Adsorption on Nanotubes Having Equilateral Triangular Geometry

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Langmuir 2008, 24, 11722-11727

Adsorption on Nanotubes Having Equilateral Triangular Geometry with First- and Second-Neighbor Interactions: Attractive First Neighbors Alain J. Phares,*,† David W. Grumbine, Jr.,‡ and Francis J. Wunderlich† Department of Physics, Mendel Science Center, VillanoVa UniVersity, VillanoVa, PennsylVania 19085-1699, and Department of Physics, St. Vincent College, Latrobe, PennsylVania 15650-4580 ReceiVed June 5, 2008. ReVised Manuscript ReceiVed July 23, 2008 The recently published general method of studying adsorption on terraces and nanotubes is applied to adsorption on nanotubes in hollows or on-tops, which form a zigzag equilateral triangular adsorbate lattice. We consider adsorbate-adsorbate first- and second-neighbor interactions, with attractive first neighbors. In addition to empty and full coverage, there are three phases of occupational characteristics which are independent of the diameter of the nanotube. The low temperature energy phase diagram is the same for nanotubes of increasing diameters and, thus, is also valid in the infinite-diameter limit. As expected, this diagram is the same as that of the infinite-width limit of an equilateral triangular terrace, on which we have recently reported. The current study also includes temperature effects on the phases and the transition between phases.

1. Introduction We model adsorption on the surface of a nanotube for which adsorption takes place on sites forming a zigzag equilateral triangular lattice. This is the case when adsorption takes place in the hollows of a zigzag hexagonal nanotube such as zigzag single walled carbon1-5 or boron nitride (BN) nanotubes,6 as in Figure 1a. It is also the case when adsorption on a zigzag BN nanotube takes place on either the B-sites or the N-sites, shown in Figure 1b. First-neighbor adsorbate lattice sites are separated by a distance of a
3, and second neighbors, by a distance 3a, where a is the distance between carbon neighbors or between boron and nitrogen neighbors. This study also applies to on-top adsorption on a nanotube whose atomic sites form a zigzag equilateral triangular lattice, as is the case for platinum,7 palladium,8 and gold9,10 nanotubes. The only difference is that first-neighbor adsorbate sites are separated by a distance a and second-neighbor sites are sparated by a distance a
3. In this article we study the effect of adsorbate-adsorbate first- and second-neighbor interactions, focusing on attractive first neighbors. The partition function of the system of monomers adsorbed on infinitely long terraces, or nanotubes with the same geometry, is obtained in terms of the largest eigenvalue of a recursively constructed transfer matrix,11 from which all pertinent thermodynamic properties of the system are derived. The transfer matrix Tterrace for a terrace M atomic sites in width and the transfer * To whom correspondence should be addressed. Phone: +1 610 519 4889. E-mail: [email protected]. † Villanova University. ‡ St. Vincent College.

(1) Iijima, S. Nature 1991, 354, 56. (2) Iijima, S.; Ichihashi, T. Nature 1993, 363, 603. (3) Iijima, S. Physica B 2002, 323, 1. (4) Dai, H. Surf. Sci. 2002, 500, 218. (5) Terrones, M. Annu. ReV. Mater. Res. 2003, 33, 419. (6) Loiseau, A.; Willaime, F.; Demoncy, N.; Hug, G.; Pascard, H. Phys. ReV. Lett. 1996, 76, 4737. (7) Oshima, Y.; Koizumi, H.; Mouri, K.; Hirayama, H.; Takayanagi, K.; Kondo, Y. Phys. ReV. B 2002, 65, 121401(R). (8) Yu, S.; Welp, U.; Hua, L. Z.; Rydh, A.; Kwok, W. K.; Wang, H. H. Chem. Mater. 2005, 17, 3445. (9) Oshima, Y.; Onga, A.; Takayanagi, K. Phys. ReV. Lett. 2003, 91, 205503–1. (10) Senger, R. T.; Dag, S.; Ciraci, S. Phys. ReV. Lett. 2004, 93, 196807–1. (11) Phares, A. J.; Grumbine, Jr.; D, W.; Wunderlich, F. J. Langmuir 2007, 23, 558.

