Misfit strain energy

Let us examine the instability oi strained thin films. In Fig. 3, thin films of30 ML are coherently bonded to the hard substrates. The film phase has a misfit strain, e = 0.01, relative to the substrate phase, and the periodic length is equal to 200 a. The three interface energies are identical to each other = yiv = y = Y Both phases are elastically isotropic, but the shear modulus of the substrate is twice that of the film (p = 2p). On the left-hand side, an infinite-torque condition is imposed to the substrate-vapor and film-substrate interfaces, whereas torque terms are equal to zero on the right. In the absence of the coherency strain, these films are stable as their thickness is well over 16 ML. With a coherency strain, surface undulations induced by thermal fluctuations become growing waves. By the time of 2M, six waves are definitely seen to have established, and these numbers are in agreement with the continuum linear elasticity prediction [16]. 

The instability of the two lamellar structures may be understood in terms ofEshelby s inclusion theory [6,7]. According to the theory, a hard coherent precipitate with a dilata-tional misfit strain is elastically stable when it takes on a spherical shape in an infinite matrix. A soft coherent precipitate, on the other hand, takes on a plate-like shape as the minimum strain energy shape. Thus, the soft-hard-soft layered structure of Fig. 7 is simply a... 

Boundaries between solids transmit shear stress, particularly if they are coherent or semicoherent. Therefore, the strain energy density near boundaries changes over the course of solid state reactions. Misfit dislocation networks connected with moving boundaries also change with time. They alter the transport properties at and near the interface. Even if we neglect all this, boundaries between heterogeneous phases are sites of a discontinuous structural change, which may occur cooperatively or by individual thermally activated steps. 

Analyses of the plastic strains caused by matrix cracks, combined with calculations of the compliance change, provide a constitutive law for the material. The important parameters are the permanent strain, e0 and the unloading modulus, E. These quantities, in turn, depend on several constituent properties the sliding stress, r, the debond energy, T, and the misfit strain, il. The most important results are summarized below. 

In layered misfit structures of the type we are discussing, bonds at the layer surfaces (within and between the layers) will be strained periodically along a non-commensurate lattice direction parallel to the layers after a certain number of subcells there is a near match of the layers. Clapp has pointed out that, for a simple case, layer mismatch will cause tension in one layer type and compression in the other. The resulting strain energy may be relieved by the introduction of periodic antiphase boundary (apb) planes so that alternate contraction and extension occurs in all layers (Fig. 22) and hence cancels out (at the price of a small deformation of coordination polyhedra). 

In systems with significant Me-S lattice misfit, the 2D Meads overlayers and/or 2D Me-S surface alloys formed in the UPD range have a different structure in comparison with the 3D Me bulk phase, and contain considerable internal strain (cf. Section 3.4). Thus, the nucleation and growth kinetics in the OPD range will be strongly influenced by the internal strain energy of 2D Me UPD phases. 

From the standpoint of the quantitative model used to describe this system, we assume that both the matrix and precipitate are elastically isotropic and characterized by the same elastic moduli and that the misfit strain associated with the precipitate is purely dilatational and is given by Further, it is assumed that the interfacial energy y,n (n) is isotropic (i.e. yint(n) = yo). Recall from... 

The primary difficulty inherent in this issue is the small niunber of materials with suitable crystal structures and lattice constants. Some transition metals and ceramics, such as Ni, Cu, Fe, and cBN (Table 5, Ch. 3), are the few isostructural materials with sufficiently similar lattice constants (mismatch <5%). In addition, the extremely high surface energies of diamond (ranging from 5.3 to 9.2 J m for the principle low index planes) and the existence of interfacial misfit and strain energies between diamond films and non-diamond substrates constitute the primary obstacles in forming oriented two-dimensional diamond nuclei. Earlier attempts to grow heteroepitaxial diamond on the transition metals were not successful. The reasons may be related to the high solubility/ mobility of C in/on the metals (for example, Fe, Co, or the... 

The primary difiBculty in diamond epitaxy is the small number of materials (Ni, Cu, Fe, Co, Si, and cBN) with suitable ciystal structure and lattiee constants. The extremely high surface energies of diamond and the existence of interfacial misfit and strain energies between diamond films and non-diamond substrates constitute the primary obstacles in forming oriented... 

An epitaxial film is strained in the initial stages of film growth. The strain energy increases with film thickness and may eventually be relaxed by the introduction of misfit dislocations [14.32-14.35], see Fig. 14.7, or by formation of (110) twins in the YBCO [14.36]. The critical thickness at which the misfit dislocations form depends on the lattice mismatch and the elastic properties of the film. The misfit in epitaxial c-axis-oriented YBCO films is accommodated by the formation of twins and edge dislocations with Burgers vectors [100]ybco and [010]ybco [14.37],... 

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