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Journal of Membrane Science 355 (2010) 186–199

Contents lists available at ScienceDirect

Journal of Membrane Science

journa l homepage: www.e lsev ier .com/ locate /memsci

rediction of carbon dioxide permeability in carbon slit pores

oung-Il Lima,b,∗, Suresh K. Bhatiaa, Thanh Xuan Nguyena, David Nicholsona

School of Chemical Engineering, The University of Queensland, Brisbane, Queensland 4072, AustraliaDepartment of Chemical Engineering, Hankyong National University, Gyonggi-do, Anseong-si 456-749, Republic of Korea

r t i c l e i n f o

rticle history:eceived 22 December 2009eceived in revised form 19 March 2010ccepted 20 March 2010vailable online 25 March 2010

a b s t r a c t

A one-site spherical CO2 model and five three-site linear CO2 models are compared for adsorption incarbon slit pores, using grand canonical Monte Carlo (GCMC) simulation. A three-site CO2 model vali-dated with bulk density experimental data is used to predict adsorption isotherms and diffusivities inthe slit pore with the width from 0.65 nm to 0.75 nm. In the range of temperature from 298 K to 318 Kand pressure from 0.01 bar to 20 bar permeability is calculated from the GCMC adsorption isotherms and

eywords:arbon dioxide (CO2)ermeabilityarbon slit porerand canonical Monte Carlo (GCMC)

the collective diffusivity obtained from equilibrium molecular dynamics (EMD). The permeability withinthe pore widths exceeds by three orders of magnitude that of reported macroscopic measurements. Thesimulation permeability obtained in this study is comparable to that derived from modified Knudsendiffusion with an energy barrier of 3.5–5 kJ/mol. The simulation results give an upper bound of the per-meability for an ideal carbon membrane without pore mouth resistance and complex pore network.

quilibrium molecular dynamics (EMD)

. Introduction

The separation of CO2/CH4 is industrially important especiallyor natural gas processing. In natural gas treatment applications,he feed gas is usually obtained directly from gas wells with a wideange of pressures (normally 20–70 bar) and compositions (5–50%O2). Its typical product is a CH4-rich residue stream containing lesshan 2–3% CO2 [1]. It has been experimentally confirmed that theO2 adsorption saturation capacity of coal is about two times thator CH4, because of the higher adsorption energy of CO2 in micro-ores [2]. Gases are readily adsorbed in the micropores of coal andther carbons, where the relative adsorbate molecular size and poretructure strongly affect selective gas transport. In practice, slit poreodels have been most commonly used to characterize porous car-

ons as well as to predict gas adsorption equilibrium on carbon [3].he critical pore size that allows molecular sieving may be deter-ined by using the effective dimensions of the penetrant molecule

uch as kinetic diameter (�k) [4]. Due to its smaller kinetic diam-

ter (�k,CO2

= 0.33 nm) CO2 can permeate micropores which areikely to block CH4, as the latter has a slightly larger kinetic diame-er (�k,CH4

= 0.38 nm), resulting in 10–100 times higher diffusivityf CO2 than that of CH4 [2].

∗ Corresponding author at: Department of Chemical Engineering, Hankyongational University, Jungangno 167, Gyonggi-do, Anseong-si 456-749, Republic oforea. Tel.: +82 31 670 5207; fax: +82 31 670 5445.

E-mail addresses: limyi@hknu.ac.kr (Y.-I. Lim), s.bhatia@eng.uq.edu.auS.K. Bhatia), T.Nguyen@eng.uq.edu.au (T.X. Nguyen), d.nicholson@uq.edu.auD. Nicholson).

376-7388/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.memsci.2010.03.030

© 2010 Elsevier B.V. All rights reserved.

Gas separation membranes have been widely used for CO2/CH4separation, due to their inherent advantages of low capital cost,operational simplicity, space and weight efficiency, easy scalabil-ity and (in some applications) low power consumption [1]. Carbonmolecular sieve (CMS) membranes show excellent separation per-formance and stability at high temperature and pressure, andappear to be a promising candidate for CO2/CH4 separation [5,6].The carbon materials not only have the ability to perform molecularsieving, but may also allow a considerably higher flux of the pene-trant through the carbon material as compared to zeolites (havingmono-dispersed micropores) with similar selectivity properties.

The CMS membrane can be tailored to acquire a distribution ofselective micropores in the order of 0.3–0.6 nm, which is similar tothe molecular size of the diffusing gas [5]. The carbon of CMS is anamorphous, microporous material, which makes it difficult to char-acterize compared to crystalline materials. From separation studiesof penetrants on these materials it appears that these materialshave both ultramicropores (<0.7 nm) and larger micropores. Theultramicropores are believed to be mainly responsible for molec-ular sieving, while the micropores provide negligible resistance todiffusion but provide high capacity sorption sites for penetrants[4]. In CMS membranes with pore spacings (d) from 0.35 nm to0.42 nm derived from polyimide precursor, the carbon dioxide per-meates selectively compared to methane [4–6]. The smaller and

more polar species (CO2) of a gas mixture is selectively adsorbedinto the carbon pores and diffuses across the membrane.

Transport of confined molecules in microporous materials isquite different from bulk phase transport. Transport of a confinedfluid is characterized by the coupling of adsorption and diffusion

brane Science 355 (2010) 186–199 187

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Table 1Indicative accessible pore width (Ha) and porosity (εp) based on the pore width (H).

H (nm) Ha (nm)a εp

0.65 0.31 0.450.68 0.34 0.47

Y.-I. Lim et al. / Journal of Mem

referred to as configurational diffusion, when this coupling isery strong, for example, in ultramicropores) [2,7]. Gas transportn some microporous materials is controlled by activated diffu-ion and/or a molecular sieve-like mechanism, which is mainlynfluenced by the characteristics of sorption, adsorbate moleculareometry, and pore structure [2].

It is well established that potential energy wells in pores becomerogressively deeper as the pore is narrowed, due to overlap, untillimiting width is reached when repulsive interactions begin to

ominate [8]. Internal pore widths that are slightly larger than aiven molecular probe, therefore, present very strong adsorptionites, and give rise to the phenomenon of hindered diffusion inher-nt to confined fluids. Quantitative prediction of hindered diffusions important in guiding experiments and improving engineeringesigns in diversified applications, such as molecular sieve oper-tions, gel permeation chromatography, and permeation throughatural and synthetic membranes [9].

Molecular modeling starting at the atomic scale is an effi-ient tool for quantitatively and qualitatively understandingtructure–property relations, and elucidating the mechanismsf microscopic phenomena [10]. There has been considerableesearch on adsorption and/or diffusion of carbon dioxide inmooth-walled slit pores [8] and two plane-parallel hard walls9,11]. It has been reported that the atomistic structure of thealls (smooth or hard walls) has little effect on the transport and

eparation properties [12]. Several studies have shown pore sizeependent adsorption density due to packing transitions in alleometries. The transitions, occurring continuously with increas-ng pore width, allow rearrangement of adsorbed molecules inores in order to attain a minimum free energy [13]. Zhou andang [9] investigated adsorption diffusion of CO2 for a wide range

f temperature (T = 273–348 K), pressure (P = 10–100 bar) and car-on slit pore width (H = 0.744–3.72 nm), using a one-site carbonolecular Lennard-Jones (LJ) model [9]. For carbon dioxide, two-

ite and three-site models have been compared for adsorptionn carbon slit pores [11], and it has been demonstrated that thehree-site model is accurate and reliable for predicting adsorp-ion and diffusion, as carbon dioxide is a polar and non-spherical

olecule. Accordingly, three LJ-center models with partial chargeslaced on three sites (three-site CO2 model) have often beensed.

