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C: Surfaces, Interfaces, Porous Materials, and Catalysis

Influence of Morphology on Transport Properties and Interfacial Resistance in Nanoporous Carbons Lang Liu, and Suresh K Bhatia J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b06270 • Publication Date (Web): 05 Aug 2019 Downloaded from pubs.acs.org on August 10, 2019

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The Journal of Physical Chemistry

Influence of Morphology on Transport Properties and Interfacial Resistance in Nanoporous Carbons Lang Liua,b* and Suresh K. Bhatiaa* a School of Chemical Engineering, The University of Queensland Brisbane, QLD 4072, Australia b Key laboratory of low-grade Energy Utilization Technologies and Systems, Ministry of Education, School of Energy and Power Engineering, Chongqing University, Chongqing 400044, China ABSTRACT The drive for enhancing process efficiency by decreasing system length scale to reduce transport resistance in nanomaterials brings into prominence the interfacial resistance, the relative importance of which scales inversely with system size; this critically limits the extent of efficiency enhancement possible. We investigate here the effects of morphology on the interfacial resistance for methane transport in carbons, by comparing the transport properties of a finite sized ordered carbon nanotube, disordered activated carbon fibre (ACF-15), and silicon carbide derived carbon (SiC-DC), using molecular dynamics simulations. We find that while the ordered CNT, having a smooth energy landscape, provides the largest transport coefficient, the relative interfacial resistance in this material is also high and exceeds that in the disordered carbons, which have intra-crystalline diffusivities that are more than two orders of magnitude smaller. This behaviour is traced to the existence of large entry and exit zones, in which the fluid motion remains correlated with that at the ends. The sizes of these zones are influenced by the small interfacial momentum accommodation coefficient (or Maxwell reflection coefficient) characteristic of carbons, and the level of disorder in the material. However, the interfacial resistivity in the CNT is smaller than that in the disordered carbons, due to dramatically reduced contribution from bending of fluid streamlines at the interfaces. Nevertheless, the interfacial resistance of the CNT is larger than that of the disordered carbons, ACF-15 and SiC-DC, due to the reduced cross sectional area of the CNT. It is found that the resulting overall corrected diffusivities in the ACF-15 and SiC-DC having thickness around 50 nm are comparable to that in the corresponding CNT, indicating that at nanoscale thicknesses morphology is not a significant consideration from a transport viewpoint, and CNTs offer little advantage compared to disordered carbons. *To whom correspondence may be addressed. Email: [email protected] and [email protected]. 1

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1 INTRODUCTION Nanoporous carbons are used in in a myriad of applications, such as heterogeneous catalysis1-2, fuel cells3, membrane separation4-6 and nanofluidics7. Understanding the transport properties of fluid molecules in nanoporous materials is of great significance, as diffusion phenomena critically influence the efficiency of these processes. However, as an inherent feature of nanoporous carbons, their morphology can vary significantly depending on the synthesis procedure, ranging from extremely disordered materials, such as silicon carbide derived carbon (SiC-DC)8 and activated carbon fibre ACF-159, to materials that are intrinsically well ordered such as carbon nanotubes (CNTs)10. Altering the disordered nature of the nanoporous carbons inevitably induces differences in the surface roughness and corrugation, tortuosity of the structure and could introduce bottle-necks. When the morphology is altered, fluid molecules experience a variety of different potential energy surfaces and energy barriers along their diffusion paths, leading to unexpected transport properties in nanoporous materials. Therefore, knowledge of the effect of disordered nature on diffusion in nanoporous carbons is essential to assess their performance in realistic applications. 1-7 Considerable effort has been made in investigating the effect of morphology on the diffusion of fluid molecules in nanoporous materials.8,11-19 The self-diffusivities of benzene in ideal graphite slit pores, having widths ranging from 0.8 to 1.6 nm, and in a disordered porous carbon with pore size distribution ranging from around 0.6 to 1.55 nm at 298 K, were determined by Coasne et al.11 in their simulation work. They reported that the disordered nature reduces the diffusion coefficient by around one order of magnitude. Based on the uptake kinetics of methane and carbon dioxide12,13, it was recently observed in our laboratory that the experimentally measured diffusion coefficients of methane in disordered SiC-DC were orders of magnitude smaller than those calculated for an ideal graphite slit model using molecular dynamics simulations.14 While the pore network topology of the structure is partially responsible for this large discrepancy, the morphology, more specifically the distorted nature, is considered to be a major contributor15.

