Microstructure actuation and gas sensing by the Knudsen thermal force

Temperature inhomogeneities in rarefied gases give rise to thermal stresses and lead to bulk fluid flows that exert forces on immersed structures.¹ Manifestations of thermally driven flows include thermal creep in a capillary from cold to hot and particle thermophoresis. Interest in such flow phenomena by physicists such as Maxwell,² Reynolds,³ and Einstein⁴ was kindled by Sir William Crookes in the 1870s after the development of his famous radiometer.⁵ Crookes' radiometer consists of a series of vanes similar to that shown in Figure 1(a) revolving around a central spindle in a partial vacuum. One side of vanes is painted black and has a higher emissivity than the opposing face. When illuminated by a light source, the vanes develop a thermal gradient through their thickness and begin to revolve around the spindle as if pushed on the hotter face. Initially, this effect was attributed to photon pressure; however, after experiments by Schuster⁶ and theoretical treatment by Reynolds,³ it was proven that force generation manifests within surrounding low-pressure gas. Later studies (see a recent comprehensive review by Ketsdever et al.⁷) have shown that the exact mechanism governing force production is dependent on the ambient gas composition, pressure, as well as thermal gradient magnitude and direction. The force produced by thermal stresses on bodies immersed in a gas is often referred to as Knudsen thermal force to signify the underlying non-continuum flow effects with the onset at non-negligible Knudsen numbers. Since very large thermal gradients, on the order of 10⁶K/m and higher, are easily achievable at the microscale, exploiting Knudsen forces offers alternative mechanisms for actuation, sensing, and energy harvesting in nano/microsystems. This letter presents the development of an in-plane microelectromechanical device based on the Knudsen force generated on a radiometer-like microstructure array with significantly enhanced force output and tunability for gas sensing as compared with the earlier microscale implementations.

For uniformly heated bodies, the Knudsen force is repulsive in nature, having peak magnitude between the free-molecule and continuum flow regimes.^8,9 Using this effect for pressure measurement was first realized by Knudsen in 1910 and later more rigorously investigated by Lockenvitz and Wu.^10–12 A typical Knudsen gauge consists of a uniformly heated vane or cantilever held adjacent to a colder substrate, as shown in Figure 1(b). Force magnitude is measured via the change in the separation between bodies and can be used to indicate ambient pressure within the sealed vessel if the gas species is known. The Knudsen gauge concept is, however, limited to only very low pressures when the flow of gas is free molecular and deflection varies linearly with pressure. At higher pressures, the response becomes non-linear and non-monotonic due to increased influence of intermolecular collisions and thus reduced thermal stresses. It has been shown recently that combining thermal gradients between several solid bodies it is possible to amplify the Knudsen force or reverse its direction.¹³ The thermal gradient was controlled by switching the bias polarity across a thermoelectric heating element.

The potential of exploiting the Knudsen force at the microscale was first demonstrated by Passian et al. whereby a microcantilever (Figure 1(b)) was heated by a chopped laser and the corresponding out-of-plane deflection was sensed capacitively at different pressures.⁹ This concept was extended by Sista and Bhattacharya using a suspended Joule heated proof mass.¹⁴ Actuation of a cool hinged flap structure away from a heated substrate has also been demonstrated as a viable method of force generation.¹⁵ The deflection, however, was measured only qualitatively by microscopy.

The present microscale Knudsen thermal force concept is shown in Figure 1(c). Microelectromechanical In-plane Knudsen Radiometric Actuator (MIKRA) consists of two microbeam arrays, one movable and the other fixed, separated by a gap on the order of microns. The heater arm is anchored to the substrate at its base using thermal oxide, whereas the shuttle arm is suspended by a serpentine spring element, permitting displacement away from the heater due to Knudsen forces. This motion is captured via a series of comb sense capacitors at one end of the shuttle. A 100 nm thick platinum resistor is used on the heater arm, while the shuttle and underlying substrate remain at or very near the ambient temperature.

For low ambient pressures (P < 100 mTorr), the mean-free path of the gas molecules is larger than the gap separating the shuttle and heater arms and it can thus be assumed that molecules impinging at the shuttle surface followed a collisionless or ballistic trajectory. An illustration of this concept is provided in Figure 2.

The relative momentum exchanged between the shuttle and the gas on all shuttle arm surfaces is represented by the length of the arrows in Figure 2. Within the gap, high kinetic energy molecules emitted from the heater arm strike the interior shuttle surface (1) with greater momentum than those approaching the opposing exterior surface (2). The incident molecules to the upper (3) and lower (4) surfaces of the shuttle arm impart equal kinetic energy, since the substrate temperature can be assumed to be very near that of the far-field. Considering only the influence of these incident species to the shuttle arm surfaces, there exists a net momentum imbalance in the x-direction, leading to a repulsive force between the arms.

