Structural Transitions of Nitrogen Confined in Slit Graphite Pores

Langmuir 2001, 17, 5245-5255

5245

Structural Transitions of Nitrogen Confined in Slit Graphite Pores J. Kamakshi and K. G. Ayappa* Department of Chemical Engineering, Indian Institute of Science, Bangalore 560012, India Received September 20, 2000. In Final Form: March 27, 2001 The structure of ordered phases that are formed when nitrogen is confined in slit graphite pores of height h is investigated using Monte Carlo simulations. The pore wall consists of a single-structured graphite sheet. Canonical ensemble simulations are carried out for temperatures ranging from 15 to 70 K with layer density distributions, in-plane, out-of-plane angular distributions and snapshots evaluated at different temperatures. At each pore height the pore densities are obtained from independent grand ensemble simulations. At the smallest pore height studied (h ) 7 Å), where a single layer of molecules is accommodated at the center of the pore, the orientations are predominantly wall parallel, forming a biaxially incommensurate herringbone structure. When two or more fluid layers are formed in the slit pore, the orientation of molecules adsorbed next to the wall can exist in either the herringbone or hexagonal phases. In all the multilayered cases studied, with the exception of the h ) 10 Å pore, where both wall layers form a commensurate herringbone structure, the low-temperature wall structures are incommensurate, possessing 6-fold hexagonal symmetry. The presence of the pinwheel structures, which were observed at low temperatures in the h ) 12 Å and h ) 14 Å pores, is determined by the pore height or the proximity and/or density of the adjacent fluid layers when inner layers are present.

1. Introduction The behavior of fluids confined in nanopores is found to be discussed in many areas of science and engineering and is important in elucidating the molecular mechanisms for processes such as adsorption, lubrication, wetting, and catalysis. With advances in both experimental and theoretical methods our understanding of molecularly thin films has been significantly enhanced. Among many of the properties of adsorbed molecular films, one of the aspects that has received a lot of attention is the formation of ordered phases when a fluid is adsorbed on a substrate. In this regard the adsorption of molecular gases on the basal plane of graphite has been extensively investigated1,2 using theory, experiments, and detailed molecular simulations. Theoretical3 and simulation4-7 studies have been carried out on the 2-D solid phase of nitrogen adsorbed on a graphite sheet. Molecular simulations have shown the presence of the two-sublattice herringbone configuration for both commensurate and incommensurate structures at low temperatures with an order-disorder transition occurring between 28 and 33 K depending upon the parameters used.7 Both theory and experiments have now confirmed that the transition is of the first order. Molecular simulations of nitrogen on graphite are in agreement with the low-temperature ordered phases observed experimentally using a variety of experimental techniques.8-15 * To whom correspondence should be addressed. E-mail: [email protected], Fax: 011-91-80-3600683 (3600085). (1) Steele, W. A. Chem. Rev. 1993, 93, 2355. (2) Marx, D.; Wiechert, H. Adv. Chem. Phys. 1996, XCV, 213. (3) Mouristen, O. G.; Berlinsky, A. J. Phys. Rev. Lett. 1982, 48, 181. (4) Talbot, J.; Tildesley, D. J.; Steele, W. A. Mol. Phys. 1984, 51, 1331. (5) Talbot, J.; Tildesley, D. J.; Steele, W. A. Surf. Sci. 1986, 169, 71. (6) Lynden-Bell, R. M.; Talbot, J.; Tildesley, D. J.; Steele, W. A. Mol. Phys. 1985, 54, 183. (7) Peters, C.; Klein, M. L. Mol. Phys. 1985, 54, 895. (8) Eckert, J.; Ellenson, W. D.; Hastings, J. B.; Passell, L. Phys. Rev. Lett. 1979, 43, 1329. (9) Diehl, R. D.; Toney, M. F.; Fain, S. C. Phys. Rev. Lett. 1982, 48, 177.

Although monolayer and multilayer16,17 adsorption of nitrogen on the basal planes of graphite has been the subject of extensive investigation, diatomic molecules under confinement, which is the focus of this paper, have received little attention. On the basis of the formation of ordered phases of nitrogen on graphite, it is natural to enquire into the nature of the phases that might be formed under confinement. Advances in experimental techniques, such as the surface force apparatus, now permit the measurement of forces on molecules that are confined to a few molecular layers. The structure of the confined fluid is expected to play an important role in interpreting data from these experiments. Experimentally, the orientation of nitrogen in intercalated graphite (C24K and C24Rb) has been studied using the nuclear resonance photon scattering technique,18,19 where the tilt angle of the nitrogen axis with the planar graphite layers is measured. In both cases, the molecular axis of the intercalated nitrogen was found to be nearly parallel to the graphite sheets. In C24K, nitrogen remains intercalated until around 190 K, and in C24Rb due to larger interlayer spacing the nitrogen remains intercalated till 130 K. The success of this technique opens up the possibility of studying orientations of adsorbed nitrogen molecules under confinement in systems such as exfoliated graphite. The adsorption of oxygen in slit graphite pores20 has been studied using grand canonical Monte Carlo (GCMC) (10) Moreh, R.; Shahal, O. Surf. Sci. 1986, 177, L963. (11) Sullivan, N. S.; Vaissiere, J. M. Phys. Rev. Lett. 1983, 51, 658. (12) Zhang, Q. M.; Kim, H. K.; Chan, M. H. W. Phys. Rev. B 1986, 33, 413. (13) Wang, R.; Wang, S. K.; Taub, H.; Newton, J. C.; Shechter, H. Phys. Rev. B 1987, 35, 5841. (14) Volkmann, U. G.; Knorr, K. Phys. Rev. Lett. 1991, 66, 473. (15) Burns, T. E.; Dennison, J. R.; Ehrlich, S. N. Langmuir 1999, 15, 1423. (16) Vernov, A.; Steele, W. Surf. Sci. 1986, 171, 83. (17) Vernov, A. V.; Steele, W. A. Langmuir 1986, 2, 219. (18) Moreh, R.; Pinto, H.; Finkelstein, Y.; Volterra, V.; Birenbaum, Y.; Beguin, F. Phys. Rev. B 1995, 52, 5330. (19) Moreh, R.; Melloul, S.; Zabel, H. Phys. Rev. B 1993, 47, 10754. (20) Sokolowski, S. Mol. Phys. 1992, 75, 999.

