Substrate curvature

What is it that makes these weak forces strong enough to crack hydrocarbons, breaking the carbon-carbon bonds It is clear that it must be a collective effect. The (interior) surface of the zeolite imparts a cooperative effect to the adsorbate. Some conclusions can be drawn from a simple analysis of adsorption phenomena as a function of the substrate curvature. 

Mamatkulov, S.I. and KhabibuUaev, P.K. Water at hydrophobic substrates curvature, pressure, and temperature, Langmuir, 20, 4756, 2004. 

Freund, L.B., Floro, J.A., and Chason, E. (1999) Extensions of the Stoney formula for substrate curvature to configurations with thin substrates or large deformations. Appl Phys. Lett., 74 (14), 1987. 

Flat n -type (100) Si wafers doped with antimony up to 4T0 cm were used as initial substrates. Curvature radii of the Si wafers were measured to be more than... 

Figure 3.8. In situ measurements of mechanical stress in a-Si H thin films during plasma CVD from SiH4 with He or Ar as buffer gas compressive stress with He and tensile stress with Ar (left part) and substrate curvature and stress with changing buffer gas (right part)/ ...

We note here that, though the substrate curvature due to the converse giant flexoelectric effect could be observed, detection of the converse flexoelectric effect of calamitic nematics using the same geometry cannot be... 

Smeal, R.M., Tresco, P.A., 2008. The influence of substrate curvature on neurite outgrowth is cell type dependent. Exp. Neurol. 213, 281-292. 

Freund, L. B. Substrate curvature due to thin film mismatch strain in the nonhnear deformation range. J. Mechanics Phys. Solids 2000,48,1159-1174. 

Figure 4.1 The stress in an aiumina partide thick fiim uniformiy saturated with water during drying as a function of time determined using an interference technique to measure the substrate curvature. The rapid rise in stress occurs as the meniscus enters the fiim.

The thickness, density, and residual stress in sintered films can be determined by the methods described above. The X-ray diHfaction method of determining stress is generally more suitable for sintered films since it does not require the removal of the film, as do the substrate curvature methods. The surface roughness can be measured with a profilometer, whereas the adhesion of the film to the substrate can be determined by a pull test, in which a wire is bonded to the film and then pulled with the force needed to remove the film from the substrate. The amount of camber or warpage can also be determined with a profilometer. Grain and pore sizes can also be determined by the same techniques used for bulk ceramics. 

Photolithography enables complicated transducer shapes and the manufacture of many transducers together on the same substrate. Flexible photomasks will easily accomodate to the substrate curvature. 

Note that the center-to-center spacing of the islands is p = L/N. The relationship between substrate curvature and mismatch strain is readily obtained by appeal to the principle of minimum potential energy. The elastic energy stored in the beam at uniform curvature k is - Esh n L. Suppose that positive curvature corresponds to a reduction in extensional strain on the beam surface to which the... 

Fig. 1.34. A model configuration of a distribution of islands on the substrate. The islands are subject to an elastic mismatch strain with respect to the substrate, which gives rise to substrate curvature.

Detailed analyses of the effects of film thickness on substrate curvature in bimaterials date back to the early twentieth century, when interest in the use of thermostatic bimetals began to expand rapidly, as described in the historical note on thermostatic bimetals in Section 2.2.3. Timoshenko (1925) and Rich (1934) derived thermoelastic solutions for curvature and stress evolution in a bimetallic strip as a function of temperature change, for arbitrary variations in the relative thickness and elastic properties of the... 

The issue of film thickness effects on substrate curvature evolution is pursued now by recourse to the energy minimization method which was introduced in Section 2.1 for the derivation of the Stoney formula. All other features of the system introduced in that section are retained in this discussion, which follows the work of Freund et al. (1999). It is assumed that the film material carries an elastic mismatch strain in the form of an isotropic extension em (or contraction if Cm is negative) in the plane of the interface the physical origin of the mismatch strain is immaterial. The mismatch strain is spatially uniform throughout the film material. In this case, em is... 

The influence of film thickness or, more precisely, the influence of the ratio /if//is on substrate curvature, is considered here in two ways. First, the leading two terms in a series expansion of curvature k in powers of /if//is... 

Fig. 2.4. With reference to the expression (2.19) for substrate curvature for any values of hf/hs and M /Mg, the error in the Stoney formula for a system corresponding to parameter values in the region bounded by the two curves and the coordinate axes is less than 10%.

A second way to represent the influence of film thickness on substrate curvature is to establish the range of both parameters Mf/Ms and hf/hs for which the error incurred in using the Stoney formula is less than some prescribed value, say 10%. For example, the range of these parameters for which... 

The change in substrate curvature induced during film deposition or temperature excursion provides valuable insight into the evolution of mismatch stress in the thin film. As noted earlier, a particularly appealing feature of curvature measurement is that extraction of the membrane force / from substrate curvature by recourse to the Stoney formula (2.7) does not involve the material properties of the film, provided that the film is sufficiently thin compared to the substrate. Substrate curvature measurements also provide a means to assess the functional properties of thin films in photonic and microelectronic applications. For example, strain in films can modify the electronic transport characteristics of layered semiconductor systems through modification of the bandstructure of the material (Singh 1993). 

Methods to measure changes in substrate curvature during stress evolution in a layered material can be broadly classified into the following... 

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