Sub-nanometer characterization of activated carbon by inelastic neutron scattering

CARBON

4 9 ( 2 0 1 1 ) 1 6 6 3 –1 6 7 1

available at www.sciencedirect.com

journal homepage: www.elsevier.com/locate/carbon

Sub-nanometer characterization of activated carbon by inelastic neutron scattering Raina J. Olsen a,*, Lucyna Firlej Carlos Wexler a a b c

b,a

, Bogdan Kuchta

c,a

, Haskell Taub a, Peter Pfeifer a,

Department of Physics and Astronomy, University of Missouri, Columbia, MO 65211, USA LCVN, Universite` Montpellier 2, 34095 Montpellier, France Laboratoire Chimie Provence, Universite` Aix-Marseille 1, 13396 Marseille, France

A R T I C L E I N F O

A B S T R A C T

Article history:

Inelastic neutron scattering spectra are calculated for hydrogen molecules adsorbed on

Received 31 August 2010

activated carbon. The slit-shaped pore model is used to calculate the adsorption potentials

Accepted 17 December 2010

of a hydrogen molecule in pores of variable width. The motion of the hydrogen molecules is

Available online 24 December 2010

quantized perpendicular to the plane of the pore and both rotational and vibrational transition energies are found. The perturbation of adjacent hydrogen molecules on the transition energies is discussed. Form factors and Debye–Waller factors are calculated for momentum transfer parallel and perpendicular to the pore; this anisotropy is particularly pronounced in pores where there is nearly enough room for two adsorbed layers. A spectrum composed of a uniform distribution of pore sizes agrees only qualitatively with experimental results for activated carbon. Reasons for this disparity are discussed, including the possibility that transitions in the translational motion parallel to the adsorption plane contribute significantly to the spectra. In addition, the positioning of the center of gravity of the first rotational transition is discussed, with several factors contributing to shift it from the 14.7 meV of the gas phase. Our results suggest that inelastic neutron scattering can be valuable as a complementary sub-nanometer pore characterization technique. Ó 2011 Elsevier Ltd. All rights reserved.

1.

Introduction

Hydrogen is desirable as an energy storage medium not only because it is clean burning and efficient, but because it can store nearly three times as much energy as an equal mass of gasoline. However, its small mass also means that it occupies several orders of magnitude more volume at ambient conditions than an equal mass of gasoline, and has generally required low temperatures and/or high pressures to store efficiently. If hydrogen is to be a viable energy storage medium, particularly for mobile applications, techniques must be

found to efficiently store hydrogen at reasonable pressures and temperatures. Storage by adsorption in porous carbon materials such as activated carbon (AC), amorphous carbon nanotubes, singlewall carbon nanotubes (SWCNT) and porous carbon has been previously studied [1]. The Alliance for Collaborative Research in Alternative Fuel Technology (ALL-CRAFT)1 at the University of Missouri has recently produced competitive activated carbons with surface areas of up to 3100 m2/g and storage capacities of up to 100 g H2/kg C at temperatures of 77 K and pressures of 90 bar [2]. Traditionally AC has been modeled

* Corresponding author. E-mail addresses: [email protected], [email protected] (R.J. Olsen). 1 http://all-craft.missouri.edu/ 0008-6223/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.carbon.2010.12.051

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as being composed of slit-shaped pores, composed of two flat parallel graphene sheets, both for adsorption simulations [3,4] and for obtaining pore size distributions from nitrogen adsorption data [5]. Simulations [4,6–8] show that slit-shaped ˚ have the widest range of pores with widths between 6–10 A properties for hydrogen adsorption, as the depth and width of the potential well varies significantly. As the width in˚ , the pore begins to behave simply as creases beyond 10 A two independent sheets. Commonly used methods of pore characterization, such as nitrogen adsorption [5], are unable to probe the smallest pores because of the larger size of N2 as compared to H2 . Conversely, small-angle X-ray scattering [9] characterizes all of the pore volume, including pores which H2 is unable to access. To characterize samples as accurately as possible, it is desirable to use H2 itself as the probe. In addition, both of these commonly used techniques rely on a large number of approximations to interpret the results. A more robust understanding of a sample is gained by comparing results from several different techniques. Incoherent inelastic neutron scattering (IINS) has been used as just such a complementary characterization technique on samples of H2 adsorbed on SWCNT [10–13], AC [14], other carbon materials [15,16], and various other adsorbents [17–25]. Because of the high incoherent cross section of H2 , IINS is able to detect the hydrogen molecules as they occupy the variety of adsorption sites present in each sample, offering information about energy levels and quantum states occupied. This information is a helpful complement to techniques which probe the carbon itself. All previous work using this method has been focused on an analysis of the rotational transitions, which give information about the anisotropy of the H2 adsorption potential. The rotational energy levels are given by 2

