Adsorption of light alkanes and alkenes onto single-walled carbon nanotube bundles: Langmuirian analysis and molecular simulations

Colloids and Surfaces A: Physicochem. Eng. Aspects 357 (2010) 43–52

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Adsorption of light alkanes and alkenes onto single-walled carbon nanotube bundles: Langmuirian analysis and molecular simulations Fernando J.A.L. Cruz, Isabel A.A.C. Esteves, José P.B. Mota ∗ Requimte/CQFB, Departamento de Química, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829-516 Caparica, Portugal

a r t i c l e

i n f o

Article history: Received 26 June 2009 Received in revised form 31 August 2009 Accepted 2 September 2009 Available online 8 September 2009 Keywords: Single-walled carbon nanotubes Alkanes Alkenes Adsorption Molecular simulation Henry constant

a b s t r a c t Grand canonical Monte Carlo (GCMC) simulations are employed to study the adsorption equilibrium properties of methane, ethane, ethylene, propane, and propylene onto homogeneous bundles of single-walled carbon nanotubes, at room temperature, from 10−4 bar up to 90% vapor pressure. Individual adsorption isotherms for the internal volume of a bundle and for its external adsorption sites are separately calculated for individual nanotube diameters in the range 11.0 Å ≤ D ≤ 18.1 Å. External adsorption is further decomposed into the contributions from its two main adsorption sites – external grooves and exposed surfaces of the peripheral tubes – based on a geometrical model for the average groove volume that takes into account the molecular nature of the adsorbate. Both intrabundle confinement and adsorption onto the grooves lead to type I isotherms, which are modeled with Langmuirian-type equations. Adsorption on the exposed surfaces of the peripheral tubes in a bundle gives rise to a type II isotherm, which is described by the BET model with a finite number of adsorbed layers. The linear combination of the Langmuir isotherm model for adsorption onto groove sites and the BET isotherm model produces a composite isotherm that is in good agreement with the GCMC isotherm for overall adsorption onto the external sites of a bundle. The influence of adsorbate molecular length and existence of an unsaturated chemical bond in its molecular skeleton are studied by monitoring the dependence of the Henry constant and zero-coverage isosteric heat of adsorption with the dispersive energy for the solid–fluid pair potential of each adsorbate. Our results show that the adsorptive properties are especially influenced by the presence of a double bond in the case of small molecules, such as the ethane/ethylene pair. © 2009 Elsevier B.V. All rights reserved.

1. Introduction Carbon nanotubes are a fascinating new class of nanomaterials, which have been the subject of much interest ever since their first laboratory preparation in the early 90s [1–3]. Their structure bears resemblance to a circularly folded graphene sheet, or several of them aligned in a concentric way, giving rise respectively to single- (SWCNTs) or multi-walled carbon nanotubes (MWCNTs). SWCNTs are usually produced via electric-arc discharge (EA) or chemical vapor deposition (CVD) techniques [4,5], and their physico-chemical properties, such as pore diameter [6] and chemical purity [7,8], can be engineered according to specific needs. Exhibiting a cylindrical geometry, SWCNTs can be regarded as model nanopores, and as such they have been receiving considerable attention due to their unique and exciting features, such as optical [9] and electronic [10] properties. Amongst other applications, SWCNTs have been proposed as storage nanomaterials for hydrogen and methane [11]. The latter adsorptive is particularly

∗ Corresponding author. Tel.: +35 1212948385; fax: +351 212948385. E-mail address: [email protected] (J.P.B. Mota). 0927-7757/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2009.09.002

relevant for industrial applications, for it is the major component of natural gas. Natural gas is commonly considered as a suitable, nonpolluting, energy source for the future [12], whose typical feedstock is a gaseous mixture composed mainly of CH4 (70–90 wt.%) and other light organics such as C2 H6 (5–15 wt.%) and C3 H8 (<5 wt.%) [13]. SWCNTs are usually produced as a collection of individual tubes aggregated into more or less complicated heterogeneous bundles, due to the strong van der Waals interactions [14] holding together the individual tubes. When SWCNTs exist in the form of bundles, these multi-tubular structures exhibit a high adsorption area with various types of adsorption sites (Fig. 1), both within the interior porous volume (intratubular and interstitial adsorption) and external surface (grooves and exposed surfaces of peripheral tubes) [15,16]. Depending on the individual tube diameter, arrays of grooves and interstitial sites are suitable candidates to study the realization of matter in one dimension [17]. As far as adsorption onto individual SWCNTs and SWCNT bundles is concerned, methane has been one of the most commonly employed probes [17–24], because it is a simple, quasi-spherical molecule, exhibiting small deviations from ideality, and whose physico-chemical properties are well known over a wide range of temperature and pressure conditions. Nonetheless, the adsorption

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isotherm into contributions from groups of different adsorption sites, which are characterized by the same specific nature of the solid–fluid interaction potential thus giving rise to the same isotherm shape. Moreover, we systematically explore fluid adsorption at very low pressures, thus producing a microscopic picture of the adsorption phenomena involved in the Henry’s law region. In the next section we introduce the intermolecular potentials employed in the calculations as well as the molecular simulation methodology. Subsequently, the main results are presented and discussed in light of an average nanotube diameter extracted from the experimental tube diameter distribution for a wellcharacterized SWCNT sample, and finally conclusions are drawn. 2. Molecular simulations

Fig. 1. Different adsorption sites in a homogeneous bundle of SWCNTs with tube diameter D: (1) intratubular, (2) interstitial channel, (3) external groove, and (4) exposed surface of peripheral tube. Sites 1 and 2 comprise the internal porous volume of the bundle, whereas sites 3 and 4 are both located on the external surface of the bundle.

