Characterization of Carbon Molecular Sieves Using Methane and

Separation of CO 2 /CH 4 using carbon molecular sieve (CMS) at low and high pressure ... E. P. Favvas , E. P. Kouvelos , G. E. Romanos , G. I. Pilatos...
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Langmuir 2003, 19, 8335-8342

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Characterization of Carbon Molecular Sieves Using Methane and Carbon Dioxide as Adsorptive Probes S. W. Rutherford,*,† C. Nguyen,‡ J. E. Coons,† and D. D. Do§ Engineering Sciences and Applications Division, Los Alamos National Laboratory, MS C930, Los Alamos, New Mexico 87545, CSIRO Manufacturing and Infrastructure Technology, Gate 4 Normanby Road, Clayton 3168, VIC Australia, and Department of Chemical Engineering, The University of Queensland, St. Lucia 4072, QLD Australia Received March 18, 2003. In Final Form: June 25, 2003 Nitrogen adsorption at 77 K is the current standard means for pore size determination of adsorbent materials. However, nitrogen adsorption reaches limitations when dealing with materials such as molecular sieving carbon with a high degree of ultramicroporosity. In this investigation, methane and carbon dioxide adsorption is explored as a possible alternative to the standard nitrogen probe. Methane and carbon dioxide adsorption equilibria and kinetics are measured in a commercially derived carbon molecular sieve over a range of temperatures. The pore size distribution is determined from the adsorption equilibrium, and the kinetics of adsorption is shown to be Fickian for carbon dioxide and non-Fickian for methane. The non-Fickian response is attributed to transport resistance at the pore mouth experienced by the methane molecules but not by the carbon dioxide molecules. Additionally, the change in the rate of adsorption with loading is characterized by the Darken relation in the case of carbon dioxide diffusion but is greater than that predicted by the Darken relation for methane transport. Furthermore, the proposition of inkbottleshaped micropores in molecular sieving carbon is supported by the determination of the activation energy for the transport of methane and subsequent sizing of the pore-mouth barrier by molecular potential calculations.

1. Introduction Porous adsorbent materials play a crucial role in many applications within the fields of separation and purification technology and chemical reaction engineering.1 In recent decades, significant advances in material synthesis have allowed porous adsorbents to be designed and tailored to suit the application, allowing significant improvements in process technology.2 Current synthesis techniques have allowed the pore size of materials such as molecular sieves to be engineered to suit the application of interest. The production of these materials with a defined pore size on the order of nanometer dimensions has provided challenges for methods, theories, and techniques that allow the determination of the dimensions of these pores. Adsorption of nitrogen at 77 K is currently employed as a standard technique for the size determination of pores of nanometer dimensions.3 Argon adsorption at 77 K is also employed for characterization; however, pores of subnanometer size known as ultramicropores pose difficulty for these techniques because nitrogen and argon are adsorbed at unreasonably slow rates at 77 K. At ambient temperature, the adsorption rates are measurable but extraordinarily high pressures are required to probe a reasonable range of pore sizes with argon and nitrogen. These limitations, coupled with the fact that the scale of the design and development of nanomaterials is reaching molecular dimensions, generate a need for standard

techniques that can probe and characterize ultramicropores. Carbon dioxide adsorption at ambient temperature has shown promise as a means for the characterization of ultramicropores in a carbon molecular sieve (CMS)4 and offers the advantage of measurable uptake rates coupled with the ability to probe a wide range of pore sizes from several angstroms to nanometer dimensions within reasonable pressure bounds. This has been demonstrated by the characterization of a commercially derived molecular sieving carbon manufactured by Takeda Chemical Co.5-7 In this investigation, previous efforts to characterize this material are expanded to include carbon dioxide and methane as probe molecules. Methane is chosen not only because of the reasonably small molecular size but also because it is a near spherical probe molecule, and the results are amenable to molecular simulation analysis. It is the intention in this investigation to span the adsorption equilibrium isotherm from high vacuum conditions to high pressures to determine the pore size distribution (PSD). Adsorption uptake kinetics will be simultaneously obtained, and the dependence of the rate of adsorption upon time, adsorbed phase concentration, and temperature will be elucidated. It is the ultimate goal to size the “pore mouth” of the molecular sieving carbon, and this will be estimated from a structural-based molecular model. 2. Experimental Section

* Corresponding author. Phone: (505)6676124. Fax: (505)6657836. E-mail: [email protected]. † Los Alamos National Laboratory. ‡ CSIRO Manufacturing and Infrastructure Technology. § The University of Queensland.

To allow the determination of the pore size at the ultramicropore level, an apparatus capable of the measurement of gas uptake from high vacuum conditions to high pressures was constructed. Operating on the volumetric, batch adsorption

(1) Ruthven, D. M. Principles of Adsorption and Adsorption Processes; Wiley: New York, 1984. (2) Yang, R. T. Gas Separation by Adsorption Processes; Imperial College Press: London, 1997. (3) Gregg, S. J.; Singh, K. S. W. Adsorption, Surface Area and Porosity; New York, 1982.

