Measuring and Modeling Supercritical Adsorption in Porous Solids

The balance allows for in situ measurement of the fluid phase bulk density ... We adopted the common approach to estimate V0 from adsorption ... In Fi...
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Measuring and Modeling Supercritical Adsorption in Porous Solids. Carbon Dioxide on 13X Zeolite and on Silica Gel Thomas Hocker, Arvind Rajendran, and Marco Mazzotti* ETH Swiss Federal Institute of Technology Zurich, Institut fu¨ r Verfahrenstechnik, Sonneggstrasse 3, CH-8092 Zu¨ rich, Switzerland Received October 1, 2002. In Final Form: November 25, 2002 Supercritical adsorption of carbon dioxide on 13X zeolite pellets and on silica gel is studied experimentally and theoretically. A gravimetric apparatus with provision for in situ density measurement is used for the measurement of excess adsorption isotherms. The volume of the adsorbent and the other solid parts in the measuring cell, which in turn affects the accuracy of the excess isotherms, is estimated by accounting for helium adsorption on the adsorbent. A model based on lattice density functional theory is introduced and used to analyze the effect of geometric confinement on excess adsorption isotherms under supercritical conditions. This is then used to describe adsorption on adsorbents with different pore size distributions, in particular for the systems that have been experimentally studied. The experimental data and the model results are compared, and their satisfactory agreement is discussed. Features such as “bumps” on the descending part of the excess adsorption isotherms in the CO2-13X zeolite system are discussed and explained using the model.

1. Introduction The adsorption behavior of high-pressure, supercritical fluids in porous sorbents has received widespread attention in recent years.1 Data from adsorption experiments have been correlated with models based on statistical thermodynamics to understand the microscopic picture of the fluid behavior in pores of widely different size, that is, from micro- to macropores.2-9 In these experiments, the truly measurable quantity is the adsorption excess, nex(Fb, T), which is proportional to the difference between the average of the local densities inside the pores and the bulk density, Fb.10,11 The experimental results clearly show that the adsorption behavior of supercritical fluids differs fundamentally from that of the standard types of adsorption isotherms according to the IUPAC classification, which is based on the concept of absolute adsorption.12,13 * To whom correspondence should be addressed. Phone: +411-632 2456. Fax: +41-1-632 1141. E-mail: [email protected]. ethz.ch. (1) Gelb, L. D.; Gubbins, K. E.; Radhakrishnan, R.; SliwinskaBartkowiak, M. Rep. Prog. Phys. 1999, 62, 1573-1659. (2) Marconi, U. M. B. Phys. Rev. A 1988, 38, 6267-6279. (3) Tan, Z. M.; Gubbins, K. E. J. Phys. Chem. 1990, 94, 6061-6069. (4) Malbrunot, P.; Vidal, D.; Vermesse, J.; Chahine, R.; Bose, T. K. Langmuir 1992, 8, 577-580. (5) Schoen, M.; Thommes, M. Phys. Rev. E 1995, 52, 6375-6386. (6) Aranovich, G. L.; Donohue, M. D. J. Colloid Interface Sci. 1998, 200, 273-290. (7) Kiselev, S. B.; Ely, J. F.; Belyakov, M. Y. J. Chem. Phys. 2000, 112, 3370-3383. (8) Soule, A. D.; Smith, C. A.; Yang, X. N.; Lira, C. T. Langmuir 2001, 17, 2950-2957. (9) Di Giovanni, O.; Hocker, T.; Rajendran, A.; Do¨rfler, W.; Mazzotti, M.; Morbidelli, M. Measuring and describing adsorption under supercritical conditions. In Fundamentals of Adsorption, Vol. FOA7; Kaneko, K., Kanoh, H., Hanzawa, Y., Eds.; IK International: Chiba, Japan, 2001. (10) Sircar, S. J. Chem. Soc., Faraday Trans. 1 1985, 81, 1527-1540. (11) Findenegg, G. H.; Thommes, M. High-pressure physisorption of gases on planar surfaces and in porous materials. In Physical Adsorption: Experiment, Theory, and Applications; Fraissard, J., Ed.; Kluwer Academic: Dordrecht, 1997. (12) IUPAC Commission on Colloid and Surface Chemistry Including Catalysis Pure Appl. Chem. 1985, 57, 603. (13) IUPAC Commission on Colloid and Surface Chemistry Including Catalysis Pure Appl. Chem. 1994, 66, 1739.

In particular, typical experimental isotherms under supercritical conditions go through distinct maxima with increasing bulk density (or pressure).14,15 For highly attractive surfaces, these maxima occur at rather low densities and are associated with the filling of the micropores and with the saturation of the fluid layers directly adjacent to the walls of the meso- and macropores. At higher densities and for temperatures well above the critical temperature, that is, for T . Tc, the isotherms then fall off almost linearly when further increasing the density.4,16 However, excess isotherms exhibit a rather peculiar behavior when lowering the temperature to values only slightly above Tc.17 For a flat and open surface, for example, Fisher and de Gennes showed that when Fb ) Fc and as T f Tc from above, the adsorption excess diverges as the correlation length of the fluid goes to infinity.18 This phenomenon, which reflects the “collective behavior” of a fluid under near-critical conditions, is termed “critical adsorption” and has been confirmed by experiments.14,17 In this work, we explore the high-pressure adsorption behavior of supercritical CO2 on 13X zeolite pellets, that is, on a commercial sorbent that consists of well-defined zeolite crystals, embedded into a clay binder with a rather broad pore size distribution. Adsorption excesses are measured gravimetrically using a magnetic suspension balance with an absolute accuracy of 0.01 mg. The balance allows for in situ measurement of the fluid phase bulk density through measurement of the weight of a calibrated titanium sinker. Special care is taken in the determination of the volumes of the solid parts inside the measuring cell that are needed to account for the buoyancy correction. To correlate the data, a model within the framework of (14) Specovious, J.; Findenegg, G. H. Ber. Bunsen-Ges. Phys. Chem. 1980, 84, 690-696. (15) Di Giovanni, O.; Do¨rfler, W.; Mazzotti, M.; Morbidelli, M. Langmuir 2001, 17, 4316-4321. (16) Aranovich, G. L.; Donohue, M. D. J. Colloid Interface Sci. 1997, 194, 392. (17) Thommes, M.; Findenegg, G. H.; Lewandowski, H. Ber. BunsenGes. Phys. Chem. 1994, 98, 477-481. (18) Fisher, M. E.; de Gennes, P. C. R. Acad. Sci. Ser. B 1978, 287, 207-209.

10.1021/la0266379 CCC: $25.00 © 2003 American Chemical Society Published on Web 01/17/2003

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lattice density functional theory (DFT) is used that incorporates information on the adsorbent geometry, that is, pore size and shape, as well as information about the fluid-fluid and fluid-surface energetic interactions. When the model is used to analyze the experimental data, direct connection is made between the shape of the adsorption isotherms and the microscopic structure of the sorbent material. Such a model is used to describe and analyze not only the data on carbon dioxide adsorption on 13X zeolite pellets but also the previously reported measurements of carbon dioxide adsorption on silica gel.15 2. Experimental Section 2.1. Materials. Zeolite 13X pellets (Z10-02, lot number 16296) were obtained from Chemie Uetikon AG, Zu¨rich, Switzerland. The particle size range of the pellets was 1.6-2.6 mm. The adsorbent was regenerated at 300 °C for 8 h before being used in the experiments. Carbon dioxide with a purity of 99.995% and helium with a purity of 99.999% were obtained from PanGas AG, Luzern, Switzerland. The critical properties of the adsorbates are as follows: Tc(CO2) ) 304.1 K, pc(CO2) ) 73.7 × 105 Pa, and Fc(CO2) ) 467.6 kg/m3; Tc(He) ) 5.26 K, pc(He) ) 2.26 × 105 Pa, and Fc(He) ) 69.3 kg/m3. 2.2. Setup. A Rubotherm magnetic suspension balance was used for measuring the excess adsorbed amounts. The balance is designed to measure excess amounts up to an operating pressure of 450 bar and a temperature of 250 °C with an absolute accuracy of 0.01 mg. The temperature in the cell is controlled to a precision of 0.1 K by the use of a heating jacket. The temperature in the jacket is measured with a PT-100 probe with an accuracy of 0.1 K. The balance is provided with a titanium sinker whose volume has been calibrated independently. Hence, under experimental conditions, measuring the mass of the sinker allows for the in situ measurement of the fluid phase bulk density. This is particularly useful when operating under near-critical conditions, where even a slight error in the measurement of the pressure or temperature could lead to a severe error in the estimation of the fluid phase bulk density. The details of the balance and the experimental procedure have been reported elsewhere.15,19 However, for the sake of completeness, a brief summary of the procedure follows. The regenerated adsorbent is weighed and placed in a basket that is fixed to the measuring apparatus. The measuring cell in which the basket and the titanium sinker are housed is then closed, and a vacuum is applied. The weight of the balance parts plus the weight of the basket and the sorbent are measured under a vacuum. In the next step, helium adsorption measurements are carried out to estimate the volume of all solid parts inside the measuring cell; this volume is needed for the buoyancy correction. The final step involves charging the cell with the adsorbate which in the present case is carbon dioxide. The balance is operated in two positions, where the excess amount adsorbed and the fluid phase density, respectively, can be measured.

