Analytical, Numerical, And Experimental Studies Of Fluid Damping In MEMS Devices
Pandey, Ashok Kumar
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Fluid damping arising from squeeze ﬁlm ﬂow of air or some inert gas trapped between an oscillating micro mechanical structure, such as a beam or a plate, and a ﬁxed substrate often dominates the other energy dissipation mechanisms in silicon based MEM devices. As a consequence, it has maximum eﬀect on the resonant response or dynamic response of the device. Unfortunately, modelling of the squeeze ﬁlm ﬂow in most MEMS devices is quite complex because of several factors unique to MEMS structures. In this thesis, we set out to study the eﬀect of these factors on squeeze ﬁlm ﬂow. First we list these factors and study each of them in the context of a particular application, using experimental measurements, extensive numerical simulations, and analytical modelling for all chosen factors. We consider ﬁve important factors. The most important factor perhaps is the eﬀect of rarefaction that is dominant when a device is vacuum packed with low to moderate vacuum, typical for MEMS packaging. The second problem is to investigate and model the eﬀect of perforations which are usually provided for eﬃcient etching of the sacriﬁcial layer during fabrication of the suspended structures. The third problem is to consider the eﬀect of non-uniform deﬂection of the structure such as those in MEMS cantilever beams and model its eﬀect on the squeeze ﬁlm. The fourth eﬀect studied is the inﬂuence of diﬀerent boundary conditions such as simple, fully open and partially closed boundaries around the vibrating structure on the characteristics of the squeeze ﬁlm ﬂow. The ﬁfth problem undertaken is to analyze the eﬀect of high operating frequencies on the squeeze ﬁlm damping. In the ﬁrst problem, the rarefaction eﬀect is studied by performing experiments under varying pressures. Depending on the ambient pressure or the size of the gap between the vibrating and the ﬁxed structure, the ﬂuid ﬂow may fall in any of the ﬂow regimes, ranging from continuum ﬂow to molecular ﬂow, and giving a wide range of dissipation. The relevant ﬂuid ﬂow characteristics are determined by the Knudsen number, which is the ratio of the mean free path of the gas molecule to the characteristic ﬂow length of the device. This number is very small for continuum ﬂow and reasonably big for molecular ﬂow. Here, we study the eﬀect of ﬂuid pressure on the squeeze ﬁlm damping by carrying out experiments on a MEMS device that consists of a double gimbaled torsional resonator. Such devices are commonly used in optical cross-connects and switches. We vary ﬂuid pressure to make the Knudsen number go through the entire range of continuum ﬂow, slip ﬂow, transition ﬂow, and molecular ﬂow. We experimentally determine the quality factor of the torsional resonator at diﬀerent air pressures ranging from 760 torr to 0.001 torr. The variation of this pressure over six orders of magnitude ensures the required rarefaction to range over all ﬂow conditions. Finally, we get the variation of the quality factor with pressure. The result indicates that the quality factor, Q, follows a power law, Q P-r, with diﬀerent values of the exponent r in diﬀerent ﬂow regimes. To numerically model the eﬀect of rarefaction, we propose the use of eﬀective viscosity in Navier-Stokes equation. This concept is validated with analytical results for a simple case. It is then compared with the experimental results presented in this thesis. The study shows that the eﬀective viscosity concept can be used eﬀectively even for the molecular regime if the air-gap to length ratio is suﬃciently small (h0/L < 0.01). However, as this ratio increases, the range of validity decreases. Next, a semianalytical approach is presented to model the rarefaction eﬀect in double-gimballed MEMS torsion mirror. In this device, the air gap thickness is 80 µm which is comparable to the lateral dimension 400 µm of the oscillating plate and thus giving the air-gap to length ratio of 0.2. As the ratio 0.2 is much greater than 0.01, the conventional Reynolds equation cannot be used to compute the squeeze ﬁlm damping. Consequently, we ﬁnd the eﬀective length of an equivalent simple mirror corresponding to the motion about the two axes of the mirror such that the Reynolds equation still holds. After ﬁnding the eﬀective length, we model the rarefaction eﬀect by incorporating eﬀective viscosity which is based on diﬀerent models including the one proposed in this paper. Then we compare the analytical solution with the experimental result and ﬁnd that the proposed model not only captures the rarefaction eﬀect in the slip, transition and molecular regimes but also couples well with the non-ﬂuid damping in the intrinsic regime. For the second problem, several analytical models exist for evaluating squeeze ﬁlm damping in rigid rectangular perforated MEMS structures. These models vary in their treatment of losses through perforations and squeezed ﬁlm, in their assumptions of compressibility, rarefaction and inertia, and their treatment of various second order corrections. We present a model that improves upon previously reported models by incorporating more accurate losses through holes proposed by Veijola and treating boundary cells and interior cells diﬀerently as proposed by Mohite et al. The proposed model is governed by a modiﬁed Reynolds equation that includes compressibility and rarefaction eﬀect. This equation is linearized and transformed to the standard two-dimensional diﬀusion equation using a simple mapping function. The analytical solution is then obtained using Green’s function. The solution thus obtained adds an additional term Γ to the damping and spring force expressions derived by Blech for compressible squeeze ﬂow through non-perforated plates. This