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Chacabuco 917, 5700-San Luis, Argentina*. Vincent H. Crespi† and Milton W. Cole. Department of Physics and Center for Materials Physics, 104 Davey ...
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PHYSICAL REVIEW B

VOLUME 58, NUMBER 20

15 NOVEMBER 1998-II

Heat capacity and vibrational spectra of monolayer films adsorbed in nanotubes Ana Maria Vidales Department of Physics and Center for Materials Physics, 104 Davey Laboratory, Pennsylvania State University, University Park, Pennsylvania 16802-6300 and Laboratorio de Ciencias de Superficies y Medios Porosos-CONICET, Departamento de Fı´sica, Universidad Nacional de San Luis, Chacabuco 917, 5700-San Luis, Argentina*

Vincent H. Crespi† and Milton W. Cole Department of Physics and Center for Materials Physics, 104 Davey Laboratory, Pennsylvania State University, University Park, Pennsylvania 16802-6300 ~Received 5 August 1998! Carbon or boron nitride nanotubes can adsorb condensed-phase monolayer films on their inner surfaces. Examples include inert gas or hydrogen films, which form at low temperatures in nanotubes with radius >5 Å. We study the phonon modes and thermodynamic properties of such films. Azimuthal quantization yields quasi-one-dimensional behavior at low temperature, manifested by a heat capacity linear in T, with quasi-twodimensional behavior at higher temperature; the crossover between these regimes has a universal form, depending only on the ratio of the film radius to the thermal phonon wavelength. The film radius and mean sound speed can be extracted from the temperature dependence of the heat capacity. @S0163-1829~98!52644-6#

The circumferential periodicity of the electronic and vibrational states in carbon1 or boron nitride2 nanotubes induces particularly tractable nanoscale boundary conditions in which transverse excitations can acquire gaps. These same boundary conditions can be transferred to films which are adsorbed on the inner or outer surfaces of the tubes, thereby inducing this interesting physical regime within systems which do not themselves form nanotubes. For inert gases and hydrogen, the graphite basal plane provides the most attractive physisorption potential of any flat surface.3 These gases exhibit high density monolayer phases on graphite up to rather high temperatures; these monolayers are strongly confined perpendicular to the surface.4–10 Similarly, a sufficiently wide nanotube can imbibe various gases to form condensed monolayer phases under a variety of thermodynamic conditions, even when the ambient vapor pressure is low. This pronounced capillary condensation occurs when the adsorption potential within the tube greatly exceeds the gas’ cohesive self-interaction, as is the case for inert gases and hydrogen exposed to carbon nanotubes of radius R>5 Å.11–16 An isolated tube could also form a film on its outer surface; geometrical terms disfavor such a film due to the reduced coordination with the substrate,14 but partial dangling-bond character on the substrate could favor such a configuration in small-radius tubes for certain adsorbates. We show that the heat capacity due to phonons in these films crosses over from low-temperature quasi-onedimensional behavior to two-dimensional behavior when the typical thermal phonon wavelength l[\s b approaches 2 p R, the circumference of the cylindrical surface defined by the adsorbates’ mean positions (s is the sound speed and b 21 5k B T). Figure 1 depicts these two regimes for a liquid He film @with an extremely low sound speed s591 m/s ~Ref. 10!#, a higher-coverage solid He film,17 and a solid Xe film.18 For the liquid, the one-dimensional regime occupies, for ex0163-1829/98/58~20!/13426~4!/$15.00

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ample, T,0.2 K for R.5 Å. For other materials ~including solid He and other solid films! the sound velocity and hence crossover temperature are typically larger by an order of magnitude or more. Adsorbed monolayers have both surface-normal and inplane vibrational modes. The surface-normal vibrations are gapped at roughly the first surface-normal excitation energy of the isolated adsorbed atom ~roughly 60 K for 4He on graphite!. Vibrations in liquid monolayer films have a single longitudinal in-plane polarization; solid films also possess an in-plane transverse shear mode. In the classical limit, the in-plane modes in commensurate solids are gapped due to

FIG. 1. Regimes for the vibrational thermodynamics of a liquid He, solid He, and Xe films adsorbed on a nanotube substrate. In each case, radii above the curve show quasi-two-dimensional behavior, while those below the line exhibit one-dimensional behavior with a crossover when R/l'0.15 ~see text!. The sound velocities are taken for densities of 0.0443/Å2 ~fluid He! ~Ref. 10!, 0.1/Å2 ~solid He! ~Ref. 17!, and 0.061/Å2 ~solid Xe! ~Ref. 18!. Acoustic dispersion is assumed. R13 426

©1998 The American Physical Society

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HEAT CAPACITY AND VIBRATIONAL SPECTRA OF . . .

