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A Molecular Dynamics Study of Water Flow Across Multiple Layers of Pristine, Oxidized, and Mixed Regions of Graphene Oxide: Effect of Graphene Oxide Layer-to-Layer Distance Jon Alexander Lewis Willcox, and Hyung J Kim J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b06063 • Publication Date (Web): 02 Oct 2017 Downloaded from http://pubs.acs.org on October 2, 2017

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The Journal of Physical Chemistry

A Molecular Dynamics Study of Water Flow Across Multiple Layers of Pristine, Oxidized, and Mixed Regions of Graphene Oxide: Effect of Graphene Oxide Layer-to-Layer Distance Jon A. L. Willcox† and Hyung J. Kim∗,†,‡,¶ †Department of Chemistry, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA ‡School of Computational Sciences, Korea Institute for Advanced Study, Seoul 02455, Korea ¶Permanent address: Carnegie Mellon University E-mail: [email protected] Phone: 412-268-6489

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Abstract Recent studies revealing exceptionally rapid water flow across graphene oxide membranes have highlighted them for potential filtration and separation applications. The physical and chemical features in graphene oxide membranes are heterogeneous, and there remains a great deal of speculation as to what is responsible for the facile water percolation. One potential contributing feature is the variation of interlayer spacing, which can occur naturally or be artificially induced. Herein, water flow across pristine, oxidized, and mixed membranes with interlayer distances of 0.7, 0.9, and 1.2 nm, corresponding respectively to the formation of discrete mono, bi, and trilayer water structures, is studied via molecular dynamics (MD) simulations. The interlayer spacing of 0.7 nm results in the formation of “square” ice for the pristine graphene membrane, which leads to collective motion, inhibiting equilibrium transport, but allowing for rapid non-equilibrium flow comparable to that in the membranes with larger interlayer distances. A four-point time correlation function analysis of water structural relaxation reveals that collective water motions are responsible for rapid non-equilibrium flow for the interlayer spacing of 0.7 nm. Meanwhile, the central water layers formed in an interlayer spacing of 1.2 nm lead to almost entirely decoupled structure and dynamics between outer water layers.

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Introduction Graphene oxide (GO) has been implicated as a potential sieving material for small molecule liquids and gases. 1,2 Water has been shown to traverse GO membranes with minimal resistance, while the same membranes can be virtually impenetrable to other species, including helium gas. 1,3 This anomalous behavior has been largely attributed to the formation of hydrogen-bonded networks between water molecules and functional groups in the oxidized regions of the membranes. The water molecules between oxidized regions are believed to expand the separation between GO sheets and allow for greater penetration. Mechanistic details, however, remain the subject of significant speculation and are complicated by the heterogeneous nature of GO. It has been theorized that regions of pristine graphene provide a “capillary network,” in which water experiences minimal friction – an analogous phenomenon to that found in carbon nanotubes. 4 Other studies suggest that defects in the membrane – nanopores, slits, wrinkles, etc. – are responsible. In all likelihood, fast water percolation across GO membranes is the result of numerous coexistent factors. In a previous study 5 we examined water flow across different membrane types—pristine, oxidized, and mixed—by employing a model membrane system displayed in Figure 1 with interlayer spacing, d = 0.9 nm. We noted that the mixed region – a region of pristine/oxidized graphene overlap – could contribute significantly to the lateral water flow via a layer of water adjacent to the pristine graphene surface. In the present article we alter the interlayer distance, d, between pristine (P) or oxidized (O) graphene sheets of the model membrane with the objective of modeling expanded and constricted regions. As in our prior study, 5 20% of carbon atoms of O-graphene are oxidized by hydroxyl groups. (The terminology and notation used throughout the paper are similar to those in ref 5 and are compiled in Table 1.) Natural distortions such as wrinkles and folds in graphene sheets are believed to result in variations of the interlayer distance, and recent studies report methods of manipulating the spacing to optimize membranes for filtration 3