EJ ¼ JðJ þ 1Þ

 h ; 2la2

ð1Þ

where J is the molecular angular momentum quantum number of the molecule, l ¼ 1:008u is the reduced mass of the ˚ is the distance between the two molecule, and a = 0.741 A atomic nuclei. Experiments are typically done at temperatures near 14 K, the triple point of bulk para-hydrogen, so that

the majority of the hydrogen is adsorbed and the high-density 2D liquid phase constrains movement of a hydrogen molecule along the adsorption plane. The low temperature and the use of para-hydrogen ðJ ¼ 0; 2; 4 . . .Þ instead of ortho-hydrogen ðJ ¼ 1; 3; 5 . . .Þ ensures that all molecules are initially in the ground J ¼ 0 rotational state. Transitions to even J states are proportional to the coherent cross section of hydrogen, which is quite small. Transitions to odd J states are proportional to the incoherent cross section of hydrogen and are easily visible. In the J ¼ 0 state, denoted as the a ¼ s state, the molecule axis is equally likely to be in any direction. In the J ¼ 1; Jz ¼ 0 state, denoted as the a ¼ pz state, the molecular axis tends to be aligned perpendicular to the graphene plane. In the J ¼ 1; Jz ¼ 1 states, denoted as the a ¼ px ; py states, the molecular axis tends to be aligned parallel to the graphene plane. In the gas phase, the first rotational transition occurs at DEs!p ¼ 14:7 meV and is triply degenerate. For adsorbed H2 , the rotational anisotropy of the adsorption potential partially removes this degeneracy, and the transitions now occur at a 2/1 intensity ratio at energies which are highly dependent on the slit characteristics. Roto-vibrational transitions, where the rotational state as well as the vibrational state changes, are also possible. Up until now, most of these IINS experiments on adsorbed hydrogen have been done with an adsorbent which is more ordered, such as SWCNT. A comparison of a typical spectra at near monolayer coverage, Fig. 1, for SWCNT [11] and AC [14] reveals a possible reason for the dearth of experimental results for AC. The spectrum from SWCNT, after the subtraction of the free recoil contribution, shows a sharp peak at about 14.5 meV, the first rotational transition for which the spectrum can be easily analyzed, as well as a much smaller, but still well-defined peak at around 30 meV, which is likely the first roto-vibrational transition (but could also be due to multiple scattering). In contrast, the spectrum from AC shows a sharp peak at 14.6 meV as well as a broad complex peak extending from 10 to 60 meV. This complex peak, which is likely due to the heterogeneity of activated carbon, makes the analysis of the spectrum more difficult. However, it might also offer a complementary characterization technique.

Fig. 1 – IINS spectra measured at 13 K from a nearly complete monolayer of adsorbed p  H2 on SWCNT (left, after Ref. [11]) and AC (right, after Ref. [14]). The solid lines are a fit of the Young and Koppel [26] model to experimental data (85–600 meV). The inset on the right shows the spectrum after subtraction of the solid line. The fits and subtraction were done by Georgiev et al. [11,14]. Both spectra show a peak around 14.7 meV, which can be attributed to the first rotational transition. The SWCNT spectrum also has a peak at around 30 meV, which is likely the first roto-vibrational transition. The AC spectrum has rotovibrational transitions at a variety of energies (10–60 meV).

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Previous work has calculated transition energies and/or spectra using a cylindrical pore model [27–29]. This work calculates transition energies and spectra using the slit-shaped pore model. Using this model, in Section 2 we calculate the energy levels and wavefunctions of an H2 molecule adsorbed in pores of variable width. The energy levels are used to calculate the transition energies between different states. The perturbation of intermolecular interactions is also considered, both its effect on the transition energies and its tendency to force bilayer adsorption in larger slits. In Section 3, the wavefunctions are used to calculate the double differential scattering cross section. In contrast to previous work, the scattering cross section is not treated as isotropic, but form factors and Debye–Waller factors are calculated for momentum transfer perpendicular and parallel to the adsorption plane. In Section 4, these results are combined to calculate IINS spectra for hydrogen adsorbed on graphite and activated carbon composed of a uniform distribution of slit-shaped pores, where the direction of the momentum transfer has been isotropically averaged. The calculated spectra are compared with previously collected experimental data. A shift in the center of gravity of the first rotational transition is observed in the calculated spectra, similar to the shift observed experimentally. Several different factors contribute to this shift. The usefulness of the slit-shaped pore model is discussed.

