J Low Temp PhysDOI 10.1007/s10909-008-9812-8

Review: Novel Physics of Gases Near CarbonNanotubes and Buckyballs

S.M. Gatica · M.M. Calbi · R.D. Diehl · M.W. Cole

Received: 1 April 2008 / Accepted: 5 May 2008© Springer Science+Business Media, LLC 2008

Abstract Nanotubes and buckyballs represent novel substrates for gas adsorption.This situation has attracted significant experimental and theoretical attention, result-ing in a wide variety of intriguing behaviors, including phases of unconventional ef-fective dimensionality. This article reviews this field. Many of the unusual predictionshave yet to be tested. In some cases, heterogeneity plays a significant role in actualexperiments. This research area therefore presents many unanswered questions offundamental significance.

Keywords Carbon nanotubes · Adsorption · Buckyballs

1 Introduction

Carbon nanotubes and buckyballs (C60 molecules) have been intensively exploredin recent years because of both their potential applications and the novel physicalphenomena they might manifest [1–4]. These systems display a host of diverse prop-erties, which have been reviewed in many journal articles and several monographs.The present review describes predictions and observations concerning simple gasesthat are adsorbed on these fullerenes or within them. The motivations for such studyare diverse, including the following:

S.M. Gatica (�)Department of Physics and Astronomy, Howard University, 2355 Sixth Street NW, Washington, DC20059, USAe-mail: sgatica@howard.edu

M.M. CalbiDepartment of Physics, Southern Illinois University, Carbondale, IL 62901-4401, USA

R.D. Diehl · M.W. ColeDepartment of Physics, Penn State University, University Park, PA 16802, USA

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(a) The fullerenes, especially nanotubes, have been proposed as substrates capable ofstoring certain gases, e.g. hydrogen, because they have a large surface-to-volumeratio [5].

(b) Because of the dominance of the surface, they may serve as sensitive probes ofambient gases and/or isotopic separators [6].

(c) These materials’ surfaces expose nanoscopic length scales, resulting in unusualphenomena associated with gas commensuration [7].

(d) Because nanotubes can exhibit enormous aspect ratios (>1,000) and nanoscalecross-section, adsorbed gases can exhibit quasi-one-dimensional (1D) phases,evolving with coverage to phases of higher dimensionality D [8–30].

The last two of these topics are the focus of the present article. There have beenseveral reviews written [5, 9, 10, 31–34] about these adsorbed films and their phasetransitions, many fewer than the number devoted to nanotube adsorption in general.Much of the research to be discussed here is that of our own groups, extending overthe last decade. In these studies, we have examined many possible phase transitions ofadsorbed gases. Some of these are simple and obvious consequences of the geometry.For example, the peapod phase of matter consists of a “gas” of buckyballs confinedwithin nanotubes, first observed and explored by Luzzi’s group [35, 36] and John-son’s group (see Fig. 1).

In the approximation that the nanotubes are infinitely long and structurally per-fect, the adsorbed molecules provide a realization of a 1D fluid [27–29], a subject ofconsiderable fundamental interest [37, 38]. The fact that transmission electron micro-graphs distinguish individual balls means that the predictions (e.g. correlation func-tions) of idealized (or realistic) 1D models can be tested directly without perturbingthe system.

The outline of this paper is the following. Section 2 describes quasi-1D phasesof matter predicted for gases near nanotubes. These phases exhibit dramatic con-

Fig. 1 (Color online)Transmission electronmicrograph of a peapod system,showing the C60 moleculesencapsulated by a single-walledcarbon nanotube (contributed byA.T.C. Johnson)

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sequences of their dimensionality. In Sect. 3, we treat the problem of the couplingbetween the gas and the tubes. There are predicted to arise significant cooperative ef-fects of this coupling, quite different from behavior arising in the conventional modelsin which the nanotubes are rigid. Section 4 describes another phenomenon, which isequally dramatic, and unexpected. This is a prediction of a 4D Bose-Einstein conden-sation (BEC) [39, 40]. Section 5 describes gas adsorption on a layer of buckyballs.This system is expected to exhibit an unusual lattice-gas transition, in which the lat-tice represents the array of adsorption sites provided by the balls.

The many predictions described here have been tested to just a small extent, par-tially due to the problem of heterogeneity, an unavoidable aspect of nanotube physicsat present [41]. Interestingly, the BEC transition requires this heterogeneity for it tooccur.

Experimental data relevant to the various phenomena are briefly discussed inSect. 6. Our outlook concerning this interesting area of research is summarized inSect. 7.

