Low coverage adsorption in cylindrical pores

surface science ELSEVIER

Surface Science 395 (1998) 280-291

Low coverage adsorption in cylindrical pores George Stan *, Milton W. Cole The Pennsylvania State UniversiO', The Physics Department, 104 Da~ +~vLab, Universi O' Park. PA 16802. USA Received 3 April 1997; accepted for publication 14 July 1997

Abstract

We present a theoretical exploration of the adsorption of rare gases in carbon nanotubes. In both the classical and the quantum cases, nanotube adsorption provides a nearly ideal realization of quasi-one-dimensional ( 1D) matter. We have studied the adsorption potentials, the gas surface virial coefficient and the isosteric heat of adsorption. Comparison shows a much stronger binding of the adsorbate in the tubes than at the planar surface of graphite. As a consequence, one can easily adsorb sufficiently many atoms to be measurable in a thermodynamic or scattering experiment. In studying the low coverage adsorption we find great sensitivity to the species, the assumed potential model, and the radius of the tubes. The effect of interactions between the adsorbed particles is evaluated in the ID classical case. ~'~ 1998 Elsevier Science B.V.

Kevwords: Adatoms; Carbon; Nanotubes: Physical adsorption

1. Introduction

The physics of nanotubes is a subject of growing interest in materials science (for recent reviews of nanopores see Ref. [1]). Among the intriguing properties are those pertaining to the structural, elastic, and electronic behavior of these materials [2-5]. There has been relatively little attention paid thus far to the subject of adsorption in nanopore materials [6+ 7]. Nevertheless, the subject has considerable fundamental interest because it corresponds essentially to a one-dimensional (1D) environment, with particularly high surface to volume ratio and (as shown here) a correspondingly high coverage compared with conventional high specific area substrates, such as graphitized carbon black. The fact that the pores tend to be * Corresponding author. Fax: (+ l ) 814 865.3604; e-mail: [email protected] 0039-6028/98/$19.00 ~') 1998 Elsevier Science B.V. All rights reserved. PH S0039-6028 ( 9 7 ) 0 0 6 3 2 - 8

very long and regular makes them nearly ideal substrates for such adsorption studies. Recently, it was reported that single-wall carbon nanotubes were produced in yields of >70%. The (10,10) tube is the dominant component,+ with a pore diameter estimated to be 13.6A [8]. Other quasi-lD systems are zeolite materials like A1PO4-5 [9] and MCM-41 [10]. AIPO4-5 has narrow cylindrical pores of diameter 0.73 nm, and the cross-section is a two-dimensional (2D) hexagonal closed-packed structure. MCM-41 exhibits a hexagonal array of uniform mesospores, with an approximate pore diameter range of 3.0-3.7 nm. In the present paper, we consider the problem of the adsorption of both classical and quantum gases in such pores. The first aspect of this study is the determination of the adsorption potential. Here, we apply a method of pairwise summation of gas-surface interactions, a procedure which has been widely used in studying both flat surface and

G. Stan. M. W. Cole / SurJace Science 395 (1998) 280 291

porous media adsorption [11]. In so doing, we make the further approximation of omitting the periodic structure of the substrate. The reasons for using this model are its simplicity (of the resulting mathematics) and the fact that the results are not too inaccurate. In the following we calculate primarily the adsorption in the regime of low coverage, i.e. the Henry's law regime of coverage proportional to pressure. This treatment omits the interactions between adsorbed atoms, of course, and hence much of the potentially interesting quasi-lD physics. Nevertheless, our results are useful in that they show great sensitivity of the adsorption to the species, the assumed potential model, and the radius of the tubes. Thus, they should serve as a characterization tool analogous to techniques based on other model isotherms, such as Langmuir and Brunauer-Emmett-Teller, used traditionally in the analysis of adsorption [11]. The results of these studies include qualitative behavior of the adsorption as a function of adatom size and temperature T. In the quantum case, for 4He, we find significant deviation from the classically computed behavior. In both the classical and the quantum cases, depending on the diameter of the nanotube and the temperature, we find 1D, 2D or three-dimensional (3D) behavior. In the 1D classical case we study the effect on the adsorption of Lennard Jones (L J) interactions between the adsorbed particles.

