Prediction of a measurable sign change in the Casimir force using a magnetic fluid

Long Ma and Dai-Nam Le and Lilia M. Woods (Department of Physics, University of South Florida, Tampa), Larissa Inácio and Mathias Boström (Centre of Excellence ENSEMBLE3, Warsaw, and the Chemical and Biological Systems Simulation Lab, Centre of New Technologies, University of Warsaw), Prediction of a measurable sign change in the Casimir force using a magnetic fluid, Physical Review Materials 10, 025002, 11 February 2026. Preprint dated 5 January 2026. The published article is at doi.org/10.1103/33cl-d2wn; the CC BY author copy reproduced below is at arxiv.org/abs/2601.00483.

(On this site, the repulsive-Casimir work this paper builds on is at /library/stm-7bee2ee082, the first measurement of a long-range repulsive Casimir–Lifshitz force is at /library/stm-bd8d775581, Casimir repulsion switched electrically in a semiconductor is at /library/stm-46be43d5f1, non-monotonic Casimir forces shaped by nanostructure are at /library/stm-5d1ea02d6d, and the time-domain formulation of Casimir forces is at /library/stm-2fbc259fbe.)

Abstract

We demonstrate quantum levitation controlled by Casimir forces acting between a polystyrene surface and a Teflon-coated metallic substrate immersed in a mixture of Toluene and magnetite particles. This system experiences repulsion-attraction transitions in the Casimir interaction for distances where the effect is measurable. This Casimir trapping can be controlled by clever choices of metallic and ferrofluid materials, which are directly linked to the emergence of the trapping effect. Thermal and quantum contributions are investigated in detail, showing how the optical and magnetic properties of the ferrofluid and other materials affect the magnitude of the trapping and its distance range of observability.

I. Introduction

Understanding the Casimir entropic equilibria in parallel plate systems is crucial in Casimir-related quantum electrodynamics and applications, including Casimir switches, Casimir self-assembled structures, and mitigation of stiction in micro- and nano-electromechanical devices (MEMS and NEMS). Casimir attraction-repulsion transition can be achieved in various ways, including by using microstructured geometry, phonon-assisted lattice vibration, twisted angle in anisotropic plates, suspended graphene in fluids, dielectric metamaterials on Mie resonance, and in Casimir-Lifshitz torques between anisotropic phosphorene sheets, among others. Casimir trapping has also been shown to play a role in geophysics of water and ice equilibria.

Here, we present an alternative way to effectively control sign changes and Casimir quantum trapping by utilizing magnetic materials in clever combination with more conventional dielectric systems. The interplay between the optical and magnetic properties, together with the dimensions of solid film coatings, proves to give a viable pathway towards Casimir force attraction-repulsion transitions. The proposed method for Casimir trapping builds on previous work, which showed that repulsive interaction between planar substrates separated by a ferrofluid gap is possible, but it did not explore its tuning capabilities. The ultimate goal achieved here is to establish ways to control interactions in NEMS via tunable magnetic Casimir effects. Our focus is on layered systems where one can have measurable multiple sign changes in the force controlled by the magnetic permeability of an intermediate ferrofluid layer and the optical properties of the fluid and metallic layers in the system.

The theoretical background of the proposed method lies within the Lifshitz formalism, established as a general theory for dispersion forces between planar surfaces, that linked the Casimir interaction to the dielectric properties of the involved materials. Furthermore, the frequency-dependent dielectric functions are related to the refractive index and adsorption coefficients of the media, which clarifies the relationship between forces and radiation processes proposed before the birth of quantum mechanics by Lebedev.

Following the original derivation, Ninham, Parsegian and co-workers showed that the Lifshitz theory can be simplified using semi-classical scattering theory; they also extended the description to account for the magnetic response properties of the materials. The Lifshitz theory has been extensively explored, so one might have expected that fundamentally new insights should be difficult to find. However, we demonstrate new ways to control multiple repulsion-attraction transition distances in a four-layered system containing a magnetic fluid composed of a solution with dispersed magnetic nanoparticles. By tuning the properties of the ferrofluid in combination with clever selection of substrate dielectrics and thickness of thin surface coatings, the Casimir interaction experiences stable equilibrium potentially measurable in the laboratory via optical interferometry measurements, for example. The ferrofluid plays a key role in this process, in which all types of quantum and thermal excitations are found to actively participate in forming the trapping regime at sufficiently small separations where the force is large enough. The proposed method is beneficial in broadening the applications of NEMS and MEMS for sensors in liquid environments and biological media.