2. Energy adsorbed H2

levels

and

wavefunctions

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of

We wish to calculate the energy levels and wavefunctions wa;n ðx; yÞ of an adsorbed H2 molecule in a quantum state ða; nÞ, where n is the quantum number for motion perpendicular to the adsorption plane. To do this, it is necessary to calculate the potential between an H2 molecule and a graphene sheet for each position ~ r of the molecular center of mass and orientation h; / of the molecular axis relative to the graphene plane. We fully quantize the rotational molecular motion, and the vibrational motion perpendicular to the adsorption plane (in the z-direction). The rotational and vibrational degrees of r; h; /Þ ¼ freedom are approximated as independent with Wa;n ð~ rÞ. Lu et al. [27] similarly separate rotational and Ua ðh; /Þwn;a ð~ translational coordinates. The molecule is assumed to be in its molecular vibrational ground state. Lennard–Jones (LJ) potentials are commonly used for inter-atomic potentials.    r 12 r6  ; ð2Þ VLJ ðrÞ ¼ 4 r r where  is the depth of the potential and r is the distance between atoms at which the potential is equal to zero. Xu et al. [30] have proposed a form for the potential between a classical hydrogen molecule and carbon atom that takes into account the fact that most of the electronic density is centered between the two protons.     ~ ~ a a ^ Þ ¼ VLJ j~ r; n r þ j þ VLJ j~ r  j þ vVLJ ðj~ rjÞ; ð3Þ VH2 C ð~ 2 2 ^ Þ give the position and orientation of the hydrogen where ð~ r; n molecule relative to the carbon atom and ~ a is the vector between the two hydrogen atom centers. Xu et al. determined ˚ , and v = 7.5 (v is a dimensionless  = 0.371 meV, r = 2.95 A

parameter that gives the proportion of the H2 -C interaction that is centered between the two hydrogen atoms) by matching experimental and theoretical values for the roto-vibrational transitions of an H2 molecule inside of a C60 fullerene [30]. Using the Xu H2 -C potential, the matrix element of the potential between a graphene sheet and an H2 molecule using the rotational states as a basis set can be calculated as H2 g ð~ rÞ ¼ hUa ðh; /Þj Va;a0

N X

^ i ÞjUa0 ðh; /Þi; VH2 C ð~ ri ; n

ð4Þ

i¼1

where N is the number of carbon atoms in the graphene sheet and ð~ ri ; hi ; /i Þ give the center-of-mass position and orientation of the hydrogen molecule relative to the ith carbon atom. The rotational wavefunctions are given by the spherical harmonics, where the px ; py states have been formed from linear combinations of the J ¼ 1; Jz ¼ 1 states. 1 Us ðh; /Þ ¼ pffiffiffiffiffiffi ; 4p 3 Upz ðh; /Þ ¼ pffiffiffiffiffiffi cos h; 4p 3 Upx ðh; /Þ ¼ pffiffiffiffiffiffi sin h cos /; 4p 3 Upy ðh; /Þ ¼ pffiffiffiffiffiffi sin h sin /: 4p

ð5Þ

The off-diagonal matrix elements vanish due to symmetry except between the px and py states, where the amount of coupling varies depending on the position above the graphene plane. This means that the molecular axis of a H2 molcule in a p rotational state does indeed tend to align in a direction perpendicular or parallel to the graphene plane. In the center of a hexagonal cell of graphite, the breaking of the rotational symmetry due to the external potential results in different energies for p states aligned parallel ðpx ; py Þ or perpendicular ðpz Þ to the adsorption plane. The potential was also calculated at several different (x,y) locations relative to the graphene plane. At these sites, the px and py states do not have the same potentials. However, the differences were small enough that all further calculations use the potential above the center of a hexagonal cell. We now consider a hydrogen molecule in a particular rotational state adsorbed inside of a slit-shaped pore composed of two graphene sheets separated by a width w. The potential can be calculated by simply adding the potential from each of the two graphene sheets. Fig. 2 shows the potentials for ˚ and w = 8 A ˚ , where each rotational state for w = 6 A 14.7 meV has been added to the potentials for molecules in p states to account for their rotational energy [Eq. (1)]. Notice that the potentials for the p states split, with the px ; py states having a deeper potential than the pz state. Because the molecular axis of a molecule in the px ; py states tends to be parallel to the graphene plane, it can approach the graphene plane closer resulting in an enhanced interaction. We can also see from these potentials that coupling between the rotational and vibrational degrees of freedom should not be significant. If, for instance, the pz potential was above the px potential at some distances and below the px potential at other distances, coupling would be an important consideration. Referring to our Fig. 2, one can see that for these poten-

