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Chemical Physics Letters 397 (2004) 233–236

Shear viscosity of simple fluids in porous media: moleculardynamics simulations and correlation models (II) –

methane in silicate pores

Hui Zhang, Bing-jian Zhang *, Jing-jun Zhang

Department of Chemistry, Zhejiang University, Hangzhou, Zhejiang 310027, PR China

Received 11 July 2004; in final form 26 August 2004

Available online 11 September 2004

Abstract

As a continuation of our last paper [Chem. Phys. Lett. 350 (2001) 247], the shear viscosity of fluid methane in silicate porous

media have been calculcated using equilibrium molecular dynamics simulations, and two similar correlation models are also pre-

sented which can describe the viscosity value under different densities, temperatures and pore widths.

� 2004 Elsevier B.V. All rights reserved.

1. Introduction

There continues to be growing interests in study of

the properties and behavior of fluids confined in

nano-scale pores. Improved understanding of confine-

ment effects is essential both fundamentally and practi-

cally in areas relating to fluxion, lubrication, adhesion,nanomaterials, nanotribology and related industries.

And the understanding of transport properties of fluid

confined in nanoporous materials is also important to

processes such as gas separations, catalysis and en-

hanced oil recovery. But it is quite difficult to run nor-

mal experiments because of the complexity of such

systems and such extreme conditions. Molecular simu-

lation plays an important role in the determination ofthe transport properties in confined fluids as well as

the development of transport theories at the micro-

scopic level. The molecular simulation approaches to

transport properties have been generally reviewed by

Cummings and Denis [2].

0009-2614/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.cplett.2004.08.122

* Correspondig author. Fax: +86 571 8785110.

E-mail address: zbj@mail.hz.zj.cn (B. Zhang).

In the recent years, molecular dynamics simulations

have been successfully employed for the investigation

of transport properties of confined fluids [3–6]. Hoo-

genboom et al. [7] used equilibrium molecular dynam-

ics simulation (EMD) to study transport diffusion of

argon in the unidirectional channels of the molecular

sieve AlPO4-5 and introduced a new method to getthe non-equilibrium property from the EMD simula-

tion. There are also many papers regarding the fluid

diffusivity in porous materials [8–10], but much less

work has been reported on the viscosity in fluid/pore

system. Bitsanis et al. [11] presented a local average

density model (LADM) of viscosity and diffusivity

using non-equilibrium molecular dynamics (NEMD).

But they only concentrated on one state point. In ourprevious work, we calculated series of viscosity values

of argon under various state points and found that

the viscosity sharply increases at very small pore width

[1]. In this Letter, we performed the EMD simulations

for the viscosity of fluid methane in the mica-like pores

in different temperatures, densities and pore widths,

and introduced a correlation model which can describe

the viscosity of methane in pores as a continuation ofour previous work [1].

Fig. 1. The slit-pore model.

234 H. Zhang et al. / Chemical Physics Letters 397 (2004) 233–236

2. Theoretical

Following our previous work, Chapman–Enskog

(CE) theory and the Heyes relations [12] are used in this

Letter. The viscosity of LJ fluids in macrovolume system

can be calculated from the following equations:ffiffiffiffiffiT �p

g�m¼ 1:2450�

ffiffiffi2

p

q� � 1:384

!

� a1 þ a2T � þ a3T �2 þ a4T �3 þ a5T �4� �2; ð1Þ

where g�m ¼ gr2ðmeÞ�1=2is the reduced viscosity of LJ

fluids in macrovolume system, q* is the reduced density,

T*kBT/e is the reduced temperature, e is the poten-tial parameter, a1 = 1.079278, a2 = �0.120154, a3 =

0.0347497, a4 = �0.00500299 and a5 = 0.0001952.

The reduced viscosity in porous media g�p is the func-tions of the reduced temperature T*, the reduced density

q*, and the reduced pore width H* = H/r. We get:

g�p / q�l; g�p /1

T �m ; g�p /1

H �n ; ð2Þ

where l, m, n are the exponents that will be determined

later. From these relationships, a new correlation model

which can describes the viscosity of the simple liquids in

porous media can be obtained

lnðg�p=g�mÞ ¼ Aq�l

T �mH �n

� �þ B; ð3Þ

where A and B are the spring constants.

