1 MOSCOW INSTITUTE OF ELECTRONIC TECHNOLOGY Zelenograd Igor V. Lavrov CALCULATION OF THE EFFECTIVE DIELECTRIC AND TRANSPORT PROPERTIES OF INHOMOGENEOUS TEXTURED MATERIALS

2 I. PERMITTIVITY OF TEXTURED COMPOSITES – permittivities of matrix and inclusions – volume fraction of r - type inclusions – effective permittivity of a composite sample: Fig.1. Composite with ellipsoidal inclusions (1.1) – distribution density of r -type inclusions; – average electric displacement and electric field of composite sample – average electric fields in matrix and in the inclusion with parameters – set of inclusion’s parameters – volume of a composite sample (1.2) (1.3) (1.4) (1.5)

3 Maxwell-Garnett approximation: (1.8) (1.9) (1.6) (1.7) – principal values of tensors of depolarization and permittivity of inclusion with parameters – tensor related with -inclusion and having the principal values – identity tensor; – tensor with principal values averaged on all inclusions of r –type tensors

4 Matrix system with ellipsoidal inclusions: 1. One-type inclusions of similar form : 2. Inclusions with a casual form : 3. Composite with a complex texture: Fig.2. Description of orientation of inclusion (1.10) (sample)(inclusion) – semiaxes of inclusions; – distribution density of orientations of inclusions, – laboratory system;

5 (1.12) (1.13) in system Components – generalized spherical functions, 1.One-type inclusions of similar form (1.11) or

6 2. Composite with a casual ellipsoidal form of inclusions, close to the sphere form (1.14) (1.15) (1.16) Form is a random vector with components: – distribution densities of (1.11) – consecutive averaging on orientations and forms of inclusions

7 3. Composite with a complex texture (1.17) (1.18) (1.19) – volume fraction, tensor of permittivity and distribution density of orientations of r –type inclusions Superposition of distributions: – coordinate system, related with orientations’ distribution of r -type inclusions – laboratory coordinate system; – matrices of rotations, (1.12) – tensors in laboratory system;– tensors in system

8 II. EFFECTIVE CONDUCTIVITY OF TEXTURED POLYCRYSTALS – conductivity tensor of i -th crystallite – electric field applied at the boundary of polycrystalline sample of volume – tensor of effective conductivity of polycrystalline sample : (2.1) (2.2) Ellipsoidal crystallites, effective-medium approximation: (2.3) (2.4) – average current density and electric field of polycrystalline sample – Green function of problem (2.2); – external normal to scalar potential – surface of i -th crystallite;

9 Spherical one-type crystallites, axial texture Kind of orientations’ distribution density: 1. Uniaxial crystallites : 2. Biaxial crystallites : Tensors in system of sample : Conductivity tensor of crystallite in system of principal axes : Conductivity tensor of crystallite in system of principal axes : (2.6) (2.7) (2.8) (2.5)(2.5) (sample)(crystallite)

10 1. Uniaxial crystallites, analitical decisions: 1. P oorly anisotropic crystallites : 2. Small disorder in orientations of crystallites: (2.9) (2.11b) (2.11a) 3. Weak macroscopic anisotropy of the polycrystal: (2.12) (2.13) (2.10)

11 2. Biaxial crystallites, analitical decisions : 1. Poorly anisotropic crystallites : 2. Small disorder in orientations of axes of crystallites and affinity of two principal values of conductivity tensor of crystallites: (2.14) (2.15)

12 Some results of numerical simulation Fig.3. Dependences of effective conductivity components of gallium polycrystal on disorder value for two variants of distribution of crystallites’ orientations. Light-green and dark-green curves correspond to a case of rotational symmetry to crystallographic axis c, blue and violet curves – to axis b Distribution density of angles : Fig.4. The area on the plane of parameters, in which relative distortion of analitical decision (2.14) is less then 1% in comparison with the numerical decision of system (2.3). (2.16) Polycrystal of gallium Dispersion Level curve of 1% - distortion