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Inhibitory Effect of Adsorbed Water on the Transport of Methane in Carbon Nanotubes Lang Liu, Chunxia Hu, David Nicholson, and Suresh K Bhatia Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b01070 • Publication Date (Web): 01 Jun 2017 Downloaded from http://pubs.acs.org on June 12, 2017

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Inhibitory Effect of Adsorbed Water on the Transport of Methane in Carbon Nanotubes Lang Liu, Chunxia Hu, David Nicholson and Suresh K. Bhatia* School of Chemical Engineering, The University of Queensland Brisbane, QLD 4072, Australia

We investigate the transport diffusion of methane at 300 K and pressures up to 15 bar in dry and wetted carbon nanotubes (CNTs) having diameters ranging from 0.95 to 2.034 nm, using non-equilibrium molecular dynamics (NEMD) simulation. Due to their strong hydrogen bonding, pre-adsorbed water molecules transport in the form of clusters and block the diffusion of methane, reducing the Onsager coefficient of methane dramatically compared to that in dry CNTs. The reduction in the methane Onsager coefficient is greater in narrower CNTs or at higher water densities. Since the diameter of the water clusters is almost invariant with water density, the Onsager coefficient of water in the (10, 10) CNT increases linearly with the water density. It is further found that while decreasing the CNT diameter from 2.03 nm to 0.95 nm enhances the Onsager coefficient of pure methane by about one order of magnitude, the Onsager coefficient of water is almost independent of the CNT diameter, at a water density of 0.05 g/cm3. We propose a theoretical model for the strong dependency of methane diffusion in wetted CNTs on the Onsager coefficient of water, the pre-adsorbed water density and the CNT diameter. The model predicts the Onsager coefficients of the methane/water mixture from the Onsager coefficients of the pure components. Our study provides a basic understanding of the coupled diffusion of immiscible components in nanochannels, and will facilitate progress in gas storage and carbon capture, as well as nanofiltrations and biomedical and biotechnological applications.

*To whom correspondence may be addressed. Email: [email protected].

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1 Introduction The transport of molecules through nanopores is integral to many processes for gas separation and in biological systems,1-5 and in recent years there has been considerable interest in understanding molecular transport in carbon nanotubes, which possess many attractive features in this context. Besides the high adsorption capacity of carbon nanotubes (CNTs), which is a consequence of their high surface area and strong overlapping force field,6-8 several simulation studies have shown that adsorbed fluid phases in carbon nanotubes, including methane, exhibit extremely high diffusion rates, attributed to their atomically smooth carbon wall.9-14 The transport of water in CNTs has attracted particular attention. Both simulation and experiment have found large diameter-dependent enhancement of water flow rate in CNTs compared to that predicted by the Hagen-Poiseuille relation.15-18 However, a wide range of enhancement factors have been reported, and the structure and dynamics of simulated water in CNTs have been found to vary considerably, depending on the water model, the temperature, and the diameter of the nanotubes; and may even be artefacts of the simulation technique.19, 20 For example, spiral columns were observed for the modified TIP3 water model in (8,8) and (9,9) CNTs,21 while n-gonal rings of water were reported by Koga et al.22 for the TIP4P model in wider CNTs at high pressure. Wang et al.19 (TIP3P), and Mashl et al.23 (SPC/E) found that water generally formed clusters in (n, n) CNTs, with n increasing from 6 to 10, the exception being the (9, 9) CNT where they observed ordered structures. Similarly, we have also found clusters, resembling those in liquid water, in CNTs containing pre-adsorbed water,6, 24 and shown that the presence of these clusters has a significant effect on the adsorption capacity of CO2 and CH4. Besides the differences in simulation-based results, large inconsistencies in experimental results are also found in the literature. In CNTs with pore sizes less than 2 nm, flow rates measured experimentally by Qin et al.25 and by Holt et al.15, deviate from each other by an order of magnitude, whilst the results from the simulation work of Thomas and McGaughey17 are more than 2 orders of magnitude lower than experimental values reported by Majumder et al.18. Consequently, there is considerable interest in developing a better understanding of the factors affecting the transport of water and other molecules in carbon nanotubes. Although there have been a number of molecular dynamic simulations of water transport in CNTs, most have been restricted to narrow diameters, where only a single file water chain can 2 ACS Paragon Plus Environment

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flow through the channel26-29, or to self-diffusion30-32. Little is known about the mechanism of transport diffusion of clusters confined in CNTs and its dependency on the CNT diameter. Both experiment and simulation have confirmed that the interior of a CNT though hydrophobic, is wettable by water.33, 34 Pascal et al.33 carried out simulations for water in CNTs with diameters ranging from 0.8 to 2.7 nm, at 1 bar and 300 K. They found that spontaneous entry of water into the hydrophobic CNT is favoured either by entropic gain, or by enthalpic loss, when the water molecules entering a CNT are liberated from their bulk phase tetrahedral hydrogen bonded framework.33 Here we have used non-equilibrium molecular dynamics (NEMD) simulation35 to investigate the transport diffusion of methane and of methane in the presence of pre-adsorbed water in CNTs at 300K. After establishing a basic understanding of the transport diffusion of pure methane and of water clusters, we have studied the transport diffusion of methane in the presence of water clusters. We propose a theoretical model to predict quantitatively the transport of two immiscible components in a confined space, based on the diffusion coefficients of the pure components. The coupled diffusion of gaseous species with water through a CNT is, in some respects, analogous to the transport of aqueous solutions in biological cells.20, 36 Therefore, advancing the basic understanding of the transport of gas species in the presence of water in CNTs will contribute to developments in nanofiltration and biomedical and biotechnology applications, besides natural gas storage and carbon capture. 2 Simulation Details Water does not spontaneously fill CNTs until its vapor pressure reaches a critical value, so adsorption isotherms are not available to calculate thermodynamic factors for use in the Darken equation.37, 38 We have therefore chosen water densities of interest and investigated the Onsager coefficients of CH4 in CNTs containing pre-adsorbed water using NEMD simulations. Accordingly, CNTs containing pre-adsorbed water are referred to as wetted CNTs in this work. We investigated the transport of methane in dry and wetted (7, 7), (10, 10), and (15, 15) CNTs. Our simulation system is set as a CNT located on the centre line of a simulation box with dimensions W × H × L = 10 ×10 × LCNT nm3, where LCNT is the length of the CNT. Periodic 3 ACS Paragon Plus Environment

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boundary conditions are applied in all the dimensions. The dimensions of the nanotubes are summarised in Table 1. The center to center diameters of the CNTs are used to determine the volume of the CNTs and the number of water molecules in the system. Three pre-adsorbed water contents of 0.025, 0.05 and 0.1 g/cm3 were chosen.

