Fluid sensitive nanoscale switching with quantum levitation controlled by α -Sn/ β -Sn phase transition
Mathias Boström, Maofeng Dou, Oleksandr I. Malyi, Prachi Parashar, Drew F. Parsons, Iver Brevik, Clas Persson
aa r X i v : . [ c ond - m a t . o t h e r] M a r Fluid sensitive nanoscale switching with quantum levitation controlled by α -Sn/ β -Snphase transition Mathias Boström,
1, 2, ∗ Maofeng Dou, † Oleksandr I. Malyi, ‡ PrachiParashar, § Drew F. Parsons, ¶ Iver Brevik, ∗∗ and Clas Persson
2, 5, †† Department of Energy and Process Engineering,Norwegian University of Science and Technology, NO-7491 Trondheim, Norway Centre for Materials Science and Nanotechnology,University of Oslo, P. O. Box 1048 Blindern, NO-0316 Oslo, Norway Center for Green Research on Energy and Environmental Materials,National Institute for Materials Science, Tsukuba, Ibaraki 305-0044, Japan School of Engineering and Information Technology,Murdoch University, 90 South St, Murdoch, WA 6150, Australia Department of Physics, University of Oslo, P. O. Box 1048 Blindern, NO-0316 Oslo, Norway (Dated: September 18, 2018)We analyse the Lifshitz pressure between silica and tin separated by a liquid mixture of bromoben-zene and chlorobenzene. We show that the phase transition from semimetallic α -Sn to metallic β -Sncan switch Lifshitz forces from repulsive to attractive. This effect is caused by the difference in di-electric functions of α -Sn and β -Sn, giving both attractive and repulsive contributions to the totalLifshitz pressure at different frequency regions controlled by the composition of the intervening liq-uid mixture. In this way, one may be able to produce phase transition-controlled quantum levitationin liquid medium. I. INTRODUCTION
Nanoelectromechanics have by now developed intoquite a mature subject, where one deals routinely withseparations between bodies of the order of a few nanome-ters. For these structures, the Lifshitz forces due to quan-tum fluctuations become accordingly important. Thisforce typically causes attraction between surfaces andthus contributes to stiction, leading to collapse of de-vices when surfaces approach each other.
It has beenshown, however, that the Lifshitz force may be repulsive, or even in an intricate way change sign as separationincreases. The development of direct measurementsof Lifshitz forces has provided a major thrust in search-ing viable systems for device engineering.
Controlled nanomechanical devices could be designedby tailoring the magnitude of the intermolecular interac-tions between surfaces. Several studies have investigatedhow this may be achieved through optical excitations andtemperature dependent phase change materials.
Forexample, crystallization of amorphous Ag-In-Sb-Te filmhas been predicted to increase the Lifshitz force up to20 % between gold and the alloy surface. However, aphase-transition controlled sign reversal of the Lifshitzforces is a novel idea that has not to our knowledge beenproposed yet.In this paper, we introduce systems where a phasetransition, induced by temperature or other environmen-tal factors, can switch the sign of the Lifshitz force be-tween surfaces of a phase transition material and a thinsolid layer across a very thin ( <
40 Å) liquid film. Themodel system we have in mind is shown in Fig. 1. We pro-pose to use a common phase transition material, solidtin, which has a phase transition temperature at T = 286.5 K. One of its two phases (grey tin; α -Sn) isa semimetal, while its other phase (white tin; β -Sn) isa metal; they have therefore very different dielectric re-sponses. In order to obtain a phase-dependent transitionfrom attractive to repulsive Lifshitz forces, the dielec-tric functions of the thin solid layer and the interven-ing liquid must be close and must cross over. One canachieve this requirement by constructing a system withsilica (SiO ) as thin solid layer and a mixture of two (ormore) liquids whose dielectric function matches thatof silica. A key element in our proposed model is theinfluence of the intervening liquid medium between theplates; the effect does not exist if the medium becomesreplaced by a vacuum (or air). The transition distancefrom an attractive to a