Water adsorption on carbon molecular sieve membranes: Experimental data and isotherm model

Water adsorption on carbon molecular sieve membranes: Experimental data and isotherm model

Carbon 43 (2005) 2769–2779 www.elsevier.com/locate/carbon Water adsorption on carbon molecular sieve membranes: Experimental data and isotherm model ...

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Carbon 43 (2005) 2769–2779 www.elsevier.com/locate/carbon

Water adsorption on carbon molecular sieve membranes: Experimental data and isotherm model S. Lagorsse, M.C. Campo, F.D. Magalha˜es, A. Mendes

*

LEPAE, Departamento de Engenharia Quı´mica, Faculdade de Engenharia, Universidade do Porto, 4200-465 Porto, Portugal Received 28 March 2005; accepted 23 May 2005 Available online 20 July 2005

Abstract Water vapor adsorption on carbon molecular sieve membranes is studied. Structural characterizations and pore size distribution determinations were performed on three different samples. Water adsorption isotherms displayed type V behavior and evidence of desorption hysteresis. An isotherm equation for water adsorption in microporous carbons was derived from the basic formulation of the Do model, originally developed for activated carbons. A mechanism for water desorption was also proposed. For each adsorption and desorption isotherm equation there are four parameters, three of which are identical for adsorption and desorption. Good description of water adsorption experimental data was obtained. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Porous carbon; Molecular sieves; Modeling; Adsorption properties; Porosity

1. Introduction The vulnerability of microporous carbons to humidity has driven several research works towards studying the adsorption of water vapor on such systems [1]. This is a complex phenomenon, considering the weak character of the water–carbon dispersion forces and the tendency of water molecules to form hydrogen bonds within the bulk phase. Water will initially adsorb onto hydrophilic groups, existing in the form of functional surface groups associated to non-carbon species. These are probably located at the edges of the carbon layers. These edge sites are much more reactive than the atoms in the interior of the graphene sheets, and chemisorb foreign elements, in particular oxygen, which is the most

*

Corresponding author. Tel.: +351 225081695; 225081449. E-mail address: [email protected] (A. Mendes).

fax:

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0008-6223/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.carbon.2005.05.042

frequent non-carbon element found in the surface of these materials [2]. Once the first water molecule is adsorbed, adsorbate–adsorbate interactions will promote the adsorption of further molecules. Examples of oxygen functional groups that might be found onto the carbon surface are carboxyl, carboxylic anhydride, phenol, carbonyl, lactone, ether and quinone groups [3]. Each type of functional group is capable of forming a certain maximum number of hydrogen bonds directly with water molecules, through unshared electron pairs associated to oxygen atoms or hydrogens bonded to oxygen. The nature and concentration of surface functional groups will depend on the carbon material production process and may be modified by suitable thermal or chemical post-treatments [4]. A variety of experimental techniques has been used to characterize the surface chemistry of carbons, such as chemical titration, temperature-programmed desorption (TPD), Xray photoelectron spectroscopy (XPS) and infra-red spectroscopy method (FTIR, DRIFT) [5]. Research on water adsorption has been mainly focused on activated carbons. A more detailed review of

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Nomenclature d aw ca cd E0 K Ka Kd m P Pv q qa

apparent density of water (g cm3) adsorbed water concentration via dispersion forces during adsorption (mol kg1) adsorbed water concentration via dispersion forces during desorption (mol kg1) characteristic energy (J mol1) hydrogen bonding equilibrium constant adsorption constant on the hydrophobic space desorption constant on the hydrophobic space number of water molecules in the cluster pressure (bar) saturation pressure (bar) concentration of the adsorbed phase (mol kg1) total adsorbed water concentration during adsorption (mol kg1)

the progress in this area, up to 2001, can be found in a paper by Brennan et al. [1]. The most commonly reported water isotherms in porous carbons are of Type V. Adsorption is essentially zero up to P/Pv  0.3 and a significant uptake occurs at higher relative pressures. As the number of hydrophilic centers increases, the inflexion point is shifted towards lower relative pressures. It is also possible that the isotherm gradually changes to a type IV. A marked type IV shape has been obtained by treating the porous surface with strong oxidizing agents [6]. Another common feature of the water isotherm is the hysteresis loop. Explanations for this remain somewhat speculative. The hysteresis may be attributed to differences in the micropore filling and emptying processes along adsorption and desorption [7]. It is believed that the extent of hysteresis depends on the pore width and that there is no hysteresis in extremely small micropore systems [8]. Since 1954 many models have been developed to describe water vapor adsorption on porous carbons. The earlier attempts to quantitatively describe adsorption are the semi-empirical models proposed by Dubinin and coworkers [7,9]. Based on simple kinetic theories of water adsorption onto hydrophilic groups, they assumed that primary adsorption centers (hydrophilic groups) in the carbon surface are the main cause for the starting up of the adsorption mechanism. Then, each adsorbed water molecule will constitute a secondary adsorption centre for other water molecules, thanks to the formation of hydrogen bonds. It has been shown by Stoeckli [10,11] that water adsorption isotherms, which fulfill the requirement for temperature invariance, can be described within the

qd qm S0 R T Vp wa wd x

total adsorbed water concentration during desorption (mol kg1) adsorption capacity (mol kg1) hydrophilic groups concentration (mol kg1) gas constant (J K1 mol1) absolute temperature (K) micropore volume per unit membrane/carbon mass (cm3 g1) adsorbed water concentration via hydrophilic groups during adsorption (mol kg1) adsorbed water concentration via hydrophilic groups during desorption (mol kg1) relative pressure