Upon collision with the shuttle arm, incident molecules accommodate to the surface, reflecting with equal probability in all directions with a kinetic energy corresponding to the wall temperature. Since shuttle arm temperatures are nearly uniform this results in zero net momentum flux in both the x and y directions among the emitted molecules. The Knudsen force production in the free-molecule regime is driven by a momentum imbalance acting over the vertical surfaces, (1) and (2), of the shuttle arm, a result which is stimulated by the presence of the high temperature heater arm. Using kinetic theory, it can be shown that force magnitude in this regime varies linearly with pressure.^10,12

As pressure is increased, ballistic trajectories of molecules are interrupted by intermolecular collisions leading to the exchange of kinetic energy before reaching the shuttle surface. The gap separating the heater and shuttle arms acts as a thermal insulator between the bodies mimicking hot/cold faces of Crookes' radiometer vanes. The thermal stresses are in the same direction as that of the radiometer vane; however, rather than inducing rotation about a spindle, the forces act to displace the shuttle away from heaters. The mechanism of force production in this high pressure regime is significantly more complicated than that of the free-molecule limit. The stresses induced by this large gradient can be qualitatively evaluated, through second-order expansion of the Boltzmann equation in the Knudsen number.¹ As the pressure is further increased, the thermal stresses vanish in the continuum regime.¹ 

To implement the geometry described in Figure 1(c), the sensor is based on a floating in-plane shuttle mass. Figure 3 shows a SEM image of the device and dimensions. The MEMS device is fabricated using a low-resistivity SOI wafer having box and device thicknesses of 4 μm and 50 μm, respectively. The shuttle is suspended on four serpentine springs, admitting large compliance along its axial dimension. Spring linearity with displacement has been verified numerically in ANSYS APDL. Extending laterally from the shuttle are a series of 16 reaction arms which serve to either transmit or measure the effects of the Knudsen force. Running adjacent to the innermost shuttle arms spaced 20 μm away are a series of immovable anchored arms. These fixed features contain a platinum heating element evaporated onto a 100 nm electrically insulating silicon nitride layer coating the upper surface. Resistive heating of these filaments allows the temperature gradient within the shuttle-heater gap to be precisely controlled. The small Biot number in the out-of-plane direction (Bi < 0.1) ensures that heater arm temperature is uniform through the thickness of the device layer.

The outermost arm pairs contain a series of comb capacitors, one for position measurement and the other for electrostatic actuation. During the measurements shown in Fig. 4, the electrostatic actuation combs remain grounded; however, they permit dynamic tuning of the sensor such that the arm separation can be adjusted to maximize sensitivity at the desired operating pressure. These actuators are also used in sense comb calibration as well as the estimation of the system's spring constant. Position sensing is carried out capacitively using a charge integrator circuit in combination with a lock-in amplifier driving a 1 Vrms carrier waveform at 10 kHz. Using this technique, the capacitive change due to shuttle deflection under the action of Knudsen forces has been shown to produce a linear change in integrator output amplitude, having a sensitivity of 182.41 μV/μm.

All experiments are carried out in small (418 cm³) stainless steel vacuum chamber backed by a dry scroll pump. Pressure is sensed using both capacitance manometer and piezo pressure transducers in parallel for pressures below and above 9.4 Torr, respectively. Pressure is regulated using a Proportional-Integral (PI) controller and solenoid valve of a mass flow controller. Air and helium are used as working gases to demonstrate sensitivity to the fluid medium. Heater current measurements are performed using shunt resistors in series with the filaments and corresponding voltages via a 4-wire technique.

Prior to measurements, the chamber is allowed to outgas to eliminate adsorbed vapors. Pressure tracking is then initiated and the chamber pressure stabilizes around the first set point. To minimize the effect of thermal spreading from the heater to the substrate and shuttle assembly, the heaters are pulsed at a 12% duty cycle of 5 s on and 55 s off. A total of 5 heating and cooling cycles are performed for each of the 17 logarithmically spaced target pressures from 75 mTorr to 113 Torr. During the bias phase, the heaters are individually regulated with a PI controller at 75, 100, or 125 mW using the measured current and voltage. After the 5 measurement cycles, the pressure increases to the next set point and stabilizes for 5 min before the next measurement cycle.