3. Experimental Results 3.1. Volume of Solid Parts from Helium Adsorption Experiments. Since in gravimetric experiments the measurement of the excess amount adsorbed depends on the buoyancy correction term, FbV0 (where Fb is the bulk density and V0 is the volume of all solid parts inside the measuring cell), the accuracy with which V0 needs to be obtained increases when increasing the density at which the adsorption experiments are carried out. We adopted the common approach to estimate V0 from adsorption measurements using helium as a weakly adsorbing reference material. Until recently, in the literature it was assumed that helium adsorption could be neglected at the temperature at which V0 was estimated. However, a (19) Dreisbach, F.; Staudt, R.; Tomalla, M.; Keller, J. U. Measurement of adsorption equilibria of pure and mixed corrosive gases: The magnetic suspension balance. In Fundamentals of Adsorption, Vol. FOA6; Meunier, F., Ed.; Elsevier: Paris, 1996.

Figure 1. Comparison of isochoric data for the excess adsorption of helium on 13X zeolite pellets with modeling results that account for mobile adsorption of helium and for thermal expansion of the solid parts inside the measuring cell (see Appendix A and ref 22 for the modeling details). Symbols: (0) FbHe ) 2.88 mol/L; (4) FbHe ) 2.35 mol/L; (O) FbHe ) 1.06 mol/L; (dashed-dotted line) FbHe f 0; (dashed line) V0(T). The last curve shows that V0(310 K) ≈ 4.285 mL. Table 1. Model Parameters and Physical Properties Used to Correlate the Mobile Adsorption Model Presented in Reference 22 (and Summarized in Appendix A) with Isochoric Data for the Excess Adsorption of Helium on Zeolite 13X Pellets model parameters

helium properties

)4 Ufs/k ) -240K ν⊥ ) 1.7 × 1012 Hz V0(310 K) ≈ 4.285 mL

rj(He) ) σ(He)21/6 ) 2.869 Åa ff/k ) -10.22 Ka

z2D

zeolite 13X pellets (Uetikon Z10-02) msorb ) 4.8681 g Rsorb ) 10-6 K-1 A ) msorbasorb ) 2677 m2 (asorb ) 550 m2/g) a

metal parts in cell mmet ) 15.9158 g Rmet ) 11 × 10-6 K-1 Fmet(300 K) ) 7.65 g/cc

Reference 27.

careful analysis by Sircar shows that this assumption breaks down at higher densities or pressures.20,21 Thus, a new protocol for estimating V0 was developed which takes into account mobile adsorption of helium in the nonlinear range of the isotherm, as well as thermal expansion of the various solid parts inside the measuring cell.22 A summary of the necessary steps for obtaining V0 is given in Appendix A. In Figure 1, isochoric data for the excess adsorption of helium on 13X zeolite pellets (Chemie Uetikon Z10-02) are reported and compared with modeling results that account for mobile adsorption of helium and for thermal expansion of the solid parts inside the measuring cell. For the sake of convenience, data are plotted as V0 - ΓHe/FbHe, that is, the quantity that can be directly measured versus temperature. The parameters and material properties used to correlate the model with the experimental data are given in Table 1. Besides the value of V0 at the reference temperature, T* ) 323 K, the fluid-surface interaction energy, Ufs, and the bonding frequency of adsorbed helium atoms normal to the surface, ν⊥, have been treated as (20) Sircar, S. AIChE J. 2001, 47, 1169-1176. (21) Sircar, S. Role of helium void measurement in estimation of Gibbsian surface excess. In Fundamentals of Adsorption, Vol. FOA7; Kaneko, K., Kanoh, H., Hanzawa, Y., Eds.; IK International: Chiba, Japan, 2001. (22) Rajendran, A.; Hocker, T.; Di Giovanni, O.; Mazzotti, M. Langmuir 2002, 18, 9726-9734.

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Figure 2. Excess adsorption isotherms for helium on 13X zeolite pellets; the isotherms were extracted from the isochoric results shown in Figure 1. The symbols represent the experimental data, whereas the dashed lines represent the model predictions that account for mobile adsorption of helium and for thermal expansion of the solid parts inside the measuring cell (see Appendix A and ref 22 for the modeling details).

adjustable parameters to fit the data. Note, however, that the assumed values for Ufs and ν⊥ lie within the physically meaningful range. For helium adsorption on graphite, a value of Ufs/k ) - 247 K (Ufs ) -491cal/mol) for the Hecarbon surface interaction energy was reported,23 which is in close agreement with the fitted value of Ufs/k ) -240 K. A rough estimate of the bonding frequency as a function of the fluid-surface interaction energy can be obtained from ν⊥ ≈ 0.762[-Ufs/(rj2m)]1/2.22,24 For Ufs/k ) -240 K, rj(He) ) 2.869 Å, and m ) 6.64 × 10-27 kg, the above expression gives ν⊥ ) 1.87 × 1012 Hz, which agrees well with the fitted value of ν⊥ ) 1.7 × 1012 Hz (see Table 1). Using the model to extrapolate the data to ΓHe/FbHe f 0, a value of V0(310 K) ≈ 4.285 mL is estimated for the volume of the solid parts. This value is about 4% higher than what one would obtain by neglecting He adsorption at 410 K; neglecting He adsorption at a lower temperature, namely, 310 K, would result in an underestimation of V0 by 5%. Note also that the thermal expansion of the solid parts inside the measuring cell causes only a minor temperature dependence of V0 due to the small values for Rsorb and Rmet, that is, the linear thermal expansion coefficients of the sorbent particles and of the metal parts, respectively. For example, the temperature increase from 310 to 410 K causes an increase in V0 by thermal expansion of only 0.15%. As seen from Figure 1, the model describes the shape of the experimental isochores well and provides a satisfactory prediction of the density dependence. This is further shown in Figure 2, which illustrates helium excess adsorption isotherms that were extracted from the isochoric results shown in Figure 1. Also shown in Figure 1 is the model prediction in the limit FbHe f 0, that is, in the Henry’s law region where Γ(Fb, T)/Fb|Fbf0 ) H(T) (dashed-dotted line, see Appendix A). Note that the experimental data are clearly outside the region where Henry’s law is valid, that is, where a helium adsorption protocol such as the one proposed by Sircar could be applied.20,21 To check our findings about the extrapolated prediction of the Henry’s law region’s behavior, we compared our model results with the experimental data of Springer et al.25 for helium adsorption (23) Hellemans, R.; Itterbeek, A. V.; Dael, W. V. Physica 1967, 34, 429-437. (24) Hill, T. L. Adv. Catal. 1952, 4, 211-258. (25) Springer, C.; Major, C. J.; Kammermeyer, K. J. Chem. Eng. Data 1969, 14, 78-82.