10.1021/la001348a CCC: $20.00 © 2001 American Chemical Society Published on Web 07/27/2001

5246

Langmuir, Vol. 17, No. 17, 2001

Kamakshi and Ayappa

(µVT) simulations for several values of the slit height. For narrow pores the fluid molecules were oriented parallel to the walls, and for larger pores both wall-parallel and wall-perpendicular orientations were observed. Orientational effects were largely restricted to the fluid layers adjacent to the wall, and no significant in-plane order was observed. µVT simulations for nitrogen and bromine molecules in single-walled carbon nanotubes21 show that the molecules adjacent to the nanotube walls orient themselves parallel to the wall and that at high densities the wall layer contains a mixture of wall-parallel and wallperpendicular orientations. The wall-perpendicular orientations were more significant for the longer bromine molecules. In this paper, we investigate structures that are formed when nitrogen is confined in a slit graphite pore whose structured walls consist of a single graphite sheet. Where possible we compare and contrast the structures with those observed in simulation studies of monolayer and multilayer adsorption on graphite. A series of canonical ensemble (NVT) simulations are carried out for temperatures ranging from 14.7 to 70 K. The pore densities that are used in the NVT simulations correspond to nearly fully loaded pores. The pore density for a particular pore height is determined from its corresponding adsorption isotherm at 298 K using µVT simulations. At different temperatures we monitor the density distributions, energies, angles that the diatom makes with the graphite surface (the wall-parallel or in-plane angle φ and the wallperpendicular or out-of-plane angle θ), out-of-plane order parameter, and snapshots of the molecules. To study the effect of confinement, simulations are carried out at different slit heights of 7, 8, 10, 12, and 14 Å, which give rise to one, two, and three adsorbed layers as the pore height is increased.

Figure 1. Schematic of a slit pore. Periodic boundary conditions are imposed in x and y directions. The pore height, h, is the center to center distance between carbon atoms on opposite graphite sheets. Table 1. Parameters for Potentials Used for Nitrogen and Carbon in the Simulations16,17 parameter

nitrogen

carbon

nitrogen-carbon

/kB, K σ, Å l, Å Q × 1040, Cm2

36.40 3.318 1.098 -3.91

28.0 3.40

31.92 3.360

angles21 θ and φ of a dumbbell, and the new configuration is accepted with the same probabilities as in eq 2 where ∆U is due to a change in the Euler angles. µVT simulations which are based on the grand canonical ensemble22 consist of addition, removal, rotation, and displacement moves. Periodic boundary conditions are applied in the x and the y directions of the simulation cell as shown in Figure 1. The lengths of the simulation cell are chosen so as to ensure no mismatch at the periodic boundaries for the in-plane commensurate herringbone structures at low temperatures. A typical GCMC simulation consists of a total of 3 × 106 moves with 1.2 × 106 equilibration moves. A typical NVT simulation consists of a total of 8 × 106 moves with 1 × 106 equilibration moves. In all cases a cutoff of half box length based on the smaller dimension of the rectangular simulation box is used. 2.2. Interaction Parameters. Fluid-Fluid. Nitrogen is modeled as a rigid homonuclear dumbbell with the interaction between sites treated by using a 12-6 Lennard-Jones (LJ) potential

2. Theory and Simulation Procedure 2.1. NVT Simulations. NVT simulations are based on the canonical ensemble partition function. For N identical homonuclear dumbbells with bond length l,

ΘN Q(NVT) ) N!Λ3N

∫‚‚‚∫ exp[-βU(r, ω)]d

3N

r d2Nω (1)

Umn(rmn) ) 4

2

Uijlj )

-

σ rmn

6

(5)

2

∑ ∑U

mn(rmn)

(6)

m)1n)1

2

l , Λ ) h/x(2πmkT) 8Λ2

12

σ rmn

where rmn is the center to center distance between the site m on molecule i and site n on molecule j. The total interaction energy between two dumbbells is

where

Θ)

[( ) ( ) ]

(2) The quadrupole-quadrupole interaction between two rigid dumbbells is

and the volume elements

d3Nr )

∏

dxidyidzi and d2Nω )

i

∏

sin θidθidφi

(3)

i

Since details of the Monte Carlo moves for diatomic molecules under confinement have been described in an earlier work,21 we only give a brief description here. In a translational move, a new configuration is generated by changing the Cartesian coordinates x, y, and z of each atom on the dumbbell by the same random displacements ∆x, ∆y, and ∆z. The maximum displacements are adjusted such that 50% of the attempted translational moves are accepted. The new configuration is accepted with probability

p)

{

1 if ∆U e 0 exp(-β∆U) if ∆U > 0

(4)

where ∆U ) Un - Uo, where Un is the potential energy of the new configuration and Uo the potential energy of the old configuration. A rotational move is carried out by randomly changing the Euler (21) Khan, I. A.; Ayappa, K. G. J. Chem. Phys. 1998, 109, 4576.

