«UNIVERSITY OF CALIFORNIA Santa Barbara A Micro/Nano-Fabricated Gecko-Inspired Reversible Adhesive A Dissertation submitted in partial satisfaction of ...»
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To qualify the micromechanical testing used in this work, initial tests were performed on more standard MEMS shapes. Though not immediately applicable to this project, the work described in this section was one of the first explorations of the use of nanoindentation instrumentation for the characterization of micromechanical structures and has not been published elsewhere. Tested were a fixed-fixed cantilever, a fixed-free cantilever and in-plane resonator.
Operating in load control, a desired load function was programmed, Fig. A-1, for a 5 μm 60˚ conical indenter tip to deflect the center portion of a fixed-fixed beam, Fig. A-2. The beam deflected a maximum of 525 nm, with a force of just over 11 μm. To extract a more accurate spring force the data was fit with a linear curve fit to determine a spring constant of 20.9 N/m.
Figure A- 1 A typical load-control function showing the ramp rate of the load.
Figure A- 2 Schematic of a nanoindenter tip pressing against the center portion of a fixed-fixed beam (top). A load vs. deflection curve for the indenting tip, the slope of the line represents the spring constant of the beam (bottom) This spring constant value can be compared with an analytical calculation of the beam. Using standard beam theory the applied normal force can be related to the deflection, and relating the two the spring constant is obtained.
Where P is the load, δ is the deflection, KTH is the spring constant, and L is the length. I is the moment of inertia about the bending axis described in terms of thickness, t, and width, b. Estimated values can then be taken to be t = 2μm, b = 5μm, E = 150 GPa, and L = 150 μm. This gives an estimated spring constant of 28.4 N/m. While in the nanoindenter experiment a value of 20.9 N/m was obtained.
Several factors may effect the analytical approximation. The beam may have slightly different geometries due to microfabrication error. The material properties of the polysilicon beam may vary from those used. And likely the largest contribution may be from the boundary conditions on the actual beam. The analytical model assumes fixed conditions, while the actual structure may have a less-fixed make-up due to undercutting of the supports during microfabrication (this issue will be discussed momentarily).
To make another empirical comparison a laser Doppler vibrometer was used to measure the frequency response of the beam. The idea was then that the resonance value could be compared with an analytical value to see if the error corresponds to that of the nanoindenter. Using the vibrometer the resonance was found to be 650
kHz. While the resonance of the ideal beam can be calculated by:
Calculating the volume using the length, width and thickness, the theoretical resonance is found to be 947 Hz. Comparing this with the experimental value a deviation of a factor of 1.46 is found. Whereas the square root deviation (frequency is related to the square root of the spring constant) between calculated spring constant and that of the nanoindenter experiment gives and factor of 1.17 difference.
While this experiment gives far from conclusive evidence of the accuracy of the two test techniques, it does give some initial proof of principle of the nanoindentation measurement technique.
Similar to the above experiment a test was performed on a fixed-free cantilever.
The cantilever was placed in the nanoindenter and the end of the cantilever displaced by the indenter tip. Again monitoring the force versus displacement it was possible to determine the spring constant of the beam, Fig. A-3. Unfortunately the flexible nature of the cantilever eluded the sensitivity of the indenter. While the spring constant was estimated to be 1.5 N/m this can not be stated conclusively.
Figure A- 3 Indentation schematic of a nanoindenter tip pressing against the end of a cantilever (top). Nanoindenter force versus displacement data for a typical experiment (bottom). Note the large amount of scatter in the data corresponding to the sensitivity limit of the tool.
As a fix to this problem the experiment was repeated, however this time the indenter tip was pressed into the middle of the beam instead of the end, Fig. A-4.
This effectively increased the stiffness of the beam by a factor of 8 and moved the spring constant within the sensitivity of the instrumentation, Fig. A-4. Using this data a spring constant of 7.3 N/m was determined. Dividing by a factor of eight the spring constant of the entire beam can be estimated to be 0.9 N/m.
Figure A- 4 Schematic of the indentation of the central portion of the beam (top). Nanoindenter data of the force versus displacement for the mid-beam indent (bottom), the slope of the curve yields a spring constant of 7.3 N/m for the halfway point of the beam.
To give a more accurate spring constant and resonant frequency prediction a model was constructed in ANSYS, Fig. A-5. This model predicts a spring constant of 1.3 N/m and a resonance at 68.3 kHz. Again using a laser vibrometer the resonance of the structure was measured to be 50.8 kHz. Comparing the empirical values with the finite element model values the resonance varies by a factor of 1.3, while the square root of the spring constant varies by a factor of 1.2. Thus using the nanoindenter it is possible to corroborate data taken using the laser vibrometer.
Figure A- 5 Finite element model of a cantilever made in Ansys. Note the quasi fixed boundary condition where the undercutting of the beam support has been taken into account.