Hocker et al.

on 13X zeolite and with independent results reported by Sircar based on the assumption that the isosteric heat of adsorption at zero coverage is independent of temperature (see eq 20 and Table 2 in ref 20). The results are illustrated in Figure 3, where it can be readily observed that Sircar’s model, the model fitted on experimental data, and our model are consistent for temperatures above 300 K, the model to model variation being of a smaller magnitude than the experimental uncertainty. 3.2. Adsorption of Carbon Dioxide on 13X Zeolite Pellets. The experimentally measured excess isotherms for adsorption of carbon dioxide on 13X zeolite pellets are reported in Table 2 and shown in Figure 4. Experiments have been carried out in the reduced temperature range 1.00246 e T/Tc e 1.282, that is, as low as 0.75 K above the critical temperature. Due to the nature of the definition of the excess amount, that is, the truly experimentally measurable quantity, all excess isotherms, in general, increase with increasing bulk density, reach a maximum, and fall off with a further increase in the bulk density.14,15 Accordingly, all experimental isotherms in Figure 4 show a sharp rise, reach a maximum, and fall off with a further increase in density. It can be seen that as the temperature is decreased, the location of the maximum slightly shifts to lower densities. The most striking feature of the isotherms can be seen in their descending part. The isotherms corresponding to T/Tc g 1.06 fall off almost linearly with the density, whereas the isotherms corresponding to T/Tc ) 1.0133 and 1.00246 exhibit a characteristic “bump” at densities slightly below the critical density. More in detail, these two isotherms decrease linearly after the maximum at low density, but while approaching the critical density there is a rise in the excess amount adsorbed, leading to the appearance of a bump at a density slightly below the critical density. After the bump, the expected linear fall of the isotherm with increasing bulk density is observed again. The bump is first observed at T/Tc ) 1.0133, that is, at T ) 308.14 K, and is very evident in the isotherm corresponding to T/Tc ) 1.00246, that is, at T ) 304.85 K. The correctness of these experimental observations has been confirmed by carrying out isothermal runs, in both the adsorption and the desorption modes, as well as isochoric runs at densities slightly below the critical density. In all cases, the same values of the excess amount adsorbed have been measured. A similar behavior, that is, the appearance of a bump at a density close to the critical density, has been observed previously for the adsorption of supercritical carbon dioxide on Calgon F400 activated carbon.26 However, in that case the observed bumps occur at densities slightly above the critical density. In Figure 5, the isotherms shown in Figure 4, that is, correctly measured by accounting for both helium adsorption and thermal expansion, are compared with the ones obtained assuming that helium adsorption is negligible at T ) 295.05 K and FbHe ) 6.1875 mol/L, that is, following a protocol that is not consistent with the evidence reported in the previous section. At low bulk densities, the corrections due to helium adsorption and thermal expansion are negligible, whereas at higher densities they lead to significant deviations between the corrected and the uncorrected results, as expected. Even though the error in the adsorption excess becomes as large as 30%, all the qualitative features of the isotherms remain unchanged. (26) Humayun, R.; Tomasko, D. L. AIChE J. 2000, 46, 2065-2075.

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Table 2. Experimental Excess Adsorption Data of CO2 on 13X Zeolite Pellets T [K]

F [g/L]

nex [mmol/g]

T [K]

F [g/L]

nex [mmol/g]

T [K]

F [g/L]

nex [mmol/g]

304.888

10 11 26 51 69 76 96 144 165 165 210 250 295 309 361 431 453 462 472 516 529 552 569 593 594 609 615 626 637 645 663 669 683 703 709 726 734 770 793 834

6.00 6.09 6.41 6.52 6.56 6.60 6.56 6.51 6.50 6.47 6.38 6.30 6.20 6.16 6.03 6.03 5.73 5.59 5.53 5.24 5.02 4.93 4.72 4.66 4.64 4.60 4.51 4.45 4.43 4.36 4.26 4.20 4.15 4.04 3.99 3.94 3.89 3.73 3.63 3.45

304.888

842 892 898 947 7 18 60 86 107 138 183 221 243 295 399 418 432 432 456 474 487 532 553 568 599 621 636 644 651 651 8 15 35 55 78 116 160 222 323 435

3.42 3.23 3.20 3.04 5.62 6.03 6.36 6.39 6.38 6.35 6.28 6.19 6.13 5.99 5.65 5.58 5.45 5.45 5.32 5.23 5.22 4.96 4.84 4.76 4.60 4.49 4.43 4.39 4.38 4.38 5.37 5.68 5.99 6.10 6.14 6.13 6.05 5.88 5.51 5.00

322.388

481 687 761 8 21 38 58 79 101 123 158 194 325 419 576 6 19 34 48 65 83 97 118 141 164 190 235 254 269 293 308 321 346 379 409 465 517 570

4.80 3.97 3.70 4.62 5.18 5.43 5.55 5.60 5.62 5.60 5.54 5.46 5.03 4.67 4.07 3.65 4.49 4.81 4.97 5.07 5.12 5.14 5.14 5.12 5.09 5.04 4.93 4.87 4.82 4.75 4.67 4.65 4.57 4.45 4.35 4.14 3.98 3.78

308.185

322.388

Figure 3. Henry’s law predictions of the model used to fit the data reported in Figures 1 and 2 for nonlinear adsorption of He on 13X zeolite pellets. Also shown are Henry coefficient measurements by Springer et al. (ref 25), a best fit to these data, and independent results reported by Sircar (based on eq 20 and Table 2 in ref 20).

4. Lattice DFT Model for Adsorption in Single Pores 4.1. Theory. Any model that aims at capturing the large variety of phenomena of near-critical fluids adsorbed in porous sorbents needs to incorporate information about the adsorbent geometry, such as pore sizes and shapes, as well as information about the fluid-fluid and fluidsurface interactions. The simplest possible model description approximates the sorbent material as a discrete distribution of properly weighted model pores with structureless walls of simple, cylindrical or slitlike shape. To describe the fluid behavior, in the simplest case, the pore

355.844

389.907

space is discretized so that fluid molecules inside the pores are confined to sites that form a regular pattern, that is, a lattice. Models that are based on the above assumptions belong to the class of lattice-gas models or lattice DFT.27 In this study, we apply the Aranovich-Donohue formalism6 within the framework of lattice DFT in order to model the supercritical adsorption behavior of a pure fluid confined in pores of different size and shape. It has been shown by Aranovich and Donohue that the statistical theory of Ono and Kondo28-30 can be generalized by employing a “thermodynamic” treatment that is based on identical assumptions. This latter formalism has the advantage of being much more flexible than the original derivation in that it allows one to model systems with complex boundaries (porous surfaces,31 energetically heterogeneous surfaces32), molecules with directional bonds,33 and molecules of different size and shape.34-36 (27) Hill, T. L. An Introduction to Statistical Thermodynamics; Dover: New York, 1986. (28) Ono, S.; Kondo, S. Molecular Theory of Surface Tension in Liquids. In Handbuch der Physik, Vol. X; Flu¨gge, S., Ed.; SpringerVerlag: Go¨ttingen, 1950; Chapters 50 and 51. (29) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Clarendon Press: Oxford, 1982. (30) Lane, J. E. Adsorption from Mixtures of Miscible Liquids. In Adsorption from Solution at the Solid/Liquid Interface; Parfitt, G. D., Rochester, C. H., Eds.; Academic Press: London, 1983. (31) Aranovich, G. L.; Donohue, M. D. J. Colloid Interface Sci. 1998, 205, 121-130. (32) Aranovich, G. L.; Donohue, M. D. J. Chem. Phys. 1996, 104, 3851-3859. (33) Aranovich, G. L.; Donohue, P.; Donohue, M. D. J. Chem. Phys. 1999, 111, 2050-2059.

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where ∆u and ∆s denote the energy and entropy changes associated to such pore/bulk swap of the molecule. To proceed further, expressions for ∆u and ∆s must be derived. For calculating the entropy change when energy fluctuations are small, the microcanonical formalism can be applied.27 Therefore, ∆s can be written as

( )

∆s ) k ln

Figure 4. Supercritical isotherms for the excess adsorption of CO2 on 13X zeolite pellets as a function of the reduced bulk density, Fb/Fc. The symbols represent experimentally measured values and are connected by lines to guide the eye. Note that for the two lowest temperatures, i.e., for T/Tc ) 1.00246 and 1.0133, the isotherms exhibit a bump on their descending branches at densities slightly below Fc.

