Langmuir 2004, 20, 4289-4294
4289
Adsorption/Desorption Hysteresis in Inkbottle Pores: A Density Functional Theory and Monte Carlo Simulation Study B. Libby and P. A. Monson* Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 01003-9303 Received November 7, 2003. In Final Form: February 7, 2004 The mechanisms of adsorption and desorption in inkbottle-shaped pores are considered for lattice models using grand canonical mean field density functional theory and Monte Carlo simulation. We find that they depend significantly on the particular pore geometry, the nature of the fluid-solid interaction, and the temperature. We find two mechanisms for desorption. One mechanism involves the emptying of the main cavity even as the density of fluid in the necks remains high, a mechanism observed recently in studies of an off-lattice model of an inkbottle. The other is a simultaneous desorption from the entire pore space, behavior that is more closely related to the traditional picture of pore blocking in the inkbottle system.
I. Introduction One of the earliest concepts used in understanding hysteresis between adsorption and desorption in gas sorption experiments is that of pore blocking. At the simplest level, this concept can be described in terms of a pore geometry that has come to be known as the “inkbottle”.1 This geometry consists of a larger pore space in contact with a bulk vapor through one or more narrower pore spaces or “necks”. The classical explanation presented by Kraemer2 and McBain3 for the phenomenon of hysteresis in such pores assumes that desorption from the larger cavity is retarded by the presence of regions of high adsorbate density in the smaller necks. When the density of adsorbate is low, there is free access to all of the internal volume, which permits adsorbed layer formation on the pore surfaces followed by filling of the pores in order of increasing size. During desorption, it is hypothesized, fluid in the larger cavity is blocked from evaporating, even below pressures low enough for it to be thermodynamically unstable, until all of the channels connecting it to the bulk region empty, giving rise to hysteresis. Thus, differences in pore connectivity should primarily affect the desorption process, leaving adsorption comparatively unaffected. The pore blocking concept is at the center of theories of hysteresis employing the properties of networks and percolation theory. Examples include the work of Mason,4 Neimark,5 and Seaton.6 In recent work the microscopic behavior associated with the inkbottle and other simple pore geometries was investigated using grand canonical molecular dynamics (GCMD) and grand canonical Monte Carlo (GCMC) simulations.7 An important result of that work was the observation that, in the case of the inkbottle geometry, the larger pore could empty during desorption even as the smaller pores remained filled. (Similar behavior has also (1) Everett, D. H. Adsorption Hysteresis. In The Solid-Gas Interface; Flood, E. A., Ed.; Dekker: New York, 1967; Vol. 2, pp 1055-1113. (2) Kraemer, E. O. In Treatise on Physical Chemistry; Taylor, H. S., Ed.; Van Nostrand: New York, 1931; p 1661. (3) McBain, J. J. Am. Chem. Soc. 1935, 57, 699-700. (4) Mason, G. Proc. R. Soc. London, Ser. A 1988, 415, 453-486. (5) Neimark, A. V. In Characterization of Porous Solids II; Stud. Surf. Sci. Catal.; Rodriguez-Reinoso, F., et al., Eds; Elsevier: Amsterdam, 1991; p 67. (6) Seaton, N. A. Chem. Eng. Sci. 1991, 46, 8, 1895-1909. (7) Sarkisov, L.; Monson, P. A. Langmuir 2001, 17, 7600-7604.
