Journal of Membrane Science 458 (2014) 236–244
Contents lists available at ScienceDirect
Journal of Membrane Science
journal homepage: www.elsevier.com/locate/memsci
Structure and dynamics of water confined in a polyamide
reverse-osmosis membrane: A molecular-simulation study
Minxia Ding a, Anthony Szymczyk a, Florent Goujon b, Armand Soldera c, Aziz Ghoufi d,n
a
Institut des Sciences Chimiques de Rennes, CNRS, UMR 6226, Université de Rennes 1, 35042 Rennes, France
Institut de Chimie de Clermont-Ferrand, ICCF, UMR CNRS 6296, BP 10448, F-63000 Clermont-Ferrand, France
c
Department of Chemistry, Université de Sherbrooke, Sherbrooke, Québec, Canada J1K 2R1
d
Institut Physique de Rennes, UMR CNRS 6251, Université Rennes 1, 35042 Rennes cedex 05, France
b
art ic l e i nf o
a b s t r a c t
Article history:
Received 21 October 2013
Received in revised form
20 January 2014
Accepted 25 January 2014
Available online 4 February 2014
Molecular dynamics simulations were carried out to investigate both the structural and dynamical
properties of water trapped inside a highly cross-linked polyamide RO membrane. The heterogeneous
structure of the membrane was characterized through local water density and cavity size distributions.
Interactions between water molecules and the polyamide membrane were investigated. Water structure
and dynamics were explored and correlated with the heterogeneous distribution of the free volumes
inside the membrane.
& 2014 Elsevier B.V. All rights reserved.
Keywords:
Molecular dynamics simulations
Polyamide
Reverse-osmosis
Water
1. Introduction
The availability of potable water has become nowadays a global
problem due to the continuous growth in water demand not
balanced by an adequate recharge. The United Nations predict
that by 2025, two-third of the world's population will live in areas
of significant water stress, lacking sufficient safe water for drinking, industry or agriculture [1]. Some methods to increase water
supply beyond what is available from the hydrological cycle are
desalination and water reuse [2]. Membrane separation processes
are recognized worldwide as promising tools for addressing the
growing concern about water availability in a process intensification strategy, i.e. by developing methods aiming at decreasing raw
materials utilization, energy consumption, equipment size, and
waste generation [3]. Reverse osmosis (RO) process is particularly
well suited for desalination and it is now the leading desalination
technique used worldwide. This pressure-driven process makes
use of thin film composite membranes made up of three layers: an
ultra thin (100–300 nm) and relatively dense polyamide active
layer that controls the separation performances of the membrane,
an intermediate mesoporous polysulfone layer (30–50 μm) and a
polyester backing material (100–200 μm) providing mechanical
n
Corresponding author.
E-mail address: aziz.ghoufi@univ-rennes1.fr (A. Ghoufi).
http://dx.doi.org/10.1016/j.memsci.2014.01.054
0376-7388 & 2014 Elsevier B.V. All rights reserved.
strength to the membrane [4]. Currently, the active layer of most
RO membranes used for desalination purposes is made from
interfacial polymerization between meta-phenylene diamine
(MPD) (Fig. 1a) and trimesoyl chloride (TMC) (Fig. 1b), which leads
to the formation of a very thin and dense layer of fully aromatic
polyamide (Fig. 1c) [4–6]. Despite years of intense research on the
transport of water and solutes through RO membranes, the
physical phenomena that control transport through the active
layer are not yet fully understood, particularly at the atomistic
level. The molecular dynamics (MD) technique is a potential
method for gaining insights into the polymeric membrane characteristics, including the polymer configuration, free volume, and
transport phenomena at a microscopic scale [7–10]. This powerful
tool has already been applied extensively to investigate water and
ion transport through model porous systems such as e.g. carbon
nanotubes [11] or mesoporous and microporous silica [12–15].
