 0.31 13.65 0.31 15.3 5 0.32 13.56 0.32 14.9 Average 14.3 � 0.37 0.31 14.9 Simulation IV8 0.32 14.3 1 0 8.89 0.31 15.5 2 0 9.510 0.32 12.6 3 0 10.311 0.31 16.2 4 0 8.912 0.32 15.6 5 0 9.7Average 14.5 � 0.3 Average 9.4 � 0.3Simulation II Simulation V1 0.31 13.5 1 0 9.92 0.32 13.4 2 0 9.63 0.31 15.3 3 0 9.14 0.31 15.3 4 0 9.65 0.32 13.9 5 0 9.5Average 14.3 � 0.4 Average 9.5 � 0.1
Fig. 4 The standard deviation of vCOMx (0) as a function of the
number of block operations.
3646 Phys. Chem. Chem. Phys., 2012, 14, 3640–3650 This journal is c the Owner Societies 2012
The EMD simulation data19 (obtained using Green-Kubo
formula) allow us to compare the efficiency of both methods
by considering uncertainties of results (uncertainties of the
present NEMD method were estimated only for simulations
given in Table 2, as mentioned above). If one wants to
compare uncertainties from two simulations of different
length, the t�1/2 decrease of the deviation of the mean values
must be taken into account. Considering that our NEMD
simulations are 4–10 times longer than EMD simulations, the
accuracy of both methods from simulation runs of the same
length would be comparable. In our case, there is an additional
factor affecting the accuracy of the NEMD method, namely
the magnitude of external forces. Increasing this magnitude
within the Newtonian regime might further improve the
accuracy of NEMD simulations due to the increased streaming
velocity to thermal velocity ratio. This effect is evident in
results of simulations IV and V (Table 2), where in the latter
simulation a 3� higher applied force resulted in 3� smaller
uncertainty. On the other hand, using an NEMD method in
order to get ‘zero shear’ viscosity requires extrapolating results
in the limit of vanishing external forces or checking that
the systematic error due to applying finite forces is small
compared to statistical uncertainties.
Distance-dependent viscosity profile
So far we have discussed viscosities in the bulk to be able to
compare our results with other numerical and experimental
data which were obtained for homogeneous systems. How-
ever, one of the key advantages of the presented method is that
it enables prediction of distance-dependent viscosity through-
out the whole slab, including inhomogeneous regions close to
the surfaces where strong viscosity inhomogeneity can be
expected due to fluid–surface interactions and inhomogeneous
structure. Such a distance-dependent profile together with
profiles of all quantities needed for calculation of viscosity is
shown in Fig. 7 for bulk concentration xMeOH = 0.31.
The distance-dependent viscosity profile was calculated
according to eqn (1) via two approaches differing in calcula-
tion of the shear rate. Since the shear rate is a derivative of the
streaming velocity, the numerical velocity data obtained in
narrow bins (B0.15 A) need to be properly smoothed in order
to reduce the statistical noise which would cause large scatter
in their derivative. In the first approach, streaming velocity
data were smoothed locally using a numerical smoothing
technique and then the derivative of the smoothed velocity
profile was determined numerically. We refer to the result of
this approach as numerical viscosity.
Table 4 Bulk values of shear viscosity Z and molar density r of themixture water + methanol under ambient conditions for various bulkcompositions xMeOH
xMeOH [mol mol�1] r [mol l�1] Z [10�4 Pa s]
SPC/E + methanol0.00 55.30 7.70.10 49.82 9.70.21 44.67 10.40.32 40.68 10.40.42 36.86 10.40.63 31.22 8.80.83 27.23 6.5TIP4P/2005 + methanol0.00 55.30 9.40.10 49.82 13.40.21 45.00 15.30.31 41.02 14.50.41 37.53 14.00.64 31.38 11.00.82 27.40 8.3Pure methanol1.00 24.41 5.5
Fig. 5 Concentration dependence of shear viscosity of a water +
methanol mixture under ambient conditions based on an SPC/E water
model. Present NEMD simulation results (full squares) are compared
to EMD simulation data (open squares) and to experimental data
(triangles).
