MEMS spring calibration with polystyrene sphere size standards
Also, use a neighboring reference spring with no beads affixed (for measuring gravitational deflection of spring alone)
2. Record images under three conditions: “Horizontal” for no gravitational effect, and “Up” and “down.” We used a 45 degree prism and a reflecting piece of polished silicon as a mirror as a quick way of performing these experiments. This setup was sufficient for our purposes in this report, but had drawbacks of (a) poor image quality and (b) we were unable to shield air currents as well as in our other work, so increased noise. Below are two images, one “up” and one “down” showing how one can even visually detect the deflection of the sensor under gravity. [[Image:GC2 deflection images and diagram.PNG|center]] Because of the aforementioned increased noise from air currents, we recorded 30 images under each gravitational condition. (Note, one image has been rotated 180 degrees for comparison.) 3. For each image series, measure the phase of the reference grating relative to the fixed grating. To do this, we: a. Define a region of interest where the movable grating is. Perform line scans for each row of pixels, and average values. Repeat for ROI over fixed grating. This results in two waveforms, as shown below: [[Image:GC3 phase readings.PNG|center]] b. Cross-correlate the two waveforms, using circular boundary conditions. Using a cubic spline fit, find the absolute maximum peak in the cross-correlation, which reveals the relative phase shift between the two gratings, in fractions of a pixel. An example image is shown below: [[Image:GC4 cross correlation cubic spline fit.PNG|center]] c. Repeat (a) and (b) for each of the 30 images for a given condition, and measure the mean and standard deviation of the phase difference. The results of one image series are shown below, with a mean shift of 1.72 pixels, +/- 0.09 pixels: [[Image:GC5 spline peak versus frame number.PNG|center]] 4. Repeat for each gravitational condition and for sensor with and without beads (6 total conditions). The results are shown in the graph below: [[Image:GC6 final results calibration.PNG|center]] You can see that the sensor moves under its own weight, but significantly more under the weight of the two beads. 5. Calculate unidirectional deflection, dx = [(up deflection) – (down deflection)] / 2.
dx no beads = 263 nanometers
dx with two beads = 655 nanometers
6. Calculate unidirectional deflection caused by a single bead alone:
'''196 nm +/- 6%''' = single bead deflection = (655 nm – 263 nm) / 2 beads
(Error calculation not shown above, but derived from standard deviation of multiple images)
7. Calculate weight of a single beads. From Polysciences certificate of analysis and tech support:
Mean diameter = 30.25 +/- 0.05 microns (1.6%) [*** Diamter, not radius]
Density of polystyrene = 1.05 g / cm3
g = 9.806 m / s2
'''149 piconewton +/- 4.8%''' = average weight of single microsphere in vacuum (same in air)
'''8.''' From weight of bead and deflection by single bead, calculate spring constant: '''0.76 pN / nm +/- 8%''' = spring constant = 149 pN / 196 nm. Additional notes: We will describe this technique in a future publication (SJK Note 3/7/2010: Probably no future pub), as we expect it may have broad applicability to the calibration of both in-plane springs as well as out of plane cantilevers (as in AFM). We used a separate method, based on image pattern matching, to analyze the data, and obtained a spring constant of 0.9 pN / nm. Slightly larger but consistent with the phase grating method. We use the phase grating method because of the poor image quality and the assumption that the phase method should be more precise and less sensitive to biases from defocusing and overall image drift.
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