Diffusion, adsorption, and desorption of molecular hydrogen on grapheneand in graphiteJustin Petucci, Carl LeBlond, Majid Karimi, and Gianfranco Vidali Citation: J. Chem. Phys. 139, 044706 (2013); doi: 10.1063/1.4813919 View online: http://dx.doi.org/10.1063/1.4813919 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v139/i4 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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THE JOURNAL OF CHEMICAL PHYSICS 139, 044706 (2013)

Diffusion, adsorption, and desorption of molecular hydrogen on grapheneand in graphite

Justin Petucci,1 Carl LeBlond,2 Majid Karimi,1,a) and Gianfranco Vidali31Department of Physics, Indiana University of Pennsylvania, Indiana, Pennsylvania 15705, USA2Department of Chemistry, Indiana University of Pennsylvania, Indiana, Pennsylvania 15705, USA3Department of Physics, Syracuse University, Syracuse, New York 13244, USA

(Received 25 April 2013; accepted 25 June 2013; published online 25 July 2013)

The diffusion of molecular hydrogen (H2) on a layer of graphene and in the interlayer space betweenthe layers of graphite is studied using molecular dynamics computer simulations. The interatomicinteractions were modeled by an Adaptive Intermolecular Reactive Empirical Bond Order (AIREBO)potential. Molecular statics calculations of H2 on graphene indicate binding energies ranging from41 meV to 54 meV and migration barriers ranging from 3 meV to 12 meV. The potential energysurface of an H2 molecule on graphene, with the full relaxations of molecular hydrogen and carbonatoms is calculated. Barriers for the formation of H2 through the Langmuir-Hinshelwood mechanismare calculated. Molecular dynamics calculations of mean square displacements and average surfacelifetimes of H2 on graphene at various temperatures indicate a diffusion barrier of 9.8 meV and adesorption barrier of 28.7 meV. Similar calculations for the diffusion of H2 in the interlayer spacebetween the graphite sheets indicate high and low temperature regimes for the diffusion with barriersof 51.2 meV and 11.5 meV. Our results are compared with those of first principles. © 2013 AIPPublishing LLC. [http://dx.doi.org/10.1063/1.4813919]

I. INTRODUCTION

The study of carbon-based materials, especially thosewith sp2 hybridization, such as graphite, carbon nanotubes,fullerene, and graphene has attracted a lot of attention in re-cent years and consequently a great deal of progress has beenmade.1–6 The discovery in 20047 that a single layer of graphite(graphene) could be stable, resulted in renewed interest incarbon materials. From a scientific standpoint, this new sys-tem provides an ideal environment for research in two di-mensions, such as catalytic processes, surface diffusion, novelelectronic properties, and molecular hydrogen formation fromchemisorbed atomic hydrogen. Graphene is thought to be ableto trap H atoms more effectively than graphite on account ofthe different phonon spectrum.8 From a technological stand-point, graphene is a suitable material for gas storage (specif-ically hydrogen)9 and its semimetal nature could allow it tochange into a semiconductor, with a desired band gap, whensaturated by hydrogen.10

An important property of a graphene layer is the unifor-mity of its surfaces, as any hydrogen adsorbed onto one sidecould be re-adsorbed onto the other side without any signif-icant change to its environment. One possible route for theadsorption of molecular hydrogen on graphene is that it willfirst physisorb and then dissociate into chemisorbed atomichydrogen. Insight into the relation between molecular hydro-gen physisorption and atomic hydrogen chemisorption couldshed light in the study of hydrogen storage materials. In suchcases, one should not only consider atomic hydrogen but also

a)Author to whom correspondence should be addressed. Electronic mail:karimi@iup.edu

molecular hydrogen that is physisorbed to those surfaces.11–13

In the study of diffusion of H2 in graphite, the experimen-tal techniques seem to probe its diffusion only in the re-gions confined to the boundaries of the crystal and, there-fore, a simulation technique that probes the interlayer space isimportant.

The interaction of hydrogen atoms/molecules with thegraphite surface or with graphene has been studied experi-mentally and theoretically.14 Using atomic and molecular hy-drogen beams the bound states of the atom/molecule–graphitelaterally averaged potential were measured. The ground stateenergies of 31.6 meV and 41.6 meV were obtained for atomicand molecular hydrogen, respectively.15, 16 When hydrogenatoms with higher kinetic energy (∼2000 K) are used, hydro-gen atoms chemisorb on graphite.17 DFT calculations showthat chemisorption occurs on top of the carbon atoms whichpucker out by ∼0.3 Å as a result of the interaction.18 Thereis an entrance barrier of about 0.2 eV for hydrogen atoms tochemisorb on graphite. A recent experiment by Areou et al.19

confirmed the presence of this barrier.Atomic hydrogen adsorbed on graphite can react with

other hydrogen atoms to form H2. There are three basic mech-anisms: Langmuir-Hinshelwood (L-H), Eley-Rideal (E-R),and “hot atom” (H-A). In the first, hydrogen atoms becomeaccommodated with the surface; they diffuse and upon en-countering they form H2 that might leave the surface in anexcited ro-vibrational state. In the second and third mech-anisms, an H atom from the gas-phase reacts directly witha chemisorbed H atom, making a bond and leaving the sur-face as an H2 molecule in high ro-vibrational state. The thirdmechanism is similar to the second, but the atom first lands onthe surface and maintains a high superthermal kinetic energy.

0021-9606/2013/139(4)/044706/12/$30.00 © 2013 AIP Publishing LLC139, 044706-1

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044706-2 Petucci et al. J. Chem. Phys. 139, 044706 (2013)

Eventually it encounters another H atom and forms an H2

molecule. This problem has generated considerable theoreti-cal work since it is germane to the formation of molecular hy-drogen in space, where it occurs on the surface of submicron-sized dust grains.14 Semi-classical trajectory methods20 andDFT18, 21–24 were used to calculate the interaction of H atomson graphite and graphene23 via the Eley-Rideal mechanism.Morisset et al.,22 Martinazzo and Tardini,25 Bonfanti et al.,26

and Ferullo et al.27 using DFT calculated the physisorptionand diffusion of H atoms, and the formation of H2 fromphysisorbed hydrogen atoms. Arelliano et al.28 used DFT tostudy the physisorption of H2 on graphene.