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Potentials in 8 A slit

Potentials in 6 A slit 0

0 p z

px /py −40 −60

potential (meV)

potential (meV)

s −20

−20 −40 s −60

pz px /py

−80

−80 −1

0

−1

1

0

1

distance from pore center (A)

distance from pore center (A)

˚ (left) and 8 A ˚ (right) slit. Fig. 2 – Calculated potentials and energy levels for the s, pz , and px ; py rotational states in a 6 A tials this is not the case, and thus coupling is not a significant issue, supporting our separation of these degrees of freedom. rÞ, was then The vibrational part of the wavefunction, wn;a ð~ calculated for each slit width by numerically solving the one dimensional time-independent Schro¨dinger equation for the corresponding potential. Fig. 2 also shows the first few energy ˚ slit, these are levels for each rotational state. In the w = 6 A ˚ slit, the first two states are widely spaced. In the w = 8 A nearly degenerate. Based on these calculations, for a flat graphene sheet one would expect rotational transitions at energies DEs;1!px ;py ;1 ¼ 14:34 meV and DEs;1!pz ;1 ¼ 15:37 meV with an intensity ratio of 2:1. However, we have not yet taken into account the effects of the adjacent hydrogen molecules. For the liquid densities at which IINS experiments are performed, the intermolecular interactions are a non-negligible perturbation. ˚ for Wang et al. [31] give values of  ¼ 2:94 meV and r ¼ 2:96 A the LJ H2  H2 interaction. In order to calculate the potential between two H2 molecules in different rotational states, it is again necessary to have an H2  H2 potential that depends on the orientation of the H2 molecules. The H-H interaction was calculated by fitting the Wang potential to a potential between two hydrogen molecules composed of two hydrogen atoms in the s state, given by H H02

Vj;j02

ð~ rÞ ¼

2 X

hUa ðhi ; /i ÞUa0 ðhk ; /k Þj  VHH ð~ ri

i;k¼1

~ rk ; hi ; hk ; /i ; /k Þ jUa ðhi ; /i ÞUa0 ðhk ; /k Þi;

ð6Þ

where ð~ ri ; hi ; /i Þ give the position and orientation of the ith atom in the first molecule and ð~ rk ; hk ; /k Þ give the position and orientation of the kth atom in the second molecule. The H-H potential is also approximated by a LJ potential. The best-fit parameters were calculated to be  ¼ 1:0 meV and ˚ , using v ¼ 0. A more accurate determination of r = 2.73 A these parameters would require experimental data which probes the H2  H2 interaction. If we assume that all of the neighbors of a particular hydrogen molecule are in the s state, we can also calculate the average potential between a molecule in the state a and its six neighbors, assuming hexagonal close packing with dH2 H2 being the average in-plane distance between neighbors. Similar to the H2 -graphene potential, the potentials for the p states split, except that now the px ; py states have the shallower potential and the pz state has the deeper potential than

