Solid State Proton Conductors E-Book

Philippe Knauth

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 731

Veröffentlichungsjahr: 2012

Beschreibung

Proton conduction can be found in many different solid materials, from organic polymers at room temperature to inorganic oxides at high temperature. Solid state proton conductors are of central interest for many technological innovations, including hydrogen and humidity sensors, membranes for water electrolyzers and, most importantly, for high-efficiency electrochemical energy conversion in fuel cells. Focusing on fundamentals and physico-chemical properties of solid state proton conductors, topics covered include: * Morphology and Structure of Solid Acids * Diffusion in Solid Proton Conductors by Nuclear Magnetic Resonance Spectroscopy * Structure and Diffusivity by Quasielastic Neutron Scattering * Broadband Dielectric Spectroscopy * Mechanical and Dynamic Mechanical Analysis of Proton-Conducting Polymers * Ab initio Modeling of Transport and Structure * Perfluorinated Sulfonic Acids * Proton-Conducting Aromatic Polymers * Inorganic Solid Proton Conductors Uniquely combining both organic (polymeric) and inorganic proton conductors, Solid State Proton Conductors: Properties and Applications in Fuel Cells provides a complete treatment of research on proton-conducting materials.

Leseprobe

Contents

Cover

Title Page

Copyright

Preface

About the Editors

Contributing Authors

Chapter 1: Introduction and Overview: Protons, the Nonconformist Ions

1.1 Brief History of the Field

1.2 Structure of This Book

References

Chapter 2: Morphology and Structure of Solid Acids

2.1 Introduction

2.2 Crystal Morphology and Structure of Solid Acids

References

Chapter 3: Diffusion in Solid Proton Conductors: Theoretical Aspects and Nuclear Magnetic Resonance Analysis

3.1 Fundamentals of Diffusion

3.2 Basic Principles of NMR

3.3 Application of NMR Techniques

3.4 Liquid Water Visualization in Proton-Conducting Membranes by Nuclear Magnetic Resonance Imaging

3.5 Conclusions

References

Chapter 4: Structure and Diffusivity in Proton-Conducting Membranes Studied by Quasielastic Neutron Scattering

4.1 Survey

4.2 Diffusion in Solids and Liquids

4.3 Quasielastic Neutron Scattering: A Brief Introduction

4.4 Proton Diffusion in Membranes

4.5 Solid State Proton Conductors

4.6 Concluding Remarks

References

Chapter 5: Broadband Dielectric Spectroscopy: A Powerful Tool for the Determination of Charge Transfer Mechanisms in Ion Conductors

5.1 Basic Principles

5.2 Phenomenological Background of Electric Properties in a Time-Dependent Field

5.3 Theory of Dielectric Relaxation

5.4 Analysis of Electric Spectra

5.5 Broadband Dielectric Spectroscopy Measurement Techniques

5.6 Concluding Remarks

Acknowledgements

References

Chapter 6: Mechanical and Dynamic Mechanical Analysis of Proton-Conducting Polymers

6.1Introduction

6.2 Methodology of Uniaxial Tensile Tests

6.3 Relaxation and Creep of Polymers

6.4 Engineering Stress–Strain Curves of Polymers

6.5 Stress–Strain Tensile Tests of Proton-Conducting Ionomers

6.6 Dynamic Mechanical Analysis (DMA) of Polymers

6.7 The DMA of Proton-Conducting Ionomers

6.8 Glossary

References

Chapter 7: Ab Initio Modeling of Transport and Structure of Solid State Proton Conductors

7.1 Introduction

7.2 Theoretical Methods

7.3 Polymer Electrolyte Membranes

7.4Crystalline Proton Conductors and Oxides

7.5 Concluding Remarks

References

Chapter 8: Perfluorinated Sulfonic Acids as Proton Conductor Membranes

8.1 Introduction on Polymer Electrolyte Membranes for Fuel Cells

8.2 General Properties of Polymer Electrolyte Membranes

8.3 Perfluorinated Membranes Containing Superacid –SO3H Groups

8.4 Some Information on Dow and on Recent Aquivion® Ionomers

8.5 Instability of Proton Conductivity of Highly Hydrated PFSA Membranes

8.6 Composite Nafion Membranes

8.7 Some Final Remarks and Conclusions

References

Chapter 9: Proton Conductivity of Aromatic Polymers

9.1 Introduction

9.2 Synthetic Strategies of the Various Acid-Functionalized Aromatic Polymers with Proton Transport Ability

9.3 Approaches to Enhance Proton Conductivity

9.4 Balancing Proton Conductivity, Dimensional Stability, and Other Properties

9.5 Electrochemical Performance of Aromatic Polymers

9.6 Summary

Acknowledgements

References

Chapter 10: Inorganic Solid Proton Conductors

10.1 Fundamentals of Ionic Conduction in Inorganic Solids

10.2 General Considerations on Inorganic Solid Proton Conductors

10.3 Low-Dimensional Solid Proton Conductors: Layered and Porous Structures

10.4 Three-Dimensional Solid Proton Conductors: “Quasi-Liquid” Structures

10.5 Three-Dimensional Solid Proton Conductors: Defect Mechanisms in Oxides

10.6 Conclusion

References

Index

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Library of Congress Cataloging-in-Publication Data

Di Vona, Maria Luisa.

Solid state proton conductors: properties and applications in fuel cells /Maria Luisa Di Vona and Philippe Knauth.

p. cm.

Includes bibliographical references and index.

ISBN 978-0-470-66937-2 (cloth)

1. Solid state proton conductors. 2. Solid state chemistry. 3. Fuel cells. I. Knauth, Philippe. II. Title.

QC176.8.E4D56 2012

621.31′2429–dc23                                                                          2011037228

A catalogue record for this book is available from the British Library.

HB ISBN: 9780470669372

Preface

Solid state proton conductors are of central interest for many technological innovations and, most importantly, for high-efficiency electrochemical energy conversion in fuel cells working at low or intermediate temperature.

The most recent textbook on all aspects of solid state proton conductors was published in 1992. Although some excellent review papers have been published since then, an updated textbook summarizing the current knowledge on solid state proton conductors seemed worthwhile.

This book presents review chapters on selected characterization techniques, modelling and properties of solid state proton conductors written by us and some of the leading experts in the field. It focuses on fundamentals and physico-chemical properties; synthesis procedures are only marginally addressed. Most chapters discuss first and foremost the basics that require a decent level of abstraction, before presenting detailed descriptions of solid state proton conductors.

We are confident that this book will close a gap in recent textbook literature.

Writing and editing a book are difficult and time-consuming tasks, but they also comprise a rewarding adventure and we hope the readers will consider their “journey” through the pages of this book a gratifying experience as well.

We want to thank all authors and friends, who contributed their knowledge in a timely manner. Without their commitment and hard work, this book would not have been possible.

We also gratefully acknowledge the financial support by many institutions which helped to finance our research in the field of solid state proton conductors over the years, including the European Hydrogen and Fuel Cell Technology Platform (FP7 JTI-FCH), the Italian Ministry of Education, Universities and Research (MIUR) and the Franco-Italian University.