However, the limited informationabout the three-dimensional
molecular structure of RO polyamide membranes makes the use
of this computational method much more challenging. Indeed, a
prerequisite for MD simulations is to have a complete threedimensional atomic model of the polyamide active layer. That is
challenging because the cross-linking of the MPD and TMC
monomers is random and the resulting polymer matrix is highly
disordered [16]. Consequently, the identification and understanding at a microscopic level of the underlying mechanisms of water
and ion transport through RO membranes is proceeding very
M. Ding et al. / Journal of Membrane Science 458 (2014) 236–244
237
Fig. 1. Illustration of the polymerization of meta-phenyldiamine and benzene-1,3,5-tricarboxylic acid chloride to produce polyamide: (a) component: meta-phenyldiamine
(MPD) molecule; (b) component: benzene-1,3,5-tricarboxylic acid chloride (TMC) molecule; (c) product: polyamide with degree of cross-linking n (n has value in between
0 and 1, n¼ 1 for fully cross-linked polyamide and n¼ 0 for fully linear one).
slowly and only few studies reported on the attempt of building
full atomistic models of RO polyamide membranes [16–20]. In the
pioneer work of Kotelyanskii et al. [17,18], a single chain with 62
repeat units was generated and further cross-linked artificially by
connecting some TMC and MPD fragments. A decade later, Hughes
and Gale [20] succeeded in building a fully aromatic linear
polyamide composed of 24 chains of 23 repeat units each. Once
equilibrated, the un-crosslinked polymer was cross-linked according to the procedure followed by Kotelyanskii et al. In both cases
the degree of cross-linking achieved was less than 20%. Harder
et al. [19] and Luo et al. [16] used a different methodology, starting
from TMC and MPD monomers which were polymerized during
the course of the simulation on the basis of a distance criterion
between the nitrogen of a free amine group and the carbonyl
carbon of a free acyl chloride group. This procedure resulted in a
ratio of cross-linked to linear segments of 37:63 [19]. The crosslinking degrees obtained in the above-mentioned computational
studies appear, however, quite low with respect to those of actual
RO polyamide membranes. Indeed, recent experimental results
obtained from either X-ray photoelectron spectrometry (XPS) [6,21–
23] or Rutherford backscattering spectrometry (RBS) [24,22] suggest
that the percentage of cross-linked repeat units is in the range 60–
100%.
In this work we used a previously developed approach [25] to
build realistic all-atom models of highly cross-linked RO membranes. In our previous work we discussed on the overall translation dynamics, the hydrogen bonds per water molecule, the
water–water interactions and the dielectric permittivity of water
[25]. In this work, we accurately investigate the interactions
between water molecules and the polyamide membrane, the
structure of confined water and the mechanism of water transport.
2. Computational details
2.1. Polyamide membrane construction
As mentioned previously, most current reverse osmosis membranes used for desalination purposes are synthesized from polymerization between TMC and MPD monomers (see Fig. 1). In the
first stage of our construction protocol several linear chains of
polyamide are packed into a simulation box according to a wellestablished method combining Monte Carlo (MC) and molecular
dynamics (MD) simulations and yielding a fairly relaxed linear (i.e.
un-crosslinked) polymer [26]. An additional MD simulation is then
performed by randomly adding a given number of MPD monomers
inside the free volumes of the un-crosslinked network. At the end
of this simulation the polymer chains are cross-linked artificially
by bridging free carboxylic acid groups on the polymer chains and
some of the added MPD monomers on the basis of a heuristic
distance criterion. Proceeding this way the degree of cross-linking
can be easily controlled by tuning the number of additional MPD
monomers inserted in the simulation. Thus our cross-linking
method is based on two stages (i) construction of a linear
polyamide and (ii) cross-linking of the linear structure.
Studying fluid transport across the membrane involves a
consideration of the explicit polyamide/water interface. Therefore
we began to build a 2D periodic linear polyamide membrane by
removing the periodic boundary conditions (PBCs) along the z
direction during the construction. The generation of the uncrosslinked polymer was made from the Amorphous Cell package
of Material Studio software [27]. The model building uses a
modified Markov process with biased conformational probabilities
chosen to account for both intramolecular and intermolecular
non-bonded interactions [26]. Thus, an orthorhombic cell containing five polymer chains of 50 monomer units each was constructed under 2D periodic boundary conditions. Afterwards, a
standard minimization protocol was used to minimize the energy
structure via a Polak–Ribiere algorithm. Kotelyanskii reported that
the experimental density of a commercial polyamide membrane
(FT30) made from MPD and TMC monomers was 1.38 g/cm3 in the
hydrated state with 23 wt% water content at ambient conditions
[17]. In our simulations the target density of the un-crosslinked
membrane was set at 1.0 g/cm3. This lower initial density allows
increasing the insertion probability of MPD monomers during the
subsequent cross-linking process. The initial configuration was
built with two water reservoirs surrounding the polyamide
membrane. Details of the MD procedure will be discussed in
the computational procedure section (Section 2.4). Fig. S1 of the
supporting information (SI) shows the initial configuration of the
atomistic model of the un-crosslinked polyamide network confined along the z-axis. It contains 5 identical chains containing 50
repeat units each. The final dimensions of the orthorhombic cell
were Lx ¼Ly ¼38.3 Å and Lz ¼80 Å.
