Kinetic Model of Gas Transport in Carbon Nanotube Channels

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Kinetic Model of Gas Transport in Carbon Nanotubes Aleksandr Noy J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/jp4005407 • Publication Date (Web): 14 Mar 2013 Downloaded from http://pubs.acs.org on March 16, 2013

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The Journal of Physical Chemistry

Kinetic Model of Gas Transport in Carbon Nanotube Channels Aleksandr Noy⇤ Biology and Biotechnology Division, Physics and Life Science Directorate, Lawrence Livermore National Laboratory, Livermore, CA 94550 School of Natural Sciences, University of California Merced, Merced, CA 95344 E-mail: [email protected]

⇤ To

whom correspondence should be addressed

1

ACS Paragon Plus Environment


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Abstract Carbon nanotubes represent a rare experimental realization of a nanofluidic channel, which has molecularly smooth walls and nanometer scale inner diameter. This unique combination of properties gives carbon nanotube channel an ability to support enhanced transport of water and gases with flows often exceeding those of conventional channels by several orders of magnitude. Surprisingly, most of these transport enhancement phenomena can be explained using very simple mechanisms that hardly go beyond classical physics concepts. Here we present a simplified analytical model that uses classic kinetic theory formalism to describe gas transport in carbon nanotube channels and to highlight the role of surface defects and adsorbates in determining transport efficiency. We also extend this description to include the possibility of gas molecule diffusion along the nanotube walls. Our results show that in all cases the conditions at the nanotube channel walls play a critical role in determining the transport efficiency, and that in some cases obtaining efficient transport has to involve optimization of flows from diffusion through the gas phase and along the nanotube surface. Keywords: carbon nanotubes; gas transport; diffusion; nanopore transport.

Nanotechnology gave physicists a number of materials and objects that enable experimental realization of concepts that they previously considered to be the exclusive domain of the thought experiments. None of the nanomaterials has proven more valuable in this regard than a carbon nanotube. 1 A combination of smooth graphitic surface of carbon nanotubes and their nanometer scale inner diameter come very close to approximating an ideally smooth nanoporous channel. It is not thus surprising that they generated intense interest for gas transport studies. 2–4 Remarkably, MD simulations by Sholl and co-workers 2 showed that the transport diffusivity of gases inside carbon nanotube pores is anomalously high, and it approaches the same order of magnitude as the diffusion in pure gas phase. This phenomenon of the enhanced transport in carbon nanotubes has been subsequently confirmed when carbon nanotube membranes demonstrated ca.100⇥ gas–transport rate enhancement and over 1, 000⇥ water–transport enhancement. 5–7 MD simulations indicated that the main reason for the transport enhancement was the smoothness of the carbon nanotube walls, which produced specular reflectance conditions in molecule-wall collisions. Indeed, even 2


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nanotubules coated inside with amorphous carbon showed evidence of higher fraction of specular gas–wall collisions. 8 MD simulations by Sholl and co-workers also showed that even when the nanotube walls are allowed some flexibility, the transport efficiency of those channels remain extraordinary high. 9 Here we propose a rather simple model that uses classic kinetic theory to describe gas transport in carbon nanotube channels. This model uses only several basic assumptions about the nature of gas–wall collisions and the influence of defects and adsorbates on the nature of those collisions. It also provides close-form solutions if one side of the nanotube channel opens into vacuum. In this case the one-way kinetic flow through pore is equal to the net (diffusive) flow. Note that the model should be applicable to the other types of nanoporous channels as long as they maintain smooth surface that produces specular reflections off the clean pore wall. We first describe the case of passive gas adsorption on the pore walls, and then introduce the possibility of adsorbate diffusion along the pore walls.

Transport of ideal gas in carbon nanotube pores A classic 1909 work by M Knudsen 10,11 described gas flow in very narrow capillaries where the distance between the capillary walls is smaller than the mean free path of the gas molecules and gas–wall collisions dominate the transport dynamics. Since the mean free path of the gas molecules is typically very short (⇡65nm at atmospheric pressure), Knudsen transport regime is usually restricted to very dilute gases flowing through very narrow pores. Nanometer-scale pores of carbon nanotubes introduce an interesting possibility of observing purely Knudsen transport at atmospheric pressures. Significantly, this analysis assumes that the capillary walls are microscopically rough, and thus the molecular velocity randomizes at each collision (Figure 1a). In the idealized case of completely elastic specular collision in a completely smooth nanotube pore, gas molecules will translocate through the pore in a billiard-ball like series of collisions, and the pore will introduce no resistance to transport (Figure 1b). In reality, carbon nanotube pores have surface defects