For realistic nanoporous carbons, surface roughness and

surface irregularities are generally enhanced compared to ideal graphene sheets8,13,16. However, as reviewed by Bhatia15, there exist extensive studies illustrating that surface roughness and irregularities play a smaller role in determining the diffusion coefficients in the presence of bottle necks that generate energy barriers in disordered carbons20. Nevertheless, most of the works focus on the intra-crystalline 2


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self-diffusivities of molecules in nanoporous carbons, which provide only limited information about the effects of the disordered nature on transport diffusivities, while neglecting the impact of molecular exchanges at the entrance and exit guest-host interfaces. As reviewed above, the intra-crystalline diffusion has long been the focus of studies, with much less attention paid to the transport of fluid molecules between the bulk phase and the crystalline structure at the entrance and exit boundaries. Nevertheless, the resistances arising from the molecular transport at the interfacial boundaries (collectively defined as interfacial resistance) have been recognised as comprising an important part determining the permeance through nanoporous materials. The Dual Control Volume-Grand Canonical Molecular Dynamics (DCV-GCMD) method has been used to study interfacial barriers by several workers. Using this method, Arya et al.21 showed a strong exit interfacial barrier in AlPO4-5, while Ahunbay et al.22 observed that the sum of entrance and exit interfacial barriers was significantly higher than the intra-crystalline resistance in silicalite membranes for thicknesses up to 64 nm. Although the flexibility of CNTs reduces the diffusivity in these materials by a significant factor, albeit within an order of magnitude, when the adsorbate loading is low, this effect becomes negligible when adsorbate-adsorbate interactions assume importance.23 Consequently, in comparison to zeolites, the intra-crystalline diffusion in carbon nanotubes, inherently having smooth walls, is generally orders of magnitude faster24,25, and the interfacial resistance dominates over the internal resistance26. We proposed an equilibrium molecular dynamics (EMD) simulation method to quantitatively measure interfacial resistance in nanoporous materials, and estimated that this resistance accounted for more than 99% of the transport resistance for methane permeating in (10, 10) carbon nanotubes of length as much as 100 nm26. Further, the permeance of fluid molecules, such as methane and water, in carbon nanotubes demonstrates a strong dependency on the carbon nanotube length, which is confirmed to be a consequence of strong interfacial resistance at the boundaries27-28. Nevertheless, Zimmermann et al.17 reported that while interfacial barriers were significant for ethane in AFI-type zeolite, they had negligible effect on the permeance of methane, suggesting that the importance of interfacial barriers in determining permeability is dependent on the adsorbate-host interaction. Using interference microscopy to track the evolution of transient concentration profiles of adsorbates in different nanoporous materials, Heinke et al.29 and Remi et al.30 found that altering the configuration of host structures could significantly alter the role of interfacial resistance in determining 3



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the permeance. They showed that the permeance process could range from interfacial diffusion controlled to intra-crystalline diffusion controlled. Since the disordered nature and interfacial resistance could both potentially play a critical role in determining the diffusion coefficient, it would be interesting to understand the correlation between these in determining transport properties of nanoporous carbons. Therefore, we report here a study on the transport diffusion of methane in carbon nanotubes, ACF-15 and SiC-DC to investigate the effects of disordered nature of nanoporous carbons on transport diffusivities of fluid molecules, in the presence and absence of interfacial resistance. In our previous study26 we developed an EMD method to calculate the diffusivity of molecules in finite carbon nanotubes in the presence of interfacial resistance; this method has been used here, and validated by comparing simulation results with those using the Zhu et al. method.31 2. SIMULATION DETAILS We used the EMD method to investigate the transport of methane in the infinite and finite (10, 10) CNT, ACF-15 and SiC-DC, for pressures up to 15 bar, at 300 K. The thicknesses of the finite CNT, ACF-15 and SiC-DC are 50, 50.15 and 52 nm. The center-to-center diameter of the (10, 10) CNT is 1.356 nm. The atomistic structures of ACF-15 and SiC-DC were previously reconstructed in our laboratory, using the hybrid reverse Monte Carlo (HRMC) method8,9. The HRMC method is based on the conventional reverse Monte Carlo method while simultaneously minimizing energy, and uses both energy-based and structure constraints to generate the atomistic configurations of nanoporous carbons. The configurations obtained from HRMC method are therefore energetically stable and physically meaningful, in addition to satisfying the experimental criteria. For the ACF-15, the experimental material was ACC-5092-15 activated carbon fiber, provided by Kynol Corporation. The SiC-DC was synthesised in our laboratory by oxidation of a  SiC precursor in a pure chlorine atmosphere at 1073K. The reconstructed atomistic structures for ACF-15 and SiC-DC have been validated against experimentally measured isotherms and diffusion coefficients of light gases8-9. Our simulation systems for the finite carbon structures, illustrated in Figure 1, are divided into three parts, two bulk reservoirs and a carbon structure located in the central region of the simulation box connecting these two bulk reservoirs. For the (10, 10) CNTs, graphite flanges are located exactly at 4