Fig. 6 Concentration dependence of shear viscosity of a water +
methanol mixture under ambient conditions based on a TIP4P/2005
water model. Present NEMD simulation results (full squares) are
compared to EMD simulation data (open squares) and to experi-
mental data (triangles).
This journal is c the Owner Societies 2012 Phys. Chem. Chem. Phys., 2012, 14, 3640–3650 3647
In the second approach, the streaming velocity profile was
fitted globally by 8th order symmetric polynomial vCOMx (z) =
c0 + c2z2 + c4z
4 + c6z6 + c8z
8 and its derivative was
calculated analytically, g(z) =dvCOMx (z)/dz= 2c2z + 4c4z
3 +
6c6z5 + 8c8z
7. In this case, the resulting viscosity is denoted as
analytical viscosity. The number of terms in the fitting poly-
nomial was chosen based on our former experience.14 It was
found that including more terms had a negligible effect on the
resulting viscosity profile.
In addition, in Fig. 7d we also indicate the bulk value of
viscosity obtained from the parabolic fit of the streaming
velocity in a bulk phase region as described in the previous
section. As can be seen, viscosities from both approaches
reproduce the bulk value in the bulk region. However, the
numerical viscosity is subject to increasing fluctuations when
approaching the center of the slab. The reason is that the
numerically obtained curve of g(z) appears in the denominator
of the viscosity formula (eqn (1)); therefore, the resulting
viscosity profile features diverging oscillations at points where
g(z) approaches zero, which happens close to the center of the
slab. The numerical viscosity is thus inaccurate in this interval.
However, with increasing the distance from the center getting
closer to surfaces, the accuracy of the numerical viscosity
increases. Moreover, we consider numerical viscosity near
the surface superior to analytical viscosity, as it reproduces
better sudden changes in numerical data, which are likely to
occur at the ends of the streaming velocity profile. Analytical
viscosity, being a smooth curve, does not suffer from fluctuations
close to the center of the slab, but its ability to capture strong
inhomogeneities near the surface is generally limited. Therefore,
numerical viscosity is more reliable in the inhomogeneous region
while analytical viscosity is more suitable to describe the bulk
behavior.
Fig. 8 shows typical behavior of properties of the studied
system at the interfacial region in detail. As can be seen from the
density profiles, surface–liquid interactions cause strong order-
ing of the liquid creating an inhomogeneous region, which
typically extends up to 15 A from a surface. In this region
significant changes of dynamical properties of the liquid take
place. A velocity profile is not parabolic but passes through an
inflection point identified by the extreme of the shear. Closer to
the surface, the absolute value of the shear decreases but the
pressure tensor Pxz monotonically increases in agreement with
eqn (2), see Fig. 7c. This behavior combined leads, according to
eqn (1), to a viscosity increase by a factor of order 1–10 relative
to the bulk value at a distance corresponding to the second
liquid layer. The streaming velocity and accordingly the derived
properties (shear rate, viscosity) were not considered closer to
the surface than the distance of the second liquid layer because
of poor statistics resulting from the lack of molecules in a gap
between first and second liquid layers. However, the streaming
velocity of the first layer is zero in the case of rutile surfaces,
indicating strong adsorption, no-slip boundary conditions and
virtually infinite or huge viscosity of the first layer.
Note that the behavior of viscosity and velocity profiles in
the interfacial region qualitatively depends on the strength of
Fig. 7 (a) Center of mass streaming velocity, (b) shear rate, (c) densities of both components (right axis) and shear stress (left axis), and (d)
resulting viscosity profiles under ambient conditions for mole fraction xMeOH = 0.31. The z-coordinate is centered in the middle of the slab. The
coordinate z0 = Lz/2 � |z|, which gives the distance from a surface, is given on top of graphs.