WI WII

(2)

where k is Boltzmann’s constant, and WI is the number of configurations of the system in state I, which is subject to the constraint that the macroscopic variables (T, V, N) are kept constant and moreover that a “microscopic constraint” applies as well, namely, that site (i, j, k) is occupied. Similarly, WII is the number of configurations for the system in state II, subject to constant (T, V, N) and to the microscopic constraint that the molecule that previously occupied site (i, j, k) is now in the bulk. Although it is difficult to calculate WI and WII separately, the ratio, WI/WII, can be estimated as follows. Let us define Wtot as the number of possible configurations for the system subject only to constant (T, V, N). Furthermore, let us assume that the probability (or lattice-site occupancy) of having a molecule on site (i, j, k), θi,j,k, is independent of its given bulk value, θb. Then, the probabilities of having the system in states I and II can be approximated as

PI )

WI ≈ θi,j,k(1 - θb) Wtot

(3)

PII )

WII ≈ θb(1 - θi,j,k) Wtot

(4)

and

Figure 5. Supercritical isotherms for the excess adsorption of CO2 on zeolite 13X pellets as a function of the reduced bulk density, Fb/Fc. The symbols represent the results corrected for helium adsorption and thermal expansion of the solid materials inside the cell, i.e., they are identical with the results shown in Figure 4, whereas the dashed curves represent the uncorrected data.

Combining eqs 2, 3, and 4 leads to the following expression for the entropy change:

∆s ) k ln

(

b θi,j,k (1 - θ ) θb (1 - θi,j,k)

)

(5)

(1)

Calculating the energy change, ∆u, requires specification of the lattice geometry and the lattice boundary conditions. However, it is assumed that only the configurational contribution to the energy is affected by the aboveconsidered pore/bulk swap of a molecule. To calculate the configurational energy, let us assume pairwise additivity, which means that the total interaction energy is given by the sum of the simultaneous interactions among all pairs of molecules. These pair interactions shall be restricted to those between nearest neighbors, designated by ff. This assumption can be relaxed to allow for longer-range interactions, but the mathematical complexity increases.37 Consider again Figure 6 and the molecule on site (i, j, k). Assuming for instance a cubic lattice structure with a three-dimensional coordination number z3D ) 6, corresponding to the number of nearest neighboring lattice sites, with which the molecule can interact, the configurational energy change, ∆u, is given by the following relationship:

(34) Aranovich, G. L.; Hocker, T.; Wu, D.-W.; Donohue, M. D. J. Chem. Phys. 1997, 106, 10282-10291. (35) Aranovich, G. L.; Donohue, M. D. J. Colloid Interface Sci. 1999, 213, 457-464.

(36) Wu, D.-W.; Aranovich, G. L.; Donohue, M. D. J. Colloid Interface Sci. 1999, 212, 301-309. (37) Runnels, L. K. Lattice gas theories of melting. In Phase Transitions and Critical Phenomena, Vol. X; Domb, C., Green, M. S., Eds.; Academic Press: Go¨ttingen, 1972.

Figure 6. Exchange of a molecule on a lattice site (i, j, k) inside a single three-dimensional pore with a vacancy in the bulk. Solid symbols represent occupied sites, whereas open symbols represent unoccupied lattice sites, i.e., vacancies.

To derive the local density equations for a pure fluid in a three-dimensional pore of arbitrary geometry, let us consider the exchange of a molecule on a lattice site (i, j, k) inside the pore with a vacancy in the bulk, as illustrated in Figure 6. Let us assume that this exchange occurs under local equilibrium. Therefore, from the uniformity of the chemical potential, such an exchange has to obey the condition

∆u - T∆s ) 0

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∆u ) -ff(θi+1,j,k + θi-1,j,k + θi,j+1,k + θi,j-1,k + θi,j,k+1 + θi,j,k-1 - 6θb) (6) where a random distribution of molecules in the bulk has been assumed. Insertion of eqs 5 and 6 into eq 1 results in

ff(θi+1,j,k + θi-1,j,k + θi,j+1,k + θi,j-1,k + θi,j,k+1 + θi,j,k-1 - 6θb) + kT ln

[

]

b θi,j,k (1 - θ ) ) 0 (7) θb (1 - θi,j,k)

Similarly, for the exchange of a molecule on a site (i, j, k), which is now directly adjacent to a portion of the sorbent walls at location (i - 1, j, k), one obtains

∆u ) -Ufs - ff(θi+1,j,k + θi,j+1,k + θi,j-1,k + θi,j,k+1 + θi,j,k-1 - 6θb) (8) where the fluid-wall interaction energy, Ufs, has been introduced. Substituting eqs 5 and 8 into eq 1 yields

Figure 7. Density profiles of a pure lattice gas adsorbed in a one-dimensional, 20-layer slit-pore. Shown are results for a constant ratio of fluid-surface to fluid-fluid interaction energies, Ufs/ff ) 6. The curves represent lattice DFT predictions based on the model presented in section 4.1. They are compared with lattice Monte Carlo simulations, whose results are shown as open circles.

Ufs + ff(θi+1,j,k + θi,j+1,k + θi,j-1,k + θi,j,k+1 + θi,j,k-1 6θb) + kT ln

[

]

b θi,j,k (1 - θ ) ) 0 (9) θb (1 - θi,j,k)

Consequently, eq 7 can be applied to each lattice site inside the pore that is not at a boundary, whereas an equation similar to eq 9 applies to sites on the pore boundaries. This leads to a system of coupled, nonlinear algebraic equations that can be solved numerically for the local occupancies, θi,j,k. Note that due to the random distribution assumption made for molecules in the bulk, the threedimensional phase behavior of the model is that of a “regular solution”, with a “mean-field” critical temperature of Tc ) z3D|ff|/(4k).27 Furthermore, the model has a critical density of θc ) 0.5, which is independent of the dimensionality. This is a general feature of Ising type lattice models based on simple nearest-neighbor interactions and a fixed lattice spacing.38 4.2. Adsorption of Supercritical Fluids inside Single Pores. Figure 7 illustrates the solution of the lattice DFT model, eqs 7 and 9, for a pure fluid adsorbed in a one-dimensional slit-pore with a width of 20 molecular layers and the fluid/wall interaction energy 6 times larger than the fluid/fluid interaction energy, that is, Ufs/ff ) 6. The figure reports the local density profile between layer 1 and layer 20, which are adjacent to the two faces of the slit-pore, for three different pairs of temperature and bulk density values, namely, T/Tc ) 1.3, 2.0, and 3.0 and θb ) 0.43, 0.24, and 0.08, respectively. The lines, representing the lattice DFT results, are compared with lattice Monte Carlo simulations, shown as open circles. As expected for temperatures well above Tc, as in this case, the predictions of the DFT model agree well with the essentially exact Monte Carlo simulations. At T/Tc ) 3.0, the fluid density remains at its bulk value up to the second fluid layer away from the walls, namely, layers 2 and 19. This implies that under these conditions, the monolayer adsorption assumption is well satisfied. When the temperature is lowered to T/Tc ) 2.0 and 1.3, the fluid density still stays at its bulk value in the center of the pore, that is, in the core of the pore. However, the number of fluid layers close (38) Hill, T. L. Statistical Mechanics; Dover: New York, 1987.