Philippe Knauth and M. Luisa Di Vona Marseille and Roma, June 2011

About the Editors

Philippe Knauth was the recipient of a doctorate in sciences (Doctor Rerum Naturalium) in 1987 and the Habilitation à diriger des recherches in 1996. He has been a professor of materials chemistry at Aix-Marseille University since 1999. Awarded the CNRS Bronze Medal in 1994, he was an Invited Scientist at the Massachusetts Instuitute of Technology, United States from 1997--1998 and an Invited Professor at the National Institute of Materials Science (NIMS), Tsukuba, Japan in both 2007 and 2010. He is currently director of the Laboratoire Chimie Provence (UMR 6264), which includes 130 academic staff working in all fields of chemistry. He has been an elected member of France's Conseil National des Universités for materials chemistry since 2003 and president of the Provence-Alpes-Côte d'Azur regional section of the Société Chimique de France since 2010. His principal research topics are ionic conduction at interfaces, electrochemistry at the nanoscale and materials for energy and the environment. He is currently mainly working on solid state proton conductors for fuel cells and micro-electrodes for lithium-ion batteries, and he is a member of the editorial board of the Journal of Electroceramics.

Maria Luisa Di Vona obtained a doctorate in chemistry cum laude in 1984. In 1987 she became a researcher in organic chemistry at the Faculty of Science of the University of Rome Tor Vergata. She was visiting professor at the Laboratoire Chimie Provence, Université de Provence, Marseille, France, in 2007 and 2009, and at the National Institute for Materials Science (NIMS), Tsukuba, Japan in 2010. She is the author of about 100 papers in international journals on materials synthesis and characterization, multifunctional `inorganic and organic–inorganic materials, the formation and study of nanocomposite materials and characterization by means of multinuclear NMR (nuclear magnetic resonance) spectroscopy. Her current research interest is in the field of proton exchange membranes. She is a project leader and recipient of research grants from the ASI, Italian Ministry, Franco-Italian University (Vinci program) and European Union (the European Hydrogen and Fuel Cell Technology Platform, or FP7 JTI-FCH). She is a member of the organizing and scientific committees of several conferences and was the principal organizer of the 2009 European Materials Research Society (E-MRS) symposium “Materials for Polymer Electrolyte Membrane Fuel Cells” as well as the 2011 Materials Research Society (MRS) symposium “Advanced Materials for Fuel Cells”.

Contributing Authors

Giulio Alberti, Department of Chemistry, University of Perugia, Via Elce di Sotto 8, I-06123 Perugia, Italy

Jean-François Chailan, Laboratoire MAPIEM, Université du Sud Toulon-Var, F-83957 La Garde, France

Jeffrey K. Clark II, Department of Chemical and Biomolecular Engineering, University of Tennessee, Knoxville, TN 37996, USA

Vito Di Noto, Department of Chemical Sciences, University of Padua, Via F. Marzolo 1, I-35131 Padova, Italy

Maria Luisa Di Vona, Dipartimento di Scienze e Tecnologie Chimiche, University of Rome Tor Vergata, Via della Ricerca Scientifica, I-00133 Roma, Italy

Guinevere A. Giffin, Department of Chemical Sciences, University of Padua, Via F. Marzolo 1, I-35131 Padova, Italy

Michael D. Guiver, National Research Council Canada, Institute for Chemical Process and Environmental Technology Ottawa, ON, K1A 0R6, Canada and WCU, Department of Energy Engineering, Hanyang University, Seoul 133–791, Republic of Korea

Rolf Hempelmann, Physical Chemistry, Saarland University, D-66123 Saarbrücken, Germany

Mustapha Khadhraoui, Laboratoire Chimie Provence-Madirel, Aix-Marseille University - CNRS, Centre St Jérôme, F-13397 Marseille, France

Philippe Knauth, Laboratoire Chimie Provence-Madirel, Aix-Marseille University - CNRS, F-13397 Marseille, France

Sandra Lavina, Department of Chemical Sciences, University of Padua, Via F. Marzolo 1, I-35131 Padova, Italy

Baijun Liu, Alan G. MacDiarmid Institute, Jilin University, Changchun 130012, P.R. China

Riccardo Narducci, Department of Chemistry, University of Perugia, Via Elce di Sotto 8, I-06123 Perugia, Italy

Stephen J. Paddison, Department of Chemical and Biomolecular Engineering, University of Tennessee, Knoxville, TN 37996, USA

Matteo Piga, Department of Chemical Sciences, University of Padua, Via F. Marzolo 1, I-35131 Padova, Italy

Oliver Schäf, Laboratoire Chimie Provence-Madirel, Aix-Marseille University, Centre St Jérôme, F-13397 Marseille, France

Emanuela Sgreccia, Dipartimento di Scienze e Tecnologie Chimiche, University of Rome Tor Vergata, Via della Ricerca Scientifica, I-00133 Roma, Italy

Sebastiano Tosto, ENEA Centro Ricerche Casaccia, Via Anguillarese 301, I-00123 Roma, Italy

Keti Vezzù, Department of Chemistry, University of Venice, Via Dorsoduro, 2137, I-30123 Venice, Italy

Chapter 1

Introduction and Overview: Protons, the Nonconformist Ions

Maria Luisa Di Vona and Philippe Knauth

“The Nonconformist Ion” is the title of a review article on proton-conducting solids by Ernsberger in 1983 [1]. Indeed, many proton properties are peculiar. First of all, the very particular electronic structure is unique: its only valence electron lost, the proton is exceptionally small and light and polarizes its surroundings very strongly. In condensed matter, this will lead to strong interactions with the immediate environment and very strong solvation in solution.

Second, two very particular proton migration mechanisms are well established. In “vehicular” motion, a protonated solvent molecule is used as a vehicle. This mechanism is typically characterized by higher activation energy and lower proton mobility. In structural motion, the so-called Grotthuss mechanism involves site-to-site hopping between proton donor and proton acceptor sites with local reconstruction of the environment around the moving proton. This mechanism is related to lower values of activation energy and higher proton mobility.

Proton conduction can be found in many very different solid materials, from soft organic polymers at room temperature to hard inorganic oxides at high temperature. The importance of atmospheric humidity for the existence and stability of proton conduction is another common point, which goes with experimental difficulties for measuring proton conductivity in solids.

Proton-conducting solids are the core of many technological innovations, including hydrogen and humidity sensors, hydrogen permeation membranes, membranes for water electrolyzers, and most importantly high-efficiency electrochemical energy conversion in fuel cells working at low temperature (polymer electrolyte membrane or proton exchange membrane fuel cells, PEMFC) or intermediate temperature (proton-conducting ceramic fuel cells, PCFC).

1.1 Brief History of the Field

Proton mobility is a special case in the field of ion transport. In early textbooks on the electrochemistry of solids, proton-conducting solids are not even mentioned [2], except ice [3].

Historically, the existence of protons in aqueous solutions had already been conjectured by de Grotthuss in 1806 [4]. The study of proton-conducting solids started at the end of the nineteenth century, when it was noticed that ice conducts electricity, with the investigation of the electrical conductivity of ice single crystals [5]. A first mention of “vagabond” ions in an inorganic compound, hydrogen uranyl phosphate (HUP), was due to Beintema in 1938 [6]. However, it was not until the 1950s that the study of solid proton conductors started in earnest: Bjerrum's fundamental study on ice conductivity led the way in 1952 [7], and Eigen and coworkers discussed the proton conductivity of ice crystals in 1964 [8]. Nevertheless, these investigations were fundamental studies and the materials could still be considered only laboratory curiosities.