2. The adsorption potential Our main interest is the nature of physically adsorbed films in carbon nanotubes. In this case, the adsorbate and the substrate are negligibly perturbed by each other because the energy scale (milli-electron-volts) is small compared with the cohesive energies of the tube. We may then assume that the structure of the substrate is unchanged by the adsorbate and that the interactions between adsorbed molecules are those of their 3D phases. The film coexists with a vapor phase, which provides the thermodynamic parameters of the film. At equilibrium, the chemical potential /~ of the film equals that of the vapor, which is a known function of the pressure p and T. One can use the

281

Maxwell relations and the dependence of the chemical potential on T, p, and coverage to characterize the thermodynamic properties of the film. The adsorption potential was calculated by summing pair interactions between individual carbon atoms and the adatom:

V(r)=~ u(lr-Ril)

(1)

where r is the adatom's position and R~ is that of a carbon atom, while the sum is over the tube's atoms. The pair potential was taken as an isotropic interaction of the LJ form: u ( x ) =

4el(a/x) 1 : - (o-/x)6]. For simplicity, we have ignored the atomistic details in the potential inside the tube by smearing out the carbon atoms, based on the comparison between the carbon lattice constant (1.4 A) and the expected equilibrium distance of the adatom ( ~ 3 A).~ This approximation simplifies the result, yields convenient cylindrical symmetry, and is adequate for the qualitative features of the adsorption. Considering first a single-wall nanotube, the potential at distance r = R x from the axis of the cylinder is MII(X)

L32 \ R /

5,x,l

(2)

where 0--0.38 A-2 is the surface density of C atoms and R is the radius of the cylinder. Here we use the integrals M.(x)=

;o

do

( 1+ X 2

--

'

2x cos

~°)~j2

(3)

where n is a positive integer. Fig. 1 shows how the shape of the potential varies with R. For large R the potential minimum rmin occurs near the surface while for small R, rmi,=0. The crossover value R c is discussed below. Calculations were performed for the rare gas series (He, Ne, Ar, Kr, Xe), which yield an increasing equilibrium distance with increasing c~and a greater well depth for increasing

1 Recently, we have explored the magnitude of the corrugation (in the hydrogen case) by including atomicity; see G. Stan and M.W. Cole, J. Low Temp. Phys., in press.

G. Stan, M. W. Cole / Sur[ace Science 395 (1998) 280-291

282 10

5 0

0

~0

-5

-10

-10

-20

~ -20

-15

-25

-30

-30 -40 0.0

0.2

0.4 r/R

0.6

-35

0.8

~ 3.0

2.5

'

' 3.5

'

~ 4.0

'

4.5

z(A)

Fig. I. The reduced potential as a function of the reduced distance from the axis of the cylinder for He inside a single-wall nanotube with R = 8 A (11), R = 5 A. (H). R=3.33 A (+), and R = 3 ~, (A).

Fig. 3. The reduced potential as a function of the distance = from the surface for Xe inside (D) and outside (+) a single wall nanotube with R - 5 A. and near a single-layer flat surface (- -).

0 -5

;<

-10

¢0

-2

W

> -15 -4

-2O -25

I

2.0

2.5

,

I

I

3.0

3.5

J

4.0

z(A) Fig. 2. The reduced potential as a function of the distance from the surface for He inside ([2) and outside (+) a singlewall nanotube with R = 5 A, inside a five-shell tube ( ~2 ) with the inner radius 5 A, near a single-layer flat surface ( ), and near a half-space surface (~).

e ( F i g s . 2 a n d 3). T h e v a l u e s o f t h e p a r a m e t e r s 0and e were available from previous calculations. 2 Apart from the varying equilibrium distance, the reduced potential depends on the identity of the adatom through the number of substrate atoms ZThe parameters used were: for He, ~r=2.74A and E= 1.40meV; for Ar, a=3.10,~ and e=5.10meV; for Xe, (r= 3.36,~ and E=7.17 meV. See Ref. [12].

-6 0.8

I

I

I

i

1.0

1.2

1.4

1.6

i

1.8

R/c~ Fig. 4. The reduced potential on the axis of the cylinder (V(0)/Ea z, R) and the reduced well-depth ( D/Ea ~', H) of the potential as a function of the reduced radius for a gas inside a single-wall nanotube. The two graphs merge when the minimum of the potential occurs on the axis of the tube (R/cr R~, retaining only the lowest order terms

V(r; R) ~- V(0; R~) + ~ ' ( R c ) ( R - Rc)rZ + /3(Rc)r 4. (8)

~

(9)

The equilibrium distance measured from the cylindrical wall (Zmin, z = R - - r ) is thus equal to R if R < Re, and decreases with R for R > Re until it reaches the R = oc equilibrium distance (flat surface) Zmin=a (Fig. 5). The discontinuity in the slope of the graph is consistent with the result in Eq. (9). The potential in the case of a single-layer flat surface is

with a minimum D 1= 6/5~Oea 2 at the equilibrium distance :ram = o. Figs. 2 and 3 contrast this result with that of a semi-infinite graphite solid and with potentials in the nanotubes and outside of these t u b e s ) The potentials inside the tube are significantly more attractive than the other cases.

