carbon molecular sieve membranes: sorption, kinetic and structural characterization

Journal of Membrane Science 241 (2004) 275–287

Carbon molecular sieve membranesSorption, kinetic and structural characterization

S. Lagorsse, F.D. Magalhães, A. Mendes∗

LEPAE, Departamento de Engenharia Qu´ımica, Faculdade de Engenharia, Universidade do Porto, 4200-465 Porto, Portugal

Received 17 December 2003; received in revised form 8 April 2004; accepted 28 April 2004

Abstract

Carbon molecular sieve membranes (CMSM) from Carbon Membranes Ltd. (Israel) were characterized. These membranes possess a verynarrow pore system, in the range of 0.3–0.5 nm, and a sharp pore size distribution, allowing for high selectivities and permeabilities in gasseparations.

Penetrants O2, N2, CO2, SF6, Xe and He were used in monocomponent adsorption, permeability and uptake experiments for obtainingadsorption and diffusion data, leading to new insights on the internal pore structure of CMSM. A number of complementary morphologicalcharacterizations were also performed.

The permeability and adsorption data were obtained over a temperature range from 273 to 323 K and over a pressure up to 5 bar. A relationbetween the adsorption and permeability data was used, allowing for the estimation of the diffusivity coefficients, which were further comparedwith those directly obtained from the uptake experiments. Permeation and uptake experiments showed a similar strong dependence of CO2

diffusivity on concentration.© 2004 Elsevier B.V. All rights reserved.

Keywords:Carbon molecular sieve membrane (CMSM); Adsorption; Permeation; Gas transport

1. Introduction

Nowadays, in the area of membrane research, concen-trated efforts are focused on the development of novel mem-branes able to overtake the productivity/selectivity trade-offlimit attained with the commercially available polymericmembranes for gas separation. Ultramicroporous (in the0.3–0.5 nm range) membranes have shown to be promisingin this context. Because these have pores with dimensionsclose to those of the diffusing species, small variations(0.02–0.03 nm) in the molecular or atomic dimensions mayimply a variation of several orders of magnitude in diffusionrates[1]. These materials are therefore described as havingmolecular sieving capabilities. That’s the case of zeoliteand carbon molecular sieve membranes.

Carbon molecular sieve membranes (CMSM) are brittleand difficult to handle, however they still offer advantages

∗ Corresponding author.E-mail address:[email protected] (A. Mendes).

over zeolite membranes. Indeed, although good progress isbeing made, the latter are difficult to obtain in large dimen-sions and in a crack-free, continuous form[2]. CMSM, onthe other hand, have been successfully manufactured in dif-ferent configurations, showing good performances in termsof having higher permeabilities than zeolite membranes, forthe same selectivity values[3].

Soffer and Koresh were the first to successfully preparecarbon molecular sieve membranes by pyrolysis of organicprecursors (cellulosic and phenolic resins)[4,5]. They stud-ied how high temperature treatments influence the develop-ment of the pore structure. High temperature (up to 700◦ C)evacuation and mild temperature (up to 450◦ C) air oxida-tion treatments lead to the opening of the pore structure. Afurther increase in temperature, under vacuum or inert at-mosphere, results in a gradual pore closure due to sintering.This way, by appropriately choosing a sequence of ther-mal treatments, the membrane’s pore aperture can be nearlycontinuously tuned. The application of these carbon mem-branes to gas separation processes was patented in 1987[6]by the same authors. A few years later, Soffer co-authored

0376-7388/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.memsci.2004.04.038

276 S. Lagorsse et al. / Journal of Membrane Science 241 (2004) 275–287

some other related patents: in 1995, a full description of theprocess of production of hollow fiber membranes from ther-mosetting polymers such as cellulose[7]; in 1996, a methodfor repairing defective hollow fiber membrane modules, byselectively clogging, with a sealant, the cracked fibers[8];in 1997, a method of improving membrane selectivity usingchemical carbon vapor deposition[9]. These patents endedup being bought by an Israeli company named Carbon Mem-branes Ltd. This was the first and, so far, the only companyworldwide to produce, at an industrial scale, high-qualityhollow fiber membranes. Commercialization started by theend of the 1990s, but the company closed in 2001.

Since 1987 that many other researchers have been syn-thesizing microporous carbon membranes, from severalprecursors such as polyimides[10–16], coal tar pitch[17],condensed polynuclear aromatic resins[18] and, recently,from ultra-filtration fiber membranes[19]. There have alsobeen reports of supported carbon membranes[20–25] (toimprove mechanical stability), composite carbon mem-branes (to reduce adverse humidity effects)[26] and, morerecently, mixed matrix membranes (using CMS to counterthe costly processing of purely homogeneous CMSM[27]).A more detailed review of the carbon membranes developeduntil 2001 can be found in a paper by Ismail and David[28].