the s state. The pz state has the molecular axis tending to be perpendicular to the adsorption plane, which means it is parallel to the other hydrogen molecules. This means it can approach adjacent molecules more closely and feels a deeper ˚ , the average distance between potential. Using dH2 H2 ¼ 3:38 A adjacent hydrogen molecules found with high-density molecular dynamics simulations, we can now correct the rotational transition energies for the perturbation of adjacent H2 molecules and find them at energies DEs!px ;py ¼1 ¼ 14:51 meV and DEs!pz ¼ 14:92 meV at a 2:1 intensity ratio. While the numerical results of this calculation are highly dependent on the H-H potential used and the positioning of adjacent H2 , it is interesting to note that this perturbation has the ability to shift the mean of the first rotational transition for adsorbed H2 from the 14.7 meV at which it is observed for gaseous hydrogen. This issue will be discussed further in the results section. The same correction can be applied for all slit widths, and now the transition energies for s; 1 ! a0 ; n0 can be plotted for each slit width, as shown in Fig. 3. At a width of about ˚ , the n0 ¼ 1 and n0 ¼ 2 energy levels differ by less than 8.4 A 0.8%, so close that we consider them degenerate. For slits above this size, the intermolecular interactions will result in pffiffiffi ‘‘left-adsorbed’’ ½wl ¼ ðw1 þ w2 Þ= 2 or ‘‘right-adsorbed’’ ½wl ¼ pffiffiffi ðw1  w2 Þ= 2 wavefunctions being lower in energy; essentially molecules are adsorbed in two layers. Because the intermolecular interactions are added only as a perturbation, this cutoff cannot be calculated exactly. However, these results match the findings of classical Monte Carlo simulations ˚ [4]. Below which show two complete layers form at w = 8.5 A this size, the transition energies are very dependent on slit width.

3.

Scattering cross section

˚ -1, the double differFor momentum transfers up to q = 4–5 A ential scattering cross-section is given by [10] 2

kf d r 0 0 ð~ ¼ F qÞe2W dðE  DEÞ; dXdE ki j;n!j ;n

ð7Þ

where ki and kf give the initial and final neutron wave vector, qÞ is the molecular form factor, and 2W is the Debye– Fa;n!a0 ;n0 ð~ Waller factor. Given our separation of the rotational and vibrational parts of the wavefunction, the form factor, which is the overlap integral between the momentum transfer and

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4 9 ( 20 1 1 ) 1 6 6 3–16 7 1

60

transition energy (meV)

55 50

s−>p

s−>p ,p , x y 1−>3

z

,

1−>4

45 40 35 30

s−>p ,p

25

1−>2

x

20

s−>p , z 1−>2

s−>p ,p

,

1−>4

y

x y

s−>p , z 1−>3

s−>p ,p ,1−>1 x

15

,

y

s−>p ,1−>1 z

10

6

6.5

7

7.5

8

8.5

9

9.5

slit width (A) Fig. 3 – Neutron energy losses for s; 1 ! a0 ; n0 transitions of adsorbed H2 as a function of slit-width. The solid lines are the s ! px ; py transitions, while the dashed lines are the s ! pz transitions. The bottom pair of curves represent the pure rotational transitions, while the other lines are rotovibrational transitions. The energies are quite variable up to ˚ . Above this range, the energy levels slit widths of 9 A become degenerate and the slits behave more as two independent graphene sheets.

the inital and final wavefunctions, can be further broken up into rotational and vibrational parts: 2

qÞ ¼ binc jhwi jei~q~r jwf ij2 ¼ Fa!a0 ð~ qÞFn!n0 ð~ qÞ: Fa;n!a0 ;n0 ð~

ð8Þ

If Jq is defined as the component of the angular momentum aligned with the momentum transfer, Young and Koppel [26] give the rotational part of the form factor for s ! p transitions of hydrogen in the gas phase as 2

Fs!J0 ¼1;J0q ðqÞ ¼ binc jhws ðh; /Þj sin

~ q ~ a jwJ0 ¼1;J0q ðh; /Þij2 ; 2

ð9Þ

where ~ a is the vector between the two atomic nuclei. Yildirim and Harris [28] give a similar expression for the s ! p scattering cross section following the formalism of Elliott and Hartmann [32]. If the wavefunctions are given by the spherical harmonics, the form factors for Jq ¼ 0 and Jq ¼ 1 can be calculated as pffiffiffi Z 2   qa cos h 2 3 cos h dX sin Fs!J0 ¼1;J0q ¼0 ðqÞ ¼ binc 4p 2 2 2 qa ð10Þ ¼ 12binc j1 ð Þ; 2 pffiffiffi Z 2   qa cos h 2 3 Fs!J0 ¼1;J0q ¼1 ðqÞ ¼ binc sin h ei/ dX ¼ 0; ð11Þ sin 4p 2

where j1 is the first order spherical Bessel function. Based on these results, one can see that the neutron tends to push the H2 molecule into a rotational wavefunction where the molecule axis is aligned with the momentum transfer. Classically, this can be understood by imagining the H2 molecule as a dumbell. If the neutron hits the dumbell perpendicular to its axis, the dumbell will gain rotational energy and tend to rotate to align with the transferred momentum. However, if the neutron hits the dumbell parallel to its axis, it will not

