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Measurement of the Casimir torque

Nature volume 564, pages 386–389 (2018)Cite this article

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Abstract

Intermolecular forces are pervasive in nature and give rise to various phenomena including surface wetting1, adhesive forces in biology2,3, and the Casimir effect4, which causes two charge-neutral, metal objects in vacuum to attract each other. These interactions are the result of quantum fluctuations of electromagnetic waves and the boundary conditions imposed by the interacting materials. When the materials are optically anisotropic, different polarizations of light experience different refractive indices and a torque is expected to occur that causes the materials to rotate to a position of minimum energy5,6. Although predicted more than four decades ago, the small magnitude of the Casimir torque has so far prevented direct measurements of it. Here we experimentally measure the Casimir torque between two optically anisotropic materials—a solid birefringent crystal (calcite, lithium niobite, rutile or yttrium vanadate) and a liquid crystal (5CB). We control the sign and strength of the torque, and its dependence on the rotation angle and the separation distance between the materials, through the choice of materials. The values that we measure agree with calculations, verifying the long-standing prediction that a mechanical torque induced by quantum fluctuations can exist between two separated objects. These results open the door to using the Casimir torque as a micro- or nanoscale actuation mechanism, which would be relevant for a range of technologies, including microelectromechanical systems and liquid crystals.

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Fig. 1: Apparatus measuring the Casimir torque.
Fig. 2: Measured sin(2θ) angular dependence of the Casimir torque.
Fig. 3: Distance dependence of Casimir torque.

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Data availability

The data that support the findings of this study are available from the corresponding author on reasonable request.

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Acknowledgements

This work was supported by the National Science Foundation under grant numbers PHY-1506047 and PHY-1806768. We acknowledge support from the FabLab at the Maryland NanoCenter and thank M. S. Leite for discussions and comments on the manuscript.

Reviewer information

Nature thanks O. Lavrentovich, S. Zumer and the other anonymous reviewer(s) for their contribution to the peer review of this work.

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Authors and Affiliations

  1. Department of Physics, University of Maryland, College Park, MD, USA

    David A. T. Somers, Joseph L. Garrett, Kevin J. Palm & Jeremy N. Munday

  2. Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, MD, USA

    David A. T. Somers, Joseph L. Garrett, Kevin J. Palm & Jeremy N. Munday

  3. Department of Electrical and Computer Engineering, University of Maryland, College Park, MD, USA

    Jeremy N. Munday

Authors
  1. David A. T. Somers
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Contributions

J.N.M. conceived and supervised the project. D.A.T.S. designed the apparatus, performed experiments and analysed the resultant data. J.L.G. and K.J.P. performed AFM experiments and analysis, and K.J.P. performed spectroscopic ellipsometric measurements and analysis. All authors discussed and interpreted the data and wrote the manuscript.

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Correspondence to Jeremy N. Munday.

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Extended data figures and tables

Extended Data Fig. 1 Evidence for amorphous, isotropic deposition of Al2O3.

Atomic layer deposition of Al2O3 on planar substrates at low temperatures (in our case 150 °C) generally results in isotropic, amorphous films38. To confirm this expectation, we performed three different experiments to characterize the films. a–c, First, TEM measurements at different magnifications show that the Al2O3 layer is amorphous (a, b), whereas the LiNbO3 layer is crystalline (a, c), as expected. d, Second, spectroscopic ellipsometry data are found to be consistent with an isotropic layer of Al2O3 existing on top of the birefringent LiNbO3 crystal. We first determined the optical properties for the birefringent LiNbO3 crystal without any Al2O3 coating. Ψ and Δ data (ellipsometric amplitude and phase ratio, respectively) for 55°, 60° and 75° measurement angles are performed and fitted to a Tauc Lorentz biaxial (anisotropic) model, yielding a good ellipsometric fit. e, Third, spectroscopic ellipsometry was performed on a LiNbO3 crystal with a 5-nm-thick Al2O3 coating. The previously determined model for LiNbO3 was used together with an isotropic model for Al2O3, yielding a similarly good fit with no increase in the mean squared error. A birefringent model for Al2O3 could also be used; however, this results in additional fit parameters that artificially reduce the mean squared error below that obtained with the LiNbO3 crystal alone, providing strong evidence that the isotropic model of Al2O3 is more physical. The resulting indices of refraction n are shown for both the ordinary (‘Ord’) and extraordinary (‘Ext’) axes of LiNbO3 and for isotropic Al2O3. f, g, Fourth, to rule out potential in-plane anisotropy due to correlation in the orientation of the surface roughness, we performed atomic force microscopy (AFM) topography scans of the Al2O3 surface as-deposited for our glass control samples (f) and the LiNbO3 crystal (g). The roughness is randomly oriented with no preferred direction in both cases. We further considered the widths of the roughness peaks in the two perpendicular directions (x and y in the insets, which show the topography scans). For the Al2O3 film on the glass substrate, the average peak widths in the x and y directions are 1.2 ± 0.3 nm and 1.0 ± 0.2 nm, respectively. Similarly, for the Al2O3 film on the LiNbO3 substrate, the average peak widths in the x and y directions are both 1.0 ± 0.2 nm. Although small variations between the x and y scans may be expected due to sample drift or tip geometry, no substantial difference is measured.