Uijq )

3Q2 [1 - 5(cos2 θi + cos2 θj) + 2(sin θi sin θj cos φij 4rij5 (7) - 4 cos θi cos θj)2 - 15 cos2 θi cos2 θj]

(8)

where Q is the quadrupole moment, rij is the distance between the centers, and θi and θj are the angles between the axes of the molecules i and j, respectively, with the vector rij joining the centers of mass of the two molecules. φij is the angle between the projections of the two molecules on a plane perpendicular to the vector rij. The LJ parameters and the quadrupole moment used for the dumbbell models shown in Table 1 are similar to those used in previous studies of nitrogen on graphite,16,17 where the particular choice of interaction parameters and in particular the quadrupole moment are discussed. The total potential energy of interaction between two diatomic molecules is as follows: (22) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, 1987.

Structural Transitions in Slit Graphite Pores

Langmuir, Vol. 17, No. 17, 2001 5247

Uij ) Uijlj + Uijq

(9)

Fluid-Wall. The slit pore consists of two parallel graphite sheets separated by a distance h in the z direction as shown in Figure 1. The height h is the center to center distance between carbon atoms on the graphite sheets. The graphite sheets are positioned such that the carbon atoms on opposite walls are in perfect registry as seen in Figure 1. Each wall of the slit pore is made up of a single graphite sheet where the carbon atoms interact with the atoms on the dumbbell via a 12-6 LJ potential. Lorentz-Berthelot mixture rules are used for the nitrogencarbon interaction parameters. Fluid-wall quadrupole-quadrupole interactions are neglected. The LJ parameters for the carbon atoms are assumed to be those of graphite (Table 1). 2.3. Density Distributions. The density distribution of the nitrogen molecules in the slit pore is calculated by dividing the pore of height h into bins of thickness ∆z. The density distribution for a slit pore of length Lx and Ly is calculated using

F(z) )

〈N(z - ∆z/2, z + ∆z/2)〉 ∆z Lx Ly

(10)

The density distributions are calculated for both the center of mass and the individual atoms of the molecules. 2.4. Distributions of θ and O. Angle distributions are computed within the fluid layers that are formed in the slit pore. The angle θ, 0 e θ e 180°, is the angle formed between the axis of the dumbbell and the axis perpendicular to the graphite sheet (z-axis). θ ) 0° and 180° indicates orientations that are perpendicular to the graphite sheet, and θ ) 90° indicates parallel configurations. Noting that sin θ dθ is the appropriate volume element (eq 3), we compute the distribution

f(cos θ) )

〈

〉

N(cos θ - ∆ cos θ/2, cos θ + ∆ cos θ/2)

∫

+1

-1

N(cos θ) d cos θ

Figure 2. Schematic showing angles θ and φ used for calculating the distributions f(cos θ) and n(φ). The acute angle θ between the molecule axis and the graphite sheet is shown. -90° < φ < 90° is the angle between the projection of the molecular axis on the graphite sheet (x-y plane) and the x-axis.

(11)

by dividing the layer into bins of width ∆ cos θ. During the simulation, only the acute angle θ made by the dumbbell axis and the normal to the graphite sheet (Figure 2) is computed using

θ ) arccos |ez|

0° e θ e 90°

(12)

where ez is a unit vector of the dumbbell in the z direction. In all our discussion only the above acute angle will be used to discuss the tilt angle θ. With the underlying assumption that the distribution is symmetric about cos θ ) 0, the distributions f(cos θ) are shown for -1 e cos θ e 0. A value of f(cos θ) ) 0.5 indicates a completely random distribution. The angle φ is the angle formed between the projection of the dumbbell on the x-y plane and the x-axis as shown in Figure 2.

()

ey φ ) arctan ex

-90° e φ e 90°

(13)

where ex and ey are the unit vectors of the dumbbell along the x and y directions, respectively. The distribution for φ is calculated by dividing the layer into bins of width ∆φ.

n(φ) ) 〈N(φ - ∆φ/2, φ + ∆φ/2)〉

(14)

2.5. Order Parameter. An order parameter OP(z), dependent on the z coordinate, is defined by using the out-of-plane acute angle θ as follows

OP(z) )

1 Nz

Nz

〈

∑ (cos (θ ; z - ∆z/2, θ ; z + ∆z/2))〉 i

i

(15)

i)1

where Nz is the total number of molecules in the shell of thickness ∆z. 0 e OP e 1, a value close to 1.0 indicating that the fluid molecules are oriented perpendicular to the pore walls and a value close to 0 indicating a wall-parallel orientation of the molecules.

