Grand Canonical Monte Carlo Simulation Study of Hydrogen Storage in Ordered Mesoporous Carbons at...

Reprinted from

Adsorption Science & Technology2006 Volume 24 Number 5

Multi-Science Publishing Co. Ltd.5 Wates Way, Brentwood, Essex CM15 9TB, United Kingdom

Grand Canonical Monte Carlo SimulationStudy of Hydrogen Storage in Ordered

Mesoporous Carbons at 303 K

Piotr Kowalczyk, Mietek Jaroniec, Lech Solarz, Artur P. Terzyk and Piotr A. Gauden

AST 24-5-Kowalczyk 18/1/07 8:55 am

411

*Author to whom all correspondence should be addressed. E-mail: [email protected].

Grand Canonical Monte Carlo Simulation Study of Hydrogen Storagein Ordered Mesoporous Carbons at 303 K

Piotr Kowalczyk1, Mietek Jaroniec2, Lech Solarz3, Artur P. Terzyk4* and Piotr A. Gauden4

(1) Department III, Institute of Physical Chemistry, Polish Academy of Science, Kasprzaka Street 44/52, 01-224 Warsaw,

Poland. (2) Department of Chemistry, Kent State University, Kent, OH 44242, U.S.A. (3) Department of Technical Physics,

Military Technical Academy, Kaliski Street 2, 00-908 Warsaw, Poland. (4) Physicochemistry of Carbon Materials

Research Group, Faculty of Chemistry, Nicolaus Copernicus University, 7 Gagarina Str., 87-100 Torun, Poland.

(Received 13 September 2006; revised form accepted 4 October 2006)

ABSTRACT: Results of Grand Canonical Monte Carlo (GCMC) simulations ofhydrogen storage at 303 K in ordered mesoporous carbons (OMCs) which areinverse replicas of cubic Im3

—m silica are presented. Of the ones investigated

here, the highest gravimetric total storage of hydrogen (ca. 2 wt% at P = 22 MPaand 303 K) was observed for the OMC replica of the cubic Im3

—m material which

was characterized by the largest lattice parameter and internal pore radius of thecarbon sphere (a = 9.98 nm and Rint ≈ 3.0 nm). However, the correspondingvolumetric density of hydrogen did not exceed 17 kg H2/m

3. The highest excessstorage was observed for a threshold value given approximately by a ≈ 7.98 nmand Rint ≈ 2.4 nm. Above this value, the excess storage and volumetric densityof hydrogen did not change.

The adsorption was dominated by the bulk storage, with the adsorbent serving asa tank whose storage properties were governed by the available space for hydrogenmolecules (bulk storage) and not by physical adsorption (i.e. densification of thesupercritical fluid on the walls of the porous material). The calculated average valuefor hydrogen adsorption in internal spaces was twice as high as in the internalspaces of the spherical cavities, i.e. in the lattice parameter range of the Im3

—m OMC

replicas that were studied experimentally at 303 K. Our calculations indicate that the calculated maximum gravimetric weight per

cent and volumetric density of hydrogen lay well below the target of 2010 for6.0 wt% and 45 kg H2/m

3 indicated by the United States Department of Energy.The storage of hydrogen at 303 K and P < 22 MPa in the OMC replicas of thecubic Im3

–m silica was effective against compression of the hydrogen fluid.

However, it was not sufficiently effective to justify the application of thesematerials for the development of fuel cells for the automobile industry.

INTRODUCTION

Various forms of carbon, including graphite and diamond, have been studied intensively since thebeginning of the 20th century. However, by the end of the last century, the discovery of the nextcarbon allotrope, buckminsterfullerene (C60), opened up a novel and distinct field of carbonchemistry (Kroto et al. 1985). As a result, in the early 1990s, elongated cage-like carbon structures(known as carbon nanotubes) were produced and characterized (Iijima 1991; Iijima and Ichihashi

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1993; Terrones and Terrones 2003a). Carbon or boron nanotubes seem to be promising materialsas catalyst supports, connectors, thermal materials, structural composites, fibres, fabrics and so on.They are also promising for field emission, molecular electronics, energy storage, sensing andbiomedical applications (Saito et al. 1998). It is believed that carbon nanotubes can be used forthe development of containers for gases and metals (Terrones et al. 2000; Terrones and Terrones2003b; Chen et al. 1999). Demonstration of the use of carbon nanotubes for enhanced storage ofgases would pave the way for the development of fuel cells for the automobile industry.