3. Classical gases Once the adsorption potentials are known, one can calculate the equilibrium properties of the adsorbed film using a surface virial expansion [15]. As in the 3D case, the second virial coefficient provides information about the interactions between the adsorbed molecules, while the first coefficient is determined by the adsorption potential. The low coverage assumption ignores the interaction between the adatoms. In that case, we obtain Henry's law of proportionality between the surface excess coverage N ~s~ and the equilibrium vapor pressure p

N ~s~=/3pBAs

( 11)

where /3-1 is Boltzmann's constant times T. The gas-surface virial coefficient BAS in the classical

s Both the inside and outside potentials can be expressed in terms of the reduced distance of closest approach between the adatom and the cylindrical surface (z/a), making use of the hypergeometric series; e.g. see Ref. [14].

284

G. Start, M. W. Cole ' Sur/ace Science 395 (1998) 280 29l

case is BAs = f {dr e x p [ - fl V(r)] - 11

(12)

where the volume of integration is the domain accessible to the adsorbate. Since the potential depends only on r, the virial coefficient in this geometry can be simplified to ( 13)

BAS = 2 ~ L fd," r { e x p [ - f i V ( r ) ] - 11

particles adsorbed in nanotubes with that in the case of a flat surface, consider N~ single-wall nanotubes with length L, radius R, and having the same total area as the flat surface, A = N , 2 z R L . The total n u m b e r of particles adsorbed, Nin is Ni, = N t f d r ,, e x p [ - f l V(r)]

(16)

where n is the v a p o r density, ;7=tiP. The corresponding n u m b e r of particles adsorbed on a flat surface is

where L is the length of the tube. If we restrict our discussion to the regime of low T the virial coefficient, to a good approximation, is

Nsurf = A

BAS "--27rL f d r r e x p [ - f i V ( r ) ] .

The ratio of the n u m b e r of particles adsorbed in single-wall nanotubes to the n u m b e r of particles adsorbed at a single-layer flat surface is

(14)

A crude estimate of the T and R dependence can be obtained by a Taylor expansion of the potential around its m i n i m u m BAs ~ (2~)3:2RL e~O(fi,#)- 12

(15)

where , # is the force constant at the m i n i m u m position. Fig. 6 (computed from Eq. (13)) shows that this Arrhenius behavior is manifested in the exact calculations. To c o m p a r e the n u m b e r of

i[

dr ;7 exp[-/~Vn.dz)].

(17)

)

Nin

1 fo: dr r e x p [ - fi V(r)]

--

-

Nsurf

(18)

R £[

dz

exp[ -- fl Vt.lat(-7)]

Owing to the larger well-depth, the adsorption inside small nanotubes is m u c h greater than at the flat surface Fig. 7). The m a x i m u m occurs at the

20 35 30

15

\

25 20

~a',.w

z~lo

10 5 0 0

i

4

8

i

I

12 T(K)

i

I

16

I

0.8

i

20

Fig. 6. The reduced virial coefficient for He, computed classically, for the case of a tube with five concentric shells (~) and of a one-shell tube of radius 5 A (inside: I1: outside: +) and 8 A (inside: ~: outside: A ). The quantity B L has been expressed in units of (fingstrOms)2.

i

1.2

I

1.6

i

2.0

R/o Fig. 7. The ratio of the number of Ne atoms adsorbed inside a single-wall nanotube (R=5 ,~) to the number of Ne atoms adsorbed at a single-layer flat surface, computed classically, as a function of the reduced radius at temperature T-10 K. The surface area is the same in the two cases. Non-interacting atoms have been assumed.

285

G. Start, k,t. W Cole Sud~we Science 395 (1998) 280 291

300

V about I'min: this yields

275

qst - ~ D + - -

1

2D

lilliiiili Ilillilli

++++++++++++

250

+

+

(21)

2#

which coincides with the numerical calculation (Fig. 8).