As shown in Fig. S2 of the SI the construction of the crosslinked polyamide is based on a random insertion of 250 MPD
molecules into the un-crosslinked polyamide network. In order to
model explicitly the water/polyamide interface we surrounded the
un-crosslinked polyamide network by two water reservoirs along
the normal z-direction of the interface (Fig. S2). Since water
molecules can diffuse into the polymeric matrix and are likely to
modify the interactions between MPD monomers and the polyamide chains, we checked that the chemical nature of the
molecules inside the external reservoirs did not impact the
cross-linking process. That was done by performing another
simulation in which the external reservoirs were filled with MPD
monomers instead of water molecules. No impact of the chemical
nature of the reservoir-filling molecules was observed (actually,
only a very small amount of these molecules diffuse into the
polyamide network during the cross-linking process). Molecular
dynamics simulation of 2 ns was performed in NpT ensemble at
T¼ 300 K (both the force field and the computational procedure
are described below). Afterwards, the Carbon–Nitrogen (C–N)
radial distribution function (RDF) was estimated between nitrogen
atoms of the added MPD molecules and the carbon atoms of the
pending carboxylic acid groups on the polyamide chains. From
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M. Ding et al. / Journal of Membrane Science 458 (2014) 236–244
these RDFs we determined the most probable distance between
these C and N atoms, dp. Our cross-linking protocol consisted in
the calculation of all dC–N so that if the distance between the
carbon and nitrogen atoms was smaller than dp an artificial
‘bridge’ (covalent bond) was built between these atoms while a
hydrogen atom of the amine group of the MPD monomer as well
as the –OH group of the carboxylic acid function were removed
(see illustration in Fig. S3a of the SI). Let us note that in our
construction protocol only the MPD monomers for which the two
amine groups satisfied the cross-linking criterion were allowed to
‘react’ with the pending carboxylic acid functions. This procedure
and the chains entanglement led to both inter-chains and intrachains cross-linking (Fig. S3b). It is worth noting that the ‘speared
rings’ cases must be checked carefully in order to avoid the
unrealistic formation of bridges through aromatic rings. In our
work, that was done by checking the existence of an intersection
point between the artificial bridge and the plane of the aromatic
ring. From the radial distribution function between C and N atoms
we found dp ¼3.1 Å. However, the cross-linking criterion had to be
set to a larger value so as to increase the degree of cross-linking.
From the above-described protocol we obtained a 3D crosslinked membrane with a cross-linking degree of 80.8% by setting
dp to 7.6 Å (102 out of the 250 MPD monomers met this crosslinking criterion). At the end of the cross-linking process the
unreacted MPD monomers were removed. It can be noted that
the cross-linking degree obtained in this work is much higher than
those reported in previous works of Harder et al. [19] and Gale
et al. [20] and seems to be in line with experiments performed
with RO polyamide membranes. Indeed, recent experimental
results obtained from either X-ray photoelectron spectrometry
(XPS) [6,21–23] and Rutherford backscattering spectrometry (RBS)
[24,22] suggest that the percentage of cross-linked repeat units for
RO polyamide membranes is in the range 60–100%.
2.2. Partial charges calculation
Calculation of the partial charges of the polyamide membrane
was carried out from DMol3 [27] by considering the repeat unit
shown in Fig. S4 of the SI. The gradient-corrected correlation
functional of PW91 (Perdew/Wang 91) was used. Additionally, the
double-ξ numerical polarization (DNP) basis set was adopted to
account for the d-type for heavier atoms and p-type polarization
for hydrogens atoms. This basis is similar to the 6–31G(d,p)
Gaussian-type basis set. In the simplest electrostatic model, atom
centered charges were derived to reproduce the molecular electrostatic potential (ESP charges) using a fitting procedure. We
chose this basis in order to be in line with the original parametrization of the AMBER force field. The partial charges were
calculated from the Mulliken population analysis where the ESP
charges are checked i.e. that atomic centered charges that best
reproduce the DFP electrostatic potential. The cleaved bonds of the
considered cluster in Fig. S4 of the SI were saturated by methyl
groups to maintain a standard hybridization. The partial charges of
the linear polyamide are reported in Fig. S4 of the SI while the
charges of MPD in cross-linked process and the cross-linked
polyamide are given in Fig. S5a and b of the SI, respectively.
