Study of molecular shape and non-ideality effects on mixture adsorption isotherms of small molecules...

Chemical Engineering Science 57 (2002) 2439–2448www.elsevier.com/locate/ces

Study of molecular shape and non-ideality e&ects on mixtureadsorption isotherms of small molecules in carbon nanotubes:

A grand canonical Monte Carlo simulation studyAndreas Heyden ∗, Tina D0uren, Frerich J. Keil

Department of Chemical Engineering, Hamburg University of Technology, Eissendorfer Str. 38, D-21073 Hamburg, Germany

Received 13 September 2001; received in revised form 20 December 2001; accepted 13 February 2002

Abstract

The sorption isotherms for binary mixtures of methane, ethane, propane and tetra7uoromethane have been determined in carbonnanotubes using con8gurational bias Monte Carlo simulation techniques. At high loadings, a curious maximum for equimolar gas-phasemixtures occurs with increasing pressure in the absolute adsorption isotherm of one or both adsorbing species. It was detected that thereexist two fundamentally di&erent reasons for this maximum. First, due to a higher packing e<ciency, one component is able to displacethe other component at high loadings. Here, it must be stressed that the displaced component is not necessarily the larger molecule.Second, non-ideality e&ects of the bulk gas phase can be made responsible for this maximum. The acceptance probability of a moleculeinsertion in a grand canonical Monte Carlo step is proportional to the component fugacity. If, owing to non-ideality e&ects of the gasphase, the fugacity of one component does not increase as steeply with pressure as the other component, a maximum can occur in theabsolute adsorption isotherm of this component. These 8ndings were demonstrated for various binary mixtures of CH4, CF4, C2H6 andC3H8. ? 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Adsorption; Monte Carlo; Simulation; Porous media; Carbon nanotubes; Separations

1. Introduction

Detailed knowledge and understanding of the adsorptionof hydrocarbons in porous media is of considerable prac-tical interest in petrochemical applications (Haag, 1994).Experimental data is often restricted to pure componentsand there is very little data in the literature on sorptionisotherms of mixtures. Therefore, it becomes even more im-portant to understand the physical interactions of 7uids inporous materials and their consequences. Over the last fewyears, a great deal of attention has been given to the e&ect ofnanopore shape, siting and loading e&ects on the adsorptionisotherms and selectivities. Several simulation studies (Mag-inn, Bell, & Theodorou, 1995; Ke&er, Davis, &McCormick,1996a; Ke&er, Davis, & McCormick, 1996b; Clark, Gupta,& Snurr, 1998) show the roles of energy contributions (bothadsorbate–pore and adsorbate–adsorbate) and packing, orentropic e&ects. More recently, the in7uence of the shapeof the adsorbents on the selectivity was studied (Krishna,

∗ Corresponding author. Tel.: +49-40-428782237;fax: +49-40-428782145.

E-mail address: [email protected] (A. Heyden).

Smit, & Vlugt, 1998; Vlugt, Krishna, & Smit, 1999; Mo-hanty, Davis, & McCormick, 2000). Krishna et al. (1998)found a curious maximum for equimolar gas-phase mixturesin the absolute adsorption isotherm of one component withincreasing pressure and used this phenomenon to develop anew separation method for the separation of alkane isomers(also Krishna & Paschek, 2000; Krishna, 2001; Schenk, Vi-dal, Vlugt, Smit, & Krishna, 2001). In these publications, itwas stated that the maximum occurs due to entropic e&ects,although no detailed reasoning was given. At the same time,Mohanty et al. (2000) demonstrated that shape selectivityis caused by both entropic and energetic e&ects. Essentially,Talbot (1997) was the 8rst to show theoretically, for theone-dimensional case that for hard rods adsorbing on a lin-ear substrate, the shape of a molecule and a non-ideal bulkphase can result in a maximum in an absolute adsorptionisotherm and selectivity reversal.The main purpose of this paper is to show in a system-

atic way that in most cases maxima in absolute adsorp-tion isotherms of mixtures occur owing to packing e&ectsor non-ideality of the gas phase. It will be demonstratedthat there are essentially two reasons for these maxima in

0009-2509/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved.PII: S 0009 -2509(02)00131 -8

2440 A. Heyden et al. / Chemical Engineering Science 57 (2002) 2439–2448

absolute adsorption isotherms. First, due to a higher packinge<ciency, one component is often able to displace the othercomponent at high loadings. This is an energetic and=or en-tropic e&ect. Here, it is proved that, at least in cylindricalpores, the displaced component is not necessarily the largermolecule as often stated but the one that occupies, on av-erage, less area of the cross–section of the pore. Second,non-ideality e&ects of the bulk gas phase can create thismaximum. The acceptance probability of a molecule inser-tion in a grand canonical Monte Carlo step is proportional tothe component fugacity. If, owing to non-ideality e&ects ofthe bulk gas phase, the fugacity of one component does notincrease as steeply with pressure as the other component,a maximum can occur in the absolute adsorption isothermof this component. Such non-ideality e&ects are especiallypronounced if the bulk gas mixture is in a thermodynamicstate that is close to the mixture critical point.The reason why this phenomenon was not observed ear-

lier is probably due to the impossibility of direct measuringabsolute adsorption isotherms nabsk by experiment. Only ex-cess adsorption isotherms nexk

nexk = nabsk − yk�bfV; (1)

which have for supercritical gases a maximum at high pres-sure (Coolidge, 1934; Sircar, 1999; Murata, El-Merraoui,& Kaneko, 2001) are measurable. Therefore, it is easy tomistake the maximum in the excess adsorption isotherm fora maximum owing to a high gas density. A further reasonwhy the maximum was not detected for other samples, likebimodal catalyst supports, active carbon or amorphous sil-ica, is the following. These adsorbens have a broad poresize distribution. A maximum due to the packing e&ect inthe micropores is hidden by a further ongoing adsorption inthe macropores. Furthermore, adsorption measurements arerarely executed up to such a high pressure where non-idealityof the gas phase becomes signi8cant.Various earlier works (Du, Manow, Vlugt, & Smit,