The first proton-conducting material applied in practice was a perfluorinated sulfonated polymer, Nafion, adapted by DuPont in the 1960s as a proton-conducting membrane for PEMFC, used in the Gemini and Apollo space programs. This gave important momentum to the domain of solid proton conductors. Several inorganic solid proton conductors were then reported in the 1970s and 1980s. The rediscovery of HUP was followed by the discovery by Russian groups of several acid sulfates showing structural phase transitions, such as CsHSO4 [9] and zirconium hydrogenphosphate (ZrP), by Alberti and coworkers [10]. Furthermore, oxide gels containing water show nearly always some proton conductivity [11]. However, with the exception of ZrP, the proton conductivity of these materials is limited to about 200 °C.

An important discovery was, therefore, the report by Iwahara and coworkers in the 1980s of “high-temperature” proton conduction in perovskite-type oxides in humidity- or hydrogen-containing atmosphere [12], where the maximum of proton conductivity is typically observed at temperatures above 400 °C.

Nowadays the main fields of development are proton-conducting polymer membranes for low-temperature applications and proton-conducting oxide ceramics for intermediate- and high-temperature devices. Given the current interest for the possible future hydrogen economy, the fuel cell field is mentioned in most articles of this book.

1.2 Structure of This Book

The most recent textbook on all aspects of solid state proton conductors was published in 1992 [13]. Excellent review papers have been published afterward, for example by Norby in 1999; [14] Alberti and Casciola in 2001 [15]; and Kreuer, Paddison, Spohr, and Schuster in 2004 [16], but an updated textbook summarizing the current knowledge on solid state proton conductors seemed worthwhile.

In the following chapters, some of the leading experts in the field have written authoritative review chapters on the characterization techniques, modeling, and properties of solid state proton conductors.

The chapter “Morphology and Structure of Solid Acids” shows an overview of structural analysis of some important solid acids by scanning electron microscopy. This beautifully illustrated chapter is an aesthetic pleasure, and the micrographs are complemented by polyhedral representations and a short introduction on the technique.

The chapter “Diffusion in Solid Proton Conductors: Theoretical Aspects and Nuclear Magnetic Resonance Analysis” starts with an overview on fundamentals of diffusion. Then, principles of nuclear magnetic resonance (NMR) spectroscopy are introduced. Nuclear magnetic resonance is a very powerful technique for investigation of structure and diffusion in solid proton conductors; NMR imaging is a newer development, and is also addressed on a basic level in this chapter.

The chapter “Structure and Diffusivity in Proton-Conducting Membranes Studied by Quasi-elastic Neutron Scattering” introduces the basics of neutron scattering, which is obviously of particular importance for the field. Analysis of diffusional processes in inorganic as well as organic solid proton conductors is presented and discussed.

The chapter “Broadband Dielectric Spectroscopy: A Powerful Tool for the Determination of Charge Transfer Mechanisms in Ion Conductors” is devoted to the electrical properties of ion-conducting solids, especially macromolecular systems. This chapter describes fundamentals and examples of dielectric measurements in a broad frequency domain, which can be used for a wide range of materials from insulators to “super-protonic” conductors.

The chapter “Mechanical and Dynamic Mechanical Analysis of Proton-Conducting Polymers” introduces first some basic principles of the mechanics of materials: elastic and plastic deformation, creep and relaxation, and dynamic mechanical analysis. Then, the mechanical properties of proton-conducting polymers and their durability are discussed.

The chapter “Ab Initio Modeling of Transport and Structure of Solid Proton Conductors” presents a rapid introduction on the theoretical methods of choice. Significant examples of solid proton conductors are discussed, including proton-conducting polymers; solid acids, such as CsHSO4; and proton-conducting perovskite oxides.

Two chapters are devoted to polymeric proton conductors. The chapter “Perfluorinated Sulfonic Acids as Proton Conductor Membranes” introduces the field and presents recent progress for the improvement of the oldest but still leading ionomer, Nafion. This chapter reviews a physicochemical approach and strategies for future enhancement of the durability of Nafion membranes.

The chapter “Proton Conductivity of Aromatic Polymers” discusses a main family of alternative ionomers based on fully aromatic polymers. Their synthesis and electrical properties and further possibilities for improvement, such as hybrid organic–inorganic ionomers and cross-linked systems, are discussed.

The last chapter reviews “Inorganic Solid Proton Conductors.” The chapter recalls fundamentals of ionic conduction in inorganic solids and presents the main classes of proton-conducting materials, including layered and porous solids, “quasi-liquid” structures, and defect solids, especially perovskite oxides.

References

1. Ernsberger, F.M. (1983) The nonconformist ion. Journal of the American Ceramic Society, 66, 747.

2. Rickert, H. (1982) Electrochemistry of Solids, Springer, Berlin.

3. Kröger, F.A. (1974) The Chemistry of Imperfect Crystals, North-Holland, Amsterdam.

4. Grotthuss, C.J.T.d. (1806) Mémoire sur la décomposition de l'eau et des corps qu'elle tient en dissolution à l'aide de l'électricité galvanique. Annales de Chimie, LVII, 54.

5. Ayrton, W.E. and Perry, J. (1877) Ice as an electrolyte. Proceedings of the Physical Society, 2, 171.

6. Beintema, J. (1938) On the composition and the crystallography of autunite and the meta-autunites. Recueil des Travaux Chimiques des Pays-Bas, 57, 155.

7. Bjerrum, N. (1952) Structure and properties of ice. Science, 115, 385.

8. Eigen, M., Maeyer, L.D. and Spatz, H.C. (1964) Kinetic behavior of protons and deuterons in ice crystals. Berichte der Bunsengesellschaft für Physikalische Chemie, 68, 19.

9. Baranov, A.I., Shuvalov, L.A. and Shchagina, N.M. (1982) Superion conductivity and phase-transitions in CsHSO4 and CsHSeO4 crystals. Jetp Letters, 36, 459.

10. Alberti, G. and Torracca, E. (1968) Electrical conductance of amorphous zirconium phosphate in various salt forms. Journal of Inorganic and Nuclear Chemistry, 30, 1093.

11. Livage, J. (1992) Sol-gel ionics. Solid State Ionics, 50, 307.

12. Takahashi, T. and Iwahara, H. (1980) Protonic conduction in perovskite type oxide solid solutions. Revue Chimie Minérale, 17, 243.

13. Colomban, P. (1992) Proton Conductors: Solids, Membranes and Gels - Materials and Devices, Cambridge University Press, Cambridge.

14. Norby, T. (1999) Solid-state protonic conductors: principles, properties, progress and prospects. Solid State Ionics, 125, 1.

15. Alberti, G. and Casciola, M. (2001) Solid state protonic conductors, present main applications and future prospects. Solid State Ionics, 145, 3.

16. Kreuer, K., Paddison, S., Spohr, E. and Schuster, M. (2004) Transport in proton conductors for fuel-cell applications: simulations, elementary reactions, and phenomenology. Chemical Reviews, 104, 4637.

Chapter 2

Morphology and Structure of Solid Acids1

Habib Ghobarkar, Philippe Knauth and Oliver Schäf

2.1 Introduction

The objective of this chapter is to introduce some important solid acids from a structural, and also morphological, point of view. The micrographs were obtained by scanning electron microscopy (SEM) on samples prepared in situ, according to the techniques described in the following section.

