CHINESE JOURNAL OF CHEMICAL PHYSICS
VOLUME 25, NUMBER 4
AUGUST 27, 2012
ARTICLE
Absorption and Structural Property of Ethanol/Water Mixture with Carbon
Nanotubes
Sheng-ping Du, Wen-hui Zhao, Lan-feng Yuan ∗
Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of
China, Hefei 230026, China
(Dated: Received on June 1, 2012; Accepted on June 8, 2012)
Molecular dynamics (MD) simulations are performed to study the structure and adsorption of
ethanol/water mixture within carbon nanotubes (CNTs). Inside the (6,6) and (10,10) CNTs,
there are always almost full of ethanol molecules and hardly water molecules. Inside wider
CNTs, there are some water molecules, while the ethanol mass fractions inside the CNTs are
still much higher than the corresponding bulk values. A series of structural analysis for the
molecules inside and outside the CNTs are performed, including the distributions of radial,
axial, angular density, orientation, and the number of hydrogen bonds. The angular density
distribution of the molecules in the first solvation shell outside the CNTs indicates that the
methyl groups of ethanol molecules have the strongest interaction with the carbon wall, and
are pinned to the centers of the hexagons of the CNTs. Based on the understanding of the
microscopic mechanism of these phenomena, we propose that the CNTs prefer to contain
ethanol rather than methanol.
Key words: Ethanol/water, Carbon nanotube, Adsorption, Molecular dynamics simulation
lations [16]. Methanol molecules confined in CNTs
tend to form a single file in thinnest CNTs or helices
in slightly wider tubes, and when the CNT diameter is larger than 14 Å, the distribution of methanol
molecules is close to that in bulk [17]. Liu et al. studied the microscopic structure and transport behavior
of methanol-water solutions within an armchair CNT
[18]. They found that CNTs preferentially absorbed
methanol over water molecule for an equimolar mixture
of methanol and water, and demonstrated nanoimmiscibility of these molecules in CNTs. Zheng et al. studied the transport of methanol-water mixtures through
CNTs under a chemical potential gradient [19]. By comparing flux through hydrophobic and hydrophilic nanotubes, they showed that solutions entered hydrophilic
nanotubes more easily and the diffusion in hydrophobic
pores was faster, and methanol had higher flux than
water through both hydrophobic and hydrophilic nanotubes. The effects of cosolvents on hydration of interior of CNT with a small diameter have been studied by Yang et al. [20]. They found that methanol
or trimethylamine N -oxide dehydrated the CNT, while
urea improved hydration of the CNT interior.
Recently, we have found that the CNTs showed very
high adsorption selectivity of methanol over water for
the methanol/water mixture by MD simulations [21].
By analyzing microscopic structures of the fluid mixtures, we find that the selective adsorption roots in a
cooperative effect of the van der Waals (vdW) interaction between CNT and the methyl groups of CH3 OH
molecules as well as the hydrogen bonding interaction
I. INTRODUCTION
Carbon nanotubes (CNTs) are discovered in 1991 [1],
and have been paid more and more attention because
of their unique electrical, optical, and mechanical properties [2, 3]. In the last decade, many interesting properties of matter confined in carbon nanotubes which
are very different from those in bulk have been found
[4, 5]. Although CNT is always hydrophobic, many
studies have demonstrated that water can fill and pass
through the inner space of open-ended CNTs quickly
[6−12]. Water solution with CNTs is a popular system,
and many special behaviors of water have been found
[13]. Therefore, water-filled CNTs hold the promise for
applications in nanofluidic devices and proton storage
[14, 15].
The design of CNT-based nanodevices requires a
thorough understanding of the interactions between
nanotubes and their surrounding media, as well as the
ability to control these interactions. Molecular dynamics (MD) and Monte Carlo (MC) simulations are very
useful methods to study the structural and dynamical
properties of fluid confined in nanotubes. For example, Shao group found that the structure of ethanol
confined in single-walled carbon nanotubes (SWCNT)
changed with the diameters of CNTs by MD simu-
∗ Author
to whom correspondence should be addressed. E-mail:
[email protected]
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Sheng-ping Du et al.