II. Optical and magnetic properties of the considered planar system

The layered system under consideration is given schematically in Figure 1. It consists of a planar polystyrene substrate, called A, and a semi-infinite metallic layer, called B, which is coated by a Teflon film, called B1, of finite thickness b1. The A substrate and Teflon-coated metallic substrate are separated by a gap of thickness l filled with a ferrofluid material, called m. The distance-dependent Casimir interaction between the substrates in this system can be controlled in different ways, including by the choice of different metals in layer B, the coating thickness b1, and various solutions and properties of the ferrofluid in layer m. Since the Casimir interaction is linked to the response properties of the system, below we discuss the dielectric function and magnetic susceptibility of the involved materials. We use frequency-dependent data obtained by measuring the material’s refractive index and its description of the real and imaginary parts based on the Kramers-Kronig relation with an upper integration frequency bound of 100 electronvolts, as reported in the literature. Details of the theoretical modeling are given below in the imaginary frequency domain, with results shown in Figure 2.

Figure 1. Schematics of the system under consideration consisting of a ferrofluid layer, denoted m, with thickness l between a polystyrene substrate A and a metallic substrate B covered by a Teflon layer B1 of thickness b1.

As representative examples of the types of metallic B substrates, here we consider gold, silver, aluminum, and lithium. The metallic dielectric functions are modeled using the Drude model given in the imaginary frequency domain, in which the permittivity at imaginary frequency equals one plus the square of the plasma frequency divided by the product of the frequency and the sum of the frequency and the relaxation frequency; the plasma frequency and relaxation frequency are collected from experimental results for the considered materials. The dielectric functions for these metals are shown in Figure 2(a).

We also consider several solvent compositions of the ferrofluid, including Toluene, Benzene, Cyclohexane, and Octane. Their dielectric functions are parametrized using a model in which the permittivity is one plus a sum of oscillator terms, each term being a fitted oscillator strength divided by one plus the square of the ratio of the frequency to that oscillator’s characteristic frequency, with the set covering all primary absorption peaks in the infrared and ultraviolet ranges. Furthermore, the static dielectric constants of the solvents are obtained from experimental data measuring the refractive index, using the relation that the square of the static refractive index is approximately the static permittivity for negligible infrared absorption.

The ferrofluid solution contains randomly dispersed magnetic nanoparticles, here taken to be made of magnetite, whose dielectric function is also shown in Figure 2(a) using available experimental data. The effective dielectric function of the ferrofluid, composed of the solvent and the magnetite nanoparticles, is calculated based on the Rayleigh mixing model, Equation 1: the difference between the effective and solvent permittivities divided by the effective permittivity plus twice the solvent permittivity equals the volume fraction times the same combination formed from the magnetite and solvent permittivities.

From Equation 1, we find that the effective dielectric function equals the solvent permittivity multiplied by the factor one plus twice the product of the mixing parameter and the volume fraction, divided by one minus that product — where the mixing parameter is the difference between the magnetite and solvent permittivities divided by the magnetite permittivity plus twice the solvent permittivity. Figure 2(b) shows that in the low frequency regime, the effective permittivity increases as the volume fraction is increased. It can also be enhanced by taking solvents with larger static permittivity in the low-frequency range, as shown in Figure 2(c).

Figure 2 shows expected overall behavior, as the metallic permittivity of all metals diverges as the frequency goes to zero. On the other hand, the dielectric functions for the other dielectric materials have finite values consistent with the static permittivity from experiments. Nevertheless, the relative magnitude of the different layers in the four-layered system in the low-frequency regime is crucial for attraction-repulsion transitions. The different methods of property modulations discussed here show that Casimir trapping can be achieved in many ways.

