Effect of fluids on theQfactor and resonance frequency of oscillating micrometer and nanometer scale beams

Resonance oscillations of micrometer and nanometer scale beams in gases and liquids have increasingly important applications in physics and biology. In this work, we calculate fluid damping and its effect on damped resonance frequency ${\omega }_d,$ and quality factor Q, for oscillating long beams at micrometer and submicrometer scales. For beams of nanometer scale, which are smaller than the mean free path of air molecules at standard conditions, the continuum limit breaks down and the commonly used Stokes drag calculation must be replaced by the appropriate calculation for rarefied gas flow. At scales where the continuum limit holds, this quasisteady Stokes solution is often still inapplicable due to the high resonant frequency associated with small beams, typically ${10}^2\mathrm{MHz}.$ The unsteady drag can be over two orders of magnitude higher than that predicted by the quasisteady Stokes solution and the added mass is non-negligible. Here we calculate Q factors as a function of gas pressure over the range from ${10}^{-5}\mathrm{torr}$ to ${10}^5\mathrm{torr},$ corresponding to free molecular to continuum limit. The comparison of the Q factors for two typical beams at various pressures suggests an advantage of using submicrometer scale over micrometer scale beams for applications near ambient pressure.