Infrared and Raman Spectroscopies of Clay Minerals

By Elsevier Science

Infrared and Raman Spectroscopies of Clay Minerals, Volume 8 in the Developments in Clay Science series, is an up-to-date overview of spectroscopic techniques used in the study of clay minerals. The methods include infrared spectroscopy, covering near-IR (NIR), mid-IR (MIR), far-IR (FIR) and IR emission spectroscopy (IES), as well as FT-Raman spectroscopy and Raman microscopy. This book complements the succinct introductions to these methods described in the original Handbook of Clay Science (Volumes 1, 1st Edition and 5B, 2nd Edition), offering greater depth and featuring the most important literature since the development and application of these techniques in clay science. No other book covers such a wide variety of vibrational spectroscopic techniques in a single volume for clay and soil scientists.

• Includes a systematic review of spectroscopic methods
• Covers the theory of infrared and Raman spectroscopies and instrumentation
• Features a series of chapters each covering either a particular technique or application

• Language: English
• Publisher: Elsevier Science
• Release date: Oct 27, 2017
• ISBN: 9780081003596

Book preview

Infrared and Raman Spectroscopies of Clay Minerals

First Edition

W.P. Gates

Institute for Frontier Materials, Deakin University, Victoria, Australia

J.T. Kloprogge

Department of Chemistry, College of Arts and Sciences University of the Philippines, Visayas, Miagao, Iloilo, Philippines

J. Madejová

Slovak Academy of Sciences, Institute of Inorganic Chemistry, Bratislava, Slovakia

F. Bergaya

CNRS, Interfaces, Confinement, Matériaux et Nanostructures (ICMN) Orléans, France

Table of Contents

Cover image

Title page

Copyright

Dedication

Contributors

Acknowledgements

Chapter 1: General Introduction

1.1 Origin and Content of the Book

1.2 Victor Colin Farmer (1920–2006)

1.3 Perspectives and Concluding Remarks

Chapter 2: Theoretical Aspects of Infrared and Raman Spectroscopies

Abstract

2.1 Introduction

2.2 Lattice Dynamics in the Harmonic Approximation

2.3 Probing the Vibrational Modes with IR Light

2.4 Raman Spectroscopy

2.5 Modeling of Vibrational Spectra from First Principles

Chapter 3: Modern Infrared and Raman Instrumentation and Sampling Methods

Abstract

3.1 Introduction

3.2 Instrumentation

3.3 IR Sampling Techniques

3.4 Raman Sampling Techniques

3.5 Epilogue

Chapter 4: Spectral Manipulation and Introduction to Multivariate Analysis

Abstract

4.1 Introduction

4.2 Overview of Postcollection Spectral Processing

4.3 Identification and Separation of Overlapping Vibrational Transitions

4.4 Multivariate Analysis and Chemometric Quantification

4.5 Concluding Remarks

Chapter 5: IR Spectra of Clay Minerals

Abstract

5.1 Introduction

5.2 Experimental

5.3 Characteristic Vibrations of Clay Minerals

5.4 The 1:1 Clay Minerals

5.5 The 2:1 Clay Minerals

5.6 Palygorskite, Sepiolite

5.7 Conclusions

Chapter 6: Raman Spectroscopy of Clay Minerals

Abstract

6.1 Introduction

6.2 Hydroxyl Stretching Region

6.3 Theory of the Low Wavenumber Vibrational Modes

6.4 The Vibrational Modes of the Tetrahedral and Octahedral Sheets in the Low-Wavenumber Region

6.5 Concluding Remarks

Chapter 7: Applications of NIR/MIR to Determine Site Occupancy in Smectites

Abstract

7.1 Introduction

7.2 Octahedral Structures of Smectites

7.3 Effect of Chemistry on the Presence and Position of Bands

7.4 Methods to Quantify Octahedral Occupancy from IR Spectra

7.5 Conclusions and Future Directions

Chapter 8: Application of Vibrational Spectroscopy in Clay Minerals Synthesis

Abstract

8.1 Introduction

8.2 Imogolite and Allophane

8.3 1:1 Clay Minerals

8.4 2:1 Clay Minerals

8.5 Vermiculite

8.6 Chlorite

8.7 Concluding Remarks

Chapter 9: Infrared Studies of Clay Mineral-Water Interactions

Abstract

9.1 Introduction

9.2 Molecular Probes and Reporter Groups

9.3 Water Confined in Clay Mineral Interlayer Spaces

9.4 Clay Mineral-Water Interactions as Directors of Clay Mineral-Organic Adsorption Processes

9.5 Conclusions

Chapter 10: Analysis of Organoclays and Organic Adsorption by Clay Minerals

Abstract

10.1 Organoclay

10.2 Basal Spacing of Organoclay

10.3 FTIR of Organoclay Intercalates

10.4 In Situ XRD and FTIR of Organoclay

10.5 FTIR of Organoclay With Adsorbed Organic Contaminants

10.6 Concluding Comments and a Future Outlook

Chapter 11: Raman and Infrared Spectroscopies of Intercalated Kaolinite Groups Minerals

Abstract

11.1 Introduction

11.2 Group A Molecules

11.3 Group B Molecules

11.4 Group C Molecules

11.5 Concluding Remarks

Chapter 12: Infrared and Raman Spectroscopies of Pillared Clays

Abstract

12.1 Introduction

12.2 Oligomers Salts

12.3 Al PILC

12.4 Mixed (Al–Metal)13 PILC

12.5 Ti PILC and Mixed (Ti-Metal) PILC

12.6 Fe-PILC and Mixed (Fe-Metal) PILC

12.7 Si-PILC and Derived Materials

12.8 Zr-PILC

12.9 Other-Metal PILC

12.10 Concluding Remarks

Chapter 13: NIR Contribution to The Study of Modified Clay Minerals

Abstract

13.1 Introduction

13.2 Mechano-Chemical Treatment

13.3 Layer Charge Reduction

13.4 Acid Treatment

13.5 Organo-Modified Clay Minerals

13.6 Clay-Based Heterostructures

13.7 Concluding Remarks

Chapter 14: Remote Detection of Clay Minerals

Abstract

14.1 Presence of Clay Minerals in Our Solar System

14.2 Remote Detection of Clay Minerals

14.3 Characterisation of Clay Minerals on Earth

14.4 Characterisation of Clays and Clay Minerals on Mars

14.5 Characterisation of Clays in Meteorites

14.6 Characterisation of Clay Minerals at Asteroid 1-Ceres

14.7 Characterisation of Clay Minerals in Comets

14.8 Summary of Remote Observations of Planetary Clay Minerals

Bibliography

Index

Copyright

Dedication

In loving memory of my son Andrew Olav Kloprogge

Feb. 2000–May 2016

J.T. Kloprogge

Contributors

Acknowledgements

(1) Faïza Bergaya acknowledges her co-editors for accepting this non-trivial task and for the heavy work they provided during these past 2 years. The author is particularly thankful to Jana Madejová for her enthusiastic participation as co-author of several chapters, and as co-editor of the book. The author also extends her gratitude to Helena Pálková, for help with the conception and development of the cover. She also expresses her thanks to Will Gates who acted as co-author of several chapters and also, later during its final stages, as first main editor of this book. His help in sharing this last responsibility of managing this book was particularly invaluable. A special thought should be addressed to Theo Kloprogge who, despite a personal setback (a sad event that happened while editing this book), continued to be active writing important chapters for this book. Both Faïza and Will are very grateful for his effort and valuable contributions.