repulsive Lifshitz force occurswhen the attractive and repulsive contributions to thetotal Lifshitz force from different frequency regions ex-actly cancel. Thus, the engineering requirements are ap-parent: the refractive indices of the two pure liquids canlie above and below the refractive index of one of thesolid materials. It is then a matter of finding the rightcombination of the liquid mixture that yields the desiredcrossover of the dielectric functions of the liquid and thesolid. In this work, we choose a particular phase changematerial (i. e., tin) just for demonstrating the conceptof switching, and we anticipate that this concept can bedeveloped further utilizing also other types of metal/non-metal transition, for example by charge injection, chem-ical insertion, and magnetic phase transition. These sys-tems open up the possibility to make use of the Lifshitzeffect as a switch, or actuator, that can be utilized indevelopments of microelectromechanical (MEMS) or na-noelectromechanical (NEMS) systems, as well as con-trolled low friction nanomechanical devices (Lifshitz re-pulsion leads to low friction between surfaces. ) -505 E ne r g y ( e V ) β -Sn (I41/amd) Meta l AttractiveSemimetal Repulsive
Structure change290 K280 K
SiO2 β -Sn L i qu i d m i x t u r e α -Sn (Fm3m) Crystal structures Electronic structures Lifshitz interactions
SiO2 α -Sn L i qu i d m i x t u r e Property change290 K280 K Force change290 K280 K E F E F ε ε ε ε ε ε Figure 1. (Color online) Scheme of the three-layer systemwith switch from repulsive to attractive forces induced by thephase transition from α -Sn to β -Sn. Left panel: the crystalstructures of the two phases. Middle panel: tin density-of-states (DOS) where α -Sn is a semimetal with valence bandsfully occupied, whereas β -Sn is a metal with bands partiallyoccupied across the Fermi level ( E F ). Right panel: the Lif-shitz interactions between SiO and tin surfaces across liquidmixtures at temperatures T = 280 and 290 K. ε signifies thedielectric response in each layer. II. THEORY
Fundamental effects from the Lifshitz force is modelledfor the three-layer system as described in Fig. 1. Theretarded Lifshitz pressure p ( L ) , between silica and tinsurfaces separated across a liquid medium by distance L ,is given as a sum over imaginary Matsubara frequencies( ζ n = n πk B T / ~ ), p ( L ) = ∞ X n =0 ′ ( g TE + g TM ) , (1)where prime on the summation sign indicates that the n = 0 term shall be divided by two. The spectralfunctions for transverse electric and transverse magneticmodes g m ( m = TE and TM) are g m = − k B T π Z d k γ r m r m e − γ L − r m r m e − γ L . (2) r mij are the reflection coefficients, r TM ij = ε j γ i − ε i γ j ε j γ i + ε i γ j and r TE ij = γ i − γ j γ i + γ j , (3)where γ i ( iζ n ) = q k + ( ζ n /c ) ε i .To model the Lifshitz force accurately, a detailedknowledge of dielectric functions of all materials involvedis essential. ζ (rad/s)10 ε ( i ζ ) β -Sn α -SnbromobenzenechlorobenzeneSiO (dataset 1)SiO (dataset 2) Figure 2. (Color online) The dielectric functions of bromoben-zene (Bb), chlorobenzene (Cb), and silica (two slightly differ-ent datasets) are from van Zwol and Palasantzas (Ref. 11).Corresponding spectra for α -Sn and β -Sn are obtained fromDFT calculations. The average static dielectric constants are5.37, 5.75, 4.0, 27.2, . × , for Bb, Cb, SiO , α -Sn, and β -Sn, respectively. III. MODELING THE DIELECTRICRESPONSES OF MATERIALS
The primary materials considered in this work are tin( α - and β -Sn) as the phase transition material, silica asthe thin solid layer, and a liquid mixture based on bro-mobenzene (Bb) and chlorobenzene (Cb). In the nextsection, it will be demonstrated that by mixing liquid Bbwith the less polarizable Cb, one obtains the necessarycondition for a switch in the Lifshitz pressure from re-pulsion to attraction when α -Sn undergoes a phase tran-sition to β -Sn. A. Experimental dielectric functions of silica andliquid mixtures
