Stable Position of a Micro Torsion Balance under the Casimir Force

The Casimir interaction between a torsion balance and flat plate is considered using proximity force approximation (PFA) and geometric optics approximation (GOA). Both approximations predict that the parallel configuration becomes unstable as the distance between a torsion balance and flat plate decreases. However, the dependence of the tilt angle at the stable position on the separation is significantly different. The tilt angle obtained by PFA continuously changes over the entire range; however, the discontinuous transition is predicted in GOA. [DOI: 10.1380/ejssnt.2013.60]


I. INTRODUCTION
The Casimir force [1][2][3] is generated by vacuum fluctuation and acts on any material.The strength of the Casimir force between objects rapidly increases as object separation decreases, and these strong interactions often cause adhesion between surfaces in micro-assemblies such as micro-electro-mechanical-systems (MEMS) [4][5][6].Therefore, to safeguard MEMS from adhesion, the stability of moving elements under the Casimir force must be examined.However, the calculation of the Casimir force for real mechanical parts used in MEMS is very difficult because the Casimir force between two bodies depends strongly on the bodies' shapes.After Casimir's conjecture of the attractive force between perfectly conductive plates in 1948, much effort has been devoted to the calculation of the Casimir force between more complex objects.Recent theoretical developments and powerful computers enable us to preciously calculate the Casimir force for many different configurations [7].In this study, we focus on the Casimir energy between non-parallel plates.We can investigate the stability of the torsion balance above the metallic flat surface by studying the dependence of the Casimir energy between non-parallel plates on the separation between them.The torsion balance is very important in MEMS.For example, torsional MEMS is widely used in scanners for displays.In addition, torsion balance has been used to measure the Casimir force [8,9].
The parallel configuration of the torsion balance to a flat surface is one of the equilibrium positions.The principle problem to be studied is the stability of this configuration.The Casimir force is a surface force, and its strength is proportional to the projection area of torsion balance onto the flat surface.The Casimir energy between the torsion balance and flat surface is negative and the projection area takes a maximum in the parallel configuration.Therefore, if the Casimir energy per unit area is independent of the angle, the parallel configuration would be most stable.However, in real systems, where one end of the torsion balance approaches the flat surface at a tilt, the absolute Casimir energy per unit area increases.Consequently, the stable angle is determined by these contrary

effects.
We calculate the tilt angle of a stable position as a function of separation using proximity force approximation(PFA) [3,10] and geometric optics approximation(GOA) and show that a parallel configuration becomes unstable for small separations.GOA is a new approach for the calculation of the Casimir energy, one that goes beyond PFA.For the configuration of a sphere and plate, the results obtained by GOA agree with other recent numerical treatments over a wide range of separations.
This paper is structured into five sections.In Section II, we calculate the Casimir energy between non-parallel plates whose conductivities are infinity using PFA, and we discuss the tilt angle of stable positions.In Section III, we briefly explain GOA and show that a jump is observed for the tilt angle when a torsion balance approaches the flat surface.In Section IV, we consider the Casimir energy between a hinge and flat surface.In Section V, we compare the results obtained by PFA and GOA and present our conclusions.

II. CASIMIR ENERGY BETWEEN NON-PARALLEL PLATES IN PFA
Let us consider a torsion balance with length L and width W above a flat surface.The configuration of a torsion balance and flat surface is shown in Fig. 1.We assume that the conductivity of both a torsion balance and flat surface is infinity.If a torsion balance is parallel to a flat surface, i.e., θ = 0, the Casimir energy between a torsion balance and flat surface per a unit area is exactly given by and the Casimir force The Casimir force is an attractive force and its strength increases as separation d decreases.For plates of area 1 cm 2 and d = 0.5 µm, the force is 2 µN.
For non-parallel plates, no exact expression of the Casimir energy has been obtained.Thus, we apply PFA to the calculation of the Casimir energy as a first approximation.Here we introduce a coordinate system as shown in Fig. 2. In PFA, the Casimir energy between an element of a flat surface dS 1 around (x, y, 0) and that of a torsion balance dS 2 around (x, y, d + tan(θ)x) is approximated by The Casimir energy is found by integration over the projection area on to the flat surface as follows: This Casimir energy, as obtained by PFA, coincides with the exact value given by Eq. ( 1) for a parallel configuration.Thus, introducing the Casimir energy normalized by E ∞ (d) is useful for discussing stable angles.The normalized Casimir energy η = E PFA /E ∞ can be expressed as a function of a normalized separation δ ≡ d/L: ( We show the dependence of the Casimir energies obtained by PFA on inclination angle in Fig. 3 for δ = 0.9 and 0.7.The normalized Casimir energies take the minimum value at θ = 0 and 0.97 for δ = 0.9 and 0.7, respectively.This result suggests that if the separation decreases, the parallel configuration becomes instable.The absolute value of the angle Θ PFA , where the Casimir energy takes the minimum value, is given by Figure 4 shows the relation between Θ PFA and the normalized separation δ.A parallel configuration of torsion balance to the flat surface is most stable for large separations, i.e., δ > 1.As separation decreases from δ = 1, the torsion balance gradually tilts and approaches π/2 as δ → 1/2, i.e., the vertical configuration. If a restoring force acts on the torsion balance, then equilibrium position depends also on torsion elastic modulus k.We assume that restoring force is proportional to inclination angle.Total potential energy U (δ, θ), including the elastic potential energy, is given by Figure 5 shows the comparison of elastic potential energy with the Casimir energy.The absolute value of the Casimir energy always exceeds that of elastic potential energy as normalized separation approaches 1/2.Thus, a torsion balance always tilts near a flat surface if the distance between the surface and balance is sufficiently small.The critical separation across which a torsion balance tilts decreases as the torsion elastic modulus increases.

