Piezoelectric Transducers and Applications

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Piezoelectric Transducers and Applications

Antonio Arnau Vives (Ed.)

Piezoelectric Transducers and Applications Second Edition

123

Editor Prof. Dr. Antonio Arnau Vives Universidad Polit´ecnica de Valencia Departamento de Ingenier´ıa Electr´onica Camino de Vera, s/n E-46020 VALENCIA Spain [email protected]

ISBN: 978-3-540-77507-2

e-ISBN: 978-3-540-77508-9

DOI: 10.1007/978-3-540-77508-9 Library of Congress Control Number: 2008922498 c 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: eStudio Calamar Girona, Spain Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

Foreword Since the publication of the first edition, the richness of the study of piezoelectric transducers has resulted in a large number of studies dealing both with new understandings underlying the principles, with new technological advances in its applications and indeed with developing new areas of utility for these transducers. The motivations driving the publication of that first edition as described in its foreword (which follows) continues with increased validity. The value of a second edition to include these new developments has been prepared. During the interim, the contributors and their students have not only continued, but increased their mutual interactions resulting in an amazing energy and synergy which is revealed in this edition. One of the most valuable aids to those beginning to investigate a new area of study is a source which will guide them from beginning principles, through detailed implementation and applications. Even for seasoned investigators, it is useful to have a reasonably detailed discussion of closely related topics in a single volume to which one can refer. This is often difficult for many emerging areas of studies because they are so multidisciplinary. The subject matter of the principles, techniques and applications of piezoelectric transducers certainly fits into this category. The host of emerging new uses of piezoelectric devices that are being commercialized as well as the growing number of potential applications ensures that this field will encompass more and more disciplines with passing time. It is extremely fortunate and timely that this volume becomes available to the student at this time. Piezoelectricity is a classical discipline traced to the original work of Jacques and Pierre Curie around 1880. This phenomenon describes the relations between mechanical strains on a solid and its resulting electrical behavior resulting from changes in the electric polarization. One can create an electrical output from a solid resulting from mechanical strains, or can create a mechanical distortion resulting from the application of an electrical perturbation. In the former case, the unit acts as a receiver of mechanical variations, converting it into electrical output, as in the case of a microphone. In the latter case, the unit can act as a transmitter converting the electrical signal into a mechanical wave. The piezoelectric units can be used both in narrow-band or resonant modes, and under broad-band regimes for detection and imaging applications. One of the remarkable properties of these devices is the ability to use them in a viscous medium, such as a liquid. When excited sinusoidally, these devices can generate waves in

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the immersing medium. Typically, as a result of the physical size of these devices, the waves are in the ultrasonic regime. From this classical discipline, an astounding number of applications are developing. From its use as a frequency generating standard in the earlier part of the 20th century, additional uses have seen these devices used as highly sensitive mass balances for use both in vacuum deposition and in electrochemical applications, as well as chemical specific sensors, as Doppler devices for fluid velocity measurements and for ultrasonic imagery. There are many other emerging applications in the bio-sciences for example. The number of applications is astounding. It is clear that the discipline is inter-disciplinary. The authors of the contents of this book are a select group who has all been challenged by the intellectual diversity of the field. To successfully pass on such diverse information, intellectual competence is only a beginning. A devotion to, and love of clear communication is also required. These authors are members of the PETRA organization, (Piezoelectric Transducers and Applications) sponsored by the European Union, devoted to the collection and dissemination of knowledge and skills in the piezoelectric arts to students among the participating universities in Europe and Latin America. I have personally observed many of the authors interacting with students and have been very impressed by their care and mentoring. Contributions from such dedicated and seasoned teachers are now available to the student in this volume. This book fills a real need for a unified source for information on piezoelectric devices, ranging from broadband applications to resonant applications and will serve both experienced researchers and beginning students well. Kay Keiji Kanazawa Technical Director, Emeritus CPIMA Stanford University Stanford, CA 94305

Preface Following the execution of the project PETRA-I, co-financed by the European Union in the framework of the ALFA Program (America Latina Formación Académica), as coordinator of the PETRA Network (PiezoElectric TRansducers and their Applications), I edited the first edition of this book. Now, four years later, I am submitting the manuscript of this revised and enlarged 2nd edition. This edition has been, in fact, an unexpected result of the project PETRA-II, also co-financed by the European Union in the framework of the ALFA Program. Effectively, halfway through the execution of this project, Springer-Verlag informed me of the good reception that the book had received and asked me if we had thought about a new edition. This very good news meant that the work during the previous four years had been worth it. The initial idea of collecting in one single volume a set of “tutorials” covering topics spread on different disciplines and linked by the use of piezoelectric devices was therefore useful. The interdisciplinary character of the discipline was made clear and the “tutorial” based format could be useful as a guide for doctoral degree students and even researchers going into this complex and multidisciplinary issue. Now we have the opportunity of improving that first approach but without losing what we think are the keys of its success: the “tutorial” style and the multidisciplinary character of the contents. The new edition covers, in 18 chapters and two appendices, different aspects of piezoelectric devices and their applications, as well as fundamental topics of related disciplines. The contents were selected according to the different areas of research of the partners of the PETRA Network; therefore, this book does not intend to be an encyclopaedia on piezoelectric transducers and their applications, which would be completely impossible in a single-volume work. Three different parts, although not explicitly separated, can be distinguished in the book: one part corresponds to general concepts on piezoelectric devices and to the fundamentals of related topics (Chapters 1,2, 7-12 and the two appendices), another part deals with piezoelectric sensors and related applications (Chapters 1,3,5,12-14) and the other part focuses on ultrasonic transducers and systems and related applications (Chapters 4,6, 15-18). Basic concepts of piezoelectricity are presented in Chap. 1 along with an introduction into the field of microgravimetric sensors; appendices A and B, at the end of the book, include fundamental concepts of electrostatics

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and physical properties of crystals which complement this initial introduction. Chapter 2 offers an overview of acoustic sensors, their basic principles of operation, the different types and their potential applications. Recent new excitation principles for bulk acoustic wave sensors such as lateral field or magnetic excitations have been added in this edition, as well as the topic of micromachined resonators such as cantilevers (MEMS) based on silicon technologies which are attracting current interest. Chapters 3, 5, 13 and 14 delve more deeply into resonant sensors, especially bulk acoustic wave thickness shear mode resonators and their applications as quartz crystal microbalance sensors, their fundamentals and models (Chap. 3), electronic interfaces and associated problems (Chap. 5), the problems associated with the analysis and interpretation of experimental data (Chap. 14) and complementary techniques used with QCM (Chap. 13). In this 2nd edition, a thorough revision of these chapters with the addition of some important topics has been made. Sub-chapters dealing with the gravimetric and non-gravimetric regimes in QCM applications and the important aspect of kinetic analysis in acoustic wave sensor-based chemical applications have been added to Chap. 3. A comprehensive review of the different electronic interfaces for QCM sensors has been included in Chap. 5; in particular the topic of oscillators for in-liquid QCM applications is deeply treated in this edition, as well as the new interface systems based on lock-in techniques. Techniques based on impedance analysis, or adapted impedance analyzers, and decay method techniques have also been updated and interfaces for fast QCM applications, such as ac-electrogravimetry (Chap. 13), have also been included. The problem of compatibility between QCM and electrochemical set-ups is treated in Chap. 13. Chapter 14 has been thoroughly revised and a completely new section with case studies has been added to complement the complex aspect of data analysis and interpretation in real experiments. The section devoted to “other effects”, which complicates even more the interpretation of results, has been extended with the inclusion of the roughness effect. As the case studies section in Chap. 14 makes clear, acoustic wave sensors are involved in applications such as biosensors, electrochemistry and polymer properties’ characterization, which require a minimum background to deal with. This background is intended to be given in Chaps. 712. Thus, Chap. 7 introduces the concept of viscoelasticity and describes in depth the physical properties of polymers. A very important aspect in resonant sensor applications is the shear parameter determination that has been added as a new subchapter in this tutorial. Chapter 8 introduces the fundamentals of electrochemistry; in relation to the first edition, the section on “What is an electrode reaction?” has been extended with more explanation on the process of electron transfer and a

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corresponding schematic figure. The section on “Rates of electrode reactions” now includes a paragraph and figure describing the important role of the interfacial region and the definition of Faraday’s law. Additionally, the section on electrochemical techniques has been significantly enlarged with respect to steady-state, pulse and impedance techniques. The final section shows the range of possible applications of electrochemistry. Chapter 9 provides an overview of chemical sensors, which is of great interest for establishing the differences between chemical sensors based on piezoelectric transducers and those based on other techniques such as electrochemical, optical, calorimetric, conductimetric (added in this version) and magnetic techniques, with the aim of facilitating the interpretation of the different data. Chapter 10 treats the specific topic of biosensors from a biological point of view; this treatment is specifically useful to understand the mechanism of biological recognition and its potential use for the development of biosensors and especially for piezoelectric biosensors, which is a field of much current interest. A new chapter (Chap. 12) has been added which introduces the fundamentals of piezoelectric immunosensors giving the basic schemes of biosensor functioning, immunoassay formats, and the principle of competitive immunoassay. The different steps involved in the production and immobilization of immunoreagents are treated in detail in this chapter which finishes with a real example of characterization of a piezoelectric immunosensor. The processes involved in a piezoelectric immunosensor make clear the necessity of the resonator sensor surface modification. This topic is treated in depth in Chap. 11 which provides a guide to the important subject of modification of piezoelectric surfaces in piezoelectric transducers for sensor applications. Some additional examples have been added in this new version. Chapters 4, 6, 15-18 deal with ultrasonic systems and applications. Chapter 4 introduces the basic aspects and the different models of piezoelectric transducers for broadband ultrasonic applications; electronic interfaces used in broadband configurations are introduced in Chap. 6; implementations of ultrasonic schemes and electronic interfaces for nondestructive testing industrial applications are detailed and analysed in Chap. 16, and some applications of ultrasound in chemistry and in medicine are treated in Chaps. 15 (Sonoelectrochemistry), 17 (Medical imaging) and 18 (Ultrasound hyperthermia). In this edition new topics have been added in the previous chapters. In Chap. 4 three sub-chapters have been added dealing with the transfers functions and time responses at emission and reception of the transducer, the acoustic impedance matching and the electrical matching and tuning. Chap. 6 includes a new sub-chapter dealing with the analysis of electrical responses in pulse-driven piezoelectric transducers by means of linear approaches has been added, including

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the inductive tuning case. In Chap. 16 two new sections have been added dealing with the electronic sequential scanning of ultrasound beams for fast operation in non-destructive testing applications. Chap. 17 is a new chapter added to deal with the application of ultrasound systems for medical imaging and tissue characterization; a basic introduction to the ultrasound properties of biological materials with different ultrasonic imaging modes is followed by a comprehensive review of the different techniques used for medical imaging. Chapter 18 includes a concise introduction to the clinical procedure and biological basis of hyperthermia therapy. In this second edition key information concerning the technical perspective of this treatment has been added: the ultrasound field measurement by the mechanically scanning method is described. In this section, it is explained how a 3D representation of the space domain response of the transducer can be obtained by using a hydrophone. In another section, the way ultrasound produces temperature increases in tissues in described. A description of the components of a general hyperthermia ultrasound system has also been included. Superficial and deep heat systems are also depicted. Finally, the ways in which ultrasound hyperthermia systems are characterized are treated, such as in the preparation and measuring of the properties of a tissue mimicking material (phantom) for use in ultrasonic hyperthermia. Finally, Chap. 15 deals with the application of ultrasound in electrochemistry. In this edition the sections on basic consequences of ultrasound and on the experimental arrangements have been extended. In the former case, more discussion of the formation of cavitation bubbles and their collapse is included. In the latter case, the horn probes are discussed in more detail. Some more applications are referred to in particular nanomaterials (new sub-section). The present volume is therefore a revised and enlarged version of the first edition which would not have been possible without the effort and dedication of all my colleagues, who contributed with the different chapters, to all of whom I will always be in debt. I would like to take advantage of this new opportunity to thank them again for giving me their confidence as coordinator of the PETRA group. My thanks also go to Springer for undertaking this new edition. New challenges are waiting for us in the near future. I hope we will be able to face them enthusiastically and with excitement. The future is a challenge that we pose to our thoughts and makes sense of our lives. Antonio Arnau Vives November 2007

Contents Associated Editors and Contributors...............................................XXIII 1 Fundamentals of Piezoelectricity......................................................... 1 1.1 Introduction .................................................................................... 1 1.2 The Piezoelectric Effect ................................................................. 2 1.3 Mathematical Formulation of the Piezoelectric Effect. A First Approach ............................................................................ 4 1.4 Piezoelectric Contribution to Elastic Constants ............................. 5 1.5 Piezoelectric Contribution to Dielectric Constants ........................ 5 1.6 The Electric Displacement and the Internal Stress......................... 6 1.7 Basic Model of Electric Impedance for a Piezoelectric Material Subjected to a Variable Electric Field.............................. 7 1.8 Natural Vibrating Frequencies ..................................................... 12 1.8.1 Natural Vibrating Frequencies Neglecting Losses............ 12 1.8.2 Natural Vibrating Frequencies with Losses ...................... 15 1.8.3 Forced Vibrations with Losses. Resonant Frequencies..... 20 1.9 Introduction to the Microgravimetric Sensor ............................... 25 Appendix 1.A........................................................................................ 28 The Butterworth Van-Dyke Model for a Piezoelectric Resonator .......................................................................... 28 1.A.1 Rigorous Obtaining of the Electrical Admittance of a Piezoelectric Resonator. Application to AT Cut Quartz................................................................................ 28 1.A.2 Expression for the Quality Factor as a Function of Equivalent Electrical Parameters ...................................... 35 References ............................................................................................ 37 2 Overview of Acoustic-Wave Microsensors ....................................... 39 2.1 Introduction .................................................................................. 39 2.2 General Concepts ......................................................................... 40 2.3 Sensor Types ................................................................................ 42 2.3.1 Quartz Crystal Thickness Shear Mode Sensors ................ 42 2.3.2 Thin-Film Thickness-Mode Sensors ................................ 43 2.3.3 Surface Acoustic Wave Sensors........................................ 45 2.3.4 Shear-Horizontal Acoustic Plate Mode Sensors ............... 46 2.3.5 Surface Transverse Wave Sensors .................................... 47 2.3.6 Love Wave Sensors........................................................... 48 2.3.7 Flexural Plate Wave Sensors............................................. 48

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2.3.8 Other Excitation Principles of BAW Sensors ................... 49 2.3.9 Micromachined Resonators............................................... 53 2.4 Operating Modes .......................................................................... 55 2.5 Sensitivity..................................................................................... 57 References ............................................................................................ 59 3 Models for Resonant Sensors............................................................. 63 3.1 Introduction .................................................................................. 63 3.2 The Resonance Phenomenon........................................................ 63 3.3 Concepts of Piezoelectric Resonator Modeling ........................... 64 3.4 The Equivalent Circuit of a Quartz Crystal Resonator................. 69 3.5 Six Important Conclusions ........................................................... 72 3.5.1 The Sauerbrey Equation.................................................... 72 3.5.2 Kanazawa’s Equation........................................................ 73 3.5.3 Resonant Frequencies........................................................ 73 3.5.4 Motional Resistance and Q Factor .................................... 74 3.5.5 Gravimetric and Non-Gravimetric Regime....................... 74 3.5.6 Kinetic Analysis................................................................ 75 Appendix 3.A........................................................................................ 77 3.A.1 Introduction....................................................................... 77 3.A.2 The Coated Piezoelectric Quartz Crystal. Analytical Solution ........................................................... 78 3.A.3 The Transmission Line Model .......................................... 82 The piezoelectric quartz crystal ........................................ 83 The Acoustic Load ............................................................ 86 3.A.4 Special Cases..................................................................... 88 The Modified Butterworth-Van Dyke Circuit................... 88 The Acoustic Load Concept.............................................. 89 Single Film........................................................................ 90 The Sauerbrey Equation.................................................... 92 The Kanazawa Equation ................................................... 93 Martin’s Equation ............................................................. 93 Small phase shift approximation....................................... 94 References ............................................................................................ 95 4 Models for Piezoelectric Transducers Used in Broadband Ultrasonic Applications...................................................................... 97 4.1 Introduction .................................................................................. 97 4.2 The Electromechanical Impedance Matrix................................... 98 4.3 Equivalent Circuits ..................................................................... 102 4.4 Broadband Piezoelectric Transducers as Two-Port Networks. ................................................................................... 105

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4.5 Transfer Functions and Time Responses.................................... 107 4.6 Acoustic Impedance Matching ................................................... 110 4.7 Electrical matching and tuning................................................... 114 References .......................................................................................... 115 5 Interface Electronic Systems for AT-Cut QCM Sensors: A comprehensive review................................................................... 117 5.1 Introduction ................................................................................ 117 5.2 A Suitable Model for Including a QCM Sensor as Additional Component in an Electronic Circuit ......................... 118 5.3 Critical Parameters for Characterizing the QCM Sensor ........... 120 5.4 Systems for Measuring Sensor Parameters and their Limitations.................................................................................. 124 5.4.1 Impedance or Network Analysis ..................................... 124 Adapted Impedance Spectrum Analyzers ....................... 126 5.4.2 Decay and Impulse Excitation Methods.......................... 129 5.4.3 Oscillators ....................................................................... 133 Basics of LC Oscillators.................................................. 134 Oscillating Conditions..................................................... 136 Parallel Mode Crystal Oscillator ..................................... 136 Series Mode Crystal Oscillator ....................................... 138 Problem Associated with the MSRF Determination ....... 140 Problem Associated with the Motional Resistance Determination ......................................................... 142 Oscillators for QCM Sensors. Overview......................... 142 5.4.4 Interface Systems for QCM Sensors Based on Lock-in Techniques......................................................... 162 Phase-Locked Loop Techniques with Parallel Capacitance Compensation..................................... 163 Lock-in Techniques at Maximum Conductance Frequency................................................................ 169 5.4.5 Interface Circuits for Fast QCM Applications ................ 171 5.5 Conclusions ................................................................................ 173 Appendix 5.A...................................................................................... 174 Critical Frequencies of a Resonator Modeled as a BVD Circuit.............................................................................. 174 5.A.1 Equations of Admittance and Impedance........................ 174 5.A.2 Critical Frequencies ........................................................ 176 Series and parallel resonant frequencies ......................... 176 Zero-Phase frequencies ................................................... 177 Frequencies for Minimum and Maximum Admittance... 178 5.A.3 The Admittance Diagram ................................................ 178 References .......................................................................................... 180

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6 Interface Electronic Systems for Broadband Piezoelectric Ultrasonic Applications: Analysis of Responses by means of Linear Approaches....................................................................... 187 6.1 Introduction ................................................................................ 187 6.2 General Interface Schemes for an Efficient Coupling of Broadband Piezoelectric Transducers ........................................ 188 6.3 Electronic Circuits used for the Generation of High Voltage Driving Pulses and Signal Reception in Broadband Piezoelectric Applications ....................................... 190 6.3.1 Some Classical Circuits to Drive Ultrasonic Transducers ..................................................................... 190 6.3.2 Electronic System Developed for the Efficient Pulsed Driving of High Frequency Transducers ............. 192 6.3.3 Electronic Circuits in Broadband Signal Reception........ 195 6.4 Time Analysis by Means of Linear Approaches of Electrical Responses in HV Pulsed Driving of Piezoelectric Transducers........................................................... 197 6.4.1 Temporal Behaviour of the Driving Pulse under Assumption 1 .................................................................. 198 6.4.2 Temporal Behaviour of the Driving Pulse under Assumption 2 .................................................................. 200 6.4.3 Behaviour of the Driving Pulse under Assumption 3: The Inductive Tuning Case ............................................. 201 References .......................................................................................... 203 7 Viscoelastic Properties of Macromolecules .................................... 205 7.1 Introduction ................................................................................ 205 7.2 Molecular Background of Viscoelasticity of Polymers.............. 206 7.3 Shear Modulus, Shear Compliance and Viscosity...................... 209 7.4 The Temperature-Frequency Equivalence ................................. 214 7.5 Conclusions ................................................................................ 219 7.6 Shear Parameter Determination.................................................. 220 References .......................................................................................... 221 8 Fundamentals of Electrochemistry ................................................. 223 8.1 Introduction ................................................................................ 223 8.2 What is an Electrode Reaction?.................................................. 223 8.3 Electrode Potentials.................................................................... 225 8.4 The Rates of Electrode Reactions............................................... 226 8.5 How to Investigate Electrode Reactions Experimentally ........... 229

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8.6 Electrochemical Techniques and Combination with Non-Electrochemical Techniques .............................................. 231 8.7 Applications................................................................................ 236 8.8 Bibliography............................................................................... 237 8.9 Glossary of Symbols .................................................................. 238 References .......................................................................................... 238 9 Chemical Sensors .............................................................................. 241 9.1 Introduction ................................................................................ 241 9.2 Electrochemical Sensors............................................................. 243 9.2.1 Potentiometric Sensors.................................................... 244 9.2.2 Amperometric Sensors .................................................... 246 9.2.3 Conductimetric Sensors .................................................. 248 9.3 Optical Sensors........................................................................... 250 9.4 Acoustic Chemical Sensors ........................................................ 251 9.5 Calorimetric Sensors .................................................................. 252 9.6 Magnetic Sensors ....................................................................... 254 References .......................................................................................... 256 10 Biosensors: Natural Systems and Machines ................................... 259 10.1 Introduction ................................................................................ 259 10.2 General Principle of Cell Signaling............................................ 259 10.3 Biosensors .................................................................................. 263 10.3.1 Molecular Transistor ....................................................... 267 10.3.2 Analogy and Difference of Biological System and Piezoelectric Device........................................................ 267 References .......................................................................................... 269 11 Modified Piezoelectric Surfaces....................................................... 271 11.1 Introduction ................................................................................ 271 11.2 Metallic Deposition .................................................................... 271 11.2.1 Vacuum Methods ............................................................ 272 Evaporation (Metals)....................................................... 272 Sputtering (Metals or Insulating Materials) .................... 272 11.2.2 Electrochemical Method ................................................. 272 11.2.3 Technique Based on Glued Solid Foil (Nickel, Iron, Stainless Steel…).................................................... 274 11.3 Chemical Modifications (onto the metallic electrode) ............... 275 11.3.1 Organic Film Preparation................................................ 275 Polymer Electrogeneration (Conducting Polymers: Polypyrrole, Polyaniline…) .................................... 275

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11.3.2 Monolayer assemblies..................................................... 276 SAM Techniques (Thiol Molecule) ................................ 276 Langmuir-Blodgett Method ............................................ 277 Self-Assembled Polyelectrolyte and Protein Films......... 277 11.4 Biochemical Modifications ........................................................ 278 11.4.1 Direct Immobilisation of Biomolecules (Adsorption, Covalent Bonding)..................................... 279 11.4.2 Entrapping of Biomolecules (Electrogenerated Polymers: Enzyme, Antibodies, Antigens…) ................. 283 11.4.3 DNA Immobilisation....................................................... 284 References .......................................................................................... 286 12 Fundamentals of Piezoelectric Immunosensors ............................. 289 12.1 Introduction ................................................................................ 289 12.2 Hapten synthesis......................................................................... 293 12.3 Monoclonal antibody production ............................................... 295 12.4 Immobilization of immunoreagents ........................................... 296 12.5 Characterization of the piezoelectric immunosensor.................. 299 References .......................................................................................... 303 13 Combination of Quartz Crystal Microbalance with other Techniques......................................................................................... 307 13.1 Introduction ................................................................................ 307 13.2 Electrochemical Quartz Crystal Microbalance (EQCM)............ 308 13.2.1 ac-electrogravimetry ....................................................... 310 13.2.2 Compatibility between QCM and Electrochemical measurements.................................................................. 311 13.3 QCM in Combination with Optical Techniques......................... 313 13.4 QCM in Combination with Scanning Probe Techniques ........... 318 13.5 QCM in Combination with Other Techniques ........................... 321 Appendix 13.A: Determination of the Layer Thickness by EQCM ................................................................................... 322 Appendix 13.B: Fundamentals on Ellipsometry................................. 323 References .......................................................................................... 326 14 QCM Data Analysis and Interpretation ......................................... 331 14.1 Introduction ................................................................................ 331 14.2 Description of the Parameter Extraction Procedure: Physical Model and Experimental Data ..................................... 332 14.2.1 Physical Model................................................................ 333 14.2.2 Experimental Parameters for Sensor Characterization .............................................................. 334

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14.3 Interpretation of Simple Cases ................................................... 338 14.3.1 One Sauerbrey-Like Behavior Layer .............................. 339 14.3.2 One Semi-Infinite Newtonian Liquid.............................. 340 14.3.3 One Semi-Infinite Viscoelastic Medium......................... 341 14.3.4 One Thin Rigid Layer Contacting a Semi-Infinite Medium ........................................................................... 343 14.3.5 Summary ......................................................................... 345 14.3.6 Limits of the Simple Cases ............................................. 346 Limits of the Sauerbrey Regime ..................................... 346 Limits of the Small Surface Load Impedance Condition and of the BVD Approximation ..................... 349 14.4 Interpretation of the General Case.............................................. 351 14.4.1 Description of the Problem of Data Analysis and Interpretation in the General Case................................... 351 14.4.2 Restricting the Solutions by Increasing the Knowledge about the Physical Model............................. 352 Restricting the Solutions by Measuring the Thickness by an Alternative Technique .......................... 352 Restricting the Solutions by Assuming the Knowledge of Properties Different from the Thickness................................................................... 355 Restricting the Solutions by a Controlled Change of the Properties of the Second Medium......................... 355 14.4.3 Restricting the Solutions by Increasing the Knowledge about the Admittance Response................... 356 Restricting the Solutions by Measuring the Admittance Response of the Sensor to Different Harmonics........................................................ 356 Restricting the solutions by Measuring the Admittance Response of the Sensor in the Range of Frequencies around Resonance................................... 357 14.4.4 Additional Considerations. Calibration........................... 357 14.4.5 Other Effects. The N-layer Model................................... 359 Four-Layer Model for the Description of the Roughness Effect ............................................................ 360 14.5 Case Studies ............................................................................... 367 14.5.1 Case Study I: Piezoelectric Inmunosensor for the Pesticide Carbaril ............................................................ 367 Model ............................................................................. 368 Experimental Methodology............................................. 369 Calibration of the piezoelectric transducer...................... 369 Results and Discussion.................................................... 370

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14.5.2 Case Study II: Microrheological Study of the Aqueous Sol-Gel Process in the Silica-Metalisicate System............................................................................. 372 Model ............................................................................. 373 Experimental Methodology............................................. 374 Results and Discussion.................................................... 374 14.5.3 Case Study III: Viscoelastic Characterization of Electrochemically prepared Conducting Polymer Films ............................................................................... 378 Model ............................................................................. 379 Experimental Methodology............................................. 380 Results and Discussion.................................................... 380 Appendix 14.A: Obtaining of the Characteristic Parameters of the Roughness Model Developed by Arnau el al. in the Gravimetric Regime ......................................................... 391 References .......................................................................................... 393 15 Sonoelectrochemistry ....................................................................... 399 15.1 Introduction ................................................................................ 399 15.2 Basic Consequences of Ultrasound ............................................ 400 15.3 Experimental Arrangements....................................................... 402 15.4 Applications ............................................................................... 405 15.4.1 Sonoelectroanalysis ......................................................... 405 15.4.2 Sonoelectrosynthesis ....................................................... 406 15.4.3 Ultrasound and Bioelectrochemistry ............................... 406 15.4.4 Corrosion, Electrodeposition and Electroless Deposition ....................................................................... 406 15.4.5 Nanostructured Materials ................................................ 407 15.4.6 Waste Treatment and Digestion ...................................... 408 15.4.7 Multi-frequency Insonation ............................................. 408 15.5 Final Remarks............................................................................. 408 References .......................................................................................... 409 16 Ultrasonic Systems for Non-Destructive Testing Using Piezoelectric Transducers: Electrical Responses and Main Schemes ................................................................................... 413 16.1 Generalities about Ultrasonic NDT ............................................ 413 16.1.1 Some requirements for the ultrasonic responses in NDT applications ........................................................ 414 16.2 Through-Transmission and Pulse-Echo Piezoelectric Configurations in NDT Ultrasonic Transceivers........................ 415

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16.3 Analysis in the Frequency and Time Domains of Ultrasonic Transceivers in Non-Destructive Testing Processes .................. 417 16.4 Multi-Channel Schemes in Ultrasonic NDT Applications for High Resolution and Fast Operation..................................... 422 16.4.1 Parallel Multi-Channel Control of Pulse-Echo Transceivers for Beam Focusing and Scanning Purposes .......................................................................... 423 16.4.2 Electronic Sequential Scanning of Ultrasonic Beams for Fast Operation in NDT .................................. 425 A Mux-Dmux of High-voltage Pulses with Low On-Impedance ................................................................. 427 References .......................................................................................... 429 17 Ultrasonic Techniques for Medical Imaging and Tissue Characterization ............................................................................... 433 17.1 Introduction ................................................................................ 433 17.2 Ultrasound Imaging Modes ........................................................ 434 17.2.1 Basic ultrasonic properties of biological materials.......... 434 17.2.2 A-Mode ........................................................................... 435 17.2.3 B-Mode............................................................................ 436 17.2.4 Other Types of B-mode Images....................................... 439 Tissue harmonic imaging and contrast agents................. 439 3D ultrasound imaging.................................................... 440 17.2.5 Doppler Imaging.............................................................. 441 17.2.6 Ultrasound Computed Tomography (US-CT)................. 442 17.2.7 Ultrasound Elastography ................................................. 444 17.2.8 Ultrasound Biomicroscopy (UBM) ................................. 449 17.2.9 Computer-Aided Diagnosis in Ultrasound Images.......... 450 17.3 Quantitative Ultrasound (QUS) .................................................. 453 17.3.1 Speed of Sound (SOS)..................................................... 453 17.3.2 Acoustic attenuation coefficient ...................................... 454 17.3.3 Backscatter coefficient .................................................... 456 17.3.4 Periodicity Analysis: the Mean Scatterer Spacing (MSS) .............................................................................. 457 Acknowledgements ............................................................................ 459 References .......................................................................................... 459 18 Ultrasonic Hyperthermia ................................................................. 467 18.1 Introduction ................................................................................ 467 18.2 Ultrasonic Fields......................................................................... 468 18.2.1 Ultrasound Field Measurement ....................................... 470

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18.3 Ultrasonic Generation................................................................. 471 18.3.1 Piezoelectric Material...................................................... 472 18.3.2 The Therapy Transducer.................................................. 474 18.3.3 Additional Quality Indicators .......................................... 474 18.3.4 Beam Non Uniformity Ratio ........................................... 475 18.3.5 Effective Radiating Area (ERA) ..................................... 475 18.4 Wave Propagation in Tissue ....................................................... 475 18.4.1 Propagation Velocity ....................................................... 475 18.4.2 Acoustic Impedance ........................................................ 476 18.4.3 Attenuation ...................................................................... 477 18.4.4 Heating Process ............................................................... 478 18.5 Ultrasonic Hyperthermia ............................................................ 479 18.6 Hyperthermia Ultrasound Systems ............................................. 480 18.6.1 Superficial Heating systems ............................................ 482 Planar Transducer Systems ............................................. 482 Mechanically Scanned Fields.......................................... 482 18.6.2 Deep Heating Systems..................................................... 482 Mechanical Focusing ...................................................... 483 Electrical focusing........................................................... 483 18.6.3 Characterization of Hyperthermia Ultrasound Systems .... 483 Ultrasound Phantoms ...................................................... 484 Ultrasound Phantom-Property Measurements ................ 487 18.7 Focusing Ultrasonic Transducers ............................................... 489 18.7.1 Spherically Curved Transducers...................................... 489 18.7.2 Ultrasonic Lenses ............................................................ 490 18.7.3 Electrical Focusing .......................................................... 490 18.7.4 Transducer Arrays ........................................................... 490 18.7.5 Intracavitary and Interstitial Transducers ........................ 491 18.8 Trends......................................................................................... 493 References .......................................................................................... 493 Appendix A: Fundamentals of Electrostatics...................................... 497 A.1 Principles on Electrostatics ........................................................ 497 A.2 The Electric Field ....................................................................... 498 A.3 The Electrostatic Potential.......................................................... 499 A.4 Fundamental Equations of Electrostatics ................................... 500 A.5 The Electric Field in Matter. Polarization and Electric Displacement .............................................................................. 501

Contents

XXI

Appendix B: Physical Properties of Crystals ...................................... 509 B.1 Introduction ................................................................................ 509 B.2 Elastic Properties ........................................................................ 509 B.2.1 Stresses and Strains........................................................... 510 B.2.2 Elastic Constants. Generalized Hooke’s Law ................... 516 B.3 Dielectric Properties ................................................................... 520 B.4 Coefficients of Thermal Expansion............................................ 521 B.5 Piezoelectric Properties .............................................................. 521 Index........................................................................................................ 525

Associated Editors and Contributors Arnau, A. Departamento de Ingeniería Electrónica, Universidad Politécnica de Valencia Camino de Vera s/n, Valencia E-46022 Spain Bittencourt, Ch. Instituto Luiz Alberto Coimbra de Pós-Graduação e Pesquisa em Engenharia (COPPE). Universidade Federal do Rio de Janeiro – UFRJ Rio de Janeiro, Brasil Brett, C. Departamento de Química, Faculdade de Ciências e Tecnologia, Universidade de Coimbra Rua Larga, Coimbra 3004-535 Portugal Calvo, E. Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires Pabellon 2, Ciudad Universitaria, AR-1428 Buenos Aires Argentina Canetti, R. Instituto de Ingeniería Eléctrica, Facultad de Ingeniería, Universidad de la República, Montevideo Uruguay Coelho, W. Instituto Luiz Alberto Coimbra de Pós-Graduação e Pesquisa em Engenharia (COPPE). Universidade Federal do Rio de Janeiro – UFRJ Rio de Janeiro, Brasil

XXIV

Associated Editors and Contributors

Ferrari, V. Dipartamento di Electrónica per l’Automazione, Universita’ degli Studi di Brescia Via Branze 38, Brescia I-25123 Italy Jiménez, Y. Departamento de Ingeniería Electrónica, Universidad Politécnica de Valencia. Camino de Vera s/n, Valencia E-46022 Spain Kanazawa, K. Center of Polymer Interfaces and Macromolecular Assemblies, Stanford University Stanford University, North-South Mall 381, Stanford, CA 94305-5025 USA Leija, L. Departamento de Ingeniería Eléctrica, Centro de Investigación y Estudios Avanzados Avda. Instituto Politécnico Nacional Nº 2508, San Pedro Zacatenco, Mexico, D.F. 07360 Mexico, D.F. Lucklum, R. Institute for Micro and Sensor Systems, Otto-Von-Guericke Universität Magdeburg Universitätsplatz 2, Magdeburg D-39106 Germany March, C. Instituto de Investigación e Innovación en Bioingeniería, Universidad Politécnica de Valencia Camino de Vera s/n, Valencia E-46022 Spain Montoya, A. Instituto de Investigación e Innovación en Bioingeniería, Universidad Politécnica de Valencia Camino de Vera s/n, Valencia E-46022 Spain

Associated Editors and Contributors

XXV

Muñoz, R. Departamento de Ingeniería Eléctrica, Centro de Investigación y Estudios Avanzados Avda. Instituto Politécnico Nacional nº 2508, San Pedro Zacatenco, Mexico, D.F. 07360 Mexico, D.F. Negreira, C. Instituto de Física, Facultad de Ciencias, Universidad de la República, Montevideo Uruguay Ocampo, A. Departamento de Ingeniería Industrial, Escuela de Ingeniería de Antioquia, Calle 25 sur No. 42-73 Envigado Colombia Otero, M. Departamento de Fisica Aplicada. Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires. Pabellón 1, Ciudad Universitaria C1428EGA. Buenos Aires Argentina. Perrot, H. Laboratoire Interface et Systèmes Electrochimiques, Université P. et M. Curie Place Jussieu 4, Paris 75252 France Ramos, A. Departamento de Señales, Sistemas y Tecnologías Ultrasónicas. Instituto de Acústica. Consejo Superior de Investigaciones Científicas. Serrano 144, Madrid 28006 Spain San Emeterio, J.L. Departamento de Señales, Sistemas y Tecnologías Ultrasónicas. Instituto de Acústica. Consejo Superior de Investigaciones Científicas. Serrano 144, Madrid 28006 Spain

XXVI

Associated Editors and Contributors

Soares, D. Instituto de Física, Departamento de Física Aplicada, Universidade Estadual de Campinas Caixa Postal 6165, Campinas 13083-970 Brasil Sogorb, T. Departamento de Ingeniería Electrónica, Universidad Politécnica de Valencia Camino de Vera s/n, Valencia E-46022 Spain Stipek, S. Institute of Medical Biochemistry, First Faculty of Medicine, Charles University in Prague Katerinská 32. Prague CZ-121 08 Czech Republic Vera, A. Departamento de Ingeniería Eléctrica, Centro de Investigación y Estudios Avanzados Avda. Instituto Politécnico Nacional Nº 2508, San Pedro Zacatenco, Mexico, D.F. 07360 Mexico, D.F.

1 Fundamentals of Piezoelectricity Antonio Arnau1 and David Soares2 1 2

Departamento de Ingeniería Electrónica, Universidad Politécnica de Valencia Institute de Fisica, Universidade de Campinas

1.1 Introduction The topic of the following chapter is relatively difficult and includes different areas of knowledge. The piezoelectric phenomenon is a complex one and covers concepts of electronics as well as most of the areas of classical physics such as: mechanics, elasticity and strength of materials, thermodynamics, acoustics, wave’s propagation, optics, electrostatics, fluids dynamics, circuit theory, crystallography etc. Probably, only a few disciplines of engineering and science need to be so familiar to so many fields of physics. Current bibliography on this subject is vast though dispersed in research publications, and few of the books on this topic are usually compilations of the authors’ research works. Therefore, they are not thought for didactic purposes and are difficult to understand, even for postgraduates. The objective of this chapter is to help understand the studies and research on piezoelectric sensors and transducers, and their applications. Considering the multidisciplinary nature, this tutorial’s readers can belong to very different disciplines. They can even lack the necessary basic knowledge to understand the concepts of this chapter. This is why the chapter starts providing an overview of the piezoelectric phenomenon, doing consciously initial simplifications, so that the main concepts, which will be progressively introduced, prevail over the accessories. The issues covered in this chapter must be understood without the help of additional texts, which are typically included as references and are necessary to study in depth specific topics. Finally, the quartz crystal is introduced as a micro-gravimetric sensor to present the reader an application of the piezoelectric phenomenon, which will be dealt with along the following chapters.

A.A. Vives (ed.), Piezoelectric Transducers and Applications, doi: 10.1007/978-3-540-77508-9_1, © Springer-Verlag Berlin Heidelberg 2008

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Antonio Arnau and David Soares

1.2 The Piezoelectric Effect The word Piezoelectricity comes from Greek and means “electricity by pressure” (Piezo means pressure in Greek). This name was proposed by Hankel [1] in 1881 to name the phenomenon discovered a year before by the Pierre and Jacques Curie brothers [2]. They observed that positive and negative charges appeared on several parts of the crystal surfaces when comprising the crystal in different directions, previously analysed according to its symmetry. Figure 1.1a shows a simple molecular model; it explains the generating of an electric charge as the result of a force exerted on the material. Before subjecting the material to some external stress, the gravity centres of the negative and positive charges of each molecule coincide. Therefore, the external effects of the negative and positive charges are reciprocally cancelled. As a result, an electrically neutral molecule appears. When exerting some pressure on the material, its internal reticular structure can be deformed, causing the separation of the positive and negative gravity centres of the molecules and generating little dipoles (Fig. 1.1b). The facing poles inside the material are mutually cancelled and a distribution of a linked charge appears in the material’s surfaces (Fig. 1.1c). That is to say, the material is polarized. This polarization generates an electric field and can be used to transform the mechanical energy used in the material’s deformation into electrical energy. F F

F

a

b

F

c

Fig. 1.1. Simple molecular model for explaining the piezoelectric effect: a unperturbed molecule; b molecule subjected to an external force, and c polarizing effect on the material surfaces

1 Fundamentals of Piezoelectricity

3

Figure 1.2a shows the piezoelectric material on which a pressure is applied. Two metal plates used as electrodes are deposited on the surfaces where the linked charges of opposite sign appear. Let us suppose that those electrodes are externally short circuited through a wire to which a galvanometer has been connected. When exerting some pressure on the piezoelectric material, a linked charge density appears on the surfaces of the crystal in contact with the electrodes. This polarization generates an electric field which causes the flow of the free charges existing in the conductor. Depending on their sign, the free charges will move towards the ends where the linked charge generated by the crystal’s polarization is of opposite sign. This flow of free charges will remain until the free charge neutralizes the polarization effect (Fig. 1.2a). When the pressure on the crystal stops, the polarization will disappear, and the flow of free charges will be reversed, coming back to the initial standstill condition (Fig. 1.2b). This process would be displayed in the galvanometer, which would have marked two opposite sign current peaks. If a resistance is connected instead of a short-circuiting, and a variable pressure is applied, a current would flow through the resistance, and the mechanical energy would be transformed into electrical energy.

F

F i

a

b

Fig. 1.2. Piezoelectric phenomenon: a neutralizing current flowing through the short-circuiting established on a piezoelectric material subjected to an external force; b absence of current through the short-circuited material in an unperturbed state

The Curie brothers verified, the year after their discovery, the existence of the reverse process, predicted by Lippmann (1881) [3]. That is, if one arbitrarily names direct piezoelectric effect, to the generation of an electric

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Antonio Arnau and David Soares

charge, and hence of an electric field, in certain materials and under certain laws due to a stress, there would also exist a reverse piezoelectric effect by which the application of an electric field, under similar circumstances, would cause deformation in those materials. In this sense, a mechanical deformation would be produced in a piezoelectric material when a voltage is applied between the electrodes of the piezoelectric material, as shown in Fig.1.2. This strain could be used, for example, to displace a coupled mechanical load, transforming the electrical energy into mechanical energy.

1.3 Mathematical Formulation of the Piezoelectric Effect. A First Approach In a first approach, the experiments performed by the Curie brothers demonstrated that the surface density of the generated linked charge was proportional to the pressure exerted, and would disappear with it. This relationship can be formulated in a simple way as follows:

Pp = d T

(1.1)

where Pp is the piezoelectric polarization vector, whose magnitude is equal to the linked charge surface density by piezoelectric effect in the considered surface, d is the piezoelectric strain coefficient and T is the stress to which the piezoelectric material is subjected. The Curie brothers verified the reverse piezoelectric effect and demonstrated that the ratio between the strain produced and the magnitude of the applied electric field in the reverse effect, was equal to the ratio between the produced polarization and the magnitude of the applied stress in the direct effect. Consistently, the reverse piezoelectric effect can be formulated in a simple way, as a first approach, as follows:

Sp = d E

(1.2)

where Sp is the strain produced by the piezoelectric effect and E is the magnitude of the applied electric field. The direct and reverse piezoelectric effects can be alternatively formulated, considering the elastic properties of the material, as follows:

Pp = d T = d c S = e S

(1.3)

Tp = c S p = c d E = e E

(1.4)

1 Fundamentals of Piezoelectricity

5

where c is the elastic constant, which relates the stress generated by the application of a strain (T = c S), s is the compliance coefficient which relates the deformation produced by the application of a stress (S = s T), and e is the piezoelectric stress constant. (Note that the polarizations, stresses, and strains caused by the piezoelectric effect have been specified with the p subscript, while those externally applied do not have subscript. Although unnecessary, it will be advantageous later on.

1.4 Piezoelectric Contribution to Elastic Constants The piezoelectric phenomenon causes an increase of the material’s stiffness. To understand this effect, let us suppose that the piezoelectric material is subjected to a strain S. This strain will have two effects. On the one hand, it will generate an elastic stress Te which will be proportional to the mechanical strain Te = c S; on the other hand, it will generate a piezoelectric polarization Pp = e S according to Eq. (1.3). This polarization will create an internal electric field in the material Ep given by (see Appendix A):

Ep =

Pp

ε

=

eS

ε

(1.5)

where ε is the dielectric constant of the material. This electric field, of piezoelectric origin, produces a force against the deformation of the material’s electric structure, creating a stress Tp = e Ep. This stress, as well as that of elastic origin, is against the material’s deformation. Consistently, the stress generated as a consequence of the strain S will be:

T = Te + T p = c S +

⎛ e2 ⎞ S = ⎜⎜ c + ⎟⎟ S = c S ε ε ⎠ ⎝

e2

(1.6)

Therefore, the constant c is the piezoelectrically stiffened constant, which includes the increase in the value of the elastic constant due to the piezoelectric effect. This coefficient will appear later on.

1.5 Piezoelectric Contribution to Dielectric Constants When an external electric field E is applied between two electrodes where a material of dielectric constant ε exists, an electric displacement is created towards those electrodes, generating a surface charge density σ =σo+σp

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Antonio Arnau and David Soares

which magnitude is D = εE 1. If that material is piezoelectric, the electric field E produces a strain given by: Sp = d E. This strain of piezoelectric origin increases the surface charge density due to the material’s polarization in an amount given by: Pp = e Sp = e d E (Fig. 1.3). Because the electric field is maintained constant, the piezoelectric polarization increases the electric displacement of free charges towards the electrodes in the same magnitude (σp = Pp). Therefore, the total electrical displacement is:

D = ε E + Pp = ε E + e d E = ε E

(1.7)

where ε is the effective dielectric constant which includes the piezoelectric contribution.

1.6 The Electric Displacement and the Internal Stress As shown in the previous paragraph, the electric displacement produced when an electric field E is applied to a piezoelectric and dielectric material is:

D = ε E + Pp = ε E + e S p

(1.8)

Under the same circumstances we want to obtain the internal stress in the material. The reasoning is the following: the application of an electric field on a piezoelectric material causes a deformation in the material’s structure given by: Sp = d E. This strain produces an elastic stress whose magnitude is Te = c Sp. On the other hand, the electric field E exerts a force on the material’s internal structure generating a stress given by: Tp = e E. This stress is, definitely, the one that produces the strain and is of opposite sign to the elastic stress which tends to recover the original structure. Therefore, the internal stress that the material experiences will be the resultant of both. That is:

T = cSp − eE

(1.9)

The free charge density which appears on the electrodes, will be the sum of the charge density which appears in vacuum plus the one that appears induced by the dielectric effect, i.e.:

1

σ o + σ d = ε o E + χ E = (ε o + χ ) E = ε E

where εo is the vacuum dielectric permittivity and χ is the dielectric susceptibility of the material.

1 Fundamentals of Piezoelectricity

7

E p o

d

Fig. 1.3. Schematic diagram that explains different electrical displacements associated with a piezoelectric and dielectric material

Eventually, both stresses will be equal leaving the material strained and static. If a variable field is applied, as it is the common practice, the strain will vary as well, producing a dynamic displacement of the material’s particles. This electromechanical phenomenon generates a perturbation in the medium in contact with the piezoelectric material. This effect is used in transducers, sensors and actuators, as it will be seen along the following chapters.

1.7 Basic Model of Electric Impedance for a Piezoelectric Material Subjected to a Variable Electric Field In the previous section, the expressions for the electric displacement and the internal stress produced in a piezoelectric material subjected to an electric field have been obtained. The electric field is created when a voltage difference is applied between two electrodes deposited on certain surfaces of the material. If the applied voltage difference changes, the electric field as well as the electric displacement change inducing an electric current through the electrodes. The ratio between the applied voltage and the induced electric current is the electrical impedance of the piezoelectric component. For example, if it only has dielectric properties, the resulting electrical impedance corresponds to a capacitance. Piezoelectric devices are included in electric and electronic circuits to use their electromechanical properties in both direct and reverse applications. Therefore, it is important to obtain an electric

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Antonio Arnau and David Soares

model that allows including the piezoelectric component in electric circuits. This will greatly facilitate the analysis of the circuit and the understanding of its operation. Next, the basic equivalent electric model mentioned will be obtained. In the obtaining of the model some simplifications will be made to minimize the mathematical formulation. These simplifications do not essentially modify the results and let us show the qualitative physical concepts of the model. On the other hand, the obtained expressions for the electric parameters of the model will be very similar to those obtained from a more rigorous mathematical development, as it will be shown in Appendix 1.A. Figure 1.4 shows the transversal section of a bar of piezoelectric material of thickness l. Let us suppose that when applying a field in the direction of the thickness (direction Y) by the application of a voltage difference between the electrodes, the material deforms as shown in the left part of Fig. 1.4. When the field is reversed, the strain is reversed as well (right part). Y

l/2

l/2

{

Y

X

X -l/2

-l/2

As

As

Fig. 1.4. Shear strains produced in a piezoelectric material subjected to a reverting voltage

The strain is produced when displacement gradients occur, or in other words, when the particles displacement increases or decreases in one direction. Therefore, the strain S is defined as the gradient of the particles displacement in the direction considered. Thus, if the displacement that the particles experience along a distance y is ξ(y), the strain produced along this section will be:

S ( y) =

ξ ( y) y

(1.10)

Figure 1.4 shows how the particles displacement increases with the coordinate y, being null on the abscissas axis2. Consistently, the maximum This type of strain is called in thickness shear mode, and is very common. Precisely, bars of quartz crystal obtained from the AT cut (bars obtained through

2

1 Fundamentals of Piezoelectricity

9

strain is produced at y=l/2 and is the same in both ends but of opposite sign due to the change of sign in the displacement. Therefore, the strain at y=l/2 will be: S (l / 2 ) =

ξ l /2

=

2ξ l

(1.11)

where ξ is the particle displacement at the coordinate y=l/2 at a generic instant. Figure 1.5 shows the forces acting on the material ends when the electric field is applied. This electric field creates a force in the X direction which produces a piezoelectric stress given by Eq. (1.4). An elastic stress Te = c S p is against the piezoelectric stress and tries to avoid the strain of the material. The internal friction that the particles experience in their displacement is also against the piezoelectric stress since it makes the particles displacement more difficult. The stress due to internal friction is usually considered proportional to the gradient of the particle displacement velocity, as in the case of a viscous phenomenon, that is: Tv = η

dv d 2ξ dS =η =η dy dy dt dt

(1.12)

where constant η is named viscosity. Y

Te

{

e E(l/2)

l/2

Tv

{

X -l/2 As

Fig. 1.5. Shear strain and stresses produced at the end of a piezoelectric plate subjected to an electric field

cuttings done with an angle of 35º15’ in relation to the optical axis Z [4]) present a very pure shear vibration mode when an electric field is applied in the direction of the thickness. The anisotropy of the quartz is the responsible for this phenomenon. The anisotropy complicates the mathematical formulation of the elastic, dielectric and piezoelectric effects (see Appendix B). A deeper study of the piezoelectricity considering the anisotropy phenomenon can be found elsewhere [4,5].

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Antonio Arnau and David Soares

The resultant of the forces will be equal to the product of mass by the acceleration of the particles. As stresses are being considered, it will be necessary to take into account the surface mass density ρs. Therefore, Newton’s first law applied to the material surface at the coordinate y=1/2 will be:

∑

T = e E (l / 2) − c S (l / 2) − η

dS (l / 2) d 2ξ = ρs 2 dt dt

(1.13)

Considering Eq. (1.11), Eq. (1.13) and that E(l/2)=V/l, where V is the voltage difference between the electrodes, the following expression for the voltage V is obtained: V=

2η dξ ρ s l d 2ξ 2c + + ξ e dt e dt 2 e

(1.14)

On the other hand, the electric displacement on the electrodes is given by Eq. (1.8). The time derivative of the electric displacement provides the density of the induced current J given by: J=

dD (l / 2) dE (l / 2) dS (l / 2) =ε +e = Jd + J p dt dt dt

(1.15)

The first term of the second member Jd corresponds to the density of the induced current by the dielectric effect and the second term Jp to the current induced by the piezoelectric effect. Let us analyse the second term, which can be written from Eq. (1.11) as: Jp =

2 e dξ l dt

(1.16)

Taking into account that the surface density current Jp= ip /AS, where ip is the current induced by piezoelectric effect and AS is the electrodes surface, the following relationship can be obtained from Eq. (1.16): dξ l = ip dt 2eAS

(1.17)

By substituting Eq. (1.17) in Eq. (1.14), it is definitely obtained: V =

ηl AS e

2

ip +

ρ s l 2 di p 2 AS e

2

dt

+

cl i p dt AS e 2

∫

(1.18)

1 Fundamentals of Piezoelectricity

11

The voltage arising between the ends of a series circuit formed by a resistance Rm, an inductance Lm and a capacitance Cm through which an ip current flows, has the following expression: V = Rm i p + Lm

di p dt

+

1 i p dt Cm

∫

(1.19)

Therefore, the current induced by the piezoelectric effect, i.e., by the electromechanical effect, in the material is the same as the one that would flow through a series electric circuit formed by a resistor, a coil and a capacitor with the following magnitudes of resistance, inductance and capacitance, respectively: R

m

=

ρ l2 Ae2 1 = K η, L = s =K ρ , C = =K =K s R m L s m C c C 2 2 c l Ae 2Ae ηl

The former expressions make clear the relationships among the electrical parameters and mechanical properties of the material, which are: the resistive electric parameter is proportional to the viscosity and models the physical phenomenon of energy loss due to viscous effects. The inductive parameter is proportional to the surface mass density and models the energy storage by inertial effect, and the capacitive parameter which is proportional to the elastic compliance models the energy storage by elastic effect. These relationships, which settle a clear analogy between the physical properties and the electric parameters, are very useful when evaluating the physical phenomena which take place when the piezoelectric material is used as a micro-gravimetric sensor, at least in simple cases, as it will be seen in Chaps. 3, 7, and 14. Apart from the ip component, it is also necessary to consider the component id associated with the dielectric effect. In fact, it can be written from Eq. (1.15) as follows: id = AS J d = AS ε

A dV dE (l / 2) =ε S dt l dt

(1.20)

Equation (1.20) corresponds to the current induced through a capacitor Co = εAS /l when a variable voltage difference V is applied. Consistently, the circuit that models the electrical impedance of a piezoelectric and dielectric material subjected to a variable voltage difference is shown in Fig. 1.6. The electric circuit is formed by two parallel branches: one of them is the so-called motional branch formed by a series Rm Lm Cm circuit that models the motional physical phenomenon. The other is the socalled static branch formed by a capacitor Co which is associated to the

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Antonio Arnau and David Soares

electrical capacitance arising from the dielectric material placed between the two electrodes. Lm

Rm

Cm

Co Fig. 1.6. Equivalent electrical model of a piezoelectric material vibrating at frequencies near resonance

The electrical model obtained, even with the corresponding simplifications, represents the real electrical impedance of the component when it vibrates at a frequency near some of its natural vibrating frequencies or resonant frequencies (see next section). In the Appendix 1.1, a more exact calculation of the component’s electric admittance (reciprocal to the electrical impedance) is developed. The process followed in the appendix is similar to the one made in Chap. 3 (Appendix 3.A) to determine the electrical admittance associated with a piezoelectric sensor in contact with a medium and, in consequence, its reading is recommended.

1.8 Natural Vibrating Frequencies 1.8.1 Natural Vibrating Frequencies Neglecting Losses

In the previous section the concept of natural vibrating frequencies or resonant frequencies has arisen. In this section these concepts will be studied. Let us suppose that a piezoelectric material, of characteristics similar to those presented in the previous section, is subjected to a strain as the one illustrated in Fig. 1.7 (upper part). The stress that the particles present under these conditions is given by Eq. (1.6). At a certain instant, the external force which maintains the strain is removed and the material starts to vibrate freely. Let us analyse that vibration. Let us consider a slice of material of thickness dy located at the coordinate y. This slice is subjected to forces at both ends, as shown in Fig.1 7 (central part). The resultant of the forces will be equal to the product of the slice’s mass by the acceleration to which the slice is subjected. This can be mathematically written as follows:

1 Fundamentals of Piezoelectricity

∂F ( y , t ) ∂ 2ξ ( y , t ) dy = ρ v AS dy ∂y ∂t2

13

(1.21)

In the former expression, it has been assumed that the force F and the displacement ξ depend on both coordinate y and the time t. Also, the mass has been written as the product of the material’s density ρv by the slice’s differential volume AS dy; where AS is the surface perpendicular to the paper plane. Y T

l/2

X T

-l/2

As

Y l/2 dy y

F(y) dy y

F(y)

F(y)

X -l/2 As Y l/2

X -l/2 As

Fig. 1.7. Figures that explain the natural vibration of a piezoelectric resonator: The upper part shows the resonator bar subjected to an external stress, the central part shows the forces that an internal thin slice of a strained piezoelectric material experience, and the lower part shows the displacement profile of a piezoelectric material subjected to a sinusoidal electric field

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Antonio Arnau and David Soares

Equation (1.21) can be written in terms of stress dividing by the surface AS in both members. Considering Eq. (1.6) and writing S=∂ξ/∂y, Eq. (1.21) results into: ∂T ∂ 2ξ ∂ 2ξ =c = ρ v ∂y ∂ y2 ∂t2

(1.22)

Now, let us assume that the particle displacement near equilibrium has a sinusoidal dependence with time. Thus, the time derivative of particle displacement can be replaced by the product jω, where j is the complex base − 1 and ω=2πf is the oscillating angular frequency of the particles; where f is the frequency. Therefore, Eq. (22) becomes: vo

2

∂ 2ξ + ω 2ξ = 0 2 ∂y

(1.23)

where v o = c / ρ v . The resolution of the former differential equation will provide the instantaneous profiles of the particle displacement with regard to the coordinate y. These profiles correspond to those shown in Fig. 1.7 (lower part). The displacement function fulfilling Eq. (1.23) is: ⎛ 2π ⎞ y +σ ⎟ λ ⎝ ⎠

ξ = ξ o sin (2π k y + σ ) = ξ o sin ⎜

(1.24)

where λ=vo /f is the wave length, k=1/λ=f /vo=ω/2πvo is called the wave number since it corresponds to the number of complete wave lengths in the distance unit; ξo is the maximum amplitude of oscillation and σ is a constant to determine consistently with the boundary conditions. In this case, the particles displacement is null at the coordinate y=0. This condition implies σ=0. Also, the amplitude of oscillation must be a maximum at the ends where y=±l/2. Therefore, it is necessary that the following condition be fulfilled: 2πk

π l =n 2 2

⇒ n = 1, 2, 3K

(1.25)

This condition forces the oscillation at frequencies fno which have to be odd multiples of a frequency fo according to the following expression: f no = n f o = n

vo 2l

⇒ n = 1, 2, 3K

(1.26)

1 Fundamentals of Piezoelectricity

15

Frequency fo is called natural vibration fundamental frequency or resonant frequency (see following paragraphs) and its multiples are called harmonics of the fundamental frequency. Notice that vo is the perturbation’s propagation speed in the material without losses. Indeed, the speed is the ratio between the distance covered by the perturbation and the time it takes to go through that distance. From the definition of vo as a function of the wave length and the oscillation frequency we get vo = λ f. This equation indicates that the perturbation covers a space corresponding to a wave length in the time corresponding to a period of the oscillation. This is precisely the definition of the propagation speed. Equation (1.26) also indicates that the frequencies of natural vibration depend solely on the material’s physical properties and on its thickness. It also seems to indicate that the only possible vibrating frequencies are the ones that fulfil that condition. In fact, Eq. (1.26) is the result of simplifying the problem to only one dimension. When the lateral dimensions are infinite in comparison with thickness, the vibrating frequencies relative to those directions are null. However, in practice, the portions of material are three-dimensional with finite dimensions. In practice, two of the dimensions are much bigger than the third one, and they can be considered approximately as two–dimensional systems. In these cases, additional possible vibrating modes take place. Most of these vibration modes are not exact multiples of the fundamental mode; that is, they are not harmonically related to the fundamental; therefore they are called inharmonic modes. An important problem in practical applications is that these inharmonic modes can vibrate at frequencies very close to those of natural vibration. The correct application of electrodes on the piezoelectric material cancels some inharmonic modes. Other modes must be cancelled through an adequate design and additional contouring techniques which are not always applicable. Besides the inharmonic modes, the crystal’s anisotropy generates the so called coupled vibrating modes, where a determined vibrating mode also excites another one [4]. 1.8.2 Natural Vibrating Frequencies with Losses

In the previous study, the vibration caused by the initially created strain is maintained indefinitely. However, the free vibrations of any real physical system disappear with time. The reason is that any vibrating system involves phenomena which dissipate energy and eventually cause the vibration to stop. The incorporation of the loss effects to the previous vibrating system’s physical model represents an approach to reality. This can be

16

Antonio Arnau and David Soares

done by including the loss stress, modelled as a viscous effect, already formulated in Eq. (1.12). This way, the global recovering stress is: T = cS + η

∂S ∂t

(1.27)

Equation (1.22) is transformed now into: ∂T ∂ 2ξ ∂ 3ξ ∂ 2ξ η ρ =c + = v ∂y ∂ y2 ∂t ∂ 2 y ∂t2

(1.28)

In order to solve Eq. (1.28), let us assume that the displacement, which is a function of the coordinate y and the time t, can be written as the product of two functions of separate variables, i.e., ξ ( y , t ) = ζ (t ) ϕ ( y ) . Additionally, let us suppose that the particles displacement has a sinusoidal profile with coordinate y. In fact, this is the profile obtained in the previous case, when neglecting the losses and is the profile the particles very accurately follow when losses are small, as it will be seen in the next section. In this case, each partial derivate in relation to y can be replaced in Eq. (1.28) by (j2πk) (see Eq. (1.24)), giving in the following expression: ∂ 2ζ 4π 2 k 2η ∂ζ 4π 2 k 2 c + + ζ =0 ρ v ∂t ρv ∂t2

(1.29)

In the former equation the following parameters were defined to simplify the mathematical formulation and to facilitate the understanding of the physical phenomena derived from it. In first place, the attenuation coefficient or losses coefficient α will be defined as:

α=

4π 2 k 2η

(1.30)

ρv

On the other hand, the coefficient associated with the third term of Eq. (1.29), taking into account the propagation speed in Eq. (1.23), results in the squared natural vibrating angular frequency without losses, that is:

ωo2 =

4π 2 k 2 c

ρv

In this way Eq. (1.29) becomes:

=

αc η

(1.31)

1 Fundamentals of Piezoelectricity

∂ 2ζ ∂ζ +α + ωo2ζ = 0 2 ∂ t ∂t

17

(1.32)

Solving Eq. (1.32) for the displacement ξ results in the following time function:

ζ (t ) = A e j (Ωt + θ )

(1.33)

where the constants Ω and θ will have to be determined according to the boundary conditions. By substituting Eq. (1.33) in Eq. (1.32) it is obtained: ( jΩ) 2 + α j (Ω) + ωo2 = 0

(1.34)

The previous equation makes Ω be a complex number, because α is not null. If Ω is assumed to be Ω=ωp+jγ, it is obtained by substitution:

ω p2 = ωo2 −

α2 4

; γ =

α 2

(1.35)

Thus, it results from Eq. (1.33), solving for real parts of the complex exponential: −

α

t

ζ (t ) = A e 2 cos(ω p t + θ )

(1.36)

Considering that, at any moment, particle displacement at coordinate y=0 must be null, the following expression for the displacement is obtained: −

α

t

⎛ 2π ⎞ ξ ( y , t ) = A e 2 cos (ω p t + θ ) sin ⎜ y⎟ ⎝ λ ⎠

(1.37)

Thus, ωp defined in Eq. (1.35) is identified as the natural vibrating angular frequency of the damped system. This frequency must be coherent with the boundary conditions which establish that the displacements at y=l/2 must always be a maximum. Consistently, that maximum condition is formulated as follows:

∂ξ ∂y

= Ae y =l / 2

−

α

t 2 cos (ω t + θ ) 2π cos ⎛⎜ 2π l ⎞⎟ = 0 p λ ⎝ λ 2⎠

(1.38)

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Antonio Arnau and David Soares

The previous condition is fulfilled if: 2π l l ωp l π = 2πk = =n λ 2 2 vp 2 2

⇒ n = 1, 2, 3K

(1.39)

where vp is the speed propagation in the medium with losses. Therefore, the damping vibrating frequencies fnp are odd multiples of the natural vibrating fundamental frequency of the damped system fp according to the following expression: f np = n f p = n

vp 2l

⇒ n = 1, 2, 3K

(1.40)

Notice that Eq. (1.40) has the same formulation as Eq. (1.26) where the speed propagation is vp. We will see that for small losses, what is true in most practical cases, the difference between vo and vp is negligible. On the other hand, the value of θ in Eq. (1.37) must be zero since the displacement in the initial instant (t=0) at the end (y=l/2) must be a maximum, i.e., ξ (l / 2,0) = A . Equation (1.37) can be written as: α

A − t ⎡ 2π ξ ( y , t ) = e 2 ⎢sin 2 λ ⎣

ω ⎞ ⎛ 2π ⎜⎜ y − p t ⎟⎟ + sin 2π k ⎠ λ ⎝

ω ⎞⎤ ⎛ ⎜⎜ y + p t ⎟⎟⎥ 2π k ⎠⎦ ⎝

(1.41)

This expression corresponds to a damped stationary wave. Stationary waves are generated by superposing two waves. One is called progressive wave (first sine in Eq. (1.41)) which displaces in the positive direction of Y, and the other is regressive and displaces in the opposite direction (second sine in Eq. (1.41)). This superposition creates a stationary wave, named so because it seems as it does not displace in space. Its spatial profile is sinusoidal and is formed by zones which do not vibrate and are called nodes (as that slice of material located in the centre of the material) and by zones of maximum amplitude of vibration, (as those slices located at the ends). The particles vibrate around their equilibrium positions according to a sinusoidal relation with time. In the case of Eq. (1.41), the maximum amplitudes of vibration decrease exponentially with time until they disappear. The wave propagation speed corresponds to the term that goes with time t in Eq. (1.41). This velocity corresponds to that of the material with losses vp and has the following value: vp =

ωp ω α2 1 ωo2 − = = o 2π k 2π k 4 2π k

⎛ α2 ⎜1 − ⎜ 4ω 2 o ⎝

⎞ ⎟ =v ⎟ ⎠

⎛ α2 ⎞ ⎜1 − ⎟ ⎜ 4ω 2 ⎟ o ⎠ ⎝

(1.42)

1 Fundamentals of Piezoelectricity

19

It is evident that the damped vibratory movement is characterized by the two parameters ωo and α. ωo is the oscillating angular frequency without losses and α is the time needed for the energy of oscillation to decrease to 1/e of its initial value. In fact, the expression for the energy of a thin slice of material with a mass dm located at the coordinate y, as a function of time t, associated with its harmonic movement, is given by: 1 − αt Wt = dmω 2 A( y ) 2 e 2

(1.43)

Therefore, in each span of time equivalent to 1/α, the energy decreases in 1/e in relation to the one at the end of the previous span. According to Eq. (1.43) the decrease of energy by time unit will be given by: dWt 1 − αt = −α dmω 2 A( y ) 2 e = −α Wt dt 2

(1.44)

Consequently, the energy lost in a cycle corresponding to a period T will be:

ΔWt (T ) = −α Wt T

(1.45)

The two parameters that characterize the damped vibratory movement can be combined into another named quality factor Q of the oscillating system. Q is defined as the ratio between the energy stored and the one dissipated by the oscillating system during a cycle, multiplied by the factor 2π, it is to say: Q = 2π

ω Wt Energy stored per cycle = 2π = o Energy dissipated per cycle α Wt T α

(1.46)

Considering Eq. (1.31) Q = c / ωoη . According to Eq. (1.42) the propagating speed in the medium with losses becomes:

⎛ 1 ⎞ ⎟ v p = vo ⎜⎜1 − 4Q 2 ⎟⎠ ⎝

(1.47)

Let us observe that if the losses are small the system’s quality factor is high and, according to Eqs. (1.35) and (1.47), the natural vibrating angular frequency of the damped oscillations ωp as well as the propagation speed with losses vp, are very similar to those of the system without losses.

20

Antonio Arnau and David Soares

1.8.3 Forced Vibrations with Losses. Resonant Frequencies

The previous analysis provides the natural vibrating frequencies of free systems without losses and also of those which have losses. Therefore, the conclusion is that natural vibrating frequencies must follow some specific relations. In a great number of applications the piezoelectric materials are subjected to a forced vibration of certain frequency. For example when subjected to a variable field of an established frequency. It is also important to study what particle displacement is like when the frequency is different from the previously obtained natural vibrating frequencies. It is also interesting to mention that from Eq. (1.17) the displacements are directly related to the intensity induced by the piezoelectric effect. This characteristic will be of special interest. Next, the situation in which a material of piezoelectric characteristics as previously described is subjected to an alternative sinusoidal field of angular frequency ω will be analysed. The losses in the material will be taken into account. The equations for the electric displacement and for the internal stress, including the losses in the material, are obtained from Eqs. (1.8) and (1.9) giving: T = cSp − eE +η

∂S p ∂t

D = ε E + eSp

(1.48) (1.49)

The analysis of forces presented in Fig. 1.7 (central part) is still valid. From it, one comes to Eq. (1.21) that, considering Eq. (1.48), finally becomes: ∂S p ∂ 2S p ∂T ∂ 2ξ ∂E = ρv 2 = c −e +η ∂y ∂y ∂y ∂y ∂ t ∂t

(1.50)

The equation of the electric displacement can be used to establish the relation between the electric field applied and the particle displacement. Indeed, as in the inside of the piezoelectric material it is assumed that there is no free charge, the divergence of the electric displacement vector must be zero. As the piezoelectric polarization only exists in direction Y, the Gauss law for electric displacement gives: ∂S p ∂D ∂E =0 ⇒ ε = −e ∂y ∂y ∂y

Substituting Eq. (1.51) in Eq. (1.50), results in:

(1.51)

1 Fundamentals of Piezoelectricity

c

∂ 2ξ ∂ 3ξ ∂ 2ξ + = η ρ v ∂ y2 ∂y 2 ∂t ∂t 2

21

(1.52)

The former expression is the wave equation for the particle displacement. In this movement, the system, after certain transitory time where it will try to vibrate at some of its natural vibrating frequencies, will end up oscillating with a forced vibration at a frequency equal to that imposed by the external field applied. Therefore, the particles’ displacement will follow a harmonic movement of the same angular frequency as the one of the applied variable electric field. This sinusoidal variation in time lets us write the particle displacement as follows:

ξ ( y, t ) = ζ ( y ) e

jω t

(1.53)

Consequently, Eq. (1.52) is reduced to:

ρ vω 2 ∂ 2ζ = − ζ ( c + jωη ) ∂y 2

(1.53)

In the previous section, it was assumed that the particle displacement had a sinusoidal profile with regard to the coordinate y, whenever there were small losses. This assumption, done at that time to simplify the calculations, will be deduced after the following general analysis. The following solution will be tried for ξ:

ζ = Ae

γy

+ Be

−γ y

(1.54)

In first place, the condition for a zero-displacement at y=0 implies that B=-A. By substituting Eq. (1.54) in Eq. (1.3) one gets the value for γ, that results in:

γ2 =−

ω2

ρv

ω 2 ρ vω 2 c = − vo =− 1 ωη c + jωη 1+ j 1+ j 2

(1.55)

Q

c

If the quality factor of the piezoelectric material is much greater than unity, as is the usual case, the constant γ can be approximated to:

ω⎛

1⎞ γ = j ⎜⎜1 + j ⎟⎟ vo ⎝ Q⎠

−1 / 2

≈

ω 2Q vo

+ j

ω vo

=

α 2vo

+ j

ω vo

(1.56)

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Antonio Arnau and David Soares

On the other hand, considering Eq. (1.2), the strain at coordinate y= l/2 will be: Sp

y=

l 2

=

∂ξ ∂y

y=

l 2

=dE

y=

l 2

=d

Vm jω t e l

(1.57)

where Vm is the maximum voltage difference applied to the material between the electrodes located at y=±l/2. The application of this boundary condition provides the value of the constant A, which results in: A=

dVm γl

1 l γ e 2

+e

−γ

l 2

(1.58)

In this way Eq. (1.54) results in:

ζ =

dVm e γ y − e −γ y dVm sinh(γ y ) = l γ l γ 2l γl −γ ⎛ γl ⎞ cosh⎜ ⎟ e +e 2 ⎝2⎠

(1.59)

It is noticed that for small losses, that is, for high Q factors, the constant γ ≈ jω/vo, and Eq. (1.59) is reduced to:

⎛ ω ⎞ ⎛ω ⎞ sinh⎜⎜ j y ⎟⎟ sin ⎜⎜ y ⎟⎟ dV ⎝ vo ⎠ = dVm vo ⎝ vo ⎠ ζ ≈ m ω ωl ⎛ ωl ⎞ j l cosh⎛⎜ j ω l ⎞⎟ ⎟⎟ cos⎜⎜ ⎜ 2v ⎟ vo o ⎠ ⎝ ⎝ 2vo ⎠

(1.60)

The previous expression, although approximated for null losses, provides useful information. In fact, it is enough for our immediate interest to consider the particles’ displacement at the piezoelectric material’s ends. Considering Eqs. (1.53) and (1.60), the displacement for y=l/2 will be:

⎛ ωl ⎞ ⎟⎟ sin ⎜⎜ ⎛ l ⎞ dVm v o ⎝ 2v o ⎠ sin (ω t ) ξ⎜ ,t⎟ = ωl ⎛ωl ⎞ ⎝2 ⎠ ⎟⎟ cos⎜⎜ ⎝ 2v o ⎠

(1.61)

In first place, it can be observed that the approximation to a sinusoidal displacement profile in relation to the coordinate y for small losses, used in the previous section, was completely valid. In second place, it can be

1 Fundamentals of Piezoelectricity

23

noticed that if the excitation frequency applied corresponds to those frequencies equal to odd multiples of the vibrating fundamental frequency, i.e., ωn= nωo= n2πfo= nπvo /l, where n is odd (see Eq. (1.26)), particle displacement becomes infinite. This effect produces the maximum displacement, theoretically infinite, even for very small excitations. This phenomenon is known as resonance and the frequencies that cause it are called resonant frequencies. It is evident that the infinite displacement amplitudes are a consequence of disregarding the losses, but the previous result indicates that a vibration forced into frequencies near those of natural vibration causes the biggest mechanical displacements. In third place, it can be noticed that excitations at frequencies which are even multiples of the natural vibrating fundamental frequency, that is, ω2n= nωo= n2πfo= nπvo /l where n is even, do not cause a displacement at the ends of the piezoelectric material. The last two observations should be more carefully commented. In fact, the current induced by piezoelectric effect, i.e. the current that flows through the motional branch in the model shown in Fig. 1.6, is proportional to the speed of the particle displacement according to Eq. (1.17). Therefore, the current due to the piezoelectric effect will be a maximum when the excitation frequencies coincide with the natural vibration frequencies without losses. These frequencies must coincide with the series resonance frequencies of the motional branch. The circulating current is a maximum for these frequencies. Consequently, the electric model indicated in Fig. 1.6 must be applied to each frequency of natural vibration. This can be done including the motional branches in parallel as shown in Fig. 1.8.

Co

Cm1

Cm3

Cm5

Rm1

Rm3

Rm5

Lm1

Lm3

Lm5

Fig. 1.8. Equivalent electrical model of a piezoelectric resonator vibrating at frequencies near any of its resonant frequencies

24

Antonio Arnau and David Soares

It is necessary to indicate that this piezoelectric model is, therefore, an approximation of the electrical impedance response of a piezoelectric material subjected to a variable electric field, whose excitation frequencies are near the natural vibration frequencies of the material. In a similar way, when the excitation frequencies are even multiples of the natural oscillation fundamental frequency, the displacement speed is null because there is no particle displacement at the ends and, therefore, no current by piezoelectric effect is induced. Under these circumstances, the electric model of the piezoelectric material is reduced to the branch formed by the capacitor representing the current induced by the dielectric effect. This result is of practical importance, as it will be shown in Chap. 6. As it has already been mentioned, these results were obtained after neglecting the material’s losses. However, they are really true for relatively small losses such as those that occur in most practical cases. A mathematical expression which included the loss effects in a more rigorous way can be obtained from Eq. (1.59). In fact, considering the relations between the trigonometric and hyperbolic functions, the equation becomes:

αy ζ =

cos

ωy

+ j sin

ωy

dVm sinh(γ y ) d Vm ⎛ v ⎞ 2v vo vo ≈ ⎜− j o ⎟ o l ⎝ γl ω ⎠ cos ω l + j α l sin ω l ⎛ γl ⎞ cosh⎜ ⎟ 2vo 4vo 2vo ⎝2⎠

(1.62)

where the following approximations have been done:

γ≈j

ω vo

, sinh

αy αy αl αl ≈ , sinh ≈ , 2vo 2vo 4vo 4vo

αy αl ≈ 1 and cosh ≈1. 2vo 4vo Operating in Eq. (1.62) and disregarding terms equal to or higher than the second order in term α, one gets to the following expression for the displacement at y= l/2: cosh

ξ ( l 2 ,t) ≈

d Vm v o 2 lω

sin 2 cos

2

ωl 2v o

ωl vo

+

+

α 2l 2

α2l2 16vo2

4vo2 sin

2

ωl

sin (ω t + θ )

(1.63)

2vo

where θ = -arctan [(sin ωl/vo)/(αl/2vo)] is the out of phase between the electric voltage applied and the displacement produced in the particles at the coordinate y= l/2.

1 Fundamentals of Piezoelectricity

25

As it can be deduced from Eq. (1.63), the losses limit the maximum displacement at resonant frequencies and keep a small displacement for the even harmonics of the natural vibration fundamental frequency without losses. However, it can be proved that the frequencies which have the maximum displacement speed are still those that correspond to the natural vibration frequencies without losses [5]. This last result implies that the frequencies that maximize the current by the series branch of the equivalent electric model shown in Fig. 1.8, that is, those at series resonance of the motional branch, are still the natural vibration frequencies without losses. Next, the different characteristic frequencies of a one-dimensional bar of piezoelectric material, are summarized: Free natural vibration frequencies without losses: Damped natural vibration frequencies:

f no = nf o = n

vo 2l

f np = f no 1 +

1 4Q 2

Forced vibration frequencies with losses for a maximum displacement:

f nf = f no 1 +

1 2Q 2

Frequencies for maximum displacement speed in forced vibration with losses:

f nv = f no = nf o = n

vo 2l

1.9 Introduction to the Microgravimetric Sensor An AT cut quartz crystal vibrates in thickness shear mode. This vibration mode is the one chosen to develop the previous sections. However, the results obtained are general when considering a piezoelectric material where only one dimension determines the vibrating state of the bar or section of material. From the natural vibrating fundamental frequency of the material, the physical fundamentals that have permitted to use the piezoelectric crystal as a micro–gravimetric sensor can be understood. Among the piezoelectric crystals, the AT cut quartz is the most commonly used as a sensor for this type of applications. According to Eq. (1.26), the natural vibrating fundamental frequency of a piezoelectric material is given by fo = vo /2l. As a result, that frequency depends on the intrinsic properties of the material and on the dimension that determines the vibrating state, in this case the thickness. Therefore, if the physical properties of the material are considered as constant, the frequency is substantially determined by its thickness and can be written as

26

Antonio Arnau and David Soares

fo = N/l, where N is the so-called frequency constant and depends on the material and the type of cut. Thus, a change in the thickness will imply a variation in the system’s vibration frequency. This variation can be mathematically obtained in a simple way by taking logarithms and deriving the expression for the frequency fo. Consequently, the following relation is obtained: Δf Δl =− fo l

(1.64)

The change in the thickness can be written according to the mass change as:

Δl =

Δm ρ v AS

(1.65)

where AS is the surface. Considering the relation between the thickness and frequency, the variation of the frequency settled by Eq. (1.64) can be written as:

Δf = −

f o2 Δm = −C f ρ s ρ v N AS

(1.66)

The previous equation indicates that if the resonant frequency is chosen as parameter, the shift in the resonant frequency provides a measurement of the surface mass density on the sensor. An important factor to know is that it has been assumed that the frequency shift was due to an increase in the material’s thickness. The properties of the material have been used to set the relations between the changes in the thickness and in the mass. So that the previous equation is still valid for masses of different materials to that used as sensor; thus, it is necessary to assume that the effect on the vibration frequency is the result of a merely inertial perturbation; i.e., the viscoelastic properties of the material deposited must not affect the resonant frequency. This assumption assumes that the layer of material deposited on the sensor does not deform and is, therefore, an approximation fulfilled under certain conditions [6]. However, it has been proved to be precise in many practical applications. It is interesting to establish to what extent the measurement of surface mass density is sensitive. For this, a 10 MHz AT cut crystal will be used as piezoelectric material whose properties are shown in Table 1.1. A resolution of 0.1 Hz will be set for the vibration frequency measurement. Under these conditions, the maximum sensitivity for the surface mass density will be:

1 Fundamentals of Piezoelectricity

Δρ s =

ρv N f o2

ρv N

Δf =

f o2

0.1 ≈ 4pg mm -2

27

(1.67)

This great sensitivity, one million times higher than conventional static balance systems, is due to the enormous acceleration that the particles joined rigidly to the quartz surface experience. To evaluate the particles’ acceleration, firstly the vibration amplitude of a quartz crystal with a quality factor Q has to be estimated. The vibration amplitude of a system with a quality factor Q at resonance can be set according to the static amplitude, for quality factors higher than 5 as [7]:

A = Q Ao

(1.68)

where Ao is the amplitude at zero frequency, i.e, the static amplitude. The static displacement can be calculated from the thickness and the strain as:

Ao = l d E

y=

l 2

= dl

Vm = dVm l

(1.69)

where d is the piezoelectric strain coefficient for an AT cut crystal and Vm is the maximum voltage difference to what the quartz plate is subjected between its electrodes. Consequently, for a 10 MHz AT quartz crystal, with a quality factor of 80.000 (very reasonable value in practice), which has been subjected to a variable voltage difference with a maximum amplitude of Vm=250 mV, the amplitude of resonance given by Eq. (1.68) will be around 330 Å (this result is in agreement with the measurements done by some researchers [810] for a vibrating quartz in an unperturbed state). Table 1.1. Properties of typical 10MHz AT-cut quartz Quartz Parameter ε22 ηq

Value

Description

3.982×10-11 A2 s4 Kg-1 m-3 9.27×10-3 Pa s

c66

2.947×1010 N m-2

e26 ρq AS l=lq

9.657×10-2 A s m-2 2651 Kg m-3 2.92×10-5 m2 166.18×10-6 m

Permittivity Effective viscosity Piezoelectrically stiffened shear modulus Piezoelectric constant Density Effective electrode surface area Thickness

28

Antonio Arnau and David Soares

Therefore, the maximum acceleration of the oscillating system will be:

a = ω 2 A = ω 2 dVm Q = 1,3 ⋅ 108 m ⋅ s -2 ≈ 10 7 g

(1.70)

where g is the acceleration of gravity. In other words, this result means that a mouse of 100 g subjected to this acceleration would weigh one thousand tons.

Appendix 1.A The Butterworth Van-Dyke Model for a Piezoelectric Resonator 1.A.1 Rigorous Obtaining of the Electrical Admittance of a Piezoelectric Resonator. Application to AT Cut Quartz

The electrical admittance of a piezoelectric resonator considered as a onedimensional system will be obtained in this appendix. An equivalent electrical model at frequencies near the resonant frequencies of the piezoelectric resonator will be derived from the expression of this admittance. An AT-cut quartz crystal will be used to represent the piezoelectric resonator but it will not reduce the generality of result. A one-dimensional AT cut quartz plate undergoes a strain xy (see Appendix B) when an electric field in the thickness direction Y is applied. The responsible of this effect is the piezoelectric stress coefficient e26 (see Appendix B to understand the meaning of subscripts) which is not null in the quartz [4, 5]. Rigorously, a strain zx corresponding to the CT and DT cuts can arise through the coefficient d25 = -d14. However, in the high frequency range, where these resonators are used, the fundamental strain is S6 = xy. Consistently, Fig. 1.A.1 shows a cross section of an AT-cut plate vibrating in thickness shear mode. In this vibrating mode particle displacement is perpendicular to the wave propagation direction, creating a transversal wave propagating in the thickness direction. Thus the particle moves in direction X around its rest position with amplitude which depends on the coordinate y. When the electric field is variable, so is the strain. Thus, when the applied voltage is sinusoidal, it is assumed that the strain is sinusoidal as well. In this case, the transitory solution corresponding to the free oscillation state vanishes with time. The pursuit of a stationary solution assumes that the particles’ displacement is harmonic with the same angular frequency as the external phenomenon which produces the oscillation. Next, the wave equation of the movement is deduced.

1 Fundamentals of Piezoelectricity

29

Y

lQ F(y) dy y

F(y) dy

F(y) y 0

X

As

Fig. 1.A.1. Shear strain profile and forces in an internal thin slice of an AT-cut quartz plate subjected to a sinusoidal electric field

The recovery force has an elastic component through the elastic constant c66 and a component corresponding to the internal friction in the material. According to Eq. (1.48), the internal stress is: T6 = c66 S 6 − e26 E 2 + ηQ

∂ S6 ∂t

(1.A.1)

where the lossless coefficient has been represented with the viscosity of quartz and indicated with the subscript Q. Considering the following relationships between the electric field and the voltage, between the strain and the particles’ displacement and keeping in mind that only displacement in the direction X exist, that is:

E 2 ( y, t ) = −

∂V ( y, t ) ∂y

∂ξ y ⎛ ∂ξ S 6 = 2 S12 = ⎜⎜ x + ∂x ⎝ ∂y

(1.A.2)

⎞ ∂ ξ x ∂ ξ ( y, t ) ⎟⎟ = = ∂y ⎠ ∂y

(1.A.3)

∂ ξ ( y, t ) ∂V ( y, t ) ∂ 2 ξ ( y, t ) + e26 + ηQ ∂y ∂y ∂t∂ y

(1.A.4)

Equation (1.A.1) becomes:

T6 = c66

On the other hand, the electric displacement in direction Y is:

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Antonio Arnau and David Soares

D2 = ε 22 E 2 + e26 S 6 = e26

∂ ξ ( y, t ) ∂V ( y, t ) − ε 22 ∂y ∂y

(1.A.5)

Since inside the quartz there is no free charge and the electrical displacements in the X and Z directions are null, the Maxwell equation for the divergence of the electric displacement indicates that the displacement in the Y direction not being null has to be a constant (see Appendix A) and consequently:

∂ 2 V ( y , t ) e26 ∂ 2ξ ( y , t ) = ε 22 ∂ y 2 ∂ y2

∂D2 =0 ⇒ ∂y

(1.A.6)

Thus, the partial derivative of the voltage will be

∂ V ( y , t ) e26 ∂ξ ( y , t ) = + C (t ) ε 22 ∂ y ∂y

(1.A.7)

and Eq. (1.A.4) results into:

⎛ e2 T6 = ⎜⎜ c66 + 26 ε 22 ⎝

⎞ ∂ ξ ( y, t ) ∂ 2 ξ ( y, t ) ⎟ η + + e26 C (t ) = Q ⎟ ∂y ∂t∂ y ⎠

∂ ξ ( y, t ) ∂ 2 ξ ( y, t ) = c66 + ηQ + e26 C (t ) ∂y ∂t∂ y

(1.A.8)

The equilibrium of forces in a thin slice of material of thickness dy shown in Fig. 1.A.1 results in the following equation:

∂ T6 ∂ 2ξ =ρ 2 ∂y ∂t

(1.A.9)

By substituting Eq. (1.A.8) in Eq. (1.A.9) the following equation is obtained:

ρ

∂ 2 ξ ( y, t ) ∂ 2ξ ( y , t ) ∂ 3 ξ ( y, t ) = c + η Q 66 ∂t2 ∂ y2 ∂t ∂ y2

(1.A.10)

In the forced vibration in stationary state the displacements do not vanish with time. Let us assume that these displacements are sinusoidal and have amplitude dependent on coordinate y. Consequently they can be formulated as follows:

ξ ( y , t ) = ( Ae

jγ Q y

+ Be

− jγ Q y

) e jω t = Φ ( y ) e jω t

(1.A.11)

1 Fundamentals of Piezoelectricity

31

By substituting Eq. (1.A.11) in Eq. (1.A.10) the following expression is obtained: γQ =ω

ρ c 66 + jωηQ

=ω

ρ c 66

1 1+ j

=

ω ηQ

ω

1

v

1 1+ j Q

c 66

≈

ω⎛

1 ⎞ ⎜1 − j ⎟ 2Q ⎟⎠ v ⎜⎝

(1.A.12)

where it has been assumed that the quality factor given by Eq. (1.46) is much greater than unity. From Eq. (1.A.12) the real and imaginary parts of γQ are obtained as follows: Re(γ Q ) ≈

2π

λ

Im(γ Q ) ≈

≈

ω

(1.A.13)

v

ω

(1.A.14)

2Q v

Parameters A and B in Eq. (1.A.11) are determined with appropriate boundary conditions. Because we are dealing with forced vibrations driven with an alternate voltage applied between the contacting electrodes, it is appropriate to establish the boundary conditions derived from this situation. From Eq. (1.A.7) the voltage as a function of the coordinate y and time t is obtained. Considering Eq. (1.A.11) one obtains: ⎛e ⎞ V ( y , t ) = ⎜⎜ 26 Φ ( y ) + C y + D ⎟⎟e jω t ⎝ ε 22 ⎠

(1.A.15)

Consequently the problem is defined with the following equations:

ξ ( y , t ) = ( Ae

jγ Q y

+ Be

− jγ Q y

) e jω t

⎛e ⎞ − jγ y jγ y V ( y , t ) = ⎜⎜ 26 ( Ae Q + Be Q ) + C y + D ⎟⎟e jω t ⎝ ε 22 ⎠

(1.A.16) (1.A.17)

Next, parameters A, B, C and D will be determined according to the following boundary conditions: a) T6 = 0, at y = 0. b) T6 = 0, at y = lQ. c) V (0, t ) = ϕ o e jω t , voltage boundary condition at y = 0. d) V (0, t ) = −ϕ o e jω t , voltage boundary condition at y = lQ.

32

Antonio Arnau and David Soares

Applying the former conditions to Eq. (1.A.16) and Eq. (1.A.17) gives the following expressions:

jγ Q c66 [A − B ] + e26 C = 0

[

jγ Q c66 Ae

jγ Q lQ

e26

ε 22 e26

ε 22

[Ae

jγ Q lQ

− Be

− jγ Q lQ

]+ e

26

(1.A.18)

C =0

[A + B ] + D = ϕ o

+ Be

− jγ Q lQ

]+ C l

Q

(1.A.19)

(1.A.20)

+ D = −ϕ o

(1.A.21)

where c66 = c66 + jωηQ The former expressions can be formulated in a matrix as follows:

⎛ jγ Q c66 ⎜ jγ l ⎜ jγ Q c66 e Q Q ⎜ e26 ⎜ ε 22 ⎜ ⎜ e26 jγ Q lQ ⎜ ε e ⎝ 22

− jγ Q c66 − jγ Q c66 e e26 e26

− jγ Q lQ

ε 22

ε 22

e

− jγ Q lQ

e26 e26 0 lQ

0⎞ ⎟ 0 ⎟⎛⎜ A ⎞⎟ ⎛⎜ 0 ⎞⎟ ⎟⎜ B ⎟ ⎜ 0 ⎟ 1 ⎟⎜ ⎟ = ⎜ ⎟ ⎟⎜ C ⎟ ⎜ ϕ o ⎟ ⎟ ⎟⎜ ⎟ ⎜ 1 ⎟⎝ D ⎠ ⎝ − ϕ o ⎠ ⎠

(1.A.22)

Parameters A, B, C and D can be determined from the former expression following traditional methods. Their values can be found elsewhere [11]. However, as we will show, to obtain the electrical admittance of the resonator, only the parameter C is necessary and its value is: C= 2

2 e26

ε 22

2 ϕ o c66γ Q γ Q lQ tan − c66γ Q lQ 2

(1.A.23)

From Eq. (1.A.5) and Eq. (1.A.7) the electric displacement D2 is: D2 = −ε 22 C e jω t

(1.A.24)

Consequently, the current density J is: J=

∂D2 = − jωε 22 C e jω t ∂t

(1.A.25)

1 Fundamentals of Piezoelectricity

33

Therefore, if the current density is assumed uniform, the total current will be: (1.A.26)

I = J AS = − jω ε 22 AS C e jω t

Considering the voltage difference applied through the quartz which is 2ϕ o e jω t , the electrical admittance Y is: Y = − jωε 22 AS

C 2ϕ o

(1.A.27)

where it can be noticed that only the parameter C appears. Thus the final expression for the electrical admittance of the resonator is:

Y = − jωε 22

AS lQ

2

2 e26

ε 22

c66γ Q lQ γ Q lQ tan − c66γ Q lQ 2

(1.A.28)

By introducing Co as the value of the capacitor formed by the quartz as dielectric material between the electrodes given by Co = ε22AS /lQ, Eq. (1.A.28) can be rewritten as follows: Y = jω Co +

1 = − jω Co Zm

2

2 e26

ε 22

c66γ Q lQ γ Q lQ tan − c66γ Q lQ 2

(1.A.29)

where Zm is:

⎛ ⎜ c66γ Q lQ i ⎜ − 1 Zm = 2 γ l ω Co ⎜ e ⎜⎜ 2 26 tan Q Q 2 ε 22 ⎝

⎞ ⎟ ⎟ ⎟ ⎟⎟ ⎠

(1.A.30)

The former expression corresponds to the electrical impedance of the resonator as a vibrating system due to the piezoelectric effect. This is the reason why such impedance is called motional impedance. For a better understanding of such impedance, it is necessary to simplify Eq. (1.A.30). For that it will be assumed that the resonator is working near any of its resonant frequencies. At those frequencies, the product γQ lQ is approximately nπ where n = 1, 3, 5, etc., being equal to nπ when the losses are neglected (Eq. (1.A.13)). For these frequencies, the tangent in equation (1.A.30) has a pole. Such a trigonometric function can be expanded through its poles as follows [12].

34

Antonio Arnau and David Soares

tan

γ Q lQ 2

≈

4γ Q lQ

(1.A.31)

( nπ ) − (γ Q lQ ) 2 2

By substituting the previous expansion in Eq.(1.A.30) and keeping in mind that the results obtained by the application of this expansion are restricted to frequencies near resonance, one obtains: Zm =

j ω Co

⎛ ( nπ ) 2 − (γ Q lQ ) 2 ⎞ ⎜1 − ⎟ 2 ⎜ ⎟ 8 K ⎝ ⎠

(1.A.32)

where K2 has been defined as follows: K2 =

2 e26

c66ε 22

=

2 e26

1 1 = K o2 1 c66ε 22 1 + j 1 1+ j Q Q

(1.A.33)

It can be noticed that, neglecting the losses, all the terms in Eq. (1.A.33) are reactive and the motional impedance Zm is null for the following value of the product γQ lQ (which is the spatial phase of the propagating mechanical wave):

γ Q lQ =

ω s2 lQ2 v2

= ( nπ ) 2 − 8K o2

(1.A.34)

For this value, the electrical admittance of the resonator is infinite, if the losses are neglected, and the corresponding frequency is called the motional series resonant frequency fs. By substituting the complex values of γQ and K2 given by Eqs.(1.A.12) and (1.A.33) in Eq. (1.A.32), the following expression for the motional impedance is obtained: 2 2 ⎛ ⎛ 1 ⎞ ω lQ ⎜ ( nπ ) 2 ⎜⎜1 + j ⎟⎟ − 2 Q⎠ v j ⎜ ⎝ Zm = ⎜1 − 2 ω Co ⎜ 8K o ⎜ ⎝

[

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(1.A.35)

]

By substituting the ratio l Q2 / v 2 = (nπ ) 2 − 8 K o2 / ω s2 obtained from Eq. (1.A.34) in Eq. (1.A.35) and taking into account that the angular frequency ω is very near to ωs, Eq. (1.A.35) can be written as follows:

1 Fundamentals of Piezoelectricity

Zm =

( nπ ) 2 ( nπ ) 2 1 ( nπ ) 2 + + ω j jω 8K o2 Co 8K o2ω Co Q 8K o2ω s2 Co

35

(1.A.36)

It can be easily noticed that Eq. (1.A.36) is analogous to the impedance of a series R, L, C circuit whose resonant frequency is ωs, which cancels the imaginary part of the impedance. Considering the relations of the electrical parameters in Eq. (1.A.36) and the physical magnitudes of the resonator, the equivalent parameters R, L, C of the model can be obtained as follows:

R=

( nπ ) 2ηQ lQ 2 AS 8 e26

;

L=

ρ lQ3 2 8 e26 AS

;

C=

2 8 e26 AS lQ ( nπ ) 2 c66

(1.A.37)

Thus, the equivalent electrical circuit modelling the impedance of a piezoelectric resonator at frequencies near resonance is a circuit, as shown in Fig. 1.A.2, formed by two parallel branches: one being a capacitor Co which corresponds to that one arisen due to the dielectric material between the electrodes, and the other one being a series R, L, C branch modelling the motional impedance of the resonator. The values derived for the parameters of the equivalent model match with those obtained by other authorities [4, 5]. In Chapter 4, an equivalent model for a quartz sensor in contact with a viscoelastic medium will be derived following a similar approach. L

R

C

Co Fig. 1.A.2. Butterworth Van-Dyke model of a piezoelectric resonator vibrating at frequencies near resonance

1.A.2 Expression for the Quality Factor as a Function of Equivalent Electrical Parameters

Next, the quality factor of the piezoelectric resonator will be derived as a function of the equivalent electrical parameters. This expression will be of importance in some of the next chapters.

36

Antonio Arnau and David Soares

The quality factor associated to the motional impedance of the equivalent electrical model at the motional series resonant angular frequency ωs is: Q=

Lω s R

(1.A.38)

Considering the relations given by Eq. (1.A.37), the former equation can be rewritten as follows: Q=

ρ lQ2 ω s ( n π ) 2 ηQ

(1.A.39)

The motional series resonant frequencies coincide with the natural vibration frequencies without looses given by Eq. (1.26). Consequently, the motional series resonant angular frequency ωs is given by:

ωs = n

vo 2 lQ

(1.A.40)

where vo is given by Eq. (1.23) which is rewritten as follows: vo =

c66

ρ

(1.A.41)

By substituting Eqs. (1.A.41) and (1.A.40) in Eq. (1.A.39), the following expression for the quality factor of the resonator is obtained: Q=

c66

ηQ ω s

(1.A.42)

It can be noticed that the former expression coincides with Eq. (1.46) derived from a physical point of view and is consistent with it. However, it is necessary to make clear that the previous expression for the quality factor is related to the motional branch and that the electrical contribution of the parallel capacitor is not included. Therefore the expression obtained for the quality factor must be considered an approximate equation for the quality factor of the resonator at frequencies very close to the motional series resonant frequencies.

1 Fundamentals of Piezoelectricity

37

References 1. W.G. Hankel (1881) “Uber die aktinound piezoelektrischen eigenschaften des bergkrystalles und ihre beziehung zu den thermoelektrischen” Abh. Sächs. 12: 457 2. P. & J. Curie (1880) “Développement, par pression, de l'électricité polaire dans les cristaux hémièdres à faces inclinées” Comptes Rendus 91: 294-295 3. G. Lippmann (1881) “Principe de conservation de l'électricité” Annales de Physique et de Chimie, 5ª Serie 24: 145-178 4. V.E. Bottom (1982), “Introduction to quartz crystal unit design”, Van Nostrand, New York 5. W.G. Cady (1964), “Piezoelectricity: An introduction to the theory and applications of electromechanical phenomena in crystals”, Dover Publication Inc., New York, 2nd edn. 1964 (II Vols) 6. G. Sauerbrey (1959) “Verwendung von schwingquarzen zur wägung dünner schichten und zur mikrowägung” Zeitschrift Fuer Physik 155 (2): 206-222 7. A.P. French (1974) “Vibraciones y ondas” Ed. Reverté S.A 8. G. Sauerbrey (1964) “Messung von plattenschwingungen sehr kleiner amplitude durch lichtstrommodulation” Zeitschrift Fuer Physik 178: 457-471 9. L. Wimmer, S. Hertl, J. Hemetsberger and E. Benes (1984) “New method of measuring vibration amplitudes of quartz crystals” Rev. Sci. Instrum. 55: 605609 10. V.M. Mecea (1989) “A new method of measuring the mass sensitive areas of quartz crystal resonators” Journal of Physics. E: Sci. Instrum. 22: 59-61 11. C. Reed, K. Kanazawa and J.H. Kaufman (1990) “Physical description of a viscoelastically loaded AT-cut quartz resonator” J. Appl. Phys. 68:1993-2001 12. J.F. Rosenbaum (1988) “Bulk acoustic wave theory and devices” Artech House Inc., Boston 13. A. Samartin Quiroga (1990) “Curso de elasticidad” Librería Editorial Bellisco 14. W.P. Mason (1948) “Electromechanical transducers and wave filters” Van Nostrand Company, Inc. 2nd edn. 15. H. Lamb (1960) “The dynamical theory of sound” Dover Publications, Inc. New York 16. J.L. Davis (1988) “Wave propagation in solids and fluids” Springer-Verlag, Berlin, Heidelberg, New York 17. J. Zelenka (1986) “Piezoelectric resonators and their applications” Elsevier 18. R.A. Heising (1946) “Quartz crystals for electrical circuits” Van Nostrand Company, Inc. New York 19. W.P. Mason (1964) “Piezoelectric crystals and their applications to ultrasonic” Van Nostrand Company, Inc. 4th edn. 20. W.P. Mason (1981) “Piezoelectricity, its history and applications” Journal of Acoustical Society of America 70:1561-1566 21. H.F. Tiersten (1969) “Linear piezoelectric plate vibrations” Plenum Press, New York

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Antonio Arnau and David Soares

22. D. Belincourt (1981) “Piezoelectric ceramics: characteristics and applications” Journal of Acoustical Society of America 70:1586-1594 23. A.H. Love (1934) “Theory of elasticity” Cambridge University Press, 4th edn

2 Overview of Acoustic-Wave Microsensors Vittorio Ferrari1 and Ralf Lucklum2 1

Dipartimento di Elettronica per l’Automazione, Università di Brescia Institute for Micro and Sensor Systems, Otto-von-Guericke-University Magdeburg

2

2.1 Introduction The term acoustic-wave microsensor in its widest meaning can be used to indicate a number of significantly different devices. Their common characteristic is the fact that acoustic waves are involved in the operating principles. Acoustic-wave microsensors can be grouped into the following three classes. 1. Microfabricated, or miniaturized, sensors where acoustic waves, i.e. matter vibrations propagating in elastic media, are involved in the sense that they define the domain of the measurand quantity. Examples of this type of devices are accelerometers, microphones, and acoustic-emission pick-ups. The piezoelectric effect, though often used, is not necessarily required in this class of sensors. 2. Microfabricated, or miniaturized, sensors that emit and receive acoustic waves in a surrounding medium along a distance which is typically longer than several wavelengths, in order to sense the properties of the medium and/or the presence and nature of internal discontinuities. This class of devices essentially includes ultrasound transducers, both singleelement and arrays, for acoustic inspection, monitoring, and imaging in air, solids, and liquids. The majority, though not the totality, of these devices base their functioning on the piezoelectric effect, mostly because of its reversibility and efficiency. 3. Microfabricated, or miniaturized, sensors in which acoustic waves propagate and interact with a surrounding medium, in such a way that the degree of interaction or the properties of the medium can be sensed

A.A. Vives (ed.), Piezoelectric Transducers and Applications, doi: 10.1007/978-3-540-77508-9_2, © Springer-Verlag Berlin Heidelberg 2008

40

Vittorio Ferrari and Ralf Lucklum

and measured from the characteristics of the acoustic or electro-acoustic field in the sensor itself [1]. The sensors of this latter kind essentially behave as acoustic waveguides which, depending on the configurations, can be made responsive to a wide range of physical quantities, like applied stress, force, pressure, temperature, added surface mass, density and viscosity of surrounding fluids. In addition, sensors can be made responsive to chemical and biological quantities by functionalizing their surface with a coating which, depending on its composition, is (bio)chemically active and works as a “receptor” for the analytes to be detected (see Chap. 11). The coating film has the role of a (bio)chemical-to-physical transducer element, as it converts signals from the (bio)chemical domain into variations of physical parameters, typically the equivalent mass, stiffness, or damping, that the acoustic sensor can detect and measure. (see also Chaps. 3, 10, 11, 12, 13, 14) This class of acoustic-wave sensors makes an extensive use of the piezoelectric effect and comprises a number of device types that differ either in the nature of the acoustic waves involved or in configurations adopted. In the following, the main characteristics of piezoelectric acoustic-wave microsensors belonging to the class 3 will be illustrated.

2.2 General Concepts The basic principle of operation for a generic acoustic-wave sensor is a traveling wave combined with a confinement structure to produce a standing wave whose frequency is determined jointly by the velocity of the traveling wave and the dimensions of the confinement structure. Consequently, there are two main effects that a measurand can have on an acoustic-wave microsensor: the wave velocity can be perturbed or the confinement dimensions can be changed. In addition, the measurand can also cause a certain degree of damping of the travelling wave. An important distinction between sensor types can be made according to the nature of the acoustic waves and vibration modes involved in different devices. The devices usually have the same name as the wave dominant in the device. In the case of a piezoelectric crystal resonator, the traveling wave is either a bulk acoustic wave (BAW) propagating through the interior of the substrate or a surface acoustic wave (SAW) propagating on the surface of the substrate (see Fig. 2.1).

2 Overview of Acoustic-Wave Microsensors QCM

SAW

LW

FPW

41

SH-APM

T op Top

E

Fig. 2.1. Different types of acoustic-wave sensors

In the bulk of an ideally infinite unbounded solid, two types of bulk acoustic waves (BAW) can propagate. They are the longitudinal waves, also called compressional/extensional waves, and the transverse waves, also called shear waves, which respectively identify vibrations where particle motion is parallel and perpendicular to the direction of wave propagation. Longitudinal waves have higher velocity than shear waves. When a single plane boundary interface is present forming a semiinfinite solid, surface acoustic waves (SAW) can propagate along the boundary. Probably the most common type of SAWs are the Rayleigh waves, which are actually two-dimensional waves given by the combination of longitudinal and transverse waves and are confined at the surface down to a penetration depth of the order of the wavelength. Rayleigh waves are not suited for liquid applications because of radiation losses. Shear horizontal (SH) particle displacement has only a very low penetration depth into a liquid (see Chap. 3), hence a device with pure or predominant SH modes can operate in liquids without significant radiation losses in the device. By contrast, waves with particle displacement perpendicular to the device surface can be radiated into a liquid and cause significant propagation losses, as in the case of Rayleigh waves. The only exception are devices with wave velocities in the device smaller than in the liquid. Other surface waves with important applications in acoustic microsensors are Love waves (LW), where the acoustic wave is guided in a foreign layer and surface transverse waves (STW), where wave guiding is realized with so-called gratings.

42

Vittorio Ferrari and Ralf Lucklum

Plate waves, also called Lamb waves, require two parallel boundary planes. The lowest anti-symmetric mode is the so-called flexural plate wave (FPW). Acoustic plate modes (APM), although generated at the device surface, belong to BAWs. Devices based on acoustic waves shown in Fig. 2.1 are shortly described in the next section. Other types of waves or devices not described here are pseudo-SAW (or leaky SAW) [2] , surface skimming bulk waves [3], Bleustein-Gulyaev-waves [4, 5] as well as magneto-SAWs [6] .

2.3 Sensor Types 2.3.1 Quartz Crystal Thickness Shear Mode Sensors The oldest application of quartz crystal resonators (QCR) as sensors is the quartz crystal microbalance (QCM or QMB). These sensors typically consist of a thin AT-cut quartz plate with circular electrodes on both parallel main surfaces of the crystal. BAWs are generated by applying an electrical high-frequency (HF) signal to the electrodes. QCMs are operated as resonators in an almost pure thickness-shear mode, hence the sensors are also called TSM sensors. The sensor resonant frequencies are inversely proportional to the crystal thickness. For the fundamental mode, resonance frequencies of 5 to 30 MHz are typical. For higher frequencies the crystals can be operated at overtones. Nowadays high-frequency QCRs with fundamental frequencies up to 150 MHz are available. The required crystal thickness down to 1 µm is prepared by chemical milling and, for mechanical stability reasons, the etching of the crystal is limited to the region of the electrode area, leading to inverted-mesa structures. After their first use as frequency-reference elements in time-keeping applications in 1921 by W. Cady and as a microbalance in 1959 by G. Sauerbrey [8], quartz crystals have become probably the most common acoustic-wave sensors, finding application in the measurement of several other quantities and, in turn, opening the way to the development of newer and more specialized sensors. The typical configuration is as singleelement sensors, but multisensor arrays on the same crystal have been recently proposed [9, 10]. The basic effect, common to the whole class of acoustic-wave microsensors, is the decrease in the resonant frequency caused by an added surface mass in the form of film. This gravimetric effect motivates the

2 Overview of Acoustic-Wave Microsensors

43

denomination of quartz-crystal microbalance and is exploited, for instance, in thin-film deposition monitors and in sorption gas and vapor sensors using a well-selected coating material as the chemically-active interface [11, 12]. Within a certain range, the frequency shift Δf is sufficiently linear with the added loading mass Δm regardless of the film material properties, and the sensitivity Δf/Δm is proportional to f 2 [8]. For higher loading, the sensor departs from the gravimetric regime and the frequency shift becomes a function of the mass as well as of the viscoelastic properties of the film [13] (see Chaps. 3, 14). TSM quartz sensors can also operate in liquid, due to the predominant thickness-shear mode. In this case, the frequency shift is a function of liquid density and viscosity [14] (see Chap. 3), which makes it possible to use TSM quartz resonators as sensors for fluid properties [15]. In addition, the mass sensitivity and in-liquid operation can be advantageously combined, and TSM sensors coated with (bio)chemically-active films can be used for in-solution (bio)chemical analysis, for instance in the chemical, biomedical and environmental fields [16] (see Chaps. 9, 11, 12, 13, 14). Mass sensitivity and liquid density-viscosity sensitivity are two special cases of the more general sensitivity of all acoustic-wave microsensors to the so-called surface acoustic load impedance, which is discussed in Chap. 3. Because of its importance and simplicity we further limit the discussion here to mass sensitivity and applicability of the devices in a liquid environment. 2.3.2 Thin-Film Thickness-Mode Sensors These are BAW sensors based on thickness-mode waves that, as opposed to TSM quartz crystals, are of the longitudinal type, at least in the early implementations of the concept. They are made by electroded piezoelectric thin films and are therefore also termed film bulk acoustic resonator (FBAR) sensors. Films of piezoelectric materials, such as AlN or ZnO, are created in the form of diaphragms photolithographically defined and etched starting from a silicon substrate. In this way, a very low thickness can be obtained that causes a high resonant frequency, up to 1000 MHz [17] and above. This, in turn, determines a high mass sensitivity in gravimetric applications. As opposed to free-standing, or suspended, homogeneous resonators, composite resonators can also be used where the piezoelectric film is deposed on a nonpiezoelectric substrate, such as silicon, with intermediate

44

Vittorio Ferrari and Ralf Lucklum

layers with different acoustic impedances. Composite film resonators can display improved thermal stability due to the property matching that can be obtained among different layers. A significant case is when the layers have alternate high and low acoustic impedances, thereby forming a Bragg reflector which acts as an acoustic mirror that isolates the film from the substrate [18]. This configuration is often termed as solidly-mounted resonator (SMR). The structures of suspended and SMR FBARs are shown in Fig. 2.2. The SMR solution has the effect to decrease the effective thickness and is especially interesting for sensing applications, because it avoids the need of etching away the silicon to form the thin suspended diaphragm. This advantageously mitigates the problem of fragility. Composite resonators can also be made by resonant piezo-layers (RPL) of lead-zirconate-titanate (PZT) films screen printed on alumina substrate [19]. RPL sensors display a mass sensitivity comparable or slightly higher than TSM quartz sensors at the same frequency, though the thermal stability is worse. Most likely due to their porosity, thick-film RPL sensors with chemically functionalized surface apparently offer an improved sensitivity as sorption sensors in air [20]. Thickness-longitudinal-mode sensors have many analogies with TSM quartz sensors. One important difference is that, in the former ones, the vibrations normal to the sensor surface irradiate energy into a surrounding liquid, which makes thin-film thickness longitudinal mode sensors generally unsuitable for (bio)chemical applications in solutions. For this reason, efforts have been aimed to the development of shearmode FBAR sensors. Recently reported devices have a configuration similar to thickness-longitudinal-mode sensors with the difference that they exploit the oriented growth of ZnO piezoelectric films to generate thickness-shear-mode vibrations [21]. As an alternative, shear-wave generation using lateral field excitation has also been reported [22]. Shear-mode FBARs are expected to have a high potential especially for highly integrated biochemical sensor arrays, though the very high operating frequencies (in the range 1-10 GHz) can pose significant challenges to the readout electronic circuits and instrumentation. electrodes

electrodes piezo film

piezo film

} acoustic reflector

Si substrate

a b Fig. 2.2. Film bulk acoustic resonator (FBAR) sensors: a free-standing structure; b solidly-mounted resonator (SMR) structure

2 Overview of Acoustic-Wave Microsensors

45

2.3.3 Surface Acoustic Wave Sensors Surface acoustic wave (SAW) sensors are made by a thick plate of piezoelectric material, typically ST-cut quartz, lithium niobate or lithium tantalate, where predominantly Rayleigh waves propagate along the upper surface [23]. Surface wave generation is efficiently accomplished by a particular electrode configuration named interdigital transducer (IDT) (Fig. 2.3a). An IDT, in its simple version, is formed by two identical comb-like structures whose respective fingers are arranged on the surface in an interleaved alternating pattern. The IDT period length d, or pitch, is the spacing between the center of two consecutive fingers of the same comb. When an AC voltage is applied to the IDT, acoustic waves are generated which propagate along the axis perpendicular to the fingers in both directions. The maximum wave amplitude is obtained when constructive interference among the fingers occurs. This happens at the characteristic or synchronous frequency fo = v/d, where v is the SAW velocity in the material. Typical SAW characteristic frequencies are 30-500 MHz. Two basic configurations are possible: one-port SAW resonators with a single IDT, and two-port SAW delay lines with two IDTs separated by a distance L. Similarly to what happens with BAW devices, SAWs can be used as high-frequency reference elements in filters and oscillators, but they can also be made responsive to a variety of quantities to have them work as sensors [24]. The primary interaction mechanisms are those that affect the frequency by changing the wave velocity, the IDT distance, or both. Temperature, strain, pressure, force, and properties of added surface materials are examples of measurand quantities. In particular, the accumulated surface mass produces a decrease in frequency. Compared to QCMs, the higher values of the unperturbed frequency and the fact that vibrations are localized near the surface, becoming more affected by surface interactions, determine a higher sensitivity of SAWs in gravimetric applications. This fact is advantageously exploited in sorption gas and vapor sensors where SAWs coated with chemically-active films (Fig. 2.3b) can achieve significantly low detection limits [25]. Due to the configuration of the IDT electrodes, SAW sensors are also responsive to the electric properties of the coating film or the surrounding medium by means of the acoustoelectric coupling. The improvement over quartz crystal TSM sensors offered by SAW sensors in air cannot be extended in liquids because of the vibration

46

Vittorio Ferrari and Ralf Lucklum

component normal to the surface involved in Rayleigh waves, which causes acoustic energy radiation into the liquid with a consequent excess of damping. In principle, IDTs can generate a spectrum of transversal horizontally and vertically polarized waves as well as longitudinal waves, which propagate on the surface or into the volume of the piezoelectric material [7]. Material properties, crystal cut, and sensor geometry are responsible for which modes appear and in what extent. A whole family of SAW-like devices has been developed. The most important ones are further described. thin film

substrate

IDTs

a gas-phase species

film

Vi

Vo

mechanical wave input IDT

interaction region

output IDT

b Fig. 2.3. a Interdigital transducer configuration as used in SAW sensors; b structure of a SAW sensor

2.3.4 Shear-Horizontal Acoustic Plate Mode Sensors Shear-horizontal acoustic plate mode (SH-APM) sensors are quartz plates with thickness of a few wavelengths, where shear-horizontal (SH) waves are generated by means of two IDTs positioned on one surface of the plate [26] (Fig. 2.4).

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SH waves have particle displacement predominantly parallel to the plate surface and perpendicular to the propagation direction along the separation path between the two IDTs and hence are suited for operation in contact with liquid. Typical operation frequencies of SH-APM sensors are 20200 MHz. APMs are a series of plate modes with slightly different frequencies. The difference between these frequencies decreases with decreasing plate thickness. To select a dominant SH mode, material and crystal cut, IDT design and oscillator electronics must be optimized. APMs have antinodes on both device surfaces so that each of them can be used as a sensing surface. In particular, the electrode-free face can be made (bio)chemically active and analysis in solution can be performed with a complete separation between the electric side and the liquid side. viscous conductive liquid medium

input IDT

output IDT

Fig. 2.4. Structure of an APM sensor

2.3.5 Surface Transverse Wave Sensors Surface transverse wave (STW) sensors are devices in which shear vibrations are confined in a thin surface area on the face where the IDTs are placed. This wave confinement is obtained by inserting a metallic grating between the IDTs that introduces a periodic perturbation in the wave path and lowers the wave velocity at the surface [1, 27]. Since the vibration energy density is concentrated on a thin layer near the surface, the device is very responsive to surface perturbations and, in particular, it provides a high mass sensitivity. As shear vibrations are predominant, STW sensors (also called SH-SAW) are indicated for in-liquid applications and are mainly used with chemically-modified surfaces for analysis in solutions.

