We report results of diffusion Monte Carlo calculations for both $^4$He
absorbed in a narrow single walled carbon nanotube (R = 3.42 \AA) and strictly
one dimensional $^4$He. Inside the tube, the binding energy of liquid $^4$He is
approximately three times larger than on planar graphite. At low linear
densities, $^4$He in a nanotube is an experimental realization of a
one-dimensional quantum fluid. However, when the density increases the
structural and energetic properties of both systems differ. At high density, a
quasi-continuous liquid-solid phase transition is observed in both cases.Comment: 11 pages, 3ps figures, to appear in Phys. Rev. B (RC

A. Thess

C. Dillon

C. Kiang

D.S. Bethune

E. Krotscheck

F. Darkrim

G. Stan

H. Yano

J. Boronat

J. Breton

J. Casulleras

M. C. Gordillo

M.R. Pederson

N. Hamada

P.A. Whitlock

P.J. Reynolds

P.M. Ajayan

R.A. Aziz

S. Giorgini

S. Ijima

W. Teizer

English

arXiv

We report results of diffusion Monte Carlo calculations for both 4He absorbed in a narrow single walled carbon nanotube (R=3.42Å) and strictly one-dimensional 4He. Inside the tube, the binding energy of liquid 4He is approximately three times larger than on planar graphite. At low linear densities, 4He in a nanotube is an experimental realization of a one-dimensional quantum fluid. However, when the density increases the structural and energetic properties of both systems differ. At high density, a quasicontinuous liquid-solid phase transition is observed in both cases.Universidad Pablo de Olavide. Departamento de Sistemas Físicos, Químicos y NaturalesVersión del edito

Gordillo, M.C.

Boronat, J.