additional term contains several parameters related to perforations and rarefaction. Setting Γ = 0, one recovers Blech’s formulas. We benchmark all the models against experimental results obtained for a typical perforated MEMS structure with geometric parameters (e.g., perforation geometry, air gap, plate thickness) that fall well within the acceptable range of parameters for these models (with the sole exception of Blech’s model that does not include perforations but is included for historical reasons). We compare the results and discuss the sources of errors. We show that the proposed model gives the best result by predicting the damping constant within 10% of the experimental value. The approximate limit of maximum frequencies under which the formulas give reasonable results is also discussed. In the third problem, we study the eﬀect of elastic modeshape during vibration on the squeeze ﬁlm ﬂow. We present an analytical model that gives the values of squeeze ﬁlm damping and spring coeﬃcients for MEMS cantilever resonators taking into account the eﬀect of ﬂexural modes of the resonator. We use the exact modeshapes of a 2D cantilever plate to solve for pressure in the squeeze ﬁlm and then derive the equivalent damping and spring coeﬃcient relations from the back force calculations. The relations thus obtained can be used for any ﬂexural mode of vibration of the resonators. We validate the analytical formulas by comparing the results with numerical simulations carried out using coupled ﬁnite element analysis in ANSYS, as well as experimentally measured values from MEMS cantilever resonators of various sizes and vibrating in diﬀerent modes. The analytically predicted values of damping are, in the worst case, within less than 10% of the values obtained experimentally or numerically. We also compare the results with previously reported analytical formulas based on approximate ﬂexural modeshapes and show that the proposed model gives much better estimates of the squeeze ﬁlm damping. From the analytical model presented here, we ﬁnd that the squeeze ﬁlm damping drops by 84% from the ﬁrst mode to the second mode in a cantilever resonator, thus improving the quality factor by a factor of six to seven. This result has signiﬁcant implications in using cantilever resonators for mass detection where a signiﬁcant increase in quality factor is obtained only by using vacuum. In the fourth and ﬁfth problem, the eﬀects of partially blocked boundary condition and high operating frequencies on squeeze ﬁlms are studied in a MEMS torsion mirror with diﬀerent boundary conditions. For the structures with narrow air-gap, Reynolds equation is used for calculating squeeze ﬁlm damping, generally with zero pressure boundary conditions on the side walls. This procedure, however, fails to give satisfactory results for structures under two important conditions: (a) for an air-gap thickness comparable to the lateral dimensions of the micro structure, and (b) for non-trivial pressure boundary conditions such as fully open boundaries on an extended substrate or partially blocked boundaries that provide side clearance to the ﬂuid ﬂow. Several formulas exist to account for simple boundary conditions. In practice, however, there are many micromechanical structures, such as torsional MEMS structures, that have non-trivial boundary conditions arising from partially blocked boundaries. The most common example is the double-gimballed MEMS torsion mirror of rectangular, circular, or hexagonal shape. Such boundaries usually have clearance parameters that can vary due to fabrication. These parameters, however, can also be used as design parameters if we understand their role on the dynamics of the structure. We take a MEMS torsion mirror as an example device that has large air-gap and partially blocked boundaries due to static frames. Next we model the same structure in ANSYS and carry out CFD (computational ﬂuid dynamics) analysis to evaluate the stiﬀness constant K, the damping constant C, as well as the quality factor Q due to the squeeze ﬁlm. We compare the computational results with experimental results and show that without taking care of the partially blocked boundaries properly in the computational model, we get unacceptably large errors. Subsequently, we use the CFD calculations to study the eﬀect of two important boundary parameters, the side clearance c, and the ﬂow length s, that specify the partial blocking. We show the sensitivity of K and C on these boundary design parameters. The results clearly show that the eﬀect of these parameters on K and C is substantial, especially when the frequency of excitation becomes close to resonant frequency of the oscillating ﬂuid and high enough for inertial and compressibility eﬀects to be signiﬁcant. Later, we present a compact model to capture the eﬀect of side boundaries on the squeeze ﬁlm damping in a simple rectangular torsional structure with two sides open and two sides closed. The analytical model matches well with the numerical results. However, the proposed analytical model is limited to low operating frequencies such that the inertial eﬀect is negligible. The emphasis of this work has been towards developing a comprehensive understanding of diﬀerent signiﬁcant factors on the squeeze ﬁlm damping in MEMS devices. We have proposed various ways of modelling these eﬀects, both numerically as well as analytically, and shown the eﬃcacy of these models by comparing their predictive results with experimental results. In particular, we think that the proposed analytical models can help MEMS device designers by providing quick estimates of damping while incorporating complex eﬀects in the squeeze ﬁlm ﬂow. The contents of the thesis may also be of interest to researchers working in the area of microﬂuidics and nanoﬂuidics.
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