Unfortunately, the adsorption of hydrogen in those materials currently produced that arecomposed of carbon nanotubes is not sufficiently high to provide a solution for the effective storageof hydrogen (Takagi et al. 2004; Ritschel et al. 2002; Ning et al. 2004) and hence the large-scaleutilization of nanotubes in the automobile industry remains a distant notion. Thus, it is important tosearch for ordered carbon nanostructures that could meet the target for hydrogen storage set by theU.S. Department of Energy (DOE) (http://www.eere.energy.gov/). The most efficient way ofscanning the properties of a variety of carbon nanostructures of different geometries and symmetriesis through computer simulations (Catlow et al. 2004; Garberoglio et al. 2005; Kowalczyk et al.2005a,b). However, it should be noted that there are examples in the literature of experimentallyirreproducible results (Chen et al. 1999; Yang 2000). Contrary to such confusing experimentalresults, computer simulations have consistently predicted lower storage capacities for hydrogen inmany materials (Wang and Johnson 1999a,b; Guay et al. 2004; Williams and Eklund 2000).

In the search for hydrogen storage materials, ordered nanoporous carbons offer a potentialalternative to the materials composed of carbon nanotubes. These porous materials are also ofpotential technological interest for the development of electronic (Kuno et al. 2003), catalytic (Jooet al. 2001; Rodriguez et al. 1995; Shin et al. 2001; Choi and Ryoo 2003), composite (Terroneset al. 2000) or gas-storage systems (Ohkubo et al. 2002). This new class of carbonaceousmaterials can be obtained by templating syntheses in which ordered mesoporous silicas (OMSs)are used as hard templates (Ryoo et al. 1999; Kim et al. 2003). Ryoo et al. (2001) have pointedout that ordered mesoporous carbons (OMCs) exhibit BET (Brunauer–Emmett–Teller) specificsurface areas as high as 2200 m2/g, excellent thermal stability in an inert atmosphere and strongresistance to acids and bases. The synthesis of these materials is conceptually simple. Variousmolecules that contain carbon (e.g. furfuryl alcohol, phenolic resin monomers, acetylene, sucrose)can be used as carbon precursors to fill the mesopores of ordered silicas or aluminosilicates atelevated temperatures. In a similar manner to the preparation of ordinary porous carbon materials,these precursors could be subjected to polymerization and converted into pure carbon by pyrolysisif required. Finally, the silica template can be removed by using NaOH or HF solutions.

The structures of such synthesized carbons depend strongly on the geometry of the template andon the carbon precursor and carbonization conditions. For example, CMK-3 OMC obtained byusing SBA-15 OMS as the template consists of hexagonally ordered carbon rods interconnectedby carbon spacers (Ravikovitch and Neimark 2001). The use of SBA-1 silica (cubic, Pm3nsymmetry) as a template affords the CMK-2 carbon of the same symmetry (Sakamoto et al. 2000).However, impregnation of the cage-like siliceous SBA-16 meso-structure with furfuryl alcohol oracenaphthene gives OMCs with the cubic Im3

—m structure (body-centred cubic arrangement of

spherical cages) (Kim et al. 2005). The latter authors also showed that filling the mesopores ofSBA-16 with furfuryl alcohol followed by polymerization and carbonization of the latter can beeasily controlled. Thus, it is possible to control the filling process and tune the size ofincompletely filled cages of SBA-16 (Kim et al. 2005). The resulting OMCs possess pores createdby incomplete filling of the SBA-16 cages by the carbon precursor as well as pores formed by thedissolution of the pore walls of the SBA-16 template.

412 Piotr Kowalczyk et al./Adsorption Science & Technology Vol. 24 No. 5 2006

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In the present work, we have investigated the efficiency of hydrogen storage at 303 K in modelOMC materials characterized by the cubic Im3

—m symmetry. A new model for solid–fluid interactions

between quantum molecules and the surface of an OMC is introduced according to the Feynman andHibbs method. Analytical equations for the estimation of the total and excess uptake of hydrogen havebeen derived. The proposed model can be used for the investigation of adsorption, diffusion and otherprocesses at lower temperatures where the quantum effects for confined fluids may not be negligible.

On the basis of this model and experimental data, five OMC replicas of cubic Im3—

m silica havebeen constructed. The efficiency of hydrogen storage at 303 K was studied using Monte Carlosimulations in the Grand Canonical ensemble (GCMC). The GCMC simulations provide insightsinto the role of the size of the bcc unit cell on the efficiency of hydrogen adsorption in the cubicOMC materials at 303 K. Our results suggest that these materials cannot achieve the target forhydrogen storage set by the U.S. Department of Energy for the years 2007, 2010 and 2015(http://www.eere.energy.gov/) under the conditions studied. On the other hand, the proposedmethodology seems to be useful for designing carbon nanostructures with desired properties forthe adsorption and storage of hydrogen and other gases.

THE MODEL AND SIMULATIONS

Fluid–fluid interaction potential

To take quantum effects into account, all simulations were performed for hydrogen interacting via thetruncated Feynman and Hibbs effective potential (FH) (Feynman and Hibbs 1965; Feynman 1972):

(1)

where r is the distance between two interacting molecules, β = (kbT)–1, h– = h/2π, µm = m/2 isthe reduced mass of a pair of interacting fluid molecules, h– denotes Planck’s constant, kb is theBoltzmann constant and Θ is the Heaviside function. Note that, according to equation (1), thequantum fluid molecule is represented by a Gaussian wave packet of width h– /(12mkbT)1/2 , sothe FH effective potential is an average of the classical potential over the Gaussian wave packet.