For the dielectric function of SiO we consider twodatasets (i. e., set 1 and set 2) given by van Zwol andPalasantzas. The dielectric functions of Bb and Cb liq-uids are also taken from Ref. 11. The dielectric functionsfor the pure components are shown in Fig. 2. The dielec-tric function of SiO lies between those of Bb and Cb;see especially inset of Fig. 2.The mixing of two miscible liquids (Cb and Bb here)adjusts the dielectric function of the intervening medium,whereby attractive and repulsive contributions arisingfrom crossings of the dielectric functions of silica and liq-uid will occur in different ways for α -Sn and β -Sn.(Cf. theremark of Lamoreaux about the possibility to ’tune’ themixing such that the force becomes attractive at largeseparations and repulsive at short range.) It is knownthat in a mixture of Bb and Cb the dielectric constantvaries approximately linearly with the relative amountof each of the two components. For the dielectric func-tions of liquid mixtures, we use the Lorentz-Lorenz-likemodel with the susceptibility χ = X i = Bb , Cb V i ε ,i − ε ,i + 2 , (4)where V i is the volume fraction occupied by liquid i component that has a dielectric function ε ,i . The di-electric function of the liquid mixture is then given by ε = (1 + 2 χ ) / (1 − χ ) .Since the calculated transition distances depend onhow the dielectric functions are modeled, we have com-pared the model in Eq. 4 with the volume average modelthat assumes a linear dependence of ε ,i on V i . Thetwo models describe rather similar dielectric spectra, andthey both can give attraction to repulsion transitions. In-accuracies of describing the exact dielectric responses canthus in an experimental setup be compensated by adjust-ing the liquid mixture to obtain the switching. B. Calculated dielectric functions of tin
For the two tin phases we modeled the dielectric func-tions within the density functional theory (DFT), em-ploying the augmented plane wave method with local or-bitals for Sn d -like orbitals (i. e., the APW+lo method) asprovided by the WIEN2k package. The imaginary partof the dielectric tensor was calculated from the linearresponse of the momentum matrix elements describingthe transition probability between occupied and unoccu-pied states. Experimental lattice constants and two-atom primitive cells were used. The regular exchange-correlation potentials with the local density approxi-mation (LDA) or the generalized gradient approxima-tion (GGA) do not accurately describe tin, especiallysemimetal α -Sn, due to overestimated hybridization be-tween valence and conduction band states. Instead, weutilize the modified Becke-Johnson meta-GGA exchangepotential combined with the LDA correlation potential.With a small k-mesh, we have verified a good density-of-states by comparing with a corresponding hybrid func-tional calculation. A dense k-mesh is however needed todescribe details in the dielectric response accurately. We, therefore, calculate it using the regular tetrahe-dron integration of the irreducible wedge of the Brillouinzone with × × k-mesh grids and an energy gridwith step length of about 0.3 meV. The plane-wave cutoff K max was determined from K max = 8 . /R with near-touching the muffin-tin radii R . We have verified thatthe computed dielectric functions of both α -Sn and β -Snphases agree very well with ellipsometric spectra mea-sured in the energy region 1.2 to 5.6 eV. The corre-sponding dielectric functions ε as functions of imaginaryfrequency were obtained from the Kramers-Kronig rela-tion, where the intraband contribution for β -Sn assumedDrude broadening of 20 meV. The dielectric functions ofthe two tin phases are displyed in Fig. 2. Distance (Å)10 P r e ss u r e ( P a ) α -Sn, 28% Cb α -Sn, 29% Cb α -Sn, 30% Cb β -Sn, 28% Cb β -Sn, 29% Cb β -Sn, 30% Cb -10 -10 -10 Figure 3. (Color online) The Lifshitz pressure as a functionof the distance L between silica (dataset 1) and α -Sn or β -Snacross a liquid mixture (28 % , 29 % , and 30 % chlorobenzene inbromobenzene), using the dielectric functions from Fig. 2 andmixing according to Eq. 4. Positive values mean repulsion,negative values mean attraction. IV. RESULTS: FUNDAMENTAL EFFECT