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and bends that can act as scatterers that randomize the molecular velocity. Any species adsorbed on the surface can also act as similar scatterers. Therefore, the mean free path, l in the pore will not be determined by the nanotube diameter, but instead will depend on the average distance between the scatterers, d (l ⇡ d). In other words, the carbon nanotube channel of the radius a behaves as a regular Knudsen-like channel with a much bigger diameter d. The nanotube defect density and the presence of surface adsorbates will then play a critical role in determining the flow enhancement in these channels. Classic kinetic theory gives the following relationship between the diffusion coefficient of the gas molecules of mass, m, and mean free path, l : 1 l D = lv = 3 3

r

8kB T pm

(1)

where v is the average molecular velocity. For an ideal gas that flows through a pore of radius a and length l, with one side of the pore held at gas phase concentration n and the other side at vacuum, the flow, N, is given by:

l a2 N= 3l

r

8pkB T n m

(2)

Note that this formula gives the classic Knudsen flow expression (N = NKN ) when l = 2a. Thus, in a tortuous nanotube channel that has defects and bends spaced by an effective distance d, the enhancement over the Knudsen flow will simply be equal to d/2a. One of the first demonstration of the enhanced transport in carbon nanotube membranes 5 reported 100⇥ enhancement of the gas flow through 0.16 nm diameter nanotube pores, which indicates that those membranes contained velocity-randomizing sites only every 160 nm (a rather remarkable quality level!).

Transport of non-ideal gas in nanotube pores Now we consider the possibility that some of the gas–wall collisions lead to adsorption of the gas molecules onto the nanotube walls and creation of additional diffuse scattering sites that reduce

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a

d

b

5x10

-3

-1

(A ps )

4 3 2

N

eq

-2

c

1 0 0

f

e

Gas Concentration, n/n0

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10

20

30

Pressure (bar)

40

50

1.0 0.1 1 5 10 100 500

0.8 0.6

10 10

0.4

3 4

0.2 0.0 0

1000

2000

3000

Distance along CNT pore (nm)

Figure 1: (a-c) Schematic representation of different kinds of molecular collisions in carbon nanotubes showing (a) diffuse reflectance collisions of the gas molecules (circles) in a conventional pore where every collision with the rough wall randomizes the molecule velocity, (b) purely specular reflections from collisions with the smooth nanotube walls, and (c) collisions in carbon nanotubes that combine specular reflections off walls with diffuse reflectance off bends, defects, and molecular adsorbates (hexagons). (d) CH4 flow through (20,0) CNT (from 12 ). Solid line represents flow calculated from the Eq. (6) using the value of Langmuir coefficient a = 8.92 nm3 obtained from. 2 Nanotube parameters were: a = 0.8nm, l = 500nm, d/2a = 100. (e) Calculated enhancement of the flow through the nanotube over Knudsen flow (Eq. (7)) as a function of the normalized concentration an0 . Calculation assumed d/2a = 50. (f) Normalized concentration profiles along the 3000 nm-long carbon nanotube calculated using Eq. (8) for different values of the Langmuir parameter an0 , as indicated in the legend. The calculation parameters were: n0 = 2.5 · 10 2 nm 3 , a = 0.8 nm, d/2a = 100.

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transport efficiency further. In this scenario, the diffusion coefficient for the gas inside the nanotube pore is no longer constant. Instead it depends on the amount of gas molecules adsorbed on the nanotube wall at a certain position along the pore length. We can quantify this effect by noticing that in the absence of adsorbed gas the mean free path is equal to d, and when all pore surface is occupied by the gas molecules, the mean free path drops to 2a. For the subsequent derivation we assume that gas molecules adsorb to the pore walls according to the Langmuir isotherm (q = an/(1 + an), where q is the fraction of the occupied surface sites). Note that this step relies on several assumption. First, we are neglecting any potential correlation between molecular jumps in this system. Second, we are assuming that the molecules in the center of the nanotube are maintaining a gas phase equilibrium that is independent of the adsorbed layer on the pore walls. These are strong assumptions, given the small diameters of carbon nanotubes, but we can use them as first order approximations. If the gas maintains adsorption equilibrium throughout the pore, then the average mean free path, < l > can be expressed as a function of gas concentration, n, and Langmuir parameter, a, as:

< l > (n) = d(1

q ) + 2aq =

d + 2aan 1 + an

(3)