In this paper, we report on a study of the formation of anH2 molecule from hydrogen atoms physisorbed on graphene,and the energetics and diffusion of molecular hydrogen ongraphene and in graphite. We use methods that allow for thecalculation of processes that would be very complex and time-consuming using first principle approaches, yet they still re-tain the main features of the H-graphene/graphite interaction.We employ molecular dynamics (MD) and molecular stat-ics (MS)29 in conjunction with the semi-empirical AIREBOpotential.30

This paper is organized as follows: In Sec. II, compu-tational techniques are presented. In Sec. III, our results aresummarized and finally in Sec. IV a summary of the impor-tant findings is given.

II. SIMULATION TECHNIQUES

The first principles approaches are, in general, very ac-curate but have the limitation that they are computationallyintensive and as a result a large system cannot be simu-lated. The classical approaches are computationally less in-tensive at the cost of sacrificing accuracy. Using a semi-empirical potential model which is fit to results from firstprinciples as well as experimental data is a compromise,because it can capture essential physics with reasonable accu-racy and time. The calculations in this paper are achieved us-ing the classical molecular dynamics code LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator).31 Theinteraction between hydrogen and carbon atoms was mod-eled using the AIREBO potential30 and was implemented inLAMMPS. The AIREBO potential is a semi-empirical model,based on the Tersoff potential and is widely used in MD sim-ulations of carbon-hydrogen systems. The AIREBO poten-tial has the capability of modeling the bond formation, bondbreaking, and various hybridizations in carbon-hydrogen sys-tems. In fact, one of the attractions of AIREBO potential isthat it could describe chemisorption and physisorption prop-erties of atomic and molecular hydrogen to the carbon basedmaterials.

The computational box that includes graphene had an or-thorhombic shape with periodic boundary conditions in allthree directions. The graphene layer makes a rectangle of di-mensions x × y ≈ 79 Å × 68 Å that includes 2048 atomswith its x and y sides oriented along the simulation box. Thegraphene layer is located at z = 0, and a periodic length of8 Å is considered along the z-direction perpendicular to thegraphene layer. For the simulation of H2 in graphite, the com-

putational box included two layers with a total of 1024 car-bon atoms. Periodic boundary conditions were invoked alongthe x and y directions. Four corner atoms of each layer ofgraphite were frozen along the x and y directions (to pre-vent layers from shifting) but allowed to relax along the zdirection. The simulations were performed using an NVTensemble. A Berendsen32 temperature control, with a temper-ature relaxation parameter of 0.1 ps, was employed to reg-ulate the temperature during simulation runs. This thermo-stat controls the instantaneous system temperature roughly asTavg ± �T , where Tavg ≈ Tdesired and �T ≈ 0.06Tavg . Ob-viously, a higher desired temperature generates a larger fluc-tuation �T.

Molecular dynamics (MD) and molecular static (MS)techniques are employed to probe the equilibrium and dy-namical properties of H2 on graphene and in graphite. En-ergy barriers for diffusion, desorption, and dissociation ofmolecular hydrogen to atomic hydrogen were obtained us-ing the nudged elastic band (NEB) method.33 Within NEB,a series of system replicas are considered which connect theinitial and final configurations on either side of the energybarrier being investigated. These replicas are defined as 3Ndimensional points on the system’s configuration space andare initially linearly interpolated between the initial and fi-nal states. Introducing inter-replica spring forces that con-nect each atom to its own image in the two most adjacentreplicas and minimizing a specific force quantity will resultin replica configurations for the minimum energy transitionpath over the barrier. The force quantity minimized for eachreplica is a sum of the force due to the interaction potentialthat is perpendicular to the transition path and the spring forcethat is parallel to the transition path. NEB calculations wereperformed with 16–32 system replicas (including the initialand final states), with a force tolerance convergence criterionranging from 0.1 to 0.001 (eV/Å). LAMMPS also implementsa climbing image algorithm in which the replica highest in en-ergy is identified and driven to the top of the energy barrier.Even though NEB was originally applied to the calculation ofmigration barriers, its applicability has been extended to thecalculations of desorption, adsorption, and chemical reactionbarriers.

The MD code LAMMPS (in the NVT ensemble) with atime step �t = 0.2 fs and a Berendsen temperature controlwas employed to perform the dynamical calculations. Simu-lations for H2 on graphene were performed for temperaturesranging from 40 K to 1000 K. For a given temperature, a typ-ical run included a total of 6 × 104 MD steps to bring thesystem into equilibrium, followed by another 8 × 106 MDsteps to take averages. For lower temperatures higher sim-ulation times were required to reduce the statistical errors.For example, for a temperature of T = 50 K, a total num-ber of 107 MD steps were considered. The trajectories gen-erated by our MD runs were post processed to recognize de-tachment/reattachment of H2 from one face of graphene to theother as well as diffusion of H2 on either side. Because of theperiodic boundary condition along the z-direction, many de-tachment/reattachment events of H2 from one face to the otherwere observed. Simulations of H2 in graphite were performedfor temperatures ranging from 90 K to 700 K.

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044706-3 Petucci et al. J. Chem. Phys. 139, 044706 (2013)

The two-dimensional diffusion coefficient, D, at eachtemperature can be calculated through the mean square dis-placement 〈r2〉.34, 35 Using the following formulas the meansquare displacement of H2 for a time interval, �t, can becalculated,

r2i (�t) = [r(ti + �t) − r(ti)]

2 = [x(ti + �t) − x(ti)]2

+[y(ti + �t) − y(ti)]2, (1a)

〈r2(�t)〉 = 1

n

n∑i=1

r2i (�t), (1b)

where x and y denote the coordinates of the center of massof H2 and n is the number of �t intervals averaged over. Cal-culating 〈r2(�t)〉 for numerous time origins (�t, 2�t, 3�t,...), a discrete sampling of the mean square displacement as afunction of time can be found,

〈r2(t)〉 ≈ (〈r2(�t)〉, 〈r2(2�t)〉, 〈r2(3�t)〉, ...). (2)

In order to calculate D at each temperature, the mean squaredisplacement, 〈r2〉, versus time, t, is plotted and the slope ofits linear part is obtained and related to D through

D = 1

4limt→∞

〈r2〉t

. (3)

With MD simulations, a time average such as this is com-monly utilized rather than an ensemble average as per the er-godic hypothesis of statistical mechanics. It is however impor-tant to note that for any finite length simulation the uncertaintyin the mean squared displacement increases as the length ofthe time origin interval increases, unless a fixed n valued isused. This is due to the fact that there are fewer intervals, n,to average over with increasing interval length.

