Submitted to J Chem Phys
Effect of Nanotube-length on the Transport Properties of
Single-File Water Molecules: Transition from Bidirectional to
Unidirectional
Supplementary Material
Jiaye Su, Hongxia Guo*
Beijing National Laboratory for Molecular Sciences, Joint Laboratory of Polymer Sciences and
Materials, State Key Laboratory of Polymer Physics and Chemistry, Institute of Chemistry,
Chinese Academy of Sciences, Beijing 100190, China
*Author to whom correspondence should be addressed. Email: hxguo@iccas.ac.cn,
PS1. Effects of CNT length and field strength on the water dipole
orientations
In order to understand the structures and orientations of water molecules inside the CNT, we
have investigated the average dipole orientations at different CNT lengths and field strengths.
Figure S1 gives the distributions of the averaged dipole orientation of water molecules inside the
CNT with different L and E. Here,
 represents the angle between a water dipole (defined from
the oxygen atom to the center of two hydrogen atoms) and the tube axis, and the average is taken
over all water molecules inside the CNT. Note that the field strength of E=0.1, 0.5, 1.0 v/nm used
here, are larger than the critical value pointed out in our previous work,S1 which thus are strong
enough to maintain the water dipole along the field direction (+z) such that the flip events can not
occur. As shown in Figure S1, for a given length of L=2.63 nm, the peak position (most probable
angle) shifts to low values as E is increased, which is in fair agreement with the case of short
CNT.S1 For E=0.1 v/nm, the peak positions for L=1.34, 2.63, 5.13 nm are almost the same at
   =300, but the distributions become shaper and more intense as L increases. Longer CNTs
should be occupied by more water molecules, and hence the increasing hydrogen bond interaction
will reduce the rotational freedom of individual water molecules in the single-file chain, leading to
such narrow distributions.
1
Probability
Submitted to J Chem Phys
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0.12
0.10
0.08
0.06
0.04
0.02
0.00
E=0.1
E=0.5
E=1.0
L=2.63
L=1.34
L=2.63
L=5.13
E=0.1
0
20
40
60
80
100 120 140 160 180
<>
Figure S1. Probability distributions of the averaged dipole orientation of water molecules inside
the CNT,    . Top: E=0.1, 0.5, 1.0 v/nm, and L=2.63 nm. Bottom: L =1.34, 2.63, 5.13 nm, and
E=0.1 v/nm.
PS2. Water occupancy and its free energy of fluctuations
The filling/emptying of nonpolar cavities or nanopores in proteins and biological water
channelsS2-S4 can be well described by the water occupancy. Figure S2(a) shows the average water
occupancy inside CNTs as a function of the CNT length L for E=0.1, 0.5 and 1.0 v/nm. Clearly,
 N W  increases linearly with increasing L for all field strengths. It should be noted that the
slope of fitted lines slightly increases as E increases, which agrees well with the theoretical
analysis by Vaitheeswaran et al.S5 In fact, the water occupancy is almost the number of units (0.26
nm) of the CNT,S6 and indeed if we use L/0.26 as the reduced length the slopes become 1.017,
1.023 and 1.032, for E=0.1, 0.5 and 1.0 v/nm, respectively. To further understand the
filling/emptying of CNTs with different lengths, we calculated the free energy of occupancy
fluctuations shown in Figure S2(b) according to G( NW )  k BT ln p( NW ) , where p (N ) is
the probability of finding exactly NW water molecules inside the CNT.S7 The bottle-neck behavior
of the free energy will trap a given number of water molecules corresponding to the CNT length,
and thus large fluctuations of the water occupancy are forbidden. In fact, the strong
hydrogen-bonding interaction between water molecules ensures the filling of CNTs. Moreover,
2
Submitted to J Chem Phys
Hummer et alS6,S7 showed that the water-CNT interaction plays a dominant role for the
filling-emptying transition of CNTs.