properties of heavier organic fluids have also been investigated, such as, e.g., ethane [18,20,22,24–27] and propane [18,22,24,26,28]. A somewhat different situation occurs if the molecular complexity of the adsorbate is slightly increased by, e.g., the existence of a ␲ chemical bond in the carbon skeleton. The presence of an unsaturated bond makes the molecule more rigid and, due to its associated quadrupole moment, energetically different from its saturated analogue. Thus, it should respond differently when the adsorptive molecule is put into contact with a solid surface. Ethylene and propylene are the lightest unsaturated organics, but their adsorption onto SWCNTs remains largely unexplored. Recently, Cruz and Müller [27] presented molecular simulation results for intratubular adsorption of ethane and ethylene onto ideal SWCNTs. Using an atomistic intermolecular potential and an united-atom parameterization, these authors concluded that both potential models agree reasonably well, except in the very low pressure region where quantitative differences were observed. The results obtained with both intermolecular potentials showed that ethane is preferentially adsorbed at low pressures, and this finding was demonstrated by comparing the isosteric heats of adsorption calculated for different bulk pressures. However, a crossover in the adsorption isotherms was observed around 3 < p < 6 bar, above which ethylene adsorption becomes dominant over its saturated counterpart. Jacobtorweihen and Keil [26] carried out grand canonical Monte Carlo (GCMC) simulations using a united-atom description of the fluids and, not only corroborated the findings of Cruz and Müller [27], but also showed that a similar crossover behavior occurs for the pair C3 H8 /C3 H6 . Recently, our group presented detailed gravimetric measurements of the pure-component adsorption equilibria of saturated and unsaturated light organics (C1 –C3 ) onto high-purity, electricarc SWCNT bundles [29], spanning reduced temperatures between 0.8 and 1.6, and pressures from 200 Pa up to 8 MPa. However, due to intrinsic experimental constraints with our vacuum system, very low adsorbate uptakes were not explored. The measured adsorption isotherms were modeled as resulting from the contributions of two distinct zones of the bundle: intrabundle confinement and external adsorption. Using the GCMC technique as a numerical experiment, in the present work we explore the issue of splitting the adsorption

The force field adopted for the adsorbates under study – methane [CH4 (sp3 )], ethane [CH3 (sp3 )–CH3 (sp3 )], ethylene [CH2 (sp2 ) CH2 (sp2 )], propane [CH3 (sp3 )–CH2 (sp3 )–CH3 (sp3 )], and propylene [CH2 (sp2 ) CH(sp2 )–CH3(sp3 )] – is the transferable potential for phase equilibria (TraPPE) [30,31]. This force field is based on a united-atom (UA) model where the CH4 (sp3 ), CH3 (sp3 ), CH2 (sp3 ), CH2 (sp2 ), and CH(sp2 ) groups are treated as single interaction sites. The nonbonded interactions between pseudo-atoms in different adsorbate molecules, as well as the interactions between carbon atoms of a nanotube and pseudo-atoms of an adsorbate molecule, are governed by the Lennard–Jones (LJ) 12–6 potential,



u(rij ) = 4εij

ij rij

12

 −

ij rij

6 

,

(1)

where rij is the intermolecular distance between sites i and j. The potential well depths, εi /kB (kB is the Boltzmann constant), and collision diameters,  i , are given in Table 1. The cross terms were determined using the classical Lorenz–Berthelot combining rules [32]: εij = (εi εj )1/2 and  ij = ( i +  j )/2. A spherical potential truncation for pairs of pseudo-atoms separated by more than 14 Å was enforced [30], but analytical tail corrections were not applied. In the TraPPE-UA force field all bond lengths are fixed; the length of the CHx –CHy bond is 1.54 Å, whereas that of the CHx CHy bond is 1.33 Å. The harmonic bond-bending potential, ubend (), along the three pseudo-atoms of either propane or propylene is given by ubend () = k ( −  0 )2 /2. For propane, the force constant is k /kB = 62500 K/rad2 and the equilibrium bending angle is  0 = 114.0◦ ; the corresponding values for propylene are k /kB = 70420 K/rad2 and  0 = 119.7◦ . Fig. 2 shows typical examples of the unit simulation cells employed to study adsorption in the internal pore volume of the bundle and onto its external surface. The simulation box depicted in Fig. 2a is for intrabundle adsorption, whereas that shown in Fig. 2b is for adsorption over the external surface of the bundle. It is worth noting that only the grayed volume delimited by the black, solid lines represents effective volume probed during the simulation; thus, the simulation box of Fig. 2a is a parallelepiped, whereas the one in Fig. 2b is obtained by subtracting two halves of a cylinder Table 1 Lennard–Jones potential parameters employed in the simulations: TraPPe-UA force field [30,31] for the fluids and Steele’s parameterization for the carbon atoms on the SWCNT [33–35]. Pseudo-atom

εi /kB (K)

 i (Å)

C (SWNT) CH4 (sp3 ) CH3 (sp3 ) CH2 (sp3 ) CH2 (sp2 ) CH (sp2 )

28.0 148.0 98.0 46.0 85.0 47.0

3.400 3.730 3.750 3.950 3.675 3.730

F.J.A.L. Cruz et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 357 (2010) 43–52

Fig. 2. Examples of typical simulation cells for GCMC study of fluid adsorption onto different adsorption sites of a homogeneous bundle of open-ended SWCNTs: (a) intrabundle volume (intratubular volume and interstitial channels); (b) external surface (grooves and exposed surfaces of the peripheral tubes). The gray volume delimited by the black lines represents the effective volume of the simulation cell probed during the simulation.

