Effect of Temperature Variations on Equilibrium Distances in Levitating Parallel Dielectric Plates Interacting through Casimir Forces

Effect of temperature variations on equilibrium distances in levitating parallel dielectric plates interacting through Casimir forces Victoria Esteso, Sol Carretero-Palacios, a) and Hern an M ıguez b) Multifunctional Optical Materials Group, Instituto de Ciencia de Materiales de Sevilla, (Consejo Superior de Investigaciones Cientıficas - Universidad de Sevilla), Calle Americo Vespucio 49, 41092 Sevilla, Spain (Received 11 February 2016; accepted 24 March 2016; published online 8 April 2016) We study at thermal equilibrium the effect of temperature deviations around room temperature on the equilibrium distance (d eq ) at which thin films made of Teflon, silica, or polystyrene immersed in glycerol levitate over a silicon substrate due to the balance of Casimir, gravity, and buoyancy forces. We find that the equilibrium nature (stable or unstable) of d eq is preserved under temperature changes, and provide simple rules to predict whether the new equilibrium position will occur closer to or further from the substrate at the new temperature. These rules depend on the static permittivities of all materials comprised in the system (eðmÞ 0) and the equilibrium nature of d eq . Our designed dielectric configuration is excellent for experimental observation of thermal effects on the Casimir force indirectly detected through the tunable equilibrium distances (with slab thickness and material properties) in levitation mode. V C2016 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4945428] I. INTRODUCTION AftertheCasimirforcewasfirstpredictedbyCasimiras an attractive force between two ideal metallic plates at zero temperature due to vacuum fluctuations, 1 one of the most striking results that followed was the prediction by Lifshitz of a repulsive force arising between bodies of arbitrary shape and dissimilar optical properties separated by a fluid medium at any temperature. 2,3 Since then, numerous studies considering different geometries and materials have been performed with the aim of controlling the repulsive nature and intensity of the Casimir force at 4–15 or out of 16 thermal equilibrium, with potential technological applications to minimize stiction, friction, or adhesion in nano and microscale devices. 17–21 Additionally, several works considering plane-parallel systems containing metallic 22–26 or ferromagnetic 27 materials, as well as magneto-dielectric metamaterials 28 have also demonstrated that it is possible to alter the nature and intensity of the Casimir force by modifying the temperature. In the same context, theoretical predictions have shown that gravity and repulsive Casimir forces acting in plane-parallel dielectric structures can be fine-tuned and balanced at room temperature giving rise to levitation phenomena. 29,30 Temperature may affect the equilibrium position at which the dielectric films levitate in two ways: on the one hand, the contribution of thermal modes to the Casimir force at (and around) room temperature becomes significant when dielectric materials and separation distances >100 nm are considered 31,32 (in contrast to metallic plates for which this contribution is assumed to be negligible). These thermal modes 2,33,34 are related to length scales kT¼hc kBT 7–8 lm at room temperature, with cas the speed of light, Tas the temperature, and k B and has the Boltzmann and the reduced Planck’s constant, respectively. On the other hand, the density of the fluid mediating the interaction may be slightly altered with temperature, and so, the balance of Casimir, gravity, and buoyancy forces may occur at a different separation distance bringing the system into a new equilibrium state at the new temperature. Uncontrolled deviations of the equilibrium position of suspended films due to temperature variations may be detrimental for the proper operation of nano and microdevices in which these films may be incorporated. 