broken translational symmetry, however, in liquid and perfectly incommensurate solid films the in-plane modes are typically ungapped.19 For typical adsorbates on carbon or boron nitride nanotubes, the low-temperature heat capacity is dominated by adsorbate phonons, since the substrate sound velocities (;104 m/s! ~Ref. 20! greatly exceed the sound speed in the adsorbed monolayer ~e.g., ;102 m/s for liquid He!.21 We focus first on a liquid 4He monolayer adsorbed inside a nanotube; this discussion easily generalizes to other cases. Experiment shows such a liquid phase for He on H2 ~Refs. 22–24! and for the second and subsequent layers of He on graphite;25–27 theory suggests a similar He liquid monolayer on Li.28 The strong corrugation at the first monolayer of He on flat graphite induces a commensurate phase at densities less than 0.9/Å2 ~i.e. ;0.8 ML!. In a carbon nanotube, the corrugation can be somewhat smaller than on graphite because the coordination number of carbon atoms around an adsorbate is more uniform.15 Given the absence of theoretical or experimental investigations into the liquid/solid character of the first He monolayer on the surface of a nanotube, we consider both liquid and solid monolayer films. The gapped surface-normal modes freeze out at low temperature, so we concentrate on the in-plane longitudinal modes. Whereas the in-plane phonons of a planar film are characterized by a continuous two-dimensional wave vector, the analogous excitations in a cylindrical film have a continuous axial wave vector k and an azimuthal quantum number n. The azimuthal and axial sound speeds are equal at large R. We assume isotropic dispersion for these modes:

v nk 5s

A SD n R

k 21

2

~1!

,

with no deviation from acoustic behavior. At low temperatures, the azimuthal degree of freedom n freezes out due to the gaps

E 05

~2!

Hence corrections to the assumed isotropy are not necessary for the low-temperature behavior even at small R. Analogous systems showing this characteristic freezing of azimuthal degrees of freedom include ripplons in a thick liquid film inside a cylindrical pore and excitations of isolated atoms within nanotubes.29,30 The statistical treatment follows that of any phonon system. The energy is

E~ T !5

L 2p

6`

( n50,61, . . .

E

1`

2`

E nk dk e b E nk 21

~3!

(

n50,61, . . .

E e

bE

21

5

pL 6\s b 2

~5!

,

C 05

p L k 2B T . 3 \s

~6!

Since only one azimuthal mode contributes, the result is independent of R and linear in T. The 6n phonon states contribute as

E n 1E 2n 5

5

I~ y !5

2L

p \s

E

dEE 2

`

nl/R

~7!

AE 2 2E 2n0~ e b E 21 !

2L I ~ nl/R ! pbl

~8!

E

~9!

dxx 2

`

nl b /R

Ax 2 2y 2 ~ e x 21 !

which can be evaluated analytically in high T and low T limits. At k B T!E 0 5\s/R, one obtains an asymptotic expansion ~with e x @1) I~ y !5

A

S

D

p y 3 2y 7 e 1... . 11 2 8y

~10!

The n.0 contributions are exponentially suppressed at low temperatures as expected. The k B T@\s/R limit involves an exact integral I(`) 5 p 2 /6'1.645, which is valid for n!R/l. We replace the sum over n by an integral and use the relation

S D nl

S D A S D nl

( A 2 2 5 E dn ~ x 2y n ! R

H x2

R

nl

x 22

2

5

pR 2l

,

~11!

R

where H(x) is the Heaviside unit-step function, obtaining the Bose integral relation

E

dx

x2 e x 21

52 z ~ 3 ! 52.404,

~12!

which yields an energy E5 z ~ 3 !

A

pbl2

~13!

,

where A52 p RL is the film area. As expected, this high T result gives the heat capacity of a planar film,

6`

5

dE

yields a contribution to the heat capacity of

H x2

\sn . E n0 5 R

E

L p \s

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E n~ T !

Here L@R is the length of the cylinder. The n50 term,

~4! 31

C 2d ~ T ! 5

3 z~ 3 !k BA

pl2

.

~14!

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VIDALES, CRESPI, AND COLE

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Corrections in powers of l/R follow from the EulerMaclaurin sum formula. Geometry-induced anisotropy in the acoustic mode dispersion for the smallest-radius tubes will cause deviations from the pure two-dimensional limit. The ratio of the two- and one-dimensional heat capacities at low T satisfies

S S

D D

\s k BR p l . 5 18z ~ 3 ! R \s T! k BR

C 1d T! C 2d

~15!