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the edges of the CNT to block the diffusion outside the CNT. The unit cells and dimensions of the CNT and disordered carbon structures8-9 are given in Table 1, and the geometric pore size distributions of ACF-15 and SiC-DC are depicted in Figure 1(d). The pore size distributions of ACF-15 and SiCDC are determined using the method proposed by Gelb and Gubbins32, demonstrating similar pore size distributions between these two structures with the maximum pore sizes in these two structures being comparable to the diameter of the (10, 10) CNT. For the finite disordered carbon structures the sizes of the left and right bulk reservoirs in the flow direction are equal and fixed at 10 nm, and the sizes of the bulk reservoirs in the other two dimensions are set to be the same as those of the carbon unit cells. For the CNT they are equal to the dimensions of the graphite flanges ( 4  4 nm2). Periodic boundary conditions were applied in all directions, and a cut-off radius of 1.45 nm was employed in our simulations. Interfacial resistance is excluded in the infinite carbon structures due to the absence of bulk reservoirs and bulk-adsorbate interfaces. 2.5 ACF-15 SiC-DC

(d)

PSD (cm 3/(g·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


2.0 1.5 1.0 0.5 0.0 0.0

0.2

0.4

0.6

0.8

1.0

position (nm)

1.2

1.4

1.6

Figure 1. Schematic view of the simulation systems for (a) (10, 10) CNT, (b) ACF-15, and (c) SiCDC and (d) pore size distributions of ACF-15 and SiC-DC. The yellow spheres are methane molecules and cyan particles represent the carbon atoms. The EMD simulations were conducted using the LAMMPS package33, starting from configurations obtained using grand canonical Monte Carlo (GCMC) simulations at the target fugacity and 300 K. Methane-methane and methane-carbon interactions were modelled with the 12-6 Lennard-Jones (LJ) pairwise potentials in both the MD and GCMC simulations. The LJ parameters for the carbon atoms in all the carbon structures considered34 were s kB  28.0 K,  s  0.34 nm and the LJ parameters for the spherical methane molecule35 were  f k B  148.1 K and 

f

 0.381 nm. All the infinite

and finite carbon structures were treated as being rigid. A time step of 1 fs and a Nosé-Hoover thermostat with a damping coefficient of 100 time steps were used in our EMD simulations. The 5



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trajectory of methane was stored every 50 fs in 20 independent EMD simulation runs to calculate the mean square displacements of the center of mass (COM-MSD) 26of fluid and the collective coordinate of the finite system fluid31, in the carbon structures. While the COM-MSD is used to extract the corrected diffusivity in the finite nanoporous carbons based on the method proposed in our previous work26, the mean square displacement of the collective coordinate of the fluid is used to calculate the collective diffusivity in finite nanoporous materials using the method proposed by Zhu et al.31. For the infinite cases, the corrected diffusivity was calculated from the COM-MSD based on Einstein equation. All the EMD simulations were run at least 100 ns, with the first 20 ns to equilibrate the system. The EMD method proposed in our previous work to calculate the diffusivity, D0finite , of fluid in the finite nanoporous materials, is written as26

finite o

D

2   N b  c Ac 2 1 N c      Nc   | zsys ,com (t )  zsys ,com (0) (1)   lim  t 2t   N sys bulk Asys N sys  

where N c , Nb and Nsys are the numbers of adsorbate molecules in the solid structure, in the bulk reservoirs and of the simulation system, c and bulk , having unit of mol/nm3, are the number densities of adsorbate in the carbon structure and in the bulk reservoirs, Ac and Asys are the cross-sectional areas of the carbon structure and the simulation system, and zsys ,com (t ) is the centre of mass of fluid molecules of the simulation system at time t. When the chemical potential difference between the two reservoirs is very small, the flux, jc , having unit of mol/(nm2.ps), through the finite carbon structure in the presence of interfacial resistance can be obtained as:

1 jc   Lc RT

2

  ( ) D  c

finite o

1

6


(  )d 

(2)

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where, R is gas constant, Lc is the thickness of the solid structure in the flow direction,  is chemical potential, and 1 ,  2 are the chemical potentials in the reservoirs, with 1  2 . For a very small chemical potential difference,   1 ~  2 

jc 

1 ( 1   2 ) , and equation (2) can be recast as 2

J c c (  ) Dofinite (  ) ( 1  2 )  L  Ac RT c

(3)

Based on eqn.(3), the total resistance, Rtot , including the interfacial and internal resistances is defined as

Rtot 

Lc ( 1  2 ) / RT  Jc c D0finite Ac

(4)

Further, the interfacial resistance, Rinterf , is calculated as the excess resistance over the internal resistance following Rinterf  Rtot  Rinternal 

Lc  1 1   finite  infinite   c Ac  D0 D0 

where the internal resistance is written as Rinternal  Lc

 A D c

c

infinite 0

 , with

(5)

D0infinite being the corrected

diffusivity in the infinite carbon structure. Table 1. The unit cell dimensions of the carbon structures carbon structure