III. CASIMIR ENERGY BETWEEN NON-PARALLEL PLATES USING GOA
Recently, several approximation methods beyond PFA were presented [11][12][13].Here we introduce GOA.Because the Casimir energy originates in fluctuation of a vacuum, the main task in its calculation is determination of the electromagnetic field satisfying a given set of boundary conditions.If the ratio of the separation of the conducting boundaries to their curvature is large, then GOA is available.The formulation of the Casimir energy in GOA for non-parallel plates is given by where l n is the length of the closed geometric optics ray beginning and ending at the point r and reflecting n times from the flat surfaces.We illustrate an optical path with four-bounces as an example in Fig. 6.
The Casimir energy between non-parallel plates using GOA was obtained by Guilfoyle et al. as follows (see details in Ref. [14]): where and , for m 0 ≥ m 1 and m 1 even, , for m 0 ≥ m 1 and m 1 odd, (11) where R 0 and R 1 ≡ R 0 +L are defined in Fig. 7. Before we explain the integers m 1 and m 2 , we define the following inequality: The integer m 1 is defined by the smallest even integer satisfying inequality (13) or the smallest odd m 1 satisfying inequality (14).The integer m 2 is defined as the smallest integer between an integer m 0 satisfying inequality ( 12) and m 1 .
Let us compare the Casimir energy obtained by PFA with that obtained by GOA. Figure 8 shows the normalized Casimir energy η PFA and η GOA = E PFA /E ∞ for δ = 0.9 and 0.7.Although PFA and GOA give the same Casimir energy at θ = 0 and π/2, the Casimir energy obtained by PFA is always smaller that obtained by GOA, except at these angles.Figure 9 shows the dependence of the angle where the Casimir energy takes the minimum value on the separation between a torsion balance and flat surface.
In contrast to the change in Θ PFA obtained PFA, a discontinuous transition is observed near δ = 0.8.This means that the torsion balance suddenly tilts as separation decreases.The normalized separation, where a jump occurs, is a root of the equation and it is evaluated as 0.808309.As separation decreases further, Θ GOA increases and we find a cusp at δ = √ 5/8= 0.790569, where Θ PFA = π/4.

IV. CASIMIR ENERGY BETWEEN A HINGE AND FLAT SURFACE
For sufficiently small separations, the Casimir energy between a tilting torsion balance and flat surface is smaller than that in a parallel configuration.If a torsion balance tilts to the left, as shown in Fig. 1, the absolute Casimir energy under the left-wing increases.On the other hand, the absolute Casimir energy under the rightwing decreases.Thus, when the torsion balance tilts, effects opposite to the Casimir energy occur simultaneously.Let consider the change in the Casimir between a hinge and flat surface (see inset in Fig. 10).In this case, there is no negative contribution of right-wing to the absolute Casimir energy.In PFA, the angle where the Casimir energy between a hinge and flat surface takes the minimum value, Θ hinge PFA (δ) is given by Figure 10 shows the dependence of the angle taking the minimum Casimir energy on the separation obtained by GOA PFA PFA and GOA.As the distance between the hinge and flat surface d decreases, the angle Θ hinge PFA changes smoothly from zero and converges to π/2.That is, in PFA the hinge above the flat surface always tilts.This behavior is different from the change in the tilt of the torsion balance.In GOA, there is a separation distance at which the hinge tilts suddenly from the parallel configuration.

V. CONCLUSION
Using PFA and GOA, we considered the dependence of tilt angle of a torsion balance and hinge above a flat surface on separation between them.Both approximation methods predict that the tilt angle increases as separation decreases and a tilting plate becomes vertical to the plane of the flat surface of the flat surface if torsion elastic modulus is very small.However, we also found a significant difference between PFA and GOA for the dependence of tilt angle on the separation.
If the separation is very large, the torsion balance does not tilt and its configuration is unchanged until separation approaches a critical value.PFA predicts that the tilt angle increases smoothly below this critical separation.On the other hand, GOA predicts that a jump occurs when separation decreases beyond the critical separation.
It may be useful to regard the transition from a parallel to a non-parallel configuration as a phase transient, a phenomenon often discussed in a thermodynamic system.Let us consider the tilt angle as a free energy in a ther-modynamic system.The order of the phase transition is identified by the lowest derivative of the free energy that is discontinuous at the transition.A first-order phase transition is characterized by a discontinuity in the free energy at the transition point.In the second-order phase transition, the free energy is continuous at the transition point, but the first-derivative of the free energy is discontinuous.
We can summarize the results in this study using the above classification scheme.For a torsion balance, PFA predicts the second order phase transition, and GOA predicts the first order transition.For a hinge, the dependence of a tilt angle above a flat surface is different from that of a torsion balance.PFA predicts that tilt angle of a hinge is a smooth function of the separation, and no phase transition takes place.On the other hand, GOA predicts that a first order transition takes place.
Although only an experiment can determine which prediction is correct, GOA gives more accurate values than PFA for other configurations.In other words, observation of tilt angle dependence on separation distance can provide a good test of GOA.

FIG. 1 :
FIG. 1: Configuration of a torsion balance.Distance between the pivot and flat surface is d.

2 FIG. 2 :
FIG.2: Illustration of a coordinate system for the torsion balance.The origin lies on the plane of the flat surface.

FIG. 9 :
FIG. 9: Plot of the angle of a torsion balance taking the minimum Casimir energy at the normalized separation by PFA (dashed line) and GOA (solid line).

FIG. 10 :
FIG.10: Plot of the angle of a hinge taking the minimum Casimir energy at the normalized separation by PFA (dashed line) and GOA (solid line).Inset is illustration of a hinge above a flat surface.