2.1.1 Preparation Technique of Solid Acids

Almost all solid acids were prepared by rapid evaporation of highly concentrated aqueous solutions from open stainless-steel containers heated either by a gas flame or by an induction furnace. Different evaporation speeds could be obtained in this way, but over-heating had to be strictly avoided. During the cooling process, the samples were placed in the sputtering unit (low-pressure Ar-plasma atmosphere) in order to cover them with a protective gold layer (necessary for subsequent SEM observations) before rehydration occurred.

High-pressure hydrothermal processing at temperatures below 200 °C and at 100 MPa pressure as described in detail in reference [1] could be used only for the synthesis of the complex transition – metal phosphoric acids, presented in Sections 2.2.3.1 and 2.2.3.3.

Samples from both synthesis pathways were immediately transferred to the SEM in order to avoid any further degradation.

2.1.2 Imaging Technique with the Scanning Electron Microscope

X-ray diffraction is the first and standard method commonly used for the identification of crystalline phases. Ghobarkar [2, 3] developed a new method for the identification of microcrystals that allows the optical identification of crystals observed by the SEM. In contrast to the optical reflection goniometer, this method allows the measurement of crystal faces even in the micrometre range applying the crystallographic principle that the face normal angles of crystals keep constant independent from size. The face normal angles of an idiomorphous crystal phase, however, are characteristic for each crystallographic system while the axis ratios are determined. Furthermore, the calculated axis ratios can be compared to X-ray diffraction data.

The differences in depth created by object points appearing in different spherical distances with respect to the eyes are called parallaxes. Ghobarkar could show that these parallaxes can be used to quantify the relative position of a plane of a microcrystal's face relative to the next. This is done in order to obtain all angles between the appearing faces (represented by their face normal angles).

By using SEM, crystals can be indexed and their crystallographic grouping determined. Furthermore, the energy-dispersive X-ray (EDS) method allows the measurement of the chemical composition in a semiquantitative way. The two different results are based on standard measurements in chemical composition and face angles.

The stereo comparator method can be subdivided into different parts. In the electron-microscopic part, the crystalline phase under investigation is analysed by stereo imaging. The specimen containing the microcrystals is installed on the goniometer specimen stage of the SEM. In a first approximation, the SEM delivers parallel projection images of the observed objects.

Different perspectives for stereo-comparator processing are created by taking two different images, the first at a position of 0° and the second after an inclination of 12° (Figure 2.1). To get useful results, the inclination has to be done precisely in the same crystallographic zone. Two different image pairs are taken in order to reduce systematic errors introduced by mechanical movement of the specimen stage. It is important that the images are taken at the same value of magnification. Generally, the method is useful for crystals which need magnification higher than 500 times as crystals bigger in size can be analysed by other methods. The smaller the crystals are, the higher the precision of the final phase angle measurements.

Figure 2.1 Position of crystal images after inclination: L: −12° inclination, M: 0°, R: +12° (two pairs for control and accuracy purposes) [1]. Reprinted with permission from The Reconstruction of Natural Zeolites by H. Ghobarkar, O. Schäf, Y. Massiani, P. Knauth, Copyright (2003) Kluwer Academic.

1.2.1 The Calculation of x,y,z from Measured x,y, Px and Py

The calculation of the face angles is done by the determination of x,y as well as the parallaxes Px and Py for a respective point on a crystal face. Four points (three points to define a plane, plus one control point) are measured per crystal face. The co-ordinates x and y can be directly taken, while Py has to be kept constant carefully during the measurement in order to guarantee accuracy. The z value for the respective point is calculated by:

(2.1)

given that Px for both directions of inclination (−12°, 0°, 12°) gives the same value (control of accuracy).

By doing this for three points (one supplementary point for control), a plane is clearly defined; the common form of the equation of a plane is:

(2.2)

The angle between planes 1 and 2 (crystal faces) is then given by:

(2.3)

The calculation is simplified by using the vector form of the plane equation. This has the big advantage that the angle between two crystal faces is identical to the angle between their normal vectors. The determination of the angle between two faces, therefore, covers two steps.

The first step is the determination of the normal vectors of both planes: the determined three points of a plane permit one to calculate two vectors which pass within the plane. The normal vector of these planes is placed perpendicular to the plane and is the complementary angle to 180°.

Second is the determination of the angles between the normal vectors: these are the angles between the crystal faces (Figure 2.2) obtained by the cross product of the two vectors.

Figure 2.2 Angles between crystal faces are obtained by determining the face normal angles from the respective plane vectors for each face [1]. Reprinted with permission from The Reconstruction of Natural Zeolites by H. Ghobarkar, O. Schäf, Y. Massiani, P. Knauth, Copyright (2003) Kluwer Academic.

2.1.2.2 Crystal Indexing

In order to confirm the results on the face normal angles obtained by the stereo-comparator with respect to the crystal habit (crystal morphology), the values are written in the stereographic projection. At last, the stereographic projection has to be turned in such a way that a standard set-up is achieved. The final indexing has to be accomplished by trial and error, while theoretical values can be taken into account once the crystal axis ratios and the crystal axis angles have been determined. More details on this SEM observation technique of microcrystals can be found in references [4, 5].

2.2 Crystal Morphology and Structure of Solid Acids

This chapter presents acid morphologies in the crystalline state, while the respective crystal structures are directly correlated to these morphologies.

The reader may use corresponding crystal visualization software to obtain complementary three-dimensional orientations of the respective crystal lattices. Crystal structure references are indicated to facilitate this approach.

2.2.1 Hydrohalic Acids

2.2.1.1 Hydrofluoric Acid

Chemical formula: HF

Crystal morphology (Figure 2.3)

Figure 2.3 Orthorhombic (class: mmm) hydrofluoric acid (SEM, magnification: 2000×).

Crystal structure (Figure 2.4)

Figure 2.4 Polyhedral representation of orthorhombic hydrofluoric acid (space group: Bmmb). Data from Reference [6].

2.2.1.2 Hydrochloric Acid

Chemical formula: HCl

Crystal morphology (Figure 2.5)

Figure 2.5 Orthorhombic (class: mmm) hydrochloric acid (SEM, magnification: 1290×).

Crystal structure (Figure 2.6)

Figure 2.6 Polyhedral representation of orthorhombic hydrochloric acid (space group: Fmmm) [7].

2.2.1.3 Hydrobromic Acid

Chemical formula: HBr

Crystal morphology (Figure 2.7)

Figure 2.7 Orthorhombic (class: mmm) hydrobromic acid (SEM, magnification: 5000×).

Crystal structure (Figure 2.8)

Figure 2.8 Polyhedral representation of orthorhombic hydrobromic acid (space group: Fmmm). Data from Reference [7].

2.2.2 Main Group Element Oxoacids

2.2.2.1 Boric Acid

Chemical formula: H3BO3

Crystal morphology (Figure 2.9)

Figure 2.9 Triclinic (class: P) boric acid (SEM, magnification: 1290×).

Crystal structure (Figure 2.10)

Figure 2.10 Polyhedral representation of triclinic boric acid (space group: P). Data from Reference [8].

2.2.2.2 Isocyanic Acid

Chemical formula: HNCO

Crystal morphology (Figure 2.11)

Figure 2.11 Orthorhombic (class: mmm) isocyanic acid (SEM, magnification: 2000×).

Crystal structure (Figure 2.12)

Figure 2.12 Polyhedral representation of orthorhombic isocyanic acid (space group: Pnma). Data from Reference [9].