CNT

ACF-15

SiC-DC

unit cell

dimensions rCNT  0.678 nm, Lz  0.241nm

Lx  Ly  Lz =2.95 2.98 3.02 nm3

7


Lx  Ly  Lz =4.0  4.0  4.0 nm3


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Since the flowrate is a function of the cross-sectional area of the carbon structure while the crosssectional areas of the unit cells of (10, 10) CNT, ACF-15 and SiC-DC differ from each other, the determined mass transfer resistances do not reveal the intrinsic resistivities of the carbon structures. Therefore, we introduce the interfacial resistivity to characterize the effect of disordered nature, by excluding the effect of cross-sectional area of the carbon structures on the exchange dynamics of fluid molecules at the interface. Since resistance is inversely proportional to cross-sectional area, evident in eqn (5), we define the interfacial resistivity,  , as the product of interfacial resistance and crosssectional area of the carbon structure, following

  Rinterf  Ac 

1 -2 RT

jc =

Lc  1 1   finite  infinite  D0 c  D0 

(6)

The comparison with the collective diffusivity calculated from Zhu et al. 31,method is provided in detail in the supporting information. As shown in Figure S1 in supporting information, quantitative match between the corrected diffusivities of D0finite calculated from our method and the method proposed by Zhu et al. was achieved in disordered porous carbons. 3 RESULTS 3.1 Effect of morphology on internal diffusion As depicted in Figure 2, the corrected diffusivity of methane in the infinite (10, 10) CNT is more than 2 orders of magnitude higher than that in the infinite ACF-15, and 3 orders of magnitude higher than that in the infinite SiC-DC. However, as shown in Figures S2 and S3 in supporting information, both in the infinite and finite SiC-DCs, the diffusion of methane in the Y and Z dimensions is slower than in the X dimension. It is evident that the diffusion of methane in the SiC-DC is anisotropic16. Further, based on the algorithm proposed in previous work from this laboratory36, the ACF-15 structure is found inaccessible in the Z dimension and hardly accessible in the Y dimension. Therefore, only the diffusion of methane in the X dimension is considered for the ACF-15 and SiC-DC in this work. The adsorption isotherms of methane in the infinite (10, 10) CNT, ACF-15, SiC-DC are provided in Figure S4 in supporting information, from which the loading dependency of the corrected diffusivity in nanoporous carbons can be extracted. We note that, for the thicknesses of carbon structures investigated, the 8


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adsorption isotherms of methane in the infinite and finite structures are quantitatively similar for the CNT, ACF-15 and SiC-DC. corrected diffusivity (nm 2·ps-1)

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


100

infinite CNT infinite ACF-15 infinite SiC-DC

50 nm CNT 50.15 nm ACF-15 52 nm SiC-DC

1 0.1 0.01 0

2

4

6

8

10

12

fugacity (bar)

14

16

Figure 2. Variation of the diffusivity of methane with pressure in infinite and finite (10, 10) CNTs, ACF-15s and SiC-DCs, at 300 K. Falk et al.37 reported that enhancing the curvature of the carbon wall promoted overlap of the potential energy fields from opposing walls; this reduced the potential energy barrier for molecular migration from one side of the wall to the other side, and reduced the collision frequency of the fluid molecules located in the potential-well region adjacent to the carbon wall38. This effect corresponds to the wellknown levitation effect.18, 39 Comparisons between the diffusivities of methane in an infinite (10, 10) CNT and infinite slit pores having pore widths of 0.8, 1.0 and 1.356 nm, at 300 K were conducted, and the results are depicted in Figure S5. It is evident that the diffusion coefficient of methane in CNTs38 is significantly higher than that in ideal graphite slit pores of similar width. Therefore, the reduced curvature of the carbon sheets is partially accounting for the reduced corrected diffusivity of methane in the stratified ACF-15 and completely disordered SiC-DC. A second factor is that the irregularities and defects existing in the carbon sheets in the ACF-15 and SiC-DC, makes the carbon sheets more diffusely reflecting than the ideal graphene sheet; this enhances the frictional momentum loss upon fluid-wall collision, which also contributes to the reduced diffusivity of methane in amorphous carbons15. Further, due to the distortion of the carbon sheets in ACF-15 and SiC-DC, “bottle necks” that affect the pore accessibility are present, leading to strong pore mouth barriers for molecules to infiltrate through the ACF-15 and SiC-DC16. The self-diffusion coefficient of methane in the infinite