3648 Phys. Chem. Chem. Phys., 2012, 14, 3640–3650 This journal is c the Owner Societies 2012
surface–liquid interactions, geometry (smoothness) of the
surface, etc. In the case of weaker surface–liquid interactions,
the reduced density of the contact layers can lead to steep build
up of streaming velocity and decrease of the viscosity in the
vicinity of the surface, as reported by Akhmatskaya32 for WCA
fluid. On the other hand, the viscosity profile of a mixture of
neutral and positively charged LJ particles in a negatively
charged cylindrical pore shows an increase of viscosity in the
contact layer by more than an order of magnitude.31
In Fig. 9 the inhomogeneous viscosity profiles at the inter-
face are plotted for several methanol mole fractions. The
plotted curves are numerical viscosities averaged from both
interfaces. Unlike in the bulk, the largest viscosity of the
second layer of liquid, around 3–5 A, is not reached for
xMeOH = 0.3, but 0.6–0.8. While this finding is discussed in
more detail in the next section, we give here a more general
reason for the observed behavior. Bulk water creates nearly
two hydrogen bonds per molecule, saturating both its oxygen
and hydrogen atoms. Limited by the presence of only one H
atom (within the united atom model adopted), bulk methanol
can create maximally one hydrogen bond per molecule, leaving
its O atom unsaturated. However, interfacial methanol in
contact with the surface can form hydrogen bonds with both
the surface and other methanol molecules, making it better
bonded relative to the bulk and more viscous. Therefore, the
presence of methanol clearly influences the viscosity in the
vicinity of the surface more dramatically than in the bulk.
Hydrogen bonding structure
Hydrogen bonding structure was analyzed to elucidate the role
of hydrogen bonds in the increase of viscosity in the interfacial
region presented in Fig. 9. Hydrogen bonds (HBs) were
identified based on a geometric criterion, i.e. HB is assumed
to exist when O and H sites forming the bond are closer than
maximum bond length rmax. The value of maximum bond
length was determined from the position of the first minimum
in the corresponding site–site radial distribution function.
Although the sites forming the HB may belong to molecules
of any of the two components, the positions of the first minima
of the gOH radial distribution functions were found to be
almost independent of the participating molecules. Therefore,
a single geometric criterion was used with rmax = 2.5 A for all
possible types of HBs: Ow–Hw , Ow–HMeOH, OMeOH–Hw and
OMeOH–HMeOH.
From Fig. 9 we can see that the most of the increase of
viscosity takes place within 5 A from a surface, corresponding
to the positions of the first two liquid layers, cf. Fig. 2. We
understand that this viscosity increase is related to the number
of HBs formed between the first and the second liquid layers,
since the HBs increase friction between the second layer and
the immobile first layer. As the external force is given per
molecule, we analyzed the number of HBs between 1st and 2nd
layer per molecule in the second layer, denoted as nHB1–2. The
concentration dependence of nHB1–2 was found to be in qualita-
tive agreement with the concentration dependence of viscosity
in the interfacial area, Fig. 9, having minimum for pure
water (0.8 HB per molecule), raising towards the maximum
at xMeOH = 0.6–0.8 (1.1 HB per molecule) and decreasing for
higher xMeOH (0.9 HB per molecule for pure methanol).
Similar characteristics concerning the number of bonds
between the 2nd and 3rd layers, nHB2–3, were gathered. The
concentration dependence of nHB2–3 is qualitatively similar to
the bulk one. This is also in agreement with Fig. 9, where the
viscosity concentration profile has qualitatively the same
behavior (with a maximum around xMeOH = 0.3) for all
distances further than about 5 A from a surface, which can
be identified as a beginning of the 3rd layer, cf. Fig. 2.
In addition, analysis of HBs allowed us to explain the
minimum in viscosity of pure methanol at about 7 A from a
surface. In this distance the bonds between the second and the
third liquid layers are important. Due to the wide gap in the
Fig. 8 Detailed view of the properties of the interfacial region as
functions of distance from a surface for mole fraction xMeOH = 0.31.
The values of center of mass streaming velocity and numerical viscosity
are expressed in corresponding units on the left axis. Densities of water
and methanol, and the shear rate calculated as a numerical derivative of
streaming velocity are given by the right axis.
Fig. 9 Profiles of shear viscosity in the interfacial region as a function
of the distance from the surface for selected mole fractions of
methanol xMeOH.
This journal is c the Owner Societies 2012 Phys. Chem. Chem. Phys., 2012, 14, 3640–3650 3649
pure methanol density profile between 2nd and 3rd layers, nHB2–3
was found to be much smaller than number of bonds per
molecule in other layers. As a result, the third layer is poorly
bonded causing high shear and low viscosity.