(2) Will P. Gates thanks the co-editors for asking him to join the team and his co-authors for three of the chapters. As co-author of Chapter 7, he acknowledges that some of the data were collected at the Australian Synchrotron during commissioning of the far-IR beamline. The authors of this chapter acknowledge D. Appadoo and J.D. Cashion for their able assistance in collecting the data displayed in Fig. 7.9.

(3) Jana Madejová acknowledges her co-authors for three chapters of this book, and P. Komadel, for many years of close collaboration in the IR study of clays and clay minerals.

(4) Theo Klopproge extends his thanks, first of all, to Faïza Bergaya for making a long-held dream come true after his forced retirement from the Queensland University of Technology. He specially acknowledges the co-editors for their tremendous help and support in what was for him personally an extremely difficult year. Finally, Theo wants to thank, in particular, Will Gates as he took over so much of the work of the main editor in getting this book to the finish line.

(5) All four co-editors express their thanks to the external authors and co-authors of the different chapters of this volume for their useful and timely contributions which made this book possible. Last but not least, several anonymous reviewers are thanked for their precious help in improving the scientific quality of this book. The hard work of the authors and reviewers has resulted in a hopefully excellent addendum to Farmer's book.

(6) Co-author of Chapter 4, Georgios Chryssikos, thanks V. Gionis (NHRF) for many years of close collaboration in the study of clays, and G. Kacandes (Geohellas S.A.) and M. Stefanakis (S&B Industrial Minerals, now Imerys) for setting the challenge and providing the means.

(7) The authors of Chapter 14, Janice Bishop, Joseph Michalski and John Carter are grateful to R.E. Arvidson and J.J. Wray for their helpful comments that greatly improved this chapter.

Chapter 1

General Introduction

F. Bergaya; W.P. Gates; J. Madejová; J.T. Kloprogge; D. Bain

Keywords

IR; Raman; IR spectroscopy; Raman Spectroscopy; clay mineral

1.1 Origin and Content of the Book

The initiative to prepare a book on Infrared and Raman Spectroscopies of Clay Minerals was originally conceived during the XVth International Clay Conference held in Rio de Janeiro (Brazil) in 2013, by one of the co-editors (F. Bergaya, who acts also as series editor for the Developments in Clay Science). In fact, she realised that the famous book Infrared Spectra of Minerals, by V.C. Farmer (Farmer, 1974a), used as a reference for infrared (IR) studies of clay minerals throughout the world, had not been updated for 40 years. Not surprisingly, this proves that Farmer's book was, and continues to be, ‘a fundamental piece of work’. The idea for this volume was to update the available information on IR spectroscopy of the clay minerals, and quickly evolved to include data on Raman spectroscopy. Both techniques are nondestructive, are often used as complementary tools, and critically in the case of Raman, require an up-to-date assessment of the state of the science as applied to studies for the clay minerals.

As series editor, F. Bergaya invited J. Madejová who accepted without hesitation to contribute as co-editor of this volume. During initial discussions on the framework of this book, she proposed inviting also as co-editor, W.P. Gates, who immediately accepted the invitation with great pleasure. On invitation to lead the Raman spectroscopy sections, J.T. Kloprogge also accepted to join the editorial team. All co-editors discussed the final table of contents with the hope that it would cover the broad field of IR and Raman research involving clay minerals.

This book contains 14 chapters, including this chapter, which introduces its origin and a short biography of Farmer, as well as brief concluding comments of the editors. Chapter 2 introduces the theoretical basis of IR and Raman spectroscopies. Advancement in instrumentation with the most appropriate configuration and the sampling methods for IR and Raman spectroscopic study of the clay minerals, since Farmer's 1974 book, are revisited in Chapter 3. In further recognition of modern IR and Raman studies, the spectral manipulation of large datasets and application of multivariate analysis is critically presented in Chapter 4. Chapters 5 and 6 present, respectively, IR (mid-infrared and near-infrared) and Raman spectral libraries of representative clay minerals, with band assignments allowing their differentiation and identification.

The next eight chapters present critical reviews on focused topics, on the application of IR and/or Raman spectroscopy. Determination of the site occupancy in clay minerals is covered in Chapter 7. Characterisation of products in the synthesis of clay minerals is presented in Chapter 8. Studies of clay mineral-water interaction are presented in Chapter 9. Applications towards determining interlayer environments of organoclay intercalates (Chapter 10), intercalated kaolins (Chapter 11), pillared clays (Chapter 12) and modifications of smectites by different treatments (Chapter 13) are covered in detail. Finally, Chapter 14 describes the remote detection of clay minerals in the visible/NIR regions on Earth and how it is often more easily performed on Mars and other extraterrestrial entities.

As this book was conceived to be in part the continuity of Farmer's contribution to IR spectroscopy of clay minerals, it seems opportune to say at least a few words about this eminent scientist and his role in developing the science of IR spectroscopy of minerals. Colin Farmer, whom F. Bergaya met personally at each International Clay Conference from the 1970s till the 1990s, is remembered as an elegant, pleasant man, always very approachable despite his high scientific notoriety. D. Bain, Colin's long-time coworker at Macaulay Institute, kindly provided the following brief biography on Farmer.

1.2 Victor Colin Farmer (1920–2006)

Although born in Ireland, Colin Farmer was brought up in Scotland and graduated with first class honours in chemistry from the University of Glasgow in 1943. This was followed by his PhD in 1947 from the University of Aberdeen for a thesis entitled Spectroscopic investigations on the minor element content of plants and soils, the research having been carried out mainly at the Macaulay Institute for Soil Research in Aberdeen. He was then appointed to a research post at the Macaulay Institute to work on applying chemical analytical techniques to the study of plants and soils and spent his whole career at the Macaulay Institute, retiring in 1983.