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Figure 1: PPOO membrane side view with slits – water is shown in light blue, carbon in black, and hydroxyl groups in red. Periodic boundary conditions extend in all directions. For this study, the x, y, and z-axes correspond to flow between slits, parallel to slits, and across slits (perpendicular to the P/O-graphene basal plane), respectively. For notation, see Table 1. applications. 6 This has been accomplished via the introduction of spacer groups 7–9 or by altering the degree of oxidation. 10–13 We selected d = 0.7 nm, 0.9 nm, and 1.2 nm, each of which gives rise to a distinct water structure – monolayers, bilayers, and trilayers, respectively. 14,15 Structurally, the most distinctive water layer was that confined between two sheets of P-graphene for d = 0.7 nm. As has been reported previously, 15–21 a 2-dimensional lattice of “square” ice is formed. As for the dynamic behavior of water, we find that under equilibrium conditions for d = 0.7 nm, the mixed region shows the fastest lateral motions in contrast to the d = 0.9 and 1.2 nm cases. Upon applying an external force, however, water flow in the PWL greatly surpasses that in the MWL. For d = 1.2 nm, dynamics in the central hydrolayer are faster than those in outer hydrolayers in the oxidized and pristine regions. In the mixed region, central hydrolayer (MWLC ) dynamics fall between those of the outer hydrolayers. An analysis of water structural correlation dynamics via a four-point time correlation function reveals that nonequilibrium water flow in the PPPP membrane with d = 0.7 nm is highly collective.

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Table 1: Terminology and notation used in this work Term Pristine (P) graphene Oxidized (O) graphene Membrane (XXXX) Channel Interlayer distance (d) Slit Region Hydrolayer PWL OWL MPWL MOWL

Definition unfunctionalized graphene hydroxyl functionalized graphene four staggered, repeating P/O-graphene (X=P or O) sheets space between P/O-graphene sheets distance between P/O-graphene sheets lateral separation between graphene edges type of graphene overlap (pristine, oxidized, or mixed) a monolayer of water – “C” subscripts indicate a central layer hydrolayer in the pristine channel hydrolayer in the oxidized channel hydrolayer in the mixed channel adjacent to P-graphene hydrolayer in the mixed channel adjacent to O-graphene

Methods The GO membrane model used for this study is identical to that used in our previous work with the exception that the intersheet distances used are 0.7, 0.9, and 1.2 nm. 5 Briefly, the GO membrane consists of four 806-carbon (3.2 nm × 6.4 nm) periodically repeating layers of P/O-graphene with staggered, hydroxyl-functionalized 1.0 nm-wide slits. Adjacent hydroxyl group pairs – one on either face of the basal plane – are randomly distributed across Ographene sheets, oxidizing 20% of the basal carbons (156 hydroxyl groups). This corresponds to the Lerf-Klinowski model without epoxide functionalization due to its lack of stability in the presence of water. 22 The density of water in the membrane was established via the same 2-step process detailed in our previous work – equilibration with bulk water followed by equilibration in the PPOO membrane (see Table 2). 5 The densities found in the PPOO equilibration are used for non-equilibrium simulations. Table 2: Interlayer densities are measured in units of molecules per nm3 d (nm) 0.7 0.9 1.2

Bulk Pristine 17.5 23.0 23.8

Equilibration Mixed Oxidized 14.8 13.6 20.5 19.0 23.5 22.2

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PPOO Equilibration Pristine Mixed Oxidized 18.5 14.7 12.8 22.7 20.6 19.0 24.1 23.2 22.4



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All Atom Optimized Potentials for Liquid Simulations (OPLS-AA) parameters were used to model P/O-graphene sheets, 23 while incorporating the hydroxyl adjustments reported by DeYoung et al. 24,25 All carbon atoms were frozen in place to maintain a constant interlayer distance for the data reported in this work. (A test simulation in which carbon atoms were restrained to their initial positions using a harmonic potential with a force constant of 1000 kJ mol−1 nm−2 was also performed and – though the water dynamics were found to be slightly faster – revealed no change in the trends reported here.) The extended simple point charge (SPC/E) model was used for water. 26 The combination of OPLS-AA parameters and the SPC/E water model have been reported to provide results in agreement with GO aggregation, interfacial behavior, and water contact angle data. 14,23,27 All simulations were performed in the canonical (N V T ) ensemble at 300 K with Nosé-Hoover temperature coupling using the Gromacs molecular dynamics simulation package. 28–31 Electrostatic interactions were calculated using the particle mesh Ewald (PME) method with a 1.3 nm cut-off. Equilibrium simulations in the PPOO consisted of 10 ns of production time with a time step of 0.5 fs, preceded by 2 ns of equilibration. Non-equilibrium simulations were performed in the PPPP, OOOO, and POPO membranes by applying an auxiliary acceleration of 0.1 nm ps−2 perpendicular to the GO layers (along the z-axis) for all water molecules. The data displayed for non-equilibrium simulations is intended to compare relative water flow rate in different membrane types, and not as a direct comparison to experimental work. For each membrane, 30 ns non-equilibrium production runs were preceded by 2 ns of non-equilibrium simulation to ensure that the system reached a steady state. All non-specified simulation details for these runs were equivalent to those for equilibrium simulations. Additional simulations performed with a reduced acceleration of 0.05 nm ps−2 to test the dependence on the auxiliary acceleration value yield the same trends established in this work.