(n,m)

Table 1 Nanotube dimensions centre to centre diameter (nm)

Length/nm

(7,7)

0.95

25

(10,10)

1.356

25

(15,15)

2.03

15

2.1 Molecular Models Methane molecules were modelled as single Lennard-Jones (LJ) sites.7 Water molecules were represented by the SPC/E model39,

40

as an LJ dispersive-repulsive term plus Coulombic

interactions between the partial charges located on the H and O atoms: (α , β ) ij

u

= 4ε

(α , β ) ij

  σ (α , β )  (ijα , β )  rij 

12

  σ ij(α , β )  −  (α ,β )   rij

  

6

 qiα q βj + 1  4πε 0 rij(α ,β ) 

(1)

(α , β ) α In eqn. (1), rij is the distance between two sites i and j on molecules α and β , and q i and

q βj are the partial charges on sites i and j of molecules α and β and ε0 is the permittivity of free space. Ewald summations were used to correct the long range electrostatic interactions with a cut-off of 1.5 nm in the real space, and the LJ interactions were cut-off at 1.5 nm as well. The CNTs were treated as rigid, with a Lennard-Jones C-atom on each site.41 The molecular parameters are summarised in Table 2; cross parameters were estimated using the Lorentz-Berthelot mixing rules.42 Table 2. Lennard–Jones parameters, partial charges and con urational parameters for the carbon nanotube, single site methane and SPC/E water L-J parameters σ (nm)

Molecular model Y (nm) Z (nm)

Molecule

atom

ε / k B (K)

CNT CH4

C CH4

28.0 148.1

0.34 0.381

0.0

0.0

0.0

0.0

SPC/E H2O

O H

78.205 0.0

0.3166 0.0

0.0 ±0.081649

0.0 0.0577359

0.0 0.0

-0.82 0.41

X (nm)


Charge (e)

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2.2 NEMD simulations In the NEMD simulations, an external force Γ ex was imposed directly on the fluid molecules; unlike previous studies10, 38, 43, 44 the magnitude of the external force was set to achieve a given streaming velocity, according to the initial configuration and a pre-determined chemical potential gradient, following

Γex = −

dµ d ln f ln( f1 f2 ) = −kBT = kBT dz dz LCNT

(2)

where kB is the Boltzmann constant and T is the temperature of the system. The initial configuration was obtained from GCMC simulations at a mean fugacity f which is an average of the feed side and permeate side fugacities, f1 and f 2 respectively. The external force, Γ ex , applied on each molecule is determined according to equation (2), and is used in the relationship between the net fluxes and the Onsager coefficients:

 ∂µ1  L12   ∂z   L11 L12   Γex1   j1   L11 = − (3)  = j  L  L22   ∂µ2   L21 L22   Γex 2   2  21  ∂z  where, ji is the net flux of component i , Lii and Lij are the corresponding diagonal and offdiagonal Onsager coefficients, and Γ exi = − ∂µi ∂z is the corresponding external force on component i . The flux ji was determined by ji = ρi vcom,i , where ρ i is the density of component i calculated from GCMC simulations, and vcom,i is the streaming velocity of component i calculated from NEMD simulations.35 The streaming velocity was sampled every 2 ps and averaged over at least 50 ns. However, for the transport of methane in dry CNTs, the flux is determined as j1 = L1Γ ex1 . Methane is labelled as component 1 and water as component 2. The number of pre-adsorbed water molecules in wetted CNTs is pre-determined and, since there is no chemical potential gradient of water over the CNT, and the external force is only applied to methane molecules, Γ ex 2 = 0 . However, non-zero net fluxes were observed for both methane and water in the wetted tubes, which implies that the transport of methane and water are correlated, and L21 ≠ 0 . To verify the applicability of the Onsager reciprocal relationship, L12 = L21 and the 5 ACS Paragon Plus Environment

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dependency of the Onsager coefficient of water, L22 , on water content, we also applied an external force to the water molecules alone (leaving Γ ex1 = 0 ) to drive a flow in this component, enabling the Onsager coefficients, L12 and L22 to be determined from eqn. (3). While the Onsager coefficients of pure methane, L1 , were determined from NEMD simulations, the corrected diffusivities and Onsager coefficients, L2 , of pure water were extracted from the long time limit of the mean square displacement of the center of mass (COM-MSD) using the Einstein relation.35 The trajectory of water molecules was stored every 20 fs and sampled over 50 ns to calculate the COM-MSD of water in equilibrium molecular dynamics (EMD) simulations. Further details of the EMD simulations and analysis of the COM-MSD can be found in our previous work.35 Both the EMD and NEMD simulations were conducted using the LAMMPS package.45 NEMD simulations were started from GCMC simulations equilibrated at the target fugacity. The initial configurations of the water clusters were obtained from canonical Monte Carlo simulations by fixing the number of water molecules inside the CNTs. The MD simulations were run at 300 K with a time step of 1 fs and a Nosé-Hoover thermostat with a damping coefficient of 100 time steps. The streaming velocity of the fluid was subtracted when the thermostat was applied on methane at each time step, so that only the thermal velocity was adjusted. In the wetted CNTs, where methane and water can interact, the temperature of the water was adjusted by methane-water collisions alone.

No temperature difference was observed between methane and water except at

pressures below 0.5 bar, with the streaming velocity of the fluid being greater than 50 m/s. The (mean) maximum temperature difference between methane and water in these cases, was less than 4.0 K. The simulations were run for 100-150 ns, in which the first 50 ns was allowed for equilibration. When different magnitudes of the external forces were applied to methane molecules in CNTs of different sizes with different water contents in the NEMD simulations, the fluxes increased linearly with the external force, confirming that the simulations were within the linear response region. The example shown in Figure 1 (a) shows the variation of the fluxes of methane with external forces Γ ex1 > 0, Γ ex2 = 0 in a (10, 10) CNT containing 0.05 g/cm3 of pre-adsorbed water, at different pressures.

