3344 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA J. Phys. Chem. 1993,97, zyxwvu 334-3349 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK Dipole-Dipole Interactions near Interfaces M. Urbakh' and J. Klafter' School zyxwvutsr of Chemistry, Tel- Aviv University, Tel Aviv 69978, Israel Received: August 18, 1992 The nature of the pair interaction between dipoles embedded in a liquid in the vicinity of a nonmetallic interface is investigated in the continuum limit. It is shown that thedipole-dipole interaction can be significantly modified in the presence of a boundary in liquids characterized by a nonlocal dielectric function or in liquids whose dielectric properties change due to geometrical restrictions. Different limits are studied and relationships to experimental observables are discussed. 1. Introduction Recent experimental and theoretical ~tudiesl-~ demonstrated that spatial restrictions can strongly affect the dynamic and thermodynamic properties of embedded liquids and molecules. The effects observed and predicted include the influence of interfaces on the pair-interaction energy between charged or uncharged particle^,^.^ translational and rotational diffusion of probe molecule^,^^^ and chemical reactions and energy transfer properties. The influence of boundaries on molecular properties in their vicinity is of course related to the nature of the intermolecular forces and to how they are modified near an interface. These modifications can directly influence processes such as adsorption and electron, or energy, transfer at interfaces. In this paper we investigate changes in the interaction between point dipoles embedded in a liquid near a nonmetallic interface when compared to the bulk liquid. The liquid is represented in the continuum approximation, in terms of a nonlocal dielectric function t(k,w), and we analyze the dependence of the dipole- dipole interactionon the distance between thedipoles and between thedipoles and the boundary as well as on thedielectric parameters that characterize the interface region. Although the continuum framework does not explicitly include molecular level details, it enables the derivation of closed expressions that relate micro- scopic quantities to measurable observables such as dielectric functions.9-' zyxwvutsrqpo * Continuum approximations and their modifications have been shown to be powerful in unraveling leading physical processes in complex systems. The electrostatic interaction energy between two dipoles is given by where T is the dipole-dipole interaction energy tensor and pa and pd are the dipole moments of the participating molecules. When a uniform liquid is considered, this tensor is usually written in the following form Here tb(w) is the local dielectric constant of the liquid and r = (rr, r,, rz) is the vector connecting the two dipoles. The approach used here is similar to the one applied by us for studying dipole relaxation in liquids near boundaries,I3 an approach suitable for both staticanddynamic limits. Inorder toevaluate theinteraction energy near an interface, one has to find the electric field, E, produced by an oscillating dipole of moment pd at the point of location of other dipole of moment zyxwvutsr pa. The electric field, zyxwvut E, is the sum of a part arising from the dipole field directly, ECdiP), and a part, E(lnd), produced by the polarization due to the presence of an interface. The net electric field acting on the dipole pa can 0022-3654/93/2097-3344504.00/0 be written as (3) In our problem, therefore, the interaction tensor T in eq 1 is the sum (4) All the characteristic distances in the problem (the distance between the dipoles and the distances between the dipoles and the interface) are assumed to be smaller than the radiation wavelength in the liquid, so that retardation can be neglected and the Laplace equation applies for the calculation of electric field, E, and interaction tensor, T. We follow ref 13where we represented the liquid by a nonlocal dielectric function and calculated the change in the dielectric friction due to the presence of a boundary. In analogy to this previous work, the effect of the interface is introduced through the concept of additional boundary conditions.I4 The nonlocal nature of the liquid defines a length scale, A, which is a measure of spatial correlations in the liquid. This parameter can be estimated on the base of diffraction experimentsI5 and on molecular dynamic simulations of liquids.16J7 For instance, in aqueous solutions the correlation length, A, is of the order of the extension of local hydrogen-bonded clusters. The nonlocal description introduces short range order within the continuum model of the liquid, at least phenomenologically, and leads to dipole-dipole and dipole-boundary distance and pore size de- pendencies which do not appear in the case of a local dielectric function. In order to mimic the possibility that the liquid itself changes its dielectric behavior due to interaction with the bo~ndary,~%~~J~ we assume a region of modified liquid near the boundary. The influence of a modified surface layer of a liquid on ion-ion and dipole-dipole interactions at metal-electrolyte solution interface was previously considered and shown to provide interpretations to experimental observations of ionic adsorption in electrochemical systems which could not be understood in the framework of the traditional d e s c r i p t i ~ n . ~ . ~ ~ 2. The Model We now consider the model we use for the description of electromagnetic interaction of time-dependent point dipoles embedded in a liquid near a substrate. As in ref 13 we assume that the substrate is characterized by a local dielectric function, C,,b(W), and the liquid is described by the nonlocal dielectric function 1 1 1 1 1 e(k,w) e.(~) (=-=)= Here t.(~) is the short-wavelength dielectric constant of a bulk (5) -=-- zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK 0 1993 American Chemical Society