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2.3.6 Love Wave Sensors Love wave (LW) sensors are rather similar to STW sensors in that they involve shear vibrations confined in the upper surface. The wave confinement is in this case obtained by depositing a thin layer of a material with low acoustic-wave velocity over a quartz plate where two IDTs are realized. Such an added overlayer, typically of silicon dioxide or polymethylmethacrylate (PMMA), works as a waveguide and keeps most of the vibration energy localized close to the surface, regardless of the plate thickness. This has the same positive effect on the mass sensitivity as the gratings in STW sensors and, once again, in-liquid operation is permitted by the shear-mode vibrations [28, 29]. Love-mode sensors are mainly used in (bio)chemical analysis in solutions. A generalized Love-wave theory considers APMs and Love waves as the two solutions of the dispersion equation of a substrate with finite thickness [30]. 2.3.7 Flexural Plate Wave Sensors In thin plates, i.e. diaphragms with thickness smaller than the wavelength, a series of symmetric and antisymmetric plate modes can be generated. These so-called Lamb waves have a particle displacement similar to Rayleigh waves [31, 32], i.e. particle motions describe a retrograde ellipsis with the major and minor axes normal and parallel to the surface, respectively. The wave velocity depends on the plate material and the plate thickness. The advantage of the lowest antisymmetric mode, the so-called flexural plate wave (FPW) mode, is a wave velocity smaller than that of SAW devices. It decreases with decreasing plate thickness and becomes lower than the wave velocity of liquids. This determines a couple of unique features that makes FPW sensors very attractive. The first is that, for a given wavelength, the corresponding frequency is comparatively low, in the range of 5-20 MHz, which alleviates the requirements on the associated electronics. The second is that FPW sensors are best suited to the measurement of fluid properties, such as liquid viscosity, and gravimetric (bio)chemical analysis in solutions. In this latter application, the plate being very thin and significantly affected by surface perturbations, the achievable mass sensitivity can be extremely high [33]. Typically, the plate is a few-micron thick rectangular silicon-nitride diaphragm with a piezoelectric overlayer,

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such as zinc oxide, in which the waves are generated by means of IDTs (Fig. 2.5). Unfortunately, those FPW sensors are still fragile and the fabrication process must be further optimized. Another version of excitation involves a magnetic field [34]. chemically sensitive film

vapor or liquid SixNy plate

Si substrate ZnO

Fig. 2.5. Structure of a FPW sensor

2.3.8 Other Excitation Principles of BAW Sensors The most known quartz crystal microbalance may reveal some limitations when applied as chemical or biochemical sensor. Sensitivity to the mass of molecular species is a very unique advantage of acoustic sensors. However, acoustic sensors are inherently nonspecific. The core of chemical analysis involving surfaces is therefore a method for immobilization of the target molecule on the surface of the transducer (see Chaps. 11, 12), hence mainly a question of surface chemistry and application to complex (bio)molecular systems. From that point of view, the necessity of metal electrodes at the surface interacting with the medium to be investigated is a limitation of applicable surface chemistry. In addition, a simple replacement method for the sensor element, which does not require a skilled operator, is an issue of practical interest. Electrical connection to electrodes on the sensor element can therefore become a critical design factor. Two other principles can overcome these limitations, lateral field excitation (LFE) and direct magnetic generation. The classical LFE design is characterized by two electrodes covering completely the left and the right side of a quartz disc just leaving a small straight gap between them. A lateral electrical field is confined in the gap and excites acoustic vibration, thenceforth the name [35]. Magnetic excitation has been utilized for nondestructive material testing, for example in automotive industry. In a static magnetic field acoustic waves are generated and detected in the material by radio frequency (RF) coils placed next to the test sample. The device has therefore been called electromagnetic acoustic transducer (EMAT) [36]. Just recently both principles have been modified for microacoustic resonator sensors. LFE sensors utilize the same piezoelectric crystal that is

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used in QCM, namely AT-cut quartz. The electrodes are located only in the bottom surface leaving the top sensing surface blank (Fig. 2.6) [37]. The bare surface gives now access to the large variety of silicon based surface chemistry. On the other hand one loses the shielding effect of the top electrode. The aspect ratio between electrode gap distance and crystal thickness is about 3-6. The electric field is not completely confined between the electrodes. Consequently, the electric field penetrates partly into the medium adjacent to the sensing surface of the crystal. This feature can provide access to additional relevant physical material properties of the material under investigation, namely the electrical parameters permittivity and conductivity. The sensor response to electrical properties can be much larger than that to density-viscosity [38]. Electrodes

Quartz Crystal Fig. 2.6. Lateral Field Excited (LFE) sensor

For the understanding of the extraordinary sensor response to electrical properties of a liquid analyte, one must consider the change in the (electrical) boundary conditions at the sensing surface. As a result of liquid application the electrical field distribution changes depending on conductivity and permittivity of the liquid and experimental conditions (grounding). As long as the sensor faces a medium which features a relative permittivity, εr, lower than that of quartz the electrical field is distributed mainly in lateral direction. For a medium featuring a dielectric permittivity higher than that of quartz the internal lateral electric field component decreases in strength and components of the traditional thickness field excitation (TFE) will be amplified. As a consequence, the wave propagation properties of the acoustic wave change, hereby modifying the resonance frequency of the sensor. In other words, the sensitivity to electrical properties of the adjacent liquid does not directly appear in the sensor response, they become effective via changes in the acoustic wave generation scheme and acoustic

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properties of the crystal [39]. Distinction of the contributions to the sensor response from liquid density and viscosity on the one hand and permittivity and conductivity on the other requires advanced analysis. By combining magnetic direct generation with an acoustic resonator it is possible to excite a mechanical resonance in the element. The coil is driven with a stationary RF current or around mechanical resonance (Fig. 2.7). When coinciding with the electrical resonance of the coil which can be adjusted by bridging the coil with a parallel capacitance, the result is a detectable signal response that is improved by the quality factors Q of both resonances. This combination of utilizing a single planar spiral coil was termed magnetic acoustic resonator sensor (MARS) [40]. The advantage of such an acoustic sensor is the ability to utilize a large variety of different materials and material combinations which have been exempt before, i.e., there is no need for piezoelectric materials. Furthermore, a variety of different modes of vibration can be excited. The planar coil setup for magnetic direct generation can also be used to remotely excite piezoelectric transducers. A static magnetic field is not necessary here, since the excitation mechanisms are fundamentally different. This magneto-piezoelectric coupling has been successfully employed to bare, electrode-free quartz crystals [41]. Due to the absence of a large parallel capacitance an additional feature of this excitation principle is the possibility to generate evanescent waves over the megahertz to gigahertz frequency range with the unique ability to focus the acoustic wave down onto the chemical recognition layer.

spiral coil

resonator

- quartz disc - Si membrane - metal plate

Fig. 2.7. Magnetic direct generation with spiral coil. For non-piezoelectric resonators a permanent magnet below the spiral coil and a conductive lower surface is required

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The description of magnetic direct generation of acoustic waves in nonpiezoelectric plates requires Maxwell’s equations and involves two mechanisms. For the first mechanism, according to Lenz’s law, in a conductive layer placed in parallel above the coil eddy currents are generated that will flow in the opposite clockwise direction as the primary current in the coil. The second mechanism involved consists of the interaction between the eddy currents and a magnetic field. Superposing the induced movement of a charge with the magnetic field will result in the Lorentz force that is capable of exciting acoustic waves in the plate. A first option to provide such a magnetic field is to generate it externally in the form of a static field. The Coulomb force due to the electric component of the electromagnetic field created by the primary current is negligible when using an additional strong static magnetic field. For a spiral coil, and therefore circular flowing eddy currents, the direction of the Lorentz forces will be radial. Due to these forces alternating with the primary current frequency, the crystal lattice of the material will start to vibrate and an acoustic wave is generated in the sensor element. A standing acoustic wave then appears if the frequency corresponds to one of the eigenmodes of the element and if the force distribution is compatible with the mode shape of the resonance. At resonance, the vibration of the crystal lattice achieves significantly increased displacements resulting in a second perpendicular induction current component, which is superposed with the eddy currents. Both induced currents affect the mutual inductance between primary coil and the resonator element, whereas the second part only takes a measurable effect in mechanical resonance, which can thus be detected by an RF analyzer circuit monitoring the coil parameters [42]. A second option to provide the required magnetic field is to exploit the same RF field that is generated by the coil and used to produce the eddy currents. The magnetic field is assumed to be sinusoidal at frequency f. As a consequence, the interaction between the eddy currents and the magnetic field itself causes Lorentz forces at frequency 2f that can set the conductive structure into resonance if 2f coincides with the frequency of a proper vibration mode. This frequency doubling action is a distinctive consequence of the nonlinearity in the force generation mechanism [43, 44]. As a further alternative to classical solutions with quartz crystal sensors, a configuration and method has been developed for contactless readout of the resonance response of a TSM resonator array [45]. The configuration uses a crystal with a large common electrode on the front face, and a number of small equal electrodes on the back face, as shown in Fig. 2.8. This leads to localized sensing regions via the confined energy trapping under the small back electrodes. Each back electrode is capacitively coupled to a

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tip electrode separated by a stand-off distance. The tip consists of a small disc and a guard ring, which confine the electric field to the electrode area and make the measurement unaffected by the stray parallel capacitances. A localized mass load added on the front electrode can be consistently detected and measured by scanning the correspondent back electrode, irrespective of the tip-to crystal stand-off distance. The proposed method may be attractive for the perspective development of monolithic TSM sensor arrays with contactless scanning, because it avoids the problems of routing connections to multiple electrodes, at the same time minimizing the influence of stray contributions external to the crystal. Mass Mass Load load

AA00

Guard ring Guard ring

Tip Tip 1

Disc Disc Front view Front ofofthe thetip tip

Fig. 2.8. Contactless localized readout of a quartz TSM resonator

2.3.9 Micromachined Resonators Silicon technology can make a new generation of resonators available with the capability to detect even smaller masses, the capability to fabricate arrays with a much larger number of elements per unit area, the capability of monolithically integrated electronic circuitry and mass production at low costs. Magnetic direct generation applied to Si membranes is a sophisticated example of the new generation. Nowadays the most prominent example are cantilevers which are applied as chemical sensors [46]. Cantilever sensors are typically made of silicon, silicon nitride, or silicon dioxide. A great variety of dimensions and shapes is available [47]. Analog to acoustic sensors, micro electro mechanical systems (MEMS) based sensors are inherently nonspecific, consequently they also need immobilization of chemically sensitive materials on the transducer surface.

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As sensors, cantilevers can be used in the resonant mode or in a nonresonant regime. Analog to acoustic sensors, the devices are sensitive to the mass of molecular species when used in the resonant mode. The mass sensitivity depends on the force constant, which is a function of geometry and the effective Young’s modulus. One major challenging issue is improvement of the quality factor of the resonator. Values of Q of about 100 in the upper kilohertz frequency range in air enable a mass resolution in the picogram range. Calibration of the sensor is required because beam thickness is usually not precisely enough known. One example of the nonresonant application is the stress generated bending. Changes in surface stress can be the result of physical interaction, for example electrostatic forces between charged molecules on the surface, or of chemical nature, e.g. analyte absorption induced swelling of a chemically sensitive coating during chemical sensing. In liquid environment, especially in biosensing applications, the out-ofplane, or flexural, vibration of the cantilever is strongly damped and results in an essentially reduced Q of a few tens only. It can be enhanced by incorporating the cantilever in an amplifying feedback loop. Another approach avoids the out-of-plane vibration. For example, disc-shaped microstructure can operate in a rotational in-plane mode with resonance frequencies in the upper kilohertz range. The FBAR sensors described in Section 2.3.2 are another example of micromachined resonators that are attracting current interest and will probably go through further development. Another group of acoustic sensors, usually called ultrasonic sensors, shares some features with acoustic microsensors but there are also some remarkable differences. According to the device classification given in Section 2.1 they belong to the class 2. Similar to acoustic microsensors of the class 3, the acoustic wave is usually generated and detected with a piezoelectric device. By contrast, the acoustic wave in this case travels along several wavelengths through the bulk of material of interest. Level and flow meters are two famous examples of a variety of applications of ultrasonic sensors. Ultrasonic sensors have also proven their capabilities as chemical sensors. Micromachined Ultrasonic Transducers (MUT)s are the MEMS version of ultrasonic sensors [48-50]. They can be driven capacitively or piezoelectrically at radio frequencies. 1D and 2D arrays are available. MUTs are very promising for microfluidic applications.

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2.4 Operating Modes Piezoelectric acoustic-wave sensors invariably have an electrical port where a driving AC signal is applied that generates vibrations via the converse piezoelectric effect (induced strain proportional to applied voltage). Such vibrations propagate through the sensor interacting with the measurand quantity, and are transduced back to the electrical domain via the direct piezoelectric effect (induced charge proportional to applied stress). Depending on the way the electrical output signal is exploited, two categories of sensors can be distinguished. In one-port sensors the electrical output can be thought as generated across the same port, i.e. the same couple of electrodes, where the input is applied. In two-port sensors the electrical output is physically available at a second port, distinct from the input one, realized by a dedicated pair of electrodes. For both one-port and two-port sensors, the effect of the measurand quantity produced on the wave propagation can be measured in two different methods. In the first method, called the open-loop, or passive, or nonresonant method, an excitation signal coming from an external generator is applied to the sensor input and the corresponding response signal at the output is detected. Usually, the measurement is performed by a network analyzer which provides the excitation signal as a fixed-amplitude sine wave swept over a frequency range, detects the output, and directly visualizes the output/input ratio as a complex function of frequency, i.e., taking both amplitude and phase into account. The open-loop operation mode has the advantage of providing the maximum of information on the electrical behavior of the sensor and further on, via the acoustic behavior of the sensor, on the measurand/sensor interaction. The limitations are that extracting such information is not always straightforward, since it implies a certain knowledge of sensor operation and modeling. Moreover, network/impedance analyzers are typically costly instruments. An alternative to swept-frequency analysis is the use of transient analysis, in which a sinusoidal excitation at the resonant frequency is applied at the sensor input and suddenly removed, and the resulting output oscillatory damped response is examined. This method is mostly used with quartz crystal TSM sensors [51] (see Chap. 5).

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In the second method, called the closed-loop, or active, or resonant method, the sensor is configured as the feedback element of an electronic amplifier. In practice, the connection schemes are different for one-port and two-port sensors, but the principle in both cases is actually the same. By a proper choice of the amplifier it is possible to establish positive feedback around the loop and make the sensor/amplifier combination work as an oscillator, which continuously sustains and tracks oscillations in the sensor at one of its resonant frequencies (see Chap. 5). In one-port devices, like quartz crystal TSM sensors, the sensor behaves like a mechanical resonator. Conversely, SAW, FPW and APM and LW configured as twoport devices behave as acoustic delay lines. The closed-loop configuration has the advantage that it provides a continuous reading of the resonant frequency, allowing to follow the evolution of the experiment in real time without the need for repeated measurements of the sensor open-loop response. For comparatively low-frequency sensors, oscillator circuits can be relatively simple and inexpensive, while for higher-frequency sensors the design becomes less straightforward. A fundamental point to keep in mind with oscillators, that can also become their main limitation in high-accuracy applications, is that, in general, the sensor resonant frequency and the output frequency of the oscillator circuit are not exactly equal under every load conditions. This is due to the combination of the sensor and amplifier phase responses that determine an oscillation condition in the loop at a frequency which, in some cases, can be appreciably different from the sensor resonance (see Chap. 5). In particular, great care must be taken with oscillators when the sensor is heavily loaded either acoustically, due to a thick viscoelastic coating, or dielectrically, due to immersion in liquid, or both. In such cases, the oscillation frequency of the oscillator can become significantly different from the resonant frequency of the sensor, causing errors in the interpretation of the results. As a limiting case, oscillations can even stop in the circuit, though the sensor resonant frequency of course still exists, with the negative consequence of restricting the operating range. Special oscillator designs developed for heavy-load conditions should be adopted in these cases (see Chap. 5). A further limitation of oscillators is that they usually provide the measurement of a single parameter of the sensor response, namely its resonant frequency. There are oscillators that incorporate circuitry for the simultaneous measurement of the vibration amplitude in addition to its frequency,

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therefore providing information also on the amount of damping undergone by the sensor. Concepts on electronics are further discussed in Chap. 5.

2.5 Sensitivity The parameter of acoustic-wave sensors that is primarily employed for measurement is the fundamental resonant frequency f. From theory, in the case of quartz crystal TSM sensors, the series resonant frequency fs where the real part of admittance has a maximum must be measured to be in accordance with the theoretically predicted values. Additional parameters ranging from damping and phase shift, to the complete spectrum provide an increasing degree of further information. Limiting to the resonant frequency f, it can be generally expressed as:

f =

v 1 = 2l 2l

c

ρ

=

1 2π

K M

(2.1)

where l is the frequency determining dimension (e.g. the crystal thickness in a QCM), v is the wave velocity, c is the effective elastic stiffness (e.g. the shear stiffness constant in a QCM), ρ is the mass density, K and M are the lumped equivalent spring and mass associated with the particular vibration mode. Note that M is definitely different from the rest mass of the sensor. The fractional frequency variation can then be derived as a function of the variations of the individual parameters caused by an external quantity as follows: df dc dρ dl dK dM = − − = − f 2c 2 ρ l 2K 2M

(2.2)

The sensitivity towards a measurand x can be defined as the ratio df/dx. Despite its simplicity, Eq. (2.2) has a certain general validity in indicating the effect of a measurand on the resonant frequency and in finding the associated sensitivity. In particular, those measurands that increase the effective stiffness c, or equivalently the spring constant K, cause f to rise. Examples are tensile stress or bending. On the contrary, those measurands that increase either the effective density ρ, or the length l, or equivalently the equivalent mass M, cause f to decrease. A typical example is mass loading. The Sauerbrey equation for the

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mass sensitivity of a QCM can be easily derived from Eq. (2.2) by assuming that the load only changes the thickness l and leaves the average density and stiffness unaltered (see Chap. 1). In general, the higher the sensor unperturbed frequency f, the greater the frequency shift at parity of measurand value. For instance, SAW sensors in the 100 MHz range have a higher mass sensitivity than TSM sensors in the 10 MHz range. However, considering sensitivity as a benchmark to compare different sensors can be misleading. In fact, a higher value of the nominal sensitivity as apparently granted by a higher resonant frequency does not necessarily imply a higher value of the usable sensitivity in a practical device. For instance, a QCM can be operated with a sensitive coating much thicker than that on a SAW sensor, which results in a higher amount of gas absorbed in the coating. Therefore, it is more appropriate to use the reduced, or fractional, sensitivity S=(df/dx)/f to normalize for the unperturbed frequency. The typical fractional mass sensitivities, where the mass is intended for unit surface area, for different sensor types are compared in Table 2.1 [5254]. It should be noted that the sensitivity is only one factor to the ultimate goal of achieving a high resolution, i.e., a discrimination capability of small incremental values of the measurand [55]. High resolution implies good frequency stability. Table 2.1. Comparison of the characteristics of different acoustic-wave sensors Sensor type TSM quartz Thin-film BAW SAW SH-APM STW LW FPW a

FROa

Smb

5-30 900-1000 30-500 20-200 100-200 100-200 5-20

12-70 400-700 100-500 20-40 100-200 150-500 200-1000

Frequency Range of Operation [MHz] Surface mass sensitivity ([Hz MHz-1μg-1cm2]) c Frequency of Operation [MHz] d Frequency Noise [Hz] e Sensitivity-to-Noise ratio ([MHz-1μg-1cm2]) f Operation in Liquid (*) Data taken from: [57-59]. b

Examples (*) FO FNd S/Ne 10 0.2 110 c

160 100

2 4

100 5

110 5

2 1

125 450

OLf Yes No No Yes Yes Yes Yes

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Short-term frequency stability is mostly determined by the sensor, especially by the coating and the measurement environment, in combination with the oscillator electronics. Sensors with higher values of the quality factor Q for the resonance in question provide a better stability at parity of electronics. Therefore, a significant figure of merit for a sensor is actually the sensitivity-quality factor product SQ [56]. Long-term frequency stability is typically dominated by thermal drift and material aging or degradation, however, these effects must be related to the time scale of sensor signal changes. To counteract drift effects a differential configuration can be helpful, with one sensor exposed to the measurand and a second identical sensor screened from it. Both sensors are subjected to the influencing quantities, such as temperature. By taking the difference of the signals from the sensor/reference pair, the common-mode perturbing factors can be compensated to some extent.

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44. M. Baù, V. Ferrari, D. Marioli, E. Sardini, M. Serpelloni and A. Taroni (2007) “Contactless excitation and readout of passive sensing elements made by miniaturized mechanical resonators” in Proc. IEEE Sensors 2007, pp.36-39 45. L. Steinfeld, M. Ferrari, V. Ferrari, A. Arnau and H. Perrot (2005) “Contactless confined readout of quartz crystal resonator sensors” in Proc. IEEE Sensors 2005, pp.457-460 46. C. Ziegler (2004) “Cantilever-based biosensors” Anal. Bioanal. Chem., 379:946-959 47. N.V. Lavrik, M.J. Sepaniak and P.G. Datskos (2004) “Cantilever transducers as a platform for chemical and biological sensors” Rev. Sci. Instrum. 75:22292253 48. I. Ladabaum, B.T. Khuri-Yakub and D. Spoliansky (1996) “Micromachined ultrasonic transducers: 11.4 MHz transmission in air and more” Appl. Phys. Lett. 68:7-9 49. G. Perçin, A. Atalar, F. Levent Degertekin and B.T. Khuri-Yakub (1998) “Micromachined two-dimensional array piezoelectrically actuated transducers” Appl. Phys. Lett. 72:1397-1399 50. S. Doerner, S. Hirsch, R. Lucklum, B. Schmidt, P.R. Hauptmann, V. Ferrari and M. Ferrari (2005) “MEMS ultrasonic sensor array with thick film PZT transducers” in Proc. IEEE Ultrasonics Symp., pp.487-490 51. M. Rodahl, F. Höök, A. Krozer, P. Brzezinski and B. Kasemo (1995) “Quartz crystal microbalance setup for frequency and Q-factor rneasurements in gaseous and liquid environments” Rev. Sci. Instrum. 66:3924-3930 52. S.W. Wenzel and R.M. White (1989) “Analytic comparison of the sensitivities of bulk-wave, surface-wave, and flexural plate-wave ultrasonic gravimetric sensors” Appl. Phys. Lett. 54:1976-1978 53. Z. Wang, J.D.N. Cheeke and C.K. Chen (1990) “Unified approach to analyse mass sensitivities of acoustic gravimetric sensors” Electron. Lett. 26 (18):1511-1513 54. S.J. Martin, G.C. Frye, J.J. Spates and M.A. Butler (1996) “Gas sensing with acoustic devices” in Proc. IEEE Ultrasonics Symp., pp.423-434 55. Vig J.R. (1991) “On Acoustic Sensor Sensitivity” IEEE Trans. Ultrason., Ferroel., Freq. Contr. 38:311 56. E. Benes, M. Gröschl, F. Seifert and A. Pohl (1998) “Comparison between BAW ans SAW sensor principles” IEEE Trans. Ultrason., Ferroel., Freq. Contr. 45:1314-1330 57. J.W. Grate, S.J. Martin and R.M. White (1993) “Acoustic wave microsensors” Anal. Chem. 65:987A-996A 58. E. Gizeli (1997) “Design considerations for the acoustic waveguide biosensor” Smart Mater. Struct. 6:700-706 59. G.L. Harding and J. Du (1997) “Design and properties of quartz-based Love wave acoustic sensors incorporating silicon dioxide and PMMA guiding layers” Smart Mater. Struct. 6:716-720

3 Models for Resonant Sensors Ralf Lucklum1, David Soares2 and Kay Kanazawa3 1

Institute for Micro and Sensor Systems, Otto-von-Guericke-University Magdeburg 2 Institute de Fisica, Universidade de Campinas 3 Department of Chemical Engineering, Stanford University

3.1 Introduction The quartz crystal resonator (QCR), as its acronym implies, is a resonant physical device. Many of its behaviors and properties can be understood physically by examining its resonant behavior. The basic principle of operation for a generic acoustic-wave sensor is a traveling wave combined with a confinement structure to produce a standing wave whose frequency is determined jointly by the velocity of the traveling wave and the dimensions of the confinement structure. The most basic way of resonator modeling consequently requires applying the theory of wave propagation thereby considering material properties and geometric dimensions of the resonator. As another successful way, there is an electrical equivalent circuit often used to characterize the resonance. For these reasons, a closer inspection of the phenomenon of resonance is useful.

3.2 The Resonance Phenomenon On certain physical systems, the phenomenon of resonance can be used to multiply the effects of a force applied to the system. There are examples in mechanical, electrical and optical systems. When energy in a system is exchanged periodically between two forms, then resonance occurs. For example, in the case of a weight hanging on a rubber band, when the band is stretched, there is potential energy stored in the extended band. Subsequently, as the weight moves, the stored potential energy is exchanged into the kinetic energy of the weight. Following this, the kinetic energy is then transferred back into the potential energy in the band itself. If there are no A.A. Vives (ed.), Piezoelectric Transducers and Applications, doi: 10.1007/978-3-540-77508-9_3, © Springer-Verlag Berlin Heidelberg 2008

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losses in the system, then this back and forth energy transfer would continue. This is the resonance phenomenon. The energy exchange occurs periodically and is characterized by a resonance frequency. Similarly in an electrical circuit with an inductor and a capacitor, electrical energy can be stored as voltage across the capacitor. This stored energy then produces a current that flows in the inductor, exchanging the energy stored in the electrical field into a stored magnetic energy associated with the magnetic fields produced by the current in the inductor. The equivalent analogy between mechanical systems and electrical systems has been used also to describe the phenomenon of resonance of quartz crystal resonators and other acoustic-wave based sensors. In the case of the thickness shear mode quartz crystal resonator, the application of alternating voltage across the crystal results in the generation of a shear acoustic wave, causing a distortion of the crystal. When the frequency of the alternating voltage is far from the resonant frequency, the distortion, as measured by the shear displacement of the surface of the resonator, is very small. The importance of the displacement is understood when one considers the force imparted to the surface by a particle rigidly attached. The force is proportional to the acceleration of the particle. This acceleration is given by the product of the displacement and the square of the angular frequency. At resonance, the displacement can exceed the farfrom-resonance displacement by 105 or more. The force exerted by this particle on the resonator surface is multiplied by 105. This accounts for the extremely high sensitivity of the QCM to loaded mass (see Chap. 1). The multiplication factor is called the “quality factor” or Q of the resonator. It measures the ratio of the peak stored energy in the resonant cycle to the mean energy dissipated per cycle. A high Q would then imply a low loss resonance.

3.3 Concepts of Piezoelectric Resonator Modeling The quartz crystal resonator is the most common device used as acousticwave based sensor. The simple geometry of the device and the predominant thickness-shear mode of the propagating wave are propitious conditions for a comprehensive derivation of the acoustic-electrical behavior of quartz crystal devices, including the resonance phenomenon. Other acoustic microsensors introduced in Chap. 2 have more complicated wave propagation pattern; the concepts of piezoelectric resonator modeling are the same. Quartz crystal resonator sensors are therefore indicative of

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acoustic-wave based sensors to demonstrate the concepts of modeling those sensors. Quartz resonators are commonly used as frequency reference due to their very high Q-factor. A well known sensor application is the measurement of mass deposition (rates) in vacuum deposition technology (gravimetric principle). In most chemical sensor applications a chemically sensitive interface is realized with a coating; the coated quartz crystal hence can be considered as composite resonator. Analyte sorption in this sensitive layer results in a measurable change of properties of this layer, whereas the quartz crystal remains unchanged (see Chap. 11). The Q-factor of a quartz crystal with a foreign layer is still high, thus the oscillating frequency is very stable and can be measured with high resolution. Exposure of the resonator to a liquid results in energy loss caused by viscous damping. The decay length of shear waves at frequencies typically for quartz crystal resonators is so small that acoustic energy is dissipated only in a very thin liquid layer adjacent to the driving surface. However, the Q-factor is still remarkable high to ensure a significant resonance. A multilayer structure like that in Fig. 3.1 is the generalization of the single coating case. In the physical understanding acoustic waves travel back and forth in this structure and superimpose. Amplitude and phase of the traveling waves are defined by geometric and material properties of each layer. stress free

n+1 Zcn, ρn

hn

Zn

Zc2, ρ2

h2

Z2

Zc1, ρ1

h1

Z1

n

3

2

ZL

1

quartz crystal Zcq, ρq

0

hq

Zq

z

stress free

Fig. 3.1. General scheme of a quartz crystal resonator with a multilayer coating

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The electrical response of such a composite quartz crystal resonator sensor is however governed by the resulting wave in the quartz crystal. This wave is the superposition of the wave reflected off the boundary (1) to the coating and the wave transmitted into the quartz crystal through this interface (assuming the other main surface of the quartz crystal (0) is uncoated). It will be shown, that reflection and transmission in structures like in Fig. 3.1 is governed by a surface acoustic impedance acting at (1), noted as ZL. ZL links up the values of all above layers in one (complex) value. Wave propagation is defined by the (complex) wave propagation constant, k, which is a function of the angular frequency, ω, of the propagating wave, the density, ρ, of the layer and its shear modulus, G (Note that the piezoelectrically stiffened modulus of the quartz crystal is a tensor and usually denoted as c (see Appendix B)). As explained above for the boundary (1) between the quartz crystal and the first layer; the acoustic wave is partly reflected off and transmitted through at any boundary of the multilayer structure (Fig. 3.1). The effect of reflection and transmission at the boundary between two materials can be described by (complex) reflection and transmission coefficients, R and T, respectively: Ri ,i +1 = f R (Z i , Z i +1 ) =

Z i +1 − Z i Z i + Z i +1

(3.1)

Ti ,i +1 = f T (Z i , Z i +1 ) =

2Z i +1 Z i + Z i +1

(3.2)

In case of an interface between semi-infinite materials Zi is simply the characteristic impedance of the respective layers:

Z = f (ρ , G ) = ρ ⋅ G

(3.3)

The complex nature of k, Z, R, T arises from the complex nature of the shear modulus, G, of viscoelastic materials as analyzed in detail in Chap. 7. The characteristic impedance must not be mixed up with the acoustic load impedance. The acoustic load impedance is an effective acoustic impedance acting at an interface. The acoustic load impedance considers also contributions to the acoustic wave from reflection off or transmission through others than the respective interface. In other words, wave propagation in arrangements with finite dimensions modifies the characteristic impedance. The result is the so-called acoustic load impedance or shortly acoustic load.

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The real part of the reflection coefficient is the known amplitude relation of incident and reflected wave. The imaginary part of the reflection coefficient can be understood as the amplitude relation between the incident and a 90°-phase shifted reflected wave. The real part of the reflection coefficient is governed by the force proportional to particle displacement, whereas the imaginary part of the reflection coefficient is governed by the force proportional to speed of the vibrating particle. The physical model of a composite resonator considers a piezoelectric plate covered with one or a number of non-piezoelectric layers, each characterized with a set of acoustically relevant parameters. A nonpiezoelectric layer may be a thin film of a rigid, pure elastic material, a pure viscous liquid (film) or a film of a viscoelastic material (described in greater detail in Chap. 7). The set of characteristic parameters contains a geometric value, the film thickness, and material properties like film density and the (complex) shear modulus or like film density and complex viscosity. Characteristic impedance of the film material and wave propagation constant or wave velocity are other versions to describe film properties. The vibration behavior of a quartz crystal and the wave propagation in a multilayer arrangement can be derived in a one-dimensional model. This is a commonly accepted approximation. The high aspect ratio between the diameter of a quartz disc and the thickness of the crystal makes this assumption reasonable. As an additional requirement all layers must be uniform and homogeneous. Furthermore, continuity of particle displacement and shear stress at any interface is assumed. However, certain deviations from these assumptions, e.g., the shear amplitude distribution across the surface of a quartz crystal, a non-uniform film or specific interfacial phenomena are therefore not considered in this treatment. Those effects may significantly contribute to the vibration behavior of the quartz crystal and must be considered in more involved resonator models. The analytical approach to describe acoustic wave propagation (Appendix 3.A.2) uses two waves with unknown amplitudes traveling in opposite direction in each layer. The linear piezoelectric equations together with Newton’s equations of motion and Maxwell’s equations must be applied for the piezoelectric plate (Eqs. (3.A.1)-(3.A.2)). The appropriate boundary conditions (Eqs. (3.A.10a-f)) must be exploited to calculate the unknown parameters. The equivalent circuit approach (Appendix 3.A.3) describes the acoustic wave propagation in analogy to electrical waves. The matrix concept uses a three-port element and a transducer impedance matrix to represent the piezoelectric plate (Eqs. (3.A.14)-(3.A.15)). Non-piezoelectric layers are represented by a two-port element and an impedance matrix for each layer

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(Eqs. (3.A.22)-(3.A.23)). Each matrix is the result of fundamental physical equations and appropriate boundary conditions. The transmission line model and the Mason model are two versions of this approach (see Chap. 4). In contrast to the analytical approach, multilayer arrangements can be treated much easier. The major results of the physical model of quartz crystal resonator sensors can be summarized as follows: The electrical impedance or admittance is a function of the electrical capacitance of the quartz crystal formed by the electrodes and the quartz as dielectric material and the so-called motional impedance. The motional impedance contains the electrical equivalent of the acoustic load impedance, ZL, acting at the surface of the quartz plate (Fig. 3.1). The quartz crystal resonator sensor response is therefore sensitive to any change in the acoustic load impedance. The acoustic load impedance (change) can be generated from a pure mass (change) of a single rigid film (Eq. (3.10)), a semi-infinite Newtonian liquid (Eq. (3.12)), a single viscoelastic film or a multilayer arrangement. A thin rigid, purely elastic film and a semi-infinite purely viscous liquid are the two special cases, which result not only in a special form of the acoustic load but also in a distinct dependence of the resonant frequency on surface mass (density thickness product) or density viscosity product, respectively (Appendix 3.A.4). Near resonance the physical model can be developed into a special notation, where the physical parameters can be summarized in lumped equivalent electrical values: motional inductance, motional capacitance and motional resistance. This finally gives rise to the modified Butterworth-Van Dyke equivalent circuit model (see Chap. 1 and Appendix 3.A.4). This model allows the analysis of the electrical behavior of a quartz crystal resonator from electrical measurements without the need of determining the physical properties of the resonator. Some relations are analyzed in the following section. Another theoretical approach different from the acoustic-wave propagation concept is the energy transfer model. In this model the quartz crystal generates and stores acoustic energy. Acoustic energy trapped in a confined structure explains also the resonance phenomenon described in Sect. 3.2 corresponding to a harmonic oscillator of mass m = mq/2. Alternatively one can also consider an analogue electrical model consisting of a capacitance, C, an inductance, L and a resistance, R in a series circuit (see Fig. 3.2b). If the resonator is coated with another material, e.g. the chemically sensitive film, a small part of this acoustic energy is transferred into this material. This energy is stored in a purely elastic film and partly stored and partly dissipated in a viscoelastic film. Considering the high Q-factor of the quartz crystal, the electrical power applied to the crystal is equal to

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the developed mechanical power at ω = ω0. By taking this into account, the mechanical impedance is analogue to the electrical one. The material medium affects the crystal through its mechanical load impedance, ZL, a result similar to that derived with the wave-propagation concept. CS

R

L

a

CS

CP

R

L

b

Fig. 3.2. a The complete Butterworth-Van Dyke circuit is shown on the left, and b the motional branch isolated is shown on the right

3.4 The Equivalent Circuit of a Quartz Crystal Resonator The Butterworth-Van Dyke circuit (BVD) consists of two parallel branches as shown in Fig. 3.2. The right hand branch consisting of only the capacitance Cp represents the fixed dielectric capacitance of the resonator. All of the motional information is contained in the left hand branch. The right hand branch does influence the phase of the current relative to the voltage driving the circuit; therefore, to obtain an accurate representation of the motional behavior as a function of frequency, the parallel capacitance must be compensated (Chap. 5). Here the major interest is in the relation between the elements L, CS and R and the resonance characteristics, such as the resonant frequency, fs, and the quality factor, Q, so the right hand branch will be neglected and we will study only the behavior of the left hand branch. The admittance, Y, of the network shown in Fig. 3.2b is defined as the current to voltage ratio and is a function of the applied frequency, f. In terms of the angular frequency ω, defined as 2πf, Y can be expressed as:

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Y (ω ) =

1 ⎛ 1 R + j⎜⎜ ωL − ωC s ⎝

⎞ ⎟⎟ ⎠

(3.4)

and its magnitude by

Y (ω ) =

1 ⎛ ⎜ R 2 + ⎛⎜ ωL − 1 ⎜ ⎜ ωC s ⎝ ⎝

⎞ ⎟⎟ ⎠

2 1/ 2

⎞ ⎟ ⎟ ⎠

(3.5)

It is seen from Eq. (3.5) that this magnitude is a maximum at the frequency ωs where ωL = 1/(ωCs). In the most commonly encountered form, this is written:

ω s2 LC = 1

(3.6)

At this frequency, Eq. (3.4) reveals that the value of the magnitude of Y at resonance has the value Y (ω 0 ) =

1 R

(3.7)

and has only a real component, with the imaginary components cancelled. This leads to the conclusion that the phase difference between the voltage and current at this frequency is zero. This is illustrated in Fig. 3.3 where the magnitude of the admittance and its phase as a function of frequency are shown. These calculations were done taking a circuit with a resonant frequency of 5,000,000 Hz, a resistance of 100 Ω and an inductance of 0.04 H. The phase of the circuit can be seen to pass through the value of zero at the resonant frequency. Next we analyze the quality factor. It can be shown that this is related to the half power spectrum of the resonance. At two frequencies, one above (ω1/2+) and one below (ω1/2-) the resonant frequency, the power dissipated in the resonance will drop by ½. In terms of the admittance, this occurs when the magnitude of the admittance is decreased by 1 / 2 . This is also termed the “3 db points”, since the response is down from the maximum by three decibels. Some investigators use dissipation (D) to describe the losses in the resonance and it is very simply related to the quality factor by D = 1/Q.

3 Models for Resonant Sensors 100 Phase (degrees)

0.01 |Y| (Siemens)

71

0.005

0 -1000

0

0

-100 -1000

1000

Freq. from resonance (Hz)

0

1000

Freq. from resonance (Hz)

Fig. 3.3. Magnitude of the admittance (left) and its phase behavior (right)

An interesting expression for Q is given by the relation: Q=

1 L ωs L = R Cs R

(3.8)

The expression L / C s has the units of resistance, and its ratio to the resistance of the circuit yields the quality factor. Typical values for a 5 MHz AT-cut quartz crystal resonator would have L = 0.04 H, Cs = 25 fF. Thus L / C s = 1.3x106 Ohms! For a resistance of 10 Ω (typical for a resonator in air), the Q would be 1.3x105, a large multiplicative factor indeed! If inductance L changes only very little over a set of changing loading conditions (usually less than 1%) the Q is inversely proportional to the resistance R. From Fig. 3.2b, we see that there are three variables L, Cs and R required for specifying the motional impedance. The interpretation in terms of the resonant frequency, ωs and the quality factor Q has been discussed. It can be instructive to write the starting equations in terms of ωs, Q and R. While we shall not go through that exercise here, we illustrate its utility by writing the half power frequencies in the following manner:

ω

+ 1/ 2

2 ⎛ ⎞ ⎛ 1 ⎞ 1 ⎟ ⎜ ⎟⎟ + = ω s ⎜ 1 + ⎜⎜ 2Q ⎟⎟ ⎜ ⎝ 2Q ⎠ ⎝ ⎠

⎛ ⎜

⎛ 1 ⎞

2

⎞ 1 ⎟

⎟⎟ − ω1−/ 2 = ω s ⎜ 1 + ⎜⎜ 2Q ⎟⎟ ⎜ ⎝ 2Q ⎠ ⎝

⎠

(3.9a)

(3.9b)

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This exact relationship shows that the two half power frequencies are spaced equidistantly from a central frequency which is only negligibly different from the resonant frequency. For example, for a Q of 1000, the central frequency is different from the resonant frequency by only a few parts in ten million. The central basis for the extreme sensitivity of the QCM is based on its resonant behavior. It is very useful to discuss aspects of the storage and dissipation of energy in the QCM (and its overlays). The BVD-model is one convenient way, if electrical properties of the sensor are of major interest. The physical models should be applied, if the relations of electrical values to material properties of the coating(s) are needed.

3.5 Six Important Conclusions 3.5.1 The Sauerbrey Equation

As shown already in Chap. 1 quartz crystal resonators are very sensitive to mass changes at its surface. The resonator modeling based on acoustic wave propagation recovers Sauerbrey’s fundamental equation for a small phase shift of the acoustic wave while propagating through the foreign film. A small phase shift requires exactly Sauerbrey’s limitations: a thin film of a rigid material. Under those circumstances, a foreign mass, ms, uniformly distributed at the surface of a quartz crystal (equivalent to a uniform film of thickness hf, ρs = ρfhf, ρs is the surface density of the film ρs = ms/A, where A is the effective surface) generates a shift in the resonant frequency Δf s = −2 f 02

ρ f hf ρ q cq

(3.10)

which can be easily rewritten into Eq. (1.66) in Chap. 1 or in Eq. (3.A.36) in the Appendix. Furthermore, replacing f0 by v, the acoustic wave velocity, and λ, the acoustic wave length v = λ f , and considering the thickness of the quartz crystal, hq, of being λ/2 at mechanical resonance, f0, Eq. (3.10) can be rewritten as

ρ f hf Δf s =− f0 ρ q hq

(3.11)

3 Models for Resonant Sensors

73

3.5.2 Kanazawa’s Equation

A second special case is a quartz crystal in contact with a purely viscous liquid (so-called Newtonian liquid) at one surface. Due to the extremely small penetration depth of a shear wave in viscous materials, a liquid film can be considered as semi-infinite. Under those circumstances resonator modeling based on acoustic wave propagation recovers also Kanazawa’s fundamental equation: Δf s = − f 03 2

ρ liqη liq πρ q c q

(3.12)

where ρ1 and η1 are the liquid density and viscosity, respectively. This equation is equivalent to Eq. (3.A.40) in the Appendix. Equations (3.10) or (3.11) and (3.12) are very important because they show the sensor capability of quartz crystal resonators. Both equations are often applied to calculate absorbed mass in chemical sensor applications or determining density/viscosity of liquids. The modeling presented in the Appendix draws a more complete picture of how acoustic-wave based devices can be applied as sensors, when Sauerbrey’s and Kanazawa’s equations can be applied and the extended capabilities of those devices in more involved systems. 3.5.3 Resonant Frequencies

The quartz crystal as shown as equivalent circuit in Fig. 3.2a has several resonant frequencies. Oscillators can work at or near one of these resonant frequencies (see Chap. 5). They all depend on the acoustic load in a certain way, however, only the resonance of the motional arm as shown in Fig. 3.2b reflects the change of the acoustic load as predicted from models developed in the Appendix. All other resonance frequencies have a distinct dependence on the motional resistance. Only in the case of a very small acoustic energy dissipation in the sensing film, i.e. a very small motional resistance, R, (e.g. valid for a rigid film), any oscillator should respond with one and the same frequency shift to a certain mass change. Under any other conditions, the motional resonant frequency, fs, must be selected. Several electronic concepts are available. They are described in Chap. 5.

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3.5.4 Motional Resistance and Q Factor

The motional resistance and the Q-factor carry information about acoustic energy dissipation in the coating and the surrounding medium. This information is different from the frequency shift, which is related to acoustic energy storage. Therefore the measurement of the equivalent resistance or the Q-factor in addition to the frequency shift is optional for all measurements in a gaseous environment and strongly recommended for all applications of acoustic-wave devices in a liquid environment or when viscoelastic materials are used as sensitive film. Determination of R from the admittance plot is usually sufficient; however, calculation of Q from the 3 dB points is less sensitive to experimental uncertainties. Frequency shift and resistance (or Q-factor) change together allow for a much more assured data interpretation. Under certain circumstances the energy dissipation term can be even more sensitive to property changes of the analyte than the frequency shift. 3.5.5 Gravimetric and Non-Gravimetric Regime

The quartz crystal resonator sensor is commonly known as mass balance. This understanding is the most simplified case of the less obvious sensitivity to the acoustic load acting at the crystal surface. An instructive notation introduced in [1] provides a bridge between the special and the general case. As derived in the Appendix the acoustic load of a single film is given by:

⎛ ρ ⎞⎟ Z L = j ρG tan ⎜ ω h = j ρG tan ϕ ⎜ G ⎟⎠ ⎝

(3.13)

This can be expanded into:

Z L = j ωρh ⋅

tan ϕ

ϕ

= j M ⋅V

(3.14)

where, ϕ = ωh(ρ/G)1/2 is the phase shift the acoustic wave undergoes while traveling through the film. M = ωρh has been called mass factor, and V = tan ϕ/ϕ has been called acoustic factor. This notation provides most obvious insights to the working mechanism of quartz crystal resonator sensors. For V = 1 , i.e., for small phase angles, where the tan-function can be approximated by its argument, the quartz crystal acts as mass balance. This regime has been called gravimetric. For V ≠ 1 the shear modulus gets into play. The sensor now also responds to (visco)elastic properties of the film;

3 Models for Resonant Sensors

75

therefore this regime has been called non-gravimetric. It includes the socalled viscoelastic contributions to the sensor signal. Table 3.1 summarizes different cases and their potential sensor application. Table 3.1. Cases of the non-gravimetric regime Mass factor

Acoustic Factor

M → M + dM

V =1

M → M + dM

V ≈ const > 1

M → M + dM

V → V ± dV

M ≈ const

V → V ± dV

Sensor Application Mass balance Thickness monitor Acoustically amplified gravimetric sensor Mass and material effect sensor Film properties sensor

3.5.6 Kinetic Analysis

Up to now we have restricted our investigations to frequency shift and change in (acoustic energy) dissipation measurement at equilibrium. The evolution of the signals with time has not been considered. Kinetic analysis, however, has been proven as an excellent tool with other experimental methods, which provides a set of independent (kinetic) data. Next some fundamentals of kinetics are very quickly summarized, for more information see respective textbooks, e.g. [2]. Chemical reactions typically follow one of the following kinetic orders. Zero order reactions, e.g. A → B , are characterized by a constant reaction rate, v. First order reactions, e.g. A → B + C , have a reaction rate depending linearly on cA, the concentration of the component A. For the second order reaction, e.g. A + A → B or A + B → C there is a quadratic dependence of v on cA or a linear dependence on both cA and cB. The respective equations are summarized in Table 3.2. Adsorption of molecules on the sensor surface as the most common first step of the acoustic sensor principle is a first order mechanism in many cases. Assuming a monolayer formation the surface coverage, Θ, follows: v=

dΘ = k obs ⋅ (1 − Θ ) dt

(3.15)

and hence

Θ(t ) = (1 − exp(− kobs t )) where kobs is the observed adsorption rate.

(3.16)

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Ralf Lucklum, David Soares and Kay Kanazawa

Table 3.2. Summary of basis kinetic equations (3.17 – 3.24) Order 0 1

2

Reaction rate dc A v=− =k dt dc v = − A = k ⋅ cA dt dc A v=− = k ⋅ c A2 dt v=−

dc A = k ⋅ c A ⋅ cB dt

Concentration as function of timea (3.17) c A (t ) = c 0 A − k t

(3.18)

(3.19) c A (t ) = c 0 A (1 − exp(− k t ))

(3.20)

1 1 =kt− c A (t ) c0 A

(3.22)

(3.21) (3.23)

⎛ c (t ) c 0 B 1 ln⎜ A c 0 A − c 0 B ⎜⎝ c B (t ) c 0 A

⎞ ⎟ = k t (3.24) ⎟ ⎠

a

k=A exp (-Ea /RT) is the rate constant, where Ea is the Arrhenius activation energy and RT is the thermal energy.

Keeping in mind that adsorption is a reversible process one has to consider both adsorption and desorption: k1 M b + Fs → Ms ← − k 1

(3.25)

where Mb, Fs, and Ms represent the molecule in the bulk, a free space on the surface and a molecule occupying a surface site, respectively. k1 and k-1 are the adsorption and desorption rate constants, respectively. Equations (3.15) and (3.16) become: v=

dΘ = k 1cb (1 − Θ) − k −1Θ dt

Θ(t ) =

Kc b (1 − exp(− k a t )) 1 + Kc b

(3.26)

(3.27)

with kobs = k 1cb + k −1 and K = k 1 k −1 = [M s ] ([M b ][Fs ]) . The first factor Kcb (1 + Kcb ) is the so-called Langmuir isotherm. By varying the concentration of the respective molecule in the bulk at constant temperature adsorption and desorption rate constants can be determined. Assuming first order processes the general structure of the equation describing the evolution of surface coverage is: Θ(t ) = C (1 − exp(− kt ))

(3.28)

3 Models for Resonant Sensors

77

A fit procedure therefore has to fit two observable parameters: the factor C describing the coverage at equilibrium and an effective kinetic constant, k. In the gravimetric regime the transduction scheme of acoustic sensors is simple:

Θ(t ) ∝ Δm(t ) ∝ −Δf (t )

(3.29)

Acoustic transduction therefore just changes the parameter C in Eq. (3.28). In the non-gravimetric regime the situation is more involved. We here demonstrate the challenge for a sensor coated with a thin viscoelastic film in a liquid. By applying Eqs. (3.A.47) to (3.28) one finds

(

− Δf (t ) ∝ 1 − e −k t

)

ΔR(t ) ∝ 1 − C R e − k t + (C R − 1)e −2k t

(3.30) (3.31)

Whereas the evolution of Δf(t) keeps the typical first order behavior, ΔR(t) includes now a second term with the kinetic constant 2k! The contribution of this last term depends on CR, which is a function of the acoustic phase shift and approaches 1 in the gravimetric regime. A fit of Eq. (3.30) delivers Δfmax and k, and a fit of Eq. (3.31) delivers ΔRmax and CR. In this way kinetic analysis opens the capability for an unambiguous determination of film properties, assuming they do not change during adsorption [3].

Appendix 3.A 3.A.1 Introduction

Acoustic waves can be employed to measure physical or chemical values, like force, film thickness or the concentration of a certain compound in a mixture. Several kinds of devices have been used for generation and detection of acoustic waves and to pick up the relevant information (Chap. 2). The underlying transduction mechanism from the input signal to the output signal contains common and specific features. The common aspect for all devices is their sensitivity to any change of the acoustic properties of themselves or at the device surface. The acoustic properties include intrinsic material parameters (density, elastic moduli) and geometric values (thickness, length of the acoustic path). The acoustic wave traveling in a coated device especially penetrates into the adjacent film, translates and

78

Ralf Lucklum, David Soares and Kay Kanazawa

deforms the film, thereby probing its mechanical properties, its thickness and the acoustic properties at the upper film surface. Most physical sensors are based on changes of the acoustic properties of the acoustic device whereas most chemical sensors relay on changes of the acoustic properties of a coating. In any case, the acoustic waves carry the information of interest. The specific aspects are related to the kind of acoustic wave used, the wave propagation in the device and sometimes the electro-mechanical transformation. In the following a model is described which is especially useful for acoustic-wave-based chemical sensors. Those sensors obtain their chemical sensitivity and selectivity from a chemically active coating on top of the acoustic device, which interacts with the surrounding environment. This interaction leads to a change in the acoustic-wave propagation, which in turn yields a change of the electrical response of the sensor. The general concept in modeling acoustic wave sensors is based on the solution of a set of wave equations with regard to suitable boundary conditions between the sensor and the adjacent media. In consequence, the principle behavior of the different types of acoustic-wave devices is similar. Bulk acoustic wave (BAW) devices are typically realized with AT-cut quartz crystals. They vibrate in an almost pure thickness shear mode; therefore they are also called thickness-shear-mode (TSM) devices. Acoustic wave generation and propagation is most concise, therefore a coated quartz crystal resonator is used here as example to demonstrate the physical background of acoustic-wave-based sensors. Surface acoustic waves (SAWs) are coupled compressional and shear waves. Similar to BAWs, the propagation of SAWs generates a periodic displacement field into the adjacent layer. In contrast to BAWs, the surface displacement field has two distinct means of inducing strain in the coating: from the SAW-specific inplane gradients arising due to the sinusoidal variation in displacement components along the direction of SAW propagation, and analog to BAWs, from surface normal gradients arising from a phase difference between the motion of the upper surface of the layer with respect to the “driven” lower surface. 3.A.2 The Coated Piezoelectric Quartz Crystal. Analytical Solution

The linear piezoelectric equations together with the resulting system of differential equations for the unknown mechanical displacement and electrical potential have been given in Chap. 1. They describe the behavior of a piezoelectric resonator in general. The full set of differential equations is

3 Models for Resonant Sensors

79

difficult to solve for the complete three-dimensional problem. The unknown displacements and the electrical potential as well as their derivatives with respect to time and location are coupled with each other in these equations. Assumptions can be applied for special geometries which enable an approximate two- or three-dimensional solution [4, 5]. These models intend to characterize the uncoated quartz crystal. They cannot solve the problem of a coated resonator. Here a one-dimensional solution for AT-cut quartz crystal resonators is presented [6]. Due to the high ratio between the lateral dimensions and the thickness for a typical quartz resonator vibrating in the thickness shear mode it is reasonable to treat the crystal as an infinite plate with a finite thickness [7]. The thickness of the resonator is orientated in the x2-direction. By using an infinite plate one assumes that physical properties do not change along the x1 and x3-directions. The derivatives along these directions vanish, and only the derivatives along x2 remain. For the quartz crystal, starting from the general piezoelectric equations, results (see Appendix 1.A):

τ 12 = c66 u1, 2 + η q u&1, 2 + e26φ, 2

(3.A.1a)

D2 = e26 u1, 2 − ε 22φ , 2

(3.A.1b)

whereτij is the component of the mechanical stress tensor, Di is the electrical displacement (vector), cij, eij, εij are the components of the material property tensors for mechanical stiffness, piezoelectric constant, and permittivity, respectively, ui is the mechanical displacement component, and φ is the electrical potential. A colon with an index represents the partial derivative of the expression with respect to the specified index. The time derivative is marked as usual with a dot above the variable. A viscous term in the stress-strain relation has been included with the phenomenological quartz viscosity, ηq. This term accounts for losses inside the quartz crystal. The equation of motion and the Maxwell equation for the electrical displacement become

τ 12, 2 = ρ q ü1

(3.A.2a)

D2 , 2 = 0

(3.A.2b)

With harmonic time dependence, i.e., the mechanical displacement and the electrical potential vary with exp(jωt), the following differential equations can be given:

80

Ralf Lucklum, David Soares and Kay Kanazawa

⎞ ⎛ e2 ⎜ c66 + 26 + jωη q ⎟u1, 22 + ω 2 ρ q u1 = 0 ⎟ ⎜ ε 22 ⎠ ⎝

(3.A.3a)

e26 u1, 22 − ε 22φ , 22 = 0

(3.A.3b)

The first equation is a wave equation for the unknown mechanical displacement u1. The second equation couples the mechanical displacement with the electrical potential. Some abbreviations will be used in the following: eq ≡ e26 ; ε q ≡ ε 22

cq ≡ c66 +

2 e26

ε 22

(3.A.4)

+ jωη q = cq 0 + jωη q

where cq is the effective complex shear modulus. With these definitions the differential equations for the quartz crystal are written as c q u1, 22 + ω 2 ρ q u1 = 0

(3.A.5a)

eq u1, 22 − ε qφ, 22 = 0

(3.A.5b)

The solution of this wave equation can be written with two components, one wave traveling in positive and one in negative x2-direction inside the quartz crystal with unknown amplitudes B1 and B2:

(

u1 = B1e

j k q x2

+ B2 e

− j k q x2

)e

jωt

(3.A.6)

where kq is the complex wave propagation vector kq = ω (ρq/cq)1/2. For a lossy quartz crystal the imaginary part of kq represents the decay of the traveling waves. Starting from the solutions from Eq. (3.A.6), the electrical potential (Eq. (3.A.5)), the stress (Eq. (3.A.1a)), and the electrical displacement (Eq. (3.A.1b)) are calculated (with unknown parameters B3 and B4) to

(

)

⎛ eq ⎞ jk x − jk x B e q 2 + B2 e q 2 + B3 x 2 + B4 ⎟ e jωt ⎜ εq 1 ⎟ ⎝ ⎠

φ =⎜

(

(

τ 12 = j k q cq B1e

j k q x2

− B2 e

− j k q x2

)+ e B ) e q

3

jωt

(3.A.7a) (3.A.7b)

3 Models for Resonant Sensors

D2 = −ε q B3 e jωt

81

(3.A.7c)

The relations for the coating (acoustic load) are similar to those of the quartz crystal, the index i is used for the layer properties. The coating is usually non-piezoelectric; therefore there is no piezoelectric component. The differential equation for the coating becomes

ci u1, 22 + ω 2 ρ i u1 = 0

(3.A.8)

with ci and ρi being the complex shear modulus and the density of the coating, respectively. The solution again has the form of a wave propagation and is written with amplitudes C1 and C2 as

(

)

u1 = C1e j ki x2 + C 2 e − j ki x2 e jωt

(3.A.9)

where ki is the complex wave propagation vector inside the coating. The following boundary conditions have to be applied for the system quartz crystal (thickness hq) - coating (thickness hi): 1. continuous displacement u1(x2 = hq) at interface crystal-coating (3.A.10a) 2. continuous shear stress τ12(x2 = hq) (at interface crystal – coating (3.A.10b) 3. vanishing shear stress τ12(x2 = 0) = 0 at free crystal surface (3.A.10c) 4. vanishing shear stress τ12 (x2 = hq + hi) = 0 at free coating surface (3.A.10d) 5. driving electrical potential φ(x2 = hq) = -φ0 exp(jωt) at upper electrode (3.A.10e) 6. driving electrical potential φ(x2 = 0) = φ0 exp(jωt) at lower electrode (3.A.10f) These boundary conditions provide six equations for the unknown parameters, B1, B2, B3, B4, C1, and C2. The solution of this system of equations can be obtained with standard matrix methods which can be found elsewhere. The final solution of the one-dimensional problem can be calculated with these six parameters. Finally some abbreviations are introduced: 2

K =

eq2

ε q cq

; vq = cq / ρ q

(3.A.11a,b)

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Ralf Lucklum, David Soares and Kay Kanazawa

(3.A.11c,d)

k q = ω ρ q / cq = ω / vq ; γ q = j k q

α = k q hq = ωhq ρ q / c q

; Z cq = ρ q cq = ρ q vq

ϕ = k i hi ; k i = ω ρ i / ci

; Z ci = ρ i ci

(3.A.11e,f) (3.A.11g,h,i)

α and ϕ are the acoustic phase shift inside the quartz crystal and the coating, respectively. Zcq and Zci are the characteristic acoustic impedance of the quartz crystal and the coating; K2 is the electromechanical coupling coefficient of quartz. With the static quartz capacitance C0 = εq (A/hq) the following relation can be found for the electrical impedance of the coated quartz resonator after some algebraic transformations: Z α ⎛ 2 tan − j L ⎜ 2 Z cq 2 1 ⎜ K Z= 1− ⎜ Z jωC 0 α 1 − j L cot α ⎜ ⎜ Z cq ⎝

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(3.A.12)

The electrical impedance of a coated quartz crystal resonator can be calculated from quartz crystal parameters, the frequency, and the acoustic load impedance, ZL. Eq. (3.A.12) simplifies for the uncoated quartz crystal: Z=

1 jωC 0

⎛ K2 α⎞ ⎜1 − 2 tan ⎟⎟ ⎜ α 2⎠ ⎝

(3.A.13)

3.A.3 The Transmission Line Model

The transmission line model can be used to describe both the (piezoelectric) transformation between electrical and mechanical vibration and the propagation of acoustic waves in the system acoustic device-coatingmedium in analogy to electrical waves [8]. This model assumes a uniform piezoelectric device and isotropic, homogeneous, uniform layers and a sensor configuration, in which lateral dimensions have no effect on the propagation of waves. The model does not have any restrictions on the number of layers, their thickness and their mechanical properties. The characteristic acoustic parameters and the geometric values of nonpiezoelectric layers are summarized in the effective acoustic impedance, which transforms the acoustic properties at one port to the other one. It

3 Models for Resonant Sensors

83

reflects, in which manner the layer is translated and deformed by the acoustic wave. The complete transmission line model relates the overall system characteristics to the electrical impedance (or admittance) at the electrical port, starting from the front acoustic port with known acoustic properties (usually stress-free corresponding to a short-circuited acoustic port, or a semi-infinite liquid). The transmission line model allows a formal separation of the acoustic wave propagation inside the acoustic device, including the transformation of mechanical displacement into the electrical signal and vice versa, and outside the acoustic device. In this context the acoustic load (impedance), ZL, which is a complex number, represents the overall acoustic load at the interface between the acoustic device and the coating. It should not be mixed up with the characteristic impedance, Zci, which is a material constant. The acoustic load summarizes all acoustically relevant information. It does not play any role, if this load is generated by a simple mass, a single viscoelastic coating, a multilayer arrangement, or a semi-infinite material. Consequently, the acoustic load, ZL, carries all information, which is related to changes in the chemically sensitive coating, no matter if it is pure mass accumulation, mass accumulation accompanied by material property changes, or only material property changes induced by chemical (e.g. cross-linking) or physical (e.g. phase transition) effects. A change in the acoustic load impedance results in a change of the electrical impedance of the BAW device or a change of sound velocity and attenuation of the propagating SAW. Finally these changes are responsible for the frequency shift and attenuation change of the acoustic device. The piezoelectric quartz crystal

The piezoelectric transducer can be regarded as a three-port black box, where the two main surfaces form two acoustic ports whereas the electrodes form the electrical port. The independent variables are treated as “currents” and the dependent variables are treated as “voltages” with the following analogy: mechanical tension τ ⇔ U electrical voltage particle velocity v ⇔ I electrical current acoustic impedance Z = τ/v ⇔ Z = U/I electrical impedance The general behavior of a three-port element is described by a transducer impedance matrix, ZT:

84

Ralf Lucklum, David Soares and Kay Kanazawa

⎛ τ1 ⎞ ⎛ v1 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ τ 2 ⎟ = Z T ⎜ v2 ⎟ ⎜U ⎟ ⎜I ⎟ ⎝ 3⎠ ⎝ 3⎠

(3.A.14)

The transmission line model implements the fundamental physical equations as well as the boundary conditions already in the transfer matrix giving ⎛ ⎜ Z cq coth j k q hq ⎜ ⎜ Z cq ZT = ⎜ ⎜ sinh j k q hq ⎜ eq ε q ⎜ ⎜ jω ⎝

(

(

Z cq

)

(

sinh j k q hq

)

(

Z cq coth j k q hq

)

)

eq ε q jω

eq ε q ⎞ ⎟ j ωA ⎟ eq ε q ⎟ ⎟ j ωA ⎟ 1 ⎟⎟ jωC 0 ⎟⎠

(3.A.15)

One of the representations of the transmission line model is the equivalent circuit from Krimholtz, Leedom, and Matthaei, which is referred to as KLM-model [9]. It is presented in Fig. 3.A.1 (big square). The elements are defined by:

hq ⎞ 1 1 ⎛ 2eq sin( k q ) ⎟ = ⎜ 2 ⎜ 2 ⎟⎠ A ⎝ ε q ω Z cq N X=

eq

2

(3.A.16a)

2

2

Aε q ω 2 Z cq

C0 = ε q

sin( kq hq )

(3.A.16b)

A hq

(3.A.16c)

The equations for the electrical impedance at port AB can be easily derived. The impedance at position CD from the right acoustic port is:

Z r = Z cq

( tanh (γ

) 2)

Z EF + Z cq tanh γ q hq 2 Z cq + Z EF

q

hq

(3.A.17a)

3 Models for Resonant Sensors

G

hq 2

hq 2

C C0

h1

E

h2

hn

coat.(Zc2, 2)

coat.(Zcn, n)

I

85

jX 1:N

A H

D B

F

quartz (Z q ,

q)

J coat.(Z c1, 1)

Fig. 3.A.1 Representation of the transmission line model (KLM model)

Similarly, the impedance at position CD from the left acoustic port is: Z l = Z cq

( tanh (γ

) 2)

Z GH + Z cq tanh γ q hq 2 Z cq + Z GH

q

hq

(3.A.17b)

The overall impedance at position CD is the parallel arrangement of Zr and Zl. The transformer transforms the acoustic into the electrical signal at port AB: Z AB =

1 1 + j X + 2 Z CD j ωC 0 N

(3.A.18a)

1 K2 sin α jωC 0 α

(3.A.18b)

jX =

(3.A.18c)

1 1 4K 2 1 α = sin 2 2 2 ωC 0 α Z cq N

In the special case of a single-side coated sensor, i.e. a stress free surface at port GH and an acoustic load acting at port EF (ZEF = ZL), Eqs. (3.A.16) –(3.A.18) yields after some calculations: Z α ⎛ 2 tan − j L ⎜ 2 Z 2 1 ⎜ K cq Z= 1− ⎜ Z α jωC 0 1 − j L cot α ⎜ ⎜ Z cq ⎝

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(3.A.19)

which is equivalent to Eq. (3.A.12). It is possible to separate the impedance Z into a parallel circuit consisting of a static capacitance C0 and a motional impedance Zm:

86

Ralf Lucklum, David Soares and Kay Kanazawa

⎛ ⎜ 1 − j Z L cot α Z cq 1 ⎜ ⎜ Zm = jωC 0 ⎜ K 2 ⎛ ⎜ 2 tan α − j Z L ⎜⎜ 2 ⎜ Z cq ⎝ α ⎝

⎞ ⎟ ⎟ − 1⎟ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎠

(3.A.20)

This motional impedance can be split into two parts: ⎛ α ⎞ ⎟ 1 ⎜ K2 1 α ZL ⎜ Zm = − 1⎟ + 2 jωC 0 ⎜ 2 tan α C Z cq ω ⎟⎟ 0 4K ⎜ 2 ⎝ ⎠

1 1−

Z j L Z cq

= Z m0 + Z mL

(3.A.21)

2 tan α 2

The first part represents the unloaded quartz (ZL = 0) and should not change during the measurement, the other one is related to the load. This important feature allows for several simplifying approximations presented in Sect.3.A.4. The Acoustic Load

Non-piezoelectric layers are represented by a two-port circuit. In case of a multilayer loading (Fig. 3.A.1), the surface acoustic load ZL = ZEF acting at the port EF of the transmission line representing the piezoelectric quartz crystal, would be the resulting impedance from all layers placed on the quartz surface. Fig. 3.A.1 shows the respective transmission line. The impedance concept in propagation problems uses a chain matrix technique. The elements of the propagation matrix, Pi, and the transfer matrix, Ti, for each layer of thickness hi, complex wave propagation constant γi, and characteristic acoustic impedance, Zci, are calculated as follows: ⎛ e −γ i hi Pi = ⎜⎜ ⎝ 0

0 ⎞ ⎟ γ i hi ⎟ e ⎠

⎛ 1 ⎜ Z Ti = ⎜ ci ⎜ 1 ⎜Z ⎝ ci

⎞ 1⎟ ⎟ ⎟ − 1⎟ ⎠

(3.A.22)

The layers are usually acoustically impedance mismatched with respect to the adjacent layers. Considering the layer i as a quadrupole with an input mechanical tension, ui+1, and an input particle velocity, ii+1, and an output mechanical tension, σi, and particle velocity, vi, the transformation can be calculated with the transformation matrix, Mi:

3 Models for Resonant Sensors

⎛ σ i ( z )⎞ ⎛ σ ( z + hi )⎞ ⎛ σ ( z + hi )⎞ ⎜⎜ ⎟⎟ = Ti−1Pi−1Ti ⎜⎜ i +1 ⎟⎟ = M i ⎜⎜ i +1 ⎟⎟ ⎝ vi ( z ) ⎠ ⎝ vi +1 (z + hi )⎠ ⎝ vi +1 (z + hi )⎠

(

)

(

Z ci γ i hi −γ h −γ h ⎛ 1 γ i hi e +e i i e −e i i ⎜ 2 M = ⎜ 12 1 γ i hi γ i hi −γ i hi −γ h i ⎜ e +e i i ⎜ 2Z e − e 2 ⎝ ci Z sinh (γ i hi )⎞ ⎛ cosh (γ i hi ) ci ⎜ ⎟ =⎜ 1 cosh (γ i hi ) ⎟⎟ ⎜ Z sinh (γ i hi ) ⎝ ci ⎠

(

) (

)⎞⎟⎟

) ⎟⎟ ⎠

87

(3.A.23a)

= (3.A.23b)

Note that both electrodes may also be described in terms of two transmission lines, acting at port CD and EF, respectively. Since the upper electrode works toward a shear stressed layer in the case of a coated quartz crystal, one must expect a noticeable own contribution of this electrode. Nevertheless, the electrodes are acoustically thin layers; their contribution is small and does not change significantly during experiment. Thus, for simplicity, their effect is usually taken into account in an effective quartz crystal thickness. With Z = σ /v the impedance transformation performed with layer i can simply be calculated: Z i = Z ci

Z i +1 + Z ci tanh (γ i hi ) Z ci + Z i +1 tanh (γ i hi )

(3.A.24)

Equation (3.A.24) can be rearranged with (G replaces ci in equations (3.A.11h,i))

γ =j

ω G ρ

(3.A.25a)

Zc = ρ G

(3.A.25b)

⎛ ρ i ⎞⎟ Z i +1 + j ρ i Gi tan ⎜ ω hi ⎜ ⎟ G i ⎝ ⎠ Z i = Z ci ⎛ ⎞ ρ i Z ci + j Z i +1 tan⎜ ω hi ⎟ ⎜ ⎟ G i ⎝ ⎠

(3.A.26)

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Ralf Lucklum, David Soares and Kay Kanazawa

The equations in this section are exact within the one-dimensional assumptions. They should be used in all cases, where highest accuracy of the calculation is required and in all cases, where no information is available about error propagation [10]. 3.A.4 Special Cases

Although the equations in the previous section are easily to compute on a computer, their comprehensibility is limited. Therefore several approximations are applied to transform these equations into a more convenient form. Some of them are summarized in the following. The Modified Butterworth-Van Dyke Circuit

Near the resonance of the unloaded quartz sensor, the approximation tan(α/2) ≈ 4α/(π2-α2) leads to a simple expression for the unperturbed part of Zm: Z m0 = Rq + jωLq +

1 1 1 + = Rq + jωLq + jωC q jω ( −C 0 ) jωC q′

(3.A.27)

For small loads (ZL/Zcq << 2 tan(α/2)) the part of Eq. (3.A.21) representing the acoustic load simplifies to: Z mL =

1 α ZL ωC 0 4 K 2 Z cq

(3.A.28)

Here the additional impedance is direct proportional to the load ZL, thus giving for the motional impedance of the quartz: Z m = R q + j ωL q +

1 1 α ZL + jωC q ' ωC 0 4 K 2 Z cq

Z m = Rq + jωLq +

1 + Zc jωC q '

(3.A.29a)

(3.A.29b)

This expression can be transformed into a modified motional arm of the commonly used Butterworth-Van Dyke equivalent circuit (BVD) as shown in Fig. 3.A.2 lower part. The equivalent circuit elements are defined in a straightforward way; however, it needs the (practically impossible) negative capacitance [11].

3 Models for Resonant Sensors

89

Motional arm

R liq L liq ZC

Liquid

Coating

Lq Co

C'q

Quartz

Rq

Fig. 3.A.2 Extended Butterworth-van-Dyke equivalent circuit

A hq

C0 = ε q

C q = C0

8K 0

2

=

π2

1−

Rq =

2

π 2 hq cq 0

8K 0

π

π2 2

8C0 K 0 ω 0

η q cq 0 Cq

8 Aeq

(3.A.30b)

Cq

C q' =

Lq =

(3.A.30a)

=

2

2

(3.A.30c)

2

=

ρ q hq 3 8 Aeq

2

η qπ 2 2

8C0 K 0 cq 0

(3.A.30d)

(3.A.30e)

The Acoustic Load Concept

Unchanged quartz parameters allow for a direct relation between the (complex) acoustic load impedance, ZL, and two directly measurable

90

Ralf Lucklum, David Soares and Kay Kanazawa

values, the frequency shift, Δfs, and the change in the motional resistance, ΔRm in the BVD equivalent circuit, the latter equivalent to a change in the quality factor Q = ωL/R or any other measure for acoustic energy dissipation [12]. The imaginary part of ZL is related to the frequency shift taken at the inphase admittance magnitude, Re(Yel). It is equivalent to the resonance of the motional arm of the BVD equivalent circuit, not to the zero phase crossing of the quartz resonator. Δf s Im(Z L ) =− f0 π Z cq

(3.A.31)

The real part of the acoustic load is related to acoustic energy dissipation: Re(Z L ) ΔR = π Z cq 2ω 0 Lq

(3.A.32)

The motional resistance change, ΔRm, is taken as the difference in the reciprocal admittance magnitude at fs between the coated and uncoated device. The motional inductance of the quartz crystal, Lq, as well as the resonance frequency, f0 or ω0, are device-specific and can be calibrated with a separate measurement of the bare quartz. These approximations can be applied for almost all coatings typically used for chemical sensing [13]. Single Film

For a single film with a stress free upper surface (Z2 = 0), usually a quartz crystal resonator with a single coating working in air, Eq. (3.A.26) simplifies to:

⎛ ρ ⎞⎟ Z L = j ρG tan⎜⎜ ω h G ⎟⎠ ⎝

(3.A.33)

where the shear modulus, G, is a complex value in case of a coating with viscoelastic properties: G = G' + jG'' (Chap. 7). This equation is visualized as a 3-D plot in Fig. 3.A.3 for a 10 MHz quartz crystal, split into imaginary and real part. The film thickness varies between 0 and 1 µm. Typical values of a polymer in the glassy state (G' ≈ 1 GPa, G' >> G''), in the rubbery state (G' ≈ 1 MPa, G' > G'') and in the transition range (G' ≈ G'') have been taken for the two components of the shear modulus.

3 Models for Resonant Sensors

36

91

resonant frequency: 10 MHz

100000 80000 60000

0

-Im(Z L / Pa s m -1)

Δf / kHz

18

-18

40000

"Sauerbrey"dependence

20000 0 -20000 -40000 -60000

rubbery

-80000 -36

transition range

-100000 0

glassy 0.25 0.5 0.75

1

Thickness / µm

a 200000 5000

180000

2500

1000

Re(ZL / Pa s m-1)

Δ R / Ohms

160000 140000 120000 100000 80000 60000 40000

rubbery

20000 250 0

transition range

0 0

0.25 0.5 0.75

glassy 1

Thickness / µm

b Fig. 3.A.3. a Real part, and b imaginary part of Eq. (3.A.33) for a 10 MHz quartz crystal.

This treatment allows the combination of the two variables on one axis. A film density of 1000 kg m-3 has been taken as a reasonable value. At first glance, one can distinguish between two major regions in both diagrams: an (inclined) rather plane range and a range with significant extreme. The two regions are related to acoustically thin films and acoustically thick

92

Ralf Lucklum, David Soares and Kay Kanazawa

films [14]. On acoustically-thin-film conditions the entire coating moves almost synchronously with the acoustic device surface, i.e., the phase shift, ϕf, between the lower and upper film surface is very small. At acousticallythick-film conditions the upper surface significantly lags behind the device-film interface; the tan-function dominates ZL. It leads to a rapid decrease followed by a sharp increase of Im(ZL) (deep groove and high crest in Fig. 3.A.3a), and a dramatic increase of Re(ZL) (maximum in Fig. 3.A.3b). The Sauerbrey Equation

The Sauerbrey equation assumes a thin rigid film. In terms of Eq. (3.A.33) it reads a small h and a large and real G. It means, that the phase shift, ϕ, is very small and the tan-function can be approximated by its argument, or, the acoustic factor, V, (Eq. 3.14) approaches 1. The acoustic load of a thin rigid layer (density ρf, thickness hf) is hence Z L = j ωρ f h f = j M

(3.A.34)

Whereas Eq. (3.A.33) cannot be transformed into simple lumped elements of the BVD-model, the acoustic load of a rigid coating, Eq. (3.A.34), would simplify the complex impedance, Zc, (Fig. 3.A.2), to an inductance, Lrig in series with the inductance Lq:

Lrig =

1 1 π ρ f hf 4C 0 K 2 Z cq ω s

(3.A.35)

Applying Eq. (3.A.31) results in an equation similar to Sauerbrey’s fundamental equation [15]: Δf rig f0

= −ω

ρ f hf π Z cq

(3.A.36)

Furthermore, Eq. (3.A.32) also provides information about the change of the equivalent resistance: ΔRrig = 0

(3.A.37)

Equation (3.A.37) makes clear that an ideal rigid coating does not increase acoustic losses.

3 Models for Resonant Sensors

93

The Kanazawa Equation

A semi-infinite pure viscous (Newtonian) liquid, index liq, generates a surface acoustic impedance:

ω ρ liqη liq

j ρ liqωη liq = (1 + j)

Z liq =

2

(3.A.38)

where G in Eq. (3.A.31) is replaced by G' = 0, G'' = jωηliq, whereas j tan ω ( ρ liq / jωηliq ) hliq = 1 for hliq >> 2η liq / ωρ liq , the penetration depth of the shear acoustic wave in a viscous liquid. ρliq and ηliq are the liquid density and viscosity. Similar to the Sauerbrey case, Zliq also can be represented as serial elements in the motional arm of the modified BVDmodel (Fig. 3.A.2, upper part):

(

)

Lliq =

1 π 1 2 4C 0 K Z cq ω s

Rliq =

1 π 4C 0 K 2 Z cq

ρ liqη liq 2ω s ρ liqη liq 2ω s

(3.A.39a) (3.A.39b)

Applying Eq. (3.A.31) results in a relation similar to Kanazawa’s well known equation [16]: Δf liq f0

=−

1 π Z cq

ω ρ liqη liq 2

(3.A.40)

Again, Eq. (3.A.32) provides additional information about the resistance change: ΔRliq

2ω0 Lq

=

1 πZ cq

ωρ liqη liq 2

(3.A.41)

Due to the factor (1 + j) in Eq. (3.A.38) the contributions of a viscous liquid to frequency shift and resistance increase are equivalent. Martin’s Equation

If a quartz crystal with a single, thin and rigid layer works in a Newtonian liquid, the second term of the denominator of Eq. (3.A.24) is small compared to Z c1 . The acoustic load of this arrangement simplifies to:

94

Ralf Lucklum, David Soares and Kay Kanazawa

Z rig liq = jω ρ h + (1 + j)

ωρ liqη liq

(3.A.42)

2

The load from the rigid layer and the liquid load are, approximately, additive. Applying again Eq. (3.A.31) results in an equation similar to that published by Martin [17]: Δf rig liq f0

=−

2f Z cq

⎛ ⎜ ρh + ⎜ ⎝

ρ liq η liq 4π f

⎞ ⎟ ⎟ ⎠

(3.A.43)

The contribution to the resistance increase arises only from the viscous damping of the liquid, Eq. (3.A.41). Small phase shift approximation

In case of a small, but not very small phase shift, ϕ, tan ϕ can be approximated, giving a simple expression for the acoustic factor, V, and its change, Vε:

V=

tan ϕ

ϕ

≈

Vε = (V + Vε ) − V = 1 +

ϕ+

ϕ3

2 2 3 =1+ ϕ =1+ M 3 ϕ 3 Z cf2

(M + M ε )2 3Z cf2

⎛ M2 − ⎜1 + ⎜ 3Z cf2 ⎝

⎞ ⎟ ≈ 2(V − 1) M ε ⎟ M ⎠

(3.A.44a)

(3.A.44b)

where M is the mass factor calculated from density and thickness of the coating, Mε is the mass factor from the mass accumulated in the film during experiment and Zcf is the characteristic impedance of the film. With Eq. (3.A.44) the frequency shift due to a single film, Δf, and due to mass accumulation in the coating, Δf ε, can be approximated. It is more instructive to replace the shear modulus by the shear compliance (see Chap. 7):

⎛ 1 J' 2 ⎞ M ⎟⎟ Δf ∝ M ⎜⎜1 + ⎝ 3ρ ⎠

(3.A.45a)

⎛ ⎞ J' Δf ε ∝ M ε ⎜⎜1 + M 2 ⎟⎟ ρ ⎝ ⎠

(3.A.45b)

3 Models for Resonant Sensors

95

Note, that the viscoelastic contribution to the frequency shift is three times higher for mass accumulation compared to the viscoelastic contributions from a viscoelastic coating although the material properties are assumed to be unchanged [18]. Whereas viscoelastic properties of a single coating in a gaseous environment contribute to an “extra mass”, a soft thin film when adjoined to a liquid gives rise to a “missing mass” [19]. The effective acoustic factor becomes negative if the acoustic factor of the film alone approaches 1. In such a case Veff becomes (L replaces ω ρ liqη liq / 2 ): V eff ≈ 1 − j

2 L2 Z cf2

+ (1 − j)

ML Z cf2

(3.A.46)

The contribution of the liquid is unchanged, whereas the contribution of the film is reduced. Suppressing the contribution of the liquid, frequency shift and resistance increase due to a thin soft film can be approximated as follows [20]:

⎛ ⎞ L2 Δf soft in liq ∝ M ⎜⎜1 − 2 J ' ' ⎟⎟ ρ ⎝ ⎠ ⎛ L2 ⎞ ⎛L ⎞ ΔR soft in liq ∝ M ⎜⎜ 2 J ' ⎟⎟ + M 2 ⎜⎜ (J '+ J ' ')⎟⎟ ⎝ρ ⎠ ⎝ ρ ⎠

(3.A.47a)

(3.A.47b)

In contrast to Eq. (3.A.45a) the acoustic factor in Eq. (3.A.47a) does not depend on the mass factor, i.e., on the film thickness.

References 1. C. Behling, R. Lucklum, and P. Hauptmann (1998) “The non-gravimetric quartz crystal resonator response and its application for polymer shear moduli determination” Meas. Sci. Technol. 9:1886-1893 2. P. G. Ashmore (1998) “Principles of reaction kinetics”, Vol. 3, Iss. 4, Springer, New York 3. R. Lucklum, S. Doerner, T. Schneider, B. Schlatt-Masuth, Th. Jacobs, and P. Hauptmann (2006) “Real time kinetic analysis with quartz crystal resonator sensors” Proceedings of IEEE Int. Freq. Contr. Symp., pp. 528-534 4. R.D. Mindlin (1963) “High frequency vibrations of plated crystal plates” in Progress in Applied Mechanics, Macmillan, New York, pp. 73-84 5. H. Tiersten (1969) “Linear piezoelectric plate vibrations” Plenum Press, New York

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6. C, Behling (1999) “The non-gravimetric response of thickness shear mode resonators for sensor applications” Shaker-Verlag, Aachen 7. C.E. Reed, K.K. Kanazawa, and J.H. Kaufmann (1990) “Physical description of a viscoelastically loaded AT-cut quartz resonator” J. Appl. Phys. 68:1993-2001 8. G.S. Kino (1987) “Acoustic waves: devices, imaging and analog signal processing” Englewood Cliffs, N.J.: Prentice-Hall 9. R. Krimholtz, D.A. Leedom, and G.L. Matthaei (1970) “New equivalent circuits for elementary piezoelectric transducers” Electron. Lett. 6:398-99 10. R. Lucklum, C. Behling, R.W. Cernosek, and S.J. Martin (1997) “Determination of complex shear modulus with thickness shear mode resonators” J. Phys. D: Appl. Phys, 30:346-356 11. C. Behling, R. Lucklum, and P. Hauptmann (1997) “Possibilities and limitations in quantitative determination of polymer shear parameters by TSM resonators” Sensors and Actuators A 61: 260-266 12. D. Johannsmann, K. Mathauer, G. Wegner, and W. Knoll (1992) “Viscoelastic properties of thin films probed with a quartz-crystal resonator” Phys. Rev. B 46:7808-7815 13. R. Lucklum, C. Behling, and P. Hauptmann (1999) “Role of mass accumulation and viscoelastic film properties for the response of acoustic-wave-based chemical sensors” Anal. Chem. 71:2488-2496 14. S.J. Martin, G.C. Frye, S.D. Senturia (1994) “Dynamics and response of polymer-coated surface acoustic wave devices: effect of viscoelastic properties and film resonance” Anal. Chem. 66:2201-2219 15. G. Sauerbrey (1959) “Verwendung von schwingquarzen zur wägung dünner schichten und zur mikrowägung” Zeitschrift Physik 155:206-212 16. K.K. Kanazawa and J.G. Gordon (1985) “Frequency of a quartz microbalance in contact with liquid” Anal. Chem. 57:1770-1771 17. S.J. Martin, V. E. Granstaff, G.C. Frye (1991) “Characterization of a quartz crystal microbalance with simultaneous mass and liquid loading” Anal. Chem. 63:2272-2281 18. R. Lucklum, P. Hauptmann (2000) “The quartz crystal microbalance: mass sensitivity, viscoelasticity and acoustic amplification” Sensors Actuators B 70(1-3):30-36 19. M.V. Voinova, M. Jonson, B. Kasemo (2000) “Missing mass effect in biosensor’s QCM application” Biosensors Bioelectronics 17:835-841 20. R. Lucklum, P. Hauptmann (2003) “Transduction mechanism of acousticwave based chemical and biochemical sensors” Meas. Sci. Techn. 14: 18541864

4 Models for Piezoelectric Transducers Used in Broadband Ultrasonic Applications José Luis San Emeterio and Antonio Ramos Departamento de Señales, Sistemas y Tecnologías Ultrasónicas. Instituto de Acústica (CSIC).

4.1 Introduction Piezoelectric transducers are key elements of many broadband ultrasonic systems, either pulse-echo or through-transmission, used for imaging and detection purposes. In ultrasonic broadband applications such as medical imaging, or non-destructive testing, piezoelectric transducers should generate/receive ultrasonic signals with good efficiency over a large frequency range. This implies the use of piezoelectric transducers with high sensitivity, broad bandwidth and short-duration impulse responses. High sensitivity provides large signal amplitudes which determine a good dynamic range for the system and the short duration of the received ultrasonic signal provides a good axial resolution. The most important and common type of piezoelectric transducer elements used in ultrasonic broadband applications is a thin piezoelectric plate, with lateral dimensions much greater than the thickness, driven in a simple thickness extensional mode of vibration [1-2]. They usually operate in the frequency range 0.5-15 MHz. Different types of piezoelectric materials are used for the active transducer element. Ferroelectric ceramics, like lead zirconate titanate (PZT), lead metaniobate, etc., have a high piezoelectric coupling coefficient. Piezoelectric polymers like polyvinylidene difluoride (PVDF) and copolymers have useful low-acoustic impedances. Piezoelectric composites are mixtures of piezoceramics with nonpiezoelectric polymers. When designing a broadband piezoelectric transducer or when finding optimal transducer system configurations, it is useful to be able to predict the global response by means of theoretical calculations, bearing in mind that there is a large number of materials and configuration parameters involved in the global system [3, 4]. The aim of this chapter is to summarize A.A. Vives (ed.), Piezoelectric Transducers and Applications, doi: 10.1007/978-3-540-77508-9_4, © Springer-Verlag Berlin Heidelberg 2008

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José Luis San Emeterio and Antonio Ramos

the basic modeling approaches describing the electrical and ultrasonic characteristics of broadband multilayer transducers. In the active piezoelectric plates, the length and width to thickness ratios are sufficiently large so that one-dimensional models are good approximations to predict the properties of the transducer [5-10]. Modeling the transducer as a two-port network permits the use of the transfer matrix formalism of the circuit theory. In this chapter, a general methodology for the treatment of all the components of a transducer system, including acoustic matching layers and electric matching components, as a set of cascade networks [11-13], is also described. A computer program for design and optimization of transducer systems can be easily developed [11].

4.2 The Electromechanical Impedance Matrix Figure 4.1 shows a simple diagram of a broadband piezoelectric transducer. A piezoelectric layer of thickness t, with very thin electrodes of area A at its surfaces, is embedded between an attenuating backing material and the irradiated medium (load). Usually, a high attenuating material (“backing”) is bonded to the back face of the transducer element in order to enlarge the emission/reception (two-way) frequency band and therefore to shorten the impulse response (at the expense of a loss on sensitivity and signal amplitude). One or more acoustic matching layers are bonded in the front face in order to optimize the transmission of energy to the explored medium [5-11].

5 3 4 2 1 Fig. 4.1. Constructive scheme of a thin piezoelectric plate transducer: 1 matching layer; 2 piezoelectric active element; 3 casing; 4 backing; and 5 coaxial cable

Piezoelectric transducers convert electrical energy into mechanical energy and vice versa. The direct piezoelectric effect consist of the change in polarization in a material induced by an applied mechanical stress, and is used in the ultrasonic reception stage. The converse piezoelectric effect consists of the dimensional change (mechanical strain) in a material induced

4 Models for Piezoelectric Transducers in Ultrasonic Applications

99

by an applied electric field, and is used in the ultrasonic emission stage. The linear piezoelectric constitutive equations define the interrelationships among the electric displacement D, the electric field E, the mechanical stress T, and the elastic strain S. The complexity of the equations involved depends on both the symmetry of the piezoelectric material and the particular geometry (mechanical and electrical boundary conditions) of the transducer element [1, 2, 5-7]. Most of piezoelectric materials used in the fabrication of broadband ultrasonic transducers, (i.e. ferroelectric ceramics, piezoelectric polymers, and piezoelectric composites) become piezoelectric by a process of electrical poling along the thickness direction (Z ≡ 3 axis). The poled plates present a symmetry which can be associated with the crystallographic class 6 mm, reducing the number of fundamental material constants to 5 elastic, 3 piezoelectric and 2 dielectric coefficients. This type of symmetry will be considered in the following. t z

3

FB

FL

uB

uL Z

3

I V Fig. 4.2. Cross section of a thickness extensional piezoelectric transducer element

Figure 4.2 shows a diagram of the cross section of a thin piezoelectric plate, poled along the thickness t, driven by electrodes on the mayor surfaces. The electrical field is applied along the same Z ≡ 3 axis, (thickness excitation (TE)) [1, 2]. This transducer element can be regarded as a three port device, with one electrical port (electrical terminals carrying a voltage V and a current I) and two mechanical ports corresponding to the back (FB, uB) and front (FL, uL) faces developing forces F and particle velocities u at the faces, being:

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José Luis San Emeterio and Antonio Ramos

V: voltage across the electrodes of the transducer element. I: current through the electrodes of the transducer element. FL: force on the front face (radiating surface). uL: particle velocity at the front face. FB: force on the back face. uB: particle velocity at the back face. The lateral dimensions of the piezoelectric plate are much larger than thickness t, so that S3 is the only non-zero strain component and D3 the only non-zero electric displacement component. According to these electrical and mechanical boundary conditions, the piezoelectric plate will vibrate in a thickness extensional mode of vibration and the pertinent piezoelectric equations are (see Appendix B for understanding the meaning of subscripts): D T3 = c33 S 3 − h33 D3

E 3 = − h33 S 3 +

D3 S ε 33

(4.1) (4.2)

D where the sub index 3≡Z corresponds to the poling axis, c33 is the elastic S stiffened constant, h33 = e33 / ε 33 is the piezoelectric constant (e33 is the S piezoelectric stress constant), and ε 33 is the clamped (high frequency) dielectric constant. The piezoelectric material is assumed to be a perfect insulator and according to the charge equation of electrostatics the displacement component D3 is constant inside the piezoelectric plate (see Appendix A):

∂D3 =0 ∂z

(4.3)

The stress equation of motion, as a function of the particle displacement ξ3, being S 3 = ∂ξ 3 / ∂ z , is

∂T3 ρ ∂ 2ξ 3 = ∂z ∂t 2

(4.4)

Starting from Eqs. (4.1) and (4.4), the following plane-wave equation is easily obtained:

4 Models for Piezoelectric Transducers in Ultrasonic Applications

ρ ∂ 2ξ 3 ∂t 2

=

D c33 ∂ 2ξ 3

101

(4.5)

∂z 2

The solution of this plane wave equation for harmonic excitations [5, 6] is:

(

)

ξ 3 = C1 e − jβ z + C 2 e j β z e jω t

(4.6)

where the constants C1 and C2 have to be determined from the mechanical boundary conditions ( u = ∂ξ 3 / ∂t , and F=T3 A), at the transducer surfaces (z = 0, backing; and z = t, load) [6]: uB =

∂ξ 3 ∂ t z =0

FB = − A T3 z =0

∂ξ 3 ∂t z =t

(4.7a, b)

FL = − AT3 z =t

(4.7c, d)

uL = −

;

;

In addition, the expressions for the current I through the transducer and the voltage V across the transducer terminals have to be introduced:

I = j ω A D3

;

V3 =

∫

t 0

E 3 dz

(4.8a, b)

Starting from Eqs. (4.1) to (4.8), three equations, which specify the relations among the terminal variables, present at the ports of the transducer, are obtained. This set of equations can be written in matrix form:

⎛ FL ⎞ ⎛ Z 0 A / j tan β t ⎜ ⎟ ⎜ ⎜ FB ⎟ = ⎜ Z 0 A / j sin β t ⎜V ⎟ ⎜ h / jω 33 ⎝ ⎠ ⎝

Z 0 A / j sin β t Z 0 A / j tan β t h33 / jω

h33 / jω ⎞ ⎛ u L ⎞ ⎟ ⎜ ⎟ h33 / jω ⎟ ⎜ u B ⎟ 1 / jω C 0S ⎟⎠ ⎜⎝ I ⎟⎠

(4.9)

S where C 0S = Aε 33 / t is the clamped (zero strain, high frequency) capacitance of the piezoelectric plate, being A the area of the electrodes; D 1/ 2 Z 0 = ( ρ c33 ) is the characteristic stiffened acoustic impedance of the pieD zoelectric material, being ρ the density, and β = ω /( c33 / ρ )1 / 2 the propagation constant. The specific or characteristic impedance gives the ratio of the stress to the particle velocity for acoustic plane waves [7]. The characteristic impedances of the backing and the load are Z B = ( ρ B c B )1 / 2 = FB /( A u B ) and Z L = ( ρ L c L )1 / 2 = FL /( A u L ) , respectively, being cB and cL the corresponding elastic constants.

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José Luis San Emeterio and Antonio Ramos

4.3 Equivalent Circuits The electrical and acoustic characteristics of a TE piezoelectric layer can be evaluated from the previous set of Eqs. (4.9). Nevertheless, it is frequently more convenient to devise an electrical equivalent circuit using the fundamental electromechanical analogies. According to the classical analogy, the mechanical force can be considered as the analog of the electrical voltage and the particle velocity as the analog of the electrical current. The previous 3x3 electromechanical matrix in Eq. (4.9), relates the terminal variables (F, u, V, I) present at the mechanical and electrical ports. From a functional point of view, three components can be distinguished in this matrix: i) The first 2X2 sub-matrix corresponds to the equations of a mechanical transmission line of length t (thickness of the piezoelectric D plate), characteristic impedance Z0 and phase velocity ( c33 / ρ )1 / 2 , which could be thought as representing the ultrasonic propagation in the piezoelectric medium; ii) the terms including the piezoelectric constant h33 correspond to the electromechanical coupling; iii) the last matrix element is the electrical impedance of the plate capacitance. Using the electromechanical analogy, different three-port (six terminals) equivalent circuits can be obtained. Figures 4.3, 4.4 and 4.5 show Mason [14], Redwood [15] and KLM [16] equivalent circuits, which have been widely used for piezoelectric transducer simulation, design and optimization. jAZ0 tan(βt/2) jAZ0 tan(βt/2) uB uL AZ0 / j sin(βt/2)

FB

FL

1 : C0S h33 - C0S I

C0S

V Fig. 4.3. Mason’s equivalent circuit of a thickness extensional transducer

4 Models for Piezoelectric Transducers in Ultrasonic Applications

103

Figure 4.3 shows the pioneer Mason equivalent circuit for a thickness extensional transducer in which the electrical port is coupled, through an ideal transformer (transformation ratio N = C0S h33 = A e33 / t ), to a T-network which represents the acoustic propagation in the piezoelectric material (the first 2x2 sub-matrix). The mechanical ports of the transducer are located at the extremes of the T-network. It should be noted the presence of the negative capacitance - C 0S close to the electrical terminals. Both, the negative capacitance and the ideal transformer account for the electromechanical coupling. Redwood equivalent circuit shown in Fig. 4.4 is very similar to Mason circuit and can be derived from it by simply substituting the T-network with the equivalent transmission line (2x2 sub-matrix). In this circuit, the ideal transformer is connected to the external shield of the transmission line. This equivalent circuit has been used for SPICE implementations of the transducer model [17-19]. Figure 4.5 shows the KLM equivalent circuit proposed by Krimholtz, Leedom, and Matthaei in 1970 [16], and widely used since then. A mechanical transmission line represents the wave propagation in the piezoelectric material (2x2 sub-matrix). The mechanical ports are located at its extremes. The electrical port is connected through a transformer (with a frequency dependent transformer ratioΦ) to the center of the transmission line. The transformer is connected through a capacitance C 0S in series with a reactance X (also frequency dependent) to the transducer electrical terminals. uB

t ; AZ0

; β uL

FB

FL

1 : C0S h33 -C0S I

C0S

V Fig. 4.4. Redwood version of Mason’s equivalent circuit

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José Luis San Emeterio and Antonio Ramos

t/2; AZ0; β

FB

t/2; AZ0; β

uB

uL

Φ: 1

X=

jX C0

S

Φ=

I

h233 ω2AZ0 2h33 ω AZ0

FL

sin(βt)

sin(βt/2)

VV Fig. 4.5. KLM equivalent circuit of a thickness extensional piezoelectric transducer

Mason and Redwood circuits, with the explicit presence of the clamped capacitance in parallel with the transducer electrical terminals, provide a good representation of the electrical port of the transducer. On the other hand, the KLM network, with the explicit transmission line analogy, represents better the mechanical sections of the transducer. In fact this model has been used to design multiple acoustic matching layers, establishing alternative design criteria [3, 10]. Although these equivalent networks are not physically realizable with discrete components (negative capacitance − C 0S , transformer ratioΦ and reactance X with a complex dependence on frequency), and contain no information other than that contained in the 3x3 electromechanical matrix, they are particularly useful for broadband multilayer piezoelectric analysis. Different non-piezoelectric layers (i.e. matching layers) can also be represented by transmission lines and the powerful techniques of network analysis can be used. In addition, the circuit diagram helps to represent the otherwise complex mathematics [2]. All these circuits correspond exactly to the 3x3 electromechanical matrix in Eq. (4.9) and can be derived from them. Their complete equivalence can be established from the equality of the open and short circuit input electrical impedance [10]. A closed form expression for the electrical impedance at the transducer terminals Z in (ω ) , in the most general case, when the mechanical ports are terminated by arbitrary acoustic impedances

4 Models for Piezoelectric Transducers in Ultrasonic Applications

105

ZB and ZL, can be derived from any of these exact one-dimensional models: Zin (ω) =

1 j ω C0S

⎛ kt2 j (ZL + ZB )Z0 sinβ t − 2 Z02 (1 − cosβ t) ⎜1 + ⎜ β t (Z 2 + Z Z ) sinβ t − j (Z + Z )Z cosβ t 0 L B L B 0 ⎝

⎞ ⎟ ⎟ ⎠

(4.10)

S D 1/ 2 / c33 ) is the electromechanical coupling coefficient. where k t = h33 (ε 33

4.4 Broadband Piezoelectric Transducers as Two-Port Networks When the back mechanical port is closed with an absorbing material of characteristic impedance ZB, the transducer can be represented by a twoport linear and reciprocal network as shown in Fig. 4.6. Any ratio between the four remaining unknowns V, I, FL, and uL can be obtained. The transfer matrix formalism of the circuit theory (with the transfer matrix ABCD relating the parameters at the electrical port and front acoustic port), can be used for the analysis of the performance of either a transmitting or a receiver transducer:

⎛ Ag ⎛V1 = V ⎞ ⎜⎜ ⎟⎟ = ⎜⎜ ⎝ I1 = I ⎠ ⎝Cg

Bg ⎞ ⎟ D g ⎟⎠

⎛V2 = FL ⎞ ⎜⎜ ⎟⎟ ⎝ I 2 = −u L ⎠

(4.11)

The transfer matrix elements A, B, C, and D of any two port network can be obtained from the short and open circuit conditions (matrix coefficients and terminal variables are functions of frequency): I1 ≡ I

I 2 ≡-uL Ag

Bg

Cg

Dg

V1 ≡ V

V2 ≡ FL

Fig. 4.6. Two-port representation of a broadband transducer system when the back mechanical port is closed with a mechanical impedance (A ZB). The electrical and mechanical variables at the terminals are related by the ABCD matrix

106

Ag=

José Luis San Emeterio and Antonio Ramos

V1 V I I ; Bg = 1 ; Cg = 1 ; Dg = 1 V 2 I =0 I 2 V =0 V 2 I =0 I 2 V =0 2

2

2

(4.12a-d)

2

In particular, the transfer matrix coefficients of the piezoelectric transducer element, as represented in Fig. 4.6, can be obtained from circuit analysis, dividing an equivalent circuit into a set of simple cascade networks. Using the KLM circuit model [11], the resulting coefficients can be obtained as the product of the following elementary transfer matrices:

⎛ Ap ⎜ ⎜C p ⎝

⎛ Bp ⎞ ⎜1 ⎟=⎜ D p ⎟⎠ ⎜ ⎝0

2 ⎛ h33 sin β t 1 ⎜ − j⎜ 2 S ⎝ ω Z 0 A ωC0 1

1 ⎛ ⎜ Z j Z + 0 B tan ( β t / 2) ⋅⎜ ⎜ Z A ( Z + j Z tan ( β t / 2)) B 0 ⎝ 0

⎛ cos( β t / 2) ⎜ ⋅ ⎜ j sin ( β t / 2) ⎜ Z0 A ⎝

⎞ ⎞ ⎛φ 0 ⎞ ⎟⎟⎜ 1 ⎟⋅ ⎟⎟⎜ ⎟⎟ ⎠ ⎜0 ⎟⎝ φ ⎠ ⎠ 0⎞ ⎟ 1 ⎟⎟ ⋅ ⎠

(4.13)

jZ 0 A sin ( β t / 2) ⎞ ⎟ cos ( β t / 2) ⎟⎟ ⎠

where these elementary matrices correspond to the series impedance in front of the transformer, the transformer itself, the transmission line of length t/2 terminated by the backing impedance, and the transmission line of length t/2 terminated by the load. The transformer ratio of the KLM equivalent circuit is Φ = 2h33 sin( β t / 2) / ω Z 0 A . Generally, the piezoelectric active element has a characteristic impedance much greater than those of the usual loads (i.e. water, tissue, metal). In addition, the electrical impedance of a piezoelectric transducer usually presents a notable miss-match with driver-receiver electronics. Consequently, different acoustic and electrical impedance matching procedures have to be used, looking for a compromise between efficiency and bandwidth. The transfer matrix formalism can be used for the treatment of all components of a transducer system as a set of cascade networks, with the final overall matrix being the product of the elementary ones. Figure 4.7 presents a global scheme for an ultrasonic pulse-echo process, including an acoustic matching section in the front face and an electrical matching network at the electrical terminals of the transducer. Different configurations of impedance matching schemes can be evaluated

4 Models for Piezoelectric Transducers in Ultrasonic Applications

107

using the transmission matrix formula [12, 13]. As an example, the matrix elements for simple series ZS and parallel ZP impedances, and mechanical transmission lines, are shown in Table 4.1. The overall Ag, Bg, Cg and Dg matrix coefficients are obtained by the product of the individual matrices corresponding to the piezoelectric element, the matching layers and the electrical matching. I ZG VG

V

Electrical Matching Network

Piezoelectric Active Element

-uL Acoustic Matching Layers

FL

Electrical Matching Network

Vr

ZL

Backing ZB

FL ZL 2FL

FL

Backing ZB

V

Acoustic Matching Layers

Piezoelectric Active Element

Zr

Fig. 4.7. Global scheme for the evaluation of broadband transducers in pulse-echo applications, including electrical and acoustical matching sections Table 4.1. ABCD-matrix coefficients for simple circuit elements Series Impedance ZS

⎛1 Zs ⎞ ⎜⎜ ⎟⎟ ⎝0 1 ⎠

Parallel impedance ZP

⎛ 1 ⎜ ⎜1 / Z p ⎝

0⎞ ⎟ 1 ⎟⎠

Mechanical transmission line

⎛ cos β l ⎜⎜ ⎝ j sin β l / Z l

jZ l sin β l ⎞ ⎟ cos β l ⎟⎠

l is the transmission line length, Zl is the mechanical impedance and β is the propagation constant

4.5 Transfer Functions and Time Responses Three of the possible ratios between the terminal variables, the electrical impedance, and the emission and reception transfer functions, are particularly

108

José Luis San Emeterio and Antonio Ramos

useful in describing the performance of a transducer system, and can be derived from equation (4.11), using the overall matrix coefficients. The electrical input impedance at the transducer terminals, Z in (ω ) = V (ω ) / I (ω ) , accounting for electrical matching networks and acoustic matching layers, is: Z in (ω )

=

Ag Z L + B g C g Z L + Dg

(4.14)

The transducer performance can be analyzed separately for emission and reception operations by means of the emission and reception transfer functions. Conventional approaches to piezoelectric emission stage analysis, usually assume a waveform generator with resistive output impedance. If the characteristics of the source VG, ZG are known, the emission frequency response FL/VG, can be computed: FL VG

=

ZL ( Ag Z L + B g ) + Z G (C g Z L + D g )

(4.15)

Emission characteristics of piezoelectric transducers are frequently analyzed with reference to the transducer terminals. The emission transfer function in this case FL /V, is obtained making ZG = 0 in equation 4.15. The reception transfer function, Vr /FL can be computed from the following expression: Vr FL

=

2Z r ( A g Z L + B g ) + Z r (C g Z L + D g )

(4.16)

The emission and reception impulse responses of the transducer (response to a Dirac delta function) can be obtained by means of the inverse Fourier transform of the previous emission and reception frequency responses. The time response to an arbitrary voltage “spike” excitation at the transducer terminals can be obtained by convolution of the impulse response with the “spike” waveform. Figure 4.8 illustrates the time and frequency characteristics of the ultrasonic emission of a broadband piezoelectric transducer [20]. A typical PZT piezoelectric transducer has been considered, with the following characteristics: fundamental mechanical resonance f0 = 1.093 MHz; diameter φ = 20 mm; Z0 = 30 Mrayls; v = 3900 m/s; kt = 0.47; Qm = 138; C0S = 1.2 nF; backing ZB = 5.4 Mrayls. Water is considered as the irradiated medium with a characteristic impedance, ZL = 1.5 Mrayls. Figure 4.8b shows the emission transfer function at the transducer terminals and

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Fig. 4.8e the corresponding impulse response; Fig. 4.8d shows a typical time domain voltage “spike” applied to the transducer terminals and Fig. 4.8a represents its frequency contents obtained by means of its Fourier transform; Fig. 4.8c shows the product of the frequency spectra of Figs. 4.8a and b; and finally Fig. 4.8f is the time domain response of the transducer obtained by the inverse Fourier transform of Fig. 4.8c. -4

-4

x 10

2

1.4

(a) a 1.5

(b) b

1.2

x 10

1

(c) c 0.8

1 0.8

0.6

0.6

0.4

1 0.4

0.5

0.2

0.2 0

0

2

f

4 f (MHz) (MHz)

6

0

0

2

f

4 f(MHz) (MHz)

6

0

0

2

4

6

(MHz) f f(MHz)

6

100

(d) d

0

1.5

x 10

(e) e

1

80

0.5

40

-100

0

20

-200

-0.5

0

-1

-20

-1.5

-40

-300 -400

0

5 (us) tt (us)

10

-2 0

5 (us) t t (us)

10

(f) f

60

-60 0

5

10

(us) t t (us)

Fig.4.8. Time and frequency emission responses of a typical piezoelectric transducer (described in the text), driven by a voltage spike.

When the same piezoelectric transducer is used to transmit and receive, ultrasonic pulses, the two-way voltage transfer function Vr /VG is the product of the previous emission and reception transfer functions. In this simplified approach, a perfect reflection of the ultrasonic emitted pulse is assumed and diffraction effects are neglected. The emission/reception (E/R) impulse response of the transducer at the electrical terminals, can be obtained by means of the inverse Fourier transform of the overall frequency response Vr /V (obtained by making ZG = 0 in Eq. 4.15). Since the described system model is linear, the response to an arbitrary voltage excitation of the transducer can be obtained by convolution of the global E/R

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José Luis San Emeterio and Antonio Ramos

impulse response with the specific voltage driving waveform applied at the transducer terminals. Figure 4.9 shows the pulse-echo time responses and frequency bands computed for the same piezoelectric transducer, after a perfect reflection with propagation through water, when it is driven by a 2V peak-to-peak sinusoidal tone burst of 15 cycles [21]. The frequency bands and time waveforms of the driving pulse are presented in Figs. 4.9a and d; the transducer two-way transfer function in Fig.4.9b; the impulse response in Fig. 4.9e; and the global echographic responses in Figs. 4.9c and f. -6

8

-6

x 10

a (a)

6

0.5

4

b (b)

0.4

x 10

c (c)

3

0.3 4

2 0.2

2

1

0.1

0 0

2

0 0

4

2

0 0

4

f (MHz)

f (MHz)

2

4

f (MHz)

5

1

(d) d

0.5

x 10

(e) e

2

0.5

f (f)

1

0

0

0

-1

-0.5 -1 0

3

-2 20

40

t (us)

60

80

3

0

20

40

60

80

-0.5 0

20

40

60

80

t (us)

t (us) Fig. 4.9. Pulse-echo time and frequency responses of a typical piezoelectric transducer (described in the text), driven by a 2V peak-to-peak sinusoidal tone burst of 15 cycles

4.6 Acoustic Impedance Matching Piezoelectric ceramics usually have a specific acoustic impedance much greater than those of the usual loads present in diagnostic ultrasound and/or non-destructive testing (i.e. water, tissue, metal). As a consequence,

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an important amount of ultrasonic energy is reflected back at the piezoelectric / load interface. In order to optimize the broadband response of a piezoelectric transducer, different acoustic impedance matching procedures are used, looking for a compromise between efficiency and bandwidth. The basic procedure consists of incorporating one or more matching layers between the radiating face of the piezoelectric element and the acoustic load (see Figs. 4.1 and 4.7). These matching layers, usually with a thickness equal to a quarter wavelength at the fundamental mechanical resonance (λ0/4, being λ0 the wavelength in the layer at f0), act as mechanical transformers increasing the mechanical load at the piezoelectric interface. The characteristic impedance Zc of these λ0/4 matching layers, are usually determined by means of the classical approach of Collins [22] to impedance matching by means of λ/4 transmission line sections in microwave applications. Ultrasonic transducers use the same principles, bearing in mind that impedance mismatch is much greater in this case and that the necessary material characteristic impedances may be difficult to obtain. Following this approach, the “optimum” characteristic impedance Zc of one quarter wave matching section, placed between a piezoelectric ceramic of characteristic impedance Z0 and an irradiated medium of characteristic impedance ZL, is: Z C =( Z 0 Z L ) 1 / 2

(4.17)

If two quarter wave matching sections are used, the “optimum” characteristic impedances of the first layer ZC1 (close to the piezoelectric element) and of the second layer ZC2 (close to the irradiated medium) are determined by: Z C1 = Z 0 3 / 4 Z L

1/ 4

(4.18a)

Z C 2 = Z 0 1/ 4 Z L

3/ 4

(4.18b)

Alternative expressions for the characteristic impedances of the quarter wavelength matching layers were proposed by Desilets et al. [3], derived from the KLM equivalent circuit, adapting to the center of the piezoelectric element. Following this approach, the expression for the characteristic impedance in the case of a single matching layer is: Z C = Z 0 1/ 3 Z L

2/3

(4.19)

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If two quarter wave matching sections are used the following expressions are considered: Z C1 = Z 0 4 / 7 Z L

3/ 7

(4.20a)

Z C 2 = Z 0 1/ 7 Z L

6/7

(4.20b)

As in the case of the piezoelectric element, matching layers of broadband piezoelectric transducers usually have lateral dimensions much greater than the thickness. The ultrasonic wave propagation in these matching layers can be analyzed using a one-dimensional model, in a similar way to that presented in Sect. 4.2, bearing in mind that now all electrical and piezoelectric magnitudes are null. The stress equation of motion Eq. (4.4) and the wave equation Eq. (4.5) can be applied using the material constants and dimensions of the layer: thickness tC; phase velocity vC; characteristic impedance ZC; propagation constant βC = ω / vC. Two equations, which specify the relations among terminal variables present at the input and output ports of the layer, are obtained, and can be written in matrix form as follows:

⎛ FI +1 ⎞ ⎛ Z c A / j tan β c t c ⎜⎜ ⎟⎟ = ⎜⎜ ⎝ FI ⎠ ⎝ Z c A / j sin β c t c

Z c A / j sin β c t c ⎞ ⎟ Z c A / j tan β c t c ⎟⎠

⎛ u I +1 ⎞ ⎜⎜ ⎟⎟ ⎝uI ⎠

(4.21)

where F and u are the forces and particle velocities at the layer surfaces; FI and uI correspond to the right front face of the layer and FI+1 and uI+1 correspond to the left back face of the layer (see Figs. 4.2 and 4.6). The same sign conventions as in Fig. 4.2 are used. These equations correspond to a mechanical transmission line. The transfer matrix elements Ac, Bc, Cc, and Dc of a matching layer, relating the parameters at the back and front ports, can be obtained from the short and open circuit conditions (Eqs. 4.12a-d), resulting (with the sign conventions of Fig. 4.6): ⎛ Ac ⎜⎜ ⎝ Cc

Bc ⎞ cos β c t c ⎛ ⎟⎟ = ⎜⎜ Dc ⎠ ⎝ j sin β c t c / AZ c

jAZ c sin β c t c ⎞ ⎟ cos β c t c ⎟⎠

(4.22)

These equations were already presented in Table 4.1, where the mechanical impedance Zl corresponds to AZC. Several matching sections connected in cascade can be evaluated from the product of the corresponding matrices. A simulation study, showing the effects of varying the characteristic impedance ZC of a λ0/4 matching layer on the transmitting voltage transfer

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functions FL / V and impulse responses, is presented in Fig. 4.10. A PZT piezoelectric transducer with the same characteristics as in Sect. 4.5 has been considered. 0.020

1

(a) a

0.000

-0.020

0

0.020

1

(b) b

0.000

-0.020

0

0.020

1

c (c) 0.000

-0.020

0 0

1 f (MHz)

2

0

2

4

6

t (us)

Fig. 4.10. Computed frequency bands FL /V and time impulse responses in emission, of a typical piezoelectric transducer (described in the text), with a λ0/4 matching layer of different characteristic impedances: a without matching layer; b Z01/2 ZL1/2 and c Z01/3 ZL2/3

The characteristic impedance of the λ0/4 matching layer has been varied according to the previously design criteria. Three cases are presented: a), transducer without matching layer; b) transducer with a single matching layer according to Eq. (4.17), with ZC = 6.7 Mrayls; and c) transducer with a single matching layer according to Eq. (4.19), with ZC = 4.1 Mrayls. It can be appreciated in this example that case c) presents a better performance in pulse compactness, with lesser ringing, than the classical criterion case b).

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4.7 Electrical matching and tuning The analysis of the electrical matching/tuning between piezoelectric transducers and driver/receiver electronics is difficult for high frequency broadband ultrasonic applications. Different electrical components and networks can be used, looking for a compromise between bandwidth and efficiency. The most frequently used electrical component is a parallel coil canceling out the clamped capacitance of the piezoelectric transducer at central working frequency. Linear tuning/matching networks can be analyzed by means of the transfer matrix approach previously described, connecting in cascade the different network components. The matrix expressions for simple series Zs and parallel Zp impedances were included in Table 4.1. Figure 4.11 illustrates the influence of a parallel tuning inductance Lp at the reception stage of a piezoelectric transducer with the same characteristics as in Sect. 4.5, without matching layers. 2 (c) c

(b) b

1

a (a) 0 0

1 f (MHz)

2

Fig. 4.11. Computed reception transfer function Vr /FL of a typical piezoelectric transducer (described in the text). a without tuning; b with a shunt coil Lp = 22 μH and c with a shunt coil Lp = 49 μH

The reception frequency responses Vr /FL, Eq. (4.16), have been computed using now different shunt coils. Three cases are included: a) transducer without electrical tuning; b) transducer with a parallel tuning coil Lp = 22 μH (this inductance approximately compensates the clamped capacitance C0S = 1.17 nF of the transducer at working frequency); and c) transducer with a higher inductance Lp = 49 μH (which in this case results

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more adequate for an optimized pulse shaping at the emission electrical stage). A strongly distorted frequency band, with a sharp peak at low frequency, can be appreciated in case c), but this distortion is compensated with the emission contribution of LP giving a rather plane global response. The analysis of electrical matching/tuning at the emission stage of broadband transducers presents especial difficulties. The main matching effect of a parallel coil Lp at the emission stage is the fitting of the spike temporal width to the impulse response of the transducer. It could be analyzed from the emission transfer function ETF1 = FL /VG if the characteristics of the source (VG, ZG) were known. Nevertheless, this tuning coil produces an oscillation in the driving voltage, which is usually avoided by non-linear rectifier devices included in the pulser circuit [4]. The presence of these non-linear components makes difficult the analysis of the emission stage. In fact, electronic “spike” generators with non-linear behavior, and based on capacitive discharges, are commonly used in practical cases of ultrasonic imaging. In these cases, the matching network topology of such generators (including semiconductor devices) specially conditions the electrical driving pulse at the transducer terminals and therefore the overall response [4]. The driving voltages at the transducer terminals are determined in this case by the interactions among the pulser circuits, the matching network and the piezoelectric transducer, as will be analyzed in more detail along the Chap.6.