2.3. Potential
The polyamide material was described from AMBER99 force
field (Assisted Model Building with Energy Refinement) [28]
developed from AMBER94 force filed [29] and re-parameterized
by Wang et al. [28] for molecular dynamics of biomolecules and
many organics: proteins and nucleic acids. Water was modeled
with the non-polarizable rigid TIP4P/2005 model [30]. The total
configurational energy U is defined as U ¼ U INTRA þ U INTER where
U INTRA and U INTER are the intramolecular and intermolecular
contributions, respectively. The intramolecular interactions
include contributions from stretching, bending, torsion energy
and non-bonded LJ interactions. The non-bonded Lennard-Jones
intramolecular interactions were considered between atoms separated by more than three bonds. The bonds between hydrogen and
carbon atoms and oxygen and hydrogen atoms of hydroxide
groups were kept constant. Electrostatic contribution between
atoms separated by more than three bonds was calculated from
the Ewald summation [31]. The intermolecular interactions are
composed of repulsion–dispersion and electrostatic contributions
that are represented by Lennard-Jones (LJ) and Coulombic (ELE)
potentials, respectively. The LJ interactions were truncated at 12 Å.
The electrostatic interactions were calculated using the Ewald sum
method [31]. The reciprocal space sum was truncated at an
max
ellipsoidal boundary at the vector jh j where the reciprocal
lattice vector h is defined as h ¼ 2πðl=Lx u; m=Ly v; n=Lz wÞ where u,
v, w are the reciprocal space basis vectors and l, m, n take values of
0; 7 1; 7 2; …; 7 1. The convergence factor α was calculated from
2π=Lx . The maximum reciprocal lattice vectors parallel to the
max
max
max
surface jhx j and jhy j were fixed to 7 and jhz j to 49. The
max
increase in jhz j is required to account for the long range
electrostatic interactions accurately in the direction normal to
the interface due to a larger dimension box in this direction.
2.4. Computational procedure
As shown in Fig. 2 the cross-linked membrane was surrounded
by two water reservoirs. Box lengths are reported in Fig. 2.
Considering the water content reported in the literature for an
extensively used RO polyamide membrane (FT-30 from Dow
Filmtech) under standard conditions [17] the polyamide membrane was hydrated to obtain a water uptake of 23 wt% [17].
Periodic boundary conditions were applied in the three directions.
MD simulations were performed in the NpT statistical ensemble.
MD simulations were performed at 300 K and 1 bar using a time
step of 0.001 ps to sample 30 ns (acquisition phase). The equilibration time corresponds to 10 ns. All MD simulations were
Fig. 2. Snapshot of the initial configuration of hydrated polyamide surrounded by two water reservoirs. Carbon atoms are in black, hydrogen atoms in white, oxygen atoms in
red and nitrogen atoms in blue. The dashed yellow lines represent the extremities of the PA membrane. The origin of box is located at the center of the box. (For
interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
M. Ding et al. / Journal of Membrane Science 458 (2014) 236–244
239
carried out with the DL_POLY package [32] using a combination of
the velocity-Verlet algorithm [31] and the Nose–Hoover thermostat and barostat [31,33]. The bonds between hydrogen and carbon
atoms of –CH and oxygen and hydrogen atoms of hydroxyl groups
were kept constant from the SHAKE algorithm [34]. In this work
we fixed the water content in the matrix to the experimental
value. Let us note that the usual GCMC simulations failed to model
the real density of the matrix because this latter is maintained
rigid with this method. Indeed, the structure of the polyamide is
too complex to be efficiently sampled in Monte Carlo even with
the statistical bias [31,35]. The III method [36] is an interesting
alternative to compute the equilibrated water content in PA.
However, the diffusion time of water across the interface is too
long to be captured in a reasonable time.