The diffusion coefficient, D, can also be related to thetemperature, T, through the empirical formula:34, 35

D = Doe− Ea

kB T , Do = r2o γo

4, (4)

where r0 is the distance to nearest neighbor adsorption sitesand γ 0 is the attempt frequency of the diffusing H2 molecule.A linear fit to the data ln(D) versus 1/T, generated from MD,gives the activation energy Ea and pre-exponential D0. An-other approach for determining Ea is to calculate the averagelifetime of H2 on graphene for each temperature. The lifetimeis defined as the time interval between an H2 trapping (ad-sorbing) and detrapping (desorbing) event. From Eq. (4), onecan easily derive the following formula for the dependence ofaverage life time τ on temperature T:36

τ = τoeEa

kB T . (5)

The average lifetime (τ ) at each temperature (T) is then cal-culated and a linear fit to ln(τ ) versus 1/T will yield the acti-vation energy for detrapping.

A potential energy surface (PES) calculation comple-ments those of migration energy barriers. A PES calculationof H2 on graphene graphically maps the energy barriers that amigrating H2 molecule experiences when diffusing along dif-ferent directions. A surface potential of a single H2 moleculeon graphene, Vmin(x,y), is obtained by minimizing the poten-tial energy, V(x,y,z,Rc), with respect to z and Rc for variousvalues of (x, y):

Vmin(x, y) = min[V (x, y, z, Rc)]z,Rc, (6)

where x, y, and z are the coordinates of the center of mass ofH2 and Rc denotes the 3N coordinates of the carbon atoms ofthe underlying graphene surface.

To find Vmin(x,y) for a specific point, H2 is placed at the(x, y) location on the surface and the total potential energyV(x,y,z,Rc), is minimized with respect to z and Rc. The pro-cedure is repeated for other points (x, y) on the surface.

In order to gain some insight into the orientation of H2

in graphite, a probability density f(θ ), where θ is the anglebetween H–H bond and the graphite layer, is calculated. Thefunction f(θ ) is required to be related to a probability functionP(θ ), such that

P (θ1, θ2) =θ2∫

θ1

f (θ )dθ, (7a)

π/2∫

−π/2

f (θ )dθ = 1, (7b)

where P(θ1,θ2), in Eq. (7a) gives the probability for findingan angle between θ1 and θ2 and Eq. (7b) is the normalizationcondition of the probability density.

The potential energy V of H2 molecule (at the equilib-rium distance zmin) to various adsorption sites on graphene isobtained using the formula,

V (zmin) = Etot − (Esub + EH2 ), (8)

where Etot, Esub, and EH2 are the total minimized energiesof H2 plus substrate, substrate, and H2, respectively. zmin isthe vertical equilibrium distance from the center of mass ofH2 to the graphene layer provided that all the energies areminimized.

III. RESULTS

A. Static properties of H2 on graphene

In this section, some equilibrium properties of molecularhydrogen on graphene are calculated. Three adsorption sitesof H2 on graphene; H (hollow site), T (top site), and B (bridgesite) are investigated. Several orientations of the H2 moleculewith respect to the graphene surface are considered at each ad-sorption site, as presented in Figure 1. In configurations H1,H3, B2, B3, T1, and T2, the axis of the H2 molecule (H–Haxis) is parallel to graphene surface. Configurations H2, B1,and T3 correspond to the cases where the H–H axis is per-pendicular to the graphene plane. The values of the potential

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044706-4 Petucci et al. J. Chem. Phys. 139, 044706 (2013)

H1 H2 H3

B1 B2 B3

T1 T2 T3

FIG. 1. The various molecular hydrogen adsorption configurations studied:hollow (H), bridge (B), and top (T).

energy and vertical distances of H2 (distance from the cen-ter of mass of H2 to the graphene surface) for these varioussites are obtained using (8) and reported in Table I along withthe first principles calculations of Costanzo et al.37 For com-parison with the first principle results, zero point energies E0

and binding energies EB are calculated, for all the adsorptionsites, and reported in Table I. The full adsorption potentials ofH2 to the H1, B2, and T2 adsorption sites are also calculatedusing NEB and presented in Figure 2. It is worth mention-ing that the horizontal axis of Figure 2 is the reaction coordi-nate. This is a measure of the distance along the reaction path-way from the initial state “0” to final state “1,” rather than anactual H2-graphene separation distance. Energies at the po-

0.0 0.2 0.4 0.6 0.8 1.0

50

40

30

20

10

0

reaction coord in ate

Em

eV

Molecu lar Hyd rogen d esorp tion ad sorp tion

TBH

FIG. 2. Adsorption potentials of molecular hydrogen to hollow (H), bridge(B), and top (T) adsorption sites.

tential minima and distances of the H2 center of mass fromthe graphene sheet for sites H1, B2, and T2 are (54.3 meV,2.97 Å), (52.2 meV, 3.02 Å), and (51.4 meV, 3.04 Å), respec-tively. The zero point energy is calculated to be 11.1 meV atthe hollow site using a harmonic approximation. These val-ues are in very good agreement with the value of the po-tential minimum (51.7 meV) of the laterally averaged poten-tial deduced from measurements of the H2 scattering fromgraphite.16 The measured ground state is at 41.6 meV and thezero-point energy is 10.1 meV.16 The additional layers whencomparing graphite vs. graphene increases the well depth by acouple of meV. From Figure 2, it is also observed that there isno barrier to the adsorption of H2 molecule to these sites. Ourresults for the potential minima are in agreement with thosefor the binding energies of Costanzo et al.37 The calculationsof Costanzo et al.37 are DFT-based with van der Waals correc-tions and give a well depth of 54 meV (H2) and a zero-pointenergy of 6 meV.