According to Sese (1994, 1995), the series given by equation (1) should be cut off at k = 1 forbetter approximation of the rigorous Path Integral Monte Carlo Simulation (PIMC). Afterincorporation of this cut-off, the final formula for the calculation of the fluid–fluid interactions isgiven by the second-order Feynman–Hibbs effective potential:

(2)

where the classical potential is represented by the one-centre Lennard-Jones (LJ) equation:

(3)V rr rff ffff ff( ) =

−

412 6

εσ σ

V r V rd

drV r

r

d

drV rFH

ffff

mff ff( ) = ( ) +

( ) + ( )

βµ

2 2

224

2

−( )Θ r rcut

h

V rn

V r rFHff

m

n

nff

n

k

cu( ) =

∇ ( )

=∑ 1

24

22

0 !

βµ

Θ tt r−( )h

GCMC Simulation Study of Hydrogen Storage in Ordered Mesoporous Carbons 413

AST_24(5)_460-32 18/1/07 8:38 am 13

Here, σff denotes the LJ fluid–fluid collision diameter, εff is the depth of the LJ fluid–fluidpotential well and rcut = 5σff is the cut-off distance. For hydrogen we used σff = 0.2958 nm andεff/kb = 36.7 K (Levesque et al. 2002). We wish to stress that the spherical symmetry of theadopted Lennard-Jones potential may not be adequate for the case of high density and lowtemperature. The description of the experimental equation of state for hydrogen at 303 K and 77K with the use of the intermolecular potentials given above [equations (2) and (3)] has beenpresented in detail previously (Kowalczyk et al. 2005). Moreover, Tanaka et al. (2005a) haveverified this model experimentally.

Solid–fluid interaction potential

The cage-like OMC studied is characterized by the cubic Im3—

m structure (body-centred cubicarrangement of spherical cages connected by short cylindrical necks). The simplified model of thebasic unit of ordered mesoporous carbon is presented in Figure 1. For simplicity in the presentmodel, we assume that interconnecting cylindrical pores (short necks) can be neglected, since thekey sites for hydrogen adsorption are (i) the internal surfaces of the spherical cavities and


Figure 1. Simplified model of the ordered mesoporous carbon used in the current study. The cubic Im3—

m structure (body-centred cubic arrangement of unconnected carbon spherical cages) is characterized by a, the cubic lattice parameter, Rint

and Rext, the internal and external pore radius of the carbon spherical cavities, respectively, and ρs, the surface density ofa single graphite layer.

AST_24(5)_460-32 18/1/07 8:38 am Page 414

(ii) interstitial spaces between the neighbouring carbon spheres. A similar treatment of nitrogenadsorption at the boiling point in ordered cage-like silicas and three-dimensional hexagonalstructures has been developed by Ravikovitch and Neimark (2002).

The hydrogen uptake obtained in the present simplified model of the cubic Im3—

m OMC materialcan be treated as a lower limit for hydrogen storage in these carbonaceous materials. However,large differences due to the presence of the short cylindrical necks are not anticipated. In oursimulations, we have neglected details of the atomic structure of the carbon surfaces, i.e. allspherical cavities have been modelled as continuous surfaces. We represent the solid–fluid potentialby integrating the quadratic Feynman–Hibbs effective potential over a smooth spherical surface.An individual spherical pore is characterized by the internal, Rint, and external, Rext, radii of the pore(cavity), by the surface density of carbon atoms (ρs = 38.2 nm–2) and by the thickness of the porewall, t = Rext – Rint. We assume that the wall of each individual cavity consists of three concentricgraphite shells, t = 3∆ � 1 nm (∆ = 0.335 nm). Note that this value is close to the experimentallymeasured thickness of the carbon walls in the cubic Im3

—m OMC materials, viz. t � 0.63–0.91 nm

(Kim et al. 2005). In accordance with these experimental results, we assume that Rint = 7.2/χ nmand that the cubic lattice parameter a = 11.97/χ nm. For χ = 1, these parameters correspond to thecubic Im3

—m OMC sample (CFA1 – 12 h) synthesized using the furfuryl alcohol carbon precursor and

SBA-16 ordered silica as the template (Kim et al. 2005). Reducing the parameter χ corresponds tothe shrinkage that accompanies the removal of the template. In addition, we wish to investigate thehydrogen storage in hypothetical OMC replicas of the cubic Im3

—m materials which contain smaller

pores that prevent the experimental ratio Rint/a � 0.3 (Kim et al. 2005) from being achieved. Thus,in the present work, we have investigated the following range of the structural parameters (Rint ∈[0.9–3.0] nm and a ∈ [2.99–9.98] nm).

