Mismatch stress

Another consideration is the difference in thermal expansion between the matrix and the reinforcement. Composites are usually manufactured at high temperatures. On cooling any mismatch in the thermal expansion between the reinforcement and the matrix results in residual mismatch stresses in the composite. These stresses can be either beneficial or detrimental if they are tensile, they can aid debonding of the interface if they are compressive, they can retard debonding, which can then lead to bridge failure (25). 

At the same time, the compressive stress developed in GaN films grown on PSC substrates is not a great disadvantage. This stress helps to compensate the tensile mismatch stress, which is normally observed for the GaN/SiC system, and thus reduces the probability of cracking. For instance, no cracking was observed, when 1 pm thick epitaxial films were grown over the substrates containing 4-8 pm thick PSC layers. In contrast, GaN-on-SiC reference samples with similar thickness of the epitaxial films were heavily cracked. 

The characteristic size of single voids observed on different specimens ranged from 3 up to about 30 p,m-. jLarge voids were found, for example, on the specimen shown in Figs. 7c and d. The specimen exhibited also a few spontaneous spallations. If one inserts the thermal expansion mismatch stress of 2.7 GPa for cooling from 1223 K to RT into the lhs of formula (10), a value of 15 p,m is obtained for the critical radius where the scale buckles (h = 1.1 xm). This value is compatible with the observed size of the largest voids and explains therefore the appearance of a few spontaneous spallations. 

Inserting, for example, the thermal expansion mismatch stress of 2.7 GPa into formula (11), one obtains a spalling radius of 25 (xm. Typical spalling radii observed on the specimen in Fig. 7c were in the range from 10 to 25 xm. The somewhat smaller radii observed can be explained by a higher stress value due to the bending deformation, which initiates spalling. 

A serious problem with a SiC coating is spallation and crack formation due to thermal mismatch stresses. Applying a graded coating pure carbon on the inside and pure SiC coating on the outside—helps to minimize spallation [116]. 

Mitigation of CTE mismatch stresses or CTE < matching, typically 9.5-12 x 10 K for YSZ -based electrolytes and ferritic stainless-steel window frame or interconnect... 

A periodic arrangement of many epitaxially grown thin layers with lattice mismatch constitutes a strained-layer superlattice. An example of such a superlattice structure can be found in the vertical-cavity surface-emitting laser (VCSEL). As discussed by Choquette (2002) and Nurmikko and Han (2002), the control of layer thickness, elastic strain due to LAN to us mismatch, stress-driven crack formation and processing induced defects in the superlattice presents major scientific and technological challenges in the development of these devices. 

In the interpretation of observational data in the context of the type of model represented by (1.23), an issue that is commonly overlooked is the importance of surface area coverage or density of islands on the growth surface. The stress field associated with an isolated island on the substrate surface is self-equilibrating overall, the stress field has no resultant force or moment and its magnitude decays to very small values within a short distance from the island. The prospect that a dilute distribution of tiny islands on a relatively thick substrate of comparable stiffness could result in a significant curvature is remote, no matter how large the mismatch stress might be. For a distribution of islands to lead to a perceptible curvature of the substrate, the distribution must be sufficiently dense so that the fields of the islands... 

The sign of curvature is the same as the sign of /. For the case of a tensile mismatch stress, the face of the substrate bonded to the film becomes concave whereas, for a compressive mismatch stress, the face of the substrate bonded to the film becomes convex. If hi is the thickness of the film, then... 

Consider a film-substrate bilayer system of circular geometry, where the film and the substrate have the same thicknesses and biaxial moduli h /hs = 1 and Mf/Mg = 1, and the bilayer diameter, d (hg + hf). Let the mismatch strain in this case be a consequence of a temperature change from an initial, stress-free temperature To to another temperature T, and let the thermal expansion coefficients of the film and the substrate be denoted by Of and Og, respectively, (a) Determine the variation of the radial stress and circumferential stress across the thickness of the film and the substrate, (b) Find the magnitude and sign of the thermal mismatch stress as the interface is approached from the film and from the substrate. Show that the magnitude of the stress at the interface is independent of the thickness of the film or the substrate for a fixed thickness ratio. 

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). 

A 0.614/xm thick tetraethylorthosilane (TEOS) film is deposited on a relatively thick Si wafer. The biaxial stress in the TEOS film, say is estimated from wafer curvature measurements to be — 114 MPa. A 0.6 pm thick silicon nitride film is then deposited on the TEOS film. The average stress in the bilayer film (composite TEOS and silicon nitride layers) is estimated from curvature measurements to be — 190 MPa. (a) If a 0.6 pm thick silicon nitride film is then deposited on a Si wafer, estimate the mismatch stress (Tsin in this film, (b) Suggest an experimental strategy for determining the stress in each film for a two-film stack deposited on a substrate, without relying on the superposition formula given in (2.74). 

A thin film of W is deposited on a Si substrate with an unknown mismatch stress. When this film-substrate system is heated from 100 °C to 300 °C, it is estimated by curvature measurement that the film stress decreases by 120 MPa. Another thin film of W is now deposited on a GaAs substrate, again at unknown mismatch stress. When this system is heated from 100°C to 300 °C, the film stress is observed to increase by 205 MPa. The coefficients of thermal expansion of the substrate materials are si = 3.5 x 10 ... 

A thin film made of a new polymeric material is bonded to a metallic substrate. Prolonged exposure to a moist atmosphere causes the polymer to swell. The moisture intake increases the stress-free volume of the polymer by 3%. Derive an expression for the biaxial mismatch stress in the polymer due to this swelling under the assumption that the biaxial modulus Mf is not affected by the swelling. 

This provides an expression for the six independent components of mismatch stress in terms of the six independent components of mismatch strain in the global coordinate system. At this point, only three components of are known. However, the conditions (3.32), when applied to (3.37), provide three linear equations for and in terms of the known components... 

Once these linear equations are solved, and the results are substituted back into (3.37), the mismatch stress is known completely. 

It is possible to express the mismatch stress in term of mismatch strain for general anisotropy without reference to the substrate because the issue has been pursued under the same set of assumptions that underlie the derivation of the Stoney formula as outlined in Section 2.1. Recall the earlier assumptions that the film is very thin compared to the substrate and that the change in film strain associated with curvature of the substrate is small compared to the mismatch strain itself. Under these conditions, the film strain and film properties determine the stress which results in substrate curvature. The details of the resulting curvature do indeed depend on the properties of the substrate, and this issue will be taken up in Section 3.7. 

Once again, it is possible to determine mismatch stress from the three known... 

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