Quantifying the Knudsen force on heated microbeams: A compact model and direct comparison with measurements

At the microscale, even moderate temperature differences leading to thermal nonequilibrium can result in significant Knudsen forces generated by the energy exchange between gas molecules and solids immersed in a gas. Experimental measurements of the microscale Knudsen force have been reported by Passian et al., Phys. Rev. Lett. 90, 124503 (2003) using heated microcantilevers of atomic force microscope probes. The present study investigates the mechanism and magnitude of Knudsen forces in detail based on numerical solution of the Boltzmann kinetic equation with the ellipsoidal statistical Bhatnagar-Gross-Krook approximation for the collisional relaxation process. A direct comparison between the numerical simulations and experimental measurements is presented. We show that, assuming a fully diffuse interaction of gas molecules with the surfaces of the heated cantilever, simulations agree with measurements for different operating pressures in argon and nitrogen ambients. For the helium ambient the simulations agree with measurements only when an incomplete accommodation is used. A closed-form model for the nondimensional Knudsen force coefficient on a heated microbeam is obtained that can be used for quantifying such forces in analysis and design of microsystems under a wide range of geometrical, thermal, and pressure conditions.

Figure 1

Two-dimensional schematic of the Knudsen force generation on a heated beam near a substrate. The gradient of temperature in the $y$ direction is higher on the bottom of the beam than on the top. Also shown are two molecules at a distance of one mean free path away from the top and bottom of the beam, which, on collision with the beam, impart a downward force on the beam. Please note that in reality the beam moves upward because of the difference in the number of molecules hitting the top and bottom surfaces.

Figure 2

Computed nondimensional pressure distributions. (a) Pressure flow fields and streamlines at (left) Kn $=$ 0.45 and (right) Kn $=$ 5. The vortex at the corner of the beam results in lowering the normal flux on the top surface and increasing the upward force on the beam. (b) Pressure profile along the beam at Kn $=$ 0.45. (c) Pressure profile along the beam at Kn $=$ 5. The pressure on the bottom surface is greater than that on the top surface, thereby resulting in an upward force.

Figure 3

Temperature profiles above and below the beam along the symmetry line $x=0$ at $\text{Kn}=0.48$. The temperature jump and gradient in the $y$ direction are higher on the bottom surface than on the top surface.

Figure 4

Comparison of Knudsen force simulations with experimental data [5]. (a) Argon, (b) nitrogen, and (c) helium. The error bars were calculated using Richardson extrapolation on three different meshes.

Figure 5

Comparison of the force coefficient based on simulations with experimental results. The experimental values deviate when $\text{Kn}>1$, that is, mean free path is larger than set of 2-$\mu m$ holes. Though the dimensional force in newtons is different for the three gases (argon, nitrogen, and helium), the coefficient of Knudsen force collapses neatly into a single line with the compact form as in Eq. (3).