1998; Vlugt et al., 1999; Schenk et al., 2001) have clearlydemonstrated the power of calculating mixture adsorptionisotherms of alkanes by con8gurational bias Monte Carlomethods in the grand canonical ensemble. This approach isfollowed in this work.

2. Model

In the present systematic study, the binary adsorption be-haviour of methane, ethane, propane and tetra7uoromethanein carbon nanotubes was examined. Carbon nanotubes(CNT) are ultra thin carbon 8bres with a nanometer-sizediameter (1.0–4:0 nm) and mircometer-size length (up to103 nm). The structure of CNT consists of enrolled graphiticsheets (see Fig. 1a). Depending on the way the sheets arerolled, zig-zag, armchair and helical CNTs are distinguished.Carbon nanotubes are further classi8ed into multi-walledor single-walled CNTs (Harris, 1999). The graphitic sheets

show the hexagonal arrangement of graphite (see Fig. 1a).Nanotubes can be used as adsorbents or molecular sieves.In the present paper, the CNTs consisted of three en-

rolled graphitic sheets (see Fig. 1b) which had a distance of0:353 nm from each other (carbon centre to carbon centre).The CNTs layers were positioned in zig-zag con8guration,and the carbon atoms were kept 8xed in space. CNTs werechosen as adsorbent since they are straight cylindrical poresand have no preferred adsorption sites in axial direction. Toillustrate the di&erent adsorption behaviour in microporousand mesoporous materials, all simulations were performedin CNTs with a pore radius of 0:705 nm (micropore) and1:489 nm (mesopore). The pore diameter is measured fromcarbon centre to opposite carbon centre. The length of thenarrower CNTwas chosen to be 9:805 and 7:247 nm, respec-tively, for the wider CNT. The common periodic boundaryconditions were applied in z-direction along the axis of thepore (Frenkel & Smit, 2001; Allen & Tildesley, 1989).The interaction of the carbon atoms with the adsorbents

and the interaction of the 7uid molecules among each otherare simulated by a Lennard–Jones (L–J) potential

Uij(rij) = 4�ij

[(�ijrij

)12

−(�ijrij

)6]: (2)

The parameters �ij and �ij are obtained from the Lorentz–Berthelot mixing rules:

�ij = 12(�ii + �jj); �ij =

√�ii�jj (3)

with potential parameters given in Table 1. The L–J potentialis an e&ective pairwise additive potential. The interactionenergy between every L–J site with all other L–J sites mustbe summed up:

U (r) =∑j

∑i¡j

Uij(rij): (4)

To shorten the computation time considerably, a cut-o&radius of 1:175 nm was chosen. This corresponds to 2.5times �CF4 . To compensate for this potential truncation, theso-called tail-corrected L–J potential was used.

Ui = U trunci (rij) +

∑j

U tailij ;

U tailij =

163��j�ij�3ij

[13

(�ijrc

)9

−(�ijrc

)3]: (5)

By using the tail-correction it is assumed that the carbonnanotubes are lying next to each other and that the error byusing an isotropic tail-correction is small for the cylindricalpore models used.Methane, ethane, propane and tetra7uoromethane were

chosen as adsorbents, since they are non-polar molecules andtheir external potential can be simulated as a L–J potential(potential parameters given in Table 1). In addition, thesefour molecules have di&erent shapes (see Fig. 1c). Methaneand tetra7uoromethane are spherical molecules with

A. Heyden et al. / Chemical Engineering Science 57 (2002) 2439–2448 2441

Fig. 1. (a) Carbon nanotube model with one layer in zig-zag con8guration. The pore illustrated has a length of 7:247 nm and an inner carbon centre tocarbon centre diameter of 2:978 nm. The hexagonal arrangement of the graphitic sheets can be observed. (b) Simulation model of a carbon nanotubewith three layers in zig-zag con8guration. The enrolled graphitic sheets have a distance of 0:353 nm from each other (carbon centre to carbon centre).(c) Relative cross–sectional sizes of CH4, CF4, C2H6 and C3H8.

Table 1Lennard–Jones parameters used to calculate the potential energy of theinteraction of the united atoms. Data for carbon, methane and tetra7uo-romethane from Reid, Prausnitz, and Poling (1986). Data for CH2, CH3from Martin and Siepmann (1998)

�= KA (�=kB)=K

CH4 3.81 148.1CH3 3.75 98CH2 3.95 46CF4 4.70 152.5C 3.40 28.0

di&erent radii. Furthermore, both can be simulated as singleL–J spheres. Tetra7uoromethane is larger than methane andadsorbs stronger at low pressure. Ethane is a short chainmolecule, and is represented by two united atoms in whichthe CH3 groups are considered as single interaction centres,which are separated by a 8xed bond length of 1:54 KA. Theradius of one CH3 group is approximately as big as theradius of methane. Propane is the largest of the moleculeswe considered and is represented by three united atoms,CH3–CH2–CH3. The bond lengths were 8xed to 1:54 KA anda bond bending potential with an equilibrium angle, �eq, of114◦ was added.