Casulleras, Joaquim

RIO: Repositorio Institucional Olavide

Quasi one dimensional 4He inside carbon nanotubesM.C. Gordillo, J. Boronat and J. CasullerasDepartament de F́ısica i Enginyeria Nuclear, Campus Nord B4-B5, Universitat Politècnica deCatalunya. E-08034 Barcelona, Spain(February 7, 2008)AbstractWe report results of diffusion Monte Carlo calculations for both 4He ab-sorbed in a narrow single walled carbon nanotube (R = 3.42 Å) and strictlyone dimensional 4He. Inside the tube, the binding energy of liquid 4He isapproximately three times larger than on planar graphite. At low linear den-sities, 4He in a nanotube is an experimental realization of a one-dimensionalquantum fluid. However, when the density increases the structural and en-ergetic properties of both systems differ. At high density, a quasi-continuousliquid-solid phase transition is observed in both cases.PACS numbers:05.30.Jp,67.40.KhTypeset using REVTEX1Since their discovery by Ijima [1] in 1991, carbon nanotubes have received a great dealof attention. Basically, they are the result of the seamless rolling up of one or severalgraphite sheets over themselves [2–4]. Depending on the relative orientation of the rollingaxis with respect underlying graphite structure, one can have different types of nanotubes[5]: armchair, zig-zag and chiral with different radii and different mechanical and electricalproperties. Nowadays, it is possible to obtain high yields of nanotubes (single and multiplewalled), with a variety of diameters ranging from 7 to 40 Å [6] and lengths up to ∼ 1000times larger.One of the most attractive features of carbon nanotubes is the possibility of filling withdifferent materials both their inner cavities and the interstitial channels among them [7,8].The interest in this field is twofold. On one hand, the expected increase in the particle-substrate potential energy with respect to a flat carbon surface has suggested the use ofnanotubes as storage devices for molecular hydrogen in fuel cells [9,10]. On the other, moretheoretical, nanotubes provide a reliable realization of one-dimensional systems in the sameway that a substance adsorbed on graphite manifests trends that are characteristic of a two-dimensional medium. If the nanotubes are filled with light atoms (He) or molecules (H2) andthe temperature is low enough, one is dealing with quasi-one dimensional quantum fluids.Such an experimental realization has been carried out for the first time by Yano et al. [11]in a honeycomb of FSM-16. This is a mesoporous substrate with tubes approximately 18 Åin diameter. Using a torsional oscillator, this group proved the existence of superfluidity ofthe 4He atoms absorbed in the pores below a critical temperature of ∼ 0.7 K. More recently,Teizer et al [12] have studied experimentally the desorption of 4He previously absorbed in theinterstitial sites of carbon nanotube bundles. In this case, the data points unambiguouslyto the one-dimensional nature of the helium inside the nanotubes.From a theoretical point of view, it has been recently established using both the hyper-netted chain (HNC) variational approach [13] and the DMC method [14] that strictly onedimensional (1D) 4He is a self-bound liquid at zero temperature. However, contrary to thesituation for dilute classical gases [14–16], there are no many body calculations of quantum2fluids inside nanotubes yet. In this work, we address the question of the quasi-one dimen-sionality of 4He absorbed in a tube by a direct comparison between the results of diffusionMonte Carlo (DMC) for strictly 1D 4He and 4He inside a nanotube of radius equal to 3.42Å, which corresponds to a (5,5) armchair tube in the standard nomenclature [2].The DMC method [17,18] solves stochastically the N -body Schrödinger equation givingresults that are exact for bosonic systems as liquid 4He, provided that the interatomic po-tential is known. In the present calculation, we have used the HFD-B(HE) Aziz potentialfor the He-He pair interaction [19], and the potential given by Stan and Cole [15] in theirstudy of Lennard-Jones fluids in tubes for the He-tube one. Basically, they consider thenanotubes as smooth cylinders by making a z-average of the corresponding sum of all theC-He interactions. Thus, the potential felt by a particle only depends on its distance to thecenter of the cylinder. This is a simplification, but one would expect the error involved to besmall since the helium atoms are much larger than the C-C distance. In fact, the differencesin energy and position