Figure 8. Excess isotherms of a pure lattice gas adsorbed in a one-dimensional, 20-layer slit-pore. The model parameters and the symbols are the same as in Figure 7.

to the walls that are affected by the surface potential increases, namely, to three at T/Tc ) 2.0 and to five at T/Tc ) 1.3. Once the local occupancies inside a model pore have been calculated, the adsorption excess based on lattice occupancies, Γlat, follows as Mlat

Γlat )

(θi,j,k - θb) ∑ i,j,k

(10)

where Mlat denotes the total number of lattice sites in the pore. Introducing the mean occupancy inside a pore,

θ hp )

1

Mlat

∑ θi,j,k

Mlat i,j,k

(11)

eq 10 can be recast as

h p - θb) Γlat ) Mlat(θ

(12)

Figure 8 presents results for Γlat as a function of θb for the same parameters used in Figure 7, that is, for adsorption of a lattice gas in a 20-layer slit-pore and Ufs/ff ) 6. Note that all isotherms satisfy the limit that Γlat f 0 as θb f 0 or θb f 1. Moreover, they go through maxima that shift

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Figure 9. Lattice-gas model predictions for supercritical adsorption in one-dimensional slit-pores of different widths, corresponding to micro-, meso-, and macropores. Γlat denotes the total excess inside the pores, whereas Γ1lat and Γ1-3 lat denote the contributions to the excess of the first and the first three fluid layers, respectively, adjacent to the sorbent walls. The ratio of fluid-surface to fluid-fluid interactions, Ufs/ff, is held constant at a value of 5. The upper series of graphs show the results for a temperature well above the critical temperature, i.e., T/Tc ) 1.2, whereas in the lower series of graphs, T is only slightly above Tc, i.e., T/Tc ) 1.0001.

to lower values of θb with decreasing temperature. After reaching their maximum value, the isotherms fall off almost linearly. To analyze the effect of pore size on the adsorption behavior of supercritical fluids, Figure 9 shows lattice DFT model predictions for adsorption in one-dimensional slit-pores of three different widths, corresponding to micro(3 layers in all), meso- (20 layers in all), and macropores (200 layers in all). In the figures, Γlat denotes the total excess, as defined by eq 10, whereas Γ1lat and Γ1-3 lat denote the contributions to the total excess from the first and the first three fluid layers adjacent to the sorbent walls, respectively. By definition, Γlat ) Γ1-3 lat for micropores with only three layers in all. The ratio of fluid-surface to fluidfluid interactions, Ufs/ff, is kept constant at a value of 5. Two temperature levels are considered, namely, rather far and very close to the critical temperature. At the higher temperature, that is, T/Tc ) 1.2, the adsorption behavior for all three pore sizes is characterized by a steep increase in the excess at low bulk densities, a maximum when the saturation of fluid molecules in the near-wall regions is reached, and a monotonic decrease for higher bulk densities. For the meso- and macropores, the values for Γ1-3 lat and Γlat almost coincide, and there is very little adsorption in the core of the pores; that is, the density there equals the bulk density. In fact, being far from the critical temperature, the effect of the wall attraction is essentially local and does not extend far into the pore. Consequently, for temperatures well above Tc, the pore size has only a small effect on the observed adsorption behavior. Now let us consider the isotherms in Figure 9 for a temperature only slightly above the critical temperature, that is, for T/Tc ) 1.0001. It can be readily observed that the isotherm corresponding to the microporous regime has the same shape as the one at a higher temperature, and it attains only slightly larger values, due to the lower temperatures. The situation is rather different for larger pore sizes. In the case of adsorption in meso- and macropores, Γlat exhibits a sharp peak that occurs at densities slightly below the bulk critical density, θc ) 0.5. Moreover, Γlat for density values close to θc is much larger

than Γ1-3 lat , thus indicating that the peaks are caused by fluid-density values in the core of the meso- and macropores that exceed the bulk density. We believe that the peaks in the isotherms observed for adsorption on meso- and macropores represent degenerated forms of a true divergence of the excess that occurs for adsorption on flat open surfaces when the critical point is approached from above along the critical isochore.18 This phenomenon, termed “critical adsorption”, reflects the collective behavior of a fluid under near-critical conditions. In other words, it is due to the increase of the correlation length in a fluid that approaches the critical temperature from above, along a critical isochore. For porous sorbents, this results in a propagation of the effect of the attractive potential exerted by the sorbent walls into the core regions of the pores which causes the excess isotherm to go through a maximum. In the following discussion, the value of Γlat corresponding to this maximum is indicated as Γc, that is, the “criticality peak”. Due to geometrical confinement, this effect is more evident in macropores than in mesopores, whereas it disappears in micropores, as shown in Figure 9. To analyze in more detail the fluid behavior inside a mesopore, let us consider the lattice-gas model predictions for supercritical adsorption in a 20-layer slit-pore at different temperatures that are shown in Figure 10. The ratio of fluid-surface to fluid-fluid interactions has been fixed at a value of 5; thus, the isotherms at T/Tc ) 1.0001 and 1.2 are identical with those already shown in Figure 9 for mesoporous adsorption. Note also that the isotherm at T/Tc ) 1.0001 exhibits a criticality peak at a bulk density of about 0.43. To illustrate the effect of near-critical fluid behavior on the adsorption excess, consider the increase in Γlat at θb ) 0.43 induced by temperature decreases when starting from different levels above Tc. Let us consider temperatures well above Tc, in particular the temperature change from T/Tc ) 1.7 to T/Tc ) 1.2, that is, a decrease of about 30%, which corresponds to a certain increase in the adsorption excess, Γlat. As shown in Figure 10, the same absolute change in the adsorption excess, Γlat, can be obtained when decreasing the temperature from T/Tc

Supercritical Adsorption in Porous Solids

Figure 10. Lattice-gas model predictions for supercritical adsorption in a 20-layer slit-pore. As in Figure 9, the ratio of fluid-surface to fluid-fluid interactions, Ufs/ff, is held constant at a value of 5.

Langmuir, Vol. 19, No. 4, 2003 1261

Figure 12. Lattice-gas model predictions of reduced bulk densities, θb/θc|Γc, at which the corresponding excess isotherms exhibit a maximum associated with the phenomenon of critical adsorption. For a fixed temperature of T/Tc ) 1.0001, θb/θc|Γc is plotted as a function of the width, i, of one-dimensional slitpores, for different ratios of fluid-surface to fluid-fluid interactions, Ufs/ff.

θb/θc|Γc, for which Γlat ) Γc, at a fixed reduced temperature T/Tc ) 1.0001. The results are shown in Figure 12, where θb/θc|Γc is plotted as a function of the pore width, i, for different values of Ufs/ff. As expected from the model predictions for an open surface, where Γc f ∞ for T f Tc+ and Fb ) Fc,18 all curves asymptotically approach the limit of θb/θc|Γc ) 1 for large values of i, that is, in the macropore limit. Furthermore, for ratios of fluid-surface to fluidfluid interactions greater than 5, the curves practically are indistinguishable for all but very narrow pores.

Figure 11. Lattice-gas model predictions for the local density profiles inside a mesosize, 20-layer slit-pore at a fixed bulk density of θb ) 0.43. See Figure 10 for the corresponding excess values that follow from integration over the pore width of the difference between the local densities and the bulk density, which is represented by the dashed horizontal line.

) 1.005 to T/Tc ) 1.0001, which corresponds to a temperature change of only about 0.5%. To better understand the strong increase in Γlat at θb ≈ 0.43 when approaching the critical temperature from above, Figure 11 shows the corresponding local density profiles inside the mesopore considered in Figure 10. Here, the bulk density is represented by the dashed horizontal line. Therefore, according to eq 10, for each temperature the adsorption excess is given by the area between the relevant local density curve and the bulk density line. When decreasing the temperature from T/Tc ) 1.7 to 1.2, the resulting gain in Γlat is solely caused by a rise of the fluid density in the first five layers adjacent to the walls, whereas the fluid density in the core region remains at its bulk value. However, the opposite behavior is observed when lowering the temperature from T/Tc ) 1.005 to 1.0001. The local density values adjacent to the walls are unaffected by the temperature change, whereas the fluid density in the core of the pore rises considerably. This confirms that for a fluid under near-critical conditions the attractive potential exerted by the sorbent walls propagates into the core regions of the pores. From the discussion of Figures 9-11, it is clear that the excess value where the isotherm exhibits a criticality peak, Γc, is a function of both temperature and pore width. To investigate the influence of the latter in a systematic fashion, we have calculated reduced bulk densities,

