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Effects of Flange Adsorption Affinity and Membrane Porosity on Interfacial Resistance in Carbon Nanotube Membranes Lang Liu, David Nicholson, and Suresh K Bhatia ACS Appl. Mater. Interfaces, Just Accepted Manuscript • DOI: 10.1021/acsami.8b08886 • Publication Date (Web): 11 Sep 2018 Downloaded from http://pubs.acs.org on September 16, 2018

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ACS Applied Materials & Interfaces

Effects of Flange Adsorption Affinity and Membrane Porosity on Interfacial Resistance in Carbon Nanotube Membranes Lang Liu, David Nicholson and Suresh K. Bhatia* School of Chemical Engineering, The University of Queensland Brisbane, QLD 4072, Australia

ABSTRACT We have used non-equilibrium molecular dynamics (NEMD) simulations to investigate the transport diffusion of methane, at 300 K and pressures of up to 15 bar, in 30 nm long (10, 10) CNTs held between two flanges mounted at the ends to represent the surface layers of an embedding matrix material. Strong interfacial resistance to the entry and exit of molecules is found in the 30 nm long CNTs, which reduces their permeability by more than two orders of magnitude. Increasing the adsorption affinity and surface area of the flange reduces the interfacial resistance and consequently enhances the methane diffusivity in CNT membranes. Curved streamlines near the flange surface make a significant contribution to the permeability, even when the adsorption on the matrix surface is negligible. We propose a model to calculate the separate components of the interfacial resistance, the flange resistance, which increases with increase in the membrane porosity, and the entrance-exit resistance which is independent of the membrane porosity. While the flange resistance accounts for the reduction of interfacial resistance with decrease in the membrane porosity, the entrance-exit resistance is responsible for the reduction of interfacial resistance with increase in the flange adsorption affinity. The flange resistivity demonstrates a complex dependency on the flange adsorption affinity, which is attributed to the competition between the enhanced adsorption and the enhanced migration time of the molecules on the flange. It is concluded that the embedding matrix adsorption affinity and membrane porosity separately play critical roles in determining the interfacial resistance and permeability in CNT membranes. Our simulation results can help reduce the interfacial resistance and improve the permeance in CNT membranes by appropriate choice of inter-tube spacing and flange material, and are readily applied to all nanoporous membranes with a passive matrix. KEYWORDS: interfacial resistance, membrane, carbon nanotubes, permeability, adsorption affinity, membrane porosity *To whom correspondence may be addressed. Email: [email protected].

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1 Introduction Porous graphene and carbon nanotubes (CNTs) are promising materials for membrane separation1-4 due to their atomically smooth carbon walls, their tuneable pore size and because they can be chemically functionalized. In the past decade, considerable effort has been devoted to investigating molecular exchange at the interface between the intracrystalline space of a nanoporous carbon membrane and its surroundings, addressing the fact that the barriers to mass transport at the entrance and exit interfaces play an essential role in determining the permeance.5-11 The barriers to mass transport at the entrance and exit interfaces are collectively defined as the interfacial resistance. Permeance was quantitatively calculated in our previous study5, which showed that interfacial resistance accounted for more than 99% of the transport resistance for methane penetrating a (10, 10) CNT of length 100 nm, and reduced the corrected diffusivity by more than 2 orders of magnitude with respect to that in the absence of interfacial resistance. Although both simulations and experiments have confirmed the high rate of transport in CNTs,12-15 there are large discrepancies between reported measurements. In the experiments of Holt et al., the measured flow rates of water in CNTs, having an average pore diameter of 1.6 nm and a length of 2  m , embedded in a silicon nitride matrix, were reported to be 560-8400 times of those predicted by the no-slip Hagen-Poiseuille equation.13 Later, Qin et al.15 used a transistor array to monitor the current jumps occurring at different positions along individual, millimetre long CNTs, with diameters ranging from 0.81 to 1.59 nm, rather than measuring the average flow rate of water in CNT membranes. They found that enhancement factors for water flow rates, relative to the predictions of continuum theory, were between 51 and 882, suggesting that the flow rates reported by Holt et al.13 were greatly overestimated. A possible cause of this discrepancy is that their CNTs were embedded in silicon nitride which is hydrophilic and which may have significantly modified the permeability. Therefore, understanding the influence of the interplay between the matrix material, the fluid molecules, and of the porosity of CNT membranes, in determining the permeability of embedded CNTs is important in estimating membrane performance. A related influence on membrane performance will come from the end flanges of the matrix material, which may affect the interfacial resistance to transport. Although little is directly reported on the effect of the matrix material, some understanding of the influence of flanges may be obtained from studies of transport through nanoporous graphene where the role of 2