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Results and Discussion Water layer structure The MD results for water density along the z-axis for each interlayer distance are plotted in Figure 2a, Figure 2b, and Figure 2c. As has been reported in previous studies, 14,15 we find a

Figure 2: Number density of water (calculated from oxygen) with respect to bulk water density in the GO membrane vs. z-axis in membranes with interlayer spacing equal to 0.7 nm (a), 0.9 nm (b), and 1.2 nm (c). Dashed lines signify the locations of pristine (black) and oxidized (red) graphene. water monolayer at d = 0.7 nm, a water bilayer at d = 0.9 nm and a trilayer at d = 1.2 nm. The density is greater in the pristine regions for each interlayer distance, and water bi or trilayers in the mixed region show greater water density in hydrolayers adjacent to pristine graphene than in hydrolayers adjacent to oxidized graphene. For the purposes of this study, in the d = 1.2 nm membrane, each P/O-graphene adjacent hydrolayer is considered to be contained within 0.44 nm of the adjacent P/O-graphene sheet, while the central layer is considered to be the central 0.32 nm in the channel – values corresponding to the minima in Figure 2c. In each region, the central layer is lower in density than the outer layers but – due to its greater volume – the OWLC contains more water molecules than the OWL and the MWLC contains more water molecules than the MOWL. The number of water molecules per unit area in the plane of the hydrolayer, nxy , for d = 1.2 nm for the PWL, PWLC , MPWL, MWLC , MOWL, OWL, and OWLC are 9.8, 9.4, 9.9, 9.8, 8.1, 8.2, and 10.4 nm−2 ,

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respectively. The lateral water structures for d = 0.9 nm and d = 1.2 nm in the pristine region display a non-repetitive assortment of hexagonal, pentagonal, rhombic, and amorphous clusters. The presence of hydroxyl groups on the adjacent graphene disrupts water structures and leads to a more intermittent network of water-water hydrogen bonds. Turning to the pristine region for d = 0.7 nm, we find a highly ordered water lattice – predominantly with the puckered rhombic morphology with no net polarity – shown in Figure 3a. This planar formation of

Figure 3: Random snapshots of water in the PWL (a) and the OWL (b) for d = 0.7 nm and the 2D-RDF for water oxygen for the different region types for d = 0.7 nm (c). nanoconfined water – sometimes referred to as “square ice” – has been reported in numerous MD simulations, 16–21,32 as well as with quantum chemistry calculations 20,21,33 and experimentally 32 (though there is some unresolved controversy regarding the experimental study 34,35 ). Analogous to the d = 0.9 nm and d = 1.2 nm cases, the oxidation of graphene surface(s) by the introduction of hydroxyl groups disrupts the ordered water structure (Figure 3b). To further illustrate the influence of the hydroxyl groups, we introduce a 2-dimensional radial distribution function (2D-RDF) as follows N X δ(rij − r) h i g2D (r) = nxy N i,j 2πr

1

(1)

where N is the number of water molecules in the hydrolayer, nxy is the water number density in the xy-plane introduced above, rij is the distance between molecule i and molecule j in the xy-plane and h. . .i denotes an ensemble average. Figure 3c shows the 2D-RDFs for water 8


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oxygen in each region of the d = 0.7 nm membrane. The prominent oscillations for PWL are the result of its nearly crystalline structure, whereas their absence in MWL and OWL is indicative of the disruption of ordered water structure by GO hydroxyl groups. None of the 2D-RDFs for the d = 0.9 and 1.2 nm membranes show the prominent oscillations indicative of rhombic ice (results not presented here).

