Quantitative theory–experiment comparison for fluid-mediated repulsive Casimir–Lifshitz forces: broadband dielectric data for the intervening fluid and removal of the small-separation and force-sensitivity limits

Munday, Capasso and Parsegian (Nature, 2009) demonstrated that the long-range quantum-fluctuation force between a gold sphere and a silica plate immersed in bromobenzene is repulsive, in agreement with the sign predicted by Lifshitz theory, but the quantitative comparison between theory and experiment was limited to the level of tens of percent. The authors attributed this discrepancy to the incomplete knowledge of bromobenzene's dielectric response over the full frequency spectrum entering the Lifshitz integral, to the insufficiency of the commonly used two-oscillator model for this fluid, to the breakdown of surface-roughness corrections at the smallest separations, to possible modification of optical properties in very thin films, and to a force-sensitivity floor of roughly 10 pN in fluid AFM measurements. The open problem is to convert this qualitative verification into a precise, predictive quantitative description of fluid-mediated repulsive Casimir–Lifshitz forces: determine the dielectric response of the intervening fluid over the entire relevant spectral range, validate material models beyond the two-oscillator approximation, control roughness and thin-film uncertainties at small gaps, and extend fluid force sensitivity below ~10 pN, so that Lifshitz theory and measured repulsive forces can be compared at the precision achieved in vacuum experiments and used predictively for material selection in quantum-levitation and ultra-low-friction applications. No measured broadband optical data for bromobenzene and no percent-level repulsive-regime agreement are known to date, and the problem remains open.

Background

Quantum and thermal electromagnetic fluctuations generate dispersion forces between neutral bodies. At separations of a few nanometres these are the non-retarded van der Waals forces; at larger separations the finite speed of light modifies the interaction, a retardation effect recognized in the theories of Casimir, Polder and Lifshitz. Lifshitz theory, as extended by Dzyaloshinskii and Pitaevskii, computes the force between real macroscopic bodies from their frequency-dependent dielectric response functions evaluated at imaginary frequencies, $\varepsilon(i\xi)$, which are obtained from ordinary optical absorption data by a Kramers–Kronig transformation. For two solids (materials 1 and 2) separated by a fluid (medium 3), the force per unit area is a sum of terms proportional to $-(\varepsilon_1-\varepsilon_3)(\varepsilon_2-\varepsilon_3)$ over the whole frequency spectrum, so the interaction is repulsive when $\varepsilon_1(i\xi)>\varepsilon_3(i\xi)>\varepsilon_2(i\xi)$ over the relevant frequencies. A fluid-mediated system satisfying this ordering is gold–bromobenzene–silica. Before 2009, only attractive Casimir–Lifshitz forces had been measured directly between macroscopic bodies; repulsion had been inferred only in the non-retarded regime, where molecular polarity, solvation and surface chemistry complicate interpretation. Munday, Capasso and Parsegian (2009) used an atomic force microscope (AFM), in which a gold-coated sphere on a soft cantilever is brought toward a plate inside a bromobenzene-filled cell, and measured the force as a function of separation: the force was attractive toward gold and repulsive toward silica, increasing in magnitude as the gap narrowed, in qualitative agreement with temperature-dependent Lifshitz theory. The quantitative comparison, however, showed a significant discrepancy between theory and experiment, which the authors traced to several specific causes: the dielectric (optical) properties of bromobenzene had not been measured over the large spectral range needed for the Lifshitz integral, so the fluid's $\varepsilon(i\xi)$ was represented by a crude two-oscillator model (a small set of damped harmonic-absorption resonances) that earlier attractive-force measurements in ethanol suggested was insufficient; the optical properties of very thin films (nanometre-scale coatings and interfacial layers) may differ from bulk values; the standard multiplicative roughness correction to Lifshitz theory begins to fail when the roughness amplitude becomes comparable to the surface separation, i.e. at the smallest, most force-rich gaps; and the spread in the fluid data set a floor of roughly 10 pN below which force magnitudes could not be accurately determined, limiting the accessible separation range. Repulsive Casimir–Lifshitz forces are the physical basis proposed for 'quantum levitation' of one surface above another in a fluid, for suppression of stiction (spontaneous adhesion of contacting micro-parts) in micro- and nanoelectromechanical systems (MEMS/NEMS), and for ultra-low static friction devices — applications that all require predictive, quantitative control of the force, not merely the correct sign. Subsequent work sharpened the diagnosis: van Zwol, Palasantzas and De Hosson (2009) showed that scatter in the dielectric data of the solids and the fluid alone can change the Lifshitz prediction by up to a factor of two when the liquid's $\varepsilon(i\xi)$ approaches one solid's, so that even the sign of the agreement test is uncertain without full-spectrum data for the actual samples; they and van Zwol and Palasantzas (2010) introduced Kramers–Kronig-consistency schemes to construct usable dielectric functions for liquids from partial literature data, noting significant variation between published sources; and van Zwol, Svetovoy and Palasantzas (2011) documented how roughness corrections grow at small separations and how the 'distance upon contact' (the unknown true zero of the gap between rough surfaces) limits absolute separation determination. Obtaining a usable full-spectrum response for the fluid therefore remains a methodological question — direct broadband optical measurement across the spectrum (far-infrared/microwave through ultraviolet) or a Kramers–Kronig-consistent construction from partial data validated to the precision the Lifshitz integral demands — as does reaching the smallest separations, through smoother surfaces or a validated small-gap roughness treatment that carries the distance-upon-contact uncertainty.