In 1955 a decision was taken to purchase an infrared (IR) spectrometer for the Macaulay Institute, and Colin was given the responsibility of investigating the possible applications of IR spectroscopy to soil science. This was an inspired decision that was to prove the start of establishing IR spectroscopy as a powerful tool for the study of minerals and establish Colin as a research scientist of outstanding international repute. He was ably assisted by Jim Russell and Tony Fraser, and this team produced an amazing flow of publications of high quality. Their reputation and willingness to collaborate resulted in a steady flow of visiting researchers to the Macaulay Institute from all over the world. Working with Colin and his team was highly sought after as they established themselves as the world's leading experts on IR spectroscopy.

Colin in effect established IR spectroscopy as a powerful tool for investigating the chemistry of minerals, particularly clay minerals. He showed that it could be used for studying mineral surfaces and their reactions in addition to simply identifying the mineral phases. In relation to soil science, it had the advantage over the more commonly used techniques such as X-ray diffraction of being able to identify and characterise X-ray amorphous phases. Much pioneering work on amorphous soils constituents was carried out in Colin's lab, and much attention was paid to the tubular paracrystalline mineral imogolite. This had first been recorded in Japanese soils derived from volcanic material but Colin found that it occurred, albeit in smaller amounts, in podzolic soils in Scotland derived from other parent materials. Subsequently it was discovered in other similar soils in other countries, and Colin proposed a new theory of podzolisation based on the transport of iron and aluminium down through soil profiles as inorganic complexes as opposed to organic complexation as in the conventional theory. This new theory was never generally accepted by the soil science community but kindled much discussion and debate in the scientific literature. Colin believed in the serendipity approach to research and was fortunate to have been active when this was possible, but science itself benefited from the research he conducted as a result.

As recognition of his outstanding ability in scientific research, Colin received a special merit promotion, a rare award in agricultural research in the United Kingdom, to enable him to pursue his research at a higher grade without having to be promoted to an administrative position such as head of a department.

One of Colin's characteristics was a real passion and profound interest in scientific matters, and he was not afraid to point out, through publications or verbally at meetings, anything that he considered to be incorrect or unsubstantiated. He never did this as a means of embarrassing anyone, but only in pursuit of the scientific truth.

After retiring from his post at the Macaulay Institute in 1983, Colin was in demand as a scientific researcher and spent time as a visiting professor or research fellow in several research establishments across the world such as the University of Adelaide and the CSIRO Division of Soils in Australia; Instituto di Chimica Agraria, Portici in Italy; University of Saskatchewan in Canada and INRA, Versailles, in France. He also continued research at the Macaulay as an honorary research fellow as long as he was able and was involved in approximately 40 publications after his retirement, mostly concerned with amorphous and poorly ordered materials in soils.

Of all Colin's publications, the one which ranks the highest is the monograph published by the Mineralogical Society of Great Britain and Ireland in 1974 on "Infrared Spectra of Minerals." Colin wrote many of the chapters in this book as well as being the overall editor. For many researchers, this is the ‘bible’ of IR spectroscopy, and it is nothing less than astonishing that it is still widely used over 40 years later.

Colin's work was recognised with a number of honours and awards. In the United Kingdom he was made a Distinguished Member of the Clay Minerals Group at a special dinner in Aberdeen organised in his honour for his lifetime research into clay mineralogy and IR spectroscopy. In the United States, the Clay Minerals Society granted him the Pioneer in Clay Science Award for his research contributions that led to important new directions in clay mineral science.

1.3 Perspectives and Concluding Remarks

The chapters presented in this book clearly indicate the actually wide range of applications that IR and Raman spectroscopies bring to the study of clay minerals and clay polymer nanocomposites. While perhaps the development of vibrational theory has not advanced much since the 70's, our ability to use it to interpret and predict IR and Raman spectra of clay minerals has; largely because of the advances in computing over the intervening decades, the potential of this technique is not exhausted yet. Certainly improvements in instrumentation, computing and methods for presenting samples to IR or Raman instrumentation have changed markedly since Farmer's book. These advances have resulted in the availability of a rather complete set of instruments for probing nearly any sample in any condition, anywhere.

Clay scientists have not been slow to employ these new instruments and methods and have adapted them into their research as they have become available. Detailed interpretation of the IR and Raman data on clay minerals obtained from the spectra taken in the traditional mid-IR (MIR) region, but also in the less common near- and far-IR (NIR and FIR) regions, supported by advanced theoretical studies, applications of different accessories for specific purposes, or real-time IR measurements are allowing us to continue to probe unanswered questions. Significant progress in instrumentation in connection with the fiberoptic probes for portable Fourier transform infrared (FTIR) spectrometers operating in both NIR and MIR regions, allowing complex characterisation of soils and clay deposits, has occurred. Utilisation of synchrotron radiation for far-infrared (FIR) region or for IR microscopy has significantly improved the detection limit of the method. Combination of FTIR and Raman spectrometers with a variety of other analytical instruments will provide more complex information on the clay minerals alone and in materials with other substances. Enormous progress in orbiter-carried visible and near-infrared (VIS-NIR) spectrometers performing global mapping of the surface of Mars and elsewhere has brought, nearly daily, marvellous details on the occurrence of clay minerals on the red planet. It follows clearly that vibrational spectroscopy will remain one of the most important experimental methods for clay minerals investigation over the next 40 years.

The editors hope that this volume fulfils its promise to update and expand on Farmer's landmark book. The editors recognise that this is a hard task to achieve, given the almost universal uptake of Farmer's book by clay scientists worldwide, but are hopeful that it will be as useful for clay researchers and teachers.

Bibliography☆

Farmer, V.C., 1974a. The Infrared Spectra of Minerals. Mineralogical Society Monograph 4, London, 539 pp.

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☆ To view the full reference list for the book, click here

Chapter 2

Theoretical Aspects of Infrared and Raman Spectroscopies

E. Balan; J.T. Kloprogge

Abstract

This chapter introduces the physical basis of infrared and Raman spectroscopies. Starting from standard harmonic lattice dynamics, a special attention is given to the relationship between the atomic-scale vibrational properties of the clay mineral and the spectroscopic quantities actually measured. The expression of the low-frequency dielectric permittivity tensor of the clay mineral is recalled, and the specifics related to measurements on powder samples are underlined. Standard approaches accounting for the composite nature of the measured samples are discussed. Finally, quantum mechanical tools that can be used to compute the vibrational spectra from clay mineral structures are briefly exposed.

