Journal of Low Temperature Physics, Vol. 110, Nos. 1/2, 1998

Hydrogen Adsorption in Nanotubes

George Stan and Milton W. Cole

Department of Physics, The Pennsylvania State University,104 Davey Lab, University Park, PA 16802, USA

Model calculations are presented of the adsorption of hydrogen in carbonnanotubes. Using a phenomenological interaction potential, we compute thelow coverage thermodynamic properties, showing explicitly the quantum ef-fects in the Feynman (semiclassical) effective potential approximation. Theeffects of interactions are evaluated with a quasi-one dimensional classicalLennard-Jones approximation.

PACS numbers: 33.15.F, 36.40-c, 61.46,+w, 67.70. +n

1. INTRODUCTION

Adsorption of atoms and molecules in fine pores has been recognizedto have both fundamental interest, because of the reduced dimensionality,1and technological importance, for many reasons, including separation of mix-tures, hydrogen storage, etc.2 We have embarked on a study of such prob-lems, using a variety of models appropriate to different systems. In thisreport we consider the case of hydrogen adsorption in carbon nanotubes, asystem of recent experimental interest.3 The present study is designed toexplore qualitative aspects of the problem.

In a previous paper, called I, we evaluated simple adsorption modelsappropriate to both ultralow-coverage helium and a quasi-one dimensionalclassical fluid, respectively.4 These models are conveniently solved if oneassumes that the potential is cylindrically symmetric, which was the caseconsidered in I. Here we extend that study by assessing the effect of atom-icity of the carbon tube. For the specific case considered here, we find thedifference in V to be small (especially compared to the uncertainty in thecalculation).

Of course hydrogen molecules move according to the laws of quantum

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mechanics. We address here the question of whether a classical approxima-tion suffices at the temperature T used in typical experiments. Specifically,we compute the first quantum correction using the Feynman “effective po-tential method” .5 In this, and all of our calculations, we use the “physicists’model” of hydrogen, taken as a spherically symmetric entity.

Finally, we further extend the description to finite coverage by evaluat-ing the adsorption equation of state with the quasi-one dimensional Lennard-Jones (LJ) classical approximation, as in I.6 Although this and the otherapproximations used here are all fairly crude, the results provide a usefulguide to understanding qualitative trends and experimental data.

2. POTENTIAL

In this paper we describe calculations based on a sum of isotropic LJinteractions U(|r - Rj|) between the molecule and the C atoms on the tube,which we take to have radius R. The LJ potential U(x) involves the usualparameters a and e. We take these to have values 2.97 A and 42.8 K,respectively,7 based on experience with the flat surface problem8

where the C atoms are situated at positions Rj. Experience with a flatgraphite surface suggests that the true potential is more accurately describedby an anisotropic interaction9,10, a treatment which we defer until a morecomprehensive study is carried out. Such an anisotropic potential is expectedto yield a more corrugated total potential than that derived here.

For specificity, we consider the case of a zigzag tube (n,0) with n=13.This is a non-helical tube,11 with radius R = na\/3/(27r)=5.09 A; a=1.421A is the lattice constant of graphite. The positions S and A on the cylinderare defined, by analogy to the graphite case, as the hexagon center and theC atom. In this case they are separated by a longitudinal distance equal toa.

Fig.l presents the potential’s variation along radial lines connecting theaxis to positions S and A on the tube’s surface. As for graphite, S is themost attractive and A is the least attractive potential energy position forthe molecule. Note that the magnitude of the potential energy minimum,D, is a larger multiple (~22) of the e parameter than is the case (~13) foradsorption of hydrogen on a flat surface. The reason is simply that moreneighbors contribute to the attraction. For the same reason, the relative cor-rugation of the potential is smaller inside the tube than on the flat surface.That corrugation arises primarily from the nearby atoms, which comprisea relatively small fraction of the total number of C atoms which contribute

540 George Stan and Milton W. Cole

Hydrogen Adsorption in Nanotubes 541

Fig. 1. Adsorption potential (reduced by energy scale e) of a hydrogenmolecule as a function of distance from the axis, along radial lines leadingto points S (dashes) and A (dots), as discussed in the text. Full curve isthe potential computed with a hypothetical continuum distribution of the Catoms. The squares (dash-dots) represent the effective potentials at T=50K (10 K).

strongly to the attraction in the tube. One observes in Fig. 1 that the poten-tial derived by replacing the carbon tube by a continuum wall is intermediatebetween the S and A potentials.