2 Quasi-1D Phases

One-dimensional physics is extremely interesting and diverse; this article can addressjust a few aspects relevant to adsorption near nanotubes. Model calculations of 1Dsystems are natural extensions of those applied to higher dimensional systems, forwhich physical realizations abound in both 3D and 2D (monolayer films), permit-ting testing of the calculations. Nanotubes provide opportunities to explore the 1Dmodels.

Such testing of models is particularly intriguing for a 1D fluid because (unlike thecase of D > 1) the equation of state is an explicit function of the two-body interac-tion V (x) between particles [37, 38]. In particular, the 1D density ρ, at 1D pressureP1D, equals the following ratio of two integrals at temperature T :

ρ = I0

I1(1)

In ≡∫ ∞

0xn exp{−β[V (x) + xP1D]}dx (2)

where β = (kBT )−1. This expression is derived from two assumptions: the validityof classical mechanics and the neglect of interactions beyond those between nearestneighbors. For the peapod problem, the balls are quite massive and T is high, sothat quantum statistical effects are assuredly negligible. As to the second assump-tion, the long-range interactions produce just small corrections to (2) because thevan der Waals interaction falls off rapidly with distance (V (x) ∼ x−6) at large sepa-ration [42]. (These long-range interactions provide relatively larger contributions inhigher D because the phase space volume scales like rD−1, emphasizing larger sepa-rations for higher D.)

An intriguing feature of (1) is that this equation of state exhibits no singulari-ties, meaning that no phase transitions occur at any density or T . That conclusion

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is robust and quite general (i.e., beyond the preceding approximations). It is a con-sequence of the destruction of long-range order by fluctuations in 1D. In systems ofdimensions 2 and higher, there is no corresponding restriction. In speaking of “dimen-sion” here, we employ the standard definition in statistical mechanics: a 1D systemhas just one space coordinate becoming infinite. Thus, a gas in a nanotube is inher-ently 1D, with the infinite coordinate along the tube’s axis, even though the tube’stransverse dimensions are nonzero. Similarly, a monolayer film on a highly corru-gated C60 layer is a 2D system. While fluctuations can destroy some ordered phasesin 2D (e.g. Bose-Einstein condensed phases), they play a relatively smaller role anddo permit transitions to occur in many model systems (e.g. 2D Ising).

2.1 Quasi-Transitions in 1D Pores

In spite of the constraints on genuine phase transitions for 1D systems, there can oc-cur two kinds of phase change of gases within nanotubes. One type might be calleda “quasi-transition”. This term refers to a physical change that can be quite abrupt,although not singular, so that even a very careful experiment might not exhibit muchdifference between this behavior and that occurring at a genuine thermodynamic tran-sition. Figure 2 displays an example of such a quasi-transition found in simulationsof adsorption within a small pore, of radius R = 1 nm [43]. For strongly attractivepotentials, the adsorption isotherm exhibits two steps. The first represents adsorptionof a “shell” phase, coating the wall of the pore. The second step corresponds to theformation of an “axial” phase, filling the pore. The curves in the case of weak gas-surface attraction, in contrast, show that pore-filling (capillary condensation) occursin a single, nearly vertical, step. Hysteresis is seen to occur in those cases of veryweak substrate attraction (for which the pore-filling occurs only above saturated va-por pressure). This behavior in simulations reflects a lack of equilibrium because ofinadequate phase space sampling; similar hysteresis is observed in laboratory exper-iments for the same reason- persistent metastability.

Another example of this two-step adsorption is seen in Fig. 3, for the case ofan R = 0.8 nm nanotube [44]. Here, H2 isotherms, computed with the path integral

Fig. 2 (Color online) Computersimulation isotherms for Ar atits triple temperature insidepores of radius R = 1 nm.Curves show coverage(dimensionless units) as afunction of pressure relative tosaturated vapor pressure,computed with adsorptionpotentials of well-depth D = 50,40, 30 and 20 meV (from left toright). Only the two cases ofweakest attraction exhibithysteresis. From Gatica andCole [43]

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Fig. 3 H2 simulated adsorptionwithin a nanotube, R = 0.8 nm,at T = 10 K, as a function ofchemical potential μ. Squaresdenote the 2D density θ of theshell phase coating the nanotubewall (left ordinate scale), whilecircles represent the 1D density(right scale) of the axial phase.From Gatica et al. [44]

Monte Carlo method, exhibit two steps, because the gas-surface interaction is verystrongly attractive. The first step, manifesting the so-called “shell phase”, occurs atpressure values (chemical potentials) lower than those shown in Fig. 3, so it doesnot appear in the figure. The second step is seen to be nearly vertical, reflecting aquasi-transition, pore-filling. No hysteresis was found in the adsorption isotherm, asexpected from the analogous behavior in Fig. 2 for a strongly attractive substrate in apore of small R.