Permeability is the well-known product of the solubility factorS) and the transport diffusion coefficient (Do) [10,14–18]. The for-

er is related to the adsorption isotherms as a function of pressureor fugacity), and often predicted by grand canonical Monte CarloGCMC) simulation [19,20]. The latter is calculated from molecu-ar dynamics simulation such as equilibrium molecular dynamicsEMD), nonequilibrium molecular dynamics (NEMD [21,22]), andual control volume grand canonical molecular dynamics (DCV-CMD [23,24]). The transport diffusion coefficient (or collectiveiffusivity, Do) can be considered to comprise a self-diffusion partDs) corresponding to the movement of a tracer molecule throughhe fluid, and a cross-term (Dc) that accounts for interactionsetween distinct molecules [25]. Comparative studies have demon-trated that NEMD and DCV-GCMD give the same results as EMDithin error limits [21,25–27].

The selectivity (or separation factor) is obtained from the ratioetween two permeabilities in a binary mixture. The selectivitiesredicted by molecular simulations have been found to be in rea-onable agreement with experiment [10]. However, the absolutealues of the predicted permeability were higher by a factor of

20–340 compared to the macroscopically measured experimentalalues [10]. These discrepancies in absolute values were attributedo the reasons: (1) external heat and mass transfer limitations mayimit transport rates in a macroscopic membrane experiment or inn adsorption and desorption measurement, (2) the complex mor-

0.72 0.38 0.500.75 0.41 0.52

a Ha = H − �ss , where �ss = 0.34 nm.

phology of the experiment membrane may lead to a decrease in themobility of sorbates [10].

Transport in a carbon molecular sieve is characterized by twoseries resistances: (1) that associated with an entry barrier at thepore mouth, and (2) a diffusional resistance in the pore body. It wasreported that the rate-limiting process in carbon molecular sieveswas entry through the pore mouth because of the large energy bar-rier [28–30]. Thus, the micropore diffusion was assumed to occurin series with the pore mouth entry [29].

In this study, the permeability of CO2 in a carbon slit poreis predicted within nominal operating temperature and pressureranges of a CO2/CH4 separation membrane. The carbon slit porewidth ranges from 0.65 nm to 0.75 nm, as measured between thecarbon-atom centers on opposite walls of the pore, where CO2/CH4separation is achieved effectively. One-site and three-site CO2 LJmolecular models are compared for adsorption and diffusion inthe slit pore. A three-site CO2 model, validated with pressure–bulkdensity–temperature (P–�–T) experimental data around the criti-cal temperature (300–340 K), is selected among the CO2 LJ modelsto examine adsorption isotherms, diffusivity and permeability. Themicropore diffusivity, without pore mouth resistance, is deter-mined using EMD. The reason for the disparity of permeabilitybetween experimental data and molecular simulation results isdiscussed.

2. Molecular simulation methods

The 10-4-3 Steele potential [31] was applied to the slit pores inthis simulation study. The carbon parameters of the Steele potentialhad the values �ss = 0.34 nm, εss/kB = 28 K, � = 0.335 nm (interplanardistance of carbon layers), and �s = 114 nm−3 (carbon site density)[11,13,32]. The potential parameters between adsorbent and adsor-bate were determined using the Lorentz–Berthelot mixing rules�sf = 1/2(�ss + �ff), and (εsf /kB = √

εssεff /kB) [13,33]. One one-siteand five three-site CO2 LJ models were considered in order to selectan appropriate CO2 model (see the next section). The cutoff dis-tance was set to 2.5 nm [13], as Coulombic interactions (for threesited models) are long-ranged. Because of the relatively large cut-off, no long-range corrections were employed. The adsorbate andadsorbent molecules (CO2 and carbon) were represented as rigid,which would not be expected to make a significant difference inadsorption and diffusivity for small molecules [10]. The wall of theslit pore was considered to reflect molecules diffusely.

In each GCMC simulation, 4 × 107 configurations were gener-ated. The first 2 × 107 configurations were discarded to guaranteeequilibration and the last 2 × 107 configurations were used to aver-age the adsorption isotherms in the slit pore. All GCMC simulationswere performed using a box size of 7 × 7 × H (nm3). The indicativeaccessible pore width (Ha) and pore porosity (εp) based on the slitpore width (H) are shown in Table 1 for the pore of a single layerwall.

For the EMD calculations, the equations of motion were solvedby the fifth-order Gear predictor–corrector algorithm for 10 ns.The initial configuration of an EMD system was obtained followingGCMC simulation performed in advance. The total EMD simulationstep number was 1 × 107 with a time step of 1.0 × 10−15 s (1 fs).

1 brane Science 355 (2010) 186–199

TtomuTt2

3

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eatt(uoaspwt

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dCLivitaCwmm

88 Y.-I. Lim et al. / Journal of Mem

he first 1 × 106 steps were used to equilibrate the system, whilehe last 9 × 106 steps were used to sample the diffusion propertiesf interest. To obtain reliable EMD results, the more than 150 CO2olecules were used for all EMD simulations, by adjusting the sim-

lation box size varying from 7 × 7 × H (nm3) to 25 × 25 × H (nm3).he calculation was performed on a Sun Microsystems X2200 clus-er (30 nodes of a dual quad-core AMD OpteronTM processor at.3 GHz and 8GB RAM per node).

. Molecular models of carbon dioxide

There have been a number of molecular models proposed forO2 [34]. Table 2 shows LJ parameters for five three-site and onene-site CO2 models. The first three-site model (termed 3-CO2-GU) used by Nguyen et al. [3,35] has shown good agreementith experimental adsorption isotherms on an activated carbon at

upercritical temperatures (313–343 K) over a wide range of pres-ure (0–500 bar). The second model (named 3-CO2-MAK [10]) haslightly higher LJ energy parameters than the first one. The thirdodel (3-CO2-HAR) proposed by Harris and Yung [36] was able to

escribe accurately vapor–liquid equilibrium (VLE) of CO2 in theulk fluid below the critical temperature (Tc = 304.2 K). 3-CO2-HARas been used to theoretically analyze adsorption characteristics ofO2 at T = 273 K and pressure up to 1 atm in carbon slit pores of theidth between 0.365 nm and 0.899 nm [32]. Adsorption and diffu-

ion of CO2 were investigated at 298 K up to 100 bar in single walledarbon nanotube as a function of pore diameter (1.08–5.42 nm),howing that representing CO2 as a spherical (one-site LJ model)ad only a slight impact in the computed diffusion coefficients [7].he fourth model (3-CO2-POT [34,37]) has similar LJ parameters to-CO2-HAR.

