Inclusions ellipsoidal

Conceptually, the self-consistent reaction field (SCRF) model is the simplest method for inclusion of environment implicitly in the semi-empirical Hamiltonian24, and has been the subject of several detailed reviews24,25,66. In SCRF calculations, the QM system of interest (solute) is placed into a cavity within a polarizable medium of dielectric constant e (Fig. 2.2). For ease of computation, the cavity is assumed to be spherical and have a radius ro, although expressions similar to those outlined below have been developed for ellipsoidal cavities67. Using ideas from classical electrostatics, we can show that the interaction potential can be expressed as a function of the charge and multipole moments of the solute. For ease... 

These problems were partially solved through the inclusion of multipole expansions in ellipsoidal cavities [23] or through the use of the polarizable continuum method... 

We take as our model of an inhomogeneous medium a two-component mixture composed of inclusions embedded in an otherwise homogeneous matrix, where e and are their respective dielectric functions. The inclusions are identical in composition but may be different in volume, shape, and orientation we shall restrict ourselves, however, to ellipsoidal inclusions. The average electric field (E) over a volume V surrounding the point x is defined as... 

Figure 19.7 Elastic strain energy function E c/a) for an incoherent ellipsoid inclusion of aspect ratio c/a.

Eshelby treated systems that are both elastically homogeneous and elastically isotropic [7]. Some results for the ellipsoidal inclusion described by Eq. 19.23 are given below. 

The elastic energy of inhomogeneous, anisotropic, ellipsoidal inclusions can be studied using Eshelby s equivalent-inclusion method. Chang and Allen studied coherent ellipsoidal inclusions in cubic crystals and determined energyminimizing shapes under a variety of conditions, including the presence of applied uniaxial stresses [11]. 

This method relies on the exact solution of the elastic problem for an inclusion of known geometry (an ellipjsoid) surrounded by an infinite matrix. The composite problem to be solved is that in which the included phases are ellipsoidal in shap>e. Selecting one as the reference ellipwoid, the effect of the remainder is approximated by a continuum surrounding the reference ellip>soid, thus reducing the problem to one for which there is an... 

Laws and McLaughlin30 solve the problem of the viscoelastic ellipsoidal inclusion in anisotropic materials and then use the self consistent method to calculate the overall viscoelastic compliances for a composite. 

The racemic alicyclic diol 10 can form two types of lattice inclusion compounds on crystallisation, ellipsoidal clathrates (Section 3.1.2) or helical tubulates (Section 3.2.1). Small guests give the former, and large guests give the latter, lattice. 

J.D. Eshelby The elastic field outside an ellipsoidal inclusion. Proc. Royal Soc. Lond. A Mathl. Phys. Sci. 252(1271) 561-569 (1959)... 

The calculation of the effect of a liquid on the dielectric properties of foams requires repre ntation of the structure as a polymeric matrix with two-layer ellipsoidal inclusions, one of the layers being a film of sorbed moisture. 

In this expression, similar to Equation (84a), the first term is the strain of the isotropic matrix given by Equation (94). The second term is the strain induced in crystallite by the matrix and is given by the Eshelby" theory for an ellipsoidal inclusion. The tensor lifg) accounts for the differences between the compliances of the inclusion and of the matrix and has the property ty = 0. To calculate the peak shift. Equation (105) is replaced in Equation (67b), which is further replaced in Equation (83). Analytical calculations can be performed only for a spherical crystalline inclusion that has a cubic symmetry. For the peak shift an expression similar to Equation (91) is obtained but with different compliances. According to Bollenrath el the compliance constants in Equation (91) must be replaced as follows ... 

As for the linear properties, numerous approaches have been proposed to predict and explain the nonlinear optical response of nanocomposite materials beyond the hypothesis leading to the simple model presented above ( 3.2.2). Especially, Eq. (27) does not hold as soon as metal concentration is large and, a fortiori, reaches the percolation threshold. Several EMT or topological methods have then been developed to account for such regimes and for different types of material morphology, using different calculation methods [38, 81, 83, 88, 96-116]. Let us mention works devoted to ellipsoidal [99, 100, 109] or cylindrical [97] inclusions, effect of a shape distribution [110, 115], core-shell particles [114, 116], layered composites [103], nonlinear inclusions in a nonlinear host medium [88], linear inclusions in a nonlinear host medium [108], percolated media and fractals [101, 104-106, 108]. Attempts to simulate in a nonlinear EMT the influence of temperature have also been reported [107, 113]. 

The elastic solution upon which many approaches to the consideration of microstructure are built is that of the so-called Eshelby inclusion in which one considers a single ellipsoidal inclusion in an otherwise unperturbed material. From the standpoint of the linear theory of elasticity, this problem is analytically tractable... 

Fig. 10.13. Schematic of an ellipsoidal inclusion within an elastic medium.

What we have learned is that the elastic energy associated with a spherical inclusion characterized by an eigenstrain e = coSij scales in a simple way with the volume of the inclusion itself. Similar analytic progress can be achieved for other simple geometries such as ellipsoidal and cube-shaped inclusions. 

One of the fundamental solutions from which much is made in the evaluation of microstructure is that of the elastic fields due to ellipsoidal inclusions, (a) In this part of the problem, revisit the problem of the elastic fields due to inclusions by deriving the expression given in eqn (10.12). Further, perform the integrations required to determine the displacements, and show that the displacements external to the inclusion are given by... 

---