35–38 Therefore, a detailed analysis of the effect of temperature variations on the equilibrium position of suspended films in levitating plane-parallel configurations is desired. Here, we analyze theoretically the effect of temperature variations on the equilibrium distance (d eq ) at which thin films levitate over a substrate due to the balance of Casimir (FC !), gravity ( Fg !), and buoyancy ( FB !) forces. We consider plane-parallel structures that have already been shown to display levitation at thermal equilibrium 30 that comprise dielectric materials, for which temperature effects are expected to be much larger than for metals. 39 In contrast to other approaches in which a system in horizontal arrangement is brought to a certain equilibrium position by an external harmonic-oscillator force acting on the system in addition to the Casimir force, 39,40 in our setup, thermal effects can be analyzed through the influence of gravity and buoyancy forces in vertical arrangement by evaluating the equilibrium distances at which thin dielectric films levitate over a substrate at different temperatures. In particular, we analyze free-standing thin films made of Teflon, silica (SiO 2 ), or polystyrene (PS), immersed in glycerol suspended over a silicon substrate, yielding short (<100 nm) or large (>600 nm) d eq values depending on the selected material. Our choice of materials is not arbitrary: we consider dielectrics whose optical properties have been extensively studied in literature and do not change with temperature, that are chemically stable when combined together, that can be easily implemented for experimental realization, and with densities such that gravity and buoyancy forces can balance the a) Electronic mail: [email protected] b) Electronic mail: [email protected] 0021-8979/2016/119(14)/144301/7/$30.00 V C2016 AIP Publishing LLC119, 144301-1 JOURNAL OF APPLIED PHYSICS 119, 144301 (2016) 15 May 2025 16:21:23 Casimir force at a given separation distance. We find that the stability nature of d eq is preserved under temperature changes, and that the effect of temperature variations on the Casimir force, and thus on d eq , is dominated by transversemagnetic modes at zero frequency. We provide simple rules to predict whether the equilibrium position takes place closer to or further from the substrate at the new temperature that depends on the static permittivity contrast of the materials involved, as well as on the stability nature of d eq . A schematic of the plane-parallel structure under study is shown in Fig. 1. It consists of a free-standing thin film (of thickness d 1 )immersedinafluidfacinga semi-infinite substrate. The separation distance between the thin film and the substrate at which the total force ( FT !¼FC !þFg !þFB !)cancels, defines d eq . The two possible scenarios of the forces acting on the system, which determine the stability nature of d eq ,arepresented in panels (a) and (b). In panel (a), a repulsive Casimir force (FC>0) is acting on the planar arrangement and it dominates at short separation distances (d 0 ), providing stable d eq when the total force is canceled. This means that any slight deviation from the equilibrium position will lead to forces pointing to that stable location. In panel (b), an attractive Casimir force (FC<0) dominating at short separation distances is contemplated, giving rise to unstable equilibrium positions that will cause that the film will float or sink if it deviates from d eq . II. THEORETICAL FORMALISM In order to determine the equilibrium distance at which free-standing thin films immersed in a fluid levitate over a substrate, it is necessary to compute the total force acting on the system and find the separation distance at which FT !ðd0¼deqÞ¼0. In our model, the plane-parallel system is represented as a multilayer structure in which each layer is made of a material m. 32 The material mediating the interaction, in this case a fluid, is denoted by m¼0. Positive and negative mvalues stand for materials above and below the fluid, correspondingly. Thus, m¼þ1, m¼þ2, and m¼1 account for the free-standing film, the fluid over it, and the substrate, respectively. The thickness of each layer is indicated by d m . For gravity and buoyancy forces, we take Fg !þFB !