Dependence of chaotic behavior on optical properties and electrostatic effects in double-beam torsional Casimir actuation

Abstract

We investigate the influence of Casimir and electrostatic torques on double-beam torsional microelectromechanical systems with materials covering a broad range of conductivities of more than three orders of magnitude. For the frictionless autonomous systems, bifurcation and phase space analysis shows a significant difference between stable and unstable operating regimes for equal and unequal applied voltages on both sides of the double torsional system giving rise to heteroclinic and homoclinic orbits, respectively. For equal applied voltages, only the position of a symmetric unstable saddle equilibrium point is dependent on the material optical properties and electrostatic effects, while in any other case stable and unstable equilibrium points are dependent on both factors. For the periodically driven system, a Melnikov function approach is used to show the presence of chaotic motion rendering predictions of whether stiction or stable actuation will take place over long times impossible. Chaotic behavior introduces significant risk for stiction, and it is more likely to occur for the more conductive systems that experience stronger Casimir forces and torques. Indeed, when unequal voltages are applied, the sensitive dependence of chaotic motion on electrostatics is more pronounced for the highest conductivity systems.

Sensitivity of chaotic behavior to low optical frequencies of a double-beam torsional actuator

“…Figure 2 depicts δ Cas vs ϕ for the two Au samples by taking into account the plasma and Drude models for both equal or p = 1 (V R = V L ), and unequal or p = 1 (V R = V L ), electrostatic potentials. The strong effect of the applied electrostatic potential on the stability of the double beam has already been shown in [20]. In fact, when the electrostatic torque has equal magnitude at both ends of the beam (…”

“…with γ = Iω o /Q, and H (• • • ) denoting the Hilbert transform [20,33]. Figures 6-8 show the threshold curves α = γ ω 0θ 0 /τ 0 vs the driving frequency ω/ω o .…”

“…So far several investigations have been conducted to study the actuation dynamics of torsional actuators under the influence of Casimir and electrostatic forces for a wide range of material optical properties [8,20,21]. From these studies it has emerged that chaotic behavior is unavoidable during actuation dynamics, which leads to increased possibility for stiction and consequently limiting the long term prediction of devices to perform stable operation.…”

Maxwell eigenmode approach to the Casimir-Lifshitz torque

“…The expression for p 3L found in this way is equal to p 3b with the eigenvalue relations Eq. (13). The detailed proof that P b = P L can be found in Appendix A 5.…”

“…This fuels the desire to come to precise comparisons between theoretical predictions and experimental data [6]. More practically, Casimir interactions affect the actuation dynamics of nano-and micro-mechanical systems, such as switches, cantilevers, and actuators at a sub-micrometer length scale [7][8][9][10][11][12][13][14].…”