References 1. 2. 3. 4. 5. 6.

IEEE Standard on Piezoelectricity (1987), ANSI/IEEE Std 176 A. Ballato (2001) “Modeling piezoelectric and piezomagnetic devices and structures via equivalent networks”, IEEE Trans. Ultrason. Ferroelect, Freq. Contr. 48(5):1189-1240 C.S. Desilets, J.D. Fraser and G.S. Kino (1978) “The design of efficient broad-band piezoelectric transducers” IEEE Trans. Sonics and Ultrason. 25(3):115-125 A. Ramos, J.L. San Emeterio and P.T. Sanz (2000) “Improvement in transient piezoelectric responses of NDE transceivers using selective damping and tuning networks” IEEE Trans. Ultrason. Ferroelec. Freq. Contr. 47: 826-835 G.S. Kino (1987) “Acoustic waves: devices, imaging, and analog signal processing” Prentice Hall, Englewood Cliffs, NJ V.M. Ristic (1983) “Principles of acoustic devices” John Wiley and Sons

116 7. 8. 9. 10.

11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

José Luis San Emeterio and Antonio Ramos B.A. Auld (1990) “Acoustic fields and waves in solids”, Krieger Publishing Company M.G. Silk (1984) “Ultrasonic transducers for nondestructive testing” Adam Hilger Ltd J.W. Hunt, M. Arditi, F.S. Foster (1983) “Ultrasound transducers for pulseecho medical imaging” IEEE Trans. Biomed. Eng. 30(8):453-481 J.L. San Emeterio, A. Ramos, P.T. Sanz, E. Riera (1988) “Modelling of multilayer piezoelectric transducers for echographic applications. I.-Analysis in the frequency domain; and II.- Equivalent circuits” Mundo Electrónico 186:85-90; and 187:159-187 J.L. San Emeterio, P.T. Sanz, E. Riera and A. Ramos (1988) “Una implementación del modelo KLM para transductores piezoelectricos en modo espesor” Anales Fisica B 84:48-55 G.R. Lockwood, F.S. Foster (1994) “Modeling and optimization of highfrequency ultrasound transducers” IEEE Trans. Ultrason. Ferroelec. Freq. Contr. 41:225-230 J.L. San Emeterio, A. Ramos, P.T. Sanz, A. Ruiz (2002) “Evaluation of impedance matching schemes for pulse-echo ultrasonic piezoelectric transducers” Ferroelectrics 273:297-302 W.P. Mason (1948) “Electromechanical transducers and wave filters” Van Nostrand, New York M. Redwood (1961) “Transient performance of a piezoelectric transducer” Journal of the Acoustical Society of America 33(4):327-336 R. Krimholtz, D.A. Leedom, G.L. Mathaei (1970) “New equivalent circuits for elementary piezoelectric transducers” Electronic Letters 6(13):398-399 S.A. Morris, C.G. Hutchens (1986) “Implementations of Mason’s model on circuit analysis programs” IEEE Trans. Ultrason. Ferroelec. Freq. Contr. 33:295-298 W.M. Leach (1997) “Controlled-source analogous circuits and SPICE models for piezoelectric transducers” IEEE Trans. Ultrason. Ferroelec. Freq. Contr. 44:60-66 A. Ramos, J.L. San Emeterio, P.T. Sanz (2000) “Dependence of pulser driving responses on electrical and motional characteristics of NDE ultrasonic probes” Ultrasonics 38:553-558 J. L. San Emeterio, A. Ramos, P.T. Sanz, A. Ruiz, A. Azbaid (2004) “Modeling NDT piezoelectric ultrasonic transmitters” Ultrasonics 42:277-281 J. L. San Emeterio, A. Azbaid, A. Ramos (2005) “Computer modeling and simulation of thickness mode piezoelectric transducers under different driving conditions” Ferroelectrics 320:153-159 R.E. Collin (1955) “Theory and design of wide-band multisection quarter-wave transformers” Proc. IRE 43:179-185

5 Interface Electronic Systems for AT-Cut QCM Sensors: A comprehensive review Antonio Arnau1, Vittorio Ferrari2, David Soares3 and Hubert Perrot4 1

Departamento de Ingeniería Electrónica, Universidad Politécnica de Valencia Dipartimento di Elettronica per l’Automazione, Università di Brescia 3 Institute de Fisica, Universidade de Campinas 4 Laboratoire Interface et Systèmes Electrochimiques, Université P. et M. Curie 2

5.1 Introduction AT quartz crystal microbalance sensors (QCMS) are becoming into a good alternative analytical method in a great deal of applications [1 – 16], with a resolution comparable, in many cases, to chemical techniques used for detecting species and suitable for fluids physical properties characterization [17, 18], though simpler and much less expensive. However, an appropriate evaluation of this analytical method requires recognizing the different steps involved in order to be conscious of their importance and to avoid the possible error propagation if the appropriate care is not taken. The three steps involved in a QCM system are: 1. Measurement of the appropriate parameters of the resonator. This includes a suitable electronics and cell interfaces for a specific application; 2. Extraction of the effective parameters related to the model selected for the application (2-, 3- or 4-layer model). This is one of the most complicated steps, including mathematical algorithm combined with appropriate measurements in step 1. The most typical situations correspond to 3layer model (Fig. 5.1), although the analysis of 4-layer model is interesting for understanding the measurements resulting from intermediate processes which occur during the experiment; and 3. Interpretation of the physical, chemical or biological phenomena responsible for the change in the effective equivalent parameters of the selected model. This is the last step and the final role of the system.

A.A. Vives (ed.), Piezoelectric Transducers and Applications, doi: 10.1007/978-3-540-77508-9_5, © Springer-Verlag Berlin Heidelberg 2008

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In this chapter we will focus on step 1, explaining the limitations of different electronic interfaces in relation to the application. In this case, the application will be evaluated as a function of a change in the measuring parameters. Thus, it is first necessary to define the parameters of the quartz crystal resonator (QCR) system to be measured.

Layer 3

Semi-infinite Medium

Layer 2

Sensitive Layer

Layer 1

G'2 G''2 G'1 G''1

2

h

1 1

Quartz Sensor

c66

q

q

Fig. 5.1. The three-layer model

5.2 A Suitable Model for Including a QCM Sensor as Additional Component in an Electronic Circuit As mentioned in previous Chaps. 1-3, a shear strain is induced in an AT quartz crystal when an alternating-current (AC) voltage is applied across it through opposing electrodes deposited on its surfaces. It generates a transversal acoustic wave propagating through the quartz to the contacting media. The mechanical interaction between the resonator and the contacting media influences the electrical response of the device. This permits the use of the resonator as a sensor device to detect changes in the physical properties of the contacting media (see Chap. 3). In order to treat the sensor as a component included in electronic circuits and to be able of analyzing its performance in relation to the external circuitry, an electrical model appropriately representing its impedance would be very useful. The loaded quartz can be appropriately described by an extended Butterworth Van Dyke (BVD) equivalent circuit (Fig. 5.2a) and, in this way, be included in electronic circuits as an additional component. This model is in fact an approximation of the TLM (Transmission Line Model – Chap. 3, Sect. 3.A.3), but it is enough to describe the problems associated to the different systems in relation to specific applications.

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For our purposes, it is not necessary to know the expressions relating Rmq , Lqm , C mq (unperturbed quartz resonator – Chap. 1, Sect. 1.A.1 and Chap. 3, Appendix 3.A, Sect. 3.A.4) and RmL , LLm and C mL (loading contribution – Chap. 3, Sect. 3.A.4) to the physical and geometrical properties of the quartz and load and they can be found elsewhere [19,20]. The equivalent circuits in Fig. 5.2 correspond to the typical configuration of only one face of the sensor in contact with the load, which is common for most of in-liquid applications. A conductance in parallel with the parallel capacitance must be included when the two faces of the sensor are in contact with the same medium, and this conductance can be very high (low resistance) when electrolytic or conductive solutions are used as contacting medium. This conductance strongly influences the characterization electronic interface and, therefore, this configuration is only used in specific cases where the conductance of the medium is one of the physical parameters of interest. Different equivalent models have been described depending on the specific electrode shape and experimental setup [21]; however the equivalent circuits in Fig. 5.2 are the most popular and can be used to represent the most common sensor set-ups as well as for modeling the behavior of the sensor in an electronic circuit like, for instance, an oscillator. Throughout this chapter we will make use of a BVD equivalent model to study the driver/sensor combination but it will not affect the generality of the results.

Co* ,

q

q

Lm q

Lm UNPERTURBED QUARTZ-CRYSTAL

Cp

Co

q

Rm

q

Cm

Cm Co*

L

q

Rm

Co*

L

Lm Z Lm LOADING CONTRIBUTION

a

C R

L Cm L Rm

b

Fig. 5.2. Equivalent circuit models for loaded QCR: a general model - extendedBVD model, and b BVD-model

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5.3 Critical Parameters for Characterizing the QCM Sensor To discuss the problem associated with the different electronic systems used to characterize the sensor, it is necessary first to define the parameters to be measured for an appropriate evaluation of the sensor response. The necessity to know the magnitude of the different parameters will depend on the specific application and on the electronic interface used. When a complete characterization of the sensor is necessary, the different parameters have to be measured and appropriate electronic interfaces must be available, for instance, impedance or network analyzers. Fortunately, there are a great deal of applications where a complete characterization of the sensor is not necessary, and only “key” parameters of the sensor need to be monitored in order to obtain the desired information, for example, applications where the use of a simple oscillator and the monitoring of the oscillating frequency shift is enough. With the aim of covering general cases the different steps for a complete characterization of the sensor parameters are introduced next: STEP 1: Determination of the appropriate reference parameters for a specific application. Some times the determination of the elements Co* , Rmq , Lqm and C mq for an evaluation of the unperturbed device response is necessary. Generally, values of Co* , Rmq , Lqm and C mq are provided by the manufacturer, but sometimes they are not accurate enough and must be obtained prior to the experiment as references. Normally, they can be determined with impedance or network analyzers by measuring the electrical response of the unperturbed resonator over a range of frequencies near resonance, and fitting the equivalent-circuit model to these data. If an impedance analyzer is not available, the corresponding standard [22], or an alternative method described elsewhere [23], can be used. A more accurate determination of Co* can be made at a frequency as high as the double of the resonant frequency [24]. From these values, the following parameters of interest can be extracted: fs: motional series resonant frequency (MSRF). It is defined as the frequency at which the motional reactance vanishes and corresponds to the following expression for the BVD equivalent circuit: fs =

1 2π Lqm C mq

(5.1)

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In practice, the frequency corresponding to the maximum conductance typically monitored in impedance analysis is very close in most practical cases to the MSRF (Sect. 5.A.3) [18]. hq: quartz thickness. It can be determined from (Chap. 1, Eq. (1.A.34):

hq ≈

_

1

ωs

c66

(nπ )2 − 8K 02

ρq

(5.2)

_

2 where c 66 = c66 + e26 / ε 22 is the piezoelectrically stiffened elastic constant, c66 is the elastic constant, e26 is the piezoelectric stress constant, ε22 is the permittivity, Ko is the lossless effective electromechanical coupling factor, n (n = 1, 3, 5,..) is the harmonic resonance of quartz, ρq is the quartz density and ωs=2π fs. The quartz thickness calculated through Eq. (5.2) is actually an “effective” quartz thickness that includes the effect of the electrodes, since the frequency resonance ωs is the measured frequency of the sensor with electrodes.

Co: static capacitance, shown in Fig. 5.2a, arises from electrodes located on opposite sides of the dielectric quartz resonator. This capacitance does not include parasitic capacitances external to the resonator (Cp) which do not influence the motional parameters [20]. Static capacitance can be determined from the values of C mq or Lqm through the following relationships:

C mq =

Lqm =

8K 02 C 0

(nπ )2

(5.3)

1

(5.4)

ω

2 q s Cm

AS: effective electrode surface area. It can be determined from C0 and hq along with the quartz permittivity from: AS =

hq

ε 22

C0

(5.5)

Its value is necessary in applications involving film thickness. The determination of the effective electrode area is not a simple task, since it does not necessarily correspond to the electrode area. Actually the effective sensitive area of the sensor could be smaller than the electrode

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area – in case of a non-perfect overlapping of the faced electrodes – or even bigger – in case of contacting with liquid media [25]. It also comes from the fact that the magnitude of C0 is not either easy to evaluate (see Chap. 14 and [26]). Cp: parasitic parallel capacitance, external to the resonator. Cp = Co*– Co. Its value is useful in applications where the influence of the dielectric properties of the load have to be accounted for [21]. Some times especial electrode configuration [27] or different excitation principles [see Sect. 2.3.8] are used to enhance relevant physical properties of the material under investigation, namely the electrical parameters permittivity and conductivity. In these configurations both the static and parasitic capacitances change and their magnitudes strongly depend on the electrical properties of the material under investigation. These parameters of the unperturbed resonator are useful in some applications where the unperturbed state of the quartz is the reference state like, for instance, in liquid property characterization. However, in most applications the reference state is not the unperturbed state, and because these parameters can change with the load [26, 28] it is better to take as reference the sensor parameters just before the beginning of the process to be monitored. For example, in electrochemical applications the sensor is in contact with an electrolytic solution, it is to say with a Newtonian liquid of known characteristic impedance and, therefore, this is the state to take as reference. This can be done by calibrating the “unperturbed” sensor parameters for assuring the best fitting between the conductance computed from the TLM, for the known characteristic impedance corresponding to the contacting semi-infinite Newtonian medium, and the experimental conductance plot taken from an impedance or network analyzer [26]. In some other applications such as in piezoelectric biosensors (see Chap. 12) where the losses, mainly due to the contacting solution, are expected to be maintained constant and only the resonance frequency shift is the parameter of interest, the resonant frequency of the sensor in contact with the solution prior to the beginning of the detection process is taken as reference. This value establishes a reference base line accounting for all the non-ideal effects, and allows the evaluation of very tiny frequency changes due to the biological interaction in the coating-liquid interface if appropriate environmental control is held. STEP 2: Measuring of those parameters of the loaded thickness shear mode (TSM) resonator, which can more appropriately characterize the physical and/or geometrical properties of the load on the basis of current models.

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From Fig. 5.2a, it is shown that the loading contribution is characterized by the elements RmL , LLm and C mL of the motional branch. A change in both LLm and C mL produces a change in the MSRF. On the other hand, changes in the loading properties are also reflected on changes in the motional resistance RmL , which does not produce MSRF changes. Thus, both MSRF and motional resistance are useful and necessary parameters for sensor characterization. In addition, it is important to state that the majority of the simpler models derived from the most comprehensive TLM, such as the Lumped Element Model (LEM) [29], or the extended BVD model [20] (Chap. 3), assume that the resonator operates around the MSRF. Furthermore, it is important to mention that most of simpler equations used to relate frequency and resistance shifts to the properties of the load have been derived assuming that the resonator is oscillating at its true MSRF. Thus, measurements of loading-induced frequency changes made with the resonator operating at a frequency different from the true MSRF could not agree with the models derived for QCM sensors. This discrepancy is specially pronounced when the resonator is loaded with heavy damping media. Another characteristic which makes the MSRF more interesting than other frequencies is that its value is independent of parallel capacitance changes. For all that mentioned, the MSRF and the motional resistance are parameters of the loaded resonator to be measured. However, it is important to notice that only these two parameters are not always enough for a complete determination of physical parameters of interest of the load under study. In general, more than two unknowns are present in quartz sensor applications (see Chap. 14); in these cases a complete characterization of the admittance spectrum of the sensor can be useful. This characterization of the complete admittance spectrum can only be done with admittance spectrum analyzers like impedance or network analyzers, or specially adapted circuits which can operate in a similar way (see Sect. 5.4.1 below); but even with a complete characterization of the sensor admittance spectrum around resonance, the resolution of the problem of load physical parameters extraction is not clearly solved, depending on different aspects like the accuracy of the experimental data, the suitability of the physical model selected to describe the real process, the accuracy of the electrical model used for describing the physical model, the fitting algorithms used, etc, [26]. Therefore, since the admittance spectrum characterization does not clearly solve the complete problem of physical parameters extraction and requires a more sophisticated instrumentation, the researchers have made great efforts to design electronic interfaces for

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monitoring, as accurately as possible, the two important parameters of interest already mentioned: the MSRF and the motional resistance. Then, the problem associated with the measuring system will be discussed in relation to the accuracy in the determination of these parameters of interest.

5.4 Systems for Measuring Sensor Parameters and their Limitations We will focus this discussion on the interface circuits currently used for sensor characterization which are based on the following principles: network or impedance analysis, impulse excitation or decay methods, oscillators and lock-in techniques. Finally, some requirements and interfaces for fast QCM sensor applications are described. 5.4.1 Impedance or Network Analysis

Since the problems associated with oscillators for the accurate monitoring of the right frequency of the QCR sensors were described [21, 30, 31] (see Sect. 5.4.3), the use of admittance spectrum analyzers to characterize the quartz sensor was extended [32-36]. Nowadays this technique is habitually used for sensor analysis under laboratory conditions having its advantages and disadvantages. Impedance or network analyzers measure the electrical impedance or admittance of the quartz sensor over a range of frequencies near resonance for a complete characterization of the device response. As a test instrument, an impedance analyzer has the following advantages in evaluating the sensor response: 1. The device is measured in isolation and no external circuitry influences the electrical behavior of the sensor. 2. Parasitic influences can be excluded by calibration due to passive operation of the sensor. 3. Differentiated information in relation to diverse contributions of the load can be obtained by measuring both the conductance and the susceptance of the sensor over a range of frequencies around resonance. However, several inconveniences remain when using this technique for sensor applications [37]:

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1. Its high cost and large dimensions of the associated equipment prevent its use for in situ or remote measurements. 2. The connection between the sensor and the equipment is sometimes difficult to accomplish such as in electrochemical or biological applications where it is convenient to ground one of the quartz electrodes. 3. It is not suitable for simultaneous multiple sensor characterization. Sometimes a multiplexing interface is used for a sequential connection of different sensors to the same analyzer, but it can perturb the device response. On the other hand, the impedance analyzer can determine with high accuracy the MSRF and motional resistance of the unperturbed quartz sensors as reference values. The MSRF is obtained by measuring the frequency corresponding to the conductance peak around resonance. The motional resistance is determined as the reciprocal of the conductance peak value. The evaluation of the MSRF and the motional resistance in this way is based on the suitability of the BVD model for characterizing the sensor response. In BVD circuits the relationships between MSRF and maximum conductance frequency and between the motional resistance and the reciprocal of the conductance peak value are exact (Sect. 5.A.3). For an unperturbed resonator, the BVD circuit can very accurately represent the device response. Additionally, the range of frequencies in which the resonance happens is very narrow and therefore the frequency resolution of the instrument is very high. However, for heavy damping loads the quality factor of the device is considerably reduced and the resonance range broadens; this reduces the frequency resolution as well as the suitability of the BVD circuit for representing the sensor response. The determination of the MSRF and motional resistance by using the mentioned relationships is not as accurate as for the unperturbed situation, but remains accurate enough for these applications in which the sensor can be used [18]. On the contrary, the determination of BVD parameters for high damping loads does not give additional information apart from the parallel capacitance, which can be measured more appropriately at double of the resonant frequency. Furthermore, the MSRF determination from the motional components (Eq. 5.1) can produce great errors depending on the algorithm used for the motional parameters extraction. An option better than a direct reading of the conductance peak and its frequency from the discrete data of the conductance plot measured by the impedance analyzer, is to fit these data to a Lorentzian curve and to obtain this information from the maximum of the curve; this provides more accurate results in case of heavy loaded quartz sensors. Additionally, an alternative parameter, probably more accurate than the motional resistance for

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measuring the damping and quality factor Q of the loaded sensor is the half power spectrum of the resonance (Chap. 3, Sect. 3.5.4). Adapted Impedance Spectrum Analyzers

To circumvent the drawbacks of classical impedance or network analyzers, important efforts have been made to adapt the principle of operation of these reference instruments into smaller electronic boards configured for the specific features of QCR sensors [38-43]. These adapted systems maintain similar specifications to high performance impedance analyzers in the range of frequencies and impedances of typical QCR sensors while improving the portability. The characteristics of these circuits make them appropriate for most common QCM applications. These adapted circuits are optimized for fast data measuring and acquisition in comparison with classical analyzers and allow to measure and acquire each datum of impedance between 1 and 5 ms [39]; this means that each complete impedance spectrum can be recorded between 1 and 5 s, assuming 1000 frequency points. This acquisition time is enough for most applications, at least in the case of 1 s. However, they are not appropriate for fast QCM applications where a very fast changing frequency is necessary to monitor [44, 45]. Alternatively, simpler systems have been developed following the principle of passive interrogation of the sensor used in impedance analyzers; it is to say, by sweeping the frequency of the interrogating signal around the frequency range under study. However, instead of being designed for recovering the information associated with the conductance and susceptance spectra of the sensor around resonance, which is more involved in terms of electronic design, they acquire the representative magnitudes of voltages associated with a “voltage transfer function” in which the impedance of the sensor takes part. Then, the impedance of the sensor is replaced by an appropriate model whose parameters are fitted to the experimental voltage transfer function acquired [46, 47]. In the transfer function method used by Calvo and Etchenique, for instance, the sensor takes part of an impedance divider where the other impedance in the divider is known [47]. Figure 5.3 shows the impedance divider where the sensor has been characterized by its equivalent BVD circuit (Fig. 5.2b). A sinusoidal signal applied at the input sweeps a range of frequencies around the resonant frequency of the sensor; the average values of the input and output voltages are measured and acquired with an analog/digital converter and captured with a computer. Thus, an experimental measure of the absolute value of the voltage transfer function is obtained in the range of frequencies considered.

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Zm Cm L *

Co

ui

uo

C R

Zq

Fig. 5.3. Impedance divider for the transfer function method used in reference [47]

The theoretical transfer function is used to find the set of parameters that best fit to the experimental data. It is assumed that the motional capacitance remains constant, the rest of parameters C0, L and R are obtained by a non-linear fitting. Finally the authors of this technique noticed the advantage of using a capacitor Cm instead of a resistor as the other branch of the voltage divider, as shown in Fig. 5.3. In this context the theoretical transfer function is: 2

⎛ 1 ⎞ ⎜⎜ ω L − ⎟⎟ + R 2 ω C ⎝ ⎠

uo = ui ⎛ ω LC o Co 1 1 ⎜⎜ ωL − + − − ωC Cm ω CC m ω C m ⎝

2

⎞ ⎛ RC o ⎞ ⎟⎟ + ⎜⎜ R + ⎟ C m ⎟⎠ ⎠ ⎝

2

(5.6)

A very interesting approach has been recently described by Kankare et al [48]. The set-up includes the QCR sensor in series with a capacitor, as described in the previous approach; however, the signal used to interrogate the voltage divider is a double-sideband suppress carrier amplitude modulated signal whose carrier is swept around the resonance frequency range. This strategy confers special characteristics to the system that are worth to treat in more detail. The working principle of the interface is depicted in Fig. 5.4 where the voltages at the inputs of the multiplier are given by the following expressions: u1 = U 0 sin ω m t cos ω c t =

U0 (sin ω + t − sin ω − t ) 2

(5.7)

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Antonio Arnau, Vittorio Ferrari, David Soares and Hubert Perrot

u1 Cm

k

BPF

uout

u2 ZQ Fig. 5.4. Schematic diagram of the interface. Adapted from [48]

u2 =

U0 ( f (ω + ) sin(ω + ) + g (ω + ) cos(ω + ) ) + 2

(5.8)

U + 0 ( f (ω − ) sin(ω − ) + g (ω − ) cos(ω − ) ) 2

where ω+= ωc+ωm, ω-= ωc-ωm and the functions f (ω) and g(ω) are the real and imaginary parts of the impedance divider transfer function given by de following expressions: ⎛ ⎜ 1 ⎜ f (ω ) = Re⎜ Y ⎜1 − j Q ⎜ ωC m ⎝

⎛ ⎞ ⎜ ⎟ 1 ⎜ ⎟ ⎟ ; g (ω ) = Im⎜ Y ⎜1 − j Q ⎟ ⎜ ⎟ ωC m ⎝ ⎠

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(5.9)

where YQ is the admittance of the sensor at the interrogating frequencies. As can be seen the sensor is simultaneously interrogated at two frequencies ω+= ωc+ωm and ω-= ωc-ωm which move together with the sweeping of the carrier. The frequency shift between the interrogating signals (2ωm) can be easily controlled by appropriate selection of the frequency of the modulating signal that is maintained constant during the impedance test. After multiplying the input signals and removing the high frequency and dc components by appropriate filtering (demodulation of u2), the following low frequency signal remain at the output: u out = (1 / 8) khU 02 (− ( f (ω + ) + f (ω − ) ) cos 2ω m t + + (g (ω + ) − g (ω − ) ) sin 2ω m t )

(5.10)

The demodulated signal is formed by a couple of two coherent terms of frequency 2ωm whose amplitudes have the information about the sensor impedance through Eqs. (5.9). By modeling the motional impedance of the sensor with the Lumped Element Model (LEM)[29] and estimating the

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parameters of the resonator in the unperturbed state, the authors can obtain the real and imaginary parts of the surface load impedance by non-linear fitting of the data corresponding to the amplitude of the component in quadrature g(ω+)-g(ω-). This configuration has three advantages in comparison with the classical impedance analysis operation: 1. The information of the phase and magnitude of the sensor impedance is carried out in the amplitude of low frequency signals; this makes easier and more accurate their acquisition. 2. Because the signal of interest is formed as the difference between two coherent signals, any additive source of noise is cancelled. 3. The differential form of the signal permits to increase the sensitivity in case of heavy loaded resonators. Effectively, within a certain range, an increase in the modulating frequency increases the difference g(ω+)g(ω-); it creates an amplifying effect while maintaining the noise and then the signal to noise ratio is improved. This original method could be in fact considered as an improvement of classical impedance analysis operation and can be used for any QCR sensor application with the exception of fast QCM at high operation rates. 5.4.2 Decay and Impulse Excitation Methods

Impulse excitation and decay methods are based on the same principles. To illustrate the principle let us consider the electrical circuit in Fig. 5.5. t=0

Quartz Sensor

i(t)

iCo* (t)

im(t) Lm

r E

Co*

Cm Rm

Fig. 5.5. Diagram for illustrating the principle of the impulse excitation method

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At t = 0 the switch connects a voltage source (E) with a very low output resistance (r) to the sensor modeled in the figure as a BVD circuit. From this moment the current i(t) changes according to the following expression (assuming r << Rm): t

E − * i (t ) = iC * (t ) + im (t ) ≈ e rCo + o r E Lm

1 ⎛ R ⎞ 1 − ⎜⎜ m ⎟⎟ Lm C m ⎝ 2 Lm ⎠

2

e

−

Rm t 2 Lm

2

⎛ R ⎞ 1 − ⎜⎜ m ⎟⎟ t sin Lm C m ⎝ 2 Lm ⎠

(5.11)

The first term associated with the current through the parallel capacitance iC * (t ) , has a time constant τ C * = rCo* while the time constant of the 0 0 second term associated with the current through the motional branch of the sensor, im(t), is τ m = 2 Lm / Rm . Assuming typical values for a 10 MHz loaded AT cut quartz resonator (Table 5.1) and r = 2 Ω, the decay time constant τ C * is about 106 times smaller than τ m . Then, the total current i(t), 0 is dominated by the second term in Eq. (5.11) once the first instant has passed. With a proper interface circuit the damped oscillation can be recorded on a digitizing oscilloscope and subsequently transferred, e.g. via GPIB, to a computer. A numerical fitting of the recorded curve permits to obtain the time constant τ m and the frequency of the damped oscillation given by: 1 fm = 2π

⎛ R 1 − ⎜⎜ m Lm C m ⎝ 2 Lm

⎞ ⎟⎟ ⎠

2

(5.12)

The time constant τ m can be related to the quality factor QL by using the approximate expression: QL ≈ 2πfsLm /Rm = πτm fs (Chap. 1-Eq. (1.A.38)), where fs is the MSRF of the loaded QCR. Then, the frequency of the damped oscillations can be written as a function of fs and QL as follows: fm ≈ fs 1−

1 4Q L2

(5.13)

Thus, by measuring the time constant and the frequency of the damped oscillations, the quality factor and the MSRF can be determined. It can be considered, without incurring a significant error, that the reduction in the quality factor is proportional to an increase in the motional resistance. In such a way a measurement of this magnitude can be obtained.

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Table 5.1. Typical values for the BVD parameters of a 10MHz AT-cut QCR C0* [pF] Lm [mH] Cm [fF] Rm [Ω]

10 7.5 33.8 10

In practice, the impulse excitation method is difficult to apply for two reasons: 1. Ideal pulse front slopes are difficult to achieve. 2. Other harmonics different from that desired can be excited. Thus, in order to avoid them, additional circuitry which perturbs the sensor response is necessary. Instead of the impulse method, the decay method is used in practice [49, 50]. The schematics of the experimental set-ups for the decay method are depicted in Figs. 5.6 and 5.7 for the excitation of the parallel and series resonant frequencies, respectively. The measuring principle is very similar to the one described. A piezoelectric resonator is excited with a signal generator approximately tuned to the frequency of the desired harmonic. Then, at t = 0 the signal excitation is eliminated by opening the appropriate relay. At this moment, the voltage or current, depending on whether the parallel or series resonant frequency is excited according to the electrical setup [51], decays as an exponentially damped sinusoidal signal, mathematically expressed by:

A(t ) ≈ A0 e

−

t

τm

sin(2πft + ϕ ),

t≥0

(5.14)

where Ao is the amplitude of the magnitude at t = 0; ϕ is the phase and f is the frequency given by: f ≈ fi 1 −

1 4Q L2

(5.15)

where fi is the MSRF of the loaded QCR, fs, or the parallel resonant frequency, fp, given by: f p = fs 1 +

Cm C0*

(5.16)

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In Eq. (5.15), it has been implicitly assumed that the quality factor is the same for both series and parallel mode, and is given by (Chap. 1, Eq. (1.A.38)): QL =

Lmω s Lmω p ≈ Rm Rm

(5.17)

In the series excitation mode (Fig. 5.7), the parallel capacitance effect is eliminated by short-circuiting and the frequency of the damping oscillations is very close to the true MSRF. This is one of the principal advantages of the method. The accuracy of the decay method is high, provided that the measurement of the frequency and the envelope are obtained with high accuracy, which becomes complicated for strongly damping loads. This technique reduces the cost of the instrumentation in comparison with network analysis; however, the quality and dimensions of the required equipment still remain high, mainly if an accurate determination of the frequency and the envelope of the exponentially damped sinusoidal is necessary. Therefore, this method is more appropriate for laboratory environment than for sensor applications and becomes more sophisticated when simultaneous multiple sensor characterization at high sampling rates have to be performed.

OSCILLOSCOPE QCM

Probe

GPIB

Relay COMPUTER Signal Generator

GPIB

Fig. 5.6. Experimental setup used to measure the parallel resonant frequency and the parallel dissipation factor (Adapted from [51])

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133

Short-Circuit

QCM OSCILLOSCOPE toroid transformer

Switch GPIB Relay COMPUTER Signal Generator

GPIB

Fig. 5.7. Experimental setup used to measure the series resonant frequency and the series dissipation factor (Adapted from [51])

Recently, the decay method has been introduced with two simultaneous excitation frequencies corresponding to two harmonics of the resonance frequency. One of the harmonics is used for a continuous evaluation of the frequency and damping of the sensor, while the other harmonic is used for perturbation purposes by changing the driving amplitude. This approach allows the analysis of binding reactions under controlled perturbation conditions, while the sensor is simultaneously characterized by monitoring the resonance frequency and the damping with the other testing signal [52]. 5.4.3 Oscillators

In this section the field of oscillators for in-liquid quartz sensor applications is covered. First, some fundamentals on LC oscillators are introduced with the typical modes of operation of a crystal controlled oscillator: parallel and series modes. This introduction is useful for understanding the problems of quartz sensor controlled oscillators for an accurate monitoring of the MSRF and motional resistance of the sensor. Finally, a review of the different oscillators proposed to drive quartz sensors in contact with liquid or with heavy loads is included.

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Antonio Arnau, Vittorio Ferrari, David Soares and Hubert Perrot

Basics of LC Oscillators

An LC oscillator consists of an amplifier with positive feedback including an LC resonant circuit as a frequency control element. In the most general case, the resonant circuit can involve transformers and piezoelectric resonators. In the last case, when the oscillator is well-designed, the piezoelectric resonator controls the oscillating frequency and the oscillator is called “Crystal Oscillator”. In order to understand the basic operation of an LC oscillator, let us consider the circuit shown in Fig. 5.8. In the initial time the switch S1 is closed and the capacitor is charged at the value of the power supply. Once the capacitor has been charged the switch S1 is opened and the switch S2 is closed. The circuit in Fig. 5.9a shows this situation where ig=0. The current i in the circuit obeys to the following differential equation: i ′′ +

1 i=0 LC

(5.18)

This current corresponds to a sinusoidal signal whose angular frequency is ω s = 1 / LC . This frequency is equal to the electrical resonant frequency of the LC circuit The phasorial diagram in Fig. 5.10a offers a different point of view. Taking the voltage u as reference, the currents iL and iC through the coil and capacitor, respectively, have the same magnitude but opposite sign. Hence: Cω u =

u Lω

(5.19) LC CIRCUIT

S1

S2

r C

L

E

Fig. 5.8. Diagram for illustrating the operation principle of a resonant circuit

5 Interface Electronic Systems for AT QCM Sensors LC CIRCUIT

S2

i g (t)

LC CIRCUIT WITH LOSSES

u(t)

S2

i g (t)

i(t)

C

135

i L (t) L

u(t)

L u L(t)

C

iC (t)

iC (t)

r u r (t)

i L (t)

a

b

Fig. 5.9. Part of the circuit in Fig. 5.8 for explaining the electric resonant phenomenon: a without losses, and b including losses

The former equation is only valid for the electrical series resonant frequency ωs. As a consequence, the LC circuit described, which provides a constant amplitude sinusoidal voltage, would constitute the simplest oscillator. Next, let us consider the unavoidable losses in energy modeled by means of a small resistance in series with the coil, as shown in Fig. 5.9b. The new phasorial diagram in Fig. 5.10b makes clear the necessity of an incoming current ig, provided from an external generator such as an amplifier, in order to maintain the magnitude of the voltage u constant. On the contrary, the amplitude of this voltage would diminish following an exponentially damped sinusoidal signal. Consequently, it is necessary to supply the current by means of an external circuit for covering the energy losses due to the resistive element. A typical way to connect an amplifier to the resonant LC circuit is shown in Fig. 5.11a, where the capacitor C has been included as two series capacitors C1 and C2. In Fig. 5.11a the amplifier could be replaced by a transistor where B=Base, C=Collector and E=Emitter. The resultant circuit and its possible combinations would provide the typical well-known configurations: Colpitts, Hartley, Pierce, Clapp [53, 54]. Despite the simplicity of the circuit shown in Fig. 5.11a, it is extremely useful for making the basic operation of an LC oscillator clear and understanding its limitations.

136

Antonio Arnau, Vittorio Ferrari, David Soares and Hubert Perrot iC

uL

u

= u

iC

=

ig ur

iL

iL

a

b

Fig. 5.10. Phasorial diagrams corresponding to the magnitudes involved in the respective circuits in Fig. 5.9

Oscillating Conditions

The circuit shown in Fig. 5.11a can be schematically represented as indicated in Fig. 5.11b, which makes the fulfillment of the following equation for oscillation clear: Aβ = 1

(5.20)

The previous equation involves two conditions: 1. The loop gain condition: Aβ = 1

(5.21)

arg A + arg β = 0

(5.22)

2. The phase condition: This condition is strongly related to the concept of stability, as we will see later. The previous conditions are named “Barkhausen conditions” and their application allows determining the relations among the different oscillator’s parameters, including those of the resonator [55]. The circuits in Figs. 5.11a-b can be used for an easy understanding of two typical crystal oscillators configurations: the parallel and series mode crystal oscillators. Parallel Mode Crystal Oscillator

In this configuration the piezoelectric crystal behaves like an inductance and is used in place of the coil in the circuit shown in Fig. 5.11a. This is possible since the impedance of the piezoelectric crystal has an inductive character in the range of frequencies between the two phase-null frequencies corresponding to a specific resonant frequency (see Fig. 5.A.1). Thus,

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137

in the parallel mode the phase of the crystal is positive but, in general, it is not accurately known. B

C

A

A

E

L C1

C2

a

b

Fig. 5.11. Schematic circuits for illustrating the basic operation of an oscillator: a the feedback network is a tank circuit, and b the feedback network is considered as a black-box

In the parallel mode the oscillating frequency corresponds to the electrical resonant frequency of the series circuit formed by the capacitors C1, C2 and the equivalent inductance Leq of the piezoelectric crystal at the oscillating frequency ωo, given by: Leq =

X eq

(5.23)

ωo

where Xeq is the lossless equivalent reactance of the crystal at the oscillating frequency ωo, given by (see Eq. (5.A.2)):

ω o2 L C − 1

X eq =

⎛

ω o2 (C + Co )⎜⎜1 − ω o2 ⎝

⎞ CC o L ⎟⎟ C + Co ⎠

(5.24)

where L, C and Co are parameters of the BVD circuit modeling the vibrating crystal in a range of frequencies around a certain resonance. Consequently, the expression for the oscillating frequency is:

ωo =

1 Leq C t

(5.25)

where C t = C1C 2 /(C1 + C 2 ) . Using Eqs. (5.23) and (5.24) in Eq. (5.25) and solving for ωo, the final expression for the oscillating angular frequency is:

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Antonio Arnau, Vittorio Ferrari, David Soares and Hubert Perrot

1

ωo = L

C (C o + C t ) C + Co + C t

(5.26)

The former expression is equal to the expression for the parallel resonant frequency of a BVD circuit with a parallel capacitance given by: Co + Ct; this is the reason for denominating parallel crystal oscillator to that configuration where the crystal behaves like an inductance. However, the oscillating frequency is not the electrical parallel frequency of the crystal at resonance (see ωp in Eq. 5.A.3), but a frequency at which the resonator acts like a coil. Such frequency depends on the specific values of capacitors C1 and C2, and must be placed between the phase-null frequencies of the crystal at the selected resonance. Series Mode Crystal Oscillator

In series mode crystal oscillators, the piezoelectric resonator is placed between two points of the feedback path which must be short-circuited for fulfilling the oscillating phase conditions. Thus, the resonator must behave like a short circuit for oscillation. The signal crossing a short circuit does not experience phase shift; therefore, if a series mode crystal oscillator is well designed, the piezoelectric resonator controls the oscillating frequency in such a way that the frequency is maintained near the phase-zero frequency of low impedance of the resonator. Let us consider Fig. 5.11b. This scheme will permit to make clear both the operation of a series mode crystal oscillator and the stability concept. To make the explanation easier let us assume that the amplifier has a real and positive voltage gain transfer function1. As a consequence, the circuit in Fig. 5.11b will fulfill the phase condition for oscillation at that frequency where the phase-shift through the circuit β is zero. Now, let us suppose that this frequency coincides with the zero-phase frequency corresponding to the lowest impedance of the piezoelectric resonator to be used as frequency control element in the series mode crystal oscillator. The scheme in Fig. 5.12a is a simplified diagram of this oscillator, where the black-box X in series with the circuit β in the feedback path represents the This ideal condition is not fulfilled in real oscillators where the phase shift through the amplifier depends on the frequency as well. In fact the gain of the amplifier in Fig. 5.11a must be negative since the circuit β in the feedback path provides a 180º phase–shift between the output and input terminals of the amplifier. Then the amplifier must provide an additional phase–shift of 180º for oscillation.

1

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resonator. The phase condition for oscillation will be fulfilled when the total phase-shift through the series circuit formed by the resonator and the circuit β is zero. Because the circuit β and the resonator have the same zero-phase frequency, this will be the oscillating frequency; in any case the gain of the amplifier will increase slightly in order to compensate the low losses introduced by the resonator. Consequently, in this ideal situation, the oscillating frequency will be the zero-phase frequency of low impedance of the resonator. Let us discuss the previous situation when the phase-shift through the circuit β at the zero-phase frequency of the resonator is not zero. This case additionally permits to discuss the concept of stability and to understand the problem associated to the use of oscillators for monitoring the appropriate frequency of the piezoelectric sensor. Fig. 5.12b illustrates this situation where the continuous line represents the impedance phase of the resonator in a range of frequencies around the zero-phase frequency of lowest impedance of the resonator. Let us suppose now that the phase–shift through the circuit β at the zero-phase frequency of the resonator is –α. Because the phase condition for oscillation requires that arg X + arg β = 0 , it is necessary that arg X = α . Consequently, the oscillating frequency shifts an amount Δf 1 with regard to the zero-phase frequency of the resonator, as described in Fig. 5.12b. X

A

X a

-

{ {

1 f2

2

f0 f f1

b

Fig. 5.12. Schematic circuits for illustrating the basic operation of a crystal controlled oscillator: a the circuit includes a resonant device X in the feedback circuit, and b plot for illustrating the high relationship between the frequency stability and the quality factor of the resonant feedback network

When the quality factor Q of the resonator is very high the frequency– shift is very low, which is apparent with the sharp slope of the phasefrequency transfer function of the resonator. However, the lower the Q the greater the frequency-shift, as can be observed in curve 2 in Fig. 5.12b.

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Antonio Arnau, Vittorio Ferrari, David Soares and Hubert Perrot

These fundamental concepts will permit the understanding of the problem associated with the monitoring of sensor parameters, MSRF and motional resistance, with an oscillator. Problem Associated with the MSRF Determination

The systems described in Sects. 5.4.1-5.4.2 passively interrogate the quartz resonator and, with an appropriate interface, the desired characteristic parameters of the sensor are measured in isolation, i.e., the external circuitry does not interfere with the sensor response. This is the greatest advantage of the methods described. However, in oscillators the sensor takes part of the feedback path and the oscillating frequency is characterized by a certain phase of the quartz impedance, which depends on both the quartz resonator and the external circuitry to the sensor. When real characteristics of the amplifier and the rest of the components in the oscillator must be taken into account, it is difficult to know precisely the phase of the circuitry external to the sensor as a function of the frequency. Therefore, in general, the phase of the sensor for oscillating condition can not be accurately known, which makes the problem associated to oscillators more critical. When the phase shift provided by the circuit external to the resonator, in the range of frequencies around resonance, can be considered constant, the phase provided by the quartz impedance at which the oscillation takes place can be considered constant as well. Thus, when the resonator properties are altered due to loading effects, the phase of the resonator changes and the oscillating frequency shifts to find the new frequency at which the phase of the sensor fits the oscillating condition. Therefore, due to the operation principle of the oscillator, the true MSRF can not be tracked continuously when an oscillator is used for monitoring load–induced frequency changes, at least in an oscillator in which the phase of the quartz impedance for oscillating condition is maintained constant. This can be easily illustrated using the BVD circuit as sensor–model (Fig. 5.2b). It can be understood that the MSRF only depends on L and C components of the motional branch, but the phase of the complete resonator also depends on the specific values of the motional resistance R and on the parallel capacitance Co* . Thus, if the motional resistance changes while remaining constant L and C, the MSRF does not change but the quartz phase changes; then the oscillating frequency has to change in order to find the phase of the resonator corresponding to the oscillating condition, and this frequency shift supposes a frequency error, which can be significant when the MSRF is taken as a parameter of interest. Figure 5.13 shows how the oscillating

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141

Sensor phase condition for oscillation

frequency and the frequency shift regarding the frequency of the resonator in the unperturbed state depend on the oscillating phase condition in an oscillator. Therefore, different oscillators can provide different frequency shifts for the same resonator in the same loading conditions; of course this can lead to important false conclusions and interpretations. Moreover, the oscillation eventually ceases because the oscillation phase condition can not be reached. Another consideration which makes clear the selection of the MSRF as a parameter of interest is its independence of the parasitic capacitances in parallel with the sensor. However, due to the operation principle of the oscillator, the stray capacitances in parallel with the sensor always have important effects on the oscillating frequency. This is because an oscillator looks for a certain phase and not for a certain frequency of the sensor. Thus, although a change in the parallel capacitance due to parasitic effects does not change the MSRF, the phase of the sensor changes and, consequently, a shift in the oscillating frequency takes place [56-58]. As described, the oscillator frequency is dependent not only on “mass loading” (motional inductance) but on “dissipation” (motional resistance). This must be taken into account and carefully considered when using an oscillator in a specific application where the physical properties of the load can change, producing changes in both the mass loading and dissipation. a

90°

f1 b

f3

0°

f2

f2 f1

c

f

f3

-90°

Fig. 5.13. Plot for illustrating the different frequency shifts, with regard to the resonant frequency of the resonator in the unperturbed state, associated with different sensor phase conditions for oscillation in an oscillator

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Antonio Arnau, Vittorio Ferrari, David Soares and Hubert Perrot

Problem Associated with the Motional Resistance Determination

In a great deal of applications in which the quartz resonator is used as a sensor, an additional magnitude to the MSRF shift is necessary in order to discriminate different physical contributions [20]. In what concerns to oscillators, many designs incorporate an automatic gain control (AGC) system for the measurement of the activity [59] of the quartz sensor at resonance [60-65]. The AGC system tries to maintain the level of the signal constant in a selected point of the oscillator. With this purpose, it provides a voltage (AGC voltage) that modifies the gain of the amplifier so as to maintain the signal level in the selected point constant in relation to a reference voltage. In many of these designs proportionality between the change in the AGC voltage and the change in the motional resistance is claimed [37, 62-63, 65]. In some of these AGC systems the proportionality is justified only from a physical point of view [63]. In others, mathematical expressions in which some simplifications were made show this proportionality [62], although it is also shown that this proportionality is lost in some cases [66]. It can be shown that even in the most ideal situation it is very difficult to ensure that the voltage shift provided by an AGC system included in an oscillator is proportional to the change in the motional resistance; unless the parallel capacitance of the sensor is suppressed or compensated. This demonstration can be found elsewhere [67]. The most important aspects concerning the problem associated with oscillators as electronic drivers for QCMS have been treated. It has been shown that although the simplicity of an oscillator makes this device very attractive for sensor applications, some limitations remain. These limitations have to be taken into account in the interest of accuracy and they should be kept in mind in any new work that looks for the simplicity and autonomy of oscillators for sensor applications. Oscillators for QCM Sensors. Overview

In spite of the drawbacks of oscillator circuits for QCR sensor applications mentioned above, their low cost of the circuitry as well as the integration capability and continuous monitoring are some features which make the oscillators to be a good choice for most chemical sensor applications. In this section an overview of the different oscillator approaches proposed in the last two decades for improving the accuracy in the tracking of the appropriate frequency and damping of the resonator sensor is presented.

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For in gas/vapor phase applications the resonator maintains a high Q factor and oscillators are the best choice for sensor monitoring; no special requirements are necessary different from classical quartz oscillators based on the well-known Pierce, Colpitts, Miller, etc., configurations. For sensorarray it would be advantageous to have the sensing face of the resonator grounded to avoid the coupling or “cross-talk” among the oscillators, but even this recommendation has been demonstrated to be unnecessary if a certain level of isolation is maintained between the circuits [68]. The use of oscillators in in-liquid phase QCR sensor applications is very much challenging and this overview will be focused on these applications. For in-liquid phase or heavy loaded sensor applications, the Q factor of the resonator is strongly reduced; for instance, for AT QCR at 10MHz the Q factor is reduced from 80.000 to 3.000 with only one face in contact with water. Moreover, the damping, and then the Q factor, can change during the experiment. This reduction implies that special care must be taken in the design of the oscillator: on one part in the selection of the more suitable configuration, which will be treated in detail throughout this section, and on the other part in the selection of the components of the oscillator circuit, mainly in terms of stability of their electrical characteristics as a function of changes in external variables such as temperature, humidity, etc. The reason has been explained before (Fig. 5.12b); for high Q factor, changes in the phase response of the sensor due to external conditions are easily compensated with very small changes in the frequency of the resonator, and appear in the signal frequency as a small noise and/or drift. However, for low Q factors small changes in the phase response of the rest of the components in the loop of the oscillator need to be compensated with bigger shifts in the oscillator frequency. In these cases, the noise and drift are not negligible and a good control of the external variables has to be done to minimize this problem. Therefore, extreme care must be taken into account in the design of the cell as well as in the control of the surrounding; temperature has to be maintained as stable as possible by thermostatic shields, the electronic noise of the circuit should be reduced as much as possible even by cooling the circuit at low temperatures, the sensor should be protected against vibrations with appropriate inertial systems, etc. This aspect, which is not normally taken into account, is very important for decreasing the level of noise in oscillators and is one of the most important aspects for increasing the resolution and sensitivity of QCR sensors by increasing the frequency [69-71]. Letting this aside, the selection of the appropriate oscillator configuration is, of course, another very important aspect. In 1980 the work of Konash and Bastiaans [72], demonstrating that the QCM could be also used for liquid-phase, and that it was possible to

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maintain the stability of a crystal oscillator with the resonator one-face in contact with a liquid medium, paved the way for numerous applications in different fields like electrochemistry and biology. This work, although with some poor results in terms of comparison with well-supported techniques, opened the way of using the quartz crystal as sensor in fluid media. The physical explanation of why the resonator could maintain the oscillation under the tremendous load of the contacting liquid was given later on by the well-known work of Kanawaza and Gordon [73] (see Chap. 3). Until the important work of Reed et al [32], in 1990, extending the application of the AT-cut quartz crystal to contacting viscoelastic media, few works described the use of the quartz sensor under liquid conditions [3031, 74-75]. The use of oscillators was a common practice on these days and the attention was quickly paid on these interfaces to explain some inconsistencies in the experiments; for instance, different frequency shifts were obtained with different oscillators in apparently the same resonator conditions [30-31]. The need for a common reference of the working phase of the QCR sensor in the oscillators used for in-liquid applications was posed, and 0º for the phase of the sensor under oscillating conditions was proposed [31]. It was also probed that the zero-phase condition not always exits for the loaded sensor [57] and then, the use of impedance analysis was generalized as an accurate but expensive interface for sensor characterization [32-36]. In the beginning of 90’s Barnes analyzed most of the typical oscillators used until then for in-liquid sensing [21]. In this work a clear explanation is included about the reasons of why different oscillators can provide different monitoring frequencies under the same sensor conditions. The two typical operational modes of oscillators are discussed: the parallel mode has a less restrictive range of operation than the series mode and it can be designed to force the resonator to oscillate under heavy load conditions, on the other hand the sensor phase condition is more difficult to control. Barnes work introduced the main aspects, which would be in the near future the “key” points in the design of oscillators for QCR sensors: a) one face of the resonator should be grounded for electrochemical or biological applications and for a better control of the parallel capacitance, b) the evaluation of the motional series resistance would be very useful in many applications, c) an automatic gain control should be implemented to adjust the gain loop for stable operation, and d) the parallel capacitance (static and parasitic components) plays an important role in determining the oscillation frequency, specially in parallel mode configurations [56]. Effectively, the evaluation of the sensor damping played an important role after the work of Martin and Granstaff in 1991 [20], showing that the

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simultaneous measuring of the frequency shift and motional resistance allowed the discrimination of different contributions on the sensor response: mass and liquid effects (see Chap.3, Sect. 3.A.4). Thus, automatic gain control systems were implemented in oscillators, not only for stabilization purposes but as a mean to evaluate the damping of the sensor, with the limitations mentioned in the previous section. Finally, the parallel capacitance compensation has been one of the key aspects in the last approaches of electronic interfaces for QCM sensor characterization (see next section). From then on great efforts were made in the design of appropriate oscillators in in-liquid applications. Parallel mode oscillators, operating at strong negative phase conditions (≈-76º) to force the oscillation of the resonator under heavy load conditions were used [60-61]. However, the major efforts were made in the design of series oscillators with the resonator working at zero phase condition and with one face grounded: Emitter coupled oscillators [37, 65, 76-77], Lever oscillator [62, 78], Active bridge oscillator [79, 80], Balanced bridge oscillators [81, 82]. As it has been mentioned the frequency at which the crystal oscillator is driven depends on the resonator phase condition in the loop. Moreover, the resonator phase condition depends on the working phase of the rest of the components of the circuit in the loop. Since it is not possible to choose the characteristic frequency of the resonator at which the oscillator must oscillate, the other alternative, although not optimal, is to maintain the phase of the sensor in the oscillator as constant as possible for a wide range of loads, at least in this way one would have a reference point in the response of the sensor. To this aim the phase of the rest of the components in the loop should be also maintained as constant as possible, in the frequency range of operation, for a wide dynamic range of loads. A good selection of components and appropriate configurations must be chosen for this aim [58, 62]. Once this requirement is covered, the matter is to decide at what phase the sensor should work for a more ideal operation. The series configuration, at which the sensor ideally works at zero-phase condition, was initially selected. The reason for developing series oscillators working near resonator zero-phase condition was the assumption that the parallel capacitance has a lower effect on the oscillating frequency near the zero-phase frequency of the sensor. In fact, the MSRF does not depend on the parallel capacitance, and for small loads the sensor zero-phase frequency is very near the MSRF. Moreover, if the parallel capacitance could be compensated, for instance by parallel resonance with a coil at the working frequency, the oscillator would be operating at the true MSRF. However, it is not know “a priori” what will be the oscillating frequency, and the tuning out of the parallel capacitance is not easy to do without additional more involved

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instrumentation. Thus, the use of a coil for tuning out the parallel capacitance, although theoretically described, is not habitually used. Under these conditions it was found, with some series configurations, that the initial design at zero-phase was not the optimal condition for the usual range of loads. Different works found that the resonator phase condition at which the frequency was reasonably closer to the MSRF, in the most usual range of loads (for motional resistances ranged from 100 to 700 Ω), was -38º±3º [37, 76-77, 83]. In other cases, although the zero-phase condition of the sensor was desired, the non-ideal characteristics of the circuit components result in a phase condition near the zero-phase condition (≈-6º for the Lever oscillator [62] and ≈-3,5º for the active bridge oscillator [79, 80]. The balanced bridge oscillator is, in principle, able to compensate the parallel capacitance, and then the oscillating frequency is theoretically driven by the condition of zero-phase of the motional impedance [81-82]. Therefore, in ideal conditions the oscillating frequency would be the MSRF. A brief description of the abovementioned oscillator configurations will be next introduced; more details can be found in the given references. Emitter coupled crystal oscillator is one of the configurations which best covers the requirements for driving QCM resonators under liquid conditions. The basic operation of the emitter coupled oscillator for series resonance condition can be understood with the basic structure showed in Fig. 5.14a. +Vcc R1

RC1

∞

R2

R1

+Vcc RC2

R2

RE1 RE2

RC2

∞ ZQ

R2 Option 1

a

R1

RC1

∞

∞ RE1

R1

R2 RE2 Option 2

ZQ

b

Fig. 5.14. Schematics for conceptual operation of the emitter coupled oscillator configuration for series resonance condition: a basic schema; and b resonator connection for series operation

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Two cascade inverting amplification stages provide, ideally, a total zero-phase shift and a loop gain given by the product of the gains of the two inverting stages: ALoop = A1 A2 ≈

Rc1 Rc 2 R E1 R E 2

(5.27)

Therefore, for ALoop = 1 with a total zero-phase shift of the loop the circuit could, in principle, oscillate. However, there is no special component in the circuit of Fig.5.14a with a phase-frequency response strong enough to determine the frequency at which the loop gain and phase conditions for oscillation are fulfilled. Thus, the oscillation of the circuit in Fig. 5.14a, if any, would strongly depend on all the components of the circuit and their specific phase-frequency response as well as the transconductance of the transistors, which is strongly dependent on the bias point and temperature. However, if a crystal resonator, which has very strong phase-frequency response around resonances, were strategically included in a well designed configuration based on the operational schema indicated in Fig. 5.14a, the resonator would control the oscillation conditions. For a series resonance condition, the resonator could be connected in two different places as indicated in Fig. 5.14b. In option 1 the sensor is in the feed-back path of the loop and controls the oscillation frequency around its zero-phase frequency. Some oscillators have been designed following this approach and successfully proved under liquid operation [58, 84]. However, the resonator connected in the circuit under option 1 has no electrode grounded and, therefore, does not comply with one of the requirements for QCM under liquid conditions, especially for electrochemical applications (see Chap. 13, Sect. 13.2.3). Option 2 is a more attractive approach since covers the previous requirement. In a well-designed crystal controlled oscillator, the gain of the oscillator must be maximized at the desired operating frequency and must be minimized at other frequencies [85]. For option 2 connexion the gain of the loop will be given, according to the simplified Eq. (5.27) by: ALoop = A1 A2 ≈

Rc1 Rc 2 Z Q RE 2

(5.28)

In the former equation, it has been assumed that at the oscillating frequency RE1//ZQ ≈ ZQ. On the contrary a RF-choke in series with RE1 can be included for decoupling RE1 at high frequencies. Therefore, the gain is maximized near the minimum impedance of the crystal around resonance and among resonances the resonance of minimum

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Antonio Arnau, Vittorio Ferrari, David Soares and Hubert Perrot

impedance is preferable. Additional methods to avoid spurious oscillations can be implemented in practical realizations. Assuming ideal behaviour of the circuit, the phase loop condition is also controlled around the zero-phase frequency of low impedance of the resonator according to the previous equation. For slight loads and small parallel capacitances the impedance of the quartz at the low impedance zerophase frequency is close to the motional resistance (Rm); under these conditions and making Rc1 = RE2 in Eq. (5.28), the loop gain of the system will be: ALoop = A1 A2 ≈

Rc 2 Rm

(5.29)

Thus, the loop gain condition for oscillation (ALoop = 1) will be covered for Rc2 = Rm and the measurement of Rc2, when the system starts the oscillation, provides a method to evaluate the damping of the sensor. A similar approach has been successfully implemented for liquids characterization and electrochemical applications [76-77, 83]. However, it was observed that the zero-phase condition was not the optimal condition for oscillation closer the MSRF. A capacitor in parallel with Rc1 allowed to control the phase conditions around -38º±3º which was theoretically demonstrated to be a more optimal phase condition in the usual range of loads [83]. This was also later confirmed in other works [37], although it is clear that the precise manner in which the oscillation frequency tracks better the MSRF depends on the combination of net parallel capacitance, loading and phase-angle [66]. On the other hand, it is clear that when a frequency different from the MSRF is monitored, the frequency and damping changes obtained are not suitable to be used in the resonance models described in Chap. 3, such as acoustic load concept and Kanazawa or Martin models. For that, it is advantageous to know the oscillation phase of the resonator and a measurement of the damping at this phase, because it could permit to recover the true MSRF and motional resistance [62, 65, 77]. The most critical components in the emitter coupled oscillator, and in general in any oscillator configuration, are the transistors; their temperature-dependent parameters, parasitic capacitances and non-linear amplifications characteristics are some inconveniences which need to be avoided. Fortunately the use of a voltage-controlled current source usually known as Operational Transconductance Amplifier (OTA) can be used almost as an ideal transistor. An OTA, like a transistor, has three terminals: a high-impedance input (base-B), a low impedance input/output (emitter-E), and the current output (collector-C). The OTA is self biased and bipolar; an ac voltage centred

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about zero provides a bipolar current centred about zero at the output. The transconductance can be adjusted with an external resistor allowing gainbandwidth trade-off. Moreover, the transconductance of the OTA is constant over a wide range of collector currents, which allows having a bandwidth almost independent of the gain; also the phase shift is constant over a wide frequency range. Due to these excellent characteristics the OTA has received the name of diamond transisitor (DT) [86, 87]. The common-E amplifier configuration, in analogy with the commonemitter configuration of a bipolar transistor, is depicted in Fig. 5.15a-b. Ideally, the voltage at the base-B is transferred to the emitter-E and is available there in low-impedance form. The current through the emitter-E is mirrored to the collector by a fixed ratio. The different operation in relation to the common bipolar transistor is that a positive B-E voltage causes a positive current flowing out of the collector-C; therefore, the common-E amplifier based on OTA is non-inverting in comparison with commonemitter bipolar transistor. Additionally, OTA maintains the transconductance constant over temperature. +Vcc R1

RL

uout

uout ui

Inverting Gain

ui

RB

B

C

Non-inverting Gain

OTA

RL

E R2

RE

a

RE

b

Fig. 5.15. Analogy between a common emitter configuration of a bipolar transistor and b common-E configuration of an OTA

For a non-bypassed emitter resistor (emitter degeneration), as shown in Fig. 5.15b, the equivalent transconductance of the OTA is given by [86]: g ′m =

1 rE + R E

(5.30)

where rE = 1/gm, being gm the transconductance of the OTA. Hence the gain of the common-E amplifier in Fig. 5.15b is given by:

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Antonio Arnau, Vittorio Ferrari, David Soares and Hubert Perrot

uo RL = g m′ R L = ui rE + R E

(5.31)

Emitter coupled oscillators have been designed with OTA following the option 1 crystal connexion mentioned in Fig. 5.14b [58, 84]. In Fig.5.16 the basic schema corresponding to the connexion showed as option 2 in Fig. 5.14b is depicted [37, 65]. A compact design is accomplished by using the OPA660 integrated circuit which includes a diamond buffer (DB) based on an abridged version of the OTA [87]. The non-inverting common-E amplifier based on the OTA and the buffer provide, ideally, a zerophase rotation of the loop and a loop gain given, according to Eq. (5.31), by: A = g m′ R4 =

R4 Z Q // R3 + rE

(

)

(5.32)

With the appropriate selection of R4 to cover wide dynamic load ranges, the loop gain is in general greater than the unity and the amplitude stabilization is provided by the antiparallel connected Schottky diodes D1 and D2. The R2-C1 high pass filter allowed the control of the total phase rotation of the loop, once the common-E amplifier gain has been selected with R4 and the OTA transconductance has been fixed by appropriate selection of the biasing point through R1. Finally, the phase rotation was selected at -40º in coincidence with previous works as mentioned above [77, 83]. The tank circuit L1-C2 is parallel tuned around the desired resonance of the sensor and strongly reduces the gain for undesired frequencies, avoiding parasitic oscillation. R1

R5

DT

B

C

D1 D2 R4

E

R2

C1

R3

C2

DB

R6 L1

ZQ

Fig. 5.16. Practical realization of an emitter coupled crystal oscillators with OTA. Adapted from [37]

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The basic circuit in Fig. 5.16 can include a buffer at the output for impedance matching with test instruments. The amplitude stabilization based on diodes can be substituted by an automatic gain control (AGC) system capable to provide, in principle, an evaluation of the damping of the resonator at the resonator phase oscillating condition [65]. The lever oscillator [62, 66, 78] and the active bridge oscillator [79, 80, 88] are two different approaches of a more general oscillator configuration, the so-called bridge oscillator. For a better understanding of the specific configurations, a brief summary of the simplified schematics and the main equations governing the operation of bridge-type oscillators is very advisable. Early references about bridge oscillators (BO) can be found elsewhere [89, 90]. For a deeper study of the first part of this section a careful reading of the Wessendorf works [79-80] is highly recommendable. Figure 5.17 is a schematic of a standard bridge oscillator (SBO) configuration. The loop-gain governing the oscillating condition is easily derived and found to be: ALoop = Av ( β p − β n )

(5.33)

where the positive feed-back ratio βp=R2 /R1+R2 and the negative feed-back ratio βn=ZQ /Rf+ZQ. For oscillation A = 1 and βp>βn. The phase-loop condition is mainly determined by the difference (βp-βn), assuming that the amplifier does not introduce additional phase shift. The amplifier must provide enough gain for sustaining oscillation; the excess loop-gain is limited under operation by the nonlinear characteristics of the active devices.

R1 + u+ R2

Av

Rf uZQ

Fig. 5.17. Schematic of a standard bridge oscillator (SBO) configuration

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For zero-phase resonator oscillating condition, the positive feed-back is designed to provide zero-phase shift. Under these conditions the resonator impedance ZQ must be real; therefore, the resonator low impedance zerophase frequency is excited by the circuit. It should be noticed that at the high impedance zero-phase frequency the negative feed-back ratio would increase near to βn ≈ 1, making improbable the oscillation at this frequency. Under small loading condition, at resonator zero-phase operation, ZQ can be approximated to the motional resistance Rm, but for heavy loads the parallel capacitance effects have an important contribution on the phasefrequency combination. This negative influence must be avoided by appropriate tuning out of the parallel capacitance with, for example, a parallel inductance. In fact, the operating loading range is drastically improved in this way; therefore, from now on, in this section, the impedance of the resonator will be substituted by the motional resistance Rm in the equations. How the oscillator affects the phase-frequency response of the sensor is important to know as well. For a steeper response, namely a higher Q factor, the system improves the frequency stability against changing conditions. The ratio of the slope of the phase-frequency response to the frequency, evaluated at the motional series resonant frequency ωs, is proportional to the resonator series Q factor. Then, by taking the quotient between the slopes of the phase-frequency responses of the loop-gain and the resonator at MSRF, a characteristic parameter MQ (Eq. 5.34) is obtained which gives information about the enhancement or worsening of the phase frequency response of the oscillator with regard to that of the resonator. The slope of the phase-frequency response of the oscillator at the MSRF is obtained by evaluating the phase of (βp-βn) with respect to ω; it is assumed that the phase shift of the amplifier in the narrow frequency range of operation around ωs is negligible. Therefore, MQ is found to be given by the following expression [79]: MQ =

dθ loop dω dθ res dω

= ω =ω s

( β n − 1) β n (β p − β n )

(5.34)

The former expression indicates that by appropriate selection of the parameter βp in relation to βn a magnitude of MQ greater than 1 is possible if βp-βn is small enough. However, the larger the MQ magnitude the larger Av is required for maintaining the oscillation condition, Av=1/(βp-βn). It is also noticed that for a fixed value of βp, MQ increases with βn; therefore, the Q factor of the oscillator is enhanced in relation to the Q factor of the resonator for increasing values of Rm, which is recommendable for heavy loads.

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For a fixed value of βp=1 (half-bridge oscillator) the Q of the oscillator is limited by the Q of the resonator. For a practical application of the previous equations, the schematic realization of a standard bridge crystal oscillator is depicted in Fig.5.18a, whose small signal equivalent circuit is represented in Fig.5.18b. In the small signal equivalent circuit, the following considerations were taken into account: the emitter-follower formed by transistor Q3 operates as an ideal follower; the parallel circuit Cc-Lc is resonant at the oscillating frequency and drastically reduces the loop-gain at undesired frequencies; L1 is a RF-choke that provides equal biasing to transistors Q1 and Q2; the coil L0 tunes out the parallel capacitance and then the remaining impedance of the resonator at the resonator zero-phase frequency is Rm. Additional considerations when deriving the following equations will be common throughout this section: high β transistors are considered with identical characteristics and then the currents through the bases can be neglected in relation to collector or emitter currents, as well as in relation to currents through a path having a node in a base. The following analysis is similar to others which will be studied later on in this section and then it will be useful for understanding and avoiding repetition in the future. +Vcc

∞

R1 u+ R2

a

Rc

Q3

Cc

Lc

uo Q1

Rf

Q2 L1

∞ ZQ

Rc

uo R1

uLo

R2

u+ +

ube1 rπ -

ube1 ube2 rE rE

Rf

rπ

uRm + ube2 -

b

Fig. 5.18. Schematic of a practical realization of a standard bridge crystal oscillator configuration: a real schematics, and b small signal equivalent circuit.

Under the abovementioned simplifications, the following equations govern the operation of the small signal equivalent circuit in Fig.5.18b. u be1 u be 2 + ≈0 rE rE

(5.35a)

u + − u − = u be1 − u be 2

(5.35b)

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Antonio Arnau, Vittorio Ferrari, David Soares and Hubert Perrot

⎛u uo u o ≈ −⎜ be 2 + ⎜ rE R f + Rm ⎝

⎞ ⎟ Rc ⎟ ⎠

(5.35c)

u+ ≈

R2 uo R1 + R2

(5.35d)

u− ≈

Rm uo R f + Rm

(5.35e)

By combining the former equations the direct gain Av is obtained: Av =

R f + Rm uo R = c u + − u − 2rE R f + Rm + Rc

(5.36)

By substituting u+ and u- given by Eqs. (5.35d-e) in Eq. (5.36) the following expression for the loop-gain A is obtained:

A=

Rc 2rE

1 Rc 1+ R f + Rm

⎛ R2 Rm ⎜ − ⎜ R1 + R2 R f + Rm ⎝

⎞ ⎟ = 1∠0º ⎟ ⎠

(5.37)

The former equation allows making the following considerations for SBO under liquid loading conditions: 1. Rf must be selected is such a way to appropriately cover the dynamic range of loads expected in the experiment. Then, according to the maximum value of βn, the magnitude of βp must be selected, but keeping in mind that a very low magnitude of MQ (Eq. 5.34) will require a high value of Av whose limited value is Rc/2rE. 2. For heavy loads, the maximum direct gain Av = Rc/2rE must be chosen high enough for exciting oscillation at steady-state. 3. The excess of gain under operating conditions will be limited by nonlinearities of the amplifiers; in this case the intrinsic emitter resistance rE which is strongly dependent on the signal amplitude and bias point. For SBO the bias point of the active devices is similar and the amplitudes of the signals at the bases, which are forced externally by the feed-back paths βp and βn, are similar as well for high values of MQ. Therefore, changes in the value of rE can only occur with drastic changes in the oscillating amplitude or by changing the bias point of the active devices under controlled conditions, for instance by automatic gain control

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(AGC) systems. Because strong changes in amplitude oscillation induce poor frequency stability and noise, it is more appropriate to implement AGC systems to maintain the gain-excess at minimum and to drive the circuit with small amplitude of the oscillation signal; this improves the linearity, the signal noise and the amplitude and frequency stability. Additionally, these AGC circuits allow monitoring of the resonator losses, although increase the complexity of the system. The lever oscillator, whose practical realization is depicted in Fig. 5.19, is in fact a half-bridge oscillator. The loop-gain governing the circuit is, according to Eq. (5.37): Rc 2rE

1 Rc 1+ R f + Rm

⎛ ⎜1 − R m ⎜ R f + Rm ⎝

⎞ ⎟ = 1∠0º ⎟ ⎠

(5.38)

As mentioned above, βp=1 and the oscillation phase condition is controlled by the negative feed-back path βn which increases with Rm; this also increases MQ which is limited, in this case, to 1. Therefore, in this configuration the Q factor of the oscillator never improves the Q factor of the resonator. The direct gain Av is also “leveraged” by the value of Rm. Effectively, Av increases with Rm; however, this increment is only significant for values of Rf and Rc of similar magnitude to Rm. Even with this increasing effect on the magnitude of Av, strong changes of rE are necessary when resonator losses change along a relative wide loading range. As mentioned, if an AGC is not used to stabilize the oscillating amplitude by appropriate change of the bias point, the oscillator suffers of stability problems, mainly at high frequencies of oscillation [37]. On the other hand, the oscillator ideally works at resonator zero-phase condition; however, when parasitic effects are taken into account the resonator phase at oscillating condition deviates to ≈-6º under usual liquid loading conditions, (Rm>200Ω) [62]. Furthermore, for lower impedances the loop gain is poorly controlled by the resonator and the phase condition is determined by parasitic effects. The circuit has been successfully proven for in-liquid operation with AGC and it has been noticed that an accurate tuning out of the parallel capacitance is necessary for a proper operation in the typical range of loads [66]. It was also found that when non-perfect parallel capacitance compensation is achieved, a negative resonator phase for oscillating condition fits better the MSRF in a wider range of loads than the zero-phase condition, in coincidence with other in-parallel works [37, 76-77, 83].

156

Antonio Arnau, Vittorio Ferrari, David Soares and Hubert Perrot +Vcc ∞

C c Lc

Rc

Q3

Rf

uo ≈uo

L1

u+ Q1

R2

∞

uQ2

∞ ZQ

AGC

uRm

Lo

RE

Fig. 5.19. Practical realization of a lever oscillator configuration

The active bridge oscillator (ABO)[79-80, 88], whose simplified schematics of a practical realization is depicted in Fig 5.20a, suggests, at first sight, to be a standard half-bridge oscillator; however, the substitution of the current source by a lower emitter impedance RE “degenerates” the halfbridge configuration in a full-bridge oscillator with βp<1 and strongly dependent on the intrinsic emitter resistance rE. Effectively, the analysis of the small signal equivalent circuit in Fig. 5.20b provides the following equations governing the operation of the system: u be1 u be 2 u E + ≈ rE rE RE

(5.39a)

u + − u − = u be1 − u be 2

(5.39b)

uo ≈ −

u− ≈

Rc u be 2 rE

Rm u+ Rm + R f

u E = u − − u be 2

where Rc =R2//R3.

(5.39c)

(5.39d) (5.39e)

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+Vcc ∞

uo R4 u+ C1 RL

a

R3

u+ Rc=R2//R3 u-

Rf

Q1 RE ∞

Lc

∞

∞

≈uo R5

Cc

R2

Q3

Q2 ZQ

ube1 ube2 rE rE

u+≈uo ∞

+

ube1

rπ

-

Lo

uE

Rf

uo u+

ube2 -

rπ

Rm

RE

R1

b

Fig. 5.20. Schematics of the active bridge oscillator configuration: a practical realization, and b small signal equivalent circuit

By combining the former equations the following loop-gain equation is obtained: ALoop = Av ( β p − β n ) = 1∠0º

(5.40)

where Av =

Rc Rm RE ;βp = ; βn = rE (1 + β p ) RE + re R f + Rm

(5.41)

In the ABO the bias points of transistors Q1 and Q2 are similar as in the case of SBO (R4 ≈Rf in Fig. 5.20a); however under oscillation conditions, the signal amplitudes at the bases are completely different, the base of Q1 has amplitude of oscillation similar to uo while the base of Q2 is operating at a fraction of the output signal amplitude uo. This makes the magnitudes of Av and βp to be a strong function of rE. For heavy loads, high Rm, Rf must be selected appropriately to cover the loading range, for instance Rf = 2500 Ω; βp must be chosen smaller than 1 with RE of adequate small value to take advantage of having MQ>1 for high values of Rm. Once Rf, RE and a representative value of Rm have been determined, the value of rE for A=1 is calculated and the corresponding bias point provided. Under these conditions, an increase of Rm increases u- and reduces the amplitude of oscillation while increases βn; on the other part, the new oscillating condition requires a smaller value of rE to sustain the loop-gain A= 1 which, in turn, increases βp allowing a certain degree of adaptation to a wider range of loads. The relative increase of βn and βp also increases MQ allowing a better performance of the frequency stability. The

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amplitude stability is improved as well in comparison with a SBO since relative small changes in the amplitude of the oscillation signal are enough to reach the new value of rE for sustaining oscillation (loop-gain condition). Therefore, as Rm increases the amplitude of oscillation decreases smoothly and with almost linear dependence with the increase of resonator loss [80]. This approach has been successfully proven under heavy loads (Rm as high as 3500 Ω for 5MHz AT cut quartz). The oscillating resonator phase condition under real experiment separate from the zero-phase condition around -10º in the mentioned range of loads (200
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capacitance at different frequencies (a parallel inductance only tunes out the capacitance at the selected resonant frequency). The principle of the balanced bridge oscillator is depicted in Fig. 5.21. Two ideally equal branches form a differential configuration; the input signal u1 is transferred to the emitters of Q1 and Q2 and the emitter currents are voltage-converted at the collectors and fed-back to the input after differential amplification. For a perfect compensation (Cv =C0*), the ideal loop-gain of the oscillator circuit is: A=

Rc AD = 1∠0º Zm

(5.42)

Assuming no phase shift of the differential amplifier, the zero-phase of the resonator motional branch governs the oscillation phase condition of the circuit. The minimum gain AD for oscillation would be AD = Rm/Rc; therefore, by appropriate selection of Rc a wide loading range could be driven with this type of oscillator. For a better operation, minimizing nonlinearities of the active devices, an AGC can be implemented which, in turn, provides the information about resonator losses. +Vcc Rc Rc Cc

Cc

Lc

Lc

ZQ

1

∞

u1 Ic

AGC

AD

Q2

Q1 ∞

uRm

Ic

Cv

Fig. 5.21. Schematics of the operation principle of a balanced bridge oscillator configuration

As in all oscillator configurations the appropriate component selection is a crucial matter for an accurate zero-phase loop condition at the desired phase of the resonator. In this configuration, the use of “diamond transistors” (OTA) [87] can greatly improve the performance of the system.

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Figure 5.22 presents a circuit proposal for a balanced bridge oscillator using diamond transistors and an AGC system implemented with a controlled gain multiplier based on AD835. The phase-frequency responses of the OTA and the multiplier are almost ideal for a wide frequency range. The operation of the circuit is very similar to the one previously described. The input voltage is transferred to the emitters and the emitter currents are non-inverted voltage-converted to the collectors. The voltages at the collectors are differentially amplified with one of the high input impedance differential amplifiers of the AD835. The output signal is level controlled with a multiplier following a typical AGC schema and fed-back to the input. The adequate selection of the VREF and Rc allows driving a wide range of loads. A small deviation on the loop-phase from the resonator zero-phase is provided by the emitter intrinsic resistance rE (see Eq. 5.30); with a value as small as 8Ω would provide a deviation of about 0.28º for a compensation capacitance of 10pF at 10MHz. Also the parallel circuit Lc-Cc is included to drastically reduce the loop-gain for undesired frequencies. Under these considerations the ideal loop-gain equation of the circuit in Fig. 5.22 is given by: A=

Rc AD k = 1∠0º Zm

(5.43)

where k is the gain of the multiplier.

DT1 Lc Rc

Cc

ZQ

AD

AD835

k AD

Rc u Zm 1

u1 Cv

1

Cc

Rc

k

Lc DT2 uRm

∫

VRef

Fig. 5.22. Proposal of an interface circuit based on a balanced bridge oscillator using diamond transistors-OTA

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The balanced bridge oscillator uses a capacitor for implementing the parallel capacitance compensation; this type of capacitance compensation can be extended to other type of oscillator configurations by using the circuit shown in Fig. 5.23 [91, 92]. The major problem of this technique is the lack of ideality of the transformers and the selection of the capacitance Cv for compensating the parallel capacitance, i.e., the calibration of the parallel capacitance compensation. QCM

A Compensating Capacitor

Level Detector

Frequency Counter

R

Fig. 5.23. Schematic circuit for parallel capacitance compensation based on a three-winding transformer

A comprehensive overview of different oscillator configurations proposed for driven QCM sensors under heavy loading condition has been presented. As it can be noticed emitter coupled oscillators designed, in principle, for a zero-phase oscillating condition of the resonator, were finally adjusted to a resonator phase of around -40º. The frequency shift and damping monitoring, together with the knowledge of the phase at which the resonator is operating in the oscillator, allow recovering the true MSRF and motional resistance if previous characterization of the unperturbed resonator and careful calibration of the oscillator circuit are made [65]. For that, expensive instrumentation, like impedance or network analyzers, could be necessary. Thus, the advantage of the simplicity of oscillator circuits is, after all, broken. Bridge and active bridge oscillators are simple and accurate enough to be successfully used to drive heavy loaded resonators in a wide dynamic range of loads, but the accuracy of the tracking frequency and resonator loss information depends strongly on the ideal characteristics of the components and, even more importantly, on the fidelity with which the parallel capacitance is tuned out with a parallel inductance, which is not an easy task and is only valid at a certain frequency not known “a priori”. Apparently balanced bridge oscillators have been proposed as excellent alternative and, appropriately designed, could compensate the parallel capacitance

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in a wide range of frequencies and then tracking the MSRF and the resonator loss Rm more accurately than the rest of the configurations. However, until now no oscillator configuration has been described able to accurately drive the resonator at MSRF on a wide range of loads. For this goal oscillator-like operating circuits based on phase locked loop techniques (next section) could be a good solution. Fortunately, there are many applications in which oscillators, although not rigorously working at MSRF, are the best option. Effectively, for applications in which the resonator loss and the parallel capacitance are maintained relatively constant during experiment, the frequency shift of the resonator sensor, which is the parameter of interest, is practically independent on the resonator phase under oscillating conditions. These applications as in most of piezoelectric biosensors and in some electrochemical applications like ac-electrogravimetry (see Chap.11), among others, can be appropriately monitored with oscillators described in this section. The relative simplicity and the small size of these circuits allow a very good control of the environmental conditions, what can drastically reduce the noise and improve the stability. On the other hand, when the resonator losses and parallel capacitance change during experiment, the deviations in the measurements should be carefully evaluated to be conscious of the error propagation in the interpretation of results. In general, oscillators are not a good interface in these cases. Passive measurements based on network or impedance analyzers could be used or, alternatively, simpler and cheaper systems which operate similarly to oscillators but implement a passive interrogation of the sensor, allowing an easy and accurate calibration of the system. These configurations are based on lock-in techniques, and will be described in the next section. 5.4.4 Interface Systems for QCM Sensors Based on Lock-in Techniques

These techniques aim at the simplicity of oscillators while overcoming their limitations. From the previous analysis on oscillators, it can be noticed that their limitations come from the fact that the oscillation depends on a loop condition in which the sensor is integrated, then the resonator phase condition for oscillation is the one necessary to compensate the phase response of the rest of the components of the loop. Any no ideality in the phase-frequency response of the external circuit, mainly due to active devices, is transferred to the sensor which changes the phase and then the frequency in order to comply with the loop oscillation condition. It has

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been shown how a good selection of components and a zero-phase condition of the external circuit to the sensor under parallel capacitance compensation can provide good results. Then, the problem of dependence of the no idealities of the active components can be solved if the resonator is passively interrogated with an external oscillator whose frequency locks the MSRF. To this aim two techniques have been proposed: a) the oscillator locks the zero-phase frequency of the resonators under parallel capacitance compensation conditions and b) the oscillator finds or locks the maximum conductance frequency of the sensor. Next some different approaches for these techniques will be introduced. Phase-Locked Loop Techniques with Parallel Capacitance Compensation

In these techniques an external voltage-controlled oscillator locks at the zero-phase frequency of the motional branch of the sensor. For that aim this technique must simultaneously accomplish accurate parallel capacitance compensation which should be controlled by an easy calibration procedure. A circuit implementing a phase locked loop (PLL) technique with parallel capacitance compensation is shown in Fig. 5.24 [67]. SENSOR CIRCUIT

Capacitance Compensation Circuit Cv Jp 1

2

AD8307

SENSOR

fexp

L1

OP27

Rv

R i =25Ω u i 2u i Ri

A1

uA

A CLC449

L2 AD8307 IC1

A2

Ra

A'

uB

IC2

B CLC449

B'

LPFs

20 MHz

uϕ

DN

MC12061 10 MHz

DA1

MCH12140

BUFFER

50Ω

OP27

UP

PFD

MCE1651

Rv

u Rm

DA2

Filter

A

vcxo

Filter

A

OSC

OP27

IF 20 MHz

Fig. 5.24. Circuit for parallel capacitance compensation based on a phase-locked loop technique

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Similar previous approaches do not maintain one face of the resonator connected to real ground, and more complex calibration procedures are required to assure that the system locks at MSRF and that the parallel capacitance is accurately compensated [93-95]. On the contrary, the system in Fig. 5.24 allows a very easy and effective calibration without the need of additional expensive instrumentation. Effectively, when the sensor and the capacitance compensation system are removed, (see Fig. 5.24), two ideally equal input branches forming a differential system remain. A very simple calibration procedure can be performed by applying a testing signal to the input; it can be easily made by opening the loop at the output of the integrator filter and connecting the control voltage of the VCO to ground. The system is calibrated when a stable dc voltage is maintained at the output of the integrator (ideally a zero voltage at the input). Because a phasefrequency-detector (PFD) has been used, which provides null voltage at the output uφ for equal phases of the signals at its inputs, it assures that the loop will lock at the frequency for which the phases of the signals at points A and B are equal. After the frequency loop has been calibrated, the sensor and the capacitance compensation circuit (for example a variable capacitor) are incorporated in the circuit. Then, a testing signal of appropriate frequency for which the resonator has no motional behavior (only behaves like the parallel capacitance) is applied to the input, for instance a signal of low frequency far from resonance or at even harmonics of the fundamental resonance frequency, in this case 20MHz. Under these conditions a stable dc voltage signal will be maintained at the integrator output when the parallel capacitance is accurately compensated by trimming the variable capacitor. If no changes in the parallel capacitance occur during experiment, the MSRF will be locked continuously by the system when closing the loop with the 10MHz VCO. In the circuit shown in Fig. 5.24, A1 and A2 should be wide bandwidth operational amplifiers; current feedback is recommended for the best phase-frequency response at different gains. IC1 and IC2 are comparators used to increase de slope of the signals at the input of the PFD to improve the phase detection. DA1 and IF form a charge pump circuit for controlling the VCO. L1 and L2 are logarithmic amplifiers which permit, with the differential amplifier DA2, an evaluation of the resonator loss Rm without the need of and AGC to maintain constant the amplitude of the signal at the input 2ui. Equations of the circuit are obtained similarly to the improved approach which is described next and they will be introduced there to avoid repetition. An improved version of the system previously described is depicted in Fig. 5.25 [96].

5 Interface Electronic Systems for AT QCM Sensors 38.88 MHz

165

OSC VH

MAX038

f out VL

L1

R1 VHLα

A1

AD835

VHLα

VCA

R2 K=1

LMH6702

Cc

Rv

SENSOR

C*0

CT LT

HPF 36-40 MHz

HPF

VC

LPFs

DA2

Vϕ L

UP

IC1

OP27

PFD IC2

DA1

MAX9383

V1H

V2H

LPFs DN

IC2

MAX9601

RT

OP27

PFD MAX9601

V2

UP

IC1

LPF

VRm

L2 AD8307

9-10 MHz

A2

VCA

KC

LPF V1

OP27 DA2

V2L

Rv

I1 AD8307

V1L

VHL

OP27

vco

Vϕ H

DN

MAX9383 OP27

I2

Fig. 5.25. Schematics of a practical realization of an interface electronic system for QCM series operation, with automatic parallel capacitance compensation, based on phase-locked loop techniques

The aim of the system is to perform a continuous parallel capacitance compensation which is very interesting as it permits the monitoring of an additional important parameter in some applications. The concept behind the design was introduced elsewhere [97, 98], that is, by simultaneous exciting the quartz crystal at two frequencies, and assuming linear behavior of the resonator at the driving level of the signals, the compensation of the parallel capacitance C0* is automatically and simultaneously made with the locking of the MSRF of the sensor. The new approach follows the concept previously described. In the present case two PLLs are used for both frequency tracking and parallel capacitance compensation. The PLL in charge of tracking the MSFR of the sensor is based on a phase-frequency detector (PFD) instead of a multiplier as it is the case in references [97] and [98]. In this configuration the non inverter amplifier in charge of driving the sensor has, ideally, the same response at different frequencies inside its bandwidth of linear operation, since only a resistance, Rv, is included in the feedback loop unlike [97, 98], where a resistance in parallel with a capacitance is proposed to obtain a 90º

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phase-shift of the high frequency component of the composed driving signal, necessary for a proper operation of the multiplier as a phase detector (Fig.3 in [97]). Therefore, an easier and, in principle, a more accurate calibration of the PLL can be performed in this new configuration by following the procedure introduced above [67]. The parallel capacitance compensation in the circuit shown in Fig. 5.25 is made following the same concept described in Fig. 5.24, based on phase detection, but the testing frequency is selected here at 4 times the fundamental frequency. The magnitudes governing the operation of the system are the phaseshifts between the signals V1 and V2 at the two frequencies, fH and fL, corresponding in this case to the signal of fix frequency equal to 4-times the fundamental resonant frequency of the sensor and to the signal which sweeps the frequency around the series resonant frequency of the sensor, respectively. The phase-shift at the lower frequencies around the series resonant frequency of the sensor controls the PLL in charge of the frequency tracking, while the phase-shift at the auxiliary higher frequency controls the PLL in charge of the parallel capacitance compensation. Therefore, the equation governing the control of the system is:

(

)

V2 HL = 1 + Rv YT + jωRv (C 0* − C c ) V1HL

(5.44)

where V1HL = VHL α; Cc = [KC /α – 1]; α=R1/(R1+R2); and YT = [jωLT + RT + 1/jωCT]-1. The subindex HL in the previous equations means that the voltage waveform considered is the sum of the two sinusoidal signals VH , with fix frequency fH generated by the auxiliary oscillator whose frequency is around four-times the fundamental resonant frequency of the sensor, and VL with frequency fL generated by the VCO around the series resonant frequency of the sensor. At the auxiliary frequency, fH , where only capacitive behaviour of the sensor is expected, Eq. (5.44) is reduced to:

(

)

V2 H = 1 + jωRv (C 0* − C c ) V1H

(5.45)

and the phase shift between the signals V2H and V1H is given by: Φ(V2H , V1H ) = arctan[2πfH Rv Cr]

(5.46)

where Cr = C0*- Cc is the residual uncompensated parallel capacitance. As it can be noticed when Cr = 0 the phase shift is zero and the differential amplifier DA2 gives zero voltage at its output that makes the integrator I2 to maintain a continuously stable dc voltage at its output. This is the only stable condition for the loop out of saturation. In a different condition, Cr ≠ 0, the amplifier DA2 gives a signal which is integrated by the integrator

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I2 until a new stable condition is reached for Cr = 0. The output VC of the integrator I2 can be used to monitor the changes in the parallel capacitance of the sensor and also for a continuous monitoring of its magnitude. The sensitivity of the capacitance compensation is limited by the sensitivity of the phase detector. Assuming a sensitivity of the phase detector of 0.1º, for the frequency fH around 40 MHz and Rv = 237Ω, the uncompensated residual capacitance obtained by solving Cr from Eq. (5.46) is near 30fF, which is enough for most cases. At the frequency fL, and assuming that the parallel capacitance has been compensated (Cr = 0), (Eq. 5.44) is reduced to: V2 L = (1 + Rv GT + jRv BT )V1L

(5.47)

where GT = RT /( RT2 + X T2 ) ; BT = − X T /( RT2 + X T2 ) and XT = ωLT – 1/ωCT As it can be noticed from Eq. (5.47) signals V2L and V1L will be in phase when XT is null and this only happens at series resonant frequency, fs = (2πLTCT)-1/2. At this frequency the differential amplifier DA1 gives a zero voltage at its output that makes the integrator I1 to maintain a continuously stable dc voltage at its output. This is the only stable condition for the loop out of saturation. If the series resonant frequency changes a phase shift between the signals will arise at the locked frequency and the amplifier DA1 will give a signal that will be integrated by the integrator I1 until a new stable condition is reached for the locking frequency at the new frequency fs. It can be noticed that the voltage at the input of the VCO can be used as a direct measurement of the shift in the series resonant frequency if the frequency range of the VCO is narrow enough to have good frequency/voltage sensitivity. A narrow bandwidth of the VCO can reduce the dynamic range of the PLL, however this can be avoided with the system described elsewhere [45]. At the locking series resonant frequency Eq. (5.47) is reduced to: ⎡ R ⎤ V 2 L = ⎢1 + v ⎥ V1L ⎣ RT ⎦

(5.48)

and by measuring the voltage levels V2L and V1L the magnitude of the motional resistance RT is obtained. In the circuit in Fig. 5.25 the VCA connected to the output of the resistive divider formed by R1 and R2, is only included for symmetry. A different parallel compensation concept is used at low frequencies very far from resonance, for instance 50 KHz, in references [97-98], where the parallel compensation detects a null voltage instead of a phase condition; the schema is depicted in Fig. 5.26 where the low frequency component VL is now in charge of the parallel capacitance compensation.

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Antonio Arnau, Vittorio Ferrari, David Soares and Hubert Perrot OSC

50 KHz

VL VH KC VHL

ZQ

VCA

AD835

R1

Rv

Cc

A1

VHLα

VHLα

V2HL

LPF

LMH6702

V2L DA2

50 KHz

R2

AD835

A2

LPF

Rv

PHASE SHIFTER 90º V1L

x y

I2 M1

V1L

V1HL

Fig. 5.26. Parallel capacitance compensation technique using null-voltage detection instead of null-phase detection

The signal difference V2L-V1L given by Eq. (5.49) (see Eq. 5.45) is 90º phase shifted and applied to one of the inputs of the multiplier. The signal V1L is applied to the other input; therefore, the multiplier will provide 0 voltage at the integrator input only when Cr = 0, namely when the signal V2L-V1L is made null. V2 L − V1L = jωRv (C 0* − C c ) V1L

(5.49)

By applying the same concept of simultaneously exciting the sensor at different frequencies, different harmonics can be monitored at the same time. Recently a very nice and simple design has been introduced for a dual-harmonic oscillator-like operating circuit [99] (Fig. 5.27). VA C V1

V3

x

R

y M1

Quartz Sensor

V13 Cc

x C

y M3

R

∫ I1

∫ I3

VCO1

f1

VCO3

f3

VB

Fig. 5.27. Dual-harmonic interface system for QCM sensors based on phaselocked loop technique

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The schematic depicted in Fig. 5.27 uses a phase detection based on multipliers, like in references [97] and [98], and includes parallel capacitance compensation as well, although not automatic in this case for simplicity purposes. The key concept of the PLL techniques is the accurate compensation of the parallel capacitance. An alternative technique is to lock at the frequency at which the conductance of the sensor reaches a maximum, which coincides with the MSRF in most of cases [18]. Because the conductance of the sensor is not influenced by the parallel capacitance, this technique avoids the necessity of its compensation. Lock-in Techniques at Maximum Conductance Frequency

The schematic concept of these techniques is depicted in Fig. 5.28. The sensor is passively interrogated by a signal whose frequency sweeps the resonance frequency range. The current through the sensor is voltageconverted and multiplied with the interrogating signal. The low frequency component, VG, at the output of the multiplier is proportional to the conductance. Conductance follower um

VCO

u1

LPF

VG

IQ

I-U Converter

u2=kcIQ=kcYQu1

IQ

IQ=u1/ZQ

ZQ

Fig. 5.28. Schematics of the basic operational concept of the lock-in techniques at maximum conductance

The problem now is how to detect or lock at the maximum of this signal. Some efforts have been addressed to this aim. Nakamoto and Kobayashi [100] proposed a circuit based on this concept which used a peak detector for detecting the maximum of this signal; at the peak detection additional digital circuitry allowed the acquisition of the corresponding voltage at the input of the VCO and the value of the voltage VGmax. This sweeping was repeated cyclically around each second. The acquired values of the VCO input voltage and the conductance voltage VG at the peak give information about the frequency and resonator loss. In order to obtain

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enough frequency resolution, a VCXO was used which greatly limits the dynamic range of the system (see next section). The design worked fine for high Q sensors but the flatness of the conductance peak for low Q resonators made the detection of the conductance peak difficult and the accuracy of the measurements to decrease. Recently, a method for continuously locking at the maximum conductance frequency has been introduced by Jakoby et al [101]. In this method the block depicted in Fig. 5.28 is included inside another loop as indicated in Fig. 5.29. According to this schema, the output signal of the VCO is frequency modulated by two signals: one coming from the integrator, uc, and another auxiliary low frequency signal coming from an external function generator, uaux, for example a sinusoidal signal. The signal from the integrator determines the central frequency, namely the carrier at the output of the VCO in Fig. 5.28, and the low frequency signal provides a carrier frequency modulation with a deviation which depends on the signal amplitude. Therefore, this system implements a frequency modulation whose carrier frequency shifts until the maximum conductance frequency is locked. Effectively, the modulating voltage of the external generator is connected to one input of the multiplier of the external loop M2; this modulating voltage provides a frequency deviation around the carrier which produces a corresponding change in the voltage VG through the conductance-frequency response. This ac voltage reduces its amplitude to 0 at the flat slope of the point of maximum conductance, providing zero voltage at the output of the multiplier M2 (input of the integrator) and stabilizing the frequency of the carrier at the maximum conductance frequency.

uaux um

Conductance Follower

VG

LPF M2

uc

∫

Fig. 5.29. Automatic lock-in technique at maximum conductance

In this technique the signal that interrogates the sensor is a frequency modulated signal, therefore the information on the maximum conductance frequency is provided through the output voltage of the integrator under locking conditions; in this case the frequency resolution depends on the

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resolution in measuring the voltage, which depends on the frequencyvoltage response of the VCO; for high resolution, narrow frequency sweeping ranges must be covered with wide voltage ranges, then a VCXO can be a good solution, although the dynamic frequency range is drastically reduced. 5.4.5 Interface Circuits for Fast QCM Applications

Fast QCM can be defined as a QCM in which the frequency changes to be monitored can occur at a very high rate, for instance 1000 times per second. It means that the characteristic resonant frequency of the resonator sensor is changing at rate of 1000 times per second. At this moment no interface circuit, with exception of oscillators, is able to monitor the frequency of the resonator sensor at such rate; PLL techniques, as those mentioned above, could be an alternative solution providing a voltage directly proportional to the frequency change. However, to have enough F-V sensitivity very narrow locking ranges to be covered with wide voltage ranges would be necessary; however, it would reduce the dynamic frequency range of the system and therefore, special PLL techniques have to be developed for this aim. The fact that oscillators are able to track the high-rate frequency shifts of the resonator does not solve the problem, because these very quickly changing frequency shifts are not easy to measure and special techniques must be developed for monitoring them. A system proposed for solving this problem in ac-electrogravimetry applications (see Chap.13) has been introduced elsewhere [45, 102]. A brief description of the problems and of the circuit concept to overcome them will be next introduced. ac-electrogravimetry is based on an electrochemical quartz crystal microbalance (EQCM) used in dynamic regime. In ac-electrogravimetry the output signal of the fast QCM system can be considered like a frequency modulated signal, in which the modulating signal is the superimposed sinusoidal voltage applied to the electrochemical cell. It is important to notice that the QCM resonant frequencies are in the megahertz range meanwhile the perturbation signal applied has a frequency ranged between 1 mHz and 1 KHz. Experimental frequency shifts, in relation to the central frequency of the QCM signal, in the range of 10 to 50 Hz are typically found when the sinusoidal voltage perturbation is applied to the electrochemical cell. For a reliable electrogravimetric transfer function (EGTF) the ratio Δm/ΔE must be obtained very accurately. The problem is that Δm is related to the frequency shifts in the QCM, which must be obtained with high resolution, around 0.1-0.5 Hz, and can occur at high rate, around

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1 KHz. It is well known that the measurement of the frequency of a signal, which is around 10 MHz, with an accuracy of 0.1 Hz and with a time gate of less than 0.1 ms is a common problem in frequency measurements. Alternatively, since we are mainly interested in an accurate tracking of the frequency shifts and not necessarily in the absolute frequency value, an accurate frequency voltage converter could provide a voltage signal directly related to the frequency shifts which occur during the electrochemical experiments. This voltage can be used in conjunction with the perturbation voltage at the inputs of a transfer function analyser (TFA), for example Solartron 1254, in order to obtain the EGTF. Therefore, the goal is to obtain a frequency-voltage converter sensitive enough and with very low amplitude and phase distortion in the frequency range from 1 mHz to 1 KHz of the perturbation signal. As it is well-known, a PLL based system is ideal for frequency to voltage (F-V) conversion, but in this case a very high F-V sensitivity, non common in typical frequency demodulation circuits, and a quick frequency tracking are necessary. Low frequency PLLs are currently used in acelectrogravimetry, however, in these systems the low pass frequency filter requirements make the system response to be slow and the amplitude and phase distortion of the demodulated signal occur for modulating frequencies much slower than 1 KHz [44] A new PLL based system has been proposed whose block diagram is shown in Fig. 5.30 [45, 102].

EQCM

TFA

Main Mixer Signal Conditioning

Loop Filter

fT fR

fM

fVCXO

Main Loop Filter

Secondary Loop FPGA

VCXO

fNCO

Secondary Mixer

NCO

Fig. 5.30. Schematics of an interface electronic system for fast QCM operation. Adapted from [45]

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In the circuit a main mixer working as a phase detector, a low pass filter followed by a signal conditioning circuit and a voltage controlled crystal oscillator (VCXO) are included in the main loop of the PLL. A VCXO, with a very high frequency-voltage gain conversion (V/F), instead of a simple voltage control oscillator (VCO), is used in the main loop to improve the F-V sensitivity; however it reduces the dynamic range and the PLL lock range. This is a new problem as it is not known “a priori” the central frequency of the QCM system (fT). A digitally controlled feed-forward correction based on a numerically controlled oscillator (NCO) is used to overcome this problem. The frequency of the NCO (fNCO) is selected in such a way that the sum of the frequencies coming from the QCM system and from the NCO (fT + fNCO) falls within the dynamic range of the VCXO and in the lock range of the PLL; with this purpose the secondary mixer in the loop is followed by a filter which selects the signal whose frequency is the difference between the frequency of the VCXO and the frequency of the NCO (fR = fVCXO-fNCO). A purposed algorithm implemented in a field programmable gate array (FPGA) integrated circuit performs a rough measurement of the central frequency of the QCM system during a time gate of 100ms, with a maximum error of 10 Hz, and controls the frequency of the NCO in the appropriate way. The FPGA only changes the NCO frequency when the deviation of the central frequency of the QCM system leaves the hold range of the PLL main loop. Thus, the NCO is used to place the lock range of the PLL around the appropriate frequency and, once the PLL is locked, to expand the PLL hold range. In other words the NCO performs the coarse tuning meanwhile the main loop performs the fine tracking of the PLL. The proposed system has been successfully implemented with sensitivity higher than 15mV/Hz and with flat amplitude-frequency and phasefrequency responses in a bandwidth bigger than 2,5KHz. It means that the system can follow frequency changes at rate as high as 2,5KHz with negligible amplitude and phase distortion. The system has been also implemented successfully in the characterization of electroactive polymers demonstrating that high rate responses in conductive polymers are possible [103]. 5.5 Conclusions

The problem associated with QCRS interface circuits reveals that an optimal device for sensor characterization does not exist at present. The systems that better characterize the sensor are those which passively interrogate the quartz resonator so that the sensor is measured in isolation.

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However, these systems based on impedance analyzers or decay/impulse excitation methods include expensive equipment and do not fulfill the autonomy normally required for portable applications and are not appropriate for fast QCM measurements. On the other hand, those systems which fulfill the simplicity, autonomy and low price requirements appropriate for sensor applications have less accuracy in the determination of sensor parameters; mainly, because the external circuitry influences the sensor response which is not measured in isolation. This problem becomes more critical due to the difficulty of calibrating the response of the external circuitry in typical oscillators. Thus, the monitored parameters can not be accurately associated to a certain phase of the sensor. Consequently, when using an oscillator to measure the frequency shift, it is necessary to consider the change in the motional resistance as well. If the motional resistance and/or the parallel capacitance change during experiment, the frequency shift monitored with the oscillator could not be equal to the MSRF shift. Thus the associated physical parameters extracted from the monitored frequency shift could include an error which is necessary to consider. New techniques are being implemented which reasonably combine the accuracy of expensive instrumentation and the simplicity of oscillators. The overview of these lock-in techniques has shown that they should be considered as a good alternative in the future.

Appendix 5.A Critical Frequencies of a Resonator Modeled as a BVD Circuit 5.A.1 Equations of Admittance and Impedance

The expressions for the admittance and impedance of the equivalent electric circuit shown in Fig. 5.2b, modeling the resonator in a range of frequencies around resonance, are: Y = G + jB =

R X ⎛ ⎞ − j⎜ 2 − ω C o* ⎟ 2 2 R +X ⎝R +X ⎠ 2

(5.A.1)

5 Interface Electronic Systems for AT QCM Sensors

Z = Re + jX e = + j

(

(1 − ω

C o*

X

)

)

R

2

+ ω 2 C o*2 R 2

+

(5.A.2)

X 1 − ω C o* X − ω C o* R 2

(1 − ω C X ) * o

2

175

+ ω 2 C o*2 R 2

where ⎞ ⎛ 1 ⎞ 1 ⎛ω2 ⎜ ⎟⎟ = X = ⎜⎜ ω L − − 1⎟⎟ ; 2 ⎜ ω C ⎠ ω C ⎝ ωs ⎝ ⎠

(5.A.3)

C* + C 1 ω s2 = ; ω 2p = ω s2 o * LC Co

Typical plots for real and imaginary parts as well as the phase-frequency response of the impedance and admittance are shown in Figs. 5.A.1 and 5.A.2, respectively. 80

25000

60

Re(ZQ) Im(ZQ)

Re (ZQ) ; Im (ZQ) / Ω

15000

40

Phase (ZQ)

20

10000

0 5000 -20 0

-40

-5000

-60

-10000 -15000 9980

Phase (ZQ) / º

20000

-80

9990

10000

10010

10020

10030

-100 10040

Freq. / KHz

Fig. 5.A.1. Plots for the real and imaginary parts as well as phase-frequency response of the impedance of a typical 10 MHz quartz crystal resonator, one face in contact with water, around the fundamental resonance

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100

0,006

80 60 40 0,002 20 0 0,000 -20 -40

Re(YQ)

-0,002

Im(YQ) Phase (YQ)

-0,004 9980

9990

10000

10010

10020

10030

Phase (YQ) / º

Re (YQ) ; Im (YQ) / S

0,004

-60

-80 10040

Freq. / KHz

Fig. 5.A.2. Plots for the real and imaginary parts as well as phase-frequency response of the admittance of a typical 10 MHz quartz crystal resonator, one face in contact with water, around the fundamental resonance

5.A.2 Critical Frequencies Series and parallel resonant frequencies

The series and parallel electric resonant frequencies are here define for convenience as those which cancel the imaginary term in the expressions for admittance and impedance when the losses are neglected; this is, when the parameter R is zero. The condition for canceling the imaginary part in admittance and impedance is:

ω 2 R 2 C 2 Co* Co* + C

⎞ ⎞⎛ ω 2 ⎛ ω2 + ⎜⎜ 2 − 1⎟⎟⎜ 2 − 1⎟ = 0 ⎟ ⎜ ⎠⎝ ω p ⎝ ωs ⎠

(5.A.4)

The angular frequencies fulfilling the former equation when the resistance R is neglected are: the series resonant frequency ωs and the parallel resonant frequency ωp. When the losses are not neglected, the impedances at resonant frequencies are:

5 Interface Electronic Systems for AT QCM Sensors

Z (ω s ) =

177

ω s C o* R 2 R j − 2 1 + ω s2 C o*2 R 2 1 + ω s C o*2 R 2

(5.A.5)

1 R − j *2 2 ω Co R ω p Co*

(5.A.6)

Z (ω p ) =

2 p

Consequently, the impedance has a capacitive character at those frequencies. Zero-Phase frequencies

The exact resolution of Eq. (5.A.4) gives two frequencies: fr and fa at which the imaginary parts of the admittance and impedance cancel. These frequencies at which the electrical behavior of the piezoelectric resonator is purely resistive are:

ωr =

ωa =

b − b 2 − 4ω s2ω 2p

(5.A.7)

2 b + b 2 − 4ω s2ω 2p

(5.A.8)

2

R 2 C 2 Co* 2 2 ωs ω p . Co* + C In order to simplify the interpretation of the relative positions among the different critical frequencies with regard to the electrical resonant frequencies fs and fp, it is usual to give the following approximate expressions for the zero-phase frequencies as a function of the ratio r = C o* / C and the quality factor Q = c66 / ω sηQ = L ω s / R (Chap. 1–Eq. (1.A.38)). These expressions are [20]: where b = ω s2 + ω 2p −

⎛

r ⎞ ⎟ 2 Q 2 ⎟⎠

(5.A.9)

1 r ⎞ ⎟ − 2 r 2 Q 2 ⎟⎠

(5.A.10)

ωr ≈ ω s ⎜⎜1 + ⎝

⎛

ωa ≈ ω s ⎜⎜1 + ⎝

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Antonio Arnau, Vittorio Ferrari, David Soares and Hubert Perrot

Frequencies for Minimum and Maximum Admittance

The exact expressions for the frequencies of maximum admittance fm and minimum admittance fn are difficult to obtain. In this case the approximations are very useful and accurate in most cases. However, their validity must be restricted to cases where the quality factor has a high value. These expressions are: ⎛

r ⎞ ⎟ 2 Q 2 ⎟⎠

(5.A.11)

1 r ⎞ ⎟ + 2 r 2 Q 2 ⎟⎠

(5.A.12)

ωm ≈ ω s ⎜⎜1 − ⎝

⎛

ωn ≈ ωs ⎜⎜1 + ⎝

The previous approximate equations for the critical frequencies permit to show graphically, the relative position among these frequencies, as illustrated in Fig. 5.A.3.

a fm

a = fs r 2 2Q

a fs

fr

a fa

fs

a fp

fn

2r

Fig. 5.A.3. Diagram for illustrating the relative positions among the electric resonant frequencies of a quartz crystal resonator

5.A.3 The Admittance Diagram

The admittance Y can be represented by a vector in the complex plane. The real component in the abscissa represents the conductance and the imaginary part in the coordinate axes represents the susceptance. Eq. (5.A.1) permits to write the admittance as the sum of the motional admittance and the admittance corresponding to the parallel capacitance Co* , as follows: Y = Ym + jω Co* = G + jB

(5.A.13)

5 Interface Electronic Systems for AT QCM Sensors

where Ym =

R X = Gm + jBm − j 2 2 R +X R + X2 2

179

(5.A.14)

As can be deduced from the former equation, the absolute value of the motional admittance describes a circumference in the complex plane. Effectively, the absolute value of the motional admittance is: Ym =

R2 X2 + = (R 2 + X 2 )2 (R 2 + X 2 )2

1 = Gm2 + Bm2 (R + X 2 ) 2

(5.A.15)

Consequently: Gm2 + Bm2 =

G 1 = m 2 2 R (R + X )

2

⎛ 1 ⎞ 1 ⎟⎟ + Bm2 = ⇒ ⎜⎜ Gm − 2R ⎠ 4 R2 ⎝

(5.A.16)

The former equation represents a circumference with the center in and a diameter equal to 1 R , as illustrated in the continuous plot in Fig. 5.A.4a. Each point in this circumference represents the affix of the vector of admittance. The affix of this vector displaces along the circumference when the frequency is varied from 0 to ∞. The susceptance is positive for frequencies between the zero conductance frequency and the maximum conductance frequency, at which Gmmax = 1 / R , while it is negative for frequencies greater than the maximum conductance frequency. It can be demonstrated that the maximum conductance frequency is equal to the electrical series resonant frequency for the BVD circuit. The admittance corresponding to the parallel capacitance, jω Co* , moves up each point corresponding to the affix of the admittance vector at the specific frequency. Because resonance occurs in a very narrow range of frequencies for high Q resonators, the admittance of the parallel capacitance can be considered constant in the range of frequencies around resonance. The result is shown in Fig. 5.A.4a where the locus representing the motional admittance is moved up by the admittance corresponding to the parallel capacitance. The final displaced locus represents the admittance corresponding to the BVD circuit. A piezoelectric resonator can be represented by several BVD circuits, each one corresponding to a specific resonance (see Chap. 1). In this case the locus of a piezoelectric resonator will be represented by a number of loci, each one corresponding to the BVD circuit representing the specific resonance of the resonator. The diameter of each locus is reduced for the resonances corresponding to higher harmonics as illustrated in Fig. 5.A.4b.

(1 / 2 R , 0)

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Antonio Arnau, Vittorio Ferrari, David Soares and Hubert Perrot

B

B |Y|

f

f

f j Co*

1 R

1 R

G

a

G

b

Fig. 5.A.4. Admittance loci of a quartz resonator: a admittance locus corresponding to the motional impedance of a BVD circuit (continuous line) and to the BVD circuit including parallel capacitance (dash line); b admittance loci corresponding to the fundamental resonance (big locus) and to a higher harmonic (small locus)

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89. A. Benjaminson (1984) “Balanced feedback oscillators” in Proceedings of the 38th Annual Symposium on Frequency Control, pp.327-333 90. A. Benjaminson (1986), “A crystal oscillator with bidirectional frequency control and feedback ALC” in Proceedings of the 40th Annual Symposium on Frequency Control, pp.344-349 91. S. Geelhood, C.W. Frank, and K. Kanazawa (2001), Acoustic Wave Sensor Worshop 3, Taos, New Mexico 92. R. Behrends and U. Kaatze (2001) “A high frequency shear wave impedance spectrometer for low viscosity liquids” Meas. Sci. Technol. 12:519-524 93. V. Ferrari, D. Marioli, and A. Taroni (2000) “Oscillator circuit configuration for quartz crystal-resonator sensors subject to heavy acoustic load” Electron. Lett. 36 (7):610-612 94. V. Ferrari, D. Marioli, and A. Taroni (2001) “Improving the accuracy and operating range of quartz microbalance sensors by purposely designed oscillator circuit” IEEE Trans. Instrum. Meas. 50:1119-1122 95. A. Arnau, T. Sogorb, Y. Jiménez (2000) “A continuous motional series resonant frequency monitoring circuit and a new method of Determining Butterworth – Van Dyke Parameters of a Quartz Crystal Microbalance in Fluid Media” Review of Scientific Instruments 71:2563-2571 96. A. Arnau, J.V. García, Y. Jiménez, V. Ferrari and M. Ferrari (2007) “Improved Electronic Interfaces for Heavy Loaded at Cut Quartz Crystal Microbalance Sensors” in Proceedings of Frequency Control Symposium Joint with the 21st European Frequency and Time Forum. IEEE International, pp.357362 97. V. Ferrari, D.Marioli and A. Taroni (2003) “ACC oscillator for in-liquid quartz microbalance sensors” in Proceedings of IEEE Sensors, Vol 2, pp.849854 98. M. Ferrari, V. Ferrari, D. Marioli, A. Taroni, M. Suman and E. Dalcanale (2006) “In-liquid sensing of chemical compounds by QCM sensors coupled with high-accuracy ACC oscillator” IEEE Trans. Instrum. Meas. 55 (3):828834 99. M. Ferrari, V. Ferrari and K.K. Kanazawa (2007) “Dual-harmonic oscillator for quartz crystal resonator sensors”, in Proceedings of Transducers & Eurosensors Conference, pp.241-244. Extended version submitted to Sensors and Actuators A 100. T. Nakamoto and T. Kobayasi (1994) “Development of circuit for measuring both Q variation and resonant frequency shift of quartz crystal microbalance” IEEE Trans. Ultrason. Ferroelect. Freq. Contr. 41 (6):806-811 101. B. Jakoby, G. Art and J. Bastemeijer (2005) “A novel analog readout electronics for microacoustic thickness shear-mode sensors” IEEE Sensors Journal 5 (5):1106-1111 102. R. Torres, A. Arnau, H. Perrot (2007) “Electronic System for Experimentation in AC Electrogravimetry II: Implemented Design” Revista EIA 7:63-73 103. R. Torres (2007) “Instrumental techniques for improving the measurements based on Quartz Crystal Microbalances” Doctoral Thesis, Universidad Politécnica de Valencia

6 Interface Electronic Systems for Broadband Piezoelectric Ultrasonic Applications: Analysis of Responses by means of Linear Approaches Antonio Ramos and José Luis San Emeterio Departamento de Señales, Sistemas y Tecnologías Ultrasónicas. Instituto de Acústica (CSIC)

6.1 Introduction There are a wide variety of applications where broadband piezoelectric systems are used, mainly in order to obtain ultrasonic information for detection or visualization of the internal parts in diverse structures. These applications require external inspections with ultrasonic waves and the use of an echo-graphic procedure. The main application areas are in industry and medicine. Most broadband piezoelectric applications require the design of very specific interface electronic systems, since the conventional continuous wave (CW) electronic schemes and the conventional analysis methods are not applicable. In order to obtain a good discrimination of internal structures, it is convenient to improve the signal to noise ratio. Therefore, it is necessary to guarantee a high efficiency in the ultrasonic process. In addition, short ultrasonic pulses must be used in order to obtain good axial resolution. The above-mentioned practical questions impose some technological requirements over the electronic systems used for broadband ultrasonic applications: 1. The use of a pulsed regime for the ultrasonic inspection process, which involves a transient electrical excitation of the piezoelectric transmitter. 2. Sensitivity considerations determine that the transducer should be excited with very short electrical pulses (spikes) of several hundreds of volts at peak amplitude.

A.A. Vives (ed.), Piezoelectric Transducers and Applications, doi: 10.1007/978-3-540-77508-9_6, © Springer-Verlag Berlin Heidelberg 2008

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3. A high efficiency in emitter and receiver piezoelectric sub-processes, under broadband conditions, is required in order to optimise the coupling between spike generators and piezoelectric emitters, as well as between piezoelectric and electronic receivers. It is not easy to attain simultaneously all these requirements because parameters influencing the response amplitudes act in a crossed way. Specifically, the optimum setting of one of the parameters depends on the settings of the others. As a typical example, to obtain a good dynamic range on the received echo-signals, by using broadband piezoelectric arrays in medical imaging [1], it is necessary to drive the array elements with electrical spikes of 300-400 volts and rise-times of less than 30 nanoseconds. In general, the time duration and frequency spectrum of the inspection signals are very relevant parameters for the quality of the result obtained in visualization processes [7]. In this chapter, the characteristics required for the high voltage stages needed for driving broadband piezoelectric transducers are explained. The dependence of the driving pulses with matching networks and the external loads is also analysed. A simple circuital block diagram for interfacing transducers with the reception stages, in ultrasonic applications, is also presented. Finally, a detailed analysis of electrical responses in high-voltage pulsed driving of piezoelectric transducers is performed in the time domain, taken into account some typical working conditions. For this purpose, several expressions related to the driving waveforms are introduced and discussed from a series of analytical linear approaches associated with practical situations in medical and industrial ultrasonic applications.

