Methods for curvature measurement

3% max), and S (0.024% max)), and Ni-Cr—Fe alloys (typically with 18-27% Ni, 3-11% Gr and the balance, Fe) as high expansivity alloys in bimetal strips for medium to high temperature regimes. 

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 

The most common mechanical method of estimating curvature involves a stylus which scans the surface along the radial direction by making physical contact with it. The resulting record of the out-of-plane displacement w(r) versus distance r from some reference point is converted to the radius of curvature p, or curvature k, using the relation 

Since the original substrate is not generally flat, it is essential to measure the substrate radii of curvature pi and p2 before and after film deposition, respectively. Use of the Stoney formula (2.7) provides the average film stress in terms of the measured values of the radii of curvature of the substrate as 

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The coherent gradient sensor (CGS) method is a full-field interferometric technique that produces fringe patterns by laterally shearing an incident wavefront. This method, developed by Rosakis et al. (1998) for curvature measurement in film-substrate systems, is amenable for use in a variety of experimental configurations in either a reflection or a transmission arrangement. 

A system based on the CGS method offers several advantages for curvature measurements for thin films and layered solids. The measurement provides all the normal and shear components of the curvature tensor. It also provides full field information from the entire area of the substrate—film system. The measurement area could also be scaled as necessary from a few millimeters to hundreds of millimeters so that large wafers and flat panels with thin film deposits are tested. The method involves non-contact measurements which are carried out with an adjustable working distance, and performed in-situ and in real time as, for example, during thermal cycling. 

This measures the curvature about an axis perpendicular to the dispersion plane, i.e. the cylindrical curvature, and it may be necessary to rotate the wafer through 90° to get the orthogonal component. This may be related to absolute stress in the wafer with knowledge of the wafer thickness, diameter and elastic modulus. The most accurate method is to measure a number of points on a wafer and use a linear regression formula for the average curvature. 

Other methods depending directly on the fundamental equation. Direct measurement of the radius of curvature of a surface, by methods similar to those used in determining the radius of curvature of mirrors, has been applied by C. T. R. Wilson1 and C. V. Boys 2 simultaneous measurement of the pressure on both sides of the surface gives the surface tension at once by (2). No convenient instrument has been designed for rapid measurement of surface tensions, on this principle, however. 

The capillary pressure was measured by two independent methods by a compensatory and closed micromanometers and by an optical method for determination of the radius of border curvature [57,58], The data in Table 5.2 indicate that at Apo = 104 Pa the final capillary pressure is equal to the applied pressure drop in all cases studied. At Ap0 = 2104 Pa the capillary pressure reaches an equilibrium value only in foams from non-ionic surfactants. For... 

Basically, all the methods for measuring interfacial tensions described so far have in common that the Helmholtz energy for extending an interface is determined. Upon this extension, the interfacial tension should not vary, otherwise the quantity y would become ill-defined. One of the changes that might be incurred could result from strong curving of the interface. In the present chapter this issue was avoided because we have only considered macroscopic interfaces with radii of curvatures above 0(10-100 nm). Already in sec. 1.2.23c we showed that y is then still independent of curvature. 

The heat capacity for KCN(cr) has been measured with a heat conduction method by Messer and Ziegler (3), 101.6-345.8 K, and adlabatlcally by Suga et al. (4), 14.23-308.5 K. The data of Suga (4) were adopted after correction for curvature. The anomaleous region between 50-105 K has been treated as second order by Inclusion of the enthalpy In the heat capacity. The C° values above 310 K were estimated by graphical extrapolation. The value of S (298.15 K) was evaluated based on S (15 K) =... 

There are several methods for interfacial tension measurement. However, at high temperatures, the choice of the measurement technique is limited. Since most high temperature liquids are corrosive and often non-transparent to visible light, the sessile drop technique can rarely be used. However, by the use of the X-ray beam, the shape of sessile drops immersed in another liquid may be determined. This technique was used by Utigard and Toguri (1985) in the measurement of interfacial tension of aluminum in cryolite melts. On the basis of the curvature of the drop and the density difference between the metal and the salt. X-rays lead to a fuzzy outline of the drop shape and together with the sensitivity of the drop outline on the interfacial tension, this technique is limited to an accuracy of about 5-10%. 

To circumvent the drawbacks of the sin2 P-technique including the problem of a non-linearity of the modulus of elasticity, attempts have been made to apply other tests to obtain estimates of the residual stresses. The curvature measurement is probably one of the most widely used method for determining residual stress and involves measuring the bending of the coated sample in response to both quenching and thermal stresses. From the measured radius of curvature, the stress can be calculated according to the Stoney equation (Stoney, 1909) as... 

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