We note that analogous layering transitions can occur in films adsorbed on flatsurfaces. Because of the higher dimensionality, D = 2, these are real thermodynamictransitions and have been described with a generalized Ising model by De Oliveiraand Griffiths [45].

2.2 True Transitions in Quasi-1D Environments

In contrast to the preceding quasi-transition within single nanotubes, a real phasetransition can occur when particles in many adjacent channels, or nanotubes, are cou-pled. Such behavior means that the physical system is just quasi-one-dimensional.One kind of geometry and low temperature phase that we envision are depictedschematically in Fig. 4. The gas atoms or molecules in this case are located within in-terstitial channels situated between the nanotubes, e.g. within a bundle of nanotubes.The adsorbed particles exert a strongly attractive interaction between their neighborswithin the same channel but just a weak attraction to neighbors in adjacent channels.This weak interaction is essential for a true transition to exist. The depicted phase isan anisotropic liquid, with properties expected to be quite sensitive to this environ-ment [46].

Quite often, hand-waving arguments are used successfully to affirm the propor-tionality of the critical temperature (Tc) of a phase transition to the strength of theinteractions. For example, in the isotropic, nearest-neighbor Ising model, this asser-tion (Tc ∝ J = ε/4, where ε is the nearest-neighbor interaction strength) is strictlyvalid for all dimensions D ≥ 2; no transition occurs for D = 1, of course. Thus, onemight assume that the value of Tc for condensation of the anisotropic liquid wouldbe proportional to the transverse interaction (Jt ) between particles in neighboring

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Fig. 4 Schematic depiction ofthe coexistence of an anisotropicliquid droplet (surrounded bydashed curve) and a gas phaseof interstitial particles

channels, with the longitudinal interaction (Jz) playing a minor role. That expecta-tion is not correct, however. As proved by Fisher and Landau [47, 48], in the case ofstrong anisotropy (weak relative coupling), α∗ = Jt/Jz � 1, Tc is a remarkable (andsingular) function of the interaction ratio α∗:

kBTc

Jz

≈ 2

ln(1/α∗) − ln[ln(1/α∗)] (3)

Thus, while the transition requires a nonzero value of Jt , the characteristic tempera-ture scale of the transition is set primarily by Jz. This unusual functional dependencecan be rationalized by recognizing that when the system undergoes the transition,the neighboring channels are individually ordered, to a spatial extent proportional tothe 1D correlation length ξ(T ) ∼ exp[Jz/(kBTc)]. Hence, the effective transverse in-teraction involves blocks of particles extending over this distance. The resulting factthat Tc has an infinite derivative with respect to α∗ (at α∗ = 0) reflects the fact thatthe 1D system at low T has a divergent correlation length and thus this low T systemis unstable with respect to weak perturbations. (These latter could include elastic ordirect van der Waals interchannel interactions.)

In Figs. 5 and 6 we present results of this predicted behavior for two differentsystems. The first figure shows the heat capacity C(T ) for a gas of particles confinedwithin interstitial channels of a nanotube bundle; this function has been calculatedwith Monte Carlo simulations of an anisotropic lattice-gas model, with couplings Jz

and Jt . The peak reveals the critical point divergence, rounded by numerical effectsof the finite simulation cell. Because this system is actually 3D, when transverse in-teractions are included, the critical point behavior (i.e., critical exponents) coincideswith that of the much-studied 3D Ising model. In Fig. 5, contrast is made with two

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Fig. 5 Specific heat at thecritical density (circles) forparticles within interstitialchannels of a nanotube bundle,as a function of reducedtemperature, kBT/ε (ε is thestrength of the nearest-neighborattraction). The dashed curve isthe corresponding heat capacityC1D(T ) of the 1D Ising model,while the full curve denotes thebackground heat capacity of thenanotubes, multiplied by 1,000for visibility. From [46].