from the bottom volume of the parallelepiped. In all cases, the nanotubes are arranged in the usual close-packed hexagonal lattice, with intertubular distance fixed at 3.4 Å to mimic SWCNTs adhering to each other via van der Waals forces forming bundles. The size of the simulation box for intratubular adsorption was adjusted by replicating the unit cell along the x and y coordinates, as many times as necessary. The size of the simulation box for external adsorption was adjusted by replicating the unit cell along the x coordinate. The faces of each simulation box implement periodic

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boundary conditions, except for the top and bottom faces of the box in Fig. 2b; the former is a reflecting wall, whereas the latter is blocked by the peripheral shell of nanotubes in the bundle. It is worth noting that the nanotubes depicted in Fig. 2b are not part of the simulation box and, therefore, molecules are not allowed to adsorb inside of them. We have explored the alternative of making the outermost shell of nanotubes accessible for internal adsorption and then correcting the final results by subtracting the average intratube adsorption in those nanotubes from the total number of adsorbed molecules. The comparison of both methods showed that external adsorption is not affected by intratube adsorption in the peripheral shell of nanotubes. At ambient temperature the SWCNTs can be approximated as smooth structureless nanocylinders; this simplifying assumption has been adopted in the present work. In this case, the SWCNTfluid potential, Usf (ı; D), for a nanotube of diameter D interacting with a pseudo-atom of an adsorbate molecule located at a nearest distance ı from the central axis of the nanotube, can be obtained analytically by integrating the LJ solid–fluid potential, usf (r), over the positions of all wall atoms of the nanotube (whose length is assumed to be infinite) [36]. Given that the SWCNT–fluid potential is a short-ranged potential function, the interactions between an adsorbate molecule and the nanotubes of a bundle may be restricted to the nearest neighbors only. For example, to calculate the overall solid–fluid potential for a pseudo-atom located inside a nanotube of the bundle, it suffices to sum the interactions with the confining tube and with its six nearest neighbors. Likewise, to determine the overall potential between a pseudo-atom and the peripheral nanotubes of the bundle it suffices to consider the interactions between its five nearest nanotubes (three on the outermost shell and two on the second shell). In both cases, including farther nanotubes has a minimum impact on the total solid–fluid interaction potential. To enhance the sampling of configurational space and increase the acceptance rate of the molecule insertion or removal step for the largest adsorbates (e.g., propane and propylene), we resorted to configurational-bias sampling techniques [37–40]. In the configurational-bias method a flexible molecule is grown atomby-atom toward energetically favorable conformations, leading to a scheme which is orders of magnitude more efficient than the traditional method of random growth. Each run was equilibrated for at least 2 × 104 Monte Carlo cycles followed by at least an equal number of cycles for the production period. Each cycle consisted of 0.8N attempts to translate a randomly selected molecule, 0.2N trial rotations, 0.2N attempts to change the conformation of a molecule using configurational-bias partial regrowth, and max(20, 0.2N) molecule insertion–deletion steps. Here, N is the number of molecules in the simulation box at the beginning of each cycle. The maximum displacement for translation and angle for rotation are adjusted during the equilibration phase to give a 50% acceptance rate. Standard deviations of the ensemble averages were computed by breaking the production run into five blocks. The imposed fugacity of the coexisting bulk fluid was converted into pressure by the Peng–Robinson equation of state with parameters taken from ref. [41].

3. Results and discussion GCMC simulations were run for adsorption at room temperature (T = 298.15 K) and pressures ranging from p = 10−4 bar up to 0.9p0 , where p0 is the vapor pressure of the adsorptive at the working temperature. In order to make a clearer comparison of the results obtained for the various adsorbates, all data are reported as a function of reduced pressure, p/p0 . Some of the simulations were carried out above the critical temperature Tc of the adsorbate, i.e.