Keywords

IR; Raman; IR spectroscopy; Raman spectroscopy; Clay mineral; Lattice dynamics; Vibrational modes; Harmonic approximation; Anharmonic properties; Quantum-mechanical description; Powder spectra; Effective medium theory

2.1 Introduction

Infrared and nonresonant Raman spectroscopies are based on the excitation of specific vibrational modes of the system under study. The frequency, intensity and width of the measured signals are highly sensitive to the local structure, composition and microstructure of the investigated samples, leading to a wide range of applications in mineralogy (e.g. Farmer, 1974c,d) and in clay science (e.g. Farmer, 1974c,d, 1979; Russell and Fraser, 1994; Madejová and Komadel, 2001; Madejová, 2003; Kloprogge, 2005). In this chapter, the physical relations between the sample structure and the measured spectrum are exposed with a special attention to the specificity of clay minerals. When possible, the presentation will be based on classical models, which provide a reasonable understanding of the involved phenomena and parameters. The relation with their quantum mechanical counterparts will be discussed when needed.

The chapter begins with an introduction to lattice dynamics, aiming at drawing direct relationships between the crystal structure of the clay minerals and their vibrational modes. Infrared (IR) and Raman spectroscopies will then be presented successively. The interaction of a vibrational mode with an oscillating electric field in the IR frequency range will be presented using the phenomenological 1D Drude-Lorentz model, illustrating how the low-frequency dielectric tensor of a crystal can be built from atomic-scale parameters. The specificities of IR spectroscopy applied to powder materials will then be discussed in the light of popular effective medium models. The principles of the nonresonant Raman spectroscopy will be exposed using a simplified picture. The final paragraph deals with some aspects of the atomic-scale modeling approaches applied to the prediction of vibrational spectra, a field that has greatly expanded during the last 30 years.

2.2 Lattice Dynamics in the Harmonic Approximation

2.2.1 Classical Model of Crystal Vibrations

The main features of the vibrational properties of crystals within the framework of classical mechanics are summarised in this section. The objective is to quantitatively describe the collective oscillatory motion of atoms in the vicinity of their equilibrium position. Various approaches can be used to expose this subject. For example, 1D models consisting in infinite chains of atoms can be used to capture the main physical ingredients determining the vibrational properties of an extended system and to display the specific features related to its translational invariance (Hadni, 1974; Bruesch, 1982). These models have the merit they can be solved in an analytic fashion, leading to explicit relations between vibrational frequencies and microscopic parameters such as atomic mass and force constant of bonds. In the present chapter, the three-dimensional crystal structure will be explicitly considered. Although mathematically less straightforward relations are obtained, this facilitates the application of theory to real systems. Extensive presentations of the lattice dynamic theory can be found in, for example, Born and Huang (1954), Bruesch (1982) and Böttger (1983).

(Fig. 2.1):

Fig. 2.1 Equilibrium position vector and displacement vector describing the displacement of atom k in cell a of a periodic crystal structure.

   (2.1)

is a lattice vector corresponding to the cell of integer index ais defined by the reduced atomic coordinates of atom k is its displacement from equilibrium position.

Within the Born-Oppenheimer approximation, electron and nuclear dynamics are decoupled. The force experienced by an atom k in cell a along the Cartesian direction α can be obtained from a potential energy term depending on atomic positions:

   (2.2)

is the potential energy surface of the system. By definition, the potential energy is minimal and the forces on atoms are nil for the equilibrium geometry.

The potential energy surface contains contributions from the electronic energy and nuclear–nuclear repulsion energy. Alternatively, assuming that the system under study can be described using an ionic bond formalism, the potential energy surface can be described as a balance between short-range dominantly repulsive interactions (incorporating smaller van der Waals attractive contributions) and long-range Coulombic interactions between atoms. An explicit expression of the potential energy surface as a function of atomic positions in the vicinity of the equilibrium geometry can be obtained using a Taylor expansion:

  

(2.3)

where the partial derivatives are taken at the equilibrium positions and the minimum of the potential energy is taken equal to zero. The indexes a and b span over the crystal cells, k and k′ over the atoms in a single cell, and α and β over the three Cartesian coordinates. The potential energy of the system being minimal for the equilibrium structure, the first-order terms of the development are zero and the second-order terms are positive:

  

(2.4)

The second-order derivatives thus define the coefficients of a symmetric matrix, referred to as the force-constant matrix:

   (2.5)

In the harmonic approximation, the Taylor expansion is restricted to second-order terms, leading to the following expression of the potential energy of the system:

  

(2.6)

This approximation holds for atomic displacements small enough such that third and higher orders can be neglected. The harmonic restoring force experienced by atom k in cell a along direction α when it is displaced from its equilibrium position is thus:

   (2.7)

From this expression, the application of Newton's second law leads to a system of coupled differential equations that relate the acceleration of atom k in cell a along direction α to the displacements of atoms k′ in cell b along direction β:

  

(2.8)

Adopting a complex notation for oscillatory quantities (i.e. the corresponding oscillating real quantities are obtained by adding to the complex quantity its conjugate value and dividing the sum by two), the discrete oscillatory solutions of the system given in Eq. (2.8) that are consistent with the translational invariance of the crystal are obtained under the form:

   (2.9)

where ηkα is the coordinate of the vector (referred to as the polarisation vector) describing the displacement of atom k in the cell origin along the direction αthe position of the crystal lattice node a, and ω is the angular frequency describing the time periodicity. These solutions have the form of plane waves defined over the nodes of the crystal lattice. It can be shown that the solutions of , the oscillatory motion of atoms around their equilibrium positions is thus fully described by a system of 3N equations, where N larger than π because of the periodicity of the function eixvectors to the first Brillouin zone of the crystal.

To describe the collective motion of atoms, Eq. (2.10) is then obtained by substituting Eq. (2.9) into Eq. (2.8) and by multiplying both sides by eiωt :

  

(2.10)

Because of the translational invariance, the force-constant coefficients only depend on the relative positions of cells a and b:

   (2.11)

As the sum is performed over all crystal cells, the index a can be arbitrarily chosen as corresponding to the origin cell; that is, a=0, and Eq. (2.10) can be rewritten as:

   (2.12)

is the discrete Fourier transform of the force constants:

  

(2.13)

The system of 3N equations expressed in Eq. (2.12) can be rewritten in a more tractable form using parameters obtained by multiplying the displacement of each atom k with the square root of its mass:

   (2.14)

Eq. (2.12) is then rewritten as:

   (2.15)

:

   (2.16)

It can be shown that the 3N×3N is Hermitian. Thus the nontrivial solutions of the system of Eq. , where the index m spans from 1 to 3N, are real, and the mechanical stability of the structure requires them to be positive. This leads to 3N . The eigenvectors are orthogonal and can be normalised to form an orthonormal and complete basis set. Because of the orthogonality of eigenvectors, each vibrational mode can be considered as an independent harmonic oscillator (Fig. 2.2). The corresponding atomic displacement patterns can be obtained from the corresponding 3N , where λ ), the harmonic potential energy curve follows a parabolic law as a function of λ:

Fig. 2.2 Left: Mechanical analog of a vibrational mode made of a mobile mass linked by a spring to a fixed point. Right: Potential energy curve for a displacement of the mass. The anharmonic potential corresponds to the grey curve, whereas the harmonic one is in black and displays a parabolic shape. The corresponding vibrational quantum levels are indicated by the horizontal lines and the corresponding quantum number n indicated. The harmonic levels are equidistant whereas the anharmonic ones are progressively downshifted. Note that the position of the fundamental state does not coincide with the minimum of the potential well, contributing to the zero-point energy.