We can then use Eq. (1), to write the position-dependent diffusion coefficient D(x) as follows: 1 D(x) = 3

r

 8kB T d + 2aan(x) pm 1 + an(x)

(4)

In the stationary transport regime, the transport obeys Fick’s law with the position-dependent diffusion coefficient and the following boundary conditions:

dn 3N = 2· dx pa

r

pm 1 + an(x) · 8kB T d + 2aan(x) n(o) = n0 ; n(l) = 0

6


(5)

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which produce the following expressions for the flow, enhancement over the Knudsen regime flow, and concentration profile along the nanotube: 2a3 N= 3l

r

 ✓ 8pkB T d · n0 · 1 + m 2a

✓ N d = 1+ NKN 2a ✓ x=l 1

1

◆

ln(1 + an0 ) an0

◆ ln(1 + an0 ) 1 an0

an + (d/2a an0 + (d/2a

1) ln(1 + an) 1) ln(1 + an0 )

(6)

(7) ◆

(8)

Interestingly, gas flow predicted by the Eq. (6) compares reasonably well (Figure 1d) to the flow of methane gas molecules through a (20,0) carbon nanotube pore calculated by Sholl and co-workers. 12 Some discrepancies between the calculation results and model predictions (such as the more abrupt roll-off of the flow with increasing pressure) could be caused by the deviations of the CH4 adsorption isotherm from the Langmuir shape (see Supplementary Information for the fit). In addition, MD simulations in 2,12 considered only defect-free carbon nanotubes, and the authors noted that the presence of defects, which add corrugation to the potential energy surface, could impact the diffusive transport. Note also that these simulations captured much more sophisticated transport physics (including, importantly, surface resistances at the nanotube pore entrances, which our model ignores), thus the correspondence between these data and our model can be fortuitous. Surface resistances in particular can have a significant impact on the flow through the smooth nanotube pores 12 and should not be ignored a-priori. Calculated concentration profiles (Figure 1d) show that in the low coverage limit (an ' 0) where the nanotube walls remain clean, the transport behaves similar to the ideal gas case. In high coverage limit (an ' •) where most of the walls are covered, the transport becomes purely Knudsen-like. While in both of these limiting cases the concentration profile remains linear, it becomes non-linear at intermediate values of the Langmuir parameter. Interestingly, the highest non-linearity is reached at fairly high surface coverage, an0 ⇡ 100. The calculated enhancement 7



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curve (Figure 1e) shows the expected sigmoidal shape over a wide range of concentrations. This system shows a slightly surprising behavior where we need to occupy as much as 9% of the binding sites at the tube mouth to start decreasing the enhancement, and the whole 99% of the sites at the mouth need to be occupied to bring the enhancement down to approximately 2. We can rationalize these results if we recall that the other end of the pore opens to vacuum; therefore, the gas concentration at the end of the nanotube is always very low, and the pore is able to maintain smooth walls and correspondingly fast transport. We can easily extend the analysis from the previous section to the situation where a noninteracting gas is flowing along the nanotube channel with the adsorbing gas (see Supplementary Information). Interestingly, this analysis shows that if a gas contains even a small fraction of highly adsorbing impurity, that impurity poisons the transport fairly quickly. This observation agrees with the conclusions of an MD simulation that also indicates that fluxes of different species in carbon nanotube channels can be strongly coupled. 13

Surface diffusion and transport through carbon nanotube pores Finally, we consider the case where the adsorbed gas molecules can diffuse along the walls of a carbon nanotube pore. This scenario has been observed in some MD simulations of the transport in nanotube pores that indicated deviations from the purely Knudsen-like transport. 14 Yet, the same approach that we used in the previous section can describe the transport in this system in a closed analytical form. Gas concentration profile along the nanotube in this case is a more complicated function determined both by the gas phase and the surface transport, and the overall gas flow is a sum of the surface flow and gas phase flow. As before, we assume that the equilibrium between the molecules in the gas phase and on the surface is governed by a Langmuir isotherm. If q is again the fraction of the surface sites occupied at a particular point along the nanotube, then surface concentration of the gas molecules is equal to ns (x) = q (x) · n0s , where n0s = x

1

and x is the area

occupied by one gas on the nanotube wall in a monolayer. If surface diffusion flow Ns obeys