Figure 3. Adsorption isotherms from µVT simulations at 298 K, for (a) h ) 14 Å, (b) h ) 12 Å, and (c) h ) 10 Å. Dark circles indicate the points at which NVT simulations were carried out. 2.6. Specific Heat Cv. The dimensionless specific heat Cv/ Nk, obtained from the total energy (sum of fluid-fluid and fluidwall interactions) expressed in K/molecule, is

Cv/Nk )

(〈U2〉 - 〈U〉2) T2

(16)

where T is the temperature, N is the number of molecules in the slit pore, and k is the Boltzmann constant.

3. Results and Discussion NVT simulations are carried out for pore heights h ) 7, 8, 10, 12, and 14 Å for temperatures ranging from 14.7 to 70 K. We first perform µVT simulations to determine the adsorption isotherm. On the basis of the adsorption isotherms, the number density that is chosen for NVT simulations is the one that corresponds to a nearly fully loaded pore. Figure 3 illustrates the adsorption isotherms at 298 K for three different pore heights. The loadings which are used for NVT simulations are indicated in Figure 3. We also performed µVT simulations at 77 K with adsorption isotherms obtained for a maximum pressure of 1 atm. For smaller pore heights the maximum loading at both temperatures is similar, with a higher maximum loading for the larger pores at 298 K due to the higher bulk pressures. The number of particles that were used

5248


Kamakshi and Ayappa

Figure 4. Fluid-wall potential energies for different pore heights: (1) wall-perpendicular orientation (2) wall-parallel orientation. Note that the well depth for the wall-parallel orientation decreases as the pore height is increased. At h ) 8 Å, the well depths for the two orientations are nearly similar. Table 2. Periodic Box Size and Number of Particles Used in NVT Simulations for Different Box Heights h height (Å)

〈N〉

length in x (Å)

length in y (Å)

7 8 10 10 12 14 14

23 29 40 44 91 144 163

21.30 21.30 21.30 21.30 25.56 29.82 29.82

17.22 19.68 17.22 17.22 31.97 29.51 29.51

in the NVT simulations reported here are shown in Table 2. In all cases, the initial configuration that was used for the NVT simulation was the disordered state at 298 K. To study the effect of pore density we carried out simulations for both the 10 and 14 Å pores, at two loadings. 3.1. Fluid-Wall Potential Energies. The variation of the fluid-wall potential energies with the height of the pore has been studied for both wall-parallel and wallperpendicular arrangements of the fluid molecule with the pore wall. For the wall-parallel orientation, the fluid molecule is placed at the center of the hexagonal carbon ring with a φ of 45°. For the wall-perpendicular orientation, the fluid molecule is placed at the center of the hexagon. The variation of the fluid-wall potential is illustrated in Figure 4 for both wall-parallel and wall-perpendicular orientations of nitrogen. For the smallest pore, h ) 7 Å (Figure 4a), the wall-parallel orientation of the fluid molecules is clearly more favorable over the wallperpendicular or out-of-plane arrangement. At h ) 8 Å (Figure 4b), the potential energies indicate that both the wall-perpendicular and wall-parallel orientations are energetically similar at the minimum of the potential, with the wall-perpendicular orientation being favored at the center of the pore. As the pore height is increased the position of the minima shifts toward the wall. At 10 and 12 Å, the variation of the potential energy for both the wall-parallel and the wall-perpendicular arrangements (parts c and d of Figure 4) indicates that the wall-parallel configurations are favored. However, for the larger pores, there is little difference between the potential energies at the center of the pores, with the wall-perpendicular

Figure 5. Center of mass density distributions for different pore heights (a) h ) 7 Å, (b) h ) 8 Å, (c) h ) 10 Å, and (d) h ) 12 Å. At h ) 7 and 8 Å, a single layer of molecules is accommodated in the center of the pore. At h ) 10 and 12 Å, two fluid layers are formed.

arrangements being slightly more favorable. The situation at h ) 14 Å (not shown) is qualitatively similar to the 12 Å pore. 3.2. Density Distributions. Although the density distributions predominantly yield information on the layering of nitrogen molecules in the pore, the presence of shoulders in the center of mass distribution and double peaks in the atom-atom density distribution yields preliminary information on the orientation of the molecules within the adsorbed layers. The center of mass density distribution of nitrogen in the pore is shown in Figure 5 for different pore heights. At h ) 7 Å (Figure 5a), the smallest pore height studied, the molecules arrange themselves in a single layer at the center of the pore. The sharp peak at low temperatures indicates that the molecules are arranged in a thin layer with orientations parallel to the walls. As the temperature increases, the distributions tend to broaden, but with no significant outof-plane orientations of the fluid molecules. This is evident from the absence of any shoulders in the density distributions. At h ) 8 Å (Figure 5b), the molecules are still confined to the central regions of the pore; however, in contrast to the h ) 7 Å pore, the density distribution shows a distinct shoulder at lower temperatures, indicating that configurations consisting of molecules that are oriented at an angle to the pore wall are also present. At h ) 10 Å (Figure 5c), two distinct fluid layers are formed. Similar to the h ) 7 Å pore, at both low and high temperatures, there are no distinct shoulders in the density distributions, indicating the absence of significant out-of-plane orientations of the fluid molecules. At h ) 12 Å (Figure 5d), although two distinct layers are still formed, the presence of the secondary inner peak at low temperatures and shoulders at high temperatures indicates the presence of wall-perpendicular orientations. The distinct secondary peaks at temperatures of 14.7 and 20 K indicate nearly perpendicular orientations of the fluid molecules to the wall. Figure 6 illustrates both the center of mass and atomatom density distributions at different temperatures at h ) 14 Å with 144 molecules. Here three fluid layers are accommodated within the pore. At 15 K, both in-plane