More convincingly, the reciprocal

relationship was obeyed by the Onsager cross-coefficients determined in our simulations. Figure 1 (b) depicts the variation of the Onsager cross-coefficients by applying an external force on methane only or by applying the external force on water only in the wetted (10, 10) CNT. The Onsager cross-coefficients from the NEMD method agree with each other within 6 ACS Paragon Plus Environment

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the error bar. In the present work, the error bars were calculated as the standard deviation of the (3 to 5) data points that are within the linear response region in the NEMD simulations.

0.18

0.30 at 1.0 bar at 2.0 bar at 5.0 bar

0.25

L21 L12

0.16

kBLij (nm-1ps-1K-1)

-2

-1

flux of CH4 (nm ps )

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0.20 0.15 0.10 0.05

0.14 0.12 0.10 0.08 0.06 0.04 0.02

0.00 0.00

0.05

0.10

0.15

0

0.20

2

external force Γex x10-13 (N)

4

6

8

10

12

14

16

fugacity of CH4 (bar)

Figure 1 (a) Variation CH4 flux with external force in the (10, 10) CNT in the presence of 0.05 g/cm3 preadsorbed water at 300 K, and (b) variation of the Onsager cross-coefficients determined by applying the external force on methane alone (L21) or on water alone (L12) in the (10,10) CNT with pre-adsorbed water at a density of 0.05 g/cm3 and at 300 K. Unless otherwise indicated, uncertainties are of the order of a symbol size.

3 Results and Discussion 3.1 Effect of pre-adsorbed water on the Onsager coefficients 3.1.1 Blocking effect of water clusters Pre-adsorbed water reduces the Onsager coefficient of methane in CNTs very significantly. For example, the Onsager coefficient of methane, L11 in the wetted CNT, having 0.10 g/cm3 of pre-adsorbed water, is about one order of magnitude lower than that ( L1 ) in the dry CNT, shown in Figure 2(a). Furthermore, the methane Onsager coefficient in the presence of water, L11 , decreases monotonically with increase in water loading. Initially, the pre-adsorbed water molecules always drift in the same direction as CH4 but remain as clusters due to their strong hydrogen bonding and weak water-carbon interactions. Representative snapshots of the configuration of water clusters in the CNTs having different amounts of pre-adsorbed water, depicted in Figure 3, show that each water molecule is hydrogen bonded to at least one other water molecule. The average hydrogen bond numbers per water molecule in the (10, 10) CNT at 1.0 bar are 2.69, 2.84 and 2.82 respectively corresponding to water contents of 0.025, 0.05


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and 0.10 g/cm3. The geometric criteria for the hydrogen bonding are described in detail in our previous studies.6, 24

(a)

1

-1

-1 -1

kBL1 and kBL11 (nm ps K )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0.1

L1, ρH O=0.0 g/cm3

0.01

L11, ρH O=0.025 g/cm3

2

2

3

3

L11, ρH O=0.025 g/cm

L11, ρH O=0.05 g/cm

2

2

3

L11, ρH O=0.10 g/cm3

L11, ρH O=0.05 g/cm 2

2

3

L11, ρH O=0.10 g/cm 2

0.001 0

2

4

6

8

10

12

14

16


Figure 2. (a)Variation of the Onsager coefficient L1 and L11 with pressure at different water contents in the (10, 10) CNT at 300 K, and (b) variation of the Onsager coefficient L2 with water content in the (10, 10) CNT at 300 K. In (a) symbols represent simulation results, and lines represent those predicted using eqn (7b).

Figure 3. Snapshots of methane and water transport in the (10, 10) CNTs having different water contents, at 1.0 bar and 300 K. The white and red spots represent hydrogen and oxygen atoms of water and the yellow spheres are methane molecules. Half of the CNT is removed for ease in visualization. The reason for the dramatic reduction in the Onsager coefficient of CH4 in the presence of pre-adsorbed water is that the water clusters severely block the diffusion channel for CH4 and act like freely moving flexible pistons to block the diffusion of methane. We calculated the ratio of the number of methane molecules that could hop into the narrow space between a water cluster and the carbon wall, to the total number of methane molecules. Figure 4 (a) illustrates an example of an active methane molecule that successfully hops into such a space. All methane molecules entering the cluster regions were counted in evaluating the fraction of successful methane molecules. Additionally, the ensemble-averaged ratio of the total length of the water clusters, to the length of the CNT was calculated. The length of each water cluster 8 ACS Paragon Plus Environment

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was evaluated as the distance in the axial direction between the farthest separated oxygen atoms of two water molecules belonging to the same cluster. As illustrated in Figures 4 (b) and (c), the ensemble averaged fractions of active methane molecules are effectively invariant with fugacity having a maximum value of 2.25% in the (10, 10) CNT with 0.10 g/cm3 preloaded water at 15 bar. The overall length of the water clusters increases almost linearly with the water content in the (10, 10) CNT (Fig. 4(c)), showing that the radial size of the water clusters is almost constant. Specifically, the length of the water clusters, l , defined as the number of oxygen atom diameters, can be evaluated as, l = ρ H 2OVCNT n A , where ρ H 2O is the water density, VCNT is the volume of the CNT, and n A is the number of water molecules occupying the cross-sectional area of the CNT. nA is required to stay constant at different water densities, so that the length of water clusters, l , increases linearly with water density. Considering that the mean radial distance between two hydrogen bonded water molecules is independent of the water density, the radial size of the water clusters stays constant as a result of constant n A at different water densities.