Figure 2. Dielectric functions in the imaginary frequency range as a function of the reduced frequency for the materials in the four-layer system: (a) A is polystyrene, B is gold, silver, aluminum or lithium, B1 is Teflon, m is Toluene with volume fraction 5 per cent and 20 nanometre magnetite spheres; (b) A is polystyrene, B is gold, B1 is Teflon, m is Toluene, with volume fractions of 1, 5, 10 and 15 per cent and 20 nanometre magnetite spheres; (c) A is polystyrene, B is gold, B1 is Teflon, m is Toluene, Benzene, Cyclohexane or Octane with volume fraction 5 per cent and 20 nanometre magnetite spheres.

In addition to the dielectric response properties of the materials, the magnetic susceptibility of magnetite must also be considered in the Casimir interaction. Based on the distance range of interest, the model for the magnetic susceptibility is different from one only for the zero-frequency term. The static, zero-frequency, effective permeability for the ferrofluid can be modeled as Equation 2: it equals one plus two pi squared, times the volume fraction, times the square of the saturation magnetization per unit volume, times the cube of the nanoparticle average diameter, all divided by nine times the product of the Boltzmann constant and the temperature.

The above expression suggests that different material characteristics can be used to modulate the static permeability, which ultimately can modify the Casimir pressure as shown in what follows.

III. Casimir interaction in planar polystyrene-ferrofluid-Teflon-metal systems

The Casimir pressure between the Teflon-coated metallic and polystyrene substrates across the ferrofluid filled gap can be calculated using the Lifshitz formalism given in Matsubara frequencies, where the nth Matsubara frequency is two pi n times the Boltzmann constant times the temperature, divided by the reduced Planck constant.

Equation 3 gives the pressure as minus the Boltzmann constant times the temperature, times a sum over Matsubara frequencies and an integral over the in-plane wavevector, of a term for each polarization: twice the z-component of the wavevector in the ferrofluid, times the product of the two Fresnel reflection coefficients and the exponential attenuation factor, divided by one minus that same product. Here l is the separation distance in the nanocavity, as shown in Figure 1. The prime in the sum indicates there is a one-half weight on the first term, the term with n equal to zero.

The z-component of the wavevector in the ferrofluid is the square root of the sum of the squared in-plane wavevector and the product of the effective permittivity, the effective permeability and the squared Matsubara frequency divided by the squared speed of light. The Fresnel reflection coefficient for the A and m interface describes electromagnetic scattering there, with the polarization label taking the transverse-electric or transverse-magnetic value. The effective Fresnel reflection coefficients capture transverse-electric and transverse-magnetic scattering from the metallic B substrate coated by the Teflon film B1. These are effective coefficients obtained via a multi-scattering formalism, Equation 4: the effective coefficient is the coefficient at the B1-to-m interface plus the coefficient at the B-to-B1 interface multiplied by an exponential attenuation across the Teflon thickness, all divided by one plus the product of those two coefficients and the same attenuation factor. The wavevector in the z-direction in the Teflon layer is the square root of the squared in-plane wavevector plus the Teflon permittivity times the squared Matsubara frequency divided by the squared speed of light.

By solving Maxwell’s equations under standard boundary conditions, we find Equation 5 and Equation 6: the transverse-magnetic coefficient at an interface between media i and j is the difference between the products of the opposite wavevector and permittivity, divided by their sum; the transverse-electric coefficient is the same expression with the permeabilities in place of the permittivities.

IV. Results and discussion

The theoretical framework described in the previous section can now be used to calculate the Casimir pressure between the planar materials separated by the ferrofluid, as shown in Figure 1. We present a comprehensive understanding of the Casimir interaction and its various contributing factors by investigating the role of the specific metallic and solvent materials and the various structural characteristics of the system.

Figure 3. Casimir pressures normalized to the perfect conductor limit in the four-layer system from Figure 1, where the ferrofluid is Toluene with magnetite nanoparticles of average diameter 20 nanometres and volume fraction 5 per cent. For panels (a) to (d), the thickness of the Teflon layer is 10 nanometres while the metallic material B varies in plasma frequency and relaxation frequency: (a) lithium, plasma frequency 6.45 electronvolts and relaxation frequency 0.13 electronvolts; (b) silver, 8.9 and 0.02 electronvolts; (c) gold, 9 and 0.03 electronvolts; (d) aluminum, 12.04 and 0.13 electronvolts. For panels (e) to (h), the metallic layer is gold while the Teflon thickness varies: (e) 5, (f) 10, (g) 15, (h) 20 nanometres. The different contributions from transverse-electric and transverse-magnetic modes with zero-frequency and summed non-zero-frequency terms are also displayed.