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adsorption on the graphene surface has been identified as being important. For example, simulation studies by Du et al.2 showed that the permeability of N2 through nanoporous graphene exceeds that of H2 because of the enhanced adsorption of N2 over H2 on the graphene sheet. However, the contribution of surface diffusion on the graphene remains unclear. In their simulations of CO2/N2 selectivity of nanoporous graphene, Shan et al.16 found that while enhancing the adsorption affinity of the graphene sheet facilitates the permeability of CO2, increasing the adsorption affinity of the pore rim reduces the CO2 permeability. This implies there are two separate flow resistances, one associated with radial diffusion on the graphene sheet towards the pore rim, and the other related to entry into the nanopore, and that these two resistances together determine the rate of molecular exchange at the interface. Sun et al.9 decomposed the overall flow rate through the nanopore into two components, a surface flow rate of the fluid radially diffusing on the graphene sheet and a direct flow rate for the fluid interacting only with the pore rim. However, in their method the measurement of the surface flow rate lies on a very carefully defined polar angle and the curvature of the molecular path, which requires a high frequency of molecular trajectory storage and restricts their method to spherical particles. There have been extensive studies2, 9, 16focusing on the effect of pore size, pore structure and the interactions between the pore rim and the fluid molecules on the permeability, but little is known about the influence of porosity of a membrane on the permeability. Carbon composite structures, consisting of aligned multilayer graphene planes separated by pillars of open carbon nanotube arrays between two adjacent planes, were studied theoretically by Matsumoto and Saito17 and experimentally by Kondo et al.18, and have attracted considerable interest as composite structures for gas storage and separation.19-21 Here we have used non-equilibrium molecular dynamics (NEMD) simulations to investigate a carbon composite model which comprises a nanotube with entrance and exit flanges at its ends, and which serves as a representation of a nanotube embedded in a membrane. We propose a new method to separately measure the surface and direct flow rates without introducing any artificial criteria. The barriers to mass transport at the interface are, for the first time, decomposed into components contributed by the interplay of the fluid molecules with the flange and with the pore. A novel method was proposed to accurately measure the values and the relative importances of these two components in determining the interfacial resistance. Our simulations shed some light on diffusion through membranes composed of nanotubes embedded