Results of this section suggest that analysis of HBs aided by
structural information from density profiles allows qualitative
explanation of the behavior of distance-dependent viscosity.
A similar effect of hydrogen bonds on dynamic properties
(diffusivity) has been observed for ionic liquids in the gas–liquid
interface.49 Quantitative prediction of the position dependence
of the viscosity is rather a difficult task. To this point, we tried
to compare viscosity with inverse diffusivity according to the
Stokes–Einstein (SE) relation51 Z = kBT/6prD, where r is the
radius of a molecule. Distance dependent diffusivity D was
computed bin-wise from EMD simulation by analyzing mean
square displacement (MSD) of a molecule in the time interval
2 ps for water and 2.5 ps for methanol, during which a
migration of a molecule to further bins was minimal and, at
the same time, MSD reached linear dependence (for simula-
tion details see ref. 14). Fig. 10 shows that the SE relation does
not hold in the presence of the surface, namely inverse
diffusivity increase is steeper and takes place further from a
surface compared to viscosity. While the SE relation is used
with success to link viscosity and diffusivity for homogeneous
fluid made of small molecules (it was derived for spherical
particles), e.g. to describe temperature dependence of viscosity
or diffusivity,52,53 its usage in inhomogeneous systems is not
justified because interaction with the inhomogeneous interface
affects viscosity and diffusivity differently.
4. Conclusions
We have investigated in detail the generalization of the NEMD
simulation method for determination of shear viscosities of fluid
mixtures. We have found out that away from the surfaces, the
method determines a bulk value of the viscosity in agreement
with recent EMD data19 and with comparable accuracy con-
sidering the simulation length. The independence of the results
on the magnitude of external forces acting on one or both
components of the mixture was verified. Also, within the range
of the external forces studied, no correction on the magnitude of
this force and/or extrapolation of the viscosity to zero shear
seemed necessary.
In addition to the bulk values of viscosity, the method is
capable of determining the inhomogeneous profile of viscosity
in the vicinity of the surface. The observed behavior infers that
the dynamical properties of a fluid flowing in channels or pores
of width of few nanometres cannot be automatically presumed
to adopt experimental values measured in the bulk phase. In
our case of rutile surfaces and water + methanol mixtures, the
increase of viscosity in the contact layer can be more than
10 times the bulk value. The inhomogeneous viscosity profiles
at the interface for various methanol concentrations were
interpreted using the structural data and information on
hydrogen bonding. The viscosity data around second and
third fluid layers were found to correlate with hydrogen
bonding, i.e. more hydrogen bonds per molecule leads to
steeper increase of local viscosity. The detailed behavior of
properties of fluid in an inhomogeneous region is determined
by particular fluid–surface interaction and structure of fluid
near surfaces. Different surfaces produce different inhomo-
geneous viscosity profiles. On the other hand, if one is interested
in the viscosity of the bulk fluid away from the surfaces, the
resulting value is independent of the surface type and structure
of fluid in the inhomogeneous region. In that case, the only
property of the interface we require is that the surfaces manage
to hold at least the first layer of fluid (no slip condition).
Regarding specific results we have obtained, we have found
that the density profiles of water + methanol mixtures in
contact with rutile surfaces are not affected by the choice of
SPC/E or TIP4P/2005 water models. As observed earlier,19 this
is not the case of dynamic properties, when SPC/E predicts the
concentration dependence of the water + methanol mixture
only qualitatively, but TIP4P/2005 even quantitatively.
Acknowledgements
This research was supported by the Czech Science Foundation
(project No. 203/08/0094) and by the Ministry of Education,
Youth and Sports of the Czech Republic (project No.ME09062).
The access to the MetaCentrum computing facilities, provided
under the programme ‘‘Projects of Large Infrastructure for
Research, Development, and Innovations’’ LM2010005 funded
by the Ministry of Education, Youth, and Sports of the Czech
Republic, is acknowledged. We thank Jadran Vrabec for pro-
viding us the results published in ref. 19 prior to publication.
Fig. 10 Comparison of distance dependent viscosity and inverse of distance dependent diffusivity for water (left) and methanol (right).
3650 Phys. Chem. Chem. Phys., 2012, 14, 3640–3650 This journal is c the Owner Societies 2012
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