Hydrogen-bonding structure The average numbers of intra and inter-hydrolayer water-water hydrogen bonds are listed in Table 4. We used the same geometric hydrogen bond definition as in our previous work. 5 To be specific, for an interaction to be considered a hydrogen bond, the intermolecular H · · · O distance must be less than 0.24 nm and the H-O · · · O angle, 6 HOO, must be less than 30◦ . A similar definition has been used for water systems in previous studies. 36,37 Table 3: Average numbers of water-water hydrogen bonds within a hydrolayer (intra) and between hydrolayers (inter) per water molecule in the hydrolayer. Values exclude water molecules directly above and below the slit. Subscript “C” denotes central water layers. Intra

d (nm) 0.7

PWL 3.68

MPWL 2.52

MOWL 2.52

OWL 2.02

PWLC N/A

MWLC N/A

OWLC N/A

Intra Inter

0.9 0.9

2.49 0.89

2.52 0.80

1.77 1.05

1.86 0.98

N/A N/A

N/A N/A

N/A N/A

Intra Inter

1.2 1.2

2.28 0.89

2.32 0.86

1.66 1.10

1.72 1.07

1.62 1.84

1.74 1.70

1.86 1.59

In general, the number of hydrogen bonds between water molecules within the same layer is greater for P-graphene adjacent layers and – among the interlayer distances studied – increases as d decreases. At d = 0.7 nm, the number of intra-layer hydrogen bonds in the pristine region approaches four, as is expected from the structure observed in Figure 3a. At d = 1.2 nm, the water layers bordering P/O-graphene are separated by a central layer of water so that the total number of interlayer hydrogen bonds in adjacent layers is no longer necessarily equivalent. The presence of a central water layer generally leads to an

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increase in interlayer hydrogen bonds and a decrease in intralayer hydrogen bonds for P/Ographene adjacent layers. The trends in hydrogen bonding behavior of the central layer water molecules (intralayer: PWLC < MWLC < OWLC , interlayer: OWLC < MWLC < PWLC ) are the inversion of outer water layer trends. In the OWLC , i.e. the central water layer in the d = 1.2 nm channel confined by two oxidized graphene sheets, the average number of interlayer hydrogen bond donors and acceptors (per water molecule) are 0.80 and 0.79, respectively, while the corresponding values for the PWLC are 0.95 and 0.89, respectively. This indicates that – to some degree – the central layer water molecules orient to complement the outer water layers (i.e. the central layer has a greater tendency to act as a hydrogen bond donor to the PWL than as a hydrogen bond acceptor from the PWL, while the opposite can be said for the central layer hydrogen bonding interactions with the OWL); this phenomenon is further discussed in the section Water orientation, below. Table 4 displays the number of hydrogen bonds formed between water molecules and hydroxyl groups in the OWL and MOWL regions. From d = 0.7 nm to d = 0.9 nm, the number of hydroxyl-water hydrogen bonds per water molecule increases, but remains steady as d is increased to 1.2 nm. Table 4: Average number of hydrogen bonds between water molecules in the O-graphene adjacent hydrolayer and hydroxyl groups. role of hydroxyl group donor (OWL) acceptor (OWL)

d (nm), per hydroxyl 0.7 0.9 1.2 0.30 0.53 0.51 0.66 0.69 0.63

d (nm), per water 0.7 0.9 1.2 0.11 0.20 0.20 0.24 0.27 0.25

donor (MOWL) acceptor (MOWL)

0.28 0.74

0.09 0.24

0.56 0.71

0.52 0.63

10


0.22 0.28

0.21 0.26

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Water orientation The probability distribution, P (θ), for the orientation of water molecules with respect to the membrane is calculated such that Z

P (θ) sin θ dθ = 1

(2)