9



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(10, 10) CNT was reported to be reduced by more than one order of magnitude by this pore mouth effect at low loadings40. As a result of similar pore size distributions and carbon densities (0.88 g/cm3 for the ACF-15 and 0.95 g/cm3 for the SiC-DC) between the ACF-15 and the SiC-DC, we showed in Figure S4 that the gravimetric adsorption isotherms of methane in these two structures were quantitatively similar. This, to a certain extent, implies that surface defects and irregularities are statistically distributed in these two structures in a similar manner. Therefore, surface defects and irregularities are not expected to induce great difference between the diffusion in the ACF-15 and in the SiC-DC15, 41. Indeed, the simulation work conducted by Priezjev41 for the flow of Newtonian fluids on surfaces having random atomic heterogeneities, in the form of strongly and weakly attractive sites, has shown such irregularities can only reduce the transport coefficient by a very small factor. Nevertheless, due to the enhanced “bottle neck” effect, the diffusivity in the SiC-DC is around one order of magnitude smaller than that in the ACF-15. It was estimated that the minimum free energy barrier for methane percolating the ACF-15 is 7.12 kJ/mol, while the that for SiC-DC is 10.67 kJ/mol42. In summary, while surface defects, irregularities and reduced wall curvature of the carbon sheets and “bottle necks” in the ACF15 are responsible for the reduction in the transport diffusivity compared to the (10, 10) CNT, the “bottle neck” effect, caused by distortions of the SiC-DC surface, is mainly responsible for the reduced diffusivity in the SiC-DC in comparison to ACF-15. 3.2 Effect of morphology on diffusion in the presence of interfacial resistance From Figure 2 it is found that while the interfacial resistance in the CNT reduces the diffusivity by more than 2 orders of magnitude, the diffusivities in the ACF-15 and SiC-DC are reduced by around a factor of 2 and 25%. Consequently, the diffusivities of methane in the 50 nm (10, 10) CNT and 50.15 nm ACF-15 become comparable in the presence of interfacial resistance. Nevertheless, the diffusivity in the 52 nm SiC-DC is still much lower than that in the CNT, due to the rather low internal diffusivity. On the other hand, as shown in Figure 3(a), while interfacial resistance dominates the diffusion of methane in the CNT, it accounts for around 50% and 25% of the total resistance in the ACF-15 and in the SiC-DC. This demonstrates that the interfacial resistance plays less important role in determining the diffusivity when the distortion of the structure is enhanced; this is associated with dramatically 10


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reduced internal diffusion. The strong decrease in internal diffusivity on increase in disorder is seen in Figure S6 (a) when the thickness of the CNT, ACF-15 and SiC-DC is reduced to around 10 nm.

120

Rinterf/Rtot (%)

100

(a)

80


60 40 20 0 0

2

4

6

8

10

12

14

16

interfacial resistance (x1023 ps/mol)

fugacity (bar)

interfacial resistivity (x1023 ps·nm2/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


25000 (b) 50 nm CNT 50.15 nm ACF-15 52 nm SiC-DC

20000 15000 10000 5000 0 0

2

4

6

8

10

12

14

16

fugacity (bar)

(c)

5000


4000 3000 2000 1000 0 0

2

4

6

8

10

12

14

16

fugacity (bar)

Figure 3. Variation of (a) the ratio of the interfacial resistance to the total resistance, (b) the interfacial resistivity, and (c) interfacial resistance with fugacity in the finite (10, 10) CNT, ACF-15 and SiC-DC. According to eqns. (4) and (5), the corrected diffusivity in the finite structure satisfies the relation

1 finite o

D



1 infinite o

D

   c Ac Rinterf

 L1

(7)

c

In Figure 4, we plot 1 Dofinite as a function of 1 Lc , with 1 Dofinite converging to 1 Doinfinite at 1 Lc =0.0. We linearly fitted 1 Dofinite as a function of 1 Lc for the SiC-DC, based on the data points at Lc = 8, 12, 20 and 52 nm. The intercept of the fitting curve yields an intra-crystalline diffusivity of 0.0216 11



nm2.ps-1 for SiC-DC, which quantitatively matches the simulation result of 0.025 nm2.ps-1 for the infinite SiC-DC, at 5 bar and 300 K. This implies that the interfacial resistance is independent of the thickness of SiC-DC when its thickness is larger than 8 nm, i.e. the sum of resistances of the interfacial zones extended from the entrance and exit boundaries towards the internal region of SiC-DC is around 8 nm. However, 1 Dofinite demonstrates significant nonlinearity with 1 Lc in the CNT and ACF-15. The linear fits of 1 Dofinite as a function of 1 Lc for the CNT and ACF-15 at 5 bar and 300 K are based on the data points at Lc = 10, 30, 50 and 100 nm for the CNTs and Lc = 8.85, 11.8, 20.65 and 50.15 nm for the ACF-15s. However, the intra-crystalline diffusivities of CNT and ACF-15 extracted from the intercepts of fitting curves are 0.29 and 0.078 nm2.ps-1, both of which significantly deviate from the corresponding simulation results of 64.9 and 0.144 nm2.ps-1 respectively. Therefore, it is apparent that the interfacial resistance is strongly dependent on the thickness of the structure for the CNT and ACF15, and this dependency occurs over a large region inside the CNT and ACF-15. This size dependence suggests that there is a significantly large zone near the ends, for the CNT or ACF-15 carbons, over which the fluid motion is developing, leading to variation in apparent interfacial resistance.