2.5 (b)

3

ρΗ Ο=0.025 g/cm 2

3

ρΗ Ο=0.05 g/cm

percentage (%)

2.0

2

3

ρΗ Ο=0.10 g/cm 2

1.5 1.0 0.5 0.0 0

2

4

6

8

10

12

14

16

fugacity (bar)

25

(c)

6 (d)

20

3

ρΗ Ο=0.025 g/cm

5

2

3

ρΗ Ο=0.05 g/cm

percentage (%)

percentage (%)

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2

15

3

ρΗ Ο=0.10 g/cm 2

10 5

4 3 2 1

0

0

0

2

4

6

8

10

12

14

16

0.0

fugacity (bar)

0.2

0.4

0.6

0.8

1.0

normalized postion within the cluster region

Figure 4. (10, 10) CNT with pre-adsorbed water at 300 K. (a) Snapshot of an active methane molecule in the presence of a water cluster, taken from the (10, 10) CNT in the presence of 0.05 g/cm3 water at 1.0 bar, (b) the fraction of active methane molecules, (c) ratio of the overall length of the water clusters to the length of the CNT, and (d) density distribution of the


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active methane along the axial direction of the CNT with 0.10 g/cm3 pre-adsorbed water at 15 bar, normalized by cluster length. Further, we divided each cluster region into 100 bins in the axial direction, and calculated the ensemble averaged ratio of the active methane molecules distributed in each bin to the total number of the active methane molecules in the CNT. The normalized density distribution of the active methane molecules along the axial direction is plotted in Figure 4 (d) for the (10, 10) CNT having 0.1 g/cm3 pre-adsorbed water at 15 bar, with an external force on the methane of

0.114 × 10−13 N . This figure shows that the fraction of methane molecules in the central region of the water clusters is nearly zero due to its hydrophobicity, other than a few that enter the cluster region because of thermal fluctuation. This explicitly demonstrates the strong blocking effect arising from water clusters on the diffusion of methane in the wetted CNT. Similar normalized density distributions for active methane molecules were observed in all the other cases considered. Accordingly, it is found the streaming velocities of methane and water in the wetted CNTs are always identical, showing that there is no relative diffusion between these two components. Additionally, as a matter of interest, we depict the adsorption isotherms of CH4 in dry and wetted (10, 10) CNTs in Figure S1. We find that the amount of methane adsorbed decreases monotonically with increase in water content in the (10, 10) CNT, which is mainly attributed to the depletion of available adsorption volume. Further, qualitatively similar results were observed in the experiment of Billemont et al.46, who investigated the adsorption of methane in moisture-equilibrated activated carbon. In their experiments, the carbon was first saturated with water, and the water content subsequently adjusted by desorption under water vapor at known pressure, so the system was free of influence by air. They reported the adsorption of CH4 in the moisture-equilibrated carbon was noticeably reduced by the pre-adsorbed water. On the other hand, Holt et al.15 measured the water flow rate in vertically aligned doublewalled CNT membranes having nanopores of diameter less than 2 nm, and reported the flow rate of water was more than 3 orders of magnitude higher than that predicted by the HagenPoiseuille equation. Although the transport of water in the hydrophobic CNTs is considered to be fast, our simulation results (Figure 2) directly reveal the inhibitory effect of water clusters on the diffusion of methane in the CNTs. It is seen that the pre-adsorbed water reduces the adsorption and the transport of gas species at the same time, emphasising the necessity of dehydration in adsorptive and membrane gas separation applications. Nevertheless, this being


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a pioneering simulation study on the diffusion of these immiscible binary components in hydrophobic nanopores, systematic experiments are required to support our findings. 3.1.2 Prediction of the Onsager coefficients In this section we propose a theoretical model for the relation between the Onsager coefficient L11 , the water loading and the corresponding Onsager coefficient of water L2 . Since there is no relative diffusion between methane and water, the external force applied on methane molecules, in the absence of any force on the water molecules, is balanced by the wall friction experienced by methane and water molecules together, so that

Γ ex1n1 = ε1n1u1 + ε 2n2u2

(4)

where ni is the adsorbed number of molecules of component i , ε i is the friction coefficient of component i , and ui is its streaming velocity with u1 = u2 = u . For the diffusion of a single component, we have −

d µi Γ = Γexi = ε i ui . Hence, we obtain the flux ji = − ρ i exi , where ρ i is dz εi

the density of the fluid. Finally, since, ji = − Li

d µi we can extract a friction coefficient for dz

the pure component as, ε i = ρ i Li . As an approximation, we adopted the friction coefficients calculated from each single component confined in the CNT to represent those for the methane and water mixture, and rewrite equation (4) as，

Γex1n1

( n1ρ1 L1 ) + ( n2 ρ 2 L2 ) u

(5)

Since Γ ex1 > 0 and Γ ex 2 = 0 , the flux of methane diffusing in the wetted CNTs can be determined as j1 = L11Γ ex1 = ρ 1u

(6)

Substituting eqn. (5) into eqn. (6), gives the Onsager coefficient L11 as

L11

ρ1n1

 ( n1ρ1 L1 ) + ( n2 ρ 2 L2 ) 


(7a)

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The number of molecules of component i is ni = ρiVCNT , where VCNT is the volume of the CNT, so eqn. (7a) can be rewritten as L11

1 2

1  ρ2  1 + L1  ρ1  L2

(7b)

The Onsager coefficients L22 can be similarly predicted using eqn. (7) with the subscripts exchanged, as the derivation does not depend on a specific component. The values of L2 are explicitly determined since the number of pre-adsorbed water molecules is fixed. To fit the Onsager coefficient L1 as a function of the density of pure CH4 we use a fourth order polynomial Li = ao + a1ρ i + a2 ρ i2 + a3 ρ i3 + a4 ρ i4

(8)

The fitting parameters are listed in the Table S1 in the Supporting Information. The Onsager coefficients L11 calculated using eqn. (7) and the Onsager coefficients L2 of pure water at the water loadings 0.025, 0.05, and 0.10 g/cm3 at 300 K are shown in Figures 2 (a) and (b). The Onsager coefficients estimated from eqn. (7) agree quantitatively with the results calculated from our simulations, within a relative deviation of less than 20% for most of the data points considered. The maximum deviation was 45%, in the (10, 10) CNT having 0.025 g/cm3 water, at 0.5 bar. The discrepancy between the model and simulation can be attributed to fluid-fluid interactions which also contribute to the overall potential energy experienced by the fluid molecules, leading to a reduction in the friction coefficients of the mixture components with respect to those of the pure components. A detailed study of the contribution of fluid-fluid interactions to the diagonal Onsager coefficients for mixtures in nanopores would be a suitable topic for future work. The Onsager coefficients L2 increase almost linearly with the water content. As discussed above the radial structure of water clusters is essentially identical for all the water densities considered in this work. For the single component, the flux can be either calculated by ji = − Li (d µi dz ) , or ji = −( ρi Do ,i / k BT )( d µi dz ) , where Do ,i is the corrected diffusivity of

the component i, such that we can obtain the relation between the Onsager coefficient and the corrected diffusivity as Li = ρ i Do ,i k B T . It can further be concluded that the corrected


