Universal Casimir interactions in the sphere–sphere geometry
Tanja Schoger · Benjamin Spreng · Gert-Ludwig Ingold · Astrid Lambrecht · Paulo A. Maia Neto · Serge Reynaud
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In one page
Tanja Schoger and colleagues in Augsburg, Davis, Jülich, Rio de Janeiro and Paris — including Astrid Lambrecht and Serge Reynaud, two of the people who built the modern theory of the Casimir force — take two experiments that look nothing alike and show they are the same problem. In one, two metal spheres sit in vacuum. In the other, two glass beads float in salty water. In both, once you go far enough apart, the force stops depending on what the objects are made of: it becomes universal, set only by temperature and geometry. The catch in vacuum is distance. The universal part only takes over beyond about eight micrometres, where the force is almost too faint to measure. In salt water the same regime starts at about a tenth of a micrometre, which is why it could actually be measured — with optical tweezers holding one silica bead near another. The authors also show the answer depends mainly on one shape parameter, a near conformal invariance.
Why it matters hereChapter 2 is the chapter about the vacuum being a real, structured medium, and the Casimir force is its most direct laboratory handle. This paper is where that handle becomes universal: at long range the force forgets the material entirely and reports only temperature and shape — and the salted-water version brings that regime down to a hundred nanometres, where optical tweezers can reach it.
What it claims
01In both configurations studied — Drude-model metal spheres in vacuum, and dielectric spheres immersed in a salted solution beyond the Debye screening length — the high-temperature Casimir free energy is due entirely to transverse magnetic modes, while the transverse electric contribution vanishes.Section II, Scattering formula for Casimir interaction, subsection A
Published and peer-reviewed02The two problems are dual to each other, and that is the source of the universality: in each case the static dielectric function of one medium is infinite while the other stays finite, with the Drude sphere and the salted water both carrying a finite static conductivity. This is why the results are independent of the details of the optical response of the materials involved.Section VI, Conclusion
Published and peer-reviewed03The distance at which the universal term takes over differs enormously between the two cases. In vacuum it is essentially the thermal wavelength, the reduced Planck constant times the speed of light divided by Boltzmann's constant times temperature — about 8 micrometres at room temperature, where the Casimir force is so weak that experimental detection is challenging. In salted water the crossover falls to about 100 nanometres, so the universal regime covers a far broader accessible range.Section I, Introduction; Section VI, Conclusion
Published and peer-reviewed04The universal thermal Casimir interaction between two silica microspheres in salted water has been measured. Pires and co-workers used single-beam optical tweezers over the range 0.2 to 0.5 micrometres, trapping a microsphere of radius 2.35 micrometres near a second sphere of radius 11.74 micrometres attached to the coverslip, in a 0.22 molar sodium iodide solution whose Debye screening length of 0.64 nanometres completely suppresses electrostatic interactions; the measured energies follow the curve that includes the universal thermal contribution.Section V; Figure 9, adapted from Pires et al., Phys. Rev. Research 3, 033037 (2021)
Published and peer-reviewed05The reduced free energy shows an approximate conformal invariance: it is governed mainly by a single conformally invariant geometric parameter built from the surface-to-surface separation and the effective radius, the product of the two radii over their sum, with only weak dependence on the radius-ratio parameter that runs from zero for the plane-sphere geometry to one quarter for two equal spheres. For the transverse magnetic modes the free energy can be written in terms of the determinant of the dimensionless capacitance matrix of two spherical conductors — a bridge between the electromagnetic Casimir effect and the critical Casimir effect.Section III, Universal Casimir interaction and conformal invariance
Published and peer-reviewed06The consequences run into soft matter and biology. Because immersion in a liquid imposes Brownian motion on the spheres, the interaction matters when it is a significant fraction of the thermal energy — which the authors show is met when the surface-to-surface distance is of order one tenth of the effective radius or smaller, a regime that cannot be deduced from any of the known limiting cases but can be estimated without the full numerical calculation.Section VI, Conclusion
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Abstract
We study universal Casimir interactions in two configurations which appear as dual to each other. The first involves spheres described by the Drude model and separated by vacuum while the second involves dielectric spheres immersed in a salted solution at distances larger than the Debye screening length. In both cases, the long-distance limit, equivalently the high-temperature limit, is dominated by the effect of low-frequency transverse magnetic thermal fluctuations. They are independent of the details of dielectric functions of materials, due to the finite conductivity of metals in the former case and of salted water in the latter one. They also show universality properties in their dependence on geometric dimensions, in relation to an approximate conformal invariance of the reduced free energy.
Tanja Schoger, Benjamin Spreng, Gert-Ludwig Ingold, Astrid Lambrecht, Paulo A. Maia Neto and Serge Reynaud. International Journal of Modern Physics A, volume 37, issue 19, article 2241005, published 20 May 2022.
Universität Augsburg, Institut für Physik, Augsburg, Germany; Department of Electrical and Computer Engineering, University of California, Davis, USA; Forschungszentrum Jülich and RWTH Aachen University, Germany; Instituto de Física, Universidade Federal do Rio de Janeiro, Brazil; Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-PSL, Collège de France, Paris, France.
(Abstract only. The author version is on arXiv as arXiv:2205.10812 — see the rights note above for why the full text is not reproduced here.)
The way in
https://doi.org/10.1142/S0217751X22410056Checked 2026-09-08 and no Creative Commons version was found. The published article in the International Journal of Modern Physics A carries no licence in its Crossref record and the publisher page is behind a copyright statement. The author version is arXiv:2205.10812, and arXiv's own OAI metadata gives its licence as the arXiv perpetual non-exclusive distribution licence 1.0, which is not a Creative Commons licence — Unpaywall and OpenAlex both label that copy cc-by, which the arXiv record contradicts. The HAL and repository copies carry no licence field either. The only Creative Commons statement inside the paper covers an adapted figure reused from Pires et al. Abstract reproduced here under fair quotation; the full text is at the source.
How to cite it
Tanja Schoger, Benjamin Spreng, Gert-Ludwig Ingold, Astrid Lambrecht, Paulo A. Maia Neto, Serge Reynaud (2022) Universal Casimir interactions in the sphere–sphere geometry. doi:10.1142/S0217751X22410056
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