Another similar correlation model can be expressed

as follows:

lnðg�p=g�mÞ ¼Lq�c

T �e1

H � � 1

� �f

; ð4Þ

where L, c, e and f are the spring constants.

The ability of these two models given by Eqs. (3) and

(4) to describe the viscosity of simple liquids in porous

media need to calculated the viscosity g�p as a function

of T*, q* and H* and compared with the experiment

data or simulation results. In this work, the simulationresults of viscosity for argon both in macrovolume sys-

tems and in slit porous media are presented.

Fig. 2. The wall configuration.

3. Simulations

In this Letter, we performed the EMD simulations

using Green–Kubo formula to calculate the shear vis-

cosity. It has been stated that Green–Kubo formula

can be applied to canonical ensembles to calculate trans-port coefficients [13]. There is an argument that EMD

and NEMD methods will produce different transport

properties in porous media. But recently, it has been

mentioned that both EMD and NEMD methods give

similar values for the shear viscosities [14]. And it has

even been demonstrated that theses simulation methods

yield the same transport properties (for example, diffu-

sivity coefficient) even in the presence of viscous flow

[15]. In light of the technical difficulty, we will use

EMD to measure the transport coefficients.

The slit-pore model consists of methane moleculesconfined transversely by two parallel sheets of flat walls

(see Fig. 1). The fluid and wall atoms are spherical, non-

polar, Lennard–Jones (LJ) atoms characterized by

diameters ri and interaction energies ei, where i repre-

sents fluid (f) or wall (w) atom. We choose the molecular

diameter, mass and interaction strength parameters so

that the fluid molecular represents methane, and the wall

atoms represent oxygen. Each wall is a sheet of discreteoxygen atoms in a hexagonal arrangement derived from

X-ray diffraction data for mica (see Fig. 2) [16]. The lat-

tice constant in x-direction is lx which is taken to be

1.985rw, and in y-direction is ly, taken to be 3.438rw,following Curry [17].

The EMD simulations are performed on the LJ 12-6

fluid, for which the potential is

uLJ ¼ 4err

� �12� r

r

� �6� : ð5Þ

The total potential energy of the fluid particles confinedbetween the walls is written as the sum of the interaction

energy Umo and Umm, where the Umo represents the

interaction energies between the confined methane and

Table 1

The simulation results of viscosity for methane in silicate pores (all the

variables are in reduced units)

Temperature (T*) Density (q*) Pore width (H*) Viscosity (g*)

0.81 0.855 7.0 6.678

0.81 0.855 8.0 5.612

0.81 0.855 9.0 3.635

0.81 0.855 10.0 3.094

H. Zhang et al. / Chemical Physics Letters 397 (2004) 233–236 235

the lattice oxygen, the Umm represents the interaction

energies between the confined methane.

U ¼ Umm þ Umo: ð6ÞThe interaction energy of confined methane is given by

Umm ¼Xi>j

/mmðrijÞ; ð7Þ

where the summation is over all pairs of methane mole-

cules. The methane–oxygen interaction energy is given

by

Umo ¼Xi

Xj

/moðrijÞ; ð8Þ

where the i summation is over all methane molecules and

the j summation is over all oxygen atoms in the lattice

wall. The two potential parameters of the fluid molecule

are given in the value of rm = 3.73 A, em/kB = 147.95 K,and mm = 16.0 amu, which correspond to methane, and

the methane–oxygen (mo) interactions are rmo = 3.214

A, emo/kB = 133.3 K, the cut-off radius is chosen to be

3.5 rm.The whole simulation system contains 500 methane

molecules. Periodic boundary conditions are applied in

x- and y-directions. If any particle center attempts to

cross any of the two bounding z-planes, it is elasticallyreflected from that plane. This reflection is easily accom-

plished by checking each particle�s new position, and if it

has passed a z boundary, return the particle to its previ-

ous position with a reversed velocity component. Then a

reflected particle behaves as if it had undergone an elas-

tic collision at a distance half of the current spatial incre-

ment from its previous position.