5. Modeling the Excess Adsorption on Real Adsorbents 5.1. Theory. To correlate the lattice DFT model with the experimental excess adsorption data, it is necessary on one hand to establish conversion rules between lattice fluid units and physical gas units and on the other hand to account for the complex pore structure of the adsorbent. The former issue is tackled by attributing physical meaning to specific values of the lattice occupancy, θ, and by establishing then a one-to-one mapping between θ and F, under the constraint that these specific θ values map onto the corresponding F values of the bulk density. The latter issue is addressed by considering the pore size distribution of the adsorbent particles and by properly discretizing it, that is, by substituting the experimentally measured continuous distribution with a set of a finite number of different pores, each having a given size and weight in the discretized distribution. Let us consider an adsorbent particle of mass msorb and a total pore volume of Vtot. Hence, the specific pore volume is vtot ) Vtot/msorb. According to its definition, for this particle the adsorption excess is given by

nex )

1 msorb

∫V

[F(x, y, z) - Fb] dV

tot

(13)

The pore volume, Vtot, is constituted of pores of different size, as characterized by its pore size distribution. In the framework of this modeling approach, the volume-averaged pore size distribution is discretized into K different types of pores labeled k ) 1, 2, ..., K, each with a volume Vk. Therefore,

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K

vtot )

∑ vk k)1

(14)

where vk ) Vk/msorb, and K

nex )

nex ∑ k k)1

(15)

where nex k is the adsorption excess in the pores of type k. For a lattice model and for pores of type k (constituted of Ik lattice sites), the adsorption excess is calculated through the following discretized form of the integral in eq 13:

nex k

)

vk

Ik

∑[Fk(i) - F ] b

Ik i)1

(16)

The fluid density, F, of the real adsorbate in the last equation is determined from the value of the lattice occupancy, θ, calculated using the lattice DFT model. To this aim, a functional relationship F ) g(θ) must be established, where the function g(θ) fulfills the following physical requirements. First, at zero density the site occupancy is zero, that is, g(0) ) 0. Second, the critical density of the fluid, Fc, must map onto the critical density of the lattice gas, θc ) 0.5; hence g(0.5) ) Fc. Finally, there is a maximum density, Fmax, corresponding to the total occupancy of the lattice sites, that is, g(1) ) Fmax. The following smooth invertible function, which is also shown in Figure 13, fulfills the three conditions:

FmaxFcθ F ) g(θ) ) Fmax(1 - θ) - Fc(1 - 2θ)

(17)

Substituting eq 17 into eq 16 yields

vk

nex k )

Ik

∑[g(θk(i)) - g(θb)]

Ik i)1

(18)

where θb ) g-1(Fb). Equation 18 gives the adsorption excess contribution due to the pores of type k. Substituting this into eq 15 yields the final form of eq 13, as it can be obtained from the lattice DFT model: K

vk

Ik

[g(θk(i)) - g(θb)] ∑ ∑ I k)1 i)1

nex )

(19)

k

Therefore, the calculation of nex from the model requires the calculation of the lattice occupancy, θk, for pores of type k, with k ) 1, 2, ..., K, and then the weighted summation of the obtained results according to eq 19. 5.2. Excess Adsorption of CO2 onto 13X Zeolite Pellets. The above theoretical analysis of the adsorption behavior of supercritical fluids indicates that to describe adsorption on a commercial adsorbent under near-critical conditions, its pore size distribution must be accounted for. Therefore, our approach is the following. We calculate the excess adsorption isotherm with the lattice model, as the weighted average of the excess isotherms obtained in single pores of different size and shape. In practice, the continuous experimental pore size distribution is substituted and mimicked by a discrete one. As illustrated in Figure 14a, the 13X zeolite pellets exhibit a bimodal pore size distribution, constituted of meso- and macropores in a rather broad range and of micropores of a single size of

Figure 13. Plot of the mapping function, F ) g(θ), between lattice site occupancy, θ, and density of the sorbate, F (see eq 17 for the definition).

about 12 Å, corresponding to the diameter of the R-cages inside the crystals. Figure 14b shows the discretized distribution used in the model, comprising two pore types, one in the micropore region, that is, 12 Å wide, and one in the mesopore region, that is, 120 Å wide. Since for CO2 σ ≈ 4 Å, we have chosen a two-dimensional channel-pore with a cross section of 3 × 3 layers. An underlying cubic lattice structure, which provides the most realistic description of the arrangement of the fluid molecules within such a micropore, has been chosen to approximate the connected channels in the zeolite crystals. The meso- and macropores formed by the binder material have been approximated by a one-dimensional, 30-layer slit-pore with an underlying hexagonal lattice structure. The hexagonal structure is the closest to physical reality, whereas computations not reported here demonstrate that for such big pores, one-dimensional and two-dimensional pores yield the same results. Hence, simpler one-dimensional slit-pores have been used here. All other lattice model parameters are given in Table 3; they have been selected according to the criteria discussed in Appendix B.1. The calculated individual contributions of the chosen two pore types and the resulting overall excess isotherm at T/Tc ) 1.00246 are shown in Figure 15, together with the corresponding experimental data from Figure 4, which exhibit the characteristic bump near the critical density. After the discussion of the theoretical results shown in Figures 9-12, it is not surprising that close to Tc, the fluid adsorbs on micropores very differently than on meso- and macropores, as seen in Figure 15. When the two contributions are combined, an excess isotherm in remarkable agreement with the experimental results is obtained. In particular, the characteristic bump near Fc is well described. Therefore, our analysis leads to the conclusion that the peculiar shape of the isotherm very close to the critical temperature is due to the bimodal pore size distribution with well-separated pore size values of the two peaks. We believe that this might also explain the same phenomenon observed for carbon dioxide adsorption onto activated carbon.26 In fact, activated carbons also exhibit a multimodal pore size distribution.39 The fact that in the case of carbon dioxide on activated carbon the bump in the isotherm was observed at a density value slightly higher than the critical density is not consistent with our theoretical analysis and experimental evidence. Those measurements were carried out in an experimental setup where the direct measurement of densities was not possible; thus the density was calculated based on the temperature and pressure readings.26 Since this may lead (39) Ruthven, D. M. Principles of adsorption and adsorption processes; John Wiley: New York, 1984.

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Figure 14. (a) Volume-weighted PSD of 13X zeolite pellets; the peak at 12 Å represents the diameter of the R-cages of the crystals, whereas the distribution of the meso- and macropores was provided by the supplier (ref 45). (b) Discrete bimodal pore size distribution used in the model (3 × 3 layers, 45%; 30 layers, 55%). Table 3. Values Used to Correlate the Adsorption Model with Experimental Data for the Excess Adsorption of CO2 on 13X Zeolite Compared with the Ranges of Values for the Relevant Physical Propertiesa parameter

physical properties

ff/k q0st/k Ufs/k Fmax/Fc Fmaxvtot PSD

Tc(CO2) ) 304.14 K 4450-5530 Kb -1750 to -2650 K 1.45-3.45 9.2-23.1 mmol/g Figure 14a

model values -4Tc/z3D -1825 K 3.45 12.9-15.2 mmol/g Figure 14b

a z3D ) 6 for the cubic lattice (micropores) and z3D ) 12 for the hexagonal lattice (mesopores). More details concerning how the latter were obtained are given in Appendix B.1. b References 46 and 47.

Figure 16. Temperature dependence of the pore-filling capacity, Fmaxvtot, to obtain a best fit of the adsorption model to the experimental data at different temperatures for adsorption of CO2 on zeolite 13X pellets, as shown in Figure 17.