6.2 General Interface Schemes for an Efficient Coupling of Broadband Piezoelectric Transducers It is quite frequent in broadband ultrasonic applications that the electrical excitation and the receiver electronics are not adequately coupled with the piezoelectric transducers. As a consequence, the best ultrasonic performance is not attained in many cases [2-3]. It is difficult to guarantee a high efficiency in the electromechanical conversion processes simultaneously with good characteristics in the whole frequency band, in particular a large bandwidth with a low ripple. These requirements in frequency are related to the generation of short ultrasonic pulses. There are some matching

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schemes for particular applications [4-5], but there is not a general solution for the matching problem in practical broadband conditions, because of the transient character of the pulsed processes used for transducer driving [6]. Moreover, some internal impedances of the pulser circuits used for this driving are variable in time [7]. Both aspects complicate a theoretical analysis of these coupling problems in the emitter phase. Driving stages in broadband applications are usually composed of highvoltage ramp generators (with low-value series impedance) in cascade with interface networks to electrically match with transducers. Assuming linear working conditions, the output function of such schemes is the temporal convolution of an exponential function with the impulse response of the coupling circuits IRCC : Driving output (t) ≡ IRcc (t ) ∗ V0 exp(−t / τ )

(6.1)

where, V0 and τ are the amplitude and fall-time of the real ramp function, which follows in practice a time evolution near an exponential curve. The piezoelectric receiver should also be matched to reception electronics in order to obtain either maximum amplitude in received signals or very short ultrasonic pulses, depending on the kind of application. In the time domain, the final ultrasonic pulse received depends on the convolution of the driving waveform with the impulse responses of emitting and receiving piezoelectric stages. Figure 6.1 shows a general diagram of the transmitter section. It includes coupling networks in series and in parallel to make the electronic interface, [8]. In some applications, one of these two coupling networks may be absent. The impedances of these networks may vary with time during some phases of the transient process. The series impedance ZS normally has a low resistive value and the block ZG usually is a capacitor fairly larger than the clamped capacitance of the piezoelectric element in the driven transducer. The parallel coupling circuit is formed by: a damping resistor RD, a coil L0 and a rectifier block. The reception matching circuits usually have a simpler structure, because they are often formed by a parallel combination of resistive and inductive components. In this receiving case, the effect desired with the connection of these elements use to be only of damping and tuning type. In the very popular pulse-echo configurations, with only one piezoelectric transducer working successively as a emitter and also as a receiver, these damping & tuning elements in reception stages are normally shared with the emitter stage.

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HV

VGEN

RAMP FUNCTION

ZG

ZS

RD

PARALLEL MATCHING NETWORK

COUPLING CIRCUITS

Lo

DRIVING OUTPUT

PIEZOELECTRIC TRANSDUCER

Fig. 6.1. General scheme of the interface electronics in pulsed piezoelectric transmitters

Other more complex circuital solutions for transducer coupling have also been proposed; for instance in order to improve particular aspects of the transducers band, such as the bandwidth [9].

6.3 Electronic Circuits used for the Generation of High Voltage Driving Pulses and Signal Reception in Broadband Piezoelectric Applications Most of the applications of broadband piezoelectric transducers have been developed in frequency bands comprised between 0.5 MHz and 10 MHz. As a consequence the shock excitation of the transducers has to be made by means of high voltage pulses with rise-times lower than 25 nanoseconds. In special applications of ultrasonic detection with very high resolution (for instance using 20 MHz transducers) a rise time around 10 ns is needed as an extreme case. Other condition present in many cases is a high acoustic attenuation during ultrasonic propagation, increasing strongly with frequency. For this reason, it may be necessary to drive piezoelectric transducers with spike amplitudes ranging up to 500 V. 6.3.1 Some Classical Circuits to Drive Ultrasonic Transducers To obtain high voltage pulses, with short rise-time, different circuits can be used. Typical schemes are shown in Figs. 6.2 and 6.3 giving output

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pulses with peak amplitude higher than 220 V. A common configuration [10] is detailed in Fig. 6.2. It uses a thyristor device Th for the high-voltage switching which produces a fast discharge of the capacitor C through the resistor RD and transducer. The repetition frequency for this discharge generally ranges between 0.5 and 5 kHz. The time evolution of the pulse generated across RD without transducer loading, during the switch-to-on period in Th, can be analysed by the expression:

(

Von (t ) = −(Vo / ton ) R D C 1 − e − t / CRD

)

(6.2)

where, ton represents the time necessary to get the thyristor to the saturation state. The particular circuit in [10] gives a rise-time in Von ranging between 150 and 250 ns. The associated frequency content is clearly insufficient for applications ranging above 2 MHz. This circuit limitation, depending on the effective ton, could be overcome by using a faster thyristor, but such type of devices is only encountered in the low-voltage range. This option is shown in Fig. 6.3 where, with a series connection of two thyristors of reduced voltage, a higher peak value pulse can be generated. By repeating this scheme for several series stages, the necessary high voltage pulse could be reached. Nevertheless, the time tolerances and the -always long- cut-off times in thyristors produce harmful transient impulses, which could distort the received ultrasonic signals. Another disadvantage of these devices is the high losses current in off state. +Vo

Vo

RC REPETITION FREQUENCY LOW-VOLTAGE PULSE GENERATOR

DIFFERENTIATOR OF RISING EDGES

G

Th

C RD TRANSDUCER

Fig. 6.2. Conventional generator of electrical pulses used for broadband driving of piezoelectric transducers. Circuit based on thyristor switching

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LOW-VOLTAGE SOURCE

RC

H.V. SUPPLY

DIFFERENTIATOR

C R1 R1>>RC

DIFFERENTIATOR

TRIGGER SIGNAL

RD

R1

Fig. 6.3. Conventional generator of electrical pulses for HV driving of transducers. Series connection of two SCRs for higher amplitude pulses

For these reasons, it is a better option to use other type of device to perform the high-voltage switching. The power mos-fet transistor fulfils all the requirements and does not present the above-mentioned limitations. A circuit based on such kind of device was proposed in reference [6]. This circuit can generate pulses of 240 V in peak amplitude over a damping resistance of 300 Ω. The rise-time associated with the output pulse is 100 ns and therefore it can only be used to drive transducers with resonance frequencies below 3 MHz. A faster switching device would be needed for a proper excitation of broadband transducers in most applications. 6.3.2 Electronic System Developed for the Efficient Pulsed Driving of High Frequency Transducers

Figure 6.4 shows the block diagram of a pulser, specially developed for efficient excitation of broadband transducers in a frequency range of 30 MHz [7]. This pulser is a particular implementation of the general scheme in Fig. 6.1. FAST CURRENT DRIVER

HIGH VOLTAGE MOS-FET RAMP GENERATOR

PULSE SHAPER AND SELECTIVE DAMPING NETWORKS

TRANSDUCER

Fig. 6.4. Block diagram of a spike generator developed for an efficient broadband driving of high frequency piezoelectric transducers

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The switching of the power mos-fet, used here as a ramp generator, was performed by the gate driver detailed in Fig. 6.5 where the complementary transistors T2 and T3 supply and carry to ground the bi-directional high currents needed for a fast switching of the transistor TMF (see the ramp generator circuit in Fig. 6.6). It should be noted that this type of transistors presents a gate-to-source capacitance that can be as high as 1500 pF. As a result, the gate control must be made through very low impedance circuits if an optimisation of transition times is desired. LOW-VOLTAGE SUPPLY BIDIRECTIONAL CURRENT DRIVER

100 ns. minim.

T2

TTL PULSE

T3

TO GATESOURCE PORT

Fig. 6.5. Bi-directional current driver for fast switching of the power mos-fet transistor H.V. POWER SUPPLY

RC Z D G

S

+ C

TMF

HIGH - VOLTAGE RAMP GENERATOR

Fig. 6.6. High-voltage ramp generator using a sudden capacitor discharge

In the circuit of Fig. 6.6, a zener diode Z appears between the drain of TMF and the capacitor C, producing a light inertia in the electrical discharge through TMF and transducer, in such a way that, at the beginning of the discharge, TMF will be closer to total saturation than in the absence of that diode Z. Another important effect of this device is to avoid distortion on the positive half-cycles of the echo-graphic traces collected in ultrasonic

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reception. Without this zener diode, the positive half-cycles would go to ground through diodes D1 (see Fig. 6.7), capacitor C and TMF, distorting the signal. The two previous circuital blocks are connected in cascade to a block for selective damping and coupling with transducers, whose circuit is detailed in Fig. 6.7.

D1 RL

D2

RD

Lo

ZT

SELECTIVE DAMPING NETWORK

Fig. 6.7. Network for selective damping and for the coupling of the ramp generator with the transducers

This third block includes an adjustable coil L0 in parallel with the damping resistor. The coil setting allows a fitting of the temporal shape in the driving pulse. Diodes D1 and D2 prevent the generation of oscillatory signals from the C-Lo resonant circuit, because they only permit the flow of the first negative half cycle to the transducer, and the remaining energy after the launching of this negative pulse will be dissipated abruptly through RL and Di. As the diodes Di have to be rapid, it is necessary to use groups of parallel and series signal diodes, to allow high peak currents (through D1) and reverse bias close to 500 V (across D2 ). The circuit formed by Lo, D1 and D2 in the selective damping network, improves the amplitude and shape in the output spike for two reasons: 1. It can efficiently substitute the damping effect of a resistor RD of low value in order to produce a fast electric damping of the spikes without losses in their amplitudes. 2. It can facilitate the formation of driving pulses with trailing slopes being independent of the cut-off instant in transistor TMF. This independence is useful to reduce temporal dispersions related to the usual tolerance in the switching-to-off times of these devices.

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Finally the low value resistance (RL << RD) protects the generator from an accidental short-circuit at output terminals, and reduces possible signal distortions if positive half-cycles greater than 3.5 V were received. Without RL, these half-cycles would go directly to ground through D1 and D2. Some of the circuits previously shown for HV driving purposes in broadband ultrasonic equipments have an intrinsic non-linear response, due to the semiconductor devices involved (TMF, D1, D2). For this reason, it is very difficult to describe their behaviours by means of simple analytic expressions. Nevertheless, in some especial cases, for instance when the impedance of the coil L0, in Fig. 6.7, is relatively high, ωL0 >>RD, then the behaviour of the output spike Vout (t) across the transducer terminals could be approximately predicted by the following expression:

(

Vout (t ) = CVo RT e −t / τ − e −t /( C +CT ) RT

) (C

T

1 + C ) RT − τ

(6.3)

where, RT = [A(RB+RM) / (2CT h33) 2] // RD. The symbol // indicates an electrical parallel connection; A is the area of the internal piezoelectric vibrator, and CT is the clamped capacitance in the driven transducer; RB and RM are the specific acoustic impedances of the two media loading the faces of the active piezoelectric element, and h33 is the piezoelectric constant. However, the useful approach of Eq. (6.3) only can be adequately used for particular dispositions in the electrical loading conditions at the output electrical terminals of the HV pulse generators, and in the other interface circuits (tuning and matching) normally employed in the most of the broadband applications. In Sect. 6.4, an ample analysis about this problem will be performed, looking for linear approaches of the pulse generator and interface circuits, and taken into account several levels of approximation in the circuital equivalence of the piezoelectric transducer. 6.3.3 Electronic Circuits in Broadband Signal Reception

There are some electronic alternatives to perform the analogic interfacing needed between a piezoelectric transducer operated as an ultrasonic receiver and the signal acquisition & digitising system. Figure 6.8 shows a typical interface circuit, for the reception stage, with this purpose. This circuit corresponds to a simple scheme for the efficient reception and decoupling (from the HV spikes) of the broadband echo signals produced at the transducer terminals during ultrasonic applications, either in pulse-echo or in through-transmission operation modes (see Chap. 16, for more details).

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Dz T2

D

Dz Dz D

D

LIMITER CIRCUIT

ATTENUATOR

TRANSDUCER

T1 SALIDA 1 0 dB

Broadband Amplifiers

G1B

INTERFACE ELECTRONICS IN RECEPTION

G1A

SALIDA 2 20/40 dB

40 dB 20 dB

Fig. 6.8. A typical interface circuit between transducers and broadband signal acquisition electronics

Across the T1 and T2 terminals, reactive parallel circuits are often connected in order to perform electrical damping and inductive tuning. The second stage consists of a bipolar limiter to ≈ ±5 V of three branches to protect the reception electronics from high-voltage spikes launched in transducer terminals during the driving process. This circuit is constituted by three successive parallel mono-polar limiting branches, each one with zener and signal diodes. The two additional branches are needed because of the high value of the current produced through the first branch, originating an elevated conducting on-voltage threshold in their semiconductors. The pulsed signals so limited are fed into the input of a broadband signal pre-amplifier, through an optional attenuator step. Figure 6.8 also includes a simple amplifier scheme with two differential steps. The first step performs an impedance matching between the often relatively highimpedance value in transducer terminals and the low input impedance usually selected in low-noise signal amplifier circuits.

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6.4 Time Analysis by Means of Linear Approaches of Electrical Responses in HV Pulsed Driving of Piezoelectric Transducers In this section the shape of the driving electrical pulses, generated by the high-voltage pulser configurations described in previous sections, is analysed as a function of the internal parameters of the generator and of the external electrical loads, as well as of the interface circuits usually present across the transducer electrodes. In Fig. 6.9, a typical electrical pulse (of rather moderated amplitude) is shown; it was provided by a generator circuit having a circuital topology like those depicted in Figs. 6.4 to 6.7. This pulse was measured when the pulser circuit was not loaded with the piezoelectric transducer. In the practical driving pulses, under piezoelectric loading, the waveforms include other non-ideal aspects [7] depending on an ample variety of factors, which must be taken into account for an accurate analysis of the pulsed excitation processes.

Voltage (volts)

60V

30V /div

240V 10ns

100ns / div T

992ns

( ns )

Fig. 6.9. Typical unloading pulse waveform for an efficient pulsed driving of broadband piezoelectric transducers

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The output pulse peak normally does not reach the optimal value amplitude (coincident with the nominal voltage of the HV voltage supply, V0, in the pulse generator), which uses to be ranged between 200 and 500 Volts, in most of applications. In fact, if low impedance transducers, and especially transducers with very high static capacitance, have to be driven this optimal amplitude can become clearly worsen, causing important decreases in the dynamic range disposable for the echographic signals. Nevertheless, by means of a careful fitting of the parameters RC, C, RL and RD in the circuits shown in Figs. 6.6 and 6.7, these problems can be partialy minimized. In this context of optimisation of the driving pulse the availability of some approximate expressions for a temporal analysis of the output pulse shape becomes very useful, since they can be used as a practical tool for improving the global efficiency of the ultrasonic emission process under broadband operating conditions. The time evolution of the output pulse produced by the high-voltage generator configurations considered here can be approximately predicted knowing, for each application, the values corresponding to V0, C, RD, L0 as well as the value of the transducer input impedance evaluated at the frequency point related to the series-resonance conditions [ZXT (fs)]. With this aim, the behaviour of these pulser circuits for three assumptions related to three different practical loading conditions is next analized by increasing the level of complexity: 1. Coil L0 non-connected across transducer terminals, and transducer impedance considered as a pure resistive load (RXT) during the driving interval. These rather simple conditions are just those proposed in reference [10]. 2. Coil L0 non-connected across transducer terminals and transducer impedance approximated as a parallel equivalent network RXT //CXT accounting for its normally notable capacitive character. 3. Coil L0 connected with the transducer terminals, and transducer modelled as a parallel electrical network RXT //CXT. 6.4.1 Temporal Behaviour of the Driving Pulse under Assumption 1

When it is assumed that, during the high-voltage switching [from (V = V0) in t = 0 to (V = 0) in t = ton], the drain voltage in the mos-fet transistor TMF decreases linearly with time, the generator circuit can be modelled by a simplified equivalent circuit.

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This equivalent circuit includes: an ideal ramp source VD in series with a capacitor C initially charged to V0 voltage, and a parallel-equivalent circuit of three resistive impedances, related to the electrical damping (RD), the transducer resistance under resonance conditions (RXT), and a simplified input resistance (RSR) of the electronic stage used for the signal reception (for instance, the circuit detailed in the Sect. 6.3.3)). The ramp source VD can be represented by the following expression:

[VD = V0 (1 − t / t on )] /(0 < ∀t < t on )

(6.4)

By means of this dependent-on-time source, the switching of the transistor TMF to the saturation state can be approximately modelled. The capacitor C is supposed initially charged, at t = 0, at the biasing high-voltage of the supply, V0. With regard to our analysis, the effects produced by RL, (RL << RXT), and by the network of diodes Di, in Fig. 6.7, can be disregarded, because of their very slight influence on the shape of the HV driving output Vout (t). As an important limitative aspect of this simplified equivalent circuit, it must be noted that its scope is only valid from the beginning of the switching process in the transistor TMF (t = 0) to the instant (t = ton), in a similar way to the situation present at the approach proposed in [10]. From this simple linear approach, it is possible to establish the corresponding differential equation, which, through its solution in the Laplace domain and the subsequent inverse Laplace transformation, gives an explicit expression in the time domain for the high-voltage driving function: Vout (t ) = − Vo

(

CR P 1 − exp (−t / CR p ) t on

)

(6.5)

where

R P = R XT RD RSR ( RD RSR + R XT RSR + R XT R D ) −1

(6.6)

This expression is coherent with that proposed in reference [10] for a simpler case. The values obtained for Vout (t) by applying Eq. (6.5) must be considered valid only during the short time interval in which the pulse leading edge (falling transition in the drain of TMF) occurs, and consequently, it would be completely inappropriate to deduce, starting from Eq. (6.5), anything about the pulse behaviour beyond that time instant. An alternative way to overcome this later limitative problem is to approximate the switching of the TMF device by means of a decreasing

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exponential function, with a fall timeτ coherent with ton. This option agrees better with the real behaviour of the transistor saturation and, in addition, the resulting calculation values would continue being valid after the instant ton, i.e., along all the positive times (t > 0). When an improved function of this kind is used for modelling the switching of the transistor, an analytical expression can be obtained [7] by using a similar process to that employed for generating the expression in Eq. (6.5), for the waveform of the pulse of output, Vout: Vout (t ) = Vo

(

CR P exp (−t / τ ) − exp (−t / RC p ) CR P − τ

)

(6.7)

It can be observed that Vout (t) is composed of two decreasing exponential functions, both having the same initial amplitudes but with opposite sign. The positive exponential function decreases according to a time constantτ, whereas the negative one decreases with a constant CRP. Consequently, the electrical waveform of the pulse Vout (t) starts with a zero value, at t = 0, and then quickly decreases with a time constant close to τ if the capacitor C is chosen with such a value that CRP >> τ . The resulting pulse Vout (t) will get a negative peak value (always minor in module than Vo) which will be so nearer to Vo as CRP increases attaining values much greater that τ. On the other hand, if a rather narrow pulse is generated, by using a little value for the capacitor C, in such a way that CRp << τ, the peak amplitude in the output pulse will decrease in a strong way. A similar negative effect could be observed when a very low parallel damping resistance is selected, because of its direct influence on the RP parameter value. 6.4.2 Temporal Behaviour of the Driving Pulse under Assumption 2

In the case of the hypothesis of an exponential type function for the TMF switching is maintained, but improving the correspondence of the modelling of the transducer with the real piezoelectric loading, by also incorporating the transducer clamped capacity CXT into the approximated equivalent circuit used for our analysis, a more accurate expression can be obtained for Vout (t). In fact, this capacitive component of the transducers exerts, in many applications, a direct influence on the waveform of Vout (t), just from the beginning of the electric excitation due to the static characteristic of that capacitance, which does not depends on the resonance state and therefore is

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always present across the output terminal of the pulse generator, loading their internal circuits. The expression that can be obtained for the pulse Vout (t) in this case [7], as a function of time, is the following: Vout (t ) = Vo

(

CR P exp ( −t / τ ) − exp (−t /[C + C XT ]R p ) (C + C XT ) R P − τ

)

(6.8)

In this expression, the maximum peak value expected for the Vout (t) pulse [supposing ideal conditions: (C+CXT)RP >> τ ] depends strongly on the transducer static capacitance CXT, in such a way that it only reaches the optimal value (close to the V0 voltage) when CXT << C, and decreasing as CXT increases. Then, in order to optimise the voltage amplitude in the high-voltage driving pulse, a capacitor C quite bigger than the corresponding transducer capacity CXT must be used in all cases where this is possible. But, there is an important limitation in this aspect. The problem arises when piezoelectric transducers with very high clamped capacitance must be excited, because the time required to charge the capacitor C could be longer than the interval disposable between two consecutive shots (which uses to be ranged between 100 and 500 μseconds in medical and industrial applications), so preventing a complete charge of the said capacitor in Fig. 6.6. This problem can be overcome by suitably decreasing the CRC product with by lowering the values in the resistance RC, but bearing in mind that, as a indirect consequence of this, it would be necessary the use of a high voltage source V0 capable of supplying the peak currents required when these low values of RC are selected, which could become very expensive in the multi-transducer applications. 6.4.3. Behaviour of the Driving Pulse under Assumption 3: The Inductive Tuning Case

The simplified equivalent circuit, corresponding to this third more complicated assumption, includes a parallel coil L0 in the output terminals of the pulse generator (which is usually employed for tuning or pulseshaping purposes). In addition, the effects of the diodes Di (depicted in Fig. 6.7) must be considered in this particular case, because they are connected in parallel with those output terminals. This last aspect is justified because, if the effects of these diodes are ignored in this circuital disposition, with L0 connected, the HV pulsed signal produced by the generator would be a rather long pulse with strong oscillations after its first positive

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crossing through the line of zero volts, as it was already explained in the Sect. 6.3.2. Nevertheless, if these diodes are taken into account in the pulser model, the calculated pulse waveform will correspond with the real response of the high-voltage pulse generator, because they would cancel the mentioned oscillations after the first pulse zero-crossing. Taking into account the loading conditions now considered, an analytical expression for Vout (t), planned directly in time domain, is quite difficult to achieve and, in consequence, it is easier to analyse first the resulting circuit in the frequency domain and then obtaining the time response by the inverse Laplace transformation of Vout (s): Vout ( s ) = − Vo

C τC S s + (C S + τ / R P ) s + (1 / R P + τ / L0 ) + (1 / L0 ) s −1 (6.9) 2

where C S = (C + C XT ) From the inverse transform of expression in Eq. (6.9) an oscillatory waveform is produced in the time domain, with an exponentially damped angular frequency, because the diodes Di have not been considered to obtain this latter expression but, in this step of the analysis, their effects can be just introduced, emulating the cancelling of oscillatory responses after the first zero-crossing, as it happens in the practice, so preventing an undesired lengthening on the driving pulse. Other interesting result of this non-linear selective damping circuit is that the positive edge starts with a slope smoother than in the pure exponential case, but allowing the total pulse rise toward zero to be produced in a shorter time [11]. This particular effect permits us to obtain driving pulses with relatively short total width (which is required to drive highfrequency transducers), without suffering as a negative consequence animportant waste on the resultant driving amplitude. In fact, for obtaining alternative pulses with similar width but without using the coil L0, it would be necessary to reduce strongly the value of C and RD, or to put a very low value in the resistance RC. The first possibility would reduce notably the peak amplitude in the driving pulse, whereas the second option would involve the use of special high-voltage sources with high peak current and thus making this solution prohibitive, especially for the cases of multichannel excitation in array configurations. In consequence, it seems that the use of a coil in parallel with the pulser output terminals is the most useful way to reach narrow output pulses with high efficiency. Moreover, this inductive component could furnish other advantageous effects as a “selective damping” of the secondary oscillating

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modes (for instance, those modes derived from lateral vibrations in the piezoelectric element), and also for cancelling the static transducer capacitance in those applications where this effect is desired, looking for a better electromechanical efficiency.

References 1. J.W. Hunt, M. Arditi, and F.S. Foster (1983) “Ultrasound transducers for pulse-echo medical imaging” IEEE Trans. Biomed. Eng. BME-30 (8):453481 2. G. Hayward (1985) “The influence of the pulser parameters on the transmission response of the piezoelectric transducers” Ultrasonics 23:103-112 3. M.E. Schafer, P.A. Lewin (1984) “The influence of front-end hardware on digital ultrasonic imaging” IEEE Trans. Sonics Ultrason. SU-31 (4):299-306 4. L.J. Augustine, J. Andersen (1979) “An algorithm for the design of transformerless broadband equalizers of ultrasonic transducers” J.Acoust. Soc. Am. 66 (3):629-635 5. R. Coates and R.F. Mathams (1988) “Design of matching networks for acoustic transducers” Ultrasonics 26:59-64 6. P. Mattila, and M. Luukkala (1981) “FET pulse generator for ultrasonic pulse echo applications” Ultrasonics 19:235-236 7. A. Ramos, P.T. Sanz, F.R. Montero (1987) “Broad band driving of echographic arrays using 10ns-500ns efficient pulse generators” Ultrasonics 25:221-228 8. A. Ramos, J.L. San Emeterio and P.T. Sanz (1997) “Electrical matching effects on the piezoelectric transduction performance of a through transmission pulsed process” Ferroelectrics 202:71-80 9. F. Lakestani, J.C. Baboux and P. Fleischmann (1975) “Broadening the bandwidth of piezoelectric transducers by means of transmission lines” Ultrasonics 13:176-180 10. J. G. Okyere, and A. J. Cousin (1979) “The design of a high voltage SCR pulse generator for ultrasonic pulse echo applications” Ultrasonics 17:81 -84 11. A. Ramos, J.L. San Emeterio, and P.T. Sanz (2000) “Improvement in transient piezoelectric responses of nde transceivers using selective damping and tuning networks” IEEE Trans. Ultrason. Ferroelect. Freq. Cont. 47: 826-835

7 Viscoelastic Properties of Macromolecules Ralf Lucklum1 and David Soares2 1

Institute for Micro and Sensor Systems, Otto-von-Guericke-University Magdeburg 2 Institute de Fisica, Universidade de Campinas

7.1 Introduction An acoustic wave traveling through a material deforms the material, thereby probing its mechanical properties. The classical theory of elasticity deals with the mechanical properties of elastic solids. In accordance with Hook’s law, for small deformations stress is always directly proportional to strain. Furthermore, stress is independent of the rate of strain. The classical theory of hydrodynamics deals with properties of viscous liquids. In accordance with Newton’s law, stress is always directly proportional to rate of strain but independent of the value of strain itself. Purely elastic solids and purely viscous liquids are idealizations as well as the two boundaries of system behavior, which combines liquid-like and solid-like characteristics. Those material properties have been called viscoelastic. A viscoelastic material does not maintain a constant deformation under constant stress. It continues to deform slowly with time. On the other hand, when the stress is removed, a viscoelastic material recovers part of its deformation. When a viscoelastic material is constrained at constant deformation, the stress required to hold it, diminishes with time. When a viscoelastic material is subjected to sinusoidally oscillating stress, the strain is neither in phase with stress as valid for purely elastic solids nor 90° out of phase as valid for purely viscous liquids. Stress and strain are some amount out of phase, which depends on the viscoelastic properties of the body. If both strain and rate of strain are infinitesimal, and if the time-dependent stress-strain relations can be described by linear differential equations with constant coefficients, the system shows linear viscoelastic behavior and the ratio of stress to strain is a function of time or frequency, not a function of stress magnitude [1, 2]. Viscoelastic properties of material should not be mixed up with deviations from purely elastic or viscous behavior due to finite strain A.A. Vives (ed.), Piezoelectric Transducers and Applications, doi: 10.1007/978-3-540-77508-9_7, © Springer-Verlag Berlin Heidelberg 2008

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(non-Hookean deformation) or finite strain rates (non-Newtonian flow). The necessity to distinguish between infinitesimal and finite depends on the level of precision of measurement and properties of the material under consideration. Viscoelastic anomalies are negligible or at least of minor significance for many classical engineering materials. By contrast, the mechanical behavior is dominated by viscoelastic phenomena in macromolecular materials. Due to the extreme importance of those materials for chemical and biochemical sensors, the consequences of viscoelasticity for acoustic sensors will be analyzed in some more detail. A comprehensive analysis of viscoelastic properties of polymers can be found in Ferry’s fundamental book [1]. The following text summarizes those statements about the phenomenon of viscoelasticity [1-3] which are important to understand specific features of acoustic-wave based sensors.

7.2 Molecular Background of Viscoelasticity of Polymers In solids, such as silicon or quartz, all atoms are arranged in a crystal with well defined distances and orientations, giving rise to a long scale order. The interaction forces are strong. In deformation of hard solids, atoms are displaced from equilibrium positions in fields of force with small range. The elastic constants are a result of inter-atomic potentials. The deformation energy is completely stored as potential energy and can be completely recovered. Other phenomena like structural imperfections, which contribute to mechanical properties relevant for acoustic wave devices, typically have distances discontinuously larger than atomic dimensions. In ordinary liquids, such as water, the molecules are arranged with a rapidly vanishing order. The interaction forces between the molecules are rather small. Viscous flow changes the distribution of molecules surrounding a given molecule with time. Viscosity is a result of the relevant forces between the molecules and the processes of readjustment. The deformation energy is completely dissipated as heat. Macromolecules cannot be described in a simple way. Each flexible macromolecule pervades an average volume much greater than the atomic dimensions. Furthermore, it continually changes the shape of its contour [1]. The nomenclature of the molecule reflects only the (chemical) constitution of the macromolecule: type, sequence of chain molecules, building blocks, constitutive repeating units, type and number of end groups, as well as type, number, arrangement and distribution of chain links. The configuration describes the specific arrangement of atoms or groups of

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atoms at a C = C double bond. The spatial arrangement of atoms or groups of atoms as a result of rotation along a single chemical bond is called conformation. Isomers are molecules with an equivalent number of equivalent atoms in a different spatial arrangement [4]. Conformation changes1 are most important for the understanding of the response of macromolecules to any macroscopic deformation. One has to consider more local relationships as well as long range contour relationships. Rearrangements on a local scale are relatively rapid, whereas rearrangements on a long-range scale are rather slow. Long-range rearrangements require a certain kind and number of cooperative local configuration changes to be realized. In a viscoelastic material a wide and continuous range of time scale exists covering the response of such a system to external stress. From that point of view one can expect a different response of viscoelastic materials to an external perturbation depending on the evaluating experimental method, first of all on the time scale of the applied perturbation. Another aspect is the morphology of polymers2. In principle one can distinguish crystalline and amorphous polymers. A local structure along a polymer chain with enough regularity and symmetry allows a partial ordering in a crystal lattice. The macromolecules can form thin lamellar single crystals, in which the chains run parallel to the thin direction and are folded back and forth many times. Polymers with a high degree of molecular symmetry can form lamellar structures, which are organized into larger morphological units such as fibriles, spherulites or helices. If the molecular weight is not uniform, fractionation occurs with species of lower molecular weight remaining uncrystallized. Side groups, atacticity, or other features, which diminish molecular symmetry, enhance the degree of disorder and amorphous regions within a crystalline polymer become more and more likely to occur. Amorphous polymers are completely disordered; the arrangement of the macromolecules is random. 1

2

Configuration and conformation isomers can be distinguished by the energy barrier between them. The distinction between them is not important as long as the spatial arrangement of atoms is of interest only. However, the lifetime of conformation isomers is much shorter than that of configuration isomers. As a consequence, conformation isomers can be recognized with fast probing methods like acoustic waves. Consequently configuration and conformation isomers can hardly be distinguished by those methods. Although it is not really consistent, many polymer textbooks prefer the words 'configuration changes' to describe rearrangements of the polymer backbone realized by the arrangement of atoms and groups of atoms along the polymer backbone. The word “polymers” is further used instead of macromolecules, though not limited to classical polymers.

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Polymers can be composed of linear chains, chains with small side groups, backbone chains with more or less long side chains, cross-linked backbone chains and complete networks without individual chains. Besides chemical bonds within the macromolecule and between cross-linked chains one has to consider several kinds of physical bonds between different chains and different segments of the chain itself. A special feature of macromolecules is the chain entanglement, which creates a kind of physical network (Fig. 7.1) [1].

a

b

c

Fig. 7.1. Concepts of entanglement coupling: a temporary cross-link, b local kink, and c long-range contour loop. Adapted from Ferry [1]

The viscoelastic behavior of polymers is not uniform. It can be illustrated by defining two groups of materials. The first group considers uncross-linked polymers, which can be treated as viscoelastic liquids. They do not possess any equilibrium compliance and exhibit viscous flow at sufficient times. The viscosity may be very high and hence the material may appear as a rubbery solid. Ferry [1] distinguishes between dilute polymer solutions (I), amorphous polymers of low molecular weight (II), amorphous polymers of high molecular weight (III) and amorphous polymers with high molecular weight with long side chains (IV). In dilute solutions the polymer molecules are sufficiently separated to move fairly independently without much interaction. Polymers with a low molecular weight up to 10,000 consist of about 100-200 monomer units. Effects of neighboring molecules on viscoelasticity arise from local frictional forces. Viscoelastic properties of polymers above a critical molecular weight of the order of 10,000 to 30,000 reveal a strong coupling to neighbors far more than local frictional forces. The interaction is usually called entanglement coupling. This phenomenon prolongs any molecular rearrangements, which are sufficiently long range to involve regions of a macromolecule separated from each other by one or more entanglement points. The other group considers viscoelastic solids, which do not exhibit viscous flow and reach or closely approach an equilibrium deformation under a constant stress. According to Ferry [1] amorphous polymers of high

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molecular weight below glass transition temperature (V), lightly (VI) and very lightly (VII) cross-linked amorphous polymers, and highly crystalline polymers (VIII) belong to the second group. The chain backbone configuration of an amorphous polymer is highly immobilized below the glass transition temperature. The response to external stress involves primarily very local adjustments. Short-range segmental rearrangements in a network of highly flexible threadlike strands (usually defined as segments between cross-links and entanglements) are rapid whereas long-range rearrangements are profoundly affected by the presence of linkage points. In very lightly cross-linked amorphous polymers one can expect dangling branched structures, which are extensively entangled. Finally, in crystalline materials both ordered and disordered regions will contribute to viscoelastic behavior in their own way.

7.3 Shear Modulus, Shear Compliance and Viscosity In general, the ratio of stress to the corresponding strain is called a modulus and hence, a measure of the strength of a material. The following is restricted to periodical perturbations with a sinusoidal alternation at a frequency f (cycles per second) or ω (radians per second), e.g., shear stress generated from a propagating acoustic wave, and furthermore to linear viscoelastic behavior. Such a periodic experiment at ω is qualitatively equivalent to a transient experiment at time t = 1/ω. The relation between stress, σ, and strain, γ = γ 0 sin ωt, can be written as

σ = γ 0 (G ' sin ω t + G ' ' cos ω t )

(7.1)

The superscript 0 marks the maximum amplitude. The term with sin ωt is in phase with γ and the term with cos ωt is 90° out of phase with γ. Equation (7.1) thereby defines two frequency-dependent functions, the shear storage modulus, G', and the shear loss modulus, G''. It is instructive to write the stress in an alternative form displaying the amplitude σ 0 (ω) of the stress and the phase angle δ (ω) between stress and strain:

σ = σ 0 sin (ω t + δ )

(7.2)

It is evident that each periodic measurement at a given frequency provides simultaneously two independent quantities, either G' and G'' or σ 0/γ 0 and tan δ. Sinusoidally varying quantities are usually expressed as complex values; then the modulus is also complex:

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σ = G = G '+ j G ' ' γ

(7.3)

The data from sinusoidal experiments can also be expressed in terms of a complex compliance: J=

γ σ

=

1 = J '− j J ' ' G

(7.4)

Note, that although J = 1/G, their individual components are not reciprocally related: 1 G' G' = 2 G ' +G ' ' 1 + tan 2 δ

(7.5a)

1 G' ' G' ' = 2 G ' +G ' ' 1 + 1 tan 2 δ

(7.5b)

1 J' J' = 2 J ' +J '' 1 + tan 2 δ

(7.5c)

1 J '' J '' = 2 J ' +J '' 1 + 1 tan 2 δ

(7.5d)

J'=

J ''=

2

2

or G' =

G' ' =

2

2

As an alternative to G, the relations can be expressed equally well by a complex viscosity:

η = η '− jη ' '

(7.6)

which is frequently used to describe viscoelastic liquids. The ratio of stress in phase with the rate of strain to the rate of strain is η', and η'' is the ratio of stress 90° out of phase with the rate of strain to the rate of strain. The individual components are given by:

η' =

G' '

ω

η'' =

G'

ω

(7.7a,b)

The frequency dependence of the quantities in Eqs. (7.3)-(7.7) can be imitated with mechanical models. The two basic elements are the Maxwell element (Fig. 7.2a) and the Voigt element (Fig. 7.2b), both consisting of a spring and a dashpot. For example, when a Maxwell element is subjected

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to a stress relaxation experiment, the force relaxes exponentially. Written in terms of the modulus G(t) becomes:

G (t ) ≡ G exp(− t τ )

(7.8a)

The Maxwell model depicts a viscoelastic liquid since the stress will relax to zero at long times. With a sufficient number of Maxwell elements arranged in parallel a complete relaxation spectrum can be modeled. Each Maxwell element is characterized by a certain relaxation time, τi, and a certain shear modulus, Gi: G (t ) =

∑ G exp(− t τ ) i

i

(7.8b)

The Voigt model represents a viscoelastic solid, however, both models are equivalent with an appropriate assignment of parameters. To extend the Voigt model for a set of relaxation times, an equivalent number of Voigt elements must be arranged in series.

a

b

Fig. 7.2. a The Maxwell element and b the Voigt element

For a liquid, all viscosities in the Maxwell model must be finite and one spring in the Voigt model must be zero. For a solid, one viscosity in the Maxwell model must be infinite and all springs in the Voigt model must be nonzero [1, 2]. Bulk compression and dilatation may be introduced from a propagating acoustic wave with displacement components normal to the surface of the viscoelastic material, e.g., from a Raleigh-wave propagating on SAWdevices. For this component, the bulk modulus, K, or the compliance, B, must be applied to describe the viscoelastic properties of the material. K and B are complex values in analogy to Eqs. (7.3) and (7.4). For a viscoelastic solid at equilibrium, the equilibrium Young’s modulus, Ee, is related to the (equilibrium) shear and bulk moduli, Ge and Ke, as follows [1]:

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Ee =

9Ge K e = 2Ge (1 + μ e ) Ge + 3K e

(7.9)

Here, μ = -γ22/γ11 is the relation between axial extension and lateral contraction for a perfectly elastic solid and is called Poisson’s ratio with the typical notation of indices known from “Mechanics of Materials” textbooks. For viscoelastic solids the glasslike modulus Gg and Kg may replace Ge and Ke, respectively. The shear storage modulus is a measure of the energy stored and recovered in each cycle of a periodic deformation. The frequency dependence is usually analyzed with logarithmic scales of the radian frequency, ω. One can find specific signatures where G has characteristic shapes. For acoustic sensors the most important zone is the transition zone from glasslike to rubberlike consistency. At low frequencies, G’(ω) increases rapidly for all viscoelastic liquids by several orders of magnitude. In molecular terms it corresponds to random average configurations of the macromolecular coils at low frequencies and the resumption of constraints with increasing frequency. The magnitude depends on what contour rearrangements can take place within the period of oscillatory perturbation. With increasing frequency the time necessary for long-range configuration rearrangements exceeds more and more the period of oscillation. The increase of G’ depicts the transition from rubbery-like to glassy-like consistency. Note, that this is not a transition in any thermodynamic sense. It reflects a change in the response of the system to experiments with different frequencies. Even if the magnitude of the shear modulus changes from rubbery-like to glassy-like, there is no change in the thermodynamic state of the material as it occurs when the glass transition temperature, Tg, is traversed in a thermal experiment. In the latter case the system is in equilibrium above Tg and not in equilibrium below Tg. The rise in G’ occurs in a single stage for polymers with low molecular weight (II). The transition can be separated into two stages for polymers with high molecular weight (III and IV). The main step reflects the relative motion of chain segments between the entanglement loci and corresponds to the glass-rubber transition. With decreasing frequency, the stress gradually falls as the distortion of the chain backbone adjusts itself through Brownian motion, first of the relative position segments near each other, then farther and farther apart along the backbone contour, requiring more and more mutual cooperation and therefore more and more time. This zone has been called terminal zone and the properties are dominated by the entanglement network or equivalent topological restraints, which strongly

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inhibit long-range configurational motions. Here, molecular weight or molecular weight distribution strongly influences the viscoelastic behavior. The terminal relaxation time is much longer than it would be in the absence of topological restraints and increases rapidly with molecular weight. The terminal zone of the storage modulus is displaced toward lower frequencies with increasing molecular weight. As a consequence, a plateau zone appears between the transition zone and the terminal zone, where G’ changes only slightly with frequency (G’≈105 Pa). At very high frequencies G’ approaches a limiting value of the order of 109 Pa. This value is Gg and represents the rigidity of the polymer in the absence of backbone rearrangements within the interval of the experiment. The corresponding region is called the glassy zone. Viscoelastic solids, except lightly cross-linked amorphous polymers (VI), do not show such pronounced frequency dependence. The glassy polymer (V) exhibits very little stress relaxation over many decades of frequency, since no backbone contour changes occur (G’≈109 Pa ), whereas the highly crystalline polymer (VIII) shows some relaxation only at very long times (G’≈108-109 Pa). For cross-linked rubbers (VI, VII) G’ approaches a limiting value of about 105-106 Pa at low frequencies, which is characteristic for rubber-like elasticity. The shear loss modulus is a measure of the energy dissipated or lost as heat in each cycle of sinusoidal deformation. The common feature of G’’(ω) can be stated qualitatively as follows. In frequency regions where G’(ω) changes slowly, the behavior of polymers is more or less nearly elastic. In such regions G’’(ω) tends to be considerably less than G’. Hence comparatively little energy is dissipated. Consequently the curves of the glassy polymer (V) and crystalline polymer (VIII) are relatively flat throughout the entire frequency range (G’’≈107 Pa). The flat parts of the G’-curves at frequencies below the transition zone for amorphous polymers (III) and those with long side groups (IV) or cross-linking (VI, VII), which exhibit entanglement coupling, is also accompanied by a plateau or a minimum in G’’ (≈103-105 Pa). At very high frequencies G’’(ω) does not approach zero, although one can expect perfect elastic behavior from the mechanical model. Atomic adjustments are still capable of dissipating energy within the period of deformation, although losses due to backbone configuration changes diminish (up to ≈108 Pa). At frequencies where G’(ω) changes rapidly, maxima in G’’ appear and G’ ≈ G’’. The relaxation time of configuration rearrangements corresponds to the frequency of the periodic mechanical perturbation and the energy of deformation can be efficiently dissipated and used to realize them.

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At very low frequencies, G’’ for viscoelastic liquids is directly proportional to ω with a slope approaching 1 on a logarithmic scale. This is the Newtonian steady-flow viscosity, η0. Only for a simple Newtonian liquid, G’’ = ωη0 over the entire frequency range. The in-phase component of the complex dynamic viscosity, η’ in Eq. (7.6), is most useful for viscoelastic liquids. From Eq. (7.7), in regions where G’’ is flat, η’ is inversely proportional to frequency, whereas when G’’ rises steeply η’ may flatten out, as particularly for the cross-linked rubber (VI, VII). However, η’ decreases monotonically with increasing frequency, altogether by many orders of magnitude in the entire frequency range. For the dilute solution (I), the steady-flow viscosity, η0, is about eight times that of the solvent; at high frequency it approaches a limiting value slightly higher than that of the solvent. A very useful dimensionless parameter is the loss tangent. It is a measure of the ratio of energy lost to energy stored in a cyclic deformation: tan δ =

G' ' J ' ' = G' J '

(7.10)

The loss tangent reveals several characteristics. For the dilute solution (I) tan δ is very high. Both solvent and solute contribute to G’’ but only the solute contributes to G’. The minimum is much more pronounced than the transition in η’. All uncross-linked polymers have a large loss factor at low frequency and tan δ becomes inversely proportional to the frequency. The glassy polymer (III) and the crystalline polymer (VIII) have values in the neighborhood of 0.1. All amorphous polymers, whether cross-linked or not, have values in the transition zone between the glasslike and rubberlike consistency of about 1. The loss tangent goes through a pronounced maximum, which is on the right of the maximum in J’’ and on the left of the maximum in G’’. Each of the three functions is a measure of energy dissipation, but the frequency region in which the “loss” occurs, depends on the choice of function [1]. The loss tangent determines macroscopic physical properties such as the damping of a free vibration or attenuation of a propagating wave.

7.4 The Temperature-Frequency Equivalence In Sect. 7.3 the viscoelastic functions have been discussed in terms of time and frequency. Just as the dependence on time or frequency, the dependence

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of viscoelastic functions on temperature is most spectacular in the transition range. The glass transition is the point or narrow region on the temperature scale, where the thermal expansion coefficient, α, undergoes a discontinuity and below which configuration rearrangements of the polymer backbones are extremely slow. It corresponds to a change in the slope in a plot of the specific volume against temperature, as illustrated schematically in Fig. 7.3. From the relaxation times point of view, ωτi >> 1 in the glassy state at low temperatures. Practically no configuration changes occur within the period of deformation. With increasing temperature τi decreases and ωτi ≅ 1 in the transition zone around Tg. The configurational modes of motion within the entanglement coupling points become fast enough to occur within the period of deformation. In the rubbery state at high temperatures τi is further decreased, hence ωτi << 1, and configuration rearrangements within the entanglements are fast enough to keep the material in equilibrium.

α v(T)/vg

Δα

f

v (T) vg

1 fg

αg

αg 0

αf

Tg(∞)

v0(T)/vg α0

Tg(t)

T/K

Fig. 7.3. Schematic variation of total (specific) volume, v, occupied volume, v0, and free volume, vf, (relative to specific volume, vg, at the glass transition temperature, Tg) with temperature, T. f = vf/vg is the fractional free volume, α is the thermal expansion coefficient. Adapted from Ferry [1] and Kovacs [8]

This behavior can be readily understood in terms of the free volume concept. In this concept the total volume per gram, v, is the sum of the occupied volume, v0, and the free volume, vf. The occupied volume includes

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not only the volume of the molecules as represented by their Van-derWaals radii; it also includes the volume associated with vibrational motions. The occupied volume increases with temperature, given by the thermal expansion coefficient, α0. It cannot be identified with the Van-derWaals co-volume and hence can be estimated only indirectly. The free volume can be understood as holes of the order of monomeric dimensions or smaller voids associated with packing irregularities. The increased thermal expansion coefficient above Tg represents, primarily, the creation of free volume with rising temperature. Above the glass transition temperature, Tg, αf has a magnitude characteristic for liquids in the order of 10-3 deg-1. Below Tg, αg is smaller by a factor of one-half to one-third. At temperatures high enough so that Brownian motion is rapid in a viscoelastic liquid or soft solid, a lowering of temperature is accompanied by a collapse of free volume as the molecular adjustments take place within the time scale of cooling. At lower temperatures, the adjustments are slower and the collapse can no longer occur within the time scale of cooling. The residual contraction is of solid-like character, and αg is only a little or possibly no larger than α0. It is inherent in the above discussion that Tg must depend on the time scale of the volumetric measurement. A slower measurement pattern leads to a lower value of Tg. The appearance of the glass transition results from the reduction of molecular mobility as the temperature falls, slowing the collapse of free volume. The mobility at any temperature depends primarily on the free volume remaining. The shift on the temperature scale as a result of the increased measuring frequency is a general phenomenon and can be described as the timetemperature correspondence principle of polymer relaxation:

G (T0 , t ) = G (T , t aT )

(7.11)

T0 may act as reference temperature, aT, is the shift factor. The function log aT exhibits similar behavior for all amorphous polymers and can be expressed as:

log aT = log

ω 0 − C1 (T − Tg ) = ω C 2 + T − Tg

(7.12a)

Equation (7.12a) is known as the WLF equation [5]. It is a scaled temperature-frequency relation of characteristic relaxation processes. C1 and C2 are originally thought to be universal constants (C1 = 17.4, C2 = 51,6 K). Equation (7.12a) can be rearranged using C1 = log(Ω/ω0) and C2 = T0 - T∞:

7 Viscoelastic Properties of Macromolecules

⎛Ω⎞ ⎞ ⎟ = (T0 − T∞ ) log⎜⎜ ⎟⎟ ⎝ω ⎠ ⎝ ω0 ⎠

(T − T∞ )log⎛⎜ Ω

217

(7.12b)

Equation (7.12b) defines a set of hyperbolas with common asymptotes at T∞ (the so-called Vogel temperature) and log Ω (Ω ≈ 1012...1015 Hz for simple glass formers). Figure 7.4 depicts Eq. (7.12b). The set of hyperbolas is parameterized by one value, T0 at a given mobility (log ω0 < log Ω) or ω0 at a given temperature T (T0 > T∞). C1 and C2 depend on these reference points. For ω/(2π) ≈ 10-2 Hz T0 ≈ Tg and C1 can be approximated to be in the range of 10 to 20, whereas C2 varies quite widely (e.g. PIB: C1 = 16,6 and C2 = 104 K) [6]. At low temperatures, the glass transition curve in Fig. 7.4 is very steep and effects are of thermodynamic nature. At high frequencies the curve is rather flat, effects are of kinetic nature. log ω Tg

Tα

T log Ω

g*

g

t

r

´

log ωexp

T∞

Fig. 7.4. WLF-equation where r indicates the rubbery zone, t the transition range, g the glassy consistency and g* the glassy zone. Adapted from Donth [3]

Finally, a distinction has been made in Fig. 7.4 between glassy zone and glasslike consistency. The glassy consistency is that part of the glassy zone, where τi >> texp. The mobility relevant for the glass transition is characterized by relaxation times much larger than the experimental time. The glassy zone is characterized by ωexp >> ωi >> 1/texp: The experimental time is large enough to realize molecular rearrangements; however, the probing frequency is much too high to respond to the perturbation. Performing a dynamic experiment in a large temperature range including Tg, the border

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between glassy zone and glassy consistency can be recognized if a change in certain thermal, mechanical or geometric properties have an influence on the signal [1]. The time-frequency equivalence principle is nothing else than a conclusion from the qualitatively similar behavior of experimental findings during the glass transition, e.g., the value of the shear modulus. The response keeps its characteristics when shifted along the glass transition curve in Fig. 7.4 due to a change in temperature in a very broad frequency range of several orders of magnitude. It indicates a basically similar relaxation process, independent of whether it takes place at low temperatures at a mHz frequency range or at high temperatures in the MHz frequency range. It is therefore common-sense to reduce the response curves at different temperatures, which is technically speaking a temperature dependent shift along the log ω-axis, the so-called master curve construction. The master curve is, in the optimal case, a universal curve with a reduced abscissa log (aTω). The variable log (aTω) describes the dependence on log ω in the isotherm case (T = const), and on T in the isochronous case (ω = const). After some “equalization” of the hyperbola one gets similar curves for the log ω- and the T-dependence [1, 6]. The absorption of vapor molecules, presumed to be of low molecular weight in comparison with the polymer and molecularly dispersed, causes dilution of the polymer. The effect of the diluents on polymer viscoelastic properties can be understood as a generation of additional free volume in proportion to the volume fraction, V. From that point of view solvent absorption has an effect similar to a temperature increase: f (T ) = f 0 + α f (T − T0 )

(7.13)

f (T , V1 ) = f 2 (T ) + β 'V1

(7.14)

In the former expressions, f is the fractional free volume (free volume related to the volume at Tg), the index 1 stands for the analyte, index 2 for the polymer and β ' is a parameter relating the volume fraction of the analyte to the free volume and is marginally smaller than the fractional free volume of the analyte as a liquid. At low analyte concentrations (V1 << 1) the contributions to the free volume of polymer and analyte are additive, approximately. The difference arises from the fact that the increase in volume from temperature increase directly influences the modulus, whereas solvent absorption influences the modulus in relation to the created additional free volume. Temperature effect, Eq. (7.13), and solvent absorption effect, Eq. (7.14), can be combined and result in a modified WLF-equation [7]:

7 Viscoelastic Properties of Macromolecules

⎛ α f (T − T0 ) + β 'V1 ⎞ ⎟ log aT = −C1 ⎜ ⎜ F0 + α f (T − T0 ) + β 'V1 ⎟ ⎝ ⎠

219

(7.15)

Analog to the temperature-frequency relation, changes in viscoelastic properties with vapor absorption can be expressed as a translation aT in the polymer relaxation time or the apparent probing frequency. As the probing frequency shifts the activation of a certain relaxation process to higher temperatures, solvent absorption results in the opposite shift. Hence, relaxation processes, which are too slow to respond to high frequency mechanical perturbations, may be activated after solvent absorption.

7.5 Conclusions The acoustic wave propagating in a coating is a mechanical perturbation of the film. In case of a viscoelastic film, the polymer chain segments try to relax back to equilibrium conditions. As long as the characteristic relaxation time,τ, is much longer than the period of the oscillation, the relaxation is “frozen”. Molecular rearrangements on a long-range scale are severely restricted. Distortions generated by external stress, being of a rather high energy, result in the high storage modulus. This is the glassy state. In this state the material is rigid. The shear storage modulus is about 109 Pa and decreases slowly with temperature. The shear loss modulus is significantly lower but increases with temperature. The rubbery state is characterized by a deformation on a time scale larger than the relaxation time. All configurational modes of motion within the entanglement coupling points can freely occur. Distortions generated by the external stress, being of a much lower energy and G' is in the order of 106 Pa and G'' < G'. Only at still higher temperatures more and more chains can escape from topological restraints and the material is capable to flow. The temperature of the transition from glassy to rubbery consistency is the glass transition temperature. The so-called dynamic glass transition, Tα, must be applied for acoustic measurements; at this temperature ωτ ≈ 1. Tα at probing frequencies, typically for acoustic devices, is a few tens of degrees higher than Tg, the static glass transition temperature, which is known, e. g., from differential scanning calorimeter (DSC) measurements. Changes in the intrinsic polymer properties do not appear directly as changes in the electrical properties of the composite resonator. They become effective via acoustic properties of the coating through the effective

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surface impedance experienced by the resonator at its interface to the coating. Performing a thermal experiment with a reasonably slow cooling rate and further assuming an ideal viscoelastic film one can expect a major response of an acoustic wave device around the dynamic glass transition temperature. However, due to the transduction mechanism, a strong response from acoustic devices can also be generated from rather small material property changes within the rubbery state. They usually have an acoustic origin. By contrast, changes within the glassy state usually do not appear as significant sensor responses of quartz crystal resonators but contribute to the sensor signal of surface acoustic wave devices. The change in the thermal expansion coefficient around Tg can hardly be recognized with an acoustic wave device, because the thickness-density product does not change. Effects at this temperature are mostly generated from deviations from the ideal behavior of the viscoelastic film, e.g., residual or thermally generated static stress. As discussed above, the different time scales in the experiment (acoustic wave frequency on the one hand and cooling/heating rate on the other hand; the same holds for solvent penetration, oxidation/reduction etc. in other kinds of experiment) allow for relaxation processes to take place and to be monitored, although they seem to be frozen on the time scale of the acoustic wave. In contrast to the more or less broad temperature range of the glass transition, the melting of the highly ordered regions in crystalline polymers or phase transitions in liquid crystals are not subjected to the time-frequency equivalence. They are characterized by a sharp change in G', usually recognized as a sharp frequency change at a temperature independent of the probing frequency and therefore similar to textbook values.

7.6 Shear Parameter Determination Due to the significant contribution of viscoelastic properties of thin films on the sensor response in the non-gravimetric regime (Chapter 3) acoustic sensors provide the chance to determine the complex shear modulus (compliance, viscosity) of thin films at high probing frequencies. Whereas the determination of surface mass or increase in surface mass with acoustic sensors is unambiguous in the gravimetric regime, the determination of G' and G'' (J' and J'' or η' and η'') introduces the problem of ambiguity. Acoustically speaking the sensor response is determined by the load impedance, also a complex value, i.e., by two independent values. The load impedance is, however, a function of four material and geometric properties (real and imaginary part of the modulus, density and thickness). In the

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typically measured frequency range while performing impedance analysis the frequency dependence of the acoustic load impedance is too weak to increase the number of measurable values. The problem can be overcome employing overtone measurements. However, performing the experiments at a different frequency brings the frequency dependence of the modulus into play. Rules must be introduced which describe the latter to avoid increasing the number of unknowns in the same manner as the number of measurement values. Those rules can be found based on de Genne's scaling concept [9]. The basic idea behind this methodology is a scaled dependence of the viscoelastic properties on frequency. In practice additional challenges may arise because the transduction scheme of real sensors tends to require also modified sensor properties, for example the acoustically effective area. Full determination of the elements of the equivalent circuit is required as described in Chapter 14. For a rough estimation of the film thickness, viscoelastic state of a coating (glassy, rubbery) or changes of this state due to analyte sorption or temperature change one can apply the following method to reduce the number of unknowns. Firstly, the density must be taken from literature. Secondly the loss tangent, tan δ (Eq. 7.10) must be fixed. For both material and mathematical reasons it is advised to apply tan δ = 0.1, tan δ = 1 and tan δ = 10. This range should cover typical material properties. The determination of film thickness, h, and shear modulus, G' and G'' becomes unambiguously. Of course, there is still an infinite number of possible solutions for h, G' and G''; however, all solutions are located in a range allowing for first valuable conclusions if the experimental conditions have been properly chosen.

References 1. J.D. Ferry (1980) “Viscoelastic properties of polymers” John Wiley & Sons, New York 2. J.J. Aklonis, W.J. MacKnight (1983) “Introduction to polymer viscosity”, John Wiley & Sons, New York 3. E. Donth (1992) “Relaxation and thermodynamics in polymers. Glass transition” Akademie-Verlag, Berlin 4. Elias Hans-Georg (1997) “Introduction to polymer science”, VCH, Weinheim 5. M.L. Williams, R.F. Landel and J.D. Ferry (1955) “The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids” J. Amer. Chem. Soc. 77:3701-3706 6. N.G. McCrum, B.E. Reed and G. Williams (1967) “Anelastic and Dielectric Effects in Polymeric Solids” John Wiley & Sons, New York

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7. S.J. Martin, G.C. Frye, S.D. Senturia (1994) “Dynamics and response of polymer-coated surface acoustic wave devices: Effect of viscoelastic properties and film resonance” Anal. Chem. 66:2201-2219 8. A.J. Kovacs (1964) “Transition vitreuse dans les polymères amorphes. Etude phénoménologique” Adv. Polym. Sci. 3:394-507 9. P.G. de Gennes (1993) “Scaling concepts in polymer physics” Cornell Univ. Press, Ithaca

8 Fundamentals of Electrochemistry Christopher Brett Departamento de Química, Universidade de Coimbra

8.1 Introduction Electrochemistry involves chemical phenomena associated with charge separation, usually in liquid media such as solutions. The charge separation often leads to charge transfer, which can occur homogeneously in solution between different chemical species, or heterogeneously on electrode surfaces. In order to ensure electroneutrality, two or more charge transfer half-reactions take place simultaneously, in opposing directions: oxidations (loss of electrons or increase in oxidation state) and reductions (gain of electrons or decrease in oxidation state). In the case of heterogeneous redox reactions, the oxidation and reduction half-reactions are separated in space, usually occurring at different electrodes immersed in solution in a cell. The electrodes are linked by conducting paths both in solution (via ionic transport) and externally (via electric wires etc.) so that charge can be transported and the electrical circuit completed. If the cell configuration permits, the products of the two electrode reactions can be separated. When the sum of the free energy changes at both electrodes is negative the electrical energy released can be harnessed (batteries, fuel cells). If it is positive, external electrical energy can be supplied to overcome the positive free energy and oblige electrode reactions to take place and convert chemical substances (electrolysis). It is clearly useful to be able to investigate the fundamentals of these electrode processes in the laboratory and this can be done by careful control of the electrode reactions.

8.2 What is an Electrode Reaction? An electrochemical cell contains two electrodes where reactions can occur and which form an electrical circuit linked externally through conducting A.A. Vives (ed.), Piezoelectric Transducers and Applications, doi: 10.1007/978-3-540-77508-9_8, © Springer-Verlag Berlin Heidelberg 2008

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wires and internally through a conducting solution. In order to understand electrode reactions, we focus on the processes occurring at just one of these electrodes. An electrode reaction involves the transfer of charge to or from an electrode; the electrode process may include charge transfer steps and chemical reactions of the products or intermediates formed. The reactions take place in the interfacial region between electrode and solution, the region where the charge distribution differs from that of the interior of the phases. Different types of electrode reaction are: oxidation or reduction of a solutionsoluble species, oxidation or dissolution of the electrode material, and electrodeposition of a species from solution by reduction. The simplest example of an electrode process is a one-step reaction in which the electrode material itself is inert, i.e. the electrode acts only as a source (for reduction) or a sink (for oxidation) of electrons transferred to or from species in solution. This reaction can be written O+ne− = R

(8.1)

where O and R are the oxidized and reduced species, respectively. At each electrode, charge separation can be represented by a capacitance and the difficulty of charge transfer by a resistance. For electron transfer to occur, there must be a correspondence between the energies of the electron orbital where transfer takes place in the donor and acceptor. In the electrode, this level is that of the highest filled orbital, which in a metal is the Fermi energy level, EF. In soluble species it is simply the energy of the orbital of the valence electron to be given or received, Eredox. • For a reduction, there is a minimum energy that the transferable electrons from the electrode must have before transfer can occur, which corresponds to a sufficiently negative potential (in volts). • For an oxidation, there is a maximum energy that the lowest unoccupied level in the electrode can have in order to receive electrons from species in solution, corresponding to a sufficiently positive potential (in volts). A scheme of the electron transfer at an inert metal electrode is shown in Fig.8.1. The change in charge distribution from the bulk in this region means that the relevant energy levels in reacting species and in the electrode are not the same as in the interior of the phases, and soluble species need to adjust their conformation for electron transfer to occur. These effects should be corrected for in a treatment of kinetics of electrode processes.

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EF

225

e-

Eredox

e-

Eredox

EF

Oxidation Reduction Fig. 8.1. Scheme of electron transfer between an inert metallic electrode and a redox species in solution. The externally controlled potential alters the Fermi energy, i.e. the energy of the highest occupied electronic energy level in the electrode, the Fermi energy, EF, facilitating reduction (more negative potential) or oxidation (more positive potential).

It must be remembered that there must always be a complete electrical circuit, with two electrodes. If a reduction process occurs at one of the electrodes (the cathode) then oxidation occurs at the other electrode (the anode). Note also that by external control of the potential (see below) the process occurring at the electrode of interest (the working electrode) can be changed from reduction to oxidation and at certain values of controlled potential it may be a mixture of both processes.

8.3 Electrode Potentials It is useful to have a scale which measures the potentials, i.e., energies, necessary for the transfer of electrons associated with a particular electrode reaction, such as shown in Eq. (8.1). By convention, electrode reactions are written as reductions. Each has an associated standard electrode potential, E,, measured relative to the standard hydrogen electrode (SHE) with all species at unit activity (ai = 1), at 25 ºC and at 1 bar pressure. For half-reactions at equilibrium, the potential, Eeq, is related to the standard electrode potential through the Nernst equation νi RT ΠaOi Eeq = E + ln nF ΠaνRi i

(8.2)

where νi are the stoichiometric numbers of each species in the electrode reaction and F the Faraday constant, the charge of one mole of electrons. For example, in the reaction MnO4– + 8H+(aq) + 5e- = Mn2+ + 4H2O

(8.3)

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the stoichiometric number of MnO4- is 1, of H+(aq.) 8 etc. (and n is 5). The tendency for the reduction to occur, relative to the SHE reference, is given by UG, = -nF E,

(8.4)

under standard conditions, where UG, is the change in free energy. The more negative the value of UG, the more easily the reduction occurs. Thus, for example, Group IA metals, which have very negative values of E,, tend to oxidise. It is often useful to be able to employ concentrations, ci, instead of activities, particularly since electrochemical experiments are carried out with known concentrations of species in solution and the number of electrons transferred is directly proportional to the amount (i.e. the local concentration) and not the activity. The Nernst equation is rewritten as Eeq = E

'

RT Π[O i ] i + ln nF Π[R i ]ν i ν

(8.5)

in which E,' is the formal potential, dependent on the medium since it includes logarithmic activity coefficient terms as well as E,. The relation between activities and concentrations of species is ai = γici, where the proportionality coefficient, γi, is the activity coefficient. The value of γi changes with concentration, since it reflects the result of interactions between chemical species which are moving in solution, under the influence of an electric field. It should be recognised that activities can also be influenced by “third party” species which do not form part of the reaction scheme. If the oxidised and reduced species involved in an electrode reaction are in equilibrium at the electrode surface, the Nernst equation can be applied. The electrode reaction is then known as a reversible reaction since it obeys the condition of thermodynamic reversibility. Clearly, the applicability of the Nernst equation, and therefore reversibility, depends on the time allowed for the electrode reaction to reach equilibrium.

8.4 The Rates of Electrode Reactions The rate of an electrode process is determined by the transport of species to the electrode surface and by the kinetics of electron transfer as well as being influenced by the structure of the interfacial region close to the electrode. Therefore, an electrode reaction can be divided into several steps. The first step is that the species that will react must diffuse from bulk solution,

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the rate of which is often described by the mass transfer coefficient, kd. When this mass transport is slower than the kinetics of the electrode reaction, the reaction is described as reversible. It can be seen immediately that increasing the mass transport, usually by externally imposed convection, can lead to a reversible reaction becoming influenced by the kinetics and moving into the so-called quasi-reversible regime. The kinetics is usually expressed by the following equations, for an oxidation ka = k0 exp[αanF(E - E,') /RT]

(8.6)

where k0 is the standard rate constant, the value of ka when E = Eeq; and αa is the anodic charge transfer coefficient whose value lies between 0 and 1 and is often around 0.5 for metallic electrodes, being a function of the position of the activation energy barrier. For a reduction, in a similar way kc = k0 exp[-αcnF(E - E,') /RT]

(8.7)

with αc the cathodic charge transfer coefficient. Thus, the criterion for a reversible reaction is

k 0 >> k d

(8.8)

and for an irreversible reaction

k0 << kd

(8.9)

the situation where the electrode reaction cannot be reversed. A high kinetic barrier has to be overcome, which is achieved by application of an extra potential (extra energy) called the overpotential. Quasi-reversible reactions exhibit a behaviour intermediate between reversible and irreversible reactions, the overpotential having a relatively small value, so that with this extra potential, reactions can be reversed. Rates of electrode reactions, measured experimentally as the current passed, are directly proportional to the product of rate constant and reagent concentration. The dependence of current, I, on potential is exponential when kinetically controlled, i.e., there is a linear relation between lg I and the potential - this is the Tafel relation. However, the consumption of reagents leads to concentration gradients and the rate of reaction begins to be limited by the mass transport of reagent, as the potential applied becomes more negative (reduction) or more positive (oxidation). Eventually a transport-limited current is reached. The importance of the interfacial region can be understood by referring to Fig.8.2. It can be seen that the presence of species adsorbed on the electrode surface conditions the access of other species from bulk solution as

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well as the energy necessary for electrons to be transferred. If the ionic strength of the solution is high enough, usually achieved by addition of a high concentration (often of the order of 0.1 mol dm-3) of inert electrolyte then the interfacial region is thin which confines these effects to very close to the electrode. An additional benefit is that nearly all the current passing through the electrical circuit is carried by the inert ions so that effects of migration of the electroactive species (i.e. the species that will react) can be neglected.

Electrode

Solution

IHP OHP

Fig. 8.2. Representation of the structure of the interfacial region at an electrode in aqueous solution, showing specific adsorption of unsolvated anions at the inner Helmholtz plane (IHP), adsorption of solvated cations at the outer Helmholtz plane (OHP) followed by the diffuse double layer in which there is some ordering of the ions but they are free to move.

For reversible reactions only thermodynamic and mass-transport parameters can be determined, for quasi-reversible and irreversible reactions both kinetic and thermodynamic parameters can be measured. It should also be noted that the electrode material can influence the kinetics of electrode processes; it should not be assumed that electrode materials are inert. For example, platinum and gold form oxides at potentials of ~ 0.8 V vs. the saturated calomel reference electrode. More complex electrode processes involve consecutive electron transfer or coupled homogeneous reactions. The theory of these reactions is also more complicated, but they correspond to a class of real, important reactions, particularly involving organic and biological compounds. The electrode material can play an even more important role in these cases.

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As mentioned above, the rate of an electrode reaction is measured by the current, I, which is equal to the rate multiplied by the concentration of the species being oxidised or reduced at the electrode surface. Currents due to oxidation (anodic currents) have a positive sign and currents due to reduction processes (cathodic currents) have a negative sign. The total charge passed, Q, the integral of the current over the time period considered, is therefore proportional to the total amount of species that has reacted. This is a statement of Faraday’s Law. In the case of deposition on or dissolution of a solid, such as an electrode, the charge passed corresponds to a mass increase or decrease, respectively. For metal deposition or dissolution, the charge can be related to the change in mass by

Q = −nF

Δm M

(8.10)

where (Δm/M) represents the number of moles of species deposited or dissolved with M the molar mass and Δm the mass change. The negative sign in Eq. (8.10) arises because a negative charge due to reduction of the metal ions to the metal leads to a positive mass increase, and a positive charge for metal oxidation corresponds to a decrease in mass. A faradaic efficiency of 100% means that all the current passed is used in the electrode reaction being considered. If there are alternative competing electrode reactions then the faradaic efficiency will be lower – this is extremely important in the measurement of mass changes in the electrochemical quartz crystal microbalance.

8.5 How to Investigate Electrode Reactions Experimentally In order to study electrode reactions, reproducible experimental conditions must be created which enable minimization of all unwanted factors that can contribute to the measurements and diminish their accuracy. Normally it is desirable to suppress migration effects, confine the interfacial region as close as possible to the electrode, and minimise solution resistance. These objectives are usually achieved by addition of a large quantity of inert electrolyte (around 1 mol dm-3), the electroactive species being at a concentration of 5 mM or less. The electrochemical cell contains three electrodes: the working electrode, the auxiliary electrode and a reference electrode. The auxiliary electrode completes the electrical circuit and should be inert; it is often placed

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in a separate compartment connected to the working electrode compartment by a salt bridge or a porous frit. The reference electrode serves to control the potential of the working electrode and should not pass current. Commonly used reference electrodes are shown in Table 8.1, given the impracticalities in normal use of the standard hydrogen electrode. The calomel electrode (Hg2Cl2) in the third line is usually used with saturated KCl and is then known as the saturated calomel electrode (SCE). Control of the experiment is achieved using a potentiostat instrument. The potential applied to the working electrode is controlled with respect to the reference electrode. The current which passes through the electrical circuit linking the working electrode and the auxiliary electrode as a result of the applied potential is measured by means of a current follower in the instrument and appropriate signal treatment. Currents due to oxidation (anodic currents) have a positive sign and currents due to reduction processes (cathodic currents) have a negative sign. The study of the current vs. potential profile is called voltammetry and the data is usually presented as a plot of current, I, on the y-axis vs. applied potential, E, on the x-axis, known as a voltammogram. If the working electrode is made as a conductive coating on a piezoelectric crystal, then mass changes will usually occur over the same range of potentials as oxidation and reduction. Occasionally, the current at the working electrode is controlled using a galvanostat, and the reference electrode is employed to measure the potential of the working electrode, usually as its variation with time (chronopotentiometry). Electrode processes can be modelled as electrical equivalent circuits of varying complexity. The simplest electrode process, i.e. an electron transfer, is modelled in Fig. 8.3, where Rct represents the resistance to charge transfer across the electrode-solution interface, Cd the capacity due to charge separation in the interfacial region (in solution and also within the electrode for semiconductors) and RΩ the resistance of the remainder of the electrochemical cell. Table 8.1. Reference electrode electrochemical reactions in water solvent Electrochemical reaction AgBr + e- = Ag + BrAgCl + e- = Ag + ClHg2Cl2 + 2e- = 2Hg + 2ClHgO + H2O + 2e- = Hg + 2OHHg2SO4 + 2e- = 2Hg + SO42-

E, / V 0.071 0.222 0.268 0.098 0.613

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Rct RΩ

Cd Fig. 8.3. Electrical equivalent of a simple electrode process: Rct charge transfer resistance, Cd capacity of interfacial region and RΩ the cell resistance

8.6 Electrochemical Techniques and Combination with Non-Electrochemical Techniques Different types of electrochemical techniques are necessary since usually a complete study of an electrode process requires measurement of kinetic as well as of thermodynamic parameters, and study of the details of changes occurring on the electrode surface, such as adsorption or deposition. Measurement of the natural potential of the electrode vs. a reference electrode, the open circuit potential, with time can also give extra information in the case of reactions involving adsorption, deposition or corrosion. An increasing importance of kinetic control of the electron transfer steps of the electrode process can be achieved in the following ways:

• steady state methods: microelectrodes by decreasing size; hydrodynamic electrodes by increasing convection; • linear sweep and cyclic voltammetry methods: by increasing the applied potential sweep rate; • step and pulse techniques (e.g,. potential step, pulse voltammetry, square wave voltammetry): by increasing the pulse amplitude and/or frequency, • electrochemical impedance methods: increasing the perturbation frequency, registering higher harmonics, etc. The type of technique chosen will depend on the timescale of the electrode reaction. Examples of results obtained using these four classes of technique will now be given.

Steady state methods. A steady state, referring to an electrode reaction, means that the current does not change with time, because the amount of species reacting at the electrode surface does not change. This is most easily achieved by imposed convection, such as with a rotating disk electrode

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i.e. a circular disk electrode embedded in the centre of the face of a cylinder rotating about its axis in solution. A steady state can also be achieved with microelectrodes, which are defined by possessing at least one dimension less than 50 µm (i.e. disk, band, ring etc.), since the very high diffusion field due to radial as well as perpendicular diffusion to the electrode surface can lead to concentration gradients that do not vary with time. The characteristics of a typical steady-state voltammogram are shown in Fig.8.4.They are normally recorded by slowly sweeping the applied potential (several mV s-1 to ensure steady-state conditions) whilst the convection is imposed The transport-limited current, IL, is proportional to the concentration of species and the value of the half-wave potential, E1/2, where the current reaches half of its maximum identifies the species, thus conferring selectivity. 1.0

I / IL

0.8

IL

0.6 0.4 0.2 0.0 -0.2

-0.1

0.0

0.1

0.2

n(E - E1/2) / V

Fig. 8.4 Steady-state voltammograms for oxidation of a chemical species R by a reversible reaction, showing the half-wave potential E1/2 and the transport limited current IL.

Linear sweep and cyclic voltammetry methods. In these methods, usually at stationary electrodes without forced convection, the applied potential is swept at different rates and the corresponding current is recorded. The current begins to rise as the potential is swept but then decays again as the amount of species close to the electrode that can react becomes depleted. Inversion of the direction of the potential sweep can result in the inverse reaction and a corresponding current peak. As an example, the cyclic voltammetry of ferrocyanide ion is shown in Fig. 8.5 and demonstrates an oxidation process on the forward scan (Fe(CN)64- → Fe(CN)63-) and the

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reverse reduction process on the inverse scan of potential. Information can be gained from the position of the peaks on the potential scale and from their height and width. Increasing the scan rate increases the possible influence of the electrode kinetics. Cyclic voltammetry is very much used as a diagnostic tool for investigating the mechanism, as well as the kinetics, of both simple and complex electrode processes. It is also commonly used in combination with the quartz crystal microbalance. 300 200

I / μA

100 0 -100 -200 0.0

0.1

0.2

0.3

0.4

0.5

0.6

E / V vs. SCE Fig. 8.5. Cyclic voltammograms at carbon film electrode after background subtraction for oxidation of 1 mM K4Fe(CN)6 in 0.4 M K2SO4 electrolyte at scan rates of 20, 50, 100, 200 and 500 mV s-1 (adapted from Electroanalysis, 13 (2001) 765).

Step and pulse techniques. The basis of this set of techniques is the potential step. A step in potential into or within a region where an electrode reaction can occur leads to a sudden increase in current, due to both the electrode reaction (faradaic current) and to the change in structure of the interfacial region (capacitive current). The faradaic current decays with time (usually with the square root of time) as the system relaxes and can be sampled at a pre-chosen time after the potential step, when the variation of current with time is relatively small and the capacitive current, which decays faster than the faradaic current, is very small. Potential steps can be combined to make pulse waveforms in many ways. These types of potential waveform are easily implemented with modern digital potentiostats. The most common pulse techniques are differential pulse voltammetry and square wave voltammetry. In both cases the difference between two

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sampled current measurements is recorded so that the response appears as a peak. The peak potential identifies the electrode reaction, the peak height is proportional to concentration and the peak width at half height can give information about the kinetics A square wave voltammogram of a reversible electrode reaction is depicted in Fig. 8.6. Usually the “forward” and “backward” currents are both recorded separately and separately as well as their difference, since, in this way, further information can often be obtained about the electrode process. Typical square wave frequencies vary from 10-200 Hz, with staircase step increments between 2 and 5 mV; pulse amplitudes are 50/n mV where n is the number of electron transferred in the electrode reaction. Increasing the square wave frequency decreases the timescale so that the influence of kinetics is more readily seen.

(a) 1

1

1

1

E 2

a

2

2

t

(b) 1.0 0.8

I / Ip,net

0.6 0.4

I(1)

0.2 0.0

I(2)

-0.2 -0.4 -0.4

b

-0.2

0.0

0.2

0.4

n(E - E1/2) / V

Fig. 8.6. Square wave voltammetry of a reversible reaction. a Applied potential waveform – a square wave superimposed on a staircase – showing the current sampling near the end of each potential step for “forward” (1) and “backward” (2) pulses. b Square wave voltammograms for the “forward” and “backward” pulses – the continuous line represents Inet= (I(1)-I(2)).

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Electrochemical impedance. Electrochemical impedance spectroscopy involves the measurement of the impedance of the electrochemical cell over a wide frequency range which can typically vary from 100 kHz down to 1 mHz. The interplay between the current passing through a resistor and capacitor in parallel changes with frequency as is easily seen from the simple model of the interfacial region shown in Fig. 8.3. Experimentally the spectra are usually obtained by superimposing a small amplitude alternating voltage (small to ensure a linear response in the current) on a fixed value of potential and the phase and magnitude of the current response are obtained. The data obtained are then often fitted to an electrical equivalent circuit, which must have a physical meaning. An example is shown in Fig. 8.7: the equivalent circuit (inset in Fig. 8.7a) is the same as that of Fig. 8.3 representing the electrode-solution interface, plus two more elements: a resistance representing the resistance of the rest of the electrochemical cell (conducting wires, solution etc.) and a capacitance in series which represents an adsorbed species. The physical interpretation of the equivalent circuits is particularly important or no meaningful information can be obtained. Alternative kinetic and transmission line models have also been used for spectra interpretation. Extension of the electrochemical impedance concepts includes the registering of harmonics and the use of large perturbations with non-linear responses. The technique of ac voltammetry involves scanning the applied potential on which the alternating perturbation is superimposed and registering the current – a peak is obtained which can be analysed in a similar way to that obtained from the pulse techniques. Non-electrochemical methods can, and should, be used for studying electrode surfaces and the interfacial region structure, particularly in situ in real time where this is possible. These can include surface analysis techniques such as atomic force and scanning tunnelling microscopy, infrared reflectance, ellipsometry, etc. Hyphenated techniques in which electrochemistry is used in conjunction with another technique also constitute a powerful tool. The most important of these in the context of piezoelectric transducers is the electrochemical quartz crystal microbalance (EQCM), which can be used for monitoring mass changes of the electrode surface as well as any viscoelastic effects whilst controlling the applied potential. The principles and functioning of the EQCM and how the QCM changes in frequency can be used to give information which is complementary to the electrochemical data is described in detail in other chapters. The coupling of power ultrasound with electrochemistry is called Sonoelectrochemistry, and will be described in Chapter 15.

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(a)

1000 Ω

-Z " / kΩ

1.2

1 mF

100 Ω

0.8

50 µF

0.4

0.0 0.0

a

0.4

0.8

1.2

Z ' / kΩ

(b) -60 3.0

-30 2.5 -15 0

2.0

b

Φ/º

lg (|Z| / Ω)

-45

0.1

1

10

100

1000

10000

f / Hz

Fig. 8.7. An electrochemical impedance spectrum in a complex plane and b Bode impedance magnitude and phase angle representations for the electrical equivalent circuit shown in the inset on the complex plane plot - the real axis (Z′) represents the resistive part of the impedance and the imaginary axis (-Z″) the reactive, i.e. capacitive) part of the impedance. See text for further details

8.7 Applications A detailed knowledge of the electrode reactions and processes can be used for: 1. tailoring electrode reactions so as to enhance required and inhibit unwanted electrode reactions, perhaps by changing electrode material or developing new electrode materials or modifying the electrode surface,

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2. studying complex systems in which many electrode reactions occur simultaneously or consecutively, as in bioelectrochemistry, 3. measuring concentrations of electroactive species, making use of the selectivity of the potential and of the electrode material at or outside equilibrium (as in potentiometric, amperometric, voltammetric, and enzyme sensors). There is a vast range of areas of research and applications deriving from electrochemistry, a summary of which is shown in Fig.8.8. It can be seen that these include many areas of chemistry, physics and biology as well as engineering: industrial electrolysis, electroplating, batteries, fuel cells, electrochemical machining, etc. Electrochemistry is a truly interdisciplinary subject. Structure of solutions Colloids and electrokinetic phenomena

Kinetics and mechanism of electrode processes

Organic electrolysis Inorganic electrolysis

Solid-líquid interface Liquid-liquid Interface

ELECTROCHEMISTRY

Membranes Bioelectrochemistry Electroanalysis

Sensors and biosensors

Coatings (electrodeposition, anodization etc.) Corrosion and protection methods

Batteries and fuel cells Conducting polymers

Fig. 8.8. Areas of research and applications of electrochemistry

8.8 Bibliography There is an extensive bibliography on electrochemistry and its applications. An exhaustive listing can be found in the following internet address: “http://electrochem.cwru.edu/estir/”. Texts of physical chemistry usually have a good introduction to electrochemistry. Some electrochemistry texts that go more deeply into aspects of techniques and the practice of electrochemistry are shown in the list of references.

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8.9 Glossary of Symbols Table 8.2. Units normally used in electrochemistry. Symbol ai ci Cd E E, E,' Eeq EF UG, I k k0

m M n

ka kc kd

Rct RΩ T α γi

αa αc

Definition activity of ion i concentration of ion i capacity of interfacial region electrode potential standard electrode potential formal potential equilibrium potential Fermi energy standard free energy change Current heterogeneous rate constant standard rate constant of an electrode reaction rate constant for an anodic reaction rate constant for a cathodic reaction mass transfer coefficient mass molar mass number of electrons involved in an electrode reaction charge transfer resistance cell resistance Temperature electrochemical charge transfer coefficient anodic cathodic activity coefficient of species i

Unit mol cm-3 or mol dm-3 F cm-2 V

J or eV J mol-1 A cm s-1

g or kg g mol-1 Ω Ω K -

Table 8.3. Physical constants F R

Faraday constant gas constant

9.6485x104 C mol-1 8.31451 J mol-1 K-1

References 1.

C.M.A. Brett and A.M. Oliveira Brett (1993) “Electrochemistry. Principles, methods, and applications” Oxford University Press, Oxford

8 Fundamentals of Electrochemistry 2. 3. 4. 5.

239

A.J. Bard and L.R. Faulkner (2001) “Electrochemical methods. Fundamentals and applications” 2nd edn. Wiley, New York E. Gileadi (1993) “Electrode kinetics for chemists, chemical engineers, and materials scientists”, VCH Publishers Inc., New York D.T. Sawyer, A. Sobkowiak and J.L. Roberts (1995) “Experimental electrochemistry for chemists” 2nd edn. Wiley, New York P.T. Kissinger and W.R. Heinemann (eds.) (1996) “Laboratory techniques in electroanalytical chemistry” 2nd edn. Dekker, New York

9 Chemical Sensors Ernesto Julio Calvo and Marcelo Otero Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires

9.1 Introduction We are surrounded by a complex environment and therefore need to monitor aspects of this environment in real time in connection with pollution, health and safety, etc. Like the human senses of olfaction, sight, touch, hearing and taste link us with the external world, chemical sensors allow us to monitor and record the chemical variables of the external environment in the same way we monitor temperature, pressure, etc. A chemical sensor is a device which responds to a particular analyte molecule in a selective way in the presence of other molecules through a chemical reaction and can be used for the qualitative and quantitative determination of the analyte in the complex sample. After recognition of a molecular entity, the chemical sensor transduces the signal through a physicochemical process into an electrical signal which can be further amplified and processed. While Chap. 2 deals with different types of acoustic transducers, this tutorial focuses on the chemical sensing layer that affects the molecular recognition and the different transducers that can be employed. Different types of chemical sensors include different transducing methods to obtain processable electrical signals, so we have a variety of chemical sensors: 1. Electrochemical (potential, current or conductivity). 2. Optical (optical waves). 3. Mass sensitive (acoustic wave sensing). 4. Heat (calorimetric). 5. Magnetic. The present trend is to integrate the chemical sensor with microelectronics and sample handling. This has led to impressive achievements in miniaturization using silicon micromachining methods and more recently soft lithography techniques [1, 2]. In addition, the recent explosive growth of microfluidics [3] has allowed liquid samples to be transferred A.A. Vives (ed.), Piezoelectric Transducers and Applications, doi: 10.1007/978-3-540-77508-9_9, © Springer-Verlag Berlin Heidelberg 2008

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by electrophoretic driving forces into contact with an array of chemical microsensors giving rise to the concept of lab-on-chip. This is a fully integrated chemical laboratory in a microchip with applications as wide as health care and remote medicine, environmental control and food analysis. Applications in the area of explosive early detection, chemical and biological warfare has recently emerged as a fast growing field. Patients requiring rapid blood analyses in emergency rooms, intensive or critical care units as well as in the operating theatres can undergo rapid blood tests with only a few drops of whole blood with the point-of-care systems. In 1985 Dr. I. Lauks developed the I-Stat point-of-care blood analysis system [4]. It comprises a single-use cartridge containing an array of electrochemical sensors capable of measuring blood oxygen, CO2 and pH. Also it measures electrolytes such as potasium, sodium, chloride, calcium ions, glucose, urea nitrogen and hematocrit in few minutes with only a few drops of whole blood. The development of I-Stat was made possible by the incorporation of microfabrication techniques from the microelectronic industry in a 5 x 4 mm chip. There are also examples of commercial systems to monitor glucose in a tiny drop of whole blood of a Diabetes Mellitus patient. Glucose levels in the clinical range 20-600 ng dL-1 can now be measured in 20 seconds with a drop from 0.3 to 3.5 μL of whole blood using commercial integrated chemical sensors [5, 6]. Implantable glucose sensors have also been developed. An alternative novel way of chemical sensors is the electronic nose [7]. This olfaction machine is an electronic device capable of detecting and recognizing complex odors. Unlike molecular recognition, electronic noses work on the principle of multiple signal detection and pattern recognition. The signal can be generated by chemical detection interfaces with changes of conductivity of a solid state or conducting polymer prompted by the interaction of analyte molecules from a complex sample. The signal transduction can be affected by a variety of physicochemical processes, i.e. metal-oxide gas sensors, conducting polymer gas sensors, bulk and surface acoustic wave gas sensors, field effect chemo-transistors, electrochemical gas sensors, etc. They are normally used in a multisensor array to generate a signal pattern. The recognition is not selective at the molecular identity but rather to a signal pattern recognition and multicomponent analysis. It can be applied to health care or food analysis, i.e., to very complex samples such as coffee, beer, wine, etc., freshness of fish or to smell a cow to determine its health status. The signal generation can be affected by electrochemical, electrical (resistance), acoustic (bulk acoustic and surface waves), etc.

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processes, normally with an array of detectors (metal oxide or conducting polymer detectors) each one slightly different from the rest. The general requirements that a chemical sensor must fulfill are: 1. Sensitivity The limit of detection of a given substance is related to the minimum concentration capable of producing a signal that can be discriminated from the noise level (i.e. by the signal-to-noise ratio) and the sensitivity is related to the slope of a signal-to-concentration plot at a given concentration. It is desirable that such a sensor response plot be linear. 2. Selectivity High selectivity is needed to discriminate qualitatively and quantitatively a given component in a complex sample matrix, i.e., detection of glucose in whole blood where other molecules like ascorbate, urate, etc., may interfere. Selectivity can be achieved by molecular recognition, i.e., by molecular shape as key and lock problem or by pattern recognition when a complex pattern of transduced signals is obtained like in electronic noses. 3. Stability The molecules that comprise the chemical recognition interface may not be stable enough and the chemical sensor response decays with time. If the inventory of chemo-receptor or molecules that recognize the analyte is not in excess with respect to the stoichiometric value – which happens when the sensing response is determined by kinetics – then the stability is a critical factor to guarantee a reproducible response. On the other hand, if the signal is determined by mass transport of the analyte, the decay in sensing molecules should not be determining.

9.2 Electrochemical Sensors Electrochemical reactions take place at electrode-electrolyte interfaces and provide a switch for electricity to flow between two phases of different conductivity, i.e. the electrode (electrons or holes are the charge carriers) and solid or liquid electrolyte (ions are the main charge carriers). Sensors based on the transduction of chemical information into an electrical signal by an electrochemical reaction are very well suited for chemical sensors [8].

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Among electrochemical sensors we can distinguish potentiometric, amperometric and conductimetric sensors depending on the mode of the electrochemical process as follows [9-11]. 9.2.1 Potentiometric Sensors Potentiometric sensors work at equilibrium or quasi-equilibrium conditions with very little perturbation of the analyte concentration adjacent to the sensor surface. In these sensors the molecular recognition of the analyte results in changes of surface charge at the electrode-electrolyte interface due to changes of concentration which are amplified by the high electrical capacitance of electrochemical interfaces. Usually, one obtains a logarithmic response of the electrode potential to the analyte activity, as:

E = A log a s + B

(9.1)

where A = RT/nF with R the universal gas constant, T the absolute temperature, n the number of electrons exchanged in the electrochemical recognition reaction and F the Faraday constant (98,485 coul mol-1). The ion activity, as, is related to the ion concentration by the mean ionic activity coefficient, γ ± , which itself can be dependent of the concentration cs, as = γ cs. As a simple example let us consider water sparingly soluble salt like AgCl deposited on a silver wire that undergoes a solubility equilibrium: AgCl ( s ) → Ag + ( aq ) + Cl − ( aq )

(9.2)

with an equilibrium ionic product that relates the ionic activities:

K sp = a Cl − .a Ag +

(9.3)

and therefore according to Nernst equation (Eq. 8.2) [10-11], E = E OAg / Ag + +

RT RT ln a Ag + = E OAg / AgCl − ln aCl − F F

(9.4)

O O with, E Ag/AgCl = E Ag + (RT/F) ln K sp and RT/F = 59.1 mV (at 25°C) / Ag + which results in the slope A of Eq. (9.1) if the ion activity coefficient can be regarded as independent of ion concentration. From Eq. (9.4) it can be seen that the linear relationship fails at very low ionic concentration when the ion concentration falls in the range of ionic solubility of sensing layer s = aAg+, i.e., when the analyte activity aCl ≈ Ksp/s.

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Other examples of ion specific electrodes are pH electrodes where a difference in proton activity across a special thin glass membrane (between the external analyte and an internal reference) produces a difference in electrode potential measured with respect to a pH independent reference electrode immersed in the same electrolyte solution. The membrane potential is given by [11]:

E=

a ext RT ln a iint zi F ai

(9.5)

where aiext and aiint are the activities of the ion i of charge zi respectively in the external and internal solution to the membrane. The membrane potential is measured with respect to a reference electrode immersed in the internal solution the potential of which is independent of ion i. In 1967, a liquid membrane ion selective electrode for calcium was introduced, and many other sensors followed after this one. In 1979, the liquid membrane component was incorporated as plasticizer of PVC with 1% ionophore and a solid membrane calcium ion sensor was developed. Other examples are sodium, ammonium ion specific electrodes which recognize ions by size, i.e. crown ethers or antibiotic molecules (i.e. validomicyn K complex) which can host ions (guests) of a given size in a cavity when supported in a PVC membrane. Solid state ion selective electrodes like PbS-Ag2S selective to sulphide ion were also developed. Alternatively, a pH glass electrode can be covered by a gel containing an enzyme such as urease, which catalyzes the hydrolysis of urea into ammonia which enters into an acid-base equilibrium: NH 3 + H 2O → NH 4+ + HO −

(9.6)

Therefore, a biosensor responsive to urea concentration can be built by simply measuring changes in pH with the modified glass electrode. Integration with microelectronics can be achieved with ion-selective field effect transistors (ISFET). This chemo-electronic device is an adaptation of the IGFET (insulated gate field effect transistor) and can be manufactured in very small size, which allows the fabrication of sensor arrays for multi-species sensing. An ISFET consists of a p-type semiconductor in contact with two regions of n-type semiconductor (the source and the drain). A layer of insulator is deposited on top of this structure and on top of it an ion selective membrane replaces the metal gate material of an IGFET; when this sensing layer is in contact with the sample solution, the ion recognition by the selective membrane results in a specific potential at the membrane surface, with respect to the reference electrode immersed in the solution. The current that flows between source and drain depends on

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the gate potential generated by the ion selective membrane and, therefore, senses the ion concentration integrated in the hybrid circuit. The insulating layer itself (SiO2, Si3N4) responds to changes of pH. When an enzyme like urease is used in the chemical sensing layer, the enzyme field effect transistor (ENFET) results. 9.2.2 Amperometric Sensors

In amperometric sensors, dynamic conditions prevail with a net electrochemical reaction (i.e. oxidation or reduction of sample species) taking place at the electrode-electrolyte interface:

O + ne → R

(9.7)

where oxidized species O are reduced by electrons at the electrode surface into the reduced species R and a net current flows through the external circuit. The number of moles of O, n, converted into R is related to the electrical charge, q, which flows at the interface by the Faraday law, q = nF

(9.8)

where F = 96,485 coulombs/mol is the Faraday constant. The electrical current per unit area that flows at the electrode-electrolyte interface due to reaction in Eq. (9.7) is given by: i=

I 1 dq F dn = . = A A dt A dt

(9.9)

and is proportional to the rate of the electrochemical interfacial reaction. Since removal of O molecules from the electrode surface prompts mass transport of O from the bulk electrolyte, the mass transport limited current is given by: iL =

nFADi ci

δ

(9.10)

where A is the electrode area, Di the diffusion coefficient of the reactant species of concentration ci and δ is a mass transport layer where the concentration gradient is constant. As an example, we will describe the Clark electrode or oxygen probe [12] to measure dissolved oxygen that is based on the electrochemical oxygen reduction reaction: O2 + 4 H + + 4e → 2 H 2 O

(9.11)

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or O2 + 2 H + + 2e → H 2O 2

(9.12)

with n = 4 and n = 2 respectively at appropriate electrode potentials. The electrode is separated from the sample by a Teflon membrane which is permeable to oxygen but not to water or other molecules and encloses a thin electrolyte layer where an oxygen concentration diffusion gradient determines the oxygen flux and therefore the diffusion limiting current given by Eq. (9.10) which results proportional to the dissolved oxygen concentration, and δ in that case is simply the thickness of the membrane and liquid electrolyte film. An extension of Clark’s oxygen probe is the enzyme electrode introduced later by Clark himself [13]. In this device, an enzyme (commonly an FAD oxidoreductase like glucose oxidase that catalyzes the aerobic oxidation of glucose to gluconic acid) immobilized at the inner side of the hydrophilic diffusion membrane converts glucose (or other substrate) and oxygen both diffusing from the external analyte solution through the semipermeable membrane to produce hydrogen peroxide which can be detected electro-chemically on a metal electrode. H 2O2 → O2 + 2 H + + 2e

(9.13)

This type of sensors, however, is limited in anaerobic samples by the supply of oxygen and this has been solved by using artificial redox mediators instead of the natural cofactor oxygen to shuttle the electrons into the redox enzyme and being regenerated at the electrode surface. The redox shuttle or redox mediator may be a soluble molecule added to the electrolyte or integrated into a surface layer with the enzyme. Cass et. al. reported a glucose biosensor using ferrocene to mediate the electron transfer between the redox site of glucose oxidase (GOx) and an electrode surface [14]. Since then, many efforts have been put on optimization of the mediated enzyme biosensors. Several strategies have been followed to integrate the redox mediator to the enzyme in amperometric biosensors as Schuhmann reviewed [15]. Intelligent biochips of the future will make use of the present technology of enzyme, immune antigen-antibody recognition, genomic (DNA hybridization) sensing integrated with microfluidics and microelectronics. There are already several successful commercial devices to measure glucose with a small drop sample of whole blood; these glucometers work on the principle of mediated enzyme electrodes with glucose oxidase [5, 6]. Biosensors find important applications in biomedical analysis and remote medicine (i.e. rapid glucose measurement in whole blood, detection

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of cholesterol, ethanol, etc.), food analysis (i.e. freshness of fish), and environmental control (detection of pollutants). Molecular switches or molecular transistors in amperometric mode were introduced by Mark Wrighton in 1985 [16, 17] and then the concept has been expanded into enzyme electrodes by Bartlett and Birkin in 1994 with the introduction of a microelectrochemical enzyme transistor responsive to glucose [18]. In these devices a conducting polymer like poly(aniline) thin layer bridges the gap between two electrodes (drain and source of the molecular transistor). A sensing layer on top of the conducting polymer film carries out the recognition reaction injecting charge into the conducting polymer and thus changing it from the conductor state to the insulator state or from the insulator state into the conductor. The amount of charge that circulates is a measure of the number of analyte molecules transformed and therefore, since q = i.t, the time to reach the midpoint in the signal time course measures the concentration of analyte. Bartlett et al. have described a system comprising a poly(aniline) thin film containing poly(vinylsulfonate) or poly(styren-sulfonate) as the only anion present in the polymerization solution [19]. 9.2.3 Conductimetric Sensors

Conductivity is a measurement of the ability of a solution to conduct an electric current. Instruments measure conductivity by placing two plates of conductive material with known area and distance apart in a sample. Then a voltage potential is applied and the resulting current is measured. Using Ohms law, V = iR = iG −1

(9.14)

where V is the voltage potential applied between the electrodes, i is the electric current in the external circuit, R is the solution resistance and G the conductivity of the solution, we may calculate the conductivity of a solution from the circulating current and the voltage applied to the electrodes. Water has a low conductivity and therefore the conductive ions present in the solution such as metalic ions, salts, etc., provide the conductive path between the two electrodes of the conductivity cell. A High ionic concentration yields a high conductivity and typically an AC voltage potential is applied in order to prevent reactions at the electrode surfaces. The conductivity cell: in theory, a conductivity measuring cell is formed by two 1 cm2 square electrodes spaced 1 cm apart. Cells of different physical configurations are characterized by different cell constants, K. The cell constant (K) is a function of the electrode areas, the distance between the

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electrodes and the electrical field pattern between the electrodes. A theoretical cell just as that already described has a cell constant of K = 1.0. However, for considerations having to do with sample volume or space, a cell’s physical configuration is designed differently. Cells with constants of 1.0 or greater have normally small widely spaced electrodes. In contrast, cells with constants of K = 0.1 or less have normally large closely spaced electrodes. Since K is a factor, which reflects a particular cell’s physical configuration, it must be multiplied by the observed conductance to obtain the actual conductivity reading. In a simplified approach, the cell constant is defined as the ratio of the distance between the electrodes, d, to the electrode area, A. However, this does not take into account the existence of a fringe-field effect, which affects the electrode area by the amount AR; therefore K = d/(A + AR). Because it is normally impossible to measure the fringe-field effect and the amount of AR to calculate the cell constant, K, the actual K of a specific cell is determined by a comparison measurement of a standard solution of known electrolytic conductivity. The most commonly used standard solution for calibration is 0.01 M KCl, which presents a conductivity of 1412 µS/cm at 25oC. The basic conductivity probe is comprised of two conductive surfaces separated by a given distance in a body. The body material can be anything from PVC, CPVC, PVDF, TEFLON, PEEK or even stainless steel. The measuring surfaces are typically constructed of graphite, stainless steel, titanium or platinum. The basic criteria for determining which is best are based on cost and performance requirements. We have discussed up to this point the basic 2-pin conductivity cell, but there is also a 4-pin technology cell that tries to control the field surrounding the conductivity sensor in a better way in order to improve stability. These are known as contacting type conductivity cells. Another type of technology is the non-contacting (Toroidal) cell, which uses a magnetic field to sense conductivity. A transmitting coil generates a magnetic alternating field that induces an electric voltage in a liquid. The ions present in the liquid enable a current flow that increases with increasing ion concentration. The ionic concentration is then proportional to the conductivity. The current in the liquid generates a magnetic alternating field in the receiving coil. The resulting current induced in the receiving coil is measured and used to determine the conductivity value of the solution. Conductimetric measurements are completely non-specific. Non-specific applications involve simply measuring conductivity to detect the presence of electrolytes. The majority of conductivity applications fall within this category. They include monitoring and control of demineralization, leak detection, and monitoring to a prescribed conductivity specification. In

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most instances, there is a maximum acceptable concentration of electrolyte, which is related to a conductivity value, and that conductivity value is used as an alarm point. The progress of a batch chemical reaction and the concentration of any of its products or reactants can be measured, if it is carried out in a repeatable fashion (for example, Sodium hypochlorite [bleach] production).

9.3 Optical Sensors Optical sensors rely on the optical transducer of the signal and comprise ultraviolet, visible and infrared spectrophotometry in transmission or reflectance modes. The relationship between the incident light intensity and the transmitted radiation is given by the Lambert-Beer law: Aλ = ε λ .l.c

(9.15)

where Aλ = log10 Io/I is the absorbance at a given wavelength λ (being Io and I the incident and transmitted light intensities); ε is the molar absorptivity; l is the optical path through the absorbing sample and c is the molar concentration of absorbing analyte. The transmitted light is then detected by a photodiode or phototube which transduces light into an electrical signal. More recently “optrodes” were introduced (the term optrode or optode was coined in analogy with “electrode” in electrochemical sensors) with fiber optics covered at one end with reagent immobilized in a gel or membrane which produces a change in color in the presence of the analyte. The light is then transmitted to the solution and partly reflected and detected in a bifurcated fiberoptics. The advantages of “chemosensor optodes” are that they do not require an extra reference like the reference electrode in potentiometric sensors and that are insensitive to electrical noise, but double transduction chemical-optical-electrical is needed at the end signal. In addition to absorption-transmission optical sensors, luminescent sensors detect light emission that results from the absorption of a photon by a molecule and then the excited state decays with either fluorescent or phosphorecent emission. These techniques are highly sensitive and the emitted light of higher wavelength against zero background enhances the sensitivity and signal-to-noise ratio allows extraction of chemical information from complex samples. Optical sensors can be used to monitor pH, metal ions, oxygen, glucose and other enzyme substrates, etc., by appropriate choice of the chemical

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sensing layer. They are limited, however, to transparent and colorless samples. Biosensors based on evanescent waves have had a high impact in recent years, for instance surface plasmon resonance (SPR) biosensors. Surface plasmon resonance arises from the interaction of light with suitable metal or semiconductor surface which generates a quantum optical-electrical phenomenon. Under certain conditions, the photon energy can be transferred to the surface of the metal as packets of electrons called plasmons. This energy transfer occurs at a particular wavelength of light, when the quantum energy carried by the photon exactly matches the quantum energy level of the plasmons. The incident light is almost completely absorbed at the wavelength that excites plasmons, that is the resonance wavelength. Using monochromatic light at the resonance wavelength and changing the incidence angle, the maximum absorption occurs at a particular angle of incidence. The interaction between the plasmon’s electric field and analyte molecules within this field determines the resonant condition (wavelength at constant angle or conversely plasmon resonant angle at constant wavelength). Therefore, any change in composition adjacent to the surface plasmon alters the plasmon resonance angle and the SPR angle shift is directly and linearly proportional to the change in surface composition. This has been used to detect biomolecules such as antibody-antigen or single stranded DNA hybridization with success. Optical waveguides and fiber optics work in a similar way differing only in geometry.

9.4 Acoustic Chemical Sensors Bulk acoustic wave sensors (BAW) and surface acoustic waves sensors (SAW), described in Chap. 2, can be used as chemical sensors if the acoustic wave travels through a chemical sensing layer that affects the sound propagation, when the molecules of the analyte react with this sensing layer [20]. Acoustic wave devices with an applied sorbent layer as vapor sensor represent one of the first applications of acoustic wave chemical sensors. King used quartz crystal microbalance (QCM) in 1964 [21]. In 1979 acoustic sensors were extended to surface acoustic wave devices [22]. The sorbent layer, often a polymer layer, acts to collect and concentrate vapor molecules from the gas phase onto the sensor surface [23]. The adsorbed molecules are then detected by their effect on the velocity of the acoustic wave (i.e. added mass and viscoelastic effects on the surface

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layer). The partition coefficient quantifies the distribution of vapor molecules between the gas phase and the sorbent phase,

K=

cs cv

(9.16)

where cs is the analyte concentration in the surface sensing layer and cv represents the concentration in the vapor. The sensor responds to the analyte molecules in the sorbent phase and not in the gas phase; therefore the acoustic device detects cs. It is necessary to calibrate the sensor against the gas phase with known cv and thus the partition coefficient is the calibration parameter. A number of analytes have been sensed with acoustic wave devices such as water, sulphur dioxide, ammonia, aliphatic and aromatic hydrocarbons, hydrogen sulphide, mercury, carbon monoxide, etc. An interesting example is the self-assembled monolayers coupled to SAW devices used effectively as chemical sensors for the detection of vapor phase of pesticides and chemical warefare such as organophosphonates, which proves the high sensitivity of acoustic detection [24]. Piezoelectric sensors have also been employed in contact with liquid samples in flow injection analysis, immunosensors and DNA hybridization. However, in this case the attenuation of the acoustic wave by the viscous liquid also enters into the generation of the sensor response.

9.5 Calorimetric Sensors Calorimetric sensors are based on measurement of the heat produced by the molecular recognition reaction and the amount of heat produced is correlated to the reactant concentration. The principle of the calorimetric sensors is the determination of the presence or concentration of chemical species by the measurement of the enthalpy change produced by any chemical reaction or physisorption process that releases or absorbs heat. We talk about exothermic reactions if heat is generated and endothermic reactions if it is absorbed. Calorimetric sensors have been described for enzyme reactions for detecting glucoses, urea, gases, etc. The thermal conductivity gas sensor has long been employed as a detector in gas chromatography (GC) and works on the basis of a heated W-Re wire filament to measure relatively high concentrations in gases. Calorimetric sensors or chemoresistors can be classified in low-temperature chemoresistors and high-temperature chemoresistors.

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Low-temperature chemoresistors consist of chemically sensitive layers applied over interdigitated electrodes on an insulating substrate. Examples of the chemically sensitive layers are: metal phtalocyanines, conducting polymers such as poly(pyrrol) and poly(aniline). These types of chemoresistors are used in the detection of ethanol, methanol and other organic volatile molecules. High-temperature chemoresistors consist of micromachined semiconductor hotplates with a sensing film on a thermally insulated inorganic membrane. ZnO, InO, GaO, SnO are usually employed as sensitive materials. Gaseous electron donors (H) or electron acceptors (NOx) adsorb on metal oxides and form surface states, which can exchange electrons with the semiconductor. Another way to classify the calorimetric sensors is taking into account the different ways of transducing heat variations: Catalytic sensors or pellistors, which use a platinum coil used as heater and temperature sensors (resistance thermometer) and contain a catalyst to enhance a combustion process. The heated catalyst permits gas oxidation at reduced temperatures and at concentrations below the lower explosive limit. Some of the applications are: monitoring/detection of flammable gas hazards, CH4 (methane), H2, C3H8 (propane), CO and organic volatiles. Thermistor based sensors. These devices detect with high accuracy changes in the electrical resistance that result from temperature changes. If a linear relationship between resistance and temperature is assumed (i.e. a first-order approximation), the following expression results:

ΔR = kΔT

(9.17)

where ΔR is the change in resistance, ΔT , the change in temperature and k a first-order temperature coefficient of resistance. Thermistors can be classified in two types depending on the sign of k. If k is positive, the resistance increases with increasing temperature, and the device is called a positive temperature coefficient (PTC) thermistor. If k is negative, the resistance decreases with increasing temperature, and the device is called a negative temperature coefficient (NTC) thermistor. These sensors use composite oxides. Negative Temperature Coefficient (NTC) Thermistors use sintered metal oxides like titanium oxide and Positive Temperature Coefficient (PTC) Thermistors use Ba/Pb titanate. Pyroelectric sensors, which are based on the phenomenon of pyroelectricity, the ability of certain materials to generate an electrical potential when they are heated or cooled (gallium nitride (GaN), cesium nitrate (CsNO3), polyvinyl fluorides, derivatives of phenylpyrazine, cobalt phthalocyanine, Lithium tantalate (LiTaO3), etc.).

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Seebeck-effect based sensors, which transform temperature differences directly into electricity. Thermal-flow based sensors and bimorph effect cantilevers.

9.6 Magnetic Sensors The most important magnetic sensor with chemical applications is the paramagnetic sensor and is a highly accurate measurement technique for determine paramagnetic oxygen concentration [25]. The paramagnetic oxygen analyzer is based on the scientific principle that oxygen is a paramagnetic material, which means that it can be attracted into a magnetic field, or is “magnetically susceptible.” Magnetic susceptibility is a measure of the intensity of the magnetization of a substance when it is placed in a magnetic field. Oxygen has an exceptionally high magnetic susceptibility compared to other gases, actually several hundred times greater than most other gases. Three types of paramagnetic oxygen analyzers are most used in industry [25]: the magnetodynamic, or “dumbbell” type; the magnetopneumatic and the thermomagnetic, or “magnetic wind” type. The magnetodynamic oxygen analyzer is the most popular of the three techniques. It consists on a small dumbbell-shaped body made of glass and charged with N2 or some other gas of low magnetic susceptibility, a light source, a mirror, a photodetector, and a calibrated indicating unit. The dumbbell body is suspended in an enclosed test cell by a quartz or platinum fiber within the magnetic field of a permanent magnet and is free to rotate in the space between the poles of the magnet. The dumbbell body is somewhat diamagnetic because of its N2 content; the balls of the dumbbell naturally deflect slightly away from the point of maximum magnetic field strength. When a sample containing O2 is introduced into the test cell, the O2 in the sample is attracted to the point of maximum field strength. Then the magnitude of dumbbell displacement is proportional to the amount of oxygen in the sample and the movement of the dumbbell is detected by a light beam from a light source exterior to the test cell. The light beam is reflected from a mirror on the dumbbell body to an exterior photodetector. The output of the photocell is amplified and transmitted to an indicating unit that is calibrated to read out directly the oxygen content in the test sample. Since the difference in magnetic susceptibility between the dumbbell and the gas sample is very subtle for low oxygen concentrations, this method is used only when measuring percent levels of oxygen and not for trace levels.

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The magnetpneumatic oxygen analyzer operates on the principle that a test sample containing oxygen molecules, when it is drawn into a nonhomogeneous magnetic field and mixed with a reference gas having different oxygen content, will generate a differential pressure. The sample gas is introduced into a test chamber containing a non-homogeneous magnetic field created by an electromagnet. Because of their paramagnetic properties, oxygen molecules in the sample flow toward the greatest magnetic field strength. A reference gas with known properties is introduced into the test chamber through two inlets. The reference gas from one inlet mixes with the test sample in the magnetic field and the paramagnetism difference between the two gases creates a differential pressure, resulting in a balancing flow of reference gas from the other inlet. This balancing flow is measured by a miniature flow sensor and converted into an electrical signal proportional to the differential pressure. The thermomagnetic oxygen analyzer is based on the principle that the magnetic susceptibility of oxygen decreases inversely with the square of its temperature. It consists of a test chamber containing two tubes for the test sample. The tubes are connected by a cross tube containing electrical heating filaments at each end of the crossover passage. The two filaments are the arms of a Wheatstone bridge. One end of the cross tube with its heating filament lies in a strong magnetic field created by the poles of a permanent magnet. The test sample is introduced in two equal streams through the two side tubes. Any oxygen in the sample is attracted to the magnetic field because of the heating filaments, and the magnetic susceptibility of the oxygen in the sample decreased rapidly as the temperature is increased. The heated sample is displaced by cool oxygen attracted to the magnetic field, and this flow of gas, or “magnetic wind,” cools the filament in the magnetic field, causing its resistance to be different from the heating filament at the other end of the cross tube. The difference in resistance is measured on a bridge-type instrument, and a signal proportional to the oxygen concentration in the test sample is transmitted to a recording instrument or display. The most common general applications of the paramagnetic types of oxygen analyzers include the analysis of combustion efficiency, the testing of the purity of breathing air and protective atmospheres, its use for laboratory measurements and in medical applications, and for selected industrial process monitoring and control applications.

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References 1. T. Vo-Dinh, B. Cullum, D.L. Stokes (2001) “Nanosensors and biochips: frontiers in biomolecular diagnostics” Sensors and actuators B 74:2-11 2. Y. Xia, G. M. Whitesides (1998) “Soft Lithography” Angew. Chem. Int. Ed. 37:550-575 3. A. van den Berg and P. Bergveld (eds.) (1995) “Micro total analysis systems” Kluwer, Boston 4. G. Ramsay (Ed.) (1998) “Commercial biosensors. Applications to clinical, bioprocesses, and environmental samples” J. Wiley & Sons, New York 5. www.medisense.com 6. www.therasense.com 7. J.W. Gardner, P.N. Bartlett (1999) “Electronic noses. Principles and applications” Oxford University Press, Oxford 8. E.J. Calvo (2003) “Interfacial kinetics and mass transport” in Encyclopedia of Electrochemistry, E.J. Calvo (Ed.) Wiley-VCH, Weinheim, Vol. 2, pp.3-31 9. R.W. Cattral (1997) “Chemical sensors” Oxford University Press, Oxford 10. A.J. Bard, L.R. Faulkner (2001) “Electrochemical methods. Fundamentals and applications” J. Wiley & Sons Inc., 2nd Ed 11. P. Unwin (2003) “Instrumentation and electroanalytical chemistry” in Encyclopedia of Electrochemistry, Wiley-VCH, Weinheim Vol. 3 12. L.C. Clark Jr. (1956) “Monitor and control of blood and tissue oxygen tensions” Trans. Am. Soc. Artif. Intern. Organs. 2:41-48 13. L.C. Clark Jr. and C. Lyons (1962) “Electrode systems for continuous monitoring in cardiovascular surgery” Ann. N.Y. Acad. Sci. 105: 29-45 14. A.E.G. Cass, G. Davis, G.D. Francis, H.A.O. Hill, W.J. Aston, I.J. Higgina, E.V. Plotkin, L.D.L. Scott, and A.P.F. Turner (1984) “Ferrocene-mediated enzyme electrode for amperometric determination of glucose” Anal. Chem. 56: 666-671 15. W. Schumann (2002) “Amperometric enzyme biosensors based on optimised electron-transfer pathways and non-manual immobilisation procedures” Rev. Mol. Biotechnology 82: 425-441 16. J.W. Thackeray, H.S. White, M.S. Wrighton (1985) “Poly(3-methylthiophene) coated electrodes: optical and electrical properties as a function of redox potential and amplification of electrical and chemical signals using Poly(3methylthiophene)-based microelectrochemical transistors” J. Phys. Chem. 89: 5133-5140 17. G.P. Kittlesen, H.S. White, M.S. Wrighton (1985) “A microelectrochemical diode with submicron contact spacing based on the connection of two microelectrodes using dissimilar redox polymers” J. Am. Chem. Soc. 107(25): 73737380 18. P.N. Bartlett, P.R. Birkin (1994) “A microelectrochemical enzyme transistor responsive to glucose” Anal. Chem. 66:1552-1559 19. P.N. Bartlett, J.H. Wang, E.N.K. Wallace (1996) “A microelectrochemical switch responsive to NADH” Chem. Commun. 3: 359-360

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20. D.S. Ballantine, R.M. White, S.J. Martin, A.J. Ricco, E.T. Zellers, G.C. Frye, H. Wohltjen (1997) “Acoustic wave sensors: theory, design, and physicochemical applications” Academic Press, San Diego, USA 21. W.H. King (1964) “Piezoelectric sorption detector” Anal. Chem. 36(9): 17351739 22. H. Wohltjen, R.E. Dessy (1979) “Surface acoustic wave probe for chemical analysis. II. Gas chromatography detector” Anal. Chem. 51: 1465-1470 23. J.W. Grate, M.H. Abraham (1991) “Solubility interactions and the design of chemically selective sorbent coatings for chemical sensors and arrays” Sensors and Actuators B 3: 85-111 24. L.J. Kepley, R.M. Crooks, A.J. Ricco (1992) “A Selective SAW-based organophosphonate chemical sensor employing a self assembled, composite monolayer: a new paradigm for sensor design” Anal. Chem. 64: 3191-3193 25. Delta F Corporation, http://www.delta-f.com/

10 Biosensors: Natural Systems and Machines Stanislav Stipek1 and Ernesto Calvo2 1 2

Institute of Medical Biochemistry, Charles University in Prague Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires

10.1 Introduction All living organisms are connected to the environment by some kind of chemical recognition and signal generation system. For instance, we make use of our senses: olfaction, sight, touch, taste and hearing to recognize either molecules (olfaction and taste) or physical stimuli (photons, sound waves, pressure, etc.). Likewise, each individual cell in unicellular or multicellular organisms senses stimuli from its surrounding environment (extracellular fluids, other cells and so on) through receptors on the cell surface. Humans have been able to copy these functions in machines, which can be connected to electronic circuits and computers to report on the quality of the outside world. In this chapter, the mechanisms that Nature has developed in living systems will be first addressed and afterwards the “biosensing” machines that mimic those functions will be addressed in order to analyze a variety of different types of sample (environment, clinic, food, industrial, etc.).