Since the cylindrical film’s heat capacity falls off as T instead of T 2 , the heat capacity of the cylindrical film greatly exceeds that of an equal-area planar film in the limit of low T and small R. The two temperature limits may be connected using a reduced temperature t[R/l5T/T R for T R 5(\s)/(k B R), the temperature at which the film radius equals the thermal phonon wavelength. The heat capacity @obtained by differentiating Eqs. ~5! and ~7!# at all temperatures is `

L C 5t f ~t! kB R n50 n

(

f n.0 ~ t ! 5

2

pt3

E

x 3 e x/t

`

dx

n

Ax 2 2n 2 ~ e x/t 21 ! 2

p f 05 . 3

~16!

~17!

~18!

Figure 2 shows a narrow crossover regime near t'0.15, when 2 p R'l. The crossover can be defined as the intersection of the extrapolated high and low temperature limits, 6 z (3)t' p /3 or t'0.145. Incommensurate solid films may be treated similarly, once again separating the dynamics into surface-parallel and perpendicular components.4 For simplicity, at low T we use an isotropic effective Debye model wherein the transverseand longitudinal-acoustic modes are combined into a single harmonic mean speed v , 17 2 1 1 5 1 . v s long s trans

~19!

As compared to the liquid case, we simply insert a factor of 2 and the new sound speed to obtain `

L C 52t f ~ t !. kB R n50 n

(

~20!

For both liquid and solid films, the one-dimensional lowtemperature heat capacity, being independent of R, allows extraction of the sound speed,

p L k 2B T , s5 3 \C

~ liquid!

~21!

FIG. 2. The universal function F(t)[ ( `n50 f n 5CR/Lk B t describing the specific heat C for an adsorbed monolayer as a function of reduced temperature t[T/T R defined in the text. Note the crossover from one-dimensional to two-dimensional behavior near t 50.15.

while the crossover to the quasi-two-dimensional case reveals the film radius; R'(k B T R )/(\s). A commensurate solid film can acquire a q50 gap due to broken translational symmetry; if this gap is smaller than the energy spacings induced by periodic azimuthal boundary conditions, then the transition from one-dimensional to two-dimensional behavior survives, but the second energy scale destroys the universal dependence on t. As mentioned previously, films can form on both the inside and outside surfaces of nanotubes. An outer-surface inert gas film would typically have lower density due to the lower holding potential.14 However, when a high-density film forms inside the tube, the consequent increase in chemical potential eventually induces a high-density film on the outside as well. For example, on graphite, the helium chemical potential at low temperature changes by ;50 K as the density increases from 0.1 to 0.11 Å22. 9 Since this densitydriven increase in chemical potential is comparable with the difference in chemical potentials between equal-density inner and outer films,14 one expects a significant outer film density as the inner film approaches ;0.1Å22 . The lower density of the outer film yields a smaller sound velocity s, which in turn yields a larger low-temperature heat capacity superimposed on the contributions from the inner film ~assuming that the adsorbate can access the interior!. The lower s and larger radius of the outer film also imply a lower crossover temperature. In conclusion, nanotube substrates transfer their periodicboundary conditions to adsorbate films, opening a new range of theoretically tractable finite-size effects in nanoscale systems. The vibrational heat capacity of an adsorbed film assumes a universal form as a function of the reduced temperature t[T/T R with a crossover from one-dimensional to twodimensional behavior as the typical thermal phonon wavelength exceeds the circumference of the film. We are grateful to G. V. Chester, Moses Chan, and Hyung-Kook Kim for helpful discussions. This research has been supported by the National Science Foundation Grant No. DMR-9705270, the Universidad Nacional de San Luis, and the Petroleum Research Fund of the American Chemical Society.

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HEAT CAPACITY AND VIBRATIONAL SPECTRA OF . . .

