Pumping of Confined Water in Carbon Nanotubes by Rotation-Translation Coupling
Sony Joseph and N. R. Aluru*
Department of Mechanical Science and Engineering, Beckman Institute for Advanced Science and Technology,
University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
(Received 1 August 2007; revised manuscript received 16 June 2008; published 6 August 2008)
Bidirectional single file water transport in a carbon nanotube is known to occur in ‘‘bursts’’ in short
nanotubes. Here we show that in long carbon nanotubes, when the orientation of the water molecules is
maintained along one direction, a net water transport along that direction can be attained due to coupling
between rotational and translational motions. The rotations of the water molecules are correlated more
with the translation of the neighboring water molecule with the acceptor oxygen than the neighbor with
the donor hydrogen. This mechanism can be used to pump water through nanotubes.
DOI: 10.1103/PhysRevLett.101.064502 PACS numbers: 47.61. k, 68.15.+e, 81.07.De
A rare combination of molecularly smooth walls and
hydrophobicity of the surface make carbon nanotube
(CNT) membranes a promising technology for drug delivery,
selective ion transport, desalination, nanofiltration, etc.
Water transport in carbon nanotubes has attracted considerable
attention over the years [1–5]. Though bidirectional
single file water transport in ‘‘bursts’’ through a 6; 6 CNT
[1] and collective intermittent reversing of water dipolar
orientations has been observed in molecular dynamics
(MD) simulation for short tubes, the molecular mechanism
governing the relation between the dipole orientation and
the flow direction has not been elucidated. In this Letter,
using molecular dynamics, we examine the molecular level
mechanism of transport of confined water through finite
length CNTs when the orientations are maintained along
one direction. The orientations can be maintained by applying
an electric field or by attaching chemical functional
groups at the tube ends. The interaction of a uniform
external electric field with molecules with net zero charge
is through the torques, which are exerted only on the rotational
degrees of freedom of the molecules. Any change in
the translational degrees of freedom in such a system is to
be attributed to the mechanism of coupling between rotational
and translational motions.
For the molecular dynamics simulations, the system
consisted of water molecules in a 9.83 nm long CNT
attached to a bath of size 3:2 3:2 1:6 nm3
at both
ends of the tube. Semiconducting CNTs of two different
diameters of 12.53 A˚ 16; 0 and 7.83 A˚ 10; 0 were
considered. The CNT was modeled as Lennard-Jones
atoms with parameters from Ref. [6]. Water was modeled
using the single point charge/extended model. Simulations
comparing polarizable models for CNTs with nonpolarizable
models do not show any appreciable difference for
water transport in semiconducting CNTs [7]. MD simulations
were performed with GROMACS 3.3.2 [8], using a time
step of 1 fs. The temperature of the fluid was maintained at
300 K by using a Nose-Hoover thermostat, and the pressure
was kept at 1 bar using an anisotropic ParrinelloRahman
barostat with compressibility along the Z direction.
Periodic boundary conditions were applied along X,
Y, and Z directions. The long-range electrostatic interactions
were computed by using a particle mesh Ewald
(PME) method, and the short-range interactions were computed
using a cutoff scheme. Three different external electric
field strengths (Eext 0, 0.1, and 1 V=nm) were
applied along the channel in the ve Z direction. To
consider the effect of screening by the CNT, an additional
case was considered where the field was screened inside
the CNT (ENT Eext= ) by using a dielectric constant .
For an infinitely long (3n 1; 0) CNT, 30 [9], but,
adjusting for a finite length assuming a similar trend as in
Ref. [10], we used an estimate of 24.
Figure 1 shows the cumulative flux of water molecules
passing through 10; 0 and 16; 0 tubes for the different
10 20 30 40 50 60
0
100
200
300
400
500
cumulativeflux(#)
5 10 15 20 25 30 35 40 45
0
500
1000
1500
t (ns)
cumulativeflux(#)
1.0 8.1±1.4
1.0(ε=24) 7.5±1.9
0.1 6.2±1.8
0.0 7.0±2.1
1.0 34.0±5.2
1.0 (ε=24) 15.3±4.7
0.1 16.5±6.5
0.0 0.0±0.0
Eext
(V/nm) Avg flux(#/ns)
E
ext
(V/nm) Avg flux(#/ns) (b) (16,0)
(a) (10,0)
Z
X
Y
Eext
z1 z29.83 nm
FIG. 1 (color online). The cumulative flux of water molecules
crossing the tube as a function of time for (a) 10; 0 and
(b) 16; 0 CNTs for various values of Eext.
