Misfit terms

Although only a1 and e are used as additional descriptors so far, some others might be useful in the future. One of the most apparent is the local electronic polarizability of the molecule in the vicinity of a surface segment i, which we may represent by a local refractive index n,. Such local polarizability or local refractive index would allow for a refinement of the electrostatic misfit term, which presently only takes into account an average electronic polarizability of organic molecules. It is also likely that the local electronic polarizability is of importance to the strength of hydrogen bonds, which so far in COSMO-RS are only a function of polarity. Finally, from a physical perspective, the vdW interactions should be a function of the local polarizability as well. 

Nucleation of a new phase in the solid state is more complicated than that of nucleation in freezing. Volume difference between the new and old phases causes an elastic misfit term that increases AG. Destruction of existing grain boundaries reduces AG. An expression for the free energy change during nucleation of 3 in a matrix of a is... 

Firstly, it has been found that the estimation of all of the amplitudes of the LI spectrum cannot be made with a standard least-squares based fitting scheme for this ill-conditioned problem. One of the solutions to this problem is a numerical procedure called regularization [55]. In this method, the optimization criterion includes the misfit plus an extra term. Specifically in our implementation, the quantity to be minimized can be expressed as follows [53] ... 

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... 

Here, G denotes the shear modulus, and f(c/r) is a function of the ratio c/r in which c and r are the spheroidal semiaxes of the precipitate. For spheres, f(c/r= 1) = 1 = /max. For discs as well as for rods, /< 1. In principle, shear stress energies and energies arising from misfit dislocation networks also have to be added. They influence AG by additional energy terms. 

After an introductory discussion of such misfit structures, various terms that have previously been applied are reviewed, and degrees of incommensurability are used as the basis for a systematic nomenclature. The known structures of specific examples are then discussed graphite intercalates minerals with brudte-like layers as one component (koenenite, valleriite, tochilinites) silicates heavy metal sulphides (cylindrite, incaite, franckeite, cannizzarite, lengenbachite, lanthanum-chromium sulphide) anion-excess, fluorite-related yttrium oxy-fluorides and related compounds. 

Most crystal structures have an internal periodicity that may be described in terms of a three-dimensional lattice. Some display an obvious sub-lattice with the periodicity of a simpler motif on which the structure is based. However, it is now clear that some others consist of two, more-or-less independent sub-sets of atoms, each with its own periodicity. Cases of one-dimensional and of two-dimensional misfit between the component sublattices are recognised. 

In the layered misfit structures each layer set A and B can be described in terms of three basic translations, i.e. by its own component lattice. [The existence of the third vector is contingent upon a strict sequence in the layer stacking. If this is absent, the two three-dimensional subcells/lattices will, in the following discussion, be replaced by two two-dimensional subcells, i.e. by submeshes (nets) built only on the intralayer vectors.] In normal layered structures the unit cells of A and B are commensurate, i.e. their unit vectors are commensurable and the periodicity of the entire structure may be described in terms of a single unit cell. In contrast, we deal with those less-frequent cases in which this is not so at least one of the basic periodicities of A and of B are incommensurate. Then the component unit cell of set A has at least one intralayer unit cell parameter which is not commensurable with the corresponding parameter of set B. Such structures have two incommensurate, interpenetrating, component lattices and can be defined as composite) layered structures with two incommensurate component unit cells. Intermediate cases, in which the nodes of the two component lattices coincide at relatively large... 

The terms incommensurate and semi-commensurate are analogous to incoherent and semi-coherent for interfaces - in grain boundaries, heterophase interfaces and epitaxial layers (cf. also Nabarro - with which layered misfit structures have much in common. In extreme cases noncommensurability may arise by mutual rotation (to varying degrees) of component layers with identical component lattices... 

Compound Misfit modulation type Component layers Component sublattices Sym- metry Subcell Component intralayer lattices axial ratio bic Sym- metry Unit cells subcells in terms of Ratio of unit cell areas Ref. 

Figure 5 Contours of r.m.s misfit (%) to seismological reference model akl35 of density (red) and bulk sound velocity (green) for candidate lower-mantle compositions, parametrized in terms of Mg/(Mg -f Fe) (= Xmj) and Si/(Mg -f Fe)(= Xpv), over the entirety of the lower mantle. Shaded region at Xpv > 1 indicates free silica. Triangle denotes pyrolite. Plus signs denote minima of r.m.s. misfit. Root of lower-mantle adiabat is 2,000 K at 660 km depth.

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