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for Tr > 1, where Tr = T/Tc (e.g., for methane Tr = 1.59, and for ethylene Tr = 1.06). Under these conditions p0 is at best an ill-defined quantity and must be replaced by a pseudo value. This issue has already been discussed elsewhere [29,42]. The contributions, qv and qs , from internal (intratubular and interstitial) and external (groove sites and outer surfaces) adsorption in the bundle, were explicitly calculated for every individual nanotube diameter considered; note that qv is an adsorption isotherm expressed in units of (g adsorbate)/(g bundle), whereas the external component qs is expressed as (g adsorbate)/(m2 bundle). It should be noted that, because the external surface of the bundle is an open surface, qs represents an excess adsorption isotherm, calculated as qs = (N − g V)/A, where N is the ensemble average of the number of molecules in the simulation box, g is the number density of gaseous adsorptive at the corresponding pressure and temperature conditions, V is the volume of the simulation box (gray volume in Fig. 2b) and A is the area of its top rectangular face. According to this convention, the external surface area of the bundle is equal to the lateral surface area of the prism with apexes located at the central axes of the peripheral shell of nanotubes [29]. This is an unambiguous definition of external surface area, but it does not take into account the curvature of the nanotube wall; a more explicit definition of external surface area is proposed below. The calculated values of qv and qs are shown in Figs. 3 and 4, respectively, where the inset at the bottom right of each graph shows a magnification of the low pressure region. From the inspection of the intrabundle adsorption data (Fig. 3), it can be observed that adsorption varies markedly with nanotube diameter, D, and that for the same value of D it increases with the number of carbon atoms of the adsorbate. At very low pressures, energetic interactions between the fluid and the solid walls are the dominant factors ruling over adsorption, and thus, under those conditions, molecules are preferentially adsorbed onto the narrower nanotubes: a molecule confined inside a 11.0 Å nanotube is subject to an effective potential field with a larger well depth than in a 18.1 Å tube, and therefore the van der Waals dispersive forces are higher in the former case. Interstitial confinement, on the other hand, in most practical cases accounts negligibly for adsorption inside a bundle. Cruz and Mota [36] have shown that, in the diameter range analyzed in the present work, the inter-tubular volume is only accessible to CH4 and C2 molecules when D is larger than 15.8 Å, and even then it does not account for more than 25% of total intrabundle adsorption. For C3 H8 and C3 H6 interstitial confinement is even more inhibited; it only occurs in bundles with D = 18.1 Å, where it never represents more than 15% of total adsorption. As the pressure is increased, packing efficiency and entropic effects start to play a major role, which is the reason why above a certain value of p/p0 the amount of adsorbed fluid increases directly with D, suggesting a situation of pore filling. The threshold pressure value depends on the adsorbate; it is lowest for C3 H8 (p/p0 = 0.0015), intermediate for C2 H6 (p/p0 = 0.005), and highest for CH4 (p/p0 = 0.035). After the pronounced concave shape in the medium pressure region, the isotherms flatten and approach a constant plateau. For all studied values of D, a type I adsorption isotherm is observed, as defined by the IUPAC classification [43]. In previous simulation work with methane, ethane, and propane, either in individual tubes (D = 14.89 Å) [18] or in the interior volume of homogeneous bundles (D = 13.56 Å) [22], a similar shape was reported for the adsorption isotherms of those fluids. When adsorption onto the external sites of a bundle is analyzed (Fig. 4), the very low pressure data show that adsorption varies with individual tube diameter. It has been previously suggested [45] that the total volume associated with groove sites scales linearly with D, and that therefore a bundle composed of larger nanotubes exhibits a larger coverage. This observation has been subsequently validated by simulations of propane and propylene

Fig. 3. Isotherms, qv (p; D), for intrabundle adsorption (intratubular and interstitial sites), as a function of individual tube diameter, D. The isotherms for C2 H4 and C3 H6 (not plotted) are qualitatively similar to those for their saturated analogues (see text for details). The data for propane are taken from ref. [44].

adsorption onto homogeneous SWCNT bundles [44]. The authors concluded that above D = 14.7 Å, adsorption starts in the grooves and only after complete filling of their volume it proceeds to the exposed surfaces of the peripheral nanotubes. A similar argument should hold for CH4 , C2 H6 and C2 H4 because they are either spherical or linear molecules made of pseudo-atoms with van der Waals diameters similar to those of the C3 molecules (cf. Table 1), and thus adsorption should be equally influenced by the nanotube diameter. In the medium- to high-pressure range, the isotherms become essentially independent of D. They increase monotonically with pressure and, unlike the isotherms for intrabundle adsorption, they do not approach a constant plateau for the (p, T) conditions studied. Once the grooves get filled with adsorbate, there are no geometrical restrictions on the amount of fluid that can be subsequently

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the individual isotherms for individual tube diameters [qv (p; D) for internal adsorption and qs (p; D) for external adsorption] into a global average isotherm characterizing adsorption over the whole range of explored nanotube diameters (11.0 Å ≤ D ≤ 18.1 Å). For this purpose, the adsorption isotherms were weighted according to a normalized tube diameter distribution, w(D), given in Table 2: qv (p) =

 D

w(D)qv (p; D)

qs (p) =

 D

w(D)qs (p; D).

(2)

The values listed in Table 2 were determined by Raman Spectroscopy on a well-characterized sample of high-purity, electric-arc nanotubes, referred to as EA95 in previous studies [29]; w(D) is used here to generate a realistic convolution of the different nanotube diameters when they are aggregated into bundles in a real sample. The obtained results for intrabundle adsorption are plotted in Fig. 5. From the inspection of Fig. 5 it becomes clear that the existence of an unsaturated bond leads to larger differences in smaller molecules, which in the present case are the adsorbates with two carbon atoms. The alkane molecule (C2 H6 ) exhibits higher adsorption than the corresponding unsaturated fluid (C2 H4 ), particularly in the low pressure region, where solid–fluid energetic interactions are dominant. In fact, ethane adsorption starts by being 64% higher than ethylene, whereas this difference is only 48% for the propane/propylene pair. However, as the pressure is increased, the alkene loading approaches that of the alkane, and the crossover is reached earlier in the case of propylene, at around p/p0 ≈ 0.01. At higher pressures propylene is always more adsorbed than propane. An identical behavior is observed for ethylene, although it remains the least adsorbed species until a higher reduced pressure is reached, p/p0 ≈ 0.3, above which it surpasses ethane. A similar finding has been observed before for the case of pure intratubular confinement of C2 H6 /C2 H4 , based on a more detailed potential than ours [27]; the authors observed that ethane remains the preferentially adsorbed species up to p ≈ 10 bar, and attributed this to the larger dispersive forces that characterize the molecule. Fluid adsorption in the intrabundle volume (interstitial and intratubular) leads to type I isotherms (Fig. 1), and can therefore be described by a Langmuir-type expression [46,47]. We have chosen to employ the Toth isotherm: qv =

qvw bx t 1/t

[1 + (bx) ]

,

(3)

