Stoney’s equation

Stoney s equation bending beam method Stranski, Iwan Nicola... 

The cantilever deflection, Az, generated by a surface stress change, Act, (due to a biorecognition process, for example) depends on the microcantilevers dimensions (length L and thickness h) and its material s properties (Young s modulus E and Poisson coefficient v) following the Stoney s equation ... 

Once the material is defined, the generated cantilever deflection will be larger when L/h increases (see Stoney s equation). For that reason, a thickness of only 0.334 pm was selected. As the limit of sensitivity is determined by the signal to noise... 

The stress change can be determined directly fixim the optical-beam deflection signal. The stress change is given by Stoney s equation ... 

The in-plane stress during alternating UV fight exposure and dark exposure is calculated from Stoney s equation as follows ... 

A direct conqiarison of the bending response of the SAM-coated microcantilever for Cs and Na conqilexation was also conducted for the same concentration of each ion (10 M) (Figure 6). The results indicated that the SAM-coated microcantilever was much more selective towards Cs ions than towards and Na ions. A relationship between the cantilever displacement, h, and the molar concentration of Cs can be obtained by manipulating Stoney s equation (eq 1). Since surface stress is directly proportional to ion absorption by the microcantilever, eq 2 can be rearranged to the form shown in eq 3,... 

The Stoney s equation is widely used to estimate the deflection of a cantilever [37]. According to the Stoney s equation, the deflection of a cantilever (Az) caused by surface stress (CTsurf) is written as ... 

FIGURE 4.3.10 (a) Schematic illustration of a cantilever covered by a coating film with isotropic internal strain, (b) Dependence of cantilever deflection on the thickness of coating films with various Young s moduli from 0.1 GPa to 100 GPa. The values calculated by FEA are represented with filled squares. Black and gray dashed lines correspond to the cantilever deflection calculated by the Stoney s equation and Sader s model, respectively. /= BOO im, Wc = Wf = 100 [xm, tc = 1 xm, Ec = 170 GPa, = 0.28, Vf = 0.30, and ajurf = 0.1 N/m. Reprinted from Ref. [11], with permission from the American Institute of Physics. 

Intrinsic stress cr at a temperatme of interest in the coinse of gel-to-ceramic thin film conversion can be obtained when the radii of the cmvature of the bare (R ) and coated substrate (il2) are known as well as the biaxial elastic modulus of the substrate (Es/(1 — v )) and the thickness of the substrate (fs) and film (ff), using Stoney s equation (Hoffman, 1966) ... 

On a long, narrow, and thin substrate ( beam ), the sum of all the stresses in the film causes it bend. From the degree of bending and the mechanical properties of the substrate (and of the film if it is thick), the film stress (a) may be calculated using a variation of Stoney s equation.f Figure 11.3 shows the results of a sample calculation. 

Stoney s equation An equation that relates cantilever end deflection 6 to applied stress a. 

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 ... 

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