between a 4He atom in the smooth cylinder model and the sameparticle considering its interaction with the surrounding individual carbons are about 1%for the tube considered here [20].The efficiency of the DMC method is greatly enhanced by introducing a trial wavefunction Ψ(R) that acts as an importance sampling auxiliary function. In 1D 4He we haveused a two-body Jastrow wave functionΨ1D(R) = ΨJ(R) (1)with ΨJ(R) =∏i<j exp[−12(brij)5], whereas liquid 4He inside nanotubes requires the addi-tional introduction of a one-body termΨT(R) = ΨJ(R)Ψc(R) (2)with Ψc(R) =∏Ni exp(−c r2i ) (ri being the radial distance of the particle to the center), thataccounts for the hard core of the helium-nanotube interaction. Using the variational MonteCarlo method (VMC) we have optimized the parameters b and c at low densities around3the equilibrium. The values obtained, b = 3.067 Å and c= 2.679 Å−2, show a negligibledependence with the density and therefore they have been used everywhere in the DMCcalculations. In all the simulations N = 30 atoms have been used, a number that hasproved to be large enough to reduce the size effects to the level of the statistical errorsreported.The variational HNC equation of Krotscheck and Miller [13] on 1D liquid 4He points tothe existence of a liquid-solid phase transition at high density. We have explored this feature,that is only possible at zero temperature, by using a solid trial function which results from theproduct of Ψ1D(R) and ΨT(R) by a z-localized factor Ψs(R) =∏Ni exp(−a(zi − zis)2). Thesolid sites zis are equally-spaced points along the z direction which is both the longitudinalaxes of the tube and the line of the 1D system. In both solid systems, a VMC optimizationat high densities leads to values b= 2.939 Å and a = 0.612 Å−2, with c= 2.908 Å−2 in thetube case.The energy per helium atom versus the linear coverage, λ, for the 1D (open squares,energy scale on the right) and the tube (full squares, energy scale on the left) is shownin Fig. 1. The two curves have have been drawn for the full square with the lowest λ tocoincide with the open symbol for the same He density. One observes that for λ < 0.05Å−1, both curves are similar, but for larger concentrations the tube curve is located belowthe other one. A similar phenomenology appears in the comparison between the energies ofpurely 2D 4He and 4He adsorbed in graphite. As for this system, the difference in energybetween 4He in a nanotube and 1D 4He is always negative with an absolute value thatincreases with the density. Both in graphite and in nanotubes this increase with respect tothe 2D and 1D systems is mainly due to the emergence of their actual 3D nature. Beyondthis qualitative agreement between 4He adsorbed on graphite and inside carbon nanotubes,there are significant differences in the values of the binding energies in the two systems.The binding energy of a single 4He atom in graphite is EGB= 140.74 K [21], whereas inthe nanotube we are studying is roughly three times larger, ETB = 429.97 K, a significant4difference that has been dramatically observed in the desorption experiment of Teizel etal [12]. On the other hand, the departure of the real 3D systems (4He in graphite or ananotube) from the idealized 2D or 1D liquids can be quantified by means of the parameter∆T(G) =(ET(G) − ET(G)B ) − E1(2)D(ET(G) − ET(G)B )(3)where T stands for the tube and G for the graphite adsorbents and E is the energy perparticle in the system under consideration. Around the respective equilibrium densities oneobtains ∆T = 90 % and ∆G = 6 %, a large difference that indicates that the 1D representationof 4He inside the nanotube is worse than the 2D modelization of 4He in planar graphite.Up to λ = 0.15 Å−1 the energies per particle (e = E/N) of both 1D 4He and 4He insidethe tube may be well fitted by a third-degree polynomiale = e0 + A(λ − λ0λ0)2+ B(λ − λ0λ0)3. (4)The optimal values for the parameters A, B, λ0, and e0 are reported in Table I. The linearequilibrium densities λ0 of both systems are close, λ1D0 = 0.062 Å−1 and λT0 = 0.079 Å−1,whereas the energy difference between λ = 0 and λ = λ0 is significantly different, 0.00364and 0.018 K for 1D 4He and 4He inside the tube, respectively. The latter results pointagain to a large enhancement of the binding energy of 4He inside the tube with respect tothe 1D system. On the other hand, the equilibrium density of liquid 4He inside the tube(ρ0 = 0.0022 Å−3) is much smaller than the one in homogeneous 3D liquid 4He (ρ0 = 0.022Å−3).In agreement with the DMC