New Method for Atomistic Modeling of the Microstructure ofActivated Carbons Using Hybrid Reverse Monte Carlo Simulation

Thanh X. Nguyen,† Nathalie Cohaut,‡ Jun-Seok Bae,† and Suresh K. Bhatia*,†

DiVision of Chemical Engineering, The UniVersity of Queensland, St. Lucia, Brisbane QLD 4067, Australia, andCentre de Recherche sur la Matiere DiVisee, UMR 6619 1b rue de la Ferollerie,

45071 Orleans Cedex 2, France

ReceiVed January 31, 2008. ReVised Manuscript ReceiVed April 19, 2008

We propose a new hybrid reverse Monte Carlo (HRMC) procedure for atomistic modeling of the microstructureof activated carbons whereby the guessed configuration for the HRMC construction simulation is generated using thecharacterization results (pore size and pore wall thickness distributions) obtained by the interpretation of argon adsorptionat 87 K using our improved version of the slit-pore model, termed the finite wall thickness (FWT) model (Nguyen,T. X.; Bhatia, S. K. Langmuir 2004, 20, 3532). This procedure overcomes limitations arising from the use of short-range potentials in the conventional HRMC method, which make the latter unsuitable for carbons such as activatedcarbon fibers that are anisotropic with medium-range ordering induced by their complex pore structure. The newlyproposed approach is applied specifically for the atomistic construction of an activated carbon fiber ACF15, providedby Kynol Corporation (Nguyen, T. X.; Bhatia, S. K. Carbon 2005, 43, 775). It is found that the PSD of the ACF15’sconstructed microstructure is in good agreement with that determined using argon adsorption at 87 K. Furthermore,we have also found that the use of the Lennard-Jones (LJ) carbon-fluid interaction well depth obtained from scalingthe flat graphite surface-fluid interaction well depth taken from Steele (Steele, W. A. Surf. Sci. 1973, 36, 317) providesan excellent prediction of experimental adsorption data including the differential heat of adsorption of simple gases(Ar, N2, CH4, CO2) over a wide range of temperatures and pressures. This finding is in agreement with the enhancementof the LJ carbon-fluid well depth due to the curvature of the carbon surface, found by the use of ab initio calculations(Klauda, J. B.; Jiang, J.; Sandler, S. I. J. Phys. Chem. B 2004, 108, 9842).

1. IntroductionPorous carbons have long been used for gas- and liquid-phase

adsorptive separations as well as the storage of volatile compoundsbecause of their high surface area and strong adsorption forcefield. It is well recognized that the microstructure of porouscarbons governs adsorption equilibrium and dynamic behavior.Accordingly, the synthesis and manufacture of carbons with thedesired adsorptive properties require a reliable characterizationof the internal structure of the carbon, which can be used topredict adsorption equilibrium and kinetics from a matureunderstanding of the behavior of fluids in confined spaces. Ingeneral, the microstructure of activated carbons normally indicatesthat they contain pores formed between curled carbon layershaving various degrees of curvature that can even form fullerene-like structures, as have been observed in high-temperature heated-activated carbons by high-resolution transmission microscopy(HRTEM).5–7 From this observation, the slit-pore model com-posed of a set of unconnected slitlike pores is widely acceptedfor the characterization of the microstructure of porous carbonsusing gas physisorption.1,8,9 Meanwhile, fullerene-like atomisticmodels are also used to represent the microstructure of hard

activated carbons prepared by high-temperature treatment of theactivated carbons.10,11

Despite such a crude approximation of the slit-pore model tothe real porous carbon structure, it has been shown that its recentlyimproved version in the form of our finite wall thickness (FWT)model, whereby pore size and pore wall thickness distributionsare simultaneously characterized, provides a correct predictionof the adsorption equilibrium of simple gases over a wide rangeof temperatures and pressures in several activated carbons, whichare free of pore-accessibility problems.12,13 However, the slit-pore model in general disregards surface roughness as well asthe complexities of pore network accessibility. This leads to theinappropriateness of the model when used for the study ofadsorption equilibrium and dynamics in porous carbons whosemicrostructure has a pore-accessibility problem for someadsorbates14,15 as well as the inability of the model to determinetransport diffusivity through nanoporous carbons without aknowledge of tortuosity. Finally, the representation of a realcarbon surface by a perfect graphitic one does not provide thecorrect prediction of the isosteric heat of adsorption in the lowadsorption coverage limit.16 Consequently, because of bothpractical and fundamental interest in relation to adsorptiondynamics and process design, 3D construction of the micro-structure of porous carbons is essential.

* To whom correspondence may be addressed. E-mail: s.bhatia@eng.uq.edu.au.

† The University of Queensland.‡ Centre de Recherche sur la Matiere Divisee.(1) Nguyen, T. X.; Bhatia, S. K. Langmuir 2004, 20, 3532.(2) Nguyen, T. X.; Bhatia, S. K. Carbon 2005, 43, 775.(3) Steele, W. A. Surf. Sci. 1973, 36, 317.(4) Klauda, J. B.; Jiang, J.; Sandler, S. I. J. Phys. Chem. B 2004, 108, 9842.(5) Harris, P. J. F.; Tsang, S. C. Philos. Mag. A 1997, 76, 667.(6) Harris, P. J. F.; Burian, A.; Duber, S. Philos. Mag. Lett. 2000, 80, 381.(7) Harris, P. J. F. Philos. Mag. 2004, 84, 3159.(8) Lastoskie, C. M.; Gubbins, K. E.; Quirke, N. J. Phys. Chem. 1993, 97,

4786.(9) Ravikovitch, P. I.; Vishnyakov, A.; Russo, R.; Neimark, A. V. Langmuir

2000, 16, 2311.

(10) Terzyk, A. P.; Furmaniak, S.; Gauden, P. A.; Harris, P. J. F.; Wloch, J.;Kowalczyk, P. J. Phys.: Condens. Matter 2007, 19, 406208.

(11) Petersen, T. C.; Snook, I. K.; Yarovsky, I.; McCulloch, D. G.; O’Malley,B. J. Phys. Chem. C 2007, 111, 802.

(12) Nguyen, T. X.; Bhatia, S. K. J. Phys. Chem. B 2004, 108, 14032.(13) Nguyen, T. X.; Bhatia, S. K.; Nicholson, D. Langmuir 2005, 21, 3187.(14) Nguyen, T. X.; Bhatia, S. K. Langmuir 2008, 24, 146.(15) Nguyen, T. X.; Bhatia, S. K. J. Phys. Chem. C 2007, 111, 2212.(16) He, Y.; Seaton, N. A. Langmuir 2005, 21, 8297.(17) McGreevy, R. L.; Putsai, L. Mol. Simul. 1988, 1, 359.(18) Opletal, G.; Petersen, T. C.; McCulloch, D. G.; Snook, I. K.; Yarovsky,

I. Mol. Simul. 2002, 28, 927.

7912 Langmuir 2008, 24, 7912-7922

10.1021/la800351d CCC: $40.75 2008 American Chemical SocietyPublished on Web 07/01/2008