Figure 6. Density distributions for h ) 14 Å (144 molecules) for (a) 15 K and (b) 40 K. Solid and dashed lines represent the center of mass and atom density distributions, respectively. The presence of double peaks in the wall and central layer indicates the presence of wall-perpendicular orientations at 15 K.

and out-of-plane orientations of the fluid molecules are present. Most of the fluid molecules in the wall layer are oriented nearly parallel to the wall with a few fluid molecules oriented perpendicular to the wall. The increased number of wall-perpendicular orientations at h ) 14 Å when compared with h ) 12 Å is evident from the distinct shoulders in density distribution of the two wall layers for the 14 Å pore. As the temperature increases, the splitting in both the center of mass and atom-atom density distributions decreases, indicating greater orientational disorder within the layers. The density distributions for the h ) 14 Å pore at a higher loading of 163 molecules is qualitatively similar to the density distributions shown in Figure 6. 3.3. Distributions of θ and O. A more detailed perspective on the relative orientations of the diatomic

Langmuir, Vol. 17, No. 17, 2001 5249

molecule with respect to the graphite sheet is obtained by examining the distribution of angles, f(cos θ) (eq 11) and n(φ) (eq 14) as a function of temperature. The distribution, f(cos θ), at h ) 7, 8, 10, and 12 Å are shown in Figure 7, where the side view snapshots are indicated as insets. The distribution at h ) 7 Å (Figure 7a) indicates that at low temperatures there are a maximum number of molecules with values of θ close to 90°, indicating mostly wall-parallel orientations. The fluid-wall potential energy profile (Figure 4a) also indicates that the wall-perpendicular orientations (small θ) are highly unfavorable at this small pore height. Raising the temperatures results in a broadening of the distribution with little change in the position of the peak, indicating that wall-perpendicular orientations are still favored. This is similar to the experimentally observed situation of wall-parallel configurations in intercalated graphite18,19 where the separation between graphitic planes sterically inhibits wallperpendicular orientations. From the distributions, n(φ), of the fluid molecules at h ) 7 Å (Figure 10a), it is evident that the low-temperature structures have molecules oriented at approximately (48°, suggesting the presence of the two-sublattice herringbone structure as is observed in the case of adsorption of nitrogen molecules on a graphite sheet.4 Snapshots of the x-y plane shown in Figure 10a confirm the herringbonelike structure with the molecules arranged in two rows parallel to the x-axis, each row with positive and negative φ angles. As the temperature increases, the peaks in the φ distribution located at (48° decrease in intensity and shift outward, prior to getting disordered. Snapshots and

Figure 7. Distribution f(cos θ) and insets of the side view (x-z) snapshots at low and high temperatures. Since f(cos θ) is symmetric about cos θ ) 0, only half the distribution is reported. cos θ ) 0 indicates wall-parallel orientations, cos θ ) - 1 indicates wallperpendicular orientations, and f(cos θ) ) 1/2 indicates completely random orientations: (a) h ) 7 Å, (b) h ) 8 Å, (c) h ) 10 Å, and (d) h ) 12 Å. Temperatures were (1) 14.7, (2) 32, and (3) 50 K.

5250


Kamakshi and Ayappa

Figure 8. Distribution f(cos θ) and side view (x-z) snapshots at low and high temperatures for h ) 14 Å (144 molecules): (a) positive wall layer (z ) h/2), (b) middle layer, (c) side view snapshot at 15 K, and (d) side view snapshot at 40 K. The low-temperature distribution of the wall layer indicates that the molecules have both wall-parallel and wall-perpendicular orientations.

the angular distributions indicate that the low-temperature state is orientationally ordered and biaxially incommensurate whereas the high-temperature state is incommensurate and orientationally disordered but translationally ordered in a triangular lattice. An estimate of the lattice parameters in the x and y directions at 14.7 K gives lx ) 3.72 Å and ly ) 4.23 Å. These are very close to the commensurate low-temperature herringbone structures of the single graphite sheet where lx ) 3.69 Å and ly ) 4.26 Å. At h ) 8 Å, the distribution f(cosθ) (Figure 7b) indicates that the angles are nearly randomly distributed, except at low temperatures where the distribution shows two peaks corresponding to θ values of approximately 35° and 90°. Note that a value of f(cos θ) ) 0.5 implies a completely random distribution. On examining the φ distributions shown in Figure 11a we observed that there is a broad distribution centered at zero, indicating no preferred inplane orientational order. However, the corresponding snapshots (parts b and c of Figure 11) show that the center of mass positions of the molecules arranged themselves in a nearly commensurate triangular lattice, indicating the presence of translational order in the system at low temperatures. At higher temperatures the structure is incommensurate. At h ) 10 Å, where the molecules arrange into two layers in the pore, f(cos θ) (Figure 7c) values peak around 0, indicating orientations that are parallel to the wall. As the temperature is raised the molecules predominantly retain the wall-parallel configuration. Note that although the pore height is large enough to favor wall-perpendicular orientations the proximity of the second layer decreases