U bend(�) = 12k�(�−�eq)2; k� = 62500 K=kB: (6)

3. Simulation method

The simulations were performed in the grand canonicalensemble wherein the carbon nanotube is in contact with abulk gas reservoir that 8xes the temperature and the chemi-cal potential of each component. In this respect, the simula-tions are performed under the same conditions as adsorption

Table 2Probabilities of selecting a trial move for various molecules

CH4 CF4 C2H6 C3H8

Displacement 0.2 0.2 0.1 0.1Rotation — — 0.1 0.1Insertion 0.3 0.3 0.2 0.2Removal 0.3 0.3 0.2 0.2Particle swap 0.2 0.2 0.2 0.2Particle regrowth — — 0.2 0.2

experiments. To increase the speed of our MC simulationsfor chain molecules, especially for propane mixtures, thecon8gurational bias Monte Carlo (CBMC) scheme is used,developed by Siepmann and Frenkel (1992) and explainedvery clearly by Vlugt (2000). The main feature of this moresophisticated Monte Carlo method is that the trial moves areno longer completely random; instead the moves are biasedin such a way that the molecule to be, for example inserted,has an enhanced probability to “8t” into the existing con8g-uration. In contrast, no information about the present con8g-uration of the system is used in the generation of unbiasedMC trial moves; that information is used only in the ac-ceptance criteria. To satisfy detailed balance, it is thereforenecessary to change the acceptance rules (see Vlugt, 2000).The present simulations are performed in cycles; in each

cycle, an attempt is made to perform one of the followingMonte Carlo moves with a speci8c probability (Table 2).(1) Displacement of a molecule; a molecule is selected atrandom and a random displacement is given; (2) rotation ofa molecule; a molecule is selected at random and a randomrotation around the centre of mass is given; (3) partial re-growth of a molecule; a molecule is selected at random anda part of the molecule is regrown using the CBMC scheme;(4) exchange with the reservoir; a molecule is removed oradded from the pore using the CBMC scheme; (5) change of


identity; one molecule is selected at random and an attemptis made to exchange its identity. The acceptance rules forthis type of move are given elsewhere (Vlugt, Martin, Smit,Siepmann, Krishna, 1998; Vlugt et al., 1999). A total simu-lation consisted of 250 000 Monte Carlo steps to equilibratethe system and 106 Monte Carlo steps to sample con8gu-rational space. In all simulations, it was carefully checkedthat equilibrium was reached after the equilibration period.

4. Binary adsorption

To analyse the factors which have an in7uence on theshape of an adsorption isotherm, it is necessary to look atthe factors which have an in7uence on the acceptance prob-ability of an unbiased MC molecule insertion or destructionstep. In principle, it is possible (but not recommended sincethe simulation would consume too much time) to performGCMC simulations by doing only insertion and destructionsteps. The whole con8gurational space can be sampled withjust these two MC moves. Since the formula for the accep-tance probability of a molecule destruction is essentially justthe reciprocal of the molecule insertion in the nanopore, itis su<cient to analyse the MC molecule insertion move.The acceptance probability of insertion is given by

acc(o→ n) = min{1;

fk�VNk(o) + 1

exp(−�Uk)}; (7)

where � is the reciprocal of the product of the Boltzmannfactor with the temperature. Eq. (7) demonstrates that onlythe pore volume V (measured from carbon centre to oppositecarbon centre), the temperature T (hidden in the coe<cient�), the fugacities fk of each component (here the in7uenceof the bulk gas pressure, gas composition and gas phasenon-ideality e&ects (which are temperature dependent) arecombined) and the potential energy, Uk , exerted by the in-serted molecule on the system, have an in7uence on theamount adsorbed of each component in the nanotube.It is concluded that an increase in the pore volume directly

increases the equilibrium amount adsorbed of each compo-nent. It must be stressed that in equilibrium the number ofmolecule insertions is equal to the number of molecule dele-tions. But during the transition from one equilibrium con8g-uration with a smaller pore volume to an equilibrium con-8guration with a larger pore volume more molecule inser-tions have to be accepted than molecule deletions. Further,if an increase in the simulation volume changes the shapeof the pore, e.g. the pore radius, this results in a change ofthe potential energy exerted by the inserted molecule on thesystem and our conclusion above can be false, as is knownfor low pressures.An increase in the component fugacity also directly in-

creases the equilibrium amount adsorbed of the component.As explained in the case of a volume change, as long as thenew equilibrium con8guration is not reached, the acceptanceprobability for the molecule insertion of this component is

Table 3Critical data and acentric factor of methane, ethane, propane and tetra7u-oromethane. Data from Reid et al. (1986)

