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### Summary

### Summary

The book discusses homogenisation principles and mixing rules for the determination of the macroscopic dielectric and magnetic properties of different types of media. The effects of structure and anisotropy are discussed in detail, as well as mixtures involving chiral and nonlinear materials. High frequency scattering phenomena and dispersive properties are also discussed.

The book includes analysis of special phenomena that the mixing process can cause, such as the difference in character between a mixture and its constituent parts. Mixing results are applied to different materials in geophysics and biology. Reference is also made to the historical perspectives of dielectric modelling. Examples are included throughout the text.

Aimed at students with research interests in electromagnetics or materials science, the book is also useful as a textbook in universities, as a handbook of mixing principles, and as a sourcebook for composite material design.

### Author Notes

Ari Sihvola is Professor of Electromagnetics at Helsinki University of Technology He is Vice-Chairman of the Finnish National Committee of URSI (International Union of Radio Science) and served as the Secretary of the 22nd European Microwave Conference, held in August 1992, in Espoo, Finland. Ari received the degrees of Diploma Engineer in 1981, Licentiate of Technology in 1984 and Doctor of Technology in 1987, all in Electrical Engineering, from the Helsinki University of Technology, Finland. Besides working for HUT and the Academy of Finland, he was visiting engineer at the Research Laboratory of Electronics of the Massachusetts Institute of Technology in 1985-1986, and a visiting scientist at Pennsylvania State University in 1990-1991. In 1996, he was visiting scientist at Lund University, Sweden.