It is clear that, along the last few years, attention has beenfocused mainly on the development of novel carbon mem-branes in order to improve gas separation performances, tak-ing as reference commercial polymeric membranes. How-ever, little work has been done on studying some other im-portant aspects related to these materials. Menendez andFuertes[29] studied aging effects associated to air oxidationand humidity. Jones and Koros analyzed the membrane per-formance over a range of humidity levels[30]. Some workson membrane regeneration and conditioning can also befound in the literature[31,32]. Tanihara et al. reported on themechanical stability of CMS hollow fiber membranes[13].

While there is a considerable amount of published ex-perimental data on pure component permeabilities throughCMSM, there is very little data on adsorption equilibrium,diffusion coefficients and multicomponent transport. Thesmall number of works reporting sorption and kinetic datarefer to flat membranes, using steady-state permeation flux[11,22] and transient time lag[11] techniques. Sedigh et al.[33] and Vu and Koros[14] performed some of the few mul-ticomponent permeation studies on CMSM.

The characterization of the microporosity in these mate-rials is still difficult. Recently, in a short communication,Nguyen et al. attempted to estimate the pore size distribu-tion from the adsorption equilibrium information, using themodel proposed by Nguyen and Do[34].

Because some fundamental understanding of these mate-rials is still lacking, this paper aims to give further insighton the pore structure of CMSM and to supply sorption anddiffusion data for several gaseous species. The CMSM char-acterized and tested in this study were supplied by CarbonMembranes Ltd. These industrially produced membranes

consisted of 1 m long hollow fibers, which were adapted forassembly in a lab-scale membrane module.

2. Carbon molecular sieve membranes

Carbon Membranes Ltd. has successfully prepared crack-free CMSM hollow fibers having narrow pore size distri-butions. The production process consists in the pyrolysis ofa large bundle of thermosetting polymeric precursor fibers,such as dense cellulose hollow fibers, from which all plas-ticizers and absorbed water have been previously removed.This thermal decomposition takes place in a controlled at-mosphere and involves a rigorous temperature history.Fig. 1shows an idealized CMSM pore structure. During pyrolysis,heteroatoms are progressively removed leaving a porous car-bon skeleton (Fig. 1a). This carbon skeleton is responsiblefor the membrane’s mechanical properties and, if a selectivelayer is not added subsequently, will also determine the sep-aration performance. Koresh and Soffer proposed that thepore structure is made of a series of very narrow constric-tions (<0.5 nm) linking larger pore sections in the micro-pore size range[1,4]. One designates this initially formedpore structure as “primary” microporosity of the CMSM.

Carbon Membranes Ltd. (CML) used a posterior carbonchemical vapor deposition (CVD) step in order to obtaina high performance membrane with improved selectivity.Depending on the size of the molecule used as the carbonsource, distinct carbon vapor deposition mail prevail[9]. Inthis process, the bore side of the fibers is coated with a carbonfilm, using propylene as the carbon source. This film plugs

Fig. 1. Idealized structure of the CMSM: (a) “primary” porosity is formedafter pyrolysis; (b) deposition of a carbon film on the bore side (CVDlayer); (c) activation step, opening ultramicropores across the CVD layerand throughout the “primary” porosity structure; (d) detail of the finalpore structure throughout the membrane. (Sketches (a), (b) and (c) arebased on an internal report by Carbon Membranes Ltd.)

S. Lagorsse et al. / Journal of Membrane Science 241 (2004) 275–287 277

the “primary” pores at the bore side with a so-called “CVDlayer” (Fig. 1b). The deposition occurs on the bore sideof the hollow fiber, both within the existing pore structureand as an additional layer on top of the membrane surface.Depending on the size of the molecule used as the carbonsource, each of these distinct forms of deposition will prevail[9]. The next stage, called “activation”, consists in applyinga high-temperature oxidation treatment in order to create apore structure of tailored dimensions within the CVD layer(Fig. 1c). A very narrow distribution of “selective” ultrami-cropores is obtained. Duration and intensity of this treatmentdetermines the diameter of the “selective” pores and there-fore allows for the tuning of the membrane performance to-wards the intended gas separation. Since the CVD is appliedto one side of the fiber, this process results in an asymmetricmembrane. The “primary” porosity is also affected in thisstep, as described below.

The structure of these porous inorganic membranes is notas regular as that of zeolite membranes, since they have anamorphous nature. A schematic picture of the carbon porousstructure is shown inFig. 1d. The evolution of the porestructure along the different treatments can be viewed as de-scribed next. This is an idealized model, which will be usedlater in the discussion of the experimental data. Pyrolysiscreates the “primary” microporosity. These pores are narrowand the number of those that actually connect both facesof the membrane is small, so that permeability is very lowfor all gases. It has been suggested that these pores are slitshaped[1]. The posterior activation treatment, applied afterthe carbon vapor deposition, creates ultramicropores, heredesignated as “selective” pores (with a characteristic diame-ter dc – seeFig. 1d). These contribute decisively to improvethe membrane’s selectivity performance. They are formedin different regions of the membrane: (a) across the CVDlayer, (b) connecting “primary” pores and (c) as dead-endpores. The activation treatment additionally contributes towiden the “primary” porosity, which is idealized as consist-ing of cavities (dH) and constrictions (dL).