TABLE I The Lennard-Jones parameters and partial
charges for ethanol, water, and carbon nanotubes. M is
a point on the bisector of the HOH angle, 0.015 nm from
the oxygen atom towards the hydrogen atoms.
H2 O
C2 H5 OH
CNT
Site
O
H
M
CH3
CH2
O
H
C
σ/µm
315.4
0.0
0.0
377.5
390.5
307.0
0.0
340.0
ε/(J/mol)
648.5
0.0
0.0
866.1
493.7
711.3
0.0
232.8
q/e
0.0
0.52
−1.04
0.000
0.265
−0.700
0.435
0.0
(a)
(b)
FIG. 1 Snapshots of the simulation box for CNT (6,6) (a)
side view and (b) top view.
among water and methanol molecules.
In this work, we extend the system to ethanol/water
mixture, investigating whether there is similar selectivity for ethanol. A positive answer is indeed given by our
numerical experiments. Furthermore, the mass fraction
(MF) of ethanol inside CNTs is even higher than that
of methanol under the same conditions.
II. SIMULATION DETAILS
In our simulations, an armchair CNT with open ends
is immersed into an enthanol aqueous solution. The
CNTs and their diameters are: (6,6), 8.14 Å; (7,7),
9.49 Å; (8,8), 10.85 Å; (9,9), 12.16 Å; (10,10), 13.56 Å;
(15,15), 20.34 Å; and (20,20), 27.12 Å. For the ethanol
molecules, the united-atom (UA) model with the OPLS
force field [22, 23] is used, which can reproduce well the
properties of bulk ethanol [24]. The OPLS/UA potential approximates the ethanol molecule by four interaction sites: one for the methyl group, one for the methylene group, one for the oxygen atom, and one for the
hydrogen atom. As for the water molecules, the TIP4P
model [25] is used. As for the CNT, each carbon atom
is taken as a neutral particle with a potential for the
carbon atoms of graphite. The intermolecular interactions are described by combination of Lennard-Jones
12-6 potential and Coulombic potential:
"µ
¶12 µ
¶6 #
σij
σij
qi qj
U (rij ) = 4εij
(1)
−
+
rij
rij
rij
where rij is the distance between atoms i and j in different molecules, qi is the charge assigned to atom i, and
σij and εij are the Lennard-Jones parameters with the
combination rules: εij =(εi εj )1/2 , σij =(σi σj )1/2 . The
parameters of all the interaction centers are shown in
Table I.
An ethanol/water box of 5.0 nm×5.0 nm×5.0 nm
with periodic boundary condition is employed (see
Fig.1). The ethanol mass fraction in bulk is taken
as 20%, which corresponds to mole fraction of 9%.
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FIG. 2 The mass fraction of ethanol in carbon nanotubes
with bulk mass fraction of 20% as a function of the tube
diameter.
The open-ended CNT (2.5 nm in length) is placed
along the z axis and fixed in the center of the mixture reservoir. To test whether the CNTs prefer ethanol
to methanol, we also perform a series of simulations
for methanol/ethanol mixture, where the ethanol mass
fraction is taken as 10%. In these simulations, the (6,6)
to (10,10) CNTs are used.
The simulations are performed in NVT ensemble with
the program GROMACS 3.3.1 [26]. The particle-mesh
Ewald (PME) algorithm [27] is used for long range electrostatic interactions. All the simulations last 12 ns
with a time step of 1 fs. The cutoff for Lennard-Jones
interactions is taken as 1.1 nm. The temperature is
maintained at 300 K by the Berendsen algorithm with
the time constant of 0.1 ps [28]. The SHAKE algorithm [29] is used to constrain the intramolecular bonds
of ethanol and water molecules. The carbon atoms of
CNT are kept at the initial positions to fix the nanotube. Energies and coordinates are saved every 200 fs.
III. RESULTS AND DISCUSSION
A. MF of ethanol within CNTs
Figure 2 gives the MF of ethanol in CNTs with different diameters. Inside the thinnest CNTs (6,6) and (7,7),
there is nearly 100% ethanol. This is also reflected from
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FIG. 3 The radial density distribution of the molecules in the box for (6,6), (7,7), (8,8), and (10,10).
the radial distribution function (RDF) of water which
has no peak inside these CNTs (Fig.3). As the CNT
diameter increases, the mass percentage of ethanol decreases gradually. When the diameter reaches 1.356 nm
(the (10,10) CNT), the ethanol MF is 97%, which is still
a very high ratio. Therefore, all these thin CNTs can
be perfectly used to purify ethanol from water.
