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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 Pařeza and Milan Předota*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 of the 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

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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 Å, |O–H| =

0.9451 Å 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 geo