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J. Phys. Chem. C 2010, 114, 7602–7610
Defect and Nondefect Interstitial Channel Availability in Carbon Nanotube Bundles: Comparison of Modeling with Experiments Matthew R. LaBrosse†,‡ and J. Karl Johnson*,†,‡ National Energy Technology Laboratory, Pittsburgh, PennsylVania 15236, and Department of Chemical and Petroleum Engineering, UniVersity of Pittsburgh, Pittsburgh, PennsylVania 15261 ReceiVed: NoVember 18, 2009; ReVised Manuscript ReceiVed: March 20, 2010
There is a controversy in the literature over whether small gases, such as Ne, can adsorb in the interstitial channels (ICs) of carbon nanotube bundles. We distinguish between two types of ICs: nondefect, defined as ICs in perfectly packed arrays of nanotubes having all of the same diameter (homogeneous) defined where three tubes meet, and defect ICs, present in bundles composed of nanotubes having different diameters (heterogeneous) where four or more tubes meet. We have performed grand-canonical Monte Carlo simulations of Ne on various model carbon nanotube bundles in order to explore the role of nondefect and defect ICs in adsorption. We have performed simulations on closed and partially opened homogeneous and heterogeneous bundles. We have computed the specific surface area, the isosteric heats of adsorption, and adsorption isotherms from our simulations, and we have compared these values to experimentally measured quantities for Ne adsorption on carbon nanotubes produced by the HiPco process. Analysis of our results indicates that gases do not adsorb in nondefect ICs. When considering Ne adsorption alone, there is ambiguity about whether homogeneous or heterogeneous bundles are better models. However, taken together with previous work on Ar, CH4, and Xe, we conclude that a model consisting of heterogeneous bundles with ∼11% open nanotubes, having nondefect ICs blocked, but allowing adsorption in the defect interstitials best describes the reported experimental data for these gases on HiPco nanotubes. Furthermore, we find that quantum mechanical diffraction effects must be taken into account for modeling Ne adsorption at the experimental conditions. 1. Introduction Adsorption of gases on single walled carbon nanotubes (SWNTs) has been studied both experimentally1–32 and theoretically.28,33–46 SWNTs are known to form bundles containing 10s to 100s of nanotubes.47–51 These bundles can be modeled as being homogeneous, where all of the nanotubes have the same diameter, or heterogeneous, where the bundle contains nanotubes having a distribution of diameters. The nanotubes are usually observed to pack together in two-dimensional hexagonal arrays with an observed nanotube-nanotube spacing of 3.2 Å,52 which is slightly less than the spacing between two graphene sheets in graphite. Nanotubes have been synthesized by a variety of methods, including electric arc, laser ablation, and high pressure carbon monoxide (HiPco).53 Nanotubes produced by the electric arc method have been shown to have closed ends,54 whereas asproduced HiPco nanotubes have been shown to have a percentage of the nanotubes that are open.45,55–58 Recent studies have claimed that the HiPco process can produce nanotubes with purities as high as 97%,59 making them an attractive choice for adsorption experiments. In this paper, we compare Ne simulation results with experimental data32,59 measured on nanotubes prepared by the HiPco process. Adsorption studies in the literature show general agreement over the nature of adsorption in internal sites, groove sites, and external surface sites.18,31,38,60–62 However, there is substantial disagreement over whether gases adsorb in interstitial channels (ICs) created by three or more nanotubes.9,14,18,32,42,43,45,63 Two * To whom correspondence should be addressed. E-mail:
[email protected]. † National Energy Technology Laboratory. ‡ University of Pittsburgh.
experimental studies by Migone and co-workers have shown that experimental specific surface areas (SSAs) for Ne are similar to those for other gases. In the case of nanotubes prepared by the electric arc method, Ne and Xe had SSAs of 41 and 38 m2 /g, respectively.14 Adsorption results from experiments using HiPco nanotubes show a similar trend, namely that all gases have roughly the same SSAs.32 The experimental results from these two studies strongly indicate that the same types of sites are available for adsorption on SWNT bundles. Therefore, because Xe is too large to adsorb in ICs created by three nanotubes, it is concluded that none of the gases adsorb interstitially. This is a surprising result, because the diameter of Ne is small enough that it is expected to adsorb in the ICs. LaBrosse and co-workers have recently compared the simulated SSAs, isosteric heats of adsorption, and adsorption isotherms for Ar, CH4, and Xe with experiments.64 Through simulations on various closed and partially opened homogeneous and heterogeneous bundles, it was shown that defect interstitials found in heterogeneous bundle models are critical for describing experimental results. However, Ar, CH4, and Xe are all too large to adsorb in the nondefect ICs formed by three nanotubes, so no conclusions were drawn about adsorption in these sites. Simulations show that Ne is small enough, in principle, to access nondefect IC adsorption sites that are unavailable for adsorption of Ar, CH4, and Xe. Hence, this paper aims to make a clear distinction between adsorption in nondefect and defect ICs. In doing so, we perform simulations with Ne to examine the role of both nondefect and defect interstitials for describing experimental results. As with previous work,42,64 we consider models of closed and partially opened homogeneous and heterogeneous SWNT bundles.