Fig. 1 also shows what we shall refer to as the “effective potential”,defined by the relation5

with l2 = (Bh2)/(6m), where 1/B is Boltzmann’s constant times T. Thenumerical value for H2 of the thermal length l is 2.005 A/VT. As describedby Feynman,5 the effective potential represents a smearing of the classicalpotential as a first quantum correction. This has been used previously12,13

to evaluate quantum corrections to both bulk and surface thermodynamicproperties. In the case of a cylindrically symmetric potential, assumed here,the averaging in Eq.(2) simplifies to yield:

542 George Stan and Milton W. Cole

Fig. 2. Ratio of adsorption in the tube to that on a flat surface of thesame area, computed in the effective potential approximation (full curve)and classically (dashes). Note the logaritmic scale on the vertical axis.

where I0(X) is the Bessel function of imaginary argument. Fig. 1 showsthat the effective potential resembles the classical potential V at high T butdeparts from it at low T. At high T, Eq. (3) yields a quantum correctionproportional to l2 and the second derivative of V with respect to r:

In the limit of a noninteracting gas, the adsorption is proportional to thepressure, with the proportionality coefficient given by the gas-surface firstvirial coefficient times B. In the classical (or semiclassical limit here), thisis proportional to the integral of a Boltzmann factor involving the effectivepotential.8 This regime applies to the case of large interparticle separation.

Fig. 2 presents results for the ratio of the total adsorption in the tubeand that on a flat surface of the same area. Note that the ratio is hugeat low T because the attraction is so much greater inside the tube than onthe surface. The quantum effect can be considerable in increasing this ratiobecause it “smears out” the deepest part of the potential more for the flatsurface than in the tube (since the potential’s curvature is greater).

Hydrogen Adsorption in Nanotubes 543

Fig. 3. Chemical potential as a function of density, computed with approxi-mations described in the text, in the case of an ideal gas (dashes) and for LJinteractions between the adsorbed particles (full curve). Hydrogen inside a(9,0) nanotube at 10 K was considered.

Fig. 3 presents the equation of state for hydrogen inside such a tube,computed on the basis of a set of assumptions: the molecules interact withspherical LJ interactions fit to the Silvera-Goldman14 form (3.04 A, 34.3 K)and the molecular motion is both classical and confined to the tube axis.This procedure parallels that of I apart from the treatment of the one-bodyterm. Here, a (9,0) nanotube (R=3.53 A) was chosen in order to more closelyexemplify the quasi-one dimensional approximation.

3. DISCUSSION

Our presentation has been aimed at assessing various aspects of thethermodynamic calculations. We have shown that the simple assumptionof cylindrical symmetry seems warranted in view of the large number ofneighbors which contribute to the net potential; this conclusion is tentativeinsofar as a more realistic calculation must be done to explore the effects ofanisotropy. We have found that the adsorption potential, although consid-erable in its attractive strength, is not adequate to explain the large heat ofadsorption data of Dillon et al.3 These authors found a heat of adsorptionper particle of 2360 K, while we find about half this value (1082 K). Thisconclusion is not limited by the choice of pore radius, as far as we know,

544 George Stan and Milton W. Cole

since we have chosen R to maximize (approximately) the heat of adsorption.The most likely origin of this discrepancy is the oversimplified modellingof the attraction. Our results and the experiment would agree if the pairpotential well-depth were ~ 2.5 times greater than the value assumed here.Finally, we have found that the quantum effects on a low density film arenot significant above 50 K; we expect that finite coverage situations will alsoconform to this finding. Of course, much interest is attracted to the case oflow T, for which the path integral method is both necessary and feasible.15

In future work we intend to address improvements in both the interactionand the statistical mechanics.

ACKNOWLED GMENTS

This research has been supported by the National Science Foundationand the Petroleum Research Fund of the American Chemical Society. Weare grateful to Karl Johnson for valuable discussion of this subject.

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M. J. Heben, Nature 386, 377 (1997).4. G. Stan and M. W. Cole, submitted to Surf. Sci.5. R. P. Feynman, Statistical Mechanics (W. A. Benjamin, Reading, MA 1972).6. F. Gtirsey, Proc. Cambridge Phil. Soc. 46, 182 (1950); H. Takahashi, in Mathe-

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