E=0.1 slope=3.912+0.029
E=0.5 slope=3.936+0.006
E=1.0 slope=3.968+0.009
25
<Nw>
20
15
10
5
1
2
3
4
L (nm)
5
6
7
(a)
14
12
G(NW)/kBT
10
8
6
4
2
0
0
5
10
15
20
25
Nw
(b)
Figure S2. (a) The averaged number of water molecules inside CNTs as function of the CNT
length for E=0.1, 0.5, 1.0 v/nm. The three lines with different colors correspond to their linear
fitting. (b) Free energy of occupancy fluctuations G( NW )  k BT ln p( NW ) for E=0.1 (black),
0.5 (read) and 1.0 (blue) v/nm, and L =1.34 (square), 2.63 (circular), 5.13 (triangle) nm.
3
Submitted to J Chem Phys
PS3. Asymmetric water-water potentials and density profiles
It is known that the total charge of an individual water molecule is zero, and thus its external
force contributed from a homogeneous electric field should be also zero. To understand why water
molecules can be driven along the field direction we calculated the water-water interaction UWW
as a function of a water position along the CNT axis, shown in Figure S3. For E=0 (not shown) the
potential curve should be symmetric with respect to the CNT center (z=0), and no net water flux
can be generated, discussed in our previous study.S1 As E increases, the water dipole orientation
inside the CNT can be maintained along the field direction, leading to the symmetry-breaking of
the system. When the water dipoles are along the field direction (+z), the two hydrogen atoms of
each water molecule inside the CNT face the +z reservoir as well as oxygen atoms face the -z
reservoir, resulting in the asymmetric water-water potentials. This asymmetric water-water
interaction can drive water molecules along the field direction, and thus produces net water flux.
Figure S3 also gives the water density profiles along the CNT axis. The wavelike patterns with
five similar peaks for E=0.1 v/nm, shown in Figure S3(a), are in excellent agreement with
previous studies,S7 implying the unique arrangement of water molecules inside the CNT. This
wavelike density profile induced by the tight bonding network inside the nanochannel is directly
related to the ice or solid-like properties of confined water and may have potential applications in
mass storage.S4 For large field strengths of E=0.5 and 1.0 v/nm, the water molecules inside the
CNT will reorganize their correlated single-file structures to accommodate the strong electric field,
resulting in bias density profiles. It should also be noted that as the CNT length increases, shown
in Figure S3(b) and (c), the density profiles become less sensitive to the field strength as the
increasing hydrogen bond interaction is more competitive than the electric field. Meanwhile, as
the CNT length increases, the wave pattern of density profiles becomes illegible. Presumably, due
to the increasing hydrogen-bonding potential, the rotational freedom of individual water molecules
in longer single-file water chains should be reduced, and also the thermal fluctuations from the
two reservoirs is decreased. Thus the water arrangement inside CNTs becomes more steady,
leading to less wavelike patterns.
4
UWW (kJ/mol)
Submitted to J Chem Phys
-35
-40
-45
-50
-55
-60
-65
E=0.1
E=0.5
E=1.0
L=1.34
(z)/0
4
3
E
2
1
0
-2
-1
0
Z (nm)
1
2
UWW (kJ/mol)
(a)
-35
-40
-45
-50
-55
-60
-65
E=0.1
E=0.5
E=1.0
L=2.63
(z)/0
4
3
E
2
1
0
-3
-2
-1
0
Z (nm)
1
2
3
UWW (kJ/mol)
(b)
-35
-40
-45
-50
-55
-60
-65
E=0.1
E=0.5
E=1.0
L=5.13
(z)/0
4
3
E
2
1
0
-4
-3
-2
-1
0
1
Z (nm)
2
3
5
4
Submitted to J Chem Phys
(c)
Figure S3. The asymmetric water-water interaction as a function of a water position UWW and
water density profiles along the CNT axis for (a) L=1.34 nm, (b) L=2.63 nm and (C) L=5.13 nm
with E=0.1, 0.5 and 1.0 v/nm. The two doted lines in each represent the positions of the inlet and
outlet of the CNT.