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diffusivity of water is weakly dependent on the water density as long as the water density is high enough to develop clusters large enough to nearly block the cross sectional area of the CNT. Substitution of

2 ( ρ 2 ρ1 )

L2 = ( ρ 2 ρ12 ) ( k BT / Do,2 ) into (7b) with the corrected

diffusivity Do ,2 being almost independent of density, explains why the diagonal Onsager coefficient of CH4, L11 , decreases with increase in the content of preloaded water. Since there is no relative diffusion between methane and pre-loaded water, it is expected that the diffusion of these two components is strongly coupled. To quantify the extent of the coupling, an interaction factor α12 may be introduced47, 48 defined as,

α12 =

L12 L11L22

(9)

Figure 5 confirms that in the (10, 10) CNT having 0.05g/cm3 preadsorbed water, the Onsager coefficients L12 calculated from our simulations and estimated according to L12 = L11L22 are almost identical, showing that α12

1 . As in earlier work47, 48, when correlation is dominant

the interaction factor will be ~1 and approaches zero when the fluid-fluid correlation is negligible. We note here, in previous studies in which the two components are miscible 0 < α12 < 1 .37, 48

0.20 L12

simulated L12

L11 L22

0.15

2.0

0.10

1.5

α12

kBL12 (nm-1ps-1K-1)

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1.0 0.5

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0.0 0 2 4 6 8 10 12 14 16

fugacity (bar)

0.00 0

2

4

6

8

10

12

14

16

fugacity (bar)

Figure 5. Variation of the Onsager cross-coefficients with pressure in the (10, 10) CNT in the presence of 0.05 g/cm3 water, at 300 K. The inset depicts the corresponding interaction factors determined as α12 = L12 L11L22 . The filled triangles represent the predictions based on 13 ACS Paragon Plus Environment

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L12 = L11L22 , using simulation values of L11 and L22. The dashed line is the predicted curve for L12 using eqn.(7) along with L12 = L11L22 . The Onsager coefficients L22 for different contents of pre-adsorbed water when the external force is applied to water ( Γ ex1 = 0, Γ ex2 > 0 ), are depicted in Figure 6. Figure 2 (b) shows that the Onsager coefficient L2 increases with increase in water content, but ( ρ1 ρ2 ) decreases; it 2

follows from eqn. (7) that at high water loading, the contribution from L1 in determining the Onsager coefficient L22 is weaker compared to the case where the loading of water is low. On the other hand, our simulations show that the value of ( ρ1 ρ 2 ) L1 increases with the pressure. 2

Consequently, the Onsager coefficient, L22 , increases with increase in water loading and decreases with increase in the pressure of methane. Figure 6 also confirms the accuracy of the expression, L22 = L12 2 L11 , for the estimation of L22 from NEMD simulations with Γ ex1 > 0 , Γ ex2 = 0 . The Onsager coefficients L22 for different water loadings predicted by eqn.

(7) are also given in Figure 6. At low pressure, the Onsager coefficients L22 determined from NEMD simulations converge to those of pure water, L2 , calculated from EMD simulations.

1 3

kBL22 (nm-1ps-1K-1)

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3

L22, ρΗ2Ο=0.025 g/cm

L22, ρΗ2Ο=0.025 g/cm

3

3

L22, ρΗ2Ο=0.05 g/cm

L22, ρΗ2Ο=0.05 g/cm

L22, ρΗ2Ο=0.10 g/cm3

L22, ρΗ2Ο=0.10 g/cm3

0.1

0.01 0

2

4

6

8

10

12

14

16


Figure 6. Variation of the Onsager coefficient L22 with pressure in the (10, 10) CNT, at 300 K, and different water contents. The solid symbols are calculated directly from our simulations 2 and the open symbols are approximated by L12 / L11 . Lines are the predicted curves calculated from eqn. (7). 14 ACS Paragon Plus Environment

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Since the interaction factor α12

1 in this system, all the diagonal and off-diagonal

coefficients can be estimated by a single set of simulations, Γ ex1 = 0, Γ ex2 > 0 , or Γ ex1 > 0, Γ ex2 = 0 . The Onsager cross-coefficients predicted by eqn. (7) with α12=1 are shown

in Figure 5. The agreement is satisfactory, although the fitted coefficients are underestimated compared to those from simulation; the maximum relative deviation being 28% at 1.0 bar. 3.2 CNT size effect on the Onsager coefficients Figure 7 (a), shows that for methane in the absence of water, the Onsager coefficient L1 decreases with increase in the diameter of CNTs, and the Onsager coefficients in the (7, 7) CNT are about one order of magnitude higher than those in the (10, 10) and (15, 15) CNTs. At rather low loadings, Skoulidas et al.11, 49 and Falk et al.50 reported that reducing the diameter of the CNT enhanced the smoothness and thus increased the specular reflection coefficient of the carbon wall. However, Figures 8 (a)-(c), which depict the axial potential energy profiles of methane inside CNTs with different diameters, reveal that increasing the curvature of the CNT does not directly enhance the smoothness of the CNT in the axial direction. The ensemble averaged axial potential energy is determined by placing probe molecules at radial positions one fluid-CNT interaction diameter, σ sf , away from the CNT wall, and subsequently calculating and averaging the fluid-CNT interactions over all the positions. Nevertheless, while the radial potential energy profile is a double well in (10, 10) and (15, 15) CNTs with a barrier separating the two wells, the two wells almost merge into a single well at the pore center in the (7, 7) CNT with only a small barrier separating them, evident from Figure 8(d). At the low loading, the ability for a molecule to overcome the barrier and migrate from one side of the cross-section to the other is dependent on the ratio of its radial kinetic energy to the height of energy barrier, i.e. a molecule can readily overcome the barrier when the kinetic energy is comparable to or higher than the energy barrier.51 The estimated ratios of the radial kinetic energy ( k B T / 2 ) to the energy barrier are 0.9332, 0.130 and 0.107 for methane in the (7, 7), (10, 10) and (15, 15) CNTs respectively. Consequently, the molecule oscillates within the potential wells in the (10, 10) and (15, 15) CNTs, and is unable to overcome the barrier, leading to higher frequency of fluid molecule-CNT wall collision frequencies in the (10, 10) and (15, 15) CNTs. Analogous to our explanation of the “floating molecule” or“levitation” effect51, reported by Derouane et al.52 and Yashonath and Santikary53, molecules in the (7, 7) CNT experience very weak van der Waals force in the radial direction, due to the much flatter 15 ACS Paragon Plus Environment