3. Results and discussion
3.1. Membrane properties
The density of the hydrated polyamide membrane was first
computed and a value about 1.32 g/cm3 was found. This result is in
good agreement with the experimental value of 1.37 0.1 g/cm3
reported for the commercial FT-30 polyamide membrane (with a
water content of 23 wt%) under standard conditions [17]. It is thus
argued that our force field parameters are accurate enough to
carry out MD simulations. Additionally, we report in Fig. 3a the
profile of the number of water molecules along the z-axis. We
show that the polyamide matrix is correctly sampled by the water
molecules. Furthermore, we report in Fig. 3a the profile of the free
energy ðFðzÞ ¼ kB T log ðρpH2 O ðzÞ=ρbH2 O ÞÞ where kB is the Boltzmann
constant, T the temperature, ρpH2 O ðzÞ the local water density in the
polyamide matrix and ρbH2 O the water density in the bulk phase.
As shown in Fig. 3a the difference in free energy between the bulk
and the polyamide matrix is lower than kBT, which can explain the
transport of water across the polyamide framework. The cavity
size distribution inside the membrane was computed by means of
the method proposed by Bhattacharya and Gubbins [37], i.e. cavity
sizes were estimated by probing local free space using a particle
with a specified radius size. In our work the probe size was set at
1.4 Å. Fig. 3b shows the distribution of cavity radii inside the
hydrated cross-linked membrane. The cavity size distribution was
calculated between z ¼25 Å and z¼ 25 Å. The average cavity
radius was found to be around 2.5 Å, which is in very good
agreement with experimental data reported for fully aromatic
polyamide membranes. Indeed, average cavity diameters between
5.1 and 5.3 Å have been recently reported from SAXS experiments
for polyamide RO membranes in the hydrated state [38]. Therefore
our numerical results are in fair agreement with experiments,
which validates our force field parameters and our construction
methodology of RO membranes. We report in Fig. 3c the average
cavity size as a function of the z coordinate. As shown in Fig. 3c we
observe that the average cavity size increases close to the membrane/reservoir interface.
The porosity of the polyamide membrane can be highlighted
from 2D maps of water density. We report in Fig. 4 the 2D
distribution of water in the xy directions. The low-intensity zones
represent the regions occupied by the polyamide. As the mean
cavity diameter is about 5 Å we provide 5 profiles according to x
and y directions for slabs of 5 Å in thickness. Fig. 4 shows that the
distribution of water molecules (and so the cavity size distribution) into the polyamide framework is highly heterogeneous.
The size of water-rich zones appears to be similar through the
different zones, which corroborates the results shown in Fig. 3c,
i.e. the cavity size is almost constant between 25 and 25 Å.
Fig. 3. (a) Profiles of water density (left axis) and free energy (right axis) along the
z-axis. Horizontal grey solid line is the water density in the bulk phase. Green
dotted line represents the kBT value at 298 K. (b) Cavity radius size distribution in
the polyamide membrane. 〈r c 〉 is the average cavity radius. (c) Average cavity size as
a function of z coordinate between z¼ 45 Å and z¼ 45 Å. (For interpretation of the
references to color in this figure caption, the reader is referred to the web version of
this paper.)
3.2. Interactions and water structure
Interactions between water molecules and the polyamide
membrane were explored from the radial distribution function
(RDF), g(r). g(r) between water and polyamide atoms are reported
in Fig. 5. Due to excluded volume effect gðrÞ a 1 at long distances
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M. Ding et al. / Journal of Membrane Science 458 (2014) 236–244
Fig. 4. 2D distribution of water density in xy directions in 5 slabs of 5 Å in thickness along the z direction from z ¼0 Å (left side) and z ¼25 Å (right side).
Fig. 5. (a) Radial distribution function between hydrogen atoms of water (HW) and nitrogen and oxygen atoms of PA (NPa and OPa, respectively). (b) Radial distribution
function between oxygen atoms of water (OW) and hydrogen atoms bonded to the nitrogen and oxygen atoms of PA.
Fig. 6. Radial distribution function between hydrogen atoms of water (HW) and the different oxygen atoms of PA (O1, O2, Oh). See scheme on the right for label meaning.