2.2.2.3 Nitric Acid

Chemical formula: HNO3

Crystal morphology (Figure 2.13)

Figure 2.13 Monoclinic (class: 2/m) nitric acid (SEM, magnification: 1590×).

Crystal structure (Figure 2.14)

Figure 2.14 Polyhedral representation of monoclinic nitric acid (space group: P121/a1). Data from Reference [10].

2.2.2.4 Phosphoric Acid

Chemical formula: H3PO4

Crystal morphology: modification 1 (Figure 2.15)

Figure 2.15 Monoclinic (class: 2/m) phosphoric acid (SEM, magnification: 2940×).

Crystal structure: modification 1 (Figure 2.16)

Figure 2.16 Polyhedral representation of monoclinic phosphoric acid (space group: P121/c1). Data from References [11, 12].

Crystal morphology: modification 2 (Figure 2.17)

Figure 2.17 Orthorhombic (class: 2/m) phosphoric acid (SEM, magnification: 2000×).

Crystal structure: modification 2 (Figure 2.18)

Figure 2.18 Polyhedral representation of orthorhombic phosphoric acid (space group: Pna21). Data from Reference [13].

2.2.2.5 Triarsenic Acid

Chemical formula: H5As3O10

Crystal morphology (Figure 2.19)

Figure 2.19 Triclinic (class:) triarsenic acid (SEM, magnification: 733×).

Crystal structure (Figure 2.20)

Figure 2.20 Polyhedral representation of triclinic triarsenic acid (space group: P ). Data from Reference [14].

2.2.2.6 Antimonic Acid

Chemical formula: H2Sb2O6

Crystal morphology (Figure 2.21)

Figure 2.21 Cubic (class: mm) antimonic acid (SEM, magnification: 1000×).

Crystal structure (Figure 2.22)

Figure 2.22 Polyhedral representation of cubic antimonic acid (space group: Fdmz). Data from Reference [15].

2.2.2.7 Sulphuric Acid

Chemical formula: H2SO4

Crystal morphology (Figure 2.23)

Figure 2.23 Monoclinic (class: 2/m) sulphuric acid (SEM, magnification: 2150×).

Crystal structure (Figure 2.24)

Figure 2.24 Polyhedral representation of monoclinic sulphuric acid (space group: A1a1). Data from Reference [16].

2.2.2.8 Selenic Acids

Selenic(VI) Acid

Chemical formula: H2SeO4

Crystal morphology (Figure 2.25)

Figure 2.25 Orthorhombic (class: 222) selenic (VI) acid (SEM, magnification: 667×).

Crystal structure (Figure 2.26)

Figure 2.26 Polyhedral representation of orthorhombic selenic (VI) acid (space group: 212121). Data from Reference [17].

Selenic(IV) Acid – Selenous Acid

Chemical formula: H2SeO3

Crystal morphology (Figure 2.27)

Figure 2.27 Orthorhombic (class: 222) selenic(IV) acid (SEM, magnification: 3340×).

Crystal structure (Figure 2.28)

Figure 2.28 Polyhedral representation of orthorhombic selenic(IV) acid (space group: P212121). Data from Reference [18].

2.2.2.9 Chloric Acids

Chloric(VII) Acid

Chemical formula: HClO4

Crystal morphology (Figure 2.29)

Figure 2.29 Orthorhombic (class: mm2) perchloric acid (SEM, magnification: 4900×).

Crystal structure (Figure 2.30)

Figure 2.30 Polyhedral representation of orthorhombic perchloric acid (space group: Pca21). Data from Reference [19].

Chloric(VII) Acid Trihydrate – Oxonium Perchlorate

Chemical formula: HClO4·3 H2O (see also Chapter 10)

Crystal morphology (Figure 2.31)

Figure 2.31 Orthorhombic (class: mmm) perchloric acid trihydrate (SEM, magnification: 2000×).

Crystal structure (Figure 2.32)

Figure 2.32 Polyhedral representation of orthorhombic perchloric acid trihydrate (space group: Pbca). Data from Reference [20].

2.2.2.10 Iodic Acids

Iodic(VII) Acid

Chemical formula: H5IO6

Crystal morphology (Figure 2.33)

Figure 2.33 Monoclinic (class: 2/m) iodic(V) acid (SEM, magnification: 2000×).

Crystal structure (Figure 2.34)

Figure 2.34 Polyhedral representation of monoclinic iodic(VII) acid (space group: P121/n1). Data from Reference [21].

Iodic(V) Acid

Chemical formula: HIO3

Crystal morphology (Figure 2.35)

Figure 2.35 Orthorhombic (class: 222) iodic(V) acid (SEM, magnification: 2000×).

Crystal structure (Figure 2.36)

Figure 2.36 Polyhedral representation of orthorhombic iodic(V) acid (space group: P212121). Data from Reference [22].

2.2.3 Transition Metal Oxoacids

2.2.3.1 Dodecamolybdophosphoric Acid Hexahydrate

Chemical formula: H3(PMo12O40) (H2O)6

Crystal morphology (Figure 2.37)

Figure 2.37 Cubic (class: m3m) dodecamolybdophosphoric acid hexahydrate (SEM, magnification: 5040×).

Crystal structure (Figure 2.38)

Figure 2.38 Polyhedral representation of cubic (class: m3m) dodecamolybdophosphoric acid hexahydrate (space group: Fdmz). Data from Reference [23].

2.2.3.2 Tungstic Acid

Chemical formula: H2WO4

Crystal morphology (Figure 2.39)

Figure 2.39 Orthorhombic (class: mmm) tungstic acid (SEM, magnification: 2000×).

Crystal structure (Figure 2.40)

Figure 2.40 Polyhedral representation of orthorhombic tungstic acid (space group: Pnmb). Data from Reference [24].

2.2.3.3 Dodecatungstophosphoric Acid 21 Hydrate

Chemical formula: H3PW12O40·21H2O (see also Chapter 10)

Crystal morphology (Figure 2.41)

Figure 2.41 Orthorhombic (class: mmm) dodecatungstophosphoric acid 21 hydrate (SEM, magnification: 2000×).

Crystal structure (Figure 2.42)

Figure 2.42 Polyhedral representation of orthorhombic dodecatungstophosphoric acid 21 hydrate (space group: Pcca). Data from Reference [25].

2.2.4 Carboxylic Acids

2.2.4.1 Formic Acid

Chemical formula: HCOOH

Crystal morphology (Figure 2.43)

Figure 2.43 Orthorhombic (class: mm2) formic acid (SEM, magnification: 2000×).

Crystal structure (Figure 2.44)

Figure 2.44 Polyhedral representation of orthorhombic formic acid (space group: Pna21). Data from Reference [26].

2.2.4.2 Acetic Acid

Chemical formula: CH3COOH

Crystal morphology (Figure 2.45)

Figure 2.45 Orthorhombic (class: mm2) acetic acid (SEM, magnification: 6440×).

Crystal structure (Figure 2.46)

Figure 2.46 Polyhedral representation of orthorhombic acetic acid (space group: Pna21). Data from Reference [27].

Note

1. This chapter is dedicated to the memory of Dr. Habib Ghobarkar († 2010).

References

1. Ghobarkar, H., Schäf, O., Massiani, Y. and Knauth, P. (2003) The Reconstruction of Natural Zeolites, Kluwer Academic, Dordrecht.