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radial potential energy profile, and oscillate over the whole cross-sectional area of the nanopore, colliding with the CNT wall with much lower frequency with respect to the (10, 10) and (15, 15) CNTs. Therefore, dramatically enhanced Onsager coefficients are observed in the narrow (7, 7) CNT with respect to the (10, 10) and (15, 15) CNTs, and the Onsager coefficient monotonically decreases with increase in the diameter of the CNT. Furthermore, it was shown by Jakobtorweihen et al.54 that as temperature, and thus the kinetic energy, increased, the fluid molecules eventually overcame the barrier in CNTs possessing larger diameters, and the Onsager coefficients moved towards an opposite dependency on the diameter of the CNT as the average time between successive collisions increases. It has been observed by Jakobtorweihen et al.55,

56

and Chen et al.57 that while methane

diffusivities in flexible CNTs are significantly lower than those in rigid CNTs at low loadings, they quantitatively converge at high loadings. This reveals that although the smoothness of the CNT wall is significantly enhanced by treating the CNT as rigid, it has insignificant effect on the diffusion coefficient at high loadings, where fluid-fluid interactions dominate. It is very interesting to note that the isosteric heat of methane decreases with increase in pressure in the narrow (7, 7) CNT while increasing in the larger (10, 10) and (15, 15) CNTs, evident in Figure 9(a). As a result of strong adsorbate-CNT interactions,7 the adsorption of CH4 in the (7, 7) CNT approaches saturation at very low pressure (Figure S2), leading to repulsive adsorbate-adsorbate interactions even at low pressure and the decreasing dependency of the isosteric heat on pressure. The adsorption isotherms of methane in the absence and presence of 0.05 g/cm3 water in the (7, 7), (10, 10) and (15, 15) CNTs are depicted in Figure S2. Due to the nearly flat potential energy profile of methane-CNT in the (7, 7) CNT, collision between methane molecules enhances the methane-CNT collision frequency, leading to decrease in the Onsager coefficient (and the corrected diffusivity, Figure S3) of methane with increase in pressure. The corrected diffusivities of pure methane and of pure water at a density of 0.05 g/cm3 in the (7, 7), (10, 10) and (15, 15) CNTs are depicted in Figure S3.


1

(a)

(b) -1 -1

10

kBL11 (nm ps K )

1

0.1

-1

-1 -1 -1

0.1 CH4 in (7, 7) CNT CH4 in (15, 15) CNT CH4 in (10, 10) CNT H2O in (7, 7) CNT H2O in (10, 10) CNT H2O in (15, 15) CNT

0.01

0.001

L11 in (7, 7) CNT L11 in (10, 10) CNT

0.01

L11 in (15, 15) CNT L11 in (7, 7) CNT L11 in (10, 10) CNT L11 in (15, 15) CNT

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1 (c)

ratio of L11/L1

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kBL1 and kBL2(nm ps K )

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(7, 7) CNT,ρΗ Ο=0.05 g/cm3 2

(10, 10) CNT,ρΗ Ο=0.05 g/cm3

0.1

2

(15, 15) CNT,ρΗ Ο=0.05 g/cm3 2

0.01 0

2

4

6

8

10

12

14

16

fugacity (bar)

Figure 7. Variation of the Onsager coefficients (a) L1 , and (b) L11 with pressure in the (7, 7), (10, 10) and (15, 15) CNTs in the presence of 0.05 g/cm3 water at 300 K, and (c) the corresponding ratios L11 / L1 in these CNTs. The Onsager coefficients L2 of 0.05 g/cm3 water in these three CNTs are also depicted in (a). The lines in (b) represent L11 calculated from eqn. (7).


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-15.490

(a) in (7, 7) CNT

potential energy (kJ/mol)


-22.840 -22.845 -22.850 -22.855 -22.860 -22.865

(b) in (10, 10) CNT -15.495 -15.500 -15.505 -15.510 -15.515 -15.520

-22.870 0.0

0.1

0.2

0.3

0.4

0.0

0.5

0.1

0.2

z (nm)


10


(c) in (15, 15) CNT -12.615 -12.620 -12.625 -12.630

0.2

0.5

(d) radial potential energy

0 -5 -10 -15 -20 in (7, 7) CNT in (10, 10) CNT in (15, 15) CNT

-30 -0.8

-12.640 0.1

0.4

5

-25

-12.635

0.0

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z (nm)

-12.610

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0.5

-0.6 -0.4

-0.2

0.0

0.2

0.4

0.6

0.8

radial position (nm)

z (nm)

Figure 8. (a)-(c) Axial potential energy distributions, and (d) radial potential energy profiles of methane experienced in empty CNTs.

28

60

(a)

number density (nm )

26

-3

isosteric heat of methane (kJ/mol)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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24

in (7, 7) CNT in (10, 10) CNT in (15, 15) CNT

22 20 18 16

(b)

(7, 7) CNT (10, 10) CNT (15, 15) CNT

50 40 30 20 10 0

14 12 0

2

4

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10

12

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16

fugacity (bar)

0.2

0.4

0.6

0.8

1.0


Figure 9. (a) Variation of the isosteric heat of methane with pressure and (b) the radial density distribution of methane at 15 bar, in the (7, 7), (10, 10) and (15, 15) CNTs, at 300 K. However, in the larger CNTs, the isosteric heat increases with pressure, implying that adsorption is far from saturation (Figure S2) and the adsorbate-adsorbate interactions are attractive. It is shown in Figure 9 (b), at 15 bar, the adsorption of methane in the central region of the (10, 10) and (15, 15) CNTs is weak, which also indicates that the interactions 18 ACS Paragon Plus Environment

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between the molecule adsorbed in the potential energy well and that adsorbed in the central region is insignificant compared to the adsorbate-adsorbate interactions between those that are both located in the potential energy well. It is also observed in our simulations that the peak of the radial density profiles of CH4 does not change with pressure in the (10, 10) and (15, 15) CNTs. In this regard, the adsorbate-adsorbate interactions make further contributions to strengthen the potential energy well in the (10, 10) and (15, 15) CNTs at the high pressures. Consequently, the possibility for the adsorbate to overcome the energy barrier to migrate from one side of the CNT to the other side is reduced at high pressures, leading to enhancement in the adsorbate-CNT collision frequency for the adsorbate located around the potential energy well. As a result, the corrected diffusivity of methane decreases with pressure in the (10, 10) and (15, 15) CNTs (Figure S3). However, the increasing trend of the Onsager coefficient of CH4 with pressure in these two CNTs is a result of the enhanced adsorption of CH4 at high pressures, considering Li = ρ i Do ,i k BT .

