We describe a micromachined force sensor that is able to measure forces as small as 1 pN in both air and water. First, we measured the force field produced by an electromagnet on individual 2.8 m magnetic beads glued to the sensor. By repeating with 11 different beads, we measured a 9% standard deviation in saturation magnetization. We next demonstrated that the sensor was fully functional when immersed in physiological buffer. These results show that the force sensors can be useful for magnetic force calibration and also for measurement of biophysical forces on chip. 8 While very useful, these methods have drawbacks in some cases. In the Stokes drag calibration, the particle radius and solvent viscosity must be known accurately; the magnetic particle may also travel a large distance compared with the particle diameter, leading to complications when the field gradient is large. When using Brownian dynamics, the length of the tether must be measured or deduced, and the temperature must be well controlled. To address these complications, we have designed, fabricated, and tested a compliant surface-micromachined spring with which the lateral force field of an electromagnet on a single magnetic microparticle can be calibrated. This provides a simple force calibration method that does not depend on particle shape, length of polymer tethers, or solvent conditions, and is sensitive enough to allow characterization of a single bead. As seen in Fig. 1͑a͒ , the device consists of gratings attached to a folded-beam suspension, as described in a previous surface micromachining implementation. 9 It is fabricated by surface micromachining methods 10 using the SUMMiT V™ process, with polycrystalline silicon ͑polysilicon͒ as the structural material. The spring constant of the grating can be calculated analytically as where E = 164 GPa 11 is Young's modulus, w = 0.75 m is the linewidth as measured by high resolution scanning electron microscopy ͑SEM͒, t = 2.25 m is the spring thickness, and L o and L i are the respective lengths of the outer and inner spring beams, as indicated in Fig. 1͑b͒ . For this letter, we use springs with L i = 542 m and L o = 561 m having a calculated spring constant of 0.93 pN/ nm. To check the accuracy of the calculated spring constant we used NIST traceable mass standard microspheres ͑Catalog No. 64170, Polysciences͒ and measured the deflection due to the weight of the microspheres when the chip was turned onto its side. Using this method ͑to be described in a forthcoming publication͒, we obtained a spring constant of 0.76± 0.06 pN/ nm. This value is slightly lower than the calculated value, but is reasonable given the strong dependence of the spring constant on the linewidth and the variability in linewidth from chip to chip due to differing linewidth loss across a wafer during processing. Imaging the grating with a numerical aperture of 0.50, 50ϫ objective ͑Olympus LMPlanFl͒ provides 2.5 nm in-plane displacement resolution and therefore about 1.9 pN force sensitivity. By design, the springs have a maximum deflection of 18 m, and the deflection remained below 1 m for all of the data herein. For a 561 m spring length, a 1 m deflection is expected to be well within the linear range. Using two micromanipulators and pulled glass fibers, we affixed with vacuum grease individual beads ͑Dynal M270 streptavidin, product No. 653.05, lot No. F72000͒ to a desired location on the microelectromechanical system ͑MEMS͒ force sensor with approximately 5 m precision. 12 Figure 1͑c͒ shows a single bead affixed to a sensor and positioned about 200 m away from the magnet face. The process required about 10 min"bead and was efficient enough to allow us to individually characterize 11 beads and thus obtain a distribution of saturation moments. To increase efficiency yet minimize uncertainty due to possible linewidth variations, three force sensors fabricated in close proximity were used for these measurements. A control microsphere remained affixed to a fourth sensor throughout all experiments. Figure 2 shows force versus magnet current for 11 different beads at an axial distance of 200 m from the magnet pole piece. We assumed the saturated bead moment to be proportional to bead displacement at a maximum applied current ͑20 A, field= 660 mT, gradient= 21 T / cm, from modeling described below͒, and constructed the histogram shown in the inset by averaging nine consecutive measurements after stepping currents from 0 to 20 A. For the 11 beads we measured a standard deviation of ±9%, compared with less than 1% standard deviation for the control microsphere. The variation is most likely caused by the 3% coefficient of variation in bead diameter reported by the manufacturer. Previous researchers had measured a 72% and 41% standard deviation for a similar magnetic microsphere ͑Dy-nal M280͒. 5, 12 Our much lower standard deviation could reflect improvements in the commercial preparation, or could also reflect uncertainties that arise from the two previous techniques' sensitivity to other factors besides forcea͒ Present address:
Koch et al. REF have fabricated and tested a compliant surface-micromachined spring which is able to measure forces as small as 1 pico-Newton (pN ).
Micromachined piconewton force sensor for biophysics investigations
Applied Physics Letters
Physics