Supplementary Information for:
Simulations of Structural and Dynamic Anisotropy in Nano-confined
Water between Parallel Graphite Plates
Hamid Mosaddeghi
Department of Chemistry, Isfahan University of Technology, Isfahan, 84156-83111,
Iran
Saman Alavia
Department of Chemistry, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada
M. H. Kowsari
Department of Chemistry and Center for Climate Change and Global Warming,
Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, 45137-66731, Iran
Bijan Najafi
Department of Chemistry, Isfahan University of Technology, Isfahan, 84156-83111,
Iran
a
e-mail: saman.alavi@nrc-cnrc.gc.ca
The relations between the MSD and VACF with self-diffusion coefficient
The relation of the MSD for molecules in unconstrained system is given by,
MSD 
1
N

r t   ri (0)
i 1 i
N
2
(S1)
where ri (0) is the initial position of species i and ri (t ) is its position at time t and N is
the number of diffusing molecules and the brackets   represent an ensemble average
over all time origins. Self diffusion coefficient of each system is proportional to the
long-time limit of the slope of the MSD plot using the Einstein equation,
Di 
1
d
2
lim
ri t   ri (0)
t


2d
dt
(S2)
where d is the dimensionality of the system,1 and Di is self diffusion coefficient of
species i. This formulation is written only for homogenous bulk fluids with no
constraints on motion in any direction.
The Green-Kubo relations are based on the time integrals of appropriate timecorrelation functions. The self diffusion coefficient can be calculated from the integral
of VACF using the Green-Kubo formula,2, 3
Di 
1 
v i (t ).v i (0) dt
d 0
(S3)
where d is dimensionality of the system, vi (0) is the initial velocity of species i and vi
(t) is the velocity at time t. The VACF in normalized form is defined by
VACF (t ) 
v i t .v i 0
(S4)
v i 0.v i 0
2
v i 0.v i 0  vi 2 0 
3k BT
and v0 being the thermal speed.4 The self diffusion
m
coefficient relation with free boundary conditions (d = 3) is defined as,
Di 
k BT
m