1667

be able to transfer rotational energy to the bar. These new theoretical results could be confirmed by taking neutron scattering spectra from hydrogen adsorbed on graphite with a given orientation relative to the momentum transfer at a high enough energy transfer to see the roto-vibrational transitions. If the momentum transfer is perpendicular to the adsorption plane, vibrational transitions of the adsorption potential will be observed. If the momentum transfer is parallel to the adsorption plane, vibrational transitions of the adsorption potential will not be observed. However, it is likely that transitions of the translational motion parallel to the adsorption plane will still be observed; but it is not known what form they will take. If our treatment of classical motion along the plane is an accurate approximation, these transitions may be well approximated by free recoil scattering. Q-dependent experimental data is needed to address this issue. If free recoil scattering is not adequate, a more sophisticated theoretical treatment would need to be developed. When hydrogen is adsorbed, its rotational wavefunctions are aligned in the z-direction (perpendicular to the graphite plane) or parallel to the xy plane. If the momentum transfer is at an angle # to the z-axis, the form factors for the pz and px ; py states are given by 2 2 qa Fs!pz ðqÞ ¼ 12binc j1 ð Þ cos2 #: 2

ð12Þ

2 2 2 qa Fs!px ;py ðqÞ ¼ 12binc j1 ð Þ sin #: 2

ð13Þ

The vibrational part of the form factor must be calculated numerically using the wavefunctions calculated in the previous section qÞ ¼ jhwn;a;w ðzÞjeizq cos # jwn0 ;a0 ;w ðzÞij2 ; Fn!n0 ð~

ð14Þ

where w is the width of the slit. The complete form factors as ˚ and 8 A ˚ slit are calculated using Eqs. 8, 12, 13, and 14 for a 6 A shown in Fig. 4 for the momentum transfer parallel to the zaxis. The Debye–Waller factor is given by 1 2W ¼ q2 hU2 i ¼ ðq2z hz2 i þ q2xy hx2 þ y2 iÞ: 3

ð15Þ

The value of hz2 i ¼ hwn;w ðzÞjðz  hziÞ2 jwn;w ðzÞi gives the variance of position perpendicular to the adsorption plane and is calculated based on the wavefunctions for each slit width. For 2 ˚ . Fig. 5 a single sheet, this value was calculated to be 0:066 A 2 shows hz i as a function of slit width. The discontinuity at ˚ is due to the double-layer adsorption that occurs w = 8.4 A above this size, which was discussed in the previous section. For pores below this width, the variance increases significantly with pore size. The value of 12 hx2 þ y2 i describes the variance of position parallel to the adsorption plane and we approximate it as independent of slit width. Nielsen and Ellenson [33,34] did inelastic scattering from para-hydrogen adsorbed on Grafoil oriented parallel with the scattering plane and found this va˚ 2 at high ˚ 2 at low coverage to 0.094 A lue to vary from 0.14 A coverage. We will use the latter value, as the experimental neutron scattering data for hydrogen adsorbed on activated carbon shown in Fig. 1 is shown for a nearly complete monolayer.

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relative intensity

s−>pz,1−>1 s−>pz,1−>2 s−>pz,1−>3

0

2

4

6

momentum transfer (1/A) relative intensity

s−>pz,1−>1 s−>pz,1−>2 s−>pz,1−>3

0

2

4

6

momentum transfer (1/A) Fig. 4 – Form factors for the first three roto-vibrational ˚ slit (top) and a 8 A ˚ slit (bottom) where transitions for a 6 A the momentum transfer is parallel to the z-axis. The oscillatory nature of the latter set of form factors reflects the fact that the wavefunctions are bimodal.

1

0.8

2



0.6

0.4

0.2

0

6

6.5

7

7.5

8

8.5

9

9.5

slit width (A) Fig. 5 – The variance of position perpendicular to the adsorption plane as a function of slit width. The ˚ is due to bilayer adsorption, which discontinuity at w = 8.4 A localizes the molecules motion perpendicular to the plane more strongly and results in a smaller variance.

4.