¼ðqf ilm þqf luidÞgd1, with g¼9.81 m/s 2 , and qas the density of either the film or the fluid. Please note that all the forces are expressed per unit area. The Casimir force in plane-parallel systems explicitly depends on d 0 and T, and it can be expressed as a sum of two components FCðd0;TÞ¼ X j¼½TE;TM Fn¼0 C;jðd0;TÞþ X j¼½TE;TM Fn>0 C;jðd0;TÞ;(1) where ndescribes the discrete and infinite Matsubara frequencies, nn¼2pkBT hnwith ncomprised between 0 and infinite. The two components defining the Casimir force account for zero frequency (n¼0) and positive frequency (n>0) modes of the transverse magnetic (j¼TM) and transverse electric (j¼TE) fields, whose corresponding expressions are 32 Fn¼0 C;jd0;T ðÞ ¼kBT 2pð1 0 k2 ?dk? e2k?d0 Rþ ðÞ jR ðÞ j 1 "# 1 ;(2) Fn>0 C;jd0;T ðÞ ¼kBT pX 1 n¼1ð1 0 k0 ðÞ nk?dk? e2k0 ðÞ nd0 RþðÞ jR ðÞ j 1 2 43 5 1 : (3) Note the 1 2factor in Eq. (2). In the above equations, kð0Þ nis the wavevector perpendicular to the plane of the surfaces of the fluid medium (m¼0), and k?is, in turn, the projection of the wavevector over the plane perpendicular to kð0Þ n. The multiple Fresnel coefficients Rð6Þ jðn;k?Þthat express the reflectance on either the top ðþÞ or the bottom ðÞ surface of the fluid are expressed as R6 ðÞ jn;k? ðÞ ¼r0;61 ðÞ jþr61;62 ðÞ je2k61 ðÞ nd61 1þr0;61 ðÞ jr61;62 ðÞ je2k61 ðÞ nd61 (4) with rðm;m0Þ jas the simple Fresnel coefficients at the interface between materials mand m0, and kðmÞ nas the wavevector in the corresponding layer. For TM and TE polarizations, the corresponding rðm;m0Þ jdefinitions read rm;m0 ðÞ TM n;k? ðÞ ¼em0 ðÞ nkm ðÞ nem ðÞ nkm0 ðÞ n em0 ðÞ nkm ðÞ nþem ðÞ nkm0 ðÞ n ;(5) rm;m0 ðÞ TE n;k? ðÞ ¼lm0 ðÞ nkm ðÞ nlm ðÞ nkm0 ðÞ n lm0 ðÞ nkm ðÞ nþlm ðÞ nkm0 ðÞ n ;(6) FIG. 1. Schematics of a plane-parallel system consisting of a thin film (of thickness d 1 ) fully immersed in a fluid standing over a semi-infinite substrate. The system is considered to have infinite area; i.e., the width and the length are much larger than the layer thickness. The balance of Casimir ( FC !), gravity ( Fg !), and buoyancy ( FB !) forces acting on the system gives rise to levitation at a certain, stable (a) or unstable (b), equilibrium distance (d eq ). 144301-2 Esteso, Carretero-Palacios, and M ıguez J. Appl. Phys. 119, 144301 (2016) 15 May 2025 16:21:23 where eðmÞ nand lðmÞ nare the material permittivity and permeability, respectively, evaluated at Matsubara frequencies. The wavevector kðmÞ nis defined as km ðÞ n¼k2 ?þem ðÞ n n2 n c2  1=2 (7) and the permittivity eðmÞ ntakes the form em ðÞ neinn ðÞ ¼1þ2 pð1 0 xe00 x ðÞ x2þn2 n dx(8) being e00ðxÞthe imaginary part of the dielectric function of material mat xfrequencies, e¼e0ðxÞþie00ðxÞ. Specifically, in our studies, non-magnetic materials are considered, so lðmÞ n¼1. 2,41 In addition, as the dielectric slabs are assumed to be fully immersed in a fluid, m¼þ2 accounts for the same fluid as m¼0, so we can take rjðþ1;þ2Þ ¼rjð0;þ1Þ. Finally, both a semi-infinite fluid above the suspended thin film and silicon substrate are considered, i.e., dþ2;d1!1, rendering the simplification of the multiple Fresnel coefficient of the bottom surface to RðÞ j¼rð0;1Þ j.In our calculations, the density of the materials at room temperature takes the values: q Teflon ¼2200 kg/m 3 ,qSiO2¼2650 kg/m 3 , q PS ¼1050 kg/m 3 ,andq glycerol ¼1256 kg/m 3 . Although the density of glycerol, which is the medium mediating the interaction, varies with Tand thus affects FB !when the temperature is modified, these variations might be assumed to be negligible as the density of glycerol within the temperature range considered here hardly changes 0.7%. Therefore, despite the fact that, in our calculations, proper q glycerol values will be taken at each temperature, variations of d eq with temperature will be mainly dictated by FC !. In addition, in our calculations, we have not taken into account the possible effect of Casimir–Polder interactions at intermolecular separations between glycerol molecules and solid interfaces, as their impact on the determined levitation distances is estimated to be minor. 42,43 III. RESULTS AND DISCUSSION Figure 2shows the total force (per unit area) as a function of d 0 acting on a system comprising Teflon (green line), SiO 2 (orange line), or PS (blue line) slabs of d 1 ¼100 nm, a thickness chosen according to feasible experimental realization, immersed in glycerol facing a silicon substrate. At room temperature (T¼300 K) and at thermal equilibrium, the predicted d eq values are 190 nm, 60 nm, and 667 nm for the three materials, respectively. Stable d eq positions are found for Teflon and SiO 2 , whereas for PS films, d eq is unstable. The different stability nature of the equilibrium positions attained is defined by the optical properties of the materials. 