(4) Cazorla-Amoros, D.; Alcaniz-Monge, J.; de al Casa-Lillo, M. A.; Linares-Solano, A. Langmuir 1996, 12, 2820. (5) Nguyen, C.; Do, D. D. Langmuir 2000, 16, 1868-1873. (6) Rutherford, S. W.; Coons, J. E. Carbon 2003, 41, 405-411. (7) Jayaraman, A.; Chiao, A. S.; Padin, J.; Yang, R. T.; Munson, C. L. Sep. Sci. Technol. 2002, 37, 2505.

10.1021/la034472d CCC: $25.00 © 2003 American Chemical Society Published on Web 09/06/2003

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Table 1. Properties of the CMS sample

macroporositya

pellet diameter

pellet length

macropore sizea

CMS 3A

0.29

0.18 cm

0.2 cm

0.27 µm

a

Obtained from mercury porosimetry.9

principle, the apparatus consists of a sample chamber connected by a series of valves to vessels of various sizes up to 1 gal. The apparatus operates by degassing the system to ultrahigh vacuum (less than 10-6 Torr) conditions followed by injection of pure gas into the isolated system volume. The gas is subsequently expanded into the sample chamber, and the dynamics of the pressure change is monitored using pressure transducers. The system volume is determined by helium expansion into a previously calibrated vessel. During the expansion, the system is thermostated by an oven in which the entire system is placed. All of the system components interface with Swagelok VCR face seal fittings. The gas flow is regulated using Swagelok BG series valves capable of handling up to 315 °C as well as high pressures and high vacuum conditions. The volumes are highpressure Swagelok sample cylinders that were modified by the addition of VCR end fittings and seal-welded for high vacuum holding capability. The sample chamber consists of a modified single-ended miniature Swagelok sample cylinder on which a VCR fitting was welded to interface with other VCR fittings. The system is housed inside a Despatch LAC 2-12 forced convection oven, which maintains isothermality up to 260 °C. The pressure transducers are MKS Baratron type 615A, one with a 20 000 Torr maximum pressure and one with a 100 Torr maximum pressure, so that the apparatus is capable of measuring from low vacuum conditions to high pressures. Vacuum is maintained by a Varian turbo V70LP backed by a Varian mechanical diaphragm pump MDP 30. The dynamic pressure signal is logged to a personal computer using National Instruments hardware and software. A system diagram and further details are available in an earlier publication.6 The molecular sieve material employed in this study is a commercially supplied CMS manufactured by Takeda Chemical Co. It is obtained in the form of extruded pellets of a cylindrical shape with the properties shown in Table 1. Before each set of dynamic measurements, the sample was prepared by outgassing at 90 °C and less than 10-6 Torr for at least 60 h. The batch adsorption experiment was then conducted differentially, as was outlined in Rutherford and Do.8 The increment of the pressure was chosen to ensure that the rate of temperature increase due to exothermic adsorption affects the rate of mass transfer negligibly.

3. Adsorption Equilibrium Commercially manufactured Takeda 3A CMS has been characterized by carbon dioxide adsorption from pressures well below atmospheric9 to well above.10 The average micropore size in this material has been determined at around 0.5 nm by employment of Dubinin’s analysis.10 The adsorption equilibrium of carbon dioxide in Takeda CMS has been reported previously at 20 °C,6 and in this investigation the equilibrium isotherms of carbon dioxide at 50 and 70 °C and methane at 50, 60, and 70 °C are measured. The Toth isotherm is commonly employed for the characterization of carbon adsorbents11 and has proven useful for characterizing Takeda 3A.9 The Toth isotherm can be represented mathematically as

Cµ )

CµsbtP [1 + (btP)n]1/n

(1)

(8) Rutherford, S. W.; Do, D. D. Carbon 2000, 38, 1339-1350. (9) Rutherford, S. W.; Do, D. D. Langmuir 2000, 16, 7245-7254. (10) Cazorla-Amoros, D., Alcaniz-Monge, J.; de al Casa-Lillo, M. A.; Linares-Solano, A. Langmuir 1998, 14, 4589-4596.