140 SiC-DC

120

y=46.30+585.45x

100

-2

1/Dofinite (nm .ps)

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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80 60 y=12.89+324.39x

ACF-15

40 (10,10) CNT y=3.59+330.67x

20 0 0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

1/Lc (nm-1)

Figure 4. Variation of 1 D

finite

o

versus 1 L in the (10, 10) CNT, ACF-15 and SiC-DC, at 5 bar and 300 c

K. Dashed lines are the corresponding linear fits. In the fitting equations, x and y represent 1 Lc and

1D

finite

o

, and their units are consistent with the units of the corresponding axes.

12


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To further investigate the existence of a finite zone over which the fluid motion is developing, we plot correlation functions of the center of mass velocity (COM-VCF) at various axial positions with that at the entry, for the (10, 10) CNT, ACF-15 carbons in Figure 5. Here  represents the distance between the position located inside the structure and the entrance of carbon structure. This way of calculating COM-VCF was detailed in our previous work26, and the same approach is adopted for the ACF-15 in this work. It is seen that the correlation effect is much stronger in the CNT than in the ACF-15. The correlation effect diminishes at the position  =10 nm in the ACF-15, while it retains significance even at the position  =50 nm in the CNT. One could expect that the intra-crystalline diffusivity extracted from the intercept will converge to the simulation result in the ACF-15 when all the sampled thicknesses are larger than 20 nm, assuming that the interfacial correlation effects at entrance and exit of the ACF-15 are similar. On the other hand, it requires sample length much larger than 100 nm for the CNT to yield the correct slope, and for the extracted intra-crystalline diffusivities to converge to the simulation results. Thus, we conclude that enhancing the disordered nature of nanoporous carbon reduces the interfacial correlation effect as well as the size of interfacial region in nanoporous carbons.

Figure 5. Correlation functions of the velocity of center of mass of molecules located at entrance with that in different regions inside the carbons structures for (a) 100 nm thick (10, 10) CNT26, (b) 50.15 nm thick ACF-15, and (c) 20 nm thick SiC-DC at 5 bar and 300 K. 13



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This finite correlation length is consistent with the smooth energy landscape and consequent low Maxwell reflection coefficient for carbons43-45, as a result of which nearly specular reflection of molecules colliding with the pore walls occurs. For methane transport in the (10,10) CNT at 298 K the Maxwell reflection coefficient is about 0.001 at low pressure43, 45, suggesting that on an average only one in about 1000 collisions are diffuse and 999 specular. Given the equipartitioning of energy in the three dimensions, the mean axial distance moved between wall collisions in the CNT is similar in order of magnitude to the accessible CNT diameter (~1 nm), so that a large correlation length of the order of a micron may be expected (on an average 999 wall collisions must occur for de-correlation of trajectories). This explains why the fluid motion within the (10, 10) CNT is strongly correlated with that at the ends even for the 100 nm (i.e. 0.1 m) long tube. Further, in the 100 nm long tube the diffusing molecule gains momentum over a much smaller distance compared to an infinitely long tube, in which on average about 1 micron is traversed between diffuse wall collisions, leading to a much smaller transport coefficient in the finite tube. On the other hand, the Maxwell reflection coefficient for a graphite surface is about 0.023,44 so that a correlation length of about 20 nm may be expected in a slit pore having accessible width of 0.5 nm, consistent with a value slightly larger than 10 nm evident from the COM-VCF in Figure 5b. This pore width of 0.5 nm roughly corresponds to the first peak of the accessible pore size distribution of ACF-1546. The much smaller correlation distance (~4 nm) for the 3-dimensionally amorphous SiC-DC, indicated above, is likely due to pore surface defects which lead to a significantly larger value of the Maxwell reflection coefficient than that for ACF-15. The finite length of the correlation zone is also consistent with the steep drop in the interfacial resistance on increasing bulk fugacity, seen in Figures 3(c) and S6(c). Starting from low pressure, increasing bulk fugacity will lead to large increase in adsorbate density before approaching saturation, which will enhance the frequency of wall-collisions of fluid molecules. This will lead to a rapid decrease of the length of the correlation region, and therefore a sharp decline in interfacial resistance ascribed to this region. Once saturation is approached the collision frequency will become constant, leading to little further reduction in interfacial resistance, consistent with Figures 3(b) and (c) and S6 (b) and (c). This behavior is indicative of the effect of increase in importance of fluid-fluid interaction relative to fluid-solid interactions with increase in fugacity, which reduces interfacial resistance. 14


