Stoney’s formula

Microcantilever deflection changes as a function of adsorbate coverage when adsorption is confined to a single side of a cantilever (or when there is differential adsorption on opposite sides of the cantilever). Since we do not know the absolute value of the initial surface stress, we can only measure its variation. A relation can be derived between cantilever bending and changes in surface stress from Stoney s formula and equations that describe cantilever bending [15]. Specifically, a relation can be derived between the radius of curvature of the cantilever beam and the differential surface stress ... 

C ahn and Hanneman 11341 and Finn and Gatos 11351 observed that thin annealed wafers of several 111-V compounds, like GaAs, bend spontaneously. This effect can be attributed to a difference in surface stress of the two faces. The III-V compounds have, due to their crystal structure, different opposite faces. If. for instance, the surface stress of the top side is smaller than that of the bottom side, the wafer bends downwards. The degree of bending is related to the difference in surface stress AT by Stoney s formula 136 ... 

Mezin, A. (2006) Coating internal stress measurement through the curvature method a geometry-based criterion delimiting the relevance of Stoney s formula. Surf. Coat. Technol., 200 (18/19), 5259-5267. 

The defection (8) of a microcantUever, which is caused by the surface stress difference of the top (receptor-coated) and the bottom surfaces, can be estimated according to Stoney s formula ... 

We can then use Stoney s formula to convert the stress into bending of the bimorph beam, similar to the bending of a substrate due to strain mismatch from a deposited film. Here the Young s modulus E, Poisson s ratio v, and the film thickness are for the substrate, and the strain oy is for the deposited film with a thickness tf. 

For a bimorph made out of films with different thicknesses and Young s moduli, we can estimate the bending of the bimorph actuator using Stoney s formula modified for a bimorph ... 

The expression for curvature in (2.7) is the famous Stoney formula relating curvature to stress in the film (Stoney 1909). Stoney s original analysis of the stress in a thin film deposited on a rectangular substrate was based on a uniaxial state of stress. Consequently, his expression for curvature did not involve use of the substrate biaxial modulus Mg. Consequently, (2.7) can be applied in situations in which mismatch derives from inelastic effects. However, the relationship (2.7) is based on Stoney s concept as outline in this section, and it has become known as the Stoney formula. It has the important property that the relationship between curvature k and membrane force / does not involve the properties of the film material. The elastic mismatch strain Cm corresponding to the stress <7 given in (2.8) is... 

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