Fig. 6 Reduced criticaltemperature, kBTc/ε, for thecondensation of C60 moleculeswithin a bundle of nanotubes, asa function of the relativestrength of the transverseinteraction, α∗, defined in thetext. Full curve is computedfrom perturbation theory, whilethe dashed curve is computedfrom the anisotropic lattice-gasmodel described in the text, (3).Inset magnifies the region ofsmall α∗. From Gatica et al. [52]

other curves. One is the heat capacity C1D of the 1D Ising model, which shows notransition and goes smoothly to zero at low T [49]. At high T , this function coincideswith the heat capacity of our anisotropic system. However, close to the transition,and below Tc, C(T ) departs dramatically from C1D(T ). The other curve appearingin Fig. 5 is the background heat capacity of the nanotube bundle. Its magnitude isdwarfed by the adsorbate’s transition because graphitic materials are thermodynami-cally inert at low T (because of their high speed of sound). Hence, the transition oughtto be readily observable even though the adsorbed gas represents a small fraction ofthe atoms of the system. There may exist some qualitative support for this statementin experimental data of Johnson and coworkers [50].

Figure 6 presents the transition temperature for a gas of buckyball molecules mov-ing within a bundle of nanotubes. This system is predicted to have a critical tempera-ture near 500 K, computed from realistic C60 interaction strengths [51, 52]. The figure

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shows the general dependence of Tc on the relative coupling, α∗, computed with twovery different models. One is thermodynamic perturbation theory, where the unper-turbed system is a 1D fluid that satisfies (1) and the perturbation is the energy shiftresulting from the interchannel coupling. The alternative model is the anisotropiclattice gas model, described above, with its singular dependence on α∗. Over an ex-tended range of α∗ the results of these two methods are surprisingly similar. Noteespecially the divergent slope in the limit of vanishing α∗.

The preceding discussion has addressed the problem of real 3D transitions oc-curring among atoms confined within a bundle of nanotubes. A related problem, notdiscussed in the literature (as far as we know) is an analogous transition occurringamong atoms confined to the grooves between pairs of nanotubes in a situation wherethe tubes lie mutually parallel, forming a monolayer mat on a substrate. Such a tran-sition would exhibit behavior characteristic of the 2D Ising model, an exactly solvedproblem [82], and thus be of considerable interest.

3 Effects of Gas-Tube Coupling: Dilation-Induced Adsorption

The vast majority of studies of physical adsorption assume that the substrate isrigid. However, this is just a convenient approximation; since Newton’s third law en-sures that the mutual gas-substrate forces are equal but opposite, the substrate atomsmust move. Explicit calculations of de Rouffingac, Alldredge and de Wette [53]reveal significant differences between the atomic motions and phonon spectra (forXe/graphite) computed with or without substrate motion. Nevertheless, for a light,weakly-interacting adsorbate, the error introduced by the assumption of rigidity isusually expected to be small, since the mass difference and weak-coupling tends toseparate the frequencies of the surface modes, dominated by the adsorbate motion,from those of the bulk substrate motions. However, in the case of adsorption withininterstitial channels of a nanotube bundle, it has been argued by Calbi, Toigo andCole (CTC) that there can be a large effect of the substrate’s motion, even in the caseof a light gas, a qualitative difference that means that substrate non-rigidity must betaken into account [10, 54, 55]. This is surely the case with the peapod system, atopic that is beyond the scope of this review. In that case, the binding and mass of theadsorbate atoms is nearly the same as that of the substrate, so the substrate dynamicaleffects are expected to be significant. A treatment of the dynamical effect for a singleH2 molecule was presented in [56]; this approach could be adapted to the C60 case.

The argument of CTC is conceptually simple. The energy of an H2 molecule inan interstitial channel is very sensitive to its restricted confinement between nan-otubes. As seen in Fig. 7, this means that the magnitude of the zero-point energy isquite large- roughly one-half of the potential’s well-depth. In this case, a very smallincrease of the separation between tubes can greatly lower both the potential and ki-netic energies, and thus the total energy, of the H2 molecule. The dilation of just 1% isseen in the figure to result in a greatly increased binding energy (from less than 200 Kto more than 600 K). These predictions are very sensitive to the assumed interactionmodels [55], but a large effect of dilation would seem to be a robust consequence ofthe model, independent of the details of the interaction [54].

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Fig. 7 Probability density (toppanel) and potential energy andground-state eigenvalue (bottompanel) of an H2 molecule as afunction of distance from thecenter of an interstitial channel.Dashed curves assume nodilation of the nanotube bundle,while the full-curve assumes a1% linear dilation. From [10]

The dilation effect (computed at temperature T = 0) yields a much stronger bind-ing of the quasi-1D adsorbed fluid to the nanotube bundle than would arise witha frozen substrate. This is a cooperative effect, since a single molecule could notsingle-handedly cause a dilation. Thus, it has been conjectured that there can arise ahigh temperature condensation of the fluid in this environment. A crude estimate [55]of the critical temperature for this exotic transition is a few hundred K, based on theenergy reduction of the ground state, depicted in Fig. 7. No detailed studies of thisintriguing conjecture have been made thus far.