Supercript exp experimental

framework of DubininÕs theory. The fundamental relation is the Dubinin–Astakhov equation (DA equation), which exhibits zero slope at zero loading and an inflexion point. Although this model is able to describe type IV and type V isotherms quite well, it does not explain the mechanism for water adsorption and does not have the correct Henry law when the pressure is approaching zero. In 1996, Talu and Meunier [12], based on a thermodynamic approach, presented a new model formulation for describing the behavior displayed in Type V isotherms. Their three parameter isotherm involves the clustering of water molecules around chemisorption sites, but does not explain the role of the microscopic structure of carbons in the adsorption of water. Additionally, in the case when some uptake exists at low relative-pressure, the theory significantly underestimates the amount of adsorbed water [1]. Do and Do [13] proposed a model for water vapor adsorption by activated carbon adsorbents. This assumes the growth of the water cluster at the hydrophilic groups located in the mesopores, at the entrance of the micropores, and the penetration of released water clusters into the micropores in the form of pentamers. Recently, Rutherford [14] employed the cooperative multimolecular sorption (CMMS) theory, originally proposed by Malakhov and Volkov to explain the sorption in polymers [15], to characterize water adsorption equilibrium in carbon. The four parameters CMMS theory considers a unit triad involving primary sorption on a central site (oxygen functional group) and on two side positions. In addition, the model incorporates secondary interactions arising from this primary unit, which allow for the formation of dimers, trimers, etc. The CMMS equation

S. Lagorsse et al. / Carbon 43 (2005) 2769–2779

can account for type II, III and type V isotherm. The type V isotherm is obtained by assigning to zero the constant for sorption through the side interaction. Thus, the type V equation only considers water molecules bonded onto the oxygen surface groups. Although water adsorption on porous carbon has been studied for many years, the adsorption mechanism is not yet well established. Furthermore, very few studies of water adsorption on carbon molecular sieves particles have been mentioned in the literature. Alcanˇiz-Monge and Lozano-Castello´ [16] presented experimental data for water adsorption on several carbon molecular sieves, including Takeda 3A and 5A. They obtained type V isotherms for all samples. Recently, Rutherford and Coons [17] presented adsorption and desorption experimental data and adsorption kinetic studies also for Takeda carbon molecular sieves. The adsorption isotherms did not display hysteresis and revealed a reversible type III shape that was well described by the cooperative multimolecular sorption theory. To the authorsÕ knowledge, there are no experimental data on carbon molecular sieve membranes (CMSM) reported in literature, despite having been reported [18,19] that inadvertent exposure to water can drastically reduce CMSM gas separation ability. The objectives of this work are to provide water vapor adsorption data on carbon molecular sieve membranes (CMSM) and, starting from the Do model formulation for activated carbon, to propose a new model to describe the adsorption and desorption of water vapor on porous carbons with no meso/macroporosity, as is the case for CMSM.

2. Experimental The carbon molecular sieve membranes (CMSM) characterized and tested in this study were supplied by Carbon Membranes Ltd. in Israel (sample MS1) and by Blue Membranes GmbH in Germany (sample MS2). The production steps of both CMSM involve pyrolysis followed by chemical vapor deposition (CVD) and an activation step, which consists on a high temperature oxygen treatment. Both processes were extensively described elsewhere [20,21]. MS1 consists on carbon molecular sieve hollow fibers, having a uniform wall thickness of 9 lm and an external diameter of 170 lm. MS2 are flat membranes composed of a selective film supported on a macroporous ceramic layer. Its total thickness is around 100 lm, while the ultramicroporous carbon layer has an average thickness of 15 lm. A membrane sample designated by MS1-T600 is also characterized in this study. This sample results from subjecting sample MS1 to a heat treatment at 620 °C, in a tubular furnace under a hydrogen flow of 10 cm3 min1, for at least half an hour.