First, Gubbins and workers19–23 constructed atomistic structuralmodels for porous carbons such as activated mesocarbonmicrobead (a-MCMB) carbons and conducted a series ofsacharose char simulations using reverse Monte Carlo (RMC)17

and hybrid reverse Monte Carlo (HRMC).18 Subsequently,Nguyen et al.24 employed a combination of the environmental-dependent interaction (EDIP) potential adapted for carbon byMarks,25 which covers a longer carbon-carbon interaction rangeup to the interlayer spacing distance of graphite (potential cutoff) 0.32 nm), and shorter-range reactive empirical bond order(REBO) potential26 (potential cutoff ) 0.2 nm) for the HRMCconstruction of the microstructure of saccharose char CS1000a.More recently, the HRMC technique has been used by Petersonet al.11 to construct an atomistic structural model of high-temperature heated glassy carbon. Although the RMC- or HRMC-constructed algorithms equipped with short-range carbon po-tentials (EDIP or REBO) have already been used for the atomisticconstruction of porous carbons, moderate success can be seenonly for rather high density porous carbon (>1.5 g/cm3). Thismay be due to the fact that such short-range carbon potentialsused in the HRMC algorithm are unable to capture the longer-range interatomic interaction induced by the pore structure oflow-density carbon such as activated carbon fiber ACF15 (Fc ≈0.9 g/cm3). However, the experimental dimensional pair distri-bution function does not guarantee a unique representation ofthe 3D microstructure of the carbon. Consequently, the currentHRMC method with the use of an arbitrary initial configurationis very time-consuming and even unfeasible for the constructionof low-density anisotropic carbons such as ACFs.

To improve the current HRMC method, we propose a novelHRMC-constructed algorithm whereby the initial configurationis generated on the basis of the characteristic results (PSD andpore wall thickness distribution) of porous carbons obtained bythe interpretation of Ar adsorption at 87 K using the FWT model.1

In this way, the generated initial configuration is expected to beclose to the target one and has long-range order, which speedsup the HRMC construction process significantly. The proposedmethod is applied here to construct an atomistic model of anACF15 activated carbon fiber provided by the Kynol Corporationinvestigated in our previous work.2 Finally, we validate ourHRMC construction method against experimental adsorption dataincluding the differential heat of adsorption of the simple gases(Ar, N2, CH4, CO2) in the actual ACF15 carbon over wide rangesof temperature and pressure.

2. Experimental Section

An ACC-5092-15 (ACF15) activated carbon fiber, provided bythe Kynol Corporation, was degassed at 300 °C overnight. Thedegassed carbon sample was analyzed in an ASAP2010 Micromeriticsvolumetric adsorption analyzer to obtain the N2 adsorption isothermat 77 K and the CO2 adsorption isotherm at 273 K, at atmosphericpressure. Subsequently, the degassed sample was also analyzed ona gravimetric sorption system (Rubotherm PrazisionsmesstechnikGmbH, Bochum, Germany) to obtain high-pressure adsorption

isotherms of CH4 at 310 and 353 K and CO2 at 310 and 333 K fora wide range of pressure up to 200 bar. Detailed information on thegravimetric sorption system was also presented elsewhere.27

For the determination of the structure factor of ACF15 over awide q range (q ) 4π sin(θ/2)/λ, 0.2 < q <140 nm-1), SAXS andX-ray diffraction curves were separately recorded, corrected forabsorption and polarization effects, and joined. SAXS experimentswere carried out under pinhole collimation by using a 12 kW Rigakurotating anode X-ray generator. During acquisitions, the groundACF was placed between two mylar windows inside a 5-mm-diameterhole, pierced in a brass plate. The sample-to-detector distance was300 mm, and the wavelength was 0.154 nm. Intensities were recordedwith an Elphyse linear Ar-CO2-sensitive position detector in the qrange of 0.2-7.0 nm-1. We recorded the X-ray diffraction (XRD)patterns by using a curved position-sensitive detector (INEL CPS120) at the Mo KR1 wavelength (λ ) 0.070926 nm) in transmissiongeometry. With this configuration, we are able to obtain the rangeq ) 2.3-140 nm-1. Long exposure times were needed to performthe measurements for large q values. We recorded the capillarycontribution and subtracted it during the analysis. Furthermore,intensities were corrected for geometric q-dependent artifacts suchas absorption and polarization, and then normalized before calculationof the structure factor S(q).28

The experimental pair distribution function is determined byFourier transformation using the bulk density of the ACF15 carbon(Fcb ) 0.876 g/cm3, as discussed in subsection 3.2.1). It is noted thatthe microstructure of low-density porous carbons such as the ACF15carbon is anisotropic in nature. Accordingly, the bulk carbon densitymay not be equal to the local density. This leads to the presence ofnegative values between the first nearest-neighbor peaks in theresultant pair distribution function such as the first and second aswell as the second and third peaks. Accordingly, these unphysicalnegative density fluctuations are truncated from the resultant pairdistribution function. Consequently, the pair distribution functionwith negative values truncated is used as the experimental pairdistribution function for HRMC construction of the microstructureof the ACF15 carbon in this work.

3. Mathematical Modeling

3.1. Simulation and Modeling Methods. 3.1.1. HybridReVerse Monte Carlo Simulation. In the current work, we utilizedthe HRMC simulation method18 whereby the conventional RMCmethod17 was supplemented with an energy constraint that enablesus to generate a stable carbon structure model whose appropriateobjective function matches the corresponding experimental onein order to construct the ACF15’s 3D microstructure. In general,the HRMC method employs stochastic sampling of configurationsin the multicanonical ensembles in which the number of particles,N, the volume of the simulated system, V, and the temperature,T, of each ensemble are held constant such that minimization ofboth the deviation for pair distribution function and the totalenergy of the simulated system are simultaneously achieved.The acceptance probability for a trial move, Pacc, is given as

Pacc )min{1, exp(-(∆�2 ⁄ 2+∆E ⁄ kBT))} (1)

The first term on the right-hand side (RHS) of eq. 1 is theacceptance probability used in the conventional RMC method.Here, a pair distribution function (PDF) is utilized as the objectivefunction. Accordingly, the structural quantity, �2, is defined by

�2 )∑i)1

Nexp (gsim(ri)- gexp(ri))2

σi(2)

where Nexp is the number of experimental PDF values, gexp(ri),and gsim(ri) is the corresponding simulated PDF. σi is the estimated

(19) Thomson, K. T.; Gubbins, K. E. Langmuir 2000, 16, 5761.(20) Pikunic, J.; Clinardv, C.; Cohaut, N.; Gubbins, K. E.; Guet, J. M.; Pellenq,

R. J. M.; Rannou, I.; Rouzaud, J. N. Langmuir 2003, 19, 8565.(21) Pikunic, J.; Gubbins, K. E.; Pellenq, R. J. M.; Cohaut, N.; Rannou, I.;

Guet, J. M.; Clinard, C.; Rouzaud, J. N Appl. Surf. Sci. 2002, 196, 98.(22) Pikunic, J.; Llewellyn, P.; Pellenq, R. J. M.; Gubbins, K. E. Langmuir

2005, 21, 4431.(23) Jain, S. K.; Pellenq, R. J.M.; Pikunic, J. P.; Gubbins, K. E. Langmuir

2006, 22, 9942.(24) Nguyen, T. X.; Bhatia, S. K.; Jain, S. K.; Gubbins, K. E. Mol. Simul. 2006,

32, 567.(25) Marks, N. A. J. Phys.: Condens. Matter 2002, 14, 2901.(26) Brenner, D. W. Phys. ReV. B 1990, 42, 9458.