4 Novel Bose-Einstein Condensation (BEC) Transition

As indicated previously, strictly 1D systems do not possess genuine phase transi-tions. For an ideal bose gas, this conclusion can be derived with the usual statisticalmechanics treatment based on the density of states N(E) as a function of energy. InD dimensions, the relationship is N(E) = CDE(D−2)/2, where CD is D-dependentcoefficient, which is independent of E. Consider the bose integral relating the totalnumber of particles, N , to the chemical potential μ,

N =∫

N(E)dE

exp[β(E − μ)] − 1(4)

For D > 2, t his integral yields a finite answer (Nmax) even when reaches μ its max-imum possible value (μmax = 0); hence if N > Nmax, the excess number of particles(N − Nmax) Bose condense. For D ≤ 2, however, no BEC occurs since the integraldiverges. Thus, no BEC is predicted to occur for a strictly 1D or 2D ideal Bose gas.

In recent studies, however, Ancilotto et al argued that BEC does occur in a quan-tum gas of Bose particles (parahydrogen or 4He) moving within the interstitial chan-nels (IC’s) of a bundle of polydisperse carbon nanotubes [39, 40]. Their point is thatthe density of states can be written as an inhomogeneously broadened convolution

N(E) =∫

g(E⊥){C1(E − E⊥)−1/2}dE⊥ (5)

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Fig. 8 Computed heat capacityresults showing BEC, forinterstitial H2 (solid, top panel)and 4He (solid, middle panel) innanotubes. The ideal 4D H2 and4He are also shown (dashedlines). The bottom panel showsthe specific heat of 1D 4He atlinear densities 0.001, 0.005 and0.01 atoms per nm. Thecorresponding H2 densities (notshown) would be lower by afactor 0.7. From Ancilotto etal. [39]

Here E⊥ is the transverse (to the long axis) ground state energy of a particle withinan IC and the function g(E⊥) is the distribution of that quantity, which depends onthe distribution of nanotube radii.

In the monodisperse case of a uniform array of IC’s, for which g(E⊥) ∼ δ[E⊥ −constant], one recovers the usual 1D density of states, with no phase transition, asseen in Fig. 8, bottom panel. Ancilotto et al. found, quite generally, that for a plausi-ble distribution of tubes, the behavior is g(E⊥) ∼ [E⊥ − E⊥min]3/2. Here E⊥min isthe transverse energy in the most strongly attractive IC, which is the lowest possibleenergy of a particle. The result of the integration in (5) is then N(E) ∼ [E −E⊥min]2,which has the energy dependence of a 4D Bose gas! Such a gas does exhibit BEC,as indicated in the computed specific heat shown in Fig. 8. The transition behavior isasymmetrical, in that it is singular only above the transition temperature Tc , ∼10 mK,while C(T ) is smoothly varying as T approaches Tc from below. Quite similar tran-sition behavior is seen in Figs. 8a and b to occur for an ideal Bose gas in 4D, whilethe 1D specific heat exhibits smooth behavior at all T . The curious reader may won-der why 4D behavior occurs. It arises from the expected 1D behavior associated withlongitudinal motion, plus three effective dimensions associated with transverse mo-tion, since there are 3 quantum numbers associated with the radii of the three tubesadjacent to any given channel. In detail, the argument is mathematically analogous tothat determining the phonon density of states at a van Hove singularity. The singular-ity has its origin in a minimum in the transverse confinement energy of a moleculewhen the tube radii are optimal (∼8 Å).

5 Ordering Transitions of Gases on Buckyballs

At low pressure (i.e. low gas coverage), gas atoms adsorbed on a close-packed C60layer occupy the hollow (3-fold) sites between the buckyballs (black triangles in

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Fig. 12). This gas provides a realization of the 2D Ising model, with a honeycomb ar-ray of sites that might be occupied. Usually, in describing the liquid-vapor transition,the lattice-gas model is a convenient abstraction, which nevertheless yields exact crit-ical exponents because the order-parameter (scalar), space-dimension and interactionare adequately representative of the physical liquid-vapor transition. For the adsorp-tion problem, in contrast, the model is an accurate representation of the physicalproblem, since the adsorbed particles really do occupy a specified set of sites.