Figure 1. (a) Equilibrium adsorption isotherm of carbon dioxide in Takeda 3A CMS at 20 °C (triangles; data taken from Rutherford and Coons),6 50 °C (circles), and 70 °C (squares). (b) Equilibrium adsorption isotherm of methane in Takeda 3A CMS at 50 °C (triangles), 60 °C (circles), and 70 °C (squares). The solid lines represent the fits of the Toth isotherm.

where Cµs and bt are the micropore capacity and affinity parameter, respectively, and n is the heterogeneity parameter that represents deviation from Langmuir behavior. The affinity coefficient has an Arrhenius dependence upon temperature, which can be represented mathematically as

bt ) bt0e-∆H/RT

(2)

where ∆H represents the heat of adsorption at zero loading. A Van’t Hoff plot of the dependence of the affinity coefficient upon temperature allows the determination of the heat of adsorption. In this investigation, methane adsorption is measured at 50, 60, and 70 °C, and carbon dioxide adsorption is measured at 50 and 70 °C and compared with previously obtained measurements at 20 °C to determine the heat of adsorption. 3.1. Carbon Dioxide and Methane Adsorption Equilibria. The carbon dioxide adsorption equilibrium is presented in Figure 1a and includes data obtained in (11) Do, D. D. Adsorption Analysis, Equilibria and Kinetics; Imperial College Press: London, 1998.

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Ppore ) P exp(- 〈Egpore〉/RT)

Table 2. Toth Isotherm Parameters Obtained from Equilibrium Data gas

Cµs (mmol/g)

n

CO2

3.1

0.65

CH4

4.1

0.55

T (°C) 20 50 70 50 60 70

bt (1/Torr) 10-3

7.0 × 2.2 × 10-3 1.1 × 10-3 3.4 × 10-4 2.6 × 10-4 2.0 × 10-4

a previous investigation.6 Figure 1b presents adsorption equilibria of methane in Takeda 3A CMS over a range of temperatures indicated. The fit of the Toth isotherm appears in Figure 1a,b where the micropore capacity (Cµs) and heterogeneity parameter (n) are held constant over the three temperatures. It is evident that the Toth isotherm can adequately characterize the data over the pressure range shown. The evaluated parameters from the Toth isotherm are presented in Table 2. The micropore capacity for methane is slightly higher than that obtained for carbon dioxide measurements consistent with the measurements made in microporous carbon,12 and the heterogeneity parameter n and affinity coefficient b are within an acceptable range.11 The temperature dependence of the affinity parameter is noteworthy because the heat of adsorption can be derived from a Van’t Hoff plot and is calculated as -30 and -24 kJ/mol for carbon dioxide and methane, respectively. These values compare reasonably well with those measured in a molecular sieving carbon given as -28.4 kJ/mol13 and -23.8 kJ/mol.14 3.2. Pore Size Determination from the Methane Isotherm Measurement. Despite the effectiveness of the Toth isotherm as a mathematical expression for the characterization of the isotherm, it provides no direct information about the physical properties of the solid. An alternative approach has been to attribute the heterogeneity observed in the equilibrium isotherm measurement to the inherent PSD of the solid. This has been undertaken in the analysis of subcritical carbon dioxide data by employment of Dubinin’s analysis.9,10 However, Dubinin’s analysis also reaches limitations in dealing with supercritical gas adsorption. A further alternative is to employ the structure-based model for supercritical gas adsorption proposed by Nguyen and Do.15 The model employs the concept of enhancement of the potential energy of interaction between adsorbate molecules and surface atoms within the pore interior. The primary advantage of employment of this model is that it requires only molecular properties of the adsorbate and adsorbent as input parameters, and the structural heterogeneity is accounted for using the distribution of the micropore size. In the following discussion, E and b will denote the potential energy of the gas molecules and the affinity of adsorption, respectively. The superscript g identifies the gas phase, and subscripts s and pore refer to surface and pore properties, respectively. A molecule adsorbed in a micropore is under the force field from all sides of the pore. Such interaction depends on the distance between the walls (i.e., pore size) as well as the position of the admolecule relative to the pore walls. At equilibrium, this interaction is a function of the pore size. The pressure of the occluded gas phase can be calculated from the bulk pressure as follows: (12) Lozano-Castello, D.; Cazorla-Amoros, D.; Linares-Solano, A.; Quinn, D. F. J. Phys. Chem. B 2002, 106, 9372-9379. (13) Reid, C. R.; Thomas, K. M. Langmuir 1999, 15, 3206-3218. (14) Reid, C. R.; Thomas, K. M. J. Phys. Chem. B 2001, 105, 1061910629. (15) Nguyen, C.; Do, D. D. J. Phys. Chem. B 1999, 103, 6900-6908.