Migration energy barriers of H2 from various initial tofinal states are calculated using the NEB method and are re-ported in Table II. The calculation of migration energy isdescribed pictorially in Figure 3. Forward energy barriers,E→, of the molecular hydrogen migrating from initial state on

TABLE I. Equilibrium distances and potential minima of H2 to various adsorption sites on graphene; the lasttwo columns on the right report the results of Costanzo et al.37 E0 is the zero point energy and all the energiesare in meV.

DFT with vdWAIREBO results (this calculation) results (Ref. 37)

Configuration zcm (Å) Vmin E0 EB zcm (Å) EB

H1 3.0 −54.3 11.1 −43 3.0 −48H2 3.2 −45.3 10.0 −35 3.5 −47H3 3.0 −54.3 11.1 −43 3.2 −54B1 3.3 −42.0 10.1 −32 3.3 −43B2 3.0 −52.2 11.1 −41 3.6 −35B3 3.0 −51.1 11.1 −40 3.6 −43T1 3.0 −52.0 11.1 −41 3.2 −13T2 3.0 −51.4 11.1 −40 3.4 −46T3 3.3 −41.6 10.0 −32 3.1 −38

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044706-5 Petucci et al. J. Chem. Phys. 139, 044706 (2013)

Initial Configuration NEB Minimum Energy Path Final Configuration

0.0 0.2 0.4 0.6 0.8 1.012

10

8

6

4

2

0

reaction coordinate

Em

eV

H2 migration: H1 B1

E : 12.3 meV

E : 0. meV

0.0 0.2 0.4 0.6 0.8 1.0

12

10

8

6

4

2

0

reaction coordinate

Em

eV

H2 migration: H1 T3

E : 12.7 meV

E : 0. meV

FIG. 3. Examples of configurations where the migration direction is perpendicular (top) and parallel (bottom) to the colored C–C bond.

the left to the final state on the right are presented in Figure3. Reverse energy barriers, E←, for the migration of molecu-lar hydrogen from final to initial state were extracted and re-ported as well. The main difference between the two depictedmigrations in Figure 3 is that the orientation of the migrationdirection of H2 is perpendicular to the colored C–C bond inthe top of Figure 3 but parallel to the colored C–C bond in thebottom of Figure 3. The notation introduced in the binding en-ergy section is adopted to represent the initial and final statesof H2 on graphene in Table II. Our result for the migration en-ergy of configuration 1 in Table II is in close agreement withthe value of 14 meV reported by Arellano et al.28 A value of10 meV is reported, for the migration barrier of H2 on

graphene, by Costanzo et al.37 using DFT calculations thatinclude van der Waals interactions. Furthermore, based onthe migration energy barriers, it seems that H2 on grapheneis very mobile at room temperature.

In Figures 4 and 5, two potential energy surfaces (PES)of H2 on graphene are presented. The axis of the H2 moleculeis parallel to the graphene surface and oriented either paral-lel (Fig. 4) or perpendicular (Fig. 5) to the C–C bond col-ored in the top portion of Figure 3, as indicated in the lowerleft corner of each figure. For both calculations, the carbonatoms are permitted to fully relax; however, the relaxation ofthe molecular hydrogen is constrained in order to maintain theproper orientation with the C–C bonds. From our calculations

TABLE II. Migration energy barriers of H2 on graphene from various initial to final sites.

Initial Final Migration Forward barrier Reverse barrierConfiguration state state direction E→ (meV) E← (meV)

1 H1 B1 ⊥ to C–C 12.3 02 H1 B2 ⊥ to C–C 2.1 03 H1 B3 ⊥ to C–C 3.0 04 H1 T3 ‖ to C–C 12.7 05 H2 B1 ⊥ to C–C 3.3 06 H2 B2 ⊥ to C–C 1.1 8.07 H2 B3 ⊥ to C–C 1.7 7.78 H2 T3 ‖ to C–C 3.7 0.09 H3 B1 ⊥ to C–C 12.3 010 H3 B2 ⊥ to C–C 2.1 011 H3 B3 ⊥ to C–C 3.0 012 H3 T3 ‖ to C–C 12.7 013 H1 B2 ⊥ to C–C 3.1 1.014 H3 B3 ⊥ to C–C 3.0 015 B1 T3 ‖ to C–C 0.4 016 B2 T3 ‖ to C–C 10.6 017 B3 T3 ‖ to C–C 9.7 0

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044706-6 Petucci et al. J. Chem. Phys. 139, 044706 (2013)

18 19 20 21

17

18

19

20

X

Y

Graphene PES H2 parallel to C C

0.

0.2

0.4

0.6

0.8

1.

1.2

1.4

1.6

1.8

2.

2.2

2.4

2.6

2.8

3.

meV

FIG. 4. Potential energy surface of molecular hydrogen on graphene. The H2 axis is parallel to the graphene layer and oriented parallel to the vertical C–Cbonds as depicted in the lower left corner.

18 19 20 21

17

18

19

20

X

Y

Graphene PES H2 perpendicular to C C

0.

0.2

0.4

0.6

0.8

1.

1.2

1.4

1.6

1.8

2.

2.2

2.4

2.6

2.8

3.

meV

FIG. 5. As in Fig. 4, but in this case the H2 axis is parallel to the graphene layer and oriented perpendicular to the vertical C–C bonds as depicted in the lowerleft corner.

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044706-7 Petucci et al. J. Chem. Phys. 139, 044706 (2013)

18 19 20 21

17

18

19

20

X

Y

Graphene PES H2

0.

0.2

0.4

0.6

0.8

1.

1.2

1.4

1.6

1.8

2.