Here we focus on the critical temperature Tc of this 2D condensation transition.Within the approximation that only nearest-neighbors interact, there is an exact map-ping from the lattice-gas model to the 2D Ising model [49]. Their relationship is thatif the pair interaction is −ε, the exchange parameter in the Ising model is J = ε/4.In the Ising model on a honeycomb lattice, the critical temperature satisfies [57]

(kBT

J

)honeycomb

= 2

ln[2 + √3] ≈ 1.519 (6)

This numerical coefficient would assume the value 3 (the coordination number) inmean-field theory, but 2D fluctuations reduce the critical temperature to about halfof the mean-field value. Combining these various relations one can evaluate Tc forXe/C60, for example, from the nearest-neighbor interaction, −ε. As a first estimate,one can take ε = C6/a

6 where a = 0.58 nm is the spacing between Xe atoms inneighboring sites and the van der Waals coefficient C6 = 2.01 K nm6 (or the case oftwo Xe atoms in free space [58]). Putting these expressions together yields Tc = 20 K.An estimate of the effect of longer-range interactions yields an 11% increase in thisvalue. Note added in proof: We have recently found that many-body effects furtherdepress the transition temperature; see [59]. These temperatures are unfortunatelylow, so that observing this transition (e.g. from specific heat measurements or scat-tering experiments) will be difficult.

We note in passing that similar commensurate ordering transitions, for simplegases forming a honeycomb array on Xe-plated graphite, were studied by Vilches’sgroup [60, 61]. These occur at somewhat higher temperatures because the nearest-neighbor spacing is less than half that on the C60 surface, so that the gas-gas interac-tion is larger.

6 Experiments Concerning Adsorbed Gases on Nanotubes and Buckyballs

6.1 Gases on Nanotube Bundles

Experimental Evidence of Quasi-1D Phases Ever since the discovery of carbonnanotubes, the possibility of realizing one-dimensional phases of matter by adsorb-ing gases on their surfaces prompted many experimentalists to search for evidence ofsuch unusual phases [8, 11–13, 15, 17–26, 34]. On that quest, several experimentalstudies were able to both confirm the presence of the predicted phase behavior andprovide support to different interaction models used in the calculations.

Adsorption experiments done on as-produced nanotube bundles (where the tubesare assumed to be generally closed) can in principle reveal the presence of one-dimensional phases formed on the external surfaces of the bundle or in the interstitial

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channels between the tubes. While there is an established consensus about the natureof the phases in the first case, experimental evidence on the interstitial phases is stillnot conclusive and rather controversial.

As predicted by early simulations studies [62], adsorption on the external sur-face of a bundle occurs by the successive formation of one-dimensional stripes ofatoms. Starting with a single line of atoms adsorbed along the groove that lies be-tween two adjacent tubes at the lowest pressure, adsorption continues by formingadditional lines at both sides of the groove as the pressure is increased. Eventually,a two-dimensional phase develops when the whole surface is covered by adsorbedatoms. This behavior was first confirmed experimentally for adsorbed Ar and Xe [18]and later on for 4He [21] and CF4 [63]. Similar behavior was also predicted and ob-served experimentally during the formation of a second layer with the creation of theso-called second layer groove phase [19, 20]. In every case, the adsorption isothermspresent steps corresponding to the formation of each new phase, in a similar mannerto the shell-axial “quasi-transition” inside the pores described in Sect. 2.1

While many theoretical studies have predicted the adsorption of small atoms ormolecules such as He, Ne or even H2 in the interstitial channels (ICs), the exper-imental evidence of one-dimensional phases formed in that region of a bundle hasbeen much more elusive [64]. Although there are some experiments that suggest thepresence of IC adsorption, others have not found any evidence of such behavior. Pos-sible reasons for this disagreement may include differences in the morphology of theexperimental sample (in which for example the ICs could be obstructed by impuri-ties) and also adsorption kinetic effects. We briefly comment on this last possibilityat the end of this section. Concerning the phase transitions in quasi-1D environmentsdescribed in Sects. 2.2 and 3 that involve the interaction between 1D lines of atoms,some experimental support of this behavior seems to be observed for the case ofHe [12, 13, 50] and H2 [65].

Observation of phases adsorbed in the interior of the tubes requires subjecting theas-produced sample to some kind of treatment to open the tubes’ ends. Many exper-imental studies have found evidence of phases adsorbed inside the tubes, althoughthe reported extent to which this occurs varies from one work to another. This dis-parity is very likely to have causes similar to those found for the case of interstitialadsorption, i.e. differences in the sample and/or adsorption kinetics issues. Temper-ature Programmed Desorption experiments have shown the formation of 1D phasesof He [11, 15], Xe [22], CF4 [23], CCl4 [25] and different alkanes inside openedtubes [24, 25]. Moreover, an adsorption study of Ar on open tubes [26] seems toprovide support to the shell-axial “quasi-transition” mentioned in Sect. 2.1.