30 Taking these systems as reference, in what follows we will analyze the stability and variations of the equilibrium positions under temperature changes around room temperature. Top panels in Fig. 3show FT !acting on the same three planar systems analyzed in Fig. 2, as a function of d 0 and at several temperatures at thermal equilibrium. In all cases, the trend of FT !with d 0 is preserved when temperature changes, attaining stable positions for Teflon and SiO 2 films, and unstable ones for PS slabs at thermal equilibrium. This means that for the plane-parallel arrangements considered here, the stability of the equilibrium position is not affected by the temperature variations. Calculations of the Casimir force as a function of d 0 for several temperatures are shown in Fig. S1 in the supplementary material. 44 In addition, we can see that in all cases, the total force is altered with temperature variations. Specifically, as temperature rises to 5 C, equidistant curves are attained, shifting horizontally a constant increment. Note that the scale of FT !in Fig. 3(c) is 100 times smaller than in panels (a) and (b); therefore, the variation of FT !with Tfor the PS system is much smaller than for structures incorporating Teflon or SiO 2 materials. Bottom panels in Fig. 3display the collected equilibrium distances obtained in the corresponding top panels at different temperatures. We find that in configurations containing Teflon and SiO 2 films, d eq decreases when Tincreases, with the smallest variations attained for SiO 2 films. PS films, in contrast, get away from the substrate as temperature rises, covering the largest d eq range despite this system presents the smallest variations of FT !with T(Fig. 3(c)). Therefore, changes of d eq with Tare related not only to the temperature dependence of FT !butalsotothevariationof FT !with the separation distance near d eq , and the stability nature of d eq , which is provided by the trend of FT !as a function of d 0 that specifically depends on the optical properties of the materials considered. 30 To explain the approach or distance of the dielectric slabs to the substrate due to thermal effects, we first assume that the contribution of FB !to the variations of the total force with temperature can be neglected, since the density of glycerol hardly changes 0.7% within the temperature range considered here. Therefore, we can accept that the main effect of temperature variations to the total force comes from the Casimir force, i.e., @FT @T@FC @T. Second, we analyze the different Fn C;jcontributions in Eqs. (2) and (3) defining the Casimir force under temperature changes. This kind of analysis has FIG. 2. Total force (per unit area) as a function of the separation distance (d 0 ) between a thin film of 100 nm thickness (d 1 ) made of Teflon, silica (SiO 2 ), or polystyrene (PS) (green, orange, and blue lines, correspondingly) immersed in glycerol facing a silicon substrate at 300 K of temperature (T). Values of d eq are, respectively, 190 nm, 60 nm, and 667 nm. Solid and dashed lines stand for systems of stable or unstable equilibrium positions, respectively. 144301-3 Esteso, Carretero-Palacios, and M ıguez J. Appl. Phys. 119, 144301 (2016) 15 May 2025 16:21:23 revealed that the Fn¼0 C;TE modes control the nature of the Casimir force in plane-parallel systems containing magnetic plates, 4,6,8 or that the main contribution to FC !between graphene sheets comes from the Fn¼0 C;TM modes at room temperature. 