calculation of Stan et al. [14] and the variational one ofKrotscheck and Miller [13], 4He selfbounds in a 1D array but with a binding energy (-0.0036±0.0002 K) much smaller than that in 2D (-0.897±0.002 K) [22] and 3D (-7.267±0.013K) [18]. It is worth noting that such a small total energy results from a big cancellationbetween the potential and kinetic energies. At λ0, we have T/N = 0.2706±0.0004 K andV/N = -0.2742±0.0004 K. In fact, the influence of the 4He interatomic potential in thissystem is very large. A calculation at the equilibrium density λ0 for the 1D system using5the HFDHE2 Aziz potential [23] indicates that 4He is still a liquid, but the total energy isa factor two smaller (-0.0018 ± 0.0003 K, with a potential energy -0.2724±0.0004 K andthe same kinetic energy). This sizeable differences partially explain the discrepancies of theDMC calculation of Stan et al. [14] and ours with the results of Krotscheck and Miller [13]who used the HFDHE2 Aziz potential.From the values of the energy, one can obtain the linear system pressure, pλ = λ2∂e/∂λ,and estimate the same property for helium inside the cylinder as p = pλ/πR2. Fig. 2displays this observable as a function of the 4He density. The range corresponds to a liquidstructure both in the 1D and tube cases (see discussion below). One can see that thepressure increases faster in a pure linear arrangement of atoms than in a tube. This can beunderstood if one considers that in a narrow nanotube it is possible to avoid the repulsivecore of the nearest neighbors by shifting transversely the helium positions, a situation that isobviously not possible in 1D. Also interesting is the comparison between the sound velocity,c(λ) = [1/m(∂P/∂λ)]1/2 at their respective equilibrium densities. The values are c1D = 7.98± 0.07 m/s and cT = 14.2 ± 0.8 m/s, in both cases a tiny fraction of the corresponding2D (c = 92.8 m/s) [22] and 3D (238.3 m/s) [18] 4He liquids. The spinodal point can beobtained as the density at which the speed of sound becomes zero. According to our resultsthe spinodal points are located at λ1D = 0.047 ± 0.001 Å−1 and λT = 0.059 ± 0.001 Å−1.Another aspect that has deserved our attention has been the existence of a liquid-solidphase transition at high densities. Evidences of this phase transition, that is only possible atzero temperature, appear in a variational calculation of 1D 4He [13]. A comparison betweenthe DMC energies for the liquid and solid phases is given in Table II. One can see that inboth systems, the energy per particle when localization is imposed (a 6= 0) is below thecorresponding to a liquid structure (a = 0) for lineal densities greater than 0.358 Å−1. Bymeans of the Maxwell double tangent construction, one would be able in principle to tell thesolid from the liquid and to obtain the freezing and melting densities. Unfortunately, theenergy differences between the z-localized and the liquid structures are too small to allow usto carry out a meaningful calculation. Our results indicate that for large enough densities6(λ > 0.358 Å−1) both the 1D and the tube arrangements have a localized (solid) phaseand that the discontinuity in the density (if any) is surely very small. This is in agreementwith the results discussed by Withlock et al. [21] about the reduction of the size of thisdiscontinuity from three to two dimensions: in 3D is fairly large, being considerably smallerin a purely 2D system. In 1D, we observe a further reduction towards a continuous or aquasi-continuous transition. It is also remarkable that 4He inside the carbon tube remainsa liquid up to a much larger pressure (around 5 times) than in bulk liquid 4He(∼ 2.6 MPa).Information on the spatial distribution of the 4He atoms may be drawn from the two-bodyradial distribution function along the z direction, gz(r). The functions gz(r) for 1D4He and4He in the tube are shown in Fig. 3 at several linear densities. Near the equilibrium density(λ = 0.08 Å−1, lower part of the figure) g1Dz (r) is quite similar to gTz (r), as corresponds toa quasi-one dimensional system. The same could be said in a broad range of densities, asit can be seen in the curves for λ = 0.182 Å−1 (middle part of the figure). On the otherhand, in the solid phase (λ = 0.406 Å−1), g1Dz (r) and gTz (r) are different: in this case, the3D nature of 4He inside the tubes produces a significant decrease in the localization withrespect to the 1D result.In conclusion, we have compared the properties of strictly 1D 4He with 4He inside a nar-row carbon nanotube using the