## Static Deformation Approach

If assume that no external gravitational, magnetic, and electrostatic forces exist, cantilever deformation may be related to a gradient of the mechanical stress generated in the devices. It is well known that uniform surface stress on isotropic material tends to either increase the surface area (compressive stress) or decrease the surface area (tensile stress). If one fails to compensate for this effect with an equal stress on the opposite side of the thin plate or beam, the entire structure will be permanently bent. The different stresses (o) that act on both sides of a cantilever, which may or may not be a composite, cause permanent bending. This phenomenon is well described by Stoney's formula (Stoney 1909), in which the radius of curvature of the bent cantilever is expressed as follows:

R Et1

Where, R is the cantilever's radius of curvature, v is Poisson's ratio, E is Young's modulus for the substrate, t is the thickness of the cantilever, and Act is the differential surface stress. The differential surface stress means the difference between the surface stress on the top and bottom surfaces of the microcantilever. Initially, the surface stress of each side, Acts1 and Acts2, are in equilibrium. When they become unequal, the cantilever bends. From the radius of curvature, R, the tip displacement can be expressed as

Where l is the length and Az is tip deflection and the surface stress is expressed as a function of the deflection of the cantilever Az. This equation shows a linear relation between the cantilever bending and the differential surface stress. It can be used to detect the adsorption of an extra layer on the surface of a cantilever.

The adsorption must occur on one side of the cantilever in order to maximise the relative stress between the two sides (Act) and, as a result, to maximise the output signal of the sensor. An exemplar of this kind of measurement is Berger's work on the self-assembled monolayers of alkanethiols. They monitored the phase transition of a layer of alkanethiols that was added to the cantilever: these molecules are alkane chains with a thiol group (-SH) grafted at one end. They then used an Au-coated cantilever with one side used for the formation of self-assembled monolayers, which are covalently immobilised through the thiol group. When we assume that the surface differential stress is proportional to the number of molecules absorbed, the mass added Aama to the cantilever is expressed as follows: 