Figure 1. Schematic of nanotube structures. In both parts, the vertical dashed line indicates that the terrace drawn with M sites in width has been wrapped around a cylinder to form a nanotube such that the first site becomes a first neighbor of the Mth site. In part a, the small circles are the carbon sites, or boron and nitrogen sites, that form a zigzag hexagonal nanotube M sites in circumference. The larger circles connected by dashed segments are the hollows of the hexagonal structures and form the equilateral triangular adsorbate lattice. In part b, the dark and blank circles are boron and nitrogen sites of a zigzag BN nanotube. The circles connected by dashed segments represent the on-top adsorbate lattice.

matrix Tnanotube for a nanotube of the same lattice geometry and number of sites M in circumference are derived from a common G matrix. This matrix is then modified in one way to account

10.1021/la801748t CCC: $40.75  2008 American Chemical Society Published on Web 09/04/2008

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Langmuir, Vol. 24, No. 20, 2008 11723

for the edge effect in the case of a terrace and, in another way, to account for the wrapping of the terrace on a cylinder to produce a nanotube. In the limit as M approaches infinity with constant atomic site spacing, the results obtained for the terrace and its associated nanotube must approach that of adsorption on the infinite two-dimensional surface. Section 2 provides the notation that we have used and the recursive construction of the transfer matrix for the nanotubes under consideration. Section 3 presents the occupational characteristics of the phases and their occupational configurations, including the low temperature energy phase diagram. Section 4 provides entropy curves, the study of phase transitions, and the infinite-diameter limit of the results at low temperature. In section 5, we study the effect of temperature and consider the high temperature limit. Section 6 is a summary and discussion.

side face each other. This is possible if, and only if, M is even. This wrapping introduces additional first- and second-neighbor sites and affects certain elements of the G matrix. These elements are multiplied by (ypzq) with p and q ranging from 0 to 6.11 Consequently, changing the exponents of y and z in the proper G matrix elements provides the transfer matrix (T matrix) associated with adsorption on a zigzag equilateral triangular nanotube M atomic zigzag sites in circumference when first- and second-neighbor interactions are included in the analysis. All of the elements of the T matrix are therefore real and non-negative, and as follows from the Frobenius-Perron theorem, the eigenvalue of largest modulus R(x, y, z) is real and positive. If the nanotube is assumed to be very long, as is true in this case, the partition function Z(x, y, z) is completely determined by the largest eigenvalue R(x, y, z) according to11

2. Notation, Transfer Matrix, and Computational Method

At thermodynamic equilibrium, the occupational characteristics of coverage, θ0, the number of first neighbors per site, θ, and the number of second neighbors per site, β, are given by

Section 5 of ref 11 provides the complete derivation of the transfer matrices for adsorption on a zigzag equilateral triangular terrace and on its associated nanotube. For convenience, we summarize the results here. The system consists of a terraced or nanotube surface in contact with a medium containing particles each having a chemical potential energy µ′. The system is at thermodynamic equilibrium and absolute temperature T. The adsorbate-substrate interaction energy of an adsorbed particle is V0 and the adsorbate-adsorbate first- and secondneighbor interactions are V and W, respectively. The relevant activities are

x ) exp[µ/kT],

y ) exp[V/kT],

z ) exp[W/kT] (1)

where µ ) µ′ + V0, and k is Boltzmann’s constant. The G matrix is recursively constructed from two sets of matrices AN(a, b, c; a′, b′) and BN(b, c, d; c′) of rank 2N. The arguments of AN and BN are either 0 or 1. Thus for any N, there are 25 A-type matrices and 24 B-type matrices. These are recursively related to matrices of lower rank according to

AN(a, b, c;a ′ , b ′ ) )

(

)

BN-1(b, c, 0;c ′ ) xyb+c+c
za+a
BN-1(b, c, 1;c ′ ) (2) BN-1(b, c, 0;c ′ ) xyb+c+c
za+a
+1BN-1(b, c, 1;c ′ )

and

BN-1(b, c, d;c ′ ) )

(

)

AN-2(c, d, 0;c ′ , 0) xy z AN-2(c, d, 1;c ′ , 0) (3) AN-2(c, d, 0;c ′ , 1) xyc+2dz2b+1AN-2(c, d, 1;c ′ , 1) c+d b

There are (25 + 24) initial conditions, namely

A0(a, b, c;a ′ , b ′ ) ) 1,

B0(b, c, d;c ′ ) ) 1

(4)