(3)

with 〈Egpore〉 being the average potential energy of the occluded molecules. The average potential is a function of the pore size and, in this investigation, is taken to be the same as the potential energy at the pore center, Egpore. The affinity coefficient of adsorption in the pore, bpore, can then be calculated from the affinity coefficient of adsorption on the flat surface bs by the formula

(

bpore ) bs exp

)

Epore - Es RT

(4)

with

bs )

β exp(Es/RT) xMT

where M is the molecular weight and β is a parameter characterizing the solid surface. To yield the affinity having units of inverse pressure (1/MPa), β takes a value of 0.426 or 0.02115 for adsorption on a flat surface and in a pore system, respectively. If f(z) is the distribution function of the pore size, defined in this investigation as the clearance between the two opposite outermost graphite planes, and H(P, z) is the single pore isotherm equation, the amount adsorbed at pressure P can be calculated as

Cµ(P) )

∫0∞H(P, z) f(z) dz

(5)

When the Langmuir equation is used to represent the local isotherm, the following is obtained:

Cµ(P) )

b

∫0∞ Cµs(z)1 +pore b

(z) Ppore(z)

pore(z)

Ppore(z)

f(z) dz

(6)

where Cµs(z) is the maximum capacity of all pores of size z. As a result of the small range of temperatures studied in this investigation, the maximum capacity is assumed to be temperature-independent. The PSD is found by fitting eq 6 to the equilibrium adsorption isotherm data at three experimental temperatures simultaneously. In the fitting process, the whole pore spectrum is divided into a number of pore subranges, and then the optimization procedure is invoked to find the volume of each subrange such that the calculated isotherm closely matches the experimental data. This method does not make an a priori assumption for the form of the PSD function. Therefore, the result may more accurately reflect the PSD of the solid. Additionally, the accuracy of the PSD may also be affected by the choice of probe molecule. In this investigation, methane was chosen over carbon dioxide as the probe molecule for the generation of the PSD from equilibrium analysis because methane molecules have a spherical symmetry. The employment of linear molecules such as carbon dioxide requires the ambiguous determination of shape factors that may yield a less accurate determination of the pore size.16 The fitting of methane equilibrium data was performed and the resulting PSD is displayed in Figure 2. It is important to note that the PSD is presented in terms of the adsorption capacity, which is different but related to the volumetric PSD. It appears that the majority of the (16) Reid, C. R.; O’koye, I. P.; Thomas, K. M. Langmuir 1998, 14, 2415-2425.

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Figure 2. PSD determined from the fitting of the methane adsorption equilibrium at three different temperatures.

pores are smaller than 1 nm in width with no significant contribution of the larger pores. 4. Adsorption Kinetics The uptake of gas into porous adsorbent pellets with a bimodal PSD involves transport through the macropores followed by adsorption and transport through the micropores of the material. The macropores are formed from the void generated after agglomeration of the micrometersized carbon grains, which contain micropores and which are often modeled as spherical in shape. It has been shown in a previous investigation that the resistance to mass transfer presented by the macropores of Takeda 3A CMS is negligible for the time scales considered in this investigation.9 The rate-limiting step in the transport process is then attributable only to diffusion into or through the micrograins of this material. CMS is an adsorbent material manufactured with the unique characteristic of “ink bottle” micropores.17,18 These ink bottle pores are generated through treatment of the micrograins via carbon deposition, which coats the entrance to the micropore and, hence, creates a barrier at the pore mouth.19 The barrier is smaller in size than the micropore itself and can be the ratelimiting factor in the transport of larger molecules into molecular sieving carbon.13,16,20-23 For this case, the dependence of the uptake upon the time conforms to the linear driving force (LDF) model, and activation energies for transport can be larger than the heat of adsorption. For small molecules, which are negligibly impeded by the barrier, the diffusion into the micropore has been shown to follow Fick’s law.24 For this case, the activation energies are often on the order of 1/3 of the absolute value of the heat of adsorption upward to around the absolute value of the heat of adsorption. Molecules that are sized on the order of the pore-mouth dimensions follow a time dependence given by a combination of Fick’s law and LDF models.14 An interesting result of these works is the apparent dependence of the rate of adsorption upon the (17) Koresh, J.; Soffer, A. J. Chem. Soc., Faraday Trans. 1981, 77, 3005-3018. (18) LaCava, A. I.; Koss, V. A.; Wickens, D. Gas Sep. Purif. 1989, 3, 180-186. (19) Srinivasan, R.; Auvil, S. R.; Schork, J. M. Chem. Eng. J. 1995, 57, 137-144. (20) Chagger, H. K.; Ndaji, F. E.; Sykes, M. L.; Thomas, K. M. Carbon 1995, 33, 1405-1411. (21) Rynders, R. M.; Rao, M. B.; Sircar, S. AIChE J. 1997, 43, 24562470. (22) Liu, H.; Ruthven, D. M. In Proceedings of the Fifth International Conference on the Fundamentals of Adsorption; LeVan, M. D., Ed.; Kluwer Press: Boston, 1996. (23) Loughlin, K. F.; Hassan, M. M.; Fatehi, A. I.; Zahur, M. Gas Sep. Purif. 1993, 7, 264. (24) Ruthven, D. M. Chem. Eng. Sci. 1992, 47, 4305-4308.

