Casimir Force and Pull-In

What is modelled

Quantum field theory gives every mode of the electromagnetic field a zero-point energy hbar omega / 2. Between perfectly conducting plates only modes with a whole number of half-wavelengths fit, outside every mode is allowed. Subtracting the infinite energy that is the same with and without the plates leaves a finite, negative energy that grows in size as the gap shrinks, so the plates attract. Here the plate also sits on a spring, which makes the force measurable and shows the pull-in instability used in micro-machines.

Equations

• Energy per area u(d) = −π² ħc / (720 d³), pressure P = −du/dd = π² ħc / (240 d⁴), ħc = 3.1615×10⁻²⁶ J m.
• Sphere of radius R above a plate (proximity approximation) F = 2πR u(d) in size, = π³ ħc R / (360 d³).
• Finite conductivity: P = P0(1 − 16δ/3d + 24δ²/d²) with plasma depth δ = 22 nm for gold and 12.4 nm for aluminium (extended smoothly beyond d = 10δ).
• Spring: x″ + x′/Q + x = FC(d0 − x)/k in units of the spring period; pull-in when no root of k x = FC(d0 − x) exists, for flat perfect plates at kc = 4C/(0.8 d0)⁵.

Checks run in the page

• Mode sum with its infinite part removed against −π²/720.
• Force from the slope of the energy and the sphere-plate formula.
• Pressure at 1 µm (1.3 mPa) and the 1/d⁴ scaling.
• Pull-in spring constant by bisection against the closed form and the released-plate end state.

Honest simplifications

The plates are ideal in the mode sum; real metals use a two-term correction that is only accurate for gaps above about ten plasma depths, the plate is treated as a single rigid mass on a linear spring, roughness, temperature and electrostatic forces are ignored, and the plates stick for good when the gap falls under 2 nm.

• Press Soft spring snaps shut: the pull rises faster than the spring and the plates collapse.
• Press Gold plates near pull-in and lower the spring constant a little at a time.
• Switch the geometry to sphere and plate: the force falls only as 1/d³.