Representative permeabilities for SF6 and N2 measuredon CMSM samples on two conditions, (1) right after pyrol-ysis and (2) after three different CVD/activation treatments(designated by samples CMSM1, CMSM2 and CMSM3),are listed inTable 1. The data for N2 evidences how differenttreatments may lead to a wide range of permeation perfor-mances. It also shows that the permeability associated with

Table 1Permeability (×10−8 m3 (STP)�m m−2 kPa−1 s−1) of CMSM samples be-fore and after different CVD/activation treatments (�p = 1 bar)

Pyrolyzed CMSMsamples beforeCVD/activationa

Pyrolyzed CMSM samples afterdifferent CVD/activation treatments

CMSM1 CMSM2 CMSM3

SF6 0 0 0 0N2 2.2–6.8 225 22.5 11.2

a Data supplied by Carbon Membranes Ltd.

the original “primary” porosity is quite low and thereforethe final activation treatment also plays a role in wideningthese pores. The fact that SF6 does not permeate throughany of the samples insures that these are crack-free. CarbonMembranes Ltd. did not supply details on the thermochem-ical treatment history associated to each of the three CMSMsamples. These samples (CMSM1, CMSM2 and CMSM3)are used along this study.

In the ultramicropore structure, the pore sizes are com-parable to the molecular sizes and therefore molecules maynever completely escape the potential field of the pore walls.Gas transport occurs according to an activated diffusionmechanism. Two alternative mechanisms may be responsi-ble for the separation of a gas mixture, depending on thepore size/effective molecular diameter ratio:

(1) Molecular sieving will be the preponderant mechanismfor diffusing species having approximately the same di-ameter as the pores. The diffusivity of a species throughthe membrane depends strongly on the molecular sizeand shape.

(2) The so-called selective adsorption mechanism[35] oc-curs when a diffusing species is preferentially adsorbedand, due to its size, hinders the passage of less adsorbedspecies through the pores. This may occur when thepore diameter is up to three or four times larger than thediameter of the more strongly adsorbed species[36].

3. Experimental

Different gases were selected: He, CO2, O2, N2, Xe andSF6 in order to analyze the sorption and transport proper-ties of the CMSM. Three different samples, designated asCMSM1, CMSM2 and CMSM3 (seeTable 1) were used.These correspond to different activation treatments appliedby the manufacturer (CML).

Several techniques were used in this study to provide in-sight onto the morphology of the carbon material. These in-cluded SEM pictures, wide-angle X-ray diffraction, X-raymicroanalysis, helium pycnometry and CO2 adsorption data.

A volumetric-type apparatus was used to measure purecomponent isothermal equilibrium data. The internal vol-umes of the various parts of the system were initially accu-rately measured. In the experimental runs, a high-accuracyDruck pressure transducer measured the gas-phase pressureand the temperature of the sample was monitored with a ther-mocouple. Each sample consisted of about 5 g of broken-upfibers. Equilibrium adsorption data for N2, O2, CO2, Xe andSF6 at three temperatures and up to 5 bar were determinedfor sample CMSM2.

Sorption rates of CO2 and N2 on CMSM2 and CMSM3were measured gravimetrically on a Rubotherm suspensionmagnetic balance having a 0.01 mg weighing resolution anda 0.02 mg reproducibility. The entire system is fully ther-mostated. Each uptake measurement was performed over


Fig. 2. Sketch of the membrane module.

Table 2Characteristics of the in-house assembled membranes modules

Membranemodule

Number offibers

Effectivearea (m2)

Hollow fiber dimensions

Dext (m) � (m)

M1 80 0.015 170× 10−6 9 × 10−6

M2 95 0.018

successive appropriately small increments in adsorbate pres-sure, so that the sorption equilibrium may be assumed lin-ear in each interval. This allowed for the determination ofdiffusivities as a function of loading.

Single gas permeation was measured in a dead-end perme-ation module, in a temperature-controlled system. The feedwas applied to the bore side of the membranes. The pressuregradient was generated both by pressurizing the feed and byevacuating the permeate side. The feed was high-purity gassupplied directly from compressed gas cylinders. Its pressurewas controlled with a pressure regulator installed after thetwo-stage regulator on the cylinder. The feed and permeatepressures were read on Druck absolute pressure transduc-ers. The volumetric flow rate of the permeating species wasmeasured using Bronkhorst HI-TEC flowmeters, chosen inaccordance to the flow rate range. Two membrane modules,designated as M1 and M2, containing a bundle of CMSM2fibers, were constructed with 1/4 in. stainless steel tubing of30 cm length and two Swagelok 1/4 tees (seeFig. 2). Epoxyglue was used to plug the ends of the modules. The charac-teristics of the in-house-assembled membrane modules arelisted in Table 2. A bulky molecule (SF6) has been used asa probe molecule to test the integrity of each module.

4. Results and discussion

4.1. Structural characterization

Several techniques were used in this study for a pre-liminary characterization of the CMSM structure. Scanningelectron microscopy techniques are useful for evaluating theoverall quality of the membrane, detecting the presence ofany mesoscopic cracks and determining the overall thick-ness of the hollow fiber wall. The results indicated a uniformwall thickness of 9�m and an external diameter of 170�m.