Figure 15. Comparison of the lattice model prediction (continuous line) with the experimental isotherm (symbols) for CO2 adsorption on 13X zeolite pellets at T/Tc ) 1.00246. The contribution of each pore width is shown as curves, and the corresponding pore size (in layers) is shown alongside. The model parameters are given in Table 3. The pore-filling capacity is Fmaxvtot ) 15.2 mmol/g.

to non-negligible errors, particularly close to the critical point, we conjecture that the occurrence of the bump at F > Fc is an artifact of the uncertainty in evaluating the experimental density value. Let us now consider the effect of changing temperature on the qualitative and quantitative features of the isotherms. With increasing temperature, the lattice model correctly predicts that the bump in the isotherm disappears and that the shape of the model isotherms is very similar to the experimental ones. Despite this satisfactory qualitative agreement, it was not possible to achieve a good quantitative agreement when keeping the pore-filling capacity of the adsorbent, Fmaxvtot, constant with temperature. While the lattice model presented in section 4.1 assumes a constant spacing of lattice sites and therefore is in principle not consistent with a temperature-dependent pore-filling capacity, recent theoretical results suggest that in more realistic lattice models, based on a variable

lattice spacing, the maximum adsorption capacity is not constant anymore.40 Therefore, we have calculated the experimental isotherms in Figure 4 by regarding all parameters in Table 3 as temperature independent but the pore-filling capacity. The latter has been considered as temperature dependent, and its value has been fitted to allow for a good description of the experimental isotherms at different temperature levels. How Fmaxvtot depends monotonically on temperature is illustrated in Figure 16. The comparison between the experimental data and the calculated isotherms is presented in Figure 17, which demonstrates an excellent agreement between calculated and experimental isotherms over the whole range of temperatures and densities. Note that a temperature dependence of Fmaxvtot, similar to that in Figure 16, has been reported by Be´nard and Chahine, who used the Ono-Kondo lattice theory, for describing methane and hydrogen adsorption on microporous carbons and zeolites.41,42 5.3. Excess Adsorption of CO2 onto Silica Gel. The procedure presented above bears general validity. After using it for a system with a bimodal pore size distribution, with a rather wide gap in the size between micropores and mesopores, in this section we apply it to the adsorption of CO2 on silica gel, that is, an adsorbent with a unimodal and broad pore size distribution. The experimental results about CO2 adsorption on silica gel (Kieselgel 60) were reported in a previous study.15 As shown in Figure 18a, silica gel has a broad volumeaveraged pore size distribution (PSD), containing pores ranging from micropores to macropores, with the average (40) Aranovich, G. L.; Donohue, M. D. Colloids Surf., A 2001, 187, 95-108. (41) Be´nard, P.; Chahine, R. Langmuir 1997, 13, 808-813. (42) Be´nard, P.; Chahine, R. Langmuir 2001, 17, 1950-1955.

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pellets and for sharing PSD data. Partial support of the Swiss National Science Foundation through Grant SNF21-55674.98 is gratefully acknowledged. Appendix A. Estimation of the Volume of the Solid Parts A summary of the necessary steps to obtain V0 is given in the following; further details can be found elsewhere.22 First, notice that in gravimetric measurements, the excess amount adsorbed (in units of mass of adsorbate) can be written as

Γ(Fb, T) ) M1(Fb, T) - M1(0,∞) + FbV0 Figure 17. Comparison of lattice DFT model calculations with the experimental data for CO2 adsorption on 13X zeolite pellets presented in Figure 4. The model parameters are given in Table 3, except for the pore-filling capacity, Fmaxvtot, which was allowed to depend on temperature, as shown in Figure 16.

pore size being 60 Å. Hence, unlike the 13X case, where the PSD was discretized into two pore types, a finer discretization is required to mimic the real pore size distribution of silica gel. The discretized pore size distribution used in the model is shown in Figure 18b. This is constituted of 4-layer micropores, accounting for 8% of the total pore volume, 6-layer micropores (12% of the total pore volume), 8-layer micropores (20%), 12-layer mesopores (30%), 20-layer mesopores (20%), and 40-layer macropores (10%). Such a distribution has an average pore size of 56.3 Å, which is comparable to that of the real solid, that is, 60 Å. For the lattice DFT simulations, the pores were described as one-dimensional slit-pores with an underlying hexagonal lattice structure. Other parameters used for the simulations are given in Table 4 and have been selected according to the criteria discussed in Appendix B.2. The lattice DFT calculations along with the experimental data for the isotherm at the lowest temperature, T/Tc ) 1.025, are illustrated in Figure 19. The contributions of the different pore types and the total calculated excess adsorption are shown. It can be seen that there is good agreement between the experimental isotherm, which exhibits a monotonic decrease after the maximum at Fb/Fc ) 0.6, and the calculated one. Similar to the previous case, when the pore-filling capacity was kept constant, only a qualitative agreement was obtained between the experimental and the calculated isotherms at higher temperatures. Hence, to obtain a better quantitative agreement, the pore-filling capacity, Fmaxvtot, was allowed to vary with temperature. The temperature dependence of the pore-filling capacity for the CO2-silica gel system is shown in Figure 20. The rather satisfactory comparison between the experimental data and the calculated isotherms is presented in Figure 21. The analysis indicated that the shape of the isotherms of CO2 on silica gel is typical of systems with a relatively low adsorption energy and a broad, unimodal pore size distribution. On the contrary, bumps in the descending part of the isotherm occur at temperatures very close to the critical one, for systems where the pore size distribution is bimodal or multimodal, and exhibit a marked gap between micropores and mesopores. In both cases, the lattice DFT model is able to describe the qualitative and quantitative features of the isotherms provided that the discretized PSD mimics accurately the real one. Acknowledgment. We thank Chemie Uetikon (Uetikon, Switzerland) for providing samples of 13X zeolite

(20)

where M1(Fb, T) denotes the balance reading at Fb and T, whereas M1(0, ∞) is that where Fb f 0 and T f ∞. Consider now a series of isochoric helium adsorption measurements, that is, a series of values of M1(FbHe, T). Equation 20 then can be rearranged to give

V0 -

Γ(FbHe, T) FbHe

)

M1(0,∞) - M1(FbHe, T) FbHe

(21)

where the right-hand side of eq 21 contains only quantities that can be directly measured in a gravimetric adsorption apparatus. The idea is now to use a physically sound description of the left-hand side of eq 21 to correlate V0(T) with helium adsorption experiments under rather moderate temperature and density conditions. Taking into account thermal expansion of the solid parts inside the measuring cell, that is, V0 ) V0(T), one obtains

V0(T) ) Vsorb(T*) exp[3Rsorb(T - T*)] + mmet exp[3Rmet(T - T*)] (22) met F (T*) where T* represents some (arbitrary) reference temperature, whereas Rsorb and Rmet are the linear thermal expansion coefficients of the sorbent material and the metal parts, respectively. In eq 21, the ratio of the adsorption excess of helium to its bulk density, Γ(FbHe, T)/ FbHe, needs to be determined from an adsorption model. Fortunately, since helium is an “ideal” substance from a modeling viewpoint and since we are measuring helium adsorption at temperatures well above Tc(He) ) 5.26 K, a rather simple monolayer adsorption model will suffice. Assuming monolayer adsorption, Γ(FbHe, T)/FbHe can be written as

Γ(FbHe, T) FbHe

(

) Arj

m θ -1 rj3 FbHe

)

(23)

which is applicable to any adsorption isotherm model θ ) θ(T, FbHe), where θ is the fractional surface coverage. For T . Tc(He), the adsorption behavior of helium can be properly described using an equation of state for a twodimensional fluid with additional terms accounting for adsorbate-surface interactions, as well as for the “frustration” of the translational degree of freedom normal to the sorbent surface. The Hill-de Boer model,

bm(T)Fb )

θ θ exp 1-θ 1-θ

(

)

(24)

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Figure 18. (a) Volume-weighted pore size distribution of silica gel. (b) Discrete pore size distribution used in the model (4 layers, 8%; 6 layers, 12%; 8 layers, 20%; 12 layers, 30%; 20 layers, 20%; 40 layers, 10%). Table 4. Values Used to Correlate the Adsorption Model with Experimental Data for the Excess Adsorption of CO2 on Silica Gel Compared with the Ranges of Values for the Relevant Physical Propertiesa parameter

physical properties

ff/k q0st/k Ufs/k Fmax/Fc Fmaxvtot PSD

Tc(CO2) ) 304.14 K 2000 Kb -1466 to -1580 K 1.45-3.45 11.4-30.8 mmol/g Figure 18a

model values -4Tc/z3D -1568 K 2.5 11.5-22.0 mmol/g Figure 18b

a z3D ) 12 for the hexagonal lattice (mesopores). More details concerning how the latter were obtained are given in Appendix B.2. b Reference 15.