(27) Bae, J.-S.; Bhatia, S. K. Energy Fuels 2006, 20, 2599.(28) Ergun, S.; Bayer, J.; Buren, W. V. J. Appl. Phys. 1967, 38, 3540.

Atomistic Modeling of ActiVated Carbons Langmuir, Vol. 24, No. 15, 2008 7913

experimental error for each PDF value. However, here σi isconsidered to be an adjustable parameter for the optimizationand is kept constant within an ensemble. ∆�2 is the scaled sumof the squared difference in g(r) between new and old configura-tions.

The second term on the RHS of eq 1 used conventionalMetropolis Monte Carlo (MMC) simulation to minimize theformation of highly strained rings such as three- and four-membered rings. Here, ∆E is the difference in total energybetween new and old trial configurations. A trial configurationis generated by a small displacement of a randomly selectedcarbon atom of the simulated system. Accordingly, the totalenergy of a trial configuration is evaluated using either theenvironmentally dependent interaction potential (EDIP), adaptedfor carbon by Marks,25 or the reactive empirical bond order(REBO) potentials developed by Brenner26 or a combination ofthese potentials, as described in our recent work.24 Although theEDIP potential covers a slightly longer range (0.32 nm) ofcarbon-carbon interaction than does the REBO potential (0.20nm), the former requires significantly more intensive computation.Consequently, to save computing time in this work we utilizedthe second-generation REBO potential recently developed byBrenner and co-workers.29 This revised potential is significantlyimproved in comparison with an earlier version26 by theincorporation of the improved analytic functions as well as anextended database for parametrization that leads to a significantlyimproved prediction of bond energies, lengths, and force constantsfor hydrocarbon molecules as well as elastic properties, interstitialdefect energies, and surface energies for diamond. As a resultof our new proposed method to preparethe initial configurationfor the HRMC construction method, which will be described indetail in the following section, it is also expected that the useof the short-range REBO potential in our current work does notgive rise to inaccuracy in the long-range order of the finalconstructed carbon model. Furthermore, the impact of the useof short-range order potential is also expected to be minimizedby the structural constraint imposed by the experimental pairdistribution function.

3.1.2. Grand Canonical Monte Carlo (GCMC) Simulation.Grand Canonical Monte Carlo (GCMC) simulation is widelyused to study adsorption equilibrium. In GCMC simulation basedon the grand canonical ensemble, the chemical potential, µ,volume, V, and temperature, T, are kept unchanged whereas thenumber of particles, N, and associated configurational energy,E, are permitted to fluctuate. Furthermore, the generation ofmicrostate configurations is carried out by the use of the well-established Metropolis sampling scheme for three types of moves:moving (including rotation for nonspherical molecule such ascarbon dioxide), creating, and deleting molecules. For each MonteCarlo (MC) step, a trial move is applied to a molecule.Subsequently, the probability of accepting each type of trial moveis determined using Adam’s algorithm.30

In the current work, we employed GCMC simulation to obtainadsorption isotherms as well as differential isosteric heat ofadsorption of all investigated gases (Ar, N2, CO2, CH4, and He)in the HRMC-constructed model of ACF15 carbon. Periodicboundary conditions are applied, and a cutoff radius of 1.47 nmis used for all GCMC simulations presented in this work. TheGCMC simulations were run for 2 × 107 Monte Carlo steps toobtain each absolute adsorbed quantity.

3.2. HRMC Three-Dimensional Construction Method ofthe Microstructure of ACF15 Carbon. 3.2.1. Preparation ofthe Initial Configuration. As briefly mentioned above, thegeneration of the initial configuration for the ACF15’s HRMCconstruction is based on its structural parameters (PSD and porewall thickness distribution) characterized by the interpretationof the argon adsorption isotherm at 87 K using our recentlyimproved version of the slit-pore model or the finite wall thicknessmodel.1 The characterization results of ACF15 by the finite wallthickness model are taken from our previous work.2 In particular,a two-step procedure is proposed to construct the initialconfiguration for the HRMC construction procedure of themicrostructure of the ACF15, as follows:

3.2.1.1. Generation of Graphitic Slitlike Pore Configuration.This step is completed on the basis of characteristic results (PSDand pore wall thickness distribution) of porous carbons obtainedby Ar adsorption at 87 K using the FWT model. It can be generallyobserved that the PSD commonly contains three prominent, well-separated peaks in the micropore range (<2.0 nm). The physicaldistance between these peaks is approximately double theinterlayer spacing distance (∼0.67 nm) so that their formationis interpreted as missing carbon sheets resulting from the activationprocess. Furthermore, this interpretation is practically supportedby the fact that the presence of curly black fringes indicative ofcarbon sheets is normally observed in transmission electronmicroscopy (TEM) images of microporous carbons. From theabove reasoning, the initial configuration can be generated as aslitlike pore model whose pore size distribution is made up ofthree typical geometric pore sizes (Hin ) 0.445, 0.734, and 1.05nm) located at the peak of three major peaks of the PSD of theporous carbon, as shown in Figure 1. Accordingly, it can beforeseen that the micropore size distribution of the microporouscarbon can be formed from the curvature of carbon sheets of thisinitial configuration during the HRMC construction process.Furthermore, from the pore size distribution three pore regionsassociated with these typical pore sizes are easily allocated, asseen in Figure 1. Subsequently, the specific surface area of thesepore regions (s1, s2, and s3) is calculated as

si ) 2∫hi1

hi2 f(H) dHH

(3)

where hi1 and hi2 are lower and upper bound pore sizes of theith pore size range region, as indicated in Figure 1, and f(H) isthe pore size distribution. The factor of 2 in eq 3 accounts forboth sides of a carbon sheet being considered.

Thus, the total number of perfect graphitic sheets, nT,corresponding to the total specific surface area sT and that, {ni},corresponding to the specific surface area, {si}, of the typicalpore regions in a unit cell are simply estimated as

nX )sX

so(4)

where subscript X corresponds to T or i and so is the specificsurface area of a single graphitic sheet in a unit cell of cubicshape, given as

so ) 2LxLy

FcbVcell(5)

where Vcell is the volume of the unit cell and LxLy, the productof the unit cell dimensions Lx and Ly, is the cross-sectional areaof the unit cell. The bulk carbon density, Fcb, is approximatelyestimated as

(29) Brenner, D. W; Shenderova, O. A.; Harrison, J. A.; Stuart, S. J.; Ni, B.;Sinnott, S. B. J. Phys.: Condens. Matter 2002, 14, 783.

(30) Adams, D. J. Mol. Phys. 1975, 29, 307.

7914 Langmuir, Vol. 24, No. 15, 2008 Nguyen et al.

Fcb )Xc

Vpore +1FHe

(6)

Here Xc ) 0.903 is the measured weight percentage of purecarbon in the ACF15 carbon sample, νpore is the total specificpore volume of the ACF15 carbon, estimated to be 0.542 cm3/gby an interpretation of Ar adsorption at 87 K using the FWTmodel,2 and FHe is the helium skeletal density of the ACF15carbon, measured as 2.0460 g/cm3 using a MicromeriticsAccupyc1340 helium pycnometer. From eq 6, the bulk carbondensity, Fcb, is determined to be equal to 0.87605 g/cm3.