45 To identify the effect of temperature changes in the Fn C;j contributions, the separation distance is fixed whereas temperature is varied around room temperature. As a representative structure showing large variations of FT !with T,weanalyzea SiO 2 film of d 1 ¼100 nm thickness suspended at d 0 ¼60 nm at room temperature immersed in glycerol. Notice that in this specific case, d0¼deq, but it is not a necessary request for the analysis of the temperature dependence of the Casimir force performed here. Panel (a) in Figure 4shows the Fn C;jcontributions as a function of temperature. Full symbols denote zerofrequency modes (n¼0), and empty symbols denote positive frequencies (n>0). Squares account for TE contributions and triangles for TM ones. As expected Fn¼0 C;TE ¼0, as it has been assumed that lðmÞ¼1 for all dielectric materials considered here, and so, Eq. (6) becomes zero. In addition, when all contributions are compared at a given temperature, we observe that the largest contributions come from Fn¼0 C;TM modes, as it was the case of graphene sheets at room temperature. 45 Most importantly, in panel (a), the largest variations of Fn C;jwith temperature are found for Fn¼0 C;TM contributions, while the other components remain almost unaltered, i.e., @Fn>0 C;TM @T;@Fn>0 C;TE @T; @Fn¼0 C;TE @T0. The same Fn¼0 C;TM dominance at finite temperature and large variations of Fn¼0 C;TM with temperature are found for systems with Teflon or PS thin films embedded in glycerol, as it is shown in Fig. S2 in the supplementary material. 44 We also analyzed the same dielectric thin films immersed in ethanol or water instead of glycerol. Except for Teflon films in ethanol, the other sets of materials do not give rise to levitation. Panel (b) in Fig. 4shows results attained for a silica film immersed in ethanol. We find that Fn¼0 C;TM modes are no longer the dominant ones, and instead, Fn>0 C;TM provides the largest contribution to the Casimir force. In this case, an attractive Casimir force together with gravity is always larger than the buoyancy force at all FIG. 3. (a)–(c) Total force (per unit area) as a function of the separation distance (d 0 ) between a thin film of 100 nm thickness (d 1 )madeofTeflon,SiO 2 ,or PS, respectively, immersed in glycerol facing a silicon substrate at different temperatures around room temperature (T¼293;298, 303, and 308 K). At each temperature, q glycerol takes the values q glycerol ¼1261, 1258, 1255, and 1252 kg/m 3 . (d)–(f) Equilibrium distances obtained in the corresponding top panel as a function of temperature. Solid and dashed lines stand for systems of stable or unstable equilibrium positions, respectively. FIG. 4. Fn C;jcontributions to the Casimir force as a function of temperature for a system consisting of a SiO 2 thin film of 100 nm, immersed in a fluid over a silicon substrate at d 0 ¼60 nm for T¼300 K. Panel (a) corresponds to glycerol and panel (b) corresponds to ethanol as the mediating fluid. Transverse electric (TE) and transverse magnetic (TM) modes are displayed with square and triangle symbols, respectively. Zero frequency and non-zero frequency modes are shown with full and empty symbols, accordingly. 144301-4 Esteso, Carretero-Palacios, and M ıguez J. Appl. Phys. 119, 144301 (2016) 15 May 2025 16:21:23 separation distances, preventing levitation. In these systems immersed in ethanol or water deprived of levitation, the Fn¼0 C;TM contribution also changes significantly with temperature. However, in some cases, the Fn>0 C;TM component changes in such a way that it counteracts the Fn¼0 C;TM variations, that is, @Fn¼0 C;TM @T@Fn>0 C;TM @T, impeding a simple prediction of the temperature effect in d eq in those systems (results shown in Figs. S3, S4, and S5 in the supplementary material 44 ). Therefore, in what follows, we will focus our analysis on the effect of temperature variations just in systems for which we can approximate @FC @T@Fn¼0 C;TM @T:(9) For TM polarization, Eq. (2) reads Fn¼0 C;TM ¼TkB 2pð1 0 k2 ?dk? e2jk?jd0 Rþ ðÞ TM R ðÞ TM 1 "# 1 ¼Tcd0;d1;em ðÞ 0  (10) in which the multiple (Rð6Þ TM ) and simple (rð6Þ TM ) Fresnel coefficients for n¼0 take the form RþðÞ TM ¼r0;þ1 ðÞ TM r0;þ1 ðÞ TM e2k?d1 1r20;þ1 ðÞ TM e2k?d1 ;(11) R ðÞ TM ¼e1 ðÞ 0e0 ðÞ 0 e1 ðÞ 0þe0 ðÞ 0 ;(12) r0;þ1 ðÞ TM ¼eþ1 ðÞ 0e0 ðÞ 0 eþ1 ðÞ 0þe0 ðÞ 0 ;(13) rð0;1Þ TM ¼RðÞ TM:(14) Equation (10) reveals that Fn¼0 C;TM is linearly dependent on temperature and proportional to a function that defines the sensitivity of the Casimir force to temperature changes, cðd0;d1;eðmÞ 0Þ. This function accounts for the reach of the Casimir interaction through the separation distance (d 0 ) and the thickness of the thin