We first probe into the role of the specific metal of layer B. In metallic systems, the Casimir force can be tailored by modulating the plasma frequency of the materials. The optimization of the plasma frequency is important in the design of epsilon-near-zero optical materials and for enhancing near-field radiation among other applications. In the context of Casimir interactions, surface plasmons and surface plasmon polaritons have also been instrumental in finding effective pathways of control. The considered system in Figure 1 can be constructed with different metallic substrates, and the calculated Casimir pressure is shown for several materials in Figure 3(a-d). The dielectric response for the metals is taken via the Drude model, as discussed previously, with parameters obtained from available experimental data.

Figure 3(a-d) shows that the total Casimir pressure exhibits similar behavior for all metals with a trapping region in the submicron range. To further understand the contributions from the different types of electromagnetic polarization, the Casimir pressure is decomposed into transverse-electric and transverse-magnetic modes, explicitly showing the thermal zero Matsubara frequency term and the summation of all Matsubara frequencies beyond it.

We note that the thermal transverse-electric and transverse-magnetic contributions hardly change for the different metals. This is verified by the asymptotic behavior of the Casimir pressure obtained in terms of effective permeabilities and dielectric responses, in Appendix Equations A.7 and A.8. This is also reflected in the Fresnel coefficients for the metal-Teflon interface for large and small distances, since the zero-frequency limit gives a vanishing transverse-electric coefficient and a transverse-magnetic coefficient of one, and in the large frequency limit both coefficients vanish. The material dependence upon the different metals is mostly seen in the ten to fifty nanometre window for the transverse-electric mode at non-zero frequencies, where the plasma frequency controls the attraction-repulsion transition. A larger plasma frequency increases the repulsive part from that contribution, which is the main factor in modulating the Casimir trapping region for this system.

Another factor that can affect the Casimir interaction is the thickness of the Teflon layer. In Figure 3(e-f), numerical results are shown for the pressure between the polystyrene layer and the Teflon-covered gold, with ferrite ferrofluid, as a function of the gap separation for different thicknesses of the Teflon layer. We observe that a stable trapping range is possible. As the Teflon thickness increases, the repulsive range shrinks, and at 20 nanometres the Casimir pressure exhibits no repulsion and becomes very close to the thermal limit from the zero-frequency transverse-electric contribution, with the pressure scaling as the inverse cube of the separation. This scaling law and the independence from the Teflon thickness are consistent with the analytical expression of the Casimir pressure for the zero-frequency transverse-electric term in Appendix Equation A.7.

To understand how the Teflon thickness affects the interaction, we focus on the characteristic behavior of the Fresnel reflection coefficients. From Equation 6 we find that the transverse-electric coefficient at the B-to-B1 interface vanishes at zero frequency and the coefficient at the B1-to-m interface is the static permeability minus one divided by the static permeability plus one, which means that the effective transverse-electric coefficient at zero frequency equals the B1-to-m coefficient. This is consistent with the finding that the thermal transverse-electric contribution is unaffected by the Teflon thickness. On the other hand, the transverse-magnetic coefficient at the B-to-B1 interface at zero frequency is the difference of the two static permittivities divided by their sum, so that the effective transverse-magnetic coefficient at zero frequency is that coefficient plus the exponential attenuation across the Teflon, divided by one plus their product. This shows that the attraction-repulsion transition of the thermal transverse-magnetic contribution can be shifted by changing the thickness. The particular thickness dependence of the thermal transverse-magnetic contribution is given in Appendix Equation A.9. The scaling law is the inverse cube of the separation both when the separation is much smaller than the Teflon thickness and when it is much larger, but there is a sign change on going from the first regime to the second, consistent with the numerical results shown in Figure 3.

From Equation 4 we also find that the transverse-electric and transverse-magnetic contributions with summed non-zero Matsubara frequencies are thickness-dependent in the ten to one hundred nanometre range. When the separation is much larger than the Teflon thickness, the effective Fresnel coefficient is approximately the coefficient at the B1-to-m interface, and the Casimir pressure is not affected by the thickness. When the separation is much smaller than the thickness, the effective coefficients are also independent of it, since both vanish in the high-frequency limit. Therefore, the Teflon thickness plays a role in the intermediate region where it is comparable to the separation, between about ten and one hundred nanometres, specifically around the attraction-repulsion transition point, as illustrated in Figure 3.