### Table of Contents

Preface | p. xi |

1 Introduction | p. 1 |

1.1 The philosophy of homogenisation of mixtures | p. 1 |

1.2 Historical background | p. 5 |

1.3 Literature | p. 9 |

1.4 Outline of the book | p. 11 |

References | p. 12 |

I To observe the pattern: Classical and neoclassical mixing | p. 17 |

2 Physics behind the dielectric constant | p. 19 |

2.1 Polarisation phenomena in matter | p. 19 |

2.2 Conduction and complex permittivity | p. 23 |

2.2.1 Field relations | p. 23 |

2.2.2 Time-harmonic fields | p. 25 |

2.2.3 Dispersion | p. 26 |

2.2.4 Complex resistivity | p. 28 |

2.3 Higher-order polarisation mechanisms | p. 29 |

2.3.1 Anisotropy and multipole moments | p. 29 |

2.3.2 Magnetic polarisation | p. 30 |

2.3.3 Other polarisation effects | p. 32 |

Problems | p. 35 |

References | p. 36 |

3 Classical mixing approach | p. 39 |

3.1 Average fields and Maxwell Garnett rule | p. 40 |

3.2 Polarisability of dielectric sphere | p. 41 |

3.2.1 Polarisability and dipole moment | p. 41 |

3.2.2 Consistency of the field solutions | p. 42 |

3.2.3 Dipole moment as solution for the external problem | p. 44 |

3.3 Mixture with spherical inclusions | p. 45 |

3.3.1 Clausius-Mossotti formula | p. 45 |

3.3.2 Maxwell Garnett mixing rule | p. 47 |

3.3.3 Q[subscript 2] function for mixture analysis | p. 49 |

3.4 Discussion on basic field concepts | p. 51 |

3.4.1 Macroscopic and microscopic fields | p. 52 |

3.4.2 Shape of the cavity in a crystal | p. 55 |

3.4.3 The internal dipoles | p. 56 |

3.4.4 Alternative routes to Maxwell Garnett formula | p. 57 |

Problems | p. 58 |

References | p. 59 |

4 Advanced mixing principles | p. 61 |

4.1 Multiphase mixtures | p. 61 |

4.2 Ellipsoidal inclusions | p. 63 |

4.2.1 Depolarisation factors | p. 63 |

4.2.2 Polarisability components of an ellipsoid | p. 66 |

4.2.3 Aligned orientation | p. 67 |

4.2.4 Random orientation | p. 68 |

4.2.5 Orientation distribution | p. 70 |

4.3 Inhomogeneous inclusions | p. 71 |

4.3.1 Polarisability of a layered sphere | p. 72 |

4.3.2 Continuously inhomogeneous inclusions | p. 75 |

4.3.3 Nonhomogeneous ellipsoids | p. 78 |

4.4 Lossy materials | p. 79 |

Problems | p. 82 |

References | p. 84 |

5 Anisotropic mixtures | p. 85 |

5.1 Anisotropy in dielectric materials | p. 85 |

5.2 Elementary dyadic analysis | p. 88 |

5.2.1 Notation and definitions | p. 88 |

5.2.2 Operations and invariants | p. 90 |

5.2.3 Examples | p. 93 |

5.3 Polarisability of anisotropic sphere | p. 94 |

5.3.1 Reinterpretation of scalar polarisability | p. 94 |

5.3.2 Depolarisation and the shape effect | p. 97 |

5.4 Mixtures with anisotropic inclusions | p. 101 |

5.5 Mixtures with anisotropic background medium | p. 102 |

5.5.1 Affine transformation | p. 102 |

5.5.2 Internal field and polarisability | p. 103 |

5.5.3 Homogenisation | p. 105 |

Problems | p. 107 |

References | p. 110 |

6 Chiral and bi-anisotropic mixtures | p. 113 |

6.1 Bi-anisotropic materials | p. 113 |

6.1.1 Bi-anisotropic constitutive relations | p. 114 |

6.1.2 Dissipation and reciprocity | p. 115 |

6.1.3 Renormalisation of field quantities | p. 116 |

6.2 Six-vector algebra | p. 117 |

6.3 Chiral mixtures | p. 119 |

6.3.1 Chiral and bi-isotropic materials | p. 119 |

6.3.2 Polarisability of chiral sphere | p. 121 |

6.3.3 Chiral Maxwell Garnett mixing formula | p. 122 |

6.3.4 Example: a racemic mixture | p. 123 |

6.4 Bi-anisotropic mixtures | p. 125 |

6.4.1 Polarisability of a bi-anisotropic sphere | p. 126 |

6.4.2 Bi-anisotropic mixing rules | p. 126 |

Problems | p. 127 |

References | p. 128 |

7 Nonlinear mixtures | p. 131 |

7.1 The characterisation of nonlinearity | p. 131 |

7.1.1 Examples of nonlinear mechanisms in matter | p. 131 |

7.1.2 Nonlinear susceptibilities | p. 133 |

7.1.3 Quadratic and cubic nonlinearities | p. 135 |

7.2 Mixing rules for nonlinear materials | p. 136 |

7.2.1 Polarisability components for a nonlinear sphere | p. 136 |

7.2.2 Dilute mixtures | p. 138 |

7.2.3 Towards denser mixtures | p. 139 |

7.2.4 Nonlinearity as perturbation to permittivity | p. 140 |

7.3 Characteristics of nonlinear mixtures | p. 141 |

Problems | p. 142 |

References | p. 143 |

II To transgress the pattern: Functionalistic and modernist mixing | p. 145 |

8 Difficulties and uncertainties in classical mixing | p. 147 |

8.1 Weak links in the mixing rule derivation | p. 148 |

8.1.1 Interaction between the scatterers | p. 148 |

8.1.2 Quasi-static approximation | p. 150 |

8.1.3 Correlation length | p. 151 |

8.2 Limits for the effective permittivity | p. 151 |

8.2.1 General bounds | p. 152 |

8.2.2 Hashin-Shtrikman bounds | p. 153 |

8.2.3 Higher-order bounds | p. 156 |

8.2.4 Anisotropic bounds | p. 156 |

Problems | p. 157 |

References | p. 158 |

9 Generalised mixing rules | p. 161 |

9.1 Bruggeman formula | p. 161 |

9.2 Coherent potential formula | p. 163 |

9.3 Unified mixing rule | p. 164 |

9.3.1 Spherical inclusions | p. 164 |

9.3.2 Ellipsoidal inclusions | p. 164 |

9.4 Other mixing models | p. 166 |

9.4.1 Power-law models | p. 166 |

9.4.2 Differential mixing models | p. 167 |

9.4.3 Periodical lattice models | p. 168 |

9.4.4 Random medium model | p. 169 |

9.5 Chiral and bi-anisotropic mixtures | p. 169 |

9.6 Numerical approaches for homogenisation | p. 170 |

Problems | p. 172 |

References | p. 174 |

10 Towards higher frequencies | p. 177 |

10.1 Rayleigh scattering contribution | p. 178 |

10.1.1 Rayleigh scattering of a single inclusion | p. 178 |

10.1.2 Rayleigh attenuation | p. 181 |

10.2 Mie scattering | p. 184 |

10.3 Scattering in random media | p. 188 |

10.3.1 Quasi-crystalline approximation | p. 188 |

10.3.2 Size-dependent polarisability approach | p. 189 |

Problems | p. 191 |

References | p. 192 |

11 Dispersion and time-domain analysis | p. 195 |

11.1 Constitutive relations as operators | p. 196 |

11.2 Susceptibility models | p. 198 |

11.2.1 Debye model | p. 198 |

11.2.2 Lorentz model | p. 200 |

11.2.3 Drude model | p. 201 |

11.2.4 Modified Debye model | p. 202 |

11.2.5 Other susceptibility models | p. 203 |

11.3 Mixing in time domain | p. 203 |

11.3.1 Quasi-static time-domain fields | p. 204 |

11.3.2 Deconvolution of kernels | p. 205 |

11.3.3 A mixture of two Debye materials | p. 206 |

11.4 Temporal dispersion in anisotropic and chiral materials | p. 208 |

Problems | p. 212 |

References | p. 213 |

12 Special phenomena caused by mixing | p. 215 |

12.1 Dispersion of the permittivity of mixtures | p. 215 |

12.1.1 Water and polar molecules | p. 215 |

12.1.2 Metals and Drude dispersion | p. 218 |

12.2 Polarisation enhancement | p. 219 |

12.2.1 Mossotti catastrophe | p. 220 |

12.2.2 Onsager model | p. 222 |

12.2.3 Single scattering and Frohlich modes | p. 223 |

12.3 Percolation | p. 226 |

12.3.1 Generalised mixing rule | p. 226 |

12.3.2 Effect of spatial dimension | p. 228 |

Problems | p. 232 |

References | p. 232 |

13 Applications to natural materials | p. 235 |

13.1 Water and ice | p. 235 |

13.1.1 Free and bound water | p. 235 |

13.1.2 Ice | p. 237 |

13.2 Snow | p. 239 |

13.2.1 Dry snow | p. 239 |

13.2.2 Wet snow | p. 242 |

13.3 Rocks and soil | p. 244 |

13.3.1 Porous bedrock | p. 245 |

13.3.2 Soil | p. 247 |

13.4 Rain attenuation | p. 249 |

13.5 Wood, trees, and canopies | p. 252 |

13.6 Biological tissues | p. 254 |

Problems | p. 257 |

References | p. 258 |

14 Concluding remarks | p. 261 |

Appendixes | p. 265 |

A Collection of dyadic relations | p. 265 |

B Collection of basic mixing rules | p. 267 |

Index | p. 271 |