PS4. Comparison with the continuous-time random-walk (CTRW)
model
Recently, Berezhkovskii and HummerS8 proposed a continuous-time random-walk (CTRW)
model to describe the concerted transport of single-file water molecules. Their model
quantitatively reproduces the MD simulation results.S7 The model predicts the bidirectional water
flow as f  k /( N W  1)  k / LR , where k is the hopping rate (1/13 ps-1), Nw is the water
*
*
occupancy, and LR is the reduced CNT length. Figure S4(a) shows the comparison between
water flows from our MD simulation and from the CTRW’s prediction. Clearly, the difference is
remarkable such that the CTRW model may be not suitable to predict the water flow for such a
biased transport in our present work. It seems that the decay of MD’s flow exceeds the CTRW’s
prediction. Actually, a similar discrepancy between the MD results and CTRW was previously
observed by Li et al without external fields.S9 In our manuscript, we have demonstrated that the
average translocation time of individual water molecules yields to  trans ~ L . Although the
v
water flow is not equal to 1 /  trans , it may scale as f ~ 1 /  trans , since for a given CNT the
longer time a individual water translocates through the CNT the lower flow should be. Thus we

could assume that the water flow still yields to a power law as f ~ 1 /  trans ~ L
, which can be
regarded as modified from CTRW model. Figure S4(b) presents the MD water flow fitted by such

a power law of f  aL
. Note that we only consider the bidirectional case (L<3.78 nm), as for
L≥3.78 nm the water flow is found to be a constant (~1.2 ns-1). As expected, the absolute value of
μ in Figure S4 is comparable to v in  trans ~ L
v
(listed in Figure3 in the main text). Following the

analogous procedure as in the main text, the number of N trans  f   trans  aL
6
 bLv  ab ,
Submitted to J Chem Phys
and considering the fitting constants of
a=33.8, 33.7, 34.4 ns-1, b=69.2, 81.3, 95.5 ps for E=0.1,
0.5, 1.0 v/nm, N trans =2.3, 2.7 and 3.3, respectively, which are in agreement with our simulation
results. Finally, we would like to suggest that the water flow for the present biased single-file
transport can be described by:
f  A0 exp(  L / L0 )  f 0 or
(S1)
f  aL  (L<3.78) and f=f0 (L≥3.78)
(S2)
Although two possible scaling laws are explored, the intrinsic physical mechanism should be
similar that the CNT-length effect is associated with thermal fluctuations of water molecules
outside the CNT.
16
E=0.1 CTRW
E=0.5 CTRW
E=1.0 CTRW
E=0.1 MD
E=0.5 MD
E=1.0 MD
14
10
-1
Flow (ns )
12
8
6
4
2
0
1
2
3
4
L (nm)
5
6
7
(a)
16
E=0.1 MD
E=0.5 MD
E=1.0 MD
E=0.1 Power Law
E=0.5 Power Law
E=1.0 Power Law
14
-1
Flow (ns )
12
10
8
6
4
2
0
1.2
1.6
2.0
2.4
L (nm)
2.8
7
3.2
Submitted to J Chem Phys
(b)
Figure S4. (a) Comparison of water flow from MD results and CTRW model, and (b) water flow
is fitted by a power law of f  aL  , where a=33.8, 33.7, 34.4 ns-1 and μ=2.543, 2.551, 2.642,
for E=0.1, 0.5, 1.0 v/nm, respectively.
Supplementary references
S1J.
Y. Su, and H. X. Guo, ACS Nano 5, 351 (2011).
S2A.
Kalra, S. Garde, and G. Hummer, Proc. Natl. Acad. Sci. USA. 100, 10175 (2003).
S3G.
Hummer, Mol. Phys. 105, 201 (2007).
S4J.
C. Rasaiah, S. Garde, and G. Hummer, Annu. Rev. Phys. Chem. 59, 713 (2008).
S5S.
Vaitheeswaran, J. C. Rasaiah, and G. Hummer, J. Chem. Phys. 121, 7955 (2004).
S6A.
Waghe, J. C. Rasaiah, and G. Hummer, J. Chem. Phys. 117, 10789 (2002).
S7G.
Hummer, J. C. Rasaiah, and J. P. Noworyta, Nature 414, 188 (2001).
S8A.
Berezhkovskii, and G. Hummer, Phys. Rev. Lett. 89, 064503 (2002).
S9J.
Y. Li, Z. X. Yang, H. P. Fang, R. H. Zhou, and X. W. Tang, Chin. Phys. Lett. 24, 2710 (2007)
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