225

3.1. 1D approximation

200

We address the regime when the gas may be described as I D. Suppose R is sufficiently small that the motion perpendicular to the axis of the tube is much smaller than the mean separation between particles. Then the canonical ensemble partition function Q can be factorized into a term Q . which depends only on the interaction with the substrate and a 1D partition function QID involving only the interaction U between the adsorbed atoms

, A~A~AAA4AAAAAAA 175

l

0

4

8

1

12

i

I

16

,

20

T(K) Fig. 8. The absolute value of the isosteric heat of adsorption for classical He inside a single wall with R = 5 ,~ ( ~ 1. at a singlelayer flat surface ( 5'~ ), and at a half-space suri:ace {• ). compared with the R = 5 ~, isosteric heat of adsorption treated quantum mechanically for 3He (11) and for 4He (+1. The flat surface result in the parabolic potential model ( - - ) . q= D + 7-,2. is observed to coincide nearly with the quasi-2D film result (Eq. (21)l, even though the pore radius is small.

Q(T, V, N ) = Q ± Q m , Q±(T, N ) -

~2N

Zx L x maximum of the well depth. This comparison is, of course, important for the feasibility of possible experiments. Note that the regime of validity of E q . ( l l ) is reduced to much lower p for the nanotube than for a flat surface. The isosteric heat of adsorption, a measure of the adsorption potential, is defined by

_(?lnp]

\ ?fi }.~'".:i

qst=

(19)

Q1D(T, L, N ) = - ).%N! x

~

--

#

--

f

).x N! d-.~. 1-i e x p [ - # U ( l--ijl)], (22)

where 2=(2~h2#/nl) ~'2 is the de Broglie thermal wavelength. The radial integral can be approximated with

#.~,)2

"

(23)

Then the I D approximation requires that the root mean square transverse displacement [2/(fi,Y)] ~2 be small compared with the interparticle spacing. Using the partition function we can determine the chemical potential:

_(?lnQ]

dr rV(r) exp[-fiV(r)] (20)

f

1

~_~2. exp[-#V(O, R)] }"

where .~/ is the area of the adsorbate. Using Henry's law we are able to determine a simple formula for the classical isosteric heat in terms of the adsorption potential in the case of non-interacting adsorbed particles

qst

,

ij

Q± (.

1

d--1 ...

dr r e x p [ - f i V ( r ) ]

dr r exp[ -/7 V(r)]

For R > Rc one obtains the 2D limit by expanding

-

#tt \~)s~.," -

,; (22)

l~=/lJ_ +IqD = V(0: R ) - w i n

+PtD, (24)

G. Stan, M. W. Cole/ Sur[clceScience395 (1998)280~91

286

where /IID=kBT is the 1D chemical potential. Analytically solvable cases of I D motion include non-interacting particles, hard rods and LJ interacting particles. To determine the chemical potential as a function of the particle density in the case of nearest-neighbor interactions only, we start with the equation of state [16, 17]

-14.0

-15.0

-16.0

fo

'~ dz exp{ - fi[U(z) + PIDZ]}

p=

(25) -17.0 0.00

o~ dZ z exp{ --fl[U(z)+ PIDZ]} where p = N / L is the particle density. We then integrate the 1D Gibbs-Duhem relation 0plD - -

~3Pm

-

1

(26)

p

with an appropriate choice of low pressure initial conditions (Pro,o) such as to reproduce an ideal gas. The 1D chemical potential is then obtained in a simplified form by performing an integration by parts

fllAID =In

flP~i,'°)~f°°~

-------dzexp{-fl[U(z)+PiD'°Z]}t .

[ dzexp{-fl[U(z)+P1oz]}

/ (27)

Using Eq. ( 2 4 ) dr r e x p [ - flV(r)]}.

