The quadrupolar Maxwell electrostatic equations predict several qualitatively different results compared to Poisson’s classical equation in their description of the properties of a dielectric interface. All interfaces between dielectrics possess surface dipole moment which results in a measurable surface potential jump. The surface dipole moment is conjugated to the bulk quadrupole moment density (the quadrupolarization) similarly to Gauss’s relation between surface charge and bulk polarization. However, the classical macroscopic Maxwell equations completely neglect the quadrupolarization of the medium. Therefore, the electrostatic potential distribution near an interface of intrinsic dipole moment can be correctly described only within the quadrupolar macroscopic equations of electrostatics. They predict that near the polarized interface a diffuse dipole layer exists, which bears many similarities to the diffuse charge layer near a charged surface, in agreement with existing molecular dynamics simulation data. It turns out that when the quadrupole terms are kept in the multipole expansion of the laws of electrostatics, the solutions for the potential and the electric field are continuous functions at the surface. A well-defined surface electric field exists, interacting with the adsorbed dipoles. This allows for a macroscopic description of the surface dipole-surface dipole and the surface dipole-bulk quadrupole interactions. They are shown to have considerable contribution to the interfacial tension—of the order of tens of mN/m! To evaluate it, the Maxwell stress tensor in quadrupolar medium is deduced, including the electric field gradient action on the quadrupoles, as well as quadrupolar image force and quadrupolar electrostriction. The dependence of the interfacial tension on the external normal electric field (the dielectrocapillary curve) is predicted and the dielectric susceptibility of the dipolar double layer is related to the quadrupolarizabilities of the bulk phases and the intrinsic polarization of the interface. The coefficient of the dielectro-Marangoni effect (surface flow due to gradient of the normal electric field) is found. A model of the Langevin type for the surface dipole moment and the intrinsic surface polarizability is presented.
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We adopt the following terminology: (i) Surface dipolar potential is a surface characteristic related to the surface excess of the polarization through Eq. (1); it corresponds to the potential difference at two points zW and zO such that and , where LD are the Debye lengths in the respective phases; (ii) Galvani potential difference is a thermodynamic bulk quantity related to bulk partitioning equilibria of all electrolytes present in the system; (iii) A third, double layer potential (a surface characteristic) stemming from the adsorption of charged species can be rigorously defined through the free charge surface moment (the integral of zρ). The latter two will be discussed in detail from the viewpoint of the linear quadrupolar electrostatics in a following paper. (iv) The potential stemming from TrQ (Bethe potential) is resulting in a fourth potential drop across the interface, which is a bulk quantity of little importance for electrochemical systems.74
Note that in case that , Ez(z) has discontinuity at z = 0. Therefore, the problem for water|gas surface is easier to analyze by taking the limit of the results for water|oil rather than solving it directly.