This analysis shows that the Teflon layer thickness is mostly effective when it is comparable to the separation, ten to one hundred nanometres in our case, with changes coming mostly from the transverse-electric non-zero frequencies contribution. Ultimately, the overall Casimir pressure is determined by the competition between the attractive transverse-electric thermal terms and the combined effect of repulsive thermal transverse-magnetic and quantum non-zero-frequency contributions. Increasing the Teflon thickness reduces the magnitude of the repulsive part of the pressure in the twenty to fifty nanometre range, leading to the disappearance of the Casimir trapping.

Figure 4. Casimir pressures normalized to the perfect conductor limit for the system in Figure 1. For sub-panels (a) to (d), the volume fraction of the 20 nanometre magnetite particles in the Toluene ferrofluid is (a) 1, (b) 5, (c) 10 and (d) 15 per cent. For sub-panels (e) to (h), the nanoparticles are dispersed in Toluene at a volume fraction of 5 per cent with diameters of (e) 5, (f) 10, (g) 15 and (h) 20 nanometres. For sub-panels (i) to (l), nanoparticles of 20 nanometre diameter and 5 per cent volume fraction are dispersed in (i) Toluene, (j) Benzene, (k) Cyclohexane and (l) Octane. In all cases B is gold and the Teflon thickness is 10 nanometres.

The Casimir interaction in the layered system from Figure 1 can also be modulated by changing the properties of the ferrofluid. Here we investigate the role of the nanoparticle volume fraction, their average diameter, and the type of solvent, with results summarized in Figure 4. Overall, we see that the trapping occurs for larger volume fractions, but this effect may disappear for larger diameters, due to the competition of different attractive and repulsive contributions for the Casimir pressure. The solvent substance also plays a role in the attraction-repulsion transitions for the total pressure.

As evident from Equations 1 and 2, the nanoparticle volume fraction affects both the effective dielectric and magnetic properties of the ferrofluid following a positive correlation. Figure 4(a-d) shows the zero-frequency transverse-electric contribution is always attractive following the characteristic inverse-cube scaling law; however, its magnitude depends on the volume fraction as the square of that fraction. The zero-frequency transverse-magnetic contribution exhibits attractive and repulsive regions, which can also be controlled by the volume fraction. Making the fraction larger increases the dielectric function of the ferrofluid as shown in Figure 2, thus the pressure magnitude of the thermal transverse-magnetic contribution, which is determined by the two transverse-magnetic Fresnel coefficients at zero frequency, is also enhanced.

Figure 4(a-d) also shows that the transverse-magnetic and transverse-electric terms with summed non-zero frequencies also experience significant transformations as the volume fraction increases. For a fraction of 1 per cent, the effective permittivity is close to the solvent permittivity. As discussed previously, when the Teflon thickness is comparable to the separation the two transverse-magnetic Fresnel coefficients have opposite signs in the small separation limit, so the finite frequency transverse-magnetic term is repulsive for smaller separations. This repulsion is further traced to the ordering in which the polystyrene permittivity exceeds the ferrofluid permittivity, which in turn exceeds the Teflon permittivity. As the separation increases, the role of the Teflon thickness diminishes and the Casimir pressure for the non-zero-frequency transverse-magnetic modes is mostly determined by the optical response of the polystyrene substrate and the ferrofluid. Thus that contribution becomes attractive at larger distances. A similar behavior is also observed for the transverse-electric contribution at 1 per cent, where the attraction-repulsion transition occurs at separations very similar to those for the transverse-magnetic modes.

For larger volume fractions, from 5 per cent and above, the non-zero-frequency transverse-magnetic term reverses this trend, sustaining a strong repulsive range beyond ten nanometres, also linked to the ordering in which the ferrofluid permittivity exceeds the polystyrene permittivity, which exceeds the Teflon permittivity. A similar behavior is observed for the transverse-electric contribution. Ultimately, as the volume fraction is increased, the role of the effective permeability grows, and the interplay between the attractive zero-frequency transverse-electric mode and the repulsive non-zero-frequency transverse-electric modes, together with all transverse-magnetic modes, results in a stable equilibrium region between about thirty and three hundred nanometres. Note, however, that since the effective dielectric response of the ferrofluid rises with the volume fraction, the attractive pressure from the zero-frequency transverse-electric term and the repulsive pressure from the other modes are enhanced simultaneously. Thus, further increasing the volume fraction does not result in significant changes in this stable equilibrium region.