   (2.17)

.

Accordingly, each vibration mode behaves as an isolated harmonic oscillator and can be described using its simple mechanical system counterpart, constituted by a mobile effective mass and a spring (Fig. 2.2).

, and defined as the inverse of the corresponding electromagnetic wavelength λ (λ=c/ν, with c the speed of light). The wavelength is usually expressed in cm and the corresponding cm−1 unit leads to numbers typically between 10 and 4000 cm−1.

2.2.2 Categorisation and Symmetry of Vibrational Modes

vectors. The curves reporting the frequency as a function of the q value along selected directions are referred to as dispersion curves. For q). For three vibrational modes, referred to as acoustic modes, the vibrational frequency tends to zero for vanishingly small values of q. The corresponding displacement patterns reduce to rigid crystal translation along the three Cartesian directions.

The other vibrational modes display finite frequencies at q. This oscillating field affects the ionic displacements (and in turn is affected by them) and modifies the corresponding vibrational frequency (Born and Huang, 1954; Baroni et al., 2001). Its occurrence depends on the direction of the q . The macroscopic electric field is zero when the electrostatic polarisation vector and phonon wave vectors are perpendicular, which corresponds to transverse optic (TO) modes. In a cubic crystal, the macroscopic electric field is maximal when the polarisation vector and phonon wave vector are parallel, leading to the higher vibrational frequency of the longitudinal optic (LO) modes. The distinction between transverse and longitudinal modes still applies in crystals with lower symmetry.

Beside the lattice translational invariance, crystal structures are invariant by the application of the finite number of operations belonging to their space group. The potential energy surface describing the motion of atoms in the vicinity of their equilibrium positions must also remain unchanged under these symmetry operations. This property imposes mathematical relations between the coefficients of the dynamical matrix that in turn impose specific symmetry properties to the related eigenvectors. Accordingly, the vibrational modes can be categorised by their transformation properties under the crystal symmetry operations. This procedure is efficiently achieved using the principles of group theory (e.g. Farmer and Lazarev, 1974). In highly symmetric structures, the relations imposed by symmetry constraints limit the number of independent force constants and facilitate their determination from experimental data.

The vibrational patterns of solids displaying molecular ions with strong bonds (e.g. silicates, hydroxyls) can also be conveniently described by relating them to the vibrational modes of the isolated molecule (e.g. ≈3600 cm−1) vibrations. They are useful probes of structural modifications of crystals, including phase transitions and varying composition of solid solutions (Salje and Bismayer, 1997; Salje et al., 2000).

In hydrous minerals, OH stretching modes are usually observed at a significantly higher frequency than the other modes because of the small mass of the hydrogen atom (Eq. (2.17)). In this case, the OH stretching frequency is sensitive to the molecular environment of OH groups. It depends on the H-bond strength (e.g. Ryskin, 1974; Libowitzky, 1999; Johnston et al., 2008; Welch et al., 2012) but is also sensitive to nearby cationic substitutions (e.g. Farmer, 1974b; Besson and Drits, 1997a, 1997b; Sainz-Diaz et al., 2000; Martínez-Alonso et al., 2002; Botella et al., 2004; Petit et al., 1999b, 2004a). Note that the vibrational modes of the crystal can involve coupling between the vibrational motion of nonequivalent molecular groups (Fig. 2.3). For example, the OH stretching modes of the kaolinite polymorphs (kaolinite, dickite, nacrite) display significant variations that can be ascribed to a different coupling scheme between the nonequivalent OH groups of their structure (Farmer, 1974b; Balan et al., 2001, 2005).

Fig. 2.3 Infrared absorption spectrum of OH stretching modes in kaolinite (KGa-1 sample, upper curve). The corresponding wavenumber is indicated above each band. The theoretical spectrum computed for small platy particles is reported below. Note the shift of the absorption bands with respect to the vibrational mode frequencies of bulk kaolinite (vertical bars, dotted bar: LO frequency), due to long-range electrostatic interactions in the particle. The vectors describing the atomic displacements of each mode are reported on the right (large spheres: oxygen atoms, small black spheres: H atoms, central sphere: aluminum, only a small cluster belonging to the kaolinite structure is represented). Note the coupled nature of the inner-surface OH stretching mode (involving H2, H3, H4 atoms) in kaolinite (From Balan et al. (2014)).

2.2.3 Relation to the Quantum Mechanical Description of Vibrational Properties

Although they have been obtained using a classical mechanics framework, the vibrational properties described can be related to their quantum mechanical counterparts. The energy carried by each vibrational mode is quantised (Fig. 2.2). The quantified energy of a harmonic oscillator, characterised by an angular frequency νm is

   (2.18)

where n is a positive integer and h the Planck constant. The number n is the quantum number indexing the discrete excited states of the harmonic oscillator. It can be equivalently understood as an occupation number of a vibration state by quasiparticles referred to as phonons (Bruesch, 1982). At thermal equilibrium, the mean occupation number is described by the Bose-Einstein distribution:

   (2.19)

does not coincide with the minimum of the potential energy surface and depends on atomic masses and force constants. This zero-point energy can significantly contribute to the low-temperature thermodynamic properties of the system. The corresponding ground-state nuclear wave function describes the quantum delocalisation of atoms at very low temperature.

2.2.4 Anharmonic Vibrational Properties

These developments have been obtained using the harmonic approximation, that is, by restricting the Taylor expansion of the potential energy surface to the second-order terms (Eq. 2.3). However, important properties also depend on higher-order terms of the expansion, among which are the thermal expansion of crystals and the finite lifetime of vibrational excitations (e.g. Bruesch, 1982; Menéndez and Cardona, 1984). These properties are referred to as anharmonic and are, in most insulating materials, efficiently described using the third- (cubic) and fourth- (quartic) order terms in the Taylor expansion of the potential energy.

The effect of anharmonic terms can be envisioned by considering the basis formed by the eigenvectors of the harmonic dynamical matrix. Because of the anharmonic terms, the system dynamics is no longer described by a perfectly diagonal matrix in this basis. The finite out-of-diagonal terms describe the anharmonic interaction between different phonons which contribute to the temperature dependence of spectroscopic parameters, including the position and width of vibrational signals (e.g. Lazzeri et al., 2003; Zhang et al., 2007). The temperature dependence of IR and Raman spectra can bring rich information on their vibrational properties. For example, the low-temperature shift and narrowing of OH stretching bands have been used to unravel OH signatures ascribed to specific stacking schemes in disordered kaolinite group minerals (Prost et al., 1989; Johnston et al., 2008; Balan et al., 2010, 2014).

