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: predota@prf.jcu.cz

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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