In Figure 4(e-h), we also show the effect of the average nanoparticle diameter on the Casimir pressure. The role of the diameter comes mainly through the static effective permeability, primarily affecting the thermal transverse-electric contribution, whose magnitude increases significantly as the nanoparticle size increases. This is unlike the effect of the volume fraction, whose role is felt not only in the zero-frequency transverse-electric mode but also in all other contributions, as discussed previously. Figure 4(e-h) shows that other terms, except for the zero-frequency transverse-electric contribution, have a similar behavior showing attraction-repulsion transition at about twenty nanometres, and they are largely unaffected by the diameter. As a result, a stable equilibrium region whose range along the separation axis is affected by the diameter of the nanoparticles becomes possible, as shown for 15 and 20 nanometres.

The particular ferrofluid solvent is another component that can significantly affect the Casimir interaction. In our system, the optical properties of the solvent not only determine the dielectric response of the ferrofluid dispersion but also generate the effective reflection coefficient and net Casimir pressure between the two substrates. In Figure 4(i-l), we show the results of the Casimir pressure for different solvents, including Toluene, Benzene, Cyclohexane, and n-Octane. We find that Toluene and Benzene, with commensurate dielectric response functions, generate similar behavior for the pressure with a defined trapping zone in the ten to one hundred nanometre range. However, Cyclohexane and Octane, whose dielectric functions are smaller than those for Toluene and Benzene, primarily create attractive Casimir interaction at the sub-micron range. The opposite behaviour observed in the two scenarios is consistent with the prediction from the analytical expressions in the Appendix. Particularly, the ordering in which the Teflon permittivity is below the polystyrene permittivity, which is below the ferrofluid permittivity, holds for Toluene and Benzene with concentration larger than 2 per cent of dispersed particles, and causes the attractive-to-repulsive transition when increasing the separation gap. Conversely, the ordering in which the Teflon permittivity is below the ferrofluid permittivity, which is below the polystyrene permittivity, holds for Cyclohexane and n-Octane with concentration less than 10 per cent of nanoparticles, resulting in the repulsive-to-attractive transition versus separation gap. The decomposition of the polarization mode further indicates the sign flips in both transverse-magnetic modes and the non-zero-frequency transverse-electric modes between the two scenarios, while the zero-frequency transverse-electric terms are rarely changed.

Figures 3 and 4 show that a stable equilibrium originating entirely from the Casimir interaction can be achieved in several ways in the multilayered system we have designed. Although the interplay between the dielectric and magnetic properties of the involved layers is complex with several tunable factors, the achieved trapping is in the ten to two hundred nanometre range with a magnitude of the repulsion reaching about ten to the minus seven millipascal. This falls within the range of experimental observations. In particular, the Casimir and Non-Newtonian force EXperiment (CANNEX), based on optical interferometry, has shown that Casimir forces in the ten to the minus eight through ten to the minus four millipascal range for sub-micron separations are indeed achievable in the laboratory.

V. Conclusion

In summary, we have calculated the Casimir interaction in a novel multi-layered system consisting of a polystyrene substrate and Teflon-coated metallic layer separated by a gap filled with a ferrofluid. It is found that the sign of the Casimir force can be changed in many ways by tuning various parameters of the dielectric and magnetic properties of the involved materials. From the theoretical analysis within the Lifshitz formalism, we show the direct property-interaction relationship helping us understand the underlying factors of interaction control. In particular, changing the type of metal and thickness of the Teflon layer are most effective for changes in the transverse-electric polarization contributing to the quantum mechanical Casimir force in the submicron distance range. The type of solvent in the ferrofluid, however, influences not only the quantum mechanical transverse-electric contribution but also the thermal and quantum mechanical transverse-magnetic modes. On the other hand, the volume fraction is linked to changes in both thermal and quantum contributions, including transverse-electric and transverse-magnetic modes, while the diameter of the nanoparticle spheres in the ferrofluid changes the quantum trapping only by affecting the thermal transverse-electric modes.

