Mesoscopic Treatment of a Fluid/Liquid Interface. 1. Theory Aly J. Castellanos,* ,²,‡,§ German Urbina-Villalba, § and Ma ´ ximo Garcı ´a-Sucre § Facultad de Ciencias y Tecnologı ´a, Escuela de Quı ´mica, UniVersidad de Carabobo, Edo. Carabobo, Venezuela, Facultad de Ciencias, Escuela de Quı ´mica, Postgrado, UniVersidad Central de Venezuela, Caracas, Venezuela, and Centro de Fı ´sica, Instituto Venezolano de InVestigaciones Cientı ´ficas (IVIC), Km. 11, Carretera Panamericana, Apartado 21827, Caracas 1020-A, Venezuela ReceiVed: August 20, 2002 A thermodynamic model of a fluid/liquid interface based on the redistribution of “elastic” energy as a consequence of the contact between formerly isolated phases is proposed. The interface consists of two sub phases adjacent to their respective bulks. Each sub-phase is capable of storing elastic potential energy. The adsorption isotherms are reproduced in the usual way, equalizing the chemical potential of the adsorbent between the bulk phases and the interface. In this formalism, the interfacial energy results from a sum of two terms each belonging to a subphase and can be expressed in terms of the activity of one component of the system in one bulk phase and at the interface. Introduction Still today most undergraduate textbooks of physical chem- istry avoid thermodynamic consideration of interfaces when treating the subject of phase equilibrium. The interfacial energy is introduced as an additional free energy contribution: the work that has to be done on the system at constant temperature and pressure to increase its interface. Such additional contribution is usually negligible in most systems because the free energy is an extensive thermodynamic quantity, and the size of the interface is generally small in comparison to the size of the bulk phases. Hence, the conventional approach to the problem avoids the complex definition of an interfacial chemical potential for each substance, and simplifies the analytical procedure for finding the value of their chemical potential at equilibrium (µ eq ). It is clear however that whenever the adsorption process is important a suitable definition of interfacial chemical potentials is necessary. This is the case of long-lasting emulsions for instance, where the stability of the dispersion is intimately related to the total interfacial area and the surfactant surface excess. It is also especially important for fluid/liquid systems in general since, up to this date, it is not possible to measure the interfacial composition simultaneously and independently from the bulk composition. Furthermore, the analytical form of an interfacial chemical potential is also interesting from a more fundamental point of view. If equilibrium is reached by equalization of the chemical potentials and the interfacial potential is substantially different from the bulk, the properties of the system at equilibrium could be sensibly affected by the characteristics of the interface. Other considerations regarding the appropriate definition of interfacial chemical potentials concern the nature of the state of the equilibrium itself. Starting from the Gibbs adsorption model, 1-2 almost all theoretical descriptions of surfactant adsorption to gas/liquid and liquid/liquid interfaces commence with the equalization of the chemical potentials between the bulk phase(s) and the interface. Such equilibrium condition is achieved between bulk phases counterbalancing energetic dif- ferences with configurational entropy contributions dependent on the local composition of the different constituents. The presence of an interface introduces an anisotropy in the system and generates inhomogeneities in the spatial distribution of components. This brings an additional complication to the equilibrium problem: interfaces are usually highly ordered, and such an ordered state is opposed to the necessary increase of entropy required for equilibrium. An insightful discussion along with a review of the most relevant aspects to be considered in the formulation of an interfacial chemical potential can be found in ref 3. Butler was probably the first to define such potential for the description of the adsorption process and the energy excess occurring in the interface. 4 Among other contributions, Lucassen-Reynders 3,5 introduced the concept of partial molar areas in order to ascribe the interfacial free energy γA to each contributing molecule. On the other hand, Cahn and Hilliard developed a completely different model in which the interfacial free energy is expressed as a function of the density gradient between the phases in contact. 6,7 The interface is described in this model as a diffuse and continuous zone. Starting from the calculation of the Helmholtz free energy arising from the referred model, and the momentum balance condition, Carey and Scriven proposed a formal development of the interface gradient density model, 8 which was subsequently extended to the description of binary systems. 9 The validity of this model depends on the reliability of the equations of state used, and there is no direct connection in it between activities and chemical potentials. On the basis of the definition of interfacial chemical potential given by Butler, Sugimoto 10,11 recently proposed an alternative formalism for understanding the interfacial tension in terms of molecular activities. In this case, not only is the interfacial chemical potential different from bulk but it is additionally supposed that such initial difference cannot be overcome preventing equalization of the chemical potentials. In that model, Sugimoto goes back to a description of the interface as a * To whom correspondence should be addressed. E-mail: ajcastel@ uc.edu.ve. ² Universidad de Carabobo. ‡ Universidad Central de Venezuela. § Instituto Venezolano de Investigaciones Cientı ´ficas (IVIC). 875 J. Phys. Chem. A 2003, 107, 875-882 10.1021/jp021908+ CCC: $25.00 © 2003 American Chemical Society Published on Web 01/21/2003