The fifth three-site model (3-CO2-MUR) established by Murthyt al. [38] provided a good account of ambient temperaturedsorption on a nonporous graphitic surface in combination withhe 10-4-3 graphite potential [33]. Point charges are placed onhe carbon atom (qC = +0.664) and on each of the oxygen atomsqO = −0.332) to give a quadrupole charge. 3-CO2-MUR has beensed to investigate CO2/CH4 separation by adsorption in slit poresf various widths [8,11,33], and to predict the high pressuredsorption capacity of CO2 in carbon slit pores [13]. Membraneeparation of CO2/CH4 has been investigated within carbon slitores over a wide range of temperature (298–413 K), and poreidth (0.68–2.85 nm), using the DCV-GCMD technique with this

hree-site LJ CO2 model [39].The five three-site CO2 molecular models show different poten-

ial well depths (εCC/kB = 26–28 K and εOO/kB = 75–82 K), while theollision diameters (�CC and �OO), the point charges (qC and qO), andhe distance between carbon and oxygen atoms (dC O) are almosthe same. Since the energy parameter (ε/kB) relates to the negativeell depth in the LJ potential, which is a major factor in determin-

ng adsorption strength: the higher well depth leads to the higherdsorption of CO2.

The parameters of the one-site CO2 model (1-CO2-BHA) wereetermined by matching the bulk isotherm with that of the 3-O2-MUR between 0 bar and 2500 bar at 308 K [13]. The one-centerennard-Jones (LJ) model did not correctly represent the true pack-ng behavior of adsorbate inside micropores and led to inaccuratealues of the adsorption capacity at high pressures [13]. Selectiv-ty of a membrane to achieve good CO2/CH4 separation occurs inhe pore size range of 0.60 < H < 0.70 nm in slit-shape micropores,

ssuming that the carbon collision diameter is �ss = 0.34 nm [2].O2 molecules have low accessibility below the size of H = 0.60 nm,hile for pores wider than H = 0.70 nm the permeate amount ofethane increases and the selectivity decreases rapidly. Thus,olecular simulation results for CO2/CH4 membrane separation are

Fig. 1. Comparison of adsorption density (�ads) obtained from GCMC simulationfor six CO2 molecular models at f = 0.09 bar and T = 323 K, over the pore size range0.63 nm ≤ H ≤ 1.0 nm.

sensitive to the molecular size (�CO2 ) and energy (εCO2 ) parametersof the CO2 molecular model in this pore size range. The one-site LJmodel of CO2 with a large effective spherical collision diameter ofabout 0.37 nm may be inadequate to predict adsorption isothermsof CO2 within this pore size range.

Fig. 1 compares the adsorption density (�ads) versus the slit porewidth (H) for the six CO2 molecular models, which is obtainedfrom GCMC simulation at f = 0.09 bar and T = 323 K. The two pairsof 3-CO2-NGU and 3-CO2-MAK, and 3-CO2-HAR and 3-CO2-POTshow similar adsorption behavior, due to similar LJ parameters.The adsorption density of 3-CO2-MUR is lower than that of theother three-site models, because of lower energy parameters(εCC and εOO). The one-site model (1-CO2-BHA) cannot predictwell the adsorption density within the narrow slit pore size(0.63 ≤ H ≤ 0.8 nm) because of the large spherical collision diame-ter, as mentioned above. The difference between the six adsorptiondensities diminishes, as the pore size increases. The errors between3-CO2-NGU and the others are shown in the last column of Table 2.In Fig. 1, the maximum adsorption density is observed at the poresize of H = 0.65 nm or H = 0.72 nm where the accessible pore width(Ha) is just larger than the collision diameter for �OO = 0.30 nm or�CO2 = 0.37 nm for the three-site models or the one-site model,respectively.

In Fig. 2, the adsorption density and the diffusivity versusfugacity are further investigated at T = 298 K and H = 0.72 nm for3-CO2-NGU and 1-CO2-BHA. For the whole fugacity considered,adsorption isotherms of the one-site model are overestimated byabout 14% (see Table 3). It has been argued that the adsorptionbehavior of CO2 confined in narrow micropores is related to thedifferent molecular shape represented by the three-site fluid andnot to the electrostatic effect [13]. The self- and collective diffu-sivities of the one-site model are underestimated by 43% and 33%(see Table 3), respectively, since the linear CO2 molecules movesmore freely than the spherical CO2 molecule in the confined nar-row slit pore. The self-diffusivity of the one-site model at f = 10 bar(2.3 × 10−9 m2/s) agrees well with that of Zhou and Wang [9], whichwas obtained in carbon slit pore of H = 0.744 nm. The self-diffusivityof 3-CO2-NGU (about 12 × 10−9 m2/s) at infinite dilution (f ≈ 0) ishigher than that of 3-CO2-HAR (about 8 × 10−9 m2/s [10]) obtainedfrom EMD at T = 303 K in a silicalite having pore size of about0.54 nm. It was reported that the self-diffusion coefficient of CO2

was 1.94 × 10−9 m2/s in aqueous solution at T = 298 K [40], while itwas about 0.9 × 10−6 m2/s in gas phase at T = 298 K and P = 10 bar[41]. Therefore, the diffusivity in the confined slit pore is closer tothat of aqueous solution, which was also recognized by Zhou and

Y.-I. Lim et al. / Journal of Membrane Science 355 (2010) 186–199 189

Table 2LJ model parameters of CO2 molecules and the adsorption density error between 3-CO2-NGU and the others.

Name �CC (nm) εCC/kB (K) �OO (nm) εOO/kB (K) qC (e) qO (e) dC O(nm) Errora (%)

3-CO2-NGU [3,35] 0.2824 28.68 0.3026 82 +0.664 −0.332 0.1162 –3-CO2-MAK [10] 0.2753 29.07 0.3029 83.2 +0.6466 −0.3233 0.114 0.93-CO2-HAR [7,32,36] 0.2757 28.13 0.3033 80.5 +0.6512 −0.3256 0.115 9.63-CO2-POT [34,37] 0.28 27 0.305 79 +0.7 −0.35 0.116 11.73-CO2-MUR [13,15,33,38,39] 0.2824 26.3 0.3026 75.2 +0.664 −0.332 0.1162 28.41-CO2-BHA [13] �CO2 = 0.37, εCO2 /kB = 220 53.6

a Adsorption density (�ads) error of CO2 molecular models to 3-CO2-NGU at f = 0.09 bar and T = 323 K (see Fig. 1).

Fig. 2. Comparison of adsorption density (�ads) and diffusivity (Ds or Do) between three-site CO2 model (3-CO2-NGU) and one-site CO2 model (1-CO2-BHA) at T = 298 K, forH = 0.72 nm: (a) adsorption density (�ads); (b) self- (Ds) and collective (Do) diffusivities.

F ta [433

Wto

wm3

TM0

ig. 3. Comparison of bulk density (�) versus pressure (P) between experimental da-CO2-NGU model; (b) experimental data and 3-CO2-POT model.

ang [9]. This may be because of the two-dimensional nature ofhe slit pore, which is less confined than the one-dimensional poref slicalite.

In Fig. 3, the bulk densities (�) obtained from GCMC simulationith 3-CO2-NGU and 3-CO2-POT are compared to experimentaleasurements (P–�–T) between 5 bar and 300 bar at T = 300 K,

13 K and 340 K. In the grand canonical ensemble (�, V, T) chemical

able 3ean difference of one-site CO2 model (1-CO2-BHA) from three-site CO2 model (3-CO

.01 bar ≤ f ≤ 20 bar (see Fig. 2).