C60 molecules adsorbed inside nanotubes are one of the best examples of matter inone dimension since they are tightly confined by the walls of the tubes. The presenceof such unusual phase of matter has been proved by many experiments [35, 36, 66,67].

The Problem of Kinetics: Adsorption in ICs and Inside Open Tubes As mentionedbefore, the experimental observation of pore-like phases either inside the tubes or inthe ICs has encountered many more difficulties than of external adsorption. Recently,researchers have started to look at the effects of adsorption kinetics, especially in thecase of adsorption in the ICs and inside open tubes.

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A recent theoretical study focused on the bundle’s ends to determine if the pres-ence of adsorption sites and/or potential barriers could be obstructing or slowingdown the entrance of the gas to the interior of the bundle [68]. In the case of H2 ad-sorbing in bundles of closed tubes (with radii R = 6.9 Å), a high binding energy siteright before the entrance of the IC was found, followed by a potential barrier at the be-ginning of the channel. This site, in fact, provides the strongest binding in the bundlefor this gas, thanks to the favorable combination of the largest carbon-gas attractivepotential (where three tubes meet) with a reduced confinement effect that lowers thezero point motion of the adsorbed particles. As a consequence, a dramatic slowingdown of the adsorption kinetics within the IC is expected, which would certainlyhamper the experimental observation of H2 interstitial adsorption. The developmentof barriers at the entrance of pores was also analyzed on a subsequent theoreticalstudy [69]. For example, Fig. 9a illustrates the appearance of potential barriers forthe case of C60 as the radius of the tube changes. Strong effects are expected when-ever Rpore ∼ 0.95σgc. Based on this criterion, it is possible to predict the size of thenanotube that would induce the emergence of a barrier for different gases inside thetubes and in the ICs as shown in Fig. 9b. In fact, a very recent study of the diffusionof O2 and N2 in a nanotube reports the effects of diffusion barriers that develop atcritical pore sizes that are in agreement with this general prediction [70].

It has also been shown that the coexistence of adsorption processes with verydifferent adsorption rates (external adsorption happening at a much faster rate thanthe internal one) may create the illusion of equilibrium at those early times, when theexternal chains have reached their equilibrium coverage but the pore-like phases arestill far from it [69]. A situation like that illustrated in Fig. 10 shows how a systemcould (erroneously) appear to have reached equilibrium long before the system hasactually reached that state. If such an experiment were ended at that time, significant

Fig. 9 (Color online) Energy barriers at pore entrance. Panel (a) shows the potential energy of a C60molecule near the entrance of a carbon nanotube. From bottom to top, the curves correspond to decreasingvalues of the tube radius. Panel (b) gives estimated values of the tube radius that would give rise to theformation of an energy barrier at the entrance of a tube or IC for different gases

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Fig. 10 (Color online) Uptake evolution for a system with external and pore-like adsorbed phases. Panel(b) displays the full time evolution of the adsorbed coverage while panel (a) shows only the early evolution

internal adsorption, shown in Panel b, would be undercounted. Very recently, thiseffect has been indeed experimentally demonstrated for the first time by comparingthe adsorption kinetics of Ar in open and closed-ended tubes [71].

6.2 Structure and Growth of Gases Adsorbed on C60 Films

A number of early studies of gas adsorption on C60 films concerned the intercala-tion of the gases into the films, between the C60 molecules [72–77]. Quartz crystalmicrobalance studies of the adsorption of He onto the surface of C60 films indicatedan enhanced density of He on the surface of C60 compared to flat graphite [76, 77].This was attributed to the higher corrugation of the C60 surface, consistent with theobservation of a trend to normal densities for subsequent adsorbed layers.

More recently, experiments were performed using low-energy electron diffrac-tion (LEED) to study the adsorption of Xe on a well-characterized monolayer ofC60 molecules on a Ag(111) surface [7]. A monolayer of C60 was prepared by thesublimation and subsequent annealing of C60 on a single crystal Ag(111) surface,producing a commensurate (2

√3 × 2

√3)R30 ◦ monolayer. Its LEED pattern and a

schematic drawing of the structure are shown in Fig. 11. The adsorption of Xe ontothis substrate was studied by measuring the diffraction intensities while performingadsorption isobars. The resulting intensity isobars indicated step-wise adsorption anddesorption of the Xe. No additional diffraction features were observed during Xe ad-sorption, suggesting that the Xe structure has the same unit cell as the C60 monolayer.These structures were explored further using grand canonical Monte Carlo (MC) sim-ulations, as described below. The isosteric heats of adsorption associated with thefirst two isobar steps were derived from the measured isobars to be 234 ± 8 meV and204 ± 14 meV, respectively. The first of these is similar to the value measured for aXe monolayer on a flat graphite substrate of 239 meV [78].