In the two wider CNTs, the inner ethanol MF is
74.1% for (15,15) and 54.0% for (20,20), respectively,
which are also much higher than the outer value 20%.
Afterwards, the structural analyses will be performed
mainly on the CNTs with nearly pure ethanol, i.e.,
(6,6) to (10,10). Comparing with the corresponding
results of the methanol/water-CNT systems with the
bulk methanol MF of 20% [21], we find that inner MFs
of ethanol are higher than those of methanol. Therefore, it is an interesting question to ask whether the
CNTs prefer to adsorb ethanol rather than methanol.
To examine this, we perform similar MD simulations
for methanol/ethanol mixture with ethanol MF of 10%.
The ethanol MFs within the CNTs (6,6), (7,7), (8,8),
(9,9), and (10,10) are 30%, 30%, 23.7%, 17%, and 10%,
respectively. Thus, as we expected, the CNTs indeed
prefer ethanol to methanol, although the selectivity is
much lower than the selectivity of alcohol vs. water and
drops quickly with the increase of the CNT diameter.
B. Radial density distributions function
The fluid structure within a nanotube can be well
described by the atomic density distributions. From the
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radial density distribution (RDF) (see Fig.3), we can
see that there are a few water molecules in the (8,8),
(9,9), and (10,10) CNTs. However, in the (6,6) and
(7,7) CNTs there are full of ethanol molecules and no
water molecule.
In every profile, we can find two groups of density
peaks which correspond to the molecules surrounding
the nanotubes (inside and outside). The peak of methyl
is sharper than that of hydroxyl, indicating a strong interaction between the methyl groups and the nanotube.
We also find a second peak of water close to the tube
center (r=0) in the (10,10) CNT. This layer structure
of water within wide nanotube was also observed by
Wang et al. [30] and in our work on methanol/waterCNT [21]. The distance of the nearst peak of the oxygen atom of water (OW) from the carbon wall is around
0.33 nm (close to σC−OW ), the same as the distance in
the (8,8) and (9,9) CNTs. The similar distance between
the wall and the peak of RDFs of water molecules [30]
and oxygen molecules [31] inside various CNTs suggests
that this distance has a close relationship with the parameter σC−O .
Inside the (6,6) CNT, the peak of the methyl group
is nearly located at r=0, closer to the center of CNT
than that in the other nanotubes. Then we calculate
the distance between the peak position of CH3 and the
CNT wall dCH3 and that of the OH group dOH for the
molecules inside the nanotubes. As we expected, the
values of dCH3 are always around 0.35−0.36 nm, close
to σC−CH3 (0.358 nm) except for the (6,6) CNT, which
is 0.39 nm. The exception of the (6,6) CNT comes from
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the confinement effect of this narrowest nanotube with
diameter of 4.07 nm. On the other hand, the value
of dOH changes greatly with the increase of diameter
of CNTs. These data are the same as those for pure
ethanol within CNTs by Shao et al. [16]. That is to
say, the water molecules within the nanotubes (if there
are any) have little effect on the distribution of ethanol.
Therefore, we can ascribe the behavior of the methyl
groups within nanotubes to the wall-fluid interactions,
while ascribing the behavior of the hydroxyl groups to
the fluid-fluid interactions.
Outside the CNTs, both the ethanol and water show
two groups of peaks with every CNT. For the (6.6)
CNT, the first peak of OW outside the CNT is close
to r=0.73 nm, and the second is at r=1.1 nm. We can
say that the outer molecules form a two-layer structure. The first layer consists of a strong peak of ethanol
group and a weak peak of water, which suggests that
the interaction between the wall and ethanol is stronger
than that between the wall and water. Besides, without
the confinement effect of the CNTs, the values of dCH3
for the outer molecules are all around 0.35 nm, including that for the (6,6) CNT, 0.351 nm. The distances
between the water O atoms and the tube wall are always around 0.33 nm for different CNTs, which is close
to σC−OW (0.327 nm). On the other hand, the peaks
of the second layer at r=1.1 nm are quite flat. This
is understandable, since as the molecules are far from
the carbon wall, the interaction between them becomes
weak. A graphene sheet can be regarded as a CNT with
diameter of infinity. So with the increase of CNT diameter, the distribution of outside molecules will become
similar and converge to the case of graphene.