The graphitic slit-like pore configuration can now be con-structed from three typical pore sizes (H1, H2, and H3), the numberof carbon sheets {ni} corresponding to the three typical poreregions, the total number of carbon sheets, nT, and the pore wallthickness distribution (p1,p2, p3, p4, p5+). In particular, a unit cellof graphitelike structure is first generated. Subsequently, thegraphitic sheets are alternatively shifted relative to each otherto create a buckled graphitic structure. Finally, the typical poresizes are created by the removal of a fractional or a full graphiticsheet such that the parameters nT, ni, and the pore wall thicknessdistribution are closely achieved.

3.2.1.2. Generation of Carbon Defects. The graphitic slitlikepore configuration as constructed above normally has a densitythat is higher than that of real carbon. Accordingly, the samenumber of carbon atoms per carbon sheet is randomly removedsuch that the carbon density of the unit cell is equal to that ofthe real carbon. This creates local defects in the slitlike pore

configuration. Accordingly, the resultant slitlike pore configu-ration, as illustrated in Figure 2, is utilized as an initialconfiguration for the HRMC construction of the microstructureof ACF15.

3.2.2. HRMC Simulation Details. In the current work, theHRMC construction procedure, as described in detail elsewhere,18

is employed to obtain a 3D model of the ACF15’s microstructure.In brief, the HRMC construction simulation procedure is carriedout using an annealing process31 utilizing multiple canonicalensembles that differ only in temperature. Accordingly, thetemperature of the HRMC simulated system is gradually decreasedfollowing

T) ToaIanneal (7)

where To is an initial temperature for equilibration of the systemand is taken to be 1000 K in this work. The a parameter is anannealing rate and is given a value of 0.95. Ianneal represents theannealing step used in the HRMC simulation and takes valuesof 1, 2...Nanneal. The total number of annealing steps, Nanneal, isdetermined from the final temperature of the HRMC simulation.In this work, the final temperature has a value of 100 K. Ascheme similar to that mentioned above was used earlier byPikunic et al.20

Furthermore, the adjusting parameter σ is controlled in a similarway to that of temperature as

σ) σoaIanneal⁄2 (8)

Here σo is an initial adjustable parameter used for the fittingof the target structure factor and is assigned a value of 0.06,which is within the common range of experimental error.

In summary, it is assumed that ACF15 carbon can be modeledas a periodic porous material with dimensions of its unit cellgiven as 2.95 nm × 2.98 nm × 3.02 nm. Subsequently, 1166carbon atoms are placed in the unit cell such that the bulk carbondensity, Fbc ) 0.87605 g/cm3, is obtained. The HRMC construc-tion simulation system is equilibrated at 1000 K for 500 000Monte Carlo (MC) steps and is followed by another 40 × 106

MC steps in the production stage, during which the simulationtemperature is gradually decreased to 100 K.

(31) Kilpatrick, S. Science 1983, 220, 671.

Figure 1. Characterization results of ACF15 obtained by argon adsorptionat 87 K taken from our previous work.2 (a) The PSD of ACF15 isdivided into three regions, each of which is represented by thecorresponding pore size, Hi, and surface area, si. (b) Pore-wall thicknessdistribution of ACF15.

Figure 2. Graphitic slitlike pore configuration, as discussed in section3.2.1.

Atomistic Modeling of ActiVated Carbons Langmuir, Vol. 24, No. 15, 2008 7915

4. Results and Discussions

In this section, we present the HRMC construction of ACF15as well as our discussions of the important aspects of the selectionof the converged configuration. Furthermore, our discussion alsofocuses on the validation of the HRMC-constructed microstructuremodel of ACF15 against experimental adsorption isotherms ofAr, N2, CO2, and CH4 over a wide range of temperature andpressure. For the latter species (N2 and CO2), a comparison ofthe accuracy of the adsorption prediction between their sphericaland nonspherical molecular models is also presented anddiscussed.

4.1. HRMC Construction Result of the Activated CarbonFiber ACF15. Figure 3 illustrates the variation of the total energy,E, and deviation, �2, of the simulated PDF from experiment withthe number of MC steps. From this Figure, it can be seen thatE and �2 simultaneously decrease with the number of MC steps,follow by leveling off at particular values of -6.22 eV and 10.0respectively. This indicates the convergence of the simulatedsystem to a stable structure. Figure 4a depicts the unit cell ofthe converged configuration of the ACF15 microstructure, andFigure 4b illustrates the comparison between the simulated PDFof the converged configuration and the corresponding target one.The latter illustration indicates an excellent match between thesimulated and the corresponding target PDF. From Figure 4a,it can be observed that the HRMC-constructed configurationcontains curly carbon sheets that are energetically more favorablethan defect-containing flat carbon sheets, created in the initialconfiguration. The formation of these curved carbon sheets isconsistent with curly black fringes, commonly observed in TEMimage of porous carbons. Furthermore, such formation of thecurly carbon sheets is also supported by the work carried out byMeyer,32 who also showed that even suspended graphene sheetsare not perfectly flat.

Figure 5 illustrates a comparison of pore size distribution resultsof the HRMC-constructed ACF15 model probed by the sphericalapproximation approach, described in detail by Thomson andGubbins,19 with that determined by the interpretation of Aradsorption at 87 K using the FWT model. For the former case,Ar is also used as a probing molecule whose Lennard-Jones (LJ)parameters are given in Table 1. From this Figure, it can begenerally seen that the PSD result of the HRMC-constructedmodel of ACF15 substantially overlaps with that of ACF15obtained by the use of Ar adsorption at 87 K, although the intensity

of both PSD results is not the same. In particular, the PSD resultof the HRMC-constructed model contains a certain fraction ofsmall pores of size below Hin ) 0.40 nm, which is absent in thatof ACF15 probed by Ar adsorption. This may be due to the slowAr diffusion rate at 87 K,15 which gives rise to a slightly higherapparent filling pressure and therefore a slightly larger estimatedprobed pore size. Furthermore, a very small fraction of largepores of size greater than Hin ) 1.1 nm, present in the PSD

(32) Meyer, J. C.; Geim, A. K.; Katsnelson, M. I.; Novoselov, K. S.; Booth,T. J.; Roth, S. Nature 2007, 446, 60.

Figure 3. Evolution of fitting parameter, �2, and energy, E, per carbonatom in the simulated system of 1166 carbon atoms.

Figure 4. (a) Snapshot of the converged configuration of the largesimulated system (1166 carbon atoms). (b) Comparison betweenexperimental and simulated pair distribution functions (PDFs).

Figure 5. Comparison between the PSD of the HRMC-constructedconfiguration probed by spherical approximation (solid line)19 and thatof the ACF15 carbon sample determined by the interpretation of Aradsorption at 87 K using the finite wall thickness model.1

7916 Langmuir, Vol. 24, No. 15, 2008 Nguyen et al.

probed by Ar adsorption 87 K, does not occur in that of itsHRMC-constructed configuration. This may be due to thelimitation of simulation box size used in the HRMC construction.Finally, it is interesting to see that the pore volume (Vp ) 0.53cm3/g) of the HRMC-constructed configuration determined bythe spherical approximation approach is in excellent agreementwith that of the ACF15 obtained by the use of Ar adsorption (Vp

) 0.542 cm3/g) using the FWT model, as reported in our previouswork.2 From the above comparison, it is apparent that a reliablecharacterization of porous carbon by the interpretation of Aradsorption at 87 K using the FWT model requires the adsorptionto be free of pore-accessibility problems.