2.2

2.4

2.6

2.8

meV

FIG. 6. Potential energy surface of H2 on graphene including full orientational relaxations of the molecular hydrogen.

of migration energy barriers of H2 on graphene, it is observedthat the orientation of the H2 molecule has a large effect onthe barriers. A more realistic potential energy surface of H2

on graphene should therefore include the orientations of themolecule that minimize the energy at a specific point on thesurface. It is worth noting that the six circular rings in the po-tential energy surface graphs represent the position of carbonatoms in graphene.

In Figure 6, we have recalculated the PES with the orien-tational constraints on H2 lifted, restoring degrees of freedomin the molecule’s inclination and azimuthal angles while con-tinuing to hold the xCM and yCM coordinates fixed. This wasachieved by a simulated annealing technique38 that could besummarized in the following steps; first, a small square por-tion of the graphene surface including a carbon hexagon ringwas considered and divided into a grid of 41 by 41 units or atotal of 1681 cells; second, the (xCM, yCM) coordinates of theCM of an H2 molecule were frozen in the center of the firstcell and zCM as well as the two hydrogen atoms of H2 andthe carbon atoms of graphene were allowed to relax; third,using a NVT-MD run and the Berendsen thermostat the sys-tem was heated to about 10 K using 6 × 104 MD steps andthen all the kinetic energy was drained in 2 × 105 MD stepsusing a drag force –γ v, where γ = 0.0005 eV∗ps/Å2. Theprocess is then repeated for the rest of the 1680 cells. Thesimulated annealing technique allows the system to evolveover the energy barriers of the local minima and reach theglobal minimum configuration. In comparing Figure 6 withFigures 4 and 5, it is evident that the included relaxations dueto the change in H2 orientation have not significantly alteredthe PES. This is an indication that the constrained H2 orienta-

tions used in the two initial PES calculations (Figs. 4 and 5),i.e., with the H2 parallel to the surface, were very close tothose lowest in energy. This is supported by the binding en-ergy calculations which showed that the orientations lowest inenergy occur when H2 is aligned along the graphene surface.When comparing the minimum configurations obtained ateach grid point by the two methods (constrained minimizationand annealing), the only additional relaxations that occurredvia simulated annealing were rotations about the azmithal an-gle, with the H2 remaining flat on the surface. Rotations as-sociated with this degree of freedom, which cause very smallchanges in energy, restore the expected symmetry of the PES(Fig. 6), as each of the top, bridge, and hollow sites areequivalent.

The formation of molecular hydrogen through the Lang-muir Hinshelwood (L-H)22, 26 mechanism is obtained by con-ducting three NEB runs with the appropriate initial and finalstates. The initial states of molecular hydrogen are para, meta,and nearest neighbor hollow sites (H), and the final statesare H3, H3, and B3. Initial and final states for these threeconfigurations are presented in Figure 7. Formation energiesof molecular hydrogen from atomic hydrogen physisorbedat para, meta, and hollow sites on graphene are 3.17 meV,1.54 meV, and 5.44 meV, respectively. Morisset et al.,18 basedon a 3D wave packet calculations, showed that within a colli-sion energy range of 2–50 meV, there exists a high probabil-ity for the formation of H2 through the L-H mechanism. Fur-thermore, our calculation of molecular hydrogen formationthrough chemisorbed atomic hydrogen resulted in high barri-ers that make the formation of molecular hydrogen throughthe chemisorption of atomic hydrogen very unlikely.

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044706-8 Petucci et al. J. Chem. Phys. 139, 044706 (2013)

Initial Configuration NEB Minimum Energy Path Final Configuration

0.0 0.1 0.2 0.3 0.4 0.54.508

4.509

4.510

4.511

4.512

4.513

4.514

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4

reaction coordinate

EeV

LH H2 formation para H3

E : 3.2 meV

E : 4.5 eV

0.0 0.1 0.2 0.3 0.44.5090

4.5095

4.5100

4.5105

4.5110

4.5115

4.5120

4.5125

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4

reaction coordinate

EeV

LH H2 formation meta H3

E : 1.5 meV

E : 4.5 eV

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.74.500

4.502

4.504

4.506

4.508

4.510

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4

reaction coordinate

EeV

LH H2 formation H B

E : 5.4 meV

E : 4.5 eV

FIG. 7. NEB minimum energy path calculations for the formation of molecular hydrogen via the Langmuir Hinshelwood mechanism from (top) para to H3,(middle) meta to H3, and (bottom) nearest neighbor hollow sites to B3. The insets show a zoomed in portion of the reaction path with altered vertical scaling tomake the reaction barrier visible.

B. Dynamical properties of H2 on graphene

A histogram of the orientation of H2 on graphene is plot-ted in Figure 8 for a 1600 ps simulation at 100 K where the an-gular orientation was recorded every 2 fs. A sharp peak withnarrow width at θ = 0 indicates that molecular hydrogen ismost frequently found oriented parallel to the graphene sur-face, i.e., θ = 0 plus or minus a few degrees is the most prob-able orientation. The other H2 orientations occur with aboutthe same frequency, and therefore the probabilities of findingH2 in any other orientation are all about the same. A NEBcalculation was performed in order to probe the barrier to ro-

tation at the hollow site. The energy barrier for H2 rotatingfrom θ = 0 to θ = π /2 (i.e., H1 to H2) was calculated to be9 meV.

Trajectories of unwrapped coordinates of the center ofmass of the H2 molecule on graphene are generated usingNVT-MD runs at various temperatures. Two dimensional dif-fusion coefficients of H2 on graphene, at temperatures rangingfrom 40 K to 100 K, were obtained using Eqs. (1)–(3) and re-ported in Table III. A plot of the logarithm of the diffusioncoefficient versus the reciprocal of temperature is shown inFigure 9. A linear fit to the data of Table III results ina diffusion activation energy barrier Ea = 9.8 meV and a

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044706-9 Petucci et al. J. Chem. Phys. 139, 044706 (2013)

1.5 1.0 0.5 0.0 0.5 1.0 1.50

5000

10 000

15 000

20 000

25 000

30 000

Angle radians

Feq

uenc

yof

Occ

urre

nce

H2 orienation angle with respect to the graphene surface T 100K

FIG. 8. Histogram plot of H2 angular orientation on graphene at 100 K for a1600 ps simulation.

pre-exponential Do = 3276(Å2/ps). Our energy barrier of9.8 meV is in reasonable agreement with our zero temperatureresults and the corresponding values of 10 meV and 14 meVof Costanzo et al.,37 respectively. The attempt frequency wascalculated using Eq. (4), (γ o = 4Do/ro

2), which yields a valueof γ 0 = 6.5 × 1015 s−1. For a typical adatom the attempt fre-quency is regularly cited as of the order of 1012–1013 s−1. Thisdiscrepancy is attributed to the low mass of H2 which is sev-eral orders of magnitude smaller than for a typical adatom.The neglect of quantum effects such as tunneling, which isexpected to be relevant here due to the small mass and Ediff

especially at low temperatures, could further increase the dif-fusivity of H2.

