A microscopic Kapitza pendulum

Abstract

Pyotr Kapitza studied in 1951 the unusual equilibrium features of a rigid pendulum when its point of suspension is under a high-frequency vertical vibration. A sufficiently fast vibration makes the top position stable, putting the pendulum in an inverted orientation that seemingly defies gravity. Kapitza's analytical method, based on an asymptotic separation of fast and slow variables yielding a renormalized potential, has found application in many diverse areas. Here we study Kapitza's pendulum going beyond its typical idealizations, by explicitly considering its finite stiffness and the dissipative interaction with the surrounding medium, and using similar theoretical methods as Kapitza. The pendulum is realized at the micrometre scale using a colloidal particle suspended in water and trapped by optical tweezers. Though the strong dissipation present at this scale prevents the inverted pendulum regime, new ones appear in which the equilibrium positions are displaced to the side, and with transitions between them determined either by the driving frequency or the friction coefficient. These new regimes could be exploited in applications aimed at particle separation at small scales.

Figure 1

System setup. (a) Quartic pendulum potential U(r), Eq. (3), used in the analytical calculations in the very high-frequency limit. (b) Schematic figure showing the colloidal particle confined by the (optical) pendulum potential under the downward force F, which is created in the experiment by a uniform drag. The pendulum is subject to a driving, fast vibration y_s(t), Eq. (2), in the y-direction.

Figure 2

Simulations illustrating the dynamical regimes. Units are defined such that L = m = F = 1. The panels depict trajectories from the initial condition (x₀,y₀) = (0.1,0.5) (indicated with a cross) showing the dependence of the equilibrium position with the friction coefficient value. The pendulum frequency is ω₀ = 100 and the driving parameters are ω = 20 and $a=(\sqrt{2}+\mathrm{0.05)/}\omega =0.0732$. The small disks in each panel indicate the position of the equilibrium points given by the analytical predictions. The friction coefficients are: (a) γ = 1 (inside inverted pendulum regime, defined for $0<\gamma <{\gamma }_1$), (b) γ = 30 (normal pendulum behavior, ${\gamma }_1<\gamma <{\gamma }_2$), (c) γ = 180 (friction-filter regime, ${\gamma }_2<\gamma \ll {\omega }_0^2/\omega$), (d) γ = 1000 (very high-frequency regime, ${\omega }_0^2/\omega \ll \gamma$). Thresholds are predicted at γ₁ ≈ 26, γ₂ ≈ 139, and the reference value ${\omega }_0^2/\omega =500$.

Figure 3

Diagram showing the different regimes under the same conditions as in Fig. 2, with ω₀/Ω = 100 and a/L = 0.0732.

Figure 4

Experimental equilibrium angles as a function of the vibration amplitude in the very high-frequency regime for an optical pendulum with length L = 8.23 μm. The vibration frequency was ω = 2π × 500 Hz. Filled points correspond to an optical pedulum with proper frequency ω₀ = 2π × 947 Hz and stage speed v_s = 1 μs⁻¹ —leading to Ω = 2π × 105 Hz and γ/ω₀ = 598. Open points correspond to an experimental setup with an increased laser power, ω₀ = 2π × 1490 Hz (γ/ω₀ = 380), and a reduced stage speed, such that Ω = 2π × 91 Hz. The lines are the analytical predictions of Eq. (7), with a friction value calculated assuming Stokes’ law γ = 3πζd/m, where ζ is the dynamic viscosity of water.

Figure 5

Experimental results (symbols) in the normal and friction-filter regime, with a/L = 0.25. The stage speed and laser power were varied to generate four sets of data, with varying values of γ/ω₀, but with a common ratio ω₀/Ω = 26.4. Top panel shows the equilibrium angle vs. ω for three of these sets. Solid lines indicate the friction-filter-regime prediction (19), and the horizontal dashed line the very high-frequency prediction (20). The diamonds in the bottom panel show the threshold from normal to friction-filter regime, estimated from the experimental data by assuming a regime transition when the angle crosses 20°. The solid line in the bottom panel is the threshold prediction given by (18).

Figure 6

Schematic diagram of experimental apparatus. The galvanometer mirrors are mounted in a plane conjugate to the back aperture of the objective and steer the position of the beam waist in two dimensions in the trapping plane perpendicular to the propagation direction.

Figure 7

Video tracking output showing the trajectory of a colloidal particle in the optical pendulum potential with no driving (red symbols) and high-frequency stabilization with a/L = 0.3 (blue symbols). The predicted equilibrium angles of both situations are indicated by black dashed lines. A background image of the fast scanning laser spot (dark grey) illustrates the pendulum potential. Both the direction and speed of microscope stage movement—and hence direction of the drag force—and the direction of the pendulum vibration are indicated.

Figure 8

Measuring the optical potential parameters. (a) Histogram of fluctuations in the radial direction due to Brownian motion, as a result of N = 7.2 × 10⁵ samples of the radial position in a typical experiment. The corresponding average radius is $\langle r\rangle =L=8.228\pm 0.001\mu \mathrm{m}$. (b) Potential energy in the radial direction reconstructed from the data shown in the left panel (solid black line), together with a quadratic fit to potential minimum using κ_r = 0.125 pNμm⁻¹ (dashed red line), and the associated quartic potential (dotted blue line).