Extended Data Fig. 2 Control experiment showing no measured torques from isotropic borosilicate glass.

Glass substrates are coated with 6 nm of Al2O3 (orange) or around 6 nm of PVA (purple). When fitted with a sin(2θ) function (solid lines), there is no measurable torque.

Extended Data Fig. 3 Determination of the distance offset.

We use AFM to probe the thickness of the FC-4430 surfactant layer that forms between the 5CB liquid crystal and the solid substrate, using two different methods39,40. a, The first sample was prepared by spreading a 0.5% FC-4430 in 5CB mixture (similar to those used in the torque experiments) across a glass substrate to allow for segregation of the materials. AFM scans (dynamic mode in air to ensure that the optical lever detection scheme of AFM was not distorted by the birefringence of the liquid crystal) show the surfactant monolayer (grey) on glass (black) with 5CB droplets (white). b, A line scan across surfactant layer (dashed red line in a) shows the step height of the surfactant film on glass to be 4.5 ± 0.3 nm. This measurement puts a minimum bound on the thickness of the surfactant layer, because when immersed in 5CB (as done in the torque experiments) the surfactant molecules will probably extend further from the surface. c, d, To determine the total extent of the surfactant in situ, we performed a second set of experiments, using a technique similar to that used in ref. 40. For these experiments, a gold sample is used to allow a bias voltage to be applied between the tip and the sample to more accurately determine the exact location of tip–sample contact. c, A droplet of 5CB is placed on the gold sample and a platinum AFM tip is brought towards the surface under 3-V a.c. bias, without any surfactant present. The amplitude of the piezoelectrically driven oscillation of the tip smoothly decreases as it approaches the 5CB–gold boundary, except where the tip sticks to the surface after contact (tip–sample separation of zero). d, A second gold sample is used with a droplet of the 0.5% FC-4430 in 5CB mixture (similar to that used in the torque experiments), so that the surfactant covers both surfaces. Again, the amplitude of tip oscillation is recorded during the approach and retract runs under 3-V a.c. bias. On approach, the amplitude decreases rapidly at a certain separation (about 20 nm for this scan), indicating the distance at which the surfactant layers on each surface join together (that is, ‘jump’ to contact). On retraction, multiple features are observed at locations where molecules attached to both surfaces separate (that is, ‘rupture’). e, Histogram showing separations at which jumps occur over several distinct distance sweeps. A jump is observed in 21 of 31 approach–retract curves and occurs at an average tip–sample separation of 16.6 nm, which would imply a 8.3-nm-thick surfactant layer on each surface when immersed in 5CB. Considering the measured surface roughness of 4–5 nm, we use a distance offset of 12 ± 4 nm.

Extended Data Fig. 4 Dielectric models for the relevant materials.

For birefringent materials, solid and dashed lines indicate the dielectric function ε along the ordinary and extraordinary axes, respectively. The dielectric models for the 5CB liquid crystal are from ref. 41, the dielectric functions for the solid crystals are modelled using a previous method42 and the dielectric data for YVO4 are from refs 43,44. The dielectric function of the thin FC-4430 layer is unknown, but its precise value has little effect on the torque. For our calculations, the layers between the two crystals are treated as a uniform, homogeneous medium described by the optical properties of Al2O3 alone (rather than part Al2O3 and part FC-4430). As an extreme case, we also calculate the torque with a uniform medium with the optical properties of H2O, which are probably similar to the unknown FC-4430, rather than Al2O3. The calculated torque changes by less than 6% for the distances used in our experiment and are thus within our calculation uncertainties.

Extended Data Fig. 5 Polarized spectrometry measurement of each sample.

a, Polarized white-light image of a 1-cm2 substrate (TiO2) with nine different Al2O3 thicknesses achieved through atomic layer deposition and masking. The coloured circles indicate separate polarized spectrometry measurements, two of which (bolded) are shown in more detail in b–g. b, c, Normalized, measured transmission as a function of analyser angle and wavelength for regions with thick (b) and thin (c) Al2O3 layers. The thick (thin) layer corresponds to a larger (smaller) separation d between the birefringent materials. The dashed lines indicate θrub and θrub − 90°. d, e, Results from fitting b and c using equation (1) in Methods. f, g, Transmitted intensities (dots, measured values; solid lines, theoretical fits) along θa = θrub = 135° and θa = θrub − 90° = 45° are shown for the thick (f) and thin (g) Al2O3 layers. A strong torque causes a larger difference in transmission between θa = θrub and θa = θrub − 90° (compare g and f). The slight offset between the fit and measurement in g is because only two of the 25 measurements used in the fit are plotted. Combining all measurements, the error in the angle is less than 4°.

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Somers, D.A.T., Garrett, J.L., Palm, K.J. et al. Measurement of the Casimir torque. Nature 564, 386–389 (2018). https://doi.org/10.1038/s41586-018-0777-8

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