The SEM pictures, illustrated inFig. 3, also showed a radi-ally symmetric, dense and crack-free structure.

The skeletal density,ρs, was measured by helium pyc-nometry. The densities obtained for CMSM1 and CMSM2were 1.7 and 1.6 g cm−3, respectively, against 2.3 g cm−3

for non-porous crystalline graphite density. As expected,CMSM is a highly disordered material with low density.

Fig. 3. SEM pictures of a CMSM.


X-ray microanalysis (0.5% weight detection limit) did notdetect the existence of other atoms in addition to carbon.Wide-angle X-ray diffraction indicated an average pore sizeof 0.4 nm and a highly amorphous structure. This is coherentwith the densities obtained by helium pycnometry, whichwere 20–30% lower than that of crystalline graphite.

To characterize the micropore volume of two dif-ferent samples, CMSM1 and CMSM2, subcritical car-bon dioxide adsorption was employed together with theDubinin–Radushkevich (DR) equation (Eq. (1)):

q = q∞ exp

[− 1

(βE0)2

(RTln

p

pv

)2]

(1)

whereq represents the amount of solute adsorbed at a rela-tive pressurep/pv, q∞ the limiting amount of solute fillingthe micropores,E0 the characteristic energy dependent onthe pore structure andβ the affinity coefficient which is char-acteristic of the adsorbate. The value of the affinity coeffi-cient,β, for carbon dioxide is around 0.35[37]. This equa-tion has been used to characterize the micropore volume ofmany carbon-based microporous materials[38]. It has beenshown that CO2 adsorption at 273 or 298 K is sensitive tothe presence of narrow micropores (pore size smaller than0.6 nm), which are not accessible to N2 at 77 K, due to dif-fusional limitations[37,39].

The limiting micropore volume is given by

Vp = q∞ Mr

dCO2ρs × 106(2)

wheredCO2 is the density of CO2 in the adsorbed state.There is uncertainty as to the true density of the adsorbed

phase, and it is likely that the adsorbate is denser than theliquid phase at the same temperature due to the overlap of thepotential energy surfaces in the micropores. The approachfollowed here was suggested by Cazorla-Amarós et al.[37].The values of density that can be used fall between thedensity of liquid CO2 and the value obtained from Dubinin’sapproach. The density values proposed for adsorbed CO2are 0.85 g cm−3 at 298 K and 0.97 g cm−3 at 273 K.

The characteristic curves obtained from the CO2 adsorp-tion isotherms on CMSM1 and CMSM2 are quite linear, in-dicating that the DR equation (Eq. (1)) provides a reasonable

Table 3CMSM structural parameters

Sample Micropore volume,Vp (cm3 g−1) L0 (nm) E0 (kJ mol−1) ρs (g cm−3) ρb (g cm−3) ε

CMSM1 0.28a 0.55 31.6 1.6 1.1 0.35CMSM2 0.16b 0.45 35.6 1.7 1.3 0.21

Published datac

CMS, T3A 0.20b 0.6CMS, T5A 0.30b 0.6

a CO2 adsorption at 283 and 293 K; CO2 density obtained by interpolation.b CO2 adsorption at 273 K.c Ref. [39].

Fig. 4. Carbon dioxide characteristic curve at 273 K for CMSM2.

description. A characteristic curve for CMSM2 at 273 K isshown inFig. 4.

The slope of this plot is related to the characteristic en-ergy of the micropore (E0) and the affinity coefficient (β).The intercept is related to the saturation capacity (q∞). Themicropore volume,Vp, was estimated and compared withdata published in the literature, as shown inTable 3.

Also shown inTable 3, are the average micropore widths,L0, estimated from Stoeckli equation which can be usedwhen the DR equation applies[40]:

L0 (nm) = 10 800(nm J mol−1)

E0 − 11 400(J mol−1)(3)

This simple approach provides only a rough estimation, but,as it will be seen in the next section, it is in agreementwith equilibrium adsorption experiments involving sphericalsolutes with different kinetic diameters.

The analysis methods of CO2 adsorption and helium pyc-nometry were combined and used to calculate the bulk den-sity of the material, according to the following equation:1/ρb = 1/ρs+Vp. This is shown inTable 3, together with theestimated porosities. Note that the previous equation doesnot account for the presence of isolated cavities.

It is interesting to note that CMSM1, which is 10 timesmore permeable toward N2 than CMSM2 (seeTable 1),


Fig. 5. Equilibrium adsorption isotherm on CMSM2 at 303 K. The solidline corresponds to the Langmuir–Freundlich fitting for CO2 and thedashed lines correspond to the Langmuir fitting: (�) O2; (�) CO2; (�)SF6; (�) N2.

presents higher micropore volume and higher mean porewidth. This is in agreement with a larger activation step ofthe membrane, leading to an additional enlargement of the“primary” micropore system and to wider “selective” pores.