Adsorption density (�ads)

Mean difference (%)a −14.3

a Difference (%) = 100 × ((three-site value) − (one-site value))/(three-site value).

], equation of state [42] and two CO2 molecular models: (a) experimental data and

potential (� = �o + ln f) is the natural input variable, and conversionof fugacity to pressure is the customary procedure. The fugac-ity is converted into the pressure deviated from the fugacity by

interactions between molecules, using the equation of state (EOS)expressing relationship between the fugacity and pressure. The EOSby Bender [42] presented in Appendix A tracks exactly the experi-mental data by Klimeck et al. [43]. The fugacity used in this GCMC

2-NGU) in adsorption density and diffusivity at T = 298 K and H = 0.72 nm within

Self-diffusivity (Ds) Collective diffusivity (Do)

+42.9 +32.7

190 Y.-I. Lim et al. / Journal of Membrane Science 355 (2010) 186–199

F ith du(

sBfat3iaGttsic

4

p

TBs

ig. 4. Comparison of adsorption isotherms (�ads) obtained from GCMC simulation wc) H = 0.72 nm; (d) H = 0.75 nm.

imulation was converted into the pressure on the basis of theender’s EOS in Fig. B.1. In Appendix B, the relationships between

ugacity, pressure, and bulk density are derived from the EOS. Atfugacity of f = 20 bar, the pressure deviates less than 10% from

he fugacity (see Fig. B.1(a)). The GCMC simulation results with-CO2-NGU show slightly higher bulk densities than those of exper-

ment over the pressure where an abrupt density change occursround the critical temperature of carbon dioxide (Tc = 304.2 K). TheCMC results with 3-CO2-POT show considerable mismatch with

he experimental data. The mean error of 3-CO2-POT is about threeimes greater than that of 3-CO2-NGU at the three temperatures, ashown in Table 4. Hereafter, the three-site LJ model (3-CO2-NGU)s used for the prediction of adsorption isotherms and diffusivity inarbon slit nanopores.

. Results and discussion

The adsorption density and diffusivity are required to obtainermeability. For the adsorption density, the GCMC simulation was

able 4ulk density errors between experimental data (Klimeck et al. [43]) and GCMCimulation results for two three-site CO2 molecular models (see Fig. 3).

3-CO2-NGU [3] 3-CO2-POT [34]

Mean errora (%)T = 300 K 3.2 10.1T = 313 K 7.4 19.9T = 340 K 3.5 12.9

a Error (%) = 100 × |(experimental data) − (GCMC results)|/(experimental data).

al mode isotherms model for four slit pore sizes (H): (a) H = 0.65 nm; (b) H = 0.68 nm;

performed at three temperatures (T = 298 K, 308 K, and 318 K), fourpore widths (H = 0.65 nm, 0.68 nm, 0.72 nm and 0.75 nm), and 17fugacities from 0.01 bar to 20 bar (totally 204 runs). To obtain thediffusivity, EMD simulations were carried out at the same tem-perature and width, but at densities corresponding to fugacitiesof 0.01 bar, 0.1 bar, 1 bar, 5 bar, 10 bar, and 20 bar (totally 72 runs).

4.1. Adsorption density

The adsorption density versus fugacity is shown in Fig. 4 for thefour pore widths and three temperatures. The isotherms are repre-sented well by a superposition of linear and Langmuir isotherms(dual mode model). The dual mode adsorption model (�model

ads)

[14,44] is applied for �GCMCads

obtained from GCMC simulation:

�modelads = klf + kmkbf

1 + kbf(1)

where kl (mmol/cm3/bar) is the linear adsorption constant, km

(mmol/cm3) is the saturation density, and kb (bar−1) is the affinityparameter between the adsorbate and adsorbent. The parametersof the dual mode isotherms are shown in Table 5 for each tem-perature and pore width, which are valid within 0 ≤ f ≤ 20 bar. Thetemperature dependence of the isotherms parameters (kl, km andkb) may be described by exponential expressions [14]:

kl = kl0eEl/RT (2)

km = km0eEm/RT (3)

kb = kb0eEb/RT (4)

Y.-I. Lim et al. / Journal of Membrane Science 355 (2010) 186–199 191

Table 5Dual mode isotherms parameters at T = 298, 308, and 318 K for four slit pore widths.

T = 298 K T = 308 K T = 318 K

kl km kb kl km kb kl km kb

H = 0.65 nm 0.8055 × 10−1 0.1213 × 102 0.3192 × 102 0.6724 × 10−1 0.1199 × 102 0.2001 × 102 0.6913 × 10−1 0.1175 × 102 0.1303 × 102

H = 0.68 nm 0.7124 × 10−1 0.1177 × 102 0.2890 × 102 0.7005 × 10−1 0.1156 × 102 0.1869 × 102 0.6559 × 10−1 0.1139 × 102 0.1228 × 102

H = 0.72 nm 0.6379 × 10−1 0.1132 × 102 0.1634 × 102 0.5689 × 10−1 0.1118 × 102 0.1076 × 102 0.5630 × 10−1 0.1100 × 102 0.0734 × 102

H = 0.75 nm 0.5780 × 10−1 0.1132 × 102 0.0948 × 102 0.5027 × 10−1 0.1119 × 102 0.0643 × 102 0.4597 × 10−1 0.1105 × 102 0.0451 × 102

Table 6Exponential model parameters of dual mode isotherms for four slit pore widths.

kl km kb

kl0 (mmol/cm3/bar) El (kJ/mol) km0 (mmol/cm3) Em (kJ/mol) kb0 (bar−1) Eb (kJ/mol)

H = 0.65 nm 0.5631 × 10−2 6.5260 7.3260 1.2529 1.9896 × 10−5 3.5400 × 101

H = 0.68 nm 1.9882 × 10−2 3.1814 6.9674 1.2986 3.7624 × 10−5 3.3579 × 101

H = 0.72 nm 0.7956 × 10−2 5.1233 7.1996 1.1233 4.6312 × 10−5 3.1649 × 101

H = 0.75 nm 0.1420 × 10−2 9.1701 7.6592 0.9695 6.7672 × 10−5 2.9359 × 101

Table 7Collective diffusivities and their exponential model parameters at T = 298, 308, and 318 K for four slit pore widths.

Mean diffusivities (Do × 109, m2/s) Exponential model parameters of diffusivity

T = 298 K T = 308 K T = 318 K Do0 × 109 (m2/s) ED (kJ/mol)

wepp

t(tt

4

2

Fa

H = 0.65 nm 5.96 6.10H = 0.68 nm 8.74 8.94H = 0.72 nm 11.93 12.18H = 0.75 nm 12.91 13.36

here kl0, km0, and kb0 are the preexponential factors, Em is anffective energy parameter related to thermal expansion of occu-ancy, and El and Eb are magnitudes of the enthalpy changes. Thearameter values are shown in Table 6.

As expected, the adsorption density decreases as the tempera-ure and the pore width increases. The dual mode adsorption model�model

ads) gives good agreement with the GCMC results (�GCMC

ads) with

he correlation coefficient (R1) of 0.9992 for the 204 GCMC simula-ion results, as shown in Fig. 5.

.2. Self- and collective diffusivities

In this study, the self-diffusivity has values in the range of–14 × 10−9 m2/s at T = 298 K for f = 0.01–20 bar at pore widths

ig. 5. Correlation between GCMC adsorption density (�GCMCads

) and dual mode modeldsorption density (�model

ads).