Rutherford et al.

adsorbed phase concentration. Complex behavior is observed that cannot adequately be characterized by previously defined theories of surface or micropore diffusion.25 Because there are relatively few studies that have elucidated the dependence of the rate of adsorption upon the loading, any experimental effort in this direction should be considered useful. There are even fewer studies that characterize the size of the pore mouth, but it is clear that measurements of kinetics at several temperatures are required so that an activation energy for the movement past the pore-mouth barrier and into the micropore can be obtained. Molecular level potential calculations are considered in this investigation to explain the activation energy obtained for carbon dioxide and methane in the Takeda 3A molecular sieve. 4.1. Rate of Adsorption. Carbon dioxide and methane molecules were employed in this investigation as kinetic probes capable of characterizing the rate-limiting processes of micropore diffusion and transport through the pore-mouth barrier. The activation energy and the nature of the dependence of the rate of adsorption upon loading are scrutinized. 4.1.1. Carbon Dioxide Uptake. In the case where a molecule is small enough not to be impeded by the barrier at the mouth entrance of the micropore, the process of diffusion through the micrograin can dominate transport. In previous investigations of the micropore diffusion of small molecules in CMS, uptake has been demonstrated to be Fickian and the dependence of the diffusivity upon the pressure has been described by the Darken relation.6,9,26 In this investigation, the differential uptake experiment allows the assumptions of isothermality, linear isotherm across the pressure increment, and constant boundary condition to be imposed. Additionally, the flux is assumed to be proportional to the concentration gradient (Fickian diffusion), and under these conditions, the fractional uptake, F, should be expressed as1

F)1-

6

∞

∑

1

π2n)1n2

(

exp -

)

Dµn2π2t Rµ2

(7)

where Dµ/Rµ2 represents the mobility parameter for micropore diffusion. At long times (t f ∞), the fractional uptake is given by

1-F)

(

)

Dµπ2t 6 exp π2 Rµ2

(8)

For Fickian behavior, a plot of the logarithm of 1 - F versus t should be linear at long times with an intercept of -0.50 and a constant slope. To maintain the condition of isothermality during measurement, limits on the maximum amount sorbed over each differential increment are imposed using the procedure outlined by Rutherford and Do.8 Additionally, over the small increments taken, the isotherm is effectively linearized and the volume is matched to the amount adsorbed such that a change of no greater than 10% in the boundary condition is allowed. This allows the fitting of eq 7 to the uptake measurements of carbon dioxide of each differential experiment to obtain the mobility parameter for micropore diffusion (Dµ/Rµ2). Typical uptake curves are shown in Figure 3 for uptakes at 50 and 70 °C obtained in this investigation and at 20 (25) Wang, K.; Suda, H.; Haraya, K. Ind. Eng. Chem. Res. 2001, 40, 2942-2946. (26) Kapoor, A.; Yang, R. T.; Wong, C. Catal. Rev.sSci. Eng. 1989, 31, 129-214.

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Figure 3. Dependence of the uptake with time for carbon dioxide at 20 °C and 90.6 Torr (triangles), 50 °C and 463 Torr (circles), and 70 °C at 750 Torr (squares) in Takeda 3A CMS. The solid line indicates the long-time Fickian behavior.

°C obtained in the investigation of Rutherford and Coons.6 For all differential measurements, the mobility parameter is obtained, and Figure 4 demonstrates the pressure dependence of the derived mobility parameter. This dependence has been characterized by the Darken relation, which has been successfully applied to diffusion in molecular sieving carbon.6,9,24 When coupled with the Toth isotherm, the Darken relation takes the following form:

Dµ 2

Rµ

)

Dµ0 [1 + (btP)n] Rµ2

(9)

where Dµ0 is the diffusivity at zero loading and the isotherm parameters bt and n are obtained from the equilibrium analysis. Figure 4 shows the fit of the Darken relation to the observed data, and it is evident that this relation maintains a reasonable prediction of the mobility parameter and its dependence upon pressure. The zero loading mobility constant at each temperature is determined from the one parameter fit of eq 9 to the data, and the evaluated parameter is presented in Table 3. 4.1.2. Methane Uptake. Carbon dioxide uptake in Takeda 3A appears to display characteristics of micropore diffusion as a result of the observations of Fickian uptake and the fact that the diffusivity varies with the pressure according to the Darken relation. However, methane uptake differs greatly in behavior and displays non-Fickian kinetics, as is indicated by the 0 intercept in Figure 5. A 0 intercept in the plot of the logarithm of 1 - F versus t was obtained in all the experimental runs at various pressures. All the kinetic curves for methane are fitted to the LDF model in which the fractional uptake (F) is given by

F ) 1 - exp(-kt)