To take into account the quantum effects in the solid–fluid interactions, we integrated thepairwise quadratic Feynman–Hibbs effective potential over carbon atoms dispersed on the singlesurface of a sphere of radius, Rs. In this way, we obtained the quantum-corrected solid–fluidinteraction of a hydrogen molecule placed inside or outside the structure-less spherical cavity, i.e.(r ≠ Rs):

(4)

where the coefficients may be written as

(5)

(6)

(7)ζ3

4 2 2 4

6 6

2

3

3 10 3=

+ +( )+( ) −( )

a a a b b

a b b a

ζ2 2 2

2=

+( ) −( )

a

a b b a

ζ1

4 2 2 4

5 5

2

5

5 10= −

+ +( )+( ) −( )

a a b b

a b b a

V r RSsf

sf s s sf sf sf sf( ) = − + −

81 0

4 42 12

16

214

38

4π ε ρ σ ζ σ ζ γσ ζ γσ ζ.

.


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(8)

(9)

(10)

(11)

In the above equations, r is the radial coordinate of the hydrogen atom with the centre of thecoordinate system in the centre of the spherical pore, β = (kbT)–1, h– = h/2π, µm = m is the reducedmass for the quantum interaction between hydrogen and carbon atoms, and m denotes the mass ofa hydrogen molecule (by definition, the reduced mass of carbon and hydrogen is the mass ofhydrogen [ ].

The total quantum-corrected solid–fluid interaction in the unit cell of the Im3—

m OMC is givenby (see Figure 1):

(12)

Here M – 1 denotes the number of concentric shells of an individual spherical cavity (in thepresent case, M = 3).

The Lennard-Jones solid–fluid parameters were calculated via the Lorentz–Berthelot mixingrule (Steele 1974):

(13)

(14)

The Lennard-Jones parameters for carbon are σss = 0.34 nm and εss/kb = 28 K (Do 1998). AllGCMC simulations were performed for the solid–fluid interaction potential given by equation (12)calculated explicitly without approximations. At this point, it is important to ask the question:Can we use the molecular parameters of carbon in graphite for spherical pores? From a recentcomparison of the simulation results with the experimental ones, we can state that the model of thesolid–fluid interactions can be used for the calculation of thermodynamic averages at moderatetemperatures (Tanaka et al. 2005a,b).

Efficiency of hydrogen storage results

The correct evaluation of the total equilibrium uptake of hydrogen is a well-known experimentalproblem. The quantity that is measurable experimentally is the adsorption excess, i.e. the amount

ε ε εsf ff ss= ( )1 2/

σ σ σsf ff ss= +( ) / 2

V r V R jTsf

Ssf

j

M

i

( ) = +( )=

−

=∑∑ int •∆

0

1

1

9

µ = +( ) ≅ ≅m m m m m m m mH C H C H C C H2 2• •/ /

22

b rRs= − 2

a r Rs= +2 2

γσ

βµ

=132

242

2

sf m

h

ζ4

2 2

3 3

2

3

3= −

+( )+( ) −( )

a b

a b b a

AST_24(5)_460-32 18/1/07 8:38 am Page 416

of adsorbate in excess of that of the bulk fluid at the same temperature and pressure, and inthe same available volume. Experimentally, the adsorption excess can only be calculated whenthe true density of carbon is known. The latter is usually obtained via the helium-displacementmethod, although the validity of this method has been questioned in the literature. Thus, forexample, Beenakker et al. (1995) have pointed out that the quantum effect for helium placedin small nanopores at room temperature cannot be neglected. Since the thermal de Brogliewavelength for helium at room temperature (λ = 0.1 nm) approaches the size of nanopores,the adsorbed amount depends on the quantum energy levels of helium. In other words,helium cannot access the smallest pores due to the Heisenberg uncertainty principle, with theresult that the helium-displacement method over-estimates the density of carbon in the examinedmaterial.

Contrary to experiment, GCMC simulation enables a precise determination of the total uptakeof hydrogen, i.e. the absolute value of adsorption, because the geometry of the adsorbent isspecified and the quantum effects are included in both fluid–fluid and solid–fluid interactions.

The total surface area of carbon in the simulation box is given by (see Figure 1):

(15)

Knowing SC, we can calculate the total weight of carbon (wC) in the unit cell:

(16)

Here, mC denotes the molar mass of carbon (mC = 12.01 g/mol) and Na is the Avogadro number.The total volume accessible for hydrogen molecules can be calculated from:

(17)

where VB denotes the volume of the simulation box (VB = a3) and VC is the volume occupied bythe carbon surface, i.e. VC = (8/3)π(R3

ext – R3int).