C. Hydrogen bonds inside CNTs
As in pure ethanol and water, hydrogen bonds play
an important role in the behavior of ethanol/water mixture. Here we use the geometrical definition of hydrogen bond proposed by Luzar et al. [32], i.e., a hydrogen bond between two molecules should fulfill these two
conditions: (i) the distance between the oxygen atoms
of both molecules is smaller than a threshold distance
0.35 nm, (ii) the bond angle between the O−O direction and the molecular O−H bond of the donor is less
than a threshold angle 30◦ .
Figure 4 shows the average number of hydrogen bonds
per molecule (hnHB i) of ethanol/water molecules within
CNTs as a function of the CNT diameter. The results
are close to those of pure ethanol within CNTs. The
number of hydrogen bonds per molecule for CNTs with
diameters less than 1.0 nm ((6,6) and (7,7)) is close to
1, nearly half of the value in bulk ethanol. This reflects that the ethanol molecules in these CNTs form
single-file structures (more detailed discussion will be
present in the next subsection). When the CNT diameter becomes larger, the number of hydrogen bonds per
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FIG. 4 The average number of hydrogen bonds per molecule
of the confined molecules as a function of the nonotube diameter.
(6,6)
(7,7)
(8,8)
(9,9)
(10,10)
FIG. 5 The snapshots of the molecules inside the CNTs
(6,6), (7,7), (8,8), (9,9), and (10,10).
molecule becomes close to 2, i.e., bulk-like.
D. Axial density distributions
Zhou et al. showed the axial density profile of the water within CNTs [33], and found that water molecules
confined in the (8,8) and (9,9) CNTs have marked
a wave-like pattern. Here we see that the ethanol
molecules show two different ways of filling CNTs which
are the “single-file mode” and “ring mode”, like the
cases of pure water [33]. In our simulations (Fig.5), the
ethanol molecules within the (6,6) and (7,7) CNTs form
a sigle-file structure. In the (9,9) and (10,10) CNTs,
the methyl groups of ethanol form ring-like structures
around the nanotube, consistent with the results of
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FIG. 6 The axial density distribution of molecules inside (a) (6,6) and (b) (7,7) CNTs.
FIG. 7 The orientation distribution of (a) the CH3 -CH2 bond, (b) the O−H bond, and (c) the dipole moment.
Shao et al. [16]. The (8,8) CNT is a special case, where
an ordered structure with two parallel single-file chains
takes place.
For the ethanol molecules within the (6,6) CNT, the
axial density distribution is a wave-like profile with a
quasi-period. In Fig.6(a) we see that there are four
density peaks for both CH3 and CH2 , corresponding
to four ethanol molecules. As the average number of
H-bonds per molecule within the (6,6) CNT is close
to 1, we can conclude that the single-file structure of
ethanol molecules within the (6,6) CNT is formed by
two groups of molecules with one hydrogen bond in each
group and no hydrogen bond between the two groups.
From the distance of the two peaks of the methyl groups
in middle, we deduce that the orientation of the two
molecules is head-to-head (methyl to methyl). On the
other hand, the axial density profile of hydroxyl seems
to have two peaks rather than four. This is because
the distributions of the two hydroxyl groups forming a
H-bond merge to make one peak. The density profile
for the (7,7) CNT is much less structured than that of
the (6,6) CNT, while we can still see a wave like structure, in which the five peaks correspond to the number of ethanol molecules within the nanotube. Inside
the wider CNTs, along with increase of the number of
molecules, the density profile of the molecules along the
axis becomes vague and less informative.