Both tin phases, interacting with silica across pure Bb,experience repulsion at short separation distances. Incontrast, across pure Cb, an attractive short-range forceis found between both phases of tin and silica. One op-tion to fine-tune the phase controlled quantum levitationis to use a mixture of liquids tailored experimentally. We show in Fig. 3 the Lifshitz pressure as a function of thedistance between silica (dataset 1) and tin across threedifferent liquid mixtures: Bb with either 28 % , 29 % , and30 % Cb added. As expected, the interaction becomes at-tractive at longer distances as more of the less polarizableCb is added to Bb. We observe that there is a strongphase transition dependence in the sign of the Lifshitzpressure. The range of separation distances for the tran-sition from attraction to repulsion depends both on thetin phase and on the specific liquid mixture. It is possibleto find the effect for thicker liquid films, but we focus hereon liquid mixtures that give transition from attraction torepulsion in the limit of very thin liquid films. Fig. 3 sug-gests two alternative switching applications. On the onehand, switching in response to a phase transition (e.g.change in temperature), and on the other hand switch-ing in response to a change in liquid composition (i. e.,Cb content).We illustrate the relative importance of the n = 0 and n > contributions to the Lifshitz pressure in Fig. 4. Weare facing a situation where the zero-frequency term is asimportant as it is in systems involving water, but not forthe reasons anticipated by standard wisdom (e.g. Ref. 42or Sect. 6.7 in Ref. 43). In contrast to liquid water, itis not a high dielectric constant or high dipole momentthat drives the dominance of the zero-frequency term in α -Sn. Rather, the dominance is due to a delicate can-cellation between negative and positive contributions in Distance (Å)10 P r e ss u r e ( P a ) α -Sn total α -Sn n = 0 α -Sn n > 0 β -Sn total β -Sn n = 0 β -Sn n > 0 -10 -10 -10 Figure 4. (Color online) The total Lifshitz pressure and itscontribution from n = 0 term and from n > terms in theMatsubara summation. The results are shown for α -Sn and β -Sn as a function of their separation from silica (dataset 1)across a liquid mixture (29.1 % chlorobenzene in bromoben-zene). the n > terms. We have reported this kind of relation-ship between n = 0 and n > previously in ice-watersystems. Cancellation of n > terms leads to repul-sion, due to the dominance of the zero frequency term,for both Sn phases for liquid films thicker than 50 Å. Forthinner liquid films the n > terms dominate when tinis metallic, leading to an attraction. The n = 0 termdominates when tin is semimetallic, leading to repulsion.The liquids and mixing ratio need to be chosen andoptimized for each system of phase transition consid-ered. That is, with a certain mixture one can obtainrepulsion for both phases, while another mixture yieldsonly attraction. Between these two cases, one can finda range of mixing ratios suitable for a phase dependentnano-switch. To exemplify the sensitivity of the levita-tion with respect to changes in the dielectric functionswe present in Fig. 5 the Lifshitz pressures for the two dif-ferent silica materials (datasets 1 and 2). Each requiresa different mixing ratio to work optimally as a phase-controlled nano-switch. The critical Cb concentrationshifts from 29.1% to 76.9%. However, the general be-havior is similar after the critical Cb concentration hasbeen tuned to optimise the attraction to repulsion dis-tance. Many different silica materials (and similar mate-rials, like mica or polystyrene) will, when combined witha properly tuned liquid, provide further examples wherethe phase transition from the semimetallic α -Sn to metal-lic β -phase changes the short-range Lifshitz interactionfrom repulsion to attraction. When the same silica ma-terial is combined with other liquid mixtures the sign ofthe interaction may be independent of tin phase transi-tion. The reason, of course, is the strict requirement tohave a crossing of dielectric functions for the specific sil-ica material and the liquid. When tin turns metallic, theinteraction with the second surface turns more attractive(or less repulsive). P r e ss u r e ( P a ) α -Sn (76.9% Cb, SiO dataset 2) α -Sn (29.1% Cb, SiO dataset 1) β -Sn (76.9% Cb, SiO dataset 2) β -Sn (29.1% Cb, SiO dataset 1) Figure 5. (Color online) The Lifshitz pressure as a function ofthe distance between silica dataset 1 and α -Sn or β -Sn acrossa liquid mixture (29.1 % chlorobenzene in bromobenzene). Forcomparison, we also show the corresponding pressure using analternative dielectric function for silica dataset 2 combinedwith tin and a different liquid mixture(76.9 % chlorobenzenein bromobenzene). V. RESULTS: FINITE SIZE SILICA LAYER