Problem Statement

Establish a precise, quantitative comparison between finite-temperature Lifshitz theory and measured repulsive Casimir–Lifshitz forces in a fluid-mediated solid–liquid–solid system, taking the gold–bromobenzene–silica configuration of Munday, Capasso and Parsegian (2009) as the reference case, and thereby make the repulsive interaction predictive for material selection. This requires, as jointly necessary components of the same objective: (1) determining the dielectric response of the intervening fluid, bromobenzene, over the full frequency range entering the Lifshitz integral, replacing the two-oscillator model that the source experiment found insufficient and removing the dependence on scattered, mutually inconsistent literature data; (2) quantifying or excluding the suspected modification of optical properties in very thin films (nanometre-scale coatings and interfacial layers) of the interacting bodies, so the theory uses the optical constants of the actual samples; (3) extending the comparison to the smallest separations, where the surface-roughness amplitude becomes comparable to the gap, the standard roughness correction to Lifshitz theory fails, and the absolute separation ('distance upon contact') is uncertain; and (4) improving fluid force-sensitivity beyond the ~10 pN floor of the original AFM measurement so that weaker forces and larger separations are reliably determined, widening the separation range over which theory and experiment can be compared. The aim is quantitative agreement between the computed and measured repulsive force over a separation range wide enough to test the theory at a precision comparable to that achieved in vacuum Casimir experiments — rather than the tens-of-percent agreement and qualitative sign verification available today — together with the ability to predict, from dielectric data, which solid–fluid–solid combinations produce repulsion of useful magnitude for quantum levitation, stiction suppression, and ultra-low-friction nanoscale devices.

References

1. J. N. Munday, F. Capasso, V. A. Parsegian, 'Measured long-range repulsive Casimir–Lifshitz forces', Nature 457, 170–173 (2009). DOI: 10.1038/nature07610, https://doi.org/10.1038/nature07610
2. P. J. van Zwol, G. Palasantzas, J. Th. M. De Hosson, 'Influence of dielectric properties on van der Waals/Casimir forces in solid-liquid systems', Physical Review B 79, 195428 (2009). DOI: 10.1103/PhysRevB.79.195428, arXiv:0905.0889, https://arxiv.org/abs/0905.0889
3. P. J. van Zwol, G. Palasantzas, 'Repulsive Casimir forces between solid materials with high-refractive-index intervening liquids', Physical Review A 81, 062502 (2010). DOI: 10.1103/PhysRevA.81.062502, arXiv:1004.5243, https://arxiv.org/abs/1004.5243
4. P. J. van Zwol, V. B. Svetovoy, G. Palasantzas, 'Characterization of optical properties and surface roughness profiles: The Casimir force between real materials', in Casimir Physics, Lecture Notes in Physics (Springer, 2011). DOI: 10.1007/978-3-642-20288-9_10, arXiv:1101.2224, https://arxiv.org/abs/1101.2224
5. A. Le Cunuder, A. Petrosyan, G. Palasantzas, V. Svetovoy, S. Ciliberto, 'Measurement of the Casimir force in a gas and in a liquid', Physical Review B 98, 201408(R) (2018). DOI: 10.1103/PhysRevB.98.201408, arXiv:1807.10350, https://arxiv.org/abs/1807.10350
6. Q. Hu, J. Sun, Q. Zhao, Y. Meng, 'Experimentally demonstration of the repulsive Casimir force in the gold-cyclohexane-PTFE system', arXiv:1911.09938 (2019). https://arxiv.org/abs/1911.09938
7. M. Moazzami Gudarzi, S. H. Aboutalebi, 'Self-consistent dielectric functions of materials: Toward accurate computation of Casimir–van der Waals forces', Science Advances 7, eabg2272 (2021). DOI: 10.1126/sciadv.abg2272, https://doi.org/10.1126/sciadv.abg2272
8. C. Shelden, B. Spreng, J. N. Munday, 'Opportunities and challenges involving repulsive Casimir forces in nanotechnology', Applied Physics Reviews 11, 041325 (2024). DOI: 10.1063/5.0218274, https://doi.org/10.1063/5.0218274
9. C. F. Shelden, J. N. Munday, 'Casimir-force spectroscopy of broadband optical response', arXiv:2601.10118 (2026). https://arxiv.org/abs/2601.10118
10. M. Hošková, O. V. Kotov, B. Küçüköz, C. J. Murphy, T. O. Shegai, 'Casimir self-assembly: A platform for measuring nanoscale surface interactions in liquids', Proceedings of the National Academy of Sciences USA 122 (2025). DOI: 10.1073/pnas.2505144122, https://doi.org/10.1073/pnas.2505144122
11. N. Inui, K. Goto, 'Diffusion Constant of a Gold Particle Levitated by a Repulsive Casimir Force near a Silica Plate Immersed in Bromobenzene', Journal of the Physical Society of Japan 82, 114003 (2013). DOI: 10.7566/JPSJ.82.114003, https://doi.org/10.7566/JPSJ.82.114003