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E. Orientation distribution of the molecules within CNTs
To understand the orientation distribution of the
ethanol/water molecules within CNTs, we define α, β,
and θ for the angles between the positive direction of the
z-axis (tube axis) and CH3 −CH2 , O−H, and the dipole
moment of the molecule, respectively. The orientation
distribution of the CH3 −CH2 bond in all the CNTs are
found to be in good accordance with the results of Shao
et al. [16]. For the (6,6) CNT this orientation is mainly
concentrated on 20◦ and 160◦ . It indicates that the
CH3 −CH2 bond is likely to stay along the axis, consistent with the single-file structure of ethanol within the
(6,6) CNT. As the CNT diameter increases, the orientation distribution becomes even. As for the orientation
distribution of O−H, Figure 7(b) shows that there are
four peaks for the (6,6) CNT. Interestingly, for the (8,8)
CNT we see a high peak around 150◦ which is quite different from the other CNTs. Similar abnormality is also
found by Mashl et al. [34] for confined water within the
(9,9) CNT.
Like the cases of pure water [30] and pure ethanol [16],
the dipole orientation of the ethanol/water molecules
within CNTs changes greatly with the CNT diameter.
The dipole orientation of the molecules within the (8,8)
CNT is mainly around 150◦ , while this distribution in
other nanotubes has not much structure in other CNTs
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FIG. 8 The angular density distribution of the molecules of the first layer outside the (a) (6,6) and (b) (9,9) CNT. 1. CH3 ,
2. CH2 , 3. OW, 4. OH, the dash lines express the hexagonal centers of of CNT.
except (6,6).
the nanotubes prefer to contain ethanol rather than
methanol.
F. Angular distribution
Figure 8 gives the angular distribution of the
molecules in the first solvation shell outside CNT. As
the axial distribution, the angle distribution also shows
a wave-like structure with a period that changes with
the CNT diameter. Notably, the positions of all the
methyl peaks correspond to the centers of the hexagonal cells of CNT. This beautiful profile implies that
there is a strong interaction between CH3 and hexagonal cell. On the other hand, the correlation between
the CNT hexagon centers and the other groups (CH2 ,
OH, OW) is weak. This is similar to the results of
methanol/water-CNTs in our previous study [21]. The
angular distributions of the groups inside the CNTs are
disordered. This can be explained by the following reason. When the curvature of the nanotubes is negative
(i.e., outside), the distance of the centers of two consecutive hexagon cells is greater than that of graphene,
so the “spatial resolution” felt by the methyl groups is
enhanced, and the contrary takes place for the inner
case.
IV. CONCLUSION
Systematic MD simulations are performed for
ethanol/water mixture with CNTs. Remarkably, there
is nearly pure ethanol within the narrow nanotubes
from (6,6) to (10,10). This compellent character of
these nanotubes makes them good instruments to separate ethanol from water. Based on a variety of structural analyses, we find that the ethanol/water mixture
also forms interesting structures both inside and outside
the nanotubes, including single-file and ring-like. The
strong interaction between the CNT and the methyl
groups of ethanol molecules is a key factor in determining these structures. We also run MD simulations
for methanol/ethanol-CNT systems. As we expected,
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V. ACKNOWLEDGMENTS
This work is supported by the National Natural Science Foundation of China (No.20603032 and
No.20733004), the Ministry of Science and Techology
of China (No.2011CB921400), the Foundation for the
Author of National Excellent Doctoral Dissertation of
China (No.200736), the Fundamental Research Funds
for the Central Universities (No.WK2340000006 and
No.WK2060140005), and the Shanghai Supercomputer
Center, the USTC-HP HPC Project, and the SCCAS.
[1] S. Iijima, Nature 354, 56 (1991).
[2] J. T. Sander, A. R. M. Verschueren, and C. Dekker,
Nature 393, 49 (1998).
[3] S. S. Wong, J. D. Harper, P. T. Lansbury Jr. and C.
M. Lieber, J. Am. Chem. Soc. 120, 603 (1998).
[4] K. B. Jirage, J. C. Hulteen, and C. R. Martin, Science
278, 655 (1997).
[5] S. H. Joo, S. J. Choi, O. Ilwhan, J. Kwak, Z. Liu, O.
Terasaki, and R. Ryoo, Nature 412, 169 (2001).
[6] S. Cambre, B. Schoeters, S. Luyckx, E. Goovaerts, and
W. Wenseleers, Phys. Rev. Lett. 104, 207401 (2010).
[7] H. J. Wang, X. K. Xi, A. Kleinhammes, and Y. Wu,
Science 322, 80 (2008).