where θ is the angle between the specified molecular axis and the z-axis (perpendicular to the P/O-graphene basal plane). The orientation of water molecules in the PWL of the d = 0.7 nm membrane (Figure 4a, Figure 4b, and Figure 4c) is fairly uniformly parallel to the surrounding graphene planes as was expected due to the highly ordered planar structure. In the MWL and the OWL, the orientation of water molecules is more evenly distributed than that in the PWL, but in both cases the planar orientation still remains prevalent. Water molecular orientation in the d = 0.9 nm membrane was studied in detail in our previous work. 5 For convenience, these results are included in Figure 4d, Figure 4e, and Figure 4f. Due to the codependency of the adjacent hydrolayers in the mixed region, the difference in water orientations between the MPWL and the MOWL is greater than that found between the PWL and the OWL. As the interlayer distance is increased to d = 1.2 nm, we find a “buffer” layer (viz. central hydrolayer) arising between the two P/O-graphene adjacent hydrolayers (cf. Figure 2c). The orientation of water molecules in the outer layers are displayed in Figure 4g, Figure 4h, and Figure 4i, while the central layer orientations are displayed in Figure 4j, Figure 4k, and Figure 4l. Upon the introduction of a central hydrolayer, the MPWL and PWL orientation distributions become nearly indistinguishable, as do the MOWL and OWL orientation distributions. Though water molecular orientations in the central layer are largely randomized, viz., their distribution is quite isotropic, there is some tendency for water molecules there to complement the orientations in the P/O-graphene adjacent layers. In other words, the

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Figure 4: P (θ) sin θ plotted against θ in d = 0.7 nm (a-c), d = 0.9 nm (d-f), d = 1.2 nm for the outer layers (g-i) and the central layers (j-l) where θ is the angle between the indicated molecular axis and the z-axis.

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Hydrolayer dynamics As in our previous work, 5 the residence time for a water molecule in an individual hydrolayer is measured using binary correlation functions S(t) and C(t). The function S(t) measures the probability that a water molecule is continually present in its original hydrolayer at all intermediate times between initial time, t = 0, and the observation time. The function C(t) measures the probability that a water molecule is present in the same hydrolayer at initial time, t = 0 and the observation time, disregarding its location at intermediate times. For the measurement of S(t), a buffer region between hydrolayers was introduced (0.15 nm for d = 0.9 nm and 0.14 nm for d = 1.2 nm). If a water molecule moves from the buffer region to its original hydrolayer, it is considered never to have left; otherwise it is considered to have left from the time that it entered the buffer region. The results for the d = 1.2 nm case are exhibited in Figure 6a, Figure 6b, and Figure 6c. As was discussed in ref 5 for water in the d = 0.9 nm membrane, the initial relaxation of C(t) occurs on the same time scale as S(t), and represents the decay as water molecules exchange with adjacent hydrolayers. The subsequent slow relaxation of C(t) occurs as water molecules cross the slits into neighboring channels. For each of the d = 1.2 nm systems the relaxation of the central layer occurs faster than the relaxation of the outer layers. This is to be expected as the central layer exchanges water molecules with two adjacent water layers, while each of the outer layers can – excluding the slow cross-slit passage – only exchange water molecules with the central layer. Because there are no adjacent hydrolayers for d = 0.7 nm, the decays of both C(t) and S(t) correspond to water passage across slits. Our analysis indicates that they decay on a time scale > 10 ns (results not shown). For a quantitative perspective on the relative rates of water exchange between layers, we calculate the time constants associated with S(t) for d = 0.9 and 1.2 nm as

t S(t) = exp − . τl 14


(3)

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Figure 6: Time correlation functions S(t) and C(t) for d = 1.2 nm in the PWL (a, blue), PWLC (a, cyan), the MPWL (b, dark green), the MOWL (b, purple), the MWLC (b, light green), the OWL (c, red), and the OWLC (c, yellow). The results for τl are displayed in Table 5. As was described above, the central layers equilibrate faster than the outer layers for d = 1.2 nm, while in general relaxation occurs 3–5 times faster for d = 1.2 nm than for d = 0.9 nm. Table 5: Values of τl (ns) determined via eq 3 for d = 0.9 and 1.2 nm.† d (nm) PWL 0.9 0.120 1.2 0.029 † Relaxation time is

MPWL 0.133 0.032 longer than

MOWL OWL PWLC 0.113 0.179 N/A 0.038 0.041 0.016 10 ns for d = 0.7 nm case.