(28)

Now, eliminating the ID pressure between Eqs. (25), (27) and (28) one obtains the dependence of/~ on p. This shows a nearly ideal gas at very low densities, the domination of repulsion at high densities and that of the attraction at low densities (Fig. 9). As shown by Landau [13], these 1D systems exhibit no phase transitions. The hard rod model is the limiting case when U(z) = oo for z < d and U(z) = 0 for z > d. Then, Eq. (25) yields the equation of state obtained by Rayleigh and by Tonks [18]: PID(I-d)=kBT

(29)

' 0.20

' 0.40 per

' 0.60

0.80

Fig. 9. The reduced chemical potential as a function of the adsorbed particle density in the case of an ideal gas ( ) and for LJ interactions between the adsorbed particles ( - - , cr= 3.38 ,~, E= 120 K). The study involved argon inside a singlewall nanotube of radius 3.5 A. at T=48 K.

where l = lip. In this case Eq. (27) yields fi~m = l n

~

l-d"

(30)

When d R c in order to show how a 1D limiting behavior arises even there. The eigenfunctions of the Schr6dinger equation for an atom in the nanotube are of the form t~,k~(r)=fn,.(r ) exp(ikz)exp(ivcp), where r, z, ¢p are the cylindrical coordinates of the atom's position vector r. The spectrum corresponding to this wave function is discrete in the radial (n = 1,2.... ) and azimuthal (v=0,_+ 1.... ) quantum numbers, but it is quasi-continuous in the longitudinal quantum number (k): E, kv = e,,, + h 2 k 2 / ( 2 m ) . The differential equation satisfied by f,~(r) is: dr 2 + r dr + ~ - [e,~ - V(r)]- 77 = 0.

( 31 )

G. Stan, M. W. Cole / SurJhce Seience 395 (1998) 280 291

Our simple way to determine the energy levels e,~ is to use a piecewise constant model potential constructed as shown in Fig. 10. We set V(r)= V(0) for 0 < r < a , where a is the inner distance at which the potential is [ - D + V(0)]/2. We let - D in the interval a < r b. This approach will yield qualitatively reliable results, as is consistent with the approximations present in the potential itself. We have to distinguish between two cases: e < V(0) and e > V(0). In the first case, the radial wave function has the general form

V(r)=

I AIv(rcr) ] f(r)={B[J~(xr )- N"(Ir) NJ~(gb) ~J

for

r>l) we obtain Eq. (36).

where the factor two in the right-hand side appears from the double-degeneracy of the energy for excited azimuthal states and Vmax is the maximum value of v for which E>e,.. We make only a negligible error in doubling the v = 0 contribution. The average density of states over an interval is

(2m~ 1/2 1 2gL--

(~ ,ff(E))qo,Cp -~ \ - ~ - / C/3

6# --610

Vmax

xIdEIdlI(E--6v)-[/2 Appendix B: The transition from the quasi-ID case to the quasi-2D case for large radii

For large radii, the azimuthal excitation energy is much smaller than the radial one. We can then choose to treat the azimuthal part of the Hamiltonian, h2vZ/(2mr 2) as a perturbation. The azimuthal excited levels are: 6 1 ~ q 0 +hZvZ(1/r2),/(2m), where (...)1 refers to the average in the first excited state. This averaging was chosen because the average in the ground state gives a divergence due to the properties of the modified Bessel function Io at the origin. Hence, for energies greater than the azimuthal excitation energy but smaller than the radial one, the transverse energy is well approximated by e,~=6,.~61o+h2v2/[2m(R-a)2], where we use a simple estimate of this expectation value. This perturbation result agrees well with the exact numerical solution of the Schr6dinger equation.

elo

,

(8.3)

0

rngLR (.U(E))clO,~, -

h2

( B.4 )

The quasi-2D density of states is

mgN ~/'2D(E)

= - -

2~zh2

H(E--610 )

(B.5)

where ~¢ is the area of the adsorbate and with the assumption that the radial degree of freedom is not excited. For an area corresponding to the lateral area of the cylinder (2~RL) the 2D density of states is ,#'2D(E) --

mgLR h2

H ( E - qo)

(B.6)

which is exactly the large R limit of the quasi-lD case, as expected.

G. Stan, M. W. Cole/Surfiwe Science 395 (1998) 280 291

B. 1. Heat capacity

The heat capacity per particle is determined from the additive contribution of longitudinal motion and transverse motion, evaluated from the usual fluctuation relation [13] C

1

NkB

2

. . . . __

_}_ 1~2 ( ~ __ g2 )

(B.7)

where 7 : : ( ~ e2,, e a'o,') / ( ~ e-aql,').