In a zigzag row of M sites, we may choose the first site to be at the bottom of the zigzag, as in Figure 1. Therefore, if M is odd (or even), the M th site is at the bottom (or the top) of the zigzag. The G matrix is given by BM(0, 0, 0; 0) if M is odd and by AM(0, 0, 0; 0, 0) if M is even. The elements of the G matrix are the product of the three activities, each raised to a non-negative integer power of the form xRyβzγ. Thus each matrix element is identified by its three exponents, R, β, and γ. Since there are 2M × 2M ) 4M elements in the G matrix, it is sufficient to generate and store, for computational purposes, an ordered array of 3(4M) integers. Each set of three array elements corresponds to the exponents of one matrix element. The design of the computational algorithm to construct this array of integers is straightforward and relies on the above recursion relations. This article is on nanotubes obtained by wrapping a terrace M zigzag sites wide around a cylinder such that its edges on either

Z(x, y, z) ) [R(x, y, z)]1/M

θ0 )

x ∂R , MR ∂x

θ)

y ∂R , MR ∂y

β)

(5)

z ∂R MR ∂z

(6)

The state of the system is then expressed by the set of numbers {θ0, θ, β}. Thus, the energy per site is ε ) µθ0 + Vθ + Wβ, and the entropy per site divided by Boltzmann’s constant follows as

S ) (1/M) ln R - ε/kT

(7)

A phase is a state of the system in which the occupational characteristics remain unchanged over a wide range of values of the chemical potential µ and corresponds to a crystallization pattern of the adsorbates. When the pattern is unique, the phase is perfectly ordered and the entropy is zero, otherwise the phase is partially ordered and its entropy has a nonzero local minimum. In the transitional region between two phases with differential characteristics ∆θ0, ∆θ, and ∆β, the entropy has a local maximum when11-15

µ(∆θ0) + V(∆θ) + W(∆β) ) 0

(8)

Following the sign convention adopted in the definition of the activities in eq 1, positive energies correspond to attractive forces. Since we are only considering attractive first-neighbor interactions, then V > 0. All of our computations were carried out with long double precision arithmetic, and numerical results are quoted with a precision greater than 10 significant figures.

3. Phases and Energy Phase Diagram at Low Temperature The energy phase diagram is generated when the system is at a relatively low temperature T < T0, with kT0/V ≈ 1/100, by plotting V ) µ/V versus u ) W/V. For example, with V ) 20 kcal/mol, the value of T0 is about 100 K. For a given value of u, the chemical potential energy of the particles in the medium is varied. This corresponds to varying the pressure, if the medium is a gas, or varying the concentration if the medium is a liquid. In addition to the empty and full coverage phases, E ) {0, 0, 0} and F ) {1, 3, 3}, three perfectly ordered phases have been identified for all possible values of M: A ) {1/3, 1/3, 0}, B ) {1/2, 1, 1/2}, and C ) {2/3, 4/3, 1}. Energy regions are defined by ranges of u over which the sequence of phases is the same, starting with empty, and ending with full coverage. There are (12) Phares, A. J.; Grumbine, D. W., Jr.; Wunderlich, F. J. Langmuir 2006, 22, 7646. (13) Phares, A. J.; Grumbine, D. W., Jr.; Wunderlich, F. J. Phys. Lett. A 2007, 366, 497. (14) Phares, A. J.; Grumbine, D. W., Jr.; Wunderlich, F. J. Langmuir 2007, 23, 1928. (15) Phares, A. J.; Grumbine, D. W., Jr.; Wunderlich, F. J. Langmuir 2008, 24, 124.

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three energy regions: u < -1, -1 < u < -1/2, and -1/2 < u. The sequences of phases in these regions are, respectively,

E f A f B f C f F;

E f B f F;

EfF

(9)

These regions are shown in Figure 2. The boundaries between phases in the phase diagram correspond to the set of points at which the entropy is a local maximum and eq 8 is satisfied. Therefore, as verified in all previous applications,11-15 the boundaries are straight lines. Dividing eq 8 by V gives the equation of the boundary line between two phases,

V ) -(∆β/∆θ0)u - (∆θ/∆θ0)

(10)

It follows that the equation of the boundary line between phases E and A is V ) -1; between phases E and B, V ) -u - 2; and between phases E and F, V ) -3u - 3. The equation of the boundary line between phases A and B is V ) -3u - 4; between B and C, V ) -3u - 2; between B and F, V ) -5u - 4; and between C and F; V ) -6u - 5. These boundaries meet at three triple critical points of coordinates:

(-1, 1),

(-1, -1),

(- 1⁄2, - 3⁄2)

(11)