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By heating the carbon material in a reducing atmosphere, a partial removal of oxygen groups from the surface was achieved. Carbon dioxide adsorption equilibrium data at 301 K on samples MS1, MS1-T600 and MS2 were measured gravimetrically on a Rubotherm magnetic suspension balance having a 0.01 mg weighing resolution and a 0.02 mg reproducibility. This system consisted of a fully computerized magnetic suspension balance, which automatically measures the weight of the sample as function of time, at constant temperature and pressure. Sample MS1-T600, just after the heat treatment, was carefully transferred to the container of the magnetic balance, minimizing any exposure to air. Water adsorption data were obtained at 301 K for samples MS1, MS1-T600 and MS2 also using the same magnetic suspension balance. A tank filled with water vapor was used to control the water relative pressure on the environment surrounding the sample (see Fig. 1). A small amount of pure water was introduced in the tank, which was then evacuated for air removal, leaving a saturated water vapor atmosphere. The cylinder was connected to the sample chamber through a valve, which allowed for controlling the relative humidity of the sample environment (measured using a Druck absolute pressure transducer with a ±0.1% full scale precision). The entire system was fully thermostated. Water vapor isotherms were obtained by setting pressure intervals relative to the saturation vapor pressure. Prior to the measurements, the samples were outgassed until constant weight was achieved at 0.006 bar and 343 K. Separate experiments have been performed to account for the amount of water condensation on the metallic hanger and pan system that holds the sample. The mass of the system, without sample, was monitored for different values of the relative humidity. The mass change was found negligible when compared to that obtained when the sample was present.

3. Results and discussion 3.1. CMSM characterization The adsorption equilibrium isotherms for CO2 at 301 K up to 5 bar, on samples MS1, MS1-T600 and MS2, are represented graphically in Fig. 2. These are type I isotherms. The shape is similar for all samples. MS1 presents a slightly lower adsorption capacity. Note that, because MS2 comprises a ceramic macroporous support, its results are presented both per unit membrane mass (support and microporous carbon layer included) and per unit carbon mass alone. The mass fraction of the ceramic portion of the support was obtained elsewhere, by thermo-gravimetric (TG) analysis, as being 0.40.

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Fig. 1. Experimental set up for water vapor adsorption measurements.

5

4

q / mol kg-1

Table 1 CMSM structural parameters

MS2 (per unit membrane mass) MS2 (per unit carbon mass) MS1 MS1-T600 Dubinin-Radushkevich fitting equation

Sample

Dubinin approach

Nguyen approach

Vp (cm3 g1)

E0 (kJ mol1)

Micropore range (nm)

Average micropore size (nm)

0.23 0.24 0.15a 0.25b

32.8 32.6 30.6

0.31–0.84 0.63–0.86 0.48–0.84

0.725 0.736 0.753

3

MS1 MS1-T600 MS2

2

a

1

b

Micropore volume per unit carbon mass. Micropore volume per unit membrane mass.

0 0

1

2

3

4

5

P / bar

Fig. 2. Adsorption equilibrium isotherms of carbon dioxide on sample MS1 (m), MS1-T600 (.) and MS2 (d and s) at 301 K. The solid lines correspond to Dubinin–Radushkevich fitting equations.

samples. A narrow ultramicropore size distribution was obtained for all samples, as exemplified in Fig. 3 for MS2.

0.6

MS2 0.5

Relative frequency

The Dubinin–Radushkevich equation [22] was used to fit the data and employed to characterize the micropore volume, Vp, of the CMSM samples. The density value used for adsorbed CO2 was obtained from the approach suggested by Cazorla-Amaro´s et al. [23], that is, 0.86 g cm3 at 301 K (obtained by extrapolation). The values of the micropore volume obtained by this method are listed on the first column of Table 1. The pore size distribution (PSD) for each sample was obtained using the method proposed by Nguyen and Do [24,25] for the determination of micro- and mesopore size distribution in carbonaceous materials. Carbon dioxide adsorption equilibrium data were used for the calculation. The micropore size ranges are listed in Table 1, together with the average pore size, for all

0.4

0.3

0.2

0.1

0.0 0.0

0.2

0.4

0.6

0.8

Pore size / nm

Fig. 3. Micropore size distribution for sample MS2.

1.0

S. Lagorsse et al. / Carbon 43 (2005) 2769–2779 6 desorption adsorption Talu-Meunier equation DA equation CMMS equation

5

4 q / mol kg-1

The heat treatment in a reducing atmosphere at 620 °C did not affect significantly the micropore volume and the average pore size of sample MS1. This reducing treatment mimics the final production step used by the two companies that supplied the membranes used in this work (Carbon Membranes Ltd. and Blue Membranes GmbH). Samples MS1 and MS1-T600 should differ especially on the number and type of oxygen functional groups. At this temperature, some groups, such as carboxyl, may be removed from the surface, leaving some unstable sites that will re-adsorb oxygen at room temperature. On the other hand, other functional groups may be reduced.