10.2 General Principle of Cell Signaling The living cell releases some kinds of extracellular signaling molecules or it presents them on its surface. Each other cell is able to response to a particular set of these molecular signals in a cell-specific way [15]. The term extracellular signaling molecules include proteins, peptides, steroids and eicosanoids, catecholamines, thyroxin and nitric oxide. In physiological classification they are hormones, growth factors and neurotransmitters. These signals can act over either short (neurotransmitters in the synaptic cleft) or long distance (hormones in the blood). A.A. Vives (ed.), Piezoelectric Transducers and Applications, doi: 10.1007/978-3-540-77508-9_10, © Springer-Verlag Berlin Heidelberg 2008

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There are two basic properties of any signaling molecule: 1. It is bound as a ligand with a high affinity by specific protein (receptor) in the target cell. 2. The lifetime of the signaling molecule in the functional compartment is very short. It is rapidly destroyed or removed by the neighboring cells. In this way the concentration of the signals can be adjusted very quickly. The essence of control is reversibility. The signal molecule binding to a receptor results in cellular response. The lipid-soluble hormones cross the cell membranes and bind to appropriate receptor inside the cell (in cytosol or in nucleus). Specific binding site appears on the complex of signal molecule with the receptor that recognizes a specific site of the DNA in a cell nucleus and so the hormone influences the activity of the genes modulating the expression of genetic information stored in the DNA. But hydrophilic signal molecules, which cannot cross the membrane lipid bilayer of the target cell, perform most of the cell-cell communications. These ligands do not enter the target cell but as the first messenger they interact with membrane receptors, which are embedded in the cell membrane having extracellular, transmembrane and intracellular domains, acting as signal transducers (Fig. 10.1). When the extracellular domain binds the signal the cytoplasmic domain undergoes a conformational change, which induces a cascade of intracellular signaling events in the intracellular signaling pathway. The last acceptor of such signaling pathway is a target molecule controlling appropriate cell response, such as change in metabolism, cell division, programmed suicidal death (apoptosis) etc. According to the mechanism of signal transduction there are three classes of membrane receptors. Some receptors as ligand-gated ion channels promptly change the ion permeability of a membrane. Other class of receptors is functionally linked with special GTP-binding proteins (Gproteins), which can interact only with an activated receptor conveying the signal by changing their own conformation. In this way, the GTP-binding proteins induce the production of some intracellular signaling molecules or they change the membrane permeability. Membrane receptors of the third class are the enzyme-linked receptors. When activated they can modify other molecules mostly by phosphorylation of some amino acids. The signaling pathways consist of a set of various intracellular signaling molecules. Some of them are small second messengers, which are rapidly formed (cyclic AMP, diacylglycerol) or released from cellular stores (Ca2+) in response to the receptor activation and so they multiply and convey the signal to other compartments of the cell. It is possible to cancel the signal by the rapid destruction of the messenger molecules or by removing

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these molecules from the cell compartment. The other intracellular signals are intracellular signaling proteins. Some of them are enzymes. When activated, they produce a large number of messenger molecules (amplification of the signal). Others convert the signal in another form or just convey the signal to the next component of the signaling pathway. In most cases, the activation mechanism is conformation change of the signaling protein. It can be introduced by phosphorylation catalyzed by a protein kinase (signaling enzyme transferring the phosphate from ATP to serine, threonine or tyrosine residue of the target protein). Such signal can be quickly canceled by removing the phosphate by another enzyme - phosphatase. Extracellular space Signal molecule ( ligand , Hormone )

PLASMA MEMBRANE

Adenylate cyclase

G protein

Receptor

GTP

GDP

Control of metabolic enzymes

ATP

cAMP cAMP

ProteinkinaseA (PKA)

cYTOPLASM CYTOPLASM

NUCLEAR MEMBRANE Transcription Transcription Factor (CREB) factor CREB)

+ ATP

Activated Activated Transcription transcription Factor (CREB) factor (CREB

P

+ADP

+ ADP

New binding site for DNA

CELL cELL NUCLEUS NUCLEUS

Newprotein synthesis Activated Transcription Factor(CREB)

P Gene activation (transcription)

DNA

Fig. 10.1. Beta-adrenergic G-protein signaling pathway

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One can summarize that molecules in a signaling pathway accept the signal mostly as a change in conformation resulting in modification of its affinity to substrate or to the next molecule in the cascade. In this way, the signal molecule acquires a new ability to activate (or inactivate) the next molecule in the cascade. Some of the signaling proteins behave like molecular switches. When they receive a signal they switch from an inactive to an active state, until another signal or process switches them off. The last target molecule modified by a signaling pathway is the factor controlling the cell response to the extracellular signal. When the final acceptor of information is an enzyme catalyzing the special reaction of a metabolic process, the result of the signaling is, for example, the release of a nutrient from the cellular storage. In the case that the signaling pathway ends at a transcription factor, then the information changes the activity of a gene, and consequently synthesizes a protein controlling, for example, the cell division or the cell differentiation. But the complex cell behaviors, such as cell proliferation and cell survival, are generally controlled by combinations of extra cellular signals rather than by a single signal. Nowadays several kinds of signaling pathways are known in the living cell. Interactions and cross talks of these pathways result in efficient molecular information network controlling the cell behavior and its function according to the surrounding conditions and requirements. The network enables to change cellular functions, morphology, movement, division (proliferation) and programmed death. Among the hundreds of various signaling proteins are the individual signaling proteins (mainly protein kinases) that serve as input-output devices, or “microchips”, integrating the signaling process. Such signal protein has to be phosphorylated at multiple special sites to be activated. One set of phosphorylated groups serves to activate it while other sets can inactivate it. The signaling protein can collect the information from several signal pathways and only in certain specific signal combinations the protein conveys the information downstream. It works as a protein microchip. Transfer activity, signaling specificity, binding affinity and other properties of the signaling molecules in the cell information network can be modulated by microenvironment of the cellular compartments. The metabolism of animal cell is characterized as the recast of nutrient chemical structure to simpler molecules. The released energy is then used for necessary synthetic, transport, motoric and signaling events. Protons and reducing forms of oxygen (water but also reactive oxygen species like peroxide, superoxide and free radicals) are released in addition to energy. The proteins have sensors for the concentration of protons (acid-base status) in the

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environment and sensors for the redox status in the cell. Redox status is defined by the capacity of biological antioxidant system, the accessibility of reducing equivalents and by the intensity of oxidative load to which a cell is exposed. Changes in proton dissociation of carboxyl-, amino-, amido-, and other groups influence the space conformation of signaling proteins resulting from the modulation of their functions. These reactive groups can be considered as acid-base sensors. The protein activity is modulated also by oxidation and reduction of other sensitive groups, like –SH groups and iron-sulfur complexes. These groups are redox sensors on the proteins, as they capture the information about the redox status in the cell compartment. Sulfhydryl groups oxidized to disulfide bridges provide another conformation of protein than reduced SH groups. Similarly Fe3+ in Fe-S complexes influence the protein conformation in other way than complexes with Fe2+. In response to oxidative stress (imbalance in oxidant generation, antioxidant defense and repair of oxidative damage) some bacteria use the protein redox sensors as transcription factors activating several defensive genes to synthesize antioxidant enzymes like hydroperoxidase, glutathione reductase, glutaredoxin.Binding of hormone epinephrine or glucagon to the receptor converts the G-protein to the active form and one of its subunits activates the formation of cyclic adenosine monophosphate (cAMP) by adenylate cyclase. The cAMP produced activates protein kinase A which phosphorylates enzymes metabolizing glycogen (in liver cell) or it is transported into the nucleus (in endocrine cells), where it phosphorylates (activates) cAMP-response element binding protein (CREB), which gets the right conformation to be a transcription factor for specific genes.

10.3 Biosensors Biosensors can be defined as analytical devices incorporating a biological material (e.g. enzymes, antibodies, nucleic acids, tissues, microorganisms, organelles, cell receptors, etc.), a biologically derived material or a biomimetic one intimately associated with or integrated within a physicochemical transducer or transducing microsystem [1]. Professor L.C. Clark Jr. has been the father of the biosensor concept since, in 1956, Clark published a paper on the oxygen electrode [2] and then based on this O2 sensor he made a landmark address in 1962 at a New York Academy of Sciences symposium in which he described how “to make electrochemical sensors (pH, polarographic, potentio-metric or

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conductometric) more intelligent” by adding “enzyme transducers as membrane enclosed sandwiches”. The use of some 50 oxido-reductases and 250 dehydrogenases would allow expanding the range of analytes that could be measured in the body, or body fluids. The concept was illustrated by an experimental device in which glucose oxidase (GOx) was entrapped at a Clark O2 electrode using dialysis membrane; thus a decrease in the measured O2 concentration was proportional to glucose concentration. Clark and Lyons [3], coined the term enzyme electrode, which in many reviews has been attributed to Updike and Hicks [4], who expanded on the experimental detail necessary to build functional enzyme electrodes for glucose. Guilbault and Montalvo [5] were the first to describe in detail a potentiometric enzyme electrode based on a urea sensor in which urease was immobilised at an ammonium-selective liquid membrane electrode. A commercial product based on Clark’s ideas appeared on the market in 1975 commercialized by Yellow Springs Instrument Company as a glucose analyser based on the amperometric detection of hydrogen peroxide, the product of the aerobic enzymatic reaction of glucose and GOx. Many other enzyme biosensor-based laboratory analysers followed this initiative. Biosensors have been applied to a wide variety of analytical problems such as medicine, the environment, food, the process industries, security and defense. There is at present a growing interest in antiterrorism to be able to rapidly detect dangerous material “in situ” with high sensitivity. The emerging field of bioelectronics seeks to exploit biology in conjunction with electronics in a wider context, encompassing, for example, biomaterials for information processing, information storage and actuators. An interesting source of commercial biosensors is the book edited by G. Ramsay “Commercial Biosensors” where the reader can find information on the fundamentals and applications of optical, acoustic, and electrochemical biosensors applied to health care, bio-processing and environmental samples [6]. Also, the journals Biosensors (Elsevier) and Biosensors and Bioelectronics (Elsevier) are a frequent source of updated scientific information. In general we need to achieve highly selective molecular recognition in complex analytical matrices like natural water, blood, urine, etc. with a multitude of molecules that may compete for the sensing reaction. Thus selectivity is achieved by molecular recognition with biological material in a lock-and-key fashion. Following the molecular recognition of the target analyte, we need to generate a signal with a transducer that may be optical, electrochemical, piezoelectric (acoustic), thermometric, or magnetic. Chemical reactions at

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electrodes are electrochemical transducers that turn biochemical information into an electrical signal [7]. The sensitivity of the biosensor depends on the transducer and the amplification stage of the signal, while the limit of detection depends on the signal-to-noise ratio. Depending on the nature of the molecular recognition event, biosensors can be based on: 1. Antigen/antibody (immunological) 2. Enzyme/substrate (enzymatic) 3. Receptor/hormone 4. S-DNA hybridization (genomic) 5. Interaction at the sensing layer. Biosensors usually yield a digital electronic signal which is proportional to the concentration of a specific analyte or group of analytes in a complex matrix sample. The electrical signal that results from the molecular recognition and transduction may, in principle, be continuous, devices can be configured to yield single measurements, like in glucometers to assess the glucose level in diabetes mellitus patients. In 1975, Divis [8] suggested to employ bacteria as the biological element in microbial electrodes for the measurement of alcohol. A major research effort in Japan and elsewhere sprung from Divis paper with applications in biotechnology and environment. Lubbers and Opitz [9] coined the term optode in 1975 to describe an optical fiber-sensor with immobilised indicator to measure carbon dioxide or oxygen. They extended the concept to develop an optical biosensor for alcohol by immobilising alcohol oxidase at the end of an optical fiber oxygen sensor [10]. Commercial optodes are now showing excellent performance for in vivo measurement of pH, pCO2 and pO2, but enzyme optodes have not succeeded yet. Clemens et al. [11] incorporated an electrochemical glucose biosensor in a bedside artificial pancreas in 1976 and this was later marketed by Miles as the Biostator. A new semi-continuous catheterbased blood glucose analyser has been introduced by VIA Medical. In 1982 a major breakthrough in vivo application of glucose biosensors was the first needle-type enzyme electrode for subcutaneous implantation reported by Shichiri et al. [12]. Liedberg et al. [13] described the use of optical evanescent waves in surface plasmon resonance to monitor affinity reactions in real time following attempts to fixing antibodies to a piezoelectric or potentiometric transducer. A direct immunosensor was marketed in 1990 by BIACORE™ (Pharmacia, Sweden) with a commercial device which was announced to

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be able to find out how much, how fast, and how strong the immunoaffinity reaction could detect in real time. La Roche introduced in 1976 the lactate analyzer, LA 640, with hexacyanoferrate as soluble mediator to shuttle electrons from lactate dehydrogenase to an electrode. This was a landmark in the development of amperometric enzyme electrodes. But, the greatest impact was from a much cited paper in 1984 by Tony Cass et. al. who proposed the use of ferrocene derivatives as a soluble integrated mediator for oxido-reductases [14]. This became the basis for the first disposable screen-printed enzyme electrodes launched by MediSense in 1987 with a pen-sized meter for home bloodglucose monitoring. With redesigned electronics into credit-card and computer-mouse style formats MediSense boosted sales reaching US$175 million by 1996 when Abbott bought the company. Boehringer Mannheim and Bayer have competing mediated biosensors and the combined sales of the three companies dominate 85% of the world market for biosensors and are rapidly displacing conventional reflectance photometry technology for home diagnostics. A recent “one-touch” device from Terasense employs 20 microliter blood sample from a mosquito-like bite on the forehand and utilizes an integrated osmium redox polymer mediator that “wires” the enzyme molecules to the electrode. The next generation of biosensors comprises integrated systems with electronics and fluidic separation integrated in a single device with improved sensitivity, stability and selectivity towards a multitude of analytes. The success of single analyte sensors has been followed by the formulation of arrays of sensors useful, for instance, in critical care where commercially-available hand-held instruments provide clinicians with information on the concentration of several analytes such as glucose, lactate, urea, creatinine and so on in blood samples, and bench-top instruments on the ward that can measure even more analytes. New techniques for the fabrication of integrated devices with micro fluidics arrays of thousands to a million sensors per cm2, such as photolithography, microcontact printing and layer-by-layer self-assembled techniques have emerged. Local probe techniques such as SECM and dip-pen lithography are promising in their ability to “write” proteins to surfaces with very high resolution and high density in nanoliter vials and individually addressable electrodes. The actual trend indicates that analytical chemistry and sensor technology will be increasingly associated with microelectronics. Miniaturization, massive use and simplicity of biosensors will cause, most probably, a proliferation of biosensors integrated with micro-electronics and telecommunications. An interesting example is the remote medicine, where a patient can be bio-chemically monitored remotely.

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10.3.1 Molecular Transistor Future intelligent biochips are supposed to acquire and process biochemical data and transmit them from remote locations to monitor our personal health, the food we eat and our environment. The future of biomolecular electronics is already here with bio-molecular transistors, molecular switches, microfluidic devices, integrating the molecular recognition, transduction, amplification and data processing in miniaturized devices. 10.3.2 Analogy and Difference of Biological System and Piezoelectric Device Piezoelectric biosensors and transductors accept the information from the surrounding environment, but they convert it to a change of quartz plate oscillation. The following part of the signaling pathways electronically integrating and displaying the piezoelectric information certainly differs from the signaling molecular network of the living cell. But the initial steps performed by the piezoelectric plate surface can (or should?) use similar principles as the natural cell signaling system. Self-assembled monolayer (SAM) of organic molecules containing free anchor groups such as thiols, disulphides, amines, silanes, or acids ordered in high degree on the plate surfaces mimics the cellular microenvironment of lipid bilayer structures providing novel substrates for immobilized biomolecules (antibodies, enzymes, nucleic acids) or biological systems (receptors, whole cells) [16, 17]. The term “self-assembly” involves the arrangement of atoms and molecules into an ordered or even aggregate of functional entities without the intervention of mankind towards an energetically stable form [18]. In nature, many supramolecular structures (like cellular organelles, membranes, viral particles) are formed by selfassembly process. Even the cell is able to produce appropriate sets of compounds step-by-step according to a program controlled by cell signaling network. The piezoelectric biosensors use the same molecular binding mechanism known in molecular biology. The DNA sensors estimate the same base pairing in nucleic acids (hydrogen bonds), which is used in the cell to store and/or express genetic information. In artificial immunosensors, enzyme electrodes and specific devices measuring any ligand in liquid or gas environment mimic the conditions in the tissue. Very specific and very strong interactions between substrate and active center of enzyme, antigen and antibody, ligand and receptor, protein and nucleic acid and interactions between signal proteins are based on the multiple simultaneous weak binding

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of the very well fitting surfaces of macromolecules with appropriate space conformation and arranged reacting groups. Any change from the optimal conformation of the biomolecule cancels part of the weak interaction and weakens the specific intermolecular binding. This dependence of protein affinity on the conformation of the macromolecules is the basic principle of control, signaling and specificity in supramolecular biological events. The “interpretation” of the molecular interaction can differ in signal transduction in piezoelectric device and in the cell signaling. Living cells recognize the specific interaction on the base of conformation changes in the activated molecule or complex. In the detection layer of piezoelelectric plate the interaction can be interpreted as an increase of mass and consequent change in the plate oscillation frequency, or as a change in the physical properties of the detection layer (impedance, density, friction, conductivity?). In any case, the correlation of the biological processes and technical proposals of the biosensors and transducers could be a good tool for the development of efficient devices useful in medicine and biotechnology. Table 10.1. Comparison of steps of signal pathways in cell and biosensor. Step

Biological cell

Biosensor

Signal (ligand, 1st messenger, oxidoreductase) Layer Receptor

Biomolecule (hormone, growth factor, antigen, light, NO, metabolite) Membrane (lipid bilayer) Transmembrane or intracellular protein

Very similar Molecular recognition layer

Signal transduction through the layer Signal detection and effect

Molecular recognition layer Protein, antibody or hapten, nucleic acid, microorganisms, tissue, etc. Conformation change (+ Electrochemical, optical, phosphorylation) resulting in acoustic, magnetic, thermal, change of binding ability etc. Change in metabolic enzyme Display, recording and storresults in altered metabolism age, feedback control based Gene regulation protein bind- on species concentrations. ing or release results in altered gene expression Cytoskeletal proteins result in altered cell shape or movement

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References 1. A.P.F. Turner, I. Karube and G.S. Wilson (1987) “Biosensors: Fundamentals and applications” Oxford University Press, Oxford. 770p 2. L.C. Clark Jr. (1956) “Monitor and control of blood and tissue oxygen tensions” Trans. Am. Soc. Artif. Intern. Organs. 2:41-48 3. L.C. Clark Jr. and C. Lyons (1962) “Electrode systems for continuous monitoring in cardiovascular surgery” Ann. N.Y. Acad. Sci. 105: 29-45 4. S.J. Updike and G.P. Hicks (1967) “The enzyme electrode” Nature 214: 986988 5. G.G. Guilbault and J. Montalvo (1969) “Urea specific enzyme electrode” JACS 91: 2164-2569 6. G. Ramsay (Ed.) (1998) “Commercial biosensors. Applications to clinical, bioprocesses, and environmental samples” Vol.48, Series of Monographs on Analytical Chemistry, J. Wiley & Sons, New York 7. C.L. Cooney, J.C. Weaver, S.R. Tannebaum, S.R. Faller, D.V. Shields and M. Jahnke (1974) “Thermal enzymes probe: a novel approach to chemical analysis” in “Enzyme Engineering” (Eds. E.K. Pye and L.B. Wingard Jnr.) 2, 411417. Plenum, New York 8. K. Mosbach and B. Danielsson (1974) “An enzyme thermistor” (1974) Biochim. Biophys. Acta 364: 140-145 9. C. Diviès (1975) “Remarques sur l’oxydation de l’éthanol par une electrode microbienne d’acetobacter zylinum” Ann. Microbiol. 126A, 175–186 10. D.W. Lubbers and N. Opitz (1975) “The pCO2-/pO2-optode: a new probe for measurement of pCO2 or pO2 in fluids and gases” (in German) Z. Naturforsch. C: Biosci. 30C: 532-533 11. K.P. Voelkl, N. Opitz and D.W. Lubbers (1980) “Continuous measurement of concentrations of alcohol using a fluorescence-photometric enzymatic method” Fres. Z. Anal. Chem. 301, 162–163 12. A.H. Clemens, P.H. Chang and R.W. Myers (1976) “Development of an automatic system of insulin infusion controlled by blood sugar, its system for the determination of glucose and control algorithms” Proceedings of Journees annuelles de diabetologie de l.Hotel-Dieu, pp.269-278, Paris 13. M. Shichiri, R. Kawamori, R. Yamaski, Y. Hakai and H. Abe (1982) “Wearable artificial endocrine pancreas with needle-type glucose sensor” Lancet 2: 1129-1131 14. B. Liedberg, C. Nylander and I. Lundstrm (1983) “Surface plasmon resonance for gas detection and biosensing” Sensors and Actuators 4: 299-304 15. A.E.G. Cass, G. Davis, G.D. Francis, H.A.O. Hill, W.J. Aston, I.J. Higgina, E.V. Plotkin, L.D.L. Scott and A.P.F. Turner (1984) “Ferrocene-mediated enzyme electrode for amperometric determination of glucose” Anal. Chem. 56: 666-671 16. B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter (2002) “Molecular biology of the cell” 4th edn. Garland Science, New York

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17. E.J. Calvo, C. Danilowicz and A. Wolosiuk (2002) “Molecular “wiring” Enzymes in organized nanostructures” J. Am. Chem. Soc.124:2452-2453 18. N.K. Chaki and K. Vijayamohanan (2002) “Self-assembled monolayers as a tunable platform for biosensor applications” Biosensors & Bioelectronics 17: 1-12

11 Modified Piezoelectric Surfaces Hubert Perrot1, Ernesto Calvo2 and Christopher Brett3 1

Laboratoire Interface et Systèmes Electrochimiques, Université P. et M. Curie, UPR 15 du CNRS 2 Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires 3 Departamento de Química, Universidade de Coimbra

11.1 Introduction The quartz crystal microbalance (QCM) is an attractive tool for gravimetric measurements and applications can be found in many research fields such as acoustic sensors (Chap. 2), chemical sensors (Chap. 9) and biosensors (Chaps. 10, 12). In general, QCM are covered with noble metals such as gold or silver, usually by evaporation or sputtering, and which can be used directly as electrodes or undergo further surface modification. Using such strategies, the study of processes ranging from electroplating to DNA immobilisation becomes possible. In this chapter, two separate sequential preparation steps should be distinguished. Metallic deposition (acting as the ultrasonic wave generator) is the first step as a support for modification. In the second step, organic or biochemical modifications are carried out before testing the modified QCM.

11.2 Metallic Deposition Metallic deposition can be carried out using two approaches: directly onto the quartz crystal by taking into account the crystalline orientation of the resonator (vacuum techniques) or onto a previously prepared metal layer on the quartz substrate, in general gold. First, we describe the two main vacuum methods: evaporation and sputtering. It should be noted that when high temperatures are necessary to achieve these operations, some trouble can occur with respect to the piezoelectric behaviour of the quartz crystal. Indeed, when the Curie temperature is overstepped, the piezoelectric properties can be lost. Moreover, the roughness and the porosity of these layers A.A. Vives (ed.), Piezoelectric Transducers and Applications, doi: 10.1007/978-3-540-77508-9_11, © Springer-Verlag Berlin Heidelberg 2008

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depend on the deposition conditions; it means that the QCM response can be affected by these parameters and therefore, users have to take aware of these effects (see Chap. 14). 11.2.1 Vacuum Methods Evaporation (Metals) This method is old and the first applications date from the 1950s. The system is widely used in microelectronic technology to produce thin metallic films which are necessary for interconnections: all ohmic contacts are based on this technique. Various metals can be used (chromium, copper, silver, gold…). The metal is evaporated at high temperature (500-2000°C) and under vacuum (10-6/10-7 torr). Metallic atoms are provided with sufficient energy to be dislodged from the source and to condense onto the substrate (quartz in our case). For quartz, it is necessary to use masks corresponding to the shape of the electrode and, if gold is deposited onto the quartz material, a sub-layer of chromium is necessary in order to improve adhesion. The typical thickness is 20 nm for chromium and 250 nm for gold, with total thickness rarely exceeding 500 nm. The cleaning of the surface is a crucial step as the adhesion of the metallic layer is completely dependent on the quality of the surface. Sputtering (Metals or Insulating Materials) Sputtering is a process whereby atoms or molecules of a material are ejected from a target (the material which is to be deposited) by bombardment of high-energy particles [1]. Positive ions derived from an electrical discharge in a gas at low pressure (~ 0.3 Pa) in a vacuum chamber are used and the material of the target is ejected in such a way as to be directed towards the substrate surface. Fig. 11.1 presents a schematic diagram of the device. Many materials can be used, even insulating materials, and it is easy to deposit alternate layers of different materials. The thickness range on the quartz substrate is between a few nm and 200 nm. 11.2.2 Electrochemical Method A previously prepared electrode (usually gold) on a quartz crystal is necessary and is used as working electrode in an electrochemical cell. The cell contains as electrolyte a salt of the metal to be deposited. The electrochemical process is a reduction reaction of the type:

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- High negative potential Target

Ar

Positive Terminal +

Ar

Me

Plasma region

Me

Substrates

To pumps

Argon inlet

Fig. 11.1. Schematic representation of a sputtering process chamber (according to reference [1])

M n + + ne − → M

(11.1)

The reduction is carried out either by control of applied potential (potentiostat) or of applied current (galvanostat). In the latter case, the current is usually maintained constant and the metal thickness can be determined from the quantity of electrons supplied and by using Faraday’s law. The film thickness is easily controllable by measuring the current and the time; the thickness range is between 10 nm to 100 μm. Various metals can be deposited in this way: copper, silver, iron, nickel, etc. This approach was used to perform calibration of a QCM by using silver electrodeposition [2]. Figure 11.2 shows a record of the microbalance frequency change over time during silver deposition at various current values: the slope of each curve extracted allows the mass sensitivity of the microbalance, K, to be calculated; K is defined as: K=

Δf microbalance Δm

(11.2)

Figure 11.3 shows the average K values obtained for different experimental conditions of silver deposition onto electrodes of different active surfaces. It can be seen that the theoretical value of K is reached only in the case of the largest active surface area (i.e. 0.64 cm2 for radius r = 4.5 mm). On the contrary, for smaller electrodes, lower values of K were obtained.

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500 Acm-2

10

- f /kHz

8 250 Acm-2

6 4 2 0

-2

50 Acm 5 Acm-2

0

100

200

300

400

t/s

Fig. 11.2. Time variation of the QCM frequency for different current densities S = 0.2 cm2 (from reference [2])

8

-7

-1

K x 10 /Hz g cm

2

10

6 4 2 0

1

2

3

4

5

6

-2

-log(i / Acm ) Fig. 11.3. Average sensitivity (K) calculated for different current densities for three different active surface electrodes. r = 4.5 mm, ▲ 2.5 mm and □ 1.5 mm. Theoretical value of K=8.15 107 Hz g-1 cm2 for a QCM working at 6 MHz

11.2.3 Technique Based on Glued Solid Foil (Nickel, Iron, Stainless Steel…) This technique uses a piezoelectric transducer working at low frequencies, for example, 2 MHz. This frequency is lower than that of the common QCM in order to allow the crystal to oscillate with much higher loads. The metal sample is a disk cut out from a foil of metal such as stainless steel, iron, nickel, 50 µm thick and 5 mm diameter. The glue is a 2-component epoxy resin with high chemical resistance in concentrated acid solution

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which, once polymerized, becomes extremely hard and acoustically lowabsorbing. The resin is insulating, since conductive glues were found not to have suitable acoustic and mechanical properties. Figure 11.4 shows a scheme of the crystal-glue-sample structure. The resonance frequency of the structure is about 1.75 MHz for an initial 2 MHz resonance frequency of the bare quartz. It should be pointed out that in this case the mass changes of the glued solid material can be studied directly, which opens up new possibilities for the investigation of processes on a wider variety of materials. For this technique of metallic preparation, it should be noted that the damping of the quartz crystal is drastically increased; in these conditions the piezoelectric behavior of the device can be modified (see Chaps. 1, 14) and the electronic working conditions have to be adapted (see Chap. 5).

11.3 Chemical Modifications (onto the metallic electrode) 11.3.1 Organic Film Preparation Polymer Electrogeneration (Conducting Polymers: Polypyrrole, Polyaniline…) The electrogeneration of conducting polymers is widely used in electrochemistry: a polymer film can be generated directly on the surface of the gold electrode of the quartz. Three methods of electropolymerisation are available: potentiostatic (constant applied potential), galvanostatic (constant applied current) or cyclic voltammetry (potential scanning). The thickness and the morphology of the polymer film are easily controllable by changing the value of the applied potential, the value of the applied current, or the value of the scan rate as well as by the composition of the electrodeposition bath. As an example, a schematic representation of the electropolymerisation of pyrrole can be depicted by the two following reactions [3]:

Py → Py + + e −

(2.1)

Py + + − Py → −(Py) 2 + 2H + + e −

(2.2)

Following this, the redox behavior of the deposited film, the insertion and removal of counter ions, the effect of doping, etc., can be studied through microbalance measurements: by using fast QCM. The microrheology of these materials must be well defined. Because polymers are not

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rigid materials, the QCM response can be due to the viscoelastic properties of the film as well as to the gravimetric response. Thus, interpretation of the QCM response purely in terms of mass changes may be completely wrong, i.e., the classical Sauerbrey equation is not valid [4] (see Chap. 14). The electroacoustic approach can partly solve these problems [5]. When conducting polymer films are deposited onto the quartz resonators, it should be noted that under some conditions, the viscoelastic contribution of the material can be important (see Chaps. 4, and 14). In this case, the pure mass response can be drastically affected or overestimated. sample

Au

resin

Cr

quartz

Cr

Au quartz

sample

Fig. 11.4. Structure of the crystal-glue sample resonator

11.3.2 Monolayer assemblies SAM Techniques (Thiol Molecule)

This procedure, which began to be used recently, is based on the adsorption of alkanethiols onto a gold electrode [6] as shown in Fig. 11.5. A covalent bond is created between the gold surface and the thiol groups: the alkane chains are maintained close to perpendicular over the surface owing to Van-der-Waals interactions (Fig. 11.6). The result obtained is, in general, close to a monolayer. Originally, this method was used for the QCM in order to investigate the influence of this surface treatment on molecular slipping [7]. It is now also used to investigate the formation and properties of different alkanethiols at applied potential, e.g. [8], and to prepare biosensors [9] (see Chaps. 9, 10 and 12).

11 Modified Piezoelectric Surfaces X

X

(CH2)n HS

(CH2)n

(CH2)n

S

X + Au

277

S Au

(n ≥ 1 and X = CO2H, CONH2, CN, CH2OH, CO2CH3, CH2NH2, CH3, CHCH2, etc.)

Fig. 11.5. Generation of a SAM over a gold electrode: general reaction (from reference [6]) HO O HO O HO O HO O HO O HO O C C C C C C

S

S

S

S

S

S

S

S

S

S

S

S

Au substrate Fig. 11.6. Diagram of the thioctic acid monolayer on the Au substrate (from reference [23])

Langmuir-Blodgett Method

The Langmuir-Blodgett (LB) is a simple method which allows molecular assemblies to be built up [10] through the deposition of one complete monolayer of amphiphilic molecules, i.e. molecules having a hydrophilic head and a hydrophobic chain. The film is first prepared on a water surface and then transferred to a solid substrate as shown in Fig. 11.7. Applications are various and LB films can be deposited directly onto the QCM active surface. These techniques can be coupled and lead, for example, to interesting smell sensors [11]. Self-Assembled Polyelectrolyte and Protein Films

The alternating adsorption of charged macromolecules or layer-by-layer adsorption has recently emerged as a promising simple method to fabricate controlled and highly ordered molecular assemblies in a pre-designed architecture. The QCM would, in principle, permit quantification of adsorbed protein, measurement of protein uptake kinetics in real time and

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determination of the binding constants from adsorption isotherms. The electroacoustic approach allows the range in which the Sauerbrey range is valid to be determined reasonably easily as has been shown [12, 13]. SAMPLE

MOLECULES

SOLUTION

WATER

WATER

SPREADING

FILM COMPRESSION

DEPOSITION (1st LAYER)

(2nd LAYER)

(3rd LAYER)

Fig. 11.7. Different steps of the Langmuir sequence: spreading, compression and transfer (from reference [9])

11.4 Biochemical Modifications Several techniques are available for immobilizing active biomolecules and four main strategies of immobilisation are presented in Fig. 11.8, this list is

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not exhaustive. Here, two different approaches are described: the direct method and the entrapped method. Finally, a specific part will be dedicated to DNA immobilisation methods. Adsorption Adsorption

Covalent grafting grafting Covalent

Inclusion Inclusion

Assembled layers layers Assembled

Fig. 11.8. Different methods of biomolecule immobilisations

11.4.1 Direct Immobilisation of Biomolecules (Adsorption, Covalent Bonding)

Two main types of biomolecules can be immobilized: enzymes and antibodies (or antigens). Biological assemblies can also be used, with different degrees of sophistication, and the QCM appears to be a very interesting tool to follow the various interactions. One possibility is based on direct adsorption of biomolecules onto a QCM gold electrode surface: this method leads to interesting results since the activity of the biomolecules is maintained at a high level under certain conditions. It should be noted that this procedure is widely used in ELISA tests, a classical method of biological analysis based on colorimetric detection. Small interactions will occur between the electrode and the biomolecules such as through hydrogen bonds, Van der Waals interactions, ionic interactions, etc. Moreover, with antibodies, the presence of sulfur groups can improve adhesion if the immobilisation is performed on a gold surface (see Chap. 12). The advantage is the ease of preparation of the surface even if, in general, it is difficult to regenerate properly the surface. The principle of detection is depicted in Fig. 11.9 - in this case a reference sensor is used in order to subtract out non-specific interactions [14]. Using a high sensitive QCM the limit of detection can reach 10-10 M for the corresponding detected antigen in solution with a 27 MHz quartz resonator [15]. In this case, this QCM allowed direct transduction to occur without the realization of a sandwich assay as in classical tests. The biolayer

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specificity can be tested in two ways: by saturating the QCM surface by a non active molecule such as BSA or by checking the response of the selective layer saturated with BSA against a non-complementary antigen (see Chap. 12). MEASUREMENT - CRYSTAL

REFERENCE - CRYSTAL

MEASUREMENT OF Fo

PIEZOCRYSTAL SPECIFIC ANTIBODY

UNSPECIFIC ANTIBODY ANTIGEN

Fig. 11.9. Schematic of measurement procedure for immunodetection with adsorbed antibodies onto a QCM surface (from reference [11])

Figure 11.10 (upper panel) is a preliminary negative control, in the case of SEB (Staphylococcus Enterotoxin B) detection, where a non selective layer was made by only BSA adsorption: the SEB interaction does not occur since no frequency shift (fm) is observed. Then, a second specific test was performed as is shown Fig. 11.10 (lower panel). The selective film was made by direct adsorption of anti-SEB antibodies. When peroxidase solution flows over the QCM, treated with SEB antibodies, the QCM response is negligible. Secondly, for the same treated quartz crystal, a SEB solution flows and the microbalance frequency decreases due to the interaction with the anti SEB antibodies. Direct antigen detection is feasible due to the high sensitivity and selectivity of the transducer used. Thus, it is sufficient to directly detect the binding event between small biomolecules, such as SEB, and the specific antibody layer.

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27003100

27003080

fm /Hz

27003060

PBS

27003040

PBS SEB 16 mg/ml

27003020

27003000

0

10

20

30

20 PO 22 mg/ml

0

PBS SEB 20 mg/ml

fm -fm,t=0 /Hz

-20 -40 -60 -80 PBS

-100 -120 0

10

20

30

40

50

60

70

Time/min

Fig. 11.10. Interactions between SEB antigens and a saturated surface of BSA adsorbed on the gold surface of a 27 MHz QCM (upper panel) and detection of peroxidase antigens and SEB antigens with a 27 MHz QCM modified by direct adsorption of anti-SEB

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More sophisticated strategies can also be used to achieve immobilisation such as covalent bonding of the biomolecules based on different coupling agents like NHS/EDC (Fig. 11.11a) or glutaraldehyde (Fig. 11.11b). Normally, functionalisation of the gold surface is necessary by using SAM techniques. The stability of the selective film is always an advantage but the resulting biological activity is not always guaranteed. S

+

Thioctic acid

S

SAM

S

C

Gold electrode Quartz

S

OH/O-

C2H5

N

O O C

EDC

C

+

(CH2)3N(CH3)2

N

OH/OO

S S

O C O

C2H5

+

NH C

N

HO

NHS

N O

(CH2)3N(CH3)2

S S

+

O

O

NH2

Antibody

C O

NH C2H5 O

C N

N O

(CH2)3N(CH3)2 S S

O C

O HO

O

NH

Antibody

N O

a Cysteamine +

SAM

SH CH2 CH2 NH2

S CH2 CH2 NH2

Dimerique glutaraldehyde +

CHO

(CH2)3

C

CH

(CH2)2

CHO

CHO

CH

CHO

S CH2 CH2 N CH C

N

Antibody

(CH2)3

(CH2)3

+

NH2

Antibody

S CH2 CH2 N CH C

(CH2)2

(CH2)2

CHO

CHO

b

Fig. 11.11. Methodology of covalent bounding for antibodies on gold electrode: a NHS-EDC and b glutaraldehyde coupling agents.

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11.4.2 Entrapping of Biomolecules (Electrogenerated Polymers: Enzyme, Antibodies, Antigens…)

This approach is widely used for making biosensors. Its great interest is to construct three dimensional structures and in this way, to improve the interactions and therefore the response range. Usually, the active biomolecule is entrapped inside a polymer structure. Either the biomolecules are mobile in the polymer film or are covalently bonded to the polymer chains. Various polymers can be used but mainly conducting polymers are employed. Electrochemical detection is well adapted to this situation but QCM devices can also offer interesting potentialities. An example is an avidin bioaffinity sensor based on an electropolymerized biotinylated polypyrrole film. The functionalisation of the gold electrode of the quartz resonator was achieved by controlled potential oxidation of a biotinyl pyrrole monomer in CH3CN leading to chemical grafting of the biotin molecule to the electrogenerated polypyrrole. In the presence of avidin solutions, gravimetric measurements indicated the reproducible formation of a compact avidin layer via the specific avidin-polymerized biotin interaction. Avidin concentrations in the range 1.5·10-9 - 3.7·10-5 M can be measured by this method [16] and an example is presented in Fig. 11.12.

100

PBS

0

Avidin

fm-fm, t=0 /Hz

0.5 microg/ml -100

-200

-300

-400 0

1000

2000 Time/s

3000

4000

Fig. 11.12. Interaction between pyrrole modified by biotin group and avidin molecule in solution through 27 MHz QCM measurements

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11.4.3 DNA Immobilisation

This application using QCM devices is relatively recent [17-21] and the idea is similar to the immunosensor. More precisely, a DNA probe has to be immobilized onto the quartz surface or onto one of its metallic electrodes. When contacted with the corresponding DNA target, hybridisation occurs and leads to a decrease of the microbalance frequency. A direct and fast response is normally obtained. All the chemical methods given previously are available to prepare the surface: covalent bonding, adsorption or more sophisticated methods. In the following, some pertinent examples are given. The first example is related to biotin-DNA modified probes which can be immobilized by using coupling with avidin layers already prepared on the gold surface as is shown in Fig. 11.13 [22]. The methods previously described can be used for avidin immobilisation. Another less sophisticated approach is based on SAM methodology [23] and the methodology of surface modification is given in Fig. 11.14. O Au

COOH

S

S

COOH

COOH

O + EDC

COOH

S OH

O

HO-N

S H2N

S

CONH-

CONHCH2CH2OH

CONHS

O O

S

S S

CO-N

O

S CO-N

CO-N

O

O H2NAvidin

CONH-

S S

CONHCH2CH2OH

Fig. 11.13. Immobilisation of biotin-DNA on avidin-modified QCM electrodes (from reference [22]) (CH2)6

Probe

(CH2)6 OH

Gold

OH OH OH OH OH OH OH OH OH

Fig. 11.14. Thiol-DNA immobilisation and step of saturation according to reference [23]

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285

A careful cleaning of the gold surface must be performed before the thiol-DNA modified probe interaction and piranha solutions are good for this purpose. Immobilisation on a gold surface of a 20-base DNA probe labeled with disulfide groups and its selective hybridisation with the complementary 20-base DNA strand is examined through QCM measurements. The oligonucleotide probe is the complementary strand of a partial sequence of the gene encoding for a large ribosomal RNA sub-unit which is a coding sequence of Alexandrium minutum DNA, a microalgae that produces neurotoxins responsible for paralytic shellfish poisoning on European and Asian coasts. The kinetics of DNA probe immobilisation and hybridisation were monitored in situ by using a 27 MHz quartz crystal microbalance under controlled hydrodynamic conditions [24] (Fig. 11.15). The frequency of the set up is stable to within a few Hertz, corresponding to the nanogram scale, for 3 hours and makes it possible to follow frequency change from immobilisation of the probe to hybridisation of the complementary target DNA. This setup constitutes a biosensor which is sensitive and selective, and the hybridisation ratio between hybridized complementary DNA and immobilized DNA probes is 47%. NaCl

0

NaCl + ADN-disulfide

Δfm/Hz

-100

NaCl

HEPES + ADNnc HEPES

HEPES

Δf = - 184 Hz -200

HEPES + ADNc

Δf = - 48 Hz

HEPES

-300

Δf = - 77 Hz 0

2000

4000

6000

8000

10000

t/s Fig. 11.15. Microbalance frequency changes measured after circulation of 20 μg ml-1 DNA-disulfide NaCl solution, 20 μg ml-1 non complementary DNA target HEPES solution and 20 μg ml-1 complementary DNA target HEPES solution [24]

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In Fig. 11.15, the successive steps from the DNA probe immobilisation to the DNA hybridisation are followed. Very attractive potentialities were also shown with this approach in term of in situ probe modification [25] or characterisation of DNA supramolecular polymerisations [26]

References 1. S. Swann (1988) “Magnetron sputtering” Physics in technology 19: 67-75 2. C. Gabrielli, M. Keddam and R. Torresi (1991) “Calibration of the electrochemical quartz crystal microbalance” J. Electrochem. Soc. 138: 2657-2660 3. B.R. Scharifker, E. Garcia-Pastoriza and W. Marino (1991) “The growth of polypyrrole films on electrodes” J. Electroanal. Chem. 300: 85-98. 4. R.A. Etchenique and E.J. Calvo (1999) “Gravimetric measurement in redox polymer polymer electrodes with the EQCM beyond the Sauerbrey limit” Electrochem. Commun. 1(5): 167-170 5. J.J. Garcia-Jareno, C. Gabrielli and H. Perrot (2000) “Validation of the mass response of a quartz crystal microbalance coated with Prussian Blue film for ac electrogravimetry” Electrochem. Commun. 2(3): 195-200 6. R.G. Nuzzo, L.H. Dubois and D.L. Allara (1990) “Fundamental studies of microscopic wetting on organic surfaces. 1. Formation and structural characterization of a self-consistent series of polyfunctional organic monolayers” J. Am. Chem. Soc. 112(2): 558-569 7. F.F. Ferrante, A.L. Kippling and M. Thompson (1994) “Molecular slip at the solid-liquid interface of an acoustic-wave sensor” J. Appl. Phys. 76(6): 34483462 8. C.M.A. Brett, S. Kresak, T. Hianik and A.M. Oliveira Brett (2003) “Studies on self-assembled alkanethiol monolayers formed at applied potential on polycrystalline gold electrodes” Electroanalysis 15(6-7): 557-565 9. Y.S. Fung YS and Y.Y Wong (2001) “Self-assembled monolayers as the coating in a quartz piezoelectric crystal immunosensor to detect salmonella in aqueous solution” Anal. Chem. 73(21): 5302-5309 10. A. Barraud (1987) “Langmuir-Blodgett supermolecular assemblies” British Polymer Journal 19(3-4):409-412 11. S.S. Shiratori, K. Kohno and M. Yamada (2000) “High performance smell sensor using spatially controlled LB films with polymer backbone” Sensors and Actuators B 64: 70-75 12. E.J. Calvo and A. Wolosiuk (2002) “Donnan permselectivity in layer-by-layer self-assembled redox polyelectrolye thin films” J. Am. Chem. Soc. 124: 84908497 13. E.J. Calvo, E.S. Forzani and M. Otero (2002) “Study of layer-by-layer selfassembled viscoelastic films on thickness-shear mode resonator surfaces” Anal. Chem. 74(14): 3281-3289

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14. G.G. Guilbault, B. Hock and R. Schmid (1992) “A piezoelectric immunobiosensor for atrazine in drinking water” Biosens. Bioelectron. 7: 411-419 15. K. Bizet, C. Gabrielli and H. Perrot (2000) “Immunodetection by quartz crystal microbalance. A new approach for direct detection of rabbit IgG and peroxidase” Appl. Biochem. Biotechnol. 89: 139-149 16. S. Cosnier, H. Perrot and R. Wessel (2001) “Biotinylated polypyrrole modified quartz crystal microbalance for the fast and reagentless determination of avidin concentration” Electroanalysis 13: 971–974 17. Okahata Y, Matsunobu Y, Ijiro K, Mukae M, Akira M and Makion M (1992) “Hybridization of nucleic acids immobilized on a quartz crystal microbalance” J. Am. Chem. Soc. 114(21): 8299-8300 18. A. Dupont-Filliard, A. Roget, T. Livache and M. Billon (2001) “Reversible oligonucleotide immobilisation based on biotinylated polypyrrole film” Anal. Chim. Acta 449: 45–50 19. I. Willner, F. Patolsky, Y. Weizmann and B. Willner (2002) “Amplified detection of single-base mismatches in DNA using microgravimetric quartz– crystal-microbalance transduction” Talanta 56: 847–856 20. X.C. Zhou, L.Q. Huang and S.F.Y. Li (2001) “Microgravimetric DNA sensor based on quartz crystal microbalance: comparison of oligonucleotide immobilisation methods and the application in genetic diagnosis” Biosens. Bioelectron. 16: 85-95 21. S. Tombelli, M. Mascini and A.P.F. Turner (2002) “Improved procedures for immobilisation of oligonucleotides on goldcoated piezoelectric quartz crystals” Biosens. Bioelectron. 17: 929-936 22. F. Caruso, E. Rodda and D.N. Furlong (1997) “Quartz crystal microbalance study of DNA immobilizaton and hybridization for nucleic acid sensor development” Anal. Chem. 69(11): 2043-2049 23. I. Mannelli, M. Minunni, S. Tombelli and M. Mascini (2003) “Quartz crystal microbalance (QCM) affinity biosensor for genetically modified organisms (GMOs) detection” Biosens. and Bioelectron. 18: 129-140 24. M. Lazerges, H. Perrot, E. Antoine, A. Defontaine and C. Compere (2006) “Oligonucleotide quartz crystal microbalance sensor for the microalgae Alexandrium minutum (Dinophyceae)” Biosens. Bioelectron. 21(7): 1355-1358 25. M. Lazerges, H. Perrot, N. Zeghib, E. Antoine and C. Compere (2006) “In situ QCM DNA-biosensor probe modification” Sensors and Actuators B 120: 329-337 26. M. Lazerges, H. Perrot, N. Rabehagasoa, E. Antoine and C. Compere (2005) “45- and 70-Base DNA supramolecular polymerizations on quartz crystal microbalance biosensor” Chem. Commun. 48: 6020-6022

12 Fundamentals of Piezoelectric Immunosensors Angel Montoya1, Aquiles Ocampo2 and Carmen March1 1

Instituto de Investigación e Innovación en Bioingeniería. Universidad Politécnica de Valencia 2 Departamento de Ingeniería Biomédica. Escuela de Ingeniería de Antioquia

12.1 Introduction As already mentioned in Chaps. 9 and10, a biosensor can be defined as an analytical device in which a biological active component (receptor), such as an enzyme, an antibody, a tissue portion, a whole cell, etc., is immobilized onto the surface of an electronic, optic or optoelectronic transducer. When a target analyte is recognized by the immobilized biological material (Fig. 12.1), the biochemical interaction is directly transformed into a quantifiable signal by means of the transducer. According to the nature of the transducer, the signal can be electrochemical (amperometric or potentiometric), piezoelectric, optical, etc. The primary signal provided by the transducer is subsequently converted into an electronic one, conditioned, and recorded using the suitable acquisition and recording devices and programmes. Finally, an unequivocal dose-response correlation has to be established between the variations of the analyte concentration and the signal shifts.

Fig. 12.1. General scheme of a biosensor A.A. Vives (ed.), Piezoelectric Transducers and Applications, doi: 10.1007/978-3-540-77508-9_12, © Springer-Verlag Berlin Heidelberg 2008

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An immunosensor is a particular type of biosensor in which the biological component and the target analyte are immunoreagents involved in an immunoassay. The term “immunoassay” refers to and comprises all the analytical procedures based on the specific antigen-antibody recognition. As regards to the immunoreagents, usually only one antibody takes part in the immunoassay, while several antigens (free analytes, protein-hapten conjugates) can be involved in the reaction, provided that all of them are recognized by the antibody (Fig. 12.2). Analyte

Antibody

Binding

Hapten conjugate

Fig. 12.2. Schematic representation of the different possibilities of antigenantibody binding in immunoassays

An antibody is a protein produced naturally by the immune system of mammals as a defence reaction against the exposure to an external agent (the antigen). Antibodies can also be produced in the laboratory and then used for analytical purposes in immunoassays. When analytes are lowmolecular weigth compounds, the antibody production requires the synthesis of chemically modified analytes (the haptens), and their covalent binding to proteins. The protein-hapten conjugates thus obtained are used as antigens for animal immunization and also as assay conjugates in immunoassays. In the most popular immunoassay configuration, one of the immunoreagents (the antigen or the antibody) is immobilized on a solid support. Depending on the immobilized molecule, two main solid-phase immunoassay formats can be defined: the conjugate-coated format and the antibody-coated format (Fig. 12.3). In the conjugate coated format (when the immobilized immunoreagent is the hapten-protein conjugate), the detection of the analyte is based on a binding inhibition test. Therefore, a competitive assay is performed, in which the free analyte competes with the immobilized conjugate for binding to a fixed, limited amount of antibody. As in any competitive assay, the assay signal decreases as the analyte concentration increases. This inversely proportional relationship allows us to obtain the typical competitive immunoassay dose-response standard curves (Fig. 12.4).

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Hapten conjugate IMMUNOREAGENTS Analyte

Antibody

COATING FORMATS

Conjugate - coated

Antibody - coated

Fig. 12.3. Solid-phase immunoassay formats

Analyte

Antibody

Signal

Detection

Hapten conjugate

[analyte]

Fig. 12.4. The principle of the competitive immunoassay

Currently, Enzyme Linked Immunosorbent Assays (ELISA) and Immunosensors are the most popular immunoassays. In ELISAs the detection of the analyte is always indirect because one of the immunoreagents is labelled with an enzyme. In the last step of the assay, a colorimetric signal is produced when the enzyme transforms a colourless substrate into a coloured product. In immunosensors one of the immunoreagents must also be immobilized on the surface of the transducer. When the immunochemical interaction occurs, a physical signal is directly produced at the transducer, thus allowing a direct analyte detection [1,2]. This direct, label-free detection represents undoubtedly an essential advantage of immunosensors as compared to label-dependent immunoassays. Piezoelectric immunosensors use a quartz crystal as the transducer element, working in the microgravimetric mode (QCM, Quartz Crystal Microbalance). The quartz crystal has been used widely in electric circuits

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as a frequency standard clock in computers, communication systems, and frequency measurement systems. In addition, the quartz crystal is able to oscillate when brought into resonance by the application of an external alternating electric field (see Chap. 5). Under certain conditions, the resulting oscillation frequency is determined by the mass of the crystal (see Chaps. 1, 3, and 14). The gold electrodes (see Fig. 12.10) of piezoelectric quartz crystals can be used as the support for immobilization of immunoreagents (antibodies, antigens or hapten-conjugates), in such a way that a subsequent immunoreaction (antigen-antibody binding) could be detected as a mass variation and correlated to the concentration of the analyte. The operation of piezoelectric resonators in liquid phase involves both mass change inducing frequency shifts and density or viscosity effects of the surrounding liquid layer [3-5] (see Chap. 3). For suitable QCM immunosensor applications, the properties of the surrounding liquid should not change during experiment, so as the changes in the QCM parameters are due only to the immunoreaction which occurs on the immobilization gold-liquid interface layer; additionally no changes in the resistance should be monitored during experiment assuring Sauerbrey-like behaviour of the interface layer and negligible effects of the liquid property changes. Because of their simplicity, low cost, and real-time response, piezoelectric quartz crystal sensors are gaining an increasing importance as competitive tools for bioanalytical assays and characterization of affinity interactions of biomolecules [6]. Piezoelectric immunosensors have been proposed for many applications in different fields such as: food and biomedical analysis, veterinary diagnosis, environmental monitoring, etc. The target analytes include viruses [6,7], bacteria and eukaryotic cells [8,9], proteins [6,10], nucleic acids [11], and small molecules as drugs, hormones and pesticides [6,12-15]. Specially they have been proposed as alternative or complementary methods in the monitoring of organic pollutants such as herbicides, dioxins, and polychlorinated biphenyls (PCBs) [4, 6, 14-19]. The development of a QCM immunosensor requires extensive previous work involving the production and immobilization of immunoreagents. Those steps can be basically summarized in: • • • •

Hapten synthesis. Antibody production and characterization. Immunoreagent immobilization (see also Chap. 11). Immunosensor characterization.

In this chapter we describe the development of a model of piezoelectric immunosensor based on monoclonal antibodies for pesticide analysis.

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12.2 Hapten synthesis Small organic molecules such as pesticides, drugs, etc., are usually nonimmunogenic and hence they do not elicit an immune response when injected into experimental animals. For this purpose they have to be coupled to carrier macromolecules such as proteins. Therefore, this type of analytes has first to be chemically modified to introduce in their structure the chemical groups suitable for protein conjugation. These modified compounds are called haptens, and the successful generation of specific antibodies and sensitive assays to a small molecule is greatly dependent upon the proper design of a wide panel of immunizing and assay haptens. The haptens should of course be as similar as possible to their corresponding analytes, maintaining their main structural features, chemical groups, and electronic distribution [20]. The proper covalent binding of the haptens to the carrier proteins should produce stable carrier-hapten complexes (protein-hapten conjugates), that can subsequently be used as immunizing antigens (Fig. 12.5) and as assay conjugates in competitive immunoassays or immunosensors.

Protein

X Analyte

Hapten

Immunogen (Hapten-Conjugate)

Fig. 12.5. Preparation of immunogenic conjugates from low-molecular weight analytes

As mentioned above, hapten design is a key step in the development of immunoassays for small molecules because the hapten is primarily responsible for determining the antibody recognition properties. The functional groups of the hapten govern the selection of the conjugation method to be employed. Common procedures use amine, carboxylic acid, hydroxyl, or sulfhydryl groups on the hapten and the protein [21]. There is a general agreement on the difficulty of predicting which hapten is theoretically the most appropriate for a particular analyte and, furthermore, whether this hapten will respond more adequately as immunogen

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and in the assays. If monoclonal antibody technology is going to be undertaken, as explained in next section, it should be preceded by a careful hapten synthesis work in order to take advantage of the striking differences in the properties of the antibodies eventually obtained. To explore the maximum number of possibilities to obtain high-performance antibodies, as well as for further improvements in assay sensitivity, the synthesis of a number of haptens with different spacer arms attached through different molecular sites is strongly recommended [22-24]. As an example of a successful hapten syntesis, Fig. 12.6 shows the chemical strategy followed at the Instituto de Investigación e Innovación en Bioingeniería (Universidad Politécnica de Valencia, Spain) to prepare the CNH hapten, used for antibody production and development of immunoassays to the insecticide carbaryl [25]. CNH (6-[[(1-phthyloxy)carbonyl]amino]hexanoic acid) was prepared by means of a two steps process: the first one consisted of the synthesis of 1-naphthyl chloroformate, which was obtained from 1-naphthol, sodium hydroxide, phosgene and toluene through an intermediate highly reactive radical. The second step was the CNH synthesis from the previously obtained naphthyl chloroformate, by reation with aminohexanoic acid in a basic dioxane solution. O O C

H N

CH3

Carbaryl

OH

O

-

NaOH / H2O

COCl2 / toluene O

O O C

COOH

N H

CNH

H2N

O C

COOH

NaOH / H2O / 1,4-dioxane

Fig. 12.6. Chemical synthesis of the carbaryl hapten CNH

Cl

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12.3 Monoclonal antibody production Immunoassay development requires the production of antibodies to the analytes and their incorporation into adequate assay configurations, most usually ELISAs. As mentioned above, the successful development of specific antibodies and sensitive assays to small molecules greatly depends on a proper design of haptens. Particularly, the detection of a group of compounds of similar structure can be often accomplished by judicious synthesis of hapten-protein immunogens to expose common features to all of the members of the group to the maximum while minimizing the presentation of structural differences to the immune system. It is currently well established that ELISA formats influences the sensitivity of immunoassays, but there is no general agreement on the configuration that provides the most sensitive assays. Thus, in several studies the antibody-coated format was more sensitive than the conjugate-coated one, whereas in other studies no difference between formats, or even the opposite behaviour, was found [26, 27]. Furthermore, the influence of the ELISA format on the assay sensitivity seems to depend not only on the format by itself but also on the hapten used to obtain the assay conjugate, i.e., on hapten heterology. The preparation of antibodies against haptens, designed for special applications such as pesticides and PCBs, is based on covalent binding of the hapten to a carrier protein, followed by immunization of animals with the synthesized immunogens, as depicted in Fig. 12.7. The chemical binding of the hapten to a protein determines in part the antibody specificity [16]. Spacer arm

Response

Fig. 12.7. Antibody production from haptens

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Once the immunogens for the analyte are prepared, the debate arises whether polyclonal or monoclonal antibodies are obtained. Polyclonal antibodies are produced by using traditional immunisation procedures, namely in rabbits, goats, sheep and pigs. These antibodies, especially from rabbits, are widely used as reagents in many immunochemical analyses, although a major disadvantage of this approach lies in the fact that it is not possible to produce identical antibody specificity even in two animals of the same species. If an unlimited supply of a single and homogeneous type of antibody is required, the choice is the hybridoma technology to obtain monoclonal antibodies. Standardized immunoreagents may facilitate the acceptance of immunoassays in the analytical laboratory by ensuring a long-term supply of kits with a defined performance [23, 28]. Nevertheless, a wider spreading of monoclonal antibodies is limited by the still low predictability of the hybridoma technology results. The procedure for producing monoclonal antibodies is similar for different analytes and involves the following steps: (1) immunization of 8-10 week-old female mice, (2) cell fusion of mouse spleen lymphocytes from positive mice with myeloma cells, (3) selection of high-affinity antibodysecreting hybridoma clones, which were expanded and cryopreserved in liquid nitrogen and (4) purification of monoclonal antibodies and storage at 4 oC as ammonium sulphate precipitates. The affinity and specificity of the produced monoclonal antibodies to the analyte are characterized using either antibody-coated or conjugate-coated ELISA formats [27]. The hybridoma selection and cloning is one of the more critical steps on the development of monoclonal antibodies. The common procedure used in our laboratory is described: eight to ten days after cell fusion, culture supernatants were screened for the presence of antibodies that recognized the analyte. The screening consisted of the simultaneous performance of a non competitive and a competitive indirect ELISA, to test the ability of antibodies to bind the protein conjugate of the immunizing hapten and to recognize the analyte, respectively. Selected hybridomas were cloned and the stable antibody-producing clones were expanded and cryopreserved in liquid nitrogen [23].

12.4 Immobilization of immunoreagents The immobilization of biomolecules on the solid substrate that constitutes the transducer surface is essential to ensure the sensor performance because of its role in the specificity, sensitivity, reproducibility and recycling ability. Some of the requirements that should be fulfilled by

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an immobilization process are the following: (1) retention of biological activity of biomolecules after immobilization onto the sensor surface; (2) reproducible, durable, and stable attachment with the substrate against variations of pH, temperature, ionic strength, and chemical nature of the microenvironment; and (3) uniform, dense, and oriented localization of the biomolecules. Various methods to immobilize biomolecules have been reported in the literature. They include physical adsorption [7, 29], avidin-biotin system [30] and covalent binding [13, 30-33]. Among them, covalent binding is the most promising technique because it fulfils most of the requirements mentioned above. Chapter 11 describes the main aspects concerning immobilization of molecules on a substrate. Recently, great effort has been devoted to the achievement and optimization of conditions for covalent binding. Self-assembled monolayer (SAM) technology is providing the best results (34-39). SAM is the generic name given to the methodologies and technologies that allow the generation of monomolecular layers, also called monolayers, of biological molecules on a variety of substrates. The formation of such monolayer systems is extremely versatile, allowing the in vitro development of biosurfaces which are able to mimic naturally occurring molecular recognition processes. SAMs also permit reliable control over the packing density and the environment of an immobilized recognition centre or multiple centres at a substrate surface. Many systems are adequate to carry out the process of self-assembly: long chain carboxylic acids or alcohols (RCOOH, ROH), where R is an alkyl chain, reacting with metal oxide substrates; organosilane species (RSiX3, R2SiX2 or R3SiX), where X is a chlorine atom or an alkoxy group, reacting with hydroxylated substrates, such as glass, silicon and aluminium oxide; and organosulfur-based species reacting with noble metal (gold, silver) surfaces. The latter system has been the most studied up to date and for this reason it is the best characterized in terms of stability and physicochemical properties. Moreover, this approach takes advantage of the fact that sulfur-containing compounds, e.g. alkanethiols, dialkyl disulfides and dialkyl sulfides, have a strong affinity for noble metal surfaces [34]. Thiols and sulfides are of particular interest because of their spontaneous chemisorption, regular organisation and high thermal, mechanical and chemical stability on gold surfaces. Long chain thiols and sulfides have been shown to be more thermally stable and the adsorption to the surface has been shown to proceed by two methods: (1) ionic dissociation (Eq.12.1a) and more favourably by (2) radical formation (Eq.12.1b) [35]. -

+

RSH + Au → AuRS + H

(12.1a)

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RSH + Au → AuRS· + H·

(12.1b)

Because of the stability, orientation and ability to functionalize the terminal groups on the molecules, SAM can offer a very convenient and versatile method for covalent immobilization of biomolecules on gold electrode surfaces for biosensor development. Being in intimate contact with the support surface, SAMs do not have the problems associated with mass transport, thus providing the advantage of a faster and potentially more intense response when exposed to external stimuli [34-35]. To illustrate the process of formation of a self-assembled monolayer for covalent attachment of macrobiomolecules, Fig. 12.8 shows the scheme of immobilization of a conjugate for the analysis of the model pesticide carbaryl. To form the SAM, an alkanethiol solution of mercaptoundecaonoic or thioctic acid was adsorbed onto the gold sensor surface. The activation of the carboxylic groups onto an intermediate reagent (N-hydroxysuccinimide ester) took place by using a mixture of N-hydroxi-succinimide (NHS) and carboxi-diimide (EDC). This intermediate is able to covalently attach the amine groups of the protein-hapten conjugates used as recognition element. Then ethanolamine was added in order to deactivate all the unreacted NHS-esters remaining on the sensor surface. This procedure ensures that only covalently bound analyte derivatives remained on the sensor surface [31-32, 36-39]. O N

Mercaptoundecanoic or Thioctic acid H

H

O

O O

S

H O O

S

H O O

S

H

H

O O

S

O O

S

H O O

S

H O O

S

C

NCH2CH3

CH2CH2CH2

O

O O

Au

O

NHS

O

S

CH3

C NCH2CH3

O

EDC

N

HN

H

H O

S

CH2CH2CH2N

CH3

O

O

S

N

H

OH CH3

CH3

O

S

S

Au

Au

Hapten-conjugate

NH 2 O

N

O

H N

O

O

O

S

S

Au

Au

Ethanolamine blocking

N

H

N

O S

H

N

O S

H

N

O S

H

N

O S

H

N

O S

H

N

O S

H

N

O S

H

N

O S

H

N

O S

H

N

O S

H

N

O S

H O

S

Au

Fig. 12.8. SAM immobilization protocol for an hapten conjugate (courtesy of Dr. Laura Lechuga, IMM-CSIC, Madrid, Spain).

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The use of mixed self-assembled monolayers of alkanethiols on gold is particularly recommended when the immobilized biomolecule is the antibody, in order to prevent denaturation of the protein, to minimize steric hindrance, and hence to improve its activity [38, 40-42]. Mixed SAMs are generally formed by co-adsorption of mixtures of two thiols, one of them providing a functional head group (like a carboxylic acid) at a low molar fraction, and the other one being the “diluting” thiol at a high molar fraction. The second thiol reduces the surface concentration of functional groups, thus minimizing steric hindrance, partial denaturation of the protein and non-specific interactions that could produce interference signals. Also the diluting thiol can be used to tailor the overall physico-chemical properties of the interface (such as its hydrophobic/hydrophilic character). The use of mixed SAMs may control chemical, structural, and biological surface properties of the immobilized protein. This ability to control nanoscale surface properties, including the abundance, the type, and the spatial (both normal and lateral) distribution of tail group sites, will undoubtedly facilitate the efforts to develop biocompatible biomaterials, biosensors, and molecular electronics. The control of nanoscale surface properties wil also allow to adjust the surface microenvironment to fit the various shapes and surface functionalities of biomolecules. In fact, mixed SAMs have already shown interesting adsorption behaviour through the variation on the surface composition of the tail groups [43-44].

12.5 Characterization of the piezoelectric immunosensor The following step in the immunosensor development is its characterization. Here we propose a practical example of a piezoelectric immunosensor developed for the analysis of the model pesticide carbaryl [45]. Fig. 12.9 shows the scheme of the flow-through system used to manage the immunosensor. The set up consists of a circuit in which buffers and samples flow through a piezoelectric cell, by means of two distribution/injection valves and a peristaltic pump. Piezoelectric transducer and measurement system. Gold coated AT-cut 9-10 MHz quartz crystals (0.167 mm thick, 5 mm diameter, 0.196 cm2 active area) were from International Crystal Manufacturing Company Inc., Oklahoma City, OK (USA). Once chemically functionalized, the piezoelectric quartz crystal was placed and sealed in a homemade Arnite cell with two Nitryl O-rings. Only one face of the crystal was allowed to be in contact with the reagents during the assays.

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Fig. 12.9. Scheme of the flow-through system of the piezoelectric immunosensor

The flow cell was included in a flow-through system controlled by a peristaltic pump (Fig. 12.9). The frequency and resistance of the crystal were continuously monitored and recorded during the assays with a Research Quartz Crystal Microbalance (RQCM) from Maxtek Inc. This instrument is based on the capacitance cancellation technique and designed for research applications. A scheme of the flow-cell and the measurement system is shown in Fig. 12.10. Δf & ΔR measurements

Flow Cell

Δf & ΔR acquisition

Quartz Crystal – The heart of the QCM

Gold electrode (air contact lower side) Gold electrode (coated-functionalized upper side)

Quartz plate

Fig. 12.10. Detail of the flow cell and the quartz crystal

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Data analysis. To optimize the reproducibility of the assay, three mixed analyte-antibody samples were measured for each concentration of carbaryl standard. Normalized standard curves were obtained by plotting the frequency decrease versus the logarithm of analyte concentration. The experimental points were fitted to the four-parameter logistic equation:

(

y = D + ( A − D) 1 + ( x / C ) B

)

(12.2)

where A is the asymptotic maximum (maximum signal in absence of analyte, Smax), B is the curve slope at the inflection point (related to the analyte concentration giving 50% inhibition of Smax: C, I50) and D is the asymptotic minimum (background signal). Immunoassay format. The immunoassay developed to determine carbaryl was an inhibition test based on the conjugate coated format, in which the hapten conjugate was immobilized on the sensor surface. For the inhibition assays, a fixed amount of the respective monoclonal antibody was mixed with standard solutions of the analyte, and the mixture was pumped over the sensor surface. Since the analytes inhibit antibody binding to the respective immobilized conjugates, increasing concentrations of analyte will reduce the frequency decrease of the piezoelectric QCM sensor. Different standard concentrations of carbaryl were prepared by serial dilutions in dimetylformamide. The standards were mixed with a fixed concentration of the monoclonal antibody. Analyte-antibody solutions were incubated for one hour at room temperature and then injected onto the sensor surface. The resonance frequency of the piezoelectric crystal was monitored in real-time for each analyte concentration, as the binding between free antibody and the immobilized conjugate took place. Once each assay was finished, regeneration of the biosensing surface was performed using diluted hydrochloric acid to break the antibody-analyte association. Representative standard curves were finally obtained by averaging several individual standard curves, previously normalized by expressing the frequency decrease provided by each standard concentration (∆f) as the percentage of the maximun response (frequency decrease obtained in the absence of analyte, ∆f o):

Normalized signal = 100 × ∆f /∆fo

(12.3)

Sensor characterization. As shown in Fig. 12.11, under the conditions assayed the piezoelectric immunosensor was working in the microgravimetric mode; only frequency changes were monitored while the signal corresponding to the resistance was nearly constant. Therefore, the frequency shifts obtained could be attributed to changes in the mass deposited (i.e., antibody bound) onto the sensor surface.

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Fig. 12.11. Sensor characterization. Frequency (upper record) and resistance (lower record) shifts with time during an experiment performed with the QCM immunosensor.

Pesticide determination. A reliable immunosensor should be able to precisely and accurately determine the analyte concentration in problem samples. As expected, the frequency signals provided by the carbaryl immunosensor, once normalized and fitted according to Eq. (12.2) and Eq. (12.3), followed the decreasing sigmoidal shape characteristic of competitive assays (Fig. 12.12). The calibration curve allows the analysts to calculate the analyte concentration which reduces the assay signal to 50% of the maximum (I50). This value is generally accepted as an estimate of the immunosensor sensitivity, in such a way that the lower the I50 value, the highest the assay sensitivity. For the carbaryl model presented here, the I50 value was around 25 ng ml-1 (25 ppb). The limit of detection, calculated as the pesticide concentration that provided 90% of the maximum signal, was around 7 ng ml-1 (7 ppb). The total time required for a complete assay cycle, including regeneration, was around 20 min. The standard curves were used to analyze carbaryl in commercial juice samples (apple and orange) spiked with the insecticide in the 1-10 µg ml-1 range. The recoveries obtained (80-130 %) were comparable to those of reference analytical techniques for pesticides.

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120

% SIGNAL (100 x Δf/Δf0)

100 80 60 40 20 0 0

10-2

10-1

100

101

102

103

104

[CARBARYL] (ppb)

Fig. 12.12. Standard calibration curve for carbaryl obtained with the QCM immunosensor

Sensor Reusability. The hapten-functionalized piezoelectric crystals prepared as described could be reused for carbaryl determination for more than one-hundred times, with only slight reductions of the maximum signal and without significant losses of sensitivity. Cross-reactivity. The cross-reactivity of several carbaryl metabolites and chemically related compounds was also tested with the immunosensor. The results indicated a very specific assay for carbaryl, in total agreement with previous results obtained with other immunochemical tecniques (ELISA and Flow Injection Immunoanalysis) based on the same combination of immunoreagents. Therefore, the cross-reactivity seems to be an intrinsic property of the immunoreagents, irrespective of the immunoassay technique or configuration they are used in.

References 1. J.L. Marty, B. Leca and T. Noguer (1998) “Biosensors for the detection of pesticides” Analusis Magazine 26(6): M144-149 2. M.P. Byfield and R.A. Abuknesha (1994) “Biochemical aspects of biosensors” Biosens. Bioelectron. 9. 373-340 3. K.K. Kanazawa and J.G. Gordon II (1985) “The oscillation frequency of a quartz resonator in contact with liquid” Anal. Chim. Acta 175: 99–105

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4. S. Kurosawa, J.W. Park, H. Aizawa, S.I. Wakida, H. Tao and K. Ishihara (2006) “Quartz crystal microbalance immunosensors for environmental monitoring” Biosens. Bioelectron. 22: 473–481 5. A. Janshoff, H-J. Galla and C. Steinem (2000) “Piezoelectric mass-sensing devices as biosensors- An alternative to optical biosensors?” Angew. Chem. Int. Ed. 33: 4004-4032 6. P. Skládal (2003) “Piezoelectric quartz crystal sensors applied for bioanalytical assays and characterization of affinity interactions” J. Braz. Chem. Soc. 14 (4): 491-502 7. X. Su, S.F.Y. Li, W. Liu and J. Kwang (2000) “Piezoelectric quartz crystal based screening test for porcine reproductive and respiratory syndrome virus infection in pigs” Analyst 125:725–730 8. Y.S. Fung, Y.Y Wong (2001) “Self-assembled monolayers as the coating in a quartz piezoelectric crystal immunosensor to detect Salmonella in aqueous solution” Anal. Chem. 73: 5302-5309 9. Z. Fohlerová, P. Skládal and J. Turánek (2007) “Adhesion of eukaryotic cell lines on the gold surface modified with extracellular matrix proteins monitored by the piezoelectric sensor” Biosens. Bioelectron. 22: 1896-1901 10. G. Shen, H. Wang, T. Shuzhen, J. Li, G. Shen and R. Yu (2005) “Detection of antisperm antibody in human serum using a piezoelecetric immunosensor based on mixed self-assembled monolayers” Anal. Chim. Acta 540: 279-284 11. K. Feng, J. Li, J.H. Jiang, G. Shen and R. Yu (2007) “QCM detection of DNA targets with single-mutation based on DNA ligase reaction and biocatalyzed deposition amplification” Biosens. Bioelectron. 22: 1651-1657 12. J. Halámek, A. Makower, P. Skládal and F.W. Scheller (2002) “Highly sensitive detection of cocaine using a piezoelectric immunosensor” Biosens. Bioelectron. 17: 1045-1050 13. J. Přibyl, M. Hepel, J. Halámek and P. Skládal P (2003) “Development of piezoelectric immunosensors for competitive and direct determination of atrazine” Sensors and Actuators B 91: 333-341 14. H. Sun and Y. Fung (2006) “Piezoelectric quartz crystal sensor for rapid analysis of primicarb residues using molecularly imprinted polymers as recognition elements” Anal. Chim. Acta 576: 67-76 15. N. Kim, I.S. Park, and D.K. Kim (2007) “High-sensitivity detection for model organophosphorus and carbamate pesticide with quartz crystal microbalanceprecipitation sensor” Biosens. Bioelectron. 22: 1593–1599 16. M. Franek and K. Hruska (2005) “Antibody based methods for environmental and food analysis: a review” Vet. Med. – Czech. 50 (1): 1–10 17. S. Rodriguez-Mozaz, M.P. Marco, M.J. Lopez de Alda and D. Barceló (2004) “Biosensors for environmental applications: Future development trends” Pure Appl. Chem. 76 (4): 723–752 18. S. Kurosawa, H. Aizawa and J.W. Park (2005) “Quartz crystal microbalance immunosensor for highly sensitive 2,3,7,8-tetrachlorodibenzo-p-dioxin detection in fly ash from municipal solid waste incinerators” Analyst 130: 14951501

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19. J. Halámek, M. Hepel and P. Skládal (2001) “Investigation of highly sensitive piezoelectric immunosensors for 2,4-dichlorophenoxyacetic acid” Biosens. Bioelectron. 16: 253–260 20. K. V. Singh, J. Kaur, G. C. Varshney, M. Raje and C. Raman Suri (2004) “Synthesis and characterization of hapten-protein conjugates for antibody production against small molecules” Bioconjugate Chem. 15: 168-173 21. F. Szurdoki, A. Szekacs, H.M. Le and B.D. Hammock (2002) “Synthesis of haptens and protein conjugates for the development of immunoassays for the insect growth regulator fenoxycarb” J. Agric. Food Chem. 50: 29-40 22. J.J. Manclús, J. Primo and A. Montoya (1996) “Development of enzymelinked immunosorbent assays for the insecticide chlorpyrifos. 1. Monoclonal antibody production and immunoassay design” J. Agric. Food Chem. 44: 4052-4062 23. J.J. Manclús, A. Abad, M.Y. Lebedev, F. Mojarrad, B. Micková, J.V. Mercader, J. Primo, M.A. Miranda and A. Montoya (2004) “Development of a Monoclonal Immunoassay Selective for Chlorinated Cyclodiene Insecticides” J. Agric. Food Chem. 52: 2776-2784 24. A. Abad, M.J. Moreno and A. Montoya (1998) “Hapten synthesis and production of monoclonal antibodies to the n-methylcarbamate pesticide methiocarb” J. Agric. Food Chem. 46: 2417-2426 25. A. Abad and A. Montoya (1994) “Production of monoclonal antibodies for carbaryl from a hapten preserving the carbamate group” J. Agric. Food Chem. 42: 1818-1823 26. W. Haasnoot, J. DuPre, G. Cazemier, A. Kemmers-Voncken, R. Verheijen, B.J.M. Jansen (2000) “Monoclonal antibodies against a sulfathiazole derivative for the immunochemical detection of sulfonamides” Food Agric. lmmunol. 12: 127-138 27. A. Abad, J. Primo and A. Montoya (1997) “Development of an enzyme-linked immunosorbent assay to carbaryl. 1. Antibody production from several haptens and characterization in different immunoassay formats” J. Agr. Food Chem. 45: 1486-1494 28. E.P. Meulenberg (1997) “Immunochemical detection of environmental and food contaminants: Development, validation and application” Food Technol. Biotechnol. 35: 153-163 29. K. Bizet, C. Gabrielli, H. Perrot, and J. Therasse (1998) “Validation of antibody-based recognition by piezoelectric transducers through electroacoustic admittance analysis” Biosens. Bioelectron. 13 (3-4): 259-269 30. S. Tombelli and M. Mascini (2000) “Piezoelectric quartz crystal biosensors: recent immobilization schemes” Anal. Letters 33 (11): 2129-2151 31. Ch. Duan and M.E. Meyerhoff (1995) “Immobilization of proteins on gold coated porous membranes via activated self-assembled monolayer of thioctic acid” Mikrochim. Acta 117: 195-206 32. D.M. Disley, D.C. Cullen, H-X. You and Ch. R. Lowe (1998) “Covalent coupling of immunoglobulin G to self-assembled monolayers as method for immobilizing the interfacial-recognition layer of a surface plasmon resonance immunosensor” Biosens. Bioelectron. 13: 1213-1225

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33. M. Prohanka and P. Skládal (2005) “Piezoelectric immunosensor for francisella tularensis detection using immunoglobulin m in a limiting dilution” Anal. Lett. 38: 411-422 34. S. Ferretti, S. Paynter, D.A. Russell and K.E. Sapsford (2000) “Selfassembled monolayers: a versatile tool for the formulation of bio-surfaces” Trends in Anal. Chem. 19 (9): 530-540 35. R.D. Vaughan, C.K. O’Sullivan and G.G. Guilbault (1999) “Sulfur based selfassembled monolayers (SAM’s) on piezoelectric crystals for immunosensor development” Fresenius J. Anal. Chem. 364: 54–57 36. E. Mauriz, A. Calle, A. Abad, A. Montoya, A. Hildbrandt, D. Barceló and L.M. Lechuga (2006) “Determination of carbaryl in natural water samples by a Surface Plasmon Resonance flow-through immunosensor” Biosens. Bioelectron. 21: 2129-2136 37. E. Mauriz, E. Calle, L.M. Lechuga, J. Quintana, A. Montoya and J.J. Manclús (2006) “Real-time detection of chlorpyrifos at part per trillion levels in ground, surface and drinking water samples by a portable Surface Plasmon Resonance Immunosensor” Anal. Chim. Acta. 561: 40-47 38. E. Briand, M. Salmain, J.M. Herry, H. Perrot, C. Compère and C.M. Pradier (2006) “Building of an immunosensor: How can the composition and structure of the thiol attachment layer affect the immunosensor efficiency?” Biosens. Bioelectron. 22: 440–448 39. S. Susmel, G.G. Guilbault and C.K. O´Sullivan (2003) “Demonstration of labeless detection of food pathogens using electrochemical redox probe and screen printed gold electrodes” Biosens. Bioelectron. 18: 881-889 40. E. Briand, M. Salmain, C. Compère, and C.M. Pradier (2007) “Anti-rabbit immunoglobulin G detection in complex medium by PM-RAIRS and QCM Influence of the antibody immobilisation method” Biosens. Bioelectron. 22: 2884-2890 41. A. Subramanian, J. Irudayaraj and T. Ryan (2006) “A mixed self-assembled monolayer- based surface plasmon immunosensor for detection of E. Coli O157:H7”Biosens. Bioelectron. 21: 998-1006 42. K. Bonroy, F. Frederix, G. Reekmans, E. Dewolf, R. De Palma, G. Bohrgs, P. Declerck and B. Goddeeris (2006) “Comparison of random oriented immobilisation of antibody fragments on mixed self-assembled monolayers” J. Immunol. Methods 312: 167-181 43. S. Chen, L. Li, C.L. Boozer and S. Jiang (2000) “Controlled chemical and structural properties of mixed self-assembled monolayers of alkanethiols on Au(111)” Langmuir 16: 9287-9293 44. F. Frederix, K. Bonroy, W. Laureyn, G. Reekmans, A. Campitelli, W. Dehaen, and G. Maes (2003) “Enhanced performance of an affinity biosensor interface based on mixed self-assembled monolayers of thiols on gold” Langmuir 19: 4351-4357 45. C. March, J.J. Manclús, A. Arnau, Y. Jiménez, T. Sogorb and A. Montoya (2006) “Development of piezoelectric immunosensors based on monoclonal antibodies for pesticide analysis” in Proceedings of IBERSENSOR 2006 (5º Congreso Iberoamericano de Sensores), Montevideo, UY

13 Combination of Quartz Crystal Microbalance with other Techniques Ernesto Calvo1, Kay Kanazawa2, Hubert Perrot3 and Yolanda Jimenez4 1

Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires Department of Chemical Engineering, Stanford University 3 Laboratoire Interfaces et systèmes Electrochimiques, UPR 15 du CNRS, Université P. et M. Curie 4 Departamento de Ingeniería Electrónica, Universidad Politécnica de Valencia 2

13.1 Introduction As implied by its acronym, the quartz crystal microbalance was first used to determine the mass of material deposited on its surface. A heuristic description of the linear relation between the change in its resonant frequency Δf, from its unloaded resonant frequency f0, and the mass density, m’, was recognized by Sauerbrey [1] and led to the now standard use of the QCM to measure mass deposition. This Sauerbrey relation is described by (Eq. 3.10):

Δf = −

2 f 02

ρ Q μQ

m'

(13.1)

The coefficient preceding m’ is a fixed quantity, depending only on the parameters of the unloaded quartz. Using this relation, one can determine the mass density (kg/m2) deposited onto the QCM surface. Although the range of validity of Eq. (13.1) has limits, it has created a whole industry to measure deposition. From the single measurable of Δf, one can measure the single parameter m’. As shown in Chaps. 3 and 14, it is possible to extract more information from the QCM. For example, when m’ exceeds certain values, Eq. (13.1) is no longer satisfied. A non-linear behavior is observed and the shape of the non-linearity can be used to extract information on elastic films concerning the shear modulus, G1, of the film (Chap. 14). This was put on a quantitative basis by Lu and Lewis in 1972 [2]. More recently, there are active A.A. Vives (ed.), Piezoelectric Transducers and Applications, doi: 10.1007/978-3-540-77508-9_13, © Springer-Verlag Berlin Heidelberg 2008

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studies on the determination of many other parameters using the QCM. Some of the variables affecting the measurements have been recently cited by Lucklum [3]. Even for a simple example, when there is a viscoelastic film on the QCM and the quartz/film is immersed in a liquid, there are a number of parameters which are involved in QCM measurements. In addition to the quartz parameters, there are the density of the film ρ1, the shear storage modulus of the film G’1, the shear loss modulus of the film G”1, the thickness of the film h1, the density of the liquid ρ2, the shear loss modulus of the liquid G”2 and the shear storage modulus of the liquid G’2. It is clear that there are a large number of parameters that influence the behavior of the QCM. Most of the measurable to date have focused on two values, the change in frequency and the change in dissipation as the QCM is loaded. It is not possible to deconvolute these two variables into a determination of the materials’ parameters (see Chap. 14). Therefore, it is not surprising that additional measurements in conjunction with the QCM measurements are being undertaken to increase the number of measureables.

13.2 Electrochemical Quartz Crystal Microbalance (EQCM) One of the earliest of these hyphenated techniques is the EQCM, or electrochemical quartz crystal microbalance [4, 5]. Descriptions of EQCM works are to be found in this Chapter and in other books [6, 7] and publications [8]. EQCM systems could be catalogued as a potentiostatic or galvanostatic. In the first type the potential of the specific electrode (cathode or anode) is controlled while in the second type the current through the working electrode is controlled. Figure 13.1 shows a typical experimental set-up for a potentiostatic EQCM. The system is composed by an electrochemical cell with three electrodes: working electrode WE, reference electrode RE, and counter electrode CE, as well as, according to the structure of a potentiostatic system, a potentiostat, a frequency meter, a power source, a controlled quartz sensor oscillator, and a computer. It can be observed that one of the metal electrodes on the QCM serves as a working electrode which is interfaced to a conducting solution containing electroactive moieties. For example, metal cations may be dissolved in an acidic solution. By applying a negative potential to the working electrode, the cations can be deposited onto the electrode as metal atoms and the deposited mass is sensed using the QCM. In addition, the current during the deposition is also monitored and, by integration, the

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charge per unit area Q’ can be measured. If it is known from the chemistry that all of the charge is used for the deposition, the charge-to-mass ratio can then be determined, or if some of the current goes into other reactions (for example the creation of hydrogen molecules), then the fraction of the current used to deposit the metal atoms (the Faradaic efficiency) can be measured. This relation between the charge transfer and mass deposited on the WE can be used as an alternative method for evaluating the thickness of the deposited layer (see Appendix 13.A). This reduces the number of unknowns of the layer and makes easier the determination of the rest of unknowns starting from the QCM parameters monitored. Power Source Oscillator Circuit

Frequency meter

RE CE

Quartz Crystal Sensor

Computer

Potentiostat

WE

Electro-Chemical Cell

Fig. 13.1. Typical experimental setup with EQCM

There are a host of other techniques used by the electrochemical community to study the details of the reaction at the electrode [6]. For example, the kinetic behavior of the reaction can be investigated using electrochemical impedance spectroscopy (see Sect. 8.6 in Chap.8). We use the adjective “electrochemical” to distinguish this technique from the established radio frequency impedance spectroscopy as detailed by Arnau et al [9]. In the case of electrochemical impedance spectroscopy, a small ac voltage is superimposed onto the working electrode potential and the phase and magnitude of the resulting current are recorded. The ratio of the voltage to the current defines the electrochemical impedance, and its behavior as a function of frequency can be used to study the reaction kinetics. In addition, the phase and magnitude of the mass deposited during this ac stimulation, the so-called ac-electrogravimetry (see Sec. 13.2.1 below), adds additional information to the study of the reaction. Another very useful technique for

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reaction studies has been the study of reactions under controlled flow conditions. These can be obtained using the so-called rotating disk electrode (RDE) and rotating ring disc electrode (RRDE) geometries. Both of these techniques have also been adapted for the QCM [8]. 13.2.1 ac-electrogravimetry

This recent technique combines fast QCM and electrochemical measurements under dynamic regime, i.e., under low frequency potential modulation of the working electrode deposited on the quartz resonator. Two transfer functions can be obtained: mass/potential transfer function, ∆m/∆E(ω), and classical electrochemical transfer function, ∆E/∆I(ω), also called electrochemical impedance. By this method the different kinetics of an electrochemical process can be easily separated and a clear discrimination of the ionic species involved can be determined. The principle of fast QCM techniques described elsewhere [10], also called ac-electrogravimetry, is similar to the electrochemical impedance described in Chap. 8: a perturbation signal is applied to one of the electrodes deposited onto the quartz resonator surface (acting as a working electrode) and the mass changes, related to this input signal, are detected through the microbalance response. In order to do this, a frequency/voltage converter is necessary to convert the microbalance frequency response in term of voltage changes. Details concerning the electronic conversion are given in Chap. 5. In a concrete way, the mass/potential transfer function, ∆m/∆E(ω), is determined at different polarization values of the electrochemical system. The mass/potential transfer function and the electrochemical impedance are simultaneously measured using a four channel frequency analyser (Solartron 1254). In this case, the main interest is its ability to separate the ionic contributions and free solvent during electrochemical/chemical processes. Indeed, the ionic species and the solvent can be identified and the kinetic parameters easily determined. A model adapted to these films was developed where the theoretical mass/potential transfer function were simulated [11, 12]. The loops obtained are related to the insertion of each species involved in the charge compensation process. In the case of separate contribution in term of insertion kinetic rates, each loop is related to one specific species: anion, cation, hydrated cation, free solvent. Different materials have been examined from inorganic films to organic films from an academic point of view [13, 14]. More complex structures were also studied as polypyrrole/PVC films in the fields of all solid state ion selective electrodes for potassium detection [15, 16] or polypyrrole

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films where antibodies against HAS (Human Serum Albumin) were entrapped [17]. In this case, it was shown, by coupling electrochemical impedance and ac-electrogravimetric transfer function measurements, that the interaction between a-HSA and the corresponding antigen in solution, HSA, affects mainly the electrochemical response of the polypyrrole. In KCl solution, the anion insertion is observed as presented in Fig. 13.2; a single loop related to chloride anion is observed in the first quadrant. When the corresponding antigen, HSA, is added to the solution, a new loop appears in the third quadrant of the Nyquist representation. In this case, the potassium contribution is demonstrated. On the contrary, the electrochemical impedance response was not affected by the presence of the HSA in solution. In each case, electroacoustic measurements were performed for validating the gravimetric regime of the QCM. 4

chloride ions

2

–lm (Δ m/ Δ E) / μg cm

–2

v

–1

KCI KCI+HAS and : decade frequencies

1 Hz

10 Hz

0

10 Hz

0.1 Hz

1 Hz

-2

potassium ions -4 -6

-4

-2

0 –2

Re (Δm/ΔE)/µg cm

2

–1

v

Fig. 13.2. Electrogravimetric transfer function of polypyrrole films with a-HSA antibodies entrapped. Measurements in KCl 0.5 M and in KCl 0.5 M with HSA at –0.1V/SCE according to reference [17]

13.2.2 Compatibility between QCM and Electrochemical measurements

The combination of different techniques poses some limitations on each individual method since the acoustic, optical and electrical requirements for optimal operation of each technique are sacrificed to some extent in

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benefit of the combined action. Very often in electrochemistry, dual measurements coupling microbalance frequency and current are very often used in electrochemical fields and they present attractive potentialities. For example, F × ∆m/∆q functions allow the identification of ionic species involved in an electrochemical process or intermediate species adsorbed on a working electrode. To estimate this function it is necessary to measure in situ mass changes, ∆m, through microbalance measurements under gravimetric regime, and to integrate the current for obtaining the charge balance, ∆q. Thus, the compatibility between QCM device and potentiostat equipment is questionable, overall when commercial systems are used. For example, if the working electrode deposited on the QCM is grounded as it is shown on Fig. 13.3, the potentiostat configuration must be designed to accommodate this. If the current measurement is performed in the WE circuit, the compatibility is impossible as the operational amplifier used in the WE branch is short circuited. This can be better understood through the simplified schematics of a commercial potentiostat depicted in Fig. 13.4a [18]. As can be noticed if the WE is grounded, due for instance to the specific set-up of the QCM used (Fig. 13.3), the input of the differential amplifier connected for measuring the current through the WE is shortcircuited making impossible the measure of the current. The alternative is to measure the current in the counter electrode branch as it is shown Fig. 13.4b; a differential amplifier is connected to a resistance R and the working electrode remains grounded without distorting the measurement of the current. + VCC

In Out

Buffer

Gain

- VCC

Output

- VCC

OTA Quartz HP Filter

Selective Filter

Fig. 13.3. Schematic representation of an electronic oscillator where one of the gold electrodes is grounded

13 Combination of Quartz Crystal Microbalance with other Techniques

Pilot

Power Amplifier

White Pin Differential Electrometer Jack

Differential Amplifier

R

CE

Rref

Current Measurement

Ref. Electrode

1MΩ

313

CE

WE Ref. Electrode

Jumper WE

Follower I/E Converter

a

b

Fig. 13.4. Graphical explanation on the compatibility of QCM and electrochemical measurements: a Simplified schematics of a commercial potentiostat adapted from [18] and b Electronic circuit for a potentiostat device used to control the stationary potential applied between a working electrode and a reference electrode: the current measurement is made in the counter electrode branch.

13.3 QCM in Combination with Optical Techniques As described in Sect.13.1, one of the material parameters that affects the QCM behavior is the film thickness h1, and an independent method of measuring the film thickness during deposition would be extremely useful in eliminating one of the unknown variables. The importance of this thickness determination is underscored by the fact that one of the favored techniques for deconvolving the materials parameters from the measurements is to try to fit the film behavior for a set of film thicknesses. Of course, it is assumed that the film is homogeneous and has properties unchanging with thickness. For transparent films, such as many polymer films, optical interferometry can be used for thickness determination for films which are relatively thick. Examples of these types of measurements are to be found in Forzani et al., [19] and Hinsberg et al. [20]. For very thin films in the Angstrom to nanometer scale, the technique of surface plasmon resonance (SPR) can be used. In this method, a laser probes the film/noble metal interface. This radiation generates a resonance of the free electrons in the metal, called the plasmon resonance, creating an evanescent wave which probes the film. In reflectance, θ-θ type geometry is used with the angle of incidence and reflectance being the same. Depending upon the film thickness, a minimum in reflectance is observed at a particular angle. Changes in the film thickness are tracked by tracking the angular position of the minimum.

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Hinsberg has found it useful for the purposes of chemical pathway determination of tracking the IR spectrum. This is effective only when there is a window in the neighborhood of the peak(s) of interest. It has been possible to trace the evolution of absorption bands in the film during film swelling as well as the changes in the film species and solution species during dissolution by using flow cell techniques. For example, one peak in the dry film resulting from acid groups is seen to decrease in intensity during swelling, consistent with conjectured deprotonation. An exploded view of the cell used for both these IR measurements is shown in Hinsberg et al. [20]. We shall discuss in some detail the combination of EQCM with optical techniques and electrochemical microscopy. Ellipsometry is a powerful tool to characterize thin films and is based on the analysis of intensity and phase shift of light externally reflected on a sample surface. Reflection of linearly polarized light from a surface generally produces elliptically polarized light, because the parallel and perpendicular components are reflected with different efficiencies and phase shift (see Appendix 13.B for more details). Film thickness, refractive index, and extinction coefficient can be assessed with ellipsometry using expressions derived from Fresnel equations, that relate the changes of elliptically polarized light parameters measured after incidence on the sample surface [21-23]. Other related optical methods such as surface plasmon resonance and scanning angle reflectometry have also been used for the study of layered films like thin protein films. Ellipsometry has been extensively employed to study the growth of oxide films on metals, polymer films, multilayers self-assembled composed of linear and hyperbranched polycations and polyanions, etc. Except in few cases where spectroscopic ellipsometry was employed or ellipsometric measurements were supported by X-ray reflectivity (XRR), the value of a complex refractive index has been assumed on the basis of literature data to extract the film thickness values. It should be noticed, however, that film optical properties may change during film growth. Gottesfeld et al. have shown that the combined information of ellipsometric film thickness and QCM film mass makes it possible to assess the apparent mass density of the GOx-polypyrrole films during film growth [23]. Forzani et al. combined ellipsometry and QCM under “ex-situ” and “insitu” conditions to study self-assembled multilayers comprised of alternated layers of the enzyme glucose oxidase (GOx) and an osmium complex attached to poly(allylamine) PAH-Os, deposited on a gold surface primed with mercapto propanesulfonate [19]. The ellipsometric parameters of the thiol film on gold in the first layer were analyzed in terms of an

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anisotropic single-layer model since the strong vertical interaction of the S-Au bond differs from the lateral Van der Waals interaction of thiol molecules. For the subsequent (PAH-Os)n(GOx)n multilayers on gold, a two-layer model with the anisotropic thiol film and the isotropic enzyme/polyelectrolyte film yielded identical results than an isotropic onelayer model with the substrate parameters measured after thiol adsorption to offset any effect due to the Au-S bond. Film thickness and complex refractive index for each adsorbed layer in protein-polyelectrolyte multilayers were determined for dry films and for films in contact with water, revealing the importance of water content control in these self-assembled structures. The values of complex refractive index determined from “in situ” experiments suggested that the enzyme kept its native state after the adsorption step. Plots of acoustic mass vs. ellipsometry thickness obtained for different enzyme/polycation self-assembled multilayers built up under similar conditions yield coincident results and, therefore, it is possible to estimate the average apparent film density [14]. As an example, Fig. 13.5 shows in-situ acoustic mass - ellipsometric thickness plots obtained for (PAH-Os)14(GOx)14 self-assembled multilayer in the reduced state.

hydrated GOx mass / μg cm

-2

100.0 80.0 60.0 40.0 20.0

0.0

0

100

200

300

400

500

600

700

ellipsometric thickness / nm

Fig. 13.5. Adsorbed enzyme acoustic mass vs. ellipsometric thickness for (PAHOs)14/ (GOx)14. Data taken after each enzyme adsorption and washing step under water or 10 mM Tris buffer pH = 6.4

The reduced polymer multilayer, with ellipsometric thickness ranging from 200 to 600 nm, behaves as an acoustically thin film. Therefore, Sauerbrey’s equation can be used for the gravimetric evaluation of film growth

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Ernesto Calvo, Kay Kanazawa, Hubert Perrot and Yolanda Jimenez

and QCM and ellipsometry can be used to determine the thickness and the density of the film under the gravimetric regime. It should be noticed, however, that the film thicknesses derived from acoustic and optical measurements do not necessarily coincide; effectively, water molecules which are not polarized in the visible spectral region are not detected in reflectance methods, unlike the shear wave that senses water in the film as a coupled mass. The acoustic contrast between the film and liquid depends on the difference in shear modulus (i.e. 106 Pa for the film and 104 Pa for water) and this is more pronounced than the optical contrast given by the difference in refractive index of the hydrated film and water (ca. 1.43 for the protein film and 1.33 for water). Therefore, the most highly hydrated outer regions of the film cannot be distinguished from the aqueous liquid electrolyte by the optical methods but can be sensed by the acoustic wave. An ellipsometric areal mass density, Δme can be derived from the optical film thickness (df) and film refractive index (nf) through the equation proposed by Feijter et al. [24]: Δm e = d f

n f − n0 ∂n ∂c

(13.2)

where no is the electrolyte solution refractive index, nf the film refractive index, and ∂n/∂c is the refractive index increment of the adsorbed substance measured. Höök and Kasemo have represented the protein film with an effective hydrodynamic thickness, deff and an effective density, ρeff [25]. The effective thickness can be expressed in terms of the Sauerbrey mass, ΔmQCM, the ellipsometric mass, Δme, and the respective densities of the dry protein film, ρdry = 2.0 g cm-3 and water, ρwater as follows: d eff =

ΔmQCM

ρ eff

ΔmQCM

=

ρ dry

⎛ Δme Δme + ρ water ⎜1 − ⎜ ΔmQCM ⎝ ΔmQCM

⎞ ⎟ ⎟ ⎠

(13.3)

The effective thickness results always larger than the ellipsometric thickness and smaller than the QCM thickness. In the few first layers the effect of the “hairy” external film/water structure, which cannot be seen by ellipsometry, becomes dominant and as the film grows, less important is the effect of the interface with the external electrolyte. Another related optical technique that has been employed in combination with the quartz crystal microbalance is the surface plasmon resonance

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(SPR). As mentioned surface plasmons are collective oscillations of the free electrons at the metal surface, which under appropriate conditions can resonate with light with absorption at certain resonant frequencies. This resonant condition is extremely sensitive to the optical properties of the media (metal film, adsorption film or monomolecular layer and liquid in contact), namely the refractive index of these media. Changes in refractive index due to the adsorption of molecules or conformational changes in the adsorbed molecules result in changes in the resonant condition which can be accurately measured. When the incident light reaches an appropriate angle, the reflection decreases sharply to a minimum, corresponding to the excitation of surface plasmon waves in the film. The angle of the minimum in reflectance depends upon the film thickness, so changes in the film thickness can be tracked by tracking the angular position of the minimum. The total internal reflection has been detected with a photodetector as a function of incident angle by rotating both the prism and the photodetector. The angular resolution achieved by this rotating prism approach is typically 10-2–10-3 degrees!, limited by errors in the angular position and noise in the intensity of the reflected light. For comparing different SPR detection techniques, the SPR resolution is often described in terms of the smallest detectable change in the refractive index of an analyte in refractive index units. The above angular resolution corresponds to 10-5–10-6 refractive index units at a wavelength of 632 nm. For higher angular resolution, a large distance between the prism and the photo detector is required which makes the setup not only bulky but also more susceptible to mechanical noise and thermal drift. The response time is slow because of the mechanical movements in the setup. Both SPR and QCM involve resonance phenomena that are perturbed by changes in the characteristics of an interfacial layer. These two techniques can be used to measure film thickness with resolution on the angstrom-tonanometer level. SPR is a noninvasive optical-measurement technique that probes the thickness and dielectric constant of thin films at a noble-metal surface. Thickness resolution is typically on the angstrom level, depending on the optical contrast between the film and the solvent, and films up to several hundred nanometers thick can be studied using this technique. It is worth mentioning the work of Johannsmann et al. [26] who measured the second-order stress birefringence in a 2 μm film of poly-methylmethacrylate sandwiched between the top electrode of a quartz resonator and a second, semitransparent gold layer. The gold–film–gold system formed an optical Fabry–Perot resonator with characteristic dips of reflectivity at certain angles of incidence.

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Ernesto Calvo, Kay Kanazawa, Hubert Perrot and Yolanda Jimenez

By combining SPR and QCM, Knoll and Kanawawa have been able to take advantage of the strengths of each technique to monitor “in situ” the interfacial behavior of thin organic films including perfluoro-poly-ether lubricant, electropolymerization of pyrrole and desorption of 1-octadecanethiol from a gold surface [27]. The stability of SPR measurements is generally quite good and sensitivity to environmental factors is minimal. However, there must be optical contrast between an adsorbed film and the solvent for the film to be seen by SPR. In addition, the refractive index of the film must be known regardless of whether a simple measurement is to provide the thickness and mass of an adsorbed layer. Both SPR and QCM can measure the properties of a film adsorbed on a metallic electrode in contact with a liquid electrolyte. The system that utilizes these two techniques can be also under electrochemical control in a measurement cell fitted with counter and reference electrodes. This addition opens the door to the study of a vast number of systems redox systems and chemically modified electrodes. Examples of SPR-QCM combination are recent publications by Kasemo et al. who combined SPR and QCM to study variations in coupled water, viscoelastic properties, and film thickness of a protein film during adsorption and cross-linking [25]; and Willner et al., who reported the swelling of acrylamido-phenylboronic acidacrylamide hydrogels upon interaction with glucose [28]. There are many benefits to the combination optical techniques and EQCM measurement. Using optical techniques and QCM independently to monitor interfacial processes in situ, two measures of a process can be obtained which rely on fundamentally different principles of physics. By carefully considering the nature of each measurement, our ability to interpret the results obtained from either technique is greatly enhanced and the weaknesses of assumptions inherent to data analysis become apparent, and differences measured by each technique can often be reconciled.

13.4 QCM in Combination with Scanning Probe Techniques The QCM has also been coupled with surface probe microscopes. Some beautiful work was done by Krim, Borovsky et al. using scanning tunneling microscopy (STM) at low temperatures [29], showing the amplitudes of motion on the surface of a QCM. This has been extended to the study of the dynamics of contact studies on the surface by Borovsky et al. [30]. In a similar vein, Bailey et al. have used the atomic force microscopy (AFM) to

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study the surface displacements of the QCM under liquid immersion [31] and subsequently the combination of QCM and SPR to monitor in situ the solution-phase adsorption of the perfluoropolyether lubricant Fomblin ZDOL onto a silver surface [32]. While in conventional EQCM the current that flows through the external circuit or the electrode potential is the resultant of integration at the whole electrode surface, space resolved electrochemistry can be achieved with the aid of the scanning electrochemical microscope (SECM). The SECM, introduced in 1986 by A.J. Bard, is a scanning probe technique where the probe signal is generated by an electrochemical reaction at the tip of an ultra-microelectrode [33, 34]. Several electrochemical scanning probe techniques have been described: Localized electrochemical impedance spectroscopy (LEIS) [35], scanning reference electrode (SRET) and scanning vibrating electrode (SVET) techniques [36], as well as the scanning Kelvin probe technique [37], the approach of the scanning droplet cell [38, 39] and scanning electrochemical microscopy (SECM) [40, 41], and the alternating current scanning electrochemical microscopy (AC-SECM) for measuring local interfacial impedance properties with high lateral resolution [42]. These techniques can map electrically or chemically a surface sample, depending on whether a local potential, current or impedance is recorded as a function of sample surface x-y position, or alternatively a local surface concentration is mapped. SECM can be used to study the flux of ions at membranes, polymer modified surfaces or liquid-liquid interfaces [43]. The scanning electrochemical microscope (SECM) can work in either potentiostatic [40] or amperometric modes [44]. In amperometric mode, a bipotentiostat controls the tip potential and in some cases the conductive substrate. The microelectrode (1-25 μm Pt or carbon) sealed at the moving tip is positioned by x,y,z-stepper motors and piezo-controllers and rasters the sample surface. In a redox electrolyte solution and far from interference by the sample surface, the microelectrode current iT,∞ at a disc ultra-microelectrode (UME) for the reaction:

O±e→R

(13.4)

iT ,∞ = 4nFDC * ro

(13.5)

is given by:

where ro is the radius of the tip ultra microelectrode, n is the number of electrons exchanged in the electrochemical reaction, D is the diffusion coefficient of the reacting species and C* its analytical concentration.

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When the ultra microelectrode tip is brought close to the sample substrate within few tip radii, the insulating surface blocks the diffusion of O to the tip and the UME current decreases. If the surface can regenerate O, because it is a metal where O and R can be exchanged through surface electrons, or if there is a surface catalytic process that can regenerate O from R then an enhancement in the UME current is observed. It is important to record the tip current as a function of tip-to-substrate distance (approach curves) which in the first case show a decrease and in the second one, a current increase as the tip gets closer to the substrate. Two operation modes are possible [45]: feedback mode, when regeneration of O takes place and collection mode when the tip close to the substrate surface is poised at a potential where the electroactive species R is detected at the UME tip. In addition, the SECM can operate at constant current with feedback at the z-piezo and recording the error signal of the feedback loop or at constant height recording the current or potential at the tip. In the constant-height mode of the SECM, the UME tip is slowly approached in the z-direction towards the sample surface, and the approach is stopped when a predefined decrease (negative feedback) or increase (positive feedback) in the diffusion limited current is detected. This z-height is kept constant during the following scanning leading to a convolution between the tip-to-sample distance and the surface activity which are both influencing the measured current through the microelectrode. An optical method to control the constant tip-sample distance of the SECM, based on the detection of shear forces occurring between the vibrating tip and the sample surface has been reported [46]. By focusing a laser beam on the very end of the microtip which is vibrating in its resonance frequency a Fresnel diffraction pattern is generated which is focused on a split photodiode (SPD). The difference current is amplified with respect to the vibration frequency using a lock-in amplifier [47-50]. Thus, a constant distance between any tip and sample can be kept, allowing the use of non-electrode tips in SECM. The electrochemical quartz crystal microbalance (EQCM) and the scanning electrochemical microscope (SECM) have been used independently to study the redox behavior of thin films on electrodes [51, 52]. Each technique has advantages and limitations, but if combined into one instrument they can provide complementary in situ valuable information on the electrochemical and rheological behavior of polymer films and overcome the limitation of the individual techniques. Positioning the tip of a microelectrode close to the film’s surface allows us to measure concentration changes of redox-active species crossing the film/solution interface, so we are able to identify fluxes of counter ions during oxidation-reduction cycles and to assess their influence on the

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film’s quartz impedance parameters measured by the quartz crystal microbalance. Several reports on the combination of an EQCM with a SECM have been published [53-55]. Hillier and Ward [56] used a single-potential control between substrate and microelectrode, where the tip acted as the counter electrode to study the radial sensitivity, mapping different quartz crystals. Cliffel and Bard [55] described a SECM/EQCM instrument under bipotentiostatic control, which was employed for studies of silver and C60 films. The EQCM in that case, however, only measured frequency changes and is thus not capable of distinguishing between acoustically thin and viscoelastic films or cases where mass and solution properties change simultaneously. Shin and Jeon [54] described the use of frequency changes with the EQCM as a method of calibrating the tip-substrate distance in the SECM. Gollas et al. reported a novel combination of an EQCM and a SECM with rapid in situ measurement of the modulus of the quartz crystal’s transfer function [53]. This technique based on a complex voltage divider method [57, 58] allows data analysis in the complex plane for the Butterworth-Van Dyke (BVD) equivalent circuit and yields the real and the imaginary components R (damping resistance) and XL (reactive inductance) of the crystal’s electro acoustic impedance around its resonant frequency of 10 MHz. The authors tested the method by cyclic voltammetry for copper plating/stripping experiments from copper(I) chloride solution of high concentration in 1 M HCl recording simultaneously four parameters, XL, R, the substrate, and the tip current. The amperometric response of the SECM tip positioned closely to the substrate reflects the concentration changes of electroactive ions in the diffusion layer of the substrate electrode.

13.5 QCM in Combination with Other Techniques Undoubtedly, there are more hyphenated techniques which have not been discussed. For example, the use of calorimetry with the QCM has been demonstrated by Allan Smith [59]. Various endoergic and exoergic phenomena are then studied simultaneously with the QCM measurements, providing a great deal of additional information on the state of the films. It is clear that these techniques, used in conjunction with the QCM, will play an important role in the determination of film parameters, film chemistry and film kinetics. The use of dual frequencies also permits the evaluation of additional parameters. For example the recent work of Edvardsson et al., where dual excitation of the same QCM at its harmonics is used [60].

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This might be considered as a QCM-QCM technique. At one frequency the amplitude of excitation is varied, and analysis of the loading is done using another harmonic. This has interestingly allowed the study of amplitude effects on adsorption.

Appendix 13.A: Determination of the Layer Thickness by EQCM In EQCM applications the measure of the charge by Faraday law is used to obtain an alternative and independent measure of the thickness. Bandey and Hillman [61] introduce and experiment where a layer of conductive polymer is generated by electro-polymerization. They estimated the density of the polymer layer and, by measuring the charge transferred during the process, estimated the deposited mass through Faraday law and then the thickness of the layer. The Faraday law was introduced in Chap. 8; this law states that the number of moles of the substance deposited on the electrode equals the number of moles of electrons interchanged during the reaction. If it is assumed that the efficiency is 100%, i.e., all the change of mass can be considered due to the total change of charge, and the density of the material deposited onto the quartz electrode is known, then it is possible to estimate the thickness of the layer as follows: The number of moles of the substance deposited is: N1 =

m M

(13.A.1)

where m is the mass and M is the molecular weight. On the other hand, the number of moles of electrons interchanged is: N2 =

Qc nF

(13.A.2)

where Qc is the charge involved in the process, which can be obtained through the product of the current measured and the time that the process takes; n is the number of electrons interchanged in the reaction, which is known and F = 96485 C/mol is Faraday’s constant. Equating Eqs. (13.A.1) and (13.A.2) results in: m=

MQc nF

and the thickness is obtained from Eq. (13.A.3) as follows:

(13.A.3)

13 Combination of Quartz Crystal Microbalance with other Techniques

h1 =

Qc M nFS ρ1

323

(13.A.4)

where S is the surface where the substance is deposited and ρ1 is the density of the substance.

Appendix 13.B: Fundamentals on Ellipsometry Ellipsometry is an optical technique based upon the analysis of a reflected wave of polarized light. Optical reflection is best understood in terms of wavelike r properties of light. The electric field vector associated with the wave, E , oscillates in a plane as the wave propagates (see Fig.13.B.1a), and the intensity of the light is proportional r to the square of the electric field amplitude. A magnetic field vector, H , oscillating in a perpendicular plane as shown accompanies the electric vector. In turn, r in Fig.13.B.1a, r both vectors, E and H , are perpendicular to the wave propagation direction [62]. Y

Y

Y

E

X

E

E X

H

a

Z

Z

b

c

Fig. 13.B.1. Electric field and magnetic field vectors of a light wave propagating along the Z direction, b the plane of polarization contains the electric field vector, c to an observer at a fixed point, a train of these waves, all oriented in the same way, would have electric field vectors along the indicated line. Such a light beam is linearly polarized. (Adapted from [62]).

Most light sources comprise multiple emitters r yielding rays that are uncorrelated with each other, in these cases E is contained in aleatory planes. This light is said to be unpolarized. However, the analysis of a reflected wave provides more valuable information when the incident light is partially polarized. By processing the light carefully, r one can achieve linearly polarized light in which the electric vector E is always contained in the same plane. Fig.13.B.1b shows an example of linearly polarized light,

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Ernesto Calvo, Kay Kanazawa, Hubert Perrot and Yolanda Jimenez

which is called in that way because to an observer at a fixed point, a train of these waves, all oriented in the same way, would have electric field vectors along the indicated line in Fig.13.B.1c. In some applications, the ellipsometric techniques are used to study the geometrical and optical properties of a material. These properties can be obtained from the analysis of the polarization of the reflected wave originated starting from the incidence of a beam of known polarization over the material. Measurements are always referred to the physical plane of incidence, as defined in Fig. 13.B.2a. If the polarization is parallel to this plane, then it and parameters related to it are traditionally denoted by the subscript p. For polarization perpendicular to the plane of incidence a subscripts s is employed. If some other angle of polarization with respect of the plane of incidence is employed, such as 45º (see Fig.13.B.2a), then one usually resolves the electric field vector into the parallel, Ep, and the perpendicular, Es , components. Δ

Es Ψ

Es 45º

Ep

Ep

Ep

Plane of incidence

a

Es

b

c

Fig. 13.B.2. a Plane of incidence and parallel and perpendicular components of a 45º linearly polarized light, b Elliptic polarization, c phase shift between parallel and perpendicular components. (Adapted from [62])

If a linearly polarized beam is reflected in the surface of a material, one usually finds that Ep and Es undergo different changes in amplitude and in phase in that way that the resultant Ep and Es projected over a plane do not describe a line but rather an ellipse; hence the light is elliptically polarized. The shape of that ellipse depends on the geometrical and optical properties of the material probed by the light. We can measure the changes in the amplitude and phase of the reflected wave and use them to characterize the sample. That approach is called Ellipsometry. The basic ellipsometric parameters are two angle, ψ and ∆, defined from the ratio between parallel and perpendicular component of the electric field:

13 Combination of Quartz Crystal Microbalance with other Techniques

Ep Es

=

Ep Es

e

(

j δ p −δ s

)

= tanψ e jΔ

325

(13.B.1)

where δp and δs are the phase shift between the incident and the reflected wave for the perpendicular and parallel components of the electrical field, respectively; E p and E s are the ratio of the amplitudes between the reflected and the incident wave for the perpendicular and the parallel component of the electric field, respectively. Hence, the parameters ψ and ∆ measured at a given wavelength, quantify the changes in the polarization state of the reflected light. Source (Laser) Polarizer

Detector

Analyzer Compensator

Fig. 13.B.3. Simplified block diagram of an ellipsometer.

Figure 13.B.3 shows a simplified block diagram of an ellipsometer. A coherent-light source which is polarized linearly at 45º with respect to the plane of incidence provides polarized light with the following characteristics: E p = E s and ∆ =0. The reflection of this light over the sample produces elliptically polarized light that is passed through a compensator, which is adjusted to restore the original condition of ∆ =0. The position of the compensator required for this restoration is a measure of the value of ∆ induced by reflection. The resulting linearly polarized beam is then passed through a second polarizer (analyzer) which is adjusted until no light passes through the analyzer to the detector and the condition of extinction is reached. The angular position of the analyzer then provides a measure of ψ. Changes in the polarization of the reflected beam depend on the optical and geometrical properties of the sample, but also on the optical properties of the substrate over which the sample is deposited in the ellipsometer. In this sense, some models which relate the ellipsometric angles ψ and ∆ with the optical and geometrical parameters of the sample and the substrate are available in the literature. There are very simple models that assume an isotropic behavior of the sample, and other more complexes that consider

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Ernesto Calvo, Kay Kanazawa, Hubert Perrot and Yolanda Jimenez

an anisotropic behavior. Since the substrate is perfectly characterized, its optical and geometrical properties are known; hence, using those mentioned model, the thickness and the complex refractive index can be obtained using a fitting algorithm which searches the minimum of the following error function [63]:

F (n, k , h1 ,) =

∑ (Ψ M

i =1

exp i

− Ψical

) + (Δ 2

exp i

− Δcal i

)

2

(13.B.2)

where Ψiexp and Δexp are the angles provided by the ellipsometer, and Ψical i are the angles provided by the models. y Δcal i

References 1.