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a

b

1000

n/n0

Ds/Dg=5 2 1

100

0.2 -2

0.1

Log(αno) 0

2

0.3 100

0.1 2 10

Ds/Dg=0.3

αn0 = 0.1 2 10

0.6 0.4 0.0 1.0

Ds/Dg=0.4

Ds/Dg=5

0.8 n/n0

10

αn0 =

0.2

0.05

150

Ds/Dg=0.05

1.0 0.8 0.6 0.4 0.2 0.0 1.0 0.8

n/n0

N/NKN

0.5

N/NKN

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


0.2

αn0 0.1 2 10

0.6 0.4 0.2

1

0.0 &3

&2

&1

0

1

2

3

Log(αno)

0

1000 2000 Length (nm)

3000

Figure 2: (a) Calculated enhancements over Knudsen regime flow for gas transport as a function of the value of the Langmuir parameter an0 for different values of the Ds /Dg ratio. The inset shows enhancement curves in the region of Ds /Dg values where the flow enhancement peaks at intermediate Langmuir parameter values. (b) Concentration profiles along the nanotubes. For all plots the diffusion coefficient ratio Ds /Dg is indicated on the plots. All calculation assumed d = 160nm; a = 0.8nm; x = 0.3nm2 ; n0 = 2.7 · 10 2 nm 3 .

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the Fick’s law with the diffusion coefficient Ds , then we can write a complete set of equations governing gas transport in this system. Note that surface flow and gas flow by themselves are no longer constant throughout the nanotube pore, and only the overall flow is conserved.

Ng (x) =

pa2 3

r

8kB T [2aq (x) + d (1 pm

dn dx 2pa dq Ns (x) = Ds x dx an q= 1 + an q (x))]

(9)

Ntot = Ns (x) + Ng (x) n(0) = n0 , n(l) = 0 The solution for the full flow through the nanotube pore is then: Ntot

2a3 = 3l

r

 ✓ d 8pkB T · n0 · 1 + m 2a

1

◆

ln(1 + an0 ) 2a Ds 1 + an0 x a Dg 1 + an0

(10)

We can calculate the expected transport flow enhancement as a function of two parameters: the q BT Langmuir parameter an0 and the ratio of diffusion coefficients, Ds /Dg (where Dg = 23 a 8kpm ): ✓ Ntot d = 1+ NKN 2a

◆ ln(1 + an0 ) 2an0 Ds 1 1 + an0 n0 x a Dg 1 + an0

(11)

The concentration profile is then described by:

x=l· 1

d an + ( 2a d an0 + ( 2a

Ds an 1)ln(1 + an) + 3a x a Dg 1+an

Ds an0 1)ln(1 + an0 ) + 3a x a Dg 1+an0

!

(12)

Flux enhancements calculated according to Eq. (11) show several interesting features (Figure 2a). First, as expected, surface diffusion makes negligible contribution at very low values of the Langmuir parameter. Second, if surface diffusion is slow, it can actually reduce the overall transport rate. This behavior likely originates from the gas molecules diffusing along the nan-

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otube and frustrating the specular collision conditions at the outlet end of the nanotube. Third, for certain values of the diffusion coefficient ratios, there is an optimal balance of gas and surface diffusion flows that maximizes the transport efficiency (Figure 2a, inset). Surprisingly, a comparison of normalized concentration profiles along the nanotube does not show much qualitative difference between the cases of strong surface diffusion and weak surface diffusion (Figure 2b). All profiles show how the concentration profile changes from the linear profile characteristic of the unimpeded fast transport through pristine carbon nanotube to the bow-shaped profile characteristic of the presence of the adsorbed centers that frustrate the specular collisions of the gas molecules with the surface. The bow-shaped profile establishes when the adsorbates that build up closer to the pore mouth frustrate the specular collisions and slow down the molecules moving through that section of the pore (this effect is somewhat similar to emergence of the traffic jam on a highway). Even a very simple idealized model presented here points out to some interesting phenomena that we can observe in carbon nanotube pores. Follow-up MD simulations that focus on the relationship between nanotubes defects and adsorbates and the flow through the channel should test the validity of this model and provide a much more detailed picture of the transport in this system. Several practical conclusions follow from even this simple initial analysis. First, it is hard to overstate the importance of nanotube quality for the transport efficiency; having low density of defects along the nanotube channel is likely more important than packing those channels tightly in a membrane. Second, impurity that have affinity to the nanotube surface can change the transport enhancement and reduce the flow advantage of a carbon nanotube channel. Finally, gases that are able to adsorb on the nanotube surface and diffuse along it present complex tradeoffs that need to balance the additional surface diffusion flow with the loss of gas phase transport efficiency from the increasingly frustrated specular collision conditions on the nanotube channel surface.