[39]. Fig. 5a shows interactions between the hydrogen atoms of
water (HW) and the oxygen and nitrogen atoms of polyamide (OPa
and NPa, respectively). RDF between HW and OPa shows that the
shortest distance is 2.0 Å (maximum of the first peak), which
corresponds to the formation of hydrogen bonds. This trend is
born out from the RDF between the oxygen atoms of water (OW)
and the hydrogen atoms of –COOH groups of PA (HOPa). Indeed,
Fig. 5b shows that the distance between OW and HOPa (2.0 Å) is
smaller than the one between OW and HNPa (2.4 Å). These results
show that the preferential sites of interactions are the oxygen
atoms of carboxylic acid and amide groups and the hydrogen
atoms of carboxylic acid groups. In order to discriminate the
different oxygen atoms of polyamide we report in Fig. 6a the
RDF between the oxygen atoms of –COOH and –CON groups
and HW. The same hydration shell radius is obtained, whatever
the oxygen atom of the membrane. The RDF between oxygen (OW)
and hydrogen (HW) atoms of water in bulk phase and in hydrated
matrix are reported in Fig. 7a and b. The location of the first and
second peaks in the confined phase is similar to those of the bulk
phase. Indeed, in bulk and confined phases the first and second
shells were found at 3.3 and 5.7 Å respectively. That suggests a
slight impact of confinement on the strength of interactions
between water molecules and their neighbors in the first and
second hydration shells. Even though the distance of short-range
interactions are similar in bulk and in confined phases, intensities
of RDF are different (see Fig. 7a and b). This suggests that the
number of coordination of water molecules and the number of
hydrogen bonds per water molecule are different in bulk and
confined phases. We report in Fig. 7c the integration of RDF
between the oxygen atoms of water molecules (OW) to calculate
M. Ding et al. / Journal of Membrane Science 458 (2014) 236–244
the coordination number of water molecules (nc). As shown in
Fig. 7c the coordination number of water molecules drastically
decreases from 4:4 in the bulk phase to 2:4 inside the
membrane. That suggests a decrease in the number of hydrogen
bonds (HB) per water molecule.
The HB number was calculated from the Chandler geometrical
criterion [40], i.e. the distance between the oxygen and hydrogen
atoms of two different water molecules should be less than 2.5 Å,
the distance between oxygen atoms less than 3.5 Å and the angle
between the oxygen–hydrogen bond of a water molecule and the
vector linking the oxygen atoms of two different water molecules
should be less than 301. We report in Fig. 8a the profile of the
number of HB per water molecule along the z-axis. Fig. 8a clearly
shows that the number of HB per water molecule in the polyamide
membrane (2.8) is lower than in the water reservoirs (3.9)
surrounding the membrane. If we add the number of HB formed
between the polyamide membrane and the water molecules, the
number of HB increases from 2.8 to 3.2 per water molecule, which
is still lower than the number of HB in bulk water. In bulk phase
we observed that water molecules with 3 and 4H-bonds represent
29.1% and 70.9% of the total number of water molecules, respectively. These percentages were calculated from nαw =nTw with α¼3 or
4 where nαw is the number of water molecules with 3 or 4 HB and
nTw corresponds to the total water number. Inside the membrane,
the amount of water molecules forming 4 hydrogen bonds sharply
falls to 4.95% of the total number of water molecules due to the
very small size of cavities. On the other hand, the number of water
molecules forming 3 HB increases up to 74.3% while 20.7 % of
water molecules form at most 2 HBs. This decrease in the average
number of HB per water molecule suggests a modification of the
tetrahedral rearrangement of water molecules inside the membrane. Fig. 8b shows the water density, the number of HB per
water molecule and the average cavity size profiles according to
the z-axis. It clearly puts in evidence the correlation between the
number of water molecules and HB with the cavity size
distribution.
The arrangement of water molecules was studied from the
tetrahedral order parameter (q) defined as [41] q ¼ 1 38
∑3i ¼ 1 ∑4j ¼ i þ 1 cos ðϕij þ 13 Þ where ϕij is the angle between i and j
molecules. For a tetrahedral structure q is close to 1. We report in
Fig. 8c the tetrahedral order parameter of water in both bulk and
confined phases. Inside the polyamide membrane we considered
the hydrogen bonds between water molecules and between water
molecules and the hydrogen and oxygen atoms of polyamide.
As shown in Fig. 8c the most probable value of q in bulk water is
0.83, which indicates (as expected) that water molecules tend to
organize themselves according to a pseudo-tetrahedral arrangement. Inside the membrane, the most probable value of q is 1,
which means that the small amount of water molecules having
four neighbors optimize their tetrahedral arrangement. We report
in Fig. 8d the size distribution of water clusters. Cluster size was
calculated from an adaptation of Stoddard's algorithm [42,43].