While the previous section described the underlyingphysics of the interlayer interactions for the three-layersystem tin/liquid/solid, this section discusses practicalaspects in order to detect quantum levitation in liq-uid. We investigate the thickness dependences on thesolid layer using an extended thickness model. We con-sider therefore a vertically oriented layer-structure, andthat the solid slab is able to move (or float) up anddown in the liquid, and the slab feels the buoyancy pres-sure. This can thus be regarded as a four-layer systemtin/liquid/solid/liquid containing a thin solid layer (typ-ically SiO ) with the finite thickness d in a liquid (typi-cally Bb in mixture with Cb). There is thus a thin film ofliquid (thickness L ) between tin and the solid, but alsoliquid above the solid slab. We will not allow the slabto float close to the liquid topmost surfaces, and there-fore the liquid layer can be modeled with a semi-infinitethickness without any major loss in accuracy. Moreover,the bottom tin layer is still considered thick enough tobe treated as semi-infinite. A. Thickness dependence of the Lifshitz pressure
We investigate the thickness dependence of the sil-ica film in the Sn/liquid/SiO /liquid system containinga layer of SiO with thickness d in the liquid mixture29.1 % chlorobenzene in bromobenzene. One can observein Fig. 6(a) that although the absolute values of the Lif-shitz pressures depend on the thickness of SiO , the or-der of magnitudes of the pressure is comparable. Thequantum levitation can be observed for all consideredthicknesses. Moreover, when the thickness of the SiO layer reaches 1000 Å, the distance dependence of the Lif- P r e ss u r e ( P a ) thick SiO , α -Sn100 Å SiO , α -Sn30 Å SiO , α -Snthick SiO , β -Sn100 Å SiO , β -Sn30 Å SiO , β -Sn (a) P r e ss u r e ( P a ) , α -Sn1000 Å SiO , β -Snthick SiO , α -Snthick SiO , β -Sn (b) Figure 6. (Color online) (a) The Lifshitz pressure as func-tions of the separation distance L between SiO (dataset 1)and α -Sn or β -Sn across a liquid mixture (29.1 % chloroben-zene in bromobenzene) for the four-layer system with differentthicknesses d of the SiO layer. Here, ’thick’ implies the semi-infinite layer of SiO used in the three-layer system (Fig. 1).(b) Comparison of the Lifshitz pressures for 1000 Å thick SiO layer and the semi-infinite layer of SiO . shitz pressures overlaps in a large range of liquid layerthicknesses with that from the semi-infinite SiO layerin the three-layer model in the main article, as shown inFig. 6(b). Thus, 1000 Å is large enough to be approxi-mated as a macroscopic thickness. B. Buoyancy pressure
The net buoyancy pressure b on a SiO slab in liq-uid due to gravity and difference in densities of the SiO film and the surrounding liquid mixture can be estimatedusing b = ( ρ liquid − ρ silica ) · gd where g is the gravita-tional acceleration. With typical values for the densitiesof the liquid mixture, ρ liquid , and of SiO , ρ silica , anda thickness of the SiO slab of d = 1000 Å the buoy-ancy pressure is b ≈ − . mPa, where the negative signindicates attraction. This value is negligible comparedto the Lifshitz pressure at small separation distances ( L < 20 Å) where the quantum levitation occurs as shownin Fig. 7(a). Figure 7(b) demonstrates that the attrac-tive buoyancy pressure can compensate the long-rangerepulsive Lifshitz contribution at large separations. In- P r e ss u r e ( P a ) , α -Sn1000 Å SiO , β -Snbuoyancy (negative) (a) P r e ss u r e ( - P a ) , α -Sn1000 Å SiO , β -Snbuoyancy (negative) (b) Figure 7. (Color online) (a) The Lifshitz pressure as a func-tion of the separation distance between SiO (dataset 1) and α -Sn or β -Sn across a liquid mixture (29.1 % chlorobenzene inbromobenzene) compared to the attractive buoyancy pressure(here, presented on a positive scale). (b) Magnification of thedistance region where the repulsive Lifshitz and the attractivebuoyancy pressures compensate each other. triguingly, α -Sn and β -Sn exhibit a noticeable differencein their respective equilibrium distances, where the netpressure due to the Lifshitz and buoyancy contributionsvanishes; they differ by more than 200 Å which is obviousin Fig. 7(b). However, although the effect is induced bythe phase transition, it is not linked directly to the quan-tum levitation found for the small separation distances. C. Role of dielectric properties of interactingmaterials