0
(S5)
VACF (t )dt
where m is mass of particle, kB is the Boltzmann constant and T is Kelvin temperature.
Normalized Radial Distribution Function
To better compare the position of the second RDF peaks in different systems,
we normalize the RDF curves with respect to the maximum peak heights as shown in
Figs. S2(a) to S2(d) . The normalized RDF is defined by
g (r ) normalized 
g (r )
g (r ) max
(S6)
Normalized RDF plots show the position of the second peaks move outwards
when going from H = 7 Å to H = 20 Å. The second peaks move to larger distances
(from approximately 4.5 Å to 6 Å) in gO-O(r) and (from 2.6 Å to 3.2 Å) in gO-H(r).
This is shown in Figs. S2(a) and S2(b) respectively. The positions of the second
peaks do not change with decreased temperature from 300 K to 50 K in SPC/E bulk
water. These results indicating the shift of second peaks may provide a method to
distinguish confined water from bulk water. Figures 6 and S2 show the same trend for
third peaks represented in S2(b) , but the trend is sharper for the second peaks. Other
authors reported same observations in previous works on confined water.5,6,7
Effect of confinement on the second peaks in the RDFs for various slit widths
Confinement has the effect of sharpening the second gO-O(r) and gO-H(r) peaks and
shifting them to shorter distances. This makes the confined water systems more solid-
3
like. To understand the displacement of the second peak, we investigated the
O···H···O angle, O···O and H···O radius between two water molecules in second
shell and observed that with increasing confinement, these quantities are reduced
which causes water molecules in second shell to become compact than bulk water, so
second peak is shifted to lower radius value and become sharper. In Fig. S2(b), gO-H(r),
at second peak in H = 7 Å we have two peaks with same intensity near another. The
position of second peaks with decreasing H comes back to the same position as bulk,
it means second peaks in bulk water and H = 7 Å located in same position ~ 4.5 Å
(see Fig. S2(c)).
When water density decreases from 0.8 to 0.6 g cm-3, the second peak of the
gO-O(r) RDF shift to a higher radius and becomes sharper In the case of H = 7 Å as
shown in Fig. S3, when water density decreases from 0.8 to 0.6 g cm-3, the lateral
pressure drops from 7.8 katm to 1.6 katm.
References:
1
M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, (Oxford Science
Publications, Oxford, 1987).
2
J. P. Hansen and I. R. McDonald, Theory of Simple Liquids, (Academic Press: New
York, 1986).
3
J. M. Haile, Molecular Dynamics Simulation Elementary Methods, (Wiley, New
York, 1992).
4
5
J. P. Boon and S. Yip, Molecular Hydrodynamics, (McGraw-Hill, New York, 1991).
P. Hirunsit and P. B. Balbuena, J. Phys. Chem. C 111, 1709 (2007).
6
A. K. Soper, F. Bruni, and M. A. Ricci, J. Chem. Phys. 109, 1486 (1998).
7
R. Zangi, and A. E. Mark, J. Chem. Phys. 119, 1694 (2003).
4
The ordered arrangement and higher density of water near the graphite surfaces are
clearly seen for all H values in the snapshots of Fig. S1. In the H = 20 Å snapshot,
there is no clear ordering of water layers in the center of the slit nanopore system as
observed in the 10 and 15 Å cases. For confined systems with H/σ > 5 where σ is
oxygen diameter, confined fluid behavior starts to become similar to the bulk fluid. In
the present simulations, this condition met after H > 16 Å and some bulk-like
properties begin to appear at H = 20 Å. The smallest possible pore width where the
fluid is highly confined occurs at H ~ 6.3 Å.
TABLE S1. Number of loaded water molecules in different systems.
System
Number of water molecules
ρ = 0.8 g cm-3
H=7Å
380
ρ = 0.7 g cm-3
332
ρ = 0.6 g cm-3
285
H = 10 Å
673
H = 15 Å
1010
H = 20 Å
1348
5
Figure S1. The snapshots of equilibrated systems for different confinement distances.
The oxygen atoms are shown by red spheres, hydrogen atoms by gray spheres, and
carbon atoms by dark gray spheres. The layered structures of the confined water can
be observed.
6
Figure S2. (a) The oxygen-oxygen RDFs and (b) the oxygen-hydrogen RDF for
different confinements; (c) the oxygen-oxygen RDF and (d) the oxygen-hydrogen
RDF for bulk water at 300, 265, and 50 K.
The RDFs for water at 300 K and the hypothetical supercooled water systems
at 265 and 50 K are given in Figs. S2(c) and S2(d). The location of first peak in the
RDF for the water systems do not change with decreasing temperature from 300 K to
50 K in SPC/E bulk water as shown in Figs. S2(c) and S2(d). Unlike the bulk water
systems, in Figs. S2(a) and S2(b), the position of the second maximum peaks moves
to larger radius values as the separation of the walls increases. We discuss further
details of the change in positions of the second peaks in the RDFs for various slit
widths in the Supplementary Information.
7
Figure S3. Normalized RDFs plots for (a) oxygen-oxygen for different confinements
at 300 K; (b) normalized RDF of oxygen-hydrogen for different confinements at 300
K; (c) oxygen-oxygen and (d) oxygen-hydrogen normalized RDFs in bulk water at
300, 265, and 50 K.
8
Figure S4. Traces show the motions of a selected water molecule from the water
layer adjacent to the graphite wall from several trajectories at different confinements.
The hydrogen atoms of the water molecule are shown by cyan and white spheres.
Oxygens are shown as red spheres. The translational mobility of the water molecules
in the H = 15 and 20 Å cases can be readily observed.
9
Figure S5. The ratio of calculated self diffusion coefficient in z-direction to the threedimensional and lateral (xy) self-diffusion coefficients.
10
Figure S6. The snapshots of Two systems with different geometric densities in H = 7
Å, with density 0.6 g·cm-3 (left) and 0.7 g·cm-3 (right).
11
(a)
(b)
Figure S7. (a) The effect of water geometric density on normalized RDF of oxygenoxygen and (b) the normalized RDF of oxygen-hydrogen for H = 7 Å.
12
Figure S8. Radial distribution functions of oxygen-oxygen in various geometric
densities for H = 20 and 7 Å.
13