Fig. 8d shows that the average size is around of 575 water
molecules, i.e. 93% of the water molecules (calculation was made
between z ¼25 Å and z ¼ 25 Å). A snapshot of the HB network is
provided in Fig. S6 of the SI. However 7% of water molecules are
trapped in cavities and do not participate to the main HB network.
To check the presence of these small water clusters or individual
water molecules we computed the partial structure factor [39,43–
45] of the oxygen atoms of water
SðQ Þ ¼
D
E
N
∑N
j ¼ 1 ∑k ¼ 1 expð iQrj ÞexpðiQrk Þ
N
ð1Þ
where Q is the momentum transfer vector, N the number of water
molecules, rj and rk are the position vectors of the oxygen atoms of
241
Fig. 7. Radial distribution function between (a) hydrogen and oxygen atoms of
water and (b) between oxygen atoms of water in bulk (right axis) and in confined
(left axis) phases. (c) Coordination number of water molecules (nc). nc was
calculated from the integration of the radial distribution functions shown in (b).
The legends for (b) and (c) are the same as that of (a). In (c) the vertical dotted blue
line represents the position of the first shell of hydration of water (minimum of the
first peak in figure (b)). (For interpretation of the references to color in this figure
caption, the reader is referred to the web version of this paper.)
the water molecules j and k, respectively, and the brackets stand
for an ensemble average. We normalized S(Q) by the number of
water molecules (N) so as to compare bulk and confined phases.
We report in Fig. 9a and b the partial structure factor of the oxygen
atoms of water in both bulk and confined phases. In the confined
phase the ascent in S(Q) at small Q ðQ o31A 1 Þ in confined phase
is characteristic of the polyamide porosity. For Q higher than
1.5 Å 1 S(Q) is identical for both bulk and confined phases. That
shows that the short range interactions are similar and corroborate the RDF results shown in Fig. 7a and b. However a prepeak is
observed for confined water at Q¼0.5 Å 1. This latter can be
interpreted as the consequence of the long range interactions
between individual water molecules or water clusters [43].
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M. Ding et al. / Journal of Membrane Science 458 (2014) 236–244
Fig. 8. (a) Profile (along the z-axis) of the number of hydrogen bond (HB) per water molecule (i) between water molecules themselves and (ii) between water molecules and
the polyamide membrane. The horizontal solid line represents the number of HB per water molecule in the bulk liquid phase. (b) Profile (along the z-axis) of (i) the number
of HB per water molecule, (ii) the number of water molecules and (iii) the cavity size. (c) Tetrahedral order parameter of water in the bulk liquid phase and into the PA
membrane. (d) Cluster size distribution of water trapped in the PA membrane. In our calculation clusters are linked by HB.
Fig. 9. (a) Partial structure factor of the oxygen atoms of water molecules in bulk and confined phases. (b) is an enlargement at small Q.
Fig. 10. (a) Spatial self Van Hove function between 1 and 8 ps. (b) Distribution of the variation of the angle Φ (defined as the angle between the O–H bond of water molecules
and x-axis) between 0 and 2 ps in bulk and confined phases.
M. Ding et al. / Journal of Membrane Science 458 (2014) 236–244
243
to 8 ps. The maximum of the Van Hove function remains at a
distance less than 1 Å from 1 to 8 ps, which indicates small
displacements and an absence of a translational jump into the
polyamide matrix. From a rotational standpoint we report in
Fig. 10b the distribution of the variation of angle Φ defined as
the angle between the O–H bond of water molecules and a fixed
axis (x-axis) between 0 and 2 ps in bulk and confined phases. In
the bulk phase the maximum of the angular distribution is located
at 481 while in the confined phase the maximum is shifted toward
271. This decrease is due to confinement inside the polyamide
membrane, which hinders rotation of water molecules. Nevertheless, as in bulk phase, a non-negligible amount of confined
water molecules exhibits rather large rotational jumps (see
Fig. 10b). Laage and Hynes have shown that these rotational jumps
are a prerequisite for HB formation in bulk phase [47] and this
conclusion probably holds under nanometric confinement too.