The values of the absolute (Γabs) and excess (Γexc) adsorption of hydrogen are given by:

(18)

(19)

where ρb denotes the bulk density of hydrogen, ⟨N⟩ is the ensemble average of the number ofhydrogen molecules in the simulation box and Rext is the external pore radius of the sphericalcarbon cavity.

Γexcb

ext

N V

R R=

−+( )ρ

π8 2 2int

Γabs

ext

N

R R=

+( )8 2 2π int

V V VB C= −

wN

S mCs

aC C=

ρ

S R jCj

M

= +( )=

−

∑82

0

1

π int •∆


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We calculate the gravimetric density (ρw) of hydrogen (storage capacity) from:

(20)

where MH2and MC are the total mass of hydrogen molecules and carbon atoms, respectively, in

the simulation box. Note that the gravimetric density of hydrogen is the absolute value of theadsorption and not the excess value.

Molar enthalpy of adsorption

By analogy to the specific enthalpy of a phase transition, we calculate the enthalpy released duringadsorption as:

(21)

where ⟨…⟩ denotes an average over an ensemble, Nav is the Avogadro number and E is the sum ofthe kinetic and potential energies.

According to the Feynman–Hibbs model, the total energy is given by (Sese 1994, 1995; Tanakaet al. 2005a,b):

(22)

where VffFH and Vsf

FH are respectively the fluid–fluid and solid–fluid interaction potentials based onthe FH effective potential, N is the number of adsorbed molecules in the simulation box (see unitcell in Figure 1) and β = (kbT)–1. The first term on the right-hand side of equation (22) is thekinetic energy of classical molecules, the second and third terms are required by thethermodynamic consistency, while the remaining terms are the potential energies.

Simulation details

We used the Grand Canonical bulk ensemble (i.e. a fixed system volume, V, temperature, T, andthe chemical potential of the bulk fluid, µp) (Allen and Tildesley 1987) in the simulation ofhydrogen adsorption in the nanoporous carbon materials considered. Widom’s particle insertmethod (Widom 1963) was used to calculate the excess part of the chemical potential of hydrogen.The corresponding bulk pressure of hydrogen was computed from the virial theorem (Garberoglioet al. 2005; Frenkel and Smit 1996). The tail corrections for the energy and pressure were addedafter simulation in canonical ensemble. For GCMC simulations, we used the simulation box (a ×a × a, where a denotes the cubic lattice parameter) with periodic boundary conditions in all threespecial directions as shown in Figure 1. The cut-off distance for the calculation of the fluid–fluidinteractions was assumed to be 5σff and no long-range corrections were used. The chemicalpotential of hydrogen calculated in canonical ensemble was used as an input for the GCMCsimulations. In order to mimic the experimental procedures, all the simulations were commencedfor systems without hydrogen. Then, the chemical potential of hydrogen was gradually increasedin each simulation. The systems used were equilibrated at different levels of the chemical potential

E N k TdV

d

dV

dV Vij

ff

i j

Ni

sf

i

N

ijff

i j

N

= + + + +<∑ ∑ ∑3

2 B

FH, FH,FH, Fβ

ββ

β HH, isf

i

N

∑<

q

Nk T

EN E N

N Nst

avb= −

−

−5

2 2 2

ρw

H

H C

M

M Mwt=

+2

2

100• %

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and employed as the starting configurations for subsequent calculations. In order to ensuremicroscopic reversibility, the probabilities of (i) a single displacement, (ii) creation and (iii)deletion of a hydrogen atom were assumed to be all equal to 1/3.

Typically, 8 × 107 configurations were generated for every value of the chemical potential andfor every Im3

—m OMC structure. In order to guarantee equilibration of the system before

measurement of the adsorption, the first 4 × 107 configurations in every run of the simulation wereneglected and only the thermodynamic properties of the samples in the remaining 4 × 107 stepswere considered. In addition, the value of the total internal energy at randomly selectedconfigurations during the simulation was stored and the fluctuations of this energy analyzed tocheck whether the system was truly in thermodynamic equilibrium.

RESULTS AND DISCUSSION

We start with a demonstration of the change in the solid–fluid potential introduced by the quantumeffects included in the simulations. Wang and Johnson (1999a) showed that quantum effects areimportant even at 298 K for adsorption in interstices between carbon nanotubes. The situation issimilar for the cubic Im3

—m OMC material studied: there are interstices between the spherical

carbon cavities, similar to the interstices between carbon nanotubes. We introduced the quantum-corrected solid–fluid interaction potential for carbon spheres

according to the method of Feynman and Hibbs. Figure 2 displays the solid–fluid potential insideand outside the carbon spheres computed using equation (12). As expected, the introduction ofquantum effects lowered the solid–fluid interaction with respect to the same potentials calculatedwithin the classical model. Generally, a reduction in the size of the pores (or equivalently, the sizeof the carbon spheres) resulted in an increase in the quantum effects (Figure 2). This remark isqualitatively similar to the observation made by Wang and Johnson (1999a). It should bementioned that the classical contribution arising from equation (12) is in a full agreement with theLennard-Jones potential for equivalent carbon spheres computed from the equations derived by


r* = r/Rint

0.0 0.2 0.4 0.6 0.8 1.0 1.2−50

−40

−30

−20

−10

0

10

R* = 1.5 (0.4437 nm)