MWLC N/A 0.022

OWLC N/A 0.030

Equilibrium water transport To measure the lateral water transport under equilibrium conditions, we calculate the 2dimensional mean squared displacement (MSDxy ) for water in each channel type for each interlayer spacing. Results are displayed in Figure 7a, Figure 7b, Figure 7c, and Figure 7d. Note that displacement is only calculated for a water molecule before it has left its original hydrolayer. For the d = 0.7 nm membrane, water mean square displacement parallel to the membrane in each channel type is lower than that for d = 0.9 and 1.2 nm by one order of magnitude. Furthermore, water in the pristine and oxidized regions moves significantly slower than water in the mixed region. Water-hydroxyl group interactions are responsible for the reduced 15



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Figure 7: MSDxy plots for d = 0.7 nm (a), d = 0.9 nm (b), the outer hydrolayers of d = 1.2 nm (c), and the central hydrolayers of d = 1.2 nm (d). translational motions in the OWL as was discussed for d = 0.9 nm in our previous work, 5 while in the PWL, movement is reduced by the formation of highly ordered water structure noted in the Water layer structure section above, i.e. the water molecules in the PWL move as a collective, not as individuals. The hydroxyl groups in the MWL, however, seem to sufficiently interrupt the ordered water structure (cf. Figure 3b) to allow for greater motion, while providing lower resistance in comparison with the OWL. In the d = 1.2 nm membrane, the lateral translational motions of water in the central layer are generally faster than those in the outer layers. The introduction of the central layer also promotes independent dynamics in the MPWL and MOWL. With the greater separation, the MPWL behavior approaches that of the PWL while the MOWL layer behavior approaches that of the OWL. It is worth noting that the data displayed in Figure 7 include the lateral motions of water molecules in the channel as well as water molecules bordering a slit. The latter water molecules have little impact on the MSDxy results for d = 0.9 and 1.2 nm, but are the 16


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primary contributor for the d = 0.7 nm membrane. To see this and also to ensure that the conclusions we have drawn in this section are maintained for water molecules bordering slits and for those confined between P/O-graphene sheets, we calculated the 1-dimensional MSD along the y-axis (parallel to the slits), MSDy , for slit-adjacent water molecules and confined water molecules separately; the results are shown in Figure 8a, Figure 8b, and Figure 8c. In both cases, the trends observed in Figure 7 are maintained. Nevertheless, the difference between the two measured as the ratio of their MSDy is considerably more prominent for d = 0.7 nm than that for larger channels. As was expected – the slit-adjacent water molecules move significantly faster than the confined water molecules in the d = 0.7 nm membrane.

Figure 8: 1-dimensional mean squared displacements MSDy for water transport parallel to the slits vs. time for d = 0.7, 0.9, and 1.2 nm in panels a, b, and c, respectively. The results for water molecules bordering the slit and in the confined region are plotted in dashed and solid lines, respectively.

Returning to MSDxy , we consider its time dependent behavior for each channel type as

|∆rxy (t)|2 ∝ tγ .

(4)

In the diffusion regime, MSDxy increases linearly with time, i.e. γ becomes unity, and the characteristic 2-dimensional diffusion coefficient, Dxy , is given by

|∆rxy (t)|2 = 4 Dxy t .

(5)

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water molecules and water molecules confined between P/O-graphene sheets. Within our simulation time window, γ in eq 5 did not reach unity for d = 0.7 nm, viz. the diffusion regime was not obtained. For d = 0.9 and 1.2 nm, transport is similar between slit-adjacent and confined water molecules (cf. Figure 8), and diffusive behavior is achieved. The results for Dxy for these cases can be found in Table 6. Table 6: 2D-diffusion coefficients calculated from the average MSDxy of all water molecules in each channel in units of nm2 ns−1 . d (nm) 0.9 1.2

PWL 1.5 2.5

MWL 0.80 1.5

OWL 0.35 1.0

Non-Equilibrium water transport To analyze the water flow in the presence of an auxiliary force (0.1 nm ps−2 ) perpendicular to the membrane, we introduce a binary function similar to S(t) used to calculate the hydrolayer dynamics. 5 In this case, the function decays as water molecules cross a slit. Results are displayed in Figure 9a, Figure 9b, and Figure 9c.