3

2

1

0

0.0

The adsorption equilibrium data for water vapor at 301 K on MS1, MS1-T600 and MS2 are represented graphically in Figs. 4 and 5. The adsorption of water resulted on type V isotherms, with the uptake starting at P/Pv  0.3 for MS1 and MS1-T600 and at P/Pv  0.2 for MS2. It appears that a hysteresis loop is present on all of our samples. Besides that, all adsorption experiments were reversible at the experimental temperature. Even though water adsorption resulted in type V isotherms, the type V form of the DA equation was insufficient to describe experimental data at low relative pressures. To be able to overcome this problem, the DA equation, with type I and type V contributions, was used. However, as expected, this leads to one inflection point, at low relative pressure, characteristic of type IV isotherms, which does not apply to our case. In addition, the type V forms of the CMMS equation underestimated water adsorption at low relative pressures.

8 adsorption, MS1 desorption MS1 desorption MS1-T600 adsorption MS1-T600 DA equation CMMS equation Talu-Meunier equation

q /mol kg-1

4

2

0 0.0

0.2

0.2

0.4

0.6

0.8

1.0

P / Pv

3.2. Water vapor adsorption

6

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0.4

0.6

0.8

1.0

P / Pv

Fig. 4. Water vapor adsorption and desorption equilibrium data at 301 K on MS1 (d adsorption, s desorption) and on MS1-T600 (j adsorption, h desorption). The solid, dashed and dashed-dotted lines correspond to CMMS, DA (type I and type V) and Talu–Meunier equations, respectively.

Fig. 5. Water vapor adsorption and desorption equilibrium data, per unit membrane mass, at 301 K on MS2 (d adsorption, s desorption). The solid, dashed and dashed-dotted lines correspond to CMMS, DA (type I and type V) and Talu–Meunier equations, respectively.

However, in general, contrary to the complete form of Dubinin–Serpinsky equation (DS2) [9] (not shown in Figs. 4 and 5), the DA, CMMS and Talu-Meunier fitting equations were found to give quite good results in fitting the experimental adsorption data for samples MS1 and MS2. A less accurate description of adsorption experimental data was obtained for sample MS1-T600, which may arise from the scarce number of experimental data obtained. The amount of this sample was also lower than the other ones. The parameters obtained from these last three equations are listed in Table 2. Also given in this table, are the fitting parameters found from Dubinin–Serpinsky equation (DS1) [7]. This equation is only able to describe the initial region of the isotherm and is widely used to obtain an indication of primary adsorption content, i.e., the number of hydrophilic groups capable of forming hydrogen bonds with water molecules. As expected, removal of oxygen complexes from the carbon surface of sample MS1, resulted in the MS1T600 isotherm starting at higher pressure values. Note that the content of surface oxygen groups is not the only factor that affects the water isotherm shape. The shape of the isotherm is also a function of the micropore size. The pronounced type V uptake is shifted to lower values of relative pressure as the micropore size decreases. Estimation of micropore volumes using a water vapor adsorption isotherm together with the liquid water density value generally yields lower values than when adsorption data from other species is used (like nitrogen or carbon dioxide). In such a confined space, molecular packing is not as effective as bulk liquid water. Iiyama et al. [26] showed by X-ray diffraction technique in a slit shaped carbon nanospace that the adsorbed water has a more ordered structure. This is thought to be an ice-like structure, with a lower density than liquid water.

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Table 2 Parameters obtained from Dubinin–Serpinsky (DS1), Dubinin–Astakov (DA), Talu–Meunier and CMMS fitting equations Isotherm equation

Fitting parameters

Dubinin–Serpinsky (DS1) [7]

Sample

a0 (mol kg1)

MS1 MS1-T600 MS2

1.26 0.383 0.642

Sample

q0(I)/q0(V) (mol kg1)

E(I)/E(V) (kJ mol1)

n1/n2

MS1 MS1-T600 MS2

1.24/6.32 0.256/6.98 0.736/4.74

6.76/1.95 17.0/1.50 6.81/2.04

1.87/3.65 2.02/5.48 2.02/3.81

Sample

H (Pa kg mol1)

k (kg mol1)

qm (mol kg1)

MS1 MS1-T600 MS2

6.18 9.51 9.40

Sample

k0

MS1 MS1-T600 MS2

0.0689 0.0112 0.0574

q ¼ a0 c

h 1  ch

where h ¼ P =P v

Dubinin–Astakov (DA) [10] n1    A q ¼ q0 ðIÞ exp  EðIÞ n2    A þ q0 ðV Þ exp  EðV Þ

c 1.58 1.55 1.78

where A ¼ RT lnðP v =P Þ Talu–Meunier [12]

HW expðW=qm Þ P¼ 1 þ kW pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 1 þ 4kf where W ¼ 2k

3.48 4.25 5.15

8.04 8.71 5.73

and f ¼ qm  q=ðqm  qÞ Type V form of CMMS Theory [14]  k 0 ðP =P v Þ 1 1  k 1 ðP =P v Þ where w ¼ q ¼ qm 2 k 0 ðP =P v Þ þ w 2  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ð1  k 1 ðP =P v ÞÞ2 þ 4k 0 ðP =P v Þ