2. Ghobarkar, H. (1978) Ph.D. Thesis, Free University of Berlin.

3. Ghobarkar, H. (1977) Kristy Technology, 12, K49–K51.

4. Franke, W. and Ghobarkar, H. (1983) Neues Jahrbuch Fuer Mineralogie, Monatshefte, 5, 211–218.

5. Franke, W. and Ghobarkar, H. (1981) Crystal Research and Technology, 16 (11), 1229–1232.

6. Atoji, M. and Lipscomb, W.N. (1954) Acta Crystallographica, 7, 173–175.

7. Natta, G. (1933) Gazetta Chimica Italiana, 63, 425–439.

8. Zachariasen, W.H. (1954) Acta Crystallographica, 7, 305–310.

9. von Dohlen, W.C. and Carpenter, G.B. (1955) Acta Crystallographica, 8, 646–651.

10. Luzzati, V. (1949) Comptes Rendus Hebdomadaires des Seances de l'Academie des Sciences, 229, 1349–1351.

11. Furberg, S. (1955) Acta Chemica Scandinavica, 9, 1557–1566.

12. Blessing, R.H. (1988) Acta Crystallographica. Section B, Structural Science, 44, 334–340.

13. Furberg, S. and Landmark, P. (1957) Acta Chemica Scandinavica, 11, 1505–1511.

14. Jost, K.H., Worzala, H. and Thilo, E. (1966) Acta Crystallographica, 21, 808–813.

15. Natta, G. and Baccaredda, M. (1936) Gazetta Chimica Italiana, 66, 308–316.

16. Pascard-Billy, C. (1965) Acta Crystallographica, 18, 827–829.

17. Moodenbaugh, A.R., Hartt, J.E., Hurst, J.J. et al. (1983) Physical Review B – Condensed Matter, 28, 3501–3505.

18. Larsen, F.K., Lehmann, M.S. and Sotofte, I. (1971) Acta Chemica Scandinavia, 25, 1233–1240.

19. Simon, A. and Borrmann, H. (1988) Angewandte Chemie (German edn.), 100 (10), 1386–1389.

20. Almloef, J. (1972) Acta Crystallographica. Section B, Structural Science, 28, 481–485.

21. Feikema, Y.D. (1966) Acta Crystallographica, 20, 765–769.

22. Rogers, M.T. and Helmholz, L. (1941) Journal of the American Chemical Society, 63, 278–284.

23. Clark, C.J. and Hall, D. (1976) Acta Crystallographica. Section B, Structural Science, 32, 1545–1547.

24. Szymanski, J.T. and Roberts, A.C. (1984) Canadian Mineralogist, 232, 681–688.

25. Spirlett, M.R. and Busing, W.R. (1978) Acta Crystallographica. Section B, Structural Science, 34, 907–910.

26. Albinati, A., Rouse, K.D. and Thomas, M.W. (1978) Acta Crystallographica. Section B, Structural Science, 34, 2188–2190.

27. Jones, R.E. and Templeton, D.H. (1958) Acta Crystallographica, 11, 484–487.

Chapter 3

Diffusion in Solid Proton Conductors: Theoretical Aspects and Nuclear Magnetic Resonance Analysis

Maria Luisa Di Vona, Emanuela Sgreccia and Sebastiano Tosto

Water behavior in solid proton conductors is a considerably complex phenomenon. However, the knowledge of transport and diffusivity inside electrolytes is an important requirement for many applications. Extensive efforts have been made in terms of modeling water transport and its management. Accurate measurements of diffusion are required to validate these models and to optimize the performance of solid proton conductors.

The focus of this chapter is the theoretical approach together with the use of nuclear magnetic resonance (NMR) techniques for the understanding of diffusion phenomena in solid proton conductors. The basic principles and the main NMR methods will be also discussed.

3.1 Fundamentals of Diffusion

Diffusion is the transport of matter activated by thermal motion of atoms or molecules in gas, liquid and solid phases [1]. The complexity of the microscopic kinetic mechanisms underlying these phenomena is due to the variety of driving forces and interaction forces that control the displacement rate of atoms, ions and molecules. The International Union of Pure and Applied Chemistry (IUPAC) defines self-diffusion as the transport of matter under vanishing chemical potential gradient. Equation 3.1 recalls the definition of the chemical potential μi, of species i, where Ni is the number of particles of species i, U is the internal energy, also simply called energy [2], and entropy S, volume V and a number of other particles of the system are constant.

(3.1)

In practice, this definition is often replaced by that using the Gibbs free energy G at constant temperature T and pressure p:

(3.2)

Self-diffusion involves a spontaneous mixing of atoms, molecules or ions in a chemically homogeneous phase under steady conditions of dynamical equilibrium and without net flow of matter. It can be evidenced replacing part of the diffusing species with isotopic tracers (tracer diffusion). More often, however, diffusion phenomena describe a transition from a situation out of equilibrium towards thermodynamic equilibrium; this is typically the case of chemical diffusion, which occurs with entropy increase due to net flow of matter, for example between two different phases. This kind of process is irreversible. As will be shortly sketched in the next section, these processes are described by appropriate generally different diffusion coefficients, D. In the former case, D concerns one species only, apart from isotopic usually negligible effects; in the latter case, D describes instead in general a multi-element system with correlation and interaction effects between different diffusing species. In general, D also depends on the activity of the diffusing species, that is, on its concentration. The quantum nature of the system is to be taken into account when the number of atoms, molecules and ions is so small that it prevents any statistical approach based on macroscopic average properties like temperature or pressure. When instead the number of species is sufficiently large to be described through statistical formalism (i.e. in terms of concentrations and concentration gradients regarded as properties of a continuum medium characterized by smooth changes of its thermodynamic properties), D characterizes the evolution of an unstable system towards its maximum entropy. The continuity condition and the Fick equations are the fundamental tools that account adequately for a huge amount of experimental data related to a wide variety of physico-chemical processes. These preliminary notes give an idea of the complexity of this topic, which controls however fundamental processes like microstructural changes, recrystallization, the nucleation of one phase into a parent phase, ionic conduction in electrolytes and so on. Despite the vastness and complexity of this topic, the present section aims to introduce in a deliberately intuitive and elementary way some basic concepts underlying the mass and charge transport; with this aim in mind, the mathematical difficulty that characterizes the modern theories of diffusion is intentionally waived to emphasize instead the conceptual link between diffusion physics and solid state ionics, with particular reference to the interest of the experimentalists on the essential concepts of the charge transport phenomena.

3.1.1 Phenomenology of Diffusion

The transport mechanisms differ, of course, depending on the nature of both diffusing species and the diffusion medium. Despite the inherent conceptual complexity, it is possible to identify some points that are common to the possible experimental situations. For instance, a general rule is that the transport occurs at decreasing rates in gases, liquids and solids. Also, from a phenomenological point of view, the most intuitive way to describe the displacement of matter is to define a flux having physical dimensions of matter per unit time and surface given by:

(3.3)

where c and v are the concentration and average displacement velocity of the diffusing species. This equation, although being a mere definition of flux J of matter rather than a physical law, introduces the basic ingredient to formulate the diffusion theory; see below a sketch of the “random walk” approach (i.e. the concentration); this way of defining the flux regards the amount of mass and charge actually transferred within a system of other atoms, ions and molecules not involved in the displacement.