been observed for one of the cases considered in GCMC studies of a slightly different inkbottle geometry.8) An important conclusion was that the results obtained from GCMD were identical with those from GCMC, as had been seen earlier in a study of a model silica gel.9 These observations are in contrast with the simple picture that desorption from the larger cavity in an inkbottle geometry does not occur until the smaller cavities have emptied. The simulation studies were partly motivated by recent results obtained from mean field density functional theories10-13 (MFT) and by computer simulations9 of adsorption/desorption in disordered porous structures that indicated that hysteresis could be described without explicitly invoking the concept of pore blocking. Since GCMD calculations are very computationally intensive, only a single pore geometry at a single temperature was studied.7 With this in mind, the present article reports a more extensive set of calculations for a lattice gas model of these pore geometries, to which both GCMC simulations and MFT have been applied. Although the model is simpler than that used in the earlier work, it nevertheless is able to capture the essential features of the adsorption/ desorption behavior while facilitating studies over a much wider range of conditions. Our use of lattice models and density functional theory for the inkbottle geometry follows earlier studies for simple single pore geometries using both lattice and off-lattice models. In this case one can use molecular modeling to test assumptions of classical theories of single-pore hysteresis, such as those of Zsigmondy14 and Cohan.15 For example, the prediction from Cohan’s analysis that a pore which is closed at one end should not exhibit hysteresis is supported by density functional calculations for a lattice model16 and by computer simulation studies performed by Sarkisov9 and Gelb.17 As another example, (8) Vishnyakov, A.; A. V. Neimark, Langmuir 2003, 19, 3240-3247. (9) Sarkisov, L.; Ph.D. Thesis, University of Massachusetts, Amherst, MA, 2001. (10) Kierlik, E.; Monson, P. A.; Rosinberg, M. L.; Sarkisov, L.; Tarjus, G. Phys. Rev. Lett. 2001, 87, 5. (11) Kierlik, E.; Monson, P. A.; Rosinberg, M. L.; Tarjus, G. J. Phys.: Condens. Matter 2002, 14, 9295. (12) Woo, H.-J.; Sarkisov, L.; Monson, P. A. Langmuir 2001, 17, 74727475. (13) Woo, H.-J.; Monson, P. A. Phys. Rev. E 2003, 67, 041207. (14) Zsigmondy, R. Z. Anorg. Chem. 1911, 71, 356-377. (15) Cohan, L. H. J. Am. Chem. Soc. 1938, 60, 433-435.
10.1021/la036100a CCC: $27.50 © 2004 American Chemical Society Published on Web 04/15/2004
4290
Langmuir, Vol. 20, No. 10, 2004
the influence of the presence of bulk regions on hysteresis in simulated systems has been studied using density functional calculations for a lattice model16,18 and using computer simulations19 of both lattice and off-lattice models. As further background to our work, we note a recent paper20 in which hysteresis in inkbottle geometries is discussed from the perspective of density functional theory of fluids confined in spherical cavities and experimental adsorption/desorption isotherms for fluids in ordered mesoporous silicas thought to contain inkbottle type pores. The authors suggest three mechanisms for desorption in inkbottle cavities: (i) the classical mechanism; (ii) the cavitation mechanism observed by Sarkisov and Monson;7 and (iii) near-equilibrium evaporation from the cavity. They propose that the mechanism is determined by the diameter of the smaller pore. The density functional theory component of their work was done for a closed spherical cavity approximation to an inkbottle pore assuming that the density distribution varies only along the radial direction. This differs from the computer simulations discussed above and with the results presented here for the actual inkbottle geometry, which requires two dimensions to represent the density distribution. The primary conclusion from the present work is that, after study of a variety of inkbottle systems, we find two kinds of desorption mechanism governing the hysteresis which depend on the particular geometry considered, the nature of the fluid-solid interaction and the temperature. One of these mechanisms is the behavior seen previously by Sarkisov and Monson,7 which, as we discussed above, involves emptying the large cavity while the small cavity remains filled. The other is a simultaneous emptying of the entire pore space, behavior that might more nearly be identified with the traditional picture of pore blocking in the inkbottle system. The desorption mechanism can be changed by changing the pore width, the temperature, or the strength or nature of the fluid-solid interactions. Qualitative differences in mechanism may also be seen between the results from GCMC simulations and MFT, depending on the conditions. After this work was largely completed we became aware of earlier unpublished work by Papadopoulou and van Swol21 in which MFT calculations were performed for a similar model. These workers found behavior similar to the second kind of behavior mentioned above but did not study the parameter regime of their model where the first kind of behavior is observed. The remainder of our paper is organized as follows: In the next section we describe our methodology, including the lattice models and our implementations of MFT and GCMC. We then describe our results, focusing on the spatial density distributions of the fluid in the inkbottle at various states during adsorption and desorption. We conclude the paper with a summary and discussion of our results and their significance.
Libby and Monson that determines the pore geometry. We have for the lattice gas Hamiltonian:
H ) -
∑ n n +∑ n φ i j
(2.1)
i i
i