When the dielectric function for SiO (dataset 1) is re-placed with different parameterizations corresponding toa different SiO sample (dataset 2), one can observe inFig. 8(a) that the attraction to repulsion transition dis-appears. This effect is expected as noticeable variationin the dielectric properties has been reported in previ-ous works. The difference between the two differentSiO samples (i. e., dataset 1 and dataset 2) may ap-pear to be small (see Fig. 2) but Fig. 8(b) demonstratesthat the spectral functions are very different. With thealternative dielectric function of SiO (dataset 2), therepulsive contributions to the Lifshitz pressure are en- P r e ss u r e ( P a ) β -Sn (29.1% Cb, SiO dataset 2) α -Sn (29.1% Cb, SiO dataset 2) α -Sn (29.1% Cb, SiO dataset 1) β -Sn (29.1% Cb, SiO dataset 1) (a) ζ (rad/s)-50000500010000150002000025000 g ( i ζ ) β -Sn (29.1% Cb, SiO dataset 2) α -Sn (29.1% Cb, SiO dataset 2) β -Sn (29.1% Cb, SiO dataset 1) α -Sn (29.1% Cb, SiO dataset 1) (b) Distance (Å)10 P r e ss u r e ( P a ) α -Sn, 76% Cb α -Sn, 77% Cb α -Sn, 78% Cb β -Sn, 76% Cb β -Sn, 77% Cb β -Sn, 78% Cb -10 -10 -10 (c) P r e ss u r e ( P a ) α -Sn (29% Cb, VAT) α -Sn (30% Cb, VAT) β -Sn (29% Cb, VAT) β -Sn (30% Cb, VAT) α -Sn (29% Cb, LL) α -Sn (30% Cb, LL) β -Sn (29% Cb, LL) β -Sn (30% Cb, LL) (d) Figure 8. (Color online) (a) The Lifshitz pressure as a function of the separation distance between SiO and α -Sn or β -Sn acrossa liquid mixture of chlorobenzene and bromobenzene using two different parameterizations for the dielectric function of SiO ,i. e., datasets 1 and 2. (b) Spectral functions revealing the contribution of each frequency mode to the Lifshitz pressures for thetwo different dielectric functions of SiO . The zero frequency term is divided by a factor of 2. (c) The Lifshitz pressure as afunction of the separation distance between SiO and α -Sn or β -Sn across a liquid mixture of chlorobenzene and bromobenzeneusing SiO dataset 2; this can be compared with Fig. 3, where SiO dataset 1 is used. (d) The Lifshitz pressure as a function ofthe separation distance between SiO and α -Sn or β -Sn across a liquid mixture of chlorobenzene and bromobenzene using SiO dataset 1 with two different models to describe the dielectric function of the liquid mixture, namely the Lorentz-Lorenz-like(LL) and the volume average theory (VAT) models. hanced, and the attractive contributions to the Lifshitzpressure are reduced as compared to the correspondingresults for the SiO dataset 1. To obtain attraction forthe interaction between the silica dataset 2 and tin, onemust reduce the magnitude of the dielectric function ofthe liquid. This change can be achieved by increasing theratio of chlorobenzene in bromobenzene, as described byFig. 8(c). In the region between the different limits withthe only repulsion and with only attraction, there is a ra-tio region where phase transition controlled attraction torepulsion transitions can occur. It is worth noticing thatthe utilization of different models for the dielectric func-tion of liquid mixture can result in a variation of absolutevalues of the Lifshitz pressure. Nevertheless, the quan-tum levitation can still be achieved by making a smallchange in liquid ratio as demonstrated in Fig. 8(d).Phase transition induced attraction to the repulsionof the Lifshitz pressure can also be observed for othersystems of materials. In particular, it is found for the in-teraction of a 1000 Å thick polystyrene film with α -Sn or β -Sn slab in a liquid mixture of methanol and bromoben- zene; see Figs. 9(a) and (b). The dielectric functions ofpolystyrene, methanol, and bromobenzene