Critical Critical Acentrictemperature (K) pressure (MPa) factor

Methane 190.5 4.61 0.008Ethane 305.4 4.88 0.099Propane 369.8 4.25 0.153Tetra7uoromethane 227.6 3.74 0.191

larger than for the deletion. This holds for pure components.For mixtures, the conclusion must be modi8ed since an in-crease in the fugacity of one component often occurs withan increase in the fugacity of the other component, e.g. in-crease of total bulk pressure. In addition, an increase in thenumber of molecules A in the pore in7uences the potentialenergy exerted from molecules B on the system. To studythe competition of molecules A and B in the pore it is es-pecially instructive to analyse the acceptance probability ofan identity exchange MC move:

acc(A → B) =min{1;

N (A)N (B) + 1

× fB

fAexp(−�(UB − UA))

}: (8)

According to Eq. (8), the ratio of the component fugacitiesand potential energy exerted by the molecule (and to somedegree, the temperature) decides which molecule wins thecompetition in the pore (this will be proven later by simula-tions) and whether or not the equilibrium amount adsorbedof component A increases with increasing component fu-gacity.In the following, the in7uence of the fugacity (or more

accurately, non-ideality e&ects of the gas phase), pore ra-dius, molecular shape and to some degree, the temperature,on mixture adsorption isotherms is demonstrated by simu-lations. All simulations were performed at a temperature of300 and 400 K with an equimolar bulk gas mixture. In par-ticular, to study the in7uence of the non-ideality e&ects ofthe bulk gas phase on adsorption isotherms, all adsorptionstudies in the present paper were also performed with a fu-gacity coe<cient of 1 (ideal gas phase). Otherwise, the fu-gacity was calculated with the Peng–Robinson equation ofstate, data taken from (Reid, Prausnitz, Poling, 1986) and abinary interaction coe<cient kij of zero (see Table 3).

4.1. Methane=tetra2uoromethane mixture adsorption

Fig. 2 illustrates the amount adsorbed of an equimolar gasmixture of methane and tetra7uoromethane with increasingpressure, in the mesopore (? 2:978 nm), at a temperatureof 300 and 400 K. As in all adsorption studies one 8ndsthat more molecules adsorb at low temperatures. In addition,one observes that all adsorption isotherms have a Langmuirshape in the pressure range studied; more tetra7uoromethane


0 50 100 150 200 2500.0

0.4

0.8

1.2

1.6

Am

ou

nt

Ad

sorb

ed /[

mo

l/kg

]

Total Bulk Pressure /[bar]

CH4 [300 K] CF4 [300 K] CH4 [400 K] CF4 [400 K]

Fig. 2. Simulated amount adsorbed of methane and tetra7uoromethane inthe mesopore (? 2:978 nm) at a temperature of 300 and 400 K. The bulkgas phase consists of an equimolar mixture. Illustrated are the adsorptionisotherms in the case of a real (solid symbols) and ideal (open symbols)gas phase.

0 25 50 75 100 125 1500

20

40

60

80

CH4 [300 K] CF4 [300 K] CH4 [400 K] CF4 [400 K]

Co

mp

on

ent

Fu

gac

ity

/[b

ar]


Fig. 3. Component fugacity of methane and tetra7uoromethane at a tem-perature of 300 and 400 K as a function of pressure. The gas phaseconsists of an equimolar mixture. The Peng–Robinson equation of statewas used for kij = 0. Dashed line for ideal gas phase.

adsorbs than methane at all total bulk pressures. Further-more, there is only a minor di&erence if one performs thecalculations with a real gas phase or an ideal gas phase. Thereason for the last point is explained in Fig. 3 where thecomponent fugacity is plotted over the total bulk pressure.At all total bulk pressures, there is no signi8cant deviationbetween the component fugacity and the partial pressure.This is expected since the critical temperature of methaneand tetra7uoromethane is 190:5 and 227:6 K, respectively,which is far below the temperatures where these simulationsare performed.

0 50 100 150 200 2500.0

0.1

0.2

0.3

0.4

0.5

0.6

Am

ou

nt A

dso

rbed

/[m

ol/k

g]


CH4 [300 K] CF4 [300 K] CH4 [400 K] CF4 [400 K]

Fig. 4. Simulated amount adsorbed of methane and tetra7uoromethanein the micropore (? 1:41 nm) at a temperature of 300 and 400 K. Thebulk gas phase consists of an equimolar mixture. Illustrated are both, theadsorption isotherms in the case of a real (solid symbols) and ideal (opensymbols) gas phase.

3 4 5 6 7 8

−200

0

200

400

600

800

Pot

enti

al E

nerg

y (U

/ k B

) / [

K]

Distance r / [Å]

CH 4

CF4

9

Fig. 5. Potential energy of the interaction of one tetra7ouromethanemolecule and one methane molecule with one carbon atom of the nanoporewall as a function of distance.