Figure 20. Temperature dependence of the pore-filling capacity, Fmaxvtot, to obtain a best fit of the adsorption model to the experimental data at different temperatures for adsorption of CO2 on silica gel as shown in Figure 21.

Figure 19. Comparison of the lattice DFT model simulations (thick continuous line) with the experimental isotherm (symbols) for CO2 adsorption on silica gel at T/Tc ) 1.025. The contribution of each pore type is shown as curves, and the corresponding pore size given in number of layers is shown alongside. The model parameters are given in Table 4. The pore-filling capacity is Fmaxvtot ) 22.0 mmol/g.

meets the assumptions above.24,43 Taking into account adsorbate-surface, Ufs, as well as adsorbate-adsorbate, ff, interactions, from statistical mechanics the coefficient bm(T) in units of volume per mass adsorbate becomes

jr2kT/(hν⊥)

(

)

Ufs + z2Dffθ bm(T) ) exp kT mx2πmkT/h2

(25)

Here, rj2 is the surface area occupied by an adsorbed molecule, h is the Planck constant, ν⊥ is the bonding frequency with which a molecule vibrates in the direction normal to the surface, m is the mass of a single molecule, and z2D is the number of nearest neighbors an adsorbed molecule can have. For given bulk conditions, eqs 24 and (43) de Boer, J. H. The Dynamical Character of Adsorption, 2nd ed.; Clarendon Press: Oxford, 1968.

Figure 21. Comparison of the lattice DFT model calculations with the experimental data for CO2 adsorption on silica gel. The model parameters are given in Table 4, except for the porefilling capacity, Fmaxvtot, which was allowed to depend on temperature, as shown in Figure 20.

25 can now be used to calculate θ which, when inserted into eq 23, specifies the term Γ(FbHe, T)/FbHe to be used in eq 21. Taking into account eq 22 as well, the functional form of the left-hand side of eq 21 is then fully determined and can be correlated with experimental data. Appendix B. Estimation of Adsorption Model Parameters B.1. Carbon Dioxide on 13X Zeolite Pellets. As mentioned in section 4.1, the lattice fluid model has a critical temperature given by Tc ) z3D|ff|/(4k). For a given adsorbate, the fluid-fluid interaction energy then follows as |ff|/k ) 4Tc/z3D; that is, ff is adjusted so that the critical temperatures of the lattice fluid and of the real species coincide. The choice of z3D depends on the type of lattice

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arrangement chosen. For cubic lattice structure which is used to represent the micropores, z3D ) 6, while for hexagonal lattice structure which is used to represent the macropores z3D ) 12. The fluid-surface interaction energy, Ufs, is estimated as

[ ]

Ufs q0st qb(T) ) - T ln s k k q (T)

(26)

where q0st is the isosteric heat of adsorption at zero coverage. In the last equation, qb(T) and qs(T) denote the single-particle partition functions for molecules in the bulk and in the adsorbed state, respectively. The second term on the right-hand side of eq 26 accounts for the fact that bulk molecules lose certain translational and rotational degrees of freedom during the adsorption process.44 As seen from Table 3, the range 4450 K e q0st e 5530 K was obtained from the literature, whereas 2700 K e T ln[qb(T)/qs(T)] e 2883 K was estimated from calculations based on the following assumptions: carbon dioxide molecules were treated within the rigid rotor, harmonic oscillator approximation, assuming that the internal vibrational modes are not affected by the adsorption process and that all translational and rotational degrees of freedom of CO2 molecules in the bulk are transformed into vibrational modes upon adsorption. The latter assumption corresponds to the picture of “localized adsorption”; that is, we assume that the CO2 molecules are “trapped” onto available adsorption sites within the zeolite channels. The pore volumes of the micro-, meso-, and macropores of the 13X pellets have been estimated as follows: for all the larger pores, a value of vmacro ) 0.33 mL/g has been used, as provided by the supplier based on mercury porosimetry measurements.45 The micropores of the zeolite crystals have a calculated volume per unit cell (uc) of 7832 Å3/uc.46 Taking into account the total unit-cell volume of vuc ) 15 531 Å3/uc and the crystal mass density, Fcrystal ) 1.43 g/mL, the specific volume of the micropores is 0.35 mL per gram of crystals. According to the supplier, the pellets contain between 75 and 85 wt % of crystals, so that vmicro ) 0.26-0.3 mL/g. The total pore volume per gram of sorbent, vtot ) vmicro + vmacro, then becomes vtot ) 0.590.63 mL/g. Finally, a maximum fluid density, Fmax, corresponding to pore filling (θ ) 1 in the lattice model) is assigned. To obtain an estimate for Fmax, consider the number density of close-packed spheres, x2/σ3, where σ denotes the molecule diameter. Since for CO2 σ ≈ 4 Å, Fma ≈ 1.615 g/mL. This value is slightly higher than the density of dry ice, that is, 1.503 g/mL. On the other hand, the saturated liquid density of CO2 at 300 K is 0.679 g/mL, which is about 1.45 times higher than its critical density, that is, Fc ) 0.4676 g/mL. Therefore, the range 1.45 e Fmax/Fc e 3.45 has been considered, and a value within this range has been chosen for the model calculations. B.2. Carbon Dioxide on Silica Gel. The fluid-fluid interaction parameter, ff, is chosen using the same method as explained in the previous section. The fluid-surface interaction energy, Ufs, is estimated using eq 26. Since the fluid-surface interaction energies are not as strong as compared to those in the CO2-13X system, a “mobile (44) McQuarrie, D. A. Statistical Mechanics; Harper Collins: New York, 1976. (45) Kleeb, B. Pore size distribution of zeolite Z10-02; Technical Report; CU Chemie Uetikon AG: Uetikon, Switzerland, 2001. (46) Breck, D. W. Zeolite Molecular Sieves; Wiley: New York, 1974. (47) Valenzuela, D. P.; Myers, A. L. Adsorption Equilibrium Data Handbook; Prentice Hall: Englewood Cliffs, NJ, 1989.

adsorption” model is used to evaluate the single-particle partition functions. In the mobile adsorption model, the molecules are assumed to vibrate in the direction perpendicular to the wall but are free to move in directions parallel to it. An isosteric heat of adsorption of 2000 K was calculated from the experimental data15 and was inserted in eq 26 to calculate the fluid-solid interaction energy. A range of 420 K e T ln[qb(T)/qs(T)] e 534 K was obtained. A specific pore volume in the range of 0.740.84 mL/g was obtained from the supplier’s data sheet. The estimate for the maximum fluid density, Fmax, is retained as discussed in the previous section, and a value within this range is used for the computations. The parameters that are used in the lattice DFT model are summarized in Table 4. Notation A a bm g h H Ik k K m msorb M1 Mlat N nex p P q0st rj s T T* u Ufs v V W z

surface area of sorbent specific surface area of sorbent coefficient in the Hill-de Boer isotherm based on mobile adsorption function converting lattice-site occupancies into densities Planck’s constant Henry’s constant overall number of lattice sites in a pore of type k Boltzmann’s constant overall number of pore types mass of a single atom/molecule mass of sorbent mass at measuring point 1 of the balance number of lattice sites number of atoms adsorption excess in moles of sorbate per unit mass of sorbent pressure probability isosteric heat at zero coverage intermolecular distance between two neighboring adsorbed atoms entropy per molecule temperature reference temperature internal energy per molecule total fluid-surface interaction energy specific pore volume volume number of microscopic configurations coordination number

Greek Symbols R Γ Γlat ff θ ν⊥ F σ

thermal expansion coefficient adsorption excess in mass of sorbate adsorption excess in lattice units fluid-fluid interaction potential fractional surface coverage bonding frequency normal to the surface density Lennard-Jones diameter

Subscripts and Superscripts * 0 b

reference state solid parts in the measuring cell bulk

Supercritical Adsorption in Porous Solids buo c ex He k m

buoyancy critical excess helium type of pore mobile adsorption

Langmuir, Vol. 19, No. 4, 2003 1267 max met p s tot

maximum metal pore solid (adsorbent) total LA0266379