the possibility of out-of-plane orientations within each layer. This effect is similar to the situation observed in bilayers16 on graphite where the presence of the second layer tends to flatten the orientation of the diatoms on the wall. The corresponding low-temperature commensurate herringbone structure in both layers is evident from snapshots and the distribution of φ shown in Figure 12. At low temperatures, n(φ) has two peaks at (45°, and each individual wall layer shows the commensurate herringbone orientation of the molecules as seen in the top view snapshot, which includes both wall layers. The top view snapshot of the entire pore shows an arrangement of the molecules that corresponds to the first two basal planes of the hexagonal close-packed structure. Increasing the temperature destroys the in-plane orientational order as seen from the snapshots at 40 K. The commensurate in-plane structures observed at h ) 10 Å are similar to the (x3 × x3) commensurate herringbone structure that is observed on a single graphite sheet.4 However, the 6-fold symmetry associated with the high-temperature hexagonal phase on the single graphite sheet is not clearly observed; at 20 K the φ distribution does show a weak shift to (30° and (90°. We also carried out a simulation at the same pore width at a loading of 44 molecules, the results of which are not shown. On comparison with the lower loading of 40 molecules, the following differences were observed. The low-temperature herringbone structure was retained; however, a broadening of the φ distributions around (45° was observed. The most significant difference was in the nature of the φ distributions where the higher temperature structures


Langmuir, Vol. 17, No. 17, 2001 5251

Figure 9. Distribution f(cos θ) and side view (x-z) snapshots at low and high temperatures for h ) 14 Å (163 molecules): (a) positive wall layer (z ) h/2), (b) middle layer, (c) side view snapshot at 18 K, and (d) side view snapshot at 40 K. In comparison with the low loading of 144 molecules (Figure 8), the low-temperature distribution of the wall layer indicates a smaller number of wallperpendicular orientations.

Figure 10. h ) 7 Å: (a) distribution n(φ), (b) top view snapshot at 14.7 K, and (c) top view snapshot at 40 K. The distribution and snapshots reveal the low-temperature biaxially incommensurate herringbone structure. At high temperatures translational order is retained.

Figure 11. h ) 8 Å: (a) distribution n(φ), (b) top view snapshot at 14.7 K, and (c) top view snapshot at 40 K. The snapshots reveal that the center of mass positions projected onto the graphite surface possess hexagonal structure.

(at higher loadings) clearly show the formation of the hexagonal phase, i.e., φ ) (30° and (90°. This phase is

formed around 25 K and persists until the system is orientationally disordered around 40 K. In contrast, a

5252


Figure 12. h ) 10 Å: (a) distribution n(φ), (b) top view snapshot at 14.7 K, and (c) top view snapshot at 40 K. The shaded molecules represent those adjacent to the positive wall layer (z ) +h/2), and the open molecules represent those adjacent to the negative wall (z ) -h/2). The commensurate herringbone structure similar to that observed on a single graphite sheet is formed on each wall of the pore.

comparison of the f(cos θ) distributions at both loadings showed them to be identical. The cos θ distributions at h ) 12 Å (Figure 7d), although qualitatively similar to the distributions at h ) 10 Å, have some key differences. The low-temperature distribution has a maximum near cos θ ) -1, indicating the presence of a few wall-perpendicular orientations in addition to the predominant prescence of wall-parallel configurations. At higher temperatures, a larger number of out-of-plane configurations are favored as seen in Figure 7d and the pinwheels are absent (Figure 13c). Examination of the snapshots at low temperatures (parts a and b of Figure 13) reveals the formation of a few pinwheel structures consisting of one wall-perpendicular molecule surrounded by six wall-parallel molecules. The φ distributions (Figure 13a) also reveal the 6-fold symmetry inherent in these structures, with peaks located at φ ) (60° and φ ) 0° at lower temperatures. Note that these peaks shift to φ ) (30° and (90° at 33 K due to a twisting of the glide line.16 The superimposed snapshots of both the fluid layers did not indicate any preferred relative orientation between molecules on the two layers. At higher temperatures a larger number of out-of-plane configurations are favored as seen in Figure 7d and the pinwheels are absent (Figure 13c). Both the low-temperature and high-temperature phases are incommensurate. At h ) 14 Å, where three fluid layers are formed in the pore, we studied the structures at two loadings. The f(cos θ) distributions for the lower loadings (144 molecules) for the wall layer are shown in Figure 8. At low temperatures, the f(cos θ) distribution indicates a mix of wall-parallel and wall-perpendicular orientations. This presence of wallperpendicular orientations, which are observed in the snapshots of the side view shown in Figure 8, lead to the formation of the low-temperature incommensurate pinwheel structures (Figure 15a). Unlike the situation for the h ) 12 Å pore, here the pinwheel forms the dominant

Kamakshi and Ayappa

Figure 13. h ) 12 Å: (a) distribution n(φ), (b) top view snapshot of positive wall layer at 14.7 K, and (c) top view snapshot at 40 K. A few pinwheel structures with six molecules lying flat and surrounding one central perpendicular molecule are visible at low temperatures. The wall-perpendicular molecules are shaded black with the deviation from a circle representing the degree of tilt.