To end this section, a common issue addressed here is theuniqueness of the HRMC-constructed configuration and itsreliability in the prediction of gas adsorption measurements. Thisissue will be addressed in the next section.

4.2. Validation of the HRMC-Constructed Microstructureof ACF15 against Experimental Adsorption Equilibria. Toperform adsorption simulations using the conventional grandcanonical Monte Carlo (GCMC) technique, the pore accessibilityof gases to the HRMC-constructed ACF15 structure must bechecked. In this work, we employ our recently proposed method,described in detail elsewhere,15 to fulfill this task. In brief, theproposed method investigates pore accessibility in a porous solidby grouping molecules of a closely packed adsorbed phase intoseparate clusters. The closely packed adsorbed phase is generatedusing GCMC simulation. Subsequently, a cluster associated witha blocked pore region is identified if the number of moleculesin the cluster is the same in both the unit cell and its supercell.Otherwise, the cluster is associated with an open pore region.Figures 6a,b illustrates the unit cell and supercell of the HRMC-constructed structure with a closely packed adsorbed phase ofargon at 87 K. From these Figures, it can easily be seen that twodetected clusters that contain red and green molecules areassociated with open pore regions due to an increase in theirmolecule population in the supercell compared with that in theunit cell, indicating no pore accessibility problem of Ar to theHRMC-constructed structure at 87 K. Similarly, no accessibilityproblem of N2 at 77 K and CH4 at 310 K to the HRMC-constructedACF15 carbon model was found. Consequently, adsorptionisotherms of all simple gases used here (Ar, N2, CH4, and CO2)in the HRMC-constructed carbon model investigated in this work

can be directly determined using conventional GCMC simulation,as described in subsection 3.1.2, without further correction forpore accessibility. For the sake of clarity, from here onward theLJ fluid-interaction collision diameter, σsf, and well depth, εsf,conform to a generalized Lorentz-Bertherlot mixing rule, asfollows

σsf )12

(σs + σf) (9)

εsf ) (1- ksf)√εsεf (10)

where ksf is a binary interaction parameter that is considered tobe an adjustable parameter representing different wall-fluidinteraction contributions from pure graphitic flat wall-fluiddispersion interactions.

4.2.1. Low-Pressure Adsorption Equilibrium. 4.2.1.1. Argon.Figure 7 depicts a comparison between a GCMC-simulated argonadsorption isotherm at 87 K in the HRMC-constructed model ofthe ACF15 carbon and the corresponding experimental data inthe actual ACF15 carbon sample. In the GCMC simulation, LJparameters for argon-argon and argon-carbon interactions takenfrom Steele3,33 are presented in Table 1. In our previous work,13

the binary interaction parameter ksf for the argon-carboninteraction was taken to be zero. Accordingly, our first attemptis to utilize LJ parameters for the argon-graphitic flat surfaceinteraction taken from Steele.33 From this Figure, it can be revealedthat although the use of the HRMC-constructed carbon modelprovides an excellent prediction (open circles) of experimentalargon capacity in the actual ACF15 carbon sample (filled squares)it rather significantly underestimates experimental adsorbedquantities at lower adsorption coverage. The latter observationwas also reported for carbon nanotubes by several authors whoshowed that simulated heats of adsorption and adsorptioncapacities fall far below reported experimental values.34 Kostovet al.35 argued that such a significant underestimation ofexperimental values by simulation is due to the curvature thatmakes adsorbate-carbon interaction much stronger than flat,perfect graphite sheets. In recent work, Klauda and co-workers4

utilized ab initio quantum mechanics (QM) to determinenitrogen-carbon interaction energies for different curved carbonsurfaces such as C60 fullerene (buckyball) and carbon schwarziteC168, which are considered to be nanoporous carbon models.Subsequently, the N2-carbon interaction energies obtained wereused to fit the LJ (6-12) potential. It was found that the LJnitrogen-carbon well depth obtained for C60 increases by a factorof 1.1 in comparison with that of the graphite surface taken fromBojan and Steele,36,37 with a very slight increase in the LJ

(33) Steele, W. A. J. Phys. Chem. 1978, 82, 817.(34) Shi, W.; Johnson, J. K. Phys. ReV. Lett. 2003, 91, 15504.(35) Kostov, M. K.; Cheng, H.; Cooper, A. C.; Pez, G. P. Phys. ReV. Lett.

2002, 89, 146105.(36) Bojan, M. J.; Steele, W. A. Langmuir 1987, 3, 1123.(37) Bojan, M. J.; Steele, W. A. Langmuir 1987, 3, 116.

Table 1. Lennard-Jones Parameters for Fluid-Fluid and Fluid-Carbon Interactions of All Investigated Gasesa

fluid-fluid interaction solid-fluid interaction

gas σff (Å) εff/kB(K) σsf(Å) εsf(1)/kB(K) εsf

(2)/kB(K) source

Ar 3.410 120.00 3.380 58.00 65.77 33CH4 3.751 148.00 3.576 64.37 73.00 13, 33He 2.557 10.22 2.980 15 17.01 44, 33N2_1LJ 3.750 95.20 3.360 61.40 69.63 39N2_2LJlNN(Å)q(e) 3.3101.100 N: -0.482COM: +0.964 36.00 3.360 33.40 37.88 40, 36CO2_1LJ_1 3.648 246.15 3.429 81.49 92.41 41CO2_1LJ_2 3.720 236.10 3.400 81.31 92.20 3CO2_3LJ C:2.824 C:28.68 C:3.112 C-C:28.34 32.14 13, 33

O:3.026 O:82.00 O:3.213 O-C:47.92 54.34loo(Å)q(e) 2.324C: +0.664O: -0.332a Note that εsf

(1)/kB and εsf(2)/kB discussed in the text are LJ well-depth parameters for the interaction of a flat graphitic surface and the HRMC-constructed

ACF15 carbon model with adsorbed fluid, respectively. For sets of LJ parameters that have two references, the first reference represents the source of theLJ fluid-fluid interaction, and the second reference represents the source of the LJ fluid-graphite surface interaction. COM depicts the center of mass ofN2 in its molecular model.

Atomistic Modeling of ActiVated Carbons Langmuir, Vol. 24, No. 15, 2008 7917

solid-fluid collision diameter. Accordingly, to obtain LJparameters for the argon-argon interaction for our case, weadjust the LJ argon-carbon interaction well depth while keepingthe solid-fluid collision diameter unchanged such that theGCMC-simulated Ar adsorption isotherm at 87 K in the HRMC-constructed carbon model of ACF15 carbon matches thecorresponding experimental data in the actual ACF15 carbon inthe lower pressure range (5 × 10-6 < p/po < 1 × 10-4) wheresolid-fluid interaction is dominant. We found that the fitted LJargon-carbon well depth, εsf

(2)/kB, is equal to 65.73 K and increasesby a factor of 1.134 in comparison with that of the graphitesurface (εsf

(1)/kB ) 58 K) taken from Steele.3 For the sake ofclarity, we defined this scaling factor in the LJ argon-carbonwell depth, Xε, as

Xε )εsf

(2)

εsf(1)

(11)

where εsf(1) and εsf

(2) are graphite- and the LJ curved carbonsurface-fluid interaction parameters, respectively.