are also takenfrom van Zwol and Palasantzas’s work. The interven-ing liquid dielectric function needs to have a crossoverwith the dielectric function of one of the solid materialsto obtain the levitation, but the liquid is a mixture ofthe two pure liquids whose dielectric functions can lie oneither side of the solid [Fig. 9(a)]. Thus, the engineeringrequirements are obvious: the refractive indices of thetwo pure liquids can lie above and below the refractiveindex of one of the solid materials. It is then a matter offinding the right combination of the liquid mixture thatyields the desired crossover of the dielectric functions ofthe liquid mixture and the solid. Hence, it is possible toachieve quantum levitation controlled by the α -Sn/ β -Snphase transition with combinations of different materi-als. Noticeable in Fig. 9(b) is that here the β -Sn systemyields more repulsion compared to α -Sn. This reversalof behavior occurs due to the difference in frequency re-gions that give positive and negative contributions to theLifshitz pressure. Since the zero frequency contributes ζ (rad/s)10 ε ( i ζ ) β -Sn α -Snpolystyrenebromobenzenemethanol (a) P r e ss u r e ( P a ) α -Sn β -Sn (b) Figure 9. (Color online) (a) Dielectric functions of materialsused in calculations of (b) the Lifshitz pressure as a functionof the separation distance between polystyrene (dataset 1)and α -Sn or β -Sn across a liquid mixture of 29 % methanolin bromobenzene. The static dielectric constant is 2.45 forpolystyrene and 32.9 for methanol; data for the other mate-rials are found in Fig. 2. repulsion, the resulting pressure for both systems is re-pulsive at distances beyond 760 Å for α -Sn and 470 Åfor β -Sn, where the lower frequency mode contributionsdominate. Thus, there are two crossings of zero pressure,with multiple extreme points, in the pressure curves forthe polystyrene-liquid-( β -Sn) system. D. Thermal fluctuation
We identify two distinct ways in which the system isaffected by thermal fluctuations. First, the contributionof the thermal fluctuations to the Lifshitz force is alreadyaccounted for through the use of Matsubara frequencies ζ n in Eq. 1. Second, the proposed system is also subjectto classical thermal fluctuations arising from the kineticenergy of the surfaces at temperature T . Kinetic energyof the surface (a type of Brownian motion) causes thedistance between surfaces to fluctuate. The stability ofthe system with respect to classical thermal fluctuationscan be established by considering a finite contact areabetween tin and SiO surfaces. The interaction energy isshown in Fig. 10, evaluated from the pressure assuminga 1000 Å = 0.1 µ m thick SiO slab with a 1 µ m con- I n t e r a c t i on ene r g y ( k T ) α -Sn β -Sn Figure 10. (Color online) Interaction energy between SiO (dataset 1) and α -Sn or β -Sn across a liquid mixture (29.1 % chlorobenzene in bromobenzene) as a function of the distance.The interaction energy is estimated from the pressure by con-version to a . × × µ m slab of SiO . The temperaturesare 280 and 290 K for α -Sn and β -Sn, respectively. tact area. The thermal kinetic energy is of the order of k B T , thus as a rule of thumb, an interaction of more than100 k B T is robust with respect to thermal fluctuations,while an interaction weaker than 10 k B T is reversible (notmechanically stable). The repulsive barrier indicatesthat the attractive force is stable when the separationbetween β -Sn and SiO is within L ≈ Å. At the samelength scale, the repulsive interaction found for α -Sn ex-ceeds 600 k B T , indicating stable repulsion.With these properties the system can be conceived asa trigger switch, initially set up for the β -Sn phase inclose contact ( L < Å) with the SiO slab in the liquidmixture. When a tin phase transition is triggered, α -Sn repulsion pushes the SiO slab outwards. The devicewill need to be reset when tin is transformed back tothe β -Sn phase, overcoming the 600 k B T barrier eithermechanically or by flushing with excess chlorobenzene forwhich the Lifshitz pressure is negative for all separationdistances. VI. DISCUSSION