R* = 2.0 (0.5916 nm)

R* = 3.5 (1.0353 nm)R* = 5.0 (1.479 nm)

R* = 10.0(2.958 nm)

1.0 1.1 1.2 1.3 1.4 1.5 1.6−15

−10

−5

0

5

R* = 1.5 (0.4437 nm)R* = 2.0 (0.5916 nm)

R* = 3.5 (1.0353 nm)R* = 5.0 (1.479 nm)

R* = 10.0 (2.958 nm)

r* = r/Rext

V* sf

(r*)

(B)

V* sf

(r*)

(A)

Figure 2. Reduced solid–fluid interaction potential, V*sf(r) = Vsf(r)/εff, (A) inside or (B) outside carbon spheres consisting

of three single graphite shells at 303 K: solid lines, total reduced solid–fluid interaction potential given by equation (4);dotted lines, the reduced classical contributions; dashed lines, the reduced quantum contributions; open circles, classicalsolid–fluid Lennard-Jones potential derived by Baksh and Yang (1991).

AST_24(5)_460-32 18/1/07 8:38 am Page 419

Baksh and Yang (Baksh and Yang 1991; Kaminsky et al. 1994). As can be seen from Figure 2, thequantum effect for hydrogen adsorption in cubic Im3

—m OMC materials at 303 K is not large.

However, lowering the temperature and increasing the pressure of hydrogen causes largediscrepancies between the classical and quantum-corrected solid–fluid interactions.

Figure 3(A) presents the absolute value for the adsorption of hydrogen in the OMC replicas ofthe cubic Im3

—m materials. For all the replicas (structures) examined in this study, the absolute

value of adsorption was an increasing function of the size of the unit cell: the highest value of theabsolute adsorption was obtained for the largest unit cell studied. In Figure 3(B) the same resultsare shown expressed as gravimetric weight per cent of hydrogen per carbon mass (storagecapacity). The highest storage of hydrogen (ca. 2 wt% at P = 22 MPa) was obtained for theOMC replica of the cubic material characterized by the largest size of the cubic lattice parameterand — correspondingly — the largest spherical pore radius. Moreover, as can be seen in Figure 4,the maximum value of hydrogen storage at 303 K in the OMC replicas of the cubic Im3

—m


0 4 8 12 16 20 240

2

4

6

8

10

12

(A)

Abs

olut

e va

lue

of a

dsor

ptio

n(µ

mol

/m2 )

Pressure, P (MPa)0 4 8 12 16 20 24

0.0

0.20.4

0.60.8

1.01.2

1.41.6

1.82.0

2.2

(B)A

bsol

ute

valu

e of

ads

orpt

ion

(wt%

)

Pressure, P (MPa)

0 4 8 12 16 20 240.0

0.2

0.4

0.6

0.8

1.0(C)

Exc

ess

valu

e of

ads

orpt

ion

(µm

ol/m

2 )

Pressure, P (MPa)

Figure 3. Hydrogen absolute adsorption isotherms: (A), gravimetric uptake (total hydrogen mass per mass of carbon);(B), surface excess adsorption isotherms; and (C) as simulated by the GCMC method in the OMC replicas of the cubic Im3

—m

material at 303 K. The internal pore radius, Rint, of individual carbon spheres corresponding to the displayed cubic latticeparameter are 0.9, 1.8, 2.0, 2.4, 3.0 nm, respectively. Data points in the three parts of the figure relate to the followingvalues of a, the lattice parameter: �, 2.99 nm; , 5.99 nm; �, 6.65 nm; �, 7.98 nm; �, 9.98 nm.

�

AST_24(5)_460-32 18/1/07 8:38 am Page 420

materials showed a linear dependence on the size of the lattice parameter of the unit cell.Consequently, a further increase in the basic unit of the Im3

—m OMC replica would lead to an

increase in the hydrogen storage uptake. However, we must take into account that this increase issimply associated with the increase in the volume of the unit cell and, consequently, the increasein volume available to bulk hydrogen; this does not correspond to an increase in the surface excessof adsorption. Figure 3(C) shows that the surface excess adsorption of hydrogen reachedsaturation as the size of the unit cell increased. Thus, the design of OMC replicas with unit cellslarger than a threshold value (a � 7.98 nm and Rint � 2.4 nm) does not change the surface excessof adsorption. The observed increase in stored hydrogen for a cubic lattice parameter a > 7.98 nmand an internal pore radius for an individual carbon sphere R > 2.4 nm in a b.c.c. structure is solelyconnected with an increase in the bulk density of hydrogen. In other words, materials with largemesopores serve as a tank whose storage capacity depends mainly on the available volume for thebulk fluid. This feature can be useful in the transportation of hydrogen; however, the predictedadsorption values do not justify the claim that these carbon materials can be promising candidatesfor effective hydrogen storage at room temperature and storage pressures less than 22 MPa.