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in a matrix material.22 By varying the size of the graphene flanges and tuning the interactions of the fluid molecules with their constituent atoms, we are able to systematically study the effects of porosity of a membrane and the adsorption affinity of the matrix material on interfacial resistance and the transport diffusivity of methane in CNT membranes. It is worth to note the methods proposed in this work are not specific to any particular membrane, such that our simulation results apply to any nanoporous membrane with a matrix. 2 Simulation Details Figure 1 illustrates the simulation system consisting of a finite (10, 10) CNT at the centre of a simulation box, connecting two bulk reservoirs. The internal diameter (centre-to-centre between carbon atoms) of the (10, 10) CNT is 1.356 nm and its length is 30 nm. In CNT membranes individual CNTs are separated and supported by a matrix material such as silicon nitride13 or polystyrene11, 23. In this work as in similar studies by Sholl et al.8 and Walther et al.7 we have positioned the CNT between two flanges; no molecules diffuse over the external surface of the CNT. Periodic boundary conditions were applied at the reservoirs in all the directions and the cut-off radius was 1.45 nm. In their simulation work, Chen et al.24 showed that the transport diffusivities of methane in the flexible and rigid CNTs are very comparable when the pressure is above 1.0 bar at room temperature. Consequently, since the flanges have the same atomic structure as graphene, it is expected that the flexibility of the flange should have a weak impact on the radial diffusion of methane on the flange. Since the pressures investigated in this work range from 0.5 to 15 bar, for computational efficiency the carbon nanotubes and the flange were treated as rigid. We also studied an infinite CNT, where the flanges and the bulk reservoirs were absent and periodic boundaries were applied in the axial direction; the resistances to mass transport at the entrance and exit of the CNT vanish for the infinite CNT. A force-driven NEMD simulation method was used to generate a net flux through the model system. The external force was applied to the molecules inside the CNT only. The method of determining the magnitude of the external forces applied in the NEMD simulations will be discussed subsequently. The NEMD simulations were conducted using the LAMMPS package25, initiated from GCMC simulations equilibrated at the target fugacity and 300 K with a time step of 1 fs and a Nosé-Hoover thermostat with a damping coefficient of 100 time steps. The Nosé-Hoover thermostat was applied at each time step, to the thermal velocities, obtained by subtracting the net streaming velocity of the fluid from the molecular velocities. The 4


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streaming velocity was sampled every time step. The NEMD simulations were run for at least 100 ns and up to 250 ns, with the first 50 ns being used to establish a steady state. All the NEMD simulations were run within the linear response region where the fluxes increased linearly with the external force, and the corrected diffusivities showed negligible dependency on the external force. In the present work, all the error bars were calculated as the standard deviation of five data points that are within the linear response region in the NEMD simulations. Adsorption of methane on the graphite flanges was significant and the adsorbed molecules on the flanges were found to make a substantial contribution to the flow rate through the CNT.

Figure 1. Schematic view of the simulation system. The Cyan spheres are the carbon atoms in the CNT and the graphene sheet, and the yellow particles are methane molecules. The site-site interactions for the adsorbate-adsorbate and adsorbate-adsorbent pairs were modelled with the 12-6 Lennard-Jones (LJ) pairwise potentials. The LJ size and well depth parameters for the unlike site-site interactions were calculated using the Lorentz-Berthelot mixing rules.26 It was found in our previous study27 that due to the stronger interaction with the CNT, the 1-site methane model only slightly (within ~20%) over predicts the adsorption of methane in the (10, 10) CNT compared to the 5-site model. Without the overlapping of the potential energy resulting from the wall curvature, this effect should be even less relevant on the flange. On the other hand, it was found by Bhatia and Nicholson28 that the 1-site and 5-site models provide quantitatively similar transport diffusivities of methane in silica nanopores. Therefore, the 1-site model is adopted in this work to investigate the transport of methane in CNT membranes. The LJ parameters used for the spherical methane molecule29 were

 f  0.381 nm and  f k B  148.1 K. The atomic structure of the flanges was that of a graphene sheet and their dimensions ranged from 3  3 to 10 10 nm2. The flanges at the entrance and exit of the CNT were identical. The dimensions of the bulk reservoirs were equal to the dimensions of the flanges in the X and Y directions and 10 nm in the axial direction. Further, since both the CNT and the flanges were treated as being rigid, the possible bonding 5