4.2. Adsorption equilibrium experiments

The equilibrium adsorption isotherms for N2, O2 andCO2 up to 5 bar on CMSM2 are represented graphically inFigs. 5 and 6. These are of type I. The Langmuir equa-tion was used to fit the data for CO2, O2 and N2. TheLangmuir–Freundlich equation was additionally used forCO2, giving slightly better results. The isotherm parame-ters are summarized inTable 4. The adsorption capacity de-creases strongly with temperature. Nitrogen and oxygen areboth weakly adsorbed in relation to carbon dioxide and therespective isotherms are essentially coincident. No hystere-sis is observed, as would be expected for a system with nomesopore condensation.

Table 4Equilibrium isotherm parameters for CO2, O2 and N2 on CMSM2 sample

T (K) Langmuir Qlangmuir (kJ mol−1) Langmuir–Freundlich QLangmuir–Freundlich

(kJ mol−1)−�Hads (kJ mol−1)

b (bar−1) qm (mol m−3) b (bar−1) qm (mol m−3) m

CO2 288 2.15 5537 1.49 6260 1.39 29.6 (θ = 0.05)303 1.51 4976 18.7 0.93 5928 1.34 25.2 25.2 (θ = 0.5)323 0.93 4492 0.48 5738 1.32

O2 288 0.23 3249303 0.16 2989 20.2 20.8323 0.09 3084

N2 288 0.24 2888303 0.19 2719 10.5 10.5323 0.15 2147

Fig. 6. Equilibrium adsorption isotherms on CMSM2 at three temperaturesfor CO2 (filled symbols: (�) 288 K; (�) 303 K; (�) 323 K), N2 (emptysymbols: (�) 288 K; (�) 303 K; (�) 323 K). The solid line correspondsto the Langmuir–Freundlich fitting and the dashed lines correspond to theLangmuir fitting.

The analysis of adsorption isotherms is still the most sen-sitive method for gathering useful information concerningthe CMSM structure at the ultra-microporous level. The sam-ples were probed using gas species of varying dimensions,in order to estimate the critical pore widths. Adsorption datafor Xe and SF6 were experimentally obtained on CMSM1and CMSM2. No measurable uptake was detected for SF6up to 5 bar on both samples. The exclusion of this speciesindicates that the critical pores in this material have a di-mension that is somewhere below 0.502 nm (SF6 moleculardimension based on liquid molar volume[41]).

On the other hand, xenon adsorption shows distinct be-haviours on each CMSM sample. On CMSM1, xenon wasstrongly adsorbed and the uptake was relatively fast (1 h).In contrast, on CMSM2, xenon uptake was very slow at thesame temperature, so that equilibrium was not yet attainedafter 8 days. From these observations one concludes thatat least a fraction of the pores in CMSM1 are larger than


Fig. 7. Carbon dioxide adsorption isosteres at different coverage onCMSM2 sample.

0.394 nm (Xe molecular dimension based on liquid molarvolume[41]) and that on CMSM2 the critical micropore sizeis close to 0.394 nm.

The isosteric heat of adsorption was determined at differ-ent coverages on CMSM2 using the thermodynamic van’tHoff equation:�H/RT2 = −(∂ ln p/∂T)qm. Table 4sum-marizes these results. For N2 and O2, the computed valuesof −�Hads are independent of the amount adsorbed. ForCO2, on the other hand, the isosteric heat decreases with in-creasing surface coverage, as shown inFig. 7. This may bea consequence of surface heterogeneity or repulsive sorbateinteractions.

4.3. Permeation experiments

Fig. 8 shows results for bore side feed permeation onmembrane (CMSM2) modules M1 and M2 (seeTable 2),measured at 303 and 323 K. Permeate pressure was kept at0.060 bar while feed pressure varied from about 0.400 barup to 4 bar.

The “permeability”,P, of the CMSM towards a specieswas determined as described by the following equation:

P = Flux

�p/�(4.1)

where�p is the partial pressure difference of the speciesacross the membrane and� is the thickness of the membrane.For convenience, in this work we also use the mass transfercoefficient or permeance,kp, defined as

kp = P

�= Flux

�p(4.2)

As seen fromFig. 8, the permeance and hence the permeabil-ity of all adsorbing species decreases with increasing feedpressure. This tendency seems to become more evident asthe intensity of adsorbate/adsorbent interactions increases.The strongly adsorbed CO2 shows a pronounced decrease of

pfeed / bar

1 2 3 4

k p / x

10-8

m3 (S

TP

)m-2

kPa-1

s-1

0

6

11

17

22

Fig. 8. Single gas permeance on CMSM2 (in M1 and M2 modules) as afunction of feed pressure at two different temperatures, 303 and 323 K:( ) O2 at 323 K, (�) O2 at 303 K; (�) He at 303 K, ( ) He at 323 K;(�) CO2 at 303 K, ( ) CO2 at 323 K; (�) N2 at 303 K, ( ) N2 at 323 K.Solid curves are fitting equations.

permeability with pressure, reflecting a strong concentrationdependence of the diffusion coefficient. For O2 and N2, ad-sorption interactions are weaker and the permeabilities areless pressure dependent. Helium permeability, on the otherhand, is independent of pressure. Gas molecules having adiameter larger than 0.39 nm, like Xe and SF6, do not ex-hibit measurable permeation up to 5 bar. The permeabilityof all species increases with temperature.