diffusion Monte Carlo method. For a wide range of densities,4He is a liquid in both systems, and also in both cases a quasi-continuous liquid-solid phasetransition has been observed. In accordance with recent experimental determinations, thepresent calculation evidences a quasi-one dimensional behaviour of 4He inside a nanotubebut significant differences with the ideal 1D system appear, specially when the linear den-sity is increased. The origin of these differences is mainly the existence of the additionaltransverse degree of freedom that helium atoms have inside a nanotube.One of us (M.C.G.) thanks the Spanish Ministry of Education and Culture (MEC) fora postgraduate contract. This work has been partially supported by DGES (Spain) GrantNo. PB96-0170-C03-02.7TABLESParameter 1D 4He 4He in a tubeλ0 (Å−1) 0.062 ± 0.001 0.079 ± 0.003e0 (K) -0.0036 ± 0.0002 -429.984 ± 0.001A (K) 0.0156 ± 0.0009 0.048 ± 0.006B (K) 0.0121 ± 0.0008 0.0296 ± 0.009χ2/ν 2.2 0.24TABLE I. Parameters of Eq. 4 for the two systems studied.λ (Å−1) E/N (1D, a = 0) E/N (1D, a 6= 0) E/N (T, a = 0) E/N (T, a 6= 0)0.406 123.726 ± 0.012 123.561 ± 0.012 -350.155 ± 0.030 -350.20 ± 0.020.380 67.070 ± 0.011 67.000 ± 0.009 -382.282 ± 0.016 -382.321 ± 0.0120.358 37.602 ± 0.008 37.596 ± 0.007 -401.873 ± 0.013 -401.844 ± 0.0100.338 21.881 ± 0.007 21.904 ± 0.005 -413.091 ± 0.014 -413.061 ± 0.0120.320 13.240 ± 0.005 13.258 ± 0.006 -419.551 ± 0.011 -419.493 ± 0.010TABLE II. Energies per particle at large λ for the systems studied. All the energies are in K.See text for further details8Figure captions1. Energy per particle (E/N) versus the linear concentration (λ), for the two systemswe have studied: a strictly one dimensional system (open squares, right energy scale), anda (5,5) armchair tube (full squares, left energy scale). In the first case, the error bars areless than the size of the symbols. Both energy scales are in K.2. Pressure at high helium densities for both systems (one dimensional, dashed line, leftscale; nanotube, full line, right scale).3. Pair distribution function along the z coordinate, gT(1D)z (r). Full lines correspond tothe narrow tube at 0.08 Å−1 (bottom), 0.182 Å−1 (middle) and 0.406 Å−1 (top), and dashedlines to the purely linear system at the same densities.9REFERENCES[1] S. Ijima, Nature 354, 56 (1991).[2] S. Ijima and T. Ichihashi, Nature 363, 603 (1993).[3] D.S. Bethune, C.H. Klang, M.S. de Vries, G. Gorman, R. Savoy, J. Vazquez, and R.Beyers, Nature 363, 605 (1993).[4] A. Thess, R. Lee, P. Nikolaev, H. Dai, P. Petit, J. Robert, C. Xu, Y.H. Lee, S.G. Kim,A.G. Rinzler, D.T. Colbert, G.E. Scuseria, D. Tomanek, J. E. Fisher, and R. Smalley,Science 272, 483 (1996).[5] N. Hamada, S. Sawada, and A. Oshiyama, Phys. Rev. Lett. 68, 1579 (1992).[6] C. Kiang, W.A. Goddard III, R. Beyers, J.R. Salem, and D.S. Bethune.,J. Phys. Chem.98, 6612 (1994).[7] M.R. Pederson and J.Q. Broughton, Phys. Rev. Lett. 69, 2689 (1992).[8] P.M. Ajayan and S. Ijima, Nature 361, 333 (1993) .[9] C. Dillon, K.M. Jones, T.A. Bekkedahl, C.H. Kiang, D.S. Bethune, and M.J. Heben,Nature 386, 377 (1997).[10] F. Darkrim and D. Levesque, J. Chem. Phys. 109, 4981 (1998).[11] H. Yano, S. Yoshizaki, S. Inagaki, Y. Fukushima, and N. Wada, J. Low Temp. Phys.110, 573 (1998).[12] W. Teizer, R. B. Hallock, E. Dujardin, and T. W. Ebbesen, Phys. Rev. Lett. 82, 5305(1999).[13] E. Krotscheck and M. D. Miller, Phys. Rev. B, in press.[14] G. Stan, V. H. Crespi, M. W. Cole, and M. Boninsegni, J. Low Temp. Phys. 113, 447(1998).10[15] G. Stan and M.W. Cole, Surf. Sci. 395 280 (1998).[16] G. Stan and M.W. Cole, J. Low Temp. Phys. 110 539 (1998).[17] P.J. Reynolds, D.M. Ceperley, B.J. Alder, and W.A. Lester, J. Chem. Phys. 77, 5593(1982).[18] J. Boronat and J. Casulleras, Phys. Rev. B 49, 8920 (1994).[19] R.A. Aziz, F.R.W. McCourt, and C.C.K. Wong, Mol. Phys. 61, 1487 (1987).[20] J. Breton, J. Gonzalez-Platas, and G. Girardet, J. Chem. Phys. 101, 3334 (1994).[21] P. A. Whitlock, G. V. Chester, and B. Krishnamachari, Phys. Rev. B 58, 8704 (1998).[22] S. Giorgini, J. Boronat, and J. Casulleras, Phys. Rev. B 54, 6099 (1996).[23] R.A. Aziz, V.P.S. Naini, J.S. Carley, W.L. Taylor, and G.T. McConville J. Chem. Phys.70, 4330 (1979). 1487 (1987).11−430       −429.9       −429.8       −429.7       −429.60 0.05 0.1 0.15 0.200.050.10.150.20.250.30.35E/N  (K)λ (Å−1)0 10  20  30  400.16     0.2     0.24     0.28    0.3202468101214P λ (K/Å)P (MPa)λ  (Å−1)00.511.522.533.544.50 2 4 6 8 10 12g z(1D)T(r)z (Å)

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