6.27 13.28 1.9869.04 15.07 1.346

12.57 27.12 2.03913.80 37.03 2.610

(H) between 0.65 nm and 0.75 nm. It is observed for the narrowslit pores that the collective diffusivity is weakly dependent onconcentration (or pressure in the range of 0–20 bar), except forH = 0.75 nm, while the self-diffusivity decreases with increase inpressure, because the movement of individual CO2 molecules rel-ative to the others is increasingly restricted as pressure increases.The collective diffusivity at H = 0.75 nm tends to decrease slightlywith fugacity. It is seen in Fig. 6 that the self-diffusivity and the col-lective diffusivity converge, as fugacity goes to zero (zero-loadinglimit) [7]. The self- and collective diffusivities increase with increas-ing pore width (H) and temperature (T), which is in accord with theobservation of Jepps et al. [26] for methane in carbon slit pores.As the pore width increases, the friction between the moleculesand pore wall is effectively reduced at sizes below that at whichlevitation occurs and a maximum in diffusivities is found [45,46].

For comparison, the self-diffusivity of CO2 ranged from0.7 × 10−9 to 1.0 × 10−9 m2/s over 0.1–30 bar at 273 K in aZIF (zeolitic imidazolate framework) with the pore spacing ofd = 1.03 nm [47], using a three-charged LJ potential of rigid lin-ear CO2 molecule. In a MOF (metal–organic framework) with thepore spacing of d = 1.1 nm, the self-diffusivity of CO2 ranged from8 × 10−9 to 40 × 10−9 m2/s within 0.1–50 bar at 298 K [37], usingthe three-charged LJ potential (3-CO2-POT) optimized for VLE(vapor–liquid equilibrium) calculations [34]. These values are con-sistent with the range found here for carbon slit pores with diffusereflection.

The collective diffusivity is also described by the exponentialexpression:

Do = Do0e−ED/RT (5)

The energy barrier (ED) is positive, as the diffusivity increases

with increasing temperature. Since the collective diffusivity (Do)varies little with fugacity (see Fig. 6), a mean collective diffusivity isdetermined at a given temperature and pore width. Table 7 reportsthe mean diffusivities and the exponential model parameters foreach pore width.

192 Y.-I. Lim et al. / Journal of Membrane Science 355 (2010) 186–199

F ined fH

4

amta

J

wf�f(d

˘

wtr

S

hsw

ig. 6. Variation of self- and collective diffusivities (Ds and Do) with fugacity, obta= 0.65 nm; (b) H = 0.68 nm; (c) H = 0.72 nm; (d) H = 0.75 nm.

.3. Solubility coefficient, flux and permeability

Based on the isotherms and collective diffusivities discussedbove, we have determined solubilities and permeabilities in aembrane where the governing resistance is that of transport

hrough the pore body. The steady-state molar flux (J, mol/m2/s)cross a membrane is written in terms of fugacity as:

≡ −Do�ads

RT∇� = Do

L

∫ f

f0

�adsdln f = Do

L

∫ f

f0

�ads

fdf (6)

here Do is the collective diffusivity, also termed corrected dif-usivity, corresponding to the transport rate in the slit pore,

is the chemical potential, L is the membrane thickness, and0 is the downstream fugacity of membrane. The permeability˘) is obtained, under the assumption of a density independentiffusivity [10,15]:

≡ J

�f/L= Do

�f

∫ f

f0

�ads

fdf = DoS (7)

here the solubility coefficient (S) is defined as the mean value ofhe adsorption density (�ads) per unit fugacity (f) within the fugacityange of interest [10,14,15]:

= 1�f

∫ f

f

�ads

fdf (8)

0

The permeability (˘ = DoS) curve versus fugacity appears to beighly responsive to the isotherm shape, as this is reflected by theolubility coefficient in the narrow pores [15]. Combining Eq. (7)ith Eq. (1), the permeability obtained from the GCMC/EMD sim-

rom EMD simulation, for three temperatures (T) and four slit pore widths (H): (a)

ulation (˘sim) is expressed by:

˘sim = Do

(kl + km

f − f0ln

[1 + kbf

1 + kbf0

])(9)

The downstream pressure of the membrane (f0) is assumed tobe 0.01 bar and the membrane thickness (L) is set to 40 �m [14] inthis study. The following model permeability (˘model) is expressedas a function of temperature (T) and fugacity (f) at each pore width:

˘model = Do0e−ED/RT(

kl0eEl/RT +km0eEm/RT

f − f0ln

[1+kb0eEb/RT · f

1+kb0eEb/RT · f0

])(10)

Fig. 7 compares the simulation permeability (˘sim) with themodel permeability (˘model). The model permeability agrees wellwith the permeability by GCMC/EMD simulation and their cor-relation coefficient (R1) is 0.9979. The deviation of the modelpermeability (˘model) from the simulation permeability (˘sim) iscaused mainly by the difference between the EMD results and expo-nential expressions in diffusivity.

The solubility coefficient (S) decreases with increase in tem-perature and fugacity. The permeability decreases with increasingtemperature (T) from 298 K to 318 K and fugacity (f) from 0.01 bar to20 bar, since the solubility (or adsorption isotherm) strongly affectsthe permeability in narrow pore widths (H = 0.65–0.75 nm). It hasbeen experimentally observed that permeability decreases withincrease in the feed pressure [6,14,48]. The trend of permeability

versus temperature agrees with experimental measurements in apolyimide membrane [14]. However, several experimental studies[6,49–51] in carbon molecular sieve (CMS) have shown that per-meability increases with increase in temperature. This may be dueto the fact that, for CO2 in CMS, the high activation energy for dif-

Y.-I. Lim et al. / Journal of Membrane Science 355 (2010) 186–199 193

F t of exponential model (˘model) for each pore size (H): (a) H = 0.65 nm; (b) H = 0.68 nm; (c)H

f(

miflsfAttfFih

ttHtfaiwdd[

te

Table 8Permeation properties obtained from GCMC and EMD simulation at f = 5 bar and twoexperimental measurements of permeability.

˘ × 1015 (mol/m/s/Pa)at T = 308 K

˘0 (mol/m/s/Pa) E˘ (kJ/mol)

This simulation studyH = 0.65 nm 6.538 × 105 6.835 × 104 −5.77H = 0.68 nm 9.097 × 105 7.863 × 104 −6.26H = 0.72 nm 1.075 × 106 9.565 × 104 −6.19H = 0.75 nm 1.038 × 106 9.622 × 104 −6.08

Duthie et al. [14]a 193 – −3.3

ig. 7. Comparison of fugacity variation of simulation permeability (˘sim) with tha= 0.72 nm; (d) H = 0.75 nm.

usion has stronger effect on the permeability than the adsorptionor solubility).

Fig. 8 shows adsorption density (�ads), solubility coefficient (S),olar flux (J), and permeability (˘sim) versus fugacity (or pressure)

n the four slit pore widths (H) at T = 308 K. In Fig. 8(c), the molarux is obtained for a membrane thickness (L) of 40 �m and down-tream fugacity of 0.01 bar. The permeability is represented as aunction of pressure converted by the EOS of Bender (see Appendix). The shape of the solubility coefficient (S = �ads/f) seems to be

he inverse of the adsorption isotherms. It is clearly shown thathe permeability follows the trend of solubility coefficient withugacity. The molar flux (J) increases with fugacity, as shown inig. 8(c). It is worth noting that the flux and the permeability do notncrease monotonically with the pore width, since the pore widthas opposite effects on adsorption and diffusion.