Effective diffusivity coefficients were evaluated com-bining experimental data from equilibrium adsorption andsteady-state permeation experiments. The flux of a sorbedspecies is given by[42]

J = −Dc(q)Γ∂q

∂z(5)

whereDc is the corrected diffusivity andΓ = ∂ ln p/∂ ln q

is the thermodynamic correction factor. The Fickian diffu-sivity, D, also called transport diffusivity, is therefore relatedto the corrected diffusivity,Dc, by

D = Dc(q)Γ (6)

Eq. (6)is sometimes referred to as the Darken equation[42].The corrected diffusivity may itself be dependent on

coverage[42]. The thermodynamic correction factor canbe determined from the equilibrium isotherm equation.For strongly adsorbed species, the transport diffusivity ishighly dependent on concentration via the thermodynamiccorrection factor. If the corrected diffusivity is consideredconstant, substitution of the Langmuir isotherm equationinto Eq. (6) leads to:

D = Dc1

1 − θ(7)


In this work, the corrected diffusivity was allowed to varywith coverage, according to:

Dc(θ) = D0c

(1 − θ)n−1(8)

D0c is the corrected diffusivity at zero loading.Under steady-state flow, the expression for the flux of the

adsorbed species can be obtained by solvingEq. (5)subjectto constant boundary conditions, corresponding to the twosides of the membrane. Assuming that sorption is describedby the Langmuir equation and usingEq. (8) for describingthe dependency ofDc on the coverage, one obtains

J = D0c

�

qm

1 − n[(bpl + 1)n−1 − (1 + bph)

n−1], n > 1 (9)

J = Dc

�qm ln

(1 + bph

1 + bpl

), n = 1 (10)

whereph andpl are the pressures on the feed and permeateside of the membrane, respectively.

For non-adsorbing species, the flux through CMSM issimply described by

J = Dc

�

1

RT(ph − pl) (11)

The curve fits, usingEq. (10) for N2 and O2, Eq. (9) forCO2 andEq. (11)for He, are shown inFig. 8. These agreequite well with the experimental data. The calculated cor-rected diffusivities at zero loading,D0

c andDc, are shown inTable 5. In all cases, the effective diffusivities increase withincreasing surface coverage. For O2 and N2, this dependencecan be described using a constant corrected diffusivity, i.e.using the so-called “Darken-like” dependency.

On the other hand, the data for CO2 cannot be describedby a “Darken-like” relation, as shown inFig. 9. It is im-portant to note that the use of another isotherm equation(like Langmuir–Freundlich), which fits the CO2 equilib-rium data somehow better than the Langmuir equation,

Table 5Corrected diffusivity at zero loading on CMSM2

T (K) Sample CMSM2

D0c or Dc

(×10−12 m2 s−1)Fittedequation

n Ea (kJ mol−1)

CO2 288 0.68 1.47303 1.07 9 1.52 23.4323 1.95 1.41

O2 288 1.36 1303 2.40 10 1 21.6323 4.51 1

N2 288 0.33 1303 0.42 10 1 26.5323 0.86 1

He 288 68.2303 78.5 11 8.2323 98.8

Fig. 9. Carbon dioxide permeance concentration dependence at 303 K.The solid line corresponds to theEq. (9) fitting and the dashed linecorresponds to theEq. (10)fitting.

also does not provide a good fit of the permeance data.A loading-dependent corrected diffusivity (Eq. (8)) hadtherefore to be used, with a parametern = 1.52 (at 303 K).

The corrected diffusivitiesD0c and Dc were correlated,

at three different temperatures, by the Arrhenius equation(seeFig. 10). The corresponding computed values of theactivation energies (Ea) are given inTable 5. For all gases,transport is well described as being an activated process, asexpected for a molecular sieving mechanism.

4.4. Uptake experiments

Gravimetric uptake measurements were carried out forCO2 and N2 on CMSM2 to determine the effective diffu-sivity coefficients and further compare them with the re-sults evaluated from the combination of permeation andadsorption equilibrium data. Since the CO2 diffusivity isstrongly concentration dependent, uptake experiments were

Fig. 10. Arrhenius plots for diffusivity (at zero loading) in CMSM2sample for: (�) CO2; (�) N2; (�) O2; (�) He.


Fig. 11. Carbon dioxide and nitrogen experimental uptakes and fittingcurves (Eq. (12)) at 303 K in CMSM2: ( ) CO2 at 0.048 bar; (�) N2 at1.07 bar.

performed only at very low concentration. At higher concen-trations, the combination of fast uptake rates with thermaleffects made it impossible to accurately extract the diffusiv-ity.