between the atoms on the flange and the carbon atoms in the CNT at the position where the CNT is attached to the flange was overlooked. Accordingly, the LJ parameters of the atoms on the flange are identical everywhere, which is independent of the LJ parameters of the CNT. Two sets of LJ parameters were used to model the atoms in the flanges, while the LJ parameters for the carbon atoms in the CNT were fixed as  C  0.34 nm and  C k B  28.0 K30 throughout the simulations. Either the value of A for the flange atoms was fixed and A/kB was varied, or A/kB was fixed and A was varied. The following parameter values were studied: (1) A=0.34 nm with A/kB =14.0, 28.0, 56.0 and 140.0 K. (2) A/kB = 28.0 K, with A = 0.255, 0.34 and 0.425 nm. We note that while the base case parameters of A=0.34 nm and A/kB = 28.0 K correspond to those commonly used for carbon, their values were independently varied keeping the graphene structure unchanged, so as to explore the sensitivity to these values and investigate the effect of changing the adsorption affinity. The surface areas of the flanges are equivalent to pore densities ranging from 1.1  1013 to 1.0  1012 cm-2, and provide some insight into the effect of membrane porosity on the interfacial resistance. It is noted that the flanges mounted at the tips of the CNT represent the surface layers of the matrix material that directly interact with the fluid molecules located in the bulk reservoirs while blocking the diffusion of fluid molecules in the gaps between individual CNTs. In practice, the inter-tube gaps are filled with the matrix material, and the interactions of the fluid molecules in the CNT with the matrix material around the CNT will additionally impact the adsorption and dynamics of the fluid molecules. Thus, when we vary the adsorption affinity of the flange, we only consider the effects of adsorption affinity of the surface layers of the matrix material or an additional distinct flange on the interfacial resistance and permeability of methane in CNT membranes. One could also additionally vary the affinity of the tube wall to investigate the effect of matrix material, however that is not considered here. Nevertheless, one expects that the interaction of the adsorbed molecules with the CNT wall will be significantly stronger due to the tight lattice spacing of 0.142 nm of the carbon atoms of the CNT. Figure 2 shows an example of the axial density distribution of methane at the entrance to the CNT from an NEMD simulation at a mean fugacity of 15.0 bar, with an external force of 1.614×10-13 N. The LJ parameters for the flange atoms were A=0.34 nm and A/kB =28.0 K, and the flange area was 8  8 nm2. The axial density in Figure 2 is an average over the entire 6


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flange area, and shows that methane molecules experience a strong repulsive force exerted by the flange atoms in the region 10>z>9.68 nm (z=10 nm corresponds to the entrance to the CNT). Consequently, the axial density approaches zero for 10>z>9.68 nm, with a small number of methane molecules adsorbed around the CNT mouth. The inset also illustrates the streamlines for the adsorbate molecules located in a layer of thickness 0.3605 nm ( 9.46  z  9.82 nm) on the

flange.

This

adsorption

layer

was

gridded

into

elemental

blocks

of

lx  l y  lz  0.05  0.05  0.3605 nm3, and the ensemble averaged X and Y velocities, v x and v y i

i for each block were used to plot the streamlines. The velocity components, v x and v y in the i-

th block were calculated as v xi 

Ni ( t )

v j 1

, v iy  j , x (t ) N i (t )

Ni ( t )

v j 1

j, y

(t ) N i (t ) , where N i ( t ) , v j , x (t )

and v j , y (t ) denote the number of molecules, and the X and Y components of the velocity of the j-th molecule, located in the i-th block at time t.

... is an ensemble average operator which

has sampled the molecular configuration every 0.5 ps over 100 ns in 10 independent NEMD simulations with different initial configurations.

12 10 -3

density (nm )

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


8 6 4 2 0 8.0

8.5

9.0

9.5

10.0

position (nm)

Figure 2. Axial density distribution of methane at the entrance interface positioned at 10 nm from the end plane of the reservoir (at z  0). The inset depicts the streamlines of the methane fluid on the XY plane with the dimension of 8  8 nm2, with grid spacings of 0.05 nm in the X and Y dimensions and 0.3605 nm in Z dimension. In the steady state, the driving force is balanced by the intracrystalline resistance from collisions of the fluid molecules with the carbon wall of the nanotube, as well as the interfacial resistance from the CNT ends and from transport over the graphene flanges at the entrance and