Alcanˇiz-Monge et al. [16,27] suggested, recently, that the density of adsorbed water was not a function of the fractional filling of the pore volume and that the most suitable value for water adsorbed in carbon micropores is 0.92 g cm3, which is very close to that of ice. The apparent adsorbed water densities, d aw , for samples MS1, MS1-T600 and MS2, calculated on the assumption that water completely fills the micropore volume, Vp, are shown in Table 3. The values obtained are smaller than 0.92 g cm3. Similar discrepancies have been noted in previous studies on microporous carbons [8,28]. As an example, the apparent densities of water, d aw , adsorbed on active carbon fibers with an average pore width of 0.75 nm (PIT-5) and 1.13 nm (PIT-20) [8], calculated following the same assumption, are also listed in Table 3. An explanation for these divergences may be based on CarrotÕs [29] suggestion that a minimum pore width is necessary for water vapor adsorption to occur, even though a water molecule can certainly penetrate con-

k1 2.07 1.74 2.14

qm (mol kg1) 8.22 7.59 5.80

Table 3 Micropore volume deduced from water vapor adsorption at 301 K Sample

1 qexp m (mol kg )

d aw (g cm3)

MS1 MS1-T600 MS2

7.2 7.8 9.2a 5.5b

0.57 0.58 0.66

Data from literature PIT-5 [8] PIT-20 [8] a b

0.86 0.81

Micropore volume per unit carbon mass. Micropore volume per unit membrane mass.

strictions much narrower than this (the kinetic diameter of water is 0.28 nm). This follows from the notion that a minimum width is required in order for a stable hydrogen-bonded structure (cluster) to form. The micropore size distribution measured in the present work indeed indicate sizes in the ultramicropore region, that is, some pores are too small for water cluster adsorption.

S. Lagorsse et al. / Carbon 43 (2005) 2769–2779

Even though a quite good fit of the experimental data is possible using the previously mentioned semi-empirical equations, an understanding of the water adsorption mechanism on carbon micropores is still lacking. In the next section, starting from the basic formulation of Do and Do [13] for water adsorption on activated carbons, a new model is presented for water adsorption on carbon micropores. A mechanism for water desorption is also discussed. 4. Water adsorption/desorption model on carbon micropores The model developed in this paper assumes that adsorption into micropores involves three consecutive steps, which are compatible with the widely accepted generalization of the mechanism of water adsorption on porous carbons [1]: (a) water molecules are adsorbed on hydrophilic groups via hydrogen-bonding. The number of water molecules directly attached to a group depends on its specific nature, (b) water clusters grow around hydrophilic groups until a critical size, m (number of water molecules), is reached, (c) cluster entities, composed of m water molecules, have enough dispersion energy to be released from hydrophilic groups and to be adsorbed on the hydrophobic carbon surface, as if they were a single molecule. From in situ small-angle X-ray scattering studies, Iiyama et al. [30] suggested that water would adsorb in carbon micropores when a cluster over 0.38 nm in radius is formed. This cluster size corresponds to an assembly comprising 7 molecules, assuming spherical geometry for clusters and using an adsorbed water density of 0.92 g cm3. Vagner et al. [31], by introducing a variable cluster size on a modified Do and Do equation [13], obtained values from 7 to 10 for the average value of water molecules on a cluster adsorbed on activated carbon micropores. In this paper, one will consider that a cluster must reach a critical number at m = 7 of water molecules in order to be able to adsorb by dispersion forces in the hydrophobic carbon nanospace. The following mass action law describe the adsorption of a water molecule, A, onto the hydrophilic group S, K1

A þ S  ½ASl

ð1Þ

Then, successive adsorption takes place, either directly on the hydrophilic group or on the adsorbed water, which constitutes a secondary adsorption site capable of forming hydrogen bonds with other water molecules,

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K1

A þ ½An1 Sl  ½An Sl

for n ¼ 2; 3; . . . ; 7

ð2Þ

Note that, considering the similar nature of the physical bonds involved (hydrogen bonding), one considers that the equilibrium constants corresponding to Eqs. (1) and (2) (formation of hydrophilic group/water and water/water bonds, respectively) are equivalent, i.e. equal to K1. It is assumed that adsorption within the hydrophobic micropore walls can only be attained by a cluster of 7 water molecules. The release of a heptamer from a hydrophilic site and its adsorption, via dispersion interactions onto the hydrophobic pore walls, is then given by: K2

½A7 Sl  S þ ½A7 l

ð3Þ

The equilibrium constants for adsorptions associated to hydrophilic groups, corresponding to Eqs. (1) and (2), respectively, are: ½AS ½SP ½An S K1 ¼ ½An1 SP

ð4Þ

K1 ¼

for n ¼ 2; 3; . . . ; 7

ð5Þ

where P is the water vapor pressure. The hydrophilic groups can exist in free form or with one or more water molecules attached. Let S0 be the number of hydrophilic groups, its free site concentration is obtained from S 0 ¼ ½S þ