Several NVT-MD runs on a graphene layer with periodicboundary conditions in all three directions generated equi-

TABLE III. (a) Data for molecular hydrogen diffusion on graphene; (b) datafor molecular hydrogen lifetime on graphene; and (c) data for molecular hy-drogen diffusion in graphite.

H2 on graphene H2 in graphite

(a) (b) (c)

T (K) D (Å2/ps) T (K) τ (ps) T (K) D (Å2/ps)

40 175.71 250 5.79 90 2.1650 385.92 300 4.49 100 2.9760 490.17 400 3.55 110 3.1670 656.99 500 3.03 125 3.5680 789.03 600 2.67 150 4.0890 849.91 750 2.32 175 6.10100 1082.79 900 2.22 200 10.95

1000 2.08 250 16.96300 25.76350 33.61400 38.41450 43.77500 60.75550 69.45600 82.01650 86.16700 89.95

10 12 14 16 18 20 22 24

5.5

6.0

6.5

7.0

1000 K T

lnD

2ps

Arrhenius plot for H2 diffusion on graphene

FIG. 9. Temperature dependence of the diffusion coefficient of molecularhydrogen on graphene.

librated trajectories at temperatures ranging from 250 K to1000 K. Average lifetimes of H2 on graphene at these vari-ous temperatures are calculated and reported in Table III. Theperiodic boundary condition along the z-direction effectivelyextends the simulation, allowing a sufficient number of trap-ping and detrapping events to occur during a single run for thecalculation of the mean lifetime at each temperature. A graphof the logarithm of the average lifetime versus the reciprocalof the temperature is shown in Figure 10. A linear fit to thedata of Table III results in an activation barrier of 28.7 meVand a pre exponential τ 0 = 1.52 ps. The activation barrierobtained from the on-surface lifetime of molecular hydrogenis assumed to be equal to the detrapping (desorption) activa-tion barrier. This is because the inverse of hydrogen lifetimeis equal to detrapping/trapping jump frequency. The obtaineddesorption barrier is about half of the one obtained from thestatic calculations performed in Sec. III A. As can be seenfrom Table III, an increase in the temperature of the simu-lation resulted in a decrease in the average lifetime. This isconsistent with the findings of Borodin et al.36 for atomic hy-drogen on graphene.

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.50.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

1000 K T

lnt

ps

Arrhenius plot for H2 desorption on graphene

FIG. 10. Temperature dependence of the average lifetime of H2 on graphene.

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044706-10 Petucci et al. J. Chem. Phys. 139, 044706 (2013)

1.0 0.5 0.0 0.5 1.00

1

2

3

4

Angle radians

Pro

babi

lity

Den

sity

H2 orienation angle with respect to the graphene bilayer surface

100 K

300 K

700 K

FIG. 11. Angular probability density function f(θ ) versus θ for H2 ingraphite for temperatures 100 K, 300 K, and 700 K.

C. Dynamical properties of H2 in graphite

The angular probability density f(θ ), in the case of H2

in graphite, is approximated with a Gaussian. In Figure 11plots of f(θ ) versus θ are presented for three temperatures:100, 300, and 700 K. The densities have a maximum at θ = 0◦

and a minimum at θ = ±90◦. Furthermore, as the temperatureincreases the density becomes wider and less peaked. Theseresults are consistent with those given by Herrero et al.35 Theprobability densities indicate that the most likely orientationof H2 is parallel to the graphite layers. However, in contrastto the single graphene layer result, the peaked nature of thedensity indicates that angles different from zero are not allequally likely to occur, those angles closest to zero are moreprobable.

NVT molecular dynamics simulations of H2 in the inter-layer space of a graphite system were performed, as describedin Sec. II. Using Eqs. (1)–(3), the diffusion coefficients at var-ious temperatures, ranging from 90 to 700 K, are calculatedand reported in Table III. The relaxation of the carbon atomswas found to have a substantial impact on the diffusion of H2

in the graphite interlayer space. Freezing all the graphite de-grees of freedom reduces the H2 diffusion coefficient to thepoint that no diffusion jumps were detected in 1600 ps simu-lations, even at the highest temperature considered (700 K). Instatic calculations (0 K), where the carbon atoms are allowedto relax, the interlayer separation was calculated as 3.44 Å and3.39 Å with and without the presence of H2, respectively. Theincrease in the presence of H2 is due to the significant corruga-tion that occurs in the layers near the molecular hydrogen. Theaverage interlayer spacing was also calculated during the MDsimulations for each temperature. This separation increaseslinearly with temperature over the entire temperature range(90–700 K).