*Present address. †

Electronic address:[email protected] 1 S. Iijima, Nature ~London! 354, 56 ~1991!. 2 N. G. Chopra, J. Luyken, K. Cherrey, V. H. Crespi, M. L. Cohen, and S. G. Louie, Science 269, 966 ~1995!. 3 G. Vidali, G. Ihm, H. Y. Kim, and M. W. Cole, Surf. Sci. Rep. 12, 133 ~1991!. 4 A review of the various phases on graphite appears in L. W. Bruch, M. W. Cole, and E. Zaremba, Physical Adsorption: Forces and Phenomena, International Series Monographs in Chemistry No. 33 ~Oxford University Press, Oxford, 1997!, Chap. 6; for 3He on graphite, see H. Godrfin and H. J. Lauter, in Progress Low Temperature Physics, edited by W. P. Halperin ~North-Holland, Amsterdam, 1995!, Vol. XIV, Chap. 4, p. 213. 5 M. Bretz, J. G. Dash, D. C. Hickernell, E. O. McLean, and O. E. Vilches, Phys. Rev. A 8, 1589 ~1973!; J. G. Dash, M. Schick, and O. E. Vilches, Surf. Sci. 299/300, 405 ~1994!. 6 S. V. Hering, S. W. Van Sciver, and O. E. Vilches, J. Low Temp. Phys. 25, 793 ~1976!; D. S. Greywall, Phys. Rev. B 41, 1842 ~1990!; 47, 309 ~1993!. 7 R. L. Elgin and D. L. Goodstein, Phys. Rev. A 9, 2675 ~1974!. 8 R. L. Elgin and D. L. Goodstein, Monolayer and Submonolayer Helium Films, edited by J. G. Daunt and E. Lerner ~Plenum, New York, 1973!. 9 Data exist for H2 films on graphite @e.g., H. Wiechert, Physica B 169, 144 ~1991!#; and on BN @e.g., M. D. Evans, N. Patel, and N. S. Sullivan, J. Low Temp. Phys. 89, 653 ~1992!#. 10 C. E. Campbell, F. J. Milford, A. D. Novaco, and M. Schick, Phys. Rev. A 6, 1648 ~1972!; P. A. Whitlock, G. V. Chester, and M. H. Kalos, Phys. Rev. B 38, 2418 ~1988!; P. A. Whitlock, G. V. Chester, and B. Krishnamachari, Phys. Rev. B 58, 8704 ~1998!. 11 M. R. Pederson and J. Q. Broughton, Phys. Rev. Lett. 69, 2689 ~1992!. 12 P. M. Ajayan and S. Iijima, Nature ~London! 361, 333 ~1993!. 13 Q. Wang and J. K. Johnson ~unpublished!. 14 G. Stan and M. W. Cole, Surf. Sci. 395, 280 ~1998!. 15 G. Stan and M. W. Cole, J. Low Temp. Phys. 110, 539 ~1998!. 16 This is sometimes attributed to the small surface tension of the imbibed fluid, but a more complete explanation is based upon

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the relative interaction strengths. G. A. Stewart, Phys. Rev. A 10, 671 ~1974!. 18 T. M. Hakim, H. R. Glyde, and S. T. Chui, Phys. Rev. B 37, 974 ~1988!. 19 Although a finite solid film cannot be perfectly incommensurate, quantum effects can reduce the q50 vibrational gap even in cases of full or partial commensuration; L. W. Bruch, Phase Transitions in Surface Films, edited by H. Taub, G. Torzo, H. J. Lauter, and S. C. Fain, Jr. ~Plenum, New York, 1991!, p. 67. Inclusion of a gap is straightforward, although the second energy scale destroys the simple scaling form on t[T/T R . 20 J. Yu, K. Kalia, and P. Vashishta, J. Chem. Phys. 103, 6697 ~1995!. 21 Preliminary measurements give a low-temperature heat capacity for helium adsorbed in single-walled nanotubes @M. Chan and H.-K. Kim ~private communication!# which exceeds the purenanotube heat capacity @Nassar K. Budraa and W. P. Beyermann ~private communication!# by ;1003. 22 M.-T. Chen, J. M. Roesler, and J. M. Mochel, J. Low Temp. Phys. 89, 125 ~1992!; P. W. Adams and V. Pant, Phys. Rev. Lett. 63, 2350 ~1992!; H.-J. Lauter, H. Godfrin, and P. Leiderer, J. Low Temp. Phys. 87, 205 ~1992!. 23 D. Tulimieri, N. Mulders, and M. H. W. Chan, J. Low Temp. Phys. 110, 609 ~1998!; G. Csathy, D. Tulimieri, J. Yoon, and M. H. W. Chan, Phys. Rev. Lett. 80, 4482 ~1998!. 24 Theory suggests monolayer superfluidity on hydrogen; see M. Wagner and D. M. Ceperley, J. Low Temp. Phys. 94, 185 ~1994!. 25 P. A. Crowell and J. D. Reppy, Physica B 197, 269 ~1994!; P. A. Crowell and J. D. Reppy, Phys. Rev. B 53, 2701 ~1996!. 26 F. F. Abraham and J. Q. Broughton, Phys. Rev. Lett. 59, 64 ~1987!; M. Pierce and E. Manousakis, ibid. 81, 156 ~1998!. 27 G. Zimmerli, G. Mistura, and M. H. W. Chan, Phys. Rev. Lett. 68, 60 ~1992!. 28 M. Boninsegni and M. W. Cole, J. Low Temp. Phys. ~to be published!. 29 W. F. Saam and M. W. Cole, Phys. Rev. B 11, 1086 ~1975!. 30 See the explicit crossover behavior in Fig. 14 of Ref. 15. 31 R. K. Pathria, Statistical Mechanics, 2nd ed. ~Butterworth Heineman, Oxford, 1996!. 17