7 X ½An S

ð6Þ

n¼1

or, substituting Eqs. (4) and (5) into (6), ! 7 X n1 n K1 P S 0 ¼ ½S 1 þ K 1

ð7Þ

n¼1

Combining Eqs. (4), (5) and (7) the concentration of water adsorbed via hydrophilic groups, wa, is wa ¼

7 X

n½An S

n¼1

¼ S0

1 þ K1

K1 P7

n1 n n¼1 K 1 P

7 X

nK 1n1 P n

ð8Þ

n¼1

At a given time, the available capacity for water adsorption is obtained subtracting from the total capacity, qm, the amount of water adsorbed in the micropores, both in hydrophilic groups and within the hydrophobic space: qm  wa  7[A7]. Assuming that all this ‘‘space’’ is available for adsorption of the heptamer clusters, then one can write the equilibrium equation corresponding to Eq. (3) as: K2 ¼

½A7 S

½A7 ½S q w  a m  ½A7  7

ð9Þ

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Combining Eqs. (4), (5) and (7) and substituting into Eq. (9), the concentration of water adsorbed via dispersion forces, ca, is ca ¼ 7½A7  ¼

K 2 ðK 1 P Þ7 1 þ K 2 ðK 1 P Þ

ðqm  wa Þ 7

ð10Þ

Finally, the total adsorbed water concentration is qa ¼ wa þ ca P7 n n n¼1 nK 1 P ¼ S0 P7 1 þ n¼1 K n1 P n þ

K 2 ðK 1 P Þ7 1 þ K 2 ðK 1 P Þ

7

P7 qm  S 0

1

n n n¼1 nK 1 P P7 þ n¼1 K n1 P n

! ð11Þ

The model for water desorption from the micropores follows a different mechanism, composed of two sequential steps. First, heptamer clusters adsorbed onto the hydrophobic space detach and dissociate into separate molecules, K3

½A7 l  7A

ð12Þ

Then the remaining water clusters attached to hydrophilic groups by hydrogen bonding are desorbed, K4

½An Sl  A þ ½An1 S

for n ¼ 2; . . . ; 7

K4

½ASl  A þ ½S

ð13Þ ð14Þ

Considering the equilibrium constant for the desorption of heptamer clusters (Eq. (12)) together with the available adsorption capacity, one has q  w d K 3 ½A7  ¼ P 7 m  ½A7  ð15Þ 7 where wd is the concentration of water adsorbed via hydrogen bonds on hydrophilic groups. The equation above gives the concentration of water adsorbed via dispersion forces as a function of pressure during desorption, P7 cd ¼ 7½A7  ¼ ðqm  wd Þ ð16Þ K3 þ P 7 Considering the similar nature of the physical bonds involved (hydrogen bonding), one will consider that the equilibrium constants corresponding to Eqs. (13) and (14) are equivalent, that is, equal to K4. The equilibrium relation obtained is, n 1 ½An S ¼ K 1 4 P ½An1 S ¼ ðK 4 P Þ ½S

The total adsorbed water concentration during desorption is given by, q d ¼ cd þ w d

! P7 n 7 1 K 1 P nðK P Þ 3 4 n¼1 ¼ qm  S 0 P n 7 1 þ K 1 1 þ 7n¼1 nðK 1 3 P 4 PÞ P7 n 1 n¼1 nðK 4 P Þ þ S0 P 7 n 1 þ n¼1 nðK 1 4 PÞ

ð19Þ

Comparing Eqs. (1), (2), (13) and (14), the equilibrium constant K4 are simply the inverse of equilibrium constants K1. Rewriting Eqs. (11) and (19) in terms of relative pressure x, that is P = xPv, one obtains two isotherm equations describing water adsorption and desorption as a function of the relative pressure, P7 n 7 K a ðKxÞ n¼1 nðKxÞ Adsorption : qa ¼ S 0 þ P7 n 7 1 þ n¼1 ðKxÞ 1 þ K a ðKxÞ ! P7 n n¼1 nðKxÞ  qm  S 0 ð20Þ P7 n 1 þ n¼1 ðKxÞ ! P7 n K d x7 n¼1 nðKxÞ q  S0 Desorption : qd ¼ P7 n 1 þ K d x7 m 1 þ n¼1 nðKxÞ P7 n n¼1 nðKxÞ þ S0 ð21Þ P 7 n 1 þ n¼1 nðKxÞ where Ka = K2, K = PvK1 and K d ¼ P 7v K 1 3 . Note that isotherm equations (20) and (21) are thermodynamically consistent, giving the correct Henry law limit: qjx!0  S0Kx. The parameters in the above model equations are listed in Table 4. For each adsorption and desorption branch, there are four parameters, three of which are identical for adsorption and desorption. The adsorption isotherm equation (20) can describe type IV or type V equilibrium behaviors. The adsorption capacity, qm, can be found from the adsorption data in the neighborhood of x = 1. Thus, when S0, is known, one uses the full equations (20) and (21) to fit adsorption and desorption data and find Ka, K and Kd. In order to optimize the fitting when S0 is not known, one fits the first term of the right-hand side of Eq. (20) in the range P/Pv < 0.3, obtaining K and S0. Then one uses the full equation to fit adsorption data obtaining Ka. It is, however, important to note that when the first region of the isotherm does not have a