The volumetric density of hydrogen in all investigated OMC replicas is shown in Figure 5. Forcomparison, the variation in the bulk density (red line) and the density of hydrogen adsorbed in anidealized bundle of single-walled carbon nanotubes (SWNTs) (blue line) (Levesque et al. 2002)have been added to the figure. As can be seen, the cubic lm3

—m OMC material characterized by the

smallest pore size and lattice parameter significantly enhanced the density of adsorbed hydrogen incomparison to the situation in the bulk. This value was substantially above the 2010 DOE goal forthe volumetric density of hydrogen (45 kg H2/m

3). On the other hand, the gravimetric weightpercentage of hydrogen in this replica was very low and did not exceed 0.15 wt% [see Figure 3(B)].


2 3 4 5 6 7 8 9 10 11

0.0

0.5

1.0

1.5

2.0

2.5

Abs

olut

e va

lue

of a

dsor

ptio

n (w

t%)

Lattice parameter, a (nm)

Figure 4. Dependence of the absolute value of hydrogen adsorption at 303 K upon the size of the lattice parameter of thecubic Im3

—m OMC material. Data relate to a pressure, P, of 22 MPa. The straight line depicted may be described by the

equation y = 0.2729x – 0.7176 (R2 = 0.997).

AST_24(5)_460-32 18/1/07 8:38 am Page 421

As a result, the material appeared not to be suitable for the efficient storage of supercriticalhydrogen. The volumetric density of hydrogen in the remaining replicas of the cubic Im3

—m OMC

materials studied was greater than the bulk density but did not exceed 17 kg/m3 at 22 MPa. Thedensity of adsorbed hydrogen in the cubic Im3

—m OMC materials was very close to that reported

for the idealized bundle of SWNTs (Levesque et al. 2002) (see Figure 5).Since the uptake of hydrogen was similarly low in both the bundle of single-walled carbon

nanotubes and the OMC samples, it seems plausible that it is not the curvature of the graphenelayers that is responsible for the low density of hydrogen, but rather the strength of thehydrogen–carbon interactions that dominate the extent of adsorption.

In Figure 6, we show that as a result of the high thermal motion at 303 K, the hydrogen fluidonly adsorbed weakly in the OMC replicas of the cubic Im3

—m materials. A well-defined and

ordered arrangement of hydrogen molecules adsorbed on the walls of the spherical cavities (quasi-liquid state of supercritical fluid) was not observed, but rather a random arrangement ofmolecules. Detailed analysis of these snapshots revealed that hydrogen molecules tend to formclusters, with the size of these clusters increasing with increasing pressure due to the densificationof hydrogen. As pointed out by Do and Do (2005), this lack of condensation in either two- orthree-dimensions is a characteristic of supercritical fluids. Next, the importance of two spaces foradsorption may be highlighted: (i) the internal space of spherical cavities and (ii) interstitial spacesbetween these cavities. Computer-generated configurations of hydrogen molecules were usedto calculate the average number of molecules in the internal and interstitial spaces of the cubic


18

16

14

12

10

8

6

4

2

00 2 4 6 8 10 12 14 16 18 20 22 24

Pressure, P (MPa)

Den

sity

of a

dsor

bed

hydr

ogen

(kg

/m3 )

Bulk densit

y

Figure 5. Simulated mass density of hydrogen in the OMC inverse replicas of the cubic Im3—

m material at 303 K. Theinternal pore radius, Rint, of individual carbon spheres corresponding to the displayed cubic lattice parameter are 0.9, 1.8,2.0, 2.4, 3.0 nm, respectively. For comparison, the density of bulk hydrogen (red line) and the density of hydrogen insingle-walled carbon nanotubes reported by Levesque et al. (2002) (blue line) have been added to the figure. Data pointsin the figure relate to the following values of a, the lattice parameter: , 7.98 nm; �, 6.65 nm; �, 5.99 nm; �, 2.99 nm.

�

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Im3—

m OMC materials. A cubic lattice parameter a = 5.99 nm was used in these calculationssince smaller values led to the number of hydrogen molecules in the interstitial spaces of the cubicIm3

—m OMC material being too small to allow reliable statistics to be constructed. The results are

summarized in Figure 7 from which it is seen that an increase in the lattice parameter of the OMC


Figure 6. Snapshots of hydrogen at P = 22 MPa and T = 303 K. A cubic lattice parameter, a, of 9.98 nm is depicted whilethe corresponding internal pore radius of the individual carbon sphere, Rint, was 3.0 nm.