Acknowledgement I thank Dr. H.-G. Park and Dr. O. Bakajin for discussions, Dr. H.G. Park for help with deriving some of the equations, and anonymous reviewer for helpful suggestions. Parts of this work were 11



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supported by NSF NIRT-CBET-0709090 and U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering. Parts of the work were performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

Supporting Information Available Supplementary information is available that contains the analysis of transport of two-gas mixture through the carbon nanotube pore, and the determination of the Langmuir constant for the CH4 adsorption on the nanotube walls.

This material is available free of charge via the Internet at

http://pubs.acs.org/.

References (1) H. J. Dai Carbon Nanotubes: Opportunities and Challenges, Surf. Sci. 2002, 500, 218. (2) Skoulidas, A. I.; Ackerman, D. M.; Johnson, J. K.; Sholl, D. S. Rapid Transport of Gases in Carbon Nanotubes, Phys. Rev. Lett. 2002, 89, 185901. (3) Mao, Z.; Sinnott, S. B. A Computational Study of Molecular Diffusion and Dynamic Flow through Carbon Nanotubes, J. Phys. Chem. B 2000, 104, 4618. (4) Mao, Z.; Sinnott, S. B. Predictions of a Spiral Diffusion Path for Nonspherical Organic Molecules in Carbon Nanotubes, Phys. Rev. Lett. 2002, 89, 278301. (5) Holt, J. K.; Park, H. G.; Wang, Y.; Stadermann, M.; Artyukhin, A. B.; Grigoropoulos, C. P.; Noy, A.; Bakajin, O. Science, Fast Mass Transport Through Sub-2-Nanometer Carbon Nanotubes, 2006, 312, 1034. (6) Hinds, B. J.; Chopra, N.; Rantell, T.; Andrews, R.; Gavalas, V.; Bachas, L. G. Aligned Multiwalled Carbon Nanotube Membranes, Science, 2004, 303, 62.

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(7) Majumder, M.; Chopra, N.; Andrews, R.; Hinds, B. J. Nanoscale Hydrodynamics: Enhanced Flow in Carbon Nanotubes, Nature 2005, 438, 44. (8) Cooper, S. M.; Cruden, B. A.; Meyyappan, M.; Raju, R.; Roy, S. Gas Transport Characteristics Through a Carbon Nanotubule, Nano Lett. 2004, 4, 377. (9) Chen, H.; Johnson, J. K.; Sholl, D. S. Transport Diffusion of Gases Is Rapid in Flexible Carbon Nanotubes, J. Phys. Chem. B 2006, 110, 1971. (10) Knudsen, M. The Law of the Molecular Flow and Viscosity of Gases Moving Through Tubes, Ann. Phys., 1909, 28, 75. (11) Steckelmacher, W. Knudsen Flow 75 years On: The Current State of the Art for Flow of Rarefied Gases in Tubes and Systems, Rep. Prog. Phys. 1986, 49, 1083. (12) Newsome D. A.; Sholl D. S. Influences of Interfacial Resistances on Gas Transport Through Carbon Nanotube Membranes, Nano Lett., 2006, 6, 2150. (13) Chen, H.; Sholl, D. S. Rapid Diffusion of CH4 /H2 Mixtures in Single-Walled Carbon Nanotubes, J. Am. Chem. Soc., 2004, 126, 7778. (14) Skoulidas, A. I.; Sholl, D. S.; Johnson, J. K. Adsorption and Diffusion of Carbon Dioxide and Nitrogen Through Single-Walled Carbon Nanotube Membranes, J. Chem. Phys. 2006, 124, 054708.

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Graphical TOC Entry a

b

2

n/n0

Ds/Dg=5

1 0.5

-2

0.1

Log( no) 0

2

0.2

Ds/Dg=0.3

n0 = 0.1 2 10

0.6 0.4

Ds/Dg=5

0.8 n/n0

100

0.1 2 10

0.0 1.0

Ds/Dg=0.4 0.3

n0 =

0.2

0.05

150

Ds/Dg=0.05

1.0 0.8 0.6 0.4 0.2 0.0 1.0 0.8

n/n0

0.2

N/NKN

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

Page 14 of 14

n0 0.1 2 10

0.6 0.4 0.2 0.0 0

no )

14


1000 2000 Length (nm)

3000

Kinetic Model of Gas Transport in Carbon Nanotube Channels

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