Constraints on Stable Equilibria with Fluctuation-Induced (Casimir) Forces

Abstract

We examine whether fluctuation-induced forces can lead to stable levitation. First, we analyze a collection of classical objects at finite temperature that contain fixed and mobile charges, and show that any arrangement in space is unstable to small perturbations in position. This extends Earnshaw's theorem for electrostatics by including thermal fluctuations of internal charges. Quantum fluctuations of the electromagnetic field are responsible for Casimir/van der Waals interactions. Neglecting permeabilities, we find that any equilibrium position of items subject to such forces is also unstable if the permittivities of all objects are higher or lower than that of the enveloping medium; the former being the generic case for ordinary materials in vacuum.Earnshaw's theorem [1] states that a collection of charges cannot be held in stable equilibrium solely by electrostatic forces. The charges can attract or repel, but cannot be stably levitated. While the stability of matter (due to quantum phenomena), and dramatic demonstrations of levitating frogs [2], are vivid reminders of the caveats to this theorem, it remains a powerful indicator of the constraints to stability in electrostatics. There is much current interest in forces induced by fluctuating charges (e.g., mobile ions in solution), or fluctuating electromagnetic (EM) fields (e.g., the Casimir force between metal plates). The former (due to thermal fluctuations) may lead to unexpected phenomena such as attraction of like-charged macroions, and is thought to be relevant to interactions of biological molecules. The latter (due mainly to quantum fluctuations) is important to the attraction (and stiction) of components of microelectromechanical (MEM) devices. Here, we extend Earnshaw's theorem to some fluctuation-induced forces, thus placing strong constraints on the possibility of obtaining stable equilibria, and repulsion between neutral objects.An extension of Earnshaw's theorem [1] to polarizable objects by Braunbek [3,4] establishes that dielectric and paramagnetic ( > 1 and µ > 1) matter cannot be stably levitated by electrostatic forces, while diamagnetic (µ < 1) matter can. This is impressively demonstrated by superconductors and frogs that fly freely above magnets [2]. If the enveloping medium is not vacuum, the criteria for stability are modified by substituting the static electric permittivity M and magnetic permeability µ M of the medium in place of the vacuum value of 1 in the respective inequalities. In fact, if the medium itself has a dielectric constant higher than the objects ( < M ), stable levitation is possible, as demonstrated for bubbles in liquids (see Ref. [5], and references therein). For dynamic fields the restrictions of electrostatics do not apply; for example, lasers can lift and hold dielectric beads with index of refraction n = √ µ > 1 [6]. In addition to the force which keeps the bead in the center of the laser beam, there is radiation pressure which pushes the bead along the direction of the Poynting vector. Ashkin and Gordon hav...

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