G. Sauerbrey (1959) “Verwendung von schwingquarzen zur wägung dünner schichten und zur mikrowägung” Zeitschrift Fuer Physik 155 (2):206-222 2. CS. Lu, O. Lewis (1972) “Investigation of film-thickness determination by oscillating quartz resonators with large mass load” Journal of Applied Physics 43(11): 4385-4390 3. R. Lucklum and P. Hauptmann (2006) “Acoustic microsensors - the challenge behind microgravimetry” Anal. Bioanal. Chem 384: 667-682 4. K.K. Kanazawa and J.G. Gordon II (1985) “Frequency of a quartz microbalance in contact with liquid” Anal. Chem. 57:1770-1771 5. S. Bruckenstein and M. Shay (1985) “Experimental aspects of use of the quartz crystal microbalance in solution” Electrochim. Acta 30:1295-1300 6. A.J. Bard and L.R. Faulkner (2001) “Electrochemical methods. fundamentals and applications”, 2nd. edn., John Wiley and Sons, New York 7. E.J. Calvo, C. Danilowicz, E. Forzani, A. Wolosiuk and M. Otero (2003) “Layered protein films: quartz crystal resonator frequency and admittance analysis” In: Rusling, J.F. (Ed.), Biomolecular Films: Design, Function and Applications., Marcel Dekker, New York (Chap. 7) 8. E.J. Calvo and R.A. Etchenique (1999) “Kinetic applications of the electrochemical quartz crystal microbalance (EQCM)” In: Comprehensive chemical kinetics., R.G. Compton and G. Hacock (Eds.), Elsevier, Amsterdam, Vol 37, pp. 461-487 9. A Arnau, Y. Jimenez, R. Fernandez, R. Torres, M. Otero and E.J. Calvo (2006) “Viscoelastic characterization of electrochemically prepared conducting polymer films by impedance analysis at quartz crystal. study of the surface roughness effect on the effective values of the viscoelastic properties of the coating” J. Electrochem Soc. 153; 455-466 10. R. Torres, A. Arnau and H. Perrot (2007) “Electronic system for experimentation in ac electrogravimetry II: implemented design” Revista EIA 7; 63-73

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11. C. Gabrielli; J. J. García-Jareño; M. Keddam, H. Perrot and F. Vicente (2002) “Ac-electrogravimetry study of electroactive thin films. I. Application to Prussian Blue” Journal of Physical Chemistry B. 106(12): 3182-3191 12. C. Gabrielli, J.J. García-Jareño, M. Keddam, H. Perrot, and F. Vicente (2002) Ac-electrogravimetry study of electroactive thin films. II. Application to Polypyrrole Journal of Physical Chemistry B. 106(12): 3192-3201 13. D. Giménez-Romero, P. R. Bueno, J. J. García-Jareño, C. Gabrielli, H. Perrot and F. Vicente (2006) “Kinetic aspects of ion exchange in KhFek[Fe(CN)6]l·mH2O Compounds: A combined electrical and mass transfer functions approach” Journal of Physical Chemistry 110 (39): 19352-19363 14. D. Benito, C. Gabrielli, J.J. García-Jareño, M. Keddam, H. Perrot and F. Vicente (2002) “An electrochemical impedance and ac-electrogravimetry study of PNR films in aqueous salt media” Electrochem. Comm. 4(8): 613-619 15. C. Gabrielli, P. Hémery, P. Liatsi, M. Masure and H. Perrot (2005) “An electrogravimetric study of an all-solid-state potassium selective electrode with prussian blue as the electroactive solid internal contact” J. Electrochem. Soc. 152(12): 219-224 16. C. Gabrielli, P. Hémery, P. Liatsi, M. Masure and H. Perrot (2006) “acelectrogravimetry study of an all solid state potassium selective electrode with polypyrrole as the solid internal contact” Electrochim. Acta. 51: 1704-1712 17. S. Al Sana, C. Gabrielli and H. Perrot (2003) “Influence of antibody insertion on the electrochemical behavior of polypyrrole films by using fast qcm measurements” J. Electrochem. Soc. 150 (9): 444-449 18. Princenton Applied Research. “Getting to know your potentiostat, part I” Electrochemical Instruments. Technical Note 200 19. E.S. Forzani, M. Otero, M.A. Perez, M. Lopez Teijelo and E.J. Calvo (2002) “The structure of layer-by-layer self-assembled Glucose Oxidase and Os(Bpy)2ClPyCH2NH-poly(allylamine) multilayers: ellipsometric and quartz crystal microbalance studies” Langmuir 18: 4020-4029 20. W.D. Hinsberg, K.K. Kanazawa (2002) “Determination of the viscoelastic properties of polymer films using a compensated phase-locked oscillator circuit” Anal. Chem. 74: 125-131 21. H. Arwin (2000) “Ellipsometry on thin organic layers of biological interest: characterization and applications” Thin Solid Films 377:48-56 22. R.M.A. Azzam and N.H. Bashara (1977) “Ellipsometry and polarized light” North-Holland, Amsterdam 23. S. Gottesfeld, Y.-T. Kim, and A. Redondo (1995) “Recent applications of ellipsometry in electrochemical systems” in Physical electrochemistry: principles, methods, and applications, I. Rubinstein (Ed.), Marcel Dekker, New York, Chap. 9, pp 393-467 24. J.A. de Feijter, J. Benjamins, and F. A. Veer (1978) “Ellipsometry as a tool to study the ad-sorption of synthetic and biopolymers at the air-water interface” Biopolymers 17: 1759-1772 25. F. Hook, B. Kasemo, T. Nylander, C. Fant, K. Scott and H. Elwing (2001) “Variations in coupled water, viscoelastic properties, and film thickness of a MEPF-1 Protein film during adsorption and cross-linking: A quartz crystal

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26. 27.

28.

29. 30. 31.

32.

33. 34.

35. 36. 37. 38.

Ernesto Calvo, Kay Kanazawa, Hubert Perrot and Yolanda Jimenez microbalance with dissipation monitoring, ellipsometry, and surface plasmon resonance study” Anal. Chem. 73(24): 5796-5804 A. Domack and D. Johannsmann (1998) “Shear birefringence measurements on polymer thin films deposited on quartz resonators” J. Appl. Phys. 83 (3): 1286-1295 L.E. Bailey, D. Kambhampati, K.K. Kanazawa, W. Knoll and C.W. Frank (2002) “Using surface plasmon resonance and the quartz crystal microbalance to monitor in situ the interfacial behavior of thin organic films” Langmuir 18(2): 479-489 R. Gabai, N. Sallacan, V. Chegel, T. Bourenko, E. Katz and I. Willner (2001) “Characterization of the swelling of acrylamidophenylboronic acidacrylamide hydrogels upon interaction with glucose by faradaic impedance spectroscopy, chronopotentiometry, quartz-crystal microbalance (QCM), and surface plasmon resonance (SPR) experiments” J. Phys. Chem. B. 105(34): 8196 -8202 B. Borovsky, B.L. Mason and J. Krim (2000) “Scanning tunneling microscope measurements of the amplitude of vibration of a quartz crystal oscillator” Journal of Applied Physics 88(7): 4017-4021 B. Borovsky, J. Krim, S.A. Syed Asif and K.J. Wahl (2001) “Measuring nanomechanical properties of a dynamic contact using an indenter probe and quartz crystal microbalance” Journal of Applied Physics 90: 6391-6396 LE. Bailey, KK. Kanazawa, G. Bhatara, G.W. Tyndall, M. Kreiter, W. Knoll, C.W. Frank (2001) “Multistep adsorption of perfluoropolyether hard-disk lubricants onto amorphous carbon substrates from solution” Langmuir 17: 8145-8155 L.E. Bailey, D. Kambhampati, K.K. Kanazawa, W. Knoll and C.W. Frank (2002). “Using surface plasmon resonance and the quartz crystal microbalance to monitor in situ the interfacial behavior of thin organic films” Langmuir 18: 479-489 J. Kwak and A.J. Bard (1989) “Scanning electrochemical microscopy. Theory of the feedback mode” Anal. Chem. 61(11): 1221-1227 H.Y. Liu, F.R.F. Fan, C.W. Lin, and A.J. Bard (1986) “Scanning electrochemical and tunneling ultramicroelectrode microscope for high-resolution examination of electrode surfaces in solution” J. Am. Chem. Soc. 108(13): 3838-3839 F. Zhou, D. Thierry and H. S. Isaacs (1997) “A high-resolution probe for localized electrochemical impedance spectroscopy measurements” J. Electrochem. Soc. 144: 1957-1965 J. W. H. de Wit, D. H van der Weijde, A. de Jong, F. Blekkenhorst and S.D. Meijers (1998) “Local measurements in electrochemistry and corrosion technology” Mater. Sci Forum, vol. 289-292, pp. 69-75 M. Stratmann, R. Feser and A. Leng (1994) “Corrosion protection by organic films” Electrochimica Acta 39(8-9): 1207-1214 M. M. Lohrengel, A. Moehrig and M. Pilaski (2000) “Electrochemical surface analysis with the scanning droplet cell” Fres. J. Anal. Chem. 367: 334-339

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39. T. Suter and H. Böhni (1997) “A new microelectrochemical method to study pit initiation on stainless steels” Electrochimica Acta 42(20-22): 3275-3280 40. J.V. MacPherson and P.R. Unwin (2001) In Scanning electrochemical microscopy, A. J. Bard and M. V. Mirkin. Ed., Marcell Dekker Inc., New York 41. R. H. Horrocks, D. Schmidtke, A. Heller and A.J. Bard (1993) “Scanning electrochemical microscopy. 24. Enzyme ultramicroelectrodes for the measurement of hydrogen peroxide at surfaces” Anal. Chem. 65(24): 3605-3614 42. B. Ballesteros Katemann; A. Schulte; M. Koudelka-Hep; E. J. Calvo and W. Schuhmann (2002) “Localised electrochemical impedance spectroscopy with high lateral resolution by means of alternating current scanning electrochemical microscopy” Electrochemistry Communications 4 (2): 134-138 43. B.R. Horrocks, M.V. Mirkin, D.T. Pierce, A.J. Bard, G. Nagy and K. Toth (1993) “Scanning electrochemical microscopy. 19. Ion-selective potentiometric microscopy” Anal. Chem. 65: 1213-1224 44. C. Wei, A.J. Bard, G. Nagy and K. Toth (1995) “Quantitative extraction using an internally cooled solid phase microextraction device” Anal. Chem. 67: 34-43 45. J. Kwak and A.J. Bard (1989) “Scanning electrochemical microscopy. Theory of the feedback mode” Anal. Chem. 61(11): 1221-1227 46. M. Ludwig, C. Kranz, W. Schuhmann and H.E. Gaub (1995) “Topography feedback mechanism for the scanning electrochemical microscope based on hydrodynamic forces between tip and sample” Review of Scientific Instruments 66: 2857-2860 47. A. Hengstenberg, A. Blöchl, I. D. Dietzel, W. Schuhmann (2001) “Spatially resolved detection of neurotransmitter secretion from individual cells by means of scanning electrochemical microscopy” Angew. Chem. Int. Ed. Engl. 40: 905-908 48. A. Hengstenberg, C. Kranz and W. Schuhmann (2000) “Facilitated tippositioning and applications of non-electrode tips in scanning electrochemical microscopy using a shear forced base constant-distance mode” Chemistry - A European Journal 6:1547-1554 49. G. Shi, L. F. Garfias-Mesias and W.H. Smyrl (1998) “Preparation of a goldsputtered optical fiber as a microelectrode for electrochemical microscopy” J. Electrochem. Soc. 145: 2011-2016 50. C. Kranz, H.E. Gaub and W. Schuhmann (1996) “Polypyrrole. Towers grown with the scanning electrochemical microscope” Adv. Mater. 8: 634-637 51. M.V. Mirkin (1996) “Recent advances in scanning electrochemical microscopy” Anal. Chem. 68: 177A-182A 52. A.J. Bard, F.R.F. Fan and M.V. Mirkin “Scanning electrochemical microscopy” In Electroanalytical Chemistry, A.J. Bard Ed., Marcel Dekker, New York, Vol. 18, pp 243-373 53. B. Gollas, P.N. Bartlett and G. Denuault (2000) “An instrument for simultaneous EQCM impedance and SECM measurements” Anal. Chem. 72 (2): 349-356

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54. M.S. Shin and I.C. Jeon (1998) “Frequency-Distance Responses in SECMEQCM - A novel method for calibration of the tip-sample distance” Bull. Korean Chem. Soc. 19: 1227-1232 55. D.E. Cliffel and A.J. Bard (1998) “Scanning electrochemical microscopy. 36. A combined scanning electrochemical microscope-quartz crystal microbalance instrument for studying thin films” Anal. Chem. 70 (9): 1993-1998 56. A.C. Hillier and M.D. Ward, (1992) “Scanning electrochemical mass sensitivity mapping of the quartz crystal microbalance in liquid media” Anal. Chem. 64: 2539-2554 57. E.J. Calvo, R.A. Etchenique, P.N. Bartlett, K. Singhal, and C. Santamaria (1997) “Quartz crystal impedance studies at 10 mhz of viscoelastic liquids and films” Faraday Discuss. Chem. Soc. 107: 141-157 58. E.J. Calvo, C. Danilowicz, R.A. Etchenique (1995) “Measurement of viscoelastic changes at electrodes modified with redox hydrogels with a quartz crystal device” J. Chem. Soc. Faraday Trans 91: 4083-4091 59. A.L. Smith, and H.M. Shirazi, (2000) “Quartz microbalance microcalorimetry A new method for studying polymer-solvent thermodynamics” Journal of Thermal Analysis and Calorimetry 59: 171-186 60. M. Edwardsson, M. Rodalh, B. Kasemo and F. Hook (2006) “A dualfrequency QCM-D setup operating at elevated oscillation amplitudes” Anal. Chem. 77 (15): 4918-4926 61. H.L. Bandey, A.R. Hillmann, M.J. Brown and S.J. Martin (1997) “Viscoelastic characterization of electroactive polymer films at electrode/solution interface” Faraday Discuss. Chem. Soc. 107: 105-121 62. A.J. Bard, L.R. Faulkner (2001) “Spectroelectrochemistry and other coupled characterization methods” In electrochemical methods: fundamental and applications, 2nd Ed. John Wiley & sons Ed. Chap. 17 63. M.J. Otero (2003) “Construcción y caracterización de estructuras complejas de biomoléculas con aplicación en el diseño de biosensores” Tesis Doctoral, Universidad de Buenos Aires

14 QCM Data Analysis and Interpretation Yolanda Jiménez1, Marcelo Otero2 and Antonio Arnau1 1

Departamento de Ingeniería Electrónica, Universidad Politécnica de Valencia. Departamento de Física Aplicada. Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires / CONICET. 2

14.1 Introduction The bulk acoustic wave - thickness shear mode resonator (BAW-TSM), whose major representative is the AT-cut quartz crystal, has been introduced in Chap. 1 as a microbalance sensor (QCM). In chapter 3, the basic concepts of modeling were introduced making the use of the resonator evident as a sensor device. Figure 3.1 showed the general schema of a quartz crystal resonator with a multilayer coating, which can be reduced to that of Fig. 14.1 by modeling a 3-layer compound resonator formed by the quartz crystal in contact with a finite viscoelastic layer contacting a semi-infinite viscoelastic medium. This reduced model is appropriate for representing a large number of applications. Changes in the physical properties of the coating are transferred to the electrical admittance or impedance of the resonator, through the acoustic load impedance ZL (see Chap. 3 and Appendix 3.A), thus allowing its use as a sensor device. Some typical applications of this acoustic wave sensor are the detection of special chemical species, the monitoring of electrochemical processes or the detection of biological molecules (see Chaps. 9, 10, 11, 12 and 13). In chapter 5, the three main steps involved in a TSM resonator sensor were introduced: 1) measurement of the appropriate experimental parameter values for a convenient representation of the resonant response of the sensor, including suitable electronic and cell interfaces (see Chap. 5); 2) extraction of the effective parameter values of the physical model used for representing the resonator sensor (2, 3 or n-layer model – see Chap. 3), and 3) final interpretation of the physical, chemical or biological phenomena responsible for the change in the effective equivalent parameters of the selected physical model. The final interpretation indicated in the third point depends on the application, but it must be coherent with the changes in the effective physical A.A. Vives (ed.), Piezoelectric Transducers and Applications, doi: 10.1007/978-3-540-77508-9_14, © Springer-Verlag Berlin Heidelberg 2008

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Yolanda Jiménez, Marcelo Otero and Antonio Arnau

properties of the coating, which have been used to define the physical model of the sensor. Thus, previously to the final interpretation of the physical, biological or chemical phenomena, it is necessary the extraction of the physical properties used in the definition of the model starting from the changes in the electrical experimental parameters measured; next, an adequate interpretation of the changes in those properties must be done. This process will be called data analysis. After the data analysis, the final interpretation of the physical, chemical or biological phenomena can be accomplished starting from the changes in the effective physical properties extracted in the data analysis. In many applications, the data analysis is a complex task and, in consequence, the final interpretation is difficult as well. This chapter attempts to clarify this aspect, and for that a set of different cases (from the simplest one to the most complex) will be presented in Sects. 14.3 and 14.4. In each case the data analysis will be described. Next, in Sect. 14.5 some typical applications of the sensor will be introduced as examples for the explanation of the final interpretation of the physical, biological or chemical phenomena. Before that, in Sect. 14.2, the general extraction procedure of the effective physical properties of the coating starting from the electrical parameters measured experimentally will be described. Layer 3

Semi-infinite Medium

Layer 2

Sensitive Layer

Layer 1

G'2 G''2

G'1 G''1

2

h

1 1

Quartz Sensor

c66

q

q hq

Fig. 14.1. Cross-section of a thickness shear mode resonator loaded by a two-layer viscoelastic medium

14.2 Description of the Parameter Extraction Procedure: Physical Model and Experimental Data The data analysis and the later interpretation in certain applications require first to define a physical model to characterize the compound resonator, which will be considered of the form represented in Fig.14.1; and second, to define which magnitudes of that model will be considered as unknowns and which will be experimentally determined.

14 QCM Data Analysis and Interpretation

333

14.2.1 Physical Model The most comprehensive model to characterize the electrical impedance of the compound resonator in Fig. 14.1 is the transmission line model, TLM, introduced in Chap. 3 (Sect. 3.A.2). The expression of this impedance (Eq. 3.A.12) is written again for practical reasons as follows:

Z α ⎛ 2 tan − j L ⎜ 2 Z 2 1 ⎜ K cq Z= 1− ⎜ Z j ωC 0 α 1 − j L cot α ⎜⎜ Z cq ⎝

⎞ ⎟ ⎟ ⎟ ⎟⎟ ⎠

(14.1)

When parasitic capacitances Cp are taken into account, the complete admittance is: Y=

1 1 = jωC 0* + Z Zm

(14.2)

where Co* = Co+Cp and Zm is given by the following expression (Eq. 3.A.21): Zm =

⎛ α ⎞ ⎟ 1 ⎜ K2 1 α ZL − 1 ⎜ ⎟+ jωC0 ⎜ 2 tan α C0 4 K 2 Z cq ω ⎟ 2 ⎝ ⎠

1 1−

Z j L Z cq

= Z m0 + Z mL

(14.3)

2 tan α2

ZL in the former equation is the acoustic load impedance acting at the surface of the quartz plate (Eq. 3.A.26), which corresponds with the following equation for the model in Fig. 14.1:

⎛ Z c 2 + j Z c1 tan⎜ ω ⎜ ⎝ Z L = Z c1 ⎛ Z c1 + j Z c 2 tan⎜ ω ⎜ ⎝

⎞ h1 ⎟ G1 ⎟⎠ ρ1 ⎞⎟ h1 G1 ⎟⎠

ρ1

(14.4)

where Zc1 and Zc2 are the characteristic acoustic impedances of the layers, given by Eq. (3.A.25b). Once the appropriate mathematical model for the physical representation shown in Fig. 14.1 has been explicitly stated, it is important to define which magnitudes are considered as experimental data and which as unknown magnitudes to be extracted from them. The total admittance, Y,

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Yolanda Jiménez, Marcelo Otero and Antonio Arnau

corresponds to the experimental data and the acoustic load impedance is the physical magnitude to be extracted and interpreted. Additional parameters of the sensor must be obtained with the appropriate calibration, such as: thickness hq, viscosity ηq, static capacitance Co of quartz and parasitic capacitance Cp. To a certain extent, they can also be considered as unknowns, especially Co; this aspect will be discussed in more detail later on. The complex acoustic load impedance, ZL, can be directly obtained from experimental data. However, the extraction of the physical parameters associated to ZL is a major problem [1-4, 30, 31]. Equation (14.4) depends on 7 unknowns: the density ρ1, the thickness h1 and the storage and loss shear moduli, G1′ and G1′′ , of the first layer and the density ρ2, and the storage and loss shear moduli, G2′ and G2′′ , of the medium (see Chap. 7). Thus, it is impossible, in principle, to extract 7 unknowns only from two data: the real and imaginary parts of ZL evaluated from experimental data; even though the shear moduli and density of the second medium are assumed to be known, four parameters remain. There are applications where simplified versions of the TLM (Eq. 14.1) can be used reducing the number of unknowns and making the interpretation of the data easier; these cases will be called “simple cases”. However, it is necessary to know the limits of these approximations to minimize the error in the interpretation. In other applications the TLM should not be simplified and the parameter extraction problem described above remains. In these cases an additional knowledge of the loading properties by alternative techniques [1-7] (see chapter 13), or a major definition of the experimental resonant behavior [8-12, 30, 31], are required to provide an adequate interpretation. Next, typical experimental parameters for sensor characterization and its relation to the acoustic load impedance will be analyzed, which will be very useful for an appropriate interpretation, at least in simple cases. 14.2.2 Experimental Parameters for Sensor Characterization

From Eq. (14.2), an evaluation of the total motional impedance as a function of the experimental data Y and Co* can be obtained as follows:

Z m = Z mo + Z mL =

1 Y − jωC 0*

= Zm

EXP

(14.5)

EXP

In the former equation, Zm is the motional impedance evaluated from the TLM and Zm|EXP is the motional impedance evaluated from the experimental data. If the sensor in the unperturbed state, or perturbed with a known

14 QCM Data Analysis and Interpretation

335

acoustic load impedance (see Sect. 14.4.4), has been calibrated by an adequate selection of parameters hq, Co and ηq, the motional impedance in the unperturbed state can be evaluated by Eq. (14.3) with ZL = 0. Thus, the part of the motional impedance associated to the load can be solved from Eq. (14.5) and results in Z mL

EXP

= Zm

EXP

− Z mo

CAL

=

α ZL 1 ωC 0 4 K 2 Z cq

1 j 1−

ZL Z cq

(14.6)

2 tan α2

where the first member in the previous equation has been derived from experimental data and sensor calibration and the second member is the part of the TLM to be adjusted to experimental data. Solving ZL in Eq. (14.6) results in Z L = Z cq

4 K 2ω C o

α

Z mL

EXP

1+ j

1 4 K ω Co 2

α 2 tan

α

Z mL

EXP

(14.7)

2

Thus, for an accurate evaluation of the acoustic load impedance, it is necessary to measure the admittance (both the conductance G and the susceptance B) of the sensor, at least at one frequency, the parasitic capacitance Cp and to make an appropriate calibration of the resonator (hq, Co and ηq). This can be “easily” made with a network or impedance analyzer or with special adapted impedance analyzers specifically designed for this purpose [13] (see also Chap.5). However, instead of using G and B for obtaining ZL many researches usually employ the resonant frequency and the motional resistance shifts, Δfs and ΔRm (see chapter 5). What is the reason? Near the resonance and under the “small surface condition“, Eq. (14.6) can be reduced to Eq. (3.A.28) [22]. Therefore, in many cases, the acoustic load impedance can be approximated to:

Z L = Z cq

4 K 2ω C o

π

Z mL

EXP

=

π Z cq 2ω o Lq

ΔZ m

EXP

(14.8)

In the former equation it has been assumed that the resonant frequency of the sensor ω = ωo, the ideal wave phase shift α = π, and the notation Z mL has been changed to ΔZm. Equation (14.8) can be split in two, associated with the real and imaginary parts, as follows:

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Yolanda Jiménez, Marcelo Otero and Antonio Arnau

RL =

π Z cq ΔRm 2ω o Lq

XL =

π Z cq ΔX m 2ω o Lq

EXP

(14.9a)

EXP

(14.9b)

where RL and XL are the real and imaginary parts of the acoustic load impedance, ZL, and ΔRm|EXP and ΔXm|EXP are the real and imaginary parts of the motional impedance shift evaluated from experimental data (from now on the subscript EXP will not be included for simplification). In Eqs. (14.9a-b), the motional resistance shift is directly related to the real part of the acoustic load impedance; on the other hand ΔXm can be associated with the motional frequency shift through the acoustic load concept explained in Sect. (3.A.4) if one assumes that the modified BVD circuit is appropriate for characterizing the loaded resonator (this is the main approximation which is not completely fulfilled in general, but is an optimal approach in a great deal of applications). Effectively, unchanged motional capacitance of the BVD circuit allows relating the loading contribution with the increase of the motional inductance LL (this condition is not coherently fulfilled, for instance, near the film resonance). Under these conditions, the shift in the imaginary part of the motional impedance is ΔXm = LLω, which can be related to the motional frequency shift as follows [15]: Δf s ≈ −

Δf f o LL f L ω = − o L o → ΔX m ≈ −2 Lq ω o s 2 Lq 2 Lq ω o fo

(14.10)

Thus, Eq. (14.9b) results in Eq. (3.A.31): XL =−

π Z cq fo

Δf s

(14.11)

Therefore, direct explicit relationships between the real and imaginary parts of the acoustic load impedance, ZL, and the two typical measurable values, the frequency shift, Δfs, and the change in the motional resistance, ΔRm in the BVD equivalent circuit can be obtained, simplifying the characterization interfaces and the subsequent calculations. As an example, for the typical 10 MHz AT-cut resonator with C0 = 5pF and a quartz electromechanical coupling factor K = 0.089, the following simple expressions are obtained from Eqs. (14.9a) and (14.11) with Zcq = 8,852,147 N s m-3:

14 QCM Data Analysis and Interpretation

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RL ≈ 30 ΔRm

(14.12a)

X L ≈ −3 Δf s

(14.12b)

Some researchers have used the value of the half power spectrum of resonance W = f1/2+- f1/2- (see Chap. 3) instead of ΔRm [16]. This provides coherent units for the experimental data, which makes the comparison between Δfs and W easier. However, W can only be accurately related to the motional resistance change in case of BVD circuits through the electrical quality factor (see Sect. 3.4). Thus, the use of W to obtain ΔRm is less sensitive to experimental uncertainties but only accurate in terms of BVD equivalent circuits, in extreme cases W should be used with the complete expression of the TLM, otherwise it gives a lack of accuracy in the evaluation of ΔRm. Therefore, ZL can be extracted directly from the G and B through the TLM without any additional approximation; or from Δfs and ΔRm or W through the acoustic load concept under the “small surface condition”. The main advantage of the acoustic load concept is that ΔRm and Δfs can be directly related to the real and imaginary parts of ZL and that the they can be obtained using simple characterization interfaces like well-designed oscillators or oscillator-like operating circuits (see chapter 5); these circuits are suitable for sensor applications, principally due to its fast operation, to the low expense of circuiting and to the major adaptability for remote or in situ measurements. However, there are applications where Eqs. (14.9.a) and (14.11) are not valid and Δfs and ΔRm must be related to the complete TLM and measured with more accurate instrumentation. The frequency corresponding to the maximum conductance and the magnitude of the conductance at this point, with regard to those values in the unperturbed state, allows obtaining these relationships as follows [14]: ΔR m =

1 G max

−

1 o G max

Δf s = f G max − f Go max

(14.13a) (14.13b)

where the superscript “o” indicates the unperturbed state. An important implication of these results, from a practical point of view, is that the conductance of the sensor is not affected by the parasitic capacitances in parallel. Until this point the extraction of the acoustic load impedance starting from electrical experimental data has been presented. The discussion of the

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problem associated to the extraction of effective parameters of the load, which are included in ZL, will be next analyzed. For simplification purposes the problem will be structured in a set of cases grouped in “simple cases” and in the “general case”. Simple cases will include those applications in which, due to the nature of the load, the physical model can be simplified and, in consequence, the number of unknowns reduced making the interpretation easier. On the contrary, those applications in which no additional approximation over the TLM can be done will be included in the “general case”. In all cases the analysis of the experimental data provided by the sensor characterization systems will be explained. Finally, some real examples covering the most representative cases will be studied with a possible interpretation of the physical, chemical or biological phenomena involved.

14.3 Interpretation of Simple Cases There are applications where the number of unknowns is reduced due to the nature of the load; some of these cases have already been presented in Chap. 3 as special cases. Next, these and other special cases will be discussed from the data analysis and interpretation point of view. The analysis made in the previous section has shown that, within those cases where the load acoustic approximation formalized through Eqs. (14.9a) and (14.11) is valid, the real and imaginary parts of the acoustic load impedance ZL are directly related to the motional frequency and resistance shifts (in the simple cases analyzed in the next section, this approximation is valid). Thus, for data interpretation it is exactly the same to use Δfs and ΔRm or directly RL and XL. However, the use of RL and XL has the following important advantages: 1) they have a direct physical meaning regardless of the resonator, while the values of Δfs and ΔRm for the same values of RL and XL depend on the resonator; for instance, Eqs. (14.12a-b) establish that RL /XL ≈ -10 Δfs /ΔRm, but this relation is only valid for a 10 MHz AT-cut quartz resonator. This is an important observation since the relative magnitudes between RL and XL have a relevant physical significance, as we will see next, 2) RL and XL, in contrast to Δfs and ΔRm, have coherent units and their magnitudes can be compared. This is another important observation because the direct use of Δfs and ΔRm, without performing the transformation into RL and XL, could make the evident relative weights between RL and XL unnoticed. This effect can be understood through Eqs. (14.12a-b) where, for a 10 MHz resonator, the magnitude of RL is reduced 30 times when ΔRm is used, while XL is reduced

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only 3 times when using Δfs, and 3) the use of RL and XL permits the representation of the data by displaying XL vs. RL, what clarifies the interpretation in terms of the physical properties of the model, at least in the simple cases analyzed next. 14.3.1 One Sauerbrey-Like Behavior Layer

The acoustic load impedance of a thin rigid layer (or Sauerbrey-like behavior layer) is (Eq. 3.A.34):

Z L = jωρh

(14.14)

In this case, RL = 0, which is the condition to validate the Sauerbrey-like behavior . It should be noted that a polar plot Δfs vs. ΔRm used to display the data for interpretation would reduce 10 times the relative sensitivity of RL with regard to XL for a 10 MHz AT-cut quartz resonator. Thus, erroneously one could make valid the Sauerbrey condition. In this case, the only experimental parameter needed to extract the property of interest is Δfs which is directly related to the surface mass density ρs through Sauerbrey’s equation (Eq. 3.A.36). This equation is obtained through the combination of Eqs. (14.11) and (14.14), resulting:

ρh = ρ s = −

Zq

2 f 02

Δf s ≈ CΔf s

(14.15)

For a 10MHz AT-cut quartz crystal ρs = 4 ∆fs pg cm-2 Hz-1. In order to use the relationship between ∆fs and ρs given by the Sauerbrey equation, the Sauerbrey-like behavior must be validated, but also an appropriate value for the constant C must be used. This constant could be considered as an intrinsic sensitivity factor; however, derivation of Eq. (14.15) assumes some physical approximations such as an infinite size of the crystal while, in practice, finite-size crystals are used. These approximations in the model makes that the real value of C in practice vary with regard to the theoretical one. In Sect. (14.5.1) a calibration procedure of the QCM, which provides the experimental value of the sensor sensitivity, will be described. Thus, when the Sauerbrey behavior of the coating has been validated, the interpretation of the experimental data is simple: a decrease or increase in the resonant frequency shift, with regard to the resonant frequency in the unperturbed state, means a decrease or increase in the surface mass density, ρs, deposited on the sensor. However, even in this simple case, the extraction of the two remaining parameters, density ρ and

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thickness h, starting from the surface mass density, requires the knowledge of one of them. On the other hand, when the TLM is applied to these cases, the response of the sensor is due almost completely to the inertial contribution of the coating; the contribution of the viscoelastic properties of the load to the sensor response is so small that any minimal mismatch between the experiment and the TLM, or any error due to experiment, make the error in the extracted viscoelastic properties very important [15, 19, 30]. 14.3.2 One Semi-Infinite Newtonian Liquid

The acoustic load impedance generated by a semi-infinite viscous (Newtonian) liquid is (Eq. 3.A.38): ZL =

(1 + j ) 2

ρ G ′′

(14.16)

In this case RL = XL, which is the condition to validate the Newtonian behavior. The parametric plot XL vs. RL would show a straight line of slope equal to the unit. The values of RL and XL increase proportionally to ρ G ′′ . Kanazawa’s equation (Eq. 3.A.40) gives an appropriate characterization of the medium from the experimental measure, Δfs. Equations (14.12b) and (14.16) provide a simple expression of Kanazawa’s equation for a 10 MHz AT-cut quartz as follows:

ρ G ′′ ≈ 4Δf s

(14.17)

Since RL = XL, only one experimental datum is necessary to extract ρ G ′′ , but it is necessary to measure both Δfs and ΔRm to validate the Newtonian condition which, from Eqs. (14.9a) and (14.11), is: Δf s ΔRm = 2ω o Lq fo

(14.18)

When the experimental data ΔRm and Δfs comply, with low mismatch, with the former expression, one can assume that the semi-infinite medium is Newtonian and an easy interpretation of the physical properties can be made: an increase or decrease in Δfs or ΔRm implies an increase or decrease of ρ G ′′ . The extraction of the remaining parameters ρ and G ′′ , also requires the knowledge of one of them, typically density.

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14.3.3 One Semi-Infinite Viscoelastic Medium

A semi-infinite viscoelastic medium generates a surface acoustic impedance:

Z L = ρG

(14.19)

where G = G ′ + j G ′′ = G (cos(δ ) + j sin(δ ) ) and tan δ = 1 Q L = G ′′ / G ′ . From Eq. (14.19) RL and XL can be solved as follows: RL = XL =

ρ G ′′ 2

ρ G ′′ 2

Q L2 + 1 + Q L

(14.20a)

Q L2 + 1 − Q L

(14.20b)

In this case, RL ≠ XL and both parameters increase with ρ G ′′ for QL = cte, as shown in Fig.14.2a where the Newtonian case has also been included as limit for QL = 0. Figure 14.2b shows the variation of XL and RL as QL increases and ρG″ remains constant, showing the transition between Newtonian to viscoelastic medium. On the other hand, the polar plot XL vs. RL in Fig. 14.3 shows straight lines which slopes depend on the value of QL. From Eqs. (14.9a), (14.11), (14.20a) and (14.20b) the expressions of Δfs and ΔRm can be obtained [14, 15]: fo π Z cq

ρ G ′′

2ω o Lq

ρ G ′′

π Z cq

2

Δf s = − ΔRm =

2

Q22 + 1 − Q2 = Δf N

Q22 + 1 − Q2

(14.21a)

Q22 + 1 + Q2 = ΔR N

Q22 + 1 + Q2

(14.21b)

Equation (14.21a) corresponds to an extension of Kanazawa’s equation where ΔfN is the frequency shift for a Newtonian Liquid. In Eq. (14.21b) ΔRN is the resistance shift for a given Newtonian case. The combination of Eqs. (14.20a) and (14.20b) provides an explicit expression of the quality factor QL as a function of the two magnitudes, as follows: 1 1⎛ ⎛ X QL = = ⎜1 − ⎜⎜ L tan δ 2 ⎜ ⎝ R L ⎝

⎞ ⎟⎟ ⎠

2

⎞⎛ R ⎟⎜ L ⎟⎜⎝ X L ⎠

⎞ ⎟⎟ ⎠

(14.22)

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Yolanda Jiménez, Marcelo Otero and Antonio Arnau

200

3

-3

(RL, XL )x10 (Nxsxm )

RL 150

w Ne

100

ian ton

XL 50

0 0

20

40

60

80

100

120

140

160

180

200

(ρG'')1/2 x103 (Nxsxm-3)

a 12

3

-3

(RL, XL )x10 (Nxsxm )

11 RL

10 9 8 7 6

XL

5 4

b

0.001

0.01

0.1

1

10

Q

Fig. 14.2. Plots showing the variation of RL and XL: a as a function of (ρG″)1/2 for QL = 1, and b as a function of QL for (ρG″)1/2= 104 N s m-3, where ρ = 1 Kg m-3 and G″ = 105 N m-2

For a 10MHz AT-cut quartz resonator a simple expression of XL/RL as a function of Δfs and ΔRm is: X L Δf s ΔR m = RL KR

(14.23)

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where K R = f s2 8K 02 C 0 / n 2 π ≈ 10 . In this way, QL could be expressed as a function of the ratio XL/RL or of the ratio of the two experimental magnitudes Δfs and ΔRm. Equations (14.22) and (14.23) permit a rapid interpretation of changes in QL from the ratio XL /RL and then from the experimental data |Δfs / ΔRm|: in semi-infinite viscoelastic media the quality factor QL decreases when the ratio XL /RL increases, and viceversa; on the other hand, when XL /RL is maintained constant, the quality factor remains constant as well. From Eqs. (14.20a-b), the properties of the medium are [16]: G′ =

R L2 − X L2

ρ

; G ′′ =

2RL X L

(14.24)

ρ

Notice that in this case XL and RL do not provide separately useful information of the medium properties. However, the parameter XL /RL is very useful for a simple interpretation of the relative changes between G ′ and G ′′ through the quality factor QL, as shown in Fig. 14.3. 200 QL=1

-3

RLx10 (Nxsxm )

150

100

3

Newtonian 50

0

0

50

100

150

200

XL x103 (Nxsxm-3)

Fig. 14.3. Polar plot XL vs. RL for viscoelastic media with QL = 1

14.3.4 One Thin Rigid Layer Contacting a Semi-Infinite Medium

When the coating is a thin layer (small-h1), the tangents in Eq. (14.4) can be approximated by their arguments, and then the expression of ZL can be reduced to:

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Yolanda Jiménez, Marcelo Otero and Antonio Arnau

ZL =

Z c 2 + j ρ 1ω h1 Z 1 + j c 2 ω h1 G1

(14.25)

If the coating is a thin rigid layer h1 G1 → 0 , and the term accompanying the unity in the denominator of the previous equation, which represents the interaction between the two layers, can be considered negligible, the acoustic load impedance reduces to the direct addition of the acoustic impedances of the first layer and second medium separately, and the real and imaginary parts result in: R L = R2 = X L = X 2 + ρ1ω h1 =

ρ 2 G2′′ 2

ρ 2 G2′′ 2

Q22 + 1 + Q2 Q22 + 1 − Q2 + ρ1ω h1

(14.26a) (14.26b)

Thus, in this case, RL (ΔRm) provides information only about the second medium, but it is not possible to distinguish if a change in RL (ΔRm) is due to a change in G2′′ or in Q2. Even with the evaluation of both RL and XL it is impossible, in general, to differentiate the contribution of each property to the acoustic load impedance. Only when the second medium is Newtonian Eqs. (14.26a-b) turn into Martin’s equation, (Eqs. 3.A.42 and 3.A.43), and it is possible to evaluate the contribution of G2′′ (Q2 = 0 in a Newtonian medium) from the measurements of ΔRm assuming the density of the medium ρ2 as constant. For a constant value of ΔRm the surface mass density of the sensitive layer can be extracted from Δfs (XL). Otherwise, ΔRm ≠ cte means that the properties of the second medium are changing or that the sensitive layer cannot be considered as rigid and its viscoelastic properties should be considered. In the case of a viscoelastic second medium, only when the properties of this medium can be considered as constant, which can be validated measuring ΔRm with a second reference resonator in contact with the medium, it is possible an easy interpretation of the differential frequency ΔfsD, i.e., the difference between the frequency shift provided by the sensor and the frequency shift provided by the reference, as a function of the surface mass density of the sensitive layer: a decrease or increase in the differential frequency means a decrease or increase in the surface mass density, ρs, deposited on the sensor. In this case ΔRmD ≠ cte means that the first sensitive layer does not behave as rigid and its viscoelastic properties should be considered ( ΔRmD is the difference between the shift provided by the

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sensor and the shift provided by the reference sensor). On the other hand, when the properties of the second viscoelastic medium are not constant and ΔRmD ≠ cte, an interpretation of the changes in the properties cannot be made only starting with the typical data, Δfs and ΔRm. As it can be noticed, even in this simplified model a coherent interpretation of the experimental data cannot be made without an additional reference sensor. In this case, the reference sensor, directly in contact with the viscoelastic medium, permits the evaluation of the changes in the viscoelastic properties of the medium and provides differential measurements of Δfs and ΔRm, which permit the direct interpretation of the changes in the surface mass density of the sensitive layer. 14.3.5 Summary

Table 14.1 can be used as a guide for a rapid interpretation of experimental data in the simple cases discussed. The first column indicates the type of approximation; the second column indicates the relevant physical parameters in each approximation; the third column includes the experimental parameters, and the forth column indicates the interpretation of the variation of experimental parameter as a function of the physical relevant properties. Table 14.1 Summary of the interpretation of experimental data in simplified cases Case One Sauerbrey layer One Newtonian medium One semi-infinitea One Sauerbrey + Newtonianb One Sauerbrey + viscoelasticc a

Parameters

Experim. Data

ρs = ρh

Δfs ; ΔRm = 0

Interpretation ↑↓ Δfs → ↑↓ ρ s

ρ Lη L

Δfs or ΔRm and Eq. (14.18)

↑↓ Δfs or ↑↓ ΔR →

ρ L , G L' , G L''

Δfs and ΔRm

↑↓

ρ s , ρ Lη L

Δfs ; ΔRm = cte

↑↓ Δfs → ↑↓ ρ s

ρ s , ρ L , G L' , G L''

ΔfsD; ΔRmD and ΔRmD = cte

↑↓ ΔfsD → ↑↓ ρ s

→ ↑↓ ρG ′′ Δf s → ↓↑Q ΔR m

The interpretation in this case assumes the density of the medium as constant. In this case, ΔRm ≠ cte means that the properties of the medium are changing or that the sensitive layer cannot be considered as rigid and its viscoelastic properties should be considered. c This case requires differential measures with a second reference sensor. ΔRD and ΔfD are the difference between the shift provided by the sensor and the shift provided by the reference sensor. b

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14.3.6 Limits of the Simple Cases

Loading approximations are useful for an easier interpretation of the experimental data. However, the greater the approximation the easier the interpretation and the lesser the information obtained from the data analysis. This means that when making approximations in order to reduce the number of unknowns, the contribution of these variables to the global response is also lost and, then, a loss of accuracy can occur in the interpretation as well. On the other hand, the advantage is that without these simplifications it would be very difficult, in some cases, to make any interpretation. Thus, it is important to be aware of the approximation limits; an example of these limits can be shown in piezoelectric biosensor applications. In these applications the typical simplification is to consider a 3-layer model formed by quartz crystal, a thin rigid layer (sensitive layer) and a semi-infinite Newtonian medium. Martin’s simplification can be applied in this case, in such a way that experimental Δfs data are directly related to changes in the surface mass density of the first layer; the properties of the medium are considered, in general, as constant. However, more and more sensitivity is demanded in order to detect lower and lower concentrations of analytes in solution and considerable errors can be made if the limits of the reduced model have not been taken into account. Limits of the Sauerbrey Regime

The Sauerbrey regime considers an ideal rigid film or, in other words, the film must behave rigid for the wave propagating through the layer. When the film is very thin, it is assumed that the phase shift of the acoustic wave through the layer is null, but it depends on the layer viscoelastic properties. Thus, a better approach could be to consider the film as very thin but with viscoelastic properties. The acoustic load impedance in this case fits to Eq. (14.25), but now the simplification in the second term in the denominator will not be made, giving the following expressions for RL and XL: RL =

2

+ ω h1G1′′ Z c 2

G1

2

+ 2ω h1 (R2 G1′′ − X 2 G1′ ) + ω 2 h12 Z c 2

2

X 2 G1 + ρ1ω h1 G1 − ω h1G1′ Z c 2 + ρ1ω 2 h12 (R2 G1′′ − X 2 G1′ ) 2

XL =

2

+ ρ 1ω 2 h12 ( X 2 G1′′ + R2 G1′ )

R 2 G1

2

2

G1 + 2ω h1 (R2 G1′′ − X 2 G1′ ) + ω 2 h12 Z c 2 2

(14.27a)

2

(14.27b)

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where R2 and X2 are the real and imaginary parts of the characteristic acoustic impedance of the Newtonian medium Z c 2 = ( ρ 2 G 2 )1 / 2 . When the thickness of the layer is very thin, additional simplifications in Eqs. (14.27a-b) result in: R L = R2 + ω h1 Z c 2 X L = X 2 + ρ1ω h1 − ω h1 Z c 2

2

G1′′

2

G1 G1′

G1

2

(14.28a)

2

= XL

2

+ XL

1/ 2

(14.28b)

As it can be noticed Eqs. (14.28) do not represent the load impedance as the additive contribution of the layer acoustic impedance and the medium impedance as it is the case in Martin’s approximation. Equation (14.28b) can be split into two terms: a first one that depends only on the properties of the medium X L 2 and a second one which has two contributions, one depending only on the first layer and the other depending on the interaction between the layer and the medium X L 1 / 2 . Expression Δfs|1/2 associated to the second term in Eq. (14.28b) results from Eq. (14.11) in: Δf s

12

2 ⎛ Z c2 fo G1′ ⎜ =− ωρ1 h1 1 − ⎜ π Z cq ρ1 G1 2 ⎝

⎞ ⎟ ⎟ ⎠

(14.29)

Considering the medium as Newtonian, ω ≈ ωs near resonance, the relationship between the complex shear modulus G and the shear complex compliance J (Eq. 7.4), and the relation between the resonant frequency fo and Zcq, Eq. (14.29) results in: Δf s

12

fs

=−

ρ1 h1 ρ q hq

⎛ ωη 2 ρ 2 ⎞ ⎜⎜1 − J 1′ ⎟⎟ ρ1 ⎝ ⎠

(14.30)

Equation 14.30 coincides with that obtained by Kankare [17] and Voinova [18]. These effects have also been considered in Chap. 3 as small phase shift approximation and they should be taken into account in biosensor applications since the contribution of the second term in Eq. (14.28b) can be a high percentage of XL [18]. The previous analysis shows that the additive contribution of the acoustic load impedance of the first layer and second medium, in the context of Fig. 14.1, is a simplification which must be carefully considered. Figure 14.4a shows the deviation, as the thickness of the first layer increases,

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Yolanda Jiménez, Marcelo Otero and Antonio Arnau

between the complete acoustic load impedance from Eq. (14.4) and the simplified acoustic load impedance corresponding to the additive contribution which is: ⎛ ⎞ ρ Z LAdd = Z c 2 + j Z c1 tan ⎜ ω 1 h1 ⎟ ⎜ G1 ⎟⎠ ⎝

(14.31)

40

3

-3

(RL, XL )x10 (Nxsxm )

35 30

Additive

25 20

Non additive XL RL

15 10 5 0

0

1

2

3

4

h (μm)

a

70 60 semi-infinite

-3

RLx10 (Nxsxm )

50 40

3

30 20 10

Sauerbrey

0 0

10

20

30 3

b

40

50

60

-3

XL x10 (Nxsxm )

Fig.14.4. a Plot showing the deviations between the complete acoustic impedance of a load like that described in Fig. 14.1 and the acoustic impedance simplified with the additive contribution approach; b polar plot associated with the thickness growth of a finite layer with viscoelastic properties G = 106 + j106 Nm-2

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Equation (14.4) is marked in Fig. 14.4a as “non additive” while Eq. (14.31) is marked as “additive”. It can be noticed that for small thicknesses the deviation is very small while the greater the thickness the higher the deviation. When the thickness of the first layer reaches values higher than the penetration depth length, the error is equal to the impedance of the second medium. The simple cases analyzed are extreme limits. When the thickness of a finite first viscoelastic layer grows from very small thicknesses, the behavior of the layer starts being Sauerbrey-like and finishes as semiinfinite. The parametric polar plot of Fig. 14.4b is very useful to describe the process. For small thicknesses, only inertial contribution associated with an increase of XL can be observed; as the thickness increases both inertial and viscoelastic contributions are transferred in the plot as variations both in XL and RL. Finally, when the thickness can be considered as semi-infinite the magnitudes XL and RL reach constant values. In general, Sauerbrey condition is very restrictive. When a finite first layer with viscoelastic properties is considered, the error in the evaluation of the surface mass density, from the experimental motional frequency shift through Sauerbrey’s equation, increases with the thickness. This effect is called “viscoelastic contribution“ [15, 19] and is explained in Fig. 14.5. Figure 14.5 plots in the lower panel the Δf associated with an increase in the thickness of the first viscoelastic layer and in the upper panel the viscoelastic contribution. As it can be observed, the greater the thickness the greater the viscoelastic contribution and more errors can be made when considering a direct relation between the thickness and Δfs through Suerbrey’s equation (or Martin’s equation). The deviation between Sauerbrey’s equation and the Δfs obtained from the TLM can start from very low thicknesses depending on the viscoelastic properties of the layer. When there is a contacting semi-infinite medium, the deviations from Sauerbrey begin for lower thicknesses. This occurs in biosensors applications. Limits of the Small Surface Load Impedance Condition and of the BVD Approximation

As it has been commented, the small surface load impedance condition assumes that the load surface mechanical impedance is small compared to the quartz shear impedance. Cernosek et al., determined empirically that when the impedance ratio ZL/Zq<0.1 the consideration of this assumption introduce errors smaller than 1% between the Δfs and ΔR computed from the TLM and those computed under the assumption of “small surface load” [22].

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Yolanda Jiménez, Marcelo Otero and Antonio Arnau

Viscoelastic contribution (%)

140 7/6 6/5 5/4

120 100 80 60 40 20 0 80 60 40

Δf (KHz)

20 0 -20 -40 -60 -80 -100

0.0

0.2

0.4

0.6

0.8

1.0

h (μm)

Fig. 14.5. Viscoelastic contribution (upper panel), Δf vs. the thickness h (lower panel); the properties of the medium are indicated as log (G′1/Pa)/log (G″1/Pa), in which the slash (/) represents a separator, not a quotient

With regard to the BVD approximation, two approximations must be made when the BVD model is obtained from the TLM [15, 30]: the small surface load impedance condition and frequency-independent parameters. In order to quantify the limits of these two assumptions, Arnau et al. made a study in which the values of Δfs and ΔRm provided by the Extended BVD model derived by these authors and those provided by the TLM were compared [15, 30]. Deviations between the models in those parameters of interest (Δfs and ΔRm) were smaller than 3% for a ratio |ZL/Zq| ≤ 0.025.When frequency shifts were restricted up to 2% of resonant frequency, a ratio |ZL/Zq| ≤ 0.05 with deviations smaller than 4% were achieved. When frequency dependent parameters were considered, deviations smaller than 3% for a ratio |ZL/Zq| ≤ 0.2 were obtained. Then, deviations larger than 3% for a ratio |ZL/Zq| ≤ 0.2 were due to the frequency independent parameter approximation.

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14.4 Interpretation of the General Case 14.4.1 Description of the Problem of Data Analysis and Interpretation in the General Case

In the context of Fig. 14.1, when no-approximation can be done about the first sensitive layer, i.e., when it is not reasonable to consider a Sauerbreylike behavior of the first layer, or the “small surface load impedance condition“ cannot be applied, the appropriate admittance model for characterizing the electrical response of the sensor, turns into the TLM (Eq. 14.2). The physical properties to be extracted and interpreted are concentrated in the acoustic load impedance ZL (Eq. 14.7). ZL depends on 7 unknowns which can be reduced to four: ρ1, h1, G1′ and G1′′ assuming the properties of the second medium ρ2, G2′ and G 2′′ known by, e.g., an additional reference sensor or during certain steps of a process where only the second medium is in contact with the sensor; it occurs in the first steps of EQCM or biosensor applications. The characteristic impedance of the second medium can also be obtained from theory, in case of Newtonian liquids, with low error. The remaining four unknowns can be lumped in three: ρs, Q1 and Zc1 [11, 30, 31]. Then, three unknowns is the minimum number of unknowns to which the TLM can be reduced without loading simplifications. Thus, the typical experimental measures considered until now Δfs and ΔRm, or the alternative magnitudes XL and RL, are not enough to extract without uncertainty the physical properties of the first layer. Infinite possible solutions provide the same couples of experimental data and it is impossible to make any coherent interpretation of experimental data in terms of the physical properties of the model (ρs, Q1 and Zc1) [1-4]. This is illustrated with a parametric plot in Fig.14.6. As it can be seen, infinite sets of properties match the same couple of XL and RL. Therefore, it is necessary to restrict the number of possible solutions; this can be done in two ways: a) increasing the knowledge of the physical model by using alternative techniques which permit the evaluation of certain properties of the first layer, in general the thickness (see Chap. 13) [1-4, 10, 26, 28], or by assuming the knowledge of some of them in order to reduce the set of possible solutions to a certain range [3], or by a controlled change of the properties of the second medium [5-6], and b) increasing the knowledge of the admittance response in order to obtain more than two fitting conditions [8-12, 30, 31]. In this way, the range of possible solutions is reduced by fitting the models to experimental data. Next, the different strategies to face the problem are described.

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60

Q=0.87 h=934 nm

3

-3

RLx10 (Nxsxm )

50

Q=0.50 h=1002 nm

40 30

Q=0.26 h=1053 nm

20

3

-3

3

-3

RL=15 x10 (Nxsxm ) XL=66 x10 (Nxsxm )

10 0

0

20

40

60

80

XL x103 (Nxsxm-3)

Fig. 14.6. Parametric plot showing the problem of indetermination in the properties extraction starting only from one couple of RL and XL

14.4.2 Restricting the Solutions by Increasing the Knowledge about the Physical Model

The strategies used for increasing the knowledge of the physical model can be classified into three topics: 1) by measuring or evaluating the thickness by alternative techniques [1-4, 10, 26, 28], 2) by assuming certain values for some of the properties of the first layer, different from the thickness [3], and 3) by a controlled change of the properties of the second medium [5, 6]. Restricting the Solutions by Measuring the Thickness by an Alternative Technique

One of the most typical strategies proposed to carry out the extraction of the four coating properties is by measuring the thickness with an alternative technique, while the density is obtained from theoretical calculations or from the literature. Thus, the remaining unknowns are reduced to two which can be extracted with the typical measurements of Δfs and ΔRm. The extraction of the remaining parameters G1′ and G1′′ is not a simple task despite the determination of the thickness. The TLM is not as simple as the simplified models described before; explicit expressions for G1′ and G1′′ cannot be obtained and the algorithm used to find the best fitting has to search for the best values of G1′ and G1′′ in a broad range of possible

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solutions, from 103 to 109; this makes the fitting more difficult because of the existence of local minima in the complete range of solutions. In general, this fitting is tedious and very time-consuming because of the necessity of evaluating Δfs and ΔRm obtained from the model for each tentative couple of parameters G1′ and G1′′ . Additionally, the TLM does not offer explicit expressions for Δfs and ΔRm, which have to be calculated for each couple of viscoelastic properties with an additional algorithm [20]. One possibility to reduce this second algorithm is to use the LEM or the extended BVD model to obtain Δfs and ΔRm, but it would reduce the accuracy in the properties extracted, especially when the “small surface loading” condition cannot be applied; this occurs as the thickness comes near the film resonance [22, 15]. Lucklum et al. propose a fast method for the extraction algorithm [21]. The main limitation of their fast method is that it is only valid for only one layer and in the range of thickness before the first film resonance; when a second medium exists or when the thickness is far from the first film resonance, the algorithm cannot be applied. The thickness evaluated by an alternative technique can also be used to evaluate the density of the layer. If during the first steps in a process the deposited mass on the quartz is small enough to assume that the behavior of thin layer can be considered as Sauerbrey, the surface mass obtained by the QCM can be plotted versus the evaluated thickness and then the slope corresponds to the density [1]. Next the most important techniques for measuring the thickness together with their limitations are described. Measuring the Thickness by Cooling

Behling and Lucklum [2] propose the evaluation of the thickness of a polymeric coating by cooling. In this situation, the viscoelastic properties of the coating change in such a way that the coating behaves like a rigid layer, and Sauerbrey’s equation can be used to measure the surface mass density. By assuming or evaluating the polymer density, the thickness is obtained. It is necessary to be accurate in the measure of the thickness because the authors showed that multiple solutions with similar thicknesses and very different viscoelastic properties exist. They propose an alternative to Sauerbrey’s equation for the calculation of the thickness [20]: ⎛ ⎞ ⎜ ⎟ 2 Z cq ⎜ 4K 0 ⎟ 2 (14.32) h1 = − 2πf s ρ1 ⎜ ⎛ α ( f s ) ⎞ α ( f s ) ⎟ ⎜ tan ⎜ ⎟ ⎟ ⎜ ⎟ ⎝ ⎝ 2 ⎠ ⎠

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Lucklum et al. stated that this technique is very accurate in comparison with other techniques such as ellipsometry, and estimated the error around 0.1% [21]. The main critical points to this technique are two: the density and the thickness of the coating can change with the change in temperature, and the continuous monitoring of the thickness is difficult to perform in certain applications. Measuring the Thickness by Ellipsometry

Film thickness, refractive index and extinction coefficient can be assessed with ellipsometry using the expressions derived from Fresnel equations, that relate the changes of elliptically polarized light parameters measured after incidence on the sample [23, 24]. The fundamentals of Ellipsometry were described in Appendix 13.B. Ellipsometry has been employed by several authors as an independent method for measuring the film thickness [3, 10, 26] (see Sect.13.3.1). The main problem associated with the use of ellipsometry to evaluate acoustic thickness is that the thicknesses measured by ellipsometry are different to those measured by QCM [9, 25, 27]. On the other hand, ellipsometry cannot be applied to opaque surfaces and the lack of information about the optical properties of the sample makes the use of ellipsometry inoperative [27]. Measuring the Thickness in EQCM Applications

Appendix 13.A describes how in EQCM applications the Faraday law can be used as an alternative method for evaluating the thickness of the deposited layer. Some works in which the thickness is obtained following this technique can be found elsewhere [1, 4]. This thickness can be obtained from Eq. (13.A.4), which is written below for practical reasons, by measuring the charge transferred during the process (Qc), and estimating the density of the polymer (ρ1): h1 =

Qc M nFS ρ1

(14.33)

In the former equation S is the surface where the substance is deposited, F is the Faraday’s constant, n is the number of electrons interchanged in the reaction and M is the molecular weight. In electrochemical applications, the simultaneous measuring of mass surface by Faraday Law and QCM can be used to estimate the efficiency or, alternatively, if the efficiency is assumed, they can be used to calibrate the effective surface S from Eq. (14.33) as follows:

14 QCM Data Analysis and Interpretation

S=

Qc M nF ρ s

355

(14.34)

where ρs is the surface mass density obtained from Sauerbrey’s equation. Restricting the Solutions by Assuming the Knowledge of Properties Different from the Thickness

Another strategy posed to carry out the extraction of the four coating properties is by assuming the knowledge of properties of the layer different from the thickness, while the density is obtained from theoretical calculations or from the literature. The remaining unknowns are reduced to two which can be extracted with the typical measures Δfs and ΔRm. Calvo et al. [3] introduced the characterization results of layers of Poly(aniline) and Glucose oxidase deposited by electrostatic adsorption and submitted to cyclic voltammetry. To extract the parameters from the first layer they followed two strategies: in the first one they assumed the shear storage modulus G1′ << G1′′ , and determined the remaining parameters h1 and G1′′ . In the second one, they assumed a constant QL and determined the remaining h1 and G1 . In both cases the density is estimated by the literature. The main limitation of this method is that the assumption cannot be compared with an alternative technique. For example, the quality factor can change during a cyclic voltammetry [28]. Restricting the Solutions by a Controlled Change of the Properties of the Second Medium

Another strategy to extract the properties of the first layer is by changing the acoustic load impedance of the second medium. In this case, and assuming the properties of the first layer unchanged when the second medium is altered, which is the main criticism to this method, the experiment provides two or more couples of Δfs and ΔRm, or the alternative magnitudes, XL and RL. This strategy, introduced by Lucklum et al. [5], is based on the following method: By solving the characteristic impedance of the first layer Zc1 from the Eq. (14.4), one obtains: Z c1 =

Z1 Z 2 Z L Z 2 + Z1 − Z L

(14.35)

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where Z1 is the acoustic load impedance of the first layer without contacting the second medium, which can be measured during the experiment before depositing the second medium; Z2 is the acoustic impedance of the second medium which can be measured with a second resonator in contact only with the second medium, and ZL is the acoustic load impedance of the first layer in contact with the second medium. Thus, by following the method an experimental value of the characteristic impedance of the first layer can be obtained, which permits a direct calculation of the shear moduli G1′ and G1′′ when the density of the first layer is known; the thickness is then calculated from the previous value of Z1. Lucklum et al. applied this method using a second thin layer of gold whose thickness is measured during the deposition with a third QCM sensor. Etchenique and Weisz used the same basis with a different method, in which they obtained additional information by changing the properties of the second semi-infinite medium [6]. It is, in fact, a special case of Luclum’s method. 14.4.3 Restricting the Solutions by Increasing the Knowledge about the Admittance Response

A different strategy is to increase the knowledge in the admittance response; i.e., in the electrical response of the sensor, by changing the frequency. It is the reverse case to the change of the physical properties of the second medium in order to increase the knowledge about the acoustic mechanical impedance at one frequency. Two different methods have been used: Restricting the Solutions by Measuring the Admittance Response of the Sensor to Different Harmonics

Johannsmann et al. introduced a sophisticated method to extract the properties of thin films, based on the measuring of the admittance response to different harmonics [8-10]. Starting from the complex resonant frequency shift, assuming very thin thickness and developing in Taylor series the tangent associated with the acoustic load impedance of the layer, they obtained the following approximated expressions for the frequency shift and the half power resonant spectrum W: Δf f

n

=−

2 f s ms ⎡ 4π 2 m s2 2 ⎤ f ⎥ ⎢1 + J ' (ω ) 3ρ ⎥⎦ c 66 ρ q ⎢⎣

(14.36a)

14 QCM Data Analysis and Interpretation

W f

n

=−

16 f s m s3π 2 3ρ c 66 ρ q

J ' ' (ω ) f

2

357

(14.36b)

From the previous equations, the strategy consists in measuring the frequency shift Δf and W at different harmonics. Then, assuming properties J′ and J″ (see Chap. 7) unchanged with the frequency, which is the main criticism to this method, they plotted the data vs. f 2 and fitted straight lines to the data. From Eq. (14.36a) the intersection with ordinate axis provides the measure of the surface mass, and the slope provides property J′. From Eq. (14.36b) the slope provides property J″. Restricting the solutions by Measuring the Admittance Response of the Sensor in the Range of Frequencies around Resonance

The limitation of the previous method can be avoided by measuring the admittance response in a range of frequencies around resonance. In this range it can be assumed that the viscoelastic properties remain unchanged. Recently, Jimenez et al. have introduced an algorithm for the first layer properties extraction that uses the admittance plots around resonance (conductance and susceptance) to find the properties by fitting the TLM to these experimental data [11, 30, 31]. The algorithm can solve the problem of the first layer parameter extraction in ideal conditions; i.e., when the admittance plots obtained from the TLM starting from certain properties of the first layer are considered as experimental data, the algorithm is able to extract these properties starting from the conductance with very small errors in all the cases [11, 30, 31]. The main limitation of the method is that the acoustic load impedance does not change too much in the range of frequencies considered. Consequently, the applicability of the algorithm to real experimental data depends very much on the accuracy in the obtained experimental data. There are cases where the change in the acoustic load impedance is greater than the error due to the experimental setup. In these cases the algorithm allows the extraction of the properties and an estimation of the possible error made in the extraction. 14.4.4 Additional Considerations. Calibration

Throughout the whole chapter it has been assumed that the parameters of the unperturbed sensor (hq, Co and ηq) were perfectly known. However, this is not the case, which complicates even more the problem of parameter

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extraction. A calibration is necessary to determine the best set of parameters. The main problem is that these parameters change with the load. This change affects mainly the static capacitance. The change in the static capacitance is pronounced at the beginning, when the load increases from the unperturbed state, and tends to stabilize as the load increases. An error in the characterization of the static capacitance produces an important error in the parameter extraction for small loads; the effect of an error in the static capacitance has a lower effect as the load increases [5]. Therefore, a method is necessary to calibrate the resonator parameters. The normal method is to calibrate the sensor in the unperturbed state, but recently, Jimenez has shown that it is preferable to calibrate the parameters starting from the experimental data obtained with known Newtonian liquids as second medium, for instance, water [30, 31]. Equation (14.18) can be used for this purpose when one knows that the semi-infinite medium is Newtonian. Hence, the value of Co can be extracted from Eq. (14.18) with Eq. (3.A.30d) as follows: Co =

Δf s

π

ΔRm 8 K o2 f o2

(14.37)

The former equation can provide considerable errors when the experimental setup modifies, to a large extent, the response of the resonator. It can occur, for instance, when a galvanostat is connected to the sensor for EQCM applications [30]. In this case, the complete TLM must be used to perform the calibration. Thus, the TLM is applied with a known ZL and the quartz parameters are calibrated to assure that the theoretical acoustic load impedance of the second medium is obtained from it. The larger the number of different Newtonian liquids with different loads is, the better the calibration in a concrete range of loads is. As an example Figure 14.7 shows the effect of calibration. Figure 14.7 shows different conductance plots. The open circles correspond to the experimental conductance plot of a resonator coated with a thin film. This resonator was calibrated, previously to be coated, in air and in water as second media. Figure 14.7 shows the conductance plots obtained from the TLM through the algorithm introduced by Jimenez et al. [11, 30, 31] in two cases: one with the sensor calibrated in air (dashed line), and the other with the calibration made in water (solid line). As it can be noticed, a much better fitting in the plots is obtained with the parameters calibrated in water - as shown by the corresponding fitting errors in Fig. 14.7.

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359

3.5 14 3.0

12 10 8

2.0

6

Error (%)

G (mS)

2.5

1.5 4 1.0 0.5 9.968

2

9.970

9.972

9.974

9.976

9.978

9.980

0 9.982

f (MHz)

Fig. 14.7. Conductance plots obtained from the TLM and the corresponding fitting errors with regard to experimental data (open circles), for the parameters of the resonator calibrated in air (dashed line) and water (solid line)

14.4.5 Other Effects. The N-layer Model

The three layer model considered in Fig. 14.1 can be complicated or simplified by adding or subtracting some layers, in order to obtain a more comprehensive interpretation of some specific phenomena. For example, the studies that deal with absorption processes include an additional layer whose thickness varies with time [26]. Another typical situation which requires an additional layer is rough quartz. If the quartz surface is rough, part of the coating adjacent to the crystal will be trapped and will move synchronously with the crystal surface. This new layer acts as a rigid mass and may contribute to energy storage in a different ways from the rest of the film; this implies that an extra frequency shift appears. On the other hand, a lack of homogeneity within the coating can disturb the acoustic wave propagation and contribute to acoustic energy loss that will result in a new shift in the motional resistance. Another effect introduced to improve the physical model is the slip between the quartz surface and the first layer [32]. The TLM used in the book, assumes that there is no slip between quartz and coating. This slip can be included in the model [29] and does not modify the equations in a relevant way. Thompson has studied the effect of the slip and has indicated that the slip is transferred into effective viscoelastic parameters in the context of the model of Fig. 14.1.

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The models can become very complex, although a simple model can provide an easy way to obtain some effective parameters that could be processed for a more comprehensive interpretation later. The roughness effect will be considered in the practical example described in section 14.5.3, for that reason, this effect will be commented in more detail in the next section. Four-Layer Model for the Description of the Roughness Effect

When a rough surface oscillates in contact with a liquid, the movement of the liquid generated by the oscillation is much more complicated than in the case of a smooth surface. Experimental increases in the motional frequency and resistance shifts of resonators in contact with liquids in relation to the expected ones have been reported and related to the crystal roughness [33]. It is usually interpreted that an additional inertial contribution exists due to the liquid trapped in the rough surface, while the increase in the losses is due to an increase in the friction between the liquid and the surface at the rough interface [33]. These assumptions have been supported by different models which intend to explain the effects of the roughness on the sensor response [34-43]; these models have been used to explain certain experimental phenomena [30, 7, 44-46]. However, only in very simple and limited models mathematical relationships with direct physical meaning are obtained [34], while in other more powerful and complex models [35, 40, 41] the obtained mathematical relationships, although predict the experimental results, do not permit a simple interpretation of the different physical contributions of the rough surface on the sensor response. At the present it has not been provided an exact unified description of the QCM response for non-uniform solid-liquid interfaces with an arbitrary geometrical structure. Urbakh et Daikhin and Etchenique roughness models are the most complete existent ones and it is important to make a brief description about their fundamentals and limitations. Scanning Tunnelling Microscopy (STM) measurements show that the characteristics heights of roughness of metallic films on the quartz crystal are ca. 10-100 nm. With regard to the lateral scales of roughness, they can change over a wide range, from 10-1 nm to 102-103 nm [38]. Starting from these results, Daikhin et Urbakh use the term slight roughness to name those rough surfaces in which the characteristic height is less than the lateral size, and strong roughness to name those ones in which the height is of the same order, or larger, than the lateral size [38]. Real surfaces used in QCM experiments may have a combination of both types of roughness [40]. For modelling the effect of slight roughness, Urbakh et Daikhin use the linealized Navier-Stokes equation and the perturbation theory [35, 36],

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and for the case of strong rough surfaces they use an approach based on Brickman’s equation [40]. Both methods are described in more detail bellow. z Liquid d

ξ(x,y)

Quartz Crystal 0

x Fig. 14.8. Cross section of the roughness model used by Urbakh and Daikhin for “slight” roughness

For the slight roughness these authors use a model in which the solid surface profile is characterized by the function ξ(x,y), giving the local height of the surface profile with respect to a reference plane placed in z=d (see Fig. 14.8). The plane z=d is the average thickness of the quartz crystal [35]. In their works, Urbakh and Daikhin determine the effect of roughness on the complex resonance frequency of the quartz crystal Δω r = ΔΩ + jΔΓ , where ∆Ω and ΔΓ are the shift and the broadening of the shear resonance respectively, from the energy balance in the system under consideration. The energy balance states that the rate of the change in the total energy of the crystal and the kinetic energy of the fluid should be equal to the rate of energy dissipated in the liquid, the energy losses in the quartz are neglected. These energy changes depend on the elastic displacement in the crystal, which is obtained from the wave equation, and on the fluid velocity, which is obtained from the linearized Navier–Stokes equation for an incompressible fluid. In order to solve the Navier-Stokes equation for any roughness profile ξ(x,y), Urbakh and Daikhin use the twodimensional Fourier transform of the fluid velocity, pressure and roughness profile function. The physical properties of roughness come to the problem through the boundary condition that establishes the equality of crystal an fluid velocities at the interface z=ξ(x,y). The resulting equation from the boundary condition is valid for any roughness profile function, however due to the complexity of that equation, it is solved only under some approximations [35, 36], or when considering simplified model of inhomogeneities [39]. Thus, for slight and smooth rough surfaces, that equation is solved in the framework of the perturbation theory with respect to the parameters Δξ (x, y ) << 1 and h δ 2 << 1 , being h the root mean

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square height of the roughness, and δ2 the wave penetration length into the fluid. The final expressions for ∆Ω and ΔΓ are the following: ΔΩ = −

(Ω 0 )3 / 2 (ρ 2η 2 )1/ 2 ⎧1 + π (2 ρ q c66 )1/ 2

ΔΓ =

⎨ ⎩

2δ 2−1S −1

(Ω0 )3 / 2 (ρ 2η 2 )1 / 2 ⎧1 + π (2 ρ q c66 )1 / 2

⎨ ⎩

dK ∫ (2π ) ξ (K ) k [a(kδ ) −

2δ 2−1S −1

2

2

2

dK ∫ (2π ) ξ (K ) k [ 2

2

]

⎫ 2 kδ 2 + 2 cos 2 Φ ⎬ ⎭

]

⎫ 2 kδ 2 − b(kδ 2 ) ⎬ ⎭

(14.38a)

(14.38b)

where Ω0 is the resonant angular frequency of a free quartz crystal, a(t ) = ((1 + 4t −4 ) −1 / 2 + 1)1 / 2 , b(t ) = ((1 + 4t −4 ) −1 / 2 − 1)1 / 2 , ξ(K) is the twodimensional Fourier transform of the surface profile, S is the area of the crystal surface, K the two-dimensional wave vector in the two-dimensional Fourier transform, k=|K|, and Φ is the angle between the direction of the shear oscillation and K. The first term in brackets in equations (14.38a) and (14.38b) corresponds to the shift and the broadening of the shear resonance at a smooth crystal liquid interface, respectively. In this model only geometrical but not physical properties, such as density and viscoelasticity, of the rough layer are considered. Moreover, the mathematical description of this model does not permit a simple physical interpretation of the different contributions of the rough surface on the complex frequency shift. Only when a particular model of inhomogeneities is considered (periodical corrugation, random roughness or hemispheres which rest on a plane surface), and when some approximation over the ratios between the characteristic lengths of roughness and δ2 are applied, simplified mathematical expressions that permit a physical interpretation are achieved [35, 36]. For strong roughness, the theoretical approaches based on perturbation theory can not be applied. In this case Daikhin et Urbakh propose another roughness model based on Brickman’s equation for the velocity field in the interfacial region. In this approach, the flow of the liquid through a nonuniform surface layer is treated as the flow of a liquid through a porous medium [40]. The morphology of the interfacial layer is characterized by a local permeability that depends on the porosity of the layer. Etchenique extends the Brickman’s equation giving to the porosity a complex character [41]. Daikhin et Urbakh and Etchenique use a four-layer model to describe the porous medium in contact with a liquid phase and use two parameters to define the characteristic dimensions of the roughness [40, 41]. This model is represented in Fig. 14.9.

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Newtonian Medium Newtonian Medium

ρ2 η 2

Lrr

h11 hqq

Rough Rough Layer Layer

Z22

ξ

Uniform Coating Uniform Coating

ρ1 G'1 G''1

Polish Polish Crystal Crystal

c66 ρq ηq

Fig. 14.9. Cross section of a four-layer model: quartz + uniform coating layer + rough layer + liquid of a coated TSM resonator [40].

The vertical dimension of the roughness is defined by the thickness of the rough layer Lr, while the lateral dimension is quantified by the characteristic length of the porosity, which is represented by the parameter ξ. This parameter comes from the Darcy’s law which is an empirical relation describing the flow of an incompressible viscous fluid through a porous mass [47]: v=−

ξ2 ∇P η2

(14.39)

where v quantifies the average velocity of the fluid particles, ∇P is the gradient of pressure, η2 is the fluid viscosity and ξ2 is the permeability of the porous mass which, according to Etchenique, depends on the layer microstructure and can only be predicted by approaching theories [41]. The former equation is only valid in the following conditions [37, 40]: a) a non-slip condition can be applied to the solid-liquid interface; b) turbulent flows do not exist, which is true for a Reynolds number smaller than unity, what can be applied in the case of the quartz crystal small amplitude oscillations [33], and c) the characteristic size of the inhomogeneities is much smaller than the vertical dimension Lr. In Daikhin [40] and Etchenique [41] models the rough layer is assumed to be rigid and moves in phase as a whole with the velocity vo = exp jωt; only geometrical but not physical properties, such as density and viscoelasticity, of the rough layer are considered. This introduces a limitation when the magnitude of Lr is comparable to the wave penetration depth on the coating rough layer; in this case it can not be reasonably assumed that the rough layer moves in phase as a whole. Daikhin et Urbakh and Etchenique assume that the liquid in the porous material is submitted to two different types of force: the first one is associated

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with the typical shear force exerted on the liquid by the surface, which is described by the Newton equation with a complex viscosity; and the second is the drifting force applied on the fluid particles due to the movement of the porous medium. Daikhin et Urbakh and Etchenique use the Brickman’s equation (derived on the basis of Darcy’s law) to describe this second force, which has the following expression: P = η 2 ξ −2 ( v o − v )

(14.40)

where vo is the amplitude of the driving velocity, which oscillates with frequency ω. Daikhin et Urbakh use directly the Brickman’s equation described in Eq.(14.40) to model the drifting force applied on the fluid particles by the oscillating porous medium. In Brickman’s equation the drifting force and the fluid particle velocity are in phase. This is because Brickman’s equation gives the drifting force as a viscous force starting from a modification of the Darcy’s law, where a stationary flow is considered through the porous medium; therefore, the time derivative of the velocity does not exist in stationary flow and the drifting force and the velocity are in phase. This is the reason why Etchenique extends the Brickman’s equation giving to the porosity ξ2 a complex character; the result is the addition of a reactive part, it is to say an inertial term, to the drifting force given by the Brickman’s equation. Then, according to Etchenique [41], when the wave penetration depth into the viscous medium δ2 is much bigger than the lateral dimension of the roughness ξ, the fluid particles move in phase with the drifting force applied by the porous medium; however, when δ2 decreases in relation to ξ, the movement of the liquid leaves the phase with the movement of the solid phase, and complex values for the parameter ξ are necessary to model the movement of the system formed by the liquid and the porous medium. The application of Brickman’s equation and its extension with a complex value for ξ2 permits the obtaining of the following expressions for the acoustic impedance derived for the Daikhin et Urbakh model (Eq. 14.41) and for the Etchenique model (Eq. 14.42) [30, 40, 41, 7]. ⎡ 1 ⎞⎤ ⎛ 2K 0 Lr 1 ⎜ (cosh(K 1 Lr ) − 1) + sinh(K 1 Lr )⎟⎟⎥ Z 2 = jωρ 2 ⎢ + − 2 2 2 2 ⎜ ⎢⎣ K 0 ξ K 1 Mξ K 1 ⎝ K 1 ⎠⎥⎦ ⎡ 1 L 1 ⎧ 2K 0 + r − Z 2 = jωρ 2 ⎢ ⎨ 2 (cosh (K 1 Lr ) − 1) ⎣⎢ K 0 ξK 1 Mξ ⎩ K 1

+

⎛ ⎞ ⎫⎤ 1 1 (cosh (K 1 Lr ) − 1)⎜⎜ 1 − 1⎟⎟⎪⎬⎥ sinh (K 1 Lr ) + K1 K0 ⎝ ξK 1 ⎠ ⎭⎪⎥⎦

(14.41)

(14.42)

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where M = K 0 sinh (K1 Lr ) + K1 cosh (K1 Lr ) , K12 = K 02 + ξ −2 and K 0 = jωρ 2 η 2 In the case of smooth surfaces ( Lr → 0 ), only the first term remains and the theory of Kanazawa is reproduced. It can be observed that, as in the case of the model of Daikhin et Urbakh based on the Navier-Stokes equation, the mathematical descriptions of Daikhin and Etchenique models based on Brickman’s equation, although very coherently obtained, do not permit a simple physical interpretation of the different contributions of the rough surface on the surface acoustic load impedance. On the other hand, the modeling of the vibrating rough surface with a liquid on top like a stationary liquid flow going through a porous medium is not exactly what physically happens in a coated shear resonator immersed in a liquid. Another model, very simplified, that makes easier the interpretation of the physical phenomena which happen when a rough surface oscillates in contact with a fluid has been recently introduced by Arnau et al. [46, 48, 49]. This model characterizes the roughness like a surface of spherical shells whose characteristic dimensions are, the height hr and the radius of the base rr (Fig. 14.10). The solution of the linearized Navier-Stokes equation applied to this model allows, after some simplifying assumptions, obtaining the corresponding drifting force and the acoustic impedance of the rough surface in contact with the liquid as follows: ⎡ η 3 η2 1 1 ⎛3 ⎞⎤ ΔS r + 2 + jω ⎜ ρ 2δ 2 ΔS r + ρ 2 Vr + ρ 2δ 2 ⎟⎥ Z 2 = ⎢3π η 2 (nhr ) + δ δ 2 4 2 2 ⎝ ⎠⎦ 2 2 ⎣

(14.43)

where ΔSr is the increase of the surface per unit area due to the roughness, in comparison with a flat surface, and Vr is the volume of the roughness per unit area, i.e., the average height of inhomogeneities. For the sphericalshell model the surface and the volume of one shell are: S r1 = π (hr2 + rr2 ) and Vr1 = π hr (hr2 + 3rr2 ) / 6 . The increase of the surface due to one shell in relation to a flat surface can be found making hr = 0 and subtracting the result from the total surface. Therefore ΔS r = nπ hr2 and Vr = nπ hr ( hr2 + 3rr2 ) / 6 , where n is the number of shells per unit area. For a complete coverage (Fig. 14.10) n can be calculated as the ratio between the unit area and the area of the hexagon in which the base of the shell is inscribed as follows: n = 1 /(2rr2 31 / 2 )

(14.44)

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Liquidmedium medium Liquid Rough layer layer Rough

hrr

Z2r

Layer hlp Layer Quartz Quartz crystal

Fig. 14.10. Model for surface roughness based on a rough surface covered by spherical shells of characteristic dimensions hr (height of the shell) and rr (radious of the circular base of the shell) [46]

According to Eq.(14.43), the drifting force per unit area is the sum of two components, one in phase with the oscillating velocity, the loss part, and the other one 90º out of phase, the inertial part. Each one of the parts includes two terms additional to the loss and inertial terms of a flat surface (Kanazawa terms corresponding to the third terms in both parts): in the loss part a term associated with the well-known Stokes law and an additional term proportional to the increase of surface due to the roughness (ΔSr) and inversely proportional to the wave penetration depth (δ2) in the liquid; in the inertial part the term corresponding to the increase of surface due to the roughness (ΔSr), which represents the extra-mass of liquid displaced by the penetration of the shear wave, generated by the oscillating movement of the extra-surface (ΔSr), into the liquid, and the term corresponding to the mass of liquid displaced by the oscillating movement of the volume of roughness (Vr). This simplified model ignores much of the complexity of a random rough surface but provides a mathematical description which permits a simpler physical interpretation of the different contributions of the roughness on the sensor response. This model does not take into account the effects on the velocity field created by one shell due to the surrounding shells in their vibratory movement, therefore Eq. (14.43) can not be considered valid for a rough surface of high density of shells with high interaction among them. A more rigorous model could be developed by solving the Navier-Stokes equation with harmonic boundary conditions; however, the physical interpretation that implicitly exists under the simplicity of Eq. (14.43) could be still valid if “effective” geometrical parameters, instead of the exact geometrical ones, are considered in the model. These

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effective parameters: effective surface increase due to the roughness and the effective average height of the inhomogeneities should be dependent on the distance among shells and on the wave penetration depth in the liquid. The bigger the shells in relation to the wave penetration depth and the more separated they are, the more similar the “effective” parameters in relation to the real ones are.

14.5 Case Studies 14.5.1 Case Study I: Piezoelectric Inmunosensor for the Pesticide Carbaril

In 1980 Konash and Bastiaans showed that it was possible to maintain the stability of a quartz crystal controlled oscillator in contact with a liquid media [50], this work opened the possibility of using the quartz sensor in detection processes which develop better in in-liquid phase than in gas. Among the several applications of the quartz sensor in liquid media, the piezoelectric biosensor is one of those that more expectation has created. As it was stated in Chap.12 piezoelectric biosensors are becoming into a good alternative tool for bioanalytical assays and characterization of affinity interactions of biomolecules. The QCM may give a direct response signal characterizing the binding event between a sensitive biological layer and the analyte to be detected. The possibility of having a direct response is important if it is compared with other chemical techniques such as ELISA, where several steps, which are time consuming, are necessary to obtain an optimal signal [51]. However, in the detection of small biomolecules, such as antigens, it is quite difficult to achieve the same sensitivity as in the ELISA techniques when working with classical QCM built on 5 to 10 MHz quartz resonators, being in some cases quite difficult to obtain an observable direct signal without intermediate steps of amplification. In order to maintain the important advantage of having a direct response with sensitivities comparable to ELISA techniques is necessary to work with higher frequency quartz resonators [51]. Piezoelectric inmunosensors (a definition of the term inmunosensor can be found in Sect. 12.1 of Chap.12) have been employed in different fields such as: food, medical analysis, environmental monitoring, etc. In particular, they have been proposed for the monitoring of organic pollutants such as pesticides. In this section, the application of a piezoelectric inmunosensor for the monitoring of pesticide Carbaril, introduced in Sect.12.5 of Chap.12, will be used to show a real application which can be interpreted as the simple

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case “one thin rigid layer contacting a semi-infinite medium” introduced in Sect.14.3.4. Model

The quartz crystal, with its surface chemically modified to work as a biological interface (see Chaps. 11 and 12), is placed in a flow cell where it is put in contact with an aqueous solution that contains a biomolecular complex (the details about the format of the immunoassay for the Carbaril can be found in Sect. 12.5). In the three layer model of Fig.14.1 the sensitive layer corresponds to the biological interface created on the surface of the quartz, which can be considered as a thin rigid layer; the validity of this consideration will be verified later on. Regarding the semiinfinite medium in Fig.14.1, it corresponds to the aqueous solution that contains the biomolecular complex. Because the properties of this contacting medium are similar to water, it can be considered as Newtonian. The simple case “one thin rigid layer contacting a semi-infinite medium” is generally used in biosensor applications. In this case, the shift in the resonant frequency from the unperturbed state is given by the Martin’s equation, which is an extension of the Sauerbrey equation for the case of a quartz crystal with a single, thin and rigid layer working in a Newtonian liquid. The expression of this equation (Eq. 3.A.43) is rewritten below for practical reasons:

Δf s = −

2 f s2 nZ cq

⎛ ⎜ ρ s+ ⎜ ⎝

ρ2 η2 4π f s

⎞ ⎟ ⎟ ⎠

(14.45)

In biosensor applications it is very usual to assume that the properties of the Newtonian medium do not change during the experiment, this can be checked through the motional resistance Rm throughout the different steps of the experiment. When the properties of the second medium do not change and the Sauerbrey-like behavior of the sensitive layer is fulfilled, Rm is almost constant. In these conditions it can be assumed that the second term in brackets in Eq. (14.45) does not change during the experiment, and therefore the changes in the resonant frequency are due only to a change in the mass of the first layer. This change in the mass can be measured through the first term in brackets in Eq. (14.45), which corresponds to the Sauerbrey Equation.

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Experimental Methodology

The flow cell that contains the sensor is included in a flow-through system used to manage the different steps of the process of the inmunosensor (see Fig. 12.9, and for more details about the flow-through system see Sect.12.5). The electrical parameters of the compound resonator are measured with a Research Quartz Crystal Microbalance (RQCM) from Maxtec Inc., which provides the resonant frequency fs and the motional resistance Rm of the compound resonator (see Fig. 12.10). Since this real application can be interpreted as the simple case “one rigid thin layer contacting a semiinfinite medium”, the measurement of fs and Rm are enough to accomplish the data analysis and the biological interpretation in this case. Therefore a well-designed oscillator-like circuit as can be the RQCM is the most appropriate measurement system in this application (see Sect. 14.2.2). As it was explained in Sect. 14.3.1, in order to use the relationship between ∆fs and ρs given by the Sauerbrey Equation, it is important to calibrate the mass sensitivity of the piezoelectric transducer in order to corroborate the Sauerbrey predictions. This calibration procedure will be introduced in the next section. Calibration of the piezoelectric transducer

The problems of the sensitivity of the QCM have been dealt with air [5254] and with liquid [52, 55, 56]. These works show that the boundary conditions employed for the derivation of Sauerbrey Equation are not always fulfilled [57, 58]. In 1991 Gabrielli et al. presented a work in which introduced a procedure for the calibration of the QCM in the case that ρs were uniformly spread on the active electrode surface, this is the case of biosensor applications; and also for localized mass changes, this is very important for the study of some electrochemical phenomena which produce a change of mass on a small area, e.g., in localized corrosion or gaseous bubble evolution [56]. In the case of uniformly spread masses on the active electrode, Gabrielli et al. concluded that the lineal relationship between frequency and mass density predicted by the Sauerbrey equation is valid provided the Sauerbrey-like behavior is fulfilled; however, the theoretical value of the sensitivity factor is only reached in practice for active surfaces of sufficient area in comparison to the size of the quartz crystal. Therefore, a calibration procedure of the mass sensitivity is necessary in order to corroborate the Sauerbrey predictions. The calibration procedure proposed by Gabrielli et al. was performed by means of a galvanostatic metal deposition on a range of current densities

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[51, 56]. By following the QCM frequency evolution during the reaction and using the Faraday law, ∆fs / ρs is determined with good accuracy. The electrodeposition current is carefully controlled and therefore the added mass is calculated assuming the efficiency reaction is almost of 100% (see appendix 13.A). In these conditions the surface mass density (ρs / g cm-2) of the deposited metal is obtained by the following expression (see Eq. 13.A.3):

ρs =

M Qc nF

(14.46)

where M is the molecular weight of the deposited metal, n the number of electrons interchanged in the reaction, F is the Faraday constant and Qc is the charge involved in the process at a constant current I during the time t (Qc = I t). Then, the experimental sensitivity could be calculated from Qc and ∆fs by combining the Sauerbrey equation (Eq. 14.15) and Eq. (14.46): Cs =

nF Δf s M Qc

(14.47)

For a 9 MHz AT-cut quartz crystal of 14 mm in diameter and with an active area S = 0.2 cm2 (such as the crystals used in the Carbaril experiment) the experimental value of the sensitivity factor obtained by Bizet et al by means a galvanostatic cupper deposition reaches the following magnitude: 16.31±0.32 Hz g-1 cm2 ×107 [51]. On the other hand, the theoretical sensitivity calculated through the Sauerbrey equation (see Eq. 14.48) is 18.3 Hz g-1 cm2 ×107. Cs =

2 f s2 nS μρ

(14.48)

In this case, the theoretical and experimental values of the sensitivity factor are in good agreement for the dimensions of this crystal. Results and Discussion

Figure 14.11 shows the resonant frequency and motional resistance recorded along the time for the experiment performed for the pesticide Carbaril. The figure is a detailed version of the seventy first minutes of Fig. 12.11.

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8983160 8983140

t1

t2

t4

t3

8983120

(Hz) fsfs(Hz)

8983100 8983080 8983060 8983040 8983020 8983000 8982980 264 262

Rs (Ω)

Rm (Ω)

260 258 256 254 252 250 10

20

30

40

50

60

70

80

(minutes) t t(minutes)

Fig. 14.11. Resonant frequency and motional resistance recorded along the time for the experiment performed for the pesticide Carbaril. The figure is a detailed version of the seventy first minutes of Fig.12.11

During the first instants of the experiment (before instant t1) only the aqueous solution is in contact with the chemically modified surface of the quartz. At the instant t1 a biomolecular complex with the minimum concentration of Carbaril is injected in the aqueous solution, then a decrease in the resonant frequency with regard to the resonant frequency in t1 is recorded, while the motional resistance practically does not change (a variation between 1 or 2 ohms can not be considered significant). The fact that Rm be practically constant means two things, on one hand that the injection of the biomolecular complex does not change the properties of the aqueous solution and, on the other hand, that the sensitive layer can be considered like a rigid layer and therefore, its viscoelastic properties should not contribute to the sensor response. In these conditions the change in the surface mass density of the layer can be extracted from the shift in the resonant

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frequency through Martin’s equation (Eq. 3.A.43), and the interpretation is simple: a decrease in the resonant frequency, with regard to the resonant frequency measured in t1, means an increase in the surface mass density (see Sect. 14.3.1), which is due to the binding between the free antibody in the solution and the immobilized conjugate that constitutes the sensible interface. At the instant t2 an increase in the resonant frequency is recorded, this instant corresponds to the regeneration of the sensitive layer in order to be used in other assay. This increase in the resonant frequency means a decrease in the surface mass density of the sensitive layer which is due to the regeneration of the sensitive layer is accomplished by breaking the antibody-analyte association. In this regeneration process part of the original sensitive layer is taken out, this is the reason for which the frequency in t3 is higher than in t1. In the following instants t3, t4… the same process as the one described above is repeated with higher concentrations of Carbaril. The higher the concentration of the pesticide the lower the frequency shift, this is typical in a competitive assay like the one described (see Chap.12 for more details of the competitive assays). 14.5.2 Case Study II: Microrheological Study of the Aqueous Sol-Gel Process in the Silica-Metalisicate System

The sol-gel process is a wet-chemical technique for the fabrication of materials starting from a chemical solution containing particles of a diameter of few hundred of nm suspended in a liquid phase (sol). When the sol phase undergoes hydrolysis and poly-condensation reactions, the sol evolves towards the formation of an inorganic network containing a liquid phase (gel). Drying the gel by means of low temperature treatments (25ºC-100ºC), it is possible to obtain amorphous ceramic precursors. The sol-gel approach is interesting in that it is a cheap and low-temperature technique that allows synthesizing ceramic materials of high purity and homogeneity by means of preparation techniques different from the traditional process of fusion of oxides. Sol-derived materials have diverse applications in optics, electronics, (bio) sensors, medicine, etc. The homogeneity and reactivity of the amorphous precursor, which depend on the control of the initial steps in the transformation of a sol into a gel, determine the crystallization temperature and formation of desirable phases during the calcinations step in the ceramic material preparation. Thus, a better understanding of the gelation kinetics is essential in relating

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the properties of the produced ceramic to the starting sol-gel process [6066]. Rheometry constitutes a macroscopic way of investigating gelling kinetics [66-69]. In this sense, the quartz sensor can be used as a rheometer which, in contrast to the conventional rheometers, has the capability of inducing perturbation with very small amplitudes comparable with the typical size of the microstructures of the system. In this section, the application of an Electrochemical Quartz Crystal Microbalance (EQCM) is used to the study the sol-gel process of silica gelation from an aqueous sodium metasilicate solution at 25ºC and pH 3 [59]. From the EQCM data it is possible to obtain information on the changing rheological properties of the system during the whole process. Model

The silica gelation process takes place in three stages [59, 70, 71]: 1. Polymerization of monosilicic acid (Si(OH)4) to form amorphous SiO2 spherical particles. 2. Growth of the particles. 3. Particle aggregation. The sol formation comprises stages 1) and 2) and it is achieved by a convenient choice of pH and SiO2 concentration of the metasilicate solution. The growth of the particles leads to the particle aggregation that forms progressively three-dimensional network regions in the sol. These micro gel regions grow at the expense of the sol regions until they link together to form the complete gel network. Additionally, in the first moments of the third stage a thin rigid film formation on the electrode happens [59]. In all the process described, the formation of the gel network (third stage) is the most interesting to be investigated. The process that happens during this stage can be modeled through the three layer compound resonator of Fig. 14.1, where the first layer corresponds to the thin rigid film deposited over the electrode, and the semi-infinite medium corresponds to the solution where the gelation process is happening. As it will be seen later, the semi-infinite medium change from Newtonian (sol state) to viscoelastic (gel state). Thus, the gelation process can be interpreted as the simple case “one thin rigid layer contacting a semiinfinite viscoelastic medium” (see Sect. 14.3.4). In this case, the expression of the mechanical impedance ZL is the next (see Eqs. 14.25 and 14.26): (14.49) Z L = jωρ1 h1 + ρ 2 G 2

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where ρ1 and h1 is the volumetric mass density and thickness of the first thin rigid layer, respectively, and ρ2 and G2 = G’2 + j G”2 are the volumetric density and shear modulus of the semi-infinite medium, respectively. G”2 can be expressed in terms of viscosity as G”2 = ωη2. Equation (14.49) is an extension of Martin’s equation for the case of viscoelastic semiinfinite second media. In this application the quartz sensor will be used to monitor the small mass changes as well as changes in the viscoelastic properties of the contacting fluid medium during the gelation process. To accomplish that, the measurement of the resonant frequency and the motional resistance during the process is enough. The theoretical expressions of these magnitudes can be obtained from Eqs. (14.9a), (14.11) and (14.49). Experimental Methodology

The kinetics of the gelation process is monitored by means of an EQCM containing a 6MHz polish AT-cut quartz crystal, driven by an oscillator which permits monitoring changes of both the resonant frequency and the motional resistance during the process. More details about the chemicals and the set up of the experiment can be found elsewhere [59]. Results and Discussion

Changes in the resonant frequency and motional resistance shifts, relative to air, during the gelation process (third stage) are shown in Fig.14.12 [59]. A thin rigid film deposition is clearly identified in the first moments of the gelation process (first 11 minutes), since a frequency change without any change in the resistance is measured (see the Sauerbrey-like behavior concept in Sect. 14.3.1). Some of the SiO2 particles in random motion which are near the electrode surface eventually interact with the electrode and adhere to it. Once the surface is totally covered, the particle adhesion ceases and no additional mass is deposited over the sensor. The mass of the film can be extracted from the Sauerbrey equation through the frequency shift measured from the instant t = 0, which corresponds to the beginning of the gelation process. The extracted mass as a function of time is shown in Fig. 14.13 [59]. Film mass deposition obeys an exponential law with saturation approximately from 400 s [59]. Using a model for film formation kinetics and the information of the deposited mass, the sol particle size, assuming spherical particles, can be estimated. This is of importance for the analysis of the sol viscosity behaviors during the first instants of the gelation process (see reference [59] for more details).

375

∆Rm (Ω)

∆fs (KHz)

14 QCM Data Analysis and Interpretation

t (103 s)

Fig. 14.12. EQCM resonant frequency shift relative to air as a function of time (upper panel) and resonant resistance shift relative to air as a function of time (lower panel) [59]

When no changes in the first layer take place, the shifts in the resonant frequency and resistance measured must be attributed to the changes in the properties of the semi-infinite contacting solution. Assuming no changes in its density only two unknowns remain: G’2 and G”2 (or the viscosity η2), which can be extracted from the two experimental data: the resonant frequency and the motional resistance (see Sect. 14.3.3).

Yolanda Jiménez, Marcelo Otero and Antonio Arnau

mf (μg)

376

t (s) Fig. 14.13. Mass deposited on the electrode surface as a function of time [59]

Figure 14.14 shows the viscosity η2 and the shear storage modulus G’2, as a function of time [59]. Unlike other experiments where large-amplitude motions are applied over the three-dimensional gel network, the viscosity does not change very much during the whole gelation process (see Fig. 14.14 upper panel). The reason is that the shear motion with small amplitude generated by the quartz crystal, dissipate energy essentially in the liquid retained by the elastic gel network, unlike the large amplitude motion generated by other rheometers, in which the network structure is affected by the movement as well, resulting in an increase of the dissipation of the system. The localization of the gelation point of the process from the viscosity data is difficult [71], being necessary the analysis of the shear storage modulus data to achieve it. At the first instants of the third stage, (a) in Fig. 14.14 lower panel, before aggregation becomes significant, G’2 ≈ 0, therefore the sol in contact with the sensor can be considered as a Newtonian dispersion of particles. As the particles collide and form aggregates the model of a dispersion of particles losses its validity, and when the network is completely formed other parameters, such as elasticity of the network, should be taken into account to better describe the rheology of the system. The abrupt rise of G’2 observed in Fig. 14.14 (lower panel) about 1600 s, discontinuity between points (a) and (b), indicates the gelation point, since a significant elastic response to stress appears when the last links are formed among large clusters to create a continuous 3D network extending throughout the system [59].

377

G' (KPa)

η (mPa·s)

14 QCM Data Analysis and Interpretation

t (103 s)

Fig. 14.14. Fluid viscosity (upper panel) and shear storage modulus (lower panel) as a function of time[59]

Finally, Fig. 14.15 shows the evolution of the loss tangent G”2/G’2. The decrease observed in the loss tangent suggests a process of bond formation in the network gel until t ≈ 5000s, where no further changes in the loss tangent are observed; it is an indication that the 3D network formation came to completion.

Yolanda Jiménez, Marcelo Otero and Antonio Arnau

Loss Tangent

378

t (103 s)

Fig. 14.15. Time evolution of the loss tangent (G’’2/G’2) in the gel phase [66]

14.5.3 Case Study III: Viscoelastic Characterization of Electrochemically prepared Conducting Polymer Films

Polythiophenes are high conductive polymers with high chemical and thermal stability. Due to its high electrical conductivity polythiophenes find interesting uses in different applications such as: patterned circuits (plastic circuits), photodiodes [72], biosensors [73], antistatic coatings [74], corrosion protection coatings [75], etc. However, they have not been extensively used due to the changes of their properties which have been observed with the course of time, with the use or when entering in contact with the environment. Moreover, relatively little is known about the mechanical properties, such us shear moduli, of these polymer layers whose thicknesses can range from some nanometers to micrometers depending on applications. The Quartz Crystal Resonator (QCR) technique can be useful for estimating these properties and their change during the polymer growth. Furthermore, other effects regarding the macroscopic structure of the polymeric coating can contribute to the sensor response and be analyzed; in particular the roughness effect, intimately related to the porosity and with the hydration grade. Recently, it has been shown that the electrochemical polymerization of electrically conducting polymers such as thiophenes on the gold electrode of an EQCM produces porous and fibrous surfaces whose degree of porosity

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depends on the polymer, on the technique used for the electropolymerization, the speed of the polymer growth, etc [7, 41, 44, 76]. In this practical example an Electrochemical Quartz Crystal Microbalance is used for a continuous monitoring of the growth of the polymer poly(3,4-ethylenedioxy) thiophene tetrabutylammonium perchlorate (PEDOT-TBAP), electro-polymerized in acetonitrile on a gold electrode of a 10 MHz AT-cut quartz crystal resonator. As the film thickness increases, two regimes for the same conducting polymer/electrolyte system are observed: gravimetric regime and viscoelastic regime. The mechanical properties of the coating, as well as the effect that the surface roughness has over them will be studied for both regimes. Model

In general, the studies on conductive polymers carried out through an EQCM do not include models accounting for the contribution of the polymer roughness on the sensor response. Thus, the EQCM layer-model used corresponds to the three-layer model shown in Fig. 14.1, where the polymer layer, the so-called coating, is assumed to have an effective uniform thickness h1|ef, with effective viscoelastic properties G’1|ef and G’’1|ef , storage and loss shear moduli respectively. Therefore, the threelayer model is used to represent a situation which can be modeled in a more real way as shown in Fig. 14.9, where a rough layer is included in a four-layer model where the coating layer is divided in two: one uniform layer at the bottom with uniform thickness h1 and viscoelastic properties G’1 and G’’1 and a rough layer on top with characteristic parameters of the roughness ξ (magnitude relative to the porosity) and Lr (thickness of the rough layer). Then, the physical properties of the polymer layer extracted when considering the three-layer model should be considered as effective properties which include additional effects such as roughness. In the context of Fig.14.1, when no-approximation can be done about the layer, i.e., when it is not reasonable to consider a Sauerbrey-like behavior of the coating, or the “small surface load impedance condition” can not be applied, the appropriate admittance model for characterizing the electrical response of the sensor turns into the TLM in Eq. (14.1). In these conditions the extraction of the coating properties is not an easy task. The problems associated with this extraction process, together with the different strategies proposed to face that problem, were described in Sect. 14.4. In this practical example three different strategies have been used: 1. Restricting the solutions by measuring the thickness by Ellipsometry (see Sect. 14.4.2)

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2. Restricting the solutions by measuring the thickness through the Faraday Law (see Sect. 14.4.2) 3. Restricting the solutions by measuring the admittance response of the sensor in the range of frequencies around resonance (see Sect. 14.4.3) The contribution of the surface roughness on the acoustic load impedance response is analyzed: for the gravimetric regime, through the model introduced by Arnau et al. [48, 49] (see Sect. 14.4.5); and for the viscoelastic regime through the Daikhin and Etchenique models derived from the Brickman’s equation [40, 41] (see Sect. 14.4.5). Experimental Methodology

The growing of the PEDOT polymer is accomplished by means of a threeelectrode electrochemical cell containing 10MHz polish AT-cut quartz crystals. The admittance spectra were acquired dynamically during EDOT electropolymerization with an HP5100A network analyzer. A SENTECH SE 400 variable angle rotating analyzer type ellipsometer equipped with a He-Ne laser (λ=632.8 nm) was used for the ellipsometric experiments. Scanning electron microscopy (SEM) investigations were also performed with a Philips XL30 CP SEM. More details about the chemicals and set-up of the experiment can be found elsewhere [46]. Results and Discussion

As mentioned in Sect. 14.2.2, it is necessary to make an appropriate calibration of the resonator for having an accurate evaluation of the acoustic load impedance. In this sense, admittance spectra measurements of the uncoated crystal in air and in contact with the electrolyte solution around resonance were used for sensor calibration. The sensor calibration parameters were obtained through a purpose developed algorithm described elsewhere [30, 31] and found to be: hqef = 166.315 μm, C0ef = 4.66 pF, η 0ef = 0.1948 Pa·s and Cex = 6.93 pF. Once the calibration parameters have been obtained, the acoustic load impedance at the frequency ω1 corresponding to the conductance peak of each acquired conductance spectrum can be obtained from the TLM. The real and imaginary parts, RL and XL, along with the frequency shift between the conductance peaks of the coated sensor in contact with the medium and the uncoated sensor, are represented in Fig. 14.16 for characterizing the electropolymerization process; as can be observed the frequency shift, Δf, follows nearly the opposite evolution to the imaginary part of the acoustic

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load impedance, as can be understood through the acoustic load approximation expression: X L ≈ −k Δf , where k ≈ 3 for a 10 MHz AT-cut QCR (see Sect. 14.2.2). Figure 14.16 shows the coating pass through different acoustic regimes: gravimetric or Sauerbrey-like regime and viscoelastic regime. 120

0

Film resonance

-10

80 60

XL

Gravimetric regime

RL

-20

Δf

40

-30

Δf (KHz)

RL, XL x 103 (N s m-3)

100

20 -40 0

Viscoelastic regime

0

200

400

600

800

1000

-50 1200

Q (mC cm-2)

Fig. 14.16. Evolution of the real and imaginary parts RL and XL of the acoustic load impedance, ZL, at angular frequencies ω1 near the maximum conductance [46]

Gravimetric Regime The sensor response is characterized in the initial times for an important increase of XL (decrease of Δf) in comparison with the increase of RL. This behavior could correspond, in principle, to a gravimetric regime where the frequency shifts could be directly related to the polymer mass through the Sauerbrey equation; however, some considerations should be done and carefully considered. The acoustic gravimetric regime occurs when the viscoelastic properties of the coating are not reflected on the sensor response due to the small thickness of the layer, in this case it is stated that the layer has a Sauerbrey-like behavior (see Sect. 14.3.1), and the way to validate it is to check that the changes in the resistance shift are negligible. However, in some cases one can obtain resistance changes due to contributions different from viscoelastic effects, and the sensor can follow working in the gravimetric

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regime, even with small changes in the resistance. This can happens, for instance, with a rough surface coating. Figure 14.17 shows a detailed representation of the initial times (Q<50 mC cm-2), where one could assume gravimetric regime. The surface mass density, mSB, obtained with the Sauerbrey equation starting from the corresponding frequency shifts is represented in the upper panel along with the theoretical mass density, mp = mFAR obtained from Faraday law assuming the efficiency to be 1 (see Eq. 13.A.3): mFAR (g cm-2) = 0.0737×10-5 Q (mC cm-2)

(14.50)

where Q is the charge density involved in the electropolymerization process.

-2) mS /(μg Δm μg cm-2

50.0 40.0

mSB

30.0 20.0

mpp = mFARADAY FAR

10.0

ΔR ΔRmm(Ω) /Ω

0.0 50.0 40.0 30.0 20.0 10.0 0.0 0

10

20

30

40

50

-2-2 QQ(mC ) / mCcm cm

Fig. 14.17. Detail of the evolution of the mass density corresponding to the Sauerbrey equation and of the shift in the reciprocal of the maximum conductance (motional resistance shift) up to 50 mC·cm-2 charge density [46]

The resistance shifts taken as the difference in the reciprocal of maximum conductance magnitudes between the coated and uncoated device in contact with the liquid are represented in the lower panel. According to Fig. 14.17, the sensor does not detect mass during the first seconds of the electro-polymerization process, and when the mass is detected the slopes of the mass gain are found to be around a factor of two bigger than those predicted by the Faraday law. A later decrease of the mass gain slope is predicted by the Sauerbrey equation; from 30 to

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50 mC·cm-2 the Sauerbrey mass has a lineal evolution with a slope very close to that of the faradic mass. Additionally a small, but significant, change in the resistance shift is measured during the first instants of the experiment. The absence of mass detection during the first seconds of the electropolymerization process can be due to the short chain oligomers formed in a precursor step, which diffuse away from the electrode and result in electrical charge lost into solution with no mass gain contribution. At longer times, oligomers of sufficient length are produced and precipitate onto the electrode surface giving rise to nucleation phenomena [77], which are reflected on the sensor response. The precipitation of the first oligomers onto the quartz surface could explain the greater slopes in the mass gain predicted by the Sauerbrey equation in comparison with the theoretical one derived from Faraday law; however additional effects remain since an extra-mass is predicted by the Sauerbrey equation (Fig. 14.17). If the nucleation phenomenon previously described leads to a nonhomogeneous deposition, where small polymer nuclei were spread on the surface leaving voids filled with the liquid, the interface with the liquid is not flat and extra contributions additional to the inertial contribution of the deposited mass exist [33]. The main contribution is inertial due to the extra volume of liquid displaced by the roughness and the sensor response could be mainly dominated by the imaginary part XL, but also a small effect on the real part RL could arise. This can explain the changes observed in the motional resistance, as well as the extra mass predicted by the Sauerbrey equation in Fig. 14.17. The simplicity of the equation derived by Arnau for the acoustic load impedance (Eq. 14.43) permits to separate roughness and coating mass effects in the gravimetric regime, and to obtain the characteristic parameters of this roughness model (hr, rr) and the thickness of the uniform coating layer hlf . According to Eq. (14.43) the effect of the roughness has additive contributions in both real and imaginary parts of Z2 in relation to the corresponding parts of the acoustic impedance of a flat surface (Kanazawa terms). In the gravimetric regime the viscoelastic contributions are negligible, therefore the changes in the real part will be due to the roughness and the changes in the imaginary part will be due (see Fig. 14.10) to the surface mass density of the coating, mp, that will be the addition of the surface mass density of the rough layer, mrp, and the surface mass density of the uniform coating layer, mlp, formed at the bottom with a thickness hlp (mlp = ρp hlp, where ρp is the density of the polymeric coating layer). As it was commented in appendix 13.A, in a galvanostatic electrochemical deposition

384

Yolanda Jiménez, Marcelo Otero and Antonio Arnau

the coating mass density mp = mrp+mlp can be estimated from the charge and Faraday law (see Eq. 13.A.3). The surface mass density of the rough layer mrp, from the polymer density and the volume of one shell, is given by: mrp = ρ pVr = ρ p nπhr (hr2 + 3rr2 ) / 6

(14.51)

If there was no liquid in contact with the coating, the surface mass density, mSB, obtained through the Sauerbrey equation from the corresponding resonant frequency shift, will be practically the same than that obtained through the Faraday law. However, the presence of the liquid (electrolyte) makes that an additional contribution in the surface mass effect due to roughness appears, Δmr. This effect can be obtained from the imaginary part of Eq. (14.43) as follows:

Δm r =

⎞ 3 1 1 ρ 2 ⎛⎜ 9δ 2 hr ⎟ m rp + 1 ρ 2δ 2 ΔS r + ρ 2 Vr = 2 2 ⎟ 4 2 2 ρ p ⎜⎝ hr + 3rr ⎠

(14.52)

and then mSB = mp + Δmr. With regard to the characteristic parameters of the roughness model (hr, rr) and the thickness of the uniform coating layer, the corresponding procedure and resulting equation can be found in appendix 14.A and reference [46]. The results are represented in Fig. 14.18 in which the evolution of the characteristic parameters of the roughness, rr and hr are shown, along with the thickness of the uniform bottom layer hlp. 250 r r r h hlp r

Dimensions (nm) Dimensions (nm)

200

hlp hrp

150 100 50 0 12

18

24

30

36

42

48

2 (mC/cm )) QQ(mC cm-2

Fig.14.18. Evolution of the roughness dimensions according to the spherical shell roughness model depicted in Fig.14.10 for the experimental data given in Fig.14.17 [46]

14 QCM Data Analysis and Interpretation

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The evolution of the roughness parameters indicates that in the beginning the coating is formed by a small thickness uniform layer with relatively big mountains on top (rr ≈ 200 nm, hr ≈ 100 nm and hlp ≈ 50 nm). As the coating grows the uniform bottom layer is getting thicker and the parameters of the rough layer, both rr and hr, decrease making the polymer layer more compact (rr ≈ 45 nm, hr ≈ 35 nm and hlp ≈ 227 nm); the effect of the roughness decreases as the polymer grows but maintains a non-negligible effect in the charge density interval considered. As the polymer coating is getting thicker, its viscoelastic properties influence the sensor response and both roughness and viscoelastic contributions exist. The analysis of the viscoelastic properties of the coating and the effect of the roughness during viscoelastic acoustic regime is more complex and this is performed and discussed in two steps: the effective coating properties are extracted without considering the roughness effect; and second, the effect of roughness on the effective viscoelastic properties of the coating is evaluated. Viscoelastic regime As the time increases (t >235 s) the acoustic load impedance shows a further increase of XL (a further decrease of Δf) and an important increase of RL, it is to say a decrease of the conductance peak (Fig. 14.16). It means that the coating is getting thicker and the contribution of its viscoelastic properties on the sensor response is not negligible. Figure 14.16 shows that during the viscoelastic regime, the acoustic load resistance reaches a maximum after around 500 mC cm-2 and then decreases while the resonant frequency shift shows a minimum with a later frequency increase. This behavior is similar to that of a film mechanical resonance1. In order to clarify the origin of this behavior it is necessary the extraction of the coating properties along with a further analysis during the electropolymerization process. The extraction process is made through an algorithm developed by Jiménez et al. [30, 31]. This algorithm permits an unambiguous extraction of three properties of the coating (the surface mass density, ms = ρ1h1, the 1/ 2 magnitude of the characteristic acoustic impedance, Z 1c = (ρ1 G1 ) and the quality factor, Q1 = 1/tan δ1 = G’1/G’’1) under theoretical conditions, starting from the admittance spectrum of the sensor around resonance. This algorithm provides, in a first stage, a set of possible triads of coating properties which are a solution of the problem, these triads are found before the The main characteristic which defines the film mechanical resonance is that the wave phase shift across the film must be π/2, it is to say, the thickness of the film must be equal to a quarter of the acoustic wavelength [78].