Fig. 4 illustrates the same adsorption isotherms in the mi-cropore (? 1:41 nm). The tetra7uoromethane isotherm hasa maximum at around 10 bar at a low temperature (300 K,Fig. 4) and this maximum disappears with increasing tem-perature. To proceed our argumentation from the beginningof this paper, the reason for this maximum (and also whymore tetra7uoromethane adsorbs than methane) must be thepotential energy exerted by each component on the system.All other factors in Eq. (7) are nearly identical for eachcomponent (nearly ideal gas phase). To conclude, the po-tential parameter �ii is smaller for methane than for tetra7u-oromethane (the potential depth is approximately the same).Consequently, if methane molecules are on average far apartfrom each other and from the pore wall atoms (low pressure,big pore radius), the potential energy exerted by a methanemolecule on the system is less negative than for tetra7uo-romethane (see Fig. 5). As a result, more tetra7uoromethaneadsorbs than methane. At a higher pressure and a small pore


0 50 100 150 200 2500.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Am

ou

nt

Ad

sorb

ed /[

mo

l/kg

]


CH4 [300 K] C2H6 [300 K] CH4 [400 K] C2H6 [400 K]

Fig. 6. Simulated amount adsorbed of methane and ethane in the mesopore(? 2:978 nm) at a temperature of 300 and 400 K. The bulk gas phaseconsists of an equimolar mixture. Illustrated are both, the adsorptionisotherms in the case of a real (solid symbols) and ideal (open symbols)gas phase.

radius the distance of the adsorbate and the pore atoms getssmall and Fig. 5 shows that under these conditions the po-tential energy exerted by a tetra7uoromethane molecule be-comes on average higher than the one exerted by a methanemolecule. Methane is able to displace tetra7uoromethane.The more methane adsorbs in the pore, the more crowdedthe pore becomes and the more tetra7uoromethane is dis-placed. A maximum in an absolute adsorption isotherm ap-pears with increasing pressure.Following this argumentation, it becomes obvious, why

this maximum disappears with increasing temperatureand=or emerges at higher pressures. The potential energyexerted by a molecule is scaled by the factor � in the ex-ponential in Eqs. (7) and (8). As a result, the in7uence ofthe exponent and therefore, the potential energy, decreaseswith decreasing �, or equivalently, increasing temperature.A much higher bulk gas pressure is necessary for the ap-pearance of this maximum. The last statement suggests thatin both, microporous and mesoporous cylindrical pores, amaximum in the tetra7uoromethane adsorption isothermoccurs if methane is also present. Following the argumen-tation from Krishna and Paschek (2000), this maximumappears due to a size entropy e&ect. A di&erence in theshape of a molecule results in a di&erent packing e<ciency.

4.2. Methane=ethane mixture adsorption

Fig. 6 illustrates the binary adsorption isotherms of anequimolar gas mixture of methane and ethane, in the meso-pore (? 2:978 nm), at a temperature of 300 and 400 K. Atall temperatures more ethane adsorbs than methane. Further,it is illustrated that a maximum in the ethane adsorptionisotherm appears at 300 K (Fig. 6). This maximum disap-

0 50 100 150 200 2500.0

0.2

0.4

0.6

0.8

1.0

Am

ou

nt

Ad

sorb

ed /[

mo

l/kg

]


CH4 [300 K] C2H6 [300 K] CH4 [400 K] C2H6 [400 K]

Fig. 7. Simulated amount adsorbed of methane and ethane in the micropore(? 1:41 nm) at a temperature of 300 and 400 K. The bulk gas phaseconsists of an equimolar mixture. Illustrated are both, the adsorptionisotherms in the case of a real (solid symbols) and ideal (open symbols)gas phase.

pears with increasing temperature. At the total bulk pres-sure, where a maximum in the ethane adsorption isothermwas found, the slope of the methane adsorption isothermincreases signi8cantly. Interestingly, the maximum disap-pears, if calculations with an ideal gas phase were per-formed. Further, at a pressure higher than 40 bar (after themaximum in the ethane adsorption isotherm), signi8cantlymore ethane and less methane molecules adsorb in the caseof an ideal gas phase, in comparison to a real gas phase.Fig. 7 illustrates the binary adsorption isotherms of

methane and ethane, in the micropore (1:41 nm), at varioustemperatures. Again, one observes exactly the same be-haviour for the methane and ethane adsorption isotherms inthe micropore as in the mesopore (Fig. 6). Even the maxi-mum at 300 K occurs at approximately the same total bulkpressure of 40 bar. This suggests that the potential energyterm in Eq. (7) cannot be the reason for this maximum.Otherwise, the maximum would, in the microporous case,be much more signi8cant and occur at a lower pressure.The cause for this maximum must be found in the fugacity.Fig. 8 shows the component fugacity of methane and

ethane with increasing pressure. The slope of the compo-nent fugacity of ethane decreases signi8cantly for pressuresapproximately between 40 and 75 bar with decreasing tem-perature and increasing pressure. The component fugacityof methane, on the other hand, hardly changes at all withtemperature and is very close to the partial pressure. Thisis expected since the critical temperature of ethane is 305:4and 190:5 K for methane, respectively. Eq. (8) suggests thatif the fugacity of ethane does not increase with pressure assteeply as the fugacity of methane, methane is able to dis-place ethane. In other words, due to a kink in the compo-nent fugacity of ethane at 300 K, one observes a maximum


0 25 50 75 100 125 150

0

20

40

60

80

CH4 [300 K]

C2H

6 [300 K]

CH4 [400 K]

C2H

6 [400 K]

Co

mp

on

en

t F

ug

ac

ity

/[b

ar]


Fig. 8. Calculated component fugacity of methane and ethane at a temper-ature of 300 and 400 K as a function of pressure. The gas phase consistsof an equimolar mixture. The Peng–Robinson equation of state was usedfor kij = 0.

in the absolute adsorption isotherm of ethane. This explainslikewise, why no maximum can be observed, if one assumesan ideal gas phase, where the fugacity is the same for eachcomponent.In the methane=ethane case one cannot expect to observe a

maximum in the adsorption isotherm owing to the potentialenergy term in Eq. (7) or, equivalently, owing to a di&erentpacking e<ciency. In a cylindrical pore the ethane chain isable to lie parallel to the pore wall and therefore as 7at nextto the pore wall as methane (see Fig. 1c). The diameter of aCH3-group and a methane molecule are approximately thesame (�CH4 =3:81 KA and �CH3 =3:75 KA). Consequently, thepacking e<ciency of ethane and methane is very similar incylindrical pores.