We now consider the occupational configurations of the nontrivial phases, starting with the {1/3, 1/3, 0} phase. When the number of zigzag sites M is an integer multiple of 6, there are three possible realizations of this phase. Figure 3 corresponds to stripes of M/2 occupied sites (either at the top or the bottom of a zigzag) separated by two rows of M/2 vacant sites, distributed either “horizontally” as in Figure 3a or at a “60° angle” as in Figure 3b. The horizontally occupied stripes of sites are distributed on circular cross-sections of the nanotube of M/2 sites (here again at the top or bottom of a zigzag of M sites), while the stripes at 60° are infinite and helical, with one turn containing M/2 sites. The third possibility is shown in Figure 4 and corresponds to isolated triangular clusters. However, only the stripe configurations are possible when M is 2 or 4 modulo 6. The occupational configuration of the B phase, {1/2, 1, 1/2},

Figure 3. Occupational configuration of the {1/3, 1/3, 0} or {2/3, 4/3, 1} phase, forming stripes which are either (a) horizontal or (b) at a 60° angle, when the number of atomic sites M around the circumference of the nanotube is an integer multiple of 6. The occupied sites of the {1/3, 1/3, 0} phase, represented by the dark circles, are the vacant sites of the {2/3, 4/3, 1} phase.

consists of two consecutive rows of M/2 occupied sites, separated by two rows of M/2 vacant sites, oriented either horizontally or at 60°. The C phase, {2/3, 4/3, 1}, is the complement of the A phase, {1/3, 1/3, 0}, where the vacant sites are occupied, and vice versa. The B phase is its own complementary phase, therefore “particle-hole” symmetry is satisfied for all nanotube diameters. We conclude that the lack of particle-hole symmetry in the case of the associated finite-width terrace15 may be attributed to the edge effect.

4. Entropy Curves, Phase Transitions, and Infinite-Diameter Limit

Figure 2. Low temperature energy phase diagram for a nanotube that is valid for any number M of sites in circumference.

As an example, we consider the first energy region and plot the entropy S versus coverage θ0 for various values of M at T < T0. As required by the particle-hole symmetry, these curves are symmetric with respect to θ0 ) 1/2, shown in Figure 5. As follows from eq 9, and shown in Figure 2, for T < T0, there are four phase transitions in the first energy region, two in the second region, and one in the third region. The transition from empty to full coverage that occurs in the third region is firstorder for all values of M, while the remaining transitions are second-order. The first-order transition remains for all M and at temperatures well above T0. We have tested this beyond 400 K with V ) 20 kcal/mol. This will be addressed in the next section. The discontinuity occurs at the boundary line between the two phases where the occupational characteristics are {1/2, 3/2, 3/2} with the numerical value of the entropy being zero to within the accuracy of our numerical computations. We interpret this as an equal number of very large patches of vacant and fully covered sites.

AttractiVe First Neighbors in Nanotubes

Langmuir, Vol. 24, No. 20, 2008 11725 Table 1. Low Temperature Occupational Characteristics at the Transitions between Phases Excluding the First-Order Transition from Empty to Full Coveragea transition

Figure 4. Occupational configuration of the { /3, /3, 0} or { /3, /3, 1} phase, forming isolated triangular clusters possible only when the number of atomic sites M on the circumference of the nanotube is an integer multiple of 6. The triangular clusters are the occupied sites of the {1/3, 1/3, 0} phase and the vacant sites of the {2/3, 4/3, 1} phase. 1

1

2

4

M)6

M)8

M ) 10

M ) 12

EfA

θ0 0.2256851478 0.1873653716 0.1630928711 0.1772598019 S 0.2438318414 0.2033112320 0.1792172997 0.1821153625

CfF

θ0 0.7743148522 0.8126346284 0.8369071288 0.8227401981 S 0.2438318414 0.2033112320 0.1792172997 0.1821153625

AfB

θ0 0.4203401790 0.4203401790 0.4203401790 0.4203401790 S 0.0664868594 0.0498651446 0.0398921157 0.0332434297

BfC

θ0 0.5796598200 0.5796598200 0.5796598200 0.5796598200 S 0.0664868594 0.0498651446 0.0398921157 0.0332434297

EfB

θ0 0.3016746809 0.3016746809 0.3016746809 0.3016746809 S 0.1074282053 0.0805711540 0.0644569232 0.0537141027

BfF

θ0 0.6983253191 0.6983253191 0.6983253191 0.6983253191 S 0.1074282053 0.0805711540 0.0644569232 0.0537141027

a The discontinuity occurs for any u > -1/2 at V ) -u - 2, with occupational characteristics {1/2, 3/2, 3/2 } and zero entropy. This is interpreted as a mixture of large patches of empty and covered sites.

the infinite-diameter limit of the nanotube calculations has provided additional information we were unable to obtain from the infinite-width limit of the terraces. (1) All transitions become first-order. (2) Since particle-hole symmetry was not observed in finite-width terraces, it could not be anticipated in the infinitewidth limit based solely on our numerical computations. Indeed, the trend with increasing M showed the coexistence of two configurations, stripes and isolated triangular clusters, which in the infinite-width limit provided the {1/3, 1/3, 0} phase. However, we could not definitely conclude that a complementary situation existed in those phases whose infinite-width limit led to the {2/ 3, 4/3, 1} phase.