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Eext cases considered. The cumulative flux at time t is
defined as the total number of water molecules that have
crossed one end of the tube that had previously entered
through the other end. It is defined as ve in the Z
direction. The average fluxes computed by averaging the
gradient of the cumulative flux in 10 ns intervals starting
from t 10 ns are also shown. We observe that (1) for the
10; 0 tube, the average water flux is similar regardless of
the Eext or screening effects, and (2) for the 16; 0 tube,
higher ve average fluxes are present when the applied
field is strong but decreases as the field decreases and
screening effects are included. The average flux in an
aquaporin channel ranges from 1–10 molecules=ns [11].
The 10; 0 tube has a single file water structure, and the
16; 0 tube has a structure with six files of water arranged
in a hexagonal ring. During the entire simulation time, the
direction of water dipoles was maintained in the ve Z
direction for the 10; 0 tube regardless of the field, but for
the 16; 0 tube the orientation along the ve Z direction
was maintained only when the field was applied. The mean
dipole orientation with respect to the ve Z axis hcos i,
where is the angle between the positive Z axis and the
water dipole vector, averaged over all water molecules in
the tube over the entire simulation time, was 0.83, 0.81,
0.74, and 0.00, for the 10; 0 CNT with E 1 V=nm, the
10; 0 CNT with E 0 V=nm, the 16; 0 CNT with E
1 V=nm, and the 16; 0 CNT with E 0 V=nm cases,
respectively. The net transport is clearly in the direction
of the aligned water dipoles. Single file water molecules
are known to preferentially orient along one direction and
intermittently reverse their orientations collectively in time
scales of 4–6 ns for short 1.3 nm CNTs [1,6]. For the
9.83 nm long 10; 0 CNT with E 0 V=nm case, the
orientations have not reversed their direction from the
initial orientation even after 60 ns. It is to be expected
that at longer time scales flipping would cause the orientation
to reverse for this case, so that there is a reversal in the
direction of flow and there is no perpetual motion without
any work being done. The time scales for flipping can be
expected to be much longer for a 9.83 nm single file chain
because of greater energy required to overcome a stronger
cooperative hydrogen bond chain.
To understand the water fluxes, we analyze the flux in a
10; 0 tube and then examine the differences as the tube
diameter increases. Figure 2(a) shows a 1.9 ps long trajectory
smoothed over a 0.1 ps window for water molecules in
the 10; 0 (E 0 V=nm) CNT. The thick ball and stick
models represent water molecules at the start of the trajectory,
and lines of the same color represent the trajectory of
each water molecule at intervals of 0.02 ps. A schematic of
the single water file is also shown in Fig. 2(b). The neighboring
water molecules undergo correlated rotational motions
such that the resultant trajectory has a helical
screwlike oscillatory motion. Note that the water dipoles
are not strictly aligned to the ve Z direction, but the
hydrogen-bonded OH vector [O-H1 in Fig. 2(b)] of the nth
water molecule is aligned with the dipole of the n 1 th
water molecule so as to minimize the energy of interaction
with the neighboring molecules. As the relative distance
between neighboring molecules changes, they undergo
rotational motions as well as transfer linear momentum
between them. In bulk water, these restricted rotational
motions known as librations do not cause any net translational
motion. In a long single file 10; 0 tube, or in the
16; 0 tube with electric field, under confinement, the
characteristics of the librational motions could be different,
and rotational-translational coupling (RTC) could give rise
to a net transport.
To understand the restricted rotational motions of single
file water, we analyze in Fig. 3 the velocity, angular velocity
autocorrelation functions (ACFs), and their crosscorrelation
functions (CCFs) for water molecules in the
10; 0 CNT and in the bath in the xyz-principal axes
[Fig. 2(b)] coordinate system where the moment of inertia
tensor is diagonal. The x axis is in the direction of the
dipole, the y axis is in the direction H1 to H2, and parallel to
H1-H2, where H1 is the hydrogen with a larger Z coordinate,
and the z axis is out of the H1-O-H2 plane normal to
the x and y axes. The velocity of a water molecule in a
moving frame xyz can be related to the simulation box
frame XYZ by three sets of direction cosines: vi
P
jRijVb
j , where j X; Y; Z are the simulation box axes,
i x; y; z are the moving frame axes, vi is the ith component
of the velocity vector v in the moving frame, and Vb
j is
FIG. 2 (color online). (a) Trajectory of water molecules in a
10; 0 (E 0 V=nm) CNT. Each color represents a water
molecule’s trajectory for 1.9 ps. The ball-stick models (various
colors) denote the position of water at t 0. (b) Schematic of a
single file water with body centered principal axes frame xyz and
the fixed box reference frame XYZ.