ð17Þ

where the concentration of free hydrophilic groups, [S], is given by Eq. (6). The concentration of water adsorbed via hydrogen bonds on hydrophilic groups is, P7 n 7 1 X n¼1 nðK 4 P Þ wd ¼ n½An S ¼ S 0 ð18Þ P7 n 1 þ n¼1 ðK 1 n¼1 4 PÞ

Table 4 Parameters of the model equations (20) and (21)

Adsorption/desorption constant on the hydrophobic space Hydrophilic groups concentration Water adsorption capacity Hydrogen bonding equilibrium constant

Adsorption

Desorption

Ka

Kd S0 qm K

S. Lagorsse et al. / Carbon 43 (2005) 2769–2779

The fitting curves are shown in Fig. 6 together with the contribution from the first and second term of the right hand-side of Eq. (20). A good description of adsorption and desorption experimental data was obtained for sample MS1 and MS2, but becomes less accurate for sample 8 adsorption, MS1 desorption, MS1 fitting model equations (20)and (21) st 1 term of equation (20) nd 2 term of equation (20)

q / mol kg-1

6

4

2

0 0.0

0.2

0.4

0.8

1.0

desorption, MS1-T600 adsorption, MS1-T600 fitting model equation (20) and (21) st 1 term of equation (20) nd 2 term of equation (20)

8

q / mol kg-1

0.6

P / Pv

a

6

4

2

0 0.0

0.2

0.4

0.6

0.8

1.0

P / Pv

b 6

dessorption, MS2 adsorption, MS2 fitting model equations (20) and (21) st 1 term of equation (20) nd 2 term of equation (20)

5

q / mol kg-1

pronounced type I form, the value of S0 found by fitting Eq. (20), or even using the DA equation, will not be very accurate. This happens because there is not one unique set of parameters (Na0(I), E(I), n1 for DA equation and S0 and K for Eq. (20)) that fits the experimental data well. The hydrophilic groupÕs concentration, S0 must preferentially be obtained from a direct analysis. It is important to clarify the main differences between the model developed here for microporous carbon materials and the theory developed by Do and Do for activated carbons [13]. Do and Do suggest that water adsorbs as clusters onto hydrophilic groups located on the materialÕs mesopores; a 1:1 ratio exists between the number of hydrophilic groups and water molecules attached directly to them. It is assumed that the equilibrium constant for the hydrogen bonding between a water molecule and a hydrophilic group ðK h1 Þ is different from the equilibrium constant for the hydrogen bonding among water molecules ðK w1 Þ, and generally K w1 6 K h1 . In addition, they consider that the equilibrium constant K w1 times the concentration of free water is equal to its reduced pressure, that is, K w1 P ¼ x. These authors also consider that a pentamer cluster can be released from the hydrophilic groups and then penetrate into the micropores, where no hydrophilic groups are assumed to be present. The resulting adsorption isotherm equation has four parameters plus the number of molecules in the clusters bonded to hydrophilic groups. On the other hand, the theory developed in this work for water adsorption on microporous carbons, where no mesoporous volume is present, considers that water is first adsorbed on hydrophilic groups, located in the micropore space, until a critical cluster size of seven molecules is reached. Then a heptamer cluster can be detached to be adsorbed, as a single entity, in the hydrophobic micropore space. The equilibrium constants resulting from water/water and water/hydrophilic group hydrogen bonding are assumed to be equal. The ratio between the number of hydrophilic groups and water molecules directly attached to them is not necessarily 1:1. One also complemented this theory describing the mechanism for micropore emptying, which yields the hysteresis branch in the desorption isotherm. The heptamer clusters adsorbed through dispersion forces are the first to be released from the micropores. Then the clusters with size m 6 7, which are attached to hydrophilic groups by hydrogen bonding, are desorbed. One obtains two adsorption/desorption isotherm equations with four parameters. The model developed here was tested using experimental data obtained for MS1, MS1-T600 and MS2. To fit the data, the values for the adsorption capacity were obtained experimentally. The analytical determination of the number and nature of hydrophilic groups was beyond the scope of this paper. Thus, three fitting parameters were allowed for Eq. (20).

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4

3

2

1

0 0.0

c

0.2

0.4

0.6

0.8

1.0

P / Pv

Fig. 6. Water vapor adsorption (d) and desorption (s) equilibrium data at 301 K on (a) sample MS1, (b) sample MS1-T600 and (c) sample MS2. The solid lines corresponds to fitting Eqs. (20) and (21). The dashed and dashed-dotted lines correspond to the first and second term of the right side-hand of Eq. (20), respectively.