(10)

where k is the LDF rate constant. The rate constant at each pressure is plotted in Figure 6 against the amount adsorbed. It is obvious from this plot that the measured rate constant has a strong dependence upon the amount adsorbed. The Darken relation would predict the following dependence of the LDF rate constant upon the amount adsorbed:

1 k ) k0 1 - (C /C )n µ µs

(11)

Figure 4. (a) Mobility parameter for carbon dioxide diffusion in Takeda 3A CMS at 20 °C and dependence upon pressure. Data were taken from Rutherford and Coons.6 (b) Mobility parameter for carbon dioxide diffusion in Takeda 3A CMS at 50 °C and dependence upon pressure. (c) Mobility parameter for carbon dioxide diffusion in Takeda 3A CMS at 70 °C and dependence upon pressure. The solid lines represent the fits of the Darken relation.

where k0 is the rate constant at zero loading. Figure 6 shows the best prediction of eq 11 to the kinetic data, and it is evident that the rate constant rises much faster than

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Table 3. Parameter Values Obtained from the Fitting of Kinetic Data CH4

CO2

parameter

value

w k0 at 50 °C (1/s) k0 at 60 °C (1/s) k0 at 70 °C (1/s) Dµ0/Rµ2 at 20 °C (1/s) Dµ0/Rµ2 at 50 °C (1/s) Dµ0/Rµ2 at 70 °C (1/s)

3 3.2 × 10-5 4.8 × 10-5 7.6 × 10-5 4.8 × 10-4 7.5 × 10-4 1.3 × 10-3

fitted to the data of Figure 6, and optimal values for the parameters w and k0 were obtained. The optimal values over the three temperatures are presented in Table 3. In contrast to the findings of Wang et al. who obtained a value of 2 for the exponent w, a value of 3 was obtained in this investigation for the data at 50, 60, and 70 °C. Further theoretical investigation is required to explain the nature of this dependence. However, the value for the rate constant at zero loading is on a similar order of magnitude to that obtained for methane by Reid and Thomas.14 At 70 °C, the value obtained in this work is 7.6 × 10-5 s-1, which compares with 2.6 × 10-5 s-1 at the same temperature.14 4.1.3. Dependence of the Rate of Adsorption on Temperature. Micropore diffusion has been shown to be an activated process, which obeys the Arrhenius expression:26

Dµ0

) 2

Rµ

Figure 5. Dependence of the uptake with time for methane at 50 °C and 22 Torr, 60 °C and 1616 Torr, and 70 °C at 80 Torr in Takeda 3A CMS.

Dµ0∞ -E/RT e Rµ2

where E is the activation energy for micropore diffusion and represents the energy required for diffusion though the pore interior. The value of the activation energy is obtained via the slope of the Arrhenius plot. In previous sections, it has been shown that carbon dioxide transport in Takeda 3A demonstrates characteristics of micropore diffusion. An additional characteristic is defined by the ratio of the activation energy and the absolute value of the heat of adsorption, which is generally on the order of 0.3-0.6 for micropore diffusion.26 The activation energy is obtained from the Arrhenius plot and is calculated at 16 kJ/mol, which is 0.5 times the absolute value of the heat of adsorption and close to the value obtained for micropore diffusion in activated carbon measured by Prasetyo and Do at 14 kJ/mol.27 Methane transport into Takeda 3A displays a much higher activation energy. This is evident from the calculation of the activation energy from the Arrhenius relation:

k ) k∞e-E/RT

Figure 6. Plot of the LDF rate constant for adsorption (k) of methane in Takeda 3A CMS, indicating its dependence upon the adsorbed phase concentration. The broken line represents the fit of eq 11 to the data, while the solid line represents the fit of eq 12.

that predicted by the Darken relation. This has been observed by Wang et al.25 through measurements of CMS membrane permeability. These authors have proposed the following power law dependence to account for changes in the rate:

k 1 ) k0 [1 - (C /C )]w µ µs

(12)

where w is an integer value obtained from fitting to the kinetic data. The relationship represented by eq 12 was

(13)

(14)

The activation energy is calculated at 40 kJ/mol, which is 1.7 times the absolute value of the heat of adsorption. This value is significantly larger than those of the measurements of the activation energy for the micropore diffusion of methane molecules in microporous carbon at around 12 kJ/mol.27 However, this value is smaller than the activation energy for methane transport in a CMS evaluated by Reid and Thomas14 at 50 kJ/mol. From the magnitude of the measurements, it appears possible that the methane molecules are impeded in their entry into the micrograin by the pore-mouth barrier, and carbon dioxide molecules do not experience the same impedance presented by the barrier at the pore mouth. In this case, the bulk of the resistance for the carbon dioxide transport is lumped into diffusion through the micropores. To further investigate this, molecular level calculations can assist in estimating the required activation energy for methane and carbon dioxide transport through the pore mouth. These calculations can allow the determination of an effective size for the pore-mouth restriction to be determined. 4.2. Molecular Potential Calculations. In Section 3, a distribution for the micropore size was obtained from (27) Prasetyo, I.; Do, D. D. Chem. Eng. Sci. 1998, 53, 3459-3467.