Translational diffusion of water can be evaluated from the
calculation of the self-diffusion coefficient Ds computed from the
mean square displacement (MSD) of the center of mass (com) of
water molecules as
2
∑t 0 ∑N
i ¼ 1 ½r com;i ðt þ t 0 Þ r com;i ðt 0 Þ
Ds ¼ lim
ð3Þ
t-1
6tNN 0
Fig. 11. (a) Mean square displacement of water molecules in reservoirs, in the
interfacial region and inside the membrane on a log scale. (b) MSD of water
molecules at short time on a linear scale. (c) Dipolar correlation as a function of
time in the three regions considered in Fig. 3a (on log scale). In (a) the vertical gray
lines show the boundaries for the calculation of the diffusion coefficient.
with t0 being the origin time, t the total time, N the number of
molecules, and N0 is the number of t0. We report in Fig. 11a the
MSD of water in three regions. MSD of confined water was
computed for jzjo 25 1A (i.e. far from the interfaces) while the
MSD of water molecules in the reservoirs was calculated for
jzj 460 1A, and MSD of interfacial water was calculated between
jzj 440 1A and jzj o 50 1A. Contrary to what is observed in the bulk
phase, for both confined and interfacial water a subdiffusive
regime (between 0 and 500 ps) precedes the diffusive regime
(see Fig. 11b). From the MSD the self-diffusion coefficients were
extracted from a linear fit of the part corresponding to the
diffusive regime (see Fig. 11a). These results show that even within
the membrane water molecules can diffuse without the cage
effect, which corroborates our findings regarding the large HB
network formed by water molecules. We found Ds ¼ 2:4 10 9 m2 s 1 in the water reservoirs, which is very close to the
bulk value [30], Ds ¼ 0:6 10 9 m2 s 1 in the interfacial region
and Ds ¼ 0:2 10 9 m2 s 1 for confined water (this value is in
goodagreement with other simulation results reported in the
literature [18–20,48]. Inside the membrane water diffusion is
divided by an order of magnitude with respect to the bulk phase.
Water diffusivity increases in the interfacial region because cavities are larger (see Fig. 3c). To complete our dynamics analysis
we report in Fig. 11c the dipolar correlation function ðCðtÞ ¼ 〈MðtÞ
Mðt 0 Þ〉=〈Mðt 0 Þ〉2 Þ where MðtÞ is the total dipolar moment of water
molecules at time t. Because of confinement, dipolar relaxation is
much slower inside the membrane than in the bulk phase. This
confinement effect is less accentuated in the interfacial region,
which is in good agreement with the larger cavities in this region
as shown in Fig. 3c.
3.3. Water dynamics
4. Concluding remarks
Information about the mechanism of water transport into the
polyamide membrane can be extracted from the spatial self Van
Hove function which is a correlation function of position and time
ðGs ðr; tÞÞ [46,17,18] defined as
Z
1
0
ρðr0 þr; tÞρðr0 ; 0Þ dr
Gs ðr; tÞ ¼
ð2Þ
N
In this study, we used a general method developed in our
previous work to construct an atomistic model of a highly crosslinked polyamide RO membrane. Simulations highlighted that the
oxygen and hydrogen atoms of the pending carboxylic acid
functions as well as the oxygen atoms of amide groups are the
preferential interaction sites with water molecules. The analysis of
hydrogen bond formation put in evidence the presence of a main
hydrogen bonding network involving about 90% of the water
molecules embedded in the polyamide membrane. Additionally,
where ρ is the atomic density at r and at time t and N is the
number of water molecules. We report in Fig. 10a Gs ðr; tÞ from 1 ps
244
M. Ding et al. / Journal of Membrane Science 458 (2014) 236–244
we showed the presence of some small water clusters into the
matrix allowing long range correlations. The study of the transport
mechanism of water molecules inside the membrane revealed an
absence of translational jumps. Although confinement was found
to hinder the rotational motion of water molecules, simulations
indicated that water molecules inside the membrane are still able
to exhibit large rotational jumps (as in bulk phase) which is
thought to be a prerequisite for HB formation. The translational
diffusion coefficient of confined water was found to be reduced by
an order of magnitude in the central part of the membrane (with
respect to the bulk value). Confinement effects were found to be
weaker in the interfacial region, which was correlated with the
increase in the average cavity size close to the membrane/reservoir
interfaces. This work constitutes a prerequisite to a future investigation of molecular mechanism of ion transport through RO
polyamide membranes.
Acknowledgments
The authors are grateful to the “Conseil régional de Bretagne”
for M. Ding's PhD fellowship and to the Agence Nationale de la
Recherche for its financial support through the program MUTINA
(ANR 2011 BS09 002).
Appendix A. Supplementary data
Supplementary data associated with this article can be found in
the online version at http://dx.doi.org/10.1016/j.memsci.2014.01.
054.
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