Mismatch strain

In other words, it should be possible to correlate the occurrence of one or the other mechanism (or some combination of both) with the mismatch strain energy between the lattices of the substrate and the new phase, and with chemical differences in electronegativity and polarization between the substituting ions. 

The process is diffusion-limited through a scale that is compact, fully dense, and crack-free. Such layers will occur only if the volume change upon oxidation is not too great. Otherwise, the growing oxide layer cannot accommodate the mismatch strain that develops and will tend to crack or buckle. 

One idea to relieve curvature mismatch strain is to introduce five- and seven-membered rings in regions of positive and negative Gaussian curvature. Numerous computations have been made on these structures. 

Because the deposition occurs at room temperature the thermal mismatch, strain, and other complications associated with other CVD techniques are not observed. 

Below we demonstrate that the surface PM and PE effects coupled with the surface (nanoparticles) or mismatch strains (thin films on substrates) lead to the appearance of built-in magnetic and electric fields. These fields generate the magnetization or polarization and hence alter the corresponding phase diagrams. 

The differences in toughness between the as-fired YAG and N-phase samples are consistent with the predicted differences in the thermal expansion mismatch strains at the whisker/matrix interfaces in the composites. The subsequent loss of toughness after heat treatment may be due to the formation of. strong interfacial bonds after the long-term thermal exposure. Additional analysis is currently underway to identify the specific mechanisms which control the composite properties. 

The mechanical properties of 20 vol.% SiC whisker reinforced Si3N4 composites which contain sintering aid additives which have widely different thermal expansion coefficients show a clear variation which depends on the sintering aid and the heat treatment after hot pressing. The results for the as-hot-pressed samples are consistent with the predicted thermal expansion mismatch strains at the whisker/matrix interface. A loss in both fracture strength and fracture toughness is observed after heat treatment, and may be due to the formation of strong interfacial bonds which inhibit crack deflection and whisker pullout. 

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

Among the most common mechanisms of relaxation of the elastic mismatch strain in epitaxial films is the formation of glide dislocations at the film substrate interface, the so-called misfit dislocations. An illustration of... 

In general, residual stress refers to the internal stress distribution present in a material system when all external boundaries of the system are free of applied traction. Virtually any thin film bonded to a substrate or any individual lamina within a multilayer material supports some state of residual stress over a size scale on the order of its thickness. The presence of residual stress implies that, if the film would be reheved of the constraint of the substrate or an individual lamina would be relieved of the constraint of its neighboring layers, it would change its in-plane dimensions and/or would become curved. If the internal distribution of mismatch strain is incompatible with a stress-free state, then some residual stress distribution will remain even under these conditions. 

As the crystallite continues to grow to larger radius R > i ia, the strain is prevented from relaxing further. In other words, as the grain becomes larger, it is subject to a mismatch strain... 

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.

Suppose that a thin film is bonded to one surface of a substrate of uniform thickness hs- It will be assumed that the substrate has the shape of a circular disk of radius R, although the principal results of this section are independent of the actual shape of the outer boundary of the substrate. A cylindrical r, 0, z—coordinate system is introduced with its origin at the center of the substrate midplane and with its z—axis perpendicular to the faces of the substrate the midplane is then at z = 0 and the film is bonded to the face at z = hs/2. The substrate is thin so that hs R, and the film is very thin in comparison to the substrate. The film has an incompatible elastic mismatch strain with respect to the substrate this strain might be due to thermal expansion effects, epitaxial mismatch, phase transformation, chemical reaction, moisture absorption or other physical effect. Whatever the origin of the strain, the goal here is to estimate the curvature of the substrate, within the range of elastic response, induced by the stress associated with this incompatible strain. For the time being, the mismatch strain is assumed to be an isotropic extension or compression in the plane of the interface, and the substrate is taken to be an isotropic elastic solid with elastic modulus Es and Poisson ratio Vs the subscript s is used to denote properties of the substrate material. The elastic shear modulus /Xg is related to the elastic modulus and Poisson ratio by /ig = Es/ 1 + t s). 

Fig. 2.1. Film and substrate separated, but with distributed force / acting on the film edge so that its strain is exactly the mismatch strain. This loading gives rise to an equi-biaxial state of stress at each material point in the film such a state of stress with magnitude a is illustrated on the right side.

Fig. 2.2. Free system with curvature due to mismatch strain represented as sum of two deformation states which are depicted here.

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