Figure 14. h ) 14 Å (144 molecules): distribution n(φ) of the wall layer as a function of temperature. The 6-fold symmetry associated with the pinwheel phase is clearly seen. Note that the distance between peaks at 15 K is 60°.

low-temperature phase. The 6-fold symmetry associated with the pinwheel phase is seen in the φ distributions (Figure 14). Due to a change in alignment in the molecules at lower temperatures, the peaks in n(φ) at 15 K are phase shifted leading to a disruption in the symmetry in n(φ). This change in alignment is clearly seen in the top view snapshots (Figure 15a). At 32 K, the center of mass positions are arranged in rows parallel to the x-axis. In the side view (Figure 8), the change in alignment is reflected as a clustering of atoms along the x-axis. Symmetry in n(φ) is restored at higher temperatures. Although the distribution at 15 K is asymmetric with respect to φ ) 0°, the spacing between the peak maxima is similar to that at higher temperatures, indicating that the low-temperature phase possesses the 6-fold symmetry associated with the hexagonal phase. As the temperature is raised, although the majority of the molecules in the wall layers retain their wall-parallel orientation, the


Langmuir, Vol. 17, No. 17, 2001 5253

Figure 16. h ) 14 Å (163 molecules): distribution n(φ) of the wall layer as a function of temperature. The 6-fold symmetry associated with the hexagonal phase is clearly seen.

Figure 15. h ) 14 Å (144 molecules): top view snapshots of the wall layers at (a) 15 K and (c) 32 K and the middle layer at (b) 15 K and (d) 32 K. Pinwheel structures form the dominant low-temperature-ordered phase in both the wall and middle layers. The wall-perpendicular molecules are shaded black with the deviation from a circle representing the degree of tilt. Note that the structures transform from an incommensurate phase at low temperatures to a commensurate phase at 32 K.

number of wall-perpendicular positions decreases as seen in the lower intensity of the peak around θ ) 10° (Figure 8). The f(cos θ) distribution of the middle layer indicates a bimodal distribution at low temperatures (15 K) with peaks corresponding to 10° and 65°. However, this gets rapidly disrupted with a small increase in temperature (to 20 K), leading to a near randomization indicated by f(cos θ) ) 0.5. Snapshots in the x-y plane of the middle layer (parts b and d of Figure 15) also indicate the formation of the low-temperature incommensurate pinwheel structure, which was verified by examining the corresponding φ distribution (not shown). Although the in-plane 6-fold symmetry associated with the pinwheel phase is disrupted in all the layers as the temperature is increased, translational order is still retained via a triangular lattice. It is interesting to note in Figure 15 that although the low-temperature structures are incommensurate and orientationally ordered the high-temperature structures are commensurate and orientationally disordered. Simulations at h ) 14 Å (Figures 16 and 17) at a higher loading of 163 molecules reveal some differences in the structures. At low temperatures the wall layer has a few wall-perpendicular molecules as seen in Figure 9, which give rise to a few pinwheel structures at 15 K (top view not shown). Although the pinwheel structures are observed at the higher loading, they are fewer in number when compared with that present at lower loadings. The top view snapshot shown in Figure 17 at 20 K indicates that a small increase in temperature eliminates the wallperpendicular molecules in the wall layer. However, a few pinwheel structures are still observed in the middle layer (Figure 17b). The low-temperature phase is incommensurate retaining the 6-fold symmetry associated with the pinwheel phase. This is clearly seen in the x-y plane

Figure 17. h ) 14 Å (163 molecules): top view snapshots of the wall layers at (a) 20 K and (c) 32 K and the middle layer at (b) 20 K and (d) 32 K. The few wall-perpendicular molecules are shaded black with the deviation from a circle representing the degree of tilt. The number of wall-perpendicular molecules is smaller in the wall layer when compared with the lower loading of 144 molecules at the same temperature (not shown).

snapshots and φ distributions. The change in alignment that was observed at the lower loading of 144 molecules was not observed at the higher loading, and the φ distribution retains its symmetry. At the higher loading, in addition to the surface density increasing, the density in the middle layer has the greatest enhancement. The presence of the high-density middle layer decreases the number of out-of-plane configurations in the wall layer. A similar effect, although at 73.6 K, has been observed in simulations of multilayered nitrogen on graphite where the formation of additional layers reduces out-of-plane orientations within the wall layer.17 Before concluding this section, we comment briefly on the pinwheel structures. The pinwheel structures that have been observed previously in simulations of monolayers4,5 occur above the orientational transition temperature or have been observed during the orientational order-disorder transition.6,23 Hence, pinwheels are generally associated as defects with the disordered state. Here

5254


Figure 18. Plots of the total internal energy (eq 9) as a function of temperature: (a) h ) 7 Å, (b) h ) 8 Å, (c) h ) 10 Å, and (d) h ) 12 Å. Heat capacity computations (eq 16) are shown as insets for the h ) 10 and 12 Å pores.