30 -3

number density (nm )

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(7, 7) CNT (10, 10) CNT (15, 15) CNT

25

in (7, 7) CNT

in (10, 10) CNT

20 15

in (15, 15) CNT

10 5 0 0.0

0.2

0.4

0.6

0.8

1.0


Figure 10 Radial density distributions of water in the CNTs at 300 K with the water loading being 0.05 g/cm3, obtained from EMD simulations. Further, as shown in Figures 7(a)-(c), the Onsager coefficient L11 in the binary mixture is much smaller than L1 in all the CNTs investigated due to the blocking effect of water clusters. While the Onsager coefficient L1 demonstrates strong dependency on the diameter of the CNTs, the Onsager coefficient of pure water, L2 , is almost independent of the diameter, k B L2 , being 0.114, 0.092, 0.1077 nm-1ps-1K-1 for the (7, 7), (10, 10) and (15, 15) CNTs having 0.05 g/cm3 of pre-adsorbed water. This near invariance of L2 is explained by the observation that water molecules move as hydrogen-bonded clusters rather than as individual molecules. As illustrated in Figure 10 and Figure S4, water clusters generally form planar structures in the narrow (7, 7) CNT and stratified 3-dimensional structures in larger (10, 10) and (15, 15) 19 ACS Paragon Plus Environment

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CNTs. In Figure 10, the radial density distributions and snapshots of the configuration of water clusters are obtained from our EMD simulations. The radial and axial projections of the water clusters in these three CNTs are further provided in Figure S4, which are also obtained from EMD simulations. In the narrow (7, 7) CNT all the water molecules are equally subject to the wall potential, while in the (10, 10) and (15, 15) CNTs the molecules located at the outermost layer of the water clusters interact strongest with the carbon wall. As discussed above, the water molecules near the wall surface experience higher frequency of collisions with the CNT wall in the larger CNTs. It might therefore be anticipated that at rather low density, the diffusion coefficient of water should also decrease with increase in the size of the CNT. Nevertheless, it was reported in earlier works that the Maxwell coefficient of fluid in nanotubes is dependent on the fluid density43, and at extremely high density at which fluidfluid interactions dominate the Maxwell coefficient of L-J fluid confined in the CNTs increases with the diameter of the CNTs.10 Similarly, the slip length of liquid water in a CNT was found to decrease with increase in the diameter of the CNTs, an indication that the interfacial friction coefficient between the liquid water and the carbon wall is enhanced in larger CNTs.50, 58 It is found at the water content of 0.05 g/cm3, the average hydrogen numbers per water molecule calculated from EMD configurations in the (7, 7), (10, 10) and (15, 15) CNTs are 2.11, 2.81 and 2.92 respectively. In consideration of the strong confinement, the water clusters in the CNTs must be assembled by hydrogen bonding as in bulk liquid water. The coordination number of water in the (7, 7) CNT is smaller because of the stronger geometric confinement in this CNT, as illustrated in Figure 10. Therefore, our simulations show that, at this water density, while the friction coefficient for the water molecules in the outermost layer is enhanced in the larger tubes, the fraction of molecules experiencing this larger friction force directly is reduced. Water molecules enclosed in the inner region of the cluster only exchange momentum with the molecules residing in their outermost layer, and the flow of water in the CNTs within the size range investigated is typically plug-like (the radial streaming velocity profile is flat rather than Poiseuille type), so that the water molecules enclosed in the inner region experience negligible (viscous) friction43, 59. Consequently, as depicted in Figure 7(a), in the (7, 7), (10, 10) and (15, 15) CNTs having 0.05 g/cm3 water, the values of the Onsager coefficient L2 (and the corrected diffusivity, in Figure S3) are almost constant, and much smaller than L1 , with the difference between L1 and L2 becoming smaller as the diameter of the CNTs is increased. Based on our EMD simulations, the calculated fractions of water molecules in the outermost layer of the water clusters are 1.0, 0.846 and 20 ACS Paragon Plus Environment

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0.545 respectively for the (7, 7), (10, 10) and (15, 15) CNT, at the water content of 0.05 g/cm3. Here, the outermost layer of water clusters is defined as the molecules located within the first peak region of the radial density distribution depicted in Figure 10 with a peak width within

1 1 [rp − σ O − C , rp + σ O −C ] , where σ O −C = 0.3283 nm is the effective interaction diameter of the 2 2 oxygen atom of water with the carbon wall and rp is one effective diameter, σ O −C , away from the wall. However, for the narrow (7, 7) CNT with a diameter of 0.95 nm, this region is

1 1 defined as [0, rp + σ O −C ] , since rp − σ O −C < 0. 2 2

Because L2 is very much less than L1 in the (7, 7) CNT and the densities of methane and water (H2O, 0.05 g/cm3 = 1.672 nm-3) are similar (see Figure S2 for the isotherms), the diagonal Onsager coefficient L11 in the narrow CNT is mainly determined by L2 as confirmed by eqn. (7). Consequently, the ratio of L11 L1 is generally smaller than 5% in the narrow (7, 7) CNT having 0.05 g/cm3 water, depicted in Figure 7 (c). Figure 7 (c) also demonstrates that the blocking effect of water clusters on the diffusion of methane is generally enhanced as the CNT diameter is reduced. However, when the pressure is low, below 1.0 bar, the blocking effect arising from water clusters shows much weaker dependency on the tube size. While the adsorption of methane in the (7, 7) CNT demonstrates weak dependency on the pressure as a result of the strong adsorbate-carbon wall interactions,7, 24 and are essentially saturated, the adsorption of methane in the wetted (10, 10) and (15, 15) CNTs is strongly dependent on the pressure, especially in the low pressure region, evident from Figure S2. According to this, at low pressure the term ( ρ 2 ρ1 ) L2 is dramatically enhanced in the wider (10, 10) and (15, 15) 2