Figure 9: Time correlation functions of water molecules crossing a slit into the adjacent channel for d = 0.7 nm (a), d = 0.9 nm (b), and d = 1.2 nm (c). For the d = 0.7 nm membrane (Figure 9a), we find that water in the OOOO membrane is virtually stationary and flow in the POPO membrane is severely hindered. What is striking, though, is that the PPPP membrane allows a rapid water flow-rate comparable to that for 18


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larger channel types, in contrast to equilibrium water transport (Figure 7) which differs by a factor of 20-30 between the former and latter membranes as noted above. We will return to this point below. As the interlayer distance is increased to 0.9 nm, the introduction of adjacent water layers dramatically improves the water flow in the oxidized membrane (Figure 9b). At d = 1.2 nm (Figure 9c), the behavior of the OOOO and POPO membranes approaches that of the PPPP membrane – a phenomenon similar to that noted by Wei, et al. 14 For further insight, we calculated the “water residence time” in the channel, τneq , by integrating over the curves in Figure 9 and the average number of water molecules to cross a slit per unit time, Nneq , both in the presence of an auxiliary external force. We note that τneq is the average wait time for a water molecule to exit from the channel. Results are displayed in Table 7. As is expected, Nneq increases with the interlayer distance d for all channel types. The extent of the Nneq increase with d is in the order OOOO > POPO > PPPP, indicating that the channel size influences water flow in oxidized channels more than that in pristine channels. Both τneq and Nneq results show that at given d, water flow decreases as the graphene layers become oxidized. It is interesting that in the PPPP membrane, the water residence time in the channel increases as d is increased from 0.9 to 1.2 nm despite the fact that the water exit rate from the channel, Nneq , becomes enhanced. This is attributed to the increase in the number of water molecules in the channel, which in turn increases their wait time to cross the slit to the adjacent channel. In this context, for d = 1.2 nm, the “bottle-neck” for the overall water flow in the direction of the external force (i.e. perpendicular to the membrane) is the slit crossing for the PPPP membrane, whereas it is the lateral flow of water in the channel for the OOOO membrane.

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Table 7: Time constants, τneq , associated with the functions plotted in Figure 9 and the average number of water molecules to cross the slit, Nneq . Nneq (ns−1 )

τneq (ns) d (nm) 0.7 0.9 1.2

PPPP 3.5 2.5 2.7

POPO > 10 3.0 3.0

OOOO ≫ 10 4.3 3.1

PPPP 87 193 253

POPO < 23 146 221

OOOO ≪ 21 94 205

To obtain a better understanding of the rapid water flow noted above for the PPPP membrane with d = 0.7 nm (Figure 9a), we examine relaxation of water structural correlation under non-equilibrium conditions. Specifically, we consider a four-point time correlation function g4 (r, t) = h

N X N X i

δ (r − |ri (0) − rj (0)|) δ (r − |ri (t) − rj (t)|)ine

(6)

j>i

where N is the number of water molecules in the system and h...ine represents non-equilibrium average in the presence of an external force. Four-point time correlation functions similar to eq 6 but under equilibrium conditions were employed to study dynamics, in particular, correlated motions and dynamic heterogeneity, in glassy systems. 38–40 We note that g4 (r, t)/g4 (r, 0) (cf. eq 7 below) gauges the probability that two water molecules are separated by a distance r at a later time given that they were separated by the same distance initially at t = 0. For our system, we limited our calculations to include only pairs of water molecules that remain confined between graphene basal planes for the duration of time that they are paired. In Figure 10, the results for

g4shell (r1 , r2 , t)

=

Z

r2

dr g4 (r, t) r1

Z

r2

dr g4 (r, 0)

(7)

r1

are displayed for the PPPP membrane at each interlayer distance for the first hydration shell defined by r1 = 0 and r2 = 0.35 nm, corresponding to the main peak region of the 2D-RDF in Figure 3c. For d = 0.7 nm, this peak region represents the first hydration shell of a water molecule. This shell consists primarily of four surrounding water molecules that 20


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are mostly hydrogen-bonded to the central water molecule though some water molecules located diagonally in the square ice lattice (i.e. separated by a single water molecule in the hydrogen-bonded network) are included. If we extend r2 to be 0.48 nm, then we include all water molecules located diagonally in the square ice lattice. The decay of this extended first hydration shell (purple) as well as that of the second hydration shell (r1 = 0.48 nm and r2 = 0.61 nm) for d = 0.7 nm is also displayed in Figure 10.