The first formulation of diffusion law was due to Fick, who assumed a concentration gradient-driven effect between two contiguous volumes of a sample:

(3.4)

where ∇c is the concentration gradient and D is a proportionality factor. The minus sign means that the mass flow vector J is oriented against the concentration gradient, that is, the prospective effect of diffusion is a spontaneous flow of matter from a high-concentration region to a low-concentration region that tends to flatten the initial gradient. In turn, having tacitly assumed that diffusion is allowed to occur in an isolated system, this has a clear connection with the Second Law of Thermodynamics, about the spontaneous evolution of an isolated system towards the most probable and disordered state. Merging together Equations 3.3 and 3.4 yields:

(3.5a)

At the right-hand side, the chemical potential is expressed as a function of the concentration gradient . Regarding the gradient of this potential as the driving force of diffusion and recalling that the mobility of the diffusing species times is defined as , then:

(3.5b)

as is well known. A further refinement of Fick's law involves the activity rather than the concentration; assuming a linear relationship between these quantities, one finds:

(3.6)

where is a proper proportionality coefficient called the activity coefficient. Thus one expects that these laws can be reformulated according to a more general thermodynamic point of view. Note in this respect that combining Equation 3.5 with , analogous to Equation 3.3, yields:

(3.7)

At the right-hand side appears the chemical potential of the diffusing species; the constant is the activity of the diffusing species in its standard state. The corresponding standard chemical potential is taken here as zero. Since D is linked to the driving thermodynamic force that triggers the diffusion, the most general way of defining the mass flow is

(3.8)

where L is a new proportionality factor having physical dimensions of mobility times concentration, sometimes also called diffusivity. The last way to define explicitly the mass flow through its thermodynamic driving force has an important consequence. Since Equation 3.8 reads also , Equation 3.4 and the second Equation 3.6 yield:

(3.9)

This equation holds for the chemical diffusion in a homogeneous body of matter, for instance in the case of displacement of an isotope in a matrix of the same element. L being the product of mobility and concentration, from this equation one infers:

(3.10)

The former equation is the Darken equation, and the right-hand side of the first equation is also called the thermodynamic or Wagner factor. Moreover, the second equation, which agrees with that previously found, links the diffusion coefficient to the mobility of the particles; when extended to the case of charged particles, replacing the general value of with that of the amount of charge carried by the particle (i.e. with ), one finds:

(3.11)

Relevant interest also has the relationship that links the diffusion coefficient to the electrical conductivity . An elementary way to find the link between and D exploits the second Equation 3.7, simply specifying the force as that due to an electric field E acting on the charge ; so, assuming a one-dimensional (1D) case for simplicity of explanation, the equation reads , with being the electric potential. Multiply now both sides by and note that we obtain at the left-hand side a charge flow ; recalling that according to Ohm's law, the result is:

(3.12)

A link also exists between D and the mean squared displacement traveled by any number of non-interacting particles in the absence of a net driving force. An elementary derivation of this link is carried out here considering an ideal reference plane crossed by any particles randomly moving in the presence of a concentration gradient along the x-axis; this assumption reduces for simplicity the problem to the 1D motion perpendicularly to an arbitrary section of the plane. Consider on the reference plane and the concentration difference defined on two arbitrary points apart at the opposite sides of the plane with . Write , and multiply both sides of the equation by the diffusion rate defined as ; that is, is the distance traveled by the particles diffusing during the time range , both arbitrary and fixed once for all so that is constant. One finds , being the net flux of matter crossing the plane. Take the average of both sides of the equation defining , and calculate the average concentration on the plane: at the right-hand side, one finds the term ; at the left-hand side is the term . Regard this latter as the average net flux of matter crossing the plane, and define in agreement with the conservation of mass flow ; then is twice , that is, . So, comparing with Equation 3.4 one finds the well-known result:

(3.13)

The extension of this result to the three-dimensional (3D) case is trivial in an isotropic diffusion medium where by symmetry , so that ; thus Equation 3.13 reads:

(3.14)

Of course, a two-dimensional (2D) flow would have given . These results show that the diffusion coefficient, early introduced as a mere proportionality factor between an intuitive definition of mass flow, that is, mass per unit time and unit surface, and a concentration–activity gradient enters actually into a large variety of phenomena of physical and technological interest; this is actually due to its relationship with the thermodynamic force controlling both mass and displacement rate. Actually this link between flux and energy gradient is not accidental; rather its meaning appears in a variety of physical problems of scientific and technological interest; several physical laws entail indeed a formal analogy between flux and gradient typical of the Fick law. For example, Ohm's law concerns voltage and current I. It is trivial to realize that this law can be rewritten as , where is the charge flow. This equation is very important for simulating the ion charge transport through electrolytes, as will be shortly sketched below.

A further relationship linking the viscosity η to the diffusion coefficient is the so-called Einstein-Stokes equation that exploits the Stokes law ; at the left-hand side appears the drag coefficient, inversely proportional to the mobility of a spherical particle of hydrodynamical radius moving in a fluid of viscosity ; so Equation 3.6 yields:

(3.15)

This equation is particularly interesting to describe diffusion phenomena that involve large molecules, whose geometry is likely to be nonspherical or affected by hydration; since for a spherical geometry , where is the partial specific volume and M the molecular weight of the diffusing species, one finds immediately for two different species at constant T:

(3.16)

If the friction coefficient is different from that quoted here (e.g. because the actual geometry of the molecules is not spherical), then the deviations of the experimental diffusivity from that predicted by Equations 3.15 and 3.16 provide valuable information on the shape and interaction of the diffusing molecule. This equation can be tested as a function of the temperature because of the strong dependence of the viscosity on temperature. This results in an exponential form of D [3]:

(3.17)

where is the prefactor and is the translational activation energy of diffusion; for water, this energy is about 0.18 eV, that is, similar to that needed to break the H bonds. The exponential factor appearing at the right-hand side is clearly related to the Boltzmann distribution law, for reasons that will be clearer later. Another example where the formal analogy between different gradient-driven effects is important regards Ohm's law to describe the diffusion of charged species under an electrical gradient. Since the mass flow and the charge flow are linked as , it follows that the charge flow under a potential gradient fulfills the equation . On the other hand, the linear character of the equations with respect to the quantities concerned by the respective gradients, concentration and voltage, legitimates the additivity of the whole diffusion behavior. So, the charge transport in a solid electrolyte is represented by the flux equation:

(3.18)

The second equality, inferred with the help of Equation 3.12, shows that the driving force is given in this case by ; the fact that this force yields a flux of matter when multiplied by (i.e. mobility times concentration according to Equation 3.8) shows that indeed is the total chemical potential of the charge carried in a fuel cell. Equations 3.18 and 3.3 suggest therefore the following general form of the flux:

(3.19)

where the average drift velocity denotes the effect of a force external to the diffusion system that overlaps to that due to the concentration gradient; for instance, the presence of an external electric potential like that present across an electrolyte yields:

(3.20)