A graph of ln(D) versus 1/T along with linear fits to thedata of Table III are presented in Figure 12. An interestingfeature of Figure 12 is that a single line alone cannot fit all thedata points. On the other hand, two lines with different slopescan accurately fit all the data points. These high and lowtemperature regimes occur at temperatures above or below∼155 K, respectively. The lines fitted to high and low tem-

2 4 6 8 10 12

1

2

3

4

5

1000 K T

lnD

2ps

Arrhenius plot for H2 diffusion in graphite interlayer spacing

FIG. 12. Temperature dependence of the diffusion coefficient of molecularhydrogen in graphite along with the two linear fits.

perature data resulted in activation energies, pre-exponentials,and attempt frequencies of (Ea = 51.2 meV, Do = 195 Å2/ps,γ 0 = 3.9 × 1014 s−1) and (Ea = 11.5 meV, Do = 10 Å2/ps,γ 0 = 2 × 1013 s−1), respectively. These activation energy bar-riers are larger than the value calculated for H2 on graphene.We believe the reason for this increase in energy is due tostronger bonding that occurs when H2 is located betweengraphite layers as opposed to on the surface of a singlegraphene layer. One observes that the diffusion coefficient ofan H2 molecule in graphite is, in general, smaller than the oneon graphene. For example, at a temperature of T = 100 K,values of 2.97 Å2/ps and 1.08 × 103 Å2/ps are obtained fordiffusion of H2 in graphite and on graphene, respectively.This is expected because H2 is more strongly bonded to thegraphite layers than to a single layer graphene. Our generaltrend for diffusion of H2 in graphite is similar to the result ofHerrero et al.35

In order to investigate the change in slope present in theArrhenius plot of Figure 12, the trajectory of the diffusing H2

was analyzed for various temperatures in the low and hightemperature regimes. Figure 13 depicts the typical behaviorobserved for one temperature in the low (90 K) and high(700 K) regimes. At low temperatures, the H2 follows thestructure of the graphite layers, diffusing along the hexagonrings (C–C bonds). The majority of the diffusion jumps gofrom a bridge site, to a neighboring top site, then again toa neighboring bridge site, and so on. At the high tempera-tures, this diffusion behavior is no longer reflected in the H2

trajectory. Jumps consist of the molecular hydrogen travelingrelatively long distances colliding with the graphite sheets andchanging direction before briefly attaching to a particular site.At high temperature the trajectory is more consistent with thatof the Brownian motion of a free gas particle. This indicatestwo distinct diffusion mechanisms or behaviors at low andhigh temperatures, which manifests as the change in slope inthe Arrhenius plot. At low temperatures, the diffusion is morelocal (smaller jumps along C–C) leading to smaller diffusioncoefficients (and thus slope) as compared to higher tempera-tures where the diffusion is less constrained and much longerjumps occur.

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044706-11 Petucci et al. J. Chem. Phys. 139, 044706 (2013)

0 5 10 15 0 25

0

5

0

5

0

5

0

5

0 50 1 0 1 0

0

0

50

0

0

2

1

1

2

2

3

3

x

yH2 Trajectory, T 90K

0 5

5

10

15

x

y

H2 Trajectory, T 700K

FIG. 13. The traced trajectory of molecular hydrogen in the graphite inter-layer space during the initial 400 ps (after equilibration) of two MD simula-tions at 90 K (left) and 700 K (right).

IV. SUMMARY

Several adsorption sites of molecular hydrogen ongraphene as well as their orientations with respect to the sur-face are investigated. Based on our zero temperature calcula-tions, strongest and weakest sites for the molecular hydrogenon graphene are the hollow and top sites, with potential min-ima of −54.3 meV and −41.6 meV, respectively. These valuesare in good agreement with estimate of the potential minimum(51.7 meV) obtained from experiments of H2 scattering fromgraphite16 and of experiments of adsorption of H2 on carbonnanostructures (50 meV).39 Similarly, a value of −28.7 meVwas obtained for the desorption barrier of an H2 moleculefrom graphene at finite temperature. Our calculations of mi-gration energies of molecular hydrogen from various initial tofinal sites, at zero temperature, ranges from 3.7 to 12.3 meV.Similarly, our finite temperature calculation of H2 diffusionon graphene resulted in an energy barrier of 9.8 meV. All ofour surface diffusion barriers are in good agreement with theresults given by Farebrother et al.21 Our surface potentialsgraphs also complement our calculated energy barriers. Thediffusion of an H2 molecule in the interlayer space betweentwo layers of graphite resulted in values of 51.2 meV and11.5 meV for the energy barriers for the high and low temper-ature regimes, respectively. Our diffusion barriers are differ-ent than the two values, 200 meV and 100 meV, which wereobtained by Herrero et al.35 using the tight binding method.The formation of molecular hydrogen through the physisorp-tion of atomic hydrogen on graphene is shown to be a likelymechanism for the production of an H2 molecule. On the otherhand, hydrogen atoms chemisorbed on graphene are veryunlikely to have the necessary mobility to form molecularhydrogen.

In summary, we have calculated, using classical MD inconjunction with the semi-empirical AIREBO potential, staticand dynamical properties of an H2 molecule on graphene andin graphite. These classical calculations have advantages overthose of first principles. They require modest computationalresources and van der Waals (vdW) interactions are imple-mented in the potential. Finally, our results are in reasonableagreement with those of first principles that include vdW cor-rections and with experiments.

ACKNOWLEDGMENTS

The authors are grateful to the Dean of the College ofNatural Sciences and Mathematics, Dr. Snavely, for providingpartial support towards the construction of the computer clus-ter that was built at IUP, where most of the calculations forthis work were performed. We would also like to thank JoeShyrock, for providing useful hardware for the constructionof this cluster. G.V. would like to acknowledge support fromNASA (Grant NNX12AF38G). Finally, supercomputer timethrough a grant from the Pittsburgh Supercomputing Centeris greatly appreciated.

1A. C. Dillon, K. M. Jones, T. A. Bekkedahl, C. H. Kiang, D. S. Bethune,and M. J. Heben, Nature (London) 386, 377 (1997).

2F. Darkarim and D. Levesque, J. Chem. Phys. 109, 4981 (1998).3C. Liu, Y. Y. Fan, M. Liu, H. T. Cong, H. M. Cheng, and M. S. Dresselhans,Science 286, 1127 (1999).

4Q. Wang and K. J. Johnson, J. Chem. Phys. 110, 577 (1999).5M. I. Bercu and V. V. Grecu, Rom. J. Phys. 41, 371 (1996); Y. Ferro, D.Teillet-Billy, N. Rougeau, V. Sidis, S. Morisset, and A. Allouche, Phys.Rev. B 78, 085417 (2008); D. Bachellerie, M. Sizun, F. Aguillon, D. Teillet-Billy, N. Rougeau, and V. Sidis, Phys. Chem. Chem. Phys. 11, 2715 (2009);D. W. Boukhvalov, ibid. 12, 15367 (2010); A. A. Dzhurakhalov and F. M.Peeters, Carbon 49, 3258 (2011).