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Fig. 11 (Color online)(a) LEED pattern from acommensurate(2

√3 × 2

√3)R30 ◦ C60

monolayer on Ag(111) atT = 32 K, using an incidentbeam energy of 300 eV. Thereciprocal space unit cells areshown for Ag(111) (largerhombus) and the C60 overlayer(small rhombus). (b) Schematicstructure model for the C60monolayer [79]. From [7]

In the simulations, the experimentally-derived structure of the C60 monolayer [79]was taken as the structure model. The Xe-Ag(111) interaction was modeled using theasymptotic form of the van der Waals interaction, including many-body corrections.The interaction of the Xe with the C60 monolayer was calculated as a sum of contri-butions from spherically-averaged C60 molecules [7, 80, 81]. The Xe-Xe interactionwas modeled using a Lennard-Jones potential. The adsorption isotherms calculatedusing this model indicate a step-wise adsorption of the Xe, in agreement with theexperiments. By examining the adsorbed density at different points on the isotherms,it was possible to deduce the evolution of the structure of the Xe film.

Figure 12 shows a schematic view of three different types of sites identified overthe substrate. The black triangles indicate the hollows between the C60 molecules,and the blue and red symbols indicate locations along the bridges between C60 mole-cules. The calculated potential energies for a Xe atom at these sites are −2380 K,−1681 K and −952 K for the hollow (black), center bridge (blue) and bridge (red)sites, respectively. The adsorption density plots show that after the first adsorptionstep, the hollows are occupied. Interestingly, the next sites to be filled are the bridgesites and not the center bridge sites. This is because the presence of the Xe atoms inthe hollow sites makes the center bridge sites less favorable, and in fact, the secondstep in the isotherm corresponds to adsorption in both the bridge and center bridgesites, as shown in Fig. 13. The heights of the Xe atoms adsorbed in these sites, mea-

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Fig. 12 (Color online)Schematic view of theadsorption sites in thesimulation cell: hollow sites(black triangles), bridge sites(red squares) and center bridgesites (blue circles). Thepositions of the C60 moleculesare also shown (open circles)

Fig. 13 (Color online) Contourplots of the 2D density withinthe simulation cell at pointscorresponding to the (a) first and(b) second steps of theadsorption isotherms

sured relative to the center of the C60 molecules, are 4.4 Å, 7 Å and 8 Å for hollow,bridge and center bridge, respectively.

The calculated heats of adsorption corresponding to the first two isotherm stepsare 213 meV and 153 meV, respectively, values that are 5% and 15% lower than theexperimental values. This discrepancy is attributed to a slight inaccuracy of the Xe-Xe potential. The fact that the heat of adsorption for the Xe in the hollow sites is

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similar to that on flat graphite should not be interpreted to mean that the adsorptionstrength is the same for the two cases. For Xe on flat graphite, the attractive Xe-Xeinteraction contributes about 25% of the monolayer heat of adsorption [62], whereasXe in the hollows of the C60 monolayer experience essentially no attractive Xe-Xeinteraction because they are far apart. This suggests that the substrate contribution tothe isosteric heat is about 25% larger in the C60 hollows than on flat graphite.

7 Conclusions

The field of adsorption on fullerenes is an active field of investigation. Remarkable,diverse phenomena have been predicted more than have been observed experimen-tally. This situation should encourage experimental study of these systems. Amongthe most intriguing possibilities are anisotropic fluids and bose-condensation. Neitherof these is an exactly one-dimensional behavior; the fully three-dimensional environ-ment plays a crucial role in the theories.

Hardly considered (even by theorists!) thus far are the effects of gas-fullerenecoupling, manifested in the static and/or dynamical distortion of the fullerenes. Thisproblem appears to be analogous to that found in the classic polaron problem andeven the BCS theory of superconductivity.

In summary, there exists a rich variety of properties which remain to be fullyelucidated.

Acknowledgements We are grateful for many discussions of this research area with Francesco An-cilotto, Massimo Boninsegni, Ted Einstein, Peter Eklund, Susana Hernández, Karl Johnson, Milen Kostov,José Riccardo, George Stan, Andreea Trasca and Flavio Toigo. A.T. Charlie Johnson kindly providedFig. 1. This research has been funded by NSF DMR-0505160.

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