we observe that the pinwheel configuration exists at the low-temperature-ordered state and can form the dominant phase as observed at h ) 14 Å. There is some evidence from elastic neutron scattering experiments on multilayers of nitrogen adsorbed on graphite24 for the presence of this four-sublattice pinwheel phase at low temperatures. The low-temperature pinwheel phase has also been observed in lattice model calculations for coverages below the monolayer regime.25 3.4. Heat Capacity and Order Parameter. To get further insight into the nature of the transitions in these pores we computed the heat capacity (from the fluctuations in the energy) as a function of temperature for the h ) 10 and 12 Å pores. The total energy (sum of fluid-fluid and fluid-wall interactions) is shown in Figure 18 with the heat capacity computations shown as insets. Heat capacity computations along with the increase in error bars around the transition (characteristic of a first-order transition) indicate that the transitions at h ) 10 and 12 Å could be of the first order with a transition occurring around 31 K for these two pores. However, these results are only preliminary, and more detailed simulations with system size variations are clearly required. We also computed the out-of-plane order parameter, OP(z), as defined in eq 15, which is plotted against the slit height for two different temperatures in Figure 19. A value of OP(z) close to 0 indicates wall-parallel orientations and OP(z) close to 1 indicates wall-perpendicular orientations. The order parameters, in addition to supporting the observations that were found from the f(cos θ) distributions, also indicate the presence of any significant changes in structure as the temperature is increased. At h ) 7 Å (Figure 19a), OP(z) < 0.18 at all temperatures, indicating that the molecules are restricted to wall-parallel configurations. The inability of the molecules to tilt themselves out of plane in these systems might influence the onset of rapid disorder that is required for a first-order transition. The order parameter for h ) 8 Å (Figure 19b) indicates the presence of out-of-plane configurations with (23) Hansen, F. Y.; Bruch, L. W. Phys. Rev. B 1995, 51, 2515. (24) Wang, S. K.; Newton, J. C.; Wang, R.; Taub, H.; Dennison, J. R.; Shechter, H. Phys. Rev. B 1989, 39, 10331. (25) Harris, A. B.; Mouritsen, O. G.; Berlinsky, A. J. Can. J. Phys. 1984, 62, 915.

Kamakshi and Ayappa

Figure 19. Plots of the order parameters OP(z) (eq 15) for (a) h ) 7 Å, (b) h ) 8 Å, (c) h ) 10 Å, and (d) h ) 12 Å.

little change in the variation of the order parameter with temperature. The most significant change in the order parameter was observed at h ) 10 Å (Figure 19c) where the low value of OP(z) indicates the presence of the wallparallel configurations within each fluid layer and the high values of OP(z) at 40K indicate that the in-plane structures have disordered. At h ) 12 Å (Figure 19d), the OP(z) indicates the presence of wall-parallel orientations at the wall and an inner region of wall-perpendicular orientations which make up the pinwheel low-temperature states. The order parameter plot broadens and decreases in value to about 0.8 at 40 K, indicating that there are a lower number of out-of-plane orientations. However, the disruption of the in-plane 6-fold symmetry associated with the pinwheel structure is not reflected in OP(z). 4. Summary and Conclusions We have carried out NVT Monte Carlo simulations of nitrogen confined in a slit-structured graphite pore. At a given pore height, a series of simulations at different temperatures are carried out. At each temperature we monitor the distribution of the out-of-plane angles, n(θ), and the distribution of the in-plane angles, n(φ), within each fluid layer. In addition to the layer density distribution, snapshots and out-of-plane order parameters reveal a complete picture of the structures that can be formed under confinement. Computations are carried out at pore heights that can accommodate one (h ) 7 and 8 Å), two (h ) 10 and 12 Å), and three fluid layers (h ) 14 Å). At the smallest pore height studied (h ) 7 Å), where a single layer of molecules is accommodated at the center of the pore, the orientations are predominantly wall-parallel forming an incommensurate herringbone structure. At slightly larger pore heights in the single-layer regime, the majority of molecules are found with their axes tilted toward the graphite sheet. When two distinct wall layers are formed in the slit pore, the distributions reveal that the orientation of the molecules adsorbed next to the wall can exist in two ordered phases at low temperatures. These phases are the herringbone phase or the triangular incommensurate phase with a few pinwheel defects at low temperatures. In all two-layer situations studied here, with the exception of the h ) 10 Å pore, where both wall layers form a commensurate herringbone structure, all other low-temperature wall structures observed were


triangular incommensurate structures. At h ) 14 Å, where three layers are formed, the low-temperature-ordered phase is the incommensurate hexagonal phase, which is predominantly pinwheel in nature at low loadings. The ability to form the pinwheel phase is dependent on some molecules having wall-perpendicular orientations. This is governed by the pore height, when two wall layers are formed, or the proximity and/or density of the adjacent fluid layers, when inner layers are present (h ) 14 Å). Our computations indicate that by varying the pore height, a wide variety of ordered structures can be formed when nitrogen is confined in a slit graphite pore. Some of the low-temperature structures like the herringbone phase are similar to those found on a single graphite sheet. However, the pinwheel and hexagonal phases, which occur

Langmuir, Vol. 17, No. 17, 2001 5255

at higher temperatures or higher coverages on a graphite sheet, are seen to form the low-temperature-ordered phase under confinement. We note that the registry of carbon atoms on the two graphite sheets could influence the structure particularly for the narrow pores where only a single layer of molecules are confined. Although our preliminary computations of the heat capacity do indicate the signatures of first-order transitions at certain pore heights, more detailed system size investigations are necessary to establish this. Acknowledgment. We thank Arun Yethiraj for useful discussions while this work was in progress. LA001348A

Structural Transitions of Nitrogen Confined in Slit Graphite Pores

Recommend Documents