Fig. 4. Isotherms, qs (p; D), for adsorption onto the external sites of a bundle (groove sites and external surfaces), as a function of individual tube diameter, D. The isotherms for C2 H4 and C3 H6 (not plotted) are qualitatively similar to those for their saturated analogues (see text for details). The data for propane are taken from ref. [44].

adsorbed over the external surface of the bundle; after monolayer completion, molecules may adsorb onto a second layer, and this will give rise to a multi-layer type adsorption mechanism. It is interesting to compare the adsorption of an alkane with what happens when an unsaturated ␲ bond is introduced into its molecular skeleton. The reader may notice that the internal and external adsorption isotherms calculated for ethylene and propylene are not shown in Figs. 2 and 3; this is because they are qualitatively similar to those of their corresponding saturated analogues, differing only in the magnitude of the amount adsorbed. To get an overall picture of the quantitative differences obtained between the alkene and its saturate analogue, we have combined

Fig. 5. Average adsorption isotherms for the internal porous volume of the bundle. Symbols are the data calculated by GCMC simulation and lines are fittings with the Toth isotherm model (Eq. (3)). Alkanes (䊉) and alkenes (). Note the evident crossover between the alkane and the alkene curves.

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Table 2 Normalized weight fraction distribution of individual nanotube diameter, w(D), for a high-purity sample of electric-arc nanotubes, obtained by Raman spectroscopy at an excitation wavelength  = 785 nm [44]. D (Å) w(D)

11.0 0.043

12.9 0.230

13.8 0.315

which relates the total amount adsorbed, qv , with the reduced pressure, x = p/p0 , where qvw is the limiting value of adsorption, and b is an empirical constant that accounts for the affinity between the fluid molecules and the solid adsorbent. The results of fitting Eq. (3) to the GCMC adsorption data are given in Table 3 and graphically represented in Fig. 5. From the inspection of Table 3, it is evident that both qvw and b increase with the number of carbon atoms. Moreover, the saturated fluids exhibit an enhanced b constant when compared with their unsaturated analogues, thus corroborating the previous observation that alkanes are the preferentially adsorbed species in the low pressure region. In fact, previous simulation work on adsorption of C2 H6 and C2 H4 inside isolated nanotubes [27] has shown that the relative order of the b constant should be 2.5 < b(C2 H6 )/b(C2 H3 ) < 3.9, depending on the specific force field employed to model the fluid. In the present study, we got b(C2 H6 )/b(C2 H3 ) ≈ 3.1, well in line with the previous findings. Note that when the same ratio is determined for the C3 molecules a smaller value is obtained, b(C3 H8 )/b(C3 H6 ) ≈ 1.7, reflecting the existence of an extra pseudo-atom in the molecular skeleton. This suggests that the increase in the number of carbon atoms smears out the energetic differences between the alkane and alkene. In order to study adsorption onto the external sites of the bundle, it is convenient to recall that there are essentially two individual contributions for total adsorption, which are those arising from groove and surface sites. The reasons for this have to do with energetic differences between these two adsorption sites and geometrical considerations. It was recently shown that molecules being adsorbed onto groove sites exhibit larger isosteric heats of adsorption than onto surface sites [36]. In fact, if we consider the simulation data obtained for propane onto bundles of 14.7 Å nanotubes at a high relative pressure (p/p0 = 0.9), and plot the projected molecular density onto the x × y plane, perpendicular to the central axis of the nanotubes, it is observed that groove sites correspond to the geometrical positions where molecular adsorption is higher, as indicated by the red colored region on the top graph of Fig. 6. The first layer of adsorbed molecules is also clearly visible. This contour plot provides the visual clues for our proposed definitions of groove volume and external surface area. We define the average groove volume, Vn , as the gray region depicted in the bottom schematic of Fig. 6; Vn is set equal to the area of the gray region divided by the width of the box to express the groove volume per unit of reference surface area (A) of bundle employed in the GCMC simulations of external adsorption. Using geometrical arguments it is also easy to relate the two surface areas A and A ; as stated above, the former is the reference surface area employed in the GCMC simulations for computing the excess isotherm for external adsorption, qs , whereas A is a more realistic definition of external surface area since it takes into account the curvature of the nanotube walls and excludes the contribution from the volume of the grooves. Using the proposed definition of groove volume, Vn , we make a further simplifying assumption with regard to the maximum adsorption capacity of the groove, by defining it as qs,v w = Vn /vm , which makes it dependent on the molecular size of the adsorbate through its molar volume, vm , which we take equal to that of the saturated liquid at the working temperature. By taking the low pressure data, where adsorption on the exposed surfaces of the peripheral tubes of the bundle can be safely neglected [44], it was

14.7 0.244

15.8 0.072

16.6 0.057

18.1 0.038

found that adsorption onto the groove sites is accurately described by a classical Langmuir isotherm, qs,v =

qs,v w bx 1 + bx

(4)

Since the value of qs,v w is already known from the definition of the average groove volume, the constant b of the Langmuir isotherm can be determined by a least-squares fitting. The results thus obtained are listed in Table 3 and the corresponding fittings plotted in Fig. 7.As the pressure is increased, the system moves away from the Henry’s law regime, and adsorption onto surface sites becomes important. Once adsorption in the groove sites is known, surface adsorption can be determined as the difference between total external adsorption and localized adsorption in the groove sites. This is graphically represented in Fig. 7, where it can