1

386

Yolanda Jiménez, Marcelo Otero and Antonio Arnau

first film resonance of the coating and between the first and the second film resonance of the coating. Then, in a second stage, the appropriate triad is selected between all of the possible provided by the previous stage. For more information about the algorithm see references [30, 31]. In this practical example the selection of the appropriate triad is made by using the following strategies: 1. By using an independent measurement of the coating thickness provided by ellipsometry. In this case the algorithm select the triad whose mass density is the nearest to the mass derived from the ellipsometric thickness measured in an additional experiment. The ellipsometric thickness measured has been represented in Fig. 14.19. This thickness shows a linear growth between 30 and 45 mC cm-2 of applied charge density. The probable non-uniform deposition makes the measurements of the ellipsometric thickness not very reliable during the initial times. For charge densities greater than 50 mC·cm-2 the ellipsometric thickness cannot be measured due to a considerable light absorption at 632.8 nm. Values of ellipsometric thickness for higher charge densities can be extrapolated from Eq. (14.53) if one assumes a constant growth of the polymer: h1 (nm) = −64.6972 + 3.2423 Q

(14.53)

-2

Ellipsometric thickness (nm)

where Q is the charge density in mC·cm . 100 80 60 40 20 0 0

20

40

60

Q (mC/cm2)

Fig. 14.19. Measured ellipsometric thickness as a function of the charge density [46]

14 QCM Data Analysis and Interpretation

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2. By using an independent measurement of the coating thickness provided by means of the Faraday law. In this case the algorithm select the triad whose mass density was the nearest to the Faraday derived from Eq. (14.50). As in the previous case a constant growth of the polymer is assumed as well. 3. By using the additional data provided by the admittance response of the sensor in the range of frequencies around resonance. In this case, the algorithm select the triad which substituted in the TLM provides the better fitting to the experimental admittance response. In all the cases, the fitting error between the experimental conductance spectrum and the one derived from the TLM for the optimum triad made unfeasible the application of the fitting restriction; the experimental and theoretical spectra only matched near the maximum conductance and the level of matching decreased as one went near the wings. For that reason, a nonmass-loss restriction where the physical condition that the coating can not lose mass was imposed to the algorithm for the selection of the triad. In strategies 1) and 2) it is assumed a constant growth of the polymer during all the process, this restrictive assumption is relaxed in the third strategy where the only restriction made is to assume that during the electropolymerization process the coating does not lose mass. The results for one example of strategy that assumes constant growth of the polymer, ellipsometry mass restriction in this case, and non-mass-loss restriction are shown in Figures 14.20 and 14.21, respectively. The results for the Faraday mass restriction can be found in references [30, 46]. Black symbols (square and circular) represented in the figures indicate that the triad whose mass density is the nearest to the mass density of the corresponding restriction was found before the first film resonance and white symbols indicate that it was found between the first and second film resonance. Fig.14.20 shows the properties extracted from the ellipsometric mass restriction. For charge densities smaller than 160 mC·cm-2 there were not triads whose mass densities perfectly matched the ellipsometric mass densities. This is a very important fact because it indicates that in this experiment the ellipsometry mass disagrees with the acoustic mass density; it is a well-known fact that the thickness measured by ellipsometry underestimates the acoustic thickness [79, 80], and this disagreement was confirmed in the experiment. With regard to the effective viscoelastic properties extracted, the analysis showed an important dispersion in the values of the properties up to a charge density of 128 mC·cm-2, probably due to the proximity to the gravimetric regime; in this range we should be more

388

Yolanda Jiménez, Marcelo Otero and Antonio Arnau

confident with the obtained mass densities than with the extracted viscoelastic properties. A sharp decrease in the loss shear modulus near the resonance range and a monotonous increase of the shear storage modulus are observed. The behavior of the shear storage modulus seems to be linked to the restriction imposed to the algorithm for the selection of the triad which better matched the ellipsometric mass density given by Eq. 14.53. This is better understood by comparing with the results of the non-mass-loss restrictions. On the other hand the pass through the first film resonance was obtained by the algorithm where it was expected according to Fig. 14.16. -3

Triads (ρ1=1500 kg m ) + ellipsometry restriction

1000

ms Ellipsometry

with ρ1=1500 kg m-3

ms (μg cm-2)

800

ms Faraday Before 1st resonance st After 1 resonance

600

Film resonance

400

200

0 12

st

G'1 Before 1 resonance st

G'1 After 1 resonance

10

st

G''1 Before 1 resonance st

G''1 After 1 resonance

G1 (MPa)

8 Film resonance

6 4 2

0.4 MPa

0

0

200

400

600

800

1000

1200

-2

Q (mC cm )

Fig. 14.20. Evolution of the effective coating properties extracted by the algorithm with the ellipsometric mass density restriction [46]

14 QCM Data Analysis and Interpretation

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Figure 14.21 shows the evolution of the effective properties of the coating when the non-mass-loss restriction was applied. As it can be observed in the gravimetric regime the non-mass-loss restriction predicts an overestimation in the mass density in relation to the ellipsometric and Faraday mass restriction. The overestimation in the case of ellipsometric mass has been discussed above; with regard to the overestimation in the case of Faraday mass, has already been explained as a consequence of the important contribution of the roughness in the initial times.

1000

Triads (ρ1=1500 kg m-3) + mass density increase restriction ms Faraday Before 1st resonance After 1st resonance ms Ellipsometry

-2

ms (μg cm )

800

with ρ1=1500 kg m-3

600

400

200

G1 (MPa)

0 10

G'1 Before 1st resonance

8

G''1 Before 1st resonance

G'1 After 1st resonance G''1 After 1st resonance

6 4 2 0

0

200

400

600

800

1000

1200

-2

Q (mC cm )

Fig. 14.21. Evolution of the effective coating properties extracted by the algorithm with the non-mass-loss restriction [46]

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Yolanda Jiménez, Marcelo Otero and Antonio Arnau

After the film resonance the algorithm followed finding the best triad after the first film resonance but, in contrast to ellipsometric mass restriction, without a significant increase in the mass density, and with a significant stability in the shear loss and shear storage modulus. If one accepts that saturation in the mass density can occur, the results obtained with the non-mass-loss restriction are coherent in the sense that one expects stable values for the components of the shear modulus once the coating has reached enough thickness and homogeneity. In these conditions the coating mass density does not show a significant increase when passing through the suspected resonance interval, then the behavior that suggest a film mechanical resonance seems to be more a consequence of the change in the viscoelastic properties of the polymer. A possible explanation of this change is that the film structure evolves during the film growth and one should expect the contribution of other factors on the effective viscoelastic properties extracted using the simplified model in Fig. 14.1. Among those factors the film roughness seems to be the principal in this case. Thus, in the beginning of the polymerization the coating is formed by polymer nuclei with big cavities filled of solvent, and then the viscoelasticity is dominated by the shear loss modulus since the viscosity of the solvent takes an important role. As the polymerization develops it seems to be coherent, according to the results of the SEM (Fig. 14.22), that the coating is getting more compact and the voids filled of solvent are replaced by polymer, then a reduction of the mass density gain could happen due to the replacement of the trapped solvent molecules in the voids by those of the polymer, additionally the role of the viscosity (shear loss modulus) diminishes at the same time as the rigidity of the coating arises. The same behavior as the one described above for the loss and storage shear modulus was observed in a simulation of the growth of a polymer with morphological changes similar to that observed in the SEM study. For the simulation, a four-layer model as the one depicted in Fig. 14.9 was used together with the Etchenique and Daikhin et Urbakh equations for the acoustic impedance of the rough layer in contact with the liquid. The effect of the roughness appears as a change in the effective properties of the coating, mainly in the effective viscoelastic properties. This change in the effective properties of the equivalent uniform coating layer, makes the coating layer to pass through the film resonance phenomenon [30, 46].

14 QCM Data Analysis and Interpretation

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Fig. 14.22. SEMs of PEDOT films prepared on Au electrode of a quartz taken at different charge densities: 12-20 mC cm-2 (upper left panel); 200 mC·cm-2 (lower left panel); 400 mC·cm-2 (upper right panel); and 900 mC cm-2 (lower right panel) [46]

Appendix 14.A: Obtaining of the Characteristic Parameters of the Roughness Model Developed by Arnau el al. in the Gravimetric Regime The expression of the equation for the acoustic load impedance of the roughness model proposed by Arnau et al. was presented in Sect. 14.4.5, and here is written again for practical reasons: ⎡ η 3 η2 1 1 ⎛3 ⎞⎤ ΔS r + 2 + jω ⎜ ρ 2δ 2 ΔS r + ρ 2 Vr + ρ 2δ 2 ⎟⎥ Z 2 = ⎢3π η 2 (nhr ) + δ δ 2 4 2 2 ⎝ ⎠⎦ 2 2 ⎣

(14.A.1)

The surface mass effect associated with the roughness, Δmr, can be obtained from the imaginary part of Eq. (14.A.1) as follows: Δm r =

⎞ 3 1 1 ρ 2 ⎛⎜ 9δ 2 hr ρ 2 δ 2 ΔS r + ρ 2 V r = + 1⎟ m rp 2 2 ⎜ ⎟ 4 2 2 ρ p ⎝ hr + 3rr ⎠

(14.A.2)

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Yolanda Jiménez, Marcelo Otero and Antonio Arnau

The surface mass density of the rough layer mrp, from the polymer density and the volume of one shell, is given by: m rp = ρ pVr = ρ p nπ hr (hr2 + 3rr2 ) / 6

(14.A.3)

In gravimetric regimes the surface mass density, mSB, obtained from the Sauerbrey equation ( Δf s coating = −C SB m SB ) will provide the additive contribution of the coating surface mass density and the surface mass effect per unit area due to the roughness as follows: m SB = m p + Δ m r

(14.A.4)

On the other hand, the shift in the real part of the acoustic load impedance, in relation to the value for a flat surface, can be express as a function of the surface mass density of the rough layer mrp as follows: ΔR 2

ω0

=

⎞ 9 ρ 2 ⎛ 2δ 22 + δ 2 hr 1 ⎛ 3 η2 ⎜ ⎜⎜ 3π η 2 (nhr ) + ΔS r ⎟⎟ = 2 ⎜ 2 2 δ2 ω0 ⎝ ⎠ 2 ρ p ⎝ hr + 3rr

⎞ ⎟m rp ⎟ ⎠

(14.A.5)

where ω0 is the resonant frequency of the compound resonator. From Eqs. (14.A.1) to (14.A.5) the characteristic parameters of the roughness hr and rr, and the thickness of the uniform coating layer, hlp, can be obtained as a function of ΔR2, mSB and mp, which can be obtained from the experimental magnitudes Y (admittance), Δfs (frequency shift) and I (current density of the galvanostatic experiment) respectively, as follows: hr = ⎛⎜ − b + b 2 + 4c ⎞⎟ / 2 ⎝ ⎠

(

rr = E hr 2δ 22 + δ 2 hr m rp = ρ p hlp =

(14.A.6)

)

(14.A.7)

π hr ⎛⎜ hr2 + 3rr2 ⎞⎟

2 2 3 rr ⎜⎝

1

ρp

6

(14.A.8)

⎟ ⎠

( m p − m rp )

(14.A.9)

where b and c have the following expressions: b=

6 Eδ 22 + 9δ 2 − 9 Dδ 2 18 Dδ 22 ; c= 3Eδ 2 + 1 3Eδ 2 + 1

(

)

with D = (ω 0 ΔR 2 )(m SB − m p ) and E = 9πρ 2 24 3 ω 0 ΔR2 .

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References 1. A.R. Hillman, A. Jackson and S.J. Martin (2001) “The problem of uniqueness of fit for viscoelastic films on thickness-shear mode resonator surfaces” Analytical Chemistry 73 (3): 540-549 2. C. Behling, R. Lucklum and P. Hauptmann (1997) “Possibilities and limitations in quantitative determination of polymer shear parameters by TSM resonators” Sensors and Actuators A 61: 260-266 3. E. J. Calvo, R. Etchenique, P.N. Barlett, K. Singhal and C. Santamaria (1997) “Quartz crystal impedance studies at 10MHz of viscoelastic liquids and films” Faraday Discuss 107: 141-157 4. H. L. Bandey, A. R. Hillmann, M. J. Brown and S. J. Martin (1997) “Viscoelastic characterization of electroactive polymer films at electrode/solution interface” Faraday Discuss 107: 105-121 5. R. Lucklum and P. Hauptmann (1997) “Determination of polymer shear modulus with quartz crystal resonators” Faraday Discuss 107: 123-140 6. R. Etchenique and A. Dan Weisz (1999) “Simultaneous determination of mechanical moduli and mass of thin layers using non-additive quartz crystal acoustic impedance analysis” Journal of Applied Physics 86 (4): 1994-2000 7. A. Bund, M. Schneider (2002) “Characterization of the viscoelasticity an the surface roughness of electrochemically prepared conducting polymer films by impedance measurements at quartz crystals” Journal of Electrochemical Society 149 (9): 331-339 8. O. Wolff, E. Seydel and D. Johannsmann (1997) “Viscoelastic properties of thin films studied with quartz crystal resonators” Faraday Discuss 107: 91104 9. A. Domack, O. Prucker, J. Rühe and D. Johannsmann (1997) “Swelling of a polymer brush probed with a quartz crystal resonator” The American physical society 56 (1): 680-689 10. D. Johannsmann, K. Mathauer, G. Wegner and W. Knoll (1992) “Viscoelastic properties of thin films probed with a quartz crystal resonator” Physical Review B 46 (12): 7808-7815 11. Y. Jiménez, T. Sogorb, A. Arnau, M. Otero and E. J. Calvo (2003) “Systematic error analysis in the determination of physical parameters of the coating in quartz crystal resonators sensors”, AWSWS IV, Salbris, Francia 12. V.E. Granstaff and S. J. Martin (1994) “Characterization of a thickness mode quartz resonator with multiple nonpiezoelectric layers” Journal of Applied Physics. 75 (3): 1319-1329 13. J. Schröder, R. Borngräber, F. Eichelbaum and P. Hauptmann (2002) “Advanced interface electronics and methods for QCM” Sensors and actuators A 97-98: 543-547 14. A. Arnau, T. Sogorb and Y. Jiménez (2000) “Thickness-shear mode quartz crystal resonators in viscoelastic fluid media” Journal of Applied Physics 88 (8): 4498-4506

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15. A. Arnau, Y. Jiménez and T. Sogorb (2001) “An extended butterworth-van dyke model for quartz crystal microbalance applications in viscoelastic fluid media” IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 48 (5): 1367-1382 16. A. Bund and G. Schwitzgebel (1998) “Viscoelastic properties of low-viscosity liquids studied with thickness-shear mode resonators” Analytical Chemistry 70: 2584-2588 17. J. Kankare (2002) “Sauerbrey equation of quartz crystal microbalance in liquid medium” Langmuir 18: 7092-7094 18. M. V. Voinova, M. Jonson and B. Kasemo (2002) “Missing mass effect in biosensor’s QCM applications” Biosensors and Bioelectronics 17: 835-841 19. R. Lucklum, C. Behling and P.Hauptmann (1999) “Role of mass accumulation and viscoelastic film properties for the response of acousticwave-based chemical sensors” Analytical Chemistry 71 (13): 2488-2496 20. R. Lucklum, C. Behling, P. Hauptmann, R. W. Cernosek and S. J. Martin (1998) “Error analysis of material parameter determination with quartz-crystal resonators” Sensors and Actuators A 66: 184-192 21. C. Behling, R. Lucklum and P. Hauptmann (1999) “Fast three-step method for shear moduli calculation from quartz crystal resonator measurements” IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control 46 (6): 1431-1438 22. R. W. Cernosek, S. J. Martin, A. R. Hillman and H. L. Bandey (1998) “Comparison of lumped-element and transmision-line models for thicknessshear-mode quartz resonator sensors” IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 45 (5): 1399-1407 23. R. M. A. Azzam and N. H. Bashara (1997) in “Ellipsometry and polarized light” North-Holland, Amsterdam 24. H. Arwin (2000) “Ellipsometry on thin organic layers of biological interest: characterization and applications” Thin Solid Films 48: 377-378 25. E. J. Calvo, E. Forzani and M. Otero (2002) “Study of layer-by-layer selfassembled viscoelastic films on thickness-shear mode resonator surfaces” Analytical Chemistry 74: 3281-3289 26. S. W. Lee, D. Hinsberg and K. Kanazawa (2002) “Determination of the viscoelastic properties of polymer films using a compensated phase-locked oscillator circuit” Analytical Chemistry 74: 125-131 27. F. Höök, J. Vörös, M. Rodahl, R. Kurrat, P. Böni, J. J. Ramsden, M. Textor, N. D. Spencer, P. Tengvall, J. Glod and B. Kasemo (2002) “A comparative study of protein adsorption on titanium oxide surfaces using in situ ellipsometry, optical waveguide light-mode spectroscopy, and quartz crystal microbalance/dissipation” Colloids and Surfaces B: Biointerfaces 24: 155-179 28. M. Otero (2003) “Construcción y caracterización de estructuras complejas de biomoléculas con aplicación en el diseño de biosensores” Tesis Doctoral, Universidad de Buenos Aires 29. F. Ferrante, A. L. Kipling and M. Thompson (1994) “Molecular slip at the solid-liquid interface o fan acoustic wave sensor” Journal of Applied Physics 76 (6): 3448-3462

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30. Y. Jiménez (2004) “Contribución a la resolución de la problemática asociada a la medida de las propiedades físicas de recubrimientos viscoelásticos en sensores de cuarzo” Tesis Doctoral, Universidad Politécnica de Valencia 31. Y. Jiménez, R. Fernández, R. Torres and A. Arnau (2006) “A contribution o solve the problem of coating properties extraction in quartz crystal microbalance applications” IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 53 (5): 1057-1072 32. D. Neto, D. Evans, E. Bonaccurso H. Butt and V. Craig (2005) “Boundary slip in newtonian liquids: a review of experimental studies” Reports on Progress in Physics 68: 2859-2897 33. S. J. Martin, G. C. Frye, A. J. Ricco, and S. D. Senturia (1993) “Effect of surface roughness on the response of thickness-shear mode resonators in liquids” Analytical Chemistry 65 (20): 2910-2922 34. R. Schumacher, G. Borges and K. K. Kanazawa (1985) “The quartz microbalance: a sensitive tool to probe surface reconstructions on gold electrodes in liquid” Surface Science 163: 621 35. M. Urbakh and L. Daikhin (1994) “Roughness effect on the frequency of a quartz-crystal resonator in contact with a liquid” Physical Review B 49 (7): 1866-1870 36. M. Urbakh and L. Daikhin (1994) “Influence of the surface morphology on the quartz crystal” Langmuir 10: 2836-2841 37. L. Daikhin and M. Urbakh (1996) “Effect of surface film structure on the quartz crystal microbalance response in liquids” Langmuir 12: 6354-6360 38. L. Daikhin and M. Urbakh (1997) “Influence of surface roughness on the quartz crystal microbalance response in a solution” Faraday Discuss 107: 2738 39. M. Urbakh and L. Daikhin (1998) “Surface morphology and the quartz crystal microbalancen response in liquids” Colloids and Surfaces A 134: 75-84 40. L. Daikhin, E. Gileadi, G. Katz, V. Tsionsky, M. Urbakh and D. Zagidulin (2002) “Influence of roughness on the admittance of the quartz crystal microbalance immersed in liquids” Analytical Chemistry 74: 554-561 41. R. Etchenique and V. L. Brudny (2000) “Characterization of porous thin films using quartz crystal shear resonators” Langmuir 16: 5064-5071 42. K. Wondraczek, A. Bund and D. Johannsmann (2004) “Acoustic second harmonic generation from rough surfaces under shear excitation in liquids” Langmuir 20 (23): 10346-10350 43. Susanne Wehner, Katrin Wondraczek Diethelm Johannsmann and Andreas Bund (2004) “Roughness-induced acoustic second-harmonic generation during electrochemical metal deposition on the quartz crystal microbalance” Langmuir 20 (6): 2356-2360 44. A. Bund (2004) “Application of the quartz crystal microbalance for the investigation of nanotribological processes” Journal of Solid State Electrochemistry 8: 182-186 45. K. Wondraczek, A. Bund and D. Johannsmann (2004) “Acoustic second harmonic generation from rough surfaces under shear excitation in liquids” Langmuir 20 (23): 10346-10350

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46. A. Arnau, Y. Jiménez, R. Fernández, R. Torres, M.Otero and E.J. Calvo (2006) “Viscoelastic characterization of electrochemically prepared conducting polymer films by impedance analysis at quartz crystal. study of the surface roughness effect on the effective values of the viscoelastic properties of the coating” Journal of Electrochemical Society 153 (7): 455466 47. H. C. Brickman (1947) “A calculation of the viscous force exerted by A1:a flowing fluid on a dense swarm of particles” Applied Scientific Research. 27-34 48. A. Arnau, Y. Jiménez and R. Fernández (2005) “A new roughness physical model for describing the mechanical impedance of a coated shear resonator immersed in liquids” Acoustic Wave Sensors Workshop V, Physikzentrum Bad Honnef, Germany 49. L. Ochoa, Y. Jimenez and A. Arnau (2004) “Estudio y modelado de la contribución que tiene la rugosidad superficial del sensor o del recubrimiento, sobre la respuesta del cristal de cuarzo como sensor microgravimétrico en fluidos viscosos” Electro 2004 26: 139-144 50. P. L. Konash and G. J. Bastiaans “Piezoelectric crystals as detectors for liquid chromatography” Analytical Chemistry 52: 1929-1931, 1980 51. K. Bizet, C. Gabrielli and H. Perrot (2000) “immunodetection by quartz crystal microbalance. a new approach for direct detections of rabbit IGG and peroxidasa” Applied Biochemistry and Biotechnology 89: 139-149 52. R. M. Mueller and W. White (1968) “Direct gravimetric calibration of a quartz crystal microbalance” Review of Scientific Instruments 39: 291-295 53. D. M. Ullevig and J. F. Evans (1980) “Measurement of sputtering yields and ion beam damage to organic thin films with the quartz crystal microbalance” Analytical Chemistry 52 (9): 1467-1473 54. V. M. Mecca (1989) “A new method of measuring the mass sensitive area of quartz crystal resonators” Journal of Physics E 22: 59-61 55. B. A. Martin and H. E. Hager (1989) “Flow profile above a quartz crystal vibrating in liquid” Journal of Applied Physics 65 (7): 2627-2629 56. C. Gabrielli, M. Keddam and R. Torresi (1991) “Calibration of the electrochemical quartz crystal microbalance” Jounal of Electrochemical Society 138 (9): 2657-2660 57. G. Sauerbrey (1959) “Verwendung von schwingquarzen zur wägung dünner schichten und zur mikrowägung” Zeitschrift Fuer Physik 155: 206-222 58. G. Sauerbrey (1964) “Messung von plattenschwingungen sehr kleiner amplitude durch lichtstrommodulation” Zeitschrift Fuer Physik 178:457-471 59. M. A. Tenan, D. M. Soares and C. A. Bertran (2000) “Aqueous sol-gel process in the silica-metasilicate system. A microrheological study” Langmuir 16: 9970-9976 60. S. Tursiloadi, H. Imai and H. Hirashima (1995) “Crystallization and sintering behavior of pzt prepared from metal alkoxides” Journal of the. Ceramic Society of Japan 103 (1202): 1069-1072 61. S. Vercauteren, K. Keizer, E. F. Vansant, J. Luyten and R. Leysen (1998) “Porous ceramic membranes: preparation, transport properties and applications” Journal of Porous Materials 5: 241-258

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62. G. Senguttuvan, T. Settu, P. Kuppusamy and V. Kamaraj (1999) “Sol-Gel Synthesis and Thermal Evaluation of Cordierite–Zirconia Composites (Ce– ZrO2, Y–Ce–ZrO2)” J. Mater.Synth. Process. 7(3): 175-185 63. C. Barrera-Solano, L. Esquivias and G. L. Messing (1999) “Effect of Preparation conditions on phase formation , densification, and microstructure evolution in La-β-Al2-O3/Al2O3 composites” Journal of the American Ceramic Society 82 (5): 1318-1324 64. M. J. Muñoz-Aguado and M. Gregorkiewitz (1997) “Sol–Gel synthesis of microporous amorphous silica from purely inorganic precursors” Journal of Colloid and Interface Science 185 (2): 459-465 65. D. L. Meixner and P. N. Dyer (1999) “Influence of sol-gel synthesis parameters on the microstructure of particulate silica xerogels” Journal of SolGel Science and Technology 14 (3): 223-232 66. L. R. B. Santos, C. V. Santilli and S. H. Pulcinelli (1999) “Sol–gel transition in SnO2 colloidal suspensions: viscoelastic properties” Journal of NonCrystaline Solids 247: 153-157 67. M. Paulsson, H. Hagerstrom and K. Edsman (1999) “Rheological studies of the gelation of the acetylated gellan gum (Gelrite®) in physiological conditions” European Journal of Pharmaceutical Science 9 (1): 99-105 68. H. T. Chiu and J. H. Wang (1999) “A study of rheological behavior of a polypyrrole modified UHMWPE gel using a parallel plate rheometer” Polymer Engineering and Science 39 (9): 1769-1775 69. P. Puyol, P. F. Cotter and D. M. Mulvihill (1999h) “Thermal gelation of commercial whey protein concentrate: influence of pH 4.6 insoluble protein on thermal gelation” International Journal of Dairy Technology 52 (3): 81-91 70. R. K. Iler (1979) “The Chemistry of silica” Wiley, 173p, New York 71. C. J. Brinker and G. W. Scherer (1990) “Sol-Gel SciencesThe Physicsand Chemistry of Sol-Gel Processing” Academic Press, 100p, Boston 72. A.C. Arias, M. Granström, D.S. Thomas, K. Petritsh and R.H. Friend (1999) “Doped conducting-polymer–semiconducting-polymer interfaces: Their use in organic photovoltaic devices” Physical Review 60: 1854-1860 73. B. Piro, L. A. Dang, M. C. Pham, S. Fabiano and C. Tran-Minh (2001) “A glucose biosensor based on modified-enzyme incorporated within electropolymerised poly(3,4-ethylenedioxythiophene) (PEDT) films” Journal of Electroanalytical Chemistry 512: 101-109 74. L. Goenendaal, F. Jonas, D. Freitag, H. Pielartzik and J. R. Rynolds (2000) “Poly(3,4-ethylenedioxythiophene) and Its Derivatives: Past, Present, and Future” Advanced Materials 12 (7): 481-494 75. U. Rammelt, P. T. Nguyen and W. Plieth (2001) “Protection of mild steel by modification with thin films of polymethylthiophene” Electrochimica Acta 46: 4251-4257 76. H. Pagés, P. Topart and D. Lemordant (2001) “Wide band electrochromic displays based on thin conducting polymer films” Electrochimica Acta 46: 2137-2143

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77. H. Randriamakazaka, V. Noel and C. Chevrot (1999) “Nucleation and growth of poly(3,4-ethylenedioxythiophene) in acetonitrile on platinum under potentiostatic conditions” Journal of Electroanalytical Chemistry 472 (2): 103-111 78. D. Johannsmann, J. Grunner, J. Wesser, K. Mathauer, G. Wegner and W. Knoll (1992) “Viscoelastic properties of thin-films probed with a quartz crystal resonator” Thin Solid Films 210: 662-665 79. E.J. Calvo, C. Danilowitz, E. Forzani, A. Wolosiuk and M. Otero (2003) “Biomolecular films: design, function and applications” J. Rusling (Ed), 337p, Marcel Dekker 80. F. Höok, B. Kasemo, T. Nylander, C. Fant, K. Scott and H. Elwing (2001) “Variations in coupled water, viscoelastic properties, and film thickness of a Mefp-1 protein film during adsorption and cross-linking: A quartz crystal microbalance with dissipation monitoring, ellipsometry, and surface plasmon resonance study” Analytical Chemistry, 73: 5796-5804

15 Sonoelectrochemistry Christopher Brett Departamento de Química, Universidade de Coimbra.

15.1 Introduction Sonoelectrochemistry is the study of the effects of the combination of ultrasonic radiation with electrode processes occurring at surfaces of electrodes immersed in a solution in an electrochemical cell [1-3] (see Chapter 8 for the fundamentals of electrochemistry). The benefits of this coupling have been recognised for many years, since 1934 when insonation was used to increase the rate of water hydrolysis [4]. Ultrasound is a sound wave that introduces energy into a system, and can be transmitted through any solid, liquid or gas which possesses elastic properties. In liquids and gases, longitudinal waves are produced which transmit the wave and thence the energy via alternate compression and rarefaction in each cycle. Thus, in an electrochemical scenario, energy is introduced both in homogeneous solution and on solid surfaces, which include electrodes. Resulting phenomena include cavitation, acoustic microstreaming, mixing, emulsification, and enhanced hydrodynamics, the extent of which decreases the higher the viscosity of the solution and the higher the ultrasound frequency. There are also effects on suspended solids, particles, nucleation-enhancing species and on dissolved gases. All of these can influence an electrochemical process. Ultrasound itself has many recognised applications. Examples are: 1. Biological studies: cell disruption and homogenization. 2. Chemistry: increasing rates of chemical reactions and possibly altering the mechanism. 3. Dentistry: cleaning and drilling of teeth. 4. Medicine – ultrasonic imaging (see Chap. 17). 5. Materials and engineering: non-destructive testing and flaw detection (see Chap. 16); assisting drilling, grinding and cutting; processing hard and brittle materials; welding of thermoplastics.

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6. Pulse-echo techniques: location of mineral and oil deposits, sonar (see Chap. 16). 7. Industrial cleaning and dispersion of pigments in paints; acoustic filtration. Different frequency ranges are employed according to the required purpose: 1. 20 kHz – 100 kHz: power ultrasound. 2. 1 – 10 MHz: diagnosis and chemical analysis. The reason for the choice of these different frequency ranges can be relatively easily understood. The higher the frequency, the shorter the cycle time, which is composed of compression then rarefaction, such that the rarefaction at high frequencies is unlikely to leave time for bubbles to be formed which is where most of the energy is stored (see below). In chemical processes, ultrasound irradiation can lead to increased transport, interfacial cleaning and thermal effects. In the rest of this chapter we shall be mainly concerned with the influence of power ultrasound on electrode processes, since this is the frequency region where the most evident changes to electrode processes are seen.

15.2 Basic Consequences of Ultrasound The passage of ultrasonic waves through a liquid leads to two main effects: 1. Acoustic streaming – a time-independent flow of liquid. 2. Cavitation – the generation, expansion and collapse of vapour-filled cavities in the liquid. The way in which this occurs depends on the shape, composition and size of the reactor. In general, the influence on electrode processes can be summarized as: 1. Enhancing the rate of mass transport to the electrode and the transport-limited current. This implies higher sensitivity, higher reproducibility and decreasing detection limits. 2. Reducing adsorption phenomena. 3. Enhancing the rate of many chemical reactions. 4. A cleaning and eroding effect on the electrode surface caused by cavitation. 5. Formation of radical species during cavitation which can influence the electrode reaction mechanism.

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Cavitation is a highly energetic process [3, 5, 6]. Sound waves consist of alternating compression and rarefaction cycles. During rarefaction the negative pressure is sufficient to overcome the fluid’s intermolecular forces and produce microbubbles. These bubbles grow either near inhomogeneities such as those provided by solid impurities or provide a nucleus for acoustic cavitation. In the power ultrasound region, i.e. 20 - 100 kHz, these bubbles are usually short-lived and collapse, releasing large amounts of energy locally, as shown in Fig. 15.1. The example of the solid surface is of particular interest in the electrochemical context Homogeneous liquid media Cavity Shock wave on bubble collapse

Solid/liquid interface

Powders: trapped bubbles collapse - surface cleaning and fragmentation Solid surfaces: bubble collapse near surface leads to microjets surface cleaning and erosion

Liquid/liquid interface

Disruption of phase boundary and emulsification

Fig. 15.1. Possible sites of reaction induced by cavitation

Estimates of the energy concentrated in the bubbles leads to calculations of local temperatures on collapse of up to 5000 K and pressures of 1700 bar. This may happen close to a solid surface, causing microjets to hit the surface, leading to surface erosion and cleaning. A schematic representation of how this can occur is shown in Fig.15.2. The external pressure of the liquid is hindered near the solid surface, so that the reduction in bubble size is not uniform and resulting in a net movement of solution towards the solid surface on final bubble collapse.

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Fig. 15.2. Schematic representation of cavitation bubble collapse on a solid surface

One of the earliest effects noticed resulting from these jets was the reduction in the voltage necessary to cause water decomposition, owing to the constant removal of gases as they formed. However, in most cases the dominant observed effect is an increase in mass transport and analysis has shown that it follows the same methodology in practice as imposed convection. Besides increasing the current, and thence the rate of electrolysis, this also means that steady-state voltammetric profiles can be observed instead of peak shaped waveforms on scanning the applied potential and also fast reversible reactions may move from the reversible into the quasireversible kinetic regime (see Chapter 8 for further details on these electrochemical techniques).

15.3 Experimental Arrangements There are two basic experimental arrangements that can be used in sonochemical and sonoelectrochemical experiments. Both use ultrasonic transducers, which are deigned to convert mechanical or electrical energy into high frequency sound. They can be of three types: gas driven, liquid driven and electromechanical. The last of these can be based on the piezoelectric or on the magnetostrictive effect, but piezoelectric transducers are far more common. 1. Immersion of the electrochemical cell in an ultrasonic bath with various transducers bonded to the walls and base, such that the ultrasonic radiation is relatively widely distributed and of low intensity (1-2 W cm-2 at each transducer face). Nevertheless, the power density is not easy to quantify and temperature control requires thermostatting. 2. Focussing the ultrasonic field on the electrode surface from a single transducer via an immersed horn, such that radiation intensities up to several hundred W cm-2 can be achieved. In these arrangements the horn tip is placed above the electrode, it being possible to vary the distance and usually the power of the horn. The ultrasound field

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created at the horn tip in solution leads to acoustic streaming which manifests itself as a jet of solution directed towards the electrode surface; thermostatting is once again necessary to avoid heating of the solution inside the cell. Transducers for horns are usually of the prestressed piezoelectric design, between two and four piezo elements (e.g. prepolarised lead titanate zirconate) being bolted between a pair of metal ends. The length is around 70 mm at 20 kHz and maximum peak displacements range between 15 - 20 μm. The function of the horn is to magnify or amplify this motion as a resonant element in the compression mode. It must therefore be exactly half a wavelength long although multiples are possible. Half a wavelength at 20 kHz is approximately 26 cm. Three shapes are used in commercial horns – linear taper, exponential taper or stepped, see Fig.15.3. In the case of the stepped design, the step must always be at a node in the wave (the nodal point of the horn) to avoid any vibration and thus any stress. Exponential tapers offer higher magnification factors but are less used than linear taper since they are more difficult to manufacture. An example of a horn with the tip immersed in a thermostatted electrochemical cell is shown in Fig. 15.4.

Linear taper Linear taper

Exponential taper Exponential taper

Stepped Stepped

Fig. 15.3. Three types of ultrasound horn probe tip

Since the maximum achievable power densities can be of the order of hundreds of W cm-2, as mentioned above, significant solution heating can occur. For this reason, apart from thermostatting the electrochemical cell, pulsed irradiation protocols are often used consisting of “on” and “off” insonation periods. Commercial equipment usually allows this to be preprogrammed. Whilst this is usually used without regard to the implications

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on the signals recorded during insonation, since the length of the “off” period is designed to be sufficiently short so as not to have a significant effect, recently the wavelet transform has been shown to offer an interesting approach to process the signal and measure the different current components of the sonovoltammetric signals during the pulse-on and pulse-off parts of the cycle [7]. The consequence is the possibility of accurate determination of submicromolar concentrations if electroactive species, through better signal background subtraction. Several materials for horn construction are used to be sufficiently resistant to cavitation and chemical attack, the best being titanium alloy. With a view to scale-up of sonochemical reactors, radially vibrating horns have been recently investigated, such that the irradiation will be spread over a larger area and not directed to below the horn [8].

Fig. 15.4. A 20kHz ultrasound horn titanium alloy probe linear tapered tip, immersed in a thermostatted electrochemical cell, directly above an electrode at a distance of ~1 cm. Cavitation bubbles can be seen on the probe tip

The ultrasound field created at the horn tip in solution leads to acoustic streaming, which manifests itself as a jet of solution directed towards the electrode surface; thermostatting is necessary to avoid heating of the solution inside the cell In most sonoelectrochemical experiments cavitation is avoided as much as possible, first, because it destroys the electrode surface and secondly,

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because it reduces the reproducibility of the experiments carried out since the formation and collapse of bubbles is not sufficiently controllable. Cavitation has been characterised electrochemically, by luminescence and photographically [9]. Important parameters which can be varied are, for a given ultrasonic frequency: ultrasonic power, cell geometry, distance between the horn tip and the electrode and chemical composition of the system. Another possibility is to use the horn tip itself as an electrode, a socalled “sonotrode” [10]. The enhancements in the currents are similar but electrodeposition and other processes are affected since the electrode is at the source of the ultrasound so that the conditions are more extreme.

15.4 Applications Some examples of the different types of application of sonoelectrochemistry will be briefly indicated in this section. The reader is referred to the literature for a broader account of these, together with more detailed examples and other applications [3, 11]. 15.4.1 Sonoelectroanalysis Given the fact that the horn probe increases mass transport and therefore enhances currents, convenient strategies can be developed for the measurement of electroactive species at trace level through preconcentration by deposition [12]. This can be particularly useful for the measurement of species in environmental matrices which usually contain organic components which adsorb and block the electrode surface rapidly. Although many of these problems could be solved with sample digestion, for example by ultrasonic digestion, this alters sample speciation and precludes the probing of the sample under as close to natural conditions as possible. One of the first applications was the measurement of trace cadmium and lead ions at Nafion-coated mercury thin film glassy carbon electrode using the pre-concentration technique of anodic stripping voltammetry, the Nafion serving to prevent surface fouling and prevent the mercury thin film from being washed away [13]. Since then a number of different systems have been studied at bare and protected electrodes, mainly for trace metal analysis, see [14] and references therein.

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15.4.2 Sonoelectrosynthesis Some of the earliest applications of ultrasound were in organic and organometallic electrosynthesis [15]. A well-studied example on mechanism change was the Kolbe reaction for oxidation of carboxylate anions to hydrocarbons [16]. The effect of ultrasound is to alter the relative importance of different mechanistic pathways and thus the product distribution, owing to the change in mass transport (in general, there is no evidence for intervention of radical species generated by cavitation), shown recently with oxidation of the pesticide diuron [17]. Interesting benefits are also be obtained by acoustic emulsification, which can increase yields by enhancing the contact between species soluble in two different solvents, one usually aqueous, the reaction product generally being extracted into the organic medium [18]. This strategy is also more environmentally friendly since it enables a smaller quantity of organic solvent to be employed. 15.4.3 Ultrasound and Bioelectrochemistry Biological molecules usually adsorb on electrode surfaces, making it more difficult to measure their electrochemical parameters. Insonation permits an increase of the mass transport, enhancing the current and also reduces adsorption. In one approach, application of ultrasound in differential pulse voltammetry until just before the oxidation waves of nucleosides and nucleotides appeared allowed their characterization, e.g. [19]. In a recent paper ultrasound was applied whilst recording an electrochemical impedance spectrum, in this way avoiding adsorption of the molecules and their products, and permitting the calculation of the rate constant of the oxidation of guanine and adenine nucleic acid bases in solution [20]. This has been recently extended to the study of the adsorption of proteins on metal surfaces, exemplified by bovine serum albumin on copper where specific interactions with copper can be probed [21]. 15.4.4 Corrosion, Electrodeposition and Electroless Deposition A number of recent studies have been undertaken concerning the influence of power ultrasound on the erosion and corrosion of different metal surfaces, using bursts or pulses of ultrasound over an extended period, most showing a synergistic effect between the two phenomena [22, 23]. Important aspects investigated concern the influence of ultrasound on the corrosion mechanism, particularly of mass transport, the surface cleaning effect,

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enhancement of corrosion through cavitation and the ease of re-formation of any protective oxide layer. Open circuit potential, polarization curves and electrochemical impedance spectroscopy as well as microscopy techniques have all been used to study the processes. Recently, it was found that ultrasound can have a beneficial effect on the corrosion of steels, where the rate of corrosion is determined by the rate of the metal oxidation reaction, the metallic cations possibly forming a loosely adherent oxide layer by hydrolysis. The re-formation of corrosion products after ultrasonic radiation is shown to lead to a more robust oxide layer. On other metals such as aluminium, where the rate of corrosion is limited by the rate of oxide formation rather than of metal dissolution, the rate of corrosion after insonation remains as before, the metal surface having been roughened [24]. Electrodeposition is also influenced by ultrasound, particularly affecting the balance between formation of nucleation centres and growth phases. This has been little exploited so far, but a recent example concerns the deposition of lead dioxide on glassy carbon, which is useful for the use of these electrodes in anodes for batteries, waste water treatment, ozone generation or electrosynthesis [25]. Studies have also been made on the influence of ultrasonic irradiation on electroless plating of copper and comparing this with conventional electroplating: these investigations show the clear effect of ultrasound in reducing internal stresses [26]. 15.4.5 Nanostructured Materials If the horn probe tip is made into the electrode itself – a sonotrode – the deposits formed do not have the opportunity to grow and are detached from the electrode. This is an attractive way to form nanopowders of metals or semiconductor oxides of uniform size (diameter ~ 5 nm) which can be used for a variety of applications such as in chemical synthesis, see [27] and references therein. Nevertheless, it is possible to form nanostructures without sonotrodes, as has been recently demonstrated. For example, lead-tellurium nanorods were formed by constant current electrodeposition and pulsed irradiation, the important point being the very slow rate of growth in allowing rods to be formed with a near-perfect crystal structure [28]. Making electrodes from carbon nanofibres coated onto ceramic paper to give carbon composites an using as an electrode material was found to increase the effect of insonation by an order of magnitude compared to a flat electrode equivalent

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owing to both the more intense diffusion field and also to the rough surface topography which increased the residence time of molecules that would be able to react [29]. This approach is potentially extremely promising in the future for sonoelectrolysis applications. 15.4.6 Waste Treatment and Digestion The treatment of waste by degradation of pollutants is a relatively new application of ultrasound and can be combined with electrochemical processes to increase the efficiency, usually by anodic oxidation, e.g. [30]. In another new application which demonstrates the importance of extreme conditions, there is complete destruction of nitroaromatics formed as byproducts in the industrial fabrication of trinitrotoluene (TNT), which contaminate wastewater [31]. The use of ultrasound increases the current efficiency associated with electrochemical reduction of these compounds, the products of which are treated in situ with ozone. Ultrasound is also one of the methods currently used for the digestion of complex samples prior to chemical analysis. 15.4.7 Multi-frequency Insonation The combination of two or more ultrasonic frequencies (one in the kHz power region and the other in the MHz region) enhances the cavitation yield [32]. The mechanism for this probably involves the production of new bubbles by the low-frequency stimulating field which enhance the higher frequency acoustic cavitation. There can also be a resonance effect which could have useful consequences.

15.5 Final Remarks Sonoelectrochemistry has been demonstrated in the last two decades to be a powerful tool in enhancing the rate of electrochemical processes, including the degradation of pollutants, in electroanalysis and in diagnosis of corrosion. Its continued use in the future seems certain.

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References 1. T.J. Mason and J.P Lorimer (1988) “Sonochemistry: theory, applications and uses of ultrasound in chemistry” Ellis Horwood, Chichester, UK 2. T.J. Mason TJ (1990) “Sonochemistry: the uses of ultrasound in chemistry” RSC, Cambridge, UK 3. D.J. Walton and S.S Phull (1996) in “Advances in Sonochemistry” T.J. Mason, Ed., JAI Press, London, Vol 4, p. 205 4. N. Moriguchi (1934) “The effect of supersonic waves on chemical phenomena, (III).The effect on the concentration polarization” J. Chem. Soc. Jpn. 55: 749-750. 5. C.E. Banks and E.G. Compton (2003) “Voltammetric exploration and applications of ultrasonic cavitation” ChemPhysChem 4 (2): 169-178 6. P.R. Birkin, D.G. Offin, P.F. Joseph and T.G. Leighton (2005) “Cavitation, shock waves and the invasive nature of sonoelectrochemistry” J Phys Chem B 109 (35): 16997-17005 7. X. Zhang, T. Tanaka and J. Jin (2006) “Application of wavelet transform in pulsed ultrasonic modulation voltammetry” Ultrasonics Sonochem. 13 (2): 133-138 8. O. Dahlem J. Reisse and V. Halloin (1999) “The radially vibrating horn: A scaling-up possibility for sonochemical reactions” Chem. Eng. Sci.. 54 (1314): 2829-2838 9. P.R. Birkin, J.F. Power, M.E. Abdelsalam and T.G. Leighton (2003) “Electrochemical, luminescent and photographic characterisation of cavitation” Ultrasonics Sonochem 10 (4-5): 203-208 10. J. Reisse, H. Francois, J. Vandercammen, O. Fabre, A. Kirsch-de-Mesmaeker, C. Maerschalk and J.L. Delplancke (1994) “Sonoelectrochemistry in aqueous electrolyte: A new type of sonoelectroreactor” Electrochim. Acta 39: 37-39 11. R.G. Compton, J.C. Eklund and F. Marken (1997) “Sonoelectrochemical processes: A review” Electroanalysis 9 (7): 509-522 12. N.A. Madigan, T.J. Murphy, J.M. Fortune, C.R.S. Hagan and L.A. Coury Jr (1995) “Sonochemical stripping voltammetry” Anal. Chem. 67 (17): 27812786 13. F.M. Matysik, S. Matysik, A.M. Oliveira Brett and C.M.A. Brett (1997) “Ultrasound-enhanced anodic stripping voltammetry using perfluorosulfonated ionomer-coated mercury thin-film electrodes” Anal. Chem. 69 (8): 1651-1656 14. J.L. Hardcastle and R.G. Compton (2001) “Biphasic sonoelectroanalysis: simultaneous extraction from, and determination of vanillin in food flavoring” Electroanalysis 13 (11): 899-905 15. D.J. Walton, J. Iniesta, M. Plattes, T.J. Mason, J.P Lorimer, S. Ryley, S.S. Phull, A. Chyla, J. Heptinstall, T. Thiemann, H. Fuji, S. Mataka and Y. Tanaka (2003) “Sonoelectrochemical effects in electro-organic systems” Ultrasonics Sonochem. 10 (4-5): 209-216

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16. D.J. Walton, A. Chyla, J.P. Lorimer, T.J. Mason and G. Smith (1989) “Modifying effect of ultrasound upon the electrochemical oxidation of cyclohexanecarboxylate” J. Chem. Soc. Chem. Commun. 9: 603-604 17. K. Macounova, J. Klima, C. Bernard and C. Degrand (1998) “Ultrasoundassisted anodic oxidation of diuron” J. Electroanal. Chem. 457 (1-2): 141-147 18. J.D. Wadhawan, F. Marken and R.G. Compton (2001) “Biphasic sonoelectrosynthesis. A review” Pure Appl. Chem. 73 (12): 1947-1955 19. A.M. Oliveira Brett and F.M. Matysik (1997) “Voltammetric and sonovoltammetric studies on the oxidation of thymine and cytosine at a glassy carbon electrode” J. Electroanal. Chem. 429 (1-2): 95-99 20. A.M. Oliveira Brett, L.A. Silva and C.M.A. Brett (2002) “Adsorption of guanine, guanosine, and adenine at electrodes studied by differential pulse voltammetry and electrochemical impedance” Langmuir 18 (6): 2326-2330 21. E.M. Pinto, D.M. Soares and C.M.A. Brett (2007) “Influence of ultrasound irradiation on the adsorption of bovine serum albumin on copper” J. Appl. Electrochem. 37 (11): 1367-1373 22. C.T. Kwok, F.T. Cheng and H.C. Man (2000) “Synergistic effect of cavitation erosion and corrosion of various engineering alloys in 3.5% NaCl solution” Mat. Sci. Eng. A 290(1-2): 145-154 23. A. Neville and B.A.B. McDougall (2001) “Erosion– and cavitation–corrosion of titanium and its alloys” Wear 250 (1-2): 726-735 24. N.N.P.A. Morais and C.M.A. Brett (2002) “Influence of power ultrasound on the corrosion of aluminium and high speed steel” J. Appl. Electrochem. 32 (6): 653-660 25. J. González-García, V. Sáez, J. Iniesta, V. Montiel and A. Aldaz (2002) “Electrodeposition of PbO2 on glassy carbon electrodes: influence of ultrasound power” Electrochem. Commun. 4 (5): 370-373 26. F. Touyeras, J.Y. Hihn, X. Bourgoin, B. Jacques, L. Hallez and V. Branger (2005) “Effects of ultrasonic irradiation on the properties of coatings obtained by electroless plating and electro plating” Ultrasonics Sonochem. 12 (1-2): 13-19 27. A. Durant, J.L. Delplancke, V. Libert and J. Reisse (1999) “Sonoelectroreduction of metallic salts: a new method for the production of reactive metallic powders for organometallic reactions and its application in organozinc chemistry” Eur. J. Org. Chem. 1999 (11): 2845-2851 28. X. Qiu, Y. Lou, A.C.S. Samia, A. Devadoss, J.D. Burgess, S. Dayal and C. Burda (2005) “PbTe Nanorods by Sonoelectrochemistry” Angew. Chem. Int. Ed. 44 (36): 5855-5857 29. M.A. Murphy, F. Marken and J. Mocak (2003) “Sonoelectrochemistry of molecular and colloidal redox systems at carbon nanofiber–ceramic composite electrodes” Electrochim. Acta 48 (23): 3411-3417. 30. L. Azprykowicz , C. Juzzolino and S.N. Kaul (2001) “A comparative study on oxidation of disperse dyes by electrochemical process, ozone, hypochlorite and fenton reagent” Water Research 35 (9): 2129-2136 31. V.A. Abramov, O.V. Abramov, A.E. Gekhman, V.M. Kuznetsov and G.J. Price (2006) “Ultrasonic intensification of ozone and electrochemical destruction

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of 1,3-dinitrobenzene and 2,4-dinitrotoluene” Ultrasonics Sonochem. 13 (4): 303-307 32. R. Feng, Y. Zhao, C. Zhu and T.J. Mason (2002) “Enhancement of ultrasonic cavitation yield by multi-frequency sonication” Ultrasonics Sonochem. 9 (5): 231-236

16 Ultrasonic Systems for Non-Destructive Testing Using Piezoelectric Transducers: Electrical Responses and Main Schemes Antonio Ramos and José Luis San Emeterio Dpto. Señales, Sistemas y Tecnologías Ultrasónicas. Instituto de Acústica (CSIC)

16.1 Generalities about Ultrasonic NDT In very distinct areas of industry the quality control of the fabrication products, as well as of the equipment used to fabricate them, is performed by means of Non-Destructive Testing (NDT) techniques. This control procedure acquires still more relevance when aspects of public security are present such as in the nuclear and aeronautics sectors. In many NDT applications, high frequency ultrasonic waves are employed. In most cases they are produced and detected by piezoelectric transducers operated in a pulsed mode. The ultrasonic radiation of these transducers is used as a tool to visualize internal parts. The propagation of these pulsed elastic waves can provide information about the internal structure of the medium by means of the analysis of reflections, scattering and attenuation in the mechanical discontinuities encountered along the ultrasonic path. Figure 16.1 shows a very general block diagram of NDT systems based on ultrasonic waves. It includes one or more ultrasonic transceivers depending on the number of piezoelectric channels involved. PIECE TO BE INSPECTED

MONO/MULTICHANNEL ULTRASONIC TRANSCEIVER

DISPLAY OF INSPECTION RESULTS

SIGNAL ACQUISITION & PROCESSING

N D T DATA EXTRACTION

Fig. 16.1. Functional blocks scheme of an ultrasonic NDT process A.A. Vives (ed.), Piezoelectric Transducers and Applications, doi: 10.1007/978-3-540-77508-9_16, © Springer-Verlag Berlin Heidelberg 2008

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Different acoustic propagation modes can be used but some of them are more frequent in the NDT applications. Fluids propagate longitudinal waves. Solids also propagate shear or transversal waves, which are useful for testing purposes [1]. Other secondary modes are those corresponding to Rayleigh waves (over solid surfaces), the lateral or creeping waves, the Love shear waves, and the Lamb waves [2]. In laminar and tubular industrial pieces, other complex waves can propagate, thus interfering with the basic transversal and longitudinal modes [1]. Other acoustic propagations, originated by lateral vibrations of the piezoelectric vibrators, have also been detected; they are known as “head” waves [3-5]. Longitudinal waves propagate in general with a higher velocity than transversal waves and they are the most commonly used in NDT applications. 16.1.1 Some requirements for the ultrasonic responses in NDT applications Regardless of the acoustic propagation mode, a good efficiency in pulsed regime is always required for the piezoelectric responses in ultrasonic NDT because that makes possible to obtain high amplitudes in the collected ultrasonic pulses, thus improving the dynamic range in the measured signals and consequently the sensitivity of the NDT processes. In addition, a good bandwidth in the testing signals (short time duration) would permit to improve the time-resolution related to the axial discrimination in ultrasonic testing. NDT of industrial pieces can be performed using different procedures depending on the geometry of the tested structure, the constructive materials and other characteristics. The two most common procedures are known as pulse-echo and through-transmission (T/T) testing. The first one is the most widely used in conventional inspections whereas the T/T method is applied in more specific cases, for instance in the inspection of large pieces of advanced materials (space, aircraft and nuclear materials) and in general when high acoustic attenuation in the testing pulses is produced through the propagation medium. In most applications, a time-domain analysis of the received signals is performed. Thus, it is necessary to obtain a good time resolution by using compact ultrasonic pulses. In fact, NDT systems should be able to generate ultrasonic pulses as short as possible. In those aspects that depend on the piezoelectric transducers, a shortening in the ultrasonic pulses can be attained through two ways: selecting a high oscillation frequency (which increases the acoustic attenuation in the medium, as a secondary effect) or

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increasing the transducer bandwidth (with notable internal losses in the transducer). An undesired consequence of both ways is a minor global efficiency, which produces worse signal-to-noise ratios in the testing signals. Therefore, the design of piezoelectric transducers adequate to NDT applications is mainly a question of trade-off between these opposite requirements: high amplitude and short length of the ultrasonic pulses. In applications requiring a frequency-domain analysis, a large bandwidth should be achieved in testing signals. Simultaneously, it is important to assure that enough ultrasonic energy distributed in many frequencies was supplied into the testing piece, in order to resolve flaw indications from electronic and acoustic noises induced in the ultrasonic signals. An alternative way for designing broadband transduction systems, with a minor disadvantage in efficiency, is to modify the working frequency band by means of electrical methods, in such a way that the final ultrasonic band, balancing electronic and transducer contributions, is adequately broad and smooth. There are some ways to achieve this objective [6-8]. In particular, a notable enlargement of both, amplitude and bandwidth of the testing signals, can be obtained by using matching and tuning circuits properly optimised for specific transducers and electronic transceivers [9, 10]. These types of interface circuits (matching and tuning) were amply analysed in the Chap. 6. In next sections, some cases of improvement in the global NDT emitterreceiver performance (efficiency and bandwidth) will be analysed. Other improvements of NDT results, related to the lateral resolution and the operation time, by means of multi-channel equipment, will also be described, including the consideration of their technological implications and of the electronic systems involved.

16.2 Through-Transmission and Pulse-Echo Piezoelectric Configurations in NDT Ultrasonic Transceivers Each ultrasonic channel in Fig. 16.1 usually has one of these two configurations: through-transmission (T/T) or pulse-echo. The general schemes of these testing procedures are respectively detailed in Figs. 16.2 and 16.3. In both cases, a selective damping block appears in cascade with the basic generation section. This block notably reduces certain oscillations induced in driving pulses when the emission matching circuit includes an inductive tuning [8]. In pulse-echo testing (Fig. 16.3), both piezoelectric stages share the same matching circuits (damping and tuning) because only one transducer

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is working, successively as an emitter and as a receiver. A decoupling circuit protects the signal amplifier device from high-voltage impulses during driving instants. BASIC GENERATOR

EMISSION SELECTIVE DAMPING

SIGNAL AMPLIFIER

EMISSION TUNING

TRE

RECEPTION TUNING & DAMPING

TRR PIEZOELECTRIC TRANSDUCERS

Fig. 16.2. Ultrasonic NDT inspection using a through-transmission procedure BASIC PULSER

COMMON TUNING CIRCUIT

SELECTIVE DAMPING

BROADBAND AMPLIFIER

DECOUPLING E/R PIEZOELECTRIC TRANSDUCER

Fig. 16.3. Ultrasonic NDT inspection by means of a pulse-echo procedure

The main difference between both dispositions is that matching circuits of each stage are independent in through-transmission mode (scheme in Fig. 16.2). Although the pulse-echo scheme is the most frequently used in NDT, an example of the T/T usefulness is the testing of aircraft manufacturing, where high-speed is required during the inspection of large laminates of composite material [10]. Moreover, this alternative is the unique possibility when inspecting pieces with strong acoustic attenuation. Ultrasonic transceivers adequate to implement the testing procedures of Figs. 16.2 and 16.3 can be represented by a unique equivalent circuital diagram, as shown in Fig. 16.4. This diagram includes specific circuits for the pulsed transmitter and the input impedance in reception (parallel circuit: LRTun // CR // RR). In this Fig. 16.4, two electrically independent stages were assumed, but this scheme may also represent the pulse-echo case if the amplitude of the received echo-graphic signals is always below a predetermined voltage threshold (between ± 0.6 V or ± 5 V, depending on NDT equipment). As

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this condition is often fulfilled, the representation in Fig. 16.4 describes the behaviour of most NDT piezoelectric transceivers. VEx

NDT PULSER

EMITTER

( II )

R

L

TUN

TUNING R

(I)

Piece under Testing

TRANSDUCER

PARALLEL MATCHING CIRCUITS

TUN

PIEZOELECTRIC

HV Ramp Generator

E

L

TRANSDUCER

RG

ZL

PIEZOELECTRIC

CG

TUNING E

V0

VR

FE

RECEIVER

ZIN CR RR

ELECTRONIC RECEIVER

Fig. 16.4. Schematic representation of a mono-channel NDT piezoelectric transceiver

16.3 Analysis in the Frequency and Time Domains of Ultrasonic Transceivers in Non-Destructive Testing Processes The frequency response of the transmitting stage notably influences the received testing signal VR in Fig. 16.4, because this response can be expressed as a product of the pulser response VEx (s) with successive transfer functions (TF) in cascade related to: linear emission and reception in transducers I, II, respectively, ultrasonic propagation (PR) and inductive tuning in reception (LTun).

V R ( s ) = VEx ( s ) TFI ( s ) TFPR ( s ) TFII ( s ) TFLTun ( s )

(16.1)

Figure 16.4 includes an equivalent block diagram for the pulser electronics on which the response VEx (s) depends. This block contains: a highvoltage ramp generator (with output impedance RG) of amplitude V0 in series with a capacitor CG and in parallel with a generic matching and tuning network. The value of the series impedance [RG + ZL] is usually very low in efficient driving cases [8]. In many NDT equipments, there are non-linear components (diodes) in the matching circuits altering the trailing edge in the excitation pulse VEx, but their effects may be neglected when analysing specifically the first main part of the spike VEx in the Laplace domain. Under this assumption, and taken into account Eq. (6.9) in Chap. 6, the following Laplace response can be derived from the scheme of Fig. 16.4:

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(

(

E V Ex ( s ) = − C G Vo / (1 + sτ ) s (CT + C G ) + (1 / Rα ) + (1 / sLTUN )

))

(16.2)

In this expression, a parallel circuit RT-CT has been assumed to be the equivalent pulser load related to the transducer; Rα is the equivalent parallel of RT and the damping resistor in the matching circuit, and τ is the fall time of the real exponentially decaying ramp from a Vo value. Figure 16.5a shows the frequency spectrum amplitudes for the excitation pulse under two tuning conditions, computed from Eq. (16.2) taken into account that s = j2πf, for the following typical pulser and transducer parameters: CG = 9.3 nF, V0 = 300 Volt, fall-time τ = 9.1 10-9 seconds, equivalent resistance Rα = 44.4 ohm, and CT = 1.2 nF. 1.4

×10-4

100

1.2 0

1.0 0.8

-100

0.6 0.4

-200

0.2 0.0 0.0

a

0.5

1.0

f (MHz)

1.5

2.0

-300 0

1

2

3

4

t (μs)

b

Fig. 16.5. Frequency spectra distribution a and driving waveforms in time domain b computed by using the approximate model of expression 16.2. The solid and dotted line curves correspond to the cases with and without a tuning coil of 22 μH in parallel, respectively. In dashed line, the results of suppressing the undesired oscillations are shown

The corresponding time waveforms, calculated by inverse FFT transform of these frequency spectra [11], are depicted in Fig. 16.5b where the computed voltage pulses are related to the following cases: without tuning coil; with a strong parallel tuning without additional processing; and with tuning but considering the diodes effects on the driving pulse. In the later case, the effects of the diodes are very important, and must be considered in the processing. The approximation for calculations used in this case consists of determining the first positive zero-crossing of the driving pulse and then neglecting the extra-oscillations from this time instant, so emulating as the diodes attenuate them in the real pulser circuit, as can be confirmed in the comparison between the calculated and the experimentally measured spikes shown in Fig. 16.6. A simple computer processing

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[11] was used for this purpose in the calculated curves, truncating the positive values of the spike to a limited level determined by the number of diodes and their dynamic characteristics.

Fig. 16.6. Comparison between an experimental spike (solid) and the corresponding simulated spike computed using the approximate model

Equation (16.2) could be also useful for describing the complete spike in the frequency domain and also for deriving an approximate time waveform, even without using the abovementioned truncating step, in those cases when the driving spikes are relatively wide in time, which occurs in NDT applications developed in the low MHz range where an optimisation of the overall ultrasonic responses through a fitting of the driving interface have to be employed [12]. As an example of this type of optimisation, Fig. 16.7 shows some effects of matching and tuning circuits on the frequency responses of the pulser and of the received signal VR in a T/T system formed by two low-frequency NDT transducers (1 MHz) in contact with a plate of methacrylate 20 mm thick. In these cases, the applied driving VEx were high-voltage spikes: a delta of Dirac δ(t) in a, and distinct broader pulses VSP in b and c. It can be appreciated how very important improvements, in sensitivity and bandwidth of the frequency response of a typical ultrasonic transceiver configuration, can be performed by means of a proper determination of the electrical tuning components. A very useful tool to be employed for these optimisation purposes is the analytical expression in Eq. (16.2). In those remaining cases, when the previous approach of 16.2 does not apply, the transceiver analysis can be performed by using an intermediate time-simulation step based on circuital analysis programs, like Spice.

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Antonio Ramos and José Luis San Emeterio 1

Lro = 22 H

Simulation Results

0.8

Lro =

0.6

Piezoelectric TF E/R

0.4 0.2

Amplitude (20 mV/div.)

a

A

0

Vsp ( ) = 1 , 0

0.5

1 1.5 Frequency (Hz)

180

2

6

x 10

2.5

fM

80

f0.5

Experim. Response Vsp * TF E/R

{ Leo =

0 30

2470 Frequency (244 KHz/div.) -8 2 6 Vsp ( ) - 310/[-1.02.10 + j 1.135 + 1.1662.10 ]

Amplitude (50 mV/div.)

b 400

fM

200

0 30

Vsp* ( )

Experim. Response * * TF E/R Vsp

f0.5 { L eo = 56 H

Frequency (244 KHz/div.) 310/[1.02 .10

-8

2

- j 1.14

. + j 1.79 10 . - 1.68 10 6

2470 12 -1

]

c Fig. 16.7. a - Simulated and b-c - Experimental results with parallel tuning (in emission L0e and reception L0r ) effects on transfer functions; b and c - frequency responses measured in NDT of a plastic plate by T/T testing. (a L0e = ∞, b L0r = ∞, and c L0r = 22 μH )

As an example of this option, the simulated pulser response in time and its experimental confirmation, obtained for the NDT emitter probe of Fig. 16.7 driven with a narrow electrical spike and emitting ultrasound in CFRP composite material [13], are shown in Fig. 16.8.

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0.00

Pulsed Response (V)

-30.00 -60.00 -90.00 -120.00 -150.00

Simulated Driving Spike

-180.00 -210.00 -240.00 -270.00 -300.00 0.00E+0

2.00E-6

4.00E-6

Time (s)

6.00E-6

8.00E-6

a

Pulsed Response (30 V/div)

0

-150

-300 -1 s

Experimental Driving Spike

0

Time (1 s / div.)

9 s

b Fig. 16.8. Waveforms of the driving spikes with a particular pulser setting, when this is loaded with a broadband piezoelectric NDT probe: a simulation result, b measured pulse

The force pulse emitted by the transducer face under this specific driving is shown in Fig. 16.9. From these results in the time domain, for the excitation and the emitted force, their corresponding frequency responses could be obtained by means of the calculation of the respective FFT transforms. The matching circuits in practical NDT ultrasonic channels are usually manually adjusted in order to find the best sensitivity and a good frequency response. Procedures to improve transceiver responses by electrical matching have been applied to pulse-echo [9, 16] and throughtransmission (T/T) testing [12]; in the first case, a trade-off reasonably

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well fitted to the emitter and the receiver has to be achieved because impedances presented from the transmitter and the receiver electronics are very different in many cases. On the other hand, with T/T schemes, the matching circuits can be calculated with different criteria for the emitting and receiving phases. This possibility offers some advantages to better adjust these circuits, separately in each E/R stage, in order to optimise both E/R frequency bands composing a compact time pulse for testing purposes (see Fig. 16.7c). 150.00

Force Response (N)

100.00

Emitted Force for Driving of Figure 16.8

50.00

0.00

-50.00

-100.00 0.00E+0

2.00E-6

4.00E-6

6.00E-6

8.00E-6

Time (s)

Fig. 16.9. Simulated force pulse in the probe surface FE for the driving conditions of Fig. 16.8

16.4 Multi-Channel Schemes in Ultrasonic NDT Applications for High Resolution and Fast Operation In ultrasonic NDT using high-resolution imaging or high-speed scanning of piezoelectric transducers, multi-channel schemes are adopted to perform an electronic control of the acoustic aperture in order to achieve the necessary lateral resolution and speed in the ultrasonic scanning. These electronic control procedures include rapid scanning and steering of ultrasonic beams, variable focusing of beams and synthetic aperture focusing [17, 18].

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16.4.1 Parallel Multi-Channel Control of Pulse-Echo Transceivers for Beam Focusing and Scanning Purposes

By means of a dynamic focusing effect in each of the scan lines emitted from multi-element piezoelectric transducers (arrays), the lateral resolution can be improved through an artificial reduction of the beam-width. When this variable focusing has to be made quickly, electronic methods must be used; they operate with a number of channels working in parallel equal to the number of elements used as ultrasonic aperture. This procedure (see Fig. 16.10) can produce a focal distribution in the transmitted wavefronts, varying in successive time intervals [9] by means of modifications in the driving delays associated with the aperture elements. This figure includes an expression to calculate these delay τij depending on the focal depth. Vo e1

0 - 500V -Vo'

ij

2 1/2

2

= 100/c { (L /4 + Fj ) 2 2

2

2 1/2

[(|i|-0.5) L /N +Fj ] } N/2 PULSER DYNAMIC DELAYS

ij en

Fig. 16.10. Time-Delay driving procedure for electronic focusing of transmitted wavefronts in an array ultrasonic aperture. In the expression for τij: Fj is the focal depth, i is the element number, N the total number of elements, and L the aperture size

Figure 16.11 shows the block diagram of an electronic implementation for a focusing / steering procedure of this type. Figure 16.12 shows a snapshot of the pulsed wavefront emitted by an (4 MHz, 32 elements) ultrasonic array aperture, electronically focused at 20 mm in depth. The acoustic pressure Pj produced in the area radiated by the array is modified by the elements delays because they alter the global impulse response following the expression detailed in the same figure, in which hi (x, t) is the spatial impulse response associated with the rectangular

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Antonio Ramos and José Luis San Emeterio

array elements [15], and v (t) is the velocity in the radiating surface. The depicted transient wavefront was detected when it propagates around the focal zone. It can be appreciated how, at the focal zone, the beam-width has been drastically reduced from the unfocused beam-width (approximately of the same size than the acoustic aperture).

Programming and control unit BANKS OF n/2 FAST RAM

e1

Between 10 ns and 5 µs

en

TRANSDUCERS ARRAY

Repetition Rate Freq.

HV SUPPLY

n/2 Digital Programmable Emission Delays

VARIABLE MULTI-DELAY GENERATOR

Delay Codes Multichannel pulser

Fig. 16.11. Blocks scheme for an implementation of the focusing procedure shown in Fig. 16.10

In a similar way, this type of beam focusing/steering process can be performed also in reception, by using the scheme detailed in Fig. 16.13, which permits to properly delay the echo-signals received in all the aperture ultrasonic channels. Finally, by scanning the aperture along the total array length, these emission/reception beam control procedures can be extended to all echographic lines of the explored field in testing pieces. The implementation of these procedures implies some specific design requirements: 1. The first one is due to the short time differences between two consecutive driving pulses (as short as a few nanoseconds). For this reason it is not possible to share one electronic transceiver for all the

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aperture transducers and further very accurate designs in the delayed transceiver are required [14]. 2. In addition, the number of pulsers / receivers may range between half of the aperture elements and the total number of array elements. The number of elements of a linear array is usually about one hundred. This number increases in a quadratic way when 2-D arrays are used for bidimensional focusing. As a consequence, the basic pulser and reception units must be as simple as possible.

A P E R T U R E

+N / 2

P(x, t)=

O

v(t)/ t * h i [ x, (t + ij )] i=-N / 2

FOCAL ZONE Fig. 16.12. Snapshot of the acoustic pressure field produced by a 4 MHz ultrasonic array electronically focused at 20 mm, when it propagates around the focal zone

16.4.2 Electronic Sequential Scanning of Ultrasonic Beams for Fast Operation in NDT

There are some NDT applications requiring fast methods to make a sequential scanning of the acoustic beam during the ultrasonic examination. For instance, many ultrasonic NDT procedures employed in aircraft and nuclear areas, with mechanical scanning of a single transducer, need a drastic increase in inspection speed because the testing of these structures, normally large in size, is very time-consuming. In manual testing with pieces of complex geometry, this aspect is especially relevant. In fact, many common techniques used in non-destructive ultrasonic testing applications, make use of the mechanical or manual scanning of the sensor devices. An important limitation of all these procedures is the time required to acquire the overall inspection data when only a single transducer is

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Antonio Ramos and José Luis San Emeterio

scanned over the whole area of interest. Further, in the testing applications carried out by a specialised operator, which is quite usual if the inspected sample has a complicated geometry, these problems related to the time needed to make the piece inspection increases dramatically. Moreover, the skill and subjectivity grade of the operator introduce some uncertainty on the interpretation of the echographic results. Programming and control unit

n/2 echo traces from the array elements

n/2 FAST RAMs

n/2 Programmable delay lines

VARIABLE SIGNAL DELAYER

r1

Bipolar Limiter

r2

Bipolar Limiter Bipolar Limiter

&

Focused Trace

n/2 inputs adder rn

Bipolar Limiter Bipolar Limiter

Wideband preamplifiers

Fig. 16.13. Electronic system for dynamic focusing of received wavefronts in ultrasonic arrays by properly delaying the echo-signals in the different channels of the aperture

To overcome all these problems abovementioned, a multichannel operation must be introduced. Some possible alternative solutions to this objective, by using an electronic scanning of multielement ultrasonic probes, are discussed in the following, and finally a sequential scheme based on a high-voltage demultiplexer is described in detail. Some systems of this multichannel type have been widely utilized in the medical imaging field for specific diagnostic explorations. Nevertheless, the number of special inspections to be made in NDT over different materials with distinct shapes and requirements make very difficult to develop a

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general purpose system versatile enough to be useful at a commercial level. For instance, in the industrial inspection, individual pulsers/receivers for each NDT channel have been reported in references [21, 22]. Such fullparallel systems are bulky and very expensive, being their prices prohibitive for the most of the practical situations. On the other hand, a series of factors have traditionally obstructed the use of demultiplexer circuits for this purposes. Firstly, high-voltage transmit pulses are required for the transducer driving with the aim to obtain an adequate dynamic range in the testing signals for many materials having strong acoustical attenuation; for instance, peak values of up to 400 Volts use to be employed for testing. Moreover, the low values of the input electrical impedance in high frequency piezoelectric transducers require that the switching circuits, intercalated between the pulser terminals and the transducer piezoelectric elements, have on-resistances of about ten ohms as a maximum. Unfortunately, either of these two requirements is satisfied by the commercially available demultiplexer circuits. Some acoustical imaging applications [23, 24], have employed multiple analog switches for sequential acquisition of ultrasonic signals from the medium under inspection, but this option presents limitations for our special requirements of low impedance transducers and high-voltage drivers. A Mux-Dmux of High-voltage Pulses with Low On-Impedance

More recently, other systems, valid for the switching of high-voltage pulses over low impedance transducers, have been specially developed for rapid-scan purposes, needing as external equipments only one commercial monochannel flaw detector. They use a block diagram as that shown in Fig. 16.14, where a NDT system based on a high-voltage analog Multiplexer/De-multiplexer device is depicted. This device was designed to fulfil the voltage and impedance requirements above commented [19-20]. The new analog demultiplexer is able to control the pass of broadband pulses containing large peaks of voltage and current, as it is required in the most of NDT applications. It contains a high-voltage analogue multiswitch network, with a low cross-coupling between channels, for the sequential control of multiple short electrical pulses. The device behaves as a de-multiplexer unit during the driving processes and as a multiplexer unit for the signals coming back from the tested material, during the receiver phase.

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Antonio Ramos and José Luis San Emeterio ECHO SIGNALS TRIGGER

DIGITAL CONTROLLER

ACQUISITION AND VISUALIZATION SYSTEM

Channel selection

MONO CHANNEL PULSER / RECEIVER

H.V. ANALOG MUX / DMUX

y

Array

ELECTRONIC SEQUENTIAL SCANNING

Piece to be inspected

SPIKES / ECHOES

Electronic Scanning Mechanical Scanning

Fig. 16.14. Blocks scheme of a NDT system for rapid-scan inspection, based on a digitally-controlled high-voltage Mux/DMux

A rapid scanning of the ultrasonic beam, along x axis, is made by fast electronic switching of the probe elements. While the probe is moving along the y-axis, this device sequentially switches the probe elements with a very high speed, which facilitates to obtain C-scan displays at different testing depths. Therefore, inspection times can be drastically decreased by using only one electronic pulser / receiver and this Mux-Dmux device, because the information of eight channels can be simultaneously recorded, obtaining in a very fast way eight line scans, being in consequence the time operation decreased by eight times. This Mux/Dmux network synchronizes the successive excitations instants of the different transducer elements with the pulse repetition rate of the external pulser/receiver employed, by using a digital controller, which sends the channel number, selected in each transmit-receive cycle, to the acquisition and visualization stages for performing C-scan displays at different depths of the inspected pieces, from the variable voltage amplitude at the output voltage of the gated peak detectors included in the usual NDT ultrasonic receivers. Mux-Dmux system description

In this device, the switching instants are digitally controlled by means of a programmable generator of binary codes based on a synchronous counter with parallel loading of preset code. Its clock input is excited with the triger output of the external pulser after being delayed a time, adjustable between 20 and 200 microseconds. This delay must be long enough to permit that all the interesting echoes of a given ultrasonic channel be received before switching to the next channel. The device admits also to address the different channels from an external digital controller. In both addressing cases, the channel codes are sent to a data selector, which successively activate the channels of the power

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switching section through of a binary decoder and an interface formed by 8 inverter buffers with open collector outputs. The analog part of the Mux/Dmux contains eight high-voltage channels of four terminals associated to the eight transducers, having each one an independent ground return line in the electronic card. The eight channels are connected to an input nodal point in order to avoid electrical paths in common. This precaution minimizes the effects of an important source of cross-coupling in the usual analog multiplexers when pulses with very fast transition times have to be switched. In fact, the small inductance associated to a few centimetres of a non-shielded conductor, represents an impedance value that can be not neglected for the frequency range here considered. In this context of the transient driving of transducer in the high frequency NDT range, if the demultiplexer network contains common lines for several channels, the driving current passing through the selected transducer channel would induce spurious signals of notable amplitude on the other outputs of the demultiplexer, exciting improperly the remaining transducers. Each channel includes a power analog switch controlled by current, based in a patented procedure proposed in reference [20]. All the channels sequentially share a 24 volts supply for the maintenance of the conduction state of the switch selected by the digital controller. As principal performances, it must be noted that this device can perform the bi-directional control of broadband pulses in the high frequency range, with peak amplitudes of 400 volts or more, and a very low on-resistance with typical values on the order of 1 ohm. This last characteristic permits to carry peak currents as large as 10 amperes during the initial instants of the pulsed excitation of high frequency transducers. Moreover, with a so low on-resistance, a very good sensitivity and signal-to-noise ratios can be obtained at the ultrasonic reception. The switching time between two channels depends on the load connected at their outputs, but it can be reduced to a few hundred of nanoseconds, which permits its use in real time inspections. Finally, this switching system presents a low level of electrical cross talk between channels, because the overall system has a two-ways off isolation of about 60 dBs in pulse-echo mode.

References 1. J. Krautkramer and H. Krautkramer (1983), “Ultrasonic testing of materials“ Springer-Verlag, Berlin 2. M.G. Silk (1984) “Ultrasonic transducers for nondestructive testing” Adam Hilger Ltd, Bristol

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3. H. Uberall (1973) “Physical acoustics” W. P. Mason Ed., Acad. Press, New York, Vol. 9, pp. 1-20 4. A.J. Hayman and J.P. Weight (1979) “Transmission and reception of short ultrasonic pulses by circular and square transducers” J. Acoust. Soc. Am. 66(4): 945-951 5. E. Riera, A. Ramos and F. Rodríguez (1994) “ Temporal evolution of transient transverse beam profiles in near field zones” Ultrasonics 32 (1): 47-56 6. F. Lakestani, J.C. Baboux and P. Fleischmann (1975) “ Broadening the bandwidth of piezoelectric transducers by means of transmission lines” Ultrasonics 13: 176-180 7. R. Kazys and A. Lukosevicius (1977) “Optimization of the piezoelectric transducer response by means of electrical correcting circuits” Ultrasonics 15: 111-116 8. A. Ramos, J.L. San Emeterio and P.T. Sanz (2000) “Improvement in transient piezoelectric responses of nde transceivers using selective damping and tuning networks” IEEE Trans. Ultrason. Ferroelect. Freq. Cont 47: 826-835 9. A. Ramos, P.T. Sanz and F.R. Montero (1987) “ Broad-band driving of echographic arrays using 10 ns - 500 V efficient pulse generators” Ultrasonics 25: 221-228 10. A. Ramos, J.L. San Emeterio, P.T Sanz and J. García (1999) “Sistema ultrasónico para END de composites laminados con núcleos nomex” in Proc. IX NDE Spanish Congress, Vitoria, AEEND, pp. 43-51 11. J.L. San Emeterio, A. Ramos, P.T. Sanz, A. Ruíz and A. Azbaid (2004) “Modeling NDT piezoelectric ultrasonic transmitters” Ultrasonics 42: 277281 12. A. Ramos, J.L. San Emeterio and P.T. Sanz (1997) “Electrical matching effects on the piezoelectric transduction performance of a through transmission pulsed process” Ferroelectrics 202: 71-80 13. A. Ramos, J.L. San Emeterio and P.T. Sanz (2000) “Dependence of pulser driving responses on electrical and motional characteristics of NDE ultrasonic probes” Ultrasonics 38: 553-558 14. M. Certo, D. Dotti and P. Vidali (1984) “A programmable pulse generator for piezoelectric multielement transducers” Ultrasonics 22 (4): 163-166 15. J.L. San Emeterio, L. Gómez-Ullate (1992) “Diffraction impulse response of rectangular transducers” Journal of the Acoustic Society of America 92: 651662 16. T.L. Rhyne (1996) “Computer optimization of transducer transfer functions using constraints on bandwidth, ripple, and loss” IEEE Trans. Ultrason. Ferroelect. Freq. Cont. 43 (6): 1136-1149 17. P.J. Howard and R.Y. Chiao (1995) “Ultrasonic maximum aperture SAFT imaging in Review of Progress in Quantitative Nondestructive Evaluation (QNDE)” Plenum Press, NewYork, Vol. 14, pp. 901-908 18. L.G. Ullate, A. Ramos and J.L. San Emeterio (1994) “Analysis of the ultrasonic field radiated by time-delay cylindrically focused linear arrays” IEEE Trans. Ultrason. Ferroelect. Freq. Cont. 41 (5): 749-760

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19. A. Ramos, F.R. Montero de Espinosa, P.T. Sanz and J.M. Torregrosa (1993) “A 5 MHz high-voltage demultiplexed ultrasonic array system for rapid-scan testing of advanced materials” Sensors and Actuators: A Physical 37-38: 385390 20. A. Ramos and P.T. Sanz (1996) “Fast bidirectional analog switching system for HF pulses of high instantaneous power” USA Patent No 5,592,031 21. R.C. Addison, K.A. Marsh, J.M. Richardson and C.C. Ruokancas (1982) “NDE Imaging with multielement arrays” in Acoustical Imaging, Plenum Press, London, Vol. 12, pp. 643-663 22. H.C.L. Vos, R. Schepers and J.A. Vocel (1988) “An ultrasonic circular array transducer for pipeline and borehole inspection” in Proceedings of IEEE Ultrasonics Symposium, Chicago, pp. 659-662 23. J.F. Gelly and C. Maerfeld (1981) “Properties of a 2D multiplexed array for acoustic imaging” in Proceedings of IEEE Ultrasonics Symposium, pp. 685689 24. D.K. Peterson, S.D. Bennett and G.S. Kino (1982) “Real-time NDE of flaws using a digital acoustic imaging system” Materials Evaluation 40: 1256-1262

17 Ultrasonic Techniques for Medical Imaging and Tissue Characterization Wagner Coelho de Albuquerque Pereira1, Christiano Bittencourt Machado1 Carlos Alther Negreira2 and Rafael Canetti3 1

Biomedical Engineering Program (COPPE), Federal University of Rio de Janeiro, Brazil 2 Laboratorio de Acustica Ultrasonora, Instituto de Física, Facultad de Ciencias, Universidad de la República, Uruguay 3 Instituto de Ingeniería Eléctrica, Facultad de Ingeniería, Universidad de la República, Uruguay

17.1 Introduction Nowadays, ultrasound (US) imaging plays a key role in diagnostics and follow-up in several medical domains (cardiology, obstetrics and gynecology, orthopedics, oncology, etc.). The use of US echoes in human body has been stimulated after World War II, due to the high development in sonar and radar technologies. In 1940 a supersonic reflectoscope is presented to the world by F. Firestone, using basic transmission and reception of the echoes in order to locate defects in metals [1]. The Dussik brothers (Karl Theodore and Friederich Dussik) were the first clinicians to utilize US as a diagnostic tool. They tried to localize brain tumors in brain ventricles by through-transmission US attenuation image. Dussik published his first paper in 1942 and his additional conclusions after the II World War, in 1947. Since then, US applications have been intensely developed. Different imaging properties were proposed (linear and three-dimensional arrays, dynamic focusing, dynamic apodization, etc.). Consequently, there was an increase in the variety of US imaging modes (A-mode, B-mode, M-mode, Doppler, 3D US, etc.). Even so, diagnostic image analysis from US is not a trivial task. The ultrasound (US) operator (specialist) needs high training and skill levels to acquire and interpret the image displayed [2]. The received echo is the result of a series of scattering and reflections of the propagated pulse along a complex medium: the biological tissue. This medium can undergo subtle and acoustically similar changes for different pathologies which may add A.A. Vives (ed.), Piezoelectric Transducers and Applications, doi: 10.1007/978-3-540-77508-9_17, © Springer-Verlag Berlin Heidelberg 2008

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extra difficulties to the elaboration of a diagnosis. This is the case, for example, of diffuse liver diseases (fibrosis, cirrhosis and steatosis) which stages cannot be easily differentiated only by visual inspection of US images. Consequently, some vital information, in special regarding initial pathological stages, can be inadequately discarded [3]. To address this issue, in the last three decades, investigators have been using essentially spectral analysis methods to evaluate the US radiofrequency (RF) signal (and its envelope), to extract quantitative information about tissue structure, in an attempt to obtain a more effective diagnostic tool. Thus, the so called quantitative ultrasound (QUS) proposes parameters to characterize tissue conditions based on the assumption that diseases change the acoustic properties of the medium. The most common parameters found in literature are US propagation speed, attenuation and backscatter coefficients, mean scatterer spacing (periodicity), and scatterers’ size [4-14]. This chapter aims to present an overview of the basics of US imaging systems (operation characteristics, advantages, limitations, etc.) for medical diagnosis and some signal processing methods currently used for tissue characterization. Detailed information can be found in the bibliographical references.

17.2 Ultrasound Imaging Modes 17.2.1 Basic ultrasonic properties of biological materials The main properties of biological tissues related to the propagation of ultrasonic waves are attenuation, speed of sound, acoustic impedance and elasticity. Wave attenuation is due to absorption and scattering that takes place when ultrasound is propagating in tissues. In soft tissue the absorption, which is the product of relaxation processes [15], varies linearly with the frequency in the usual frequency range in ultrasonic imaging and the wave speed dispersion can be neglected [16]. The variation of acoustic impedance between soft tissues is small, which allows obtaining small amplitude echoes and a good penetration depth. Nevertheless in the interfaces between soft and hard tissues or gases the amplitude of the reflections is very strong. On the other hand, the Young elastic modulus in soft tissues is proportional to the shear elastic modulus and presents a great variation between different soft tissues.

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17.2.2 A-Mode The A-Mode (“A” stands for “amplitude”) is the precursor of all US imaging modes. Basically, it uses the time of flight (TOF) of the echo from the transducer face to a reflector and back again to the transducer. Since the average US propagation velocity in soft tissues is approximately 1,540 m s-1, it is possible to estimate the position of the reflector by using a simple equation:

d=

c.t 2

(17.1)

where d is the depth of the reflector, c is the average US velocity, and t is the time period during echo transmission and reception by the transducer (see Fig. 17.1). Using this principle, if a medium composed by two reflectors is considered, the distance between them can be calculated from the difference between the time-of-flights of their respective echoes. Fig. 17.1 shows such situation, where the distance L between reflectors R1 and R2 can be calculated using the relation:

L=

c (t 2 − t1 ) 2

(17.2)

where t1 and t2 are the TOF of echoes from R1 and R2, respectively. The A-mode is not used in medicine anymore, essentially because biological variability generates echo lines difficult to interpret. Two last remarks should be convenient: it is generally accepted that A-mode is the trace of the envelope of the US radio-frequency wave; and US frequency for biomedical application ranges typically from 1 MHz to 20 MHz for external (on the skin) imaging and up to 100 MHz for intracavitary (transrectal, trans-vaginal, intra-vessels) or biomicroscopic imaging (see sect. 17.2.8). Transducer

R1

R2

L

Fig. 17.1. Transducer receiving echoes from two reflectors. The distance between reflector R1 and the transducer can be found by Eq. (17.1). The separation distance L between R1 and R2 can be found by Eq. (17.2)

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17.2.3 B-Mode Also known as 2D-mode, the B-Mode provides an image of a scanning plane where depth is one of the directions. It can be understood of a series of A-mode lines put one beside the other, thus, giving an idea of the relative positions of reflectors and scatterers. The amplitudes received are coded in gray-scale (512 levels, for instance), as can be seen in Fig.17.2. Initially single-element transducers were used and so it was necessary to manually and gradually displace the transducer along one direction (in the patients abdomen, for instance) to generate the image line by line. Although very useful to diagnosis, this system (also called the static BMode) had some inconveniences like a sophisticated arm to generate spatial coordinates for the transducer position and also the incapacity to see moving structures. In the seventies the linear array transducers started their era. It is composed of several elements cuts from one single piezoelectric bar (usually 128, 264 elements). The elements are excited by group (e.g. 8, each time) and electronically controlled so the 2D scan is made entirely by circuits (Fig. 17.3).

Fig. 17.2. Example of breast US imaging, showing orthogonal views of malign lesion of irregular contour and heterogenic echo texture

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US equipment

Transducer Image Body surface Scanning

Fig. 17.3. Scheme illustrating B-mode imaging principle. Structures are scanned by the US beam and then displayed in grey-scale

This new transducer opened the possibility of more uniform images with two important characteristics: (a) electronic adjustable focusing (including more than one per image); and (b) real-time scanning, which means imaging moving structures. Nowadays, virtually all ultrasonic image transducers have a linear matrix of elements and in several sizes and shapes, appropriate for each application (superficial, intra-cavitary, etc). One important aspect of all B-mode systems is the resolution, which is the ability to represent in the image structures very close to each other, and it can be divided in axial and lateral resolution. The latter one is defined by the beam focus width, so, if one scatterer has its diameter smaller then the focus width, then its image will be larger then it should (Fig 17.4). In the same way, if two scatterers are close to each other and at the same depth they will be imaged separately only if their distance is greater than or equal to the focus width [31]. The axial resolution is influenced by the transmitter pulse length, and the lateral resolution depends on the beam width. It is possible to calculate axial resolution by assuming a medium composed by two scatterers separated by Δz (Fig. 17.5). The interval between the end of the first received US echo and the start of the second one is given by:

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Δt = 2Δz / c − T

(17.3)

where Δt is the time interval between the two echoes, c is the US propagation velocity, and T is the transmitted pulse period (duration). When Δt = 0, Δz = cT/2, and that is the system axial resolution [19]. Transducer

US beam

Image

Fig. 17.4. Definition of lateral resolution. The scatterer diameter is smaller than the focus width, and its image appears artificially stretched. The proper representation happens when scatterer diameter is greater than or equal to he focus width Transducer

R1

R2

∆z

0

T

0

T

∆t

Fig. 17.5. Method for the estimation of axial resolution. The two reflectors can be identified when there is a time difference Δt between the end of the pulse reflected in R1 and the beginning of the pulse reflected in R2. When Δt = 0 then this is the resolution limit

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In practice, the axial resolution can be improved by generating shortduration pulses. However, less energy is carried during propagation. To compensate this deficit, various techniques have been developed. Pulse compression is obtained with frequency or phase modulation of the transmitted pulse, consequently increasing bandwidth [20]. Coded transmission systems using frequency modulated signals (chirps) improve signal-to-noise ratio (SNR), then allowing imaging of structures located more deeply, or even migration to higher frequency range, which results in better image quality [21]. B-mode images always present a granular texture filling the biological structures, known as “speckle” (Fig. 17.2). This phenomenon results from the interaction of ultrasonic wave and randomly distributed irresolvable particles (scatterers) in the medium. The presence of this feature mainly reduces the image contrast and the ability for detecting subtle tissues changes and boundaries. Several techniques for speckle reduction with diverse degrees of success can be found in literature [22-26]. 17.2.4 Other Types of B-mode Images

Besides the classic B-mode imaging, other types of US signal processing and image exhibition have been created along time. Two of them are worth mentioning. Tissue harmonic imaging and contrast agents

The reduction in image clutter can be obtained by restricting signal detection to the second harmonic frequency backscattered by ultrasonic microbubble contrast agents. The contrast-media imaging uses contrast agents, usually gas-filled microbubbles encapsulated in a biodegradable shell. The bubbles diameter ranges from 2 to 6 µm [19]. They are placed inside the desired organ (e.g., through a vein). The bubbles behave in a nonlinear fashion when the ultrasonic pressure amplitude is sufficiently high; however the biological soft tissues also have this property and the harmonic imaging using these properties yields a more enhanced resolution and reduced acoustic noise [27]. In fact, ultrasound backscattered by tissues has harmonic frequency components that can be detected if the incident ultrasonic pressure is sufficiently high. As with microbubble contrast agents, it is the second harmonic frequency that is most relevant. The fundamental beam is generated at the transducer, and the harmonic beam is produced along the propagation axis, following local instantaneous

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pressure amplitude. The harmonic beam presents a narrower bandwidth than the fundamental one, since its frequency is the double. In the harmonic imaging procedure the receiver is tuned to the second harmonic of the transmitted ultrasonic frequency. For this reason the echoes detected from tissues close to the transducer are weak, because second harmonic generation builds up to a higher level only if the ultrasonic wave has an adequate tissue penetration. In addition contrast resolution is improved because the side lobes of the transmitting and receiving beams have different structures [28]. In this way the spatial resolution is improved, because the distortion of the transmitted beam by the inhomogeneity of the superficial layers is small and the receiving beam can be made narrower than the transmitted one [29]. It is also worth mentioning the pulse inversion imaging technique. This technique aims at overcoming the limitations of tissue harmonic imaging. The overlap of the fundamental and harmonic spectrum results in echoes detected in the harmonic signal, then reducing the image contrast. The pulse inversion principle consists of generating two US pulses, with the second pulse inverted. When the echoes are added, the linear component will cancel, but the nonlinear component is reinforced, producing a stronger signal [28, 29]. It has been also applied in Doppler (pulse inversion Doppler) [30]. 3D ultrasound imaging

Another interesting imaging technique is the three-dimensional ultrasound (3D-US) ; which is a volume reconstruction based on a series of 2D conventional images. It exploits the real-time US ability to construct a volume and is used in many clinical areas, like pregnancy imaging, more accurate organ measurements and as a guide for interventional procedures [29]. The major reason for the increase in the use of 3D ultrasound is related to the limitations of 2D viewing of 3D anatomy, using conventional ultrasound. Different 3D ultrasound scanning techniques can be mentioned. Nevertheless the principal 3D systems make use of conventional 1D ultrasound transducers to acquire a series of 2D ultrasound images, and differ only in the method used to determine the position and orientation of these 2D images within the 3D image volume being examined. To prevent distortions it is necessary to consider three factors [15]: 1. The scanning technique must be either rapid or gated, to avoid image artifacts due to involuntary, respiratory or cardiac motion.

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2. The locations and orientations of the acquired 2D images must be accurately known, to avoid geometric distortions in the 3D image that would lead to measurement errors. 3. The scanning apparatus must be simple and convenient to use, so that the scan is not complicated or awkward to perform and hence easily included in the examination procedure. 17.2.5 Doppler Imaging

The Doppler effect was apparently first described out for both light and sound waves by Johann Christian Doppler, professor of mathematics at the Realsehule in Prague, in 1842. It was first tested for sound waves in 1845, by the Dutch meteorologist Christopher Heinrich Dietrich Buys-Ballot. This effect can be described as the frequency shift underwent by a propagating wave after being reflected by a moving target. The original emitting frequency (fo) is increased (fo+fd) or decreased (fo-fd), if the target is approaching or going away form the emitting source, respectively. The phenomenon can be equally observed if the emitting source is moving instead of the target [30]. Considering the same transducer for transmission and reception, the Doppler shift frequency is given by the following equation [29, 30]: fd =

− 2Vf 0 cos(θ ) c

(17.4)

where V is the velocity of the target (scatterer), f0 is the frequency of the emitted pulse, and θ is the angle between the US beam and the direction of movement (Fig. 17.6). Transducer

θ Scatterer

V

Direction of the scatterer movement

Fig. 17.6. The Doppler effect for a single-element transducer in transmission and reception of the US pulse

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In biomedical applications, the transducer beam is directed to the blood flow in arteries or veins. The red cells (erythrocytes) act as the moving targets and then as a moving source (which means that the phenomenon happens two times). Following Eq. (17.4), the frequency shift is proportional to the flow velocity. Since the red cells have a distribution of velocities along the cardiac cycle, Doppler signals from blood flow are a mixture of frequency shifts proportional to the number of cells in each velocity. The continuous-wave (CW) Doppler system has two piezoelectric elements in the same transducer: one continuously transmitting US waves and the other receiving the corresponding echoes. These echoes are processed by a simple electronics (amplification and demodulation) and then the Doppler shift signal (which is within the audio bandwidth), can be processed and registered and by standard means (zero-crossing technique, time-frequency methods) or simply heard with headphones. The main drawback of CW Doppler is that it cannot separate signals from vessels that are along the beam, nor different regions of the same vessel. The US pulsed Doppler was designed to overcome this limitation. It is based on the principle that the receiving circuit is time-gated, so once the pulse is emitted the gate is opened to receive only echoes coming from a certain depth. Pulses are sent in a certain repetition frequency that is enough to sample (capture) the blood flow in different moments of the cardiac cycle so as to reconstruct it. In the same way the signal can be registered and processed mainly with time-frequency methods [31, 33-35]. The use of US Doppler is concentrated in the evaluation of blood flow in human body vessels, as well as in heart chambers. In B-mode images the Doppler signal is usually transformed in a color pattern placed inside the vessels and cardiac images. More recently it is used to monitor cerebral flow of patients in critical conditions, but in that case the transducers are a bi-dimensional matrix of elements and the circuitry is designed to maintain the beam always in the position with highest receiving amplitude [36]. 17.2.6 Ultrasound Computed Tomography (US-CT)

The US-CT aims at reconstructing the acoustic properties of a medium with a sequence of measurements. The image to be analyzed is represented as a function f(x,y), and the projection (called ray-sum) is calculated as [17]:

pθ (r ) =

∫ f ( x, y)ds

r ,θ

(17.5)

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where θ is the angle of the incident beam direction, and r is the location of the scatterers. Moreover, f(x,y) can be reconstructed from the following inverse Fourier transform:

f ( x, y ) = ℑ −21D [ℑ1D {p (r , θ )}]

(17.6)

i.e., the image can be obtained by the inverse 2D Fourier transform of a sequence of Fourier transform obtained by changing the angle θ of the function pθ(r). For each angle it is obtained a profile of US received signals (Fig. 17.7). The image reconstruction theory requires the angle to vary continuously between 0-360º and the propagation to obey geometrical laws (straight-line propagation). In practice, θ could assume discrete values and, e.g., for the case of X-rays, geometrical propagation is a reasonable assumption (that is why we have X-rays Computed Tomography systems largely diffused in clinical practice). But in the case of ultrasound, the main experimental problem is that it is virtually impossible for the US wave to propagate in straight lines inside the human body, because important phenomena like diffraction, refraction and scattering compromises this assumption. Moreover, the beam should have enough intensity to cross the whole body, to overcome strong acoustic attenuation, which brings biological safety concerns. This approach is called transmission tomography. A backscattered tomography has also been proposed which was experimentally simpler (one transducer to send and receive) and less costly, but the experimental propagation problems remained essentially the same, that is why this kind of tomography has no clinical application so far [37, 38]. 1D Fourier Transform

Inverse 2D Fourier Transform Transducer

f(x,y)

Fig. 17.7. Obtaining the function f(x,y) related to the propagation medium, by the inverse 2D Fourier transform of a sequence of 1D Fourier transform of the raysum pθ(r) for each beam incidence angle θ

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17.2.7 Ultrasound Elastography

In Medicine, palpation is a very useful clinical technique to detect different abnormalities. Palpation investigates the variation in the stiffness of different tissues and that can reveal the presence of abnormalities (if they are close to the surface). The most classical example is the detection of breast cancer, because malignant tumor tissue is harder than normal glandular and fatty tissues. The goal of ultrasonic elasticity imaging is to map tissue properties such as Young’s modulus (or stiffness), Poisson’s ratio and viscosity [39]. In soft biological tissues the Young modulus is proportional to the shear elastic modulus. The general expression for the Young modulus is: E=μ

3λ + 2µ λ+µ

(17.7)

The elastic compressional modulus value in soft biological tissues is

λ = 2.5 GPa and the shear elastic modulus value is µ = 25 kPa (µ << λ). This fact implies that: E ≈ 3μ

(17.8)

In the Figs.(17.8) and (17.9) it is seen the variation of elastic volumetric and shear modulus for different tissues. It can be observed that the shear modulus presents a very strong variation. This fact is the key for the elasticity imaging techniques.

Liquids

Soft tissue

Bone

K (MPa) 104 103 Fig. 17.8. Elastic volumetric modulus of biological tissue

17 Ultrasonic Techniques for Medical Imaging Liver Relaxed muscle Fatty tissue

10

0

10

1

Connective tissue Contracted muscle Nodules

10

2

10

3

Cartilage

10

4

10

5

445

Bone

10

6

10

7

Shear modulus (kPa)

Fig. 17.9. Elastic shear modulus of biological tissues. The shear modulus shows a large range of variation

Nowadays, there are different methods of ultrasonic elasticity imaging. It can be mentioned the vibration amplitude sonoelastography where a low-frequency (20–1000 Hz) vibration is externally applied to excite internal tissue motion of which an image is produced by Doppler detection. A second method is the vibration phase gradient sonoelastography, in this case both the amplitude and the phase of the externally-excited lowfrequency internal tissue motion are measured. It is known that viscosity at low frequencies is negligible and that shear waves predominate, then it is possible to obtain phase gradient images of thigh muscle under various conditions of active muscle contraction [40]. The value of Young’s modulus increases with increasing contraction. A third imaging method is the compression strain elastography, in which the tissue is externally compressed and pre- and post-compression ultrasonic A-scan line pairs are cross-correlated to produce a set of strain profiles [41,42]. A very interesting modification of the method [43] is using an intravascular imaging system to obtain strain elastograms of diseased arteries in vitro, with the change in strain being produced by change in pressure. Presently different researchers try to use this method in vivo, by making use of the arterial pressure pulse [44]. Another group [45] have devised a method of compression strain elastography with relatively large displacements (around ten wavelengths), measuring the overall displacement by summing the small displacements resulting from incremental step loading from the A-scan cross-correlations. It is necessary also to mention the works of Greenleaf’s laboratory mainly in magnetic resonance elastography and ultrasound stimulated vibro-acoustography [46].

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Finally, it could be mentioned the significant works in transient elastography of M. Fink and his group [47-49]. Ultrasound signals are used to follow the propagation in the tissue of a low-frequency shear elastic wave (generated by transient mechanical pulse), searching to measure the shear wave speed that depends on tissue stiffness. The ultrasonic technique of interferometry speckle [50] is used to detect the local amplitude displacements of the tissue generated by the low frequency pulse. In this way it is possible to measure the shear wave speed and the attenuation coefficient. The displacements equations for a plane monochromatic wave are: ∇ 2 u p + k p2 u p = 0

(17.9a)

∇ 2 u s + k s2 u s = 0

(17.9b)

where up and us are the amplitude displacements of the longitudinal and shear waves, respectively, being k p2 =ρω2/(λ+2μ) and k s2 = ρω2/μ the corresponding wave numbers, with ω the angular frequency and λ= λ1+iωλ2 and µ=μ1+iωμ2 the complex Lamé coefficients. The shear wave speed cs and the shear attenuation coefficient αs become: cs =

ω Re(k s )

α s = Im( k s ) =

=

(

2 μ12 + ω 2 μ 22

)

(17.10a)

ρ ⎛⎜ μ1 + μ + ω μ 22 ⎞⎟ 2 1

⎝

2

⎠

ρω 2 ⎛⎜ μ12 + ω 2 μ 22 − μ12 ⎞⎟ ⎝

(

2 μ +ω μ 2 1

2

2 2

)

⎠

(17.10b)

By measuring these parameters it is possible to obtain the shear elastic µ1 and shear viscosity µ2 modulus as:

μ1 =

ρ c s2

and μ 2 =

1

ρ μ1c s2 − μ12 2 1 −η

(7.11) ω ⎛ 2 ⎞ ⎟ 1 − η 2 ⎜⎜ 1 − 2 ⎟ ⎝1 −η ⎠ 2 2 where η = (α s c s / ω ) In a typical experimental configuration of transient elastography, the acquisition of an ultrasonic signal can be regarded as instantaneous compared to the time scale of the low frequency waves. Each one of these signals is thus instantaneous which accounts for the position of the scatterers that belong to an elementary volume of the tissue (isochrones volumes [51]. The movement of the scatterers is reconstituted by comparing the

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ultrasonic signals between each other with a method of intercorrelation. This treatment can be divided in three phases. The first, the elementary phase, consists in calculating the displacement of a small group of scatterers between two consecutive ultrasonic signals. The second phase consists in calculating the displacement of several elementary volumes between two consecutive ultrasonic signals. Finally, the third one is an iteration of the first two phases. It consists in calculating the displacement of several elementary volumes between several consecutive ultrasonic signals. The final result is a matrix which contains elementary displacements of volumes in the course of time. This matrix is visualized in the form of a seismogram, and accounts for the movements of the scatterers to all depths corresponding to each ultrasonic signal. In Fig. (17.10) the shear waves (S) are visualized in the seismogram (the shear wave amplitude is much bigger than the longitudinal one, P) and the phase of the wave is calculated using the slope of the corresponding (S) line. In this way, the shear wave speed is obtained for an angular frequency ω of the mechanical excitation. The attenuation coefficient of the shear wave is also measured according to the penetration depth as: ⎛ ∂ϕ ⎞ c sϕ = ω ⎜ ⎟ ⎝ ∂z ⎠

−1

(17.12)

Csφ is the wave speed, φ is the phase and z is the depth. The ultrasonic technique of interferometry speckle thus leads to a matrix which contains the speed profile of elementary volumes of matter according to time and depth. The whole of these measurements is visualized in the form of a seismogram. One can see the attenuation of the low frequency wave there, the amplitudes are more important for the layers on the side of the piston on the left. One also distinguishes phase delaying due to the wave propagation; the sinusoids phase delays increase as they are located far from the piston (depth). The speed of these low frequency waves can be measured starting from such a seismogram. In the example of the figure it is 2.94 m/s. Such a low speed value constitutes a strong argument to determine the shear nature of the low frequency wave. The image contrast between the different types of tissues (healthy and pathological) obtained by this method is much greater in the case of the elastogram than in an echographic B-scan. In Fig. (17.11) it can be seen the results of an in vitro experiment made with a phantom of biological tissue having a spherical inclusion of higher hardness than its surroundings. The relation between elasticity modulus values of the inclusion and of the surroundings is similar to those of a cancerous versus healthy tissue. Studies based on this technique can be mentioned in addition, where the

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creation of another type of waves like head, lateral and surface waves is analyzed [53,54]. Longitudinal wave P – Shear wave S 0 10

Depth (mm)

20 30 40 50 60 70 80 0

20

40

60

80

100

120

Time (ms) Csφ = 2.94 ± 0.07 m·s-1 μ1 = 9043 ± 450 Pa

Fig. 17.10. In the seismogram the (S) line indicates the shear wave maximum amplitude propagation in depth. The (P) line indicates the amplitude of longitudinal wave. The slope of the (S) line allows obtaining the shear speed and therefore calculating the shear elastic modulus of the sample. Courtesy from MSc. N. Benech and adapted from [52] inclusion scan

scan

30 35

us transducer

Normal “tissue”

mm

40

Mechanical excitation

45 50 55

Seismogram

65

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0

5

10

15

20

25

30

35

40

45

50

15

20

25

30

35

40

45

50

Elastogram mm

30

Normal “tissue”

55

5

35

50

40

45

Pathological (hard) “tissue”

40

mm

Displacement (mm)

60

0

45

35

50

30

Normal “tissue”

25 20

0

10

20

30

40

55 50

Time (ms)

60

70

80

Echogram mm

Fig. 17.11. The figure shows an elastogram and a B-Scan of the tissue sample with a stiff inclusion. The elasticity image is much clearer that the echographic image. Courtesy from MSc. N. Benech and adapted from [52]

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Ultrasonic elasticity imaging has not yet been used, except rather crudely, in routine clinical practice. It is a very promising method, however, because it is expected to have spatial resolution at least comparable with that of real-time grey-scale imaging, as well as potentially better tissue discrimination. 17.2.8 Ultrasound Biomicroscopy (UBM)

Using high-frequency transducers (20 – 100 MHz), UBM generates biological tissue images on a microscopic level. Unfortunately, there is a price to pay: low depth of penetration (frequency-dependent attenuation), varying from 4 to 7 mm. Axial resolution lies between 20 and 200 µm [55, 56]. In comparison with conventional microscopy, UBM has many advantages: it is noninvasive, allows the study of in vivo structures in realtime, and it is a low-cost procedure and a relative simple method. Many studies have been developed in several clinical domains (ophthalmology, dermatology, intravascular imaging). For example, an evaluation of the anterior portion of the eye can be performed at 50 MHz [55, 57]; skin thickness can be imaged using transducers with up to 100 MHz [58]. In Fig. 17.12 there is a schematic representation of UBM physical principle. Figures 17.13a-b present an example of BMU images at 50 MHz from tissue samples immersed in a bath with saline solution, positioned under a sapphire plate and covered with a PVC membrane (in order to avoid specimen displacement). Transducer

Spherical Lens Water

Specimen

Specular reflector

Fig. 17.12. Representation of a scanning acoustic microscope: the transducer is attached to a spherical lens that guarantees US focusing and the specimen is placed on a specular reflector

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a

b Fig. 17.13. Example of BMU at 50 MHz from rat colon tissue: (a) without ulcer, with the presence of the muscular mucosa layer (MM), a fold (Pr) and the mucosa layer (M); (b) with ulcer (Ul) and mucosa layer (M). In both images, it can be observed the PVC membrane (Mpvc) and the sapphire plate surface (Sa). Vertical and horizontal scales are 100 micrometers/division. Courtesy from Dr. J. C. Machado and Dr. M. Soldan

17.2.9 Computer-Aided Diagnosis in Ultrasound Images

All ultrasonic imaging types that are used nowadays in clinical routine (essentially B-mode, Doppler and their modified versions) are visually interpreted by specialists and thus, strongly dependent on their experience, which brings some degree of uncertainty to the final diagnosis. One possible solution that was proposed is the image quantification by image processing techniques and the construction of an application system, commonly known as Computer-Aided Diagnosis (CAD).

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The basic proposal of CAD systems is to present additional information for the specialists to deal with in the development of final diagnosis. Ultrasonic image quantification consists basically of detecting lesion shape and defining parameters to characterize its contour and texture. This approach is based on the fact that benign lesions tend to have homogeneous textures and well-defined contours while malignant lesions usually tend to infiltrate the surrounding healthy tissue and thus have heterogeneous textures and irregular contours. CAD systems can be divided in three main tasks: lesion detection, quantification and classification, which will be briefly described as follows: 1. Detection involves: (i) pre-processing, where some kind of filtering is usually applied to homogenise image sectors; (ii) segmentation, where image is divided in regions of similar grey-scale levels and then a segmented region is selected as containing the lesion; and (iii) contour detection, which identifies the lesion boarders [59-62]. 2. Parameterization of contour [63] and texture [64-66], which is the quantification of these entities by parameters like circularity, roughness, area ratio (for contour) or entropy, contrast, angular momentum (for texture). 3. Classification of the lesion, where some linear or nonlinear technique (e.g., linear discriminant analysis, neural networks) is implemented to try to separate malignant from benign lesions, by processing their parameters [67-69]. Examples of automatic delineation of lesion contours can be seen in Fig. (17.14a-b), where it is evident the more detailed contour made by the algorithm as compared to the one made by a specialist. It is also possible to see the texture difference from the lesion to its surroundings. These details in contour and texture are quantified by parameterization. In ultrasound images, CAD systems are being developed mainly for breast cancer diagnosis as it is one of the most important health problems worldwide. Up to now the best research results point to an accuracy performance of 85% to 95%. Nevertheless, available commercial CAD versions have still a very limited success, once they frequently mark incorrect regions and/or miss the important one. One probable cause is due to its high level of automated steps, and now semi-automated CAD systems are object of research, where the specialists indicate the region to be analysed. A powerful and trustful CAD system is yet a subject of scientific investigation.

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a

b Fig. 17.14. a Example of automatic delineation for a malignant breast lesion (left) and respective manually designed contour by the specialist (right). Computer algorithms can give more contour details then the specialist. Texture inside the lesion is different from outside. These characteristics are quantified by parameterization (Courtesy from Dr. A. V. Alvarenga); b Example of automatic delineation for a benign breast lesion (left) and respective manually designed contour by the specialist (right). In this case both contours are more similar to each other as compared to the malignant case. Nevertheless important differences are evident. Texture inside the lesion is homogeneous and different from outside. These characteristics are captured by parameterization (Courtesy from Dr. A. V. Alvarenga)

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17.3 Quantitative Ultrasound (QUS) Ultrasonic imaging techniques are extremely important in nowadays clinical routine, aiding in the process of medical diagnosis. However, image analysis is highly subjective, thus, dependent on the specialist’s experience. Moreover, in some cases it is virtually impossible to visually differentiate the early stages of pathologies (beginning of tumor formation, slight contour changes, etc.), that is the main reason to develop quantitative methods for ultrasound tissue characterization. This goal has been pursued for the last four decades with still very limited application in clinical practice; nevertheless some of them are worth mentioning and it still a main subject of research. The technique aims at characterizing biological tissues by means of defining parameters related to their acoustical properties. Several US parameters have been proposed, estimated from RF backscattered signals and/or their envelopes acquired form in vitro tissue samples. 17.3.1 Speed of Sound (SOS)

The speed of sound (SOS) can be estimated using the substitution method [70, 71]. Standard experimental setup includes a water tank, containing a recipient with saline water. A biological tissue sample is placed inside it, over a perfect steel reflector. A transducer is mounted on a mechanical support connected to micrometric stepping motors for lateral (XY - plane) and vertical (Z - axis) movements. The received pulses are amplified and digitized using an oscilloscope (or a digitizing board), and then stored in a computer for posterior analysis (Fig. 17.15). Transducer Saline water

Tank Water bath

Liver sample

Steel reflector

Fig. 17.15. Experimental setup for RF backscattered signals collection, used for liver samples [14]

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Three measurements are made: the time of flight (TOF) for the transducer-steel plate distance with (t1) and without (t0) the specimen and the TOF ts for the distance between transducer and water/specimen interface. The speed of US in saline water vsw can be easily calculated once the distance transducer-steel plate is known (Fig. 17.16). To estimate SOS, the equation used is ⎛t −t SOS = v sw .⎜⎜ 0 s ⎝ t1 − t s

⎞ ⎟⎟ ⎠

(17.13)

Several works have been developed using this technique [9, 71-73] and it can be found tables with speed of sound for several tissues [17, 74]. For example, mean SOS in soft tissues is 1,540 m/s; for water (200C), 1,480 m/s; for bone, 3,500 m/s [31]. The SOS measured is the longitudinal component of velocity once shear waves are of negligible amplitude in soft tissue. Tissue

Steel plate

Transducer

ts

t1 t0

Fig. 17.16. Schematic representation of the substitution method for SOS estimation

17.3.2 Acoustic attenuation coefficient

Attenuation is defined as the amount of US energy that was somehow “lost” in the propagation path and did not reach the receiving transducer. In fact, several phenomena contribute to diminish US amplitude (refraction, scattering, absorption, reflection, etc.), but it is generally accepted by theory that attenuation is basically the result of scattering and absorption. Theoretically, the intensity I in a given propagation distance x, for incident

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intensity I0 and an intensity attenuation coefficient α, can be modeled as the following exponential equation:

I ( x) = I 0 exp(−α x)

(17.14)

where α has units Neper/cm, and when multiplied by 4.3 it becomes dB/cm [74]. Experimental procedures also use the substitution method and the estimation of the attenuation coefficient α(f) can be made by the expression [72,75, 76]:

α( f ) =

⎡ S0 ( f ) ⎤ 1 .10 log10 ⎢ ⎥ 2.L ⎢⎣ S1 ( f ) ⎥⎦

(7.15)

where L is the sample thickness, S0(f) is the power spectrum of the reflected signal from the steel plate without specimen, S1(f) is the power spectrum of the reflected signal from the steel plate with the specimen. An approximately linear attenuation curve is then obtained, and its slope is taken as the attenuation coefficient (in dB.cm-1.MHz-1) (Fig. 17.17). Actually, a mean attenuation value is estimated from signals of several different points in sample. Steel plate Tissue Transducer

Division

Fig. 17.17. Schematic representation for attenuation estimation using the substitution method. Dividing the power spectrum of the reflected signal from the steel plate without the specimen S0(f), by the power spectrum of the reflected signal from the steel plate with the specimen S1(f), it is obtained a curve which is a function of frequency α(f) and whose angular coefficient is obtained by linear regression

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Tables with attenuation values for biological tissues can be found in literature [5, 17, 74]. For instance, at 1 MHz: blood (0.20 dB/cm), skin (0.35 dB/cm) and liver (0.50 dB/cm) [17]. 17.3.3 Backscatter coefficient

This parameter aims to estimate the amount of energy that is scattered back to the transducer face. For example, for a spherical scatterer, the scattered intensity Is can be expressed as [1]: (7.16)

I s = σ t I i / 4πr 2

where σt is the scattering cross-section, Ii is the incident intensity, and r is the sphere radius. The term σtIi in Eq. (17.16) is the scattered power. The biological tissue has a density of scatterers of different shapes and sizes (fibers, vessels, cysts, ducts, etc.), then the backscatter coefficient measures the global scattering phenomena. The backscatter coefficient can be estimated by using a spectral method like a short-time Fourier transform (STFT) [72, 73], allowing the correction for beam diffraction [77]. The σ(f) can be then calculated with the help of the following equation: Ss ( f , F)

1 ⋅ ⋅ σ(f ) = S p ( f , F ) (0.63) 2

(17.17)

R p2 k 2 a 2 ⎡ ⎛ ka 2 8πd ⎢1 + ⎜⎜ ⎢ ⎝ 4F ⎣

⎞ ⎟ ⎟ ⎠

2

⎤ ⎥ ⎥ ⎦

where F is the focal length, a is the transducer radius, k is the wave number, Rp is the amplitude reflection coefficient of the reflector plane (normally assumed to be 1), S s ( f , F ) is the spatially averaged apparent backscattered power spectrum of the sample at the focal length, and S p ( f , F ) is the reference power spectrum obtained from a plane reflector at the focal length. The factor 1/(0.63)2 can be introduced to compensate the application of a Hamming gating function, used to select the region of interest (ROI) in received RF signal. The unit is (cm·Sr)-1. Several backscatter coefficients can be found in literature for biological tissues. For instance, Wear and colleagues [78] reported average value of 2.9 x 10-4 (cm·Sr)-1 for healthy human liver tissue. Another parameter derived from backscatter coefficient is the Integrated Backscatter Coefficient (IBC) [7, 71]. It is given by the Eq. 17.18, and is an integral of the backscattered coefficient given in decibels:

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[η ( f )] ∫ = f max

IBC dB

dB df

f min

457

(17.18)

f max − f min

where the term η(f) is equal to 10log10(σ(f)). The parameter σ(f) can be calculated using Eq. (17.17). IBC can also be used to characterize tissues [71, 79-81]. The human liver tissue has an IBC of -31.17 dB, for example [71]. 17.3.4 Periodicity Analysis: the Mean Scatterer Spacing (MSS)

Some biological tissues present a quasi-periodic structure (e.g, the liver and spleen), and the presence of pathologies may alter these patterns. Ultrasound backscattered signals can also play a role in the characterization the periodicity pattern of biological media. Assuming a linear system approach, the impulse response of a periodic medium can be modeled as a sum of quasi-periodic scatterers with irregularly distributed ones. Thus, the tissue microstructure through which the US pulse propagates can be modeled as [82]: x(t ) =

Nr

∑ n =1

a n δ (t − τ n ) +

Nr

∑ v δ (t − θ n

n)

(17.19)

n =1

where t is the time-axis (related to the distance by the velocity of the pulse) δ is the Dirac impulse; τn is the time-delay between neighbor regular scatterers and usually follows a Gamma distribution; in a similar way, θn is the time-delay between neighbor irregular scatterers and usually follows a uniform distribution; Nr and Nd are the total number of regular and diffuse scatterers, respectively; finally, an and vn represent the reflectivity of the nth regular and diffuse scatterers, respectively. They both follow a uniform distribution (typically between 0 and 1). The basic physical principle is that, when a US wave propagates through a medium composed of regular and diffuse scatterers (see Fig. 17.18), if the tissue structure presents a quasi-periodic pattern (i.e., distances between similar scatterers approximately equal), this characteristic will be acknowledged by the RF backscattered signal. In the example of Fig. 17.18, periodicities can be seen separated by approximately a constant time-delay. Thus, by calculating the signal power spectrum, a peak will be formed at the frequency related to that regularity. In the simple example given, no heavy processing is needed to estimate periodicity, however, in real world periodicity can be completely hidden in the RF signal and that is where mathematical methods can be helpful.