Thus, Xε ) 1.134 found for argon in this work is surprisinglyvery close to that determined for C60 fullerene (1.1) by Klauda

et al.4 using the accurate QM method. This strongly indicates thereliability of our proposed HRMC construction method becausethe area of positive curvature is available to adsorption whereasthe counterpart forms closed regions to adsorption, as seen inFigure 6a. Figure 7 showed an excellent match between theGCMC-simulated adsorption isotherm of Ar at 87 K in theHRMC-constructed carbon model (open squares) using the newvalue of the argon-carbon well depth, εsf

(2)/kB ) 65.73 K, andthe corresponding experimental one (filled squares). From nowon, we employ the factor Xε ) 1.134 as a common scaling factorto scale up the LJ fluid-flat graphitic interaction well depthtaken from Steele,3,33,36–39 εsf

(1), for the other gases investigatedin following subsections. Accordingly, the scaled LJ carbon-fluidinteraction well depth, εsf

(2), used in all GCMC simulationspresented in this work to determine the adsorption isotherms ofthe other gases is given as

εsf(2) )Xεεsf

(1) (12)

The values of εsf(2)for all of the investigated gases determined

using eq 12 are presented in Table 1. It is further noted that theconverged configuration is selected such that it provides the bestdescription of the argon adsorption at 87 K with the newcarbon-fluid well depth, εsf

(2).4.2.1.2. Nitrogen. In this subsection, we predict the experi-

mental subcritical adsorption isotherm of N2 at 77 K in the ACF15carbon. For comparison, we employ both spherical and non-spherical models of N2 for GCMC adsorption simulations usingthe HRMC-constructed model. LJ N2-N2 interaction parametersfor the spherical and nonspherical molecular models of nitrogenare taken from Steele39 and Potoff and Seipmann,40 respectively.In addition, LJ N2-carbon parameters, εsf

(1) and σsf, for sphericaland nonspherical models, respectively, are taken from Steele.36–38

All of these LJ parameters including corresponding values of εsf(2)

used in the GCMC simulations of N2 in the HRMC-constructedACF15 carbon model are presented in Table 1. Figure 8 depictsa comparison between the experimental subcritical adsorptionof N2 at 77 K in ACF15 (square symbols) and correspondingGCMC predictive isotherms using spherical and nonsphericalmolecular models of N2 (open and filled circles, respectively).

(38) Bojan, M. J.; Steele, W. A. Langmuir 1993, 9, 2569.(39) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon

Press: Oxford, 1974.(40) Potoff, J. J.; Siepmann, J. I. AIChE J. 2001, 47, 1676.

Figure 6. Illustration of pore connectivity of Ar at 87 K using ourrecently proposed method.15 (a) Snapshot of the unit cell and (b) snapshotof the supercell of the HRMC-constructed ACF15 configuration withthe closely packed adsorbed phase of argon at 87 K. Green and red ballsdepict two separate clusters of argon, as mentioned in section 4.2.

Figure 7. Comparison between GCMC-simulated argon adsorptionisotherms (open squares and circles) at 87 K in the HRMC-constructedACF15 carbon configuration and corresponding experimental data foran actual ACF15 carbon sample (filled squares). Open circles depict thesimulated argon adsorption at 87 K using the well depth εsf

(1) ) 58 Ktaken from Steele33 whereas open squares represent that obtained usingscaled well depth εsf

(2) ) 65.77 K, as mentioned in section 3.2.1.

7918 Langmuir, Vol. 24, No. 15, 2008 Nguyen et al.

From this Figure, it is clearly seen that although the use of thenonspherical molecular model for N2 provides a slightly betterprediction than the spherical molecular one (especially in thelow relative pressure range 10-4), both molecular models for N2

predict almost equally well the subcritical adsorption isothermof N2 in ACF15 at 77 K. The slight overprediction of theexperimental data of the N2 adsorption at low relative pressures(<10-5) using the nonspherical molecular model is due to theinherently slow diffusion of N2 at 77 K, as shown in our recentwork.14,15 The well-predicted results of N2 adsorption by usingboth spherical and nonspherical molecular models, as shownabove, support the use of the spherical molecular model formodeling its adsorption equilibrium. This may be due to the factthat the characteristic molecular geometric ratio of N2 () 0.75),defined by the ratio of the collision diameter of an atom siterelative to molecular length, approaches unity.

4.2.1.3. Carbon Dioxide. In this subsection, we predict theexperimental adsorption isotherms of CO2 in ACF15 at subat-mospheric pressure and temperature at 273 K by performingGCMC simulations in the HRMC-constructed model of thiscarbon. As investigated above for N2, both spherical andnonspherical molecular models are used to represent the CO2

molecule. For comparison, two sets of LJ CO2-CO2 interactionand CO2-carbon parameters for spherical model are taken fromSteele3 and Vishnyakov et al.,41 respectively, whereas LJCO2-CO2 and carbon-carbon interaction parameters for thenonspherical molecular model are taken from our previous work13

and from Steele,3 respectively. All of these LJ parameterscomprising corresponding values of εsf

(2) used in the GCMCsimulations of CO2 in the HRMC-constructed ACF15 carbonmodel are presented in Table 1. We have also found that the useof ksf ) 0 provides correct predictions of all investigatedexperimental adsorption isotherms of CO2 in ACF15 using itsHRMC-constructed model. In particular, Figure 9 illustrates thecomparison between experimental adsorption isotherms (filledtriangles) of CO2 in the ACF15 carbon sample at 273 K underthe subatmospheric condition and its corresponding GCMC-simulated adsorption isotherms in the HRMC-constructed ACF15carbon model using two spherical molecular models (open circlesand squares) and nonspherical molecular model (open triangles).

From this Figure, it can be seen that the use of the nonsphericalmodel for CO2 provides an excellent prediction of the experi-mental data of CO2 in ACF15 despite the slight underpredictionnear 1 bar. However, it is noted here that the experimental datapoints at pressures close to 1 bar are rather sensitive to the errorin the determination of the volume of cold and warm spaces inthe micromecritics ASAP2010. In comparison, the use of thespherical model for CO2 does not predict its experimentaladsorption isotherm in ACF15 equally well. Thus, whereas theuse of spherical and nonspherical models for N2, as shown in theabove subsection, provides very similarly predictive N2 adsorptionisotherms, a significant deviation in the predictive isothermsbetween the use of spherical and nonspherical models is observedfor CO2. This seems to be consistent with its much smallercharacteristic geometric ratio (0.566).

4.2.2. High-Pressure Adsorption Equilibrium. 4.2.2.1. CarbonDioxide. From the successful prediction of the experimental CO2

adsorption in ACF15 at 273 K and under the subatmosphericcondition, as shown above, we now predict the supercriticalhigh-pressure adsorption of CO2 in this carbon at temperaturesof 310 and 333 K. It is well known that the experimental high-pressure excess adsorption isotherm normally shows a maximumat moderate pressures (around 6-10 MPa). Accordingly, theexcess adsorption isotherm significantly increases in the low-pressure range (left adsorption branch) because of the predomi-nance of the adsorption contribution and decreases in the higher-pressure range (right adsorption branch) due to predominanceof the bulk contribution in this region. Accordingly, it can beforeseen that the right adsorption branch is strongly dependenton the pore volume of the porous solid but the left one is notbecause of the high bulk density in the high-pressure range,especially at low temperature. However, the pore volume usedfor the estimation of the theoretical excess adsorbed quantity isphysically ill-defined for a disordered porous carbon at the atomiclevel because of the roughness of its carbon surface. Furthermore,the adsorbent volume of porous carbon can undergo swellingand shrinking under high-pressure adsorption42 whereas thetheoretical excess adsorbed quantity is normally estimated withthe assumption of rigidity of the model carbon structure.