It is of interest to apply a more comprehensive per-spective and ask: will it be practically possible to applythis kind of phase transition system as an actuator?The timescale for the complete tin phase transitionmay be more than s. In Fig. 11 we show how theLifshitz pressure varies during the process of conversionfrom α - to β -Sn, when the two phases coexist. The di-electric functions of the α -Sn/ β -Sn mixture as well asof the liquid were evaluated using the Lorentz-Lorenz-like relation, Eq. 4, and we choose the three-layer systemwith semi-infinite SiO thickness as described in Fig. 1.The switch in pressure occurs after only 6% conversion,at 94% α -Sn. It follows that the switch from repulsionto attraction can be achieved at a faster time scale than P r e ss u r e ( P a ) α -Sn100%96%94%92%90%80%60%40%0% α -Sn fraction Figure 11. (Color online) The Lifshitz pressure as a functionof the distance L between silica (dataset 1) and an α -Sn/ β -Sn mixture across a liquid mixture (29.1 % chlorobenzene inbromobenzene) for the three-layer system (Fig. 1). The pres-sure is given for a partial tin phase transition at T = 286.5 Kindicated by the fraction of α -Sn coexisting with β -Sn. the time required for full conversion.A second consideration for the practicality of this de-sign of actuator is frictional resistance from the fluidmedium. An actuator executing oscillations will neces-sarily be exposed to viscous drag forces. In order to workproperly, the decay time from the drag has to be muchshorter than the time needed for the phase transition.In this context we may recall the instructive discussiongiven by Sedighi and Palasantzas: they replaced theupper plate by a metal sphere (gold) with mass M andradius R , elastically suspended in the gravitational field,able to move vertically with velocity ˙ z under the com-bined influence of gravity, viscous drag from the envi-ronment (in their case air), and Casimir force from theplate beneath. The key difference from our case is thatwe are considering a liquid-induced drag instead of anair-induced one. One can set up the governing equationand from that estimate the viscous decay time. We havegone through this calculation under our conditions, as-suming a gold sphere with R = 10 µ m, sphere velocity ˙ z = 3 mm / s typically, and viscosity as for water. Un-der these conditions, the Reynolds number is very small,hence the Stokes drag formula is applicable. We omitthe details here, but the result is that the decay time becomes small, of the order of milliseconds. This resultis promising as the decay time is much smaller than thetime scale needed for the full tin phase transition andthe actuator can in our case be considered to be react-ing instantaneously. We point out that such devices arenot limited to the particular materials chosen, nor thetype of metal/non-metal transition. Other combinationsof materials and phase transitions can be chosen, tailoredto the second surface in order to control switching andphase transition time. VII. CONCLUSIONS
As a conclusion, we have shown that by exploiting thecombined effect of (i) tin phase transition and (ii) fluidmixture composition, it becomes possible to fabricate aswitch operative at a moderate temperature range. Arepulsive interaction is found in semimetallic α -Sn whenthe chlorobenzene content is mixed with bromobenzene(the specific critical concentration depending on the typeof silica). The repulsive interaction switches to attrac-tion when there is as little as 6% conversion of α -Sn to β -Sn. It also switches when the chlorobenzene contentrises above the critical concentration. We have verifiedthat our model accurately can represent a 0.1 µ m thickSiO slab with a 1 µ m contact area, and that the buoy-ancy pressure and thermal effects then are negligible. Fordistances around Å between Sn and SiO surfaces,the repulsive Lifshitz free energy exceeds 600 k B T for α -Sn, and its energy barrier is sufficiently large relativethe thermal energy, indicating that the attractive interac-tion for β -Sn is stable. Hence, thermodynamically stablenano-switching can be achieved. We think that the ideais worth noticing for its possible applications in nanome-chanical systems and environmental sensors. ACKNOWLEDGMENTS
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