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exit interfaces. The flux through the nanotube,

jCNT was calculated as jCNT  CNT vcom

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where

CNT ( f ) is the ensemble averaged density of the fluid in the CNT as determined from GCMC simulations at fugacity f and

vcom is the streaming velocity of the fluid in the CNT obtained

from the NEMD simulations. In our previous study,5 the net number of molecules passing the plane at the entrance, middle and exit of the CNT was used to directly determine the net flux i following jCNT  niLTR  niRTL

i is the net flux at the specific plane, i , and ACNT , where jCNT

niLTR , niRTL denote the number of molecules passing through the plane from the left to the right

and from the right to the left per unit time, respectively. The averaged flux over the independent net fluxes at the entrance middle and exit is adopted as the net flux in the CNT. Quantitative agreements were observed in the results obtained using these two methods, confirming the finite validity of the method used in this work to calculate jCNT . The corrected diffusivity, Do , in

the finite (10, 10) CNT (including the effect of interfacial resistances) can be calculated from the net flux as

Dofinite ( f ) 

jCNT kBT v kT  com B CNT ( f )ex ex

(1)

where k B is the Boltzmann constant, T is the temperature of the system and  ex is the external force applied to methane inside the CNT. When the pressure drop between the feed reservoir and the permeate reservoir is very small, the equivalent external force  ex applied on the molecules in the CNT in the NEMD simulation can be approximated as 5, 31

ex 

where,

d d ln f f f  kBT  kBT 1 2 dz dz fLCNT

d is the chemical potential gradient in the CNT, dz

(2)

f1 and f2 are the fugacities of the

feed and permeate reservoirs, f is the mean fugacity = ( f1  f2 )/ 2, and LCNT is the length of the CNT. Here we assume that the length of the interfacial region outside the CNT is small compared to LCNT. The overall resistance, including the interfacial and intracrystalline resistances, can be obtained by combining eqn. (1) and eqn. (2), to give

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Rtot 

( f1  f2 ) fLCNT  JCNT ACNT CNT Dofinite

(3)

where J CNT  jCN T ACNT is the flow rate in the CNT, with ACNT being the cross sectional area of the CNT. ACNT was calculated based on the center-to-center diameter of the (10 10) CNT, with ACNT  1.443 nm2. Therefore, according to equations (1) and (3), for a specific CNT membrane, both Rtot and Dofinite are functions of fugacity f and temperature. The intracrystalline resistance can be obtained from NEMD simulations in an infinite CNT at a given fugacity and temperature using eqn. (3), following

Rinternal 

fLCNT ( f1  f2 )  infinite JCNT ACNT CNT Doinfinite

(4)

infinite infinite infinite where, J CNT are the flow rate and diffusivity in the infinite (10, 10)  jCNT ACNT and Do infinite CNT with fugacity gradient -(f1-f2)/LCNT, and jCNT the corresponding flux. The interfacial

resistance can be calculated as

Rinterf  Rtot  Rinternal 

fLCNT  1 1   finite  infinite  ACNT CNT  Do Do 

(5)

and represents the excess resistance of the finite CNT over the value based on the diffusivity in an infinitely long CNT. 3 Results 3.1. Effect of adsorption affinity and flange size on the interfacial resistance Figure 3(a) demonstrates that the barriers to mass transport at the entrance and exit to the CNT can reduce the diffusivity by more than 2 orders of magnitude compared to that in an infinite CNT.5 However, when the adsorption affinity of the flange is increased, i.e. the matrix adsorption affinity is increased, the diffusivity of methane in the finite (10, 10) CNT also increases, as exemplified in Figure 3a. For example: at 1 bar and 15 bar the methane diffusivity, Dofinite , increases from 0.067 and 0.162 nm2ps-1 to 0.297 and 0.439 nm2ps-1, when the LJ

energy parameter of the atoms in the flange is increased from A/kB = 14.0 K to 140.0 K. Figure 3(b) shows that increasing the adsorption affinity of the flange reduces the interfacial resistance. The isotherms for methane adsorbed in an adsorption layer of thickness 1 nm (9.0

Effects of Flange Adsorption Affinity and Membrane ... - ACS Publications

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