Results

Using the transition energies and scattering cross section calculated in the previous sections, spectra were calculated for ˚ . The meadifferent slit-shaped pores for the range w = 7–9.5 A surements shown in Fig. 1 were performed on the invertedgeometry crystal-analyzer instrument TOSCA [35] at ISIS. The instrument has a fixed final energy of 3.98 meV and collects scattered neutrons at angles of 42.5° for forward-scattering and 137.5° for back-scattering. At a neutron energy transfer of h  w ¼ 14:7 meV, where the first rotational transition for free H2 molecules occurs, the momentum transfer is

˚ -1 for back-scattering. For comparison purposes, qb = 4.1 A spectra were calculated to match these back-scattering conditions. Fig. 6 shows calculated spectra for hydrogen adsorbed in ˚ , where the relaslit shaped pores with widths w = 6, 7, 8, 9 A ~ tive orientation of q and the graphene plane has been taken to be isotropic by integrating Eqs. (12) and (13) over sin(#) = d#. The center of gravity (COG) for the pure rotational transitions is shown for each. The COG is shifted from the 14.7 meV at which it is observed for gaseous hydrogen. A similar shift of 0.05–0.2 meV downwards has been observed experimentally [10,11,14,12,13,33] for H2 adsorbed on SWCNT, AC, and Grafoil. In our calculations, this shift is attributable to several factors. Firstly, the adsorption potentials for the different rotational states have different zero-point energies. This effect is heavily dependent on pore size, but not on the magnitude of the momentum transfer. Secondly, the effects of adjacent hydrogen molecules act to further shift the transition energies of different rotational states. This effect is likely independent of pore size and the magnitude of the momentum transfer, but should be dependent on the coverage. Finally, the Debye–Waller factor for momentum transfer perpendicular to the adsorption plane is also different for different sizes of slits, and generally acts to reduce the intensity of the s ! pz transitions, lowering the COG. This effect strongly depends the on pore size and the magnitude of the momentum transfer. Experimentally, taking data which is q-dependent and coverage dependent from a variety of samples may help to differentiate between these effects. ˚ slit (Fig. 6, top), the rotational transitions For the w = 6 A are strongly split and the vibrational transitions are at a high energy transfer. This is due to the very narrow adsorption po˚ slit (Fig. 6, tential that occurs at this pore size. For the w = 7 A second from top), the s ! pz transition now occurs at a lower energy than the s ! px ; py transitions. This shift is caused by effects of the adjacent hydrogen molecules. In addition, the wider pore results in a larger Debye–Waller factor, reducing the magnitude of the s ! pz transitions. The net effect is an ˚ slit, which increase in the COG for this size slit. For the w = 8 A is nearly large enough for bilayer adsorption to occur, (Fig. 6, second from bottom), the Debye–Waller factor in z direction is so large that the s ! pz transitions are barely visible. In addition, the energy levels are so close that the first rotovibrational transition is now at nearly the same energy as the pure rotational transition, resulting in the appearance of ˚ slit (Fig. 6, bottom), a split rotational transition. For the w = 9 A bilayer adsorption now occurs. There is no observable splitting of the rotational transition, and the COG now occurs at 14.64 meV, comparable to the 14.65 meV at which Georgiev [14] finds a peaks for hydrogen adsorbed in activated carbon. The vibrational transitions occur at an average energy of 27 meV, comparable to the 27.5 meV at which Georgiev et al. [14] also finds a peak. It is likely, then, that the sample of activated carbon used for their experiment contained a significant proportion of pores that are wide enough for two full layers of hydrogen to adsorb. Fig. 7 shows the calculated spectra for a sample of activated carbon that has a uniform pore size distribution for ˚ . This spectrum again shows a large narrow w = 5.75–9.5 A peak for the ðs; 1Þ ! ðp; 1Þ transition at 14.6 meV as well as a

CARBON

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w=6 A COG=14.70

10

15

20

25

30

35

40

45

50

35

40

45

50

w=7 A

relative intensity

COG=14.73

10

15

20

25

30

w=8 A total s−>p ,p

COG=14.68

10

15

x y s−>p z

20

25

30

35

40

45

50

w=9 A COG=14.64

10

20

30

40

50

energy transfer (meV)

relative intensity

relative intensity, 30x

Fig. 6 – Calculated neutron scattering spectrum at monolayer coverage for H2 adsorbed in slit-shaped pores with widths w = 6, ˚ . The spectra, shown in solid black, is split up into s; 1 ! pz ; n0 and s; 1 ! px ; py ; n0 transitions. The center of gravity 7, 8, 9 A (COG) for the pure rotational transitions is shown on each.