The time step is chosen to be 0.007159 in reducedunits (10�14 s for methane). The initial 50000 time steps

are disregarded and then performed equilibrium MD for

1050000–2050000 time steps (10.5–20.5 ns). In the

simulations, Verlet�s leapfrog algorithm is used to solve

the equations of motion.

The viscosity is calculated by Green–Kubo formula

g ¼ 1

kBTV

Z 1

0

dt hJvð0ÞJvðtÞi; ð9Þ

where kB is the Boltzmann constant. Jv is the momentum

flux

Jv ¼Xj

pxjpyj

mþXi>j

F xijr

yij; ð10Þ

where p is the momentum and F is the force.

0.68 0.855 9.0 5.699

0.71 0.855 9.0 5.224

0.74 0.855 9.0 4.998

0.78 0.855 9.0 3.838

0.74 0.750 9.0 2.277

0.74 0.810 9.0 3.396

0.74 0.870 9.0 5.747

0.74 0.900 9.0 7.680

4. Results and discussion

The viscosities in slit-porous media for methane at

different reduced temperatures, densities and pore

widths are calculated and the result is shown in Table 1.

Clearly, the viscosity slowly increases with the density

increases when keeping the temperature constant, and

slowly decreases with the temperature increases when

keeping the density constant. But at very small pore

width, the viscosity value sharply increases. This agrees

with the results in our previous work.In order to get the two models that can calculate the

methane viscosity in silicate pores, the exponents of den-

sity, temperature and pore width should be determined

first. In Eq. (3), the density, temperature and pore width

are regarded as the only independent variables, respec-

tively, then drawing diagrams with lnðg�p=g�mÞ versus

the three variables, l, m and n can be obtained to be 2,

6 and 5, respectively. So using the date in Table 1 anddrawing the diagram with lnðg�p=g�mÞ versus q*2/(T*6H*5), the empirical constants A and B can be ob-

tained. Then Eq. (3) becomes

lnðg�p=g�mÞ ¼ 7183:5q�2

T �6H �5

� �þ 0:0837: ð11Þ

On the other hand, in Eq. (4), we use the same methodand c, e and f are obtained to be 2, 6 and 4, respectively.

Then drawing a diagram with lnðg�p=g�mÞ versus (q*2/T*6)(1/(H* � 1))4, the constant L can be obtained, Eq.

(4) becomes

lnðg�p=g�mÞ ¼594:35q�2

T �61

H � � 1

� �4

: ð12Þ

The results calculated from Eqs. (11) and (12) are shownin Fig. 3 and compared with the simulation data. It can

be seen that the calculated results agree with the simula-

tion data well.

Summarizing this Letter and our previous work, it

can be concluded that the simple fluids behave similarly

in very small pores, especially for their shear viscosity.

The correlation models of shear viscosity have the same

forms but different constants. We will investigate thechanging rules of these constants for different species

in our future work.

7.2 8.0 8.8 9.6

3.2

4.0

4.8

5.6

6.4

MD Theory1 Theory2

Red

uced

Vis

cosi

ty

Reduced Pore Width0.70 0.75 0.80

3.5

4.0

4.5

5.0

5.5

6.0

Reduced Temperature0.75 0.80 0.85 0.90

2

3

4

5

6

7

8

Reduced Density

Fig. 3. Viscosity versus pore width, temperature and density. The symbol squares (h) are our simulation data, the dashed lines are obtained from Eq.

(11), the solid lines are obtained from Eq. (12).

236 H. Zhang et al. / Chemical Physics Letters 397 (2004) 233–236

5. Conclusion

The viscosity of methane in silicate pores has been

calculated from molecular dynamics simulations under

different temperatures, densities and pore widths. As acontinuation of our previous work, two correlation

models that could predict the shear viscosity of methane

in pores are proposed in the same forms as which in our

last paper. These models just have different constants, so

it will be necessary to figure out the rules behind them.

That will be done in the future work.

Acknowledgements

This study is financially supported by the National

Major Fundamental Research and Development Project

G19990433 and the National Natural Science Founda-

tion if China Project 20277034.

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