(41) Vishnyakov, A.; Ravikovitch, P. I.; Neimark, A. V. Langmuir 1999, 15,8736. (42) Ozdemir, E.; Morsi, B. I.; Schroeder, K. Langmuir 2006, 19, 9764.

Figure 8. Comparison of GCMC-simulated isotherms (O and b) of N2

at 77 K in the HRMC-constructed ACF15 carbon model, using welldepth εsf

(2) presented in Table 1, with corresponding experimental datafor the ACF15 carbon sample (0). Isotherms predicted using spherical(O) and nonspherical (b) molecular models of N2.

Figure 9. Comparison of GCMC-simulated CO2 adsorption isotherms(open symbols) in the HRMC-constructed ACF15 carbon configurationat 273 K under the subatmospheric condition with correspondingexperimental data for the ACF15 carbon sample (-2-). Open circles andsquares in turn depict the predicted isotherms using spherical molecularmodels taken from Steele3 and Vishnyakov et al.41 with a scaled welldepth of εsf

(2), as presented in Table 1.

Atomistic Modeling of ActiVated Carbons Langmuir, Vol. 24, No. 15, 2008 7919

Consequently, these issues lead to a significant deviation of thetheoretical excess adsorption isotherm from the correspondingexperimental one in the high-pressure range. In consideration ofthe possibility of such a deformation of the solid, we provide anexplicit expression for the theoretical excess adsorbed quantity,mex

f , derived in detail in section S1 of Supporting Information,which is consistent with the experimental one obtained by thegravimetric technique, as follows

mexf )ma

f -ma

He

FaHeFb

f -∆VmaxFbf (13)

where maf and ma

He are the absolute adsorbed quantities of adsorbinggas and helium (used to measure the solid volume), respectively.Fb

f and FbHe are the bulk densities of adsorbing gas and helium,

respectively, and ∆Vmax is the maximum excess adsorbent volumeunder the experimental adsorption condition. As explained indetail in section S1 of Supporting Information, whereas the excesssolid phase volume will in principle be a function of the bulkgas density. In practice, it may be taken to be constant at itsmaximum value, ∆Vmax, in eq 13.

From eq 13, it can be seen that if the actual ACF15 carbonis an incompressible material under high pressure adsorption orexcess adsorbent volume, ∆Vmax ) 0, then eq 13 is similar tothat suggested for helium pore volume calibration by Neimarket al.43 In particular, eq. 13 rewritten for an incompressibleadsorbent is then given as

mexf )ma

f -ma

He

FaHeFb

f (14)

For comparison, here we in turn used eqs 13 and 14 to predictthe experimental adsorption isotherm of CO2 in the actual ACF15carbon sample.

To predict using eqs 13 and 14, the term maHe/Fb

He for the HRMC-constructed ACF15 carbon model is directly determined byGCMC adsorption simulation of helium adsorption in this carbonmodel at 373 K. All LJ parameters for helium are presented inTable 1. It is noted here that temperature 373 K is used to obtainhelium adsorbent volume Vads

He experimentally using eqs S3 andS4 in Supporting Information. In particular, we found that thevalue of the term ma

f/Fbf is essentially invariant for the pressure

range of 1-200 bar and so was estimated by averaging itsinstantaneous values for this pressure range. Determined in thisway, the value of the term ma

f/Fbf is 0.553 ( 0.00269 cm3/g.

Figure 10a illustrates the comparison between the GCMC-predicted isotherm of CO2 in the HRMC-constructed ACF15carbon model at 310 and 333 K using eq 14 (dashed line withopen symbols) and the corresponding experimental one in theactual ACF15 carbon (filled symbols). A three-LJ-center mo-lecular model was used to represent the CO2 molecule. It can beseen that excellent agreement between the GCMC-predicted andexperimental adsorption isotherms of CO2 is achieved only inthe left adsorption branch but significant underprediction occursin the right adsorption branch. This indicates that the actualACF15 carbon may not be strictly rigid under high-pressureadsorption. Accordingly, eq 13 must be used to predictexperimental adsorption in this ACF15 carbon.

To use eq 13, the maximum excess adsorbent volume, ∆Vmax,was determined by fitting the theoretical excess adsorbed isothermof CO2 in the HRMC-constructed ACF15 carbon model, mex

f ,defined in eq 13, to the corresponding experimental one in theactual ACF15 carbon sample at 310 K. We obtained an excellent

fit between the GCMC-predicted (solid line with open circles)and the experimental isotherms with a maximum excess adsorbentvolume, ∆Vmax, of -0.14 cm3/g, as shown in Figure 10a.Subsequently, ∆Vmax ) -0.14 cm3/g was used to predict theexperimental adsorption isotherm of CO2 at 333 K in the actualACF15 carbon using eq 13, as shown in Figure 10a. From thisFigure, it can also be seen that the use of ∆Vmax ) -0.14 cm3/gprovides excellent agreement between the GCMC-predictedisotherm of CO2 in the HRMC-constructed ACF15 carbon (solidline with open squares) and the corresponding experimental onefor the actual ACF15 carbon sample (filled squares).

4.2.2.2. Methane. With the successful prediction of experi-mental high-pressure CO2 adsorption in ACF15 using eq 13presented above, here we also utilize eq 13 for the prediction ofexperimental high-pressure methane adsorption in ACF15 at 310and 353 K over the pressure range of up to 200 bar. The maximumexcess adsorbent volume, ∆Vmax )-0.14 cm3/g, obtained in theabove subsection is used to perform this task. GCMC simulationis used to obtain the absolute adsorption isotherm of CH4 in theHRMC-constructed carbon model of ACF15 for a givenexperimental condition. All of the LJ parameters for CH4-CH4

and CH4-carbon interactions are presented in Table 1. Figure10b illustrates the comparison between experimental high-pressure adsorption of CH4 at 310 K (filled triangles) and 353K (filled squares) and the corresponding GCMC-simulated ones(solid line with open triangles and dashed line with open squares).As is evident in this Figure, excellent agreement betweenexperimental and predicted adsorption isotherms is achieved.(43) Neimark, A. V.; Ravikovitch, P. I. Langmuir 1997, 13, 5148.

Figure 10. (a) Comparison between the GCMC-simulated isotherms ofCO2 in the HRMC-constructed ACF15 carbon model using eq 14 (dottedline with open symbols) and using eq 13 (solid line with open symbols)at 310 and 333 K and the corresponding experimental data for the ACF15carbon sample (filled symbols). (b) Comparison between the GCMC-simulated CH4 adsorption isotherms (solid line with open symbols) at310 and 353 K using eq 13 and the corresponding experimental data forACF15 carbon (filled symbols).