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the jth component of the simulation box velocity vector
Vb
. The Rij’s are the components of the rotation matrix
made up of the direction cosines. The velocity autocorrelation
function (v-ACF) can be written as hvi t vi 0 i
h
P
jRij t Vb
j t
P
kRik 0 Vb
k 0 i [Fig. 3(a)]. Along with the
v-ACFs along the three principal axes, also shown is the
v-ACF along the axial Z direction of the tube [dashed red
line in Fig. 3(a)]. Note that the x component of the v-ACF
is the one that is the closest to the v-ACF along Z, which
shows that the biggest contribution to the axial Z velocity is
from that along the x direction. The components !i of the
angular velocity vector ! in the principal frame can be
obtained by solving ! I 1L, where L
P
lml rl vl
is the angular momentum vector about the center of mass
in the principal axes reference frame, l O; H1; H2, rl is
the coordinate vector of atom l in the principal axis reference
frame whose origin is at the center of mass, ml is the
mass of atom l, I is the moment of inertia tensor in the
principal axes frame with nonzero diagonal elements computed
as Iii P
lml ri
l
2
, i x; y; z, and ri
l’s are the x, y,
and z coordinates in the principal axis reference frame. The
angular velocity autocorrelation (!-ACF) is defined as
h!i t !i 0 i, and the cross correlation (v-!-CCF) is defined
as hvi t !h 0 i, where i h x; y; z. In Fig. 3(b),
the ACFs of !x and !y have similar characteristics with a
superposition of a high frequency mode and a low frequency
mode that decays much slower. Note that !z has a
different frequency of oscillation than !x and !y.
Figures 3(c)–3(f) show the four nonzero CCFs that indicate
coupling between rotational and translational motions.
Also shown is the only nonzero CCF between the angular
velocity and the box frame velocity h!z 0 VZ t i
[Fig. 3(f)]. From Figs. 3(c) and 3(e), it is seen that linear
combination of a ve !x and a ve !y can be coupled to
a ve vz (or ve !y and ve !x coupled to ve vz) that
give rise to out-of-H1OH2-plane helical motions
[Fig. 2(a)]. The rotation !z can be coupled to velocities
vx and vy [Figs. 3(d) and 3(f)]. Of all the rotations !x, !y,
and !z, the one that gives a net translation along the Z
direction is !z, as the only nonzero CCF between angular
velocities and box frame velocities is h!z 0 VZ t i
[Fig. 3(f)]. From the v-ACF in Fig. 3(a), the v-ACF closest
to the v-ACF of VZ is that of vx, and the biggest contribution
to net axial velocity from RTC is h!z 0 vx t i.
Figure 4 has the cross correlations of h!z 0 vx t i for
16; 0 tubes with and without an electric field and in the
bath. As the field strength decreases to zero, the coupling
diminishes in the 16; 0 tube, and no net flux is generated
in 45 ns, since orientations are not maintained. In the bath,
the coupling h!z 0 vx t i is zero.
In the previous paragraph, we discussed the coupling of
the rotation of one molecule about its center of mass to its
translational velocity. In order to understand the coupling
between the rotation of one molecule and the translation of
the neighboring molecules, and to distinguish between net
translation in the ve Z direction and in the ve Z
direction, we plot the CCFs between the nth and the n
j th molecules where j 2; 1; 1; 2 for the 10; 0 tube.
(n 1) is the neighboring molecule with the acceptor
oxygen, and (n 1) is the neighboring molecule with the
donor hydrogen [Fig. 2(b)]. Figure 5(a) shows the CCF
between the axial velocities, and Fig. 5(b) has the CCF
between the angular velocity of a molecule and the axial
velocity of the neighboring molecule. From Fig. 5(a), it is
0 0.1 0.2 0.3 0.4 0.5 0.6
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
t (ps)
〈ω
z
(0)⋅v
x
(t)〉
(16,0) E=1 V/nm
(16,0) E=0 V/nm
Bath
FIG. 4 (color online). Cross correlations of angular velocity !z
with vx for 16; 0 tubes for E 1 V=nm, E 0 V=nm, and in
the bath.