film (d 1 ). It also depends on the optical properties of the materials, specifically, on the static permittivity of all objects in the system (eðmÞ 0). Figure S6 in the supplemental material shows results of cðd0;d1;eðmÞ 0Þas a function of d 1 at different separation distances for Teflon, SiO 2 , and PS. We find that the largest cðd0;d1;eðmÞ 0Þvalues are attained for very thin films of high dielectric contrast at short separation distances. Next, in order to determine whether the new equilibrium position will take place closer to or further from the substrate due to temperature variations, we first assume that the total force depends linearly with d 0 around d eq , so for a given temperature T 1 , we can write FTd0;T1 ðÞ @FT @d0deq d0deq ðÞ (15) with @FT @d0as the slope of the line. In addition, since parallel and equidistant curves are attained as temperature varies (top panels in Fig. 3) for a different temperature T 2 (with T2>T1 or T2<T1), we take FTd0;T2 ðÞ @FT @d0deq d0deq ðÞ þ@FT @Tdeq DT ! :(16) We define dð1Þ eq and dð2Þ eq as the distances at which FT !ðdð1Þ eq ;T1Þ¼0 and FT !ðdð2Þ eq ;T2Þ¼0, correspondingly. Recalling that temperature variations of the total force can be approximated to be produced just by Casimir force variations, i.e., @FT @T@FC @T, and that parallel curves have been attained, i.e., @FT @d0jdð1Þ eq ¼@FT @d0jdð2Þ eq , the variation of the equilibrium distance Ddeq with temperature can be written as Ddeq ¼d2 ðÞ eq d1 ðÞ eq  @FC @T  deq DT @FT @d0  deq :(17) Combining together Eqs. (9),(10), and (17), we have Ddeq  @FC @T  deq DT @FT @d0  deq  @Fn¼0 C;TM @T  deq DT @FT @d0  deq ¼cdeq;d1;em ðÞ 0  DT @FT @d0  deq :(18) In the above expression, the sign of cðdeq;d1;eðmÞ 0Þand @FT @d0(that defines the stability nature of d eq at a given temperature) will determine whether the new equilibrium position will take place closer to (Ddeq <0) or further (Ddeq >0) from the substrate with Tvariations. The sign of cðdeq;d1;eðmÞ 0Þis in turn defined by the denominator RðþÞ TM  RðÞ TM in Eq. (10), since jRð6Þ TM j<1, and the exponential is always >1, which relies upon the contrast between eðmÞ 0of all materials involved in the levitating system (Eqs. (13) and (14)). For our systems, the fluid–slab interface is the only one modified while the fluid–substrate one is kept the same. Corresponding eðmÞ 0values are eTef lon 0¼2.1, eSiO2 0¼3.7, ePS 0¼2.45, eglycerol 0¼42.4, and esilicon 0¼11.74. In general, in a plane-parallel structure displaying levitation dominated by zero-frequency TM modes, if the static permittivities of all materials fulfill any of the following inequations: eð1Þ 0> eð0Þ 0>eð1Þ 0or eð1Þ 0<eð0Þ 0<eð1Þ 0, the sign of the denominator is negative and therefore the sign of the cðdeq;d1;eðmÞ 0Þfunc- tion. If d eq is stable (@FT @d0jdeq <0) at a given temperature, Ddeq >0 as temperature rises, and so the system reaches the equilibrium position further away from the substrate. In the opposite case, if d eq is unstable (@FT @d0jdeq >0), the system will find the equilibrium state closer to the substrate. On the other hand, if the static permittivities of all the materials in a 144301-5 Esteso, Carretero-Palacios, and M ıguez J. Appl. Phys. 119, 144301 (2016) 15 May 2025 16:21:23 plane-parallel system do not fulfill any of the above inequations, the sign of cðdeq;d1;eðmÞ 0Þfunction is positive and so d eq will become shorter with Tfor stable equilibrium positions, or larger for unstable ones. These two scenarios correspond to Teflon and SiO 2 films in Figs. 3(d) and 3(e), for which closer equilibrium positions are found as Trises, and PS thin films in Fig. 3(f), where d eq increases with temperature, accordingly. In Table I, we gather and summarize all the possible eðmÞ 0combinations, the stability of d eq , and the prediction of Ddeq with T. Note that for predicting the behavior of d eq with Tchanges, all that needs to be known is the stability of d eq at a given temperature, and eðmÞ 0of all materials in the system. This analysis is only possible because in the systems under study, the effect of temperature variations on the Casimir force is dominated by zerofrequency TM modes. Therefore, we could say that the effect of temperature variations on the equilibrium distance