5. Temperature Effects

Figure 5. Entropy curves versus coverage at T < T0, in the first energy region, for increasing values of M.

The occupational characteristics at the transition between phases listed in Table 1 were obtained at T < T0. For all M, as anticipated, there is a perfect symmetry between pairs of transitions. The sum of the coverage at the transition E f A and the coverage at the transition C f F is exactly one, with perfect matching of both entropies. The trend with increasing M shows some oscillation of the coverage and of the entropy. The full width at half-maximum of the entropy curve as a function of V decreases with increasing M. This indicates that, in the infinite-M limit, both transitions become first-order. The symmetry holds again between the pair of transitions A f B and B f C and between the pair E f B and B f F. While the coverage at these transitions is independent of M, as shown in Table 1, the entropy for the first pair varies as (1/M) ln(1.490 216 120), and the entropy for the second pair varies as (1/M) ln(1.905 166 167). The infinite-M limit is therefore zero-entropy for both pairs. At T < T0, the full width at half-maximum of the curve S versus V decreases with increasing M, giving a second indication that these transitions become first-order. The infinite-M limit of the results obtained for nanotubes matches exactly that obtained for the associated equilateral triangular terraces with attractive first neighbors.15 However,

Temperature effects, for a given M and in each of the three energy regions, have also been studied from several different points of view. For example, in the plot of S versus θ0, there is a gradual disappearance of the cusps observed in the first and second energy regions as the temperature is increased. This is exhibited in Figures 6-8 with M ) 6, 8, and 10 at u ) -13/12 (first energy region) and for increasing values of kT/V. This is similar to what we pointed out in the case of interacting dimers on a square lattice16,17 and was verified by others using Monte Carlo simulation.18-20 It should be pointed out that, while the characteristics of the phases are diameter or M-independent, gradual disappearance of the phases with increasing temperature is M-dependent. In the present situation, the high temperature limit could be approached numerically in the same manner followed in the study of the high temperature adsorption isotherms on equilateral triangular terraces that we have recently investigated.13 The chemical potential µ is allowed to vary in a range sufficiently large so that the activity x ) exp(µ/kT) can take on any non-negative value as T increases. Mathematically, this is equivalent to setting y ) z ) 1 and allowing x to vary. Consequently, the T matrix for the system is identical to the G matrix with y ) z ) 1, which is the Kronecker product of Mthorder of the following matrix

( ) 1 x 1 x

The largest eigenvalue of this matrix is R(x, 1, 1) ) (1 + x)M. This is identical to the problem of high temperature adsorption isotherms on equilateral triangular terraces that we have recently investigated.13 Its solution is that of the one-dimensional problem,21namely,

11726 Langmuir, Vol. 24, No. 20, 2008

S ) -(1 - θ0) ln(1 - θ0) - θ0 ln θ0

Phares et al.

(12)

The maximum entropy is ln(2) and occurs at half-coverage, θ0 ) 1/2. Obtaining the numbers of first and second neighbors per site as functions of coverage requires the evaluation of the partial derivatives of R(x, y, z) with respect to y and z at y ) z ) 1. The T matrices for terraces of increasing widths are different from those for nanotubes of increasing diameters and yield, in the high temperature limit, width-dependent values of θ and β.13 Unlike terraces, the numerical results for the nanotubes are diameter-independent,

θ ) β ) 3θ02

(13)

and were not derived analytically. However, as expected, they agree with the high-temperature infinite-width limit of the equilateral triangular terrace.13 The maximum entropy is reached when θ0 ) 1/2; therefore, θ ) β ) 3/4. We have reported earlier that the transition E f F, which occurs in the third energy region, remains first order well above 400 K when V is about 20 kcal/mol. A numerical estimate of the critical temperature beyond which this transition becomes second

Figure 8. Temperature dependence of the entropy curves versus coverage in the first energy region with u ) -13/12 for a nanotube with M ) 10.

order may be obtained when the entropy reaches (1/100) ln(2), which corresponds to 2280 K. From a physical point of view, we can therefore claim that this transition remains first-order.