Regardless of the chosen method of control, our calculations show that a stable regime of quantum trapping in the submicron regime with measurable magnitude is achievable and experimentally measurable. The diversity of different ways to modulate the system is also advantageous from an experimental point of view, as one can accommodate various laboratory capabilities that can yield measurable results of Casimir trapping. Our study gives useful guidelines for Casimir force control, which is useful for the development of novel micro- and nanomechanical devices.

Acknowledgments

Part of the research by L.I. and M.B. is part of project No. 2022/47/P/ST3/01236, co-funded by the National Science Centre and the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 945339. Institutional and infrastructural support for the ENSEMBLE3 Centre of Excellence was provided through the ENSEMBLE3 project (MAB/2020/14) delivered within the Foundation for Polish Science International Research Agenda Programme and co-financed by the European Regional Development Fund and the Horizon 2020 Teaming for Excellence initiative (Grant Agreement No. 857543), as well as the Ministry of Education and Science initiative "Support for Centres of Excellence in Poland under Horizon 2020" (MEiN/2023/DIR/3797). L.M.W. acknowledges financial support from the US Department of Energy under Grant No. DE-FG02-06ER46297.

Appendix: the thermal contributions to the Casimir pressure

The zero-frequency transverse-electric term is found directly from Equation 3. Equation A.7: it equals minus the Boltzmann constant times the temperature, divided by eight pi times the cube of the separation, times the trilogarithm of the square of the quantity formed by the static permeability minus one divided by the static permeability plus one. This term is always attractive and it scales as the inverse cube of the separation. Given Equation 2 in the main text, it is tuned by the concentration and diameter of the nanoparticles in the ferrofluid. When the permeability excess is much smaller than one, the term becomes proportional to the square of the volume fraction and to the sixth power of the nanoparticle diameter.

The zero-frequency transverse-magnetic term is also found from Equation 3. Equation A.8: it is approximately minus the Boltzmann constant times the temperature, divided by eight pi times the cube of the separation, times the trilogarithm of a ratio function evaluated at the exponential attenuation across twice the Teflon thickness over the separation. That ratio function is the sum of the polystyrene-to-ferrofluid static reflection term and the Teflon-to-ferrofluid static reflection term multiplied by the attenuation, divided by one plus the Teflon-to-ferrofluid term times the attenuation.

In the case where the separation is much larger than the Teflon thickness, Equation A.9 applies: the pressure is approximately minus the Boltzmann constant times the temperature, divided by eight pi times the cube of the separation, times the trilogarithm of the polystyrene-to-ferrofluid static reflection term, minus a correction built from twice the ferrofluid static permittivity divided by the Teflon static permittivity, multiplying the dilogarithm of that same reflection term scaled by the ratio of the Teflon thickness to the separation. Thus the Casimir interaction is independent of the thickness.

In the case where the separation is much smaller than the Teflon thickness, the Casimir interaction is independent of the metallic layer thickness, as found asymptotically in Equation A.10: the pressure is approximately minus the Boltzmann constant times the temperature, divided by eight pi times the cube of the separation, times the trilogarithm of the product of the polystyrene-to-ferrofluid and Teflon-to-ferrofluid static reflection terms.

From these limits, we can show that by tuning the ferrofluid’s static dielectric constant, one can manipulate the sign of the thermal transverse-magnetic Casimir pressure at small and large separation distances. In particular, since the Teflon static permittivity is 2.1 while the polystyrene static permittivity is 2.4, different situations may occur, such as:

• Teflon below polystyrene below ferrofluid: then for separations much smaller than the Teflon thickness the zero-frequency transverse-magnetic contribution to the force is attractive, and for separations much larger than the Teflon thickness the force is repulsive.
• Teflon below ferrofluid below polystyrene: then the zero-frequency transverse-magnetic contribution is repulsive for separations much smaller than the Teflon thickness, while it is attractive for separations much larger.
• Ferrofluid below Teflon below polystyrene: the zero-frequency transverse-magnetic contribution is always attractive for any separation distance.

(References omitted; the complete list of fifty items is at the source.)