10 10

15

20

20 25

30 30

40 35

50 40

60 45

50

energy (meV)

Fig. 7 – Calculated neutron scattering spectrum at monolayer coverage for H2 adsorbed on activated carbon composed of a uniform distribution of slit-shaped pores ˚ . The inset shows the same with widths w = 5.75–9.5 A spectra at a magnification of 30.

broad band from 14.7 to 60 meV that covers the ðs; 1Þ ! ðp; n0 –1Þ transitions. Comparing with the experimental data for activated carbon, shown on the right of Fig. 1, the shape appears to be similar. Differences in shape could be explained by a non-uniform pore size distribution. However, the ratio of the intensity of the second broad peak (the roto-vibrational transitions) to the intensity of the first narrow peak (the pure rotational transition) seems to be considerably larger in the experimental data than in the calculated spectra. There are some possible reasons for this disparity. Firstly, this calculation has neglected the possibility of transitions of the vibrational motion parallel to the adsorption plane. However, these transitions would likely be observed in the experimental data. There are two possibilities for these transitions. Either the approximation of classical motion parallel to the adsorption plane is accurate and free recoil scattering would be observed, or the hydrogen molecules could be localized along the adsorption plane, not just by the graphene potential, but by the effects of adjacent hydrogen molecules.

1670

CARBON

4 9 ( 2 0 1 1 ) 1 6 6 3 –1 6 7 1

The latter type of transitions would vary heavily with coverage, as adjacent molecules are squeezed more closely to one another. In addition, calculations of these transitions would be more complex, as the effects of adjacent molecules can not be treated simply as a first-order perturbation. At present, available experimental data does not support the development of a more sophisticated treatment, however obtaining coverage-dependent spectra from a variety of samples would help to address this issue. Another reason for this disparity could be the approximation that the pores are infinite and slit-shaped, i.e., they are formed of parallel planar graphene structures of relatively large dimensions. Transmission electron microscopy images of activated carbon [36] show evidence of non-hexagonal rings in the carbon structure, which results in substantial curvature of the surface in all directions. In cases where the pores are drastically different than the slit-shaped pores considered in our calculations, the rotational and vibrational degrees of freedom can no longer be treated as independent. Because the form factor depends heavily on the wavefunctions, the transition probabilities will change significantly as the structure changes. On the other hand, the energies at which the transitions occur are less dependent on the details of the structure and seem to be in the same range for the calculated and experimental spectra. Another issue that could affect the form factors significantly is the form of the potentials used. We have used the Xu H2 -C potential [30] because it allows for calculation of the potential as a function of the orientation of the hydrogen molecule. However, this potential assumes the form of a quasi Lennard–Jones potential which may result in an incorrect shape, again an issue that would affect the form factors more significantly than the transition energies. Obtaining q-dependent measurements may help to determine the accuracy of the form factors calculated here.

5.

Summary

In conclusion, we have calculated both the vibrational and rotational transition energies for slit-shaped pores having a ˚, range of widths. In pores with widths on the order of 8 A the vibrational and rotational transition energies are very close to one another, calling into question analysis techniques which consider only the rotational transitions. In contrast to previous work that has treated the problem isotropically we have calculated form factors and Debye–Waller factors as a function of the relative angle between the adsorption plane and the momentum transfer. The resulting spectra are most strongly influenced by this anisotropy when the pore is slightly too small for bilayer adsorption. We have also found a shift in the center of gravity of the rotational transitions, which has also been observed experimentally. This shift is dependent on the pore size, and results from a combination of three factors: the different shape and thus zero-point energies of the different rotational states, the effects of adjacent adsorbed hydrogen molecules , and the anisotropy of the Debye–Waller factor. A spectrum was calculated for a uniform distribution of pores and compared with previously published experimental data. Only qualita-

tive agreement was found, suggesting the theory is not yet complete. Likely explanations for the observed disparity include the neglect of transitions in the translational motion parallel to the adsorption plane, a significantly different pore shape, or incorrect potentials. It seems that even at the current level of theoretical understanding, IINS may provide useful information about important length scales for sub-nm pore characterization; experimental work is planned that will compare samples with significantly different adsorption characteristics, providing the opportunity to directly learn more about the adsorption environment of the hydrogen molecules in these samples based on the difference between their spectra. Future theoretical work will also calculate the double differential cross section for different pore models, such as the fullerene-like fragment model proposed by Terzyk et al. [37,38].

Acknowledgments This material is based upon work supported in part by the department of Energy under Award No. DE-FG02-07ER46411 and the US Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC0206CH11357. Work by H.T. was partially supported by the US National Science Foundation under Grant No. DMR-0705974.

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