10.1021/jp910966e 2010 American Chemical Society Published on Web 04/08/2010
Interstitial Channel Availability in SWNTs
J. Phys. Chem. C, Vol. 114, No. 17, 2010 7603
2. Methodology 2.1. Bundle Construction. The details of constructing the bundles used in simulations were described previously.64 To summarize, several assumptions were made to circumvent the complexities of a fully articulated nanotube bundle model. There are no interactions between nanotubes during the adsorption simulations and nanotube positions are held fixed. All nanotubes used in simulations are assumed to be straight, parallel, rigid cylinders free of chemical defects. Flexibility issues, such as thermal vibrations, are not considered, which is a reasonable assumption because previous work has shown that thermal vibrations can be safely ignored for transport properties of gases inside SWNTs at high loadings.65 Homogeneous bundles are constructed such that tube centers are arranged in a twodimensional hexagonal lattice configuration. Heterogeneous bundles are generated by optimizing the two-dimensional nanotube centers in energy to a local minimum on the potential energy surface. Nanotube centers are then randomly moved to explore the potential energy surface and locate new, possibly more favorable, local minima. During this process, the lowest energy configuration is stored until it is replaced by one even lower in energy. This technique is called the basin-hopping method.64,66–68 The result is a heterogeneous bundle that has settled into a local or global minimum configuration. After the basin-hopping algorithm was completed, the nearest-neighbor nanotube spacing was checked for consistency. The spacing between nanotubes is very close to 3.2 Å for all neighboring pairs of nanotubes in these bundles. 2.2. Interaction Potentials for Classical Fluids. The Lennard-Jones (LJ) potential was used to calculate fluid-fluid interactions. The LJ potential is given by
[( ) ( ) ]
VLJ(r) ) 4εff
σff r
12
-
σff r
6
(1)
where r is the fluid-fluid separation distance and εff and σff are the LJ parameters for the adsorbate-adsorbate (fluid-fluid) interactions. For Ne, εff/kB and σff are taken from Morales and Nuevo.69 These parameters are 36.83 K and 2.79 Å, respectively, where kB is the Boltzmann constant. Solid-fluid interactions require additional consideration due to the cylindrical geometry of the nanotubes. We assume that the interaction potential between adsorbates and nanotubes is LJ in nature. The derivation of the solid-fluid potential includes one additional assumption; nanotubes are curved sheets of graphene with a carbon atom surface density (θ) of 0.38 Å-2. We have used the hypergeometric series potential to describe the interaction between an adsorbate and an infinitely long cylinder of radius R.33,70 The hypergeometric potential expresses the interaction between an adsorbate atom/molecule and a nanotube as a function of r, the distance of the atom or molecule from the center of the nanotube. Thus, the solid-fluid potential is smoothed, lacking atomic roughness. Smoothed potentials have been used in most previous adsorption simulations for SWNTs.29–31,33,38,40–42,44,46,57,62,64,71–75 Different hypergeometric expressions are used to describe an adsorbate inside or outside of the nanotube. The potential for a molecule outside the nanotube is
[ ( )
out VLJ (r, R) ) 3πθεsf σsfR
21 σsf 32 r
11
M11/2(x) -
( )
]
σsf 5 M5/2(x) r (2)
where x ) R/r, σsf and εsf are the solid-fluid LJ parameters, and the elliptical integral, Mn(x), is given by70
Mn(x) ) π
F[1 - n, 1 - n;1;x2] (1 - x2)2n-1
(3)
The generic form of the hypergeometric series, F, is given by
F[a, b;c;z] ) 1 +
ab a(a + 1)b(b + 1) 2 z+ z + ... 1!c 2!c(c + 1)
(4) Inserting eqs 3 and 4 into eq 2 yields a potential form describing an adsorbate outside of a nanotube as a function of its distance from the center of the nanotube and the nanotube radius. A similar treatment is applied to the potential describing an adsorbate inside of a nanotube. There are two main advantages to using the hypergeometric potential: (1) correct physical description of the interaction potential at all solid-fluid separations and (2) the ability to easily generate potentials for nanotubes of any diameter. Previous studies by LaBrosse and co-workers used an eighth-order polynomial, fit for a specific (n,m) nanotube and adsorbate pairing to describe solid-fluid interactions.64 This approach requires polynomial fits to several adsorbate-(n,m) type nanotube combinations and yielded polynomials that do not give the correct asymptotic behavior as r f ∞. 2.3. Interaction Potentials for Quantum Fluids. 2.3.1. PathIntegral Formalism. Quantum diffraction effects must be taken into account in order to accurately model Ne at the temperatures of interest in this study. One method of accounting for quantum effects in molecular simulations is path-integral Monte Carlo.76 The path-integral formalism was first described by Feynman in 1948, who postulated a connection between a classical system and a quantum system through a transformation of the classical particle into a polymeric ring of P “beads”.77 In this context, each atom or molecule is replaced by a ring polymer. The potential then contains two parts, the interactions between beads of different rings (intermolecular) and the interactions between beads on the same ring (intramolecular). The intermolecular potential is given by P
V inter )
N
∑ ∑ φijR
1 P R)1
(5)
i