Fig. 6. (Top) contour plots of molecular density on the x × y plane for propane adsorbed onto the external sites of a bundle with 14.7 Å nanotubes at 298.15 K and p/p0 = 0.9. The molecular density increases from dark blue to red. (Bottom) schematic representation of the first adsorbed layer, showing the definitions of groove volume, Vn (equal to the gray volume divided by area A), and surface area, A (dotted line). Symbols in the bottom schematic are defined as follows: Rs = (D +  s )/2, Rf = Rs +  f /2 = (D +  s +  f )/2, cos  = Rs /Rf , h = (Rf sin ) + ( f /2); sin   = h/Rs , D is the nanotube diameter,  s and  f are the LJ size parameters for the SWCNT atoms and for a pseudo-atom of the adsorbate, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

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Table 3 Average isotherm parameters obtained by correlating intrabundle data (Toth model, Eq. (2)), and external adsorption in the groove sites (Langmuir model, Eq. (4)) and bundle outer surface (n-BET model, Eq. (4)). R2 is the correlation coefficient. Toth model

CH4 C2 H6 C2 H4 C3 H8 C3 H6

Langmuir model

n-layers BET model

qvw × 102 (g/g)

b × 10−2

t

R2

5 2 qs,v w (g/m ) × 10

b × 10−2

R2

4 2 qs,s m × (g/m ) × 10

c

n

R2

9.377 13.452 13.815 14.969 15.578

0.362 8.074 2.612 55.828 32.815

0.782 0.667 0.700 0.651 0.647

1.000 0.998 0.998 0.994 0.995

3.013 5.835 5.859 6.999 7.197

0.138 0.412 0.316 0.669 0.543

0.998 1.000 1.000 1.000 1.000

1.792 2.216 2.440 2.882 2.919

0.945 2.657 2.133 2.660 2.257

1.261 2.556 1.907 2.017 2.185

0.998 0.999 0.999 0.997 0.997

be observed that the corresponding isotherms belong to type II. It is worth noting that for all fluids studied, not only groove sites become saturated at very low pressures, p/p0 < 0.1, but also adsorption onto surface sites only becomes relevant after filling of the grooves (Fig. 8). Moreover, this effect increases with the number of carbon atoms on the adsorbate molecule, as clearly depicted in Fig. 7 for CH4 and C3 H8 : propane adsorption in groove sites approaches complete pore filling at a much lower pressure than methane. This shows that our approach of separating the two individual contributions to total external adsorption, treating them as independent from one another, is fairly correct. The data plotted in Fig. 7 for ethane and ethylene also indicate that, without the confinement effect, solid–fluid interactions are dominated by energetic effects. Because ethane interacts with the solid via higher dispersive forces (cf. Table 1), it is always preferentially adsorbed over ethylene. This effect is much less evident for the propane/propylene pair, for they exhibit almost an identical adsorption capacity at high pressure. In the present work, adsorption over the exposed surfaces of the peripheral tubes is interpreted using the BET model [48], which

Fig. 7. Average adsorption isotherms on the external sites of the bundle exterior volume. Symbols are the data calculated in the simulations, total adsorption (䊉), grooves sites ( ) and surface (), and lines are fittings with Eq. (6).

is known to describe rather well type II and type III adsorption isotherms [46]. If the number of molecular layers on the external surface of the bundles is limited to a finite number n, the BET treatment leads to the following expression: qs,s =

s,s m cx 1 − (n + 1)xn + nxn+1 · 1−x 1 + (c − 1)x − cx

(5)

s,s where m is the monolayer capacity of the external surface of the bundles, and c is an empirical constant related to adsorbate–adsorbent energetic and entropic interactions [46]. In accordance with the bottom schematic of Fig. 6b, the monolayer s,s capacity is expressed here as m = (A /A)/(am NA ), where am is the average area occupied by a molecule of adsorbate in the completed monolayer and NA is the Avogadro’s constant; we recall that A /A is the ratio of the surface area denoted by the dotted line in Fig. 6 to the reference area employed in the GCMC simulations. Following an early suggestion of Emmett and Brunauer [49], am (Å2 ) is calculated from the molar volume vm (cm3 /mol) of the liquid adsorbate 2/3 using the formula am = 1.091(vm /NA ) . The results obtained by fitting Eq. (5), with c and n as adjustable parameters, to the GCMC adsorption data, are given in Table 3. Once the individual contributions from groove sites and external surfaces have been determined, the overall adsorption onto the external sites of the bundle can be calculated by combining Eqs. (4) and (5) into a global isotherm able to describe fluid adsorption over the whole pressure range, with the global fitting parameters

Fig. 8. Magnification of the average adsorption isotherms for methane (blue) and propane (red) on the external sites of the bundle. Symbols are the GCMC data; total adsorption (䊉), grooves sites ( ) and surface (), and lines are fittings with Eq. (6). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

50

F.J.A.L. Cruz et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 357 (2010) 43–52

indicated in Table 3, qs = qs,v + qs,s =

s,s cx 1 − (n + 1)xn + nxn+1 qs,v w bx + m · 1−x 1 + (c − 1)x − cx 1 + bx

(6)