[8] G. Hummer, J. C. Rasaiah, and J. P. Noworyta, Nature
414, 188 (2001).
[9] T. A. Pascal, W. A. Goddarda, and Y. Jung, Proc.
Natl. Acad. Sci. USA 108, 11794 (2011).
[10] J. Köfinger, G. Hummer, and C. Dellago, Phys. Chem.
Chem. Phys. 13, 15403 (2011).
[11] H. Kyakuno, K. Matsuda, H. Yahiro, Y. Inami, T.
Fukuoka, Y. Miyata, K. Yanagi, Y. Maniwa, H. I.
Kataura, T. Saito, M. Yumura, and S. Iijima, J. Chem.
Phys. 134, 244504 (2011).
[12] J. Su and H. Guo, ACS Nano 5, 351 (2011).
c
°2012
Chinese Physical Society
Chin. J. Chem. Phys., Vol. 25, No. 4
[13] J. A. Thomas and A. J. H. McGanghey, J. Chem. Phys.
128, 084715 (2008).
[14] J. Li, X. Gong, H. Lu, D. Li, H. Fang, and R. Zhou,
Proc. Natl. Acad. Sci. U.S.A. 104, 3687 (2007).
[15] D. J. Mann and M. D. Halls, Phys. Rev. Lett. 90,
195503 (2003).
[16] Q. Shao, L. L. Huang, J. Zhou, L. H. Lu, L. Z. Zhang,
X. H. Lu, S. Y. Jiang, K. E. Gubbins, Y. Zhu, and W.
F. Shen. J. Phys. Chem. C 111, 43 (2007).
[17] G. Garberoglio, J. Phys.: Condens. Matter 22, 415104
(2010).
[18] Y. Liu, S. Consta, and W. A. Goddard, J. Nanosci.
Nanotechnol. 10, 3834 (2010).
[19] J. Zheng, E. M. Lennon, H. Tsao, Y. Sheng, and S.
Jiang, J. Chem. Phys. 122, 214702 (2005).
[20] L. Yang and Y. Q. Gao, J. Am. Chem. Soc. 132, 842
(2010).
[21] W. H. Zhao, B. Shang, S. P. Du, L. F. Yuan, J. Yang,
and X. C. Zeng, J. Chem. Phys. 137, 034501 (2012).
[22] W. L. Jorgensen, J. D. Madura, and C. J. Swenson, J.
Am. Chem. Soc. 106, 6638 (1984).
[23] W. L. Jorgensen, J. Phys. Chem. 90, 1276 (1986).
DOI:10.1088/1674-0068/25/04/487-493
Ethanol/Water Mixture with Carbon Nanotubes
493
[24] E. J. W. Wensink, A. C. Hoffmann, P. J. van Maaren,
and D. van der Spoel, J. Chem. Phys. 119, 7308 (2003).
[25] W. L. Jorgensen, J. Chandrasekhar, J. D. Madura, R.
W. Impey, and M. L. Klein, J. Chem. Phys. 79, 926
(1983).
[26] D. van der Spoel, E. Lindahl, B. Hess, G. Groenhof, A.
E. Mark, and H. J. C. Berendsen, J. Comput. Chem.
26, 1701, (2005).
[27] T. Darden, D. York, and L. Pedersen, J. Chem. Phys.
98, 10089 (1993).
[28] H. J. C. Berendsen, J. P. M. Postma, A. DiNola, and
J. R. Haak, J. Chem. Phys. 81, 3684 (1984).
[29] J. R. Ryckaert, G. Ciccotti, and H. J. C. Berendsen, J.
Comp. Phys. 23, 327 (1977).
[30] J. Wang and J. Zhou, X. H. Lu, Phys. Chem. Chem.
Phys. 6, 829 (2004).
[31] K. H. Lee and S. B. Sinnott, Nano Lett. 5, 793 (2005).
[32] A. Luzar and D. Chandler, Nature 379, 55 (1996).
[33] X. Y. Zhou and H. J. Lu, Chin. Phys. 16, 335 (2007).
[34] R. J. Mashl, S. Joseph, N. R. Aluru, and E. Jakobsson,
Nano Lett. 3, 589 (2003).
c
°2012
Chinese Physical Society