6. Summary and Discussion

Figure 6. Temperature dependence of the entropy curves versus coverage in the first energy region with u ) -13/12 for a nanotube with M ) 6.

Figure 7. Temperature dependence of the entropy curves versus coverage in the first energy region with u ) -13/12 for a nanotube with M ) 8.

We have provided the complete adsorption features on nanotubes, for which the adsorption sites form a zigzag equilateral triangular lattice, when first- and second-neighbor interactions are included with attractive first neighbors. Therefore the results may apply to zigzag single walled carbon or boron nitride nanotubes for which adsorption takes place in the hollows of the hexagonal structures. It may also apply to on-top adsorption on zigzag BN nanotubes, with adsorption on either boron or nitrogen sites or on-top adsorption on zigzag platinum, palladium, or gold nanotubes. The main features are the following: 1. There are only three nontrivial phases of 1/3, 1/2, and 2/3 coverage for all nanotube diameters. 2. Particle-hole symmetry is satisfied. The 1/3 phase and its complementary 2/3 phase exhibit the coexistence of stripes and isolated triangular clusters of occupancies in the 1/3 phase and vacancies in the 2/3 phase. The 1/2 phase corresponds to two consecutive rows of occupied sites separated by two rows of vacant sites, either circular or helical. 3. There are three triple critical points in the energy phase diagram. 4. The calculations are diameter-independent and therefore valid in the infinite-diameter limit. They provide the adsorption features on the infinite 2D equilateral triangular lattice. In a recently published article,15 the infinite-width limit of the results on finite-width terraces with the same lattice geometry and the same adsorbate-adsorbate interactions was obtained following a systematic and elaborate technique. However, we could not (16) Phares, A. J.; Wunderlich, F. J.; Grumbine, D. W., Jr.; Curley, J. D. Phys. Lett. A 1993, 173, 365. (17) Phares, A. J.; Wunderlich, F. J.; Curley, J. D.; Grumbine, D. W., Jr. J. Phys. A.: Math. Gen. 1993, 26, 6847. (18) Ramirez-Pastor, A. J.; Riccardo, J. L.; Pereyra, V. D. Surf. Sci. 1998, 411, 294. (19) Roma´, F.; Ramirez-Pastor, A. J.; Riccardo, J. L. Langmuir 2000, 16, 9406. (20) Roma´, F.; Ramirez-Pastor, A. J.; Riccardo, J. L. J. Chem. Phys. 2001, 114, 10932. (21) Baxter, R. J. Exactly SolVed Models in Statistical Mechanics; Academic Press: New York, 1982.

AttractiVe First Neighbors in Nanotubes

determine, in the infinite-width limit, whether the transitions between phases where first- or second-order or whether particle-hole symmetry was definitely satisfied. Our current results provide an independent check of the validity of the technique we have followed to extrapolate our results on finite-width terraces while adding valuable information. In all of our previous calculations for terraces, we showed how our simple model could be used to determine the interaction energies from the experimental observation of the phases and the conditions prevailing at the boundaries between phases. For example, if the coexistence of stripes and isolated triangular clusters is experimentally observed, the energy phase diagram predicts that this is possible if, and only if, u < -1, indicating that first neighbors are attractive and second neighbors are repulsive with a strength greater than the strength of the firstneighbor attractions. Another example is the case for which observation shows no intermediate phase(s) between empty and full coverage. Then the energy phase diagram again predicts that u > -1/2. Experimental knowledge of the conditions prevailing at the boundaries between phases may also be valuable since the

Langmuir, Vol. 24, No. 20, 2008 11727

model predicts the exact relationship between the energies, as follows from the equation of the straight-line boundary. Calculations for nanotube adsorption with first-neighbor repulsion (V < 0), are under way. We have completed the calculations with the number M of zigzag sites around the circumference less than or equal to 10. They show a strong dependence on M. Higher values of M will have to be considered in order to find the explicit analytic M dependence in a manner similar to what was achieved in refs 14 and 15 for finite-width terraces. The transfer matrix is exactly the same whether V is positive or negative. It does not show any particular symmetry, and it is striking to find the results to be simple for V positive but not for V negative. Acknowledgment. This research was supported by an allocation of advanced computing resources supported by the National Science Foundation. The computations were performed in part on the TG PSC TCS1 at the Pittsburgh Supercomputing Center. LA801748T