CNTs with respect to that in the (7, 7) CNT, with the result that the water blocking effect in these two CNTs is as strong as that in the (7, 7) CNT when the pressure is below 1.0 bar. As shown in Figure 7 (c), the L11 L1 ratios are all below 7.0% in these three CNTs, in the low pressure regime. Further, it is demonstrated in Figure 7(a) that unlike (7, 7) CNTs, in which the Onsager coefficient, L1 is generally two orders of magnitude higher than L2 , L2 is of the same order of magnitude as L1 at pressures up to 15 bar in the larger (10, 10) and (15, 15) CNTs. Further, the term ( ρ 2 ρ1 ) L2 is dramatically reduced in the (10, 10) and (15, 15) 2


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CNTs at high pressure, compared to that at pressures below 1.0 bar. Therefore, the ratios of L11 L1 in the (10, 10) and (15, 15) CNTs are generally much larger than that in the (7, 7) CNT,

showing that the pre-adsorbed water clusters impose a weaker blocking effect on methane diffusion in the larger CNTs. Similarly, the Onsager coefficient L1 and the ratio L11 L1 in the (10, 10) CNT are always larger than those in the (15, 15) CNT, due to the enhanced adsorption and diffusion of methane in the (10, 10) CNT compared to those in the (15, 15) CNT, evident from Figure S2 and Figure S3. 4 Conclusions The transport diffusion of CH4 at 300 K and at pressures up to 15 bar, in dry CNTs and in CNTs wetted with pre-adsorbed water, having diameters ranging from 0.95 to 2.034 nm, has been investigated using NEMD simulations. In the NEMD method proposed in this work, the force is imposed on a single species in the mixture, and is determined according to the magnitude of a virtual gradient in chemical potential; this provides an efficient way to tackle the diffusion of immiscible species that are strongly interacting. Three pre-adsorbed water contents, 0.025, 0.05 and 0.10 g/cm3, were chosen. It was found that the diffusion coefficient of pure CH4 in the dry (7, 7) CNT with pressure up to 15 bar is one order of magnitude higher than those in the (10, 10) and (15, 15) CNTs. Contrary to earlier studies,11, 50 we have found the smoothness of the CNT wall does not increase with enhancement in the curvature of the CNTs. Hence, the enhanced diffusion coefficient in the narrower CNT is a result of shallow radial potential wells in the potential field exerted by the CNT wall, so that the gas molecule in the (7, 7) CNT can overcome the energy barrier separating the two weak potential wells, which reduces the fluid molecule-CNT wall collision frequency. The corrected diffusivity of pure water in the CNTs is almost independent of the water density and the CNT size. When the density of water is high enough to develop clusters that fill the cross-sectional area of the CNT, the Onsager coefficient, L2 , increases linearly with the water content, as the diameter of the water clusters is constant in the CNT. At a fixed water loading of 0.05 g/cm3, the corrected diffusivities of pure water in the (7, 7), (10, 10) and (15, 15) CNTs are very similar, because in the larger CNTs the radial size of the water clusters expands with the CNT diameter and a larger fraction of water molecules is enclosed in the central region of the cluster in comparison to that in the narrow (7, 7) CNT. These enclosed molecules exchange momentum with molecules that are colliding


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with the wall, but do not experience wall friction directly, and the larger fraction of enclosed molecules contributes to an increased diffusion coefficient for water, overcoming the negative effect of the enhanced friction coefficient for strongly hydrogen bonded liquid water in the larger CNTs. Consequently, the diffusion of water in CNTs has a weak dependence on CNT diameter. Water clusters pre-adsorbed in CNTs act like pistons to severely block the diffusion channel of CH4. As a consequence, the diagonal Onsager coefficient L11 of methane in the wetted CNTs is dramatically reduced by water clusters, especially in the narrow (7, 7) CNT which has a much higher diffusion coefficients for CH4 than the larger CNTs. The Onsager coefficient L11 of methane in the (7, 7) CNT having 0.05 g/cm3 is only about 5% of the Onsager coefficient, L1 of pure CH4. The diffusion of methane and water are strongly coupled and no relative diffusion between these two components is observed. We propose a new model using the diffusion coefficients of the pure components to predict the diagonal Onsager coefficients, L11 , of CH4 in the wetted CNTs, and explain the underlying mechanism for the decrease in the dependency of L11 on water loading and the strong dependency on the blocking effect of water clusters on the CNT diameter. Our NEMD simulations associated with the theoretical model provide a fundamental understanding of the coupled diffusion of gas species and water confined in hydrophobic nano-channels, which may benefit emerging technologies in adsorptive gas storage and carbon capture, besides those for nanofiltration and biomedical applications.20, 24, 36

Acknowledgement This work has been supported by a grant from the Australian Research Council through the Discovery scheme (Grant No. DP150101824. Chunxia Hu gratefully acknowledges the Chinese Government for a CSC (China Scholarship Council) scholarship. This research was undertaken with the assistance of the computational resources provided at the NCI National Facility systems at the Australian National University (ANU), and at the Pawsey Supercomputing Centre in Western Australia, through their National Computational Merit Allocation Schemes supported by the Australian Government and the Government of Western Australia. 23 ACS Paragon Plus Environment

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Supporting Information Figures showing the isotherms of methane in CNTs of different diameter at various levels of water loading, the pressure variation of the corrected diffusivity of methane in the absence and presence of pre-adsorbed moisture, and snapshots of water clusters in CNTs of different size are given in the Supporting Information. This information is available free of charge via the Internet at http://pubs.acs.org.


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Table of Contents Graphic

1

-1

-1 -1

kBL1 and kBL11 (nm ps K )

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0.1

L1, ρH O=0.0 g/cm3

0.01

L11, ρH O=0.025 g/cm3

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L11, ρH O=0.025 g/cm3

L11, ρH O=0.05 g/cm3

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3

L11, ρH O=0.10 g/cm3

L11, ρH O=0.05 g/cm 2

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3

L11, ρH O=0.10 g/cm 2

0.001 0

2

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Increased pore blockage by water on increasing moisture loading, with inhibition of CH4 transport


Inhibitory Effect of Adsorbed Water on the ... - ACS Publications

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