4.3. Methane=propane mixture adsorption

Figs. 9 and 10 illustrate the binary adsorption isothermsof an equimolar gas mixture of methane and propane withincreasing pressure at various temperatures in the meso-pore and micropore. As in the case of a methane=ethanemixture, one observes, at a temperature of 300 K, a maxi-mum in the propane adsorption isotherm and a signi8cantincrease in the slope of the methane adsorption isotherm atthe same pressure, where the maximum in the propane ad-sorption isotherm occurs. Both, this maximum and the in-crease in the slope for methane, decreases with increasingtemperature. Likewise, the propane adsorption isotherm hasno maximum, if one assumes an ideal gas phase (see opensymbols in Figs. 9 and 10). Following the argumentationused in the methane=ethane case, the potential energy termin Eq. (7) cannot be the reason for this maximum but thecomponent fugacity. Fig. 11 con8rms this hypothesis, where

0 50 100 150 200 2500.0

0.5

1.0

1.5

2.0

2.

3.0

Am

ou

nt

Ad

sorb

ed /[

mo

l/kg

]


CH4 [300 K] C3H8 [300 K] CH4 [400 K] C3H8 [400 K]

5

Fig. 9. Simulated amount adsorbed of methane and propane in the meso-pore (? 2:978 nm) at a temperature of 300 and 400 K. The bulk gasphase consists of an equimolar mixture. Illustrated are both, the adsorp-tion isotherms in the case of a real (solid symbols) and ideal (opensymbols) gas phase.

0 50 100 150 200 2500.0

0.2

0.4

0.6

0.8

Am

ou

nt

Ad

sorb

ed /[

mo

l/kg

]


CH4 [300 K] C3H8 [300 K] CH4 [400 K] C3H8 [400 K]

Fig. 10. Simulated amount adsorbed of methane and propane in themicropore (? 1:41 nm) at a temperature of 300 and 400 K. The bulk gasphase consists of an equimolar mixture. Illustrated are both, the adsorptionisotherms in the case of a real (solid symbols) and ideal (open symbols)gas phase.

the component fugacity is plotted for various pressures andtemperatures. The deviation of the propane fugacity fromthe partial pressure is much larger than the deviation of themethane fugacity (the propane fugacity is much smaller thanthe methane fugacity). At a temperature of 300 K, one evenobserves a maximum in the propane fugacity with increas-ing pressure at around 20 bar. A decrease in the componentfugacity of propane and an increase in the methane fugacitymust clearly result in a maximum in the absolute adsorptionisotherm of propane.


0 50 100 150 200 250 3000

20

40

60

80 CH4 [300 K] C3H8 [300 K] CH4 [400 K] C3H8 [400 K]

Co

mp

on

ent

Fu

gac

ity

/[b

ar]


Fig. 11. Calculated component fugacity of methane and propane at atemperature of 300 and 400 K as a function of pressure. The gas phaseconsists of an equimolar mixture. The Peng–Robinson equation of statewas used for kij = 0.

0 50 100 150 200 2500.0

0.5

1.0

1.5

2.0

2.5

3.0

Am

ou

nt

Ad

sorb

ed /[

mo

l/kg

]


CF4 [300 K] C2H6 [300 K] CF4 [400 K] C2H6 [400 K]

Fig. 12. Simulated amount adsorbed of ethane and tetra7uoromethane inthe mesopore (? 2:978 nm) at a temperature of 300 and 400 K. Thebulk gas phase consists of an equimolar mixture. Illustrated are both, theadsorption isotherms in the case of a real (solid symbols) and ideal (opensymbols) gas phase.

Similar phenomena as for the methane=propane mixturewere found for the ethane=propane and tetrafuoromethane=propane mixture.

4.4. Tetra2uoromethane=ethane mixture adsorption

Fig. 12 illustrates the binary adsorption isotherms ofan equimolar gas-phase mixture of tetra7uoromethane andethane in the mesopore at a temperature of 300 and 400 K.At a temperature of 300 K, one observes a maximum in theethane adsorption isotherm at around 50 bar and an increasein the slope of the tetra7uoromethane adsorption isotherm.

0 50 100 150 200 2500.0

0.2

0.4

0.6

0.8

Am

ou

nt

Ad

sorb

ed /[

mo

l/kg

]


CF4 [300 K] C2H6 [300 K] CF4 [400 K] C2H6 [400 K]

Fig. 13. Simulated amount adsorbed of ethane and tetra7uoromethanein the micropore (? 1:41 nm) at a temperature of 300 and 400 K. Thebulk gas phase consists of an equimolar mixture. Illustrated are both, theadsorption isotherms in the case of a real (solid symbols) and ideal (opensymbols) gas phase.