The diagonal values also differ from the harmonic ones contributing to modifications of the vibrational frequencies. In many cases, however, the anharmonic features only correspond to minor variations of the harmonic scheme. An important exception concerns the OH stretching modes. Because of the small H mass, even the vibrational ground state is far above the minimum of the potential energy (Eq. 2.17), and the harmonic approximation appears a very crude one: The anharmonic stretching frequencies of OH groups are downshifted by about 200 cm−1 with respect to their harmonic counterparts. A quantitative assessment of the frequency shift can be obtained by using models based on a Morse potential description of the OH bond (Pascale et al., 2004; Tosoni et al., 2005) or by applying perturbation theory to the harmonic oscillator (Balan et al., 2007).

The significant anharmonicity of the OH stretching potential also leads to the detection of specific multiphonon bands in the near-infrared (NIR) range. The overtone bands correspond to absorptions from the ground vibrational state to the second or the third excited state of the OH stretching oscillator (Fig. 2.2). Because of the anharmonicity of the potential, these vibrational states are no longer equally spaced, and the position of the corresponding bands in the NIR range provides quantitative information on the anharmonicity parameters. In addition, bands corresponding to the combination of different vibrational modes can be observed (e.g. Petit et al., 2004a,b; Petit, 2005). The combination between OH stretching (ν(OH)~3600 cm−1) and bending (δ(OH)~1600 cm−1) vibrations in the water molecule leads to a characteristic band at ca. 5200 cm−1, whereas combination bands involving the stretching of structural OH groups and libration modes are observed at ca. 4500 cm−1. This makes it possible to make a better distinction between structural OH groups and molecular water, whose stretching spectra usually overlap in the mid-infrared (MIR) range. Applications of NIR spectroscopy range to the remote sensing of planetary surfaces (e.g. Bishop et al., 2008c) and ore prospection using portable spectrometers (e.g. Marsh and McKeon, 1983).

2.3 Probing the Vibrational Modes with IR Light

2.3.1 Drude-Lorentz Model Applied to IR Spectroscopy

The interaction between an electric field oscillating at IR frequencies and a crystal can be described by the induced macroscopic polarisation of the material due to the displacement of the charged particles from their equilibrium position. In turn, the polarisation can be decomposed in an electronic contribution at fixed ionic positions and an ionic contribution due to ion displacements. In the range of IR frequencies, far from the frequencies typical of electronic excitations, the electronic contribution can be considered to be frequency independent. In contrast, the ionic contribution in a polar material exhibits a strong dependence on the IR light frequency. This dependence can be understood within a classic mechanical framework, by considering the interaction of a damped harmonic oscillator of fundamental angular frequency, ω0, with an oscillating electric field, E. The oscillator is characterised by a mobile material point of effective mass, Meff, its electrostatic charge, Zeff. In this model, the natural angular frequency corresponds to the transverse frequency of the vibrational mode. In case of a system that can be described in terms of individual localised oscillators (e.g. OH groups), the model frequency is not that of the free oscillator but takes into account the long-range electrostatic interactions at q=0. A phenomenological damping parameter, conveniently expressed as Meff γ, is also introduced to account for the anharmonic interaction of the oscillator with the other vibrational states of the crystal. The damping parameter γ is usually much smaller than transverse frequencies, typically few cm−1.

The second-order linear differential equation describing the dynamical properties of this system results from Newton's second law and Hooke's law:

  

(2.20)

where u , a general solution of Eq. (2.17) will be of the form:

   (2.21)

where u0 is a complex displacement amplitude, which can account for a potential phase lag between the excitation and the response of the system.

Eq. (2.20) leads to the following expression of the displacement amplitude:

   (2.22)

This expression can be split into real and imaginary parts:

  

(2.23)

As expected for a driven oscillator, the system displays a resonance when the exciting frequency coincides with the eigen frequency of the oscillator. Such resonances typically occur in the far-IR and MIR range. Because of the damping term, the displacement amplitude does not diverge, but a π/2 phase lag occurs between the excitation and the displacement (Eq. 2.23).

To relate the mechanical resonance to an absorption process, it is relevant to compute the time-averaged electromagnetic power dissipated in the system. This quantity is obtained by considering the time average of the work done by the electric field on the moving charge:

   (2.24)

Using the real expression of the electric field and mass displacement (Eq. 2.24) leads to:

   (2.25)

For a weak damping, combining Eqs. (2.23) and (2.25) shows that for frequencies close to the resonance one, the electromagnetic absorption will have a nearly Lorentzian shape centered on the fundamental frequency of the oscillator. This explains why spectroscopic line shapes are most often fitted using symmetric Lorentzian functions. Voigt functions can be used to model inhomogeneously broadened bands, that is, related to a distribution of vibrational frequencies (e.g. Brauns and Meier, 2009). A different approach based on spectral autocorrelation has also been developed to extract width parameters from absorption bands displaying a more complex line profile (Salje et al., 2000).

From this model, it is also apparent that the absorption is closely connected to the polarisation induced by the ionic displacements. A straightforward consequence of this observation is that when the polarisation is nil for symmetry reasons, for example, for atomic displacements symmetric with respect to an inversion center, no absorption occurs and the mode is not IR active.

2.3.2 Low-Frequency Dielectric Permittivity Tensor of a Crystal and Born Effective Charge Tensors

The response of a crystal to an applied oscillating electric field can be modeled by considering that the crystal consists in a collection of damped oscillators characterised by their fundamental frequency, effective charge Zeff,m, and effective mass Mm. Each oscillator corresponds to an IR active mode of the crystal. The total polarisation is obtained by adding the ionic and electronic contributions, here considered as isotropic:

  

(2.26)

where Ω is the electronic electric susceptibility, considered as frequency independent in the IR range, and χ the complex electric susceptibility of the crystal. Eq. (2.26) defines the frequency-dependent susceptibility that is related to the complex and dimensionless dielectric permittivity ɛ, via:

   (2.27)

where the frequency dependence is implicitly assumed.

