The Physics of Capillary Condensation in Disordered Mesoporous Materials: A Unifying Theoretical Description
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The Physics of Capillary Condensation in Disordered Mesoporous Materials: A Unifying Theoretical Description FRANCOIS DETCHEVERRY, EDOUARD KIERLIK, MARTIN LUC ROSINBERG∗ AND GILLES TARJUS Laboratoire de Physique Th´eorique des Liquides, Universit´e Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris Cedex 05, France [email protected]
Abstract. We present a unifying theoretical approach of capillary condensation in disordered mesoporous materials. It provides a comprehensive picture of this phenomenon that accounts for processes occuring on all length scales and clarifies the relation between hysteretic and equilibrium behavior. The shape of the hysteresis loop is shown to depend on the presence or absence of out-of-equilibrium phase transitions, whose nature differs on adsorption and on desorption. Keywords: capillary condensation, hysteresis, disordered solids
1.
Introduction
Despite its widespread use for characterizing mesoporous materials, the phenomenon of capillary condensation is still debated and interpreted in a variety of ways often at odds with each other. Part of the confusion comes from the fact that disorder (both geometric and energetic), which charaterizes materials such as porous glasses and gels, brings about new physics that makes the well-established theory of capillary condensation in single pores or in ordered porous materials not transferable. As a result, there are a number of very basic points whose consequences have often not been fully appreciated; these points involve considerations of time scales and length scales. Consider first the time-scale problem. The signature of what is known as “capillary condensation” is a hysteresis in the adsorption/desorption isotherms (Gregg and Sing, 1982; Rouquerol et al., 1999). By its very nature, hysteresis means that the observed phenomenon is out of equilibrium. The fact that the location and the shape of the hysteresis loop are reproducible in experiments indicates that the observation time is larger than fast, local equilibration processes, but is smaller than ∗ To
whom correspondence should be addressed.
the time for reaching true, global equilibrium; this latter time may indeed be extremely large in the regime of interest, like in the case of glassy liquids and polymers below their glass tranformation temperature. This out-of-equilibrium character can be benign as in the liquid/gas transition in bulk fluids or fluids in single pores (Evans, 1990): there, the system can be trapped in a metastable phase that can be described by extending equilibrium concepts and relations. The situation, however, is radically different in the presence of externally imposed disorder. This is well known from the study of “dirty” magnetic systems with impurities (Fisher et al., 1988): disorder generates a very large number of metastable states (possibly exponentially large in the size of the system) in which the system can get trapped, states that are not describable by a thermodynamic-like approach. The second issue concerns length scales. Disordered mesoporou
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