0
0.05
0.1
0.15
v−ACF 〈vx
(0)⋅vx
(t)〉
〈vy
(0)⋅vy
(t)〉
〈vz
(0)⋅vz
(t)〉
〈V
Z
(0)⋅V
Z
(t)〉
−200
0
200
400
ω−ACF
〈ωx
(0)⋅ωx
(t)〉
〈ωy
(0)⋅ωy
(t)〉
〈ωz
(0)⋅ωz
(t)〉
−1
−0.5
0
0.5
v−ω−CCF
〈ω
y
(0)⋅v
z
(t)〉
−0.5
0
0.5
v−ω−CCF
〈ω
z
(0)⋅v
y
(t)〉
0 0.2 0.4 0.6
0
0.5
1
v−ω−CCF
t (ps)
〈ω
x
(0)⋅v
z
(t)〉
0 0.2 0.4 0.6
−0.3
−0.2
−0.1
0
0.1
0.2
v−ω−CCF
t (ps)
〈ωz
(0)⋅vx
(t)〉
〈ωz
(0)⋅VZ
(t)〉
(a) (b)
(c) (d)
(e) (f)
FIG. 3 (color online). Velocity autocorrelations [ nm=ps 2],
angular velocity autocorrelations [ rad=ps 2
], and nonvanishing
velocity-angular velocity cross correlations (nm=ps rad=ps) for
water molecules inside a 10; 0 tube for E 0 V=nm.
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seen that the CCF hVn
Z 0 Vn 1
Z t i is the same as
hVn
Z 0 Vn 1
Z t i and likewise for the CCFs hVn
Z 0 Vn 2
Z t i
and hVn
Z 0 Vn 2
Z t i. If the profiles for n 2 are shifted to
the left by t 0:05 ps, the locations of the peaks and
valleys match, suggesting a collision time of 0.05 ps between
neighbors. The linear momentum transfer between
the neighboring molecules is the same in both directions.
But in Fig. 5(b), the cross correlations between the rotations
of the nth and the axial velocity of n 1 th shows
a stronger correlation than that between the nth and the
n 1 th water molecule. The same trend is observed
for n 2 th and n 2 th molecules. The CCF
h!n
z 0 Vn 2
Z t i has similar features as h!n
z 0 Vn 1
Z t i
with a shift of t 0:05 ps. This shows a preferential
transfer of rotational energy of the nth molecule towards
the translation of n 1 th and n 2 th molecules as
compared to the opposite direction.
To check whether the phenomenon was caused by periodicity
in the Z direction, even with baths attached, we
performed NVT simulations with sufficient empty space
beyond the 4 nm long baths. Periodicity was applied only
along the X and Y directions using a 2D corrected PME [8].
We found that fluxes were in the direction of the electric
field (Eext 0:01, 0.1, and 1 V=nm), and the same ACFs
and CCFs as seen previously were obtained. For the 10; 0
tube with Eext 0 V=nm, the orientation flipped 3 times in
60 ns, and fluxes in both directions were obtained depending
on the direction of the orientation.
In order to ascertain whether this phenomenon is not a
result of entrance-exit effects, we also performed simulations
with infinitely long periodic tubes without baths at the
ends. The fluxes generated were in the direction of the
dipoles with the same correlations.
The cumulative fluxes do not increase uniformly with
time but in bursts in the 10; 0 tube [Fig. 1(a)]. The
variations in the fluxes are related to the fluctuations in
the occupancy of water in the tube [37–40 in the 10; 0
tube] from the fully filled state. At the highest occupancy,
the number of cooperative hydrogen bonds in the tube is
maximum, but, as occupancy decreases from the maximum
value, the hydrogen bond chain is broken and the increase
in cumulative flux is slower.
To maintain orientations without a large Eext, asymmetric
functional groups can be attached at the ends [12]. With
chemical functionalization, the mechanism of water transport
by RTC could pave the way towards developing ultraefficient,
next-generation nanofluidic devices.
This work was supported by NSF under Grants
No. 0120978 (Water CAMPWS, UIUC), No. 0325344,
No. 0328162 (nano-CEMMS, UIUC), and No. 0523435
and by NIH under Grant No. PHS 2 PN2 EY016570B.
*Corresponding author.
aluru@uiuc.edu; http://www.uiuc.edu/~aluru
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0
0.02
0.04
0.06
〈V
Z
n
(0)⋅V
Z
n±i
(t)〉 n−2
n−1
n+1
n+2
0 0.1 0.2 0.3 0.4 0.5 0.6
−0.2
0
0.2
〈ω
z
n
(0)⋅V
Z
n±i
(t)〉
t (ps)
(a)
(b)
FIG. 5 (color online). (a) Cross correlation of the axial velocity
of the neighboring n i th water molecule with axial velocity
of the nth water molecule for the 10; 0 tube. (b) Cross correlation
of the axial velocity of the neighboring n i th water
molecule with the out-of-plane angular velocity !z of the nth
water molecule. Coupling with (n 1) and (n 2) are greater
than that with (n 1) and (n 2) showing preferred transfer of
rotational energy to translational energy along the ve Z
direction.
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