On the macroscopic theory of Van der Waals forces

Abstract

A simple macroscopic derivation is given of the non-retarded Van der Waals interaction between two semi-infinite dielectric media.Neutral macroscopic bodies, when placed at a distance of the order of 1000 A, attract each other as a result of the Van der Waals forces between their atoms. Lifshitz [1] has developed a macroscopic theory for this phenomenon by introducing fluctuating terms in the Maxwell equations for a dielectric medium. For the interaction energy per unit area at T = 0 between two semi-infinite media a distance d apart IAfshitz finds in the case that d << k (;~ is the principal absorption wave length of the material); ; 1 Starting from the Drude-Lorentz model of an atom (harmonic oscillator model) Renne and Nijboer [2] gave an atomistic derivation of the non-retarded interaction between one atom and a semi-infinite medium by summing two-particle, three-particle, etc., interactions. Their derivation can be extended to the case of two semi-infinite media and for arbitrary atoms [3]. In this note we will give a simple alternative macroscopic derivation of eq. (1), which can easily be generalized to more complicated configurations.Consider two semi-infinite dielectric media a distance d apart. We look for those solutions of the equations of electrostatics: div D = 0, curl/~ = 0, which oscillate harmonically in time. It will be obvious that these equations can be satisfied by: 1°: curl E = O, ~(¢o) = 0 (longitudinal bulk modes); 20: div D = 0, e(c0) = co (transverse bulk modes); 30: div~ = 0, curl E = 0 (surface modes). At T = 0 the interaction between the media may be found from the sum of the zero point energies ~k ½~wk of all modes evaluated at a distance d minus the corresponding sum for infinite d. This procedure, due to Casimir [4], who applied it to the case of ideal conductors in vacuo, can be justified in our case by considering a macroscopic Hamiltonian of the system which is equivalent to a sum of harmonic oscillator Hamiltonians with the frequencies ¢o k as eigenfrequencies. A similar Hamiltonian was used by Hopfield [5]. Now the frequencies of the bulk modes 1 ° and 2 ° are evidently independent of the distance d. Hence for the evaluation of the Van der Waals interaction only the surface modes 3 ° need be considered.Introducing the electrostatic potential ~ (x,y,z,t) by E = -grad ~ we have to solve the equation V2~ = 0, subject to the boundary conditions q~ and ~(w)a~/Sz continuous at z = 0 and z = d (the z-axis is chosen perpendicular to the plane-parallel boundaries of the media).Assuming ¢ to be of the form

Variations of Casimir energy from a superconducting transition

“…This method was applied to dielectric cavities in Refs. [21], where it was shown that one recovers the Lifshitz result, and was later generalized to multilayer systems in Ref. [22].…”

New developments in the Casimir effect

“…In order to solve this problem we use the formalism of surface modes [106,107], which are exponentially damping for z > a 2 + d and z < − a 2 − d. These modes describe waves propagating parallel to the surface of the walls [109]. They form a complete set of solutions and this approach is widely used.…”

“…(4.1) over the solutions of Eqs. (4.7), (4.9) can be performed by applying the argument theorem which was applied for this purpose in [106,107]. According to this theorem…”

Van Der Waals Dispersion Force Contribution to the Interfacial Free Energy of Nematic Liquid Crystals

“…-First we discuss the case of an isotropic liquid as an illustration of our method. The starting point is the following equation for the van der Waals interaction free energy (per unit area) of semi-infinite media interacting across a plane parallel vacuum gap of width / [8] where H is the Planck constant divided by 2 n, and k c is a cut-off wave number of fluctuating electric field along the surfaces. A(co) is related to the complex dielectric permittivity of the liquid, a(co), in the following way, Here the zero point of the free energy is taken as Now, starting from / = oo we decrease the separation / and eventually we make / = 0 in which both media are in molecular contact.…”

“…The relevant free energy is calculated by using the method of surface mode analysis first introduced by van Kampen, Nijboer and Schram [8] and later extended by Ningam, Parsegian and Weiss [lo]. Since we are interested in the non-retarded limit, the surface mode analysis reduces to an electrostatic boundary value problem.…”