5 6 7 8 9 10 11

30

40

50

60

70

Interstitial spaces = 68%

Internal spaces = 32%

Ave

rage

num

ber

of h

ydro

gen

mol

ecul

es (

%)

Lattice parameter, a (nm)

Figure 7. Contribution of the internal and interstitial adsorption in the OMC replicas of the cubic Im3—

m materials at303 K. Data points refer to the following pressures: �, 21.8 MPa; �, 17.1 MPa; , 10.7 MPa; �, 6.0 MPa. Averagevalues of the hydrogen uptake in the interstitial and internal spaces of these carbonaceous materials were ca. 32% and68%, respectively.

�

AST_24(5)_460-32 18/1/07 8:38 am Page 423

replicas studied resulted in a rapid decrease in the role of hydrogen adsorption in the internalspaces. In contrast, adsorption in interstitial spaces dominated the total hydrogen uptake as thesize of the unit cell increased. This behaviour was a result of the decreasing role of the solid–fluidinteraction in larger unit cells and the increasing available free space for hydrogen molecules. Thistendency was observed over the whole range of pressures studied (Figure 7). The calculatedaverage value of hydrogen adsorption in the internal spaces was twice the magnitude of that forthe OMC materials studied at 303 K.

Finally, Figure 8 depicts the molar enthalpy of adsorption for the cubic Im3—

m OMC materials.It will be seen from the figure that the shape of the plot of the heat of adsorption versus pressurewas independent of the size of the OMC replicas studied. The highest value for the molar enthalpy(� 9 kJ/mol) was obtained for the replica possessing the smallest cubic lattice parameter and,consequently, the smallest radius of the spherical cavity. This (highest) value of the heat ofadsorption is comparable to the corresponding value for hydrogen adsorption in carbon slit-likenanopores of 0.69 nm width (Kowalczyk et al. 2005). An increase in the cubic lattice parameterfor the b.c.c. unit cell resulted in a gradual decrease in the enthalpy of adsorption down to ca.4 kJ/mol. The highest value of the enthalpy of adsorption occurred for small cavities when thecurvature of the carbon surface was high, with the value decreasing asymptotically to the value ofthe enthalpy of adsorption on flat surfaces as the size of the unit cell increased.

In summary, OMC materials of cubic Im3—

m symmetry cannot be treated as an effective materialfor hydrogen storage at 303 K and pressures less than 22 MPa. The highest storage ofhydrogen (ca. 2 wt% at P = 22 MPa) was predicted for the OMC replica of the cubic Im3

—m

material characterized by the largest size of the cubic lattice parameter and internal pore radius ofthe carbon sphere studied (a = 9.98 nm and Rint � 3.0 nm). Unfortunately, the density of adsorbedhydrogen was found to be low and did not exceed 17 kg/m3. Further increase in the basic unit ofthe cubic Im3

—m OMC led to an increase in the gravimetric uptake of hydrogen due to pure


0 4 8 12 16 20 240

2

4

6

8

10

Ads

orpt

ion

enth

alpy

(kJ

/mol

)

Pressure, P (MPa)

� � � � � � � � � � �

� � � � � � � � � �

� � � � � � � � ��

� � � � � � � � � � �

��

��

Figure 8. Quantum-corrected molar enthalpy of adsorption simulated by the GCMC method in the OMC replicas of thecubic Im3

—m materials at 303 K. The internal pore radius, Rint, of individual carbon spheres corresponding to the displayed

lattice parameter are 0.9, 1.8, 2.0, 2.4, 3.0 nm, respectively. Data points in the figure relate to the following values of a,the lattice parameter: �, 2.99 nm; , 5.99 nm; �, 6.65 nm; �, 7.98 nm; �, 9.98 nm.

�

AST_24(5)_460-32 18/1/07 8:38 am Page 424

geometrical factors (an increase in the available volume) and not as a result of adsorption on thecarbon surface. In contrast, the volumetric density of adsorbed hydrogen remained constant as thesize of the OMC replica increased. A comparison of the current computations with those reportedfor the bundle of SWNTs showed that the curvature of graphene was not responsible for the lowuptake of hydrogen in carbonaceous materials. This low uptake is more likely caused by therelatively weak carbon–hydrogen interactions. A potential route for increasing the hydrogenuptake in Im3

—m OMC materials would be through a reduction in temperature and/or through

doping these structures with alkali metals. Doping cubic Im3—

m OMC with alkali metals wouldincrease the mass of the material and hence its interaction with hydrogen molecules. Since themaintenance of low temperatures in practical storage applications (e.g. automobile applications)is expensive and impractical, it would appear that the quest for promising nano-architectures forcarbonaceous materials is still open.

ACKNOWLEDGEMENTS

This work is dedicated to Prof. Katsumi Kaneko, Faculty of Science, Chiba University, Chiba,Japan.

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