Neglecting effects of fringe fields, shuttle displacement due to Knudsen forces is inferred from the lock-in output by means of calibration. As a result of the inherently small capacitances associated with the microdevice (≈250 fF), the parasitic capacitances from the vacuum chamber, vias, bond wires, etc., are on the order of picoFarads, prohibiting accurate determination of absolute measurement. This difficulty is overcome, however, by assuming all parasitic capacitances to be static and considering only the relative capacitance between on/off heater states. Thus, a decrease in output voltage corresponds to a decrease in capacitance and an increase in separation between the shuttle and heater arms. To measure this relative change, the lock-in voltage was sampled 5 s prior to heater engagement through the 5 s heating cycle.

The shuttle spring constant is obtained using the actuation capacitors, biasing from 0 to 40 V in 10 V increments. Shuttle position is measured optically at each voltage using a microscope. Taking the slope of the lock-in signal vs. measured displacement for each of the actuation voltages allows the lock-in voltage to be directly correlated to shuttle displacement. With the position, capacitor geometry, and actuation voltages known, the nominal spring constant can be computed assuming negligible effects from fringe fields. Force magnitude is then evaluated from Hooke's law. The average shuttle displacement and Knudsen force magnitude over 5 cycles per pressure are shown in Figure 4.

As expected, the data behave with direct and inverse proportion to pressure in the free-molecule and continuum limits, respectively. A bell-shaped transition region branches these two regimes. The measurements for air show a peak in lock-in output at around 2.39 Torr corresponding to a Knudsen number of around 1.2. Here, the Knudsen number is defined in terms of dynamic viscosity, μ,

where $P∞$ is chamber pressure, g is the measured force-dependent gap separating the heater and shuttle arms, k_B is the Boltzmann constant, $T∞$ is the chamber temperature, and m is the molecular mass. Using the lock-in amplifier signal calibration, the peak signal corresponds to a shuttle deflection of 1.02, 1.48, and 1.92 μm for heater powers of 75, 100, and 125 mW, respectively. Peak magnitude increases with heater power due to greater kinetic energy of molecules. For helium force, peak increases by a factor of 2.7 at a pressure of 6.56 Torr due to the increased mean-free path of helium as well as its higher molecular velocity. Similar to air, the variation in peak magnitude is linear with temperature in the peak region and the peak shuttle deflection is 2.63, 3.54, and 4.45 μm for heater powers of 75, 100, and 125 mW, respectively.

The data in Figure 4 can be represented in terms of the non-dimensional force coefficient,⁸ C_Kn, defined by

where F_Kn is the Knudsen force, A is the total wet area, and $ΔT$ is the average temperature difference between all heater and shuttle arms. Heater and shuttle temperatures are quantified using a QFI InfraScope for 100 and 125 mW input powers. These measurements confirm the presence of a thermal gradient on the order of 10⁶K/m in the gap between heater and shuttle. Knudsen force coefficient results using air as the working fluid are provided in Figure 5 as well as the model and experimental data corresponding to an out-of-plane thin heated cantilever configuration^8,9 (see Figure 1(b)).

Experimental data for the in-plane MIKRA configuration demonstrate a consistently higher force coefficient than that of the out-of-plane deflecting cantilever, showing a peak enhancement of about 7 times. The measurement uncertainty at peak magnitude for the out-of-plane configuration was 5.9% whereas for in-plane MIKRA it is about 0.93%. This results stems from the larger force output leading to an enhanced signal-to-noise ratio and overall sensing performance. The force coefficients for MIKRA among applied powers agree well between Knudsen numbers of 0.22 and 8.8 with all data falling to within a 6% difference. Outside of this range, the small coefficient magnitude leads to larger variations near the continuum and free-molecule limits.

In summary, we have presented an alternative microelectromechanical system exploiting Knudsen thermal forces based on the in-plane motion of a suspended shuttle mass. Owing to its non-continuum nature, the Knudsen force was shown to achieve a maximum in the transitional rarefied flow regime, demonstrating a peak non-dimensional force coefficient magnitude around 7 times greater than that of previously studied out-of-plane configurations. These thermal stresses exhibit sensitivity to ambient pressure, temperature gradient, as well as composition of the working fluid. Range of applicability of the sensor could be tailored to specific application by changing the microstructure size and heater/shuttle gap to achieve transitional flow regime at target pressures. Combining these qualities with favorable scaling at reduced length scales and low power input with a possibility of waste heat energy harvesting make the Knudsen force phenomena especially attractive for sensing and actuation in nano/microsystems.

This work was supported by the Division of Chemical, Bioengineering, Environmental, and Transport Systems, NSF Grant No. 1055453.