As expected, the average velocity is proportional to the diffusion coefficient and to the driving force ; recalling Equation 3.11 this is nothing else but the definition of mobility. Actually, Equation 3.18 should include a correction coefficient due to correlation effects in the process of diffusion. The physical meaning of this coefficient and this aspect of the diffusion process will be very shortly sketched below. Before introducing the microscopic mechanisms that govern the mass transfer, however, at this point of the exposition we stress that for practical purposes, in particular to calculate quantities susceptible to comparison with experimental results, it is useful to introduce a further diffusion equation: the second Fick equation, a straightforward consequence of Equation 3.4 that completes the previous considerations in the particular but very important case where a non-equilibrium flux occurs without sources or sinks. In this case holds the condition described by the so-called continuity equation, important because it introduces explicitly the time into the diffusion problem. To elucidate the general character of this equation, consider first for simplicity the 1D case and define an arbitrary function ; for instance, and could be diffusion coordinate and time. Calculate the change of between any , and , , differentiating f with respect to the variables yields at the first order. Assume now that f represents a quantity that is conserved in the given range of variables; if, for instance, f represents the concentration of a given element, this simply means redistributing a fixed amount of this element within during the time range without changing its total amount. Putting, then, , and with by definition an arbitrary length not dependent on the local coordinate x, one finds , where and . In the particular case of present interest, yields , that is, according to Equation 3.3, ; eventually this result reads also . In the general case where , the result is clearly:

(3.21)

which holds of course also for the activity. This approach evidences that the continuity equation excludes the presence of sources or sinks of matter. The lack of sources or sinks of matter assumed in this derivation is further emphasized considering the total mass flow in an arbitrary volume element . The first component of flux along the -axis yields at the first order the accumulation or loss of the species across the area of a flow of matter entering at and exiting at ; so is the net mass balance in the volume . If this mass flow occurs in a time range , then the net flux must be equal to the flow mass change . Putting and equating these expressions, one finds . It is trivial to extend this result to the 3D case; one obtains obviously Equation 3.21, having explicitly emphasized, however, the lack of sinks or sources in the arbitrary volume . The second Fick equation, which in fact is a direct expression of the continuity equation, reads then:

(3.22)

However, these considerations hold in isotropic media only. In anisotropic media, the diffusion coefficient must be replaced by a tensor of rank 2; the matrix representing such a diffusion tensor is:

(3.23)

The diagonal matrix elements describe the diffusion behavior along the , and axes of the laboratory frame; the off-diagonal elements describe the correlation between the diffusion behavior in perpendicular directions. Strictly speaking, the isotropic diffusion hitherto concerned is a particular case where with the off-diagonal term identically null. Owing to Onsager's reciprocity theorem, the diffusivity tensor is symmetric; since any such tensor can be transformed to its three principal axes, the D matrix then reduces to the form:

(3.24)

There are therefore three diffusion coefficients, called principal coefficients, whereas the unique Equation 3.4 splits into three equations having the forms with . Strictly speaking, only the cubic crystal symmetry is isotropic, so that , that is, . Instead, the hexagonal, tetragonal and rhombohedral symmetries have, for instance, , that is, , with the principal axis parallel to their crystal axis. From an experimental point of view, the diffusion coefficient should be therefore evaluated in principle with 1D measurements in single crystals oriented along well-defined crystallographic directions; if one of these directions is parallel to one of the principal axes, then , or is measured directly. In any case, the Fick laws provide a continuum description of diffusion, regardless of any correlation between diffusion coefficient and microstructure of the medium; any information on the latter is hidden in the value of D, so far merely regarded as a macroscopic parameter that summarizes statistically the local microscopic details of transport mechanisms.

A better knowledge of the mechanisms that control D and its expected correlation with microstructure has importance not only from a theoretical point of view but also for the possibility to optimize the transport phenomena, for example the ion drift in solid electrolytes. The random walk theory, based on the modeling of atomic jumps, allows a comprehensive picture of the microscopic details of the diffusion process. It is worth recalling that the atomistic nature of matter was first hypothesized by Einstein just regarding the Brownian motion of mesoscopic particles, described by an equation like Equation 3.13, as due to their random interaction with atoms and molecules moving in the surrounding fluid. In the case of solids, the atom displacement occurs via jumps through lattice sites, in particular vacancies. First of all, the atomistic point of view easily justifies the fundamental Equation 3.4, as shown in [4] and also shortly reported here. Consider and particles per unit surface of mass in two neighbor crystal planes a distance apart; assuming an average jump frequency between sites in the respective planes, the net flux of particles is , that is, in terms of concentrations . Extrapolate the term in parentheses to the ideal limit case of an infinitesimal concentration gap between infinitesimal crystal plane spacing; regarding this term as , since the jumps tend to decrease and increase if initially , one finds Equation 3.4 putting . From the kinetic point of view, it is possible to define the jump frequency, of the order of the Debye frequency to , and the residence time on a lattice site, usually much longer than the reciprocal jump frequency. The macroscopic travel distance results from the sum of multiple jump events, each of length . Without going into the mathematical details of the theory, it is evident that, like in any kinetic process, the activation energy also plays a fundamental role, upon which depends the resulting jump rate . Elementary considerations show that , where the coefficient Z is the number of nearest interstices available for the jump and typically takes the values 2 or 4 or 6; for instance, in a cubic lattice . The residence time is instead . The thermal energy of atoms in a lattice is of the order of , that is, around 0.025 eV at room temperature; the activation energy is, however, much higher than this, of the order of eV [5], so the Boltzmann distribution law in Equation 3.17 shows that the atoms mostly vibrate around their equilibrium positions. Sometimes, large oscillations allow displacements by one step , after which the atom is deactivated because of the energy spent to overcome the “saddle point” energy barrier corresponding to a successful jump. After further oscillations around the new position, it has again the chance of performing a further jump, and so on. The free energy gap between barrier and equilibrium energy, defined by the migration enthalpy and entropy, describes the energy balance of each jump. As concerns the net progression as a consequence of these jumps, there are two possibilities depending on whether each jump has memory or not of the previous events. A typical example of “memory” is that of atom and vacancy that exchange their places in the lattice; that is, the atoms fills the site where was previously located a vacancy, which is therefore annihilated, whereas a new vacancy is formed in the lattice site left behind by the atom jump. If this happens. one expects a significant probability that after the first jump the atom returns back to its previous position, which of course decreases the net migration probability. In the absence of “memory,” we speak of a “Markov sequence” or “uncorrelated random walk”; a typical example of this kind of process is that of diffusion in a lattice via a direct interstitial mechanism, that is, the atom randomly moves jumping through interstitial positions. Of course, also in this case, the atom can return back in the previous position; yet one expects that the probability of the reversed path is now less significant than before. To better understand this point, let us describe quantitatively the non-Markov situation through the so-called correlation coefficient, whose meaning is shortly sketched as follows. The total path of the tracer atom after an arbitrary number n of jumps defines the scalar ; the first sum averages the single jumps, and the second one two different jumps, for example, the ith one followed by the jth one. The Markov sequence requires , where the subscript “rnd” stands for “random”; indeed, to every pair of jumps corresponds another possible sequence of jumps equal and of opposite sign, so that the contribution of the second sum to the random path is statistically null. It is not so, however, if each jump has some memory of the previous ones. So it is possible to introduce the correlation factor defined as follows:

(3.25)

which therefore consists of a Markovian jump sequence plus a correction factor controlled by the sum of jumps. To guess the sign of the non-Markovian term, consider again the atom diffusion in solids activated by vacancies and recall the nonvanishing probability of jumps that follow the direction of motion of the vacancy. This occurrence reverses the diffusion path and corresponds to a negative contribution of terms of the sum, so that is negative. Hence one concludes that