To assess the accuracy of the proposed isotherm model, the total adsorption isotherm given by Eq. (6) is plotted in Fig. 7 along with total adsorption data calculated by GCMC. A good agreement is generally observed over the whole studied pressure range. The average deviation between Eq. (6) and the GCMC data points has been calculated for the range 0.4 ≤ p/p0 ≤ 0.9, and corresponds to 1.5% in the case of C3 H8 , but is always less than 1% in all other cases (CH4 , C2 H6 , C2 H4 , C3 H8 ). One of the purposes of the present work is to study the influence of an unsaturated bond on the adsorption properties of short, linear hydrocarbons. This effect can be assessed through analysis of the Henry constant, H, which can be calculated from adsorption data in the very low pressure region. It is in that pressure region that fluid–fluid interactions can be safely neglected and attention focused on the solid–fluid energetics. It has already been demonstrated above that the simulation data are well described, over the whole pressure range, by the Toth isotherm model for intrabundle adsorption (intratubular and interstitial spaces), by the Langmuir equation for the grooves, and by the finite-layer BET equation for adsorption on the exposed surface of the peripheral tubes. Having determined the various isotherm coefficients, the Henry constants for each fluid and for each isotherm model v b/p0 (intrabundle), H s,v = s,v b/p0 (grooves), are given by H v = w w s,s s,s 0 and H = m c/p (peripheral surfaces). The calculated Henry constants are plotted in Fig. 9 against a characteristic energy, εsf /kB , for the solid–fluid pair potential, which has been approximated as the sum of the effective Lennard–Jones well depths  for the pseudo-atoms of the adsorbate, εsf /kB = kB−1 i (εi εs )1/2 (cf. Table 1). The reason for this choice is explained below. The effective well depth for each adsorbate increases according to εsf (CH4 ) < εsf (C2 H4 ) < εsf (C2 H6 ) < εsf (C3 H6 ) < εsf (C3 H8 ). Two main conclusions can be drawn from the analysis of Fig. 9: (i) solid–fluid interactions increase with the number of carbon atoms of the adsorbate, and, most noticeably, with the dispersive energy of the adsorbate, which is why the alkanes are the preferentially adsorbed species in the low pressure region; and (ii) the Henry constants for intrabundle adsorption exhibit a higher sensitivity to the molecular nature of the adsorbate. In fact, the magnifications in Fig. 2 clearly show that, at very low pressures, there are significant differences between the adsorption behavior of methane, ethane, and propane, with the heavier adsorbates reaching pore filling much earlier, regardless of the value of D. This is not surprising bearing in mind that for intrabundle confinement, Hv (C3 H8 )/Hv (CH4 ) = 1095, while the same ratio for ethane is reduced down to Hv (C2 H6 )/Hv (CH4 ) = 47. As for the influence of a double bond, an opposite tendency is observed, Hv (C3 H8 )/Hv (C3 H6 ) = 1.89 and Hv (C2 H6 )/Hv (C2 H4 ) = 3.75, showing that the existence of an unsaturated bond has a higher influence in smaller molecules. The Henry constant, H, is given by the ratio of the partition functions per unit volume for the adsorbed and vapor phases [50]. It is also directly related to the excess chemical potential [51], ex , of the adsorbed molecules by H = ˇ exp(−ˇex ), where ˇ = 1/kB T and ex can be determined in a Monte Carlo simulation from the ensembleaverage of the energy of a test particle, U+ , as exp(ˇex ) = exp(−ˇU + ) . For a LJ system, U+ can be expressed as U + =



(i) ε ϕ(|r i i sf

i runs over the pseudo-atoms of the adsorbate, (i)

(i)

− r s |/sf ), where (i) εsf

= (εi εs )1/2 ,

sf = (i + s )/2, ri is the coordinate of the ith pseudo-atom, (i)

and ϕ(|r i − r s |/sf ) results from the summation of the LJ term

Fig. 9. Henry constant, H, and isosteric heat of adsorption [36] at zero coverage, 1/2 (εi εs ) (i runs q0st , plotted against the solid–fluid effective LJ depth, εsf /kB = kB−1 i

over the pseudo-atoms of the adsorbate) for alkanes (䊉) and alkenes ( ). Symbols represent calculated data and solid lines are fittings using an exponential dependency for H and linear dependency for q0st . Data from Jiang et al. [22] are indicated by a dotted line.

(i)

(i)

(j)

u(rij /sf )/εsf (cf. Eq. (1)) over all pair separations rij = |r i − r s | between pseudo-atom i and the carbon atoms, with coordi(1) (2) nates r s = (r s , r s , . . .), of the interacting nanotubes. In the Henry’s law region, short, linear alkanes are expected to be aligned along preferential directions in the bundle, so that for molecules located close to the potential minima the value of (i) ϕ(|r i − r s |/sf ) will be weakly dependent on the i index running over the pseudo-atoms of the adsorbate. The average value of U+   (i) can thus be approximated as U + = ϕ(| r − r s |/sf ) i εsf , where r represents the coordinates of the center of mass of the adsorbate at the potential minimum and  sf is the mean value of (i) the collision diameters sf . Therefore, to a good approximation the Henry constant for small adsorbate molecules should scale with H = ˇexp(ˇεsf ) [or exp(εsf /kB ) for a fixed value of T], where  (i) εsf = ε . As shown in Fig. 9, this is indeed the case for the i sf Henry constants of the three types of adsorption sites in the bundle. The obtained fittings are: ln H v = −12.23 + 9.06 × 10−2 εsf /kB (intrabundle), ln H s,v = −18.63 + 5.15 × 10−2 εsf /kB (grooves), and ln H s,s = −18.64 + 3.81 × 10−2 εsf /kB (exposed surfaces). This behavior is well in accordance with the findings of Jacobtorweihen