Representative uptake curves, obtained for CO2 and N2at 303 K, are shown inFig. 11. The gaps in the CO2 ex-perimental points are related to the automatic base line cor-rection procedure of the microbalance used. The observedtransient behaviour seems to be consistent with microporediffusion control. Since membrane thickness is less than 6%of the fiber’s external diameter, one can assume a slab geom-etry model. The fractional uptake,F (ratio between the totalamount of solute adsorbed, at instantt, and the total amountof solute adsorbed in equilibrium att → ∞), is given by[42]

F = 1 − 8

π2

∞∑k=0

1

(2k + 1)2exp

[−D(2k + 1)2π2t

�2

](12)

From uptake measurements, one obtains a diffusivity of 0.76× 10−14 m2 s−1 for nitrogen. For carbon dioxide one obtainsa diffusivity of 3.51× 10−14 m2 s−1, for a feed pressure of0.048 bar. The corresponding values obtained from perme-ation data are 50.7× 10−14 m2 s−1 for nitrogen and 120×10−14 m2 s−1 for carbon dioxide. This large discrepancy maybe associated to the CMSM pore structure, taking into ac-count that the two methods refer to transient and steady-statemeasurements, respectively. Each method involves transportthrough different pore systems. In a transient experiment,the entire pore structure is being filled, including a fractionof dead-end pores (seeFig. 12). These pores, in accordanceto the structural model described earlier, are mostly com-posed by “selective” pores, having diameters close to thesize of the diffusing species. Therefore, transport throughthe dead-end pores represents an additional significant re-sistance. On the other hand, in a steady-state method, asis the permeation measurements, only the fraction of poreswhose pass crosses the membrane (wall to wall) contributes

Fig. 12. Description of transient uptake mass transport in a CMSM without“selective” dead-end pores (situation A) and with “selective” dead-endpores (situation B). The transient measurement corresponding to situationB (fractional uptakeF as a function of time) will yield a lower averagediffusivity.

to the observed transport performance, which excludes thedead-end pores. Note that molecular probing experimentsperformed with SF6 exclude the hypothesis of higher per-meabilities being due to the presence of cracks on the mem-brane. It is interesting to remark that a similar disagreementwas also reported by Wang et al.[11], for CMS membranes.These authors observed that CO2 diffusivities, measured bytime lag and steady-state permeation, differed over an orderof magnitude.

One might question whether the micropore diffusionmodel (Fickian diffusion) is actually adequate to describethe transport process observed. Indeed, a surface resistanceeffect, in the form of a “pore mouth” barrier has beenobserved in some systems[43,44], which would be bet-ter described by a linear driving force (LDF) type model.Fig. 13shows a plot of ln(1− F) as a function of time fornitrogen, together with the solutions of the LDF model[42]and, using one hundred terms in the rapidly converging se-

Fig. 13. Variation of ln(1− F) as a function of time for the adsorptionof nitrogen on CMSM2 at 303 K and 1.07 bar. The solid line correspondsto Eq. (12)fitting and the dashed line corresponds to LDF model fitting.


Fig. 14. Nitrogen experimental uptakes and fitting curves at 303 K inCMSM3.

ries of the Fickian solution. The micropore diffusion modelprovides indeed the best description of the data.

In order to study the concentration dependence of the dif-fusion coefficient by uptake measurements, another carbonmolecular sieve sample, CMSM3, had to be used. This mem-brane exhibits lower diffusivities, therefore preventing theintrusion of heat effects. Some experimental uptake curvesfor N2 and CO2 are shown inFigs. 14 and 15, respectively.These curves were fitted according toEq. (12). Experimen-tal data obtained for N2 show a higher scattering than forCO2, due to the first gas being less adsorbed. This causesa higher relative weighing error. The experimental diffusiv-ities of N2 and CO2 from uptake experiments and the CO2diffusivity concentration dependence parameters are givenin Table 6. As already concluded from permeation experi-ments, N2 diffusivity is weakly dependent on pressure. Thisis no longer true for CO2. The variation of the diffusivitywith coverage is shown inFig. 16, together with the best-fitcurve according to the Darken relation with a constant cor-

Table 6Carbon dioxide and nitrogen diffusivities from uptake experiments with CMSM3 sample. Carbon dioxide diffusivity concentration dependence parameters

Uptake results for sample CMSM3 Fitting curve

T (K) Feed pressure (bar) D (×10−15 m2 s−1) Darken-like relation,Dc (×10−15 m2 s−1) Eq. (8)

D0c (×10−15 m2 s−1) n

CO2 0.05 0.910.06 1.520.26 2.33

303 0.28 3.48 2.34 1.45 1.450.91 6.701.04 6.661.36 9.252.03 14.42.97 17.0

N2 303 0.07 0.380.41 0.44

Fig. 15. Carbon dioxide experimental uptakes and fitting curves at 303 Kin CMSM3.

Fig. 16. Carbon dioxide concentration dependence diffusivity at 303 K inCMSM3.


rected diffusivity and toEq. (8). The trend, for the diffusivitydependence on coverage, obtained from these uptake exper-iments (n = 1.45) agrees quite well with the one observedon the permeation experiments (n = 1.52). The optimal val-ues of the fitting parameters are listed inTable 6.