(B.8,

In the low temperature limit, [J(en-eto)>>l, we find C

1

Nk~

2

___+__

_~ [32 (Cl1 -- 6.10)2

e- a~ql qo)

(B.9)

In the high temperature limit, the density of states approaches the 2D density of states as shown above. Introducing Eq.(B.6) in Eq. (B.8), we obtain E 1 ---~- +qo. X fi

(B.10)

Thus, in this limit C ---+1. NkB

(B.11)

References [1] T.W. Ebbesen, Phys. Today, June 1996, pp. 26 32. T.W. Ebbesen, Annu. Rev. Mater. Sci. 24 (1994) 235. P.M. Ajayan, Condensed Matter News 4 (1995) 9. [2] R. Saito, F. Mitsutaka, G. Dresselhaus, M.S. Dresselhaus, Phys. Rev. B 46 (1992) 1804. [3] J.W. Mintmire, B.I. Dunlap, C.T. White, Phys. Rev. Lett. 68 (1992) 631. [4] N. Hamada, S. Sawada, A. Oshiyama, Phys. Rev. Lett. 68 ( 1992 ) 1579. [5] S. Iijima, T. lchihashi, Nature 363 (1993) 603. [6] J. Fischer, M. Bohn, B. K6rner, G. Findenegg, Ger. Chem.

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Eng. 6 (1983) 84. R. Cracknell, K.E. Gubbins, M. Maddox, D. Nicholson, Acc. Chem. Res. 28 (1995) 281. R. Radhakrishnan, K.E. Gubbins, Phase transition in narrow cylindrical pores: inter-pore correlation effects, in preparation. C. Martin, J.P. Coulomb, Y. Grillet et al., in: M.D. Le Van (Ed.), Fundamentals of Adsorption, Kluwer, Boston, 1996, p. 587. I. Derycke, J.P. Vigneron, Ph. Lambin, A.A. Lucas, E.G. Derouane, J. Chem. Phys. 94 ( 1991 ) 4620. [7] P.M. Ajayan, S. [ijima, Nature 361 (1993) 333. P.M. Ajayan, T.W. Ebbesen, T. Ichihashi, S, lijima, K. Tanigaki, H. Hiura, Nature 362 (1993) 522. S.C. Tsang, Y.K. Chen, P.J.F. Harris, M.L.H. Green, Nature 372 (1994) 159. P.M. Ajayan. O. Stephan, P. Redlich, C. Colliex, Nature 375 (1995) 564. [8] A. Thess, R. Lee, P. Nikolaev, D. Hongjie, 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. Fischer, R.E. Smalley, Science 273 (1996) 483, and references cited therein. [9] W.M. Meier, D.H. Olson, Atlas of Zeolite Structure Types, Butterworth Heinemann, Oxford, 1992, p. 26. [10] P,J. Branton, P.G. Hall, K.S.W. Sing, J. Chem. Soc. Chem. Commun. ( 1993 ) 1257. [11] L.W. Bruch, M.W. Cole, E. Zaremba, Physical Adsorption: Forces and Phenomena, Oxford University Press, 1997. [12] M.W. Cole, J.R. Klein, Surf. Sci. 124 (1983) 547. [13] L.D. Landau, E.M. Lifshitz, Statistical Physics, Pergamon, London, 1958. [14] G.J. Tjiatjopoulos, D.L. Feke, J.A. Mann, Jr., J. Phys. Chem. 92 (1988) 4006. [ 15] W.A. Steele, The Interactions of Gases with Solid Surfaces, Pergamon, Oxford, 1984. [16] M.J. Bojan, W.A. Steele, Computer simulations of sorption in pores with rectangular cross-sections, Phys. Rev. E: in press. [17] H. Takahashi, in: E.H. Lieb, D.C. Mattis, (Eds.), Mathematical Physics in One Dimension, Academic Press, New York, 1966, pp. 25-27. F. Giirsey, Proc. Cambridge Philos. Soc. 46 (1950) 182. [ 18] Lord Rayleigh, Nature (London) 45 ( 1891 ) 80. L. Tonks, Phys. Rev. 50(1936)955. [19] G. Derry, D. Wesner, W.E. Carlos, D.R. Frankl, Surf. Sci. 87 (1979) 518. [20] R.K. Pathria, Statistical Mechanics, Butterworth Heinemann, Oxford, 1996, pp. 144 146. [21] M.W. Cole, D.R. Frankl, D.L. Goodstein, Rev. Mod. Phys. 53 (1981) 199. Z.-C. Guo. L.W. Bruch, J. Chem. Phys. 77 (1982) 7748. [22] E. Dujardin, T.W. Ebbesen, H. Hiura, K. Tanigaki, Science 265 (1994) 1850.