7920 Langmuir, Vol. 24, No. 15, 2008 Nguyen et al.

Finally, a question arising here is why the use of eq 13 withthe assumption of compressible adsorbent provides an excellentprediction of the experimental high-pressure adsorption isothermswhile still utilizing the predicted absolute isotherms in a rigidHRMC-constructed ACF15 carbon model. The success of theapproach reveals that the compression of ACF15 carbon underhigh-pressure adsorption predicted by the negative value of largestexcess adsorbent volume, ∆Vmax)-0.14 cm3/g, has a negligibleimpact on micropore structure that governs the adsorption. Suchshrinkage (∆Vmax/Vext

He), estimated to be 13%, is similar to thatobtained for high-rank coal.42 This is also supported by the factthat the use of the rigid HRMC-constructed ACF15 carbon modelprovides an excellent prediction of the CO2 adsorption isothermat 273 K in the actual ACF15 carbon at low pressure, as shownin Figure 9. Accordingly, a possible answer to the above questionis that softer parts of the adsorbent may mainly contribute to itsshrinkage under high-pressure adsorption. The softer parts maybe carbon walls having at least two carbon sheets whose interlayerspacing is greater than that of perfect graphite (0.335 nm). Thisis expected for ACFs, whose averaged interlayer spacingdetermined from XRD data in our previous work2 is approximately0.36 nm. In practice, interlayer spacing may include closed poresinaccessible to investigated adsorbing gases as well as helium.Accordingly, under high-pressure adsorption open pores saturatedwith adsorbing gas increase the solvation force inside the poresand so lead to their mechanical stability. On the contrary, theclosed or inaccessible spacing may be contracted because of thefact that opposite carbon sheets of a closed pore have weakinteraction.

4.3. Validation of the HRMC-Constructed Microstructureof the ACF 15 Carbon against the Experimental DifferentialHeat of Adsorption. In this section, we perform a comparisonbetween the GCMC-simulated differential heat of adsorptiondata of CO2 at 298 K in the HRMC-constructed model of theACF15 carbon and the corresponding experimental data inactivated carbon fiber KF-1500 taken from Guillot et al.45 Ourreason for selecting activated carbon fiber KF-1500 for thiscomparison is that the BET surface area of the KF-1500 carbon(1440 m2/g)46 is very similar to that of the ACF15 carbon (1420m2/g).47 The inset of Figure 11 shows good agreement betweenthe GCMC-simulated adsorption data of CO2 in the HRMC-constructed model of the ACF15 carbon and the correspondingexperimental data in the KF-1500 carbon taken from both Guillotet al.48 and Nobuyuki et al.2 Subsequently, we determined thedifferential heat of adsorption, qd, of CO2 in the HRMC-constructed configuration of the ACF15 carbon using GCMCsimulation. For this, we used the fluctuations theory49-basedequation given by Ramirez-Pastor et al.50 as

qd )- ⟨UN⟩ - ⟨U⟩⟨ N⟩⟨N2⟩ - ⟨N⟩2

(15)

where U is the total energy of simulation system, N is the absolutenumber of particles in a microscopic configuration of the system,and ⟨⟩ is the ensemble average.

Figure 11 shows excellent agreement between the GCMC-simulated differential heat of adsorption in the HRMC-constructed

carbon model of ACF15 carbon (filled squares) and thecorresponding experimental data for KF-1500 (open symbols)given by Guillot et al.,45 especially at low adsorption coverage(<2 mmol/g). Accordingly, our use of enhanced well depth dueto positive curvature is strongly supported by this correctprediction of the experimental differential heat of adsorptiondata.

5. Conclusions

In this work, we have presented a new HRMC constructionprocedure for obtaining the 3D atomistic microstructure ofporous carbons, whereby the initial configuration generatedis based on characterization results of the porous carbon (PSDand pore wall thickness distribution) by interpretation of Aradsorption at 87 K using our finite wall thickness model. Theproposed procedure enables one to save computing timerequired for the convergence of the HRMC constructionprocedure. A most striking feature of the proposed procedureis that the initial configuration is not only sufficiently closeto the target one but also provides the necessary long-rangeorder, which helps in rapidly obtaining the reliable converged-constructed microstructure of porous carbons such as that foractivated carbons while using short-ranged carbon potentialsin the HRMC simulation. The newly proposed HRMCconstruction procedure is directly applied to construct themicrostructure of activated carbon fiber ACF15 provided bythe Kynol Corporation.2 The HRMC-constructed configurationcontains curved carbon sheets, and its pair distribution functionmatches the experimental one measured on actual ACF15carbon very well. The PSD of the HRMC-constructedconfiguration probed by the spherical molecule approximationapproach is in good agreement with that obtained by theinterpretation of argon adsorption at 87 K using the finite wallthickness model.

Furthermore, we have also presented a comprehensivevalidation of the HRMC-constructed configuration againstexperimental adsorption isotherms of simple gases (Ar, N2,CO2, and CH4) over a wide range of temperatures and pressures.We found that the use of an LJ carbon-fluid interaction welldepth for the interaction of curved carbon surfaces in the

(44) Tchouar, N.; Benyettou, M.; Kadour, F. O. Int. J. Mol. Sci. 2003, 4, 595.(45) Guillot, A.; Stoeckli, F.; Bauguil, Y. Adsorp. Sci. Technol. 1993, 10, 3.(46) Nobuyuki, T.; ABE, M.; Nitta, T.; Katayama, T. J. Chem. Eng. Jpn. 1998,

21, 315.(47) Mangun, C. L.; Daley, M. A.; Braatz, R. D.; Economy, J. Carbon 1998,

36, 123.(48) Guillot, A.; Follin, S.; Poujardieu, L. Characterization of Porous Solids;

McEnaney, B., Ed.; Royal Society of Chemistry: Cambridge, U.K., 1980.(49) Nicholson, D. ; Parsonage, N. G. Computer Simulation and the Statistical

Mechanics of Adsorption; Academic Press: London, 1982.(50) Ramirez-Pastor, A. J.; Bulnes, F. Physica A 2000, 283, 198.

Figure 11. Comparison of the GCMC-simulated differential heat ofadsorption data of CO2 at 298 K in the HRMC-constructed carbon modelof the ACF15 carbon with the corresponding experimental data inactivated carbon fiber KF1500 reported by Guillot et al.45 using TG-DSC and a Tian-Calvert calorimeter (C80). The inset depicts thecomparison between the GCMC-simulated adsorbed quantity for CO2

at 298 K in the HRMC-constructed configuration and the correspondingexperimental data in the KF1500 carbon, taken from Gulliot et al.48 andNobuyuki et al.46

Atomistic Modeling of ActiVated Carbons Langmuir, Vol. 24, No. 15, 2008 7921

HRMC-constructed microstructure of ACF15, obtained byscaling the corresponding LJ flat graphite surface-fluid welldepth by a factor of 1.134, provides an excellent predictionof all experimental adsorption isotherms of the investigatedgases in the ACF15 carbon over a wide range of temperaturesand pressures up to 200 bar. In addition, the nonrigidity ofthe actual ACF15 carbon sample was also recognized in thiswork, revealing a 13% shrinkage of the solid skeleton. Goodagreement between the GCMC-simulated and experimentaldifferential heat of adsorption data of CO2 at 298 K is seen.Finally, although we have proposed a method to generate aninitial configuration to provide long-range order for the HRMC

construction of porous carbons, it can be used for theconstruction of highly disordered porous carbons havingshorter-range order by increasing the initial temperature inthe equilibration stage of the HRMC construction.

Acknowledgment. This research has been supported by agrant from the Australian Research Council under the Discoveryscheme.

Supporting Information Available: Derivation of theoreticalexcess adsorbed amount for the gravimetric technique. This material isavailable free of charge via the Internet at http://pubs.acs.org

LA800351D

7922 Langmuir, Vol. 24, No. 15, 2008 Nguyen et al.