in systems displaying levitation relies on intrinsic properties of the materials comprising the system. This is of utmost importance, since some of the proposed routes to fine tune the Casimir force entail the design of the material optical properties of the system, 14 either by modifying the dielectric function at certain frequency ranges while maintaining the static permittivity by means of photonic crystals, 46 or by modifying the dielectric function including the static permittivity upon crystallization, 47 for instance. According to our results, the use of photonic crystals operating in the Ultraviolet–Visible-Infrared range would not have an influence on the equilibrium position under temperature variations, while the crystallization approach would bring the suspended film to a different equilibrium state, as long as zero-frequency TM modes dominate the contribution to the Casimir force. The quantification of the deviation from the equilibrium position of suspended films due to temperature variations depends on the interplay between cðdeq;d1;eðmÞ 0Þand @FT @d0, and for this, the full calculation of @FT @d0is needed. In any case, an examination of Eq. (18) indicates that for systems with @FT @d0!1 slope, unmodified equilibrium distances are predicted (Ddeq 0), whereas in the opposite limiting case, for @FT @d0!0, the equilibrium position of the levitating film will be strongly modified. These two limiting situations would correspond to SiO 2 films for which large values of @FT @d0are observed in Fig. 3(b), and almost the same d eq is obtained irrespective of the temperature considered (Fig. 3(e)), whereas for PS films, almost zero slope lines are found (Figs. 3(c) and 3(f)), and so, the largest d eq range is obtained. IV. CONCLUSION We have performed a theoretical analysis of the effect of temperature variations of suspended free-standing thin films in dielectric plane-parallel arrangements under the influence of Casimir, gravity, and buoyancy forces at thermal equilibrium. In particular, thin films made up of Teflon, SiO 2 , or PS immersed in glycerol facing a silicon substrate are considered. For all the temperatures evaluated, we find that the trend of the total force acting on the system is kept as the separation distance varies, and so the stability nature of the equilibrium distances under temperature changes is preserved. In addition, the effect of temperature variations on the total force acting on the systems is dominated by the transverse-magnetic modes at zero-frequency defining the Casimir force which depends upon intrinsic dielectric properties of the system. We provide simple rules to predict whether the new equilibrium position takes place closer to or further from the substrate in levitating arrangements at a different temperature, what can be inferred from the stability nature of the equilibrium state and the static permittivities of all materials comprising the system. Our results serve as a guide for designing planar arrangements in levitation mode to be incorporated in nano and micro-devices in which the equilibrium distance can be fine tuned with temperature simply attending to the materials’ static permittivities contrast, or for observing the dynamic Casimir effect as the dielectric slab reaches the new equilibrium position at a different temperature. ACKNOWLEDGMENTS We acknowledge support from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 307081 (POLIGHT) and the Spanish Ministry of Economy and Competitiveness under Grant No. MAT2014-54852-R. V.E. is grateful to “Obra Social La Caixa” for its support. 1 H. B. G. Casimir, “On the attraction between two perfectly conducting plates,” Proc. K. Ned. Akad. Wet. 51, 793 (1948). 2 E. M. Lifshitz, “The theory of molecular attractive forces between solids,” Sov. Phys. JETP 2, 73 (1956). 3 I. E. Dzayloshinskii, E. M. Lifshitz, and L. P. Pitaevskii, “Van der Waals forces in liquid films,” Sov. Phys. JETP 37, 161 (1960). 4 T. H. Boyer, “Van der Waals forces and zero-point energy for dielectric and permeable materials,” Phys. Rev. A 9, 2078–2084 (1974). 5 O. Kenneth, I. Klich, A. Mann, and M. Revzen, “Repulsive Casimir forces,” Phys. Rev. Lett. 89, 033001 (2002). 6 N. 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