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Figure 7. Configuration of the CMS pore entrance. Table 4. Physical Properties of Adsorbing Molecules and Carbon Atoms

Figure 8. Heat of adsorption of methane (solid line) and carbon dioxide (dotted line) as a function of the pore width.

the adsorption equilibrium data, and in this section, we aim at estimating the size of the pore-mouth restriction. This will be performed by introducing a configuration for the pore-mouth barrier and the pore-wall structure such that the resulting model is capable of estimating the repulsive force exerted by the pore-mouth atoms and yet is still simple enough for potential calculations. 4.2.1. Carbon Pore-Wall Configuration. The potential energy of a molecule in the vicinity of a carbon adsorbent is a function of its position relative to the carbon atoms, and it is ideally calculated as a sum of the interaction energies between the molecule and the individual carbon atoms of the pore wall. Various configurations are used to derive compact integral potential energy equations such as the 10-4, 10-4-3 or 9-3 models. While simple, these models suffer an inherent limitation when dealing with pores of molecular dimensions. This limitation is due to the proximity of the adsorbate molecule to the pore wall and the assumption of a homogeneous distribution of mass centers over the graphite plane or in the space of the graphite domain. Such an assumption will lead to the underestimation of the effect of the repulsive forces exerted by the individual carbon atoms nearest to the adsorbate molecule. This is particularly important in the case of adsorption in CMS, which has a sieving aperture on the order of molecular size and ultramicropores, which generate strong interactions of adsorbed molecules with pore walls. An appropriate model of the pore-wall structure has been used in previous work of equilibrium isotherm calculation28 and is based on the observed arrangement of the graphite structure in carbon.29 Here, the micropore is visualized as the space between the graphite sheets, which are stacked on top of each other with an interlayer spacing of 0.3354 nm. The stacking of the graphite layers is in a hexagonal arrangement with the layers offset from one another, as is shown in Figure 7. The number of graphite layers of each pore wall is limited to 3 or 4, corresponding to a pore-wall thickness of about 1.1-1.5 nm. This model allows for the calculation of the potential energy as the sum of the 12-6 interaction between the

admolecule and individual solid atoms. In all of the following potential energy calculations, the collision diameters and the interaction energy between the adsorbate molecules and the carbon surface are taken from the literature30,31 and are listed in Table 4 for reference. The adsorbent-adsorbate collision diameter σ12 and the interaction energy 12 are calculated by using the LorentzBerthelot mixing rule.30 4.2.2. Heat of Adsorption from Molecular Potential Calculations. When the pore-wall configuration presented previously is used, the minimal potential energies of methane and carbon dioxide molecules can be calculated. Figure 8 shows the calculated minimal potential energy of the methane and carbon dioxide molecules in pores of different sizes. This figure demonstrates that the absolute value for the heat of adsorption will reach a maximum, followed by a decrease, as pores of larger dimensions are filled. At very a large pore size (which represents adsorption onto a flat graphite slab), a constant value for the heat of adsorption is approached. The maximum corresponds to pore sizes on the order of the carbon-adsorbate collision dimension σ12 and generates the minimal potential energy. The maximum absolute values for the heats of adsorption are calculated to be 22.5 and 27.5 kJ/mol for methane and carbon dioxide, respectively. These pores should offer the greatest affinity for molecules in the gas phase and, hence, should be the first to be filled by sorbate molecules. As a result, these calculated potential energy values should be comparable to the measured heat of adsorption at zero loading. The measured absolute values for methane and carbon dioxide are calculated from the isotherm data and given as 24 and 30 kJ/mol, respectively, which compare well with the calculated values. 4.2.3. Activation Energy. To allow for the estimation of the activation energy, a structural form for the barrier constriction at the pore mouth is needed. This structure should readily allow calculation of the potential of molecules passing through the constriction at the pore mouth. In this investigation, a structural model similar to one previously proposed by Nguyen and Do5 is employed. This model assumes a relatively simple structure in which one row of hexagonal carbon rings along both sides of the pore mouth is allowed. This is proposed to mimic the carbon, which is assumed to be deposited at the mouth of

(28) Nguyen, C.; Do, D. D. Langmuir 1999, 15, 3608. (29) Introduction to Carbon Science; Marsh, H., Ed.; Butterworths: Cornwall, 1989. Chemistry and Physics of Carbon; Walker, P. L., Jr., Ed.; Marcel Dekker: New York, 1966, Vol. 2.