This maximum and the increase in the slope disappear withincreasing temperature. Interestingly, one observes at 300 Ka maximum in the tetra7uoromethane adsorption isothermand not in the ethane adsorption isotherm, if one simulatesthe gas phase as ideal. Simulations performed in the mi-cropore (Fig. 13) show the same maximum in the ethaneadsorption isotherm at a lower temperature (T =300 K). Inaddition, a maximum is observed in the tetra7uoromethaneadsorption isotherm at a low pressure (¡ 0:1 bar). Simula-tions performed with an ideal gas phase show the maximumin the tetra7uoromethane adsorption isotherm, the maximumin the ethane adsorption isotherm, in contrast, vanishes.Following the argumentation for the methane=ethane ad-

sorption isotherm, the maximum in the ethane adsorptionisotherm results from a non-ideal gas phase. A plot of thecomponent fugacity versus total bulk pressure con8rms thatat 300 K and a pressure larger than 50 bar the ethane fugac-ity increases much slower with pressure than the tetra7u-oromethane fugacity (see Fig. 14). The maximum in thetetra7uoromethane adsorption isotherm, on the other hand,must result from the potential energy exerted by each com-ponent on the system. One ethane molecule occupies ap-proximately the same volume of the pore as a tetra7uo-romethane molecule (see Fig. 1c), but since ethane is a chainmolecule and since the pore is very long, ethane is able tolie parallel to the pore wall. Consequently, to compare pack-ing e<ciencies, one must compare the potential energy ex-erted by a CH3-group with the potential energy exerted by atetra7uoromethane molecule. A plot of the potential interac-tion energy between a carbon atom and a CH3-group and atetra7uoromethane molecule looks similar to Fig. 5. Follow-ing our argumentation for tetra7uoromethane and methane,at a high pressure and in a narrow pore, the distance of the


0 25 50 75 100 125 1500

20

40

60

80

CF4 [300 K] C2H6 [300 K] CF4 [400 K] C2H6 [400 K]

Co

mp

on

ent

Fu

gac

ity

/[b

ar]


Fig. 14. Calculated component fugacity of ethane and tetra7uoromethaneat a temperature of 300 and 400 K as a function of pressure. The gasphase consists of an equimolar mixture. The Peng–Robinson equation ofstate was used for kij = 0.

adsorbate and the pore atoms (and the other molecules inthe pore) gets small. As a result, the potential energy ex-erted by tetra7uoromethane becomes on average higher thanthe one exerted by an ethane molecule; and ethane is ableto displace tetra7uoromethane. The more ethane adsorbs inthe pore the more crowded the pore will be and the moretetra7uoromethane is displaced. A maximum owing to a dif-ferent packing e<ciency or con8gurational entropy appears.This is likewise the reason for the maximum in the tetra7u-oromethane adsorption isotherm in the mesopore, if an idealgas phase is assumed (see Fig. 12). Under ideal conditionsthe ethane fugacity is higher than under real conditions. Thepore is more crowded and a maximum occurs. In the caseof a real gas phase the mesopore is not crowded enough atthe same total bulk pressure to cause this maximum in thetetra7uoromethane adsorption isotherm at a temperature of300 K. In addition, this maximum does not occur at a higherpressure, since the amount adsorbed of ethane decreases ow-ing to the non-ideal gas phase e&ect.

5. Conclusions

This study throws some new light onto mixture ad-sorption isotherm behaviour in micro- and mesoporousmaterials and is of importance in the development of sep-aration technologies. It is shown in a systematic way thatmaxima in absolute adsorption isotherms of mixtures oc-cur with increasing pressure owing to packing e&ects ornon-ideality e&ects of the gas phase. The acceptance prob-ability of a molecule insertion in a grand canonical MonteCarlo step is proportional to the component fugacity. If,owing to non-ideality e&ects of the bulk gas phase, the fu-gacity of one component is not increasing as steeply with

pressure as the other component, a maximum can occur inthe absolute adsorption isotherm of this component. Suchnon-ideality e&ects are especially pronounced if the bulkgas mixture is in a thermodynamic state close to the mix-ture critical temperature. This phenomenon was observedfor the component with the higher critical point of equimo-lar methane=ethane, methane=propane, ethane=propane,tetra7uoromethane=ethane and tetra7uoromethane=propanemixtures at a temperature of 300 K. In addition, methaneand ethane are able to displace tetra7uoromethane in mi-croporous materials owing to a higher packing e<ciency,an energetic and=or entropic e&ect. The fact that ethaneis able to displace tetra7uoromethane proves that, at leastin cylindrical pores, the displaced component is not nec-essarily the larger molecule as often stated but the onethat occupies, on average, less area of the cross–section ofthe pore.

Notation

A, B component A, BN number of moleculesT temperature in KelvinU potential energyU bend bond bending potentialUi potential energy exerted by component

i on the systemU trunci trunctated potential energy exerted by

component i on the systemU tailij tail correction of the potential energy exerted

by component i, owing to component jV simulation volume=pore volumeacc acceptance probabilityf fugacitykij binary interaction coe<cientkB Boltzmann factork� force constant of the bond bending potentialn new con8gurationnex excess amount adsorbednabs absolute amount adsorbedo old con8gurationr distancerc cut-o& radiusy mole fraction in the bulk phase

Greek letters

� 1=(kBT )� Lennard–Jones potential well depth parameter� particle density�bf bulk 7uid density� Lennard–Jones size parameter� bond bending angle�eq equilibrium bond bending angle


Subscript

i; j; k component i; j; k

Acknowledgements

A. Heyden wants to thank Professor Dr. Nigel Seaton formany helpful discussions he had when he was doing partsof this work at the University of Edinburgh, Scotland. Theauthors A. Heyden and F. J. Keil are very grateful to theMax Buchner Stiftung and Fonds der Chemischen Industrie.

