Advances in Condensed Matter Physics

Volume 2015 (2015), Article ID 198657, 10 pages

http://dx.doi.org/10.1155/2015/198657

## A Generalization of Electromagnetic Fluctuation-Induced Casimir Energy

Department of Mechanical, Industrial and Systems Engineering, University of Rhode Island, Kingston, RI 02881, USA

Received 4 February 2015; Accepted 13 March 2015

Academic Editor: Yuri Galperin

Copyright © 2015 Yi Zheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Intermolecular forces responsible for adhesion and cohesion can be classified according to their origins; interactions between charges, ions, random dipole—random dipole (Keesom), random dipole—induced dipole (Debye) are due to electrostatic effects; covalent bonding, London dispersion forces between fluctuating dipoles, and Lewis acid-base interactions are due to quantum mechanical effects; pressure and osmotic forces are of entropic origin. Of all these interactions, the London dispersion interaction is universal and exists between all types of atoms as well as macroscopic objects. The dispersion force between macroscopic objects is called Casimir/van der Waals force. It results from alteration of the quantum and thermal fluctuations of the electrodynamic field due to the presence of interfaces and plays a significant role in the interaction between macroscopic objects at micrometer and nanometer length scales. This paper discusses how fluctuational electrodynamics can be used to determine the Casimir energy/pressure between planar multilayer objects. Though it is confirmation of the famous work of Dzyaloshinskii, Lifshitz, and Pitaevskii (DLP), we have solved the problem without having to use methods from quantum field theory that DLP resorted to. Because of this new approach, we have been able to clarify the contributions of propagating and evanescent waves to Casimir energy/pressure in dissipative media.

#### 1. Introduction

The phenomena of adhesion and cohesion play an important role in many areas of science of technology; they are responsible for stiction in microelectromechanical (MEMS) devices, leading to their failure; microbial adhesion is responsible for the formation of biofilms, which have beneficial as well as detrimental effects; they contribute to friction and wear between surfaces; other areas include food science, pharmacology, and locomotion and prey capture by some animals. Adhesion and cohesion can be loosely defined as the molecular attraction that holds together surfaces of two different substances or two identical substances, respectively. Intermolecular forces responsible for adhesion and cohesion can be classified according to their origins; some forces are of pure electrostatic origin arising out of interactions between charges, ions, random dipole—random dipole (Keesom), random dipole—and induced dipole (Debye); quantum mechanical effects give rise to covalent bonding, London dispersion forces between fluctuating dipoles, and Lewis acid-base interactions; pressure and osmotic forces are of entropic origin [1]. Of all these interactions, the London dispersion (it is so-named because of its relation to the dispersion of light in the visible and UV portion of the spectrum) interaction is universal and exists between all types of atoms as well as macroscopic objects. The focus of this paper is on the contribution of dispersive interactions to cohesion and adhesion between macroscopic objects.

Dispersion forces between macroscopic objects, resulting from alteration of the quantum and thermal fluctuations of the electrodynamic field due to the presence of interfaces, play a significant role in the interactions at micrometer and nanometer length scales. Hamaker was the first to extend the concept of London dispersion forces between two atoms to forces between macroscopic spheres separated by vacuum by pairwise summation of the interaction energy between the atoms that constitute the spheres [2]. Hamaker established that the unretarded Casimir force between two half-spaces separated by a film of thickness is given by , where is a constant that is now referred to as the Hamaker constant. Since the assumption of pairwise additivity is valid only in the low density limit, alternate theories are necessary for condensed media. Lifshitz, in his seminal work [3], outlined a method based on Rytov’s theory of fluctuational electrodynamics [4] for computing the Casimir forces between two semi-infinite regions separated by a vacuum gap. It required calculation of the average value of the Maxwell stress tensor in the vacuum gap. Lifshitz theory takes many body effects into account in the continuum limit and expressions for can be derived in terms of the frequency dependent relative dielectric function, , of the materials involved. For magnetic materials frequency dependent relative magnetic permeability, of the materials also play a role. The generalization of Lifshitz’ theory to the case when the gap between the half-spaces is filled with any dissipative medium is made surprisingly difficult because of the lack of definition of the electromagnetic stress tensor for arbitrary time-varying fields in dissipative media (see p. 161 of [5] and p. 263-264 of [6] for discussions on this topic). It was eventually solved by Dzyaloshinskii et al. (DLP from now on) [5]. They used the Matsubara diagram technique which was developed for the solution of thermal equilibrium problems in quantum many-body theory [5, 7]. This assumption is most practically reflected in the usage of the so-called* Matsubara frequencies* in calculating Casimir energy and pressure. The Matsubara frequencies take on the form (), where is the Boltzmann constant, is Planck’s constant, and all objects are at temperature . Van Kampen et al. [8] and Parsegian and Ninham [9] circumvented the complications of the DLP method but, in doing so, they had to postulate that the free energy of an electromagnetic mode at frequency is given by , even though is, in general, complex for dissipative media. Barash and Ginzburg [6] argued that the above-mentioned form of the free energy is the right one for electromagnetic fields in thermal equilibrium with matter [6, 10]. While many authors have attempted to generalize Lifshitz theory to determe Casimir pressure in dissipative media, they do so by assuming an expression for the electrodynamic stress tensor [11] or by defining a Lagrangian density for the electrodynamic field [12], both of which are debatable for media with dissipation [13].