Figure 10: g4shell (r1 , r2 , t) vs. time for r1 = 0.00 nm and r2 = 0.35 nm (solid lines); r1 = 0.00 nm and r2 = 0.48 nm (dashed-dotted purple line); and r1 = 0.48 nm and r2 = 0.61 nm (dashed blue line). The most salient aspect of our results is that for the PPPP system, g4shell (r1 , r2 , t) of the first hydration shell for the d = 0.7 nm membrane decays much more slowly than that for the d = 0.9 and 1.2 nm membranes even though these three membranes are characterized by similar water residence time τneq (Table 7). Even g4shell (r1 , r2 , t) of the second hydration shell in the d = 0.7 nm channel is characterized by considerably slower relaxation than that of the first hydration shell in the larger channels. This indicates that structural correlation of water persists much longer in the PPPP membrane with d = 0.7 nm than with d = 0.9 and 1.2 nm. We calculated structural correlation persistence time, τ4pt , of flowing water by integrating g4shell (r1 , r2 , t) over t. We found that τ4pt = 0.419 and 0.071 ns for the first and second hydration shells in the d = 0.7 nm channel, while the first hydration shell (r2 = 0.35 nm) in d = 0.9 and 1.2 nm yields τ4pt = 0.014 and 0.010 ns, respectively. This shows that pairwise 21



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correlation between the central and first hydration shell water molecules in the d = 0.7 nm membrane lasts 30-40 times longer than that in the d = 0.9 and 1.2 nm membranes. (As an additional point of reference, we note that the water-water hydrogen-bond lifetime – a time scale related to g4shell (r1 , r2 , t) but accounting for relative water-water orientation – in bulk water has been found to be ∼ 1.4 ps in ref 41 though the SPC water potential model was used there.) If we extend the first hydration shell from r2 = 0.35 nm to 0.48 nm to include all diagonal water molecules in the d = 0.7 nm case, the structural correlation persistence time rises from 0.419 ns to ∼ 1.8 ns. This large increase in τ4pt indicates that the structural relaxation of a contact water pair (i.e. O-O separation ≤ 0.35 nm) in the square ice configuration occurs largely via water molecules moving to adjacent locations in the square ice lattice. Allowing for these local configurational changes in the structural correlation of water pairs results in τ4pt comparable to the time τneq associated with water flow between channels (3.50 ns from Table 7). Our analysis here indicates that to a large degree, pairwise structural correlation of water persists during its lateral flow, suggesting collective water motions as the mechanism for rapid flow in the PPPP membrane with d = 0.7 nm. It is worthwhile to note that τ4pt would become shorter if the lateral separation of the slits is decreased because the influence of disrupted water structure in the slit region on water structural correlation in the channel region would increase. We think that this dependence of τ4pt on the slit-to-slit distance is responsible for the uncorrelated water motion found in a narrow but short pristine channel. 37 For further illustration of collective water motions, sample water trajectories in the PPPP membrane with d = 0.7 and 0.9 nm are compared in Figure 11a and Figure 11b, respectively. In each case, a water molecule (displayed in red) selected randomly from the most crystalline domain of water layers was used as a reference, viz., central water molecule. Water molecules colored in blue and yellow represent, respectively, those in the first and second hydration shells of the central water molecule. Thus the molecules in blue and yellow in Figure 11a correspond to the solid and dashed blue curves in Figure 10, respectively, while those in blue in

22


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entirely disrupted in the MWL and the OWL. In general, for d = 1.2 nm there is a decrease in the intralayer water-water hydrogen bonding, while inter-layer hydrogen bonding is enhanced by the introduction of a central water layer. The hydrogen bonding data along with the molecular orientation of water for d = 1.2 nm indicate that the central layer structure adjusts slightly to compliment the outer layers and acts as a sufficient buffer to structurally decorrelate them – i.e. the MPWL and MOWL become nearly indistinguishable from the PWL and OWL, respectively. Equilibrium dynamics show an unexpected MSDxy trend of MWL>PWL>OWL for d = 0.7 nm. This was attributed to the greater number of water-hydroxyl interactions in the OWL and the highly packed structure in the PWL, which results in correlated water motion. Dynamics become accelerated with greater P/O-graphene separation. For d = 1.2 nm, dynamics are found to be fastest in the central region. Upon the introduction of an external force, water flow in the PPPP membrane with d = 0.7 nm was dramatically improved – surpassing that of the POPO membrane – when compared to the equilibrium case. This is attributed to the collective water flow in the PPPP membrane. We also find that for d = 1.2 nm, the PPPP, OOOO, and POPO membranes show similar rates of water passage across the slits, indicating that the central layer is largely responsible for flow.

Acknowledgement This work was supported in part by the National Science Foundation through NSF Grant No. CHE-1223988.

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