References

Allen, M. P., & Tildesley, D. J. (1989). Computer simulation of liquids.Oxford: Clarendon Press.

Clark, A. L., Gupta, A., & Snurr, R. Q. (1998). Siting and segregatione&ects of simple molecules in zeolites MFI, MOR, and BOG. Journalof Physical Chemistry B, 102, 6720–6731.

Coolidge, A. S. (1934). Adsorption at high pressures I. Journal of theAmerican Chemical Society, 56, 554–561.

Du, Z., Manow, G., Vlugt, T. J. H., & Smit, B. (1998). Molecularsimulation of adsorption of short linear alkanes and their mixtures insilicalite. A.I.Ch.E. Journal, 44(8), 1756–1764.

Frenkel, D., & Smit, B. (2001). Understanding molecular simulation:from algorithms to applications (2nd ed.). San Diego: Academic Press.

Haag, W. O. (1994). Studies in surface science and catalysis, In J.Weitkamp, H. G. Karge, H. Pfeifer, & W. H0olderich (Eds.), Zeolitesand related microporous materials: State of the art 1994, Vol. 84(pp. 1375–1394). Amsterdam: Elsevier.

Harris, P. J. F. (1999). Carbon nanotubes and related structures.Cambridge, MA: Cambridge University Press.

Ke&er, D., Davis, H. T., & McCormick, A. V. (1996a). The e&ect ofnanopore shape on the structure and isotherms of adsorbed 7uids.Adsorption, 2, 9–21.

Ke&er, D., Davis, H. T., & McCormick, A. V. (1996b). E&ect ofloading and nanopore shape on binary adsorption selectivity. Journalof Physical Chemistry, 100, 638–645.

Krishna, R. (2001). Exploiting con8gurational entropy e&ects forseparation of hexane isomers using silicalite-1. Transactions of theInstitution of Chemical Engineers A, 79, 182–194.

Krishna, R., & Paschek, D. (2000). Separation of hydrocarbonmixtures using zeolite membranes: A modelling approach combiningmolecular simulations with the Maxwell-Stefan theory. Separation andPuri<cation Technology, 21, 111–136.

Krishna, R., Smit, B., & Vlugt, T. J. H. (1998). Sorption-induceddi&usion-selective separation of hydrocarbon isomers using silicalite.Journal of Physical Chemistry A, 102(40), 7727–7730.

Maginn, E. J., Bell, A. T., & Theodorou, D. N. (1995). Sorptionthermodynamics, siting, and conformation of long n-alkanes in silicaliteas predicted by con8gurational-bias Monte Carlo integration. Journalof Physical Chemistry, 99, 2057–2079.

Martin, M. G., & Siepmann, J. I. (1998). Transferable potentials forphase equilibria. 1. United-atom description of n-alkanes. Journal ofPhysical Chemistry B, 102, 2569–2577.

Mohanty, S., Davis, H. T., & McCormick, A. V. (2000). Shape selectiveadsorption in cylindrical pores. Chemical Engineering Science, 55,3377–3383.

Murata, K., El-Merraoui, M., & Kaneko, K. (2001). A new determinationmethod of absolute adsorption isotherm of supercritical gases underhigh pressure with a special relevance to density-functional theorystudy. Journal of Chemical Physics, 114(9), 4196–4205.

Reid, R. C., Prausnitz, J. M., & Poling, B. E. (1986). The properties ofgases and liquids. New York: McGraw-Hill.

Schenk, M., Vidal, S. L., Vlugt, T. J. H., Smit, B., & Krishna, R.(2001). Separation of alkane isomers by exploiting entropy e&ectsduring adsorption on silicalite-1: A con8gurational-bias Monte Carlosimulation study. Langmuir, 17, 1558–1570.

Siepmann, J. I., & Frenkel, D. (1992). Con8gurational bias Monte Carlo:A new sampling scheme for 7exible chains. Molecular Physics, 75,59–70.

Sircar, S. (1999). Gibbsian surface excess for gas adsorption—revisited.Industrial and Engineering Chemistry Research, 38, 3670–3682.

Talbot, J. (1997). Analysis of adsorption selectivity in a one-dimensionalmodel system. A.I.Ch.E. Journal, 43(10), 2471–2478.

Vlugt, T. J. H. (2000). Adsorption and di?usion in zeolites: Acomputational study. Ph.D. thesis, Amsterdam.

Vlugt, T. J. H., Krishna, R., & Smit, B. (1999). Molecular simulationsof adsorption isotherms for linear and branched alkanes and mixturesin silicalite. Journal of Physical Chemistry B, 103, 1102–1118.

Vlugt, T. J. H., Martin, M. G., Smit, B., Siepmann, J. I., & Krishna,R. (1998). Improving the e<ciency of the con8gurational-bias MonteCarlo algorithm. Molecular Physics, 94(4), 727–733.

Study of molecular shape and non-ideality effects on mixture adsorption isotherms of small molecules...

Documents

Transcript of Study of molecular shape and non-ideality effects on mixture adsorption isotherms of small molecules...