Researchers have attempted to justify a loading depen-dence of the corrected diffusivity based on the system het-erogeneity[45]. On the other hand, Do et al.[46] arguedthat this cannot account for the increase of permeability withloading, implying that the so-called corrected diffusivity isa function of loading. This was interpreted by the authors interms of species sorbed on successive layers having differentmobilities. This study referred to microporous activated car-bon. In CMSM, however, it is not clear whether multilayeradsorption can occur. Xiao and Wei[47], in the context ofzeolite systems, suggested that repulsive interactions amongneighboring species at high loadings may also contributefor a concentration dependence of the transport diffusivity.Further studies are necessary for clarifying this topic.

5. Conclusions

Carbon molecular sieve hollow fiber membranes(CMSM), from Carbon Membranes Ltd. (Israel), werecharacterized. These CMSM are made from a hollow fiberprecursor of a thermosetting polymer such as cellulose.The production steps involve pyrolysis followed by carbonvapor deposition (CVD) and an activation step, which con-sists on a high-temperature oxygen treatment. Adsorption,permeability and diffusion data were collected and ana-lyzed, bringing some insights on the pore structure of thesematerials.

The porous structure of the asymmetric CMSM is hereschematically modeled as involving two main features: cav-ities linked by constrictions (designated here as “primary”pores), which are associated with the initial pyrolysis pro-cedure, and ultramicropores (<0.5 nm) (designated here as“selective” pores), formed during the activation step, whichexist across the CVD layer and as dead-end pores through-out the membrane.

Equilibrium adsorption isotherms and uptake curves wereobtained for N2, O2 and CO2. Langmuir (N2, O2 and CO2)and Langmuir–Freundlich (CO2) equations were fitted tothe equilibrium data. Despite equilibrium adsorption infor-mation for CMSM being rarely taken into account in theliterature, it can be very useful in qualitatively predictingcompetitive adsorption effects in multicomponent gas sep-arations. In addition, adsorption experiments are a simplemethod for estimating critical ultramicropore size. Ad-sorption experiments were complemented with wide-angleX-ray diffraction, used to characterize the CMSM micro-porous structure. An average pore size of 0.45 nm, with acritical pore size of 0.39 nm, was found.

Single gas permeabilities of He, N2, O2 and CO2 wereobtained as a function of feed pressure for two different

temperatures. It was observed that permeability decreaseswith feed pressure. This trend seems to follow the increaseon adsorbate/adsorbent interactions. The permeabilities alsoincrease with temperature. The transport of all gases wasfound to be an activated process, as expected for a molecularsieving mechanism.

Effective diffusivities obtained from steady-state (perme-ation method) experiments were found to be one order ofmagnitude higher than the ones obtained from transient ex-periments (gravimetric/uptake method). This discrepancy isconsistent with the idealized pore system presented here.Transport through dead-end pores does not play a role insteady-state experiments. However, these pores are presentand are expected to have a measurable contribution to theadsorption capacity of the CMSM. Additionally, they are as-sociated to a significant mass transport resistance, since theyare “selective” pores (ultramicroporous). Therefore, sincetransient uptake experiments involve diffusion through thedead-end pores, those measurements will yield lower aver-age diffusivities.

The increase of the transport diffusivity with loading wasalso studied. For weakly adsorbed species, this dependenceis quite well described by a “Darken-type” relation. How-ever, the transport diffusivity of CO2 shows strong concen-tration dependence. A mathematical relation was proposedto describe this dependence. The corresponding fittings todata from permeation or uptake experiments yielded veryclose results.

Acknowledgements

The authors acknowledge Dr. Vitaly Krakov from Car-bon Membranes Ltd., for supplying the hollow fiber CMSmembranes and for some insightful discussions.

This work was funded by FCT research project POCTI/EQU/34224/2000.

Nomenclature

b parameter of isotherm equation (bar−1)dCO2 carbon dioxide density (g cm−3)D transport diffusivity (m2 s−1)Dc corrected diffusivity (m2 s−1)Dext hollow fiber external diameter (m)D0

c corrected diffusivity at zero loading(m2 s−1)

Ea activation energy (J mol−1)E0 characteristic energy (J mol−1)F fractional uptake�Hads isosteric heat of adsorption (J mol−1)J diffusion flux (mol m−2 s−1)


kp permeance (m3 (STP) m−2 kPa−1 s−1)� membrane thickness (m)L0 average micropore width (nm)m parameter of Langmuir–Freundlich

isotherm equationMr Molecular weight (g mol−1)n fitting parameterp pressure (bar)ph feed side pressure (bar)pl permeate side pressure (bar)pv vapor pressure (bar)�p pressure difference across the membrane

(bar)P permeability (m3 (STP) m m−2 kPa−1 s−1)q concentration (mol m−3)qm saturation concentration (mol m−3)q∞ limiting amount filling the micropores

(mol m−3)Q adsorption heat (J mol−1)R gas constant (J K−1 mol−1)t time (s)T absolute temperature (K)Vp micropore volume (cm3 g−1)z distance coordinate along membrane (m)

Greek symbolsβ affinity coefficientΓ thermodynamic correction factorε porosityθ fractional coverageρb bulk density (g cm−3)ρs skeletal density (g cm−3)

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carbon molecular sieve membranes: sorption, kinetic and structural characterization

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