A quick survey of chapters 2 and 3 of [14] should convince the reader that a lot has to be learned, in comparison to what is necessary to understand Lifshitz theory of Casimir energy/pressure in vacuum, before one can truly understand the nuances of DLP’s method to calculate Casimir energy/pressure in dissipative media. Yet the expression for for Casimir/van der Waals forces between two objects separated by vacuum, obtained by Lifshitz’ method, is strikingly similar to the expression derived by DLP. Given the similarity between the expressions of Casimir forces via the two techniques, the disparity between the relative simplicity of Lifshitz’ method and the complications of DLP’s generalization to dissipative media prompted us to look at this problem more closely. The question we asked ourselves was as follows: is it not possible to obtain the Casimir force between two objects separated by a dissipative medium without having to rely on DLP’s method? We answered this question in the affirmative and obtained a more, in our opinion, transparent method for calculating Casimir energy/pressure in dissipative media [15, 16]. The hallmark of the method we developed was that we restricted all calculations of electromagnetic stress tensor to locations in vacuum, even though the eventual goal was to compute the Casimir energy/pressure in a dissipative medium.

The outline of the rest of this paper is as follows: in Section 2, the principle of conservation of energy is used to relate the Casimir energy and pressure of a multilayer system of thin films to the Casimir energy/pressure of smaller units that comprise the multilayer system. In Section 2.1, the stress tensor is related to the DGF through the fluctuation-dissipation theorem. A commonly used technique of replacing integration along the real frequency axis (-axis) by a summation along the imaginary axis (-axis) is described in Section 2.2, along with a discussion of the pros and cons of both representations. In Section 3, the method developed in Section 2 is applied to the case of a thin dissipative film sandwiched between two half-spaces. The contributions of propagating waves and evanescent waves to the Casimir energy/pressure in a multilayer system of thin films are discussed in Section 4. The main points of this paper are summarized in Section 5.

#### 2. General Formulation of Casimir/Van Der Waals Energy and Pressure in Multilayered Media

The concepts of work of adhesion and cohesion are usually explained with the aid of thought experiments involving cleaving of two contiguous half-spaces and (to be referred to as from now on) into two half-spaces and at infinite separation with vacuum between them [1, 17, 18], as shown in Figure 1(a). The work of adhesion of , , is the energy required to separate into two half-spaces and at infinite separation with vacuum between them. The superscript is used to indicate that this is the work done in separating and across a vacuum gap. The work of adhesion can be related to the three free energies of the three interfaces , ( refers to half-space of adjacent to a half-space vacuum), and as follows:where , , and are free energies of , , and , respectively. The reverse of the procedure shown in Figure 1(a), that is, starting from and and bringing them together to form , is shown in Figure 1(b). Free energy balance for this process corresponds to reorganization of (1) to the form , so that can be determined if , , and are known. While Casimir/van der Waals interactions contribute to the energy of half-spaces such as , , and , what is measurable in an experiment is the force between two objects due to Casimir/van der Waals interactions. The Casimir force between two objects exists because of the pressure due to fluctuational electromagnetic waves in the medium between the two objects. It can be shown that the Casimir pressure in a half-space of homogeneous material, such as or , is identically equal to zero. So the Casimir force between half-spaces and can only be measured when they are separated by a film of finite thickness between them. The main insight of our method is that can be calculated exactly by using Rytov’s theory of fluctuational electrodynamics and that from such calculations the Casimir pressure in a dissipative material can be determined.