Koch Lab:Publications/Supplemental info for MEMS force sensor gravitational calibration

Supplemental information for "Micromachined piconewton force sensor for biophysics investigations"
Steve Koch 16:58, 8 March 2010 (EST): We developed and used a gravitational-based calibration method to calibrate the spring used in the APL report. At the time (2006), we expected to pursue the work a bit further and write up a methods paper describing it further. However, all of the co-authors on the paper left their groups at Sandia shortly thereafter, and we were unable to take further data. To at least have a small benefit, I'm going to post the details of the method here.

Information for original publication:
 * Micromachined piconewton force sensor for biophysics investigations
 * Steven J. Koch, Gayle E. Thayer, Alex D. Corwin, and Maarten P. de Boer
 * Appl. Phys. Lett. 89, 173901 (2006); doi:10.1063/1.2364118 (3 pages)

Description of method for calibrating spring constant via gravitational force on NIST-traceable size standards
(SJK 3/7/2010: Adapted from word document sent to editor and referee for APL submission)

1. Affix Polysciences “Megabead NIST Traceable Particle Size Standard 30.0 Micron” (Catalog #64170 http://www.polysciences.com ) microspheres to the sensor, using same method as for magnetic microspheres. Picture to the right shows bright field image of sensor with two 30 micron diameter polystyrene beads affixed. (On bottom left of sensor, 3 micron magnetic bead is visible as small dark dot.) 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.

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:

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:

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:

4.	Repeat for each gravitational condition and for sensor with and without beads (6 total conditions). The results are shown in the graph below:

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.