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Arya et al.21 indicate that the difference between the potential of mean force (PMF) for fluid molecules in the solid structure and in the bulk region represents the energy barrier that the molecules need to overcome to exit the solid structure; this qualitatively reflects the magnitude of the thermodynamic resistance arising from the entropy and enthalpy changes at the interface47. It is explicitly defined in our previous work48 that the interfacial resistance, Rinterf, is contributed by the entrance-exit resistance, Rentrance-exit, arising from the thermodynamic resistance and that of the large interfacial zone, and the flange resistance Rflange, due to the bending effect of the streamlines at the interfaces, i.e. Rinterf=Rentranceexit+

Rflange. The flange resistance was defined in our previous study48 as R flange  2 flange A flange , with

 flange and A flange being the flange resistivity and flange area. Accordingly, the interfacial resistivity comprises two parts, written as

   entrance  exit   2 flange  Ac  A flange

(8)

where the entrance-exit resistivity, entranceexit , is defined as Rentrance-exit  Ac . The flange area is equal to the area of graphene flanges mounted at the edges of the (10, 10) CNT. The tortuous, bent and twisted streamlines at the entrances of ACF-15 and SiC-DC demonstrate that the flange effect also exists for the ACF-15 and SiC-DC. Here, we focus our analysis on the exit of the solid structure. We plotted the potential of mean forces (PMF) of methane at the exit of the carbon structures in Figure 6, with position 0.0 nm being the exit boundary of the carbon structures, and position -1 nm being a plane that is 1 nm away from the exit surface located inside the carbon structure. The PMF was determined as the difference between the excess chemical potentials of methane at the axial position x, and in the bulk reservoir, as ex xex  bulk  kBT ln  x bulk  , with  x and bulk being the densities of methane averaged over the

plane of the structure at position x and in the distant bulk region. The corresponding PMF profiles at the entrance are approximated as the mirror images of those depicted in Figure 6, in consideration of Figure S7. It is known that the enhanced PMF change at the exit of the CNT and the enhanced interfacial correlation effect in the CNT, compared to the ACF-15 and SiC-DC, indicate that the relative entrance-exit resistivity in the CNT is much stronger than those in the ACF-15 and SiC-DC. 15



Nevertheless, the similar PMF distributions at the exits of the ACF-15 and SiC-DC imply that the disordered nature has insignificant effect on the thermodynamic resistance. On the other hand, the flange resistivity should make a much smaller contribution to the interfacial resistivity in the CNT for its large flange area, compared to the cases for the ACF-15 and SiC-DC, evident in equation (8) and Figure 7. The ratios of cross-sectional area to the flange area, Ac A flange , are 0.09, 2.5 and 2.94 at the exits of the CNT, ACF-15 and SiC-DC. Therefore, as shown in Figure 3(b), the reduced interfacial resistivity in the CNT compared to the ACF-15 and SiC-DC is a consequence of the reduced contribution from the flange resistivity to the interfacial resistivity. Nevertheless, due to the crosssectional area of the (10, 10) CNT being much smaller than the ACF-15 and SiC-DC, the interfacial resistance in the (10, 10) CNT is subsequently observed higher than those in the ACF-15 and SiC-DC, shown in Figure 3(c). 2 (10, 10) CNT ACF-15 SiC-DC

0

PMF (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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-2 -4 -6 -8 -10 -1.0

-0.5

0.0

0.5

1.0

position (nm)

Figure 6. PMF profiles in the (10, 10) CNT, ACF-15 and SiC-DC, at 5 bar and 300 K. Position 0.0 nm is the exit boundary of the carbon structure. As stated, the entrance-exit resistivity is contributed by the thermodynamic resistance and the velocity correlation that occurs in the interfacial zone. However, the interfacial zone in the ACF-15 (around 10 nm, at 5 bar and 300 K) is larger than that in the SiC-DC (~ 4 nm, at 5 bar and 300 K). This emphasises that enhanced disordered nature could make a contribution to reducing the entrance-exit resistivity/resistance (and therefore the interfacial resistivity/resistance) by reducing the correlation between the molecules located at the entrance and exit boundaries, associated with a reduced interfacial zone, and those located deep inside the structure. Consequently, the 50.15 nm thick ACF-15 holds a 16


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larger entrance-exit resistivity compared to the 52 nm thick SiC-DC, despite the similar PMF distributions in these two structures. This behaviour is consistent with the recent finding from this laboratory that the interfacial resistance in the more heterogeneous and 3-dimensional MFI and SAS zeolites was smaller than that in PON zeolite49.