In Fig. 9, the flux and permeability versus pore width are inves-igated at f = 5 and 10 bar for the three temperatures. For all theemperatures, the flux and permeability show a maximum at= 0.72 nm (or Ha = 0.38 nm) at the given fugacity. This is due to

he fact that, at a given temperature and pressure, the effect of dif-usion on the flux and permeability dominates until H = 0.72 nm,nd the decrease in solubility coefficient overwhelms the increasen diffusivity at H = 0.75 nm. As observed in Fig. 8, the flux increases

ith the fugacity (see also Fig. 9(a) and (c)), while the permeabilityecreases with the fugacity (see Fig. 9(b) and (d)). The pressure-

ependence of permeability is in agreement with NEMD simulation52] and experiments [6,48].

Since solubility generally decreases with increasing tempera-ure, while diffusivity increases, the relative magnitudes of the twoffects will dictate the overall trend of permeability with temper-

Vu et al. [6]b 422 – −4.8

a Permeability was measured at P = 5 bar.b Permeability was measured at P = 1.4 bar for 10% CO2 of CO2/CH4 mixture.

ature [14]. In the present case, the CO2 permeability declines withincrease in temperature, consistent with the work of Duthie et al.[14]. The temperature dependency of permeability is described bythe exponential expression:

˘ = ˘0e−E˘ /RT (11)

In Table 8, the preexponential factor (˘0) and the apparent tem-perature coefficient (E˘ ) of permeability are indicated for each porewidth. The permeabilities obtained from GCMC/EMD simulation atT = 308 K and f = 5 bar are compared to two experimental measure-ments in Table 8. The simulation results exceed by three orders of

magnitude those of the two macroscopic measurements, while thetemperature coefficient of the simulation permeability is greaterby a factor of 1.5 than that of experimental values. The follow-ing section discusses the disparity between the experiment andsimulation.

194 Y.-I. Lim et al. / Journal of Membrane Science 355 (2010) 186–199

Fig. 8. Adsorption isotherms, and fugacity variation of solubility coefficient, molar flux and permeability for each pore size at T = 308 K (molar flux is obtained for L = 40 �m):(a) adsorption isotherms (�ads); (b) solubility coefficient (S); (c) molar flux (J); (d) permeability (˘).

Fig. 9. Molar flux (J) and permeability (˘) versus pore width (H) at f = 5 and 10 bar (L = 40 �m): (a) molar flux (J) at f = 5 bar; (b) permeability (˘) at f = 5 bar; (c) molar flux(J) at f = 10 bar; (d) permeability (˘) at f = 10 bar.

brane Science 355 (2010) 186–199 195

4

isct[to(Mnob

ifmiiritwdfhbe

FH

Y.-I. Lim et al. / Journal of Mem

.4. Upper bound for permeability

Evidently, the permeability predicted by GCMC/EMD simulationn the carbon slit pore shows a large discrepancy with the macro-copic experimental measurements, as mentioned above. Table 9ompares the permeability of this study with those in the litera-ure. The three simulation studies (this study, Makrodimitris et al.10], and Firouzi et al. [52]) show large permeability values. A rela-ively high permeability (about 20,000 Barrer) was experimentallybserved in a glass membrane [53]. Relatively low permeabilities224 and 336 Barrer) were reported for polymer membranes [17].

ost of the experimental measurements in CMS and polyimide doot exceed 1000 Barrer (or 335 × 10−15 mol/m/s/Pa) except thosef Steel and Koros [18] and Vu et al. [6], where the big discrepancyetween the permeabilities is caused mainly by diffusivity.

The collective diffusivity (Do) obtained from EMD simulations higher by a factor of 70–1000 than the CMS experimental dif-usivity (or effective diffusivity, Deff), which is the same order of

agnitude as the difference of permeability. For CMS and poly-mide, the solubility coefficients (S) derived from the adsorptionsotherms show relatively small differences between simulationesults and experimental data. Since the experimental permeabil-ty was measured in membranes having complex pore networks,he effects of porous structure on the membrane mass transportere taken into account for the effective diffusivity (Deff), intro-

ucing the so-called tortuosity (�) and the porosity (ε) [54]. Asar as there are impermeable obstacles, the diffusing moleculesave to follow a more tortuous path to go through the mem-rane [55]. Thus, the diffusion rate is reduced and it may bexpressed in terms of the tortuosity (�) related to morphology

ig. 11. Comparison of theoretical permeability (˘ theo) with effective model permeabilit= 0.72 nm; (d) H = 0.75 nm.

Fig. 10. Collective diffusivity (Do) of CO2 versus slit pore width (H) at T = 298 K andf = 1 bar.

and pore connectivity of membrane at the macroscopic level[56].

Since the collective diffusivity (Do) inside the narrow slit poreswith 0.31 nm ≤ Ha ≤ 0.41 nm is much larger than the effective diffu-

sivity, it is assumed that the rate-limiting step of membrane masstransport occurs in the pore mouth, as mentioned earlier [28–30].Therefore, the difference of permeability between the simulationsand macroscopic experiments may result from reduced pore acces-sibility in complex pore morphology. For example, constricted pore

y (˘modeleff

) at f = 5 bar (εp = 0.45–0.52 and � = 1): (a) H = 0.65 nm; (b) H = 0.68 nm; (c)

196 Y.-I. Lim et al. / Journal of Membrane Science 355 (2010) 186–199

Table 9Comparison of diffusivity, solubility coefficient and permeability between this study and literatures.

Reference Year T (K) P (bar) Membrane (pore sizea, nm) Deff × 109 (m2/s) Do × 109 (m2/s) S (mol/m3/Pa) ˘

(×1015, mol/m/s/Pa) (Barrer)b

This studyc 308 5.0 Slit pore (0.34) – 8.85 9.95 × 10−2 881,000 2,630,000Makrodimitris [10]c 2001 303 1.0 Zeolite (0.53) – 7.0 2.82 × 10−2 197,000 590,000Firouzi [52]c 2007 323 45.0 Slit pore (1.0) 125,000 373,000Markovic [53] 2009 308 1.0 Glass (2.3) 1.17 – 5.81 × 10−3 6800 20,310Steel [18] 2005 308 6.5 CMSd (0.42) 1.32 × 10−1 – 1.23 × 10−2 1630 4864Vu [6]e 2002 308 1.4 CMS (0.34) 422 1260Yoshimune [51] 2007 298 1.0 CMS (0.40) 6.2 × 10−3 – 2.85 × 10−2 177 529Tin [5] 2004 308 10.0 CMS (0.35) 167 499Duthie [14] 2007 308 5.0 Polyimide (–) 9.0 × 10−4 – 2.14 × 10−1 193 577Hasan [48] 2009 308 5.0 Polyimide (–) 3.9 × 10−4 – 4.72 × 10−1 184 550Ostwal [17] 2009 303 2.9 PFAPf (–) 2.71 × 10−1 – 4.19 × 10−4 110 336

a Accessible pore width (Ha) or pore spacing (d).b Barrer = 3.348 × 10−16 mol/m/s/Pa.

mhrcniure

D

p

J

˘

[toTAtd

˘

r

D

wjEhdrsT[

c Simulation studies.d Permeability for 10% CO2 of CO2/CH4 mixture and L = 30 �m.e CMS: carbon molecular sieve.f PFAP: poly(fluoroalkoxyphosphazene).

ouths played an important role in sorption in coal, resulting inysteresis of the adsorption and desorption loop [57]. Diffusionalestriction at the pore mouth led to slower mobility and largerossing time between neighboring pores [58,59]. The pore bottle-ecking effect was partially studied in a carbon slit pore connected

n series by three different pore widths (1.0 nm, 2.3 nm and 7.7 nm)sing a DCV-GCMD technique [52]. Considering the pore mouthesistance (1/Dmouth) and the pore structure effect (� and ε), theffective diffusivity may be expressed as follows:

eff = ε

�· 1

1/Do + 1/Dmouth(12)

As a result, the effective flux of membrane (Jeff) and the effectiveermeability (˘eff) are defined as:

eff = Deff S�f

L(13)

eff = Deff S (14)

The porosity (ε) of membranes is typically from 0.25 to 0.356,49], of which the pore wall thickness typically comprises aboutwo graphene layers [60]. In this simulation study, the porosityf one-layer carbon slit pore (εp) varying from 0.45 to 0.52 (seeable 1) is about twice as much as that of the practical membranes.s the pore mouth resistance is not taken into account in this study,

he effective model permeability (˘modeleff

) is obtained, consideringiffusion parallel to the pore, as follows:

modeleff = εp

�DoS = εp

�˘model (15)

For the straight pore, the tortuosity is equal to unity (� = 1).A modified Knudsen diffusivity (DK) of adsorbed molecules was

epresented in the gas phase as follows [53,61–63]:

K = kgeo

√8RT

Mwe−EDK /RT (16)

here kgeo is the geometrical probability that the diffusant makes aump whose length is , Mw is the molecular weight (kg/mol), andDK is the energy barrier of modified Knudsen diffusivity whichas strong dependency on the pore size. The modified Knudsen

iffusivity converges to Knudsen diffusivity, when the energy bar-ier is close to zero. The nanopores considered in this study aretraight and uniform, the geometrical factor is assumed to kgeo = 1/3.he diffusion length is taken as the accessible pore width ( = Ha)61,62].

In Fig. 10, Eq. (16) with EDK = 4.0 kJ/mol is compared to the EMDsimulation at T = 298 K and f = 1 bar in the slit pore widths fromH = 0.54 nm to H = 0.75 nm. The magnitude of the modified Knudsendiffusivity can be adjusted with the energy barrier (EDK) but Eq. (16)has a linear relationship with the pore width (H) at a given energybarrier. The collective diffusivity obtained from EMD simulationdecreases to 1.5 × 10−9 m2/s at H = 0.54 nm (Ha = 0.2 nm). The mod-ified Knudsen diffusivity with EDK ≤ 3.0 kJ/mol shows considerableoverprediction, as shown in Fig. 11. The theoretical permeability(˘ theo) is combined with the solubility factor (S) for the configura-tion transport in slit pore:

˘theo = εp

�· DK · S (17)

Rearranging Eq. (17) with Eq. (16), the theoretical permeabilityfor the single slit nanopore (� = 1) is defined as:

˘theo = εp

3HaS

√8RT

Mwe−EDK /RT (18)

where the solubility coefficient (S) obtained from the GCMC sim-ulation is used. Fig. 11 compares the two permeabilities (˘ theoand ˘model

eff) at f = 5 bar for the four pore widths. Since the solu-

bility coefficient (S) obtained from GCMC simulation is employedfor the calculation of the two permeabilities, both permeabili-ties decrease with increase in temperature. The energy barriers ofmodified Knudsen diffusivity (EDK) corresponding to those of simu-lation diffusivity (ED) for the four pore widths are about 4.7 kJ/mol,4.0 kJ/mol, 3.5 kJ/mol and 3.5 kJ/mol, respectively.

Taking the effective diffusivity of Yoshimune et al. [51], the poremouth resistance in Eq. (12) is approximately estimated at T = 298 Kand P = 1 bar.

Dmouth = 1(ε/�) · (1/Deff ) − (1/Do)

∼= 1.3 × 10−10 m2/s (19)

where the porosity ε = 0.29 and the tortuosity (� = 6.0) taken fromthe experimental results for CMS Takeda 3A [29] are applied, anda value of collective diffusivity, Do = 12 × 10−9 m2/s, is used forH = 0.72 nm (Ha = 0.38 nm), which is obtained from the EMD sim-ulation at the same temperature and pressure. The pore mouth

resistance (1/Dmouth) is estimated to be 100 times larger than thepore resistance (1/Do). The CO2 permeability obtained in this studyis therefore considered as an upper bound, being based on a hypo-thetical membrane composed of the straight and uniform carbonslit pores without pore mouth resistance.

brane

5

fmwtaoa(lmpa

rsdtv

TP

TC

Y.-I. Lim et al. / Journal of Mem

. Conclusions

Adsorption characteristics in carbon slit pore were comparedor a one-site spherical CO2 model and five three-site linear CO2

odels. A three-site CO2 molecular model being in good agreementith experimental (P–�–T) data within T = 300–340 K was selected

o predict adsorption isotherms, diffusivity, solubility coefficient,nd permeability. GCMC and EMD simulations were carried out tobtain adsorption isotherms and diffusivity in the range of temper-ture (T = 298–318 K), fugacity (f = 0.01–20 bar), and slit pore widthH = 0.65–0.75 nm). The adsorption isotherms were modeled by theinear and Langmuir isotherms (dual mode adsorption isotherms

odel), and an exponential expression was used to describe tem-erature dependence of parameters of the dual mode adsorptionnd diffusivity.

Since the effect of adsorption on permeability is stronger in nar-

ow slit pores than the diffusion effect, the permeability shows theame tendency as the solubility coefficient defined as adsorptionensity per unit fugacity. At the given temperature and pressure,he flux and the permeability of carbon dioxide show a maximumalue with respect to the pore width. The theoretical permeability

able A.1arameter values of equation of state for carbon dioxide [42].

n1 0.22488558 n6 0.45554393 × 10−1

n2 −0.13717965 × 103 n7 0.77042840 × 102

n3 −0.14430214 × 105 n8 0.40602371 × 105

n4 −0.29630491 × 107 n9 0.40029509n5 −0.20606039 × 109 n10 −0.39436077 × 103

Fig. B.1. Relationship between fugacity (f), pressure (P), and bulk density (�) for C

able B.1oefficients of 5th-order polynomials for pressure (P, bar) and bulk density (�, mmol/cm3

T (K) a b

P = (a + bf + cf2 + df3 + ef4)f298 1.0012 0.4620 × 10−2

308 0.9970 0.4610 × 10−2

318 1.0002 0.3632 × 10−2

� = (a + bf + cf2 + df3 + ef4)f298 0.4090 × 10−1 0.2350 × 10−3

308 0.3861 × 10−1 0.2533 × 10−3

318 0.3732 × 10−1 0.2660 × 10−3

Science 355 (2010) 186–199 197

derived from a modified Knudsen diffusion is comparable to thepermeability predicted by GCMC and EMD simulation. Compar-ing the effective diffusivity from experiment with the diffusivityfrom GCMC/EMD simulation, it is postulated that the fluid mobil-ity of membrane is limited by the pore mouth resistance relatedto pore accessibility rather than by the inside-pore diffusion. It isnecessary to take into account the pore mouth resistance, and thepore connectivity and morphology (e.g., tortuosity and porosity)to predict the macroscopic permeability by molecular simula-tion.

Acknowledgments

The first author thanks The University of Queensland for invit-ing him as a visiting scholar. This research has been funded by aDiscovery Grant from the Australian Research Council.

Appendix A. Equation of state of Bender (1970)

The equation of state proposed by Bender [42] is expressed asfollows:

n11 0.12115286 n16 −0.29186718 × 1013

n12 0.10783386 × 103 n17 0.24358627 × 108

n13 0.43962336 × 102 n18 −0.37546530 × 1011

n14 −0.36505545 × 108 n19 0.11898141 × 1014

n15 0.19490511 × 1011 n20 0.5 × 101

O2: (a) CO2 pressure versus fugacity; (b) CO2 bulk density versus fugacity.

) versus fugacity (f, bar).

c d e

0.9403 × 10−4 −0.1834 × 10−5 0.5316 × 10−7

0.7210 × 10−4 −0.2144 × 10−5 0.5199 × 10−7

0.8710 × 10−4 −0.2283 × 10−5 0.4022 × 10−7

0.2426 × 10−4 −0.7928 × 10−6 0.1754 × 10−7

0.2354 × 10−4 −0.8752 × 10−6 0.1562 × 10−7

0.2020 × 10−4 −0.9395 × 10−6 0.1671 × 10−7

1 brane Science 355 (2010) 186–199

P

weM(

l

w

Z

tc

l

Ad

gtiei

H pore width (nm)Ha accessible pore width (nm)J molar flux (mol m−2 s−1)Jeff effective molar flux (mol m−2 s−1)kb affinity parameter of dual mode adsorption model

(bar−1)kB Boltzmann constant (J K−1)kgeo geometrical probabilitykl linear adsorption constant of dual mode adsorption

model (mmol cm−3 bar−1)km saturation density of dual mode adsorption model

(mmol cm−3)L membrane thickness (nm)Mw molecular weight (kg mol−1)n constants of Bender’s equation of stateP pressure (bar)q Coulomb electrostatic charge (e)R gas constant (kJ K−1 mol−1)S solubility coefficient (mol m−3 Pa−1)T temperature (K)Tc critical temperature (K)V volume (m3)Z compressibility

Greek symbols� interplanar distance of carbon layers (nm) or differ-

ence between two valuesε energy parameter in Lennard-Jones potentials (J) or

porosityεp pore porosity� fugacity coefficient jumping distance of diffusant (nm)� chemical potential (J)˘ permeability (mol m−1 s−1 Pa−1)˘eff effective permeability (mol m−1 s−1 Pa−1)˘ theo theoretical permeability (mol m−1 s−1 Pa−1)� bulk fluid density (mmol cm−3)�ads adsorption density in slip pore (mmol cm−3)�s carbon site density for slip pore (nm−3)

98 Y.-I. Lim et al. / Journal of Mem

= �(

RT +(

n1T + n2 + n3

T+ n4

T2+ n5

T3

)� +

(n6T + n7 + n8

T

)�2

+(n9T + n10)�3 + (n11T + n12)�4 + n13�5

+[( n14

T2+ n15

T3+ n16

T4

)+( n17

T2+ n18

T3+ n19

T4

)�2

]�2 · e−n20�2

)(A.1)

here the constant is R = 0.188918 MPa cm3/g K, and the 20 param-ters are shown in Table A.1 for CO2. Pressure (P) has the unitsPa, the density (�) has g/cm3 and T has K. The fugacity coefficient

� ≡ f/P) is defined as:

n � ≡∫ P

0

(Z − 1)dP

P≡ (Z − 1) − ln Z +

∫ �

0

(Z − 1)d�

�(A.2)

here the compressibility (Z) is expressed as follows:

≡ PV

RT≡ P

�RT

={

1 + B�

RT+ C

�2

RT+ D

�3

RT+ E

�4

RT+ F

�5

RT+ [G + H�2]

�2

RTe−n20�2

}(A.3)

Here, the constants (B, C, D, E, F, G, and H) are equivalent tohe coefficients of Eq. (A.1) in terms of �. Therefore, the fugacityoefficient is obtained as follows [64]:

n � = (Z − 1) − ln Z

+ 1RT

{B� + C

2�2 + D

3�3 + E

4�4 + F

5�5 − 1

2n20

(G + H

n20+ H�2

)e−n20�2

+ 12n20

(G + H

n20

)}(A.4)

ppendix B. Relationships of fugacity, pressure and bulkensity

Using the equation of state (EOS) of Bender for CO2, the fugacityiven in molecular simulation is converted into the pressure, and

he fluid bulk density at a given fugacity is calculated, as shownn Fig. B.1. When the relationship obtained from EOS of Bender isstimated by a fifth-order polynomial, their coefficients are shownn Table B.1.

Nomenclature

d accessible pore spacing (nm)Deff effective diffusivity (m2 s−1)DK modified Knudsen diffusivity (m2 s−1)Dmouth pore mouth diffusivity (m2 s−1)Do collective diffusivity (m2 s−1)Ds self-diffusivity (m2 s−1)Eb energy factor of affinity parameter (kJ mol−1)ED energy factor of collective diffusivity (kJ mol−1)EDK energy factor of modified Knudsen diffusivity

(kJ mol−1)El energy factor of linear adsorption constant

(kJ mol−1)Em energy factor of saturation density (kJ mol−1)E˘ energy factor of permeability (kJ mol−1)f fugacity (bar)f0 downstream fugacity of membrane (bar)

� collision diameter in Lennard-Jones potentials (nm)� tortuosity

Subscripts0 indicates a preexponential factorC carbonf fluid

O oxygens solid

References

[1] R. Qi, M.A. Henson, Optimization-based design of spiral-wound membranesystems for CO2–CH4 separations, Sep. Purif. Technol. 13 (1998) 209–225.

[2] X. Cui, R.M. Bustin, G. Dipple, Selective transport of CO2, CH4, and N2 in coals:insights from modeling of experimental gas adsorption data, Fuel 83 (2004)293–303.

[3] T.X Nguyen, S.K. Bhatia, D. Nicholson, Prediction of high-pressure adsorptionequilibrium of supercritical gases using density functional theory, Langmuir 21(2005) 3187–3197.

[4] K.M. Steel, W.J. Koros, Investigation of porosity of carbon materials and relatedeffects on gas separation properties, Carbon 41 (2003) 253–266.

[5] P.S. Tin, T.S. Chung, Y. Liu, R. Wang, Separation of CO2/CH4 through carbonmolecular sieve membranes derived from p84 polyimide, Carbon 42 (2004)3123–3131.

[6] D.Q. Vu, W.J. Koros, S.J. Miller, High pressure CO2/CH4 separation using car-bon molecular sieve hollow fiber membranes, Ind. Eng. Chem. Res. 41 (2002)367–380.

brane

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