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As molecular reactions on a surface is ultimately driven by free energy reduction of the surface, the free energy reduction leads to a change in surface tension or surface stress.



FIGURE 2.2. Schematic diagram of the experimental setup used by Wu et al. [35]. A cantilever was mounted in a fluid cell which allows liquid exchange through I/O ports. A laser was reflected off the cantilever and focused onto a PSD.

While this produces no observable macroscopic change on the surface of a bulk solid, the adsorption-induced surface stresses are sufficient to bend a cantilever if the adsorption is confined to one surface of the beam. However, adsorption-induced forces should not be confused with bending due to dimensional changes such as swelling of thicker polymer films on cantilevers. The sensitivity of adsorption-induced stress sensors can be three orders of magnitude higher than those of frequency variation mass sensors (for resonance frequencies in the range of tens of kHz) [30]. Moreover, the static cantilever bending measurement is ideal for liquid-based applications where frequency-based cantilever sensors suffer from huge viscous damping.

Using Stoney's formula, the deflection at the end of a cantilever, z, can be related to the differential surface stress, a, as [20, 26]

where d and L are the cantilever beam thickness and length, respectively; E and v are the elastic modulus and the Poisson ratio of the cantilever material, respectively. Equation 2.1 shows a linear relation between cantilever bending and differential surface stress. For a silicon nitride cantilever of 200 |j.m long and 0.5 |j.m thick, with E = 8.5 x 1010 N/m2 and v = 0.27 [36], a surface stress of 0.2 mJ/m2 will result in a deflection of 1 nm at the end. Because a cantilever's deflection strongly depends on geometry, the surface stress change, which is directly related to biomolecular reactions on the cantilever surface, is a more convenient quantity of the reactions for comparison of various measurements. Changes in free energy density in biomolecular reactions are usually in the range of 1 to 50 mJ/m2, or as high as 900 mJ/m2.

The ultimate noise of a cantilever sensor is the thermal vibrational motion of the cantilever. It can be shown from statistical physics [23] that for off-resonance frequencies, thermal vibrations produce a white noise spectrum such that the root-mean-square vibra-tional noise, hn, can be expressed as

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