van der Waals equation of state for a fluid in a nanopore Guillermo J. Zarragoicoechea and Vı ´ ctor A. Kuz IFLYSIB (UNLP, CONICET, CICPBA), casilla de correo565, 1900 La Plata, Argentina Received 6 July 2001; published 24 January 2002 A generalization of the van der Waals equation of state is presented for a confined fluid in a nanopore. The pressure in the fluid, confined in a narrow pore of infinite length, has tensorial character. From this hypothesis, the Helmholtz free energy is constructed and expressions for the axial and transversal components of the pressure tensor are obtained. The equations predict liquid-vapor equilibria, and a shift of the critical point with respect to that obtained from the van der Waals bulk equation. The results are in good agreement with recent experiments. DOI: 10.1103/PhysRevE.65.021110 PACS numbers: 64.10.+h, 64.70.Nd, 05.70.Fh I. INTRODUCTION There is a vast knowledge about phase transitions in bulk fluids. When in a given thermodynamic system the volume is reduced to microscopic levels, the equilibrium between phases is no longer size independent. Confinement changes the thermodynamic character of the fluid 1. The pressure is a diagonal tensor, and should be used in the description of fluids at the microscopic and mesoscopic scales when equi- librium between phases is analyzed. The behavior of the fluid within a pore is of fundamental importance in many fields. The determination of mesopore diameters or micropore volumes in a solid 2, the pressure in a fluid confined in a cell membrane, and the behavior of water in a channel of proteins where solutes inorganic ions pass through the cell membranes 3 are problems of practi- cal and theoretical interest. In this context it is important to mention the Kelvin equation, or some of the modern versions of this equation 4–6, which predict the adsorbed layer thickness or the transition from capillary to multilayer ad- sorbed phase inside the pore. In tribology, the science of friction at the microscopic scale 7, the phenomenon of dissipation of energy is of much concern. In a mechanically confined fluid, the energy dissipated by friction can induce chemical transformations, liquid-gas phase transitions, or drastic changes of static and dynamic properties like shear stress, coefficient of friction, compressibility, and viscosity. These dynamic and static properties can no longer be described even qualitatively in terms of the bulk properties 8. Liquids confined between two surfaces or within a narrow space with dimensions smaller than 5–10 molecular diam- eters become ordered into layers, and within each layer they can also have lateral order. Across molecularly thin films of simple liquids, there is a structuring of the molecules and an exponentially decaying oscillatory force, varying between at- traction and repulsion with a periodicity of the order of the solvent molecular dimension 9. A similar result is found in polymeric thin films in relation to the density, which exhibits gradually decaying oscillations 10. A lattice-gas cellular automata model of porous media constructed at the pore scale predicts formation of a microscopic liquid film con- densed on the solid walls in equilibrium with the gas phase 11. All these analyses tacitly show that a fluid in a nanop- ore has tensorial character. Equilibrium and nonequilibrium experiments show that a confined fluid behaves differently from the corresponding bulk fluid. The relaxation rate of ethylene glycol versus tem- perature is different in a zeolite host system from the bulk fluid 12. In pure liquid sulfur hexafluoride, the experi- ments of Thommes and Findenegg 13 determined the criti- cal point shift in three kinds of controlled-pore glass. A direct determination of the phase coexistence properties of fluids by Monte Carlo simulation predicts the adsorption and cap- illary condensation of a simple fluid Ar in narrow cylindri- cal pores (CO 2 ). The gas-liquid critical temperature de- creases as the pore radius is reduced 14. The same phenomenon was observed in liquid-liquid phase equilibria by Sliwinska-Bartkowiak et al. 15. Here the effect of con- finement produced a lowering of the critical mixing tempera- ture and a shift in the critical mixing composition. In relation to this general presentation, we study the prob- lem of confined fluids in a narrow pore via an extension of the van der Waals equation. The phase transitions, shift of the critical point, and critical temperatures predicted by this theory are in good agreement with the results of experiments and numerical simulations. II. VAN DER WAALS EQUATION FOR A CONFINED FLUID We assumed that the pressure in a confined fluid is a diagonal tensor P ˆ with components p ii ( i =x , y , z ). The in- ternal energy is given by 16 dE =T dS -  i p ii d  ii V , 1 where the second term on the right-hand side represents the work done by the internal tension under a specific deforma- tion d  ii of the volume V . From the Helmholtz free energy F =E -TS we obtain dF =-S dT -  i p ii d  ii V 2 PHYSICAL REVIEW E, VOLUME 65, 021110 1063-651X/2002/652/0211104/$20.00 ©2002 The American Physical Society 65 021110-1