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The methods commonly used propose the characterization of a periodic medium by estimating its mean scatterer spacing (MSS) from US signals and that can be obtained by the following equation: MSS = c / 2 f

(17.20)

where c is the US propagation speed in the medium, and f is the frequency related to the maximum peak in the spectrum, that corresponds to the periodicity of the scatterers (as explained before). The US propagation speed can be estimated by the substitution method as in the topic 17.3.1. Thus the methods are basically used to estimate f, and then, MSS can be found using equation (17.20). Typical MSS values for biological tissues range from 0.5 to 2.00 mm. Several spectral analysis methods have been proposed: autocorrelation function [83], cepstrum [84], spectral autocorrelation - SAC [11, 82, 85], quadratic transformation of the RF signal [12], wavelet transforms [86], generalized spectrum [87], and the singular spectrum analysis – SSA [13, 88, 89]. These methods were already applied in tissues like liver, spleen and breast. All methods work well for simple signals, nevertheless, performance is progressively diminished due to (a) the growing of energy ratio for diffuse-to-regular scatterers; and/or (b) the variation in the regularity of the periodic scatterers (also know as jitter). The main difficulties in estimating periodicities is that strong isolated reflectors (like vessels or fibers) may generate stronger spectral peaks than the ones related to periodicity, thus, making more complicated the task of finding the right peak. Transducer Regular scatterers

Diffuse scatterers

Amplitude (arbitrary scale)

Time (μs)

Fig. 17.18. Schematic diagram of a US wave propagating in a medium composed of regular and diffuse scatterers. The periodicity is present in the US backscattered signal, although not always visually identified as in this case

Literature points out that none of the present parameters is able to discriminate conveniently healthy from pathological tissues, thus discrimination

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studies with more than one method, or using more than one acoustic parameter, have been developed, resulting in higher degree of accuracy (up to 85%) [14, 54, 90]. Acknowledgements

The authors would like to thank Dr. André Alvarenga, Dr. João Carlos Machado, Dr. Mônica Soldan and MSc. Nicolás Benech for the ultrasound images.

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T.L. Szabo (2004) “Diagnostic ultrasound imaging”, Oxford: Elsevier Academic Press 2. P.N.T. Wells (2000), “Current status and future technical advances of ultrasonic imaging” IEEE Engineering in Medicine and Biology 19: 14-20 3. W.C. Yeh, S.W. Huang and P.C. Li (2003) “Liver fibrosis grade classification with B-mode ultrasound” Ultrasound in Medicine & Biology 29 (9): 12291235 4. F.A. Duck, A.C. Baker and H.C. Stanitt (1998) “Ultrasound in medicine” Philadelphia: IOP Publishing Ltda 5. K.K. Shung, M.B. Smith and B. Tsui (1992) “Principles of medical imaging” San Diego: Academic Press 6. F.L. Lizzi, E.J. Feleppa, M. Astor and A. Kalisz (1997) “Statistics of ultrasonic spectral parameters for prostate and liver examination” IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 44 (4): 935-942 7. C. Fournier, S.L. Bridal, G. Berger and P. Laugier (2001) “Reproducibility of skin characterization with backscattered spectra (12–25 MHz) in healthy subjects” Ultrasound in Medicine & Biology 27 (5): 603-610 8. B.J. Oosterveld, J.M. Thijssen, P.C. Hartman and G.J.E. Rosenbusch (1993) “Detection of diffuse liver disease by quantitative echography: dependence on a priori choice of parameters” Ultrasound in Medicine and Biology 19 (1): 2125 9. J.M. Thijssen, B.J. Oosterveld, P.C. Hartman and G.J.E. Rosenbusch (1993) “Correlations between acoustic and texture parameters from RF and B-mode liver echograms” Ultrasound in Medicine and Biology 19 (1): 13-20 10. A.F.W. van der Steen, J.M. Thijssen, J.A.W.M. van der Laak, G.P.J. Ebben and P.C.M. de Wilde (1994) “Correlation of histology and acoustic parameters of liver tissue on a microscopic scale” Ultrasound in Medicine and Biology 20 (2): 177-186 11. T. Varghese and K.D. Donohue (1993) “Characterization of tissue microstructure scatterer distribution with spectral correlation” Ultrasonic Imaging 15: 238-254

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60. R.F. Chang, W.J. Wu, W.K. Moon and D.R. Chen (2005) “Automatic ultrasound segmentation and morphology based diagnosis of solid breast tumors” Breast Research and Treatment 89: 179-185 61. K. Drukker, M.L. Giger, K. Horsch, M.A. Kupinski, C.J. Vyborny and E.B. Mendelson (2002) “Computerized lesion detection on breast ultrasound” Medical Physics 29: 1438-1446 62. A.V. Alvarenga, W.C.A. Pereira, A.F.C. Infantosi and C.M. Azevedo (2007) “Complexity curve and grey level co-occurrence matrix in the texture evaluation of breast tumor on ultrasound images” Medical Physics 34: 37963. 387 Y. Chou, C. Tiu, G. Hung, S.C. Wu, T.Y. Chang and H.K. Chiang (2001) “Stepwise logistic regression analysis of tumour contour features for breast ultrasound diagnosis” Ultrasound in Medicine & Biology 27: 1493-1498 64. S. Baheerathan, F. Albregtsen and H.E. Danielsen (1999) “New texture features based on the complexity curve” Pattern Recognition 32: 605-618 65. A. Al-Janobi (2001) “Performance evaluation of cross-diagonal texture matrix method of texture analysis” Pattern Recognition 34: 171-180 66. B.S. Garra, B.H. Krasner, S.C. Horii, S. Ascher, S.K. Mun and R. K. Zeman (1993) “Improving the distinction between benign and malignant breast lesions: The value of sonographic texture analysis” Ultrasonic Imaging 15: 267–285 67. J. Kilday, F. Palmieri and M.D. Fox (1993) “Classifying mammographic lesions using computerized image analysis” IEEE Transactions on Medical Imaging 12:664–669 68. M.L. Giger, H. AI-Hallaq, Z. Huo, C. Moran, D.E. Woiverton, C.W. Chan and W. Zhong (1999) “Computerized analysis of lesions in US images of the breast” Acad Radiology 6: 665-674 69. K.G. Kim, J.H. Kim and B.G. Min (2002), “Classification of malignant and benign tumors using boundary characteristics in breast ultrasonograms” Journal of Digital Imaging 15: 224-227 70. A.F.M. van der Steen, H.M. Cuipers, J.M. Thijssen and P.C.M. de Wilde (1991) “Influence of histochemical preparation on acoustic parameters of liver tissue: a 5-MHz study” Ultrasound in Medicine & Biology 17(9):879891 71. M. Meziri, W.C.A. Pereira, A. Abdelwahab, C. Degott and P. Laugier (2004) “In vitro chronic hepatic disease characterization with a multiparametric ultrasonic approach” Ultrasonics 43(5): 305-313 72. V. Roberjot, S.L. Bridal, P. Laugier and G. Berger (1996) “Absolute backscatter coefficient over a wide range of frequencies in a tissue mimicking phantom containing two populations of scatterers” IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 43: 970-978 73. F.L. Lizzi, D.L. King and M.C. Rorke (1988) “Comparison of theoretical scattering results and ultrasonic data from clinical liver examinations” Ultrasound in Medicine & Biology 14: 377-385 74. F. W. Kremkau (2005) “Diagnostic ultrasound: principles and instruments” Saunders.

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75. R. Kuc and M. Schwartz (1979) “Estimating the attenuation slope for liver from reflected ultrasound signals” IEEE Transactions on Sonics and Ultrasonics SU-26: 353-362 76. F. Hottier and J.L. Bernatets (1984) “Estimation of ultrasonic attenuation in biological tissues” Acta Electronica 26 (1-2): 33-58 77. M. Fink and J.F. Cardoso (1984) “Diffraction effects in pulse echo measurement” IEEE Transactions on Sonics and Ultrasonics SU-31: 313-329 78. K.A. Wear, B.S. Garra and T.J. Hall (1995) “Measurements of ultrasonic backscatter coefficients in human liver and kidney in vivo” Journal of the Acoustical Society of America 98(4): 1852-1857 79. S.L. Bridal, C. Fournier, A. Coron, I. Leguerney and P. Laugier (2006) “Ultrasonic backscatter and attenuation (11-27 MHz) variation with collagen fiber distribution in ex vivo human dermis” Ultrasonic Imaging 28(1): 23-40 80. C. Fournier, S.L. Bridal, A. Coron and P. Laugier (2003) “Optimization of attenuation estimation in reflection for in vivo human dermis characterization at 20 MHz” IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 50(4): 408-418 81. Z.F. Lu, J.A. Zagzebski and F.T. Lee (1999) “Ultrasound backscatter and attenuation in human liver with diffuse disease” Ultrasound in Medicine & Biology 25 (7): 1047-1054 82. T. Varghese and K.D. Donohue (1994) “Mean scatterer spacing estimate with spectral correlation” Journal of Acoustical Society of America 96: 3504-3515 83. L. Fellingham and F. Sommer (1984) “Ultrasonic characterization of tissue structure in the in vivo human liver and spleen” IEEE Transactions on Sonics and Ultrasonics SU-31: 418-428 84. L. Landini and L. Verrazzani (1990) “Spectral characterization of tissue microstructure by ultrasound: a stochastic approach” IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 37: 448-456 85. T. Varghese and K.D. Donohue (1995) “Estimating mean scatterer spacing with the frequency-smoothed spectral autocorrelation function” IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 42(3): 86. 451-463 X. Tang and U.R. Abeyratne (2000) “Wavelet transforms in estimating scatterer spacing from ultrasound echoes” Ultrasonics 38: 688-692 87. K.D. Donohue, L. Huang and T. Burks (2001) “Tissue classification with generalized spectrum parameters” Ultrasound in Medicine & Biology 27(11): 1505-1514 88. W.C.A. Pereira, A. Abdelwahab, S.L. Bridal and P. Laugier (2002) “Singular spectrum analysis applied to 20 MHz backscattered ultrasound signals from periodic and quasi-periodic phantoms” Acoustical Imaging 26: 239-246 89. C.B. Machado, W.C.A. Pereira, M. Meziri and P. Laugier (2005) “Characterization of in vitro healthy and pathological human liver tissue periodicity applying singular spectrum analysis to backscattered ultrasound” in Proccedings of IEEE Ultrasonics Symposium 2005, pp. 1679-1682

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90. M. Meziri, W.C.A Pereira, C.B. Machado, B. Boudjema and P. Laugier (2006) “Multiparametric human liver fibrosis identification from ultrasound signals” in 6th IFAC Symposium (Modelling and control in biomedical systems), pp. 177-181

18 Ultrasonic Hyperthermia Arturo Vera, Lorenzo Leija and Roberto Muñoz Departamento de Ingeniería Eléctrica, Centro de Investigación y Estudios Avanzados (CINVESTAV)

18.1 Introduction At the beginning, ultrasonics in the field of medicine was oriented to applications in therapy rather than diagnosis, and its heating and dissociation effects were used on biological tissues. We call ultrasonic therapy to the use of high-intensity ultrasounds in order to induce changes in the state of the tissue by means of their thermal and other effects. Hyperthermia is a relatively new therapy cancer treatment. Its effects are obtained by increasing the temperature range in the tumor target up to 42-45 °C. When the rate of body temperature increment exceeds the ability of the regulation system to dissipate the heat, the cells die. The cells of a solid cancerous tumor are even more sensitive to heat than normal cells. For this therapy to be effective, the temperature increase must be maintained between 30 to 60 minutes per treatment, and it is usually preceded or followed by conventional oncology treatments such as radiotherapy or chemotherapy. The clinical results obtained by several clinical researchers are encouraging for the treatment of some kinds of tumors. The heating of cells induces conformational changes of certain proteins that depend on pH values. These conformational changes lead to alteration of multimolecular structures like cytoskeleton, membranes and also some structures in the cell nucleus. Metabolic changes like the increase of metabolic rates, lactate rates and the decrease of pH are also induced by heat. These conformational and metabolic changes lead to alteration of the microenvironment in tumors and has an impact on cellular death induced by heat [1]. The heating of cancerous tissues can be accomplished by several means, like electromagnetic and ultrasonic radiation among others. Electromagnetic radiation can be used to treat small tumors located in relatively homogeneous tissue regions such as breast, brain, and perhaps the soft tissue regions of the head and neck; yet, some hyperthermia treatments may be A.A. Vives (ed.), Piezoelectric Transducers and Applications, doi: 10.1007/978-3-540-77508-9_18, © Springer-Verlag Berlin Heidelberg 2008

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best executed by using ultrasound radiation. However, it is important to mention that the areas where ultrasound cannot be successfully applied are those where bone or air regions block its path [1]. Some of the advantages of using ultrasound include the feasibility of constructing applicators of almost any shape and size and the good penetration of ultrasound at frequencies where the wavelengths are on the order of millimeters. The small wavelengths allow beams to be focused and controlled. Hyperthermia systems need highly accurate control of the changes in the focalization of the field, which result from the changes in the properties of tissue after heating, as well as of the temperature increase in the zone treated. It is easier to focus energy with ultrasound than with other hyperthermia techniques, which allows that tumors located deeply within the body can be treated. The main problem, as mentioned before, is the accuracy in the method of directing, measuring and controlling the heat.

18.2 Ultrasonic Fields Next, some basic issues related to therapy applications will be underlined. A detailed analysis of the problem can be found in [2]. The general equation for wave propagation in liquids and isotropic solids is:

∇ 2φ −

1 ∂ 2φ =0 V 2 ∂t 2

(18.1)

where φ is the potential from which the displacement u of particles under a longitudinal wave in an isotropic medium can be derived, and V is the longitudinal wave velocity. For a wave, in spherical coordinates, whose components vary as exp( jω t ) , where k = ω V we can write Eq. (18.1) as −

1 ∂ ⎛ 2 ∂φ ⎞ 2 ⎜R ⎟+ k φ = 0 R 2 ∂R ⎝ ∂R ⎠

(18.2)

f (t ± R / V ) Ae ± jkR , or more generally φ = R R For a cylindrical planar piston transducer, considering it is placed in a rigid baffle (that is, displacement at the face u z (r ,0) = u 0 uniform over its

the solution is φ =

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radius and zero outside) and the piston transducer radiating into a liquid, Fig. 18.1, the amplitude of a wave at point z on the axis is

e − jkR r ' dr ' r '=0 R

φ (0, z ) = − u 0 ∫

a

(18.3)

that can be solved as φ(0,z) =

(18.4)

2 ka u z (0) sin k 2

Then, if z2 >> a2 the paraxial approximation (small angles from the axis) can be used. Introducing the Fresnel normalized parameter S = zλ/a2, two regions are defined: 1. Fraunhofer zone, when z >> a2/λ or S >> 1. 2. Fresnel zone, when z >> a2/λ or S >> 1. In the Fraunhofer zone I(z) decreases with 1/z2. In the Fresnel zone there are values of z where φ is maximum on the axis for z ≈ a2/[(2m+1) λ] and ka >> (2m+1) π. Figure 18.2 shows both zones for a circular planar piston transducer. The values of x, the distance z, and a/λ are the conditioning factors for the number of maxima and minima along the ultrasonic beam. Frequency of peaks increases when x decreases. The width of the beam also changes along z, narrowing towards the last axial maximum, being about one-quarter of the diameter of the transducer (-3dB beam diameter) at the last axial maximum. The ultrasonic field is diverging after the last axial maximum and, at large distances (z >> a2/λ or S >> 1), the intensity follows the inverse square law, I(x) ~ 1/x2. The intensity distribution across the beam in the far field can be approximated by a Gaussian distribution. The energy is distributed in the main lobe (about 84%) and the side lobes (rest of energy). TRANSDUCER R + dR r' + dr'

r a

r''

R Z

Fig. 18.1. Scheme for the study of the diffraction theory for the planar piston transducer. The ultrasound transducer is considered as an ideal piston from which ultrasound propagation, in a homogeneous medium, can be obtained

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Intensity I

470

1/I2

2

Near Field a /

Far Field

Fig. 18.2. Distribution of intensity along the axis of a transducer. Near and far fields of a circular planar transducer are shown. It can be observed that the last maximum corresponds to the beginning of the far field region

18.2.1 Ultrasound Field Measurement

The pattern of the ultrasound beam of a hyperthermia transducer determines the way it heats the tissue. A full knowledge of the spatial field could provide an accurate mapping of a focal region as well as it can identify unexpected intense sidelobes and other regions of high acoustic intensity away from the area of ultrasound treatment. Additionally, a plane pressure field measured near the transducer surface could provide information for precise field modeling. For these reasons, it is important to characterize ultrasonic field. For a complete characterization of the beam of a hyperthermia ultrasonic transducer, it is necessary to determine its spatial distribution of pressure amplitude in the wave field of the transducer, i.e. the space domain response of the transducer. One would like to be able to plot cross sections in space on planes parallel to the transducer face and also to view the beam from the side. If it is considered that the propagation direction is Z, then X-Y, X-Z, and Y-Z planar plots of the beam can be obtained. There are several methods used to visualize the beam pattern of hyperthermia transducers, e.g., the C-scan with ball target, C-scan with microprobe, holography with flexible pellicle, the Schlieren method, the photoelastic method, the Bragg refraction, the liquid crystal scanner and the mechanical scanning. In this section we are only to describe briefly the mechanically scanning method using a single element detector because this method is the most commonly used. In this method, the transducer is scanned over a volume by using a scan apparatus and a precision hydrophone, Fig. 18.3. The characterization is performed inside a tank filled with degassed water. Generally, the transducer is rigidly mounted with its axes of propagation parallel to the long axis of the tank. Then, the hydrophone, which is attached to a 3D stepping

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motor-guided positioning system, scans over the plane perpendicular to the acoustic axis. The hydrophone’s signal is amplified by a preamplifier and monitored by an oscilloscope. During field characterization, a high voltage driving system excites the transducer. The ultrasound transmission is synchronized with data signal acquisition system of the transducer. The hydrophone’s position and the amplified hydrophone’s signal acquisition are both computer controlled. The complete system is depicted in Fig. 18.4. An example of an X-Y and Z-Y plane ultrasonic fields are shown in Figs. 18.5a and b, respectively. Hydrophone output (To the preamplifier)

Ultrasound Transducer

Fig. 18.3. Ultrasound field measurement by using the mechanically scanning method. An ultrasound transducer is placed in front of a hydrophone so that the ultrasound pressure is measured in every position. X-Y, X-Z and Y-Z planes can be obtained by using this method

18.3 Ultrasonic Generation In the last years therapeutic ultrasound techniques have been followed with increasing interest and the future of this technique highly depends on the development of high performance transducers [3]. They must fulfill a set of requirements, some of which are common for diagnostic transducers and others are related to the specific task of therapy, i.e. transducers capable of producing high-power and focused waves for a long time.

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Fig. 18.4. Block diagram of the mechanically scanning method. The system consists of the ultrasound generation and acquisition units moved by a motor-guided position system

18.3.1 Piezoelectric Material

The piezoelectric material most widely used in medical ultrasonic transducers is piezoelectric ceramics type PZT. Among the different kinds of PZT, PZT-4 has been generally accepted in therapy applications because it is suitable for driving high level signals. Its main singular properties are the high resistance to depolarization, small dielectric losses under high electric fields, high electromechanical coupling, high resistance to depolarization under high mechanical stresses and great deformation ability. The typical value for the dissipation factor (%) is 0.40, almost four times smaller than for PZT-5, which is commonly used in diagnosis transducers and has a mechanical Q =750. Hyperthermia ultrasound applicators (planar waves) are basically in the diameter range of 3 cm2 to 10 cm2, with frequencies from 0.5 MHz to 5 MHz. Consequently, the treated region belongs to the nearby field.

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X

Y

a

Z

Y

b Fig. 18.5. a X-Y plane of the ultrasound field of a transducer and b Z-Y plane of the ultrasound field of a transducer. The plane was obtained for a circular planar ultrasound transducer by using the mechanically scanning method

Anomalous lateral lobes and ultrasonic field patterns having local energy peaks in some points inside and away the treated zone have been detected in single-aperture diathermy applicators. The origin of this behavior is related to the quasi-narrow band ultrasonic radiation associated with continuous-wave (CW) and long-burst pulsed modes usually adopted in the diathermy practice [4]. This behavior could strongly influence the treatment. However it is generally accepted that the proximity of maxima and minima results in a “homogenization” of the temperature distribution because of thermal conduction.

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18.3.2 The Therapy Transducer

Lead zirconate titanate (or PZT) is the most widely used material for therapy transducers due to its mechanical structure. Figure 18.6 shows the basic transducer (single crystal probe) for therapy. The maximum stress wave is obtained when the thickness of the plate is d = λ/2 or an odd multiple of λ/2. Acoustic and electrical properties of the PZT are very well known and there are many references that have studied the basic problem of the planar piston transducer and have proposed equations describing its behavior [2, 5]. Typical driving configurations of the transducer include a signal generator or an oscillator and an RF amplifier. RF

Housing

Air Backing

Matching & Tuning

Piezoelectric Plate

Electrodes Matching Layer ULTRASOUND

Fig. 18.6. The basic ultrasonic transducer for therapy. The housing contains a backing layer and the ultrasound transducer, which is attached to a matching layer

18.3.3 Additional Quality Indicators

It has been shown that the ultrasonic field has strong non-uniformity near the transducer surface and an important aspect to mention is that the behavior of the transducer, described until now as an ideal planar piston, is only an approximation. In practice not the whole area of the transducer is effectively radiating waves. Then, it is convenient to perform the characterization of both phenomena.

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18.3.4 Beam Non Uniformity Ratio

One further quality indicator is a value attributed to the Beam Non uniformity Ratio (BNR). This gives an indication of this nearby field interference. It describes numerically the ratio of the intensity peaks to the mean intensity, that is, BNR = spatial peak intensity/spatial average intensity

where spatial peak means a “peak dosage at any part of the sound head”, and spatial average the “average dosage over the entire sound head” (0.1 to 4.0 W/cm2 for common therapy units). A higher BNR (> 6) results in risk of “hot spot” causing periosteal pain and transient cavitation. Ratios between 2:1 and 6:1 seem to be clinically acceptable. 18.3.5 Effective Radiating Area (ERA)

It represents the portion of the transducer’s surface area that actually produces US waves. It is defined as the area receiving at least 5% of maximum intensity at a depth of 5 mm.

18.4 Wave Propagation in Tissue There are three important tissue characteristics in ultrasonic propagation through a biological medium: propagation velocity, acoustic impedance and wave attenuation. These tissue properties will be studied in the next sections because they take part in the heating process during hyperthermia treatment. 18.4.1 Propagation Velocity

The acoustic properties of soft tissue have been studied extensively [6,7]. Typical values for air, water, various soft tissues, and bone are given in Table 18.1. Most soft tissues have an acoustic velocity within ±3% with an average value of 1540 m/s. Fat is an exception with nearly 6% less. It is important to mention that velocity of most soft tissues appears to be independent of frequency, but it increases with the increment of temperature; nevertheless, in the case of fat, the velocity decreases with the increment of temperature.

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Table 18.1. Acoustic velocities and impedances for several media. Medium Air Water Fat Blood Kidney Liver Muscle Bone

Velocity (m/s) 330 1,480 1,450 1,570 1,560 1,550 1,580 4,080

Impedance (Mrayls)(Z values) 0.00004 0.148 0.138 0.161 0.162 0.165 0.170 0.780

18.4.2 Acoustic Impedance

Acoustic impedance, which is defined as the complex ratio of acoustic pressure to the particle at a specific point of the acoustic field velocity Zac = P/VS, is an important characteristic for the therapy applications of ultrasound. When ultrasound travels through the medium it can cause changes in its acoustical properties. For planar waves traveling in x direction, the general expression for the specific acoustic impedance is: ⎛ P e − jkx + P− e jkx ⎞ ⎟ Z sp = ρ 0 c ⎜⎜ + − jkx − P− e jkx ⎟⎠ ⎝ P+ e

(18.5)

For a free planar wave traveling in one direction, P- = 0 and the impedance is the characteristic impedance (ρ0 c)1/2. Acoustic impedance is important in the determination of acoustic transmission and reflection at the boundary of two materials having different acoustic impedance and in the design of ultrasonic transducers. The reflected energy is the square of the difference of the acoustic impedance divided by the sum of the acoustic impedances of the two materials through which ultrasound is transmitted. Most of the tissues have an impedance approximately equal to that of water. However, fat has a slightly lower impedance value, and bone and lung tissues have impedances that are significantly higher and lower, respectively. That is why ultrasound beam does not suffer large reflection losses while it penetrates from one soft tissue to another; whereas about 33 % of the energy is reflected at bone-soft tissue interfaces, and in tissuegas interfaces all the energy is reflected back into the tissue. Typical impedance values of some tissues are given in Table 18.1.

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18.4.3 Attenuation

It is well known that sound energy decreases with the distance traveled. Attenuation is the decrease of sound intensity with distance. In ideal materials, sound pressure is only reduced by the scattering of the wave. Biological materials, however, cause attenuation of the sound pressure. The rate of energy flow through a unit area normal to the direction of the wave propagation is called acoustic intensity (I). If no wave distortion is assumed for a planar wave, intensity I(x) at depth x is described by: I ( x) = I (0)e −2 μx

(18.6)

where I(0) is the acoustic intensity at the surface and μ is the amplitude attenuation coefficient per unit length. Thus, wave propagation in tissue shows an attenuation of ultrasonic energy according to an exponential law. For the amplitude attenuation coefficient we have Az = A0 exp(-μaz), where Az is a wave variable of the ultrasonic wave traveling in the z direction, with a starting value of A0 (in z = 0), and μa= -(1/z) ln (A0 /Az) Nepers per centimeter [Npcm-1]. This coefficient can also be expressed as α = 20 μa log10 e = 8.686 μa decibels by centimeter, [dBcm-1]. The attenuation of soft tissue is generally between (0.5 to 1) dB/(cm MHz). For a purely elastic medium the energy in an ultrasonic field is kinetic or potential with the pressure wave in phase with the particle velocity. The viscous forces between the moving particles in a real medium play a very important role in attenuation. They result in a delay between the particle pressure and velocity (change in density) and, consequently, a loss of energy during each cycle. Soft tissues have exponential attenuation coefficients that are proportional to frequency. The absorption in a viscoelastic medium should depend on the square of the frequency (f 2 ) . In the case of tissues, absorption has shown to increase almost linearly as a function of frequency; αc = αc1(f )n, where αc1 is the absorption coefficient at 1 MHz and f is the frequency in MHz. The values of αc1 and n depend on the tissue type and n varies from 1 to 1.2 for soft tissues [8]. Absorption and attenuation are topics of great importance in order to develop efficient generators for heating tissues, and model this behavior. Absorption and attenuation values measured for different tissues are summarized in Table 18.2. The divergence between the measured absorption coefficient values and the estimated values based on the classical absorption theory shows that additional absorption mechanisms exist.

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Table 18.2. Typical values for ultrasonic attenuation coefficients. Medium Air (STP) Water Fat Blood Kidney Liver Bone

Attenuation (dB/cm at 1 MHz) 12 f2 0.002 f2 0.63 0.18 1.0 0.94 15

The tissue heating process is influenced by the classical absorption, but other sources of absorption are also important because of their influence on the ultrasonic energy useful for hyperthermia. Ultrasonic attenuation in tissues is the sum of the losses due to absorption and scattering. In the scattering process, the elastic discontinuities within the tissue absorb the energy and then re-emit it in a different direction of propagation. 18.4.4 Heating Process

Ultrasound produces temperature rise because the tissue absorbs the ultrasonic waves, as mentioned before. The amount of power deposited in the tissue may be determined from Eq. (18.6), which describes the propagating wave intensity. The wave undergoes power losses according to the distance, dx, and can be calculated by differentiation of Eq. (18.6): dI ( x) = −2 μI (0)e − 2 μx , dx

(18.7)

The former equation gives the rate of power loss per unit volume at a distance x from an initial intensity of I(0). The attenuation coefficient, μ, takes into account scattering and absorption, being the absorption greatly predominant in most homogeneous tissues. Equation (18.7) allows estimating the rate of heat generation, that is, the Absorbed Power Density (APD). It can be inferred from Eq. (18.7) that for high power deposition, the attenuation should be large; however, it decreases the penetration depth, i.e., the amount of power left in the beam after penetration decreases through the tissue as stated by the negative exponential term in the Eq. (18.7). In general, for any continuous single frequency ultrasound field, when the effects of interfaces and shear viscosity are small, the time-averaged absorbed power density <APD>time is:

18 Ultrasonic Hyperthermia

APD

time

=

μ abs PA2 ρoc

479

(18.8)

where PA is the instantaneous pressure, ρo is the density of the medium and c is the wave velocity. Therefore, the pressure amplitude of the incident ultrasound beam, the ultrasound absorption coefficient and the heat transfer mechanisms of the tissue determine the amount of absorbed energy. The specific absorption rate (SAR, which is related to the APD) can be expressed by: SAR =

APD

ρo

(18.9)

where SAR is defined as the time rate of energy absorption per unit mass. SAR is a source term for the temperature distribution and it is generally the main element for controlling the temperature. In the absence of heat transport, the SAR has a simple connection to the time rate of change of temperature SAR =

dT c dt

(18.10)

where c, in J/Kg/°C, is the specific heat of the medium in question, dT is the temperature increase and t is the time. In interfaces of soft tissue-bone, the incident energy is reflected back and the amplitude attenuation coefficient is about 10-20 times higher in bones than in soft tissues. This causes the transmitted beam to be absorbed rapidly, which increases the temperature significantly. The increment of temperature can be avoided, or at least reduced to an acceptable level, if the intensities at the bone surface are only between 10% and 50% of the value at the back of the target volume. This can be achieved, in many cases, by using multiple focused beams, high frequencies or more sharply transducers. If the amount of energy transmitted into the bone through the interface by increasing the incident angle is reduced, the increment of temperature can also be decreased.

18.5 Ultrasonic Hyperthermia Hyperthermia is used in the clinical treatment of cancer and benign diseases. High-temperature hyperthermia (>42°C) alone is being used for selective tissue destruction as an alternative to conventional invasive

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surgery. Ultrasound technology has important advantages compared to other hyperthermia technologies: a higher degree of dynamic and spatial heating control. The spatial and temporal control of the increment of the temperature, which results from hyperthermia therapy, is difficult due to the often heterogeneous and dynamic properties of tissues and the blood perfusion. This is considerably overcome by the advantages of ultrasonic hyperthermia, which include a favorable range of energy penetration characteristics in soft tissue and the ability to shape the energy deposition patterns. However, the focalized region for a simple focused ultrasound beams is too small (1-3 mm of full width half maximum, FWHM) to heat the volume of a large tumor unless it is scanned. In order to get a convenient distribution of induced temperature during ultrasonic hyperthermia, we need to consider several factors in the design of therapeutic systems: 1. 2. 3. 4. 5.

The transducer properties The power deposition patterns The pattern and speed of scanning The output power The measuring of temperature increase

It is also necessary to consider the following tissue characteristics:

The thermal properties of the tissue treated (specific heat, thermal conduction and blood perfusion and flow). Thermal properties are both heterogeneous and dynamic [9] The ultrasonic properties of the tissue (absorption, density and speed of sound, nonlinear parameter and cavitation threshold). The geometrical characteristics of tumors, often irregularly shaped. The structure of tissues and tissue interfaces (gas, soft tissue-bone). The scattering and resulting reflection-refraction phenomena have an important effect on the amount of absorbed power and its distribution.

An ideal power source for hyperthermia should be able to customize power deposition to each individual tumor field.

18.6 Hyperthermia Ultrasound Systems A hyperthermia ultrasound system consists basically of two subsystems: an ultrasound generator and a thermometry unit. The generation of the RF signals that will be converted into mechanical motion is, in principle, similar in all systems; therefore, a typical system diagram is depicted in

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Fig. 18.7. A signal generator or oscillator produces the RF signal and an RF amplifier amplifies it. The forward and reflected electrical powers are measured after amplification in order to obtain the total acoustic power output. The signal enters the transducer through a matching and tuning network that couples the electrical impedance of the transducer to the output impedance of the power amplifier. The power output is controlled by the amplitude and duty cycle of the RF voltage. Commonly invasive thermometry systems are used in clinic, among which the most used are thermocouples, thermistors, and fiberoptic thermometers [10]. Several authors have produced guidelines for some sensor characteristics like diameter, accuracy, drift, insensitivity to moisture, response time, precision and passivity. Recently, nuclear magnetic resonance, microwaves, impedance tomography and ultrasound have been proposed for a non-invasive estimation of the temperature inside the tissues. The most important part of a hyperthermia system is the transducer. Basically, the hyperthermia ultrasound systems can induce superficial and deep heating. Planar transducer and mechanically scanned field systems can be used in order to induce superficial heating; meanwhile mechanical and electrical focusing, ultrasound intracavitary applicators, interstitial ultrasound arrays, high temperature and intraoperative systems and thermal surgery are used in order to induce deep heating [1].

Fig. 18.7. Block diagram of an ultrasound hyperthermia system. The system consists of an ultrasound generation unit and a thermometry unit

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18.6.1 Superficial Heating systems Planar Transducer Systems

These systems use single circular and planar transducers, which transmit ultrasound to the patient through a temperature-controlled water column, Fig. 18.8. It is simple to build and to operate this kind of systems. They have a good penetration depth, which can be controlled by using transducers with different frequencies. In addition, the beam energy is well collimated and a relatively uniform power output can be obtained over the whole transducer surface. Multielement transducers, with independent power control, can be used to heat large tumors and obtain good energy deposition.

Fig. 18.8. Planar transducer system. In this system, the ultrasound transducer is matched to the medium by using a water temperature controlled bolus

Mechanically Scanned Fields

In this case, the ultrasound transducer position is controlled by a mechanical scan system. These superficial heating systems have some theoretical advantages, e.g. the power can be controlled as a function of the scan position, which gives these systems a good spatial resolution, and the patient can identify pain locations so that power in certain treated zones can be reduced. Another advantage is that one can control the penetration depth using multiple frequencies. 18.6.2 Deep Heating Systems

Some of the requirements for deep heating ultrasound systems are: 1. The effective beam diameter has to decrease to compensate for the attenuation losses in the tissue.

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2. Intensity gain and convergence are required to obtain higher temperatures in tumor than in adjacent tissues. 3. Hot and cold spots must be reduced by controlling deposition pattern. 4. Accurate tumor localization must be achieved by using the adequate patient-system interface. Mechanical focusing and electrical focusing systems are briefly described in the next sections. Mechanical Focusing

This kind of systems can use multiple beams that overlap so that it provides a better delivery of energy into deep tumors by diminishing the effect of attenuation. Spherically curved transducers or lenses can be used for this purpose. Due to the pain produced to some patients, there are no mechanical focusing systems used in clinic at present. It has been demonstrated that these systems can locally deliver ultrasound energy and induce therapeutic temperatures, but some improvements must be made before their use in clinic. Electrical focusing

Another way to focalize ultrasound energy is by dephasing the ultrasound radiation coming from transducers of an array. The first attempt to use this kind of focusing was by using a concentric ring transducer driven by different excitation signals. This system generates secondary focus in front and behind the focal plane. To avoid this problem, a combination of electrical focusing and mechanically focused transducer has been proposed. The most outstanding way of using this system is by means of a twodimensional array of small transducers with both separate amplitude and phase controls. 18.6.3 Characterization of Hyperthermia Ultrasound Systems

There are many factors which influence the ultrasonic field produced by a hyperthermia transducer that are difficult to account for in computations. For this reason, an experimental characterization of the hyperthermia ultrasound system is needed. This can be accomplished by measuring temperature or ultrasonic field inside some materials that mimic the ultrasound properties of tissues; such materials are called ultrasound phantoms. The performance of a system can be determined under standardized and repeatable

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conditions and with a reduced number of variables by applying a phantom as a stable load. Experimental data are necessary for improvement and development of hyperthermia systems and the control of parameters such as power and phase settings. They are also essential for the development and evaluation of computer models, which describe power deposition and temperature distribution, thus leading to clinical hyperthermia planning systems. The final goal of this characterization is to determine SAR distribution inside the phantom, which gives a first approximation of heat distribution inside real tissues. Some important medical parameters, like penetration depth and effective field size, can also be obtained from this distribution. When using ultrasonic field for characterization, a mapping of the ultrasonic pressure inside the phantom is carried out. Degassed water may be used as a general phantom material. However, when using this phantom, the measured field profiles must be adjusted mathematically to account for the attenuation that would be present in tissue. The way this technique is performed is similar to the one described in Sect. 18.2.1, which is used to obtain the free ultrasonic field. In order to characterize hyperthermia systems by means of temperature measurements, the phantom is heated during enough time to produce a measurable temperature increment, and then this increment is measured. Temperature must be rapidly measured to minimize the conduction effects inside the phantom, which can distort the distribution of SAR. There are several techniques used for measuring the temperature distribution, like the power pulse technique and the infrared (IR) and liquid crystal films (LCF) thermography. The power pulse technique, Fig. 18.9, consists in measuring the temperature by using small sensors placed inside catheters that are embedded in the phantom. After heating it, the temperature inside the phantom is scanned by using a positioning system. Infrared thermography consists in heating a phantom divided in two parts, Fig. 18.10a, and then, they are split to capture an image by using an IR camera, Fig. 18.10b. LCF thermography is performed by using a transparent phantom in which a Liquid Crystal Film is placed parallel to the transducer. The image is acquired after heating the phantom, Fig. 18.11. Ultrasound Phantoms

Ultrasound phantoms mimic the ultrasonic tissue properties that determine the way the tissue is heated, that is, propagation velocity and attenuation. There are some general requirements to be accomplished for a phantom material such as reproducibility and reliability of preparation, stability, solidness, and toxicity. Phantoms are prepared with three materials: one

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that determines the propagation velocity, one that determines the attenuation and finally another that gives it solidness. For some applications, scatterers are added to phantoms. SAR distribution can be determined in these materials with similar absorptions as those of tissues; a list of mimicking materials is given in Table 18.3.

Fig. 18.9. SAR measurement by using the power pulse technique. After heating the phantom, temperature sensors are moved inside a phantom in order to obtain a temperature distribution, which is related to SAR

a

b

Fig. 18.10. SAR measurement by using the IR thermograpy. a The technique uses an ultrasound phantom divided in two parts. b After heating, the phantom is separated to get an image by using an IR camera. The temperature distribution is related to SAR

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Fig. 18.11. SAR measurement by using LCF thermograpy. After heating a transparent phantom, in which a LCF is embedded, an image is acquired. The temperature distribution is related to SAR Table 18.3. Different formulas used to prepare ultrasound phantoms [26]. Tissues simulated

Ingredients

Various soft tissues

Gelatin Graphite powder n-propanol Agar n-propanol Water Graphite Gelatin 34% (vol/vol) olive oil 3% (vol/vol) n-propanol 0.11mg/cm3 scatterers Gelatin 25% (vol/vol) olive oil 25% (vol/vol) kerosene 2.2% (vol/vol) n-propanol 0.065 mg/cm3 scatteres

Various soft tissues Breast tissue

Fat

Propagation Velocity (m/s) 1520-1650

Attenuation (db/cm/MHz)

n

≈02-1.5

≈1

1489->1600

0.04-1.4

≈1

1519

0.8

1.01

1459

0.42

1.02

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Table 18.3. Different formulas used to prepare ultrasound phantoms [26] (cont.) Tissues simulated

Ingredients

Liver

Gelatin 13.6%(vol/vol) castor oil 20.4%(vol/vol) olive oil 8.8% (vol/vol) n-propanol Gelatinalginate gel Scatterers

Various tissues

Propagation Velocity (m/s) 1539

Attenuation (db/cm/MHz)

n

0.51

1.03

1519

0.12-0.5

≈1

Ultrasound Phantom-Property Measurements

It is desirable to measure the ultrasonic properties of the tissue mimicking materials to guarantee precise data of the hyperthermia system. Those properties are propagation velocity, and attenuation. There are several methods to measure the propagation velocity. In liquid and quasi-liquid phantoms, it is measured by Interferometry. In this method, a transducer generates an ultrasound wave and a micrometer with a reflector is moved so that a standing wave is established. The distance d, between the transducer and the reflector, is related to the wavelength and a minimum in the driving voltage. The velocity is then calculated from c = fλ , where c is the ultrasound velocity in the medium, f is the frequency and λ is the wavelength. The accuracy of this method depends on the parallelism between the reflector and the transducer, and also on the mechanical precision of the arrangement. A typical arrangement is depicted in Fig. 18.12.

Fig. 18.12. Block diagram of a system to measure propagation velocity by using the interferometry technique. Distance between the transducer and reflector is adjusted according to a minimum in the driving voltage, which is related to the wavelength in the medium

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The pulse transit time technique consists in measuring the pulse transit time for a known propagation distance. One way to measure it is by using one transducer that emits and other that receives the ultrasonic waves, which are the two axes of the transducers coincident. Other approach is by using the pulse-echo technique, as illustrated in Fig. 18.13, in which the same transducer emits and receives the ultrasonic waves, in this technique, a reflector is positioned normal to the ultrasonic beam axis. The sing-around technique is similar to that of the pulse transit time technique, but instead of measuring the transit time of a pulse across a known length of phantom, it counts the number of times that the pulse travels backwards and forwards through the phantom over a known period of time. A typical arrangement is illustrated in Fig. 18.14.

Fig. 18.13. Block diagram of a system to measure propagation velocity by using the transit time technique. The time of flight of an ultrasound wave is measured in a known propagation distance

Fig. 18.14. Block diagram of a system to measure propagation velocity by using the sing-around technique. In this technique the number of times the pulse travels backwards and forwards through the phantom is counted over a known period

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Concerning the measurement of attenuation, it can be carry out by using either the pulse-echo or the through transmission technique. The measurement is performed by calculating the rate of decay of echo amplitude within distance or by using frequency spectrum analysis.

18.7 Focusing Ultrasonic Transducers As mentioned before, transducer focusing is a very important factor in ultrasonic therapy. The goal is to obtain highly focused transducers having an f-number (focal distance/active diameter relation) of the order of f≤1. Focusing changes the shape of an ultrasound beam emitted by a transducer. This shape can be achieved by using self-focusing radiators, lenses or electrical focusing (i.e., transducer arrays that are driven by signals having the proper phase difference to obtain a common focal point). The wavelength and the shape restrict the size of the focal region by the ratio between the apertures of the array to the wavelength. Fig. 18.15 shows the ultrasonic focusing transducers.

a

b

c

d

Fig. 18.15. Ultrasonic focusing transducers: a spherical transducer, b lens, c reflector and d electrical focusing

18.7.1 Spherically Curved Transducers

The focused acoustic field of this kind of transducers is similar to that of the near field of a plane wave transducer. Beyond the focus, it is similar to that of the far field of a plane transducer, except that the divergence of the beam is dominated by the geometrical divergence angle of the transducer. The approximate half-intensity beam width (dxy) at the focus in spherically curved radiator can be obtained from [11] ⎛ R⎞ d xy = 1.417 ⎜ ⎟ ⎝ 2a ⎠

(18.11)

where R is the radius of curvature and 2a is the diameter of the transducer. The axial length of the focus (dz) is

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⎛ R⎞ d z = 7.17 ⎜ ⎟ ⎝ 2a ⎠

2

(18.12)

Then, the shape of the focus is long and narrow with dimensions depending on the focusing properties of the transducer, i.e., the transducer diameter, radius of curvature and frequency. 18.7.2 Ultrasonic Lenses

Acoustic lenses are made of materials in which the speed of sound (cL) is different from that in the coupling medium (czm) so that ultrasound beam can be focused. Planar concave is a desirable shape for a lens whose generating curve of the concave surface is elliptical and cL > cm. One of the advantages of lenses over the spherically curved transducers is that it is possible to use one transducer and modify its focus by selecting the appropriate lens. 18.7.3 Electrical Focusing

The beam convergence can also be obtained by using phased array applicators. The ultrasonic beams can be focused by using arrays of transducers. Each element is driven by a signal of certain specific phase so that the waves generated by each element are in phase at the focal point. The elements must be small enough, compared with wavelength, in order to act as a point source. It is possible to use one or two-dimensional arrays of transducers. Fig. 18.16 shows a one-dimensional array. The center-to-center spacing between elements is a crucial factor in the construction of a phased array (maximum size of the elements). This spacing must be, as maximum, one half of a wavelength in order to avoid grating lobes [12]. The use of curve arrays increases the element size even up to two wavelengths [13-15]. 18.7.4 Transducer Arrays

The array theory requires that the spacing of the array elements be less than half a wavelength to avoid grating lobes, which for CW excitation can be as intense as the main lobe. In order to localize a heated volume at depth in attenuating tissue, large aperture arrays are necessary. In particular,

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two-dimensional arrays are desirable due to their ability to control a beam in three dimensions. Complex beam-forming techniques can be applied to electronically focus and to drive the ultrasound energy in 3-D or to conform complex beam patterns. Versions of this technology include: 1. Spherical and cylindrical section arrays [13-14]. 2. Concentric ring arrays [16]. 3. Tapered phased arrays [17]. 4. Sector-vortex phased arrays [18]. These driving and control techniques and offer significant advantages over mechanically scanned and focused systems [13, 14].

FOCUS

Fig. 18.16. Focusing by one-dimensional array. Ultrasound signals are electronically delayed so that the energy is concentrated

18.7.5 Intracavitary and Interstitial Transducers

The piezoelectric elements are manufactured in the shape of a cylinder with electrodes on its inner and outer surfaces. If an RF voltage is applied to the electrodes, the cylinder’s wall thickness will expand and contract with the voltage. This generates a cylindrical ultrasound wave, which propagates radially outward. The cylinder has to be on the order of ten wavelengths in order to obtain a well-collimated beam. Intracavitary techniques are useful for applying conventional hyperthermia to deep-seated tumors that are close to a body cavity. Thus, design and construction of linear arrays of PZT tubes was the initial development of intracavitary ultrasound applicators [19-23], being the most important ones for the treatment of carcinoma of the prostate and benign prostatic hyperplasia (BPH), but they can also be used for treatment of vaginal and rectal tumors. Fig. 18.17 shows an intracavitary transducer mounted on a

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plastic structure that facilitates support and placement in the cavity when the tumor is located only on one side [24]. The radial emission of heating energy from the length of each transducer segment and the power applied along the length of the applicator is adjustable to the heating distribution. It is important to mention that each element of the array can be driven independently at different power levels to get a better temperature distribution pattern. As for other hyperthermia systems, a water bolus, which also allows temperature control inside the cavity by circulating the water through a heat exchanger, does coupling between the transducer and the cavity. Linear phased arrays of tubular sections (Fig. 18.18) have been considered in order to increase spatial control and depth of penetration [22-23]. Cooling Water Output

Solid PVC Body

Cooling Water Input

RF Signal Lines (Individual)

Latex Membrane

Transducer Elements (1/2 Cylinders)

Rubber Seals

Fig. 18.17. Intracavitary applicator. This kind of transducers is introduced inside natural cavities of the body for the treatment of deep-seated tumors Prostate RF Power Lines

Outer Catheter Water

Inner Catheter Temperature Regulated Water Tubular Piezoelectric Transducers

Fig. 18.18. Transrectal applicator, arrays of tubular section

Interstitial methods are used for treating tumors or sites that are difficult to reach by external or intracavitary methods. Despite the invasive nature of the techniques, the heating sources are implanted directly into the tumor, surgically or percutaneously, thereby localizing heat in the target volume. There are two approaches for this method. The first one consists of using a wave-guide to deliver external ultrasound energy into the tumor from a planar transducer. The second approach deals with the use of small cylindrical ultrasound sources interstitially placed. Several types of interstitial ultrasound applicators are currently being developed and they can be

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classified as catheter-cooled and direct-coupled devices. When using a catheter, it is possible to circulate water through it so better depth of penetration can be obtained.

18.8 Trends The technological advances in electronic and computer science would allow the potential development of improved and effective ultrasonic systems for new therapy applications. Efforts should be addressed to the determination and evaluation of computational tools and specific procedures [25] for: i) the precise analysis of the propagation in biological tissues and of their thermo mechanical effects, ii) improving by electronic means the capacity to concentrate high-frequency ultrasonic energy in a controlled way and selective on the target of interest and iii) minimizing potential collateral damages in the neighbor tissues.

References 1. M. H. Seegenschmiedt, P. Fessenden and C.C. Vernon (1995) “Thermoradiotherapy and thermo-chemotherapy”, Vol. 1: Biology, Physiology and Physics, Springer 2. G.S. Kino (1987) “Acoustic Waves: devices, imaging, and analog signal processing” Prentice Hall, Inc 3. M.H. Lente, A.L. Zanin, D. Garcia, L. Leija, A. Vera, H. Calas and H. Moreno (2006) “Field induced piezoelectric transducer for generation of controlled diffraction ultrasonic field” in Proceeding of IEEE Ultrasonic Symposium, pp. 2210-2213 4. G. González, A. Azbaid, L. Leija, A. Ramos, X. Rami, J.L. San Emeterio and E. Moreno (2002) “Experimental evaluation of some narrow-band ultrasonic transducers as therapy applicators” in Forum Acusticum Sevilla, Spain. PACS Reference 43.35-43.38. ISBN: 84-87985-06-8 5. P.N.T. Well (1977) “Biomedical ultrasound” Academic Press 6. S.A. Goos, R.L. Johnson and F. Dunn (1978) “Comprehensive compilation of empirical ultrasonic properties of mammalian tissues” J. Acoustical Soc. Am. 64: 423-457 7. S.A. Goos, R.L. Johnson and F. Dunn (1980) “Compilation of empirical ultrasonic properties of mammalian tissues. II” J. Acoust. Soc. Am. 68: 93-108 8. S.A. Goos, L.A. Frizzell and F. Dunn (1979) “Ultrasonic absorption and attenuation of high frequency sound in mammalian tissues” Ultrasound Med. Biol. 5: 181-186

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9. D.P. Anhalt et al. (1995) “Patterns of changes of tumor temperatures during clinical hyperthermia: implications for treatment planning, evaluation and control” Int. J. Hyperthermia 11: 425-36 10. M. Vázquez, A. Ramos, L. Leija and A. Vera (2006) “Noninvasive temperature estimation in oncology hyperthermia using phase changes in pulse-echo signals” Jpn. J. Appl. Phys. 45: 7991-7998 11. J.W. Hunt (1987) “Principle of ultrasound used for hyperthermia” NATO ASI Series E: No. 127, Martinus Nijhoff Publisheers, Boston 12. B.D. Steinberg (1976) “Principles of aperture and array system design” John Wiley, New York 13. E.S. Ebbini and C.A. Cain (1991) “Experimental evaluation of a prototype cylindrical section ultrasound hyperthermia phased-array applicator” IEEE Trans. Ultrasonics Ferroelectrics Frequency Control 38: 510-520 14. E.S. Ebbini and C.A. Cain (1991) “A spherical section ultrasound phased array applicator for deep localized hyperthermia” IEEE Trans. Ultrasonic Ferroelectrics Frequency Control 38: 634-643 15. E.S. Ebbini, S.I. Umemura, M. Ibbini and C. Cain (1988) “A cylindricalsection ultrasound phased array applicator for hyperthermia cancer therapy” IEEE Trans. Ultrasonic, Ferroelectrics and Frequency Control 35: 561-572 16. M.S. Ibbini and C.A. Cain (1990) “The concentric-ring array for ultrasound hyperthermia: combined mechanical and electrical scanning” Int. J. Hyperthermia 6: 401-419 17. P.J. Benkeser, I.A. Frizzel, K.B. Ocheltree and C.A. Cain (1987) “A tapered phased array ultrasound transducer for hyperthermia treatment” IEEE. Trans. Ultrasonic Ferroelectrics and Frequency Control 34: 446-453 18. S. Umemura and C.A. Cain (1989) “The sector-vortex phased array: acoustic field synthesis for hyperthermia” IEEE Trans. Ultrasonic. Ferroelectrics. Frequency Control 36: 249-257 19. J.Y. Chapelon et al. (1993) “The feasibility of tissue ablation using high intensity electronically focused ultrasound” in Proceedings of IEEE Ultrasonic Symposium 2: 1211-1214 20. A. Gelet et al. (1996) “Treatment of prostate cancer with transrectal focused ultrasound: early clinical experience” Eur. Urol. 29: 174-183 21. E.B. Hutchinson and K. Hynynen (1996) “Intracavitary ultrasound paced arrays for noninvasive prostate surgery” IEEE. Trans. Ultrasonic Ferroelectrics and Frequency Control 43: 1032-1042 22. E.B. Hutchinson et al. (1996) “Design and optimization of an aperiodic ultrasound paced array for intracavitary prostate thermal therapies” Med. Phys. 23: 767-776 23. M.T. Buchanan and K. Hynynen (1994) “The design and evaluation of an intracavitary ultrasound phased array for hyperthermia” IEEE Trans Biomed Eng. 41: 1178-1187 24. C. Diederich and K. Hynynen (1991) “The feasibility of using electrically focused ultrasound arrays to induce deep hyperthermia via body cavities” IEEE Trans. Ultrasonic Ferroelectrics and Frequency Control 38: 207-219

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25. “Determination of efficient ultrasonic patterns for safe therapy by control of distributed pulsed radiation” in Pulsets, Cyted international cooperative project, vol. I, II and III, ISBN: 90-4054-2-7, (2002-2005) 26. J.J.W. Lagendijk (1992) “Treatment planning and modelling in hyperthermia: a task group report of the European society for hyperthermic oncology” in Tor Vergata Medical Physics Monograph Series 27. E.P. Papadakis (1999) “Ultrasonic Instruments and Devices: Reference for a Modern Instrumentation, Techniques, and Technology” Academic Press

Appendix A: Fundamentals of Electrostatics Antonio Arnau and Tomás Sogorb Departamento de Ingeniería Electrónica, Universidad Politécnica de Valencia

A.1 Principles on Electrostatics It is usually attributed to Thales of Miletus (600 B.C.) the knowledge of the property of amber (in Greek elektron) of attracting very light bodies, when rubbed. This phenomenon is called electrification and the cause of the phenomenon electricity. The electric manifestations of electrification phenomena on different materials can be opposite and, in these cases, it is said that both materials have acquired qualities of opposite sign. The quality that a material acquires when electrified is called charge, establishing arbitrarily the sign of this quality in relation to the phenomenon that it manifests. In addition, it could be experimentally demonstrated that the electrified bodies interact with each other, and generate repulsive forces when the charges are of the same sign and, of attraction when they are of opposite sign. The fact that the charge is manifested in two opposite ways involves that if a body has as many positive charges as negative, its external electric manifestations are balanced, and we say that it is electrically neutral. Matter is mainly neutral and it is not common to find bodies whose net charge has a considerable value, therefore, the electrical interactions between bodies are in general quite weak. For this reason, for a material to manifest electric properties it is necessary to subject it to some type of action, for example mechanical, as in the case of amber. This external action produces a loss of balance between the positive and negative charges of the rod and allows the electric interaction with other bodies. At this time, it is necessary to introduce the principle of charge conservation in an isolated system. This experimental principle states that in a system in which charge cannot enter or leave, the positive and negative charges can vary with time but its net charge (positive + negative) remains constant. Therefore, when the amber rod is rubbed with a cloth, the charges of the cloth “arbitrarily called negative” pass to the amber rod that now has an excess of negative charge. A redistribution of the charges has taken place between the cloth and the rod but the net charge of the system cloth-rod has not been altered. A.A. Vives (ed.), Piezoelectric Transducers and Applications, doi: 10.1007/978-3-540-77508-9_19, © Springer-Verlag Berlin Heidelberg 2008

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If the rubbing action on the rod is vigorous, it can be noticed that the electric interactions are stronger. That is to say, the small bodies that are used to show the electric interaction are attracted with more strength. This means that the amber acquires a larger negative charge. However, this charge increase is always in integer multiples of a minimum amount. This minimum amount of electric charge is the negative charge of the electron or the positive charge of the proton. This hypothesis establishes the principle of charge quantization as well as the existence of a natural unit of electric charge.

A.2 The Electric Field It is important to notice that it has been necessary to bring the electrified amber rod close to small bodies in order to see the electric interactions. That is, it has been necessary to introduce a test element to check that the electrified amber rod had modified the electric characteristics of its immediate environment in such a way that other bodies, when coming closer to this environment, were subjected to an external force. However, the electrified amber rod modifies the electric characteristics of its environment even if no light bodies, which allow seeing this interference, are placed nearby. For example, we can use an apple to show the gravitational force but this still exists even without the apple. This concept is very important because it indicates that, similarly to the gravitation example, regions of the space are created around the positive and negative charges where they can show electric interactions. If these regions of the space appear due to the presence of punctual charges or superficial or volumetric distributions of static charge, the electric field created is called electrostatic field and the forces on the charges located in this field are determined by Coulomb’s law1. This electric field is of special importance in the study of piezoelectricity. Coulomb’s law provides the force that acts on a charge q’ due to the presence of another punctual and fixed charge q. Its mathematical formulation is:

1

F=

1 qq' 4πε r 2

where ε is the permittivity of the medium and r is the distance between charges. Consequently, the force is proportional to the product of the charges and inversely proportional to the square of the distance that separates them. The direction is the straight line that links the charges. The direction is repulsion for charges of the same sign and attraction for charges of opposite sign.

Appendix A: Fundamentals of Electrostatics

499

Intensity of the field, or simply field, at a point, is the force that acts on the unit of positive charge placed at this point. Consequently, a charge with value q located at a point, P, of this field would be subjected to a force given by:

F ( P) = q E ( P)

(A.1)

where E(P) is the intensity of the electric field at point P. Similarly to the force, the field is a vector magnitude with a direction. Generally, electric fields are represented by their so-called lines of force. These lines are a graphic representation of the trajectories that a positive charge would follow, subjected to the influence of the field, in a succession of elementary paths, starting every time from rest. Let us imagine a surface immerse in an electric field. Infinite lines of force will cross this surface. At each point of the surface, there will be a line of force that, at that point, will be characterized by one value of field intensity and one direction. It is called flux Φ of an electric field through this surface: r r Φ = E o dA (A.2)

∫

where dA is a differential element of the surface under study. The product indicated in the previous equation is a scalar product of the two vectors. The previous definition provides an intuitive idea of the electric field as the density of electric flux per area unit. The value of the intensity of electric field at a point can be graphically represented by the number of lines of force that cross the surface unit at this point. Thus, there will be more lines of force in those points where the field intensity is stronger and more lines of force will cross the surface increasing the flux.

A.3 The Electrostatic Potential If the potential energy of a punctual charge inside an electrostatic field is defined as a dependent function of point U(P) so that the difference between its values in the initial and final positions is similar to the work acting on the charge, by the force of the field. This potential energy would be formulated as: 2

r r 2 W 1 = qE o dr = U 1 − U 2

∫ 1

and its differential expression would be:

(A.3)

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r r dW = qE o dr = −dU

(A.4)

From the previous expression, results:

r r dU E o dr = − = −dV q

(A.5)

where dV=dU/q is called potential difference. V is called electrostatic potential, a scalar function dependent on the considered point, and it represents the work carried out to transfer the charge unit from infinite (where it is considered that V(∞)=0) to this point. From the previous expression it is deduced that the field is equal and with opposite sign to the gradient of the potential, that is to say:

r ∂V r ∂V r ∂V r E = − grad V = − i− j− k (A.6) ∂x ∂y ∂x r r r where i , j and k are the unitary vectors in the directions of the coordinated axes.

A.4 Fundamental Equations of Electrostatics The main problem in Electrostatics is the calculation of the fields produced by different charge distributions. Gauss’ law, which provides the value of the flux of the field through a closed surface, is one of the fundamental vector relationships that allow solving part of the problem. Gauss’ law states that the flux of an electric field through a closed surface is equal to the sum of all the charges enclosed in this surface divided by the permittivity of the vacuum εo. Its mathematical formulation is as follows: r

r

Q

∫ E o dA = ε A

(A.7)

o

where Q represents the total inner charge to surface A. If the distribution of the charge were a volumetric distribution, defined by a charge density ρv, the previous equation would be written:

∫

A

r r 1 E o dA =

εo

∫ρ

V

v

⋅ dV

(A.8)

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501

The integral of the first member of the previous expression can be written as an integral of volume keeping in mind the theorem of divergence2, thus Eq. (A.8) is: r

r

1

∫ divE ⋅ dA = ε ∫ ρ

V

v

⋅ dV

(A.9)

o V

or its differential expression that is:

r ρ div E = v

(A.10)

εo

Notice that if there is no charge, or there are as many positive charges as negative, inside the closed surface, the field in the inside is constant. Substituting Eq. (A.6) in Eq. (A.10) one reaches Poisson’s Equation, which is the fundamental equation of electrostatics:

r ρ ( x, y , z ) ∂ 2V ∂ 2V ∂ 2V div E = div grad V = 2 + 2 + 2 = − v εo ∂x ∂y ∂z

(A.12)

Notice that if the volumetric density of net charge is null, or the positive and negative charges are balanced, inside a volume, the potential inside this volume keeps a linear relationship with the distance.

A.5 The Electric Field in Matter. Polarization and Electric Displacement From the electrostatic point of view, two types of charges can be considered in the substances: free charges which are likely to move from The theorem of divergence establishes that the flux of the vector field through a closed surface is equal to the integral of the field divergence extended to the inner volume of such surface. Its mathematical formulation is as follows: r r r ∫ E o dA = ∫ div E ⋅ dV V A

2

where r ∂E x ∂E y ∂E z divE = + + ∂x ∂y ∂z

r

and Ex, Ey and Ez are the components of the vector E

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one place to another inside the material, and linked charges whose movement within the material is limited to small changes of position around its equilibrium positions. The substances with free charge are called conductors. Metals, for example, are substances where the charges move easily, that is to say, the electrons, at least one per atom, can move throughout the solid and they are not bound to their corresponding atom. On the contrary, in dielectrics, such as paraffin, the electrons are firmly bound to their positive ions and they cannot move freely. From the definition of conductor it is deduced that in a conductor in static state, or in equilibrium, i.e., when all its charges are at rest, the electric field is always null in its inside. Since the charges can move freely, if the field in the inside were not null the charges would move and the conductor would not be in equilibrium. Indeed, when a finite conductor, with the same number of positive and negative charges, is introduced in an electric field, E, the negative charges of one side will move in the opposite sense to the electric field to the conductor's surface, since it is supposed that they cannot abandon it, leaving positively charged ions in the other end (Fig. A.1); this creates a superficial distribution of charge that produces an induced field Ei in the opposite direction to the external field applied, reducing the total field inside the conductor to zero.

E

Ei = - E

Fig. A.1. Conductor in equilibrium subjected to an external field

The total field in the conductor's limits should be perpendicular to its surface at each point, otherwise the charges would move tangentially, and the conductor would not be in equilibrium. Therefore, if the field is perpendicular to the surface, the variation of the electrostatic potential on the conductor's surface is null (see Eq. (A.5)) and this surface is an equipotential surface. The distribution of charge induced in the conductor does not create any external field to the conductor since the conductor's overall charge is null, although there is a certain distribution of charge densities. Therefore, the application of Gauss’ law to any external closed surface to the conductor would allow verifying this result. However, it creates an internal field to the conductor that compensates, at each inner point, the external field. In

Appendix A: Fundamentals of Electrostatics

503

fact, the field created by the polarization of the charge at a nearby interior point to the conductor's surface can be calculated by applying Gauss’ law to the closed surface shown in Fig. A.2. Since the conductor does not create any field outside and the flux through the lateral surfaces is null, there is only flux through the surface A1. By applying Eq. (A.7) one obtains that the internal field generated is proportional to the superficial density of induced charge σi according to the following expression:

Ei =

σi εo

(A.13)

In the case of one conductor Ei=-E. That is, the charges move freely in the solid and generate the distribution of necessary charge so that this condition is fulfilled.

AL A1

i

A2

AL

Fig. A.2. Application of Gauss’ law to the closed surface of a conductor

When a dielectric material is in presence of an electric field, charges also appear on the surface of the dielectric. This phenomenon is called polarization of the dielectric and is similar to the polarization seen previously in the conductor, but its origin is completely different. In the conductor, polarization takes place by the migration of the charges, while in the dielectric, elementary dipoles are generated that try to align in the direction of the field, thus causing a bound charge density to appear on the surfaces of the dielectric. Polarization takes place for polar substances as well as for non-polar ones. In the polar substances, the molecules are true dipoles that are distributed at random making the dielectric, as a whole, to be discharged (Fig. A.3a). When subjecting it to a field, the dipoles are guided and the actions of the opposite close poles inside the dielectric are cancelled, thus a superficial density of charge appears in the sides of the

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Antonio Arnau and Tomás Sogorb

dielectric (Fig. A.3b). In case of non-polar substances, the centers of gravity of the positive and negative charges in the molecules coincide (Fig. A.4a). However, in presence of an electric field, these centers of gravity separate and originate small dipoles that are guided in a similar way to the previous case, causing superficial densities of charge, similar to the previous case (Fig. A.4b).

a

b

Fig. A.3. a Polar substance before being subjected to an electric field; b polar substance polarized by an electric field

a

b

Fig. A.4. a Non polar substance before being subjected to an electric field; b non polar substance polarized by an electric field

As a result, when introducing a dielectric material in an electric field, for example between the plates of a charged and insulated capacitor, this material undergoes an initial electric field Eo (between the plates of the capacitor in absence of dielectric) (Fig. A.5) and the dielectric is polarized. This generates an electric field in the inside Ei that opposes to the initial field. The result of both is a new field E with the same direction as the initial field Eo (if the material is homogeneous and isotropic) and of smaller value. In fact, the alignment is never complete; the thermal

Appendix A: Fundamentals of Electrostatics

505

agitation that increases with temperature is opposed to any type of alignment that tends to disarray the directions of the dipoles. Therefore, the dielectric properties depend strongly on the temperature. In any case, it is necessary to introduce some magnitude that allows measuring the state of polarization of the dielectric material when it is subjected to an electric field. Eo

o

i

Ei E = Eo - E i

Fig. A.5. Dielectric material subjected to a field between the plates of a charged and isolated capacitor

The superficial density of charge induced on the surfaces of a dielectric by the action of an external field is called density of dielectric polarization. The vector whose module has the value of the surface density charge due to dielectric polarization and has a direction perpendicular to the surface considered is called polarization vector P. Consequently, if the polarization takes place on different surfaces, the polarization vector may have more than one component. Polarization P can be expressed as a linear function of the field (the same conclusion can be reached experimentally for most of the dielectric materials where higher order effects can be neglected as it is the case of quartz). Therefore:

P = χE

(A.14)

where the constant χ is called dielectric susceptibility and it is, by definition, the polarization per unit of the existing electric field in the dielectric. Let’s call Ei the field nearby the dielectric due to the bound charges that have appeared in their surface. Gauss’ law provides the value of this field according to the superficial density of charge and the permittivity in the vacuum as (Eq. A.13):

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Antonio Arnau and Tomás Sogorb

Ei =

σ i P χE = = εo εo εo

(A.15)

Therefore, the existing field after the insertion of the dielectric will be:

E = Eo − Ei = Eo −

χE εo

(A.16)

That is to say,

E=

εo

εo + χ

Eo =

εo E Eo = o ε εr

(A.17)

where ε = εo + χ is the dielectric permittivity of the material and εr the relative permittivity of the material to the vacuum. Therefore, the field in the dielectric decreases with regard to the existing field in its absence. Suppose that the dielectric material has been introduced between the plates of a capacitor that are subjected to a constant potential difference (Fig. A.6); under these circumstances, the field between the plates of the capacitor will be constant with or without the dielectric whenever it keeps the distance between plates. The charge density in the plates of the capacitor before introducing the dielectric will be σo = εo E, in this case E being the constant field between the plates of the capacitor. When introducing the dielectric, its polarization causes an increase in the charge density in the plates due to the bound charge of the polarized dielectric. This increase of density corresponds to polarization. That is to say, there is an electric displacement, of free charges, from the conductors of the circuit towards the electrodes in order to compensate the effect of the polarization of the dielectric and to maintain the field inside the dielectric constant (Fig. A.6). Maxwell, in order to facilitate the calculation in a great number of problems and allow for a more compact writing of certain expressions of electromagnetism, introduced the displacement vector D, and defined it as:

D = εoE + P

(A.18)

In the case of a homogeneous and isotropic dielectric, the polarization is P=χE and the previous expression becomes:

D=ε E

(A.19)

Appendix A: Fundamentals of Electrostatics

507

In isotropic materials, the dielectric constant or permittivity is a characteristic of the medium. If the material is not homogeneous, it may be a function of the point; if it is anisotropic, it may depend on some direction, and if it is not linear, it may even depend on the field applied. E o= E

o

Ei

i i

Fig. A.6. Dielectric material subjected to a constant field between the plates of a capacitor subjected to a constant potential difference

An important equation is obtained when the flux of the vector displacement is calculated in a closed surface A. That is:

r

r

r

∫ D o dA = ε ∫ E o dA + ∫ P o dA o

A

A

(A.20)

A

The first term of the second member is the flux of the electric field, which is equal to the sum of all the charges inside the surface under study. That is to say:

r

ε o ∫ E o dA = Q f − Ql

(A.21)

A

where Qf is the total free charge inside the surface A, and Ql is the charge bound in A. The second term is the flux of the polarization vector that is, in fact, the total bound charge inside A. Therefore, Eq. ( A.20) results:

r D ∫ o dA = Q f − Ql + Ql = Q f = ∫ ρ vf ⋅ dV A

(A.22)

V

where ρvf is the volumetric density of free charge. Thus, keeping in mind the theorem of divergence, one obtains:

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r divD = ρ vf

(A.23)

The previous expression is known as first Maxwell Equation and it indicates that if there is no free charge inside a closed surface, the vector displacement in its interior is constant.

Appendix B: Physical Properties of Crystals Antonio Arnau, Yolanda Jiménez and Tomás Sogorb Departamento de Ingeniería Electrónica, Universidad Politécnica de Valencia

B.1 Introduction Crystals are, in general, anisotropic materials, that is, many of their physical properties depend on the direction in which they are considered. It is therefore not possible to understand their behavior and performance against external, mechanical or electrical effects or against changes in the environmental conditions such as changes in temperature, without due analysis of at least those properties most closely related to the object under study. The analysis of such properties provides the physical factors that relate a magnitude considered as the cause to another magnitude whose effect is the consequence of such magnitude. In the case under study, the physical factors include elastic, dielectric, and piezoelectric properties and thermal expansion coefficients.

B.2 Elastic Properties The elastic properties characterise the behaviour of deformable solids studied in the theory of Elasticity. Any mathematical formulation for modelling physical phenomena is limited in the sense that it is necessary to develop some approaches to obtain a simple model that accurately represents the behavior of the phenomenon. One of the most common approaches is to consider physical phenomena as behaving linearly. In this case, linear elasticity studies the behavior of the elastic solid as a deformable, continuous, elastic, homogeneous and isotropic system of material points. In this theory, however, it is possible to include cases of heterogeneity corresponding to elastic constants dependent on point coordinates and anisotropy involving changes of the constants at a given point with respect to the direction.

A.A. Vives (ed.), Piezoelectric Transducers and Applications, doi: 10.1007/978-3-540-77508-9_20, © Springer-Verlag Berlin Heidelberg 2008

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The elastic constants are calculated by the constitutive equations that express the behavior of the material, i.e., they relate stresses and strains. These equations are phenomenological, their parameters are the elastic constants of the material, and they can only be obtained by experimental tests. However, their formulation requires knowing the physical constraints established by rational mechanics and by the equations of static equilibrium, which relate the acting forces with the static magnitudes called stresses, and by the equations of cinematic compatibility that represent the conditions between the displacement of the solid and the cinematic magnitudes called strains. In addition, the condition of linearity implies, on one hand, that the equations between strain and displacement be of first order, which means to neglect terms of a higher order; and on the other hand, that the material be elastic and Hookean, i.e., that the equations between stresses and strains be linear. Further, the elastic condition requires the solid to return to its original state without deformation when the forces acting on the material are removed. B.2.1 Stresses and Strains In order to obtain the equilibrium conditions at a generic interior point of the solid it is necessary to consider a rectangular contour with the sides parallel to the coordinate planes (Fig. B.1). X3 T3 + dT3 T1

T2

T2 + dT2 X2

T1 + dT1 T3 X1

Fig. B.1. Generic point of a solid subject to mechanical stresses

On the sides of the elemental parallelepiped convergent with such planes, stresses T1, T2 and T3 occur. Each stress can be broken down into its corresponding components depending on the direction of the coordinates as shown in Fig. B.2. Thus, stress Ti, will have as its components Ti1 for direction X1, Ti2 for direction X2 and Ti3 for direction X3. The stresses of

Appendix B: Physical Properties of Crystals

511

each corresponding parallel plane will consider the changes in the components according to the direction of the coordinates. Therefore, the plane of the parallelepiped parallel to the coordinate plane on which stress T1 is acting, will have as components T11+T11,1 dx1, T12+T12,1 dx1, T13+T13,1 dx1. The other components of the different planes are obtained in the same way. X3 T33 +T33,3 dx3 T32 +T32,3 dx3 T31 +T31,3 dx3 T12

T21 T22

T11

T13 +T13,1 dx1 T13 T23

T23 +T23,2 dx2 T22 +T22,2 dx2

T12 +T12,1 dx1 T21 +T21,2 dx2 T11 +T11,1 dx1 T32 T31

X2

T33

X1

Fig. B.2. Decomposition of the stresses acting on a section of the solid

The equilibrium of the forces acting on the volume of the elemental parallelepiped gives the following equations: Resultant of null forces:

F1 − (T11,1 + T21, 2 + T31,3 ) dx1 dx2 dx3 = 0 F2 − (T12,1 + T22, 2 + T32,3 ) dx1 dx2 dx3 = 0

(B.1)

F3 − (T13,1 + T23,2 + T33,3 ) dx1 dx2 dx3 = 0 Null moment with respect to the gravity centre of the elemental volume:

∑M ∑M ∑M

1

= 0 → T23 = T32

2

= 0 → T13 = T31

3

= 0 → T12 = T21

(B.2)

This result is obtained neglecting the terms of higher order. Therefore, from the static equilibrium conditions the stress tensor is considered to be symmetrical. In this way, any element of a material system may experience 6 types of stresses, three longitudinal depending on the directions of the

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coordinates, and three transversal or torsor type around the corresponding axes as shown in Fig. B.3. 3 6 =12 2

4 =23 5 =31 1

Fig. B.3. Different types of stresses or strains acting on a material

The stress components in the previous equations have been written following the typical tensor notation. However, other types of notation are often used whose equivalence is shown next ⎛ T11 T12 ⎜ ⎜ T12 T22 ⎜T ⎝ 31 T23

T13 ⎞ ⎛ T xx ⎟ ⎜ T23 ⎟ = ⎜ T xy T33 ⎟⎠ ⎜⎝ Tzx

T xy T yy T yz

Tzx ⎞ ⎛ X x ⎟ ⎜ T yz ⎟ = ⎜ X y T zz ⎟⎠ ⎜⎝ Z x

Xy Yy Yz

Z x ⎞ ⎛ T1 ⎟ ⎜ Y z ⎟ = ⎜ T6 Z z ⎟⎠ ⎜⎝ T5

T6 T2 T4

T5 ⎞ ⎟ T4 ⎟ T3 ⎟⎠

(B.3)

The first matrix follows the tensor notation, the second and third matrices show the directions of the stresses both longitudinal and torsors, whereas the fourth matrix is a tensor notation with reduced indices from 1 to 6 following the relationship previously established and considering Fig. B.3. When a force is acting, all the points of an elastic solid experience a displacement. However, the stresses are not caused by the absolute movements of the points but by the separations and approximations between its particles, i.e., by the deformations. To obtain the expression of the strains in terms of the displacements, let us consider a point P located at the origin of the coordinates, and a point Q with coordinates dx1, dx2, and dx3 (Fig. B.4). During the deformation of the body, both points move to new positions P’ and Q’. In order to estimate the strain, the difference in length before and after the deformation will be calculated, i.e., P’Q’- PQ.

Appendix B: Physical Properties of Crystals

Q'

X3 Q dr

513

P' P

dr' X2

X1 Fig. B.4. Deformation analysis of a segment PQ

Let u1, u2 and u3 be the displacements for axes X1, X2 and X3 respectively. If ξ1, η1 and ζ1 are the coordinates of point P’, the coordinates of Q’ will be dx1 + ξ 2, dx2 + η2 and dx3 + ζ2. Since the displacements are continuous functions of the coordinates, the relations between the displacements of each point will be given by:

ξ 2 = ξ1 +

∂u1 ∂u ∂u dx1 + 1 dx2 + 1 dx3 ∂x1 ∂x 2 ∂x3

η2 = η1 +

∂u2 ∂u ∂u dx1 + 2 dx2 + 2 dx3 ∂x1 ∂x 2 ∂x3

ζ 2 = ζ1 +

∂u3 ∂u ∂u dx1 + 3 dx2 + 3 dx3 ∂x1 ∂x2 ∂x3

(B.4)

As a consequence, the displaced differential vector dr’ = P’Q’ can be related to the differential vector previous to the displacement dr = PQ according to the following expression: (B.5)

dr ' = dr + J dr

where J is the Jacobian matrix defined as:

⎛ ∂ u1 ⎜ ⎜ ∂ x1 ⎜ ∂u J =⎜ 2 ⎜ ∂ x1 ⎜ ∂ u3 ⎜ ∂x ⎝ 1

∂ u1 ∂ x2 ∂ u2 ∂ x2 ∂ u3 ∂ x2

∂ u1 ⎞ ⎟ ∂ x3 ⎟ ⎛ u u1, 2 1,1 ∂ u2 ⎟ ⎜ ⎟ = ⎜ u2,1 u2, 2 ∂ x3 ⎟ ⎜ ∂ u3 ⎟ ⎝ u3,1 u3, 2 ∂ x3 ⎟⎠

u1,3 ⎞ ⎟ u2,3 ⎟ u3,3 ⎟⎠

(B.6)

which can be broken down into two matrices, one symmetrical matrix S and one asymmetrical matrix A defined as:

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⎛ ∂ u1 ⎜ x1 ∂ ⎜ ⎜ 1 ⎛ ∂u ∂u ⎞ S = ⎜ ⎜⎜ 1 + 2 ⎟⎟ ⎜ 2 ⎝ ∂ x 2 ∂ x1 ⎠ ⎜ 1 ⎛ ∂u ∂u ⎞ ⎜ ⎜⎜ 1 + 3 ⎟⎟ ⎜ 2 ∂x ∂x 1 ⎠ ⎝ ⎝ 3

1 ⎛ ∂ u1 ∂ u 2 ⎞ ⎟ ⎜ + 2 ⎜⎝ ∂ x 2 ∂ x1 ⎟⎠ ∂u2 ∂ x2 1 ⎛ ∂ u3 ∂ u2 ⎞ ⎜ ⎟ + 2 ⎜⎝ ∂ x 2 ∂ x 3 ⎟⎠

1 ⎛ ∂ u1 ∂ u3 ⎞ ⎞ ⎜ ⎟⎟ + 2 ⎜⎝ ∂ x 3 ∂ x1 ⎟⎠ ⎟ ⎛ S11 1 ⎛ ∂ u3 ∂ u 2 ⎞ ⎟ ⎜ ⎜⎜ ⎟⎟ ⎟ = ⎜ S12 + 2 ⎝ ∂ x2 ∂ x3 ⎠ ⎟ ⎜ ⎟ ⎝ S13 ∂ u3 ⎟ ⎟ ∂ x3 ⎠

S12 S 22 S 23

S13 ⎞ ⎟ S 23 ⎟ S 33 ⎟⎠

(B.7)

A13 ⎞ ⎟ A23 ⎟ A33 ⎟⎠

(B.8)

and ⎛ 1 ⎛ ∂ u1 ∂ u2 ⎞ ⎜ ⎜ ⎟ 0 − 2 ⎜⎝ ∂ x 2 ∂ x1 ⎠⎟ ⎜ ⎜ 1 ⎛ ∂u ∂u ⎞ A = ⎜ − ⎜⎜ 1 − 2 ⎟⎟ 0 ⎜ 2 ⎝ ∂ x 2 ∂ x1 ⎠ ⎜ 1 ⎛ ∂u ∂u ⎞ ∂u ⎞ 1 ⎛ ∂u ⎜ − ⎜⎜ 1 − 3 ⎟⎟ − ⎜⎜ 2 − 3 ⎟⎟ ⎜ 2 ∂x ∂x ∂ ∂ 2 x x2 ⎠ 1 ⎠ ⎝ 3 ⎝ 3 ⎝

1 ⎛ ∂ u1 ∂ u3 ⎞ ⎞ ⎜ ⎟⎟ − 2 ⎜⎝ ∂ x3 ∂ x1 ⎟⎠ ⎟ ⎛ A11 1 ⎛ ∂ u 2 ∂ u3 ⎞ ⎟ ⎜ ⎟⎟ ⎟ = ⎜ A12 ⎜⎜ − 2 ⎝ ∂ x3 ∂ x2 ⎠ ⎟ ⎜ ⎟ ⎝ A13 ⎟ 0 ⎟ ⎠

A12 A22 A23

Matrix S is symmetrical, since Sij = Sji. In addition, it possesses tensor properties and its components represent the different deformations experienced by the planes of the elemental parallelepiped when external forces act on it. Figure B.5 shows the side of the elemental parallelepiped located on axes X1 and X2. Thus, the unitary longitudinal deformation in direction X1 is defined as:

S1 =

O ' A'−OA ∂ u1 = = S11 OA ∂ x1

(B.9)

In this result, the effects of higher order corresponding to second partial derivatives have been neglected. On the other hand, transversal strain in plane X1X2 is defined as the variation in the angle formed by two infinitely small segments when a solid experiences deformation, in both directions. Fig. B.5 shows that this variation, for small deformations, is: S 6 = 〈 BOA − 〈 B ' O ' A' = θ1 + θ 2 ≈

∂ u1 ∂ u2 + = 2 S12 ∂ x2 ∂ x1

(B.10)

Total transversal strain in plane X1X2 will be the sum of the tangential deformations or glidings according to the perpendicular directions and thus, S12 = S21 = S6 /2. The transversal deformations and the glidings or tangential deformations will be obtained in a similar way for the other planes.

Appendix B: Physical Properties of Crystals

u1 dx 2 x2

X2 u2 dx 2 x2 u

u2 + u2 dx2 x2

C'

B'

θ

2

B dx2

C

O'

u2

A' 1

u

u1 dx 1 x1

A

u1

u2 dx 1 x1

θ

u O

515

X1

Fig. B.5. Deformation of the elemental parallelepiped

The previous notation corresponds to the reduced indices used for the stress tensor. Other common ways to express the strain tensor are:

⎛ S11 ⎜ ⎜ S12 ⎜S ⎝ 31

S12 S 22 S 23

⎛ ⎜ xx S13 ⎞ ⎜ ⎟ ⎜ xy S 23 ⎟ = ⎜ 2 S 33 ⎟⎠ ⎜ z x ⎜ ⎜ 2 ⎝

xy 2 yy yz 2

⎞ ⎛ ⎟ ⎜ S1 ⎟ ⎜ ⎟ ⎜ S6 ⎟=⎜ 2 ⎟ ⎜S zz ⎟ ⎜ 5 ⎟ ⎝ 2 ⎠

zx 2 yz 2

S6 2 S2 S4 2

S5 ⎞ ⎟ 2 ⎟ S4 ⎟ 2 ⎟ ⎟ S3 ⎟ ⎠

(B.11)

On the other hand, each component of the antisymmetrical tensor A represents half turning of the bisectrix of each side of the elemental parallelepiped on the axis perpendicular to that side, the whole turn corresponding to the sum of the asymmetrical components. In this sense, when there is only deformation, the turnings are null. The total internal energy stored in a general strain can be calculated as the sum of the energies caused by the different modes of distortion. In this way, the work performed in a longitudinal strain of the elemental parallelepiped in direction X1 will be the product of the force in that direction by the displacement, that is, T11 dx2 dx3 S11 dx1. Hence, the variation in the internal energy caused by the variation of the displacement in that direction is: T11 dS11 dx1 dx2 dx3. The variation of internal energy due to the work performed by the transversal forces that cause an elementary shear deformation according to a turning around axis X3 will be T12 dx2 dx3 dS12 dx1 + T21 dx1 dx3 dS21 dx2 = 2 T12 dS12 dx1 dx2 dx3 = T6 dS6 dx1 dx2 dx3. Therefore, the internal energy stored for all the modes will be:

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Antonio Arnau, Yolanda Jiménez and Tomás Sogorb

dU = [T1dS1 + T2 dS 2 + T3 dS 3 + T4 dS 4 + T5 dS 5 + T6 dS 6 ]dx1 dx 2 dx 3

(B.13)

The application of rational mechanics has allowed establishing constraints that involve the static and cinematic behavior of an elastic solid subjected to the action of external forces. However, to characterise the material it is necessary to know the parameters that relate the stresses caused by external forces with the deformations generated. These parameters result from the formulation of constitutive or phenomenological equations that relate stresses and strains and are based on experimental results. B.2.2 Elastic Constants. Generalized Hooke’s Law

The experiments performed by Hooke show that for small displacements the more general strains in an elastic solid can be formulated through linear combinations of longitudinal and transversal deformations. Longitudinal deformations can occur in three directions parallel to the orthogonal axes. Similarly, any transversal deformation corresponds to the turnings around such axes (see Fig. B.3). In an isotropic material subjected to longitudinal stress, the unitary strain in the direction of the force for small displacements is proportional to the force per unit area or stress applied. This proportionality coefficient is referred to as Young's modulus and has the following expression: F Y= A Δl l

(B.14)

where A is the transversal section on which force F acts, l is the original length of the material in the longitudinal direction of the force, and )l is the lengthening of the material in that direction. The application of the law to transversal stresses relates shear stress with the resulting strain, defined as the shear angle, by the transversal stiffness coefficient or transversal stiffness modulus or shear modulus. It is easy to understand that a stress in the longitudinal direction acting on an isotropic material may cause not only a longitudinal strain but also a deformation in the direction perpendicular to it. The relationship between longitudinal and lateral strains is referred to as Poisson coefficient and is dimensionless. The situation in anisotropic materials is much more complex. In addition to the lateral and longitudinal strains, a longitudinal force stress can also cause transversal deformations. For instance, a stress acting on a thin

Appendix B: Physical Properties of Crystals

517

bar of quartz cut in such a way that its length is parallel to axis X, not only lengthens it and makes it thinner (longitudinal and lateral deformation) but also tends to rotate it around that axis (transversal strain). Therefore, to thoroughly describe the relationship between stresses and deformations in crystalline materials, it is necessary to consider that a stress may produce any kind of deformation. Since there are 6 independent ways of causing stresses and of indicating deformation (Fig.B.3), 36 constants are needed to describe the general behavior of the solid. Usually there are two sets of equations depending on whether we want to express the stresses in relation to the deformations or vice versa. In the former case, the constants are called stiffness coefficients or constants and elastic constants, and in the latter compliance coefficients or constants. These equations are:

T1 = c11 S1 + c12 S 2 + c13 S 3 + c14 S 4 + c15 S5 + c16 S 6 T2 = c21 S1 + c22 S 2 + c23 S 3 + c24 S 4 + c25 S5 + c26 S 6 T3 = c31 S1 + c32 S 2 + c33 S 3 + c34 S 4 + c35 S5 + c36 S 6 T4 = c41 S1 + c42 S 2 + c43 S 3 + c44 S 4 + c45 S5 + c46 S 6

(B.15)

T5 = c51 S1 + c52 S 2 + c53 S 3 + c54 S 4 + c55 S5 + c56 S 6 T6 = c61 S1 + c62 S 2 + c63 S 3 + c64 S 4 + c65 S5 + c66 S 6 where coefficient cij is an elastic constant that expresses the proportionality between deformation Sj and stress Ti. Using Einstein agreement the equations can be written as:

Ti = cij S j

i, j = 1 a 6

(B.16)

The principle of energy conservation requires that cij = cji. From the equation that provides the variation of internal energy in terms of the strains it can be inferred that the stress component Ti is:

Ti =

∂U ∂ Si

and then cij =

∂ Ti ∂2 U ∂2 U = = = c ji ∂ Si ∂ Si ∂ S j ∂ S j ∂ Si

since the order of differentiation does not vary the result. This reduces the number of independent elastic constants to 21. The Generalized Hooke's Law can be formulated taking the stresses as independent variables. Then, we obtain:

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Antonio Arnau, Yolanda Jiménez and Tomás Sogorb

S1 = s11 T1 + s12 T2 + s13 T3 + s14 T4 + s15 T5 + s16 T6 S 2 = s21 T1 + s22 T2 + s23 T3 + s24T4 + s25 T5 + s26 T6 S 3 = s31 T1 + s32 T2 + s33 T3 + s34 T4 + s35 T5 + s36 T6 S 4 = s41 T1 + s42 T2 + s43 T3 + s44 T4 + s45 T5 + s46 T6

(B.17)

S5 = s51 T1 + s52 T2 + s53 T3 + s54 T4 + s55 T5 + s56 T6 S 6 = s61 T1 + s62 T2 + s63 T3 + s64 T4 + s65 T5 + s66 T6 The values of coefficients sij can be obtained from the cij using the following relationship: sij =

( −1) i + j Δcij Δc

where )c is the determinant corresponding to the matrix of coefficients cij of (B.17) and Δcij is the adjunct of term ij in the same matrix. Constants cij and sij define the elastic behavior of the material. However, they do not possess tensor character since the deformation components S1, S2 ... S6 do not have it. The expressions used in the formulation of the physical properties of crystals are greatly simplified when the tensor notation is used. In addition, the calculation of the constants that characterise the properties for the orthogonal axes forming specific angles with the reference axes of the crystal becomes simpler due to the transformation laws of the tensors. On the other hand, the elastic and compliance constants provided by the researchers correspond to the expressions mentioned above. Therefore, it is necessary to establish the relationship between these constants and their corresponding coefficients in tensor notation. The lack of tensor character of the strains defined in the expressions above is solved using as deformation tensor the symmetrical matrix S (Eq. (B.7)). The set of the nine elements that forms the matrix has a tensor character, like the symmetrical stress matrix (Eq. (B.3)). The relationship between both tensors of second order can be established using certain coefficients, depending on whether we want to express the stresses as a function of the strains or vice versa, which must form a tensor of fourth order. Then, the Generalized Hooke's Law can be written as:

Tij = cijkl S kl

(B.18)

Appendix B: Physical Properties of Crystals

519

or either its inverse form:

Sij = sijkl Tkl

(B.19)

The elastic coefficients in tensor notation form a tensor of fourth order with 81 terms. However, as a result of the symmetry of the stress tensors Tij and deformations Sij, the tensor of elastic coefficients is also symmetrical. That is, its components fulfil the following equivalencies:

cijkl = cijlk = c jikl = c jilk The relation between the tensor coefficients and the elastic constants is determined extending the tensor notation and comparing the equations obtained for each stress component Tij with its corresponding one in Eq. (B.15) and considering that Sij = Sk/2 when i ≠ j and k = 4, 5 or 6 (Eq. (B.11)). The equivalencies obtained can be summarised as follows:

cijkl = cλμ considering that subindex λ takes the values 1, 2 and 3 when the subindex pair ij takes the values 11, 22 and 33 respectively, and 4, 5 and 6 for 23, 31 and 12 and its permutations. The same happens with subindex µ in relation to the pair kl. The equivalencies between the coefficients sijkl, of the equations that express the components of the deformation tensor, and the compliance constants in Eq. (B.17) are calculated in the same way and can be summarised as follows:

s λμ = sijkl

when i = j and k = l

s λμ = 2 sijkl

when i ≠ j or k ≠ l

s λμ = 4 sijkl

when i ≠ j and k ≠ l

with the same relations as for subindices λ and µ in relation to the pairs ij and kl. Only 21 of the original 81 components are independent, like in crystals belonging to the triclinic system. However, this number decreases as crystal symmetry increases. The independent components correspond to half of the symmetrical matrix obtained from Eq. (B.15) (equation of the elastic constants) and are:

520

Antonio Arnau, Yolanda Jiménez and Tomás Sogorb c1111

c1122 c 2222

c1133 c 2233 c 3333

c1123 c 2223

c1131 c 2231

c11

c1112 c 2212

c 3323

c 3331

c 3312

c 2323

c 2331

c 2312

c 3131

c 3112 c1212

c12 c 22

c13 c 23 c 33

=

c14 c 24

c15 c 25

c16 c 26

c 34

c 35

c 36

c 44

c 45

c 46

c 55

c 56 c 66

For the case of the compliances the following relationship results: s11 s1111

s1122

s1133

s1123

s1131

s1112

s2222

s2233

s2223

s 2231

s 2212

s3333

s3323

s3331

s3312

s2323

s 2331 s3131

s 2312 s3112

=

s1212

s12

s13

s 22

s 23 s33

s14 2 s 24 2 s34 2 s 44 4

s15 2 s25 2 s35 2 s45 4 s55 4

s16 2 s26 2 s36 2 s46 4 s56 4 s66 4

B.3 Dielectric Properties In an anisotropic medium, the dielectric constant is actually a tensor that relates the components of the electric field with the vector of electric displacement according to the following expression: D1 = ε 11 E1 + ε 12 E 2 + ε 13 E3 D2 = ε 21 E1 + ε 22 E 2 + ε 23 E3

(B.20)

D3 = ε 31 E1 + ε 32 E 2 + ε 33 E3

where subscripts 1, 2 and 3 refer to axes X, Y and Z respectively. Similarly, the susceptibility tensor results: P1 = χ11 E1 + χ12 E 2 + χ13 E3 P2 = χ 21 E1 + χ 22 E 2 + χ 23 E3

(B.21)

P3 = χ 31 E1 + χ 32 E 2 + χ 33 E3

These tensors represent the most common case. The transversal permittivities and susceptibilities (referring to the constants relating the

Appendix B: Physical Properties of Crystals

521

displacement vector and the polarization vector in one direction with the field in one perpendicular direction) are only different from zero in the case of triclinic or monoclinic crystals. In the case of quartz, for example, the independent dielectric constants reduce to two.

B.4 Coefficients of Thermal Expansion Another magnitude that depends on the direction is the strain generated by a change in temperature. In isotropic materials, thermal expansion is similar in all directions; however, in crystals, particularly in quartz, it depends on the direction. It is important to know the variation in length in a given direction due to changes in temperature since such variation will affect other physical properties, such as the temperature coefficients of the resonance frequencies. In addition, it may also affect the value of the constants that relate other magnitudes with the strain and may even make it difficult to determine the origin of certain electrical behaviors (e.g. piroelectricity). By definition, the thermal expansion coefficients provide the deformation per temperature unit and can be expressed as the partial derivative of the strain with respect to temperature. It could be formulated as follows:

αi =

∂ Si ∂θ

(B.22)

B.5 Piezoelectric Properties As we have already mentioned, in a deformable solid there are three fundamental directions according to axes X, Y, Z and six possible types of deformations and stresses, polarization in one direction may be due to the contributions of each strain or stress. In this way, a first approximation of the piezoelectric phenomenon can be made as a tensor of third order that relates polarization in one direction with each of the possible stresses or strains, that is:

Pi = eijk S jk Pi = d ijk T jk

i, j, j = 1, 2, 3

(B.23)

The use of the first or second former expressions depends on whether we want to relate strains or stresses to polarizations.

522

Antonio Arnau, Yolanda Jiménez and Tomás Sogorb

Coefficients eijk and dijk represent the piezoelectric properties of the material and are called stress or strain piezoelectric coefficients or constants respectively. A reduced notation can be used, like in the case of the elastic constants, to simplify the mathematical formulation. Then, it can be written as: P1 = d11 T1 + d12 T2 + d13 T3 + d14 T4 + d15 T5 + d16 T6 P2 = d 21 T1 + d 22 T2 + d 23 T3 + d 24 T4 + d 25 T5 + d 26 T6

(B.24)

P3 = d 31 T1 + d 32 T2 + d 33 T3 + d 34 T4 + d 35 T5 + d 36 T6 or P1 = e11 S1 + e12 S 2 + e13 S 3 + e14 S 4 + e15 S 5 + e16 S 6 P2 = e21 S1 + e22 S 2 + e23 S 3 + e24 S 4 + e25 S 5 + e26 S 6

(B.25)

P3 = e31 S1 + e32 S 2 + e33 S 3 + e34 S 4 + e35 S 5 + e36 S 6

Considering the symmetry of the stress and strain tensors and their relations with the components of the reduced indices, the following relationship between the piezoelectric coefficients in tensor and reduced notation is obtained: eiλ = eijk

and

d iλ = d ijk d iλ = 2d ijk

for for

j = k = λ = 1, 2, 3 j ≠ k ; λ = 4, 5, 6

Equations (B.24) and (B.25) formulate the direct piezoelectric phenomenon. Lippmann predicted the existence of the inverse piezoelectric phenomenon by the application of the thermodynamic potentials to the piezoelectric effect. His reasoning was as follows: If a piezoelectric crystal is located in an electric field of intensity E and subjected to a mechanical stress T, the crystal is deformed by an amount S and a polarization P appears. If under these circumstances both the field and the stress experience a small variation in the values dE and dT respectively, the overall variation of the internal energy can be expressed as an exact differential dU = PdE+SdT. If the process is assumed to be reversible, then it can be written as: ⎛∂P⎞ ⎛∂S ⎞ ⎟⎟ = ⎜⎜ ⎟⎟ = d ⎜⎜ ⎝ ∂ T ⎠E ⎝ ∂ E ⎠T

(B.26)

This equation expresses the fact that the relationship between the polarization generated by a stress is equal to the strain caused by an electric

Appendix B: Physical Properties of Crystals

523

field. This constant has been called d and corresponds to the strain piezoelectric coefficient previously mentioned. The Curie brothers soon proved Lippmann's predictions, and since then, a number of studies on the topic appeared that generalised Lippmann's theory and used it to formulate their theories of piezoelectricity. Most of these studies have assumed that the phenomena are reversible and that all the relationships are linear. In addition, the theories based on thermodynamic potentials also allow the formulation of elastic, dielectric and piezoelectric phenomena as wells as their relationships. Thus, considering the relations for the dielectric and piezoelectric polarizations due to field and stress, the total electric displacement and total deformation in an anisotropic piezoelectric material can be written as: S i = sij T j + d mi E m Dm = d mj T j + ε mn E n

i, j = 1 to 6 m, n = 1 to 3

(B.27

The first equation corresponds to the reverse piezoelectric phenomenon and the second equation to the direct one. It is possible to obtain two additional formulations. By multiplying both equations by the elastic stiffness matrix cij and considering that this matrix is the inverse of the compliance matrix, we obtain: cij S j = cij s ji Ti + cij d mj E m

i, j = 1 to 6 m = 1 to 3

(B.28)

Then: Ti = cij S j − cij d mj E m

i, j = 1 to 6 m = 1 to 3

(B.29)

Considering that the piezoelectric constant that relates the field to the stress is e, the stress can be written as: Ti = cij S j − emi E m

i, j = 1 a 6 m =1 a 3

(B.30)

Therefore there is a relationship between the stiffness coefficients and the piezoelectric constants given by: emi = cij d mj = d mj c ji

(B.31)

Multiplying the expression above by sji a new relationship is obtained:

524

Antonio Arnau, Yolanda Jiménez and Tomás Sogorb

s ji emi = s ji cij d mj

⇒ d mj = s ji emi = emi sij

i, j = 1 a 6 m=1a 3

(B.32)

Replacing dmj in the second equation of (B.27) and considering that Si = cij Tj, a new expression is obtained for the direct piezoelectric effect: Dm = emi S i + ε mn E n

i =1 a 6 m, n = 1 a 3

(B.33)

Thus, two equations are obtained for the direct effect and two equations for the reverse effect, shown next: Dm = d mj T j + ε mn E n ⎫ ⎬ Direct Effect Dm = emi S i + ε mn E n ⎭ Ti = cij S j − emi E m ⎫ ⎬ Inverse Effect S i = sij T j + d mi E m ⎭ emi = d mj c ji d mi = emj sij

i, j = 1 a 6 m, n = 1 a 3

Index ac-electrogravimetry, 171, 310 Acoustic attenuation coefficient, 454 Acoustic energy dissipation, 74 - motional resistance, 74 Acoustic energy storage, 74 - frequency shift, 74 Acoustic factor, 74 Acoustic impedance, 476 Acoustic impedance matching, 106, 110 Acoustic load concept, 89 Acoustic load impedance, 66, 68, 83, 333, 335 - impedance concept, 86 - Newtonian liquid, 340 - special cases, 68 - viscoelastic medium, 341 Acoustic properties, 77 Acoustic wave propagation, 67 - analytical approach, 67 - Mason model, 68 - matrix concept, 67 - transmission line model, 68 Acoustic-wave sensors, 39, 63 - acoustic plate wave, 42 - bulk acoustic wave, 40, 78 - cantilever sensors, 53 - characteristics, 58 - chemical sensors, 78 - flexural plate wave, 42, 48 - lateral field excitation, 49 - Love wave, 41, 48 - magnetic acoustic resonator, 51 - magnetic generation, 49 - magneto-surface, 42 - micro-electro mechanical systems, 53 - micromachined resonators, 53 - operating modes, 55 - operation principle, 40, 63 - pseudo-surface, 42 - sensitivity, 57

- shear horizontal, 41 - shear-horizontal acoustic plate mode, 46 - surface acoustic wave, 40, 45, 78 - surface skimming bulk, 42 - surface transverse wave, 41, 47 - thickness shear mode, 42 - thin-film thickness-mode, 43 Antibody production, 295 - haptens, 295 AT cut crystal, 26, 331 - properties, 27 Atomic force microscopy, 318 Attenuation, 477 - attenuation coefficients, 478 - specific absorption rate, 479 Backscatter coefficient, 456 Beam focusing, 423 Beam scanning, 425 Biochemical modification, 278 Biosensors, 259, 263, 290 - cell signaling, 259 - DNA sensors, 267 - entrapping, 283 - enzyme electrode, 264 - functionalisation, 282 - general scheme, 290 - immobilisation of biomolecules, 279 - immunosensors, 265 - membrane receptors, 260 - molecular recognition, 264 - molecular switches, 262 - molecular transistor, 267 - piezoelectric immunosensors, 289 - redox sensors, 263 - selectivity, 264 - sensitivity, 265 Bleustein-Gulyaev waves, 42

526

Index

Bragg reflector, 44 Brickman’s equation, 362, 364 Broadband models, 97 Broadband piezoelectric applications - broadband signal reception, 195 - electronic circuits, 190 Broadband piezoelectric systems, 187 Broadband piezoelectric transducers, 188 - efficient coupling, 188 Broadband signal reception, 195 - interface circuits, 195 Broadband ultrasonic systems, 97 Bulk acoustic waves, 41 - longitudinal waves, 41 - shear waves, 41 Cavitation, 401 Characteristic impedance, 66, 333 Chemical sensors, 241 - acoustic sensors, 251 - amperometric sensors, 246 - calorimetric sensors, 252 - conductimetric sensors, 248 - electrochemical sensors, 243 - integration, 245 - magnetic sensors, 254 - optical sensors, 250 - potentiometric sensors, 244 - selectivity, 243 - sensitivity, 243 - stability, 243 Clamped capacitance, 101 Complex compliance, 210 Compliance coefficient, 5 Composite resonator, 44, 67 - physical model, 67 - PZT sensors, 44 - RPL sensors, 44 Corrosion, 406 Coulomb’s law, 498 Crystal oscillator configurations - active bridge oscillator, 151, 156 - balanced bridge oscillator, 158

- bridge oscillator, 151 - emitter coupled oscillator, 146 - lever oscillator, 151, 155 Crystal oscillators, 133 - automatic gain control, 142 - critical aspects, 140 - design key points, 144 - LC oscillators, 134 - liquid phase, 143 - oscillating conditions, 136 - parallel mode, 136 - series mode, 138 - stability, 139 Crystals, 509 - dielectric properties, 520 - elastic properties, 509 - piezoelectric properties, 521 - thermal expansion coefficients, 521 Darcy’s law, 363, 364 Decay method, 129 - time constant, 130 Dielectric constant, 5 Displacement vector, 506 Elastic constant, 5 Elastic constants, 516 Elastic properties, 509 Electric displacement, 6 Electric field, 498 Electrical admittance, 28, 68 - conductance, 335 - piezoelectric resonator, 28 - susceptance, 335 Electrical impedance, 7, 68, 108 - basic model, 7 - coated resonator, 82 - equation, 105 - equivalent model, 24 - model parameters, 11, 35 - motional impedance, 33, 68 Electrical impedance matching, 106 Electrical matching, 114 Electrical tuning, 114 Electrochemical impedance, 231, 309

Index Electrochemical impedance spectroscopy, 235 Electrochemical quartz crystal microbalance, 308 Electrochemical techniques, 231 Electrochemistry, 223 - applications, 236 - electrochemical cell, 229 - electrochemical impedance, 231 - electrode reactions, 223, 224, 226 - galvanostat, 230 - oxidation, 223, 224 - potentiostat, 230 - reduction, 223, 224 - voltammetry, 230 Electrode potentials, 225 Electrogravimetric transfer function, 171 Electromagnetic acoustic transducer, 49 Electromechanical coupling factor, 105 Electromechanical impedance matrix, 98, 101 Electronic focusing, 423, 426 Electronic noses, 242 Electropolymerisation, 275 Electrostatic potential, 499 Electrostatics, 497 - free charges, 501 - linked charges, 502 - polarization, 503 - principle of charge conservation, 497 - principle of charge quantization, 498 Ellipsometry, 314, 354 - ellipsometric thickness, 315 - fundamentals, 323 Energy transfer model, 68 Entrapping, 283 Equivalent circuits, 102, 118 - Butterworth-Van Dyke, 28, 68, 69

- extended Butterworth-Van Dyke, 88, 118 - KLM, 84, 103 - lumped element model, 123 - Mason, 102 - Mason model, 102 - Redwood, 103 Faraday constant, 244 Faraday’s law, 229 Fast QCM applications, 171 Fermi energy level, 224 Ferroelectric ceramics, 97 - PZT, 97 Figure of merit, 59 Film bulk resonator, 43 Focused ultrasonic field, 426 Fraunhofer zone, 469 Frequency constant, 26 Fresnel zone, 469 Gauss’ law, 500 Gravimetric regime, 43, 74 - acoustic factor, 74 Hooke’s Law, 516 Hyperthermia, 467 - deep heating systems, 482 - focusing, 483 - superficial heating systems, 482 - ultrasound systems, 480 Hyperthermia transducers, 470 Immobilisation, 278, 279 - DNA, 284 - immunoreagents, 292, 296 - methods, 297 Immunoassay, 290 Immunosensors, 289 - antibody production, 295 - hapten synthesis, 293 - monoclonal antibody, 295 Impedance analyzers, 124 Impedance matching, 110

527

528

Index

Inductive tuning, 201 Interface electronics, 117, 187 - broadband applications, 187 - crystal oscillators, 133 - decay method, 129 - fast QCM, 171 - impedance analyzers, 124 - lock-in techniques, 162 - parallel capacitance compensation, 161, 163 - phase locked loop techniques, 163 - transfer function method, 127 Inverted-mesa, 42 Kanazawa equation, 73, 93, 340 Kinetic analysis, 75 - kinetic equations, 76 - rate constants, 76 Lamb waves, 42, 48 Lambert-Beer law, 250 Langmuir-Blodgett, 277 Lock-in techniques, 162 - maximum conductance, 169 - phase locked loop, 163 Long-term frequency stability, 59 Loss tangent, 214 Love waves, 41 Macromolecules, 205 - conformation changes, 207 - isomers, 207 - morphology, 207 - polymers, 207 Magnetic excitation, 49 Magneto-piezoelectric coupling, 51 Martin’s equation, 93, 368 Matching layers, 98, 107 Maxwell equation, 508 Mechanical transmission line, 107, 112 Medical imaging, 433 - backscattered tomography, 443 - computed tomography, 442 - computer-aided diagnosis, 450

- Doppler imaging, 441 - dynamic apodization, 433 - dynamic focusing, 426, 433 - imaging properties, 433 - intravascular imaging system, 445 - quantitative ultrasound, 453 - tissue harmonic imaging, 439 - transmission tomography, 443 - ultrasound biomicroscopy, 449 - ultrasound elastography, 444 Metallic deposition, 271 - electrochemical deposition, 272 - sputtering, 272 Microbalance sensors, 117 - critical frequencies, 174 - critical parameters, 120, 123 - effective electrode area, 121 - electronic interfaces, 117 - loading contribution, 123 - measuring parameters, 124 Micro-electro mechanical systems, 53 Microgravimetric sensor, 11, 25 - sensitivity, 27 Micromachined ultrasonic transducers, 54 Models for piezoelectric transducers, 97 - broadband, 97 - electromechanical impedance matrix, 98 - equivalent circuits, 102 - KLM model, 104 - Mason model, 102 - Redwood model, 103 - transmission line analogy, 104 Monoclonal antibody, 295 Monolayer assemblies, 276 Motional impedance, 33, 85, 334 - series resonant frequency, 34 Motional resistance, 74, 123 - determination, 142 Motional series resonant frequency, 120 Mux-Dmux of transducers, 427

Index Navier-Stokes equation, 360 Nernst equation, 225, 244 Non-destructive testing, 413 - beam focusing, 423 - dynamic focusing, 426 - electrical responses, 413 - electronic sequential scanning, 425 - fast operation, 425 - focused ultrasonic field, 426 - frequency domain analysis, 417 - high resolution, 422 - high speed scanning, 422 - multichanel schemes, 422 - pulse-echo, 415 - time domain analysis, 417 Optical sensors - optrodes, 250 Parasitic capacitance, 334 Partition coefficient, 252 Phantom, 483 Piezoelectric effect, 2 - converse piezoelectric effect, 98 - current induced, 10 - direct piezoelectric effect, 98 - effective dielectric constant, 5, 6 - elastic constants, 5 - electric displacement, 6 - mathematical formulation, 4 - piezoelectric polarization, 5 - piezoelectric polarization vector, 4 - piezoelectric strain coefficient, 4 - piezoelectric stress constant, 5 - piezoelectrically stiffened constant, 5, 66 Piezoelectric equations, 79, 99 Piezoelectric immunosensors, 289 - characterization, 299 Piezoelectric material, 12, 99, 472 - equivalent model, 12 - frequency constant, 26 - resonant frequencies, 12

529

Piezoelectric polymers, 97 - PVDF, 97 Piezoelectric transducers, 97 - modeling, 97 Piezoelectricity, 1, 2 Poisson coefficient, 516 Poisson’s equation, 501 Poisson’s ratio, 212 Polarization vector, 505 Polymer electrogeneration, 275 Polymers, 208 - complex compliance, 210 - complex viscosity, 210 - dynamic glass transition, 219 - entanglement coupling, 208 - glass transition, 215 - glass-rubber transition, 212 - loss tangent, 214 - macromolecules, 205 - modified WLF-equation, 218 - morphology, 207 - shear modulus, 209 - shear parameter determination, 220 - temperature-frequency equivalence, 214 - transition temperature, 209 - viscoelastic behavior, 208 - viscoelastic properties, 206, 208 - viscoelastic solids, 213 - viscosity, 208 - WLF equation, 216 Propagation speed, 15 - with losses, 18 Pulse-echo applications, 107 PZT, 474 Quality factor, 19, 35, 64, 70, 74 Quantitative ultrasound, 434 Quartz resonator - contactless, 52 Quartz-crystal microbalance, 43 - atomic force microscopy, 318 - calibration, 339, 357, 369 - calorimetry, 321

530

Index

- combination with other techniques, 307 - data analysis, 331 - ellipsometric thickness, 315 - ellipsometry, 314 - EQCM, 308, 373, 379 - experimental parameters, 334 - impedance spectroscopy, 309 - interpretation, 331 - layer thickness, 322 - measuring the thickness, 353 - Newtonian liquid, 340 - optical interferometry, 313 - parameter extraction, 332 - physical model, 333 - porous medium, 362 - roughness effect, 360 - Sauerbrey-like behavior, 339 - scanning electrochemical microscope, 319 - scanning probe techniques, 318 - scanning tunneling microscopy, 318 - sensitivity, 43 - small surface condition, 335 - small surface load, 351 - surface modification, 271 - surface plasmon resonance, 313, 316 - viscoelastic contribution, 349 - viscoelastic medium, 341 Rayleigh waves, 41, 45 Resonance phenomenon, 23, 63 - energy transfer model, 68 Resonance spectrum, 70 - dissipation factor, 70 - half power spectrum, 70 Resonant frequencies, 12, 20, 73 - characteristic frequencies, 25 - coupled vibrating modes, 15 - fundamental frequency, 15 - harmonic modes, 15 - inharmonic modes, 15 - maximum admittance, 178 - minimum admittance, 178

- motional resistance, 73 - oscillators, 73 - parallel frequency, 176 - phase-zero frequencies, 177 - propagation speed, 15 - series frequency, 176 - stationary waves, 18 - wave length, 14 - wave number, 14 - with losses, 15 Resonant phenomenon - quality factor, 64 Resonant sensors, 63 - acoustic load, 86 - acoustic load concept, 89 - coated quartz crystal, 78 - equivalent circuit, 69 - gravimetric regime, 74 - Kanazawa equation, 93 - kinetic analysis, 75 - Martin’s equation, 93 - mass factor, 94 - modeling, 64 - models, 63 - multilayer structure, 65 - non gravimetric regime, 74 - quartz crystal, 63 - Sauerbrey equation, 92 - small phase shift approximation, 94 - transmission line model, 82 - viscous damping, 65 Resonator admittance, 174 - diagram, 178 - equivalent-circuit equations, 174 Resonator impedance, 174 - equivalent-circuit equations, 174 Sauerbrey equation, 72, 92, 339, 368 Scanning electrochemical microscope, 319 Scanning tunneling microscopy, 318 Sensitivity, 57 - nominal, 58 - usable, 58 Sensor characterization, 334

Index Shear loss modulus, 209, 213 Shear modulus, 66 - frequency dependence, 210 Shear storage modulus, 209, 212 Shear-horizontal waves, 46 Short-term frequency stability, 59 Sonoelectroanalysis, 405 Sonoelectrochemistry, 399 - bioelectrochemistry, 406 - corrosion, 406 - electrodeposition, 407 - waste treatment, 408 Sonoelectrosynthesis, 406 Specific absorption rate, 479 Sputtering, 272 Static capacitance, 121, 334 Stationary waves, 18 ST-cut quartz, 45 Stokes law, 366 Surface acoustic impedance, 66 Surface modification, 271 - biochemical modification, 278 - electropolymerisation, 275 - Langmuir-Blodgett, 277 - metallic deposition, 271 - SAM techniques, 276 Surface plasmon resonance, 251, 313 Tafel relation, 227 Therapy transducer, 474 Thickness extensional transducer, 99, 102, 104 Thickness shear mode, 8 Thickness-longitudinal mode, 44 Time analysis of transducer driving, 197 Tissue characterization, 433 - acoustic impedance, 434 - attenuation, 434 - tissue acoustic properties, 434 - Young modulus, 434 Transmission coefficients, 66 Transmission line model, 82, 103

531

Ultrasonic application, 97, 187, 413 - acoustic properties of tissues, 475 - axial resolution, 187 - electrical focusing, 490 - hyperthermia, 467 - phantom, 483 - wave propagation in tissue, 475 Ultrasonic fields, 468 - beam non uniformity ratio, 475 - effective radiating area, 475 - measurement, 470 Ultrasonic generation, 471 Ultrasonic hyperthermia, 467, 479 - acoustic impedance, 476 - focusing, 489 - therapy transducer, 474 Ultrasonic pulses, 188 - generation, 188 - matching, 188 Ultrasonic systems, 413 - non-destructive testing, 413 Ultrasonic transducers - focusing, 489 - hyperthermia, 470 - impulse responses, 108 - interstitial transducers, 491 - intracavitary transducers, 491 - pulse-echo, 107 - time responses, 107 - transducer arrays, 490 - transfer functions, 107 Ultrasound elastography, 444 - compression strain elastography, 445 - magnetic resonance elastography, 445 - phase gradient sonoelastography, 445 - sonoelastography, 445 - transient elastography, 446 Ultrasound imaging, 433 - 3D ultrasound imaging, 440 - A-Mode, 435

532

Index

- B-Mode, 436 - computed tomography, 442 - Doppler imaging, 441 - electronic adjustable focusing, 423, 437 - imaging modes, 433 - interferometry speckle, 446 - periodicity analysis, 457 - pulse inversion imaging technique, 440 - quantitative ultrasound, 453 - real-time scanning, 428, 437 - resolution, 422, 437 - speckle phenomenon, 439 - ultrasound biomicroscopy, 449 Ultrasound phantoms, 484 - property measurement, 487

Viscoelasticity, 206 - Maxwell model, 211 - Voigt model, 211 Viscosity, 9 Viscous phenomenon, 9 Voltammetry, 230 Wave length, 14 Wave number, 14 Wave propagation, 66 Wave propagation constant, 66, 86 Wave propagation vector, 80 WLF equation, 216 X-rays computed tomography, 443 Young’s modulus, 516

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