A more realistic expression of the dielectric permittivity should, however, take into account the tridimensional nature of the crystal structure, leading to the dielectric permittivity tensor (Gonze and Lee, 1997):

  

(2.28)

where α and β vary over the Cartesian coordinates, and Sm,αβ, γm and ωm are the oscillator strength, damping coefficient and transverse angular frequency of mode m, respectively. The mode oscillator strength is defined by:

  

(2.29)

where Zk,ββ′ is the Born effective charge tensors of atom k and ηm,kα is the displacement of atom k along the direction α for the mode m. The atomic displacement patterns ηm,kα verify the following condition due to the orthonormality of the eigenvectors of the dynamical matrix:

   (2.30)

where Mk is the mass of atom k, and δnm=1 if n=m, δnm=0 if n≠m.

The macroscopic dielectric properties of a crystal, here described by its dielectric permittivity tensor, can thus be obtained from microscopic parameters, such as electrostatic charges, atomic masses and force constants. At this stage, the interaction of IR light with the system, including propagation, absorption and scattering, can be determined knowing the complex dielectric permittivity tensor of the material and using Maxwell's laws of electromagnetism. In Eq. (2.29), the expression of the charges differs from the simplified static and isotropic effective charge used to characterise the coupling of the mechanical oscillator with the oscillating electric field in Eq. (2.20). The static ionic charges already differ from the formal ones because of the partially covalent character of chemical bonds in most clay minerals. In addition, the anisotropic and dynamical nature of the charges also have to be considered for a more realistic description of the low-frequency dielectric permittivity of crystals. The Born effective charge tensors (Born and Huang, 1954; Pasquarello and Car, 1997; Gonze and Lee, 1997) are defined at the centre of the Brillouin zone (q=0) and describe the linear relation between the polarisation per unit cell along the direction β, and the displacement of the atom k along the direction α:

   (2.31)

where e is the electron charge. Equivalently, they describe the linear relation between the force acting on the atom k in the direction β and a homogeneous (i.e. constant over the crystal cell) electric field along the direction α. The Born effective charges are dynamical quantities because they incorporate the changes of the electronic structure induced by the atomic displacements and related to the local polarisability of atoms or interatomic charge transfers (Ghosez et al., 1998). When a positive atom is displaced toward a negative atom, the electronic shell around a polarisable negative atom can move toward the positive atom or a change in the bond hybridisation can cause the transfer of electrons from the negative to the positive atom. As a consequence, the polarisation due to the ion displacement is dynamically augmented with respect to static charges. This property has strong consequences on IR absorption intensity (e.g. Pasquarello and Car, 1997) because the absorption scales as the squared electric charge.

Note that a different formulation of the dielectric tensor can be adopted to account for a different damping of TO and LO modes (e.g. Gervais and Piriou, 1974). Such type of formulation is required in materials exhibiting a large difference between the LO and TO frequencies, which implies that the anharmonic mode coupling to the vibrational density of states of the TO and LO modes can be significantly different.

2.3.3 IR Spectroscopy of Powder Materials

2.3.3.1 Light Reflection and Transmission by an Isotropic Dielectric Slab

Various experimental geometries can be used to record an IR spectrum, depending on the nature of the sample. The measured signal can correspond, for example, to the fraction of incident light intensity reflected or transmitted by a macroscopic sample (Fig. 2.4). In the case of powder material, such as clay minerals, the spectroscopic information is usually not recorded on a single crystal. Transmission measurements are performed on composite samples consisting in a small quantity of powder diluted in a nonabsorbing matrix (Fig. 2.5), whereas the IR spectrum of pure powders can be recorded using other geometries (e.g. attenuated total reflection (ATR) spectroscopy; photoacoustic spectroscopy). In this latter case, the powder sample displays some porosity and actually consists in a composite material made of mineral and air. Therefore, although individual particles are characterised by a significant anisotropy, the actual macroscopic sample usually displays isotropic properties, provided that the particle size can be considered as small with respect to the IR wavelength and sufficient statistical orientation homogeneity is ensured.

Fig. 2.4 Flowchart relating the atomic-scale structure of a mineral to its measured powder infrared spectrum. Note that all the atomic-scale parameters are in the dielectric permittivity tensor of the crystal whereas the effective dielectric function of the measured sample also incorporates macroscopic parameters. The interpretation of experimental spectra has to consider these various aspects and can be sustained by direct modeling strategies.

Fig. 2.5 (A) Usual geometry of powder transmission measurements. The sample consists in dilute particles in a nonabsorbing matrix. (B) For platy particles, the influence of the particle shape on powder IR spectra is related to the oscillating charges appearing at the surface of the polarised particles when the polarisation field is perpendicular to the plate. In this case, the absorption spectrum can be related to the orientation and time average of the electromagnetic power dissipated in a particle. Models explicitly accounting for the fraction of substance in a composite sample are illustrated below. In the Maxwell Garnett model (C), the particles of one substance A are considered as embedded in a matrix B whereas in the Bruggeman model (D) the two substances, A and B, are considered as embedded in the effective medium (in grey).

Considering an electromagnetic wave perpendicularly incident on a slab of nonopaque isotropic body with parallel faces, a part of the radiation will be reflected off the surface, whereas another one is transmitted through the slab (Fig. 2.4). Both parts can be actually detected and measured. Neglecting any light scattering, a balance of energy can relate these two parts to a last one, which corresponds to the radiation absorbed on its way through the slab:

   (2.32)

where rν=reflectance, tν=transmittance, αν, which describes the propagation of the electromagnetic wave through the material:

   (2.33)

where n1 and n2 are the real and imaginary parts, respectively, and the frequency-dependence implicitly considered. For a normal incidence on a slab of thickness d with isotropic properties, the transmittance t and reflectance r are given by (Bruesch, 1986):

  

(2.34)

  

(2.35)

In these expressions, R, ϕ, α and T are frequency-dependent quantities defined by:

   (2.36)

   (2.37)

   (2.38)

   (2.39)

In this last relation, the absorption coefficient K describes the absorption of the electromagnetic wave propagating through the sample and is related to the imaginary part of the refractive index by:

   (2.40)

The IR spectrum of a material measured in transmission or reflection reports the fraction of transmitted or reflected intensity as a function of the light frequency, usually expressed in wavenumber units. It is thus fully determined at a macroscopic level by the experimental geometry and the refractive index of the sample. In turn, Maxwell's equations imply that the refractive index is related to the dielectric function of the material ɛ(ω) (which links the macroscopic polarisation of the material to the macroscopic electric field). For an isotropic material, this relation is

   (2.41)

In the case of a crystalline material, its dielectric function is directly determined by the microscopic (atomic-scale) properties of the crystal (Eq. (2.28)). In the case of a powder crystalline sample, the long-range nature of electrostatic interactions and the finite size of particles can lead to a significant dependence of the measured spectra on the microstructure of the actual sample. Thus an additional step has to be considered to relate the dielectric function, and corresponding IR spectrum, of the composite sample to the bulk dielectric properties of the crystal (Fig. 2.4). Different strategies can be used to achieve this goal, depending on the characteristics of the powder and related composite sample.