6H. McKay, D. J. Wales, S. J. Jenkins, J. A. Verges, and P. L. deAndres,Phys. Rev. B 81, 075425 (2010); Z. SIjivancanin, M. Andersen, L.Horneker, and B. Hammer, ibid. 83, 205426 (2011); L. Hornekaer, E.Rauls, W. Xu, Z. SIjivancanin, R. Otero, I. Stensgaard, E. Laegsgaard, B.Hammer, and F. Besenbacher, Phys. Rev. Lett. 97, 186102 (2006).

7A. K. Geim and K. S. Novoselov, Nature Mater. 6, 183 (2007).8B. Lepeit, D. Lemoine, Z. Medina, and B. Jackson, J. Chem. Phys. 134,114705 (2011).

9K. S. Novoselov, Science 306, 666 (2004).10D. C. Elias, R. R. Nair, T. M. G. Mohiuddin, S. V. Morozov, P. Blake, M.

P. Halsall, A. C. Ferrari, D. W. Boukhvalov, M. I. Katsnelson, A. K. Geim,and K. S. Novoselov, Science 323, 610 (2009).

11Y. Ferro, F. Marinelli, and A. Allouche, J. Chem. Phys. 116, 8124(2002).

12G. Stan and M. W. Cole, J. Low Temp. Phys. 110, 539 (1998).13M. Mirnezhad, R. Ansari, M. Seifi, H. Rouhi, and M. Faghihnasiri, Solid

State Commun. 152, 842 (2012).14G. Vidali, J. Low Temp. Phys. 170, 1 (2013); “H2 formation on interstellar

grains,” Chem. Rev. (submitted).15E. Ghio, L. Mattera, C. SaIvo, F. Tommasini, and U. Valbusa, J. Chem.

Phys. 73, 556 (1980).16L. Mattera, R. Rosatelli, C. Salvo, F. Tommasini, U. Valbusa, and G. Vidali,

Surf. Sci. 93, 515 (1980).17T. Zecho, A. Guttler, X. Sha, D. Lemoine, B. Jackson, and J. Kuppers,

Chem. Phys. Lett. 366, 188 (2002).18X. Sha, B. Jackson, and D. Lemoine, J. Chem. Phys. 116, 7158

(2002).19E. Areou, G. Cartry, J. M. Layet, and T. Angot, J. Chem. Phys. 134, 014701

(2011).20P. Parneix and Ph. Bréchignac, Astron. Astrophys. 334, 363 (1998); M.

Rutigliano, M. Cacciatore, and G. D. Billing, Chem. Phys. Lett. 340, 13(2001).

21A. J. Farebrother, A. J. H. M. Meijer, D. C. Clary, and A. J. Fisher, Chem.Phys. Lett. 319, 303 (2000); L. Jeloaica and V. Sidis, ibid. 300, 157 (1999);A. Meijer, A. Farebrother, D. Clary, and A. Fisher, J. Phys. Chem. A105, 2173 (2001); B. Jackson and D. Lemoine, J. Chem. Phys. 114, 474(2001).

22S. Morisset, F. Aguillon, M. Sizun, and V. Sidis, Phys. Chem. Chem. Phys.5, 506 (2003); Chem. Phys. Lett. 378, 615 (2003); J. Chem. Phys. 121,6493 (2004); 122, 194702 (2005).

23D. Bachellerie, M. Sizun, D. Teillet-Billy, N. Rougeau, and V. Sidis, Chem.Phys. Lett. 448, 223 (2007).

24M. Rutigliano, and M. Cacciatore, AIP Conf. Proc. 1125, 123 (2009); S.Casolo, R. Martinazzo, M. Bonfanti, and G. Tantardini, J. Phys. Chem.A 113, 14545 (2009); M. Bonfanti, S. Casolo, G. F. Tantardini, and R.Martinazzo, Phys. Chem. Chem. Phys. 13, 16680 (2011).

25R. Martinazzo and G. Tantardini, J. Chem. Phys. 124, 124702 (2006).

Downloaded 25 Jul 2013 to 108.3.93.119. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

044706-12 Petucci et al. J. Chem. Phys. 139, 044706 (2013)

26M. Bonfanti, R. Martinazzo, G. Tantardini, and A. Ponti, J. Phys. Chem. C111, 5825 (2007).

27R. M. Ferullo, N. F. Domancich, and N. J. Castellani, Chem. Phys. Lett.500, 283 (2010).

28J. S. Arellano, L. M. Molina, A. Rubio and J. A. Alonso, J. Chem. Phys.112, 8114 (2000).

29M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (ClarendonPress, Oxford 1994).

30S. J. Stuart, A. B. Tutein, and J. A. Harrison, J. Chem. Phys. 112, 6472(2000).

31S. Plimpton, J. Comput. Phys. 117, 1 (1995).32H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, A. DiNola, and

J. J. R. Haak, J. Chem. Phys. 81, 3684 (1984).

33G. Henkelman and H. Jonsson, J. Chem. Phys. 113, 9978 (2000); D.Sheppard, R. Terrell, and G. Henkelman, ibid. 128, 134106 (2008).

34O.-E. Hass, J. M. Simon, and S. Kjelstrup, J. Phys. Chem. C 113, 20281(2009).

35C. P. Herrero and R. Ramirez, J. Phys. D: Appl. Phys. 43, 255402(2010).

36V. A. Borodin, T. T. Vehvilamen, M. G. Ganchenkova, and R. M. Nieminen,Phys. Rev. B 84, 075486 (2011).

37F. Costanzo, P. L. Silvestrelli, and F. Ancilotto, J. Chem. Theory Comput.8, 1288 (2012).

38S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, Science 220, 671 (1983).39H. G. Schimmel, G. Nijkamp, G. J. Kearley, A. Riveraa, K. P. de Jong, and

F. M. Mulder, Mater. Sci. Eng., B 108, 124 (2004).

Downloaded 25 Jul 2013 to 108.3.93.119. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions