Fluid Phase Equilibria 382 (2014) 116–126

Modeling of molecular gas adsorption isotherms on porous materialswith hybrid PC-SAFT–DFT

Gulou Shen a,b, Xiaohua Lu a,*, Xiaoyan Ji b,**a State Key Laboratory of Materials-Oriented Chemical Engineering, Nanjing Tech University, Nanjing 210009, PR ChinabDivision of Energy Science/Energy Engineering, Lulea University of Technology, Lulea 97187, Sweden

A R T I C L E I N F O

Article history:Received 19 June 2014Received in revised form 28 August 2014Accepted 1 September 2014Available online 6 September 2014

Keywords:PC-SAFTDFTAdsorptionPore size distribution

A B S T R A C T

The developed hybrid PC-SAFT–DFT model, a coupling of density functional theory (DFT) with perturbed-chain statistical associating fluid theory (PC-SAFT), was used to study the adsorption of pure- and mixed-fluids on nano-porous materials, and carbons and zeolites were chosen as examples of nano-porousmaterials in this work for model performance evaluation. In the PC-SAFT–DFT model, the modifiedfundamental measure theory was used for the hard sphere contribution, the dispersion free energyfunctional was represented with a weighted density approximation, and the chain free energy functionalfrom interfacial SAFT was used to account for the chain connectivity. The fluid was modeled as a chainmolecule with molecular parameters taken from those in the bulk PC-SAFT. The external force field wasused to describe the interaction between the solid surface of a nano-porous material and fluid.Application of this model was demonstrated on the gas adsorption on porous carbons and zeolites whichwere assumed to have slit- and cylinder-shaped pores with mean pore sizes, respectively. The parametersof the adsorption model were obtained by fitting to the pure-gas adsorption isotherms measuredexperimentally. With parameters of the model fitted to the pure-gas adsorption at one temperature, themodel was used to predict the pure-gas adsorption at other temperatures as well as the adoptionisotherms of mixtures. The model prediction was compared with the available experimental data, whichshows that the predictions are reliable for most of the systems studied in this work. The effect of the poresize distribution on the model performance was further investigated, and it was found that theconsideration of the pore size distribution (PSD) can improve the accuracy of the model results butthe PSD analysis requires much more computing time.

ã 2014 Published by Elsevier B.V.

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1. Introduction

Inhomogeneous fluid confined in nanopores is of considerableimportance in various industrial processes and scientific fieldssuch as adsorption, polymers, lubrication and tribology [1–3]. Theproperties of fluid confined in nanopores or nanochannels aredifferent from those in the bulk phase due to wall-moleculeinteractions and wall-geometry. A lot of research has beenaddressed to represent the inhomogeneous behavior of confinedfluids from experimental measurements, molecule simulations totheoretical models, and the classical density functional theory(DFT) has been proved to be successful in explaining and predictingthe behavior of inhomogeneous fluids [4].

Meanwhile, fluid adsorption on nano-porous materials (nano-adsorbents) is of importance for both gas separation and

* Corresponding author. Tel.: +86 25 83588063.** Corresponding author. Tel.: +46 920492837.

E-mail addresses: xhlu@njtech.edu.cn (X. Lu), xiaoyan.ji@ltu.se (X. Ji).

0378-3812/$ – see front matter ã 2014 Published by Elsevier B.V.http://dx.doi.org/10.1016/j.fluid.2014.09.002

production. For example, a lot of research work is ongoing todevelop novel nano-porous materials, such as metal organicframework-based materials and new generation of zeolites [5], forbiogas/natural-gas upgrading or CO2 capture. Meanwhile, theprocess of enhanced coal-bed methane or shale gas productionalso relates to the gas adsorption on nano-porous materials.Although a lot of experimental measurements have been carriedout, the prediction of gas adsorption with modeling is still achallenging task [6–8].

A number of approaches can be used to describe the adsorptionof pure gases or mixtures from empirical models, (e.g. theLangmuir equation) to molecule interaction-based theories (e.g.DFT [9]) as well as molecule simulations. The original or modifiedideal adsorption solution theory (IAST) [10], the real adsorptionsolution theory (RAST) [11], the non-local density functionaltheory (NLDFT) [12–14], the simplified local density model (SLD)[15–17] and the extended empirical models (multicomponentpotential theory of adsorption (MPTA) [18,19] and extendedLangmuir model [20]) are typical examples that have beenproposed to represent the adsorption of multicomponent

Nomenclature

Symbolse Attractive pair potential, Jm Chemical potentialG Excess adsorption quantityV Grand potentialA Helmholtz free energy, Jr Number density of molecule, 1/Å3

m Number of segment per molecular chains Segment diameter, Å

Fig. 1. Schematic of a molecular chain comprised of m spherical segments.

G. Shen et al. / Fluid Phase Equilibria 382 (2014) 116–126 117

mixtures. Among these models, NLDFT is the most rigorous one,and it has been used as a standard method for pore sizedistribution analysis. However, the application of NLDFT is limiteddue to the fact that only the free energies of short range repulsionand Van der Waals attraction were accounted for.

Meanwhile, DFT is a useful tool for representing the behaviorof confined fluids including the gas adsorption on nano-adsorbents. DFT methods can give detailed information of thestructure of inhomogeneous fluid but require much lesscomputing resource than molecular simulation. Recently, a classof DFTs was developed based on Wertheim's first orderthermodynamic perturbation theory (TPT) [21–24], and thesemolecular DFTs reduce to SAFT equation of state [25–33] in thebulk limit. As such DFTs also account for the excess free energiesof chain connectivity and association, their application can beextended to a wide range of substances, [34–38] and it allowsdescribing homogeneous and inhomogeneous properties in aconsistent manner. Several of these DFTs have been applied forrealistic fluid interface properties, among them, the modelsproposed by Xu et al. [39], Hu et al. [40], Llovell et al. [41], Kahlet al. [42], von Müller et al. [43] and Klink et al. [44] have beenused to represent the interfacial properties of fluids such ashydrocarbons, CO2, H2O and polymers. However, the applicationof such molecular DFTs for realistic fluid confined in pores or neara solid surface is still limited.

The long-term goal of our work is to study the confinedbehavior of realistic fluids including chain molecule withinteractions other than repulsion and Van der Waals attraction(e.g. associative interactions). To achieve the goal, in our previouswork [45,46], a hybrid DFT model — PC-SAFT–DFT model wasdeveloped to represent the properties of confined fluid innanopores on the basis of PC-SAFT because of its well-knownperformance for bulk fluids. In the developed PC-SAFT–DFT, theinteractions of hard-chain, dispersion and association wereconsidered, and the extension to including the ionic term canbe conducted easily if necessary. In our previous work, it has beenshown that the developed PC-SAFT–DFT model can reproduce thedensity profiles of chain fluids compared to molecular simu-lations, and it is feasible to represent the adsorption isothermsfor real substances. This observation reveals that the developedPC-SAFT–DFT is promising to study the confined behavior for allkinds of realistic fluids. To further verify the model performance,considering the available experimental data, in this work themodel was used to represent the adsorption of pure- and mixed-gases on nano-porous materials. In principal, as DFT methodshave been successfully applied in the characterization ofmicroporous and mesoporous materials with different poregeometry and surface chemistry [47], PC-SAFT–DFT model canbe applied for different nano-porous materials including carbons,zeolites, silicas and metal–organic frameworks. Considering the

computation efficiency and the available experimental data, thepure- and mixed-gas adsorption on porous carbons and zeolitesin a wide temperature and pressure range will be the focus in thiswork, and the effect of pore size distribution on modelperformance will be further discussed. The chosen nano-porousmaterials in this work was specific, while the description of themodel was general discarding the materials.

2. Theory

The developed PC-SAFT–DFT model is a coupling of PC-SAFT[26] and DFT, i.e. the Helmholtz free energy required in DFT wascalculated from PC-SAFT. In PC-SAFT–DFT, we consider a molecularmixture where the molecule is coarse-grained as a chain made upof spherical segments with attractive sites as illustrated in Fig. 1.The chain is formed by bonding between the attractive sites. Themolecule could have additional association sites accounting forassociating between molecules. Therefore, the hard-chain, disper-sive and associative interactions were considered for fluidmolecules.

In DFT, for an open system, the grand potential V at fixed m,V and temperature T is given by the equation

V riðrÞ½ � ¼ A riðrÞ½ � �Xi

Zdr0riðr0Þðmi � miVi;extðr0ÞÞ (1)

where A is the Helmholtz free energy, ri(r) is the density ofcomponent i, mi is the chemical potential, mi is the number ofsegments in a chain, and Vi,ext(r) is the external field acting on thesegment of component i in order to describe the interactionbetween the solid surface of a nano-porous material and the fluidsegment.

Following PC-SAFT, the Helmholtz free energy A can beexpressed as

A riðrÞ½ � ¼ Aid riðrÞ½ � þ Ahs riðrÞ½ � þ Achain riðrÞ½ � þ Adisp riðrÞ½ �þ Aassoc riðrÞ½ � (2)

where Aid is the ideal free energy, Ahs, Achain, Adisp and Aassoc are theexcess free energies due to hard sphere, chain, dispersive andassociative interactions, respectively. The extension of PC-SAFT[26] to inhomogeneous fluid has been described in our previouswork [45,46].

2.1. Numerical procedure

Minimization of the grand potential with respect to the densityprofile yields the following Euler–Lagrange:

dV rðrÞ½ �driðrÞ

¼ dA rðrÞ½ �driðrÞ

� ðmi � miVi;extðrÞ ¼ 0 (3)

At equilibrium, mi is equal to the bulk chemical potential andcalculated from the temperature, pressure and homogeneouscomposition. The equilibrium density profile can be solved usingsimple Picard iteration.

118 G. Shen et al. / Fluid Phase Equilibria 382 (2014) 116–126

2.2. Gas adsorption description

In modeling gas adsorption on nano-porous materials, thesolid-fluid (gas) interaction is the key and directly relates to thepore structure and surface. For activated carbons, the pore can becommonly modeled as a slit-like pore. For materials with complexstructures, for example, zeolites, silicas or metal–organic frame-works, if the pore geometry is simple, the pore can beapproximately modeled as a cylindrical or spherical shape, or itcan be represented by an assembly of cylindrical and sphericalpores; if the pore geometry is complex, the pore has a threedimensional connected structure, and the solid-fluid interactioncan be considered as the summation of the interaction potential ofeach atom of the solid with fluid. In general, for most of nano-porous materials, the pore can be simplified as one-dimensionalstructure, and then the PC-SAFT–DFT can be simplified to be aone-dimensional case. Subsequently, the solid–fluid interactionVext(z) can be represented with a Lennard-Jones potential.

In this work, the activated carbon was modeled as the materialwith slit-shaped pores, and the interaction between the surfaceand the segment of component i (Vext,i(z)) was represented by Lee's10-4 Lennard-Jones potential [15,48] and given by

Vext;iðzÞ ¼ 2pratomesis2si

25ssi

z

� �10�X4i¼1

s4si

ðz þ ði � 1ÞDÞ4" #

(4)

where ratom (= 0.382 atoms/Å) is the solid atom density, ssi is theeffective solid–fluid diameter, (= 3.35 Å) is the distance betweencarbon planes [15], and esi is the potential representing theinteraction between surface and fluid segment.

The following mixing rule was used to represent the size andinteraction parameters between the segment on the solid surfaceand the segment of fluid:

ssi ¼ss þ si

2(5a)

esi ¼ ð1 � kiÞffiffiffiffiffiffiffiffiesei

p(5b)

In Eq. (5), ss is the diameter of solid atom (e.g. carbon incarbonaceous materials, and oxygen in zeolites) and it was set to be3.4 Å for carbon in this work, es is the solid interaction parameter,and ki is used to adjust the interaction between the segments offluid and solid surface and can be obtained from the fitting ofpure-gas adsorption if necessary.

For zeolites studied in this work, the pore was modeled as acylinder, and the external interaction potential for the cylindricalpore was given by [49]

Vext;iðr; RcÞ ¼ p2ratomesis2si

6332

rssi

2�rRc

� �� ��10

F �92; �9

2; 1; 1 � r

Rc

� �2� �"

�3rssi

2�rRc

� �� ��4

F �32; �3

2; 1; 1 � r

Rc

� �2� �

� (6)

where Rc is the radius of the pore, F is the hypergeometric function,and ssi was calculated by Eq. (5) in which the diameter of oxygenatom ss in zeolites was set to be 3.0 Å in this work [50].

If the pore size is uniform, the excess adsorption (G) of gas(fluid) for a slit pore can be calculated by

Gi ¼A�

2

Z H�0:5ss

0:5ss

dzðriðzÞ � rbulk;iÞ (7a)

And for a cylindrical geometry we have

Gi ¼2A?

H � ss

ZRc�ss=2

0

drðriðrÞ � rbulk;iÞ (7b)

where A* is the effective surface area of the adsorbent(nano-porous materials), and H is the effective size of the pore.The lower and upper limit of the integration (Eq. (7)) iscommonly used and different from that in our previous work(lower limit ssi) [46].

In fact, the pore size of nano-porous materials is a distributioninstead of a pore with a single value. With the consideration of poresize distribution, the adsorption isotherm was expressed as

GðT; PÞ ¼ZHmax

Hmin

GlocalðT; P; HÞf ðHÞdH (8)

where G(T,P) is the adsorption isotherm, Hmin and Hmax arethe smallest and largest effective sizes of pores in PSDanalysis, respectively, Glocal(T,P,H) is a local adsorption isothermin a pore with a diameter H at T and P (similar to Eq. (7)). f(H) isthe PSD.

In our previous work [46], the PSD was assumed as a gammadistribution. Since the actual PSD may not agree with anydistribution function, in this work, the PSD was analyzed in ageneral way without any presumed distribution function.f(H)dH was the volume with width between H and H + dH,i.e. f(H)dH = f(H)DH, and it was determined similarly to theoptimization method suggested by Nguyen and Do [51]. Specifi-cally, the total pore range was divided into small intervals, so the(total) adsorption (G(T,P)) is the summation of the adsorption ineach small pore interval. Then the integral in Eq. (8) was replacedby a summation over discrete pore intervals. In practice, thevolume f(H)DH can be determined from the fitting of experimentalpure-gas adsorption isotherms, and the objective function E can bewritten as

E ¼ 1Nexp

XNexp

i¼1

ðXNH

j¼1

GlocalðT; Pi; HjÞfðHjÞDH � GexpðT; PiÞÞ2 (9)

where NH is the total number of discrete pore intervals.As mentioned in the forgoing text that due to the available

systematical experimental data and the effective computationefficiency, the carbons and zeolites were chosen as examples ofnano-porous materials in this work for investigating the modelperformance in predicting the gas adsorptions on nano-porousmaterials. Following this, the model for representing the gasadsorption on the activated carbons, carbon molecular sieveand zeolites was discussed in more detail in the following text.

With the assumption that the porous materials have simplepore geometries with an average width H, in the PC-DAFT–DFTthere are two material-specific parameters, i.e. effective pore sizeH and effective area A*, and one solid–fluid interaction parameteresi. These parameters can be obtained from the fitting of theexperimental adsorption isotherms of pure gas, and no additionalparameters are needed for representing the adsorption isothermsof gas mixtures. The parameter set is different from that inprevious work [45,46]. The preliminary investigation shows thatthe parameter set used in this work is effective under variousconditions.

For the model with a pore size distribution, f(H) (or the volume f(H)DH) can be simultaneously regressed from the pure-gasadsorption isotherms together with es and ki. However, thepreliminary study reveals that the parameter fitting takes a verylong time for such a strategy and the fitting itself is unstable.Therefore, in this work, when considering PSD, the solid-fluidinteraction was still represented by Eqs. (4) and (6), but es/k wastaken from the literature (e.g. es/k = 28 for carbon in carbonaceousmaterials [52] and es/k = 93.53 for oxygen in zeolites [50]) insteadof an adjustable parameter and ki was set to be zero.

Table 2Parameters of fluids used in this work [26].

Substance m sf (Å) ef/k (K)

CH4 1.0 3.7039 150.03CO2 2.0729 2.7852 169.21N2 1.2053 3.3130 90.96C2H6 1.6069 3.5206 191.42

G. Shen et al. / Fluid Phase Equilibria 382 (2014) 116–126 119

3. Available experimental data of gas adsorption on carbons andzeolites

Experimental data plays a significant role in model develop-ment. The model parameters can be obtained from the fitting ofexperimental data, and the model prediction can be verified by theavailable experimental data. Experimental gas adsorption mea-surement has been studied a lot, but the systematic research workis still limited. The available systematic experimental gas adsorp-tion data used in this work is summarized in the following text.

For pure-gas adsorption on carbon, Choi et al. [53] measuredthe adsorption of pure CH4, N2, C2H4 and C2H6 on activated carbonPCB at 293.15, 303.15 and 313.15 K and at pressures up to 2 MPa.Zhou et al. [54] measured CH4 adsorption on activated carbon AX-21 in a wide temperature range from 233 to 333 K and up to 10 MPa.Watson et al. [55] measured the adsorption of pure CH4, N2 and CO2

on a carbon molecular sieve MSC 3K-161 at temperatures from115 to 323 K and up to 5 MPa. Meanwhile, Savitz et al. [56]measured the adsorption isotherms of CH4, C2H6 and C3H8 on high-silica zeolites. Since the adsorption of CH4, N2, C2H6, and CO2 isrelated to the application of biogas/natural gas upgrading andenhanced coal-bed methane, the adsorption isotherms of pureCH4, N2, C2H6 and CO2 were chosen for model verification in thiswork as summarized in Table 1.

For mixtures, there are fewer high-pressure experimentaladsorption data. Considering the application of biogas/natural gasupgrading and enhanced coal-bed methane, four systems werestudied in this work for model verification, i.e. mixtures of CH4 andC2H6 on activated carbon BPL at 301.4 K [57], mixtures of CH4, N2

and CO2 on activated carbon Calgon F400 at 318.2 K [58], andmixtures of CH4/N2 and CO2/N2 on zeolite silicalites [59,60].

4. Results and discussion

In modeling, to simplify the calculation, firstly (Sections 4.1 and4.2), the porous materials were assumed to be a slit or cylinderwith an average size, and then the model parameters of H, esi and A*

were obtained from the fitting of experimental data of pure-gasadsorption at one temperature. The model prediction of pure-gasadsorption at other temperatures and of mixed-gas adsorptionisotherms was verified by comparing with the available experi-mental data. Secondly (Section 4.3), the effect of PSD on modelperformance was further studied in which the parameters of solidsurface were taken from the literature and the f(H) (or the volume f(H)DH) was obtained from the fitting of experimental data of pure-gas adsorption at one temperature. The model performance withthe consideration of PSD was compared with those without theconsideration of PSD (results in Sections 4.1 and 4.2).

Table 1Summary of systems investigated in this work.

Fluid (gas) Material T,K Pmax, MPa Ref.

CH4 PCB 293.15–313.15 2 [53]CH4 AX-21 233–333 10 [54]CH4, CO2 MSC 3 K – 161 248–323 5 [55]CH4, C2H6 Zeolite TON 309.05 0.1 [56]CH4, C2H6 BPL 301.4 4 [57]CH4 + C2H6

CH4, N2, CO2 Calgon F400 318.2 13.6 [58]CH4 + N2

CH4+ CO2

N2 + CO2

CH4, N2 Zeolite silicalite 313.15, 343.15 0.5 [59]CH4 + N2

CO2,N2 Zeolite silicalite 313.15 0.5 [60]CO2 + N2

4.1. Prediction of pure-gas adsorption isotherms in a wide temperaturerange

In this section, the model was used to study the adsorptionisotherms of pure-gases with the assumption that the porousmaterial was a slit or cylinder pore with an average size. Inmodeling, the parameters of each gas were taken from theliterature which had been obtained from the fitting of theirproperties in the bulk phase. The parameters of CH4, N2, C2H6 andCO2 used in this work were taken from literatures and are listed inTable 2. The model parameters of H, ei and A* were fittedsimultaneously to the adsorption of pure-gas i at one temperature,generally near room temperature, on a type of porous material. Themodel with the parameters was then used to predict the pure-gasadsorption at other temperatures on the same porous material. The

average relative deviation ARD ¼ 100Nexp

PNexp

i¼1jG

cali �Gexp

i

Gexpi

j� �

was used to

illustrate the model performance.The adsorption of CH4 on activated carbon PCB was studied in

which the model parameters were fitted to the experimentaladsorption data at 293.15 K. The fitted parameters are listed inTable 3. The adsorption isotherms at 303.15 and 313.15 K werepredicted and compared with the available experimental data. Thedeviations of ARDs are listed in Table 3 and the comparison isillustrated in Fig. 2. The illustration in Fig. 2 and ARDs listed inTable 3 show that the model predictions are in good agreementwith experimental data.

The adsorption of CH4 on activated carbon AX-21 has beenstudied experimentally in a wide temperature range. In modeling,the model parameters were fitted to the experimental adsorptionisotherm at 293.15 K. With the fitted parameters at 293.15 K listedin Table 3, the model was used to predict the adsorption isothermsat other temperatures from 233.15 to 333.15 K. The comparisonwith the available experimental data is shown in Fig. 3, and thecorresponding ARD values are listed in Table 3. At low temper-atures of 233.15 and 253.15 K, the adsorption isotherms showmaxima, i.e. the adsorption increases and then decreases withincreasing pressure. At higher temperatures (�273.15 K), theadsorption increases with increasing pressure. With increasingtemperature, the adsorption decreases at low pressures. The modelcaptures the effects of both temperature and pressure onadsorption, and the model results agree well with experimentaldata in a wide temperature range with the deviation ARD lower

Table 3Model parameters fitted from pure-gas adsorption and deviations.

Puregas

Adsorbent H(Å)

esi/ka

(K)A*

(m2/g)ARD (%)

Correlation Prediction

CH4 Activated carbon PCB 13.6 82.73 539.9 4.35 6.55CH4 Activated carbon AX-21 18.2 76.71 1249.3 2.10 3.78CH4 Carbon molecular sieve

(MSC 3K-161)13.8 97.18 295.6 3.19 7.38

CO2 97.38CH4 Zeolite TON 9.8 23.96 634.2 6.40 –

C2H6 26.26

a For the given solid–fluid energy parameter of zeolites, it is multiplied by densityof solid atom.

Fig. 4. Adsorption of CH4 and CO2 on a carbon molecular sieve (Shirasagi MSC 3 K –

161). Symbols: experimental data listed in Table 1.�: CH4, 298 K; &: CO2 298K; &:CO2, 323 K; ^: CO2, 273 K; ~: CO2, 248 K. Solid curves: model results with aneffective H; dashed curves: model results with PSD.

Fig. 2. Adsorption of CH4 on activated carbon PCB. Symbols: experimental datalisted in Table 1; dashed curves: model correlation results; solid curves: modelpredictions.

120 G. Shen et al. / Fluid Phase Equilibria 382 (2014) 116–126

than 4%. The accuracy of the model prediction in this work iscomparable to that in multicomponent potential theory ofadsorption (MPTA) [18].

The adsorption of pure CH4 and CO2 on carbon molecular sieveShirasagi MSC 3K-161 was measured experimentally. To representthe adsorption isotherms of different pure-gases on the sameadsorbents, the material-specific parameters H and A* are the samefor the same adsorbents, while the fluid-specific parameteresi depends on the adsorbed gas. Therefore, in total four parameterswere used in fitting for this system.

The model parameters were fitted to the experimentaladsorption isotherms for both pure-gases at 298.15 K. The fittedparameters are listed in Table 3. At 298.15 K, the model provides agood representation for CH4, while for CO2 the model results areslightly lower than the experimental data at higher pressures asshown in Fig. 4. The model with the fitted parameters was used topredict the adsorption isotherms of CO2 at 323, 273 and 248 K. Thecomparison of the model prediction for CO2 with the availableexperimental data is shown in Fig. 4, and the model prediction athigh pressures shows discrepancies from the experimental data.

Fig. 3. Adsorption of CH4 on activated carbon AX-21. Symbols: experimental datalisted in Table 1; dashed curves: model correlation results; solid curves: modelpredictions.

The deviation for the CO2 system is due to the drawback of using asingle slit pore for representing the structure of porous materials.In fact, most porous materials have a wide range of pore sizedistribution. The pores with different sizes show differentselectivity at different pressures for fluid adsorption, and thenthe excess adsorption quantity calculated by the model is sensitiveto the pore size. The investigation of the effect of the pore sizedistribution on the model performance was further discussed inSection 4.3.

The adsorption of pure CH4 and C2H6 on zeolite TON wasmeasured experimentally [56], TON has a straight channel in onedirection, which can be founded in IZA database of zeolitestructures [61]. We approximated the channel as a cylindricalpore. With this assumption, the model was used to represent thegas adsorption isotherms. Fig. 5 shows the model results of gasadsorption isotherms for pure CH4 and C2H6 as well as theexperimental data. The comparison in Fig. 5 shows that the modelresults are in good agreement with experimental data for CH4 andshow a deviation for C2H6 at pressures higher than 40 KPa.

In summary of this section, from the comparison of the modelresults of pure-gas adsorption with the experimental data, we can

Fig. 5. Adsorption of CH4 and C2H6 on zeolite TON. Symbols: experimental datalisted in Table 1; solid curves: model correlation results.

Table 4Model parameters fitted from pure-gas adsorption isotherms and the correspond-ing deviations.

Pure gas Adsorbent H (Å) esi/k� (K) A* (m2/g) ARD%

CH4 Activatedcarbon BPL 16.3 85.79 443.4 4.04C2H6 91.99CH4 Calgon F400 16.9 87.87 455.9 3.41N2 72.23CO2 91.56

Table 5Prediction errors for the adsorption quantities.

Mixture ARD (%)

CH4/C2H6 22.26CH4/N2 4.49N2/CO2 13.47CH4/CO2 26.51

G. Shen et al. / Fluid Phase Equilibria 382 (2014) 116–126 121

conclude that the model can provide a reliable prediction for theadsorption of pure CH4 in a wide temperature range and up to highpressures, while for the adsorption of pure CO2 and C2H6, themodel prediction is not as good as those for CH4, but its accuracy isstill acceptable.

4.2. Prediction of mixed-gas adsorption

The model was further used to study the adsorption of mixtureson a porous material that was modeled as a slit or cylinder with aneffective size. As the materials studied in this section are differentfrom that studied in the foregoing section, the model parameterswere firstly obtained from the fitting of pure-gas adsorptionisotherms, and then the adsorption isotherms of mixtures on thesame material were predicted without any additional parameters.The binary interaction parameters between gases (fluids) in thissection were taken from the literature, i.e. zero for CH4/C2H6 [62],0.0307 for CH4/N2 [63], 0.065 for CH4/CO2 [26] and �0.0246 forCO2/N2 [64].

4.2.1. CH4 and C2H6 on activated carbon BPLThe adsorption of CH4, C2H6 and their mixtures on activated

carbon BPL was measured by Reich et al. [57]. Using the samemethod as described in the foregoing section, the modelparameters were fitted to pure CH4 and C2H6 on activated carbonBPL at 301.4 K, respectively. The fitted model parameters are listedin Table 4 together with the corresponding ARDs. The modelresults were compared with the experimental data and areillustrated in Fig. 6. The model results of the adsorption of CH4

agree well with experimental data, while the representation of theadsorption of C2H6 is less accurate at high pressures. Similar to theabove discussion, the inaccurate performance of the model results

Fig. 6. Adsorption of CH4 and C2H6 on activated carbon BPL at 310.4 K. Symbols:experimental data listed in Table 1; solid curves: model correlation results with aneffective H; dashed curves: model results with PSD.

for C2H6 is partially due to the disadvantage of a single slit poreassumption for representing porous materials. Further investiga-tion is in Section 4.3.

Without any additional parameters, the model was used topredict the adsorption of gas mixture on activated carbon BPL, inwhich the compositions of CH4 are 74.5, 49.9 and 26.7%,respectively, in the gas phase. The deviations (ARDs) are listedin Table 5. Fig. 7 shows the comparison of the model predictionwith the experimental data. It shows that the model prediction forC2H6 agrees with the experimental data, and the effect of the bulkcomposition on the individual adsorption on BPL was reasonablycaptured by the model prediction, while the model prediction forlower adsorbed component (CH4) is less accurate at higherpressure, which is consistent with the observations in other works[65].

Among the various models for mixed-gas adsorption, IAST is anengineering thermodynamic model, and it has been widely used topredict mixed-gas adsorptions on porous materials such asactivated carbons, zeolites and MOFs. In modeling, the pure-gasadsorption isotherms at one temperature were used as input topredict the mixed-gas adsorption isotherms at the same tempera-ture. Deviations of IAST model could come from the heterogeneoussurface of materials (variation of pore shape and size, structure andchemical heterogeneity of surface) and the dissimilarity of gasmolecules (molecular size and interaction type). In IAST method,the accurate experimental pure-gas adsorption data and accuratecorrelation for pure-gas adsorption are required.

In this section, IAST was also used to study the gas adsorptionisotherms for this system to further check the performance ofdifferent models. A single-site Langmuir–Freundlich equation wasused to fit the pure-gas isotherms. Then the fitted parameters were

Fig. 7. Adsorption of CH4 (open symbols) and C2H6 (filled symbols) binary mixtureson activated carbon BPL at 310.4 K. Gas composition, symbols: experimental datalisted in Table 1.4, ~: 74.5% CH4; �, *: 49.9% CH4; &, &: 26.7% CH4; solid curves:model predictions with an effective H; dashed curves: model predictions with PSD;dashed dotted curves: IAST predictions.

Fig. 9. Adsorption of CH4 (a) and N2 (b) in the CH4/N2 mixtures on activated carbonCalgon F400 at 318.2 K. Symbols: experimental data listed in Table 1; solid curves:model predictions with an effective H.

122 G. Shen et al. / Fluid Phase Equilibria 382 (2014) 116–126

used as input to predict the mixed-gas adsorption. The modelresults of IAST theory were illustrated as dashed-dotted curves inFig. 7 for further comparison. The comparison in Fig. 7 implies thatIAST theory gives worse prediction for CH4 compared to PC-SAFT–DFT, although the correlation of IAST for pure-gas adsorptionextremely agrees well with experimental data. This observationimplies that the model of PC-SAFT–DFT is comparable to thewidely-used engineering model.

It should be pointing out that, in practice, a lot of substances arechain molecules, and association and ionic interactions may alsoinclude. In our previous work, it has been proved that the PC-SAFT–DFT can be used to represent the confined behavior of chain fluidswith association interaction. In this section, it is shown that forsimple or spherical molecules, the model performance of PC-SAFT–DFT is comparable to IAST. Thus, we may conclude that PC-SAFT–DFT can be used to represent the confined behavior of all kinds offluids with interactions of hard-sphere, chain, dispersion andassociation.

4.2.2. CH4, N2 and CO2 on activated carbon Calgon F400Sudibandriyo et al. [58] measured the adsorption of pure CH4,

N2 and CO2 and their binary mixtures on dry activated carbonCalgon F400 at 318.2 K. For this system, two material-specificparameters (H, A*) and three fluid-specific parameters (esi for CH4,N2 and CO2, respectively) were fitted simultaneously to the pure-gas adsorption of CH4, N2 and CO2, respectively, on dry activatedcarbon Calgon F400 at 318.2 K. It should be mentioned that theresults of adsorption isotherms and the parameters are differentfrom others in our previous work. This is because that theparameter set and the lower and upper limit in the integration ofadsorption quantity (Eq. (7)) used are different from those in ourprevious work [46]. The fitted results are listed in Table 4. Fig. 8shows the model representation of pure-gas adsorption isotherms.The model provides a very good representation for both N2 and CH4

while shows discrepancies for CO2. Similar deviation of CO2 wasalso observed by Monsalvo and Shapiro using MPTA [18]. Asexplained in the previous text, the reason is partially from theassumption of one-size pore structure for the porous material.However, in fact, there may be some special pores available forspecific components due to the pore confinement, and the porestructure affects the model result. It was further evaluated inSection 4.3.

Fig. 8. Adsorption of pure CH4, N2 and CO2 on activated carbon Calgon F400 at318.2 K. Symbols: experimental data listed in Table 1; solid curves: modelcorrelation results with an effective H; dashed curves: model results with PSD.

Without any additional parameters, the model was used topredict the adsorption isotherms for gas mixtures. Figs. 9–11 showthe comparison of the predicted adsorption isotherms of mixtureswith the experimental data, and the corresponding ARDs aresummarized in Table 5. The model captures the maxima in theisotherms and the negative adsorption of N2 in the N2/CO2 system.In general the model results for CH4/N2 are better than those forCO2-containing mixtures. The adsorption of binary mixtures wasalso studied by Monsalvo and Shapiro. While the model perfor-mance in this work (mean ARD is 14.8%) is better than MPTAmodels that were combined with sPC-SAFT and Steele's 10–4-3 Lennard-Jones potential (mean ARD is 18.8%) [18].

4.2.3. N2 and CO2 on zeolite silicaliteLi and Tezel [59] measured the adsorption of pure CH4 and N2

and their binary mixtures on zeolite silicalite at differenttemperatures. The parameters of two material-specific parameters(H, A*) and two fluid-specific parameters (esi/k for CH4 and N2,respectively) were determined by fitting simultaneously to thepure-gas adsorption of CH4, N2 at 313.15 K. The fitted parametersare listed in Table 6. Fig. 12(a) shows the model results of pure-gasadsorptions. The model provides a very good representation forboth CH4 and N2. Without any additional parameters, the modelwas used to predict the adsorption for gas mixtures at 313.15 K.

Fig. 10. Adsorption of N2 (a) and CO2 (b) in the N2/CO2mixtures on activated carbonCalgon F400 at 318.2 K. Symbols: experimental data listed in Table 1; solid curves:model predictions with an effective H; dashed curves: model predictions with PSD.

Fig. 11. Adsorption of CH4 (a) and CO2 (b) in the CH4/CO2 mixtures on activatedcarbon Calgon F400 at 318.2 K. Symbols: experimental data listed in Table 1; solidcurves: model predictions with an effective H; dashed curves: model predictionswith PSD.

Table 6Model parameters fitted from pure-gas adsorption isotherms on zeolite silicate.

Pure gas Adsorbent H (Å) esi/ka (K) A* (m2/g)

CH4 Zeolite silicalite 6.5 34.26 456.5N2 29.17CO2 Zeolite silicalite 8.4 40.36 368.5N2 33.87

a For the given solid–fluid energy parameter, it is multiplied by density of solidatom.

G. Shen et al. / Fluid Phase Equilibria 382 (2014) 116–126 123

Fig. 12(b) shows the comparison of the predicted adsorptionisotherms of mixtures with the experimental data. The model withparameters was also used to predict the pure- and mixed-gasadsorption at 343.15 K. Fig. 12(c) and (d) shows the comparison ofthe model prediction with the experimental data at 343.15 K. Theresults shown in Fig. 12(a–d) implies that the model givesacceptable prediction of pure- and mixed-gas adsorption atvarious temperatures.

Adsorption of pure CO2 and N2 and their binary mixtures onzeolite silicalite was also measured by Li et al. Fig. 13 illustrates thecomparison of the model results in which the pure-gas adsorptionis the model correlation and the mixed-gas adsorption is the modelperdition with the parameters obtained from the pure-gasadsorption isotherms. The model parameters fitted in this workwere listed in Table 6. The model correlation results show certaindiscrepancies for CO2. As explained before that the reason may befrom the assumption of a single pore with an effective size for thismaterial. As shown in Fig.13(b), the model prediction of the mixed-gas adsorption isotherm is also acceptable.

From the comparison of the model prediction with the availableexperimental data discussed in Section 4.1 and 4.2, we canconclude that the predictions of the adsorption of pure CH4 and N2

in a wide temperature range and of their mixtures are sufficientlysatisfied. The correlations of the adsorption of pure C2H6 and CO2 inthe range of temperature and pressure investigated are slightlyworse; however, the predictions of mixtures are acceptable. The

excess adsorption quantity is sensitive to the pore structure, andthen it will affect the accuracy of the model predictionsignificantly. In PC-SAFT, the chain-length of C2H6 and CO2 islonger than that of CH4 and N2. In nano-scale, the pore showsselective adsorption for specific fluid due to the effect of size.Therefore, it is more reasonable to combine the adsorption modelwith pore size distribution.

4.3. Effect of pore size distribution on model performance

As shown in previous sections (Sections 4.1 and 4.2), the modelwith the assumption of porous materials as a slit and cylinder withan effective size can be used to predict the gas adsorption for

124 G. Shen et al. / Fluid Phase Equilibria 382 (2014) 116–126

pure- and mixed-gas adsorption in a wide temperature andpressure range. However, for some systems, the model resultsshow discrepancies from the experimental data. The neglect ofpore size distribution (PSD) may be one of reasons. In this section,the effect of PSD on the model performance was studied. It shouldbe mentioned that the investigation in this section was only onthree systems listed in Table 7 because of the relative largediscrepancies of model prediction from experimental data.

As mentioned in Section 2.2, with the consideration of PSD, f(H)was determined from the fitting of pure-gas adsorption isothermsused in Sections 4.1 and 4.2. The model with the fitted PSD wasused to predict the pure-gas adsorption at other temperatures andmixed-gas adsorptions. The model results are depicted in thefigures as dashed curves in order to compare with the results insections 4.1 and 4.2, and the deviations (ARDs) are summarized inTable 7.

Comparing the ARDs listed in Tables 7 and 3, for the adsorptionof CH4 and CO2 on a carbon molecular sieve, the model results bothin fitting and prediction are improved significantly with theconsideration of PSD. As shown in Fig 4, with the consideration ofPSD, the model results for CO2 agree well with the experimentaldata in the whole studied temperature and pressure range withconsideration of PSD.

For the adsorption of CH4, C2H6 and their mixtures on BPL,the results of pure-gas adsorption were improved with the

Fig. 12. Adsorption of pure CH4 and N2 and CH4/N2mixtures on zeolite silicalite at 1 bar ain Table 1; solid curves: model results. (a): results of model correlation; (b)–(d): resul

consideration of PSD, especially for C2H6 at higher pressure asshown in Fig. 6. For the model prediction of mixtures, the modelresults of the stronger adsorbed component (C2H6) were improvedas shown in Fig. 7. For the adsorption of CH4, N2, and CO2 on CalgonF400, the results of pure-gas adsorption were improved, especiallythe results of CO2 are accurate in the whole range of pressure, andthe results of CO2-containing mixture systems are improved withARD down to 18.7%. In summary, with the consideration of PSD, theresults of pure- and mixed-gas adsorption isotherms can beimproved more or less, but it requires more computing resourcesas shown in Eq. (8). Considering the acceptable prediction accuracyin Sections 4.1 and 4.2, for the sake of simplification andcalculation efficiency, it is reasonable to assume the porousmaterials as a single slit or cylinder with an average effective size.

Up to here of this section, the model prediction performance ofpure- and mixed-gas adsorption on nano-porous materials hasbeen evaluated. The model prediction with a reasonable accuracyimplies the promising of the model to represent the gas adsorptionof other advanced nano-porous materials such as other zeolitesand metal–organic frameworks. The model has been developedand is ready to be extended to these advanced materials asdescribed in Section 2. However, the evaluation of the modelperformance should be the combination of experimental data aswell as the synthesis and characterization of materials, which willbe the research focus in the future.

nd at 313.15 K (a) and (b) and 343.15 K (c) and (d). Symbols: experimental data listedts of model prediction.

Fig. 13. Adsorption of pure CO2 and N2 and CO2/N2 mixtures on zeolite silicalite at1 bar and at 313.15 K. Symbols: experimental data listed in Table 1; solid curves:model results.

Table 7Deviation of model prediction with PSD for three investigated systems.

Puregas

Adsorbent ARD (%)

Correlation Prediction

CH4 Carbon molecular sieve (Shirasagi MSC 3 K– 161)

1.75 3.42CO2

CH4/C2H6

Activated carbon BPL 1.39 15.03

CH4/N2 Activated carbon Calgon F400 2.42 15.09N2/CO2 18.67CH4/CO2

18.58

G. Shen et al. / Fluid Phase Equilibria 382 (2014) 116–126 125

5. Conclusions

In this work, PC-SAFT–DFT was used to represent theadsorption of pure gases and multicomponent mixtures, and thenano-porous carbons and zeolites were chosen as examples formodel performance evaluation. A single slit-shaped or cylindricalpore was used to describe the porous materials, Lennard-Jonespotential was used to represent the solid–fluid interaction, and theadsorption quantity was calculated from the density profile. Theresults show that the model can predict the adsorption of puregases and mixtures in a wide temperature and pressure range, andthe model captures the properties of adsorption of isotherms, such

as the trends of adsorption with increasing pressure and themaximum adsorption on isotherms. The consideration of pore sizedistribution can improve the model prediction more or less butrequires much more computation resources.

Acknowledgments

G.S. and X.J. thank Swedish research council. G.S. and X.L. thankthe Key Project of the National Natural Science Foundation of China(Grant No. 21136004), and National Basic Research Program ofChina (Grant No. 2013CB733500).

References

[1] T. Koga, N. Jiang, P. Gin, M.K. Endoh, S. Narayanan, L.B. Lurio, S.K. Sinha, Impactof an irreversibly adsorbed layer on local viscosity of nanoconfined polymermelts, Phys. Rev. Lett. 107 (2011) 22901.

[2] P.T. Cummings, H. Docherty, C.R. Iacovella, J.K. Singh, Phase transitions innanoconfined fluids: the evidence from simulation and theory, AIChE J. 56(2010) 842–848.

[3] R. Maboudian, W.R. Ashurst, C. Carraro, Tribological challenges in micro-mechanical systems, Tribol. Lett. 12 (2002) 95–100.

[4] C.P. Emborsky, Z. Feng, K.R. Cox, W.G. Chapman, Recent advances in classicaldensity functional theory for associating and polyatomic molecules, FluidPhase Equilib. 306 (2011) 15–30.

[5] M.P. Suh, H.J. Park, T.K. Prasad, D.-W. Lim, Hydrogen storage in metal–organicframeworks, Chem. Rev. 112 (2011) 782–835.

[6] J.A. Swisher, L.-C. Lin, J. Kim, B. Smit, Evaluating mixture adsorption modelsusing molecular simulation, AIChE J. 59 (2013) 3054–3064.

[7] J. Keskin, J.K. Rankin, Johnson, D.S. Sholl, Progress, opportunities, andchallenges for applying atomically detailed modeling to molecular adsorptionand transport in metal–organic framework materials, Ind. Eng. Chem. Res. 48(2008) 2355–2371.

[8] K.Y. Foo, B.H. Hameed, Insights into the modeling of adsorption isothermsystems, Chem. Eng. J. 156 (2010) 2–10.

[9] Y. Liu, H.L. Liu, Y. Hu, J.W. Jiang, Density functional theory for adsorption of gasmixtures in metal–organic frameworks, J. Phys. Chem. B 114 (2010)2820–2827.

[10] A.L. Myers, J.M. Prausnitz, Thermodynamics of mixed-gas adsorption, AIChE J.11 (1965) 121–127.

[11] O. Talu, I. Zwiebel, Multicomponent adsorption equilibria of nonidealmixtures, AIChE J. 32 (1986) 1263–1276.

[12] T.X. Nguyen, S.K. Bhatia, D. Nicholson, Prediction of high-pressure adsorptionequilibrium of supercritical gases using density functional theory, Langmuir21 (2005) 3187–3197.

[13] C. Lastoskie, K.E. Gubbins, N. Quirke, Pore-size distribution analysis ofmicroporous carbons – a density-functional theory approach, J. Phys. Chem.97 (1993) 4786–4796.

[14] J.P. Olivier, Modeling physical adsorption on porous and nonporous solidsusing density functional theory, J. Porous Mater. 2 (1995) 9–17.

[15] J.E. Fitzgerald, M. Sudibandriyo, Z. Pan, R.L. Robinson Jr., K.A.M. Gasem,Modeling the adsorption of pure gases on coals with the Sld model, Carbon 41(2003) 2203–2216.

[16] J.H. Chen, D.S.H. Wong, C.S. Tan, R. Subramanian, C.T. Lira, M. Orth, Adsorptionand desorption of carbon dioxide onto and from activated carbon at highpressures, Ind. Eng. Chem. Res. 36 (1997) 2808–2815.

[17] B. Rangarajan, C.T. Lira, R. Subramanian, Simplified local density model foradsorption over large pressure ranges, AIChE J. 41 (1995) 838–845.

[18] M.A. Monsalvo, A.A. Shapiro, Study of high-pressure adsorption fromsupercritical fluids by the potential theory, Fluid Phase Equilib. 283 (2009)56–64.

[19] M.A. Monsalvo, A.A. Shapiro, Modeling adsorption of binary and ternarymixtures on microporous media, Fluid Phase Equilib. 254 (2007) 91–100.

[20] G. Li, P. Xiao, P. Webley, Binary adsorption equilibrium of carbon dioxide andwater vapor on activated alumina, Langmuir 25 (2009) 10666–10675.

[21] M.S. Wertheim, Fluids with highly directional attractive forces. I. Statisticalthermodynamics, J. Stat. Phys. 35 (1984) 19–34.

[22] M.S. Wertheim, Fluids with highly directional attractive forces. II. Thermody-namic perturbation theory and integral equations, J. Stat. Phys. 35 (1984)35–47.

[23] M.S. Wertheim, Fluids with highly directional attractive forces. III. Multipleattraction sites, J. Stat. Phys. 42 (1986) 459–476.

[24] M.S. Wertheim, Fluids with highly directional attractive forces. IV. Equilibriumpolymerization, J. Stat. Phys. 42 (1986) 477–492.

[25] J. Gross, G. Sadowski, Application of perturbation theory to a hard-chainreference fluid: an equation of state for square-well chains, Fluid Phase Equilib.168 (2000) 183–199.

[26] J. Gross, G. Sadowski, Perturbed-chain SAFT: an equation of state based on aperturbation theory for chain molecules, Ind. Eng. Chem. Res. 40 (2001)1244–1260.

[27] H. Adidharma, M. Radosz, SAFT1 for associating fluids: alkanols, J. Phys. Chem.B 105 (2001) 9822–9827.

126 G. Shen et al. / Fluid Phase Equilibria 382 (2014) 116–126

[28] H. Adidharma, M. Radosz, Prototype of an engineering equation of state forheterosegmented polymers, Ind. Eng. Chem. Res. 37 (1998) 4453–4462.

[29] A. Gil-Villegas, A. Galindo, P.J. Whitehead, S.J. Mills, G. Jackson, A.N. Burgess,Statistical associating fluid theory for chain molecules with attractivepotentials of variable range, J. Chem. Phys. 106 (1997) 4168–4186.

[30] S.H. Huang, M. Radosz, Equation of state for small, large, polydisperse, andassociating molecules: extension to fluid mixtures, Ind. Eng. Chem. Res. 30(1991) 1994–2005.

[31] W.G. Chapman, K.E. Gubbins, G. Jackson, M. Radosz, New reference equation ofstate for associating liquids, Ind. Eng. Chem. Res. 29 (1990) 1709–1721.

[32] W.G. Chapman, K.E. Gubbins, G. Jackson, M. Radosz, SAFT: equation-of-statesolution model for associating fluids, Fluid Phase Equilib. 52 (1989) 31–38.

[33] W.G. Chapman, G. Jackson, K.E. Gubbins, Phase equilibria of associating fluids,Mol. Phys. 65 (1988) 1057–1079.

[34] Y.X. Yu, J.Z. Wu, Density functional theory for inhomogeneous mixtures ofpolymeric fluids, J. Chem. Phys. 117 (2002) 2368–2376.

[35] Z.C. Ye, J. Cai, H.L. Liu, Y. Hu, Density and chain conformation profiles ofsquare-well chains confined in a slit by density-functional theory, J. Chem.Phys. 123 (2005) 194902.

[36] S. Tripathi, W.G. Chapman, Microstructure of inhomogeneous polyatomicmixtures from a density functional formalism for atomic mixtures, J. Chem.Phys. 122 (2005) 094506.

[37] J. Gross, A density functional theory for vapor-liquid interfaces using the PCP-SAFT equation of state, J. Chem. Phys. 131 (2009) 204705.

[38] G.J. Gloor, G. Jackson, F.J. Blas, E.M. del Río, E. de Miguel, Prediction of thevapor–liquid interfacial tension of nonassociating and associating fluids withthe SAFT-VR density functional theory, J. Phys. Chem. C 111 (2007)15513–15522.

[39] X.F. Xu, D.E. Cristancho, S. Costeux, Z.G. Wang, Density-functional theory forpolymer-carbon dioxide mixtures: a perturbed-chain SAFT approach, J. Chem.Phys. 137 (2012) 054902.

[40] J. Hu, J. Chen, J. Mi, Prediction of interfacial structure and tension of binarymixtures containing carbon dioxide, Ind. Eng. Chem. Res. 51 (2011)1236–1243.

[41] F. Llovell, N. Mac Dowell, F.J. Blas, A. Galindo, G. Jackson, Application of theSAFT-VR Density functional theory to the prediction of the interfacialproperties of mixtures of relevance to reservoir engineering, Fluid PhaseEquilib. 336 (2012) 137–150.

[42] H. Kahl, J. Winkelmann, Modified PT-LJ-SAFT Density functional theory: i.prediction of surface properties and phase equilibria of non-associating fluids,Fluid Phase Equilib. 270 (2008) 50–61.

[43] A. von Müller, K. Leonhard, Surface tension calculations by means of a PCP-SAFT–DFT formalism using equation of state parameters from quantummechanics, Fluid Phase Equilib. 356 (2013) 96–101.

[44] C. Klink, J. Gross, A density functional theory for vapor–liquid interfaces ofmixtures using the perturbed-chain polar statistical associating fluid theoryequation of state, Ind. Eng. Chem. Res. 53 (2014) 6169–6178.

[45] G. Shen, X. Ji, X. Lu, A hybrid perturbed-chain SAFT density functional theoryfor representing fluid behavior in nanopores, J. Chem. Phys.138 (2013) 224706.

[46] G. Shen, X. Ji, S. Öberg, X. Lu, A hybrid perturbed-chain SAFT density functionaltheory for representing fluid behavior in nanopores: mixtures, J. Chem. Phys.139 (2013) 194705.

[47] J. Landers, G.Y. Gor, A.V. Neimark, Density functional theory methods forcharacterization of porous materials, Colloids Surf. A: Physicochem. Eng.Aspects 437 (2013) 3–32.

[48] L.L. Lee, Molecular Thermodynamics of Nonideal Fluids, Butterworths, Boston,1988.

[49] G.J. Tjatjopoulos, D.L. Feke, J.A. Mann, Molecule-micropore interactionpotentials, J. Phys. Chem. 92 (1988) 4006–4007.

[50] P. Pascual, P. Ungerer, B. Tavitian, P. Pernot, A. Boutin, Development of atransferable guest-host force field for adsorption of hydrocarbons in zeolites I.Reinvestigation of alkane adsorption in silicalite by grand canonical montecarlo simulation, Phys. Chem. Chem. Phys. 5 (2003) 3684–3693.

[51] C. Nguyen, D.D. Do, Simple optimization approach for the characterization ofpore size distribution, Langmuir 16 (1999) 1319–1322.

[52] W.A. Steele, The physical interaction of gases with crystalline solids: I. Gas–solid energies and properties of isolated adsorbed atoms, Surf. Sci. 36 (1973)317–352.

[53] B.-U. Choi, D.-K. Choi, Y.-W. Lee, B.-K. Lee, S.-H. Kim, Adsorption equilibria ofmethane, ethane ethylene, nitrogen , and hydrogen onto activated carbon, J.Chem. Eng. Data 48 (2003) 603–607.

[54] L. Zhou, Y. Zhou, M. Li, P. Chen, Y. Wang, Experimental and modeling study ofthe adsorption of supercritical methane on a high surface activated carbon,Langmuir 16 (2000) 5955–5959.

[55] G. Watson, E.F. May, B.F. Graham, M.A. Trebble, R.D. Trengove, K.I. Chan,Equilibrium adsorption measurements of pure nitrogen carbon dioxide, andmethane on a carbon molecular sieve at cryogenic temperatures and highpressures, J. Chem. Eng. Data 54 (2009) 2701–2707.

[56] S. Savitz, F. Siperstein, R.J. Gorte, A.L. Myers, Calorimetric study of adsorption ofalkanes in high-silica zeolites, J. Phys. Chem. B 102 (1998) 6865–6872.

[57] R. Reich, W.T. Ziegler, K.A. Rogers, Adsorption of methane ethane, and ethylenegases and their binary and ternary mixtures and carbon dioxide on activatedcarbon at 212–301 K and pressures to 35 atmospheres, Ind. Eng. Chem. ProcessDes. Dev. 19 (1980) 336–344.

[58] M. Sudibandriyo, Z. Pan, J.E. Fitzgerald, R.L. Robinson, K.A.M. Gasem,Adsorption of methane, nitrogen, carbon dioxide, and their binary mixtureson dry activated carbon at 318.2 K and pressures up to 13.6 MPa, Langmuir 19(2003) 5323–5331.

[59] P. Li, F. Handan Tezel, Pure and binary adsorption of methane and nitrogen bysilicalite, J. Chem. Eng. Data 54 (2008) 8–15.

[60] P. Li, F.H. Tezel, Pure and binary adsorption equilibria of carbon dioxide andnitrogen on silicalite, J. Chem. Eng. Data 53 (2008) 2479–2487.

[61] www.iza-structure.org/databases/, 6 June 2014.[62] S.R. Panuganti, F.M. Vargas, D.L. Gonzalez, A.S. Kurup, W.G. Chapman, PC-SAFT

characterization of crude oils and modeling of asphaltene phase behavior, Fuel93 (2012) 658–669.

[63] F. García-Sánchez, G. Eliosa-Jiménez, G. Silva-Oliver, R. Vázquez-Román,Vapor–liquid equilibria of nitrogen–hydrocarbon systems using the PC-SAFTequation of state, Fluid Phase Equilib. 217 (2004) 241–253.

[64] D.N. Justo-García, F. García-Sánchez, N.L. Díaz-Ramírez, A. Romero-Martínez,Calculation of critical points for multicomponent mixtures containinghydrocarbon and nonhydrocarbon components with the PC-SAFT equationof state, Fluid Phase Equilib. 265 (2008) 192–204.

[65] M. Sudibandriyo, S.A. Mohammad, R.L. Robinson, K.A.M. Gasem, Ono–kondomodel for high-pressure mixed-gas adsorption on activated carbons and coals,Energy Fuels 25 (2011) 3355–3367.