2.3.3.1.1 IR Absorption by Isolated Small Particles in a Nonabsorbing Matrix

Pellets used for transmission measurements contain a small fraction of powder diluted in a nonabsorbing matrix, most often KBr (Fig. 2.5). In the limit of high dilution, the electrostatic interaction between particles can be neglected, and the sample absorption is obtained as an orientation average of the electromagnetic power dissipated in a single particle. Although not taking into account the spectral dependence on the substance fraction in the KBr pellet, this approach accounts for the relations between the absorption spectrum and the shape of particles.

For a particle size smaller than the IR wavelength and an ellipsoidal particle shape, the internal macroscopic electric and polarisation fields are homogeneous. The internal electric field differs from the external one because of the polarisation of the particle, which behaves as a capacitor (Fig. 2.5). The transitory accumulation of charges at the surface of particles, related to the oscillating motion of ions, induces a macroscopic electric field opposing their displacement. This contribution adds to the microscopic restoring force related to the interatomic potential, thus increasing the vibrational frequency. Such electrostatic effect is closely related to the electrostatic interactions giving birth to the LO-TO splitting in the bulk infinite crystals. The observed frequencies will range between the transverse and the longitudinal ones (Fig. 2.3), depending on the boundary conditions that control the charge accumulation at the particle surface. Larger effects will be observed on the strong absorption bands, corresponding to a large ionic polarisation. Interestingly, these electrostatic effects not only involve the vibrational properties of minerals and the shape of particles but also the various electronic contributions to the sample polarisability, that is, the dielectric permittivity of the nonabsorbing matrix and the high-frequency electronic permittivity of the mineral.

The influence of these macroscopic electrostatic effects on powder IR absorption spectra has been investigated in simple oxides and relatively symmetric minerals such as periclase, hematite, zircon or apatite (e.g. Rendon and Serna, 1981; Serna et al., 1987; Iglesias et al., 1990; Pecharroman et al., 1994; Pecharroman and Iglesias, 1996, 2000; Blanchard et al., 2008; Balan et al., 2011b). They also have to be considered for the interpretation of the spectrum of a number of clay minerals and related minerals, such as kaolinite group minerals (Farmer, 1998, 2000; Balan et al., 2001, 2005), serpentine group minerals (Balan et al., 2002a,b; Trittschack et al., 2012), aluminium and iron (oxy) hydroxides (Balan et al., 2006, 2008; Delattre et al., 2012; Blanchard et al., 2013). A specific dependence of the IR spectrum on the high-frequency electronic permittivity of the mineral has been demonstrated on irradiated kaolinite samples (Fourdrin et al., 2008). In this case, irradiation induces the creation of electronic defects and increases the electronic polarisability of the material, indirectly affecting the vibrational frequencies and the position of specific IR absorption bands.

2.3.3.1.2 Effective Dielectric Functions of Composite Samples

Various models provide the approximate dielectric function of a composite sample from its individual constituents in a quasistatic approach (e.g. Sihvola and Kong, 1988; Spanier and Herman, 2000; Kendrick and Burnett, 2016). They neglect scattering losses and can be extended to frequency-dependent dielectric properties when the characteristic size of individual particles is much smaller than the wavelength of the electromagnetic radiation, a condition usually approximately verified in MIR spectroscopy (λIR>10 μm). The effective dielectric function of the isotropic composite sample can then be related to the measured spectrum, either through the related absorption or reflection coefficients depending on the geometry of the experiment.

One of the simplest models was proposed by Maxwell-Garnett (Garnett, 1904), in which the particles of one of the constituents are considered as embedded in a homogeneous medium, corresponding to the other constituent for binary samples. In this model, the two constituents are therefore not treated in a symmetrically equivalent way. The Maxwell Garnett model is closely related to the Clausius-Mossotti derivation of the dielectric constant of molecular substances from the polarisability of molecules (e.g. Landauer, 1978) and is well suited to describe the dielectric properties of a sample in which a low amount of one of the constituents is diluted in a nonabsorbing matrix. In the limit of high dilution, it leads to results equivalent to those obtained by considering isolated particles. Note, however, that it also leads to the average properties of the pure material for a sample fraction tending to one. The correct asymptotic behavior at both high and low dilution limits explains its wide use for the interpretation of the electromagnetic properties of composite samples.

However, for arbitrary microstructures and intermediate volume fractions, the particles of one of the constituents cannot be considered as isolated and embedded in a homogeneous matrix of the other constituent. Correspondingly, an alternative model could be used in which each individual particle of a mixture is considered as embedded in the effective medium (Bruggeman, 1935; Landauer, 1978). At variance with the MG model, all constituents are treated in a symmetrically equivalent way and the effective dielectric properties of the sample are obtained self-consistently.

The two effective medium approaches display very distinct properties as a function of the volume fraction of the constituents ((Fig. 2.6). In the limit of a vanishingly small damping coefficient, the maximum of ɛs″ defines the TO frequency, whereas the LO frequency is defined by |ɛs|=0. The second medium of the binary composite is characterised by its dielectric constant ɛh, taken to be equal to unity in Fig. 2.6. The corresponding Maxwell Garnett effective dielectric function is defined for spherical particles by:

Fig. 2.6 Real and imaginary parts of a model single-resonance dielectric function (top) and corresponding imaginary part of the effective medium dielectric function in the Maxwell Garnett (middle) and Bruggeman (bottom) models for a composite sample made of the substance (volume fraction f) and a second substance of dielectric constant equal to unity. The effective dielectric functions have been divided by corresponding volume fraction, reported on the right. Note the significant shift of the resonance of the effective dielectric function with respect to that of the pure substance, as well as the specific lineshape in the Bruggeman model for intermediate fractions (From Aufort et al., 2016).

   (2.42)

It presents a single resonance, which progressively shifts toward the resonance frequency of the pure material when the volume fraction increases. In the limit of low volume fraction and vanishingly small damping parameter, this resonance coincides with that of the isolated sphere in the host matrix. The intensity of the resonance is not linearly related to the volume fraction of the composite.

Under the same assumptions, the Bruggeman model (Bruggeman, 1935) leads to the following relation:

  

(2.43)

In the limit cases ƒ→0 or ƒ→1, the Bruggeman dielectric function is identical to that obtained using the Maxwell Garnett model, with a single resonance at the characteristic frequency of the isolated sphere or of the pure bulk material, respectively. In contrast, the line shape of the imaginary part of the dielectric function at intermediate volume fractions significantly broadens and becomes asymmetric (Fig. 2.6). For volume fractions above 1/3, the lines extend between their TO and LO frequencies with a relatively higher intensity on the TO frequency side.

More sophisticated models can also