F.J.A.L. Cruz et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 357 (2010) 43–52

and Keil [26], who performed GCMC adsorption simulations for the intratubular volume of different nanotube diameters, and concluded that, for both alkanes and alkenes, the corresponding Henry constants increase exponentially with the number of carbon atoms of the fluid molecule. Their model is somewhat different from ours, in the sense that we are studying intrabundle adsorption, therefore accessing both intratubular and interstitial confinement, and convoluting the isotherms to describe a bundle that has a finite diameter distribution. Another work, the one of Jiang et al. [22], employed a homogeneous SWCNT bundle with D = 13.56 Å and a working temperature slightly different from ours. Although the simulations were confined to alkanes, the results pointed out for the above mentioned exponential dependence, regarding the Henry constants. Their results for intrabundle adsorption were recalculated with our energetic model, and are included in Fig. 9 as a dotted line. The agreement is quite satisfactory, bearing in mind the differences in both models. Those differences lie essentially in the individual SWCNT diameter (D ≈ 0.6 Å), working temperature, and in the calculation of the cross-collision diameter between the solid and an adsorbate pseudo-atom, which those authors assumed to be constant regardless of the fluid molecule. Recently, we have presented a detailed energetic analysis of adsorption onto SWCNT bundles [36], and the zero-coverage isosteric heats of adsorption, q0st , were calculated as a function of the number of carbon atoms in the molecule. Both intrabundle volume and external surface were probed. It is well known that in the zero-coverage region, q0st provides a good measure of the solid–fluid interactions governing over adsorption [52]. The zero-coverage isosteric heats of adsorption are plotted in the bottom graph of Fig. 9 as a function of the solid–fluid energetic parameter εsf /kB . Wood et al. [53] have shown that, in the limit of zero coverage, the heat of adsorption can be calculated using





q0st = Usf − Uf

ig

− kB T,

(7)

where Uf ig is the ensemble average of the total energy of an adsorbate molecule in the ideal gas state; in the present case, Uf ig is nonzero only for propane and propylene because of the internal contribution from bond-bending. Using an argument similar to the one employed above for the dependence of the Henry constant, it is seen that q0st should scale linearly with the solid–fluid energetic



parameter εsf /kB = kB−1 i (εi εs )1/2 . Fig. 9 shows that this is indeed the case. Fig. 9 shows that it is possible to identify two main types of adsorption mechanisms, according to the slope of q0st vs εsf /kB . Intrabundle confinement is always energetically more favorable than adsorption onto the external sites of the bundle, thus corroborating the previous discussion about the Henry constants. The q0st ratios for the alkane/alkene pairs, q0st (C3 H8 )/q0st (C3 H6 ) = 1.05 and q0st (C2 H6 )/q0st (C2 H4 ) = 1.11, show that small molecules are always energetically more affected by the presence of an unsaturated chemical bond.

51

spanned pressure range did not cover very low pressures, as low as the ones presently reported, Esteves et al. [29] still observed that saturated fluids are preferentially adsorbed over the corresponding unsaturated molecules. Nonetheless, and due to entropic effects, after a certain threshold pressure the alkene replaces its corresponding alkane as the most adsorbed fluid. The external surface of the bundle was further decomposed into its two main adsorptive sites: grooves and exposed surfaces of the peripheral tubes. This was done by defining a rigorous geometric volume for groove adsorption, based on the molecular size of the adsorbate. Using this approach we were able to determine the individual adsorption amounts for each of the three probed regions: intrabundle volume, groove sites and external surface. It was demonstrated that adsorption onto intrabundle volume and groove sites, leading to type I isotherms, could be modeled by Langmuir-type equations, indicating that although groove sites are geometrically different from the intrabundle volume, both regions can be described by a similar mechanism. Adsorption on the external surfaces, belonging to type II, was modeled by the BET isotherm with a finite number of layers. The linear combination of groove and external surface isotherms, to obtain a composite isotherm able to describe total external adsorption, was shown to be in good agreement with the GCMC simulation data for total adsorption on the external surface of the bundle. Although this has been applied to homogenous bundles, it should also be valid for heterogeneous bundles and real samples [29]. The calculation of the Henry constant H showed that adsorption onto the intrabundle volume is more dependent on the molecular nature of the adsorbate than adsorption outside the bundle, as indicated by the curves of H against the effective dispersive energy, εsf /kB . The slope of these curves for intrabundle confinement was determined to be the highest one; the same quantity for either groove or external surface adsorption was found to be quite similar. An analogous conclusion was drawn using zero-coverage isosteric heats of adsorption, q0st , as the relevant physical variable to probe the nature of solid–fluid interactions. In fact, previously determined q0st values were recalculated for the average nanotube diameter considered in the present work, and plotted as a function of εsf /kB . It was observed that the intrabundle volume was not only the energetically most favorable location for adsorption, but also that it was the most susceptible one to the adsorbate molecular characteristics. The ratios [˝(C2 H6 )/˝(C2 H4 )]/[˝(C3 H8 )/˝(C3 H6 )], where ˝ = (H, q0st ), always led to values greater than unity, supporting the conclusion that the existence of a chemical unsaturated bond affects more the adsorption properties of small molecules, such as the pair ethane/ethylene. Acknowledgements F.J.A.L. Cruz and I.A.A.C. Esteves gratefully acknowledge financial support from F.C.T./M.C.T.E.S. (Portugal) through grants SFRH/BPD/45064/2008 and SFRH/BPD/14910/2004.

4. Conclusions By separating a SWCNT bundle into its two main regions – external surface and internal volume –, we were able to individually probe the corresponding adsorption mechanisms and to establish general trends: (i) adsorption increases with the number of carbon atoms of the molecular skeleton, due to the increase in the number of interaction sites, and (ii) confined alkanes in the intrabundle volume are the preferentially adsorbed species in the low pressure region, for they possess higher dispersive energies compared to their unsaturated analogues. The latter observation has been put into evidence in the recent experiments reported by Esteves et al. [29], who performed gravimetric measurements of fluid adsorption (C1 –C3 ) onto electric-arc prepared SWCNT bundles. Although the

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