It has been known for a long time that the adsorption and condensation of gas cause elastic deformation of the porous matrix. The reversible formation of an adsorbed film, which precedes capillary condensation, results in an extension of the porous material while, in the hysteresis region, the negative liquid pressures under the concave menisci contract the porous matrix. The elastic deformation exhibits a hysteresis loop in the same pressure region as the adsorption phenomenon. These deformations have been neglected in practically all the theoretical treatments of adsorption. There are two reasons for this. First, the deformations are small in magnitude and were supposed to have small effects on the adsorption process, and second, no experiment has contradicted the existing model according to which, in systems where the pores interact, the source of interactions is pore-pore intersections. They are the experimental results obtained in SBA-15 and $p^{+}$-type porous silicon, systems in which the pores interact whereas they are not connected, which lead us to question these models and consider the elastic deformation of the pore walls as a possible coupling parameter. Based on the experimental work of C. H. Amberg and R. McIntosh [Can. J. Chem. 30, 1012 (1952)], who measured both the linear deformation of a porous glass rod and the adsorbed amount during isothermal adsorption of water, we develop a thermodynamic approach which includes the elastic energy of the solid. This approach is generic to all porous materials. In the region of reversible adsorption preceding the capillary condensation where the variation of the surface free energy can be deduced from adsorption data, the linear extension of the solid is proportional to the variation of the surface free energy and to the elastic constant of the solid. This is the crucial point of the paper: thermodynamics of adsorption is directly connected to the elastic properties of the porous solid. In the hysteresis region, this linear relation can be used to deduce the variation of the surface free energy from the deformation measurements, a calculation which cannot be done from adsorption data. We find that the surface free energy related to the elastic deformation is an important component of the total free energy. It is shown that the condensation branch represents the more stable states and that an energy barrier exists to evaporation, which depends essentially on the elastic deformation. The pores interact through the deformation of the walls. Based on this interaction mechanism and on the shape of the scanning curves, which are common to materials with interconnected pores such as porous glass or noninterconnected pores such as $p^{+}$-type porous silicon and SBA-15, we propose scenario for the filling and emptying of these porous materials.

• Received 8 April 2008

DOI:https://doi.org/10.1103/PhysRevB.78.085433