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S. Lagorsse et al. / Carbon 43 (2005) 2769–2779

Table 5 Parameters obtained from model fitting equations (20) and (21) Fitting equations (20) and (21)

MS1 MS1-T600 MS2

qm (mol kg1)

S0 (mol kg1)

K

Ka

Kd

7.2 7.8 5.5

0.46 0.16 0.38

2.39 2.64 1.99

0.23 0.04 1.03

183 72.6 268

MS1-T600. The same problem was observed in Fig. 4 using preceding fitting equations (listed in Table 2). The fitting parameters obtained are shown in Table 5. Comparing the results obtained for sample MS1 and MS1-T600, one concludes that the heat treatment has removed, as expected, some functional groups from the surface, as is revealed by the lower S0 obtained for sample MS1-T600. Parameter Ka, which reflects how easily a heptamer cluster is released from a hydrophilic site and adsorbs onto the hydrophobic pore walls, is quite low for sample MS1-T600. One suggests that during the hydrogen heat treatment some hydrophilic groups were reduced (for example, a carbonyl could be reduced to a hydroxyl group) and transformed into groups with higher polarity, making it more difficult for a heptamer to be released. The formation of stronger hydrophilic groups is also reflected on the higher constant K obtained for sample MS1-T600. Note that K d / K 1 3 , that is, a lower Kd implies an easier desorption of clusters from the hydrophobic space. One suggests that this parameter is a function of the micropore size and shape and the number of constrictions throughout the pore structure, which affects the stability of hydrogen-bonded clusters. The lower value of Kd was obtained for sample MS1-T600, which is the only sample that, apparently, does not present constrictions in the range 0.31–0.63 nm and that presents the higher pore size limit range. The determination of the number and nature of oxygen functional groups present on the CMSM surface constitute an interesting future work, which will allow for the accurate estimate of parameters K, Ka and Kd.

5. Conclusion Water vapor adsorption on two different carbon molecular sieve membranes (MS1 and MS2) was studied. In addition, a sample MS1-T600, resulting from a heat treated (at 620 °C in hydrogen) MS1 sample, was also studied. For all samples, water adsorption isotherms displayed type V behavior with evidence of desorption hysteresis. From the structural characterization of the three samples, one found similar carbon dioxide isotherm and micropore volumes. A narrow micropore size distribution below 0.86 nm, was found for all samples. These differ specially on the lower limit

of the micropore range and on the number of surface oxygen functional groups. Removal of some oxygen functional groups from sample MS1 (leading to sample MS1-T600), caused the apparent elimination of some constrictions. One suggests the existence of surface functional groups contributes to the existence of constrictions generally found on CMSM. These have an important role on the kinetic and, consequently, on the permeability of a CMSM membrane [20,21]. An isotherm equation for water adsorption in microporous carbons was derived from the basic formulation of the Do model [13], originally developed for activated carbons. The model developed assumes that water clusters grow around hydrophilic groups, via hydrogenbonding, until a critical size, m = 7 (number of water molecules), is reached. Then, a cluster has enough dispersion energy to be released from the hydrophilic group and to be adsorbed on the hydrophobic carbon surface, as if it were a single entity. A mechanism for water desorption was also proposed: first, heptamer clusters adsorbed onto the hydrophobic space detach and dissociate into separate water molecules. Finally, the remaining water clusters, attached to hydrophilic groups by hydrogen bonding, are desorbed. There are four parameters for each adsorption and desorption isotherm equation, three of which are identical for adsorption and desorption. A good description of water adsorption experimental data was obtained. Other isotherm equations known from the literature and applicable to carbon micropores were also discussed and compared. The physical interpretation of the fitting parameters K, Ka and Kd, obtained using the model developed in this paper, was in agreement with the structural characterization of the membranes. The higher K value (equilibrium constant for hydrogen bond formation onto a hydrophilic group) and the lower Ka value (equilibrium constant for the release of a heptamer cluster onto the hydrophobic space) were found for the sample that probably contains the stronger hydrophilic functional groups. The lower value of Kd (equilibrium constant associated to the detachment of the clusters from the hydrophobic space) corresponded to the only sample that did not present pores in the range 0.31–0.63 nm. When one considers the pore size distributions of these materials, it is seen that a fraction of the pore volume will not be accessible to the cluster entities, as described in the model as condition for adsorption in the hydrophobic space. Even though this fraction cannot be quantified precisely, it may be sufficient to account for the low values of apparent water densities computed.

Acknowledgments The authors gratefully acknowledge the financial support from European Growth Project GRD1-2001-

S. Lagorsse et al. / Carbon 43 (2005) 2769–2779

40257-Spec Sep and FCT research project POCTI/ EQU/34224/2000.

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