(30) Steele, W. A. The Interaction of Gases with Solid Surface; Pergamon Press: Oxford, 1974. (31) Reid, R. C.; Prausnitz, J. M.; Polling, B. E. The Properties of Gases and Liquids; McGraw-Hill: New York, 1987.

σcarbon

σcarbon dioxide

σmethane

carbon/ k

0.34 nm

0.3941 nm

0.3758 nm

28 K

carbon dioxide/ k

methane/ k

195.2 K

148.6 K

8342

Langmuir, Vol. 19, No. 20, 2003

Figure 9. Potential energy of methane (solid line) and carbon dioxide (dotted line) molecules residing at the pore entrance as a function of the aperture half width. Arrows in the inset illustrate the estimation of the pore-mouth aperture size from the methane activation energy measurement.

the micropore during the manufacture of CMS and which creates a barrier to penetration of large molecules. Consistent with the observed stacking structure of graphite,29 the row of carbon rings that forms the pore-mouth barrier is offset relative to the rings of the pore-wall graphite sheet, as is shown in Figure 7. In this configuration, the three-dimensional arrangement of the carbon atoms at the barrier creates a chain of openings. Each of these openings is defined by three carbon atoms, two from one carbon plane and the third from the opposite carbon plane. To avoid any confusion between the pore width z (i.e., the clearance of the pore interior) and the dimension of the pore mouth, “aperture half width” is used hereafter to denote the vertical distance from the pore center plane to the center plane of the hexagonal rings of deposited carbon at the pore mouth. The pore aperture half width is denoted in Figure 7 as r. The effective aperture width is defined as the clearance between the outer edge of the carbon atom of the barrier at the pore mouth and the outer edge of the opposite carbon and, as a result of the offset in the row of deposited carbon, is larger than the aperture width less the collision diameter of the carbon molecule. Having defined a structural model for CMS, it is possible to perform potential calculations to make an estimate of the effective aperture size based on the kinetic measurements. In these transport calculations, it is the kinetic diameter of carbon dioxide that is employed instead of the collision diameter that is used in the analysis of the equilibrium data in the previous sections. The value for the kinetic diameter of the linear carbon dioxide molecule is given as 0.33 nm.32 Figure 9 shows the resulting potential energies of the methane and carbon dioxide molecules at the barrier calculated as a function of the (32) Breck, D. W. Zeolite Molecular Sieves; John Willey & Sons: New York, 1974.

Rutherford et al.

aperture half width. The figure indicates that the potential energy is a strong function of the aperture size. At small sizes, the aperture acts as a repulsion to the carbon dioxide and methane molecules indicated by a positive potential energy. The repulsion diminishes very quickly with size. At larger sizes, the aperture acts as an attraction to carbon dioxide and methane indicated by a negative activation energy. The attraction effect diminishes very quickly with size, and it becomes negligible when the pore aperture half width is about 1.2 nm (the upper end of the micropore range). 4.2.4. Pore-Mouth Size Determination. The activation energy for the methane adsorption into Takeda 3A CMS was estimated by kinetic measurement to be about 40 kJ/mol. As is highlighted in the inset in Figure 9, this value corresponds to an aperture half width of around 0.29 nm. From this, the effective aperture width is calculated at around 0.32 nm. At this size, the carbon dioxide molecule experiences a significantly smaller poremouth activation energy that is estimated to be around 5 kJ/mol. This energy is outweighed by the measured activation energy for micropore diffusion, which was estimated from kinetic measurement to be 16 kJ/mol. This is in agreement with the hypothesis that the main resistance to the transport of methane is due to the poremouth barrier while the mode of carbon dioxide transport is primarily governed by micropore diffusion. 5. Conclusions The notion of “ink-bottle” shaped micropores in commercially supplied CMS has been supported by determination of the PSD and pore-mouth width. To make this determination, methane and carbon dioxide adsorption equilibria and kinetics for a range of temperatures were measured in Takeda 3A CMS. Carbon dioxide uptake exhibited Fickian behavior, the diffusivity followed a dependence described by the Darken relation, and the activation energy for diffusion was 1/2 that of the absolute value of the heat of adsorption. On the basis of these observations, it is concluded that carbon dioxide transport in Takeda 3A CMS is governed by micropore diffusion. Methane transport, however, exhibited non-Fickian kinetics but obeyed a LDF model. The change in the rate of adsorption with loading was greater than that predicted by the Darken relation, and the activation energy was 1.7 times the absolute value of the heat of adsorption. Activation energy measurements reveal that a pore-mouth restriction impedes the methane uptake but provides negligible resistance to carbon dioxide molecules. Subsequent interpretation of the methane activation energy measurement based on potential calculations allows sizing of the pore-mouth opening. As a result of the ability of carbon dioxide and methane to characterize the PSD and pore-mouth opening in CMS, these probe molecules are proposed as alternatives to the standard argon and nitrogen measurements at 77 K. LA034472D