Figure 7. Streamlines of methane on the (a) XZ plane at the entrance of CNT, and on the YZ planes at the entrances of (b) ACF-15 and (c) SiC-DC, determined from NEMD simulations, at 15 bar. The streamlines are spaced by 0 .2 nm in each dimension and the external forces applied on methane were 1.61x10-13, 1.14x10-12 and 1.25x10-11 N/molecule in the finite (10, 10) CNT, ACF-15 and SiC-DC. We cut 0.5 nm thick (in the X direction) slices of the ACF-15 and SiC-DC unit cells near their entrance and exit boundaries, with the atomistic structures being shown in Figure S8. The calculated geometric pore volumes (regions with pore size smaller than 0.32 nm are excluded) of the slices of ACF-15 and SiC-DC are 2.50 and 4.79 nm3 at the entrance, and the corresponding slices at the exit are 2.58 and 5.31 nm3. The estimated ratios of cross-sectional area of the unit cell to the flange area, Ac A flange for the ACF-15 and SiC-DC are 2.27 and 2.50 at the entrance and 2.32 and 2.94 at the exit. The flange area in the unit of nm2 is calculated as (Vsys-Vp)/0.5, where Vsys is the total volume of the carbon slice, Vp is the measured accessible pore volume of the carbon slice and 0.5 nm is the thicknesses of the carbon slice. It was observed in our previous work48 that when the ratio of Ac A flange is larger than 0.09, the ratio of flange resistance to the entrance-exit resistance becomes larger than unity and 17



Page 18 of 25

increases sharply with further increase of the ratio of Ac A flange by reducing the flange area. Therefore it is expected the external surface (i.e. flange) resistance plays a dominant role in determining the interfacial resistance, associated with the flange resistivity making the major contribution to the interfacial resistivity, in the ACF-15 and SiC-DC, and a slight enhancement in the (porosities) Ac A flange ratios in the ACF-15 and SiC-DC leads to noticeable enhancement in the interfacial

resistivity/resistance. In consideration of the enhanced tortuosity and bending effect for the streamlines at the boundaries of SiC-DC, the external surface (i.e. flange) resistivity,  flange , is expected to be enhanced for the SiC-DC compared to the ACF-15 that possesses smoother and layered boundary structures, evident in Figure 7 and Figure S8. Consequently, although the entrance-exit resistivity,

entranceexit , in the ACF-15 is expected to be larger than the SiC-DC, the interfacial resistivity in the ACF-15 is smaller than the SiC-DC due to the reduced contribution from the flange resistivity. Nevertheless, the interfacial resistances in the ACF-15 and SiC-DC are observed very comparable due to the ACF-15 having much smaller cross-sectional area, shown in Figure 3(c) and Figure S6 (c). 4. CONCLUSIONS We have used our EMD method to investigate the transport diffusion of methane in a well-defined (10, 10) CNT, stratified ACF-15 and completely disordered SiC-DC, in the presence and absence of interfacial resistance. It is found that in the absence of interfacial resistance, the transport diffusion coefficients of methane in the ACF-15 and SiC-DC are 2~3 orders of magnitude smaller than that in the (10, 10) CNT. While the energy barriers introduced by distortions and bottle necks are mainly responsible for the reduced diffusivity in the distorted ACF-15 and SiC-DC, the associated surface defects and irregularities and reduced wall curvature also make their contributions to slow down the diffusion in disordered nanoporous carbons. Nevertheless, the transport diffusion coefficients of methane in CNT and ACF-15 of finite size become very comparable, and are higher than that in the completely disordered SiC-DC by less than an order of magnitude, implying the morphology plays insignificant in determining the transport properties in the presence of interfacial resistance. However, it is noted that among the carbon structures investigated the minimum interfacial resistivity is obtained in the (10, 10) CNT due to the reduced contribution from the flange resistivity. The ratio of the 18


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interfacial resistance to the total resistance to transport reduces from around 99% in the CNT to 50% in the ACF-15 and 25% in the SiC-DC, which suggests that the interfacial resistance becomes less relevant when the disorder is enhanced in the nanoporous carbons. Further, we find that disorder in nanoporous carbons reduces the velocity correlation effect and the size of the interfacial zone, which subsequently reduces the interfacial resistance. ASSOCIATED CONTENT Supporting Information. Simulation details, Figures showing the effect of morphology on internal transport coefficients, adsorption isotherms, and effect of morphology on diffusion in the presence of interfacial resistance, for CNT, ACF-15 and SiC-DC of various lengths are available in Supporting Information. This material is available free of charge via the Internet at http://pubs.acs.org.

NOTES There are no conflicts of interest to declare.

ACKNOWLEDGEMENT

This work has been supported by a grant from the Australian Research Council through the Discovery scheme (Grant No. DP150101824). 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.

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(49) Dutta, R.C.; Bhatia, S.K., Interfacial Barriers to Gas Transport in Zeolites: Distinguishing Internal and External